Authors:
Dipl.-Ing. Thomas Halfmann
Dr.-Ing. Eckhard Hennig
Dr.-Ing. Manfred Thole
Dipl.-Math. Tim Wichmann
Copyright  2000–2001 by Fraunhofer-Institut für Techno- und Wirtschaftsmathematik (ITWM)
Gottlieb-Daimler-Strasse, D-67663 Kaiserslautern, Germany
Fax: +49-631-205-4139
Email: analog.insydes@itwm.fhg.de
WWW: http://www.analog-insydes.de
This document is furnished by Fraunhofer-ITWM to licensed users of Analog Insydes for information purposes only. All information in this
document is subject to change without notice. This document is provided as is without any express or implied warranties.
No part of this document may be reproduced or redistributed, on any media or by any means, without prior written permission of
Fraunhofer-ITWM.
This document was typeset using the WRILaTeX Document System provided under license by Wolfram Research, Inc.
The sparse matrix routines used in the Analog Insydes MathLink applications are copyright  1985, 86, 87, 88 by Kenneth S. Kundert and
the University of California.
Analog Insydes and the Analog Insydes logo are trademarks of Fraunhofer-ITWM.
Mathematica and MathLink are registered trademarks of Wolfram Research, Inc.
All other product names are trademarks of their respective owners.
Printed by Rohr Druck GmbH Kaiserslautern.
Printed in Germany.
Table of Contents
Part 1.
1.1
A Short Introduction
Welcome to Analog Insydes.........................................................................................................................6
How to Read this Book
1.2
Getting Started............................................................................................................................................8
Command Overview
Palette
1.3
Feature List
The First Step
Part 2.
New Features: A More Detailed Description
Compatibility
Preface......................................................................................................................................................32
Loading Analog Insydes
Netlists.......................................................................................................................................................33
The Netlist Format
2.3
The Analog Insydes
Tutorial
How to Read this Tutorial
2.2
The Online Help System
What’s New?.............................................................................................................................................25
New Features: An Overview
2.1
An Example Application
Circuit Elements
A Brief Preview on Circuit Analysis
More about Netlists
Circuits and Subcircuits..............................................................................................................................46
Circuits, Netlists, and Subcircuits Defining Subcircuits and Device Models Referencing Subcircuits
Expansion Subcircuit Parameters The Scope Argument The Translation Argument
2.4
Setting up and Solving Circuit Equations....................................................................................................70
Naming Conventions
CircuitEquations
2.5
Circuit Equations
Circuit Equation Formulations
Nyquist Plots
Nichol Plots
Root Locus Plots
Transient Waveforms
Modeling and Analysis of Nonlinear Circuits.............................................................................................90
Behavioral Models
Defining Behavioral Models
Referencing Behavioral Models
Multi-Dimensional Models Nonlinear DC Circuit Analysis References
2.7
Additional Options for
Graphics....................................................................................................................................................81
Bode Plots
2.6
Nonlinear Circuit Equations
Time-Domain Analysis of Nonlinear Dynamic Circuits...............................................................................104
Transient Circuit Analysis Analysis of a Differential Amplifier Flow of Transient Circuit Analysis
Options Transient Analysis with Initial Conditions TransientPlot Options
2.8
NDAESolve
Linear Symbolic Approximation................................................................................................................127
Introduction to Symbolic Approximation
Approximation Combining SBG and SAG
2.9
Subcircuit
Solution-Based Symbolic Approximation
Options for ApproximateMatrixEquation
Equation-Based Symbolic
Frequency-Domain Analysis of Linear Circuits..........................................................................................142
Transistor Models Transfer Functions Device Mismatch Input and Output Impedances
Advanced Transistor Modeling AC Analysis of the CMOS Amplifier
Two-Port Parameters
2.10 Nonlinear Symbolic Approximation..........................................................................................................164
Introduction to Nonlinear Approximation
Analyzing a Square Root Function Block
2
Part 3.
3.1
Reference Manual
Netlist Format..........................................................................................................................................178
Netlist
Circuit
Netlist Entries
Netlist Entry Options
ElementTypes Value Specifications for Independent
Sources Time and Frequency Model References Model and Subcircuit ModelParameters GlobalParameters
3.2
Subcircuit and Device Model Definition....................................................................................................190
Model
3.3
Subcircuit
Model Library Management.....................................................................................................................211
ListLibrary
FindModel
FindModelParameters
GlobalSubcircuits
LoadModelParameters RemoveSubcircuit RemoveModelParameters
3.4
GlobalModelParameters
LoadModel
Expanding Subcircuits and Device Models................................................................................................221
ExpandSubcircuits
3.5
Setting Up and Solving Circuit Equations.................................................................................................227
CircuitEquations
3.6
ACEquations
DCEquations
Solve
Circuit and DAEObject Manipulation........................................................................................................255
AddElements DeleteElements GetElements RenameNodes GetEquations GetMatrix GetVariables GetRHS
GetParameters
GetDAEOptions
SetDAEOptions
GetDesignPoint
ApplyDesignPoint
UpdateDesignPoint
MatchSymbols ShortenSymbols Statistics
3.7
Numerical Analyses.................................................................................................................................280
Analog Insydes Numerical Data Format
3.8
Parameter Sweeps
ACAnalysis
NoiseAnalysis
Pole/Zero Analysis..................................................................................................................................304
GeneralizedEigensystem
GeneralizedEigenvalues
PolesAndZerosByQZ
RootLocusByQZ LREigenpair ApproximateDeterminant
3.9
NDAESolve
PolesByQZ
ZerosByQZ
Graphics Functions...................................................................................................................................330
BodePlot
FourierPlot
NicholPlot
NyquistPlot
RootLocusPlot
TransientPlot
3.10 Interfaces.................................................................................................................................................365
ReadNetlist
WriteModel
Simulator-Specific Remarks on ReadNetlist
ReadSimulationData
WriteSimulationData
3.11 Linear Simplification Techniques...............................................................................................................385
ComplexityEstimate
ApproximateTransferFunction
ApproximateMatrixEquation
CompressMatrixEquation
3.12 Nonlinear Simplification Techniques..........................................................................................................401
NonlinearSetup
CompressNonlinearEquations
ShowLevel ShowCancellations
CancelTerms
SimplifySamplePoints
NonlinearSettings
3.13 Miscellaneous Functions...........................................................................................................................418
ConvertDataTo2D
DXFGraphics
MathLink Module MSBG
MathLink Module QZ
3.14 Global Options........................................................................................................................................425
Options[AnalogInsydes]
InstanceNameSeparator
UseExternals Option Inheritance
LibraryPath
ModelLibrary
Protocol
Simulator
3.15 The Analog Insydes Environment..............................................................................................................432
The Initialization Procedure
ReleaseInfo
ReleaseNumber
License
Version
Info
Table of Contents
Part 4.
4.1
Appendix
Stimuli Sources........................................................................................................................................438
AMWave
4.2
3
ExpWave
PulseWave
PWLWave
SFFMWave
SinWave
Netlist Elements........................................................................................................................................446
Resistor
Conductance
Admittance
Impedance
Capacitor
Inductor
OpenCircuit
ShortCircuit
CurrentSource
VoltageSource
CCCSource
CCVSource
VCCSource
VCVSource
OPAmp
OTAmp
Nullator
Norator
Fixator
SignalSource
Amplifier
Integrator
Differentiator
TransmissionLine
TransferFunction
4.3
Model Library..........................................................................................................................................460
Diode
Bipolar Junction Transistor
MOS Field-Effect Transistor
Junction Field-Effect Transistor
Credits..............................................................................................................................................................492
Index................................................................................................................................................................493
A Short Introduction
1.1
Welcome to Analog Insydes . . . . . . . . . . . . . . . . . 6
1.2
Getting Started . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3
What’s New? . . . . . . . . . . . . . . . . . . . . . . . . . 25
1. A Short Introduction
6
1.1 Welcome to Analog Insydes
Welcome to Analog Insydes, a Mathematica toolbox for symbolic modeling, analysis, and design of
analog electronic circuits. With Analog Insydes you can model and analyze linear and nonlinear
circuits, manipulate analysis results mathematically, produce graphical visualizations of circuit characteristics, exchange data with commercial circuit simulators, and document your solutions all in one
integrated environment.
You will always get the latest information concerning Analog Insydes at our home page
http://www.analog−insydes.de
There you will also find a collection of demo notebooks, which show the application of Analog
Insydes on different tasks from the field of electrical engineering.
If you have questions concerning Analog Insydes, please send an email to:
analog.insydes@itwm.fhg.de
If you want to register to the Analog Insydes newsletter, please write an email to the above mentioned email address with subject line "Newsletter " and body text "subscribe". You can find the
list of previous newsletters at:
http://www.analog−insydes.de/newsletter.html
1.1.1 How to Read this Book
This book is divided into four parts: an introduction (Part 1), a tutorial (Part 2), a reference manual
(Part 3), and an appendix (Part 4).
The introduction part gives an overview of Analog Insydes’ functionality, highlights the new features
of version 2, and shows the application of Analog Insydes on an practical example. We recommend
to read this part, especially Section 1.2.1 and Section 1.2.3, before working with Analog Insydes.
The tutorial part explains dedicated topics of Analog Insydes in detail. Although entitled Tutorial, it
is not meant to be read completely before working with Analog Insydes. Rather, each chapter can
be read separately as soon as you need a closer look at special topics.
The reference part explains each Analog Insydes command, shows its argument list, describes the
available options, and shows the application on small examples. If you need information on a certain
command, the reference manual is the preferred source.
The appendix part explains all stimuli sources defined in Analog Insydes, shows all available netlist
elements, and finally describes the content of the predefined symbolic device model library.
1.1.2 Feature List
The list below shows a summary of Analog Insydes’ main program features.
1.1 Welcome to Analog Insydes
7
Netlist-based input of electronic circuits and control systems
Available linear circuit elements: resistor, impedance, conductance, admittance, capacitor, inductor, independent current and voltage source, all four types of controlled sources (VCVS, CCVS,
VCCS, CCCS), operational amplifier (OP), operational transconductance amplifier (OTA), nullator,
norator, fixator
Available control network elements: proportional amplifier, differentiator, integrator, ideal transmission line, generic transfer function, signal source
Hierarchical netlist entry and device modeling by means of subcircuits and behavioral model
definitions
User-extendible device model library which includes bipolar (full Gummel-Poon small-signal and
large-signal) and MOS (full Level 1-3 and BSIM3v3 small-signal, Level 1 large-signal) transistor
models
Automatic setup of symbolic or mixed symbolic/numeric circuit equations by modified nodal
(MNA), sparse tableau (STA), and extended sparse tableau analysis (ESTA) from netlists for linear
and nonlinear circuits
Automatic equation setup for AC, DC, transient, and noise analysis
Symbolic calculation of transfer functions, input impedances, two-port parameters, etc.
Design formula extraction by approximation of linear symbolic circuit equations and transfer
functions
Simplification techniques for the approximation of nonlinear circuit descriptions
Support for automated generation of behavioral models for nonlinear circuits
Numerical AC, noise, and pole/zero analysis
Numerical DC, DC-transfer, and transient analysis of nonlinear circuits
Solution of circuit equations not only for currents and voltages but also for design parameters
Graphics functions for Bode, Nyquist, Nichol, Fourier, and root locus plots, transient waveforms,
DC-transfer, and parametric analyses
Netlist import for PSpice, Eldo, and Saber netlist files, including small-signal, operating-point, and
model-card parameters
Import filter for PSpice, Eldo, and Saber simulation data
Export filter of Analog Insydes simulation data for PSpice and Saber
Export filter for Saber MAST model templates
Import filter for circuit schematics and other line graphics from DXF files
1. A Short Introduction
8
1.2 Getting Started
If this is your first contact with Analog Insydes, this chapter will help you to get familiar with
the system. To get an overview of the functionality of Analog Insydes, the first section provides
a list of the most important commands (Section 1.2.1). Next, it is described how to load Analog
Insydes into the Mathematica kernel (Section 1.2.2), and an example application shows one possible
way to use Analog Insydes for solving a specific analysis task (Section 1.2.3). If you have never seen
Analog Insydes before, reading this example will give you a first impression of the system. Finally,
Section 1.2.4 explains how to access the Analog Insydes online help system.
1.2.1 Command Overview
The most important Analog Insydes commands are listed in this section. For each command, references are given where to find more information. A link to the reference manual (Part 3 of this book)
is given first, followed by links to related topics described in the tutorial (Part 2 of this book).
Interfaces
Analog Insydes provides import and export filters to PSpice, Eldo, and Saber (Chapter 3.10):
ReadNetlist (Section 3.10.1, Section 2.9.2) translates external netlist files (including model cards,
small-signal data, and operating-point information) into the Analog Insydes netlist language.
ReadSimulationData (Section 3.10.3, Section 2.10.2) allows for importing and transforming numerical simulation data into Mathematica InterpolatingFunction objects.
WriteSimulationData (Section 3.10.4) exports numerical data calculated with Analog Insydes for
further postprocessing in external waveform viewers.
WriteModel (Section 3.10.5) generates behavioral model descriptions of symbolic (approximated)
equation systems.
Netlists, Circuits, and Models
In Analog Insydes, analog circuits are expressed in terms of Netlist, Circuit, and Model objects:
Netlist (Section 3.1.1, Chapter 2.2) is the basic data structure which contains the elements of a
flat netlist.
Circuit (Section 3.1.2, Chapter 2.3, Section 2.3.1) is a data structure which combines netlists,
models, model parameters, and global parameters.
Model (Section 3.2.1, Section 2.3.2, Section 2.6.2) is a data structure which defines netlist-based or
equations-based models or subcircuits.
AddElements (Section 3.6.1, Section 2.9.4),
DeleteElements (Section 3.6.2, Section 2.9.4), GetElements (Section 3.6.3), and RenameNodes (Sec-
1.2 Getting Started
9
tion 3.6.4) allow for adding, deleting, or extracting netlist entries and changing node names,
respectively, in a comfortable way.
Statistics (Section 3.6.17) prints information on the contents of a Netlist or Circuit object.
Analog Insydes provides a predefined symbolic device model library (Chapter 4.3), which can be
extended by the user. Several commands allow for accessing the model data base (Chapter 3.3):
LoadModel (Section 3.3.6) loads a specific model from a given model library.
FindModel (Section 3.3.2) searches the default model library for a given name/selector pair.
GlobalSubcircuits (Section 3.3.4, Section 2.3.6) prints the name/selector pairs of all global
models currently loaded.
ListLibrary (Section 3.3.1) prints the contents of a specific model library.
Equations
Starting from Circuit or Netlist objects, circuit equations can be set up automatically in different formulations and analysis modes. They are stored (together with additional information) in a
DAEObject.
CircuitEquations (Section 3.5.1, Chapter 2.4, Section 2.4.2, Section 2.6.4) sets up circuit equations
from a Circuit or Netlist object and returns a DAEObject.
Solve (Section 3.5.4, Section 2.4.2) can be used to symbolically solve the equations stored in a
DAEObject.
GetEquations (Section 3.6.5, Section 2.10.2), GetVariables (Section 3.6.7, Section 2.10.2),
GetDesignPoint (Section 3.6.12, Section 2.9.6), and GetParameters (Section 3.6.9, Section 2.10.2)
give easy access to the data stored in a DAEObject.
ApplyDesignPoint (Section 3.6.13), UpdateDesignPoint (Section 3.6.14, Section 2.10.2), and
MatchSymbols (Section 3.6.15, Section 2.9.3) allow for modifying the contents of a DAEObject.
GetDAEOptions (Section 3.6.10) and SetDAEOptions (Section 3.6.11) allow for accessing or modifying the options stored in a DAEObject.
Statistics (Section 3.6.17, Section 2.9.7) prints information on the complexity of a DAEObject.
Numerical Analyses
The standard numerical circuit analyses can be carried out by the following commands (Chapter 3.7):
NDAESolve (Section 3.7.5, Chapter 2.7) is used to solve nonlinear differential-algebraic equation
systems. It calculates the DC (Section 2.6.6), DC-transfer (Section 2.7.2), and transient solution
(Section 2.7.1), also parametric (Section 2.7.2).
ACAnalysis (Section 3.7.3, Section 2.9.7) computes the small-signal solution of a linear equation
system.
1. A Short Introduction
10
NoiseAnalysis (Section 3.7.4) computes the output noise and the equivalent input noise of a
linear equation system.
Poles and Zeros
Besides the standard numerical analyses, Analog Insydes provides functions for numerically computing poles and zeros as well as root loci of linear systems (Chapter 3.8):
PolesAndZerosByQZ (Section 3.8.3), PolesByQZ (Section 3.8.4), and ZerosByQZ (Section 3.8.5) numerically compute poles and zeros of a linear system using the QZ algorithm.
RootLocusByQZ (Section 3.8.6) computes the root locus of a linear system.
Graphical Postprocessing
Analog Insydes provides special graphics functions for the most important electrical engineering
plots:
BodePlot (Section 3.9.1, Section 2.5.1)
FourierPlot (Section 3.9.2)
NicholPlot (Section 3.9.3, Section 2.5.3)
NyquistPlot (Section 3.9.4, Section 2.5.2)
RootLocusPlot (Section 3.9.5, Section 2.5.4)
TransientPlot (Section 3.9.6, Section 2.7.1, Section 2.7.6)
Symbolic Approximation
One of the most important features of Analog Insydes is its capability to reduce the complexity of
symbolic equations and expressions with automatic error control. For linear circuits, Analog Insydes
provides SBG and SAG methods (Chapter 2.8):
ApproximateTransferFunction (Section 3.11.2, Chapter 2.8, Section 2.8.2) approximates a symbolic transfer function by removing insignificant terms.
ApproximateMatrixEquation (Section 3.11.3, Chapter 2.8, Section 2.8.3) approximates a symbolic
matrix equation with respect to a certain output variable.
ApproximateDeterminant (Section 3.8.8) approximates a symbolic matrix equation with respect
to a certain pole.
As of Analog Insydes Version 2 there are also simplification routines for nonlinear equations (Chapter 3.12):
CompressNonlinearEquations (Section 3.12.2, Section 2.10.2) algebraically simplifies nonlinear
equations by eliminating irrelevant variables.
1.2 Getting Started
11
CancelTerms (Section 3.12.3, Section 2.10.2) approximates a symbolic nonlinear equation system with respect to a certain output variable. The command NonlinearSetup (Section 3.12.1,
Section 2.10.2) prepares the application of CancelTerms .
Miscellaneous
DXFGraphics (Section 3.13.2) translates DXF files into Mathematica graphics objects. It can be used
to display circuit schematics in a Mathematica notebook for documentation purposes.
Options[Analog Insydes] (Chapter 3.14) returns the list of global Analog Insydes options. See
Section 3.14.8 for a description of the option inheritance mechanism in Analog Insydes.
Info[AnalogInsydes] (Section 3.15.6) prints the exact location of your Analog Insydes installation
and lists all loaded init files. For a description of init file loading see Section 3.15.1. For further
information on the Analog Insydes environment see Chapter 3.15.
1.2.2 The First Step
To work with Analog Insydes start Mathematica and open a new notebook window. Then load
Analog Insydes by evaluating the command
<<AnalogInsydes‘
in the first command line (please do not forget to type the quote character "‘").
The above command reads in the Analog Insydes context and sets up autoload declarations for all
other Analog Insydes packages. If you have not launched a Mathematica kernel yet, you will have to
wait a few extra seconds until the kernel is loaded.
If Mathematica complains that it cannot find the master package check your context search path by
inspecting the value of the Mathematica variable $Path. The directory in which the Analog Insydes
subdirectory resides should be on this list. If this is not the case then add the package path to $Path
by typing AppendTo[$Path, pathspec] and try loading Analog Insydes again. Another common error
is to type the wrong quote character, thus be sure to use "‘".
If you are already familiar with Analog Insydes 1 you probably want to test the new version before
reading the manual. Almost all commands of version 1 are also available in version 2, but please
note that the function patterns may have changed (see Section 1.3.3 for more details). If you are not
familiar with Analog Insydes, please at least go through the example in Section 1.2.3 before using
Analog Insydes. This section will give you a first impression how to work with Analog Insydes.
Additionally, we recommend to take a look at the demo notebooks which are available for download
at
http://www.analog−insydes.de/demos2.html
There you will find a number of different problems and tasks from the field of electrical engineering
and how to solve them using Analog Insydes 2.
1. A Short Introduction
12
Once you have loaded Analog Insydes into the Mathematica kernel you can obtain detailed information concerning your local Analog Insydes installation by means of the following commands:
ReleaseInfo[AnalogInsydes]
ReleaseNumber[AnalogInsydes]
prints information about your Analog Insydes license type
returns the release number of your Analog Insydes
installation
License[AnalogInsydes]
returns your Analog Insydes license number
Version[AnalogInsydes]
prints a detailed description of your Analog Insydes
version
Info[AnalogInsydes]
prints the location of your Analog Insydes installation and
the list of loaded init files
Obtaining Analog Insydes release information.
If you encounter any bug in Analog Insydes, please send a bug report to:
analog.insydes@itwm.fhg.de
When reporting problems, please proceed as follows: Evaluate all commands in a Mathematica
notebook until your problem occurs and describe it as detailed as possible. Whenever possible,
print Netlist and Circuit objects as well as DAEObjects in InputForm . Afterwards, please add
the information concerning your Analog Insydes installation by evaluating each of the following
commands in separate cells of the Mathematica notebook and attach this notebook to your problem
report:
$VersionId
$Path
$LicenseID
ReleaseInfo[AnalogInsydes]
Version[AnalogInsydes]
License[AnalogInsydes]
Info[AnalogInsydes]
Please be sure to evaluate the commands after your problem occurs, because otherwise some modules might not be loaded during the evaluation of Version[AnalogInsydes] . If possible, attach
additional data such as netlist or data files to the email as well.
1.2.3 An Example Application
This example describes the standard way how to use Analog Insydes. It can easily be adapted to
different analysis tasks and applications. The usage of many different commands, from netlist import
to linear simplification techniques, is explained and cross references are given where to find more
information. Reading this section should give you a first overview of the capabilities of Analog
Insydes.
1.2 Getting Started
13
Analysis Task
Following, we want to analyze the ΜA741 operational amplifier. The task is to find an interpretable
symbolic formula for the corner frequency of the small-signal voltage transfer function. This is useful
to identify those circuit elements and parasitics, which have a dominant influence on it. In addition,
the symbolic transfer function can be utilized for further investigation, such as the extraction of its
poles and zeros, to find methods for the compensation of certain effects.
Importing Schematics
To import the schematics picture of the operational amplifier, we use the Analog Insydes command
DXFGraphics (Section 3.13.2). This command translates two-dimensional DXF data into Mathematica
graphics objects which can be displayed in a Mathematica notebook. Most schematics entries of
commercial circuit simulators support the export of schematics in DXF format. DXFGraphics allows
for importing your schematics into Mathematica for documentation purposes.
Before using DXFGraphics , of course we have to load the Analog Insydes master package as described
in Section 1.2.2.
In[1]:= <<AnalogInsydes‘
In[2]:= DXFGraphics["AnalogInsydes/DemoFiles/OP−741.dxf"]
+
-
+
-
5 Vic
Vos
0
1
+
4
Q8
Q9
QPNP741
QPNP741
Vid
2
Q1 QNPN741
QNPN741
6
Q3
QPNP741
Q12
Q131
QPNP741
QPNP741
Q132
QPNP741
Q15
R5
40k
7
Q4
QPNP741
Q14
22
16
Q2
QNPN741
Q18 QNPN741
Q19
QNPN741
17
23
R10
R6
27
C1
30p
V1
25
V
26 R13
1k
R7
Q21 22
QPNP741 27
Q222
24 Q20
Q221
8 Q7 QNPN741 9
R12
QPNP741
18Q16 QNPN741
QPNP741
QPNP741
15 Q11
Q17
300 Q10
Q5
Q6 QNPN741
QNPN741
QNPN741
QNPN741
19
QNPN741
10
20
21
Q23
11
12
QNPN741
14
R8
R3 R2
R4
Q24
R9 QNPN741
R1
R11
50k 1k
5k
50k 100
1k
50k
3
+
-
QNPN741
0
0
50k
+
-
V2
13
Out[2]= Graphics DXFGraphics provides many options for altering the appearance of imported DXF files. They are
described in Section 3.13.2.
Importing Netlist Data
For the application of symbolic analysis, especially symbolic approximation, the next step is to generate a netlist including the numerical reference information. The netlist of the operational amplifier
ΜA741 consists of 45 elements. Writing the complete circuit in the Analog Insydes netlist language
(which is described in Chapter 2.2 and Chapter 2.3) by hand would be a tedious and error-prone
task. Therefore, Analog Insydes provides the command ReadNetlist (Section 3.10.1), which auto-
1. A Short Introduction
14
matically translates an external netlist description into the Analog Insydes netlist format. You can
import netlists written for or generated by PSpice, Eldo, and Saber.
The first argument to ReadNetlist is the simulator input file. Since this file (*.cir in our case)
only contains the numerical values of the elements, the operating-point information and the values
for the small-signal parameters of the transistors have to be extracted from the simulator output file
(*.out in our case). This is performed automatically by ReadNetlist , too. All we have to do is
specifying the corresponding simulator output file as second argument.
In[3]:= op741 = ReadNetlist[
"AnalogInsydes/DemoFiles/OP−741.cir",
"AnalogInsydes/DemoFiles/OP−741.out",
Simulator −> "PSpice", KeepPrefix −> False]
Out[3]= Circuit As the syntax of external netlist files varies for each simulator, you have to specify from which
external format you are reading by means of the Simulator option of ReadNetlist .
The return value of ReadNetlist is a Circuit object which contains symbolic values for each netlist
element as well as the corresponding numerical reference information. By default, the Circuit object
is displayed as a single line in your output cell. To view the complete netlist apply the command
DisplayForm to the Circuit object (we will not evaluate the command since the output is too big
to be shown here):
DisplayForm[op741]
Additionally, you can use the command Statistics (Section 3.6.17) to get an impression of the
circuit complexity. This command prints the number of different element types occuring in the
circuit.
In[4]:= Statistics[op741]
Number of Resistors
: 13
Number of Capacitors
: 1
Number of VoltageSources
: 5
Number of models BJT/QNPN741 : 15
Number of models BJT/QPNP741 : 11
Total number of elements
: 45
Setting Up Circuit Equations
Once the netlist is imported, the next step is to set up the symbolic circuit equations. In Analog
Insydes this can be done automatically using the command CircuitEquations (Section 3.5.1). The
Analog Insydes netlist read by ReadNetlist contains generic model references for each transistor and
does not specifiy a concrete model implementation. Thus, we have to instruct Analog Insydes which
model to use during equation setup by means of the option ModelLibrary . Analog Insydes comes
with a predefined SPICE-compatible, symbolic model library in three different complexity levels
called "FullModels‘" , "SimplifiedModels‘" , and "BasicModels‘" (the implemented models are
explained in detail in Chapter 4.3. For special tasks the model library can be extended by the user;
this topic is discussed in Chapter 2.3). In the following, we choose a complexity-reduced BJT model
1.2 Getting Started
15
from the Analog Insydes model library by setting the option ModelLibrary −> "BasicModels‘" .
Note that a quote mark ("‘") is following the word BasicModels .
In[5]:= mnaAC741 = CircuitEquations[op741,
ElementValues −> Symbolic,
ModelLibrary −> "BasicModels‘"]
Out[5]= DAEAC, 33 33 The additional option ElementValues −> Symbolic tells Analog Insydes to set up equations using
the symbolic element values instead of the numerical ones. By default, CircuitEquations sets up
small-signal equations in matrix formulation, but DC or transient equations can be set up, too. See
Section 3.5.1 and Chapter 2.4 for details. CircuitEquations returns a DAEObject which contains
the equation system as well as additional data which is used and extracted automatically by other
Analog Insydes commands. As for Circuit objects, a DAEObject is printed in short notation only
and can be displayed using DisplayForm (we once more omit evaluating the command since the
equation system is too big to be printed here):
DisplayForm[mnaAC741]
Again, you can use Statistics (Section 3.6.17) to get an impression of the equation size.
In[6]:= Statistics[mnaAC741]
Number of equations
: 33
Number of variables
: 33
Number of entries
: 1089
Number of non−zero entries : 171
Complexity estimate
: 1.6e21
Importing Simulation Data
When using netlists from external simulators we always have to verify that the models used during
equation setup are compatible to the ones used by the simulator. This is done by comparing the
numerical solution calculated by the external simulator with the solution calculated by Analog Insydes. If both solutions do not coincide within a tolerable error range, we have to go back one step
and choose different models during equation setup before going on with the symbolic analysis. To
simplify this procedure, Analog Insydes provides the command ReadSimulationData (Section 3.10.3)
for importing numerical data generated by an external simulator. As for ReadNetlist we have to
specify from which format we are reading by means of the Simulator option. The list of supported
formats is shown in Section 3.10.1.
In[7]:= op741data = ReadSimulationData[
"AnalogInsydes/DemoFiles/OP−741.csd",
Simulator −> "PSpice"]
Out[7]=
V26 InterpolatingFunction0.1, 1. 106 , ReadSimulationData returns the result according to the Analog Insydes numerical data format
which is described in Section 3.7.1. It consists of a list of rules, where the variables (which are
returned as strings) are assigned InterpolatingFunction objects.
1. A Short Introduction
16
Graphically Displaying Simulation Data
We can use Mathematica’s ReplaceAll operator (or /. in short notation) to extract a single interpolating function from the numerical data and assign it to a new variable.
In[8]:= vout741PSpice = "V(26)" /. First[op741data]
Out[8]= InterpolatingFunction0.1, 1. 106 , The variable vout741PSpice now contains the numerical data for the output voltage at node 26.
Its waveform can be graphically displayed in a Bode diagram using the function BodePlot (Section 3.9.1).
In[9]:= BodePlot[vout741PSpice[f], {f, 0.1, 10^6}]
Magnitude (dB)
1.0E-1
1.0E3
1.0E5
1.0E1
1.0E3
Frequency
1.0E5
1.0E1
1.0E3
1.0E5
1.0E1
1.0E3
Frequency
1.0E5
80
60
40
20
0
1.0E-1
1.0E-1
0
Phase (deg)
1.0E1
100
-20
-40
-60
-80
1.0E-1
Out[9]= Graphics Analog Insydes provides several standard graphic functions for visualizing numerical analysis results,
see Chapter 2.5 and Chapter 3.9.
Performing Reference Simulation
A numerical reference simulation is performed applying the Analog Insydes function ACAnalysis
(Section 3.7.3) in the frequency range of interest. The first argument of ACAnalysis is the DAEObject
created by CircuitEquations , which contains the system of equations. The second argument denotes the output variable to solve for. We are interested in the node voltage at node and according
to the automatic naming mechanism (which is described in Section 2.4.1) the corresponding variable
is called V$26.
In[10]:= reference = ACAnalysis[mnaAC741, V$26, {f, 0.1, 10^6}]
Out[10]=
V$26 InterpolatingFunction0.1, 1. 106 , Next, the frequency response calculated with Analog Insydes and the simulation data imported from
PSpice are compared graphically, using the capability of the command BodePlot to display several
transfer functions within one plot.
1.2 Getting Started
17
In[11]:= BodePlot[reference, {vout741PSpice[f], V$26[f]},
{f, 0.1, 10^6}, ShowLegend −> False]
Magnitude (dB)
1.0E-1
1.0E3
1.0E5
1.0E1
1.0E3
Frequency
1.0E5
1.0E1
1.0E3
1.0E5
1.0E1
1.0E3
Frequency
1.0E5
80
60
40
20
0
1.0E-1
1.0E-1
0
Phase (deg)
1.0E1
100
-20
-40
-60
-80
1.0E-1
Out[11]= Graphics The curves match perfectly in the frequency range of interest. Hence, the simplified BJT model is
sufficient for being used in the next step of the symbolic analysis flow.
Calculating the Complexity of the Symbolic Transfer Function
Recall that the task is to calculate the symbolic transfer function in order to extract a formula for the
corner frequency. Thus, at this point it is useful to estimate the complexity of the symbolic transfer
function to find out whether the number of its terms is manageable. This can be done with the help
of the function ComplexityEstimate (Section 3.11.1).
In[12]:= ComplexityEstimate[mnaAC741] // N
Out[12]= 1.61868 1021
ComplexityEstimate returns an integer value. We use Mathematica‘s N operator to transform this
integer into a real number. The result shows that the number of terms forming the fully expanded
symbolic transfer function is greater than and thus too large to be handled. Therefore,
we will now apply a routine which removes all those terms whose influence on the behavior of the
transfer function is negligible. This drastically reduces the complexity.
Setting Up Circuit Equations for Approximation
To prepare the simplification process we set up the system of circuit equations again. As against
the previous call to the function CircuitEquations , where the circuit equations were given in the
modified nodal analysis format, they are set up in sparse tableau formulation this time. Sparse
tableau is the preferred equation formulation when performing linear approximation tasks. Details
about the Formulation option are given in Section 3.5.1.
1. A Short Introduction
18
In[13]:= staAC741 = CircuitEquations[op741,
ElementValues −> Symbolic, Formulation −> SparseTableau,
ModelLibrary −> "BasicModels‘"]
Out[13]= DAEAC, 402 402 The returned equation system is a DAEObject with equations and variables.
In[14]:= Statistics[staAC741]
Number of equations
: 402
Number of variables
: 402
Number of entries
: 161604
Number of non−zero entries : 1392
Complexity estimate
: 1.6e21
Although this sparse tableau system is more than 10 times bigger than the modified nodal system,
it usually produces better approximation results.
Performing Matrix Approximation
Now we call the matrix approximation routine on the symbolic circuit equations of the ΜA741 circuit.
This routine simplifies the equations based on numerical constraints with respect to a given output
variable. The required information is provided in form of several sampling points, specified in
combination with a maximum error constraint each. The approximation routine then processes the
matrix such that the sought output function is located within the defined error range around the
given sampling points. For further information please refer to Section 3.11.3.
To capture the first corner frequency, we place sampling points at Hz and at Hz with a maximum
error of each.
In[15]:= samplingpoints = {{s −> 2. Pi I 1., MaxError −> 0.3},
{s −> 2. Pi I 10., MaxError −> 0.3}}
Out[15]=
s 6.28319 I, MaxError 0.3, s 62.8319 I, MaxError 0.3
With the given setup, the approximation routine can be called. The computation is carried out
with respect to the output voltage across the load resistor R13, which in this case is equivalent
to the sought transfer function. Again, based on the automatic naming scheme (Section 2.4.1), the
corresponding variable is called V$R13 (note that sparse tableau equations consist of branch voltages
instead of node voltages).
In[16]:= op741approx = ApproximateMatrixEquation[staAC741, V$R13,
samplingpoints]
Out[16]= DAEAC, 402 402 1.2 Getting Started
19
In[17]:= Statistics[op741approx]
Number of equations
: 402
Number of variables
: 402
Number of entries
: 161604
Number of non−zero entries : 455
Complexity estimate
: 14
As another call to the function Statistics shows, the approximation routine could considerably
reduce the complexity of the DAEObject (i.e. the number of non-zero entries), although its equation
size did not change. With just a small number of terms remaining, a symbolic computation of the
transfer function is now possible.
Calculating the Transfer Function
To calculate the transfer function, we now solve the approximated system of equations for the output
voltage:
In[18]:= approximation = Solve[op741approx, V$R13]
Out[18]=
V$R13 gm$Q1 gm$Q16 gm$Q17 gm$Q2 gm$Q3
2
gm$Q4 gm$Q5 R9 Ro$Q131 Ro$Q17 Ro$Q4 Rpi$Q16 Rpi$Q17 VID gm$Q3 gm$Q1 gm$Q2 gm$Q1 gm$Q4 gm$Q5 Ro$Q131 Ro$Q17
gm$Q16 gm$Q17 R8 R9 Ro$Q4 Rpi$Q16 Rpi$Q17 Ro$Q4 R9 Ro$Q4 gm$Q17 R8 Ro$Q4 Rpi$Q17 C1 gm$Q16 gm$Q17 gm$Q3 gm$Q1 gm$Q2 gm$Q1 gm$Q4 gm$Q5 R9
2
Ro$Q131 Ro$Q17 Ro$Q4 Rpi$Q16 Rpi$Q17 s
Again, we use Mathematica’s ReplaceAll operator to extract the calculated solution and assign it to
a new variable.
In[19]:= vout741 = V$R13 /. First[approximation] // Simplify
Out[19]=
gm$Q16 gm$Q17
gm$Q2 gm$Q4 R9 Ro$Q131 Ro$Q17 Ro$Q4 Rpi$Q16 Rpi$Q17 VID gm$Q2 gm$Q4 gm$Q17 R8 Ro$Q131 Ro$Q17 Ro$Q4 Rpi$Q17 R9 Ro$Q17 Ro$Q4 gm$Q16 gm$Q17 R8 Rpi$Q16 Rpi$Q17 Ro$Q131 Ro$Q4 gm$Q16 gm$Q17 R8 Rpi$Q16 Rpi$Q17 C1 gm$Q16 gm$Q17 Ro$Q17 Ro$Q4 Rpi$Q16 Rpi$Q17 s
This expression now represents the approximated symbolic transfer function of the ΜA741 circuit. It
contains only those circuit elements and parasitics that were not removed by the matrix approximation routine. Thus, they have been identified as those elements, which have a major influence on
the behavior of the transfer function.
Verifying Approximation Results
The quality of the approximation can be determined by performing a numerical comparison between
the simplified transfer function and the data imported from the numerical simulator (PSpice in this
case). Therefore, we need to extract the design point from the DAEObject first using the command
1. A Short Introduction
20
GetDesignPoint (Section 3.6.12). The design point specifies those numerical values for each one of
the circuit elements that were given in the netlist.
In[20]:= designpoint = GetDesignPoint[staAC741]
Out[20]=
Rpi$Q1 755999.9999999999,
Ro$Q1 2.19 107 , Cbc$Q1 3.7 1013 , Cbe$Q1 1.47 1012 ,
gm$Q1 0.00029, Cbx$Q1 0., Rpi$Q2 753000.,
Ro$Q2 2.18 107 , Cbc$Q2 3.7 1013 , Cbe$Q2 1.47 1012 ,
gm$Q2 0.000291, Cbx$Q2 0., Rpi$Q3 218999.9999999999,
Ro$Q3 8.46 106 , Cbc$Q3 3.85 1013 , Cbe$Q3 1.42 1012 ,
gm$Q3 0.000287, Cbx$Q3 0., Rpi$Q4 216999.9999999999,
Ro$Q4 8.4 106 , Cbc$Q4 3.87 1013 , Cbe$Q4 1.42 1012 ,
gm$Q4 0.0002879999999999999, Cbx$Q4 0., Rpi$Q5 704999.9999999999, Ro$Q5 2.04 107 , Cbc$Q5 8.28 1013 ,
Cbe$Q5 1.47 1012 , gm$Q5 0.000285, Cbx$Q5 0., Rpi$Q6 704999.9999999999, Ro$Q6 2.04 107 , Cbc$Q6 7.93 1013 ,
Cbe$Q6 1.47 1012 , gm$Q6 0.000285, Cbx$Q6 0.,
Rpi$Q7 533000., Ro$Q7 1.55 107 , Cbc$Q7 2.97 1013 ,
Cbe$Q7 1.48 1012 , gm$Q7 0.0004469999999999999, Cbx$Q7 0.,
Rpi$Q8 89499.99999999999, 86, Cbe$Q19 1.57 1012 ,
gm$Q19 0.00787, Cbx$Q19 0., Rpi$Q20 5809.999999999999,
Ro$Q20 225000., Cbc$Q20 1.11 1012 , Cbe$Q20 4.48 1012 ,
gm$Q20 0.0111, Cbx$Q20 0., Rpi$Q21 3.34 1013 ,
Ro$Q21 6.24 1011 , Cbc$Q21 3.66 1013 , Cbe$Q21 1. 1012 ,
gm$Q21 3.43 1013 , Cbx$Q21 0., Rpi$Q24 1.86 1014 ,
Ro$Q24 9.28 1011 , Cbc$Q24 1. 1012 , Cbe$Q24 1. 1012 ,
gm$Q24 3.77 1018 , Cbx$Q24 0., Rpi$Q23 1.86 1014 ,
Ro$Q23 9.82 1011 , Cbc$Q23 7.13 1013 , Cbe$Q23 1. 1012 ,
gm$Q23 6.91 1014 , Cbx$Q23 0., C1 3. 1011 ,
Rpi$Q222 7549.999999999999, Ro$Q222 291999.9999999999,
Cbc$Q222 3.75 1013 , Cbe$Q222 1.52 1012 ,
gm$Q222 0.008449999999999999, Cbx$Q222 0.,
Rpi$Q221 5. 1013 , Ro$Q221 7.67 1011 , Cbc$Q221 3.75 1013 ,
Cbe$Q221 3.88 1013 , gm$Q221 2.68 1014 , Cbx$Q221 0.,
R13 1000., VID 1., Simulator PSpice
The numerical representation of the approximated transfer function depending on the Laplace variable s is now found by applying the numerical design point data to the symbolic transfer function.
In[21]:= vout741N[s_] = vout741 /. designpoint // Simplify
7.32152 1022
Out[21]= 2.90287 1017 1.51745 1016 s
Another Bode plot shows the PSpice simulation data compared with the approximated numeric
transfer function in the frequency range from Hz to MHz.
1.2 Getting Started
21
In[22]:= BodePlot[{vout741PSpice[f], vout741N[2. Pi I f]},
{f, 0.1, 10^6}, ShowLegend −> False]
Magnitude (dB)
1.0E-1
1.0E3
1.0E5
1.0E1
1.0E3
Frequency
1.0E5
1.0E1
1.0E3
1.0E5
1.0E1
1.0E3
Frequency
1.0E5
80
60
40
20
0
1.0E-1
1.0E-1
0
Phase (deg)
1.0E1
100
-20
-40
-60
-80
1.0E-1
Out[22]= Graphics It is apparent that although the number of terms occuring in the transfer function has been reduced from almost to (!), the curves still match perfectly in the frequency range of
interest. Therefore, the approximation is qualified to replace the actual transfer function for further
computations.
Further Processing
From the approximated transfer function we can now extract additional information, such as the
calculation and graphical representation of its poles and zeros.
The poles of the approximated symbolic transfer function can be found by calculating the zeros of
its denominator.
In[23]:= poles1 = Solve[Denominator[vout741] == 0, s] // Simplify
Out[23]=
s Ro$Q131 Ro$Q17 gm$Q17 R8 Ro$Q4 Rpi$Q17 R9 Ro$Q4 gm$Q16 gm$Q17 R8 Rpi$Q16 Rpi$Q17 C1 gm$Q16 gm$Q17 R9 Ro$Q131 Ro$Q17 Ro$Q4 Rpi$Q16 Rpi$Q17
This single pole of the approximated transfer function corresponds to the dominant pole of the
original transfer function. To achieve its value, we have to insert the given design-point information.
In[24]:= p1 = poles1 /. designpoint
Out[24]= s 19.1299
We now verify the above result by computing the exact pole locations, employing the function
PolesByQZ (Section 3.8.4) on the original equation system.
1. A Short Introduction
22
In[25]:= poles = PolesByQZ[mnaAC741]
Out[25]=
4.19822 1010 , 3.02054 1010 , 3.97873 107 7.96404 107 I,
3.97873 107 7.96404 107 I, 6.47642 106 ,
8.33125 107 1.17 108 I, 8.33125 107 1.17 108 I,
2.53983 108 , 2.04133 108 , 2.02066 108 ,
3.85479 108 , 6.26147 108 , 8.75148 108 , 19.5077,
9.68714 108 , 1.2364 109 , 2.15863 109 6.82302 108 I,
2.15863 109 6.82302 108 I, 2.63327 109 , 7.93728 109 ,
9.89724 109 , 1.58287 1010 , 1.46898 1010 Now we sort the poles in ascending order of their absolute values.
In[26]:= Sort[poles, Abs[#1] < Abs[#2] &]
Out[26]=
19.5077, 6.47642 106 , 3.97873 107 7.96404 107 I,
3.97873 107 7.96404 107 I, 8.33125 107 1.17 108 I,
8.33125 107 1.17 108 I, 2.02066 108 , 2.04133 108 ,
2.53983 108 , 3.85479 108 , 6.26147 108 , 8.75148 108 ,
9.68714 108 , 1.2364 109 , 2.15863 109 6.82302 108 I,
2.15863 109 6.82302 108 I, 2.63327 109 , 7.93728 109 ,
9.89724 109 , 1.46898 1010 , 1.58287 1010 , 3.02054 1010 ,
4.19822 1010 This yields a list, whose first entry represents the equivalent to the numerical solution which was
found before with the help of the approximated transfer function. Since both values match well, this
is another indication that the extracted formula for the pole indeed describes the dominant pole of
the operational amplifier.
For further insights, there are various graphics functions available to visualize the extracted information, such as the functions RootLocusPlot (Section 3.9.5) or NyquistPlot (Section 3.9.4). For details
please refer to the respective sections.
Conclusion
This example showed the application of Analog Insydes on the analysis of the ΜA741 operational
amplifier for extracting a formula for the corner frequency of the small-signal transfer function. In a
first step we imported the netlist and simulation data files from PSpice, set up the equations in Analog Insydes and verified the numerical solution. A complexity estimation showed that calculating the
transfer function for the original equation system is not possible, thus we applied an approximation
routine to achieve a simplified equation system. Using this system it was possible to symbolically
calculate its transfer function. Finally, we achieved a symbolic formula describing the pole of the
simplified transfer function and the dominant pole of the original transfer function. An additional
comparison of the numerical pole values showed that indeed the formula represents the dominant
pole. This symbolic formula now provides deeper insight into the corner frequency location.
1.2 Getting Started
23
1.2.4 The Online Help System
Because Analog Insydes is a complex system, it comes with a detailed online documentation: For
quick information, a compact help text is available for each Analog Insydes symbol. This text can be
displayed using the Mathematica ? operator (which is also called Information ). For Analog Insydes
commands, the purpose and function patterns are printed, for Analog Insydes options, the purpose
and possible values are given. For example:
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Get help text for the
command LoadModel .
In[2]:= ?LoadModel
LoadModel[name, selector] or LoadModel[{{name1,
selector1}, ...}] loads one or several models from the
model library. LoadModel[name, selector, Scope −>
scope] assigns local or global scope to the loaded
model.
Get help text for the
option Protocol .
In[3]:= ?Protocol
Protocol is a global Analog Insydes option used by several
functions. It specifies if and where protocols should
be printed. Option values are StatusLine, Notebook,
All, or None, or a list of these symbols.
For more detailed information, the complete Analog Insydes documentation including tutorial and
appendix is available in the standard Mathematica online help system. Launch the help browser
and select the category Add-ons. In the left column then appears a row labeled Analog Insydes. By
choosing this item you enter the Analog Insydes online documentation. For searching a certain
keyword, select the Master Index category and type the keyword in the Go To: field. One or more
hyperlinks to corresponding sections in the Analog Insydes documentation will then be given.
Note that for the documentation to appear in the help browser, Analog Insydes has to be properly
installed on your system and the help index has to be recalculated. For the latter choose the menu
item Help->Rebuild Help Index.
1.2.5 The Analog Insydes Palette
Analog Insydes provides a palette window which allows you to enter nearly all Analog Insydes
commands by simply pressing a button.
The palette is divided into subgroups which can be opened by clicking on the triangle symbol.
Clicking on a button pastes the corresponding text into your session notebook. Clicking on a button
marked with a blue star pastes a more complex skeleton into your notebook. The status line of the
palette window indicates which skeleton is pasted.
1. A Short Introduction
24
The Analog Insydes palette
By choosing the menu item File->Palettes->AnalogInsydes you launch the Analog Insydes palette
window.
Alternatively, you can find the Analog Insydes palette in the file
aidir/FrontEnd/Palettes/AnalogInsydes.nb
of your Analog Insydes installation. Use the command Info (Section 3.15.6) to obtain the path of
your Analog Insydes installation directory.
1.3 What’s New?
25
1.3 What’s New?
Since the first release in 1998, Analog Insydes has been successfully applied to real-world problems
by our customers. The key algorithms implemented in Analog Insydes – particularly the symbolic
approximation methods – proved to be very effective, but it turned out that improvements regarding
tool functionality, user-friendliness, interfaces, processing speed, and the device model library would
be desirable. After three years of development, here is the result: Analog Insydes Version 2. This
chapter guides you through the most noticable enhancements of the new version.
1.3.1 New Features: An Overview
New Interfaces for PSpice, Eldo, and Saber
Netlist import including small-signal and model-card parameters
Behavioral model writer
Simulation data import and export
New Device Model Library
PSpice and Eldo compatible models for R, L, C, D, BJT, MOS, and JFET
BJT: full Gummel-Poon small-signal and large-signal models
MOS: full Level 1-3 + BSIM3v3 small-signal models, Level 1 large-signal model
Extended Circuit Description and Modeling Language
Model parameter sets, global parameters, multiple source values
Enhanced modeling language for nonlinear dynamic systems
Improved Circuit Equation Setup
Automatic model library search
Automatic design-point extraction for linear and nonlinear circuits
Consistent data handling through DAEObjects
New and Improved Analysis Modes
AC, noise, pole/zero, DC, DC transfer, temperature, transient, parametric, root locus
1. A Short Introduction
26
New and Improved Symbolic Approximation Techniques
New and improved linear approximation techniques for AC and pole/zero analysis
New approximation techniques for nonlinear circuits
Full Integration of High-Performance MathLink Binaries
For symbolic matrix approximation and numerical pole/zero analysis (QZ algorithm)
Multi-platform support (Solaris, Linux, HP-UX, Windows)
Graphics Functions
New and improved graphics functions
Improved command interfaces for a more comfortable use
Improved User Interface
Improved user interface allows for setting up and running circuit analyses from PSpice, Eldo, or
Saber netlists with as few as three Analog Insydes commands
1.3.2 New Features: A More Detailed Description
The most noticeable changes between Analog Insydes versions 1 and 2 were made to the circuit
description and modeling language and to the way circuit equations are set up, stored, and presented
to the user:
Netlist Format
The circuit description language (netlist format) was extended by functionality for managing device
model parameter sets (model cards), global parameter settings, initial conditions, and multiple source
values for different analysis modes (AC, DC, transient). With these enhancements it is now possible to represent all netlist and device model information contained in a SPICE circuit description
consistently in an Analog Insydes Circuit object.
The netlist fragment shown below illustrates the use of some new language features. Note that
the unspecific arguments of the keywords Model, Selector, and Parameters in the value field
of the device reference Q1 permit an easy selection of device models and parameter sets at run
time using Mathematica’s pattern rewriting functionality. The model expansion mechanism of Analog
Insydes 2 makes use of this approach to automatically select appropriate device models according to
the specified analysis mode.
1.3 What’s New?
Analog Insydes 2 netlist
fragment
27
amplifier = Circuit[
Netlist[
{V1, {1, 0}, Symbolic −> V1,
Value −> {AC −> 1, DC −> 2.5,
Transient−>SinWave[Time, 2.5, 0.1, 1000]}},
{Q1, {2 −> C, 1 −> B, 3 −> E},
Model−>Model[BJT, NDEF, Q1],
Selector −> Selector[BJT, NDEF, Q1],
Parameters −> Parameters[BJT, NDEF, Q1],
GM$ac −> 0.02, RPI$ac −> 5.0*10^3, },
],
ModelParameters[Name −> NDEF, Type −> NPN,
IS −> 10^−16, BF −> 100, BR −> 1, VAF −> 150],
GlobalParameters[TEMP −> 300, GMIN −> 10^−12]
]
Circuit Equations
In Analog Insydes 1, symbolic circuit equations and corresponding numerical information (design
point, initial conditions, etc.) were not handled in a consistent fashion. To prevent errors due to potentially inconsistent data and to provide a unified and more user-friendly mechanism for managing
all additional data associated with a system of symbolic circuit equations, the internal representation
of circuit equations was fundamentally changed in version 2. Now, all equation setup and circuit
analysis functionality is centered around a common data structure, the DAEObject.
A DAEObject is generated by means of the command CircuitEquations . It represents a linear
or nonlinear system of differential-algebraic equations (DAE) along with numerical data needed for
numerical analysis or symbolic approximation. In addition, a DAEObject contains information about
the type of equations it represents (AC/DC/transient, MNA/STA) plus a snapshot of all relevant
option settings taken at the point of time when the equations were set up. This data encapsulation
approach ensures the consistency of all data belonging to a particular circuit analysis context and
reduces the number of parameters that need to be passed to functions which operate on circuit
equations.
Interfaces
Analog Insydes 2 provides several new or improved modules for interfacing with commercial circuit
design environments. Platform-independent netlist converters with small-signal data and modelcard import functionality are now available for a variety of widely used circuit simulators, including
PSpice, Saber, and Eldo. Waveforms can be imported from PSpice, Saber, and Eldo data files. Circuit
schematics can be imported into Mathematica via a DXF file converter. Moreover, Analog Insydes 2
features a model writer which is capable of generating behavioral model templates (e.g. Saber MAST)
automatically from a DAEObject.
28
1. A Short Introduction
Modeling Language
As a consequence of the requirements identified during recent work on symbolic modeling and
analysis of nonlinear circuits, the behavioral modeling capabilities for nonlinear dynamic systems
were substantially extended. With the enhanced model description format of Analog Insydes 2, it is
now possible to designate model port nodes as optional, to specify default values for model parameters as well as initial conditions and guesses for model variables, and to control how design-point
information is generated for symbolic parameters of model equations.
Library Concept
Based on the modeling language extensions described above, a completely new device model library
concept was designed for Analog Insydes 2. PSpice and Eldo compatible models are available for R,
L, C, D, BJT, MOS, and JFET devices. Three model simplification levels can be selected when setting
up circuit equations to control the complexity of symbolic analysis.
Nonlinear Symbolic Analysis
Completely new functionality has been implemented for symbolic analysis and approximation of
nonlinear circuits. Based on a flexible user interface, the approximation routines allow an arbitrary
choice of analysis and error-control functions. In combination with the new model writer, Analog
Insydes 2 can be used for automated generation of nonlinear behavioral models.
Pole/Zero Analysis
Efficient numerical pole/zero and root locus analysis is now supported through two different generalized eigenvalue problem solvers: an enhanced version of the QZ algorithm and a variant of the
Jacobi orthogonal correction method (JOCM). The introduction of the DAEObject concept reduces
the amount of parameters passed to functions. In addition, a new matrix-based method for symbolic pole/zero analysis has been implemented. The algorithm simplifies a symbolic generalized
eigenvalue problem with respect to a selected root, using the JOCM for efficient iterative error
tracking.
User Interface
The use of Analog Insydes 2 has been considerably simplified by combining many frequently used
steps in powerful macro commands and providing more default actions. With as few as three Analog Insydes commands it is possible to set up and run circuit analyses from PSpice, Eldo, or Saber
netlists. In addition, the external binaries (QZ algorithm, matrix approximation) have been fully
integrated into Analog Insydes 2 and can be used transparently like built-in commands.
1.3 What’s New?
29
Multi-Platform Support
Multi-platform support is available for all binaries: One Analog Insydes installation can be accessed
from different platforms. Transparently for the user the correct binaries are automatically chosen.
This allows for installing Analog Insydes on a site-wide file server and simultaneously using it from
different platforms.
1.3.3 Compatibility
To realize the improvements made in version 2 of Analog Insydes, drastic changes on the internal
data format had to be performed, such that we could not keep 100% compatibility with version 1.
As a consequence, version 1 notebooks may not evaluate without errors using Analog Insydes 2.
But fortunately, only some minor changes (like modifying function arguments) are usually sufficient
to evaluate with version 2. These changes are listed below.
ApproximateMatrixEquation
altered pattern according to ApproximateTransferFunction
former option PrintProtocol now called Protocol, option values changed
CircuitEquations
option DefaultModelLibrary now called ModelLibrary
option DifferentialOperator now called FrequencyVariable
option setting SparseTableauVariant −> Basic | Extended now obtained by
Formulation −> SparseTableau | ExtendedTableau
option setting SymbolicValues −> True | False now obtained by
ElementValues −> Symbolic | Value
option setting TimeDomain −> True | False now obtained by AnalysisMode −> Transient | AC
option UnitValues is obsolete
CSDFRead
former command CSDFRead substituted by new implementation called ReadSimulationData
DXFGraphics
option FillColor now called FillColors , option values modified
options LineThickness , LineAbsoluteThickness , PolylineWidthScale , PrimitivesOut ,
TextOffsetX , TextOffsetY , and TextAlignBottom are obsolete
1. A Short Introduction
30
NDAESolve, NDAESolveDC
former command NDAESolveDC substituted by additional pattern for NDAESolve
option AnalysisMethod now called AnalysisMode
modified values for option Protocol
RootLocusPlot
modified values for options PoleStyle and ZeroStyle
SolveCircuitEquations
former command SolveCircuitEquations substituted by Solve
spice2ai.exe, SpiceToAI, WhereIsSpice2ai
former external executable spice2ai.exe and wrapper function SpiceToAI substituted by new
implementation ReadNetlist
options Spice2aiOptions and Spice2aiPath are obsolete
command WhereIsSpice2ai is obsolete
CheckedCircuit, CheckedNetlist, FlatNetlist
replaced by NetlistObject[Checked | Flat]
Tutorial
2.1
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2
Netlists . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3
Circuits and Subcircuits
2.4
Setting up and Solving Circuit Equations . . . . . . . . . . 70
2.5
Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.6
Modeling and Analysis of Nonlinear Circuits . . . . . . . . 90
2.7
Time-Domain Analysis of Nonlinear Dynamic Circuits . . 104
2.8
Linear Symbolic Approximation . . . . . . . . . . . . . . 127
2.9
Frequency-Domain Analysis of Linear Circuits . . . . . . 142
. . . . . . . . . . . . . . . . . . . 46
2.10 Nonlinear Symbolic Approximation . . . . . . . . . . . . 164
2. Tutorial
32
2.1 Preface
2.1.1 How to Read this Tutorial
Whereas Part 3 of this book serves as a reference manual for all Analog Insydes commands, Part 2
explains dedicated topics of Analog Insydes in a tutorial manner. Although this part is entitled
Tutorial, it is not meant to be read entirely at once by the novice user before working with Analog
Insydes. Rather, each chapter can be read separately to get a more detailed insight into the corresponding topic. For example, Chapter 2.3 describes how to write your own subcircuits and models
and how to use them during equation setup. However, in the usual application, netlists are imported
automatically using the command ReadNetlist , and equations are set up using predefined models
of the Analog Insydes model library. Thus, in most cases there is no need to write netlists and
models by hand. Nevertheless, you may encounter a task that requires a special hand-written device
model. In this case it is recommended to consult Chapter 2.3 (and perhaps Chapter 2.2 as well) to
learn more about the fine points of Analog Insydes’ modeling capabilities. For the novice user we
recommend to read Chapter 2.4, Chapter 2.5, Chapter 2.7, and Chapter 2.9.
2.1.2 Loading Analog Insydes
The first thing we need to do before we can use any circuit analysis function is to load the Analog
Insydes master package. This file reads in the context AnalogInsydes‘Main‘ which contains all
function definitions for working with netlists. In addition, the master package sets up autoload
declarations for other Analog Insydes functions which are not defined in the Main context.
Loading of Analog Insydes is performed by evaluating the command
<<AnalogInsydes‘
in a Mathematica notebook window.
If this command fails then check if your package directory is on Mathematica’s list of search paths. For this,
evaluate the expression $Path in a Mathematica notebook and inspect the result. If the path to your Analog
Insydes directory is not listed then append it to the list with the command AppendTo[$Path, "your Analog
Insydes path"] and re-execute the above command.
Besides loading the Analog Insydes context and installig autoloaders, the above command additionally reads several init files which allow for adapting Analog Insydes to your local needs.
Section 3.15.1 describes the location of the init files, the loading order, and how to suppress loading
of init files. An init file can for example be used to permanently change default values of options.
Once Analog Insydes is loaded, you can obtain information on your local Analog Insydes installation
using the commands described in Chapter 3.15.
2.2 Netlists
33
2.2 Netlists
2.2.1 The Netlist Format
Analog Insydes provides functions which can automatically set up several types of circuit equations
from the netlist description of a circuit. Netlists are sequences of Mathematica lists encapsulated by
the Analog Insydes command Netlist (Section 3.1.1). There must be one such list, or netlist entry,
for each element in a circuit. Netlist entries are not required to be listed in any particular order.
netlistname =
Netlist[
netlist entry 1,
netlist entry 2,
]
A Netlist object is the Analog Insydes data structure for representing flat netlists. For hierarchical
circuit descriptions using subcircuits, the Circuit object is used which is discussed in Chapter 2.3.
For a simple example refer to the voltage divider circuit shown in Figure 2.1.
1
R1
V0
2
R2
Figure 2.1: Voltage divider circuit
An Analog Insydes netlist desribing the circuit looks as follows:
In[1]:= <<AnalogInsydes‘
In[2]:= voltageDivider =
Netlist[
{V0, {1, 0}, 10},
{R1, {1, 2}, R1},
{R2, {2, 0}, R2}
]
Out[2]= NetlistRaw, 3 2. Tutorial
34
The Netlist command returns a raw netlist object which prints its content in short notation, i.e.
only the number of elements is shown. To get a more detailed view of the netlist content use the
command DisplayForm :
In[3]:= DisplayForm[voltageDivider]
Out[3]//DisplayForm= Netlist Raw, 3 Entries:
V0, 1, 0, 10
R1, 1, 2, R1
R2, 2, 0, R2
The netlist entry format (see Section 3.1.3) bears some remote similarity to that of the well-known and
widely used numerical circuit simulator SPICE in that circuit elements are specified by an individual
name, a list of nodes, and a value. In Analog Insydes, netlist entries must be lists of three fields
which are called the reference designator, the connectivity list, and the value field:
{reference designator, {connectivity list}, value field}
This global scheme describes the only valid syntax for netlist entries – there is no exception to this
format.
Reference Designators
The reference designator is a unique name by which a particular circuit element can be distinguished
from all other elements in the same netlist. Typically, the leading one, two, or three characters of a
reference designator implicitly determine the type of the corresponding element. Hence, V0 denotes
a voltage source (type tag V), R1 and R2 denote resistors (type tag R), and VCCS a voltage-controlled
current source (type tag VC).
There are mechanisms to override automatic type detection from reference designators explicitly but this and
other related advanced topics will be discussed later.
Unlike Mathematica variables, type tags are not case sensitive. Therefore, two circuit elements with
reference designators R1 and r1 are both recognized as resistors. However, the symbols R1 and r1
represent two entirely different elements.
Connectivity Lists and Node Identifiers
The connectivity list specifies the nodes of the circuit to which the terminals of an element are
connected. For that purpose, every node in a circuit must be given a unique name, a node identifier,
by which it can be referenced. While some circuit simulators require the nodes to be enumerated
by consecutive nonnegative integers, Analog Insydes lets you choose node identifiers quite freely.
You do not have to number your nodes consecutively, nor do you need to use numbers as node
identifiers at all. In addition to nonnegative integers, you may also use symbols or strings as node
labels. The only requirement is that the circuit’s ground node must be identified by the label 0
(zero). Internally all node identifiers are converted to strings. Thus, the node identifiers OUT and
"OUT" refer to the same node. Moreover, node identifiers are case sensitive. Thus, OUT and Out refer
to different nodes.
2.2 Netlists
35
out1
4
1
V1
R
gm*V1
6
2
out2
Figure 2.2: Resistor and voltage-controlled current source
For a two-terminal element, such as a resistor (Section 4.2.1), the connectivity list must contain
exactly two node identifiers whereas a controlled source, i.e. a four-terminal element, requires four
node identifiers: two for the controlling branch and two for the controlled branch. Netlist entries for
the resistor and the voltage-controlled current source (Section 4.2.13) shown in Figure 2.2 thus have
to be written as follows:
{R1, {1, 2}, R}
{VC2, {4, 6, out1, out2}, gm}
Reference Directions for Currents and Voltages
In Analog Insydes netlists, the order of the nodes in a connectivity list determines the positive
reference direction for both the associated branch voltage and the branch current. In other words,
the reference directions for voltages and currents always have the same orientation.
1
I$V0
V0
2
Figure 2.3: Voltage source, reference directions for branch voltage and current
In Figure 2.3, the positive terminal of the voltage source (Section 4.2.10) is connected to node 1,
and the negative terminal is connected to node 2. The corresponding netlist entry then defines the
positive reference direction for the branch voltage as well as for the current I$V0 to be oriented from
node 1 to node 2:
2. Tutorial
36
{V0, {1, 2}, V0}
Hence, a positive value for the branch current, I$V0 , denotes a current flowing from node 1 to
node 2. So whenever a circuit analysis yields a negative result for I$V0 there is nothing wrong. This
just means that the current is in fact flowing into the opposite direction, i.e. from node 2 to node 1.
Element Values
As opposed to SPICE, the values of circuit elements need not be purely numerical quantities. Since
Mathematica is capable of performing mathematical calculations symbolically, the element values may
also be any symbolic or mixed symbolic/numeric expressions. In the voltage-divider example (see
Figure 2.1) we assigned a numerical value of 10 (Volts) to the voltage source whereas we used the
symbolic values R1 and R2 for the two resistors. (Note that you don’t have to specify any units
like Volts for numerical values.) In this case, the symbols used for the values of the resistors are
identical to the reference designators, but we could also have supplied any other valid Mathematica
expression.
1
R1=R
V0
out
R2=2*R
Figure 2.4: Voltage divider circuit with different node names and resistor values
Let’s make use of some of the facts presented in the preceding paragraphs by renaming node 2 of
the voltage divider to out, arbitrarily making the assignments R1 R and R2 R, and rewriting
the netlist accordingly (see Figure 2.4):
In[4]:= voltageDivider2 =
Netlist[
{V0, {1, 0}, 10},
{R1, {1, out}, R},
{R2, {out, 0}, 2 R}
]
Out[4]= NetlistRaw, 3 2.2 Netlists
37
2.2.2 Circuit Elements
Element Types
Having learned the basics about the netlist format we will now take a look at the circuit element
types (see Chapter 4.2) which are supported by Analog Insydes. The names of all available types
are stored in a globally accessible list named ElementTypes (Section 3.1.5):
In[5]:= ElementTypes
Out[5]=
Resistor, Conductance, Admittance, Impedance, Capacitor,
Inductor, CurrentSource, VoltageSource, CCCSource, CCVSource,
VCCSource, VCVSource, Nullator, Norator, Fixator, OpAmp, OTAmp,
ABModel, OpenCircuit, ShortCircuit, SignalSource, Amplifier,
Integrator, Differentiator, TransmissionLine, TransferFunction
To obtain information about the netlist format for a particular circuit element, type ?element. This
will print the corresponding usage message to the screen. For example, let’s find out more about
the elements Capacitor (Section 4.2.5) and VCCSource (Section 4.2.13):
In[6]:= ?Capacitor
Capacitor (type tag "C") denotes a linear capacitor. The
netlist format is {Cname, {N1, N2}, value}. Network
equation setup for capacitors is influenced by the
value field option Pattern.
In[7]:= ?VCCSource
VCCSource (type tag "VC") denotes a linear
voltage−controlled current source. The netlist format
is {VCname, {C1, C2, N1, N2}, value}. Note that the
nodes of the controlling branch, C1 and C2, are listed
before those of the controlled branch, N1 and N2. As
compared to Spice, the order is reversed!
You may have noticed that there are no such elements as diodes, bipolar or MOS transistors contained
in ElementTypes .
Nevertheless, Analog Insydes comes with a predefined library of symbolic semiconductor device models (see
Chapter 4.3), which contains full-precision SPICE-compatible models for the most important devices such as
Diodes, BJTs, MOSFETs, and JFETs.
In addition to the linear electric circuit elements like resistors and capacitors the list of element types
contains six elements from control theory, namely signal sources, amplifiers, integrators, etc. This is
because Analog Insydes is capable of analyzing linear control circuits as well. You can even perform
symbolic analyses of systems which contain both electrical and control components, allowing you to
do behavioral circuit modeling across different levels of abstraction.
2. Tutorial
38
Controlled Sources: Some Caveats
If you have worked with SPICE netlists before you will see some differences in the way you have
to formulate the netlist entries for controlled sources in an Analog Insydes netlist. Apart from the
more meaningful type tags (VV, CC, VC, CV as opposed to E, F, G, H), Analog Insydes uses a uniform
scheme for all four types of controlled sources, i.e. you have to specify two nodes for the controlling
branch and two nodes for the controlled branch for both voltage-controlled and current-controlled
sources.
N1
C1
I1
r*I1
1
3
V1
gm*V1
C2
N2
2
4
Figure 2.5: Current-controlled voltage source (left) and voltage-controlled current source (right)
For the two controlled sources shown in Figure 2.5 the netlist entries must be written as follows.
Note, that the nodes of the controlling branches are listed first.
{CV1, {C1, C2, N1, N2}, r}
{VC2, {1, 2, 3, 4}, gm}
Analog Insydes treats controlled sources as true two-port elements, so every controlled source adds
two electrical branches to a circuit, a controlling and a controlled branch. In the case of currentcontrolled sources you have to keep in mind that the controlling branch is nothing else than a short
circuit (Section 4.2.8). Therefore, do not connect the controlling branch of a current-controlled source
in parallel to another circuit element. Instead, always use a series connection of the controlling
branch and the circuit element whose branch current controls the source.
Let’s clarify this important point by writing a netlist for the circuit shown in Figure 2.6. The currentcontrolled current source is controlled by the current flowing through the resistor RB. If we wrote
the netlist entry for the CCCS (Section 4.2.11) as
{CC1, {1, 0, 2, 0}, beta}
the controlling branch would be inserted in parallel to RB and thus short-circuit the resistor.
2.2 Netlists
39
CM
2
1
V0
RL
RB
beta*IB
IB
Figure 2.6: Result of incorrectly inserted current-controlled current source
The correct solution to this problem is to add another node to the circuit and connect the controlling
branch in series with RB as shown in Figure 2.7.
CM
2
1
RB
V0
3
IB
RL
beta*IB
Figure 2.7: CCCS circuit with correctly inserted controlling branch
We can then write a correct netlist as follows:
In[8]:= cccsCircuit =
Netlist[
{V0, {1, 0},
{RB, {1, 3},
{CM, {1, 2},
{CC1, {3, 0,
{RL, {2, 0},
]
V0},
RB},
CM},
2, 0}, beta},
RL}
Out[8]= NetlistRaw, 5 2.2.3 A Brief Preview on Circuit Analysis
There is much more to learn about netlists, but before going into the details let’s get an impression of
Analog Insydes’ basic symbolic circuit analysis functionality. Computing voltages and currents in a
circuit requires essentially two steps: Setting up a system of circuit equations and then solving these
equations. Circuit equations can be set up automatically by applying the function CircuitEquations
(Section 3.5.1) to a netlist description.
We set up equations for the voltage divider circuit defined in Section 2.2.1 (see Figure 2.1):
2. Tutorial
40
In[9]:= vdeqs = CircuitEquations[voltageDivider]
Out[9]= DAEAC, 3 3 By default, CircuitEquations sets up a system of modified nodal, or MNA, equations (Modified
Nodal Analysis) in the frequency domain. The return value is a DAEObject which contains the three
components of the linear matrix equation A x b, where A is the coefficient matrix, x the vector
of unknown voltages and currents, and b the vector of independent sources. To display the system
of equations in a more readable form we can apply the command DisplayForm to the DAEObject
vdeqs:
In[10]:= DisplayForm[vdeqs]
Out[10]//DisplayForm=
1
1
1
V$1 0 R1
R1
1
1
1
0 V$2 .
0
R1
R1
R2
10
I$V0
0
0
1
Here, the vector of unknowns comprises the node voltages V$1 and V$2 at nodes 1 and 2, plus the
current I$V0 flowing through the voltage source V0.
We can solve the MNA equations by using the function Solve (Section 3.5.4). Given the system
of equations and no additional arguments, Solve solves for all voltages and currents listed in the
vector of unknowns.
In[11]:= Solve[vdeqs]
10
10 R2
Out[11]= I$V0 , V$2 , V$1 10
R1 R2
R1 R2
The values of V$1 and V$2 correspond to what we would expect from a voltage divider. If you
wonder about the negative sign of I$V0 remember what has been said about positive reference
directions of branch voltages and currents in Section 2.2.1.
As a second example we will compute the node voltage at node 2 of the CCCS circuit (see Figure 2.7)
as a symbolic function of the circuit parameters in the frequency domain.
In[12]:= cccseqs = CircuitEquations[cccsCircuit];
DisplayForm[cccseqs]
Out[13]//DisplayForm=
1
1
CM s
RB
CM s
RB
1
0
CM s
RL
CM s
1
1
0
RB
RB
1
0
0
0
0
1
1
0
0
0
0
0 V$1 0 V$2 beta 0
.
V$3
1 V0
I$V0
0 IC$CC1 0
0 0
The last variable in the vector of unknowns, IC$CC1, denotes the controlling current of the controlled
source CC1 (see Section 2.4.1). We can solve for V$2 by explicitly passing this variable as a second
argument to Solve.
In[14]:= Solve[cccseqs, V$2]
RL beta CM RB s V0
Out[14]= V$2 RB 1 CM RL s
2.2 Netlists
41
The result represents the Laplace transform of the zero-state response of the voltage at node 2,
expressed in terms of the complex frequency variable s.
The zero-state response is the response of a system which was in the zero state at the time just prior to the
application of the input signal. Zero state means that all initial conditions, i.e. capacitor voltages and inductor
currents, are zero.
2.2.4 More about Netlists
The General Value-Field Format
As indicated in the previous section, some aspects regarding the netlist format have not yet been
mentioned. So far, we have used only the simplest form of writing netlist entries (see Section 2.2.1).
In general, however, the value field of a netlist entry does not need to be a single Mathematica
expression but may also be a nonempty sequence of options. Netlist entries may thus look like this:
{name, {nodes}, option1 −> value1 , option2 −> value2 , }
There are several value-field option keywords which Analog Insydes understands (see Section 3.1.4):
Value, Symbolic , Type, Pattern, InitialCondition , Model (Section 3.2.1), and Subcircuit (Section 3.2.2). Their meanings will be discussed in the following subsections.
Value
If the value field is written in its general sequence-of-options form then all of its components must be
options. This necessitates an option which specifies the value of the circuit element, namely Value
(see Section 3.1.4). With the Value option we can rewrite a netlist entry such as
{R1, {1, 2}, R1}
in the following semantically equivalent general form:
{R1, {1, 2}, Value −> R1}
Of course, as long as the value field contains no more information than only the element value the
simple form is the preferred form. Note that if the value field is written in option form, then the
Value option must be present.
Symbolic
The Symbolic option (see Section 3.1.4) is closely related to the Value option. It allows you to
specify an alternative symbolic element value in addition to a numerical value given as argument to
the Value option, for instance:
{R1, {1, 2}, Value −> 100, Symbolic −> R1}
By default, circuit equations are set up using the arguments of the Value options as element values,
but CircuitEquations (Section 3.5.1) can be instructed to use the symbolic values instead wherever
2. Tutorial
42
the Symbolic option is given. This allows for assigning both a numerical as well as a symbolic value
to a circuit element. You will be able to assess the value of this feature better once you have learned
more about the topics described in Chapter 2.3 and Chapter 2.8.
Type
The value field option Type (see Section 3.1.4) allows you to specify the type of an element explicitly,
thus overriding the automatic type detection from an element’s reference designator. The argument
to the Type option must be the full name of a circuit element type supported by Analog Insydes
(see Chapter 4.2). The available types are listed in the global variable ElementTypes . Using the
Type option we could write a netlist entry for a resistor in the following way even though the name
Shunt does not begin with the type tag R.
{Shunt, {1, 2}, Type −> Resistor, Value −> R1}
Moreover, as Analog Insydes generates voltage and current identifiers (see Section 2.4.1) by adding
the prefixes "V$" and "I$" to the reference designators, the Type option gives us some more influence on the names of voltages and currents. In this example, the resistor current would be named
I$Shunt , whereas it would be named I$R1 for the netlist entries from the previous two subsections.
The example below shows a more convincing application of the Type option. Assume that you have
an amplifier circuit with an output load connected between nodes out and ground, and that you
wish to calculate the amplifier’s frequency responses for both resistive and capacitive loads. You
could, of course, write two separate netlists for this purpose. On the other hand, the Type option
offers you the alternative to set up only one single netlist but with a variable load type:
amplifier =
Netlist[
{Load, {out, 0}, Type −> loadtype, Value −> loadval},
]
Then, by replacing the type variable with a concrete type name, you can set up circuit equations for
a particular type of load, e.g. for a capacitor:
CircuitEquations[
amplifier /. {loadtype −> Capacitor, loadval −> CL}]
Note that in both cases the identifier for the load current will be I$Load, regardless of the load
element type eventually selected.
Finally, the Type option helps us to correct a frequently made mistake, which usually occurs
when a netlist contains a supply voltage source named VCC. If we relied on automatic type
detection from the reference designator VCC, Analog Insydes would give us the error message
"Netlist::numnode: Expected 4 nodes but received 2 for VCCSource VCC". The reason for this
error is that VC is the type tag for voltage-controlled current sources, so Analog Insydes interprets
VCC as (VC)C and not as a voltage source V(CC). This problem can be easily solved by means of the
Type directive:
2.2 Netlists
43
{VCC, {1, 0}, Type −> VoltageSource, Value −> VCC}
Pattern
The Pattern option (see Section 3.1.4) can be used only in conjunction with two-terminal immittances, i.e. impedance and admittance elements such as resistors (Section 4.2.1), conductances
(Section 4.2.2), capacitors (Section 4.2.5), and inductors (Section 4.2.6). With this directive you can
explicitly choose whether the contribution of an immittance element to a system of modified nodal
equations is entered into the matrix using the fill-in pattern for admittances or the pattern for
impedances. The two associated values of the Pattern option are Impedance and Admittance .
When setting up modified nodal equations, Analog Insydes automatically converts impedance elements such as resistors into their admittance equivalents in order to keep the matrix size as small
as possible. In other words, a resistor with the value R will be treated as an admittance with the
value R and will be entered into the modified nodal matrix using the fill-in pattern for admittances
(see the example in Section 2.2.3). However, if we are interested in computing the current through a
particular resistor it would be better to use the fill-in pattern for impedances as this would augment
the MNA system by the corresponding branch current. So if we want to calculate the load current of
the above-mentioned amplifier using the MNA formulation we would have to select the impedance
pattern for the load resistor in order to introduce the branch current I$Load:
{Load, {out, 0}, Type −> Resistor, Value −> RL,
Pattern −> Impedance}
Let’s experiment with some value-field options using the CCCS circuit from Section 2.2.2 (see
Figure 2.7). We replace the resistor RL by a variable load type and select the fill-in pattern for
impedances.
In[15]:= cccsCircuit2 =
Netlist[
{V0, {1, 0}, V0},
{RB, {1, 3}, RB},
{CM, {1, 2}, CM},
{CC1, {3, 0, 2, 0}, beta},
{Load, {2, 0}, Type −> loadtype, Value −> loadval,
Pattern −> Impedance}
]
Out[15]= NetlistRaw, 5 We choose a resistive load with the symbolic value RL and set up the MNA equations. Due to the
Pattern directive, the load current I$Load now appears as an additional variable for which we can
solve directly.
2. Tutorial
44
In[16]:= cccseqs2 = CircuitEquations[
cccsCircuit2 /. {loadtype −> Resistor, loadval −> RL}];
DisplayForm[cccseqs2]
Out[17]//DisplayForm=
1
1
RB
CM s CM s RB
CM s
CM s
0
1
1
0
RB
RB
1
0
0
0
0
1
0
1
0
1
0
0 0 V$1 0 0 beta 1 V$2
0
V$3
0
1
0 .
V0
I$V0
0
0
0 0
IC$CC1
0
0
0 0
0
0
RL I$Load In[18]:= Solve[cccseqs2, I$Load]
beta CM RB s V0
Out[18]= I$Load RB 1 CM RL s
Next, we select a capacitive load CL and solve for the load current again.
In[19]:= cccseqs3 = CircuitEquations[
cccsCircuit2 /. {loadtype −> Capacitor, loadval −> CL}];
DisplayForm[cccseqs3]
Out[20]//DisplayForm=
1
1
RB
CM s CM s RB
CM s
CM s
0
1
1
0
RB
RB
1
0
0
0
0
1
0
1
0
1
0
0 beta
0
1
0
0
0
0
0
0
0
1
0
0
0
1
CL s
0 V$1 0 V$2
0 V$3 .
V0
I$V0
0
IC$CC1
I$Load 0
In[21]:= Solve[cccseqs3, I$Load]
CL beta CM RB s V0
Out[21]= I$Load CL CM RB
InitialCondition
With the InitialCondition option (see Section 3.1.4), you can specify initial currents and voltages
for the dynamic elements Inductor (Section 4.2.6) and Capacitor (Section 4.2.5), respectively. If no
initial condition is given explicitly, it will be assumed to be zero (AC analysis) or will be calculated
automatically from the operating-point data (transient analysis).
As an example, we set an initial capacitor voltage of V0 for the netlist entry C1 of the following filter
circuit:
In[22]:= filter =
Netlist[
{V1, {1, 0}, V1},
{R1, {1, 2}, R1},
{C1, {2, 0}, Value −> C1, InitialCondition −> V0}
]
Out[22]= NetlistRaw, 3 2.2 Netlists
45
In[23]:= filtereqs = CircuitEquations[filter,
InitialConditions −> Automatic];
DisplayForm[filtereqs]
Out[24]//DisplayForm=
1
1
1
0
R1
R1
V$1 1
1
.
C1
V0 V$2
C1
s
0
R1
R1
V1 0
0 I$V1 1
See Section 2.7.5 for a description of initial condition handling during equation setup.
Model, Subcircuit
The purpose of the Model and Subcircuit directives is to mark a netlist entry as being a reference
to a device model or subcircuit definition. Chapter 2.3 deals with the details.
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46
2.3 Circuits and Subcircuits
2.3.1 Circuits, Netlists, and Subcircuits
Hierarchical Netlist Entry
The flat netlist representation we used so far is usually sufficient for small analysis tasks with no
more than a handful of circuit elements. Now imagine that we want to analyze active circuits like
the transistor amplifier shown in Figure 3.1, and want to do several transfer function calculations
with different transistor models. It would be rather inconvenient if we had to edit the netlist every
time we select another semiconductor device model. Therefore, Analog Insydes provides a hierarchical netlist entry scheme which allows us to regard the transistor object as a black box with
three terminals and lets us write a netlist for the circuit containing only a reference to the box but
no specification of what’s inside. Separately from this netlist, we can define one or more concrete
realizations of the box in the form of equivalent (sub)circuits and let Analog Insydes automatically
replace the reference by one of the subcircuit definitions.
2
RC
R1
1
3
4
V1
VCC
Q1
R2
Vout
RE
Figure 3.1: Common-emitter transistor amplifier
The Circuit Object
When structured hierarchically, a circuit description comprises a top-level netlist with subcircuit or
model references, and a set of subcircuit definitions.
2.3 Circuits and Subcircuits
47
Throughout this text, the terms "subcircuit" and "model" shall be used as synonyms because small-signal
device models are effectively implemented as subcircuits composed of circuit primitives like resistors and
controlled sources.
Keeping all netlist and subcircuit components of a circuit together is the task of the Analog Insydes
data structure Circuit (Section 3.1.2), whose content sequence may be any number of netlist and
subcircuit definitions.
Circuit[
Netlist[top-level netlist with subcircuit references],
Model[subcircuit/model definition],
Model[subcircuit/model definition]
]
A Circuit object additionally may contain model parameters and global parameters, see
ModelParameters (Section 3.1.10) and GlobalParameters (Section 3.1.11).
When a Circuit object is defined, it is printed in short notation only. To view the entire circuit
structure use the command DisplayForm .
2.3.2 Defining Subcircuits and Device Models
The Model Object
Let’s start to work with hierarchically structured netlists by learning how to define subcircuits using
the Analog Insydes objects Model (Section 3.2.1) or Subcircuit (Section 3.2.2). This section describes
how to write netlist-based models or subcircuits. Section 2.6.2 shows how to write equation-based
models.
Model and Subcircuit are semantically equivalent symbols. In fact, Subcircuit is just an alias name
for Model. You may want to use both names in a Circuit if you wish to give visual cues as to which
subcircuits represent device models by using the Model directive for these and the Subcircuit object for all
other subcircuits.
The Model object takes several arguments which must all be written in Mathematica’s option syntax
keyword −> value. Four of the arguments must always be present whereas the remaining ones are
optional or have default values. Below, the optional arguments are printed in slanted typewriter
font. For now, we will restrict ourselves to the discussion of the required arguments. The optional
ones will be introduced later in a step-by-step fashion, as this will help you to better understand
why and when they are needed.
2. Tutorial
48
Model[
Name −> subcircuit class name,
Selector −> selector,
Scope −> scope,
Ports −> {port nodes},
Parameters −> {parameters},
Defaults −> {defaults},
Translation −> {parameter translation rules},
Definition −> Netlist[subcircuit netlist]
]
The value of the Name argument must be a symbol which identifies an entire group, or class, of
different subcircuit implementations of a non-primitive object, such as a transistor. The value of the
Selector argument, which must also be a symbol, then selects one particular member from this
class. The Ports argument defines the port nodes of a subcircuit structure. Its value must be a list
of the identifiers of the subcircuit nodes which serve as external connection points.
Finally, the netlist of the subcircuit must be specified by means of the Definition argument. There
is no built-in limit for the nesting depth of subcircuits, so the netlist may itself contain references
to other Model definitions. However, you may only reference but not define subcircuits within
subcircuits.
Small-Signal Transistor Models
To fully understand this abstract description of the Model function, let’s examine a practical example.
Assume that we want to calculate the AC voltage transfer function of the common-emitter amplifier
from Section 2.3.1 (see Figure 3.1) and want to replace the transistor by either one of the two
small-signal equivalent circuits shown in Figure 3.2.
2.3 Circuits and Subcircuits
49
C
B
E
CM
B
C
B
C
IB
IB
X
X
R0
RB
RB
E
beta*IB
E
beta*IB
E
E
Figure 3.2: Different transistor small-signal models: simple (left), dynamic (right)
Both circuits have in common that they represent the same object, namely an NPN transistor. Thus,
they are two members of a subcircuit class we shall name NPNTransistor :
Name −> NPNTransistor
Within this class, each member must have a unique selector by which it can be identified. We will
denote the simple small-signal model on the left-hand side of Figure 3.2 by
Selector −> ACsimple
and the one on the right-hand side by
Selector −> ACdynamic
From now on we will adopt the notation name/selector for denoting a subcircuit definition. For instance, we will refer to the model defined with Name −> NPNTransistor and Selector −> ACsimple
as NPNTransistor/ACsimple . Both subcircuits contain four nodes: B, C, E, and X, where B, C,
and E represent the transistor’s base, collector, and emitter terminal respectively. The node X is
an internal subcircuit node which is needed for properly connecting the controlling branch of the
current-controlled current source in series with the base resistor RB (see also Section 2.2.2). Therefore,
we declare only B, C, and E as port nodes of the subcircuits in order to make X invisible from the
outside:
Ports −> {"B", "C", "E"}
Note that we use strings to denote the port nodes instead of symbols. Although this requires some
more typing this is the recommended way for specifying nodes: If we would have used symbols
2. Tutorial
50
in the above example, the symbol for the emitter node E could have been mixed up with the
Mathematica symbol E which denotes the exponential constant e.
Connections to subcircuits can only be made through designated port nodes. Any attempt to interface directly with an internal node will cause the generation of an error message when the netlist
hierarchy is flattened. However, there is one exception: since the ground node (0) is considered to
be global, elements in subcircuit definitions may be connected directly with global ground without
the need to specify 0 as a port node. In fact, 0 cannot be used as port node identifier in Analog
Insydes. Finally, all which remains to do is to write the netlists of the two subcircuits and set up the
Model definitions as follows:
Model[
Name −> NPNTransistor,
Selector −> ACsimple,
Ports −> {"B", "C", "E"},
Definition −>
Netlist[
{RB, {"X", "E"}, RB},
{CC, {"B", "X", "C", "E"}, beta}
]
]
Model[
Name −> NPNTransistor,
Selector −> ACdynamic,
Ports −> {"B", "C", "E"},
Definition −>
Netlist[
{RB, {"X", "E"}, RB},
{CM, {"B", "C"}, CM},
{CC, {"B", "X", "C", "E"}, beta},
{RO, {"C", "E"}, RO}
]
]
In this example, we used node names for all subcircuit nodes given by strings ("B", "C", "E", "X") but
we could have used positive integers just as well. However, using symbolic names is usually preferable. The
reason is that during subcircuit expansion all internal nodes of subcircuit objects will be instantiated and
labeled with unique identifiers such as X$Q1, which are automatically generated from the node names and the
reference designator of the subcircuit instance. If an internal node identifier is an integer, e.g. 2, the generated
instance identifier 2$Q1 could be confused easily with the product 2*$Q1.
2.3 Circuits and Subcircuits
51
2.3.3 Referencing Subcircuits
The Netlist Entry Format for Subcircuit References
Subcircuits or device models defined with the Model command can be referenced from a netlist on a
higher hierarchy level by a netlist entry whose value field contains the options Model and Selector:
{subcircuit instance name, {port connections},
Model −> subcircuit class name, Selector −> selector}
To reference a subcircuit object (see Section 3.1.8) the subcircuit class name and the selector must
be the same identifiers as those given as arguments to the Name and Selector parameters in the
corresponding Model statement. The subcircuit instance name may be any symbol or string which can
serve as the unique identifier of this instance of the subcircuit object.
You can even use subcircuit instance names which begin with an element type tag known to Analog Insydes.
A subcircuit reference with an instance identifier such as RX will never be confused with a resistor because
the reference will have been expanded before circuit equations are set up.
A netlist entry which marks a reference to an instance of the NPN transistor model
NPNTransistor/ACsimple , using the instance identifier Q1, would be written as follows:
{Q1, {connectivity},
Model −> NPNTransistor, Selector −> ACsimple}
This line does not yet specify how the instance Q1 of NPNTransistor/ACsimple should be embedded into the surrounding circuit structure, i.e. which nodes of the latter should be connected to the
subcircuit’s port nodes. Associating external nodes with subcircuit port nodes is done by means of
a special format of the connectivity field. All entries of this field must be written as rules of the form
external node −> subcircuit port node
In our common-emitter amplifier (see Figure 3.1 in Section 2.3.1), node 1 of the top-level netlist is
connected to the base node "B" of the transistor, node 3 to the collector node "C", and node 4 to the
emitter node "E", resulting in a node-to-port mapping such as this:
{Q1, {1 −> "B", 3 −> "C", 4 −> "E"},
Model −> NPNTransistor, Selector −> ACsimple}
Since the mapping is defined by names, the relative positions of the entries in the connectivity field
are not important. We could have written the connectivity field as
{3 −> "C", 1 −> "B", 4 −> "E"}
equally well.
A Hierarchical Netlist Description of the Amplifier
Having learned how to define and how to make references to subcircuits (see Section 3.1.8), we can
now write a hierarchically structured netlist for the common-emitter amplifier from Section 2.3.1 (see
Figure 3.1).
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52
In[1]:= <<AnalogInsydes‘
In[2]:= commonEmitterAmplifier=
Circuit[
Netlist[
{V1, {1, 0}, inputVoltage},
{VCC, {2, 0}, Type −> VoltageSource,
Value −> supplyVoltage},
{R1, {2, 1}, R1},
{R2, {1, 0}, R2},
{RC, {2, 3}, RC},
{RE, {4, 0}, RE},
{Q1, {1 −> "B", 3 −> "C", 4 −> "E"},
Model −> NPNTransistor, Selector −> BJTModel}
],
Model[
Name −> NPNTransistor,
Selector −> ACsimple,
Ports −> {"B", "C", "E"},
Definition −>
Netlist[
{RB, {"X", "E"}, RB},
{CC, {"B", "X", "C", "E"}, beta}
]
],
Model[
Name −> NPNTransistor,
Selector −> ACdynamic,
Ports −> {"B", "C", "E"},
Definition −>
Netlist[
{RB, {"X", "E"}, RB},
{CM, {"B", "C"}, CM},
{CC, {"B", "X", "C", "E"}, beta},
{RO, {"C", "E"}, RO}
]
]
]
Out[2]= Circuit Note that in the top-level netlist of this circuit description we used a generic subcircuit selector
BJTModel and not ACsimple or ACdynamic . This allows us to select the model at run time by
replacing BJTModel with ACsimple or ACdynamic just before the subcircuit hierarchy is expanded.
In addition, for reasons which will become apparent in the next section, we have used the variables
inputVoltage and supplyVoltage to denote the values of the voltage sources V1 and VCC.
Remember that the Type directive in the value field of VCC prevents Analog Insydes from falsely interpreting
the netlist entry as belonging to a voltage-controlled current source.
2.3 Circuits and Subcircuits
53
2.3.4 Subcircuit Expansion
The ExpandSubcircuits Command
Before we can set up circuit equations we must resolve all references to subcircuit objects. Subcircuit
instantiation and flattening the netlist is done by the command ExpandSubcircuits (Section 3.4.1)
which accepts a Circuit (Section 3.1.2) or a Netlist (Section 3.1.1) object as argument and returns
a flat netlist object.Such a flat netlist is similar to a Netlist, except that it is guaranteed to contain
only circuit primitives and no subcircuit references.
Please note that subcircuit expansion is taken care of automatically by CircuitEquations, so that there is
usually no need to call ExpandSubcircuits explicitly, except for inserting models into the global data base.
Section 2.3.6 deals with this topic.
We can now use ExpandSubcircuits to generate one flat netlist of the common-emmitter amplifier
from Section 2.3.1 (see Figure 3.1) for each of the two transistor models we have defined. As the
circuit description is still generic we must first personalize it by replacing its variables according to
the specific analysis task. Our task was to calculate the small-signal voltage transfer function, so
besides choosing an appropriate transistor model we must set the value of the supply voltage source
VCC to zero and the value of the input signal source to (see Section 2.9.2). Then, the node voltage
at the output node 3 will be identical to the transfer function. We apply these variable settings to the
netlist with the Mathematica command ReplaceAll (or /. in short notation) and pass the resulting
netlist to ExpandSubcircuits .
In[3]:= flatAmpSimple = ExpandSubcircuits[
commonEmitterAmplifier /.
{supplyVoltage −> 0, inputVoltage −> 1,
BJTModel −> ACsimple}];
DisplayForm[flatAmpSimple]
Out[4]//DisplayForm=
Netlist Flat, 8 Entries:
V1, 1, 0, 1
VCC, 2, 0, Type VoltageSource, Value 0
R1, 2, 1, R1
R2, 1, 0, R2
RC, 2, 3, RC
RE, 4, 0, RE
RB$Q1, X$Q1, 4, RB
CC$Q1, 1, X$Q1, 3, 4, beta
2. Tutorial
54
In[5]:= flatAmpDyn = ExpandSubcircuits[
commonEmitterAmplifier /.
{supplyVoltage −> 0, inputVoltage −> 1,
BJTModel −> ACdynamic}];
DisplayForm[flatAmpDyn]
Out[6]//DisplayForm=
Netlist Flat, 10 Entries:
V1, 1, 0, 1
VCC, 2, 0, Type VoltageSource, Value 0
R1, 2, 1, R1
R2, 1, 0, R2
RC, 2, 3, RC
RE, 4, 0, RE
RB$Q1, X$Q1, 4, RB
CM$Q1, 1, 3, CM
CC$Q1, 1, X$Q1, 3, 4, beta
RO$Q1, 3, 4, RO
As we can see from the flattened netlists, ExpandSubcircuits has not simply replaced the subcircuit references by the original netlists from the Model definitions. The function has also generated
unique symbols for all internal reference designators and node identifiers in the subcircuit netlists by
appending a separator character ($) and the subcircuit instance name (Q1) to the original identifiers.
This eliminates possible name conflicts with reference designators from the top-level netlist and
allows for using multiple instances of a subcircuit definition.
Analysis of the Common-Emitter Amplifier
Now we can analyze the circuit by means of the functions CircuitEquations (Section 3.5.1) and
Solve (Section 3.5.4). We start by setting up the modified nodal equations for the simple AC model
from the flattened netlist. Note that we could call CircuitEquations on the Circuit object equally
well. We used ExpandSubcircuits for demonstrating the subcircuit expansion mechanism.
2.3 Circuits and Subcircuits
55
In[7]:= eqAmpSimple = CircuitEquations[flatAmpSimple];
DisplayForm[eqAmpSimple]
Out[8]//DisplayForm=
1
1
1
R1
R2
R1
1
1
1
R1
R1
RC
1
0
RC
0
0
0
0
1
0
0
1
1
0
0
0
0
1 0
1
RC
1
RC
0
0
0 1
0
0 0
1
RB
1
RB
0 0
0
0
0
0
0
0
1
RB
1
RE
1
RB
0
0
0
0
0
1
0
0
0
0
0
0
0
0
beta beta .
1 0 0 0 1
0
0
V$1
0
V$2
0
V$3
0
V$4
0
V$X$Q1
1
I$V1 0
I$VCC
0
IC$CC$Q1 To compute the small-signal voltage transfer function we must solve the MNA equations for the
node voltage V$3 at the output node 3.
In[9]:= v3 = Solve[eqAmpSimple, V$3]
beta RC
Out[9]= V$3 RB RE beta RE
The following command extracts the symbolic transfer function expression from the return value of
Solve.
In[10]:= vtf = V$3 /. First[v3]
beta RC
Out[10]= RB RE beta RE
Now, let’s see what happens when we make the (usually realistic) assumption that the transistor
current gain is very large:
In[11]:= Limit[vtf, beta −> Infinity]
RC
Out[11]= RE
The above result is indeed identical to the well-known approximate formula for the voltage gain of
a common-emitter amplifier. Doing the same calculations for the other transistor model shall be left
to the reader as an exercise.
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56
2.3.5 Subcircuit Parameters
Instantiating Element Values in Subcircuit Definitions
Although reference designators and internal node identifiers in subcircuit netlists are always uniquely
instantiated we still have to cope with one difficulty that arises when working with multiple references to the same model definition. Consider the following small-signal equivalent circuit of a
Darlington amplifier (see Figure 3.3) for which we shall have to compute the overall current gain as
a function of the individual current gains of the two transistors Q1 and Q2. The gain can be obtained
by calculating the output response to a unit current I1 applied at the input.
2
1
Q1
I1
Iload
Q2
3
RL
Figure 3.3: Darlington amplifier (small-signal equivalent circuit)
Let’s decide to use the transistor model NPNTransistor/ACsimple from Section 2.3.2 (see Figure 3.2)
for both transistors. We would then write the circuit description and expand the model references
as follows.
Remember that the Pattern option in the value field of the netlist entry Load causes the current through the
load resistor to be introduced into the modified nodal equations, allowing us to compute the current directly.
Alternatively, we could have inserted a short circuit in between RL and node 2 for measuring the current.
2.3 Circuits and Subcircuits
57
In[12]:= Circuit[
Netlist[
{I1, {0, 1}, 1},
{Q1, {1 −> "B", 2 −> "C", 3 −> "E"},
Model −> NPNTransistor, Selector −> ACsimple},
{Q2, {3 −> "B", 2 −> "C", 0 −> "E"},
Model −> NPNTransistor, Selector −> ACsimple},
{Load, {0, 2}, Type −> Resistor, Value −> RL,
Pattern −> Impedance}
],
Model[
Name −> NPNTransistor,
Selector −> ACsimple,
Ports −> {"B", "C", "E"},
Definition −>
Netlist[
{RB, {"X", "E"}, RB},
{CC, {"B", "X", "C", "E"}, beta}
]
]
] // ExpandSubcircuits // DisplayForm
Out[12]//DisplayForm=
Netlist Flat, 6 Entries:
I1, 0, 1, 1
RB$Q1, X$Q1, 3, RB
CC$Q1, 1, X$Q1, 2, 3, beta
RB$Q2, X$Q2, 0, RB
CC$Q2, 3, X$Q2, 2, 0, beta
Load, 0, 2, Type Resistor, Value RL, Pattern Impedance
While ExpandSubcircuits (Section 3.4.1) has generated unique identifiers for the base resistor RB
and the current-controlled current source CC for each instance of the transistor model, it has not
made any changes to the associated symbolic element values RB and beta. Obviously, these have
been regarded as global symbols, resulting in identical base resistances and current gains of both
transistors. This is certainly not the result we wanted because the individual current gains of the
transistors in a Darlington stage may be quite different. Instead, ExpandSubcircuits should instantiate the element values as well, and thus generate the symbols RB$Q1, RB$Q2, beta$Q1 , and beta$Q2
for the two resistors and the two current gains respectively.
The reason behind the unwanted behavior exhibited here is that ExpandSubcircuits has not been
explicitly instructed to treat RB and beta as what they really are, namely as parameters of the model.
We can solve this problem by adding the parameter declaration
Parameters −> {RB, beta}
to our Model definition, which instructs ExpandSubcircuits to replace all occurrences of these symbols in the value fields of the subcircuit netlist by different identifiers for each subcircuit instance.
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58
In[13]:= darlington =
Circuit[
Netlist[
{I1, {0, 1}, 1},
{Q1, {1 −> "B", 2 −> "C", 3 −> "E"},
Model −> NPNTransistor, Selector −> ACsimple},
{Q2, {3 −> "B", 2 −> "C", 0 −> "E"},
Model −> NPNTransistor, Selector −> ACsimple},
{Load, {0, 2}, Type −> Resistor, Value −> RL,
Pattern −> Impedance}
],
Model[
Name −> NPNTransistor,
Selector −> ACsimple,
Ports −> {"B", "C", "E"},
Parameters −> {RB, beta},
Definition −>
Netlist[
{RB, {"X", "E"}, RB},
{CC, {"B", "X", "C", "E"}, beta}
]
]
]
Out[13]= Circuit In[14]:= ExpandSubcircuits[darlington] // DisplayForm
Out[14]//DisplayForm=
Netlist Flat, 6 Entries:
I1, 0, 1, 1
RB$Q1, X$Q1, 3, RB$Q1
CC$Q1, 1, X$Q1, 2, 3, beta$Q1
RB$Q2, X$Q2, 0, RB$Q2
CC$Q2, 3, X$Q2, 2, 0, beta$Q2
Load, 0, 2, Type Resistor, Value RL, Pattern Impedance
After having been defined as parameters, the base resistances and current gains appear properly
instantiated in the flat netlist. We can now complete our analysis of the Darlington amplifier by
solving the MNA equations for the load current I$Load.
2.3 Circuits and Subcircuits
59
In[15]:= eqDarlington = CircuitEquations[darlington];
DisplayForm[eqDarlington]
Out[16]//DisplayForm=
0 0
0
0
0 0
0
0
1
1
0 0 RB$Q1
RB$Q1
1
1
0 0 RB$Q1
RB$Q1
0 0
0
0
1
0
0
1
1
0
0 0
0
0
0 1
0
0
0
0
1
RB$Q2
0
1
0
1
0
0 beta$Q1 beta$Q2 1 beta$Q1
1
0 1
0
0 .
0
1
0 0
0
0 0
0
0 0
0
RL 1
V$1
0
V$2
0
V$3
0
V$X$Q1 0
V$X$Q2
0
IC$CC$Q1
0
IC$CC$Q2 0
I$Load In[17]:= Solve[eqDarlington, I$Load]
Out[17]=
I$Load beta$Q1 beta$Q2 beta$Q1 beta$Q2
For large current gains beta$Q1 and beta$Q2, the contribution of the product beta$Q1 beta$Q2
dominates this sum, resulting in the known current-gain formula for a Darlington stage. Later, in
Chapter 2.8, we will learn how to let Analog Insydes extract such dominant contributions automatically in order to provide compact and interpretable symbolic analysis results.
A Remark on Local and Global Symbols
Critical reading of the preceding text may give rise to the following question: Why are identifiers
of internal nodes always instantiated even though they are not explicitly declared as local symbols
while symbolic element values are assumed to be global symbols unless they are explicitly declared
as parameters? The logic behind this behavior is that it hardly makes sense to regard any internal
node identifier in a generic subcircuit definition as a global symbol – except for rare cases in which
the global ground node is involved. Whenever we need to connect a subcircuit to a global node
we must make the connection via a port node. Otherwise, from the point of view of the subcircuit
definition, we would have to exploit bottom-up knowledge about the set of node names in the
top-level netlist, either to be sure that a certain global node is present or in order to prevent name
clashes. Consequently, the only reasonable option is that all nodes which are not ports must be local,
hence they are instantiated individually.
On the other hand, there are some applications where the element values of multiple subcircuit
instances are supposed to be identical, for example in repetitive circuit structures like RLC ladder
networks. If the element values in the subcircuit definition were generally treated as local parameters then, after subcircuit expansion, we would have to go through the tedium of manually
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60
assigning the same global symbols to all instances of the corresponding subcircuit element values. This would not be very convenient as it would require typing something along the line of
R$X1 R$X2 R$X3 R for every group of identical elements. By explicitly declaring which
symbols in a subcircuit denote parameters and quietly assuming all others to be global quantities,
cases like that of the Darlington amplifier as well as that of the ladder circuit are both dealt with
most easily. However, Analog Insydes provides the command MatchSymbols which applies matching information on a DAEObject. Thus, you can decide which parameters to match even after setting
up the equation system.
Passing Parameter Values to Subcircuit Instances
Though intentional and useful, the automatic instantiation of symbolic subcircuit parameters is only
a side effect of the parameter declaration. The original purpose of the Parameters argument in a
Model definition is to declare the set of local model variables which can be replaced by instancespecific values given in the value fields of subcircuit references (see Section 3.1.8). In other words,
we can pass parameter values to a subcircuit in a similar way as we can pass arguments to a function
defined in Mathematica. Since we do not have to specify values for model parameters, automatic
instantiation comes into effect whenever a default value is needed for a parameter which has not
been assigned a value.
If we need to pass parameter values to instances of subcircuit objects we must append a sequence
of replacement rules of the form
parameter symbol −> value
to the value field of the corresponding subcircuit references. The full netlist format of a subcircuit
reference is thus
{subcircuit instance name, {port connections},
Model −> subcircuit class name, Selector −> selector,
parameter1 −> value1 , parameter2 −> value2 , }
To illustrate the possible cases consider the netlist of the Darlington amplifier from Section 2.3.5 (see
Figure 3.3) once again. Let’s assign the symbolic value RB1 and a numerical value of to the
parameters RB and beta of Q1, respectively. In the case of Q2, we assign the symbol beta2 to the
current gain whereas we leave the parameter RB unspecified.
2.3 Circuits and Subcircuits
61
In[18]:= Circuit[
Netlist[
{I1, {0, 1}, 1},
{Q1, {1 −> "B", 2 −> "C", 3 −> "E"},
Model −> NPNTransistor, Selector −> ACsimple,
RB −> RB1, beta −> 150},
{Q2, {3 −> "B", 2 −> "C", 0 −> "E"},
Model −> NPNTransistor, Selector −> ACsimple,
beta −> beta2},
{Load, {0, 2}, Type −> Resistor, Value −> RL,
Pattern −> Impedance}
],
Model[
Name −> NPNTransistor,
Selector −> ACsimple,
Ports −> {"B", "C", "E"},
Parameters −> {RB, beta},
Definition −>
Netlist[
{RB, {"X", "E"}, RB},
{CC, {"B", "X", "C", "E"}, beta}
]
]
] // ExpandSubcircuits // DisplayForm
Out[18]//DisplayForm=
Netlist Flat, 6 Entries:
I1, 0, 1, 1
RB$Q1, X$Q1, 3, RB1
CC$Q1, 1, X$Q1, 2, 3, 150
RB$Q2, X$Q2, 0, RB$Q2
CC$Q2, 3, X$Q2, 2, 0, beta2
Load, 0, 2, Type Resistor, Value RL, Pattern Impedance
During subcircuit expansion, the model parameters have now been replaced by the values from the
top-level netlist. As we did not assign a value to the base resistor of Q2, the automatic instantiation
mechanism has created the symbol RB$Q2 in order to provide a default value for RB.
2.3.6 The Scope Argument
Local and Global Subcircuit Definitions
In all the preceding examples, the top-level netlist is immediately followed by the definitions of
the subcircuits referenced from within the netlist. These subcircuit definitions are local and volatile
because they are neither visible outside the scope of the enclosing Circuit statement nor do they
"survive" subcircuit expansion. When doing multiple circuit analyses with a number of frequently
used device models it would be rather tedious if we always had to include all potentially selected
models in the circuit description together with the top-level netlist. Therefore, we have the option to
make any subcircuit definition permanent and accessible from within other netlists by providing an
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62
appropriate Scope argument (see Section 3.2.1). Scope has two possible values, Local and Global.
To make a subcircuit available globally, we need to set the scope as follows:
Scope −> Global
By expanding a circuit which contains only model definitions with global scope we can implement a globally accessible model library in our current Mathematica environment. For example,
let’s set up a model library from the two NPN transistor models NPNTransistor/ACsimple and
NPNTransistor/ACdynamic from Section 2.3.2 (see Figure 3.2).
In[19]:= Circuit[
Model[
Name −> NPNTransistor,
Selector −> ACsimple,
Scope −> Global,
Ports −> {"B", "C", "E"},
Parameters −> {RB, beta},
Definition −>
Netlist[
{RB, {"X", "E"}, RB},
{CC, {"B", "X", "C", "E"}, beta}
]
],
Model[
Name −> NPNTransistor,
Selector −> ACdynamic,
Scope −> Global,
Ports −> {"B", "C", "E"},
Parameters −> {RB, CM, beta, RO},
Definition −>
Netlist[
{RB, {"X", "E"}, RB},
{CM, {"B", "C"}, CM},
{CC, {"B", "X", "C", "E"}, beta},
{RO, {"C", "E"}, RO}
]
]
] // ExpandSubcircuits;
Both models are now stored in the global subcircuit database and can thus be referenced by any
other netlist. Note that simply defining the models is not sufficient for storing them in the global
database – we have to call ExpandSubcircuits in order to store them into the global database.
If the Circuit object contains a top-level Netlist , a call to CircuitEquations stores the models in the global database, too. We can check the contents of the database using the command
GlobalSubcircuits (Section 3.3.4), which will return a list of the names and selectors of the global
subcircuits.
In[20]:= GlobalSubcircuits[]
Out[20]=
NPNTransistor, ACdynamic, NPNTransistor, ACsimple
2.3 Circuits and Subcircuits
63
The global definition of the transistor models allows us to write the circuit description of the Darlington amplifier from Section 2.3.5 (see Figure 3.3) without the Model commands. Then, during
subcircuit expansion, the global database will be searched for models which are not defined locally.
In[21]:= darlWithoutModels =
Circuit[
Netlist[
{I1, {0, 1}, 1},
{Q1, {1 −> "B", 2 −> "C", 3 −> "E"},
Model −> NPNTransistor, Selector −> ACsimple,
RB −> RB1, beta −> 150},
{Q2, {3 −> "B", 2 −> "C", 0 −> "E"},
Model −> NPNTransistor, Selector −> ACsimple,
beta −> beta2},
{Load, {0, 2}, Type −> Resistor, Value −> RL,
Pattern −> Impedance}
]
];
In[22]:= ExpandSubcircuits[darlWithoutModels] // DisplayForm
Out[22]//DisplayForm=
Netlist Flat, 6 Entries:
I1, 0, 1, 1
RB$Q1, X$Q1, 3, RB1
CC$Q1, 1, X$Q1, 2, 3, 150
RB$Q2, X$Q2, 0, RB$Q2
CC$Q2, 3, X$Q2, 2, 0, beta2
Load, 0, 2, Type Resistor, Value RL, Pattern Impedance
Redefining and Removing Global Subcircuits
If ExpandSubcircuits or CircuitEquations is called on a Circuit object containing a global model
definition, any previously stored model in the global database with the same Name and Selector
arguments is replaced by the new one. To remove a particular subcircuit from the global database
we can use the command RemoveSubcircuit (Section 3.3.8). Its arguments must be the name and
the selector of the subcircuit definition to be deleted. The return value is a list of the names and
selectors of the remaining subcircuit definitions. For example, let’s remove the transistor model
NPNTransistor/ACdynamic . The only remaining model is then NPNTransistor/ACsimple :
In[23]:= RemoveSubcircuit[NPNTransistor, ACdynamic]
Out[23]= NPNTransistor, ACsimple
Locally Overriding Global Subcircuit Definitions
One further application of the Scope argument is that we can locally override a global subcircuit
definition. Local subcircuits always take precedence over global definitions with the same name and
selector without overwriting the global definition. To illustrate this, let’s temporarily replace the
NPNTransistor/ACsimple model with the subcircuit shown in Figure 3.4. The difference between
the original implementation of NPNTransistor/ACsimple and this model is that the latter contains a
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64
voltage-controlled current source VCCS (Section 4.2.13) instead of a current-controlled current source
CCCS (Section 4.2.11).
B
C
RB
gm*VBE
E
E
Figure 3.4: Transistor model with voltage-controlled current source
Although the global subcircuit database still contains a definition of NPNTransistor/ACsimple we
can safely use the same name and selector for a local subcircuit without changing the global definition, provided we specify Scope −> Local. Upon subcircuit expansion, Analog Insydes first looks
for a local subcircuit with the requested name and selector. The global definition is retrieved only if
no matching local definition is present.
In[24]:= darlWithLocalModel =
Circuit[
Netlist[
{I1, {0, 1}, 1},
{Q1, {1 −> "B", 2 −> "C", 3 −> "E"},
Model −> NPNTransistor, Selector −> ACsimple},
{Q2, {3 −> "B", 2 −> "C", 0 −> "E"},
Model −> NPNTransistor, Selector −> ACsimple},
{Load, {0, 2}, Type −> Resistor, Value −> RL,
Pattern −> Impedance}
],
Model[
Name −> NPNTransistor,
Selector −> ACsimple,
Scope −> Local,
Ports −> {"B", "C", "E"},
Parameters −> {RB, gm},
Definition −>
Netlist[
{RB, {"B", "E"}, RB},
{VC, {"B", "E", "C", "E"}, gm}
]
]
];
Now, ExpandSubcircuits will use the local definition of NPNTransistor/ACsimple . The global
definition has not been altered, as we can see by expanding the netlist of the Darlington amplifier
from Section 2.3.5 (see Figure 3.3) without models again.
2.3 Circuits and Subcircuits
65
In[25]:= ExpandSubcircuits[darlWithLocalModel] // DisplayForm
Out[25]//DisplayForm=
Netlist Flat, 6 Entries:
I1, 0, 1, 1
RB$Q1, 1, 3, RB$Q1
VC$Q1, 1, 3, 2, 3, gm$Q1
RB$Q2, 3, 0, RB$Q2
VC$Q2, 3, 0, 2, 0, gm$Q2
Load, 0, 2, Type Resistor, Value RL, Pattern Impedance
In[26]:= ExpandSubcircuits[darlWithoutModels] // DisplayForm
Out[26]//DisplayForm=
Netlist Flat, 6 Entries:
I1, 0, 1, 1
RB$Q1, X$Q1, 3, RB1
CC$Q1, 1, X$Q1, 2, 3, 150
RB$Q2, X$Q2, 0, RB$Q2
CC$Q2, 3, X$Q2, 2, 0, beta2
Load, 0, 2, Type Resistor, Value RL, Pattern Impedance
2.3.7 The Translation Argument
Implicit Translation of Parameter Values
Another argument of the Model function which remains to be discussed is the Translation keyword. We can use it to define functional relations between formal subcircuit parameters and internal
element values when there is no one-to-one correspondence between both. The syntax is:
Translation −> {symbol1 −> value1 , symbol2 −> value2 , }
The purpose of the Translation argument is best explained by an example. Assume that we wish
to analyze the Darlington amplifier from Section 2.3.5 (see Figure 3.3) using the VCCS transistor
model from Section 2.3.6 (see Figure 3.4) instead of a CCCS model. For nodal analysis, the former is
usually a better choice than the latter because the VCCS model does not increase the matrix size by
introducing additional controlling currents into the modified nodal equations. However, even with
a VCCS model we might like to describe the transistors in terms of their current gains beta instead
of their transconductances gm by making use of the well-known relation gm = beta/RB. With the
help of the Translation argument we can define a translation rule which maps the formal model
parameters RB and beta to the transconductance gm used internally as the element value of the
VCCS:
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66
In[27]:= Circuit[
Model[
Name −> NPNTransistor,
Selector −> ACgm,
Scope −> Global,
Ports −> {"B", "C", "E"},
Parameters −> {RB, beta},
Translation −> {gm −> beta / RB},
Definition −>
Netlist[
{RB, {"B", "E"}, RB},
{VC, {"B", "E", "C", "E"}, gm}
]
]
] // ExpandSubcircuits;
In[28]:= darlNumParams =
Circuit[
Netlist[
{I1, {0, 1}, 1},
{Q1, {1 −> "B", 2 −> "C", 3 −> "E"},
Model −> NPNTransistor, Selector −> ACgm,
RB −> 400.0, beta −> 150.0},
{Q2, {3 −> "B", 2 −> "C", 0 −> "E"},
Model −> NPNTransistor, Selector −> ACgm,
RB −> 100.0, beta −> 50.0},
{Load, {0, 2}, Type −> Resistor, Value −> RL,
Pattern −> Impedance}
]
];
Just before subcircuit expansion the right-hand sides of the translation rules are evaluated with the
parameter values specified in the value fields of the subcircuit references. The evaluated parameter
translation rules are then applied to all value fields in the subcircuit netlist, resulting in the replacement of all symbols appearing on the left-hand side of the translations. Thus, gm is automatically
calculated from RB and beta.
In[29]:= ExpandSubcircuits[darlNumParams] // DisplayForm
Out[29]//DisplayForm=
Netlist Flat, 6 Entries:
I1, 0, 1, 1
RB$Q1, 1, 3, 400.
VC$Q1, 1, 3, 2, 3, 0.375
RB$Q2, 3, 0, 100.
VC$Q2, 3, 0, 2, 0, 0.5
Load, 0, 2, Type Resistor, Value RL, Pattern Impedance
Delayed Translation Rules
In this simple example, using the Translation argument is not absolutely necessary. We could have
achieved the same effect without a translation rule by writing the netlist entry for the VCCS like
this:
2.3 Circuits and Subcircuits
67
{VC, {"B", "E", "C", "E"}, beta / RB}
So why should we use translation rules at all? The answer is that parameter value translations can
also be specified by means of delayed rules, i.e. the RuleDelayed lhs :> rhs instead of the Rule
lhs −> rhs. Since delayed rules are left unevaluated until the very moment they are needed, their
right-hand sides also contain calls to external functions which perform calculations depending on
the model parameter values. For instance, let’s define a function which, given values for beta and
RB, calculates the transconductance of a transistor by solving the formula gm RB beta numerically
for gm, using gm as initial guess.
In[30]:= Transconductance[b_Real, r_Real] :=
Module[ {g},
Print["Computing gm for beta = ", b, " and RB = ", r];
g /. FindRoot[g * r == b, {g, 1.0}]
]
By means of a delayed translation rule for gm we can instruct Mathematica not to execute the function
call at the time the transistor model is defined but to wait until the rule is used to compute gm for a
particular subcircuit instance.
In[31]:= Circuit[
Model[
Name −> NPNTransistor,
Selector −> ACgm,
Scope −> Global,
Ports −> {"B", "C", "E"},
Parameters −> {RB, beta},
Translation −> {gm :> Transconductance[beta, RB]},
Definition −>
Netlist[
{RB, {"B", "E"}, RB},
{VC, {"B", "E", "C", "E"}, gm}
]
]
] // ExpandSubcircuits;
If we had used an immediate rule above, gm −> Transconductance[beta, RB], Mathematica would
have already called Transconductance with the symbolic arguments beta and RB at the time of
model definition. On the other hand, the delayed rule permits Analog Insydes to replace the arguments with instance-specific values before making the function call. This is done once per subcircuit
instance as can be seen from the following output:
In[32]:= ExpandSubcircuits[darlNumParams] // DisplayForm
Computing gm for beta = 150. and RB = 400.
Computing gm for beta = 50. and RB = 100.
Out[32]//DisplayForm=
Netlist Flat, 6 Entries:
I1, 0, 1, 1
RB$Q1, 1, 3, 400.
VC$Q1, 1, 3, 2, 3, 0.375
RB$Q2, 3, 0, 100.
VC$Q2, 3, 0, 2, 0, 0.5
Load, 0, 2, Type Resistor, Value RL, Pattern Impedance
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An advanced application of such a parameter translation scheme would be the calculation of transistor smallsignal parameters from given operating-point voltages and currents via a set of predefined device equations.
Alternative Ways to Assign Values to Parameters
In combination with the Parameters setting there is one further useful application of the argument
Translation . Specifying a translation rule for a symbol which is also designated as a parameter of
the same subcircuit definition allows us to choose whether we let the instance value of the symbol be
calculated from other parameters or whether we assign a value to this symbol directly. For example,
let’s add the transconductance gm to the list of model parameters while leaving the translation rule
for gm unchanged:
Parameters −> {RB, beta, gm},
Translation −> {gm :> Transconductance[beta, RB]},
The value field of a reference to the transistor model can now be written in two alternative ways.
Just as in the previous subsection we can assign values to RB and beta only and let Analog Insydes
calculate gm from the two quantities automatically. Alternatively, we can bypass the translation rule
by directly assigning a value to gm, such as gm −> 0.5. Any value given for beta is then effectively
ignored because it is not needed anywhere else than in the argument list of the translation rule. In
the netlist below, we let gm$Q1 be computed from RB and beta by means of the translation rule. In
the case of Q2, however, we do not specify a value for beta but make a direct assignment to gm:
gm −> gm2. Therefore, the parameter translation function Transconductance is not called for gm$Q2:
2.3 Circuits and Subcircuits
69
In[33]:= Circuit[
Netlist[
{I1, {0, 1}, 1},
{Q1, {1 −> "B", 2 −> "C", 3 −> "E"},
Model −> NPNTransistor, Selector −> ACgm,
RB −> 400.0, beta −> 150.0},
{Q2, {3 −> "B", 2 −> "C", 0 −> "E"},
Model −> NPNTransistor, Selector −> ACgm,
RB −> 100.0, gm −> gm2},
{Load, {0, 2}, Type −> Resistor, Value −> RL,
Pattern −> Impedance}
],
Model[
Name −> NPNTransistor,
Selector −> ACgm,
Scope −> Global,
Ports −> {"B", "C", "E"},
Parameters −> {RB, beta, gm},
Translation −> {gm :> Transconductance[beta, RB]},
Definition −>
Netlist[
{RB, {"B", "E"}, RB},
{VC, {"B", "E", "C", "E"}, gm}
]
]
] // ExpandSubcircuits // DisplayForm
Computing gm for beta = 150. and RB = 400.
Out[33]//DisplayForm=
Netlist Flat, 6 Entries:
I1, 0, 1, 1
RB$Q1, 1, 3, 400.
VC$Q1, 1, 3, 2, 3, 0.375
RB$Q2, 3, 0, 100.
VC$Q2, 3, 0, 2, 0, gm2
Load, 0, 2, Type Resistor, Value RL, Pattern Impedance
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2.4 Setting up and Solving Circuit Equations
2.4.1 Naming Conventions
Automatically Generated Voltage and Current Identifiers
From the small circuit analysis examples already presented in the previous chapters you will have
noticed that Analog Insydes automatically creates variables for voltages and currents when setting
up circuit equations from a netlist. In order to interpret the equations correctly you need to know
everything about the identifier naming scheme employed by Analog Insydes.
By default, identifiers for branch voltages and currents associated with two-terminal elements are
generated by attaching the prefixes V$ and I$ to the reference designator (Section 3.1.3) of the corresponding circuit element, respectively. For instance, the branch voltage across and branch current
through a resistor R1 would be named V$R1 and I$R1.
Similarly, identifiers for node voltages are generated by prefixing the node names with the tag V$,
so the node voltage at node 1 would be named V$1. Although the prefix is used for branch voltages
you will not experience any name clashes among branch and node voltage identifiers, provided that
your netlists do not contain any symbolic node identifiers (Section 3.1.3) which are identical to some
reference designators. However, if you would rather like different prefixes to be used for branch
and node voltages then read the following sections to learn how to change the standard settings.
In the case of a controlled source we need to distinguish between the voltages and currents at the
controlling and at the controlled branch. While the latter quantities are given names according to the
same convention as applied to two-terminal elements, controlling voltages and currents are marked
with the prefixes VC$ and IC$. For example, the voltage across and current through the controlled
branch of a controlled source VC1 would be called V$VC1 and I$VC1 whereas the corresponding
quantities at the controlling branch would be called VC$VC1 and IC$VC1.
Changing the Default Identifier Prefixes
All identifier prefix settings can be customized by changing the corresponding option. The current
option settings can be displayed using the command Options[CircuitEquations] and can be
changed using the Mathematica command SetOptions . For instance, the command
SetOptions[CircuitEquations, BranchVoltagePrefix −> "E$"]
would change the prefix used for branch voltages from V$ to E$. The names of the prefix options
and their default settings are listed below.
NodeVoltagePrefix −> "V$"
BranchVoltagePrefix −> "V$"
BranchCurrentPrefix −> "I$"
ControllingVoltagePrefix −> "VC$"
ControllingCurrentPrefix −> "IC$"
2.4 Setting up and Solving Circuit Equations
71
The values of all these options must be strings which can be converted to a single symbol by means
of the command ToExpression . Strings such as "V_" or "@I" cannot be converted to valid symbols
and may therefore not be used for this purpose.
Changing the Instance Name Separator for Reference Designators
Related to the identifier prefix options is the option InstanceNameSeparator (Section 3.14.2). This
option determines the character (or string) which Analog Insydes uses to separate name components
of reference designators generated for instantiated subcircuit elements. With the default setting
InstanceNameSeparator −> "$"
a resistor RB in a subcircuit instance Q1 would be named RB$Q1. Using SetOptions you can change
the separator character to (almost) anything you like, with the same restrictions that apply to the
choice of the prefix options.
2.4.2 Circuit Equations
The Command CircuitEquations
Setting up systems of symbolic circuit equations is done by the Analog Insydes command
CircuitEquations (Section 3.5.1), which takes a Netlist or Circuit object as first argument. In
addition, the function may be called with a variety of options – to be introduced later – with which
many aspects of equation setup can be influenced individually:
CircuitEquations[circuit, options]
If the type of the netlist argument is not a flat netlist, then the function ExpandSubcircuits (Section 3.4.1) is automatically applied to the circuit description in order to produce a flat netlist object
before proceeding with equation setup.
This section describes how to set up equations of linear circuits in the Laplace domain.
CircuitEquations can also be used to set up DC or transient equations for nonlinear circuits. This
topic is discussed in Section 2.6.4.
1
V0
RA
2
L1
3
C1
Figure 4.1: RLC filter circuit
C2
RB
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To illustrate equation setup let’s write down the netlist of the RLC filter circuit displayed in Figure 4.1.
In[1]:= <<AnalogInsydes‘
In[2]:= rlcfilter =
Netlist[
{V0, {1,
{RA, {1,
{C1, {2,
{L1, {2,
{C2, {3,
{RB, {3,
]
0},
2},
0},
3},
0},
0},
V0},
RA},
C1},
L1},
C2},
RB}
Out[2]= NetlistRaw, 6 By default, CircuitEquations sets up a system of modified nodal (MNA) equations in the frequency domain, or, more precisely, the Laplace domain. In the Laplace domain, all linear dynamic
elements are described by frequency-dependent complex admittances or impedances, obtained by
Laplace-transforming the corresponding constitutive equations. For instance, a linear capacitance C
with a constitutive equation it C dutdt is represented by its complex admittance s C, where s
denotes the complex Laplace frequency.
Unless specified otherwise (see netlist option Pattern in Section 3.1.4), all impedances, i.e. resistors
and inductors, are automatically converted to their admittance equivalents. Therefore, the complex
impedance s L of an inductor L will be treated as an admittance with the reciprocal value s L.
In[3]:= rlceqs = CircuitEquations[rlcfilter]
Out[3]= DAEAC, 4 4 The return value of CircuitEquations is a DAEObject which contains the three components of
the linear matrix equation A x b, where A is the MNA matrix, x the vector of unknown node
voltages and impedance branch currents, and b the vector of independent source voltages and source
currents. To display the system of equations in a more readable form we can apply the command
DisplayForm to the DAEObject rlceqs:
In[4]:= DisplayForm[rlceqs]
Out[4]//DisplayForm=
1
1
RA
RA
1
1
1
RA
RA
L1 s C1 s
1
0
L1 s
0
1
0
1
L1 s
1
1
RB
L1 s 0
1
V$1 0 0 V$2
0
.
0
V$3 C2 s 0 I$V0 V0 0
Solving Circuit Equations
Systems of circuit equations generated by Analog Insydes can be solved symbolically by means of
the command Solve (Section 3.5.4). The sequence of arguments may have one of the following three
forms, depending on which variable or set of variables the equations should be solved for:
Solve[dae]
Solve[dae, var]
Solve[dae, {var , var , }]
2.4 Setting up and Solving Circuit Equations
73
If Solve is called with a DAEObject only and no second argument is given then the solutions for
all variables contained in the vector x are computed. If, in addition to the DAEObject, one single
variable of x is specified as second argument then only this variable is solved for. To compute the
solutions for a subset of x containing more than only one variable, the second argument must be a
list of the variables of interest.
Solve finds all solutions of an equation system only for linear equations. For nonlinear (dynamic)
equations it is in general not possible to find symbolic solutions. Thus, Analog Insydes provides the
command NDAESolve (Section 3.7.5) to compute the solution of nonlinear DAE systems numerically.
See Chapter 2.7 for details.
Just like Mathematica’s built-in Solve function, Solve[dae] returns a list of rules
{{var −> solution, † † † }} which associate the variables with the corresponding solutions. Individual
solutions can be extracted from the list by using Mathematica’s ReplaceAll operator (or /. in short
notation).
Let’s solve the MNA equations of the RLC filter for the node voltages at nodes 1 and 3. Since we
are only interested in solving for two out of all four variables, we must use the third calling format:
In[5]:= rlcsol = Solve[rlceqs, {V$1, V$3}]
Out[5]=
V$3 RB V0 RA RB L1 s C1 RA RB s C2 RA RB s C1 L1 RA s2 C2 L1 RB s2 C1 C2 L1 RA RB s3 ,
V$1 V0
As indicated above, we can retrieve the solution for a particular variable using the /.-operator. The
following input line serves to extract the value of V$3. The Mathematica command First removes the
outer level of list brackets from the set of solutions, so the result is returned as a single expression
and not as a list.
In[6]:= rlcv3 = V$3 /. First[rlcsol]
Out[6]=
RB V0 RA RB L1 s C1 RA RB s C2 RA RB s C1 L1 RA s2 C2 L1 RB s2 C1 C2 L1 RA RB s3 Quite frequently, the raw solutions computed from symbolic circuit equations are not presented in
an immediately comprehensible form. It usually takes some additional mathematical postprocessing
to transform the results into something which is easier to read or technically more meaningful.
Mathematica provides several commands for rearranging expressions, such as Simplify , Factor,
Together , Apart, or Collect. The choice of which function to use depends heavily on the particular application, and often some experimenting is required until the results are satisfactory.
For example, in the case of the above expression for V$3, we may want to combine all coefficients
belonging to the same power of s into one single coefficient each. For this purpose, we apply a
rewrite rule to all Plus subexpressions (i.e. sums) which groups the terms by corresponding powers
of s and then pulls out the si . This yields a transfer function with coefficients in the canonical
sum-of-products form.
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In[7]:= rlcv3 /. p_Plus :> Collect[p, s]
Out[7]=
RB V0 RA RB L1 C1 RA RB C2 RA RB s C1 L1 RA C2 L1 RB s2 C1 C2 L1 RA RB s3 2.4.3 Circuit Equation Formulations
Modified Nodal Analysis
The modified nodal analysis method (MNA) has already been introduced as the default formulation by which Analog Insydes sets up circuit equations. The MNA formulation is general, yields
relatively compact systems of equations, and is easy to implement on a computer. These and a few
other favorable properties have made it the most widely used analysis method in electrical circuit
simulators, including Analog Insydes.
There are a number of user-settable options which have an influence on the way Analog Insydes
sets up MNA equations. You have already encountered the netlist option Pattern which specifies
explicitly whether to use the impedance or admittance fill-in pattern for an immittance element, i.e.
a two-terminal admittance or impedance. This option allows, for instance, to prevent the implicit
conversion of an impedance to an admittance, thereby augmenting the MNA system by the branch
current of the impedance. However, the Pattern option pertains only to the element in whose
value field it is given. To prevent implicit impedance conversion globally you can set the value
of the CircuitEquations option ConvertImmittances to False, either permanently by modifying
Options[CircuitEquations] using the SetOptions directive, or temporarily by passing the option
ConvertImmittances −> False to the function CircuitEquations :
In[8]:= CircuitEquations[rlcfilter,
ConvertImmittances −> False] // DisplayForm
Out[8]//DisplayForm=
0
0
0
0 C1 s
0
0
0
C2 s
1
0
0
1
1
0
0
1
1
0
1
0
1 1
0
0 V$1 0 0 1
1
0 0 V$2 0 0
1
1 0
V$3
0 0
0
0 .
V0
I$V0
0 RA
0
0 0
I$RA 0 0 L1 s 0 0
I$L1
0 0
0
RB I$RB 0 The Formulation Option
With modified nodal analysis being the default analysis method there is usually no need to request
this formulation explicitly. However, if you have changed the default settings in
Options[CircuitEquations] or if you wish to use one of the analysis methods to be discussed in
the following subsections you need to specify the formulation as an option to CircuitEquations .
The option keyword is Formulation , and the option value can be either
ModifiedNodal, SparseTableau , or ExtendedTableau . Hence, modified nodal analysis can be explicitly selected by this command:
2.4 Setting up and Solving Circuit Equations
75
In[9]:= CircuitEquations[rlcfilter,
Formulation −> ModifiedNodal] // DisplayForm
Out[9]//DisplayForm=
1
1
RA
RA
1
1
1
RA
RA
L1 s C1 s
1
0
L1 s
0
1
0
1
L1 s
1
1
RB
L1 s 0
1
0 V$1 0 V$2
0
.
0
V$3 C2 s 0 I$V0 V0 0
Sparse Tableau Analysis: The Basic Formulation
In addition to nodal analysis, Analog Insydes also offers the possibility to set up circuit equations
according to two variants of the sparse tableau formulation (STA). For an electrical network containing n electrical branches, the basic variant of the sparse tableau comprises the n topological node
and loop equations resulting from Kirchhoff’s voltage law (KVL) and current law (KCL), plus the n
voltage-current relations of the circuit elements. Hence, the basic sparse tableau forms the following
n n matrix equation in terms of the branch voltages v and the branch currents i.
0 B 0 v
0
A
0 i
Y
Z
s
Here, A denotes the nodal incidence matrix of the network graph and B a loop incidence matrix of
maximum rank. B is derived automatically from A by an algorithm which searches for a tree in the
network graph and then extracts the fundamental loop matrix defined by that tree. The matrices Y
and Z in the third row of the tableau represent the coefficient matrices of the element relations. On
the right-hand side, s denotes the contribution of the independent sources. All remaining tableau
entries are structurally zero (0).
2. Tutorial
76
In[10]:= CircuitEquations[rlcfilter,
Formulation −> SparseTableau] // DisplayForm
Out[10]//DisplayForm=
1 1
1
0
0
0
1 1
0
1
1
0
1 1
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0 1
0
0
0
0
0
0 C1 s 0
0
0
0
0
0
1
0
0
0
0
0 C2 s 0
0
0
0
0
0
1
0
0 0
0
0
0
0 V$V0 0 0
0
0
0
0 V$RA 0 0
0
0
0
0 V$C1
1 1
0
0
0
0 V$L1
0 1 1
1
0
0 V$C2 0 0
0
1
1
1 V$RB
.
0 0
0
0
0
0 I$V0
0 RA 0
0
0
0 I$RA 0 0 1
0
0
0 I$C1 0 0
0 L1 s 0
0 I$L1
0 0
0
0
1 0 I$C2 0 0
0
0
0 RB I$RB 0 0 0 0
0
0 V0
0
0 0
0
0 Sparse Tableau Analysis: The Extended Formulation
Analog Insydes also supports an extended sparse tableau formulation in which the vector of unknowns comprises the branch voltages v, the branch currents i, and the node voltages vn . With
I denoting an identity matrix of rank n and AT the transpose of the nodal incidence matrix A,
Kirchhoff’s voltage law is expressed indirectly in terms of the node voltages as I v AT vn 0.
Therefore, the extended sparse tableau formulation does not require the calculation of a loop matrix,
though at the expense of a larger matrix size than that of the basic variant:
T
I 0 A v 0 0 I 0 0 A
0 vn s Y Z
The extended variant is selected by the option setting Formulation −> ExtendedTableau :
2.4 Setting up and Solving Circuit Equations
77
In[11]:= rlcstaext = CircuitEquations[rlcfilter,
Formulation −> ExtendedTableau];
DisplayForm[rlcstaext]
Out[12]//DisplayForm=
1 0
0
0
0
0
0 1
0
0
0
0
0 0
1
0
0
0
0 0
0
1
0
0
0
0
0
0
1
0
0 0
0
0
0
1
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
1 0
0
0
0
0
0
1
0
0
0
0
0
0
0 0 C1 s 0
0 0
0
1
0
0
0
0
0
0
C2
s
0
0
0
0
1
0 0
0 0
0
0
0
0 1 0
0 0 0
0
0
0
0 1 1
0 0 0
0
0
0
0
0 1 0 0 0
0
0
0
0
0 1 1 0 0
0
0
0
0
0
0 1 0 0
0
0
0
0
0
0 1 1 1
0
0
0
0
0
0
0 0 1 1
1
0
0
0
0
0 .
0 0
0
1
1
1
0
0
0 0 0
0
0
0
0
0
0
0 0 RA 0
0
0
0
0
0
0 0 0 1
0
0
0
0
0
0 0 0
0 L1 s 0
0
0
0
0 0 0
0
0
1 0
0
0
0 0 0
0
0
0 RB 0
0
0 0 V$V0 0 V$RA 0
V$C1
0
V$L1
V$C2 0
V$RB 0
0 I$V0 0
I$RA
I$C1 0
V0 I$L1 0
I$C2
0
I$RB
0 V$1 0
V$2
0 V$3 Of course, the solution for a particular quantity in a circuit does not depend on the type of equations
from which the solution is computed. So the value for V$3 obtained by an extended sparse tableau
analysis is identical to the one computed by modified nodal analysis:
In[13]:= V$3 /. First[Solve[rlcstaext, V$3]]
Out[13]=
L1 RB s V0
RA RB RA L1 s 1 C1 RA s RB L1 s 1 C2 RB s
2. Tutorial
78
2.4.4 Additional Options for CircuitEquations
All Analog Insydes options which have an influence on circuit equation setup by the function
CircuitEquations can be accessed through Options[CircuitEquations] . The current option settings can be inspected by retrieving the value of this variable:
In[14]:= Options[CircuitEquations]
Out[14]=
AnalysisMode AC, BranchCurrentPrefix I$,
BranchVoltagePrefix V$, ControllingCurrentPrefix IC$,
ControllingVoltagePrefix VC$, ConvertImmittances True,
DefaultSelector Automatic, DesignPoint Automatic,
ElementValues Value, Formulation ModifiedNodal,
FrequencyVariable s, IgnoreMissingGround False,
IndependentVariable Automatic, InitialConditions None,
InitialGuess Automatic, InitialTime 0,
InstanceNameSeparator InheritedAnalogInsydes,
LibraryPath InheritedAnalogInsydes, MatrixEquation True,
ModelLibrary InheritedAnalogInsydes, ModeValues None,
NodeVoltagePrefix V$, Protocol InheritedAnalogInsydes,
Symbolic All, TimeVariable t, Value None
Some of these options are discussed in more detail below.
AnalysisMode
CircuitEquations sets up circuit equations for the analysis modes AC, DC, and transient. Which
mode to use is determined by the option AnalysisMode , whose value is one of the symbols AC, DC,
and Transient . The default value is AC. Up to now, we used CircuitEquations to create smallsignal equations. Section 2.6.4 describes how to set up DC equations and Section 2.7.1 describes how
to set up transient equations.
MatrixEquation
Using the option MatrixEquation −> False, we can force CircuitEquations not to use a matrix
representation when setting up small-signal circuit equations but to return a list of scalar equations
instead.
In[15]:= CircuitEquations[rlcfilter,
MatrixEquation −> False] // DisplayForm
Out[15]//DisplayForm=
V$1 V$2
V$1 V$2
V$2 V$3
I$V0 0, C1 s V$2 0,
RA
RA
L1 s
V$3
V$2 V$3
C2 s V$3 0, V$1 V0,
RB
L1 s
V$1, V$2, V$3, I$V0, DesignPoint When the matrix representation is turned off, the value returned by CircuitEquations is a DAEObject containing the circuit equations as a list of two elements, the first being the system of circuit
equations and the second the vector of unknowns. The advantage of the scalar representation is that
it consumes less memory than the matrix representation because Mathematica does not have a special
2.4 Setting up and Solving Circuit Equations
79
storage mechanism for sparse matrices. Nevertheless, some commands (e.g. ACAnalysis ) rely on the
fact that the equation system is formulated in matrix representation.
FrequencyVariable
The option FrequencyVariable −> symbol specifies the symbol which Analog Insydes uses to denote
the differential operator, i.e. the complex frequency, in the Laplace domain. By default, this is the
symbol s, but if you prefer to use another symbol for this purpose, for instance p, you can replace
s by p as follows:
In[16]:= CircuitEquations[rlcfilter,
FrequencyVariable −> p] // DisplayForm
Out[16]//DisplayForm=
1
1
RA
RA
1
1
1
RA
L1 p C1 p RA
1
0
L1 p
0
1
0
1
L1 p
1
1
L1 p C2 p RB
0
1
0 V$1 0
0 V$2 .
0
V$3
0
I$V0 V0
0
ElementValues
With ElementValues −> Symbolic we can instruct CircuitEquations to use the symbolic values
from the value fields of netlist entries in which a Symbolic option is given. To demonstrate the effect
of this option let’s make some enhancements to the netlist of the RLC filter circuit from Section 2.4.2
(see Figure 4.1). Using the extended value-field format (see Section 3.1.4) we specify a numerical as
well as a symbolic value for the circuit elements.
In[17]:= rlcfilter2 =
Netlist[
{V0, {1, 0},
{RA, {1, 2},
{C1, {2, 0},
{L1, {2, 3},
{C2, {3, 0},
{RB, {3, 0},
];
Value
Value
Value
Value
Value
Value
−>
−>
−>
−>
−>
−>
V0, Symbolic −> V0},
1000., Symbolic −> RA},
4.7*10^−6, Symbolic −> C1},
1.0*10^−3, Symbolic −> L1},
2.2*10^−5, Symbolic −> C2},
1000., Symbolic −> RB}
Note that the value of the Symbolic option needs not to coincide with the reference designator
as this is the case in the example above. You can use any Mathematica expression as value to the
Symbolic field.
A call to CircuitEquations without any additional options (i.e. the default value of the
ElementValues option is used, which is Value) will then cause the numerical values to be entered
into the matrix as these are given as arguments to the Value rules.
2. Tutorial
80
In[18]:= rlc2eqs = CircuitEquations[rlcfilter2];
DisplayForm[rlc2eqs]
Out[19]//DisplayForm=
0.001
0.001
0
6
1000.
1000.
0.001
0.001
4.7
10
s
s
s
1000.
1000.
0.001
0.000022 s
0
s
s
1
0
0
1
0
.
0
0
V$1 0 0 V$2 0
V$3
I$V0 V0 Alternatively, we can select the symbolic values by means of the option ElementValues −> Symbolic .
Note that ElementValues affects only those netlist entries in whose value field a Symbolic rule is
present.
In[20]:= rlc2eqs2 = CircuitEquations[rlcfilter2,
ElementValues −> Symbolic];
DisplayForm[rlc2eqs2]
Out[21]//DisplayForm=
1
1
RA
RA
1
1
1
RA
RA
L1 s C1 s
1
0
L1 s
0
1
1
0 V$1 0 V$2 0
.
1
1
0
V$3
RB
L1 s C2 s 0 V0
I$V0
0
0
0
1
L1 s
2.5 Graphics
81
2.5 Graphics
Mathematica has proven to be a powerful problem-solving environment for a large variety of engineering tasks. One of many reasons which make Mathematica such a useful tool is that it provides
excellent support for visualizing the results of technical computations. Due to its freely programmable
graphics functionality the flexibility of adapting the graphics display to individual applications is
virtually unlimited. Analog Insydes extends Mathematica’s basic graphic capabilities by adding special plotting functions for circuit analysis and design, including Bode, Nyquist, Nichol, root locus
plots, and transient waveforms.
2.5.1 Bode Plots
The Bode plot is perhaps the most commonly used graphing scheme for visualizing frequency responses of linear analog systems. It consists of two separate charts which display magnitude and
phase of a transfer function on a logarithmic and a linear scale vs. frequency, the latter being scaled
logarithmically. The magnitude values are usually given in decibels (dB) and the phase values in
degrees.
In Analog Insydes, Bode plots are displayed using the command BodePlot (Section 3.9.1). With the
basic syntax
BodePlot[tfunc, frange]
you can plot the transfer function tfunc within the frequency range frange. The transfer function
must be a complex-valued function of the frequency variable fvar, and the frequency range a list of
the form {fvar, fstart, fend}. BodePlot also allows to plot several transfer functions simultaneously.
For this purpose, the first argument of BodePlot must be specified as a list of transfer functions.
BodePlot[{tfunc , tfunc , }, frange]
To demonstrate the Bode plot facility we define the following transfer function in terms of the
frequency variable s.
In[1]:= <<AnalogInsydes‘
In[2]:= H1[s_] := 1000./(1. + 10.1*s + s^2)
We then display a Bode plot of the frequency response by evaluating H1 along the imaginary axis,
s I Ω, for a frequency range of sec to sec.
2. Tutorial
82
Magnitude (dB)
In[3]:= BodePlot[H1[I w], {w, 0.001, 1000.}]
1.0E-3
60
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E3
1.0E-2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
1.0E3
20
0
-20
-40
-60
1.0E-3
Phase (deg)
1.0E-2
40
1.0E-3
0
-25
-50
-75
-100
-125
-150
-175
1.0E-3
1.0E-2
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E3
1.0E-2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
1.0E3
Out[3]= Graphics BodePlot Options
The output generated by BodePlot is influenced by a number of options listed in
Options[BodePlot] , many of which are inherited from Options[LogLinearListPlot] from Mathematica’s standard package Graphics‘Graphics‘ . Here, we will discuss only some of those options
which pertain exclusively to BodePlot or which must be used with a different syntax as compared
to LogLinearListPlot . The relevant options and their default settings are listed below. Defaults
are listed first, followed by the possible alternatives printed in slanted typewriter font.
MagnitudeDisplay −> Decibels | AbsoluteValues | Linear
PhaseDisplay −> Degrees | Radians
PlotRange −> {Automatic, Automatic}
The option MagnitudeDisplay controls the display of the magnitude values. When set to
AbsoluteValues , magnitude values are not converted to decibels. Instead the absolute numerical
values are displayed on a logarithmic scale. When MagnitudeDisplay is set to Linear, magnitude
values are displayed on a linear scale. In a strict sense, the latter does no longer constitute a true
Bode plot, but there exist a few applications where linear magnitude scaling is useful.
With the value of the option PhaseDisplay we can choose whether phase values are shown as
degrees or as radians.
The meaning of the PlotRange option is slightly different for BodePlot than for other Mathematica
plotting commands. Here, the value of PlotRange must be a list of two elements, {rangem , rangep },
specifying the plot ranges of the y-axes of the magnitude and the phase chart respectively. PlotRange
has no effect on the ranges of the x-axes as these are determined by the given frequency range. The
format for both y-ranges is the usual one as documented in the Mathematica manual, so any of the
forms below are valid syntax:
2.5 Graphics
83
Automatic
{ymin, ymax}
{ymin, Automatic}
Let’s graph H1 together with a second transfer function H2 and watch the effects of some BodePlot
options. Other common graphics options such as PlotStyle can be applied just as usual.
In[4]:= H2[s_] := 100./(1.6 + 8.2*s + s^2)
Phase (rad)
Magnitude (dB)
In[5]:= BodePlot[{H1[I w], H2[I w]}, {w, 0.001, 1000.},
PhaseDisplay −> Radians,
PlotRange −> {{−60, 80}, Automatic},
PlotStyle −> {{RGBColor[0, 0, 1], Dashing[{0.05}]},
{RGBColor[0, 1, 0]}}]
1.0E-3
80
60
40
20
0
-20
-40
1.0E-2
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E3
1.0E-3
1.0E-2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
1.0E3
1.0E-3
0
-0.5
-1
-1.5
-2
-2.5
-3
1.0E-3
1.0E-2
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E3
1.0E-2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
1.0E3
Out[5]= Graphics 2.5.2 Nyquist Plots
Another widely used graphical representation of transfer functions is the Nyquist plot. It is a parametric plot of the real and imaginary part of a transfer function in the complex plane as the frequency
parameter sweeps through a given interval. Nyquist plots are particularly useful for stability analysis
in control system design because one can immediately check whether a negative feedback loop meets
Nyquist’s stability criterion: If the Nyquist curve of the open loop system wraps around the point on the real axis then the corresponding closed loop system is unstable.
Nyquist plots are produced by the command NyquistPlot (Section 3.9.4) whose argument sequence
is the same as that of BodePlot (Section 3.9.1). Just as in the case of the latter you can plot one
single transfer function or several transfer functions simultaneously.
Let’s make a Nyquist plot of the frequency response of a system which is formed by a series connection of the two transfer functions H1 and H2, followed by a dB attenuator ( dB ). The
overall transfer function of the resulting system is given by the product of the individual transfer
functions.
2. Tutorial
84
In[6]:= H12a[s_] := H1[s] * H2[s] * 0.002
To make the curve look smoother we increase the number of plot points to .
In[7]:= NyquistPlot[H12a[I w], {w, 0.001, 1000.},
PlotPoints −> 100]
Im
20
40
60
80
100
120
Re
-20
-40
-60
-80
Out[7]= Graphics We determine whether the corresponding closed-loop system is stable or unstable by taking a closer
look at the region around the point using the PlotRange option.
In[8]:= NyquistPlot[H12a[I w], {w, 0.1, 1000.},
PlotPoints −> 100, PlotRange −> {{−3, 1}, {−1, 1}}]
Im
1
0.75
0.5
0.25
-3
-2.5
-2
-1.5
-1
-0.5
-0.25
0.5
1
Re
-0.5
-0.75
-1
Out[8]= Graphics This plot reveals that the point is located to the right of the Nyquist curve, i.e. the curve
wraps around the critical point. Hence the closed-loop system is unstable.
2.5 Graphics
85
NyquistPlot Options
NyquistPlot inherits its plot options, which are accessible through
Options[NyquistPlot] , from Mathematica’s standard function ListPlot . See the Mathematica book
for detailed explanations of these options. NyquistPlot has additional options like, for example
FrequencyScaling which controls the spacing of the frequency points at which the transfer function
is sampled. FrequencyScaling can be set to Linear or Exponential , resulting in equidistant or
exponentially spaced sampling points respectively. The default is Exponential .
2.5.3 Nichol Plots
Yet another method of visualizing frequency responses is Nichol’s chart. A Nichol plot is similar to
a Nyquist plot but shows gain on a logarithmic scale (dB) vs. phase on a linear scale (degrees), with
an axis origin at the point dB . The advantage of Nichol’s chart is the ease by which gain
and phase margins can be determined graphically. The gain margin is simply the negative value of
the gain axis intersect. The phase margin is equal to the distance between the axis origin and the
phase axis intersect.
Nichol charts are computed by the function NicholPlot (Section 3.9.3). The argument sequence is
the same as for NyquistPlot (Section 3.9.4) or BodePlot (Section 3.9.1).
Let’s use a Nichol chart to determine the gain and phase margins of a system which is characterized
by the following transfer function.
In[9]:= H3[s_] := 20*(3 + s)/(s*(5 + s)*(20 + 5*s + 2*s^2))
We draw a Nichol chart of H3[I w] for an angular frequency Ω varying from sec to sec .
Again, we increase the number of plot points to produce a smooth curve.
In[10]:= NicholPlot[H3[I w], {w, 0.1, 5.},
AspectRatio −> 0.8, PlotPoints −> 100]
dB
15
10
5
-360 -300 -240
-120 -60
0
deg
-5
-10
-15
-20
Out[10]= Graphics Now, in order to read off the margins better, we zoom in on the part of the curve located in the
fourth quadrant.
2. Tutorial
86
In[11]:= NicholPlot[H3[I w], {w, 0.1, 5.}, AspectRatio −> 0.8,
PlotPoints −> 100, PlotRange −> {{−200, −80}, {−15, 2}}]
dB
-150
-120
-90
deg
-2.5
-5
-7.5
-10
-12.5
-15
Out[11]= Graphics The curve crosses the gain axis at a gain of dB, so we have a gain margin of dB. At the
phase axis intersect we have a phase value of approximately which is away from the axis
origin. Hence, the phase margin is .
NicholPlot Options
Like NyquistPlot , NicholPlot inherits its options from ListPlot . NicholPlot has additional options like, for example, PhaseDisplay (see Section 2.5.1) and FrequencyScaling (see Section 2.5.2).
2.5.4 Root Locus Plots
Let Hs k denote a rational transfer function whose coefficients depend on the real parameter k. A
root locus plot shows the locus of the poles and zeros of Hs k in the complex plane as k varies
within an interval k k .
To draw a root locus plot use the command RootLocusPlot (Section 3.9.5). The calling format is
RootLocusPlot[tfunc, {k, k , k }]
where tfunc is a transfer function in the frequency variable s and one real parameter k.
Of course, the parameter does not necessarily have to be named k. We can also use any other symbol
to denote the parameter. Below, we define the transfer function H4[s, a] with coefficients which are
functions of the parameter a.
In[12]:= H4[s_, a_] := (a + 2*s + s^2)/(10 + 3*a*s + 4*s^2 + s^3)
Then we graph the root locus of H4 as a varies from to . By default, RootLocusPlot samples
the parameter interval at five equally spaced points. This number can be increased or decreased by
2.5 Graphics
87
means of the PlotPoints option. The poles and zeros of the transfer function are displayed as red
crosses and green circles, respectively.
In[13]:= RootLocusPlot[H4[s, a], {a, 3, 5}]
a = 3.000e0 .. 5.000e0 (0% .. 100%)
Im s
Poles
0%
2.0E0
1.
2.0E0
-1.
100
Re s
-0.5
-1.
Zeros
0%
-2.0E0
100
Out[13]= Graphics RootLocusPlot Options
RootLocusPlot inherits its options from Graphics and adds its own options like PoleStyle and
ZeroStyle to Options[RootLocusPlot] . With these options you can customize the display of poles
and zeros, e.g. for better black-and-white printouts. The syntax of the PoleStyle option is
PoleStyle −> mark[size, colorfunc, grstyle]
where mark specifies the marker symbol (e.g. CrossMark , PlusMark , CircleMark , SquareMark , and
PointMark ), size denotes the size of the marker in scaled coordinates, colorfunc is a pure function that
returns a color value for an argument between and , and grstyle is a Mathematica graphics style
or a sequence of styles, such as Hue or Thickness . Note that everything mentioned above applies
to the ZeroStyle option as well.
In the following command line we instruct RootLocusPlot to modify the styles for the pole and
zero markers:
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In[14]:= RootLocusPlot[H4[s, a],{a, 3, 5},
PoleStyle −> PlusMark[0.03, Hue[1−0.3*#] &,
Thickness[0.012]],
ZeroStyle −> PointMark[0.03, Hue[0.5−0.3*#] &,
Thickness[0.005]]]
a = 3.000e0 .. 5.000e0 (0% .. 100%)
Im s
Poles
0%
2.0E0
1.
2.0E0
-1.
100
Re s
-0.5
Zeros
0%
-1.
-2.0E0
100
Out[14]= Graphics Pole-Zero Plots of Transfer Functions Without Parameters
The RootLocusPlot function can also be used to display the pole and zero locations of transfer
functions without parameters. For this purpose, RootLocusPlot must be called with a univariate
transfer function, and no parameter interval must be passed as a second argument.
In[15]:= RootLocusPlot[H3[s]]
Im s
2.0E0
1.
.0E0
-2.0E0
-1.
Re s
-0.5
-1.
-2.0E0
Out[15]= Graphics 2.5 Graphics
89
Animating Root Locus Plots
Sometimes it is helpful to animate a root locus plot because this shows the orientations of the
pole and zero trajectories better than a static plot. Let’s demonstrate root locus plot animation on
the transfer function H4[s, a] from above. To prepare the animation we compute a sequence of
pole-zero plots with identical plot ranges by mapping the following RootLocusPlot function to a
table of parameter values ranging from to in steps of . This is necessary because we want to
produce one separate plot in every parameter step instead of one single plot in which the solutions
from all steps are superimposed.
In[16]:= RootLocusPlot[H4[s, a], {a, #, #},
PoleStyle −> CrossMark[0.03, Hue[0.7] &,
Thickness[0.007]],
ZeroStyle −> CircleMark[0.03, Hue[0.3] &,
Thickness[0.007]],
PlotRange −> {{−6, 1}, {−4, 4}},
ShowLegend −> False,
LinearRegionStyle −> RGBColor[1, 1, 1]
] & /@ Table[x, {x, −3, 5, 2}]
a = 5.000e0
Im s
2.0E0
1.
-5.0E0
-2.0E0
-1.
1.
Re s
-1.
-2.0E0
Out[16]=
Graphics , Graphics , Graphics , Graphics , Graphics We show only one plot here; several plots are generated if the command is evaluated in a Mathematica
notebook.
To animate the root locus plot double-click on one of the images. Alternatively, you can select
the resulting group of notebook cells containing the images with your mouse and then click on
Cell | Animate Selected Graphics in Mathematica’s frontend menu.
2.5.5 Transient Waveforms
For displaying transient waveforms, Analog Insydes provides the command TransientPlot (Section 3.9.6). The usage of this function is demonstrated in Section 2.7.1 and Section 2.7.6.
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2.6 Modeling and Analysis of Nonlinear Circuits
2.6.1 Behavioral Models
In Chapter 2.3 we implemented small-signal equivalent circuits for semiconductor devices as subcircuit objects composed of the available linear circuit primitives. We cannot use this approach to
model the electrical characteristics of nonlinear elements, such as diodes or transistors. However,
particularly for nonlinear circuit analysis, Analog Insydes provides special support for modeling
arbitrary circuit element characteristics by specifying device equations directly.
Models which are based on sets of mathematical equations describing the behavior of a circuit
element or an analog building block are frequently called (analog) behavioral models, or ABMs. A
behavioral model is similar to a subcircuit object in that it constitutes a box which is connected to a
circuit through electrical ports. As indicated above, the difference is that the interior of a behavioral
model box is implemented in terms of symbolic equations rather than a netlist. With Analog Insydes’
behavioral modeling capabilities you can model a large variety of nonlinear element characteristics.
You can then use the function CircuitEquations (Section 3.5.1), with which you are already familiar,
to set up symbolic systems of nonlinear modified nodal or sparse tableau equations.
2.6.2 Defining Behavioral Models
The Argument Sequence of the Model Command for ABM Definitions
Behavioral models are defined using the Model (Section 3.2.1) command, just like subcircuits (see
Section 2.3.2). Analog Insydes recognizes a request to define a behavioral model by the presence of
the named parameter Variables and the absence of a subcircuit Netlist on the right-hand side of
the Definition argument in a Model call. The full syntax for an ABM definition is shown below.
Again, slanted typewriter font denotes optional arguments.
Model[
Name −> subcircuit class name,
Selector −> selector,
Scope −> scope,
Ports −> {port nodes},
Parameters −> {parameters},
Symbolic −> {symbolic parameters},
Defaults −> {defaults},
InitialConditions −> {initial conditions},
InitialGuess −> {initial guess},
Translation −> {parameter translation rules},
Variables −> {variables},
Definition −> Equations[equations]
]
2.6 Modeling and Analysis of Nonlinear Circuits
91
Except for the additional arguments and the keyword Definition , all other named arguments of
the Model function have the same meaning as for subcircuit definitions. Following, we discuss the
two arguments Definition and Variables .
Example: A Diode
Let’s illustrate the steps which are necessary to define a behavioral model by implementing a
nonlinear device model for a junction diode.
A
vd/vt
id(vd)=Is (e
VD
-1)
ID
C
Figure 6.1: Junction diode
The diode shown in Figure 6.1 has two terminals, the anode and the cathode, denoted by the node
identifiers A and C, respectively. The branch voltage vD and the branch current iD satisfy the device
equation
iD IS evD Vt where IS denotes the reverse-bias saturation current and Vt the thermal voltage. Typical values for IS
are A A. Vt is given by k Tq where k denotes Boltzmann’s constant (k WsK).
T is the absolute temperature (in Kelvin), and q is the charge of an electron (q As).
In[1]:= <<AnalogInsydes‘
In[2]:= ThermalVoltage[T_] := 1.381*10^−23 * T / (1.602*10^−19)
At a temperature of K ( C), Vt is approximately mV:
In[3]:= ThermalVoltage[300.]
Out[3]= 0.0258614
Since the thermal voltage is only defined in terms of constants and the global quantity temperature,
Vt is a global parameter. Thus, we have a local parameter of the diode device equation which is
IS , and a global parameter which is Vt . Note that a global parameter param is defined by setting
Global[param] in the Parameters argument. Assuming Scope −> Local and no Translation rules,
the first half of the Model definition is thus given by
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Model[
Name −> Diode,
Selector −> DC,
Ports −> {A, C},
Parameters −> {Is, Global[Vt]},
]
The Definition Argument
For behavioral model definitions, the Definition parameter must be of the form
Definition −> Equations[eq1 , eq2 , ]
where eq1 , eq2 , denote the symbolic device equations. (For compatibility reasons the right hand
side of the Definition parameter may also be a list of equations.) When specifying an arbitrary
set of equations we must tell Analog Insydes how the unknowns of the equations relate to the port
currents and voltages of a model object.
node+
Voltage[node+, node-] Current[node+, node-]
nodeFigure 6.2: Voltage and current identifiers at a model port branch
Ports of behavioral models are defined in terms of electrical port branches. Every unique pair
of node identifiers [node+, node−] introduces an electrical port branch in between the model port
nodes node+ and node−. For the associated port voltage and current the positive reference direction
is defined to be oriented from node+ to node−, which is illustrated in Figure 6.2.
Port variables in behavioral model equations can be referred to by means of two special keywords
which are recognized by Analog Insydes and replaced by the corresponding global current or voltage
identifiers during model instantiation and expansion. The keywords provided to make references to
port currents and voltages are, respectively:
Current[node+, node-]
Voltage[node+, node-]
Note that the above definition of positive reference directions implies that Current[nodeA, nodeB] and
Current[nodeB, nodeA] are two different quantities and refer to different port branches. Similarly,
Current[nodeA, nodeB] and Voltage[nodeB, nodeA] do not refer to the same port branch.
2.6 Modeling and Analysis of Nonlinear Circuits
93
In the case of the diode we have two port nodes A and C, which are connected by an electrical port
branch [A, C]. The port current, i.e. iD , is thus designated by Current[A, C] and the port voltage,
i.e. vD , is designated by Voltage[A, C]. Hence, the model equation must be written as
Definition −>
Equations[Current[A, C] == Is (Exp[Voltage[A, C]/Vt] − 1)]
In this example, the Definition contains only one single equation, but there is no limit to the
number of equations. We will discuss some more advanced examples in which this will become
apparent later in this chapter.
A note on interfacing: Some numerical circuit simulators with behavioral model simulation capabilities use the
currents into the pins (port nodes) of a model object and the node voltages at the pins as symbolic quantities
by which the model equations are interfaced with the exterior circuit. This approach is not well suited for other
analysis methods than modified nodal analysis. Using port branches with associated currents and voltages
facilitates setting up other formulations which involve loop equations, such as the sparse tableau.
The Variables Argument
The Variables argument serves to specify the symbols which are unknowns of the model equations.
The diode equation contains two unknowns, Current[A, C] and Voltage[A, C], so the Variables
argument is
Variables −> {Current[A, C], Voltage[A, C]}
The need for the Variables specification may not be entirely obvious here. One might argue that
Analog Insydes should know that port current and voltage identifiers are unknowns. However, port
currents and voltages are not always the only variables of a set of model equations. Similarly to
a subcircuit which may have internal nodes, a behavioral model may also contain internal variables which are not of the form Voltage[node+, node−] or Current[node+, node−]. Without the
Variables argument, Analog Insydes would not be able to distinguish between internal variables
and global parameters, such as Vt in the diode equation. So, by convention, you must specify all
identifiers in a set of model equations which are neither local nor global parameters, including the
port branch quantities. The complete ABM definition for the diode is then given by
Model[
Name −> Diode,
Selector −> DC,
Ports −> {A, C},
Parameters −> {Is, Global[Vt]},
Variables −> {Current[A, C], Voltage[A, C]},
Definition −>
Equations[Current[A, C] == Is (Exp[Voltage[A, C]/Vt] − 1)]
]
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2.6.3 Referencing Behavioral Models
Behavioral models are referenced exactly like subcircuits. Since we have already learned in Chapter 2.3 how to write hierarchical netlists we can start immediately with an example. Let’s write a
netlist for the diode circuit in Figure 6.3 using the diode model developed above.
1
R1
V0
2
D1
Figure 6.3: Diode circuit
In[4]:= diodeNetwork =
Circuit[
Netlist[
{V0, {1, 0}, V0},
{R1, {1, 2}, R1},
{D1, {2 −> A, 0 −> C},
Model −> Diode,
Selector −> DC}
],
Model[
Name −> Diode,
Selector −> DC,
Ports −> {A, C},
Parameters −> {Is, Global[Vt]},
Variables −> {Current[A, C], Voltage[A, C]},
Definition −>
Equations[
Current[A, C] == Is (Exp[Voltage[A, C]/Vt] − 1)
]
]
]
Out[4]= Circuit 2.6.4 Nonlinear Circuit Equations
Setting up Nonlinear Equations
Using the command CircuitEquations you can set up modified nodal or sparse tableau equations
for nonlinear circuits just as for linear circuits. The only restriction is that nonlinear systems of
2.6 Modeling and Analysis of Nonlinear Circuits
95
equations cannot be set up in matrix form. Therefore, whenever a netlist contains behavioral models,
you must call CircuitEquations with the option MatrixEquation −> False, or as in our case,
you need to switch the AnalysisMode to e.g. DC for setting up nonlinear static equations which
automatically implies the setting MatrixEquation −> False.
In[5]:= dnwmna = CircuitEquations[diodeNetwork,
AnalysisMode −> DC];
DisplayForm[dnwmna]
Out[6]//DisplayForm=
V$1 V$2
V$1 V$2
I$V0 0, I$AC$D1 0, V$1 V0,
R1
R1
V$2
Vt Is$D1, V$1, V$2, I$V0, I$AC$D1,
I$AC$D1 1 E DesignPoint Here, we have set up a system of nonlinear modified nodal equations in the unknowns V$1, V$2,
I$V0, and I$AC$D1 . The latter symbol has been created automatically from the port current identifier
Current[A, C] in the definition of the diode model. All port branch voltages have been replaced
by corresponding differences of node voltages.
Generally, a port current Current[x, y] in a model instance MX will be denoted by symbols of the
form I$xy$MX. A port voltage Voltage[x, y] will be denoted by V$xy$MX, provided that branch
voltages appear as unknowns in the selected analysis method. To examine both effects we set up
the sparse tableau equations.
In[7]:= CircuitEquations[diodeNetwork,
Formulation −> SparseTableau, AnalysisMode −> DC
] // GetVariables
Out[7]= V$V0, V$R1, V$AC$D1, I$V0, I$R1, I$AC$D1
We use the command GetVariables (Section 3.6.7) to extract the list of variables from the equation
system. As you can see, the corresponding variables are called V$AC$D1 and I$AC$D1
Solving Nonlinear Equations
Solving nonlinear circuit equations analytically is, unfortunately, mathematically impossible in the
general case. However, in many applications it is possible to reduce the original set of equations by
eliminating a number of variables. This may already yield some qualitative insight into the behavior
of a nonlinear circuit. On the other hand, we can always assign values to symbolic parameters and
solve the equations numerically using NDAESolve (Section 3.7.5).
Let’s derive an expression which relates the diode current
I$AC$D1 to the input voltage V0 by eliminating all other variables. For this task, Analog Insydes
provides the function CompressNonlinearEquations (Section 3.12.2) which removes equations and
variables from a nonlinear DAEObject that are irrelevant for solving for a set of given variables. The
option setting EliminateVariables −> All additionally allows for eliminating variables that occur
linear somewhere in the equations.
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In[8]:= dnwsol = CompressNonlinearEquations[dnwmna, I$AC$D1,
EliminateVariables −> All];
DisplayForm[dnwsol]
Out[9]//DisplayForm=
I$AC$D1 R1V0
Vt
I$AC$D1 1 E Is$D1, I$AC$D1, DesignPoint The implicit equation is what we have been looking for. Without resorting to approximation methods such as Taylor series we cannot simplify the result any further and solve for the diode current
analytically.
2.6.5 Multi-Dimensional Models
The Ebers-Moll Transistor Model
The procedure for modeling one-dimensional nonlinear element characteristics can be easily extended
to the multi-dimensional case. Let’s demonstrate multi-dimensional device modeling on a practical
example by defining nonlinear DC models for a bipolar junction transistor (see Figure 6.4).
IC
VBC
C
B
VCE
IB
E
VBE
IE
Figure 6.4: NPN transistor
Our considerations will be based on the BJT model introduced by Ebers and Moll which expresses
the relations between the transistor currents and voltages IC , IE , VBE , and VBC as
IC IS eVBE Vt IE IS
eVBC Vt Αr
IS
eVBE Vt IS eVBC Vt Αf
In these equations, the parameter IS represents the transport saturation current, Αr and Αf denote the
large-signal reverse and forward current gains of the common base configuration, and Vt designates
the thermal voltage. In integrated circuit design, the saturation current is usually expressed as the
product of the transport saturation current density JS , which is a process parameter, and the emitter
area A, which is a design parameter:
2.6 Modeling and Analysis of Nonlinear Circuits
97
IS JS A
The remaining two unknown voltages and currents at the transistor’s terminals, IB and VCE can be
computed from KCL and KVL (see Figure 6.5) as
IB IC IE VCE VBE VBC
IC
VBC
IB
C
B
VBE
VCE
E
C
Voltage[B,C]
Current[B,C]=-IC
B
Current[B,E]=-IE
Voltage[B,E]
E
IE
Figure 6.5: NPN transistor and corresponding ABM port branch definition
In most linear applications, bipolar transistors are typically operated in the forward active region
which is roughly characterized by VBE V and VBC V. Under these assumptions we can
simplify the Ebers-Moll model to
IC IS eVBE Vt
IS
IE eVBE Vt
Αf
because as compared to the terms involving eVBE Vt , all other terms are smaller by several orders of
magnitude.
A BJT operating in the forward active region acts as a good current amplifier from the base current IB
to the collector current IC with the gain relationship
I C Βf I B
The parameter Βf denotes the large-signal forward current gain in common-emitter configuration.
It is often more convenient to think in terms of forward and reverse current gains with respect to
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the base current, i.e. Βf and Βr , than in terms of the model parameters Αf and Αr in the original
Ebers-Moll equations. The relationship between Αf and Βf , as well as between Αr and Βr is given by
Β
Α
Α
Α
Β
Β
or, equivalently,
By making use of parameter translation rules in the model definition (see Section 2.3.7), we can let
Analog Insydes compute the values for Αf and Αr automatically from given values for Βf and Βr .
Implementation of Nonlinear DC Transistor Models
With the above background information on transistor models we are now ready to define a largesignal BJT model for DC circuit analysis. We start by implementing a global DC model for an NPN
transistor using the unsimplified device equations after Ebers and Moll from the previous subsection.
Name −> NPNBJT,
Selector −> EbersMoll,
Scope −> Global,
Ports −> {C, B, E},
Let’s decide that we do not want to characterize a transistor by the Ebers-Moll model parameters Is, alphaf, alphar, and Vt but want to use the parameters Js, Area, betaf, betar, and
Temperature instead. We do not need to rewrite the model equations if we specify translation rules
which translate our model parameters to those of the Ebers-Moll model. Remember that the function
ThermalVoltage has already been defined in Section 2.6.2.
Parameters −> {betaf, betar, Js, Area, Global[Temperature]},
Translation −>
{alphaf −> betaf/(1+betaf), alphar −> betar/(1+betar),
Is −> Js*Area, Vt :> ThermalVoltage[Temperature]},
The Ebers-Moll equations describe the functional relations between the voltages and currents at
two electrical branches, the base-collector branch and the base-emitter branch, which we use as the
port branches of our model. Associated with the port branches are the variables Voltage[B, E],
Current[B, E], Voltage[B, C], and Current[B, C], which must be specified as
2.6 Modeling and Analysis of Nonlinear Circuits
99
Variables −>
{Current[B, C], Voltage[B, C],
Current[B, E], Voltage[B, E]},
in the model definition.
It remains to write the model equations in terms of the port branch variables, paying attention to
the different positive reference directions assumed for the port currents as compared to IC and IE in
the Ebers-Moll equations. While the latter are defined to be positive for currents flowing into the
collector and emitter, positive values of Current[B, C] and Current[B, E] denote currents flowing
out of these terminals. We account for this difference by changing the sign on the right-hand side of
the model equations.
Definition −>
Equations[
Current[B, C] == −Is*(Exp[Voltage[B, E]/Vt] − 1)
+ Is/alphar*(Exp[Voltage[B, C]/Vt] − 1),
Current[B, E] == Is/alphaf*(Exp[Voltage[B, E]/Vt] − 1)
− Is*(Exp[Voltage[B, C]/Vt] − 1)
],
To store the transistor model in the global database we wrap the complete Model definition into a
Circuit structure and expand it with ExpandSubcircuits (Section 3.4.1).
In[10]:= Circuit[
Model[
Name −> NPNBJT,
Selector −> EbersMoll,
Scope −> Global,
Ports −> {C, B, E},
Parameters −>
{betaf, betar, Js, Area, Global[Temperature]},
Translation −>
{alphaf −> betaf/(1+betaf),
alphar −> betar/(1+betar),
Is −> Js*Area, Vt :> ThermalVoltage[Temperature]},
Variables −>
{Current[B, C], Voltage[B, C],
Current[B, E], Voltage[B, E]},
Definition −>
Equations[
Current[B, C] == −Is*(Exp[Voltage[B, E]/Vt] − 1)
+ Is/alphar*(Exp[Voltage[B, C]/Vt] − 1),
Current[B, E] == −Is*(Exp[Voltage[B, C]/Vt] − 1)
+ Is/alphaf*(Exp[Voltage[B, E]/Vt] − 1)
]
]
] // ExpandSubcircuits;
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Similarly, we define a simplified Ebers-Moll model for transistors operating in the forward active
region.
In[11]:= Circuit[
Model[
Name −> NPNBJT,
Selector −> EbersMollForwardActive,
Scope −> Global,
Ports −> {C, B, E},
Parameters −> {betaf, Js, Area, Global[Temperature]},
Translation −>
{alphaf −> betaf/(1+betaf),
Is −> Js*Area, Vt :> ThermalVoltage[Temperature]},
Variables −>
{Current[B, C], Voltage[B, C],
Current[B, E], Voltage[B, E]},
Definition −>
Equations[
Current[B, C] == −Is*(Exp[Voltage[B, E]/Vt]),
Current[B, E] == Is/alphaf*(Exp[Voltage[B, E]/Vt])
]
]
] // ExpandSubcircuits;
Both models are now stored in the global database:
In[12]:= GlobalSubcircuits[]
Out[12]=
NPNBJT, EbersMoll, NPNBJT, EbersMollForwardActive
2.6.6 Nonlinear DC Circuit Analysis
Calculating the Bias Point of a Transistor Circuit
Analog Insydes’ nonlinear modeling capabilities and numeric equation solver allow for solving circuit analysis problems quickly and reliably without tedious and error-prone hand calculations. Let’s
demonstrate one such application:
Determine the operating-point currents and voltages IB, IC, VBE, VCE for the circuit shown in Figure 6.6, if (a) RB k and (b) RB k . Assume that the transistor has an emitter area of
mil , Βf , Βr , JS ΜAmil , and that it operates at a temperature of K. The
values of the supply voltage and the collector resistance are VCC V and RC k .
2.6 Modeling and Analysis of Nonlinear Circuits
101
1
RC
RB
IB
IC
VCC
3
Q1
2
VCE
VBE
Figure 6.6: Bipolar transistor circuit
For didactic reasons we do not use ReadNetlist to automatically import the netlist, but rather
write the circuit by hand. Moreover, instead of using a BJT transistor model from the predefined
Analog Insydes model library, we select a suitable model from the global database which contains
the two NPNBJT models defined in the previous section. By leaving the model selector unspecified,
Selector −> BJTModel, we can postpone the final choice of the underlying model equations until
we set up circuit equations. The option Pattern −> Impedance in the value field of RB introduces
the current I$RB into the MNA equations. The current I$RB is identical to the base current IB, so we
can compute the latter directly from the equations without having to determine it manually from IC
and IE by KCL.
In[13]:= bjtbias =
Circuit[
Netlist[
{VCC, {1, 0}, Type −> VoltageSource,
Symbolic −> VCC, Value −> 12.},
{RB, {1, 2}, Pattern −> Impedance,
Symbolic −> RB, Value −> 500.*10^3},
{RC, {1, 3}, Symbolic −> RC, Value −> 4000.},
{Q1, {3 −> C, 2 −> B, 0 −> E},
Model −> NPNBJT, Selector −> BJTModel,
betaf −> 100., betar −> 0.2,
Js −> 6.*10^−16, Area −> 4., Temperature −> 300.}
]
]
Out[13]= Circuit In order to keep model complexity as low as possible, we make the initial assumption that Q1 operates
in the forward active region and set up a system of symbolic static circuit equations using the simplified Ebers-Moll model by setting the options AnalysisMode −> DC and ElementValues −> Symbolic.
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In[14]:= mnaeqsfa = CircuitEquations[
bjtbias /. BJTModel −> EbersMollForwardActive,
AnalysisMode −> DC, ElementValues −> Symbolic];
DisplayForm[mnaeqsfa]
Out[15]//DisplayForm=
V$1 V$3
I$RB I$VCC 0, I$BC$Q1 I$BE$Q1 I$RB 0,
RC
V$1 V$3
I$BC$Q1 0, V$1 VCC,
RC
I$RB RB V$1 V$2 0, I$BC$Q1 2.4 1015 E38.6676 V$2 ,
I$BE$Q1 2.424 1015 E38.6676 V$2 ,
V$1, V$2, V$3, I$VCC, I$RB, I$BC$Q1, I$BE$Q1,
DesignPoint VCC 12., RB 500000., RC 4000.
This system of equations can be solved by Analog Insydes’ numerical root finding function NDAESolve
(Section 3.7.5), which starts from an initial guess for the values of the variables. By default, initial
values of for all unknowns are chosen which is very often close enough to ensure convergence.
In[16]:= NDAESolve[mnaeqsfa, {t, 0}]
Out[16]=
V$1 12., V$2 0.712996, V$3 2.9704, I$VCC 0.00227997,
I$RB 0.000022574, I$BC$Q1 0.0022574, I$BE$Q1 0.00227997
From the result we can immediately read off the operating point. We have
VBE
VCE
IB
IC
V$2 V
V$3 V
I$RB mA
I$BC$Q1 mA
Next, we repeat the same calculation for RB k using the command UpdateDesignPoint
(Section 3.6.14) to alter the numeric element value for RB.
In[17]:= mnaeqsfa = UpdateDesignPoint[mnaeqsfa, RB −> 1.*10^5];
NDAESolve[mnaeqsfa, {t, 0}]
Out[18]=
V$1 12., V$2 0.75452, V$3 32.9819, I$VCC 0.0113579,
I$RB 0.000112455, I$BC$Q1 0.0112455, I$BE$Q1 0.0113579
This time, we obtain a physically nonsensical value of V for VCE which indicates that our
initial assumption about the operating condition of Q1 has been violated. Indeed, for RB k ,
Q1 operates in deep saturation. We must therefore select the full Ebers-Moll model to take these
effects into account.
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103
In[19]:= mnaeqssat = CircuitEquations[
bjtbias /. BJTModel −> EbersMoll,
AnalysisMode −> DC, ElementValues −> Symbolic];
DisplayForm[mnaeqssat]
Out[20]//DisplayForm=
V$1 V$3
I$RB I$VCC 0, I$BC$Q1 I$BE$Q1 I$RB 0,
RC
V$1 V$3
I$BC$Q1 0, V$1 VCC,
RC
I$RB RB V$1 V$2 0, I$BC$Q1 2.4 1015 1 E38.6676 V$2 1.44 1014 1 E38.6676 V$2V$3 , I$BE$Q1 2.424 1015 1 E38.6676 V$2 2.4 1015 1 E38.6676 V$2V$3 ,
V$1, V$2, V$3, I$VCC, I$RB, I$BC$Q1, I$BE$Q1,
DesignPoint VCC 12., RB 500000., RC 4000.
Solving these enhanced equations will now deliver a proper result for RB k .
In[21]:= mnaeqssat = UpdateDesignPoint[mnaeqssat, RB −> 1.*10^5];
NDAESolve[mnaeqssat, {t, 0}]
Out[22]=
V$1 12., V$2 0.720901, V$3 0.13522, I$VCC 0.00307899,
I$RB 0.000112791, I$BC$Q1 0.0029662, I$BE$Q1 0.00307899
We finally obtain:
VBE
VCE
IB
IC
V$2 V
V$3 V
I$RB mA
I$BC$Q1 mA
2.6.7 References
The BJT circuit DC analysis example was adapted from the book by R. L. Geiger, P. E. Allen, and
N. R. Strader: VLSI Design Techniques for Analog and Digital Circuits, Mc-Graw Hill, 1990.
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2.7 Time-Domain Analysis of Nonlinear Dynamic
Circuits
In Chapter 2.6 it was shown how to model and analyze circuits with nonlinear element characteristics. In this chapter, we extend these considerations to the analysis of circuits containing not only
nonlinear but also dynamic components, such as capacitors and inductors. From a mathematical
point of view this means that a system of equations which describes a nonlinear dynamic circuit involves linear and nonlinear algebraic constraints as well as time derivatives of some circuit variables.
Equations with these properties are called system of nonlinear Differential and Algebraic Equations, or
DAE.
When analyzing a dynamic circuit we are usually interested in finding the response of the circuit to
a given input signal as a function of time. This analysis mode is commonly referred to as transient
analysis. Computing a transient response requires solving an implicit DAE system numerically, starting from an initial solution for the voltages and currents known as the DC operating point. With
Mathematica’s built-in command NDSolve we cannot handle this type of problem because solving differential equations with algebraic constraints is not supported. Therefore, Analog Insydes includes a
special DAE solver for circuit simulation called NDAESolve which will be introduced in the following
section.
2.7.1 Transient Circuit Analysis
A Diode Rectifier Circuit
Let’s start to demonstrate transient analysis on the half-wave diode rectifier circuit shown in Figure 7.1. This circuit contains both nonlinear as well as dynamic components, namely the diode D1
and the capacitor C1. Given numerical element values and input voltage waveform V Vin t we
shall compute the transient response Vout t across the load resistor R1.
D1
1
V0
2
C1
R1
Figure 7.1: Diode rectifier circuit
Vout
2.7 Time-Domain Analysis of Nonlinear Dynamic Circuits
105
As usual, we prepare our circuit analysis by writing a netlist description of the circuit. For the diode,
we use the model from the Analog Insydes model data base (Chapter 4.3). The model parameters
Is (saturation current) and Vt (thermal voltage) are set to Is pA and Vt mV. The values of
the circuit elements are assumed to be R1 and C1 nF.
In[1]:= <<AnalogInsydes‘
In[2]:= rectifier =
Circuit[
Netlist[
{V0, {1, 0}, Vin},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*10^−7},
{D1, {1 −> A, 2 −> C},
Model −> "Diode", Selector −> "Spice",
IS −> 1.*10^−12}
]
]
Out[2]= Circuit Next, we use the function CircuitEquations (Section 3.5.1) to set up a system of nonlinear differential MNA equations in the time domain. In oder to express all voltages and currents as functions
of time f t and to include time derivatives f ! t introduced by dynamic components the option
AnalysisMode −> Transient must be specified in the function call. AnalysisMode −> Transient
implies MatrixEquation −> False, therefore the equations are written as list of equations regardless of the current setting of MatrixEquation . CircuitEquations returns a Transient DAEObject
which can be displayed via the command DisplayForm .
In[3]:= rectifierMNA = CircuitEquations[rectifier,
AnalysisMode −> Transient];
DisplayForm[rectifierMNA]
Out[4]//DisplayForm=
I$AC$D1t I$V0t 0,
I$AC$D1t 0.01 V$2t 1. 107 V$2 t 0,
V$1t Vin, I$AC$D1t 1. 1012 1 E38.6635 V$1tV$2t 1. 1012 V$1t V$2t,
V$1t, V$2t, I$V0t, I$AC$D1t, DesignPoint This set of modified nodal equations is a typical DAE system. It comprises implicit differential
equations as well as both linear and nonlinear algebraic constraints.
The NDAESolve Command
Analog Insydes provides the function NDAESolve (Section 3.7.5) for solving DAE systems numerically.
NDAESolve[dae, {tvar, t , t } , params
, opts
]
where dae is a DC or Transient DAEObject containing the circuit equations and variables. The
argument tvar denotes the time variable for which the solutions are computed in the time interval
tvar " t t , or at the time instant tvar t . Additionally, params allows for carrying out a parametric analysis. For possible parameter specifications please refer to Section 3.7.2. Finally, opts is a
sequence of zero or more solver options of the form option −> value (see Section 2.7.4).
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Let’s use NDAESolve to compute the transient response Vout t of the diode circuit to a sinusoidal
input voltage Vin t V sinΩt. We choose Ω s , replace the symbol Vin which denotes the
input voltage by the sine function, and integrate the circuit equations numerically with respect to
the time variable t from t to t Μs.
In[5]:= solutions = NDAESolve[rectifierMNA /. Vin −> Sin[10^6 t],
{t, 0., 2.*10^−5}]
Out[5]=
V$2 InterpolatingFunction0, 0.00002, ,
V$1 InterpolatingFunction0, 0.00002, ,
I$V0 InterpolatingFunction0, 0.00002, ,
I$AC$D1 InterpolatingFunction0, 0.00002, Like Mathematica’s NDSolve, NDAESolve returns the solutions for the node voltages and currents
in terms of InterpolatingFunction objects. This allows for accessing the numerical values of all
variables at any point of time in the simulation interval. To obtain the voltages and currents at a
certain point of time ti , we first have to rewrite the interpolating functions as functions of time.
In[6]:= functions =
First[solutions] /. Rule[a_, b_] −> Rule[a, b[t]]
Out[6]=
V$2 InterpolatingFunction0, 0.00002, t,
V$1 InterpolatingFunction0, 0.00002, t,
I$V0 InterpolatingFunction0, 0.00002, t,
I$AC$D1 InterpolatingFunction0, 0.00002, t
Then we replace t by a particular value, here t Μs. This yields the solution vector as a list of
rules.
In[7]:= functions /. t −> 10^−5
Out[7]=
V$2 0.337585, V$1 0.543881, I$V0 1.88147 1012 ,
I$AC$D1 1.88147 1012 From the result we can immediately extract the voltage values for V (t Μs) and Vout (t Μs),
as well as the value for the diode current ID1 (t Μs)
V V$1 V
Vout V$2 V
ID1 I$AC$D1 pA
Plotting Transient Waveforms
For graphical display of transient waveforms computed with NDAESolve , Analog Insydes provides
the function TransientPlot (Section 3.9.6). With TransientPlot , several interpolating functions
can be plotted in a single diagram or displayed as graphics array. The command syntax is
TransientPlot[numericaldata,
vars, {tvar, t , t } , opts
]
where numericaldata is a list of rules of interpolating functions. The format is compatible to the return
value of e.g. NDSolve and NDAESolve (see Section 3.7.1). The argument vars is a list of the variables
to be plotted. Besides the variables themselves, this list may also contain mathematical expressions
2.7 Time-Domain Analysis of Nonlinear Dynamic Circuits
107
in terms of these variables. The symbol tvar denotes the time variable for which the waveforms are
plotted from tvar t to tvar t . Zero or more options opts can be specified as sequence of rules of
the form option −> value (see Section 2.7.6).
To visualize the transient waveforms of the node voltages V$1 and V$2 in the simulated time interval
we can use TransientPlot as shown below. By default, all waveforms are superimposed in one
single graph.
In[8]:= TransientPlot[solutions, {V$1[t], V$2[t]},
{t, 0., 2.*10^−5}]
1
0.5
V$1[t]
-6
5. 10
0.00001
0.000015
t
0.00002
V$2[t]
-0.5
-1
Out[8]= Graphics Quasi-DC Analysis
To demonstrate the influence of the capacitor C1 on the output voltage we rerun the simulation,
but this time we neglect the dynamic components of the circuit equations. This can be achieved
by setting the NDAESolve option AnalysisMode from its default value Transient to DC. The DC
analysis method replaces all time derivatives in the circuit equations by zero. Thus, inductors are
replaced by short circuits because all inductor voltages are set to zero, and capacitors are replaced
by open circuits because all capacitor currents are set to zero.
Inductor:
Capacitor:
#IL t
#t
#UC t
IC t C
#t
VL t L
$
DC
VL t DC
IC t $
%
%
Short Circuit
Open Circuit
Let’s apply quasi-DC analysis to our rectifier circuit by specifying the above-mentioned option in the
call to NDAESolve .
In[9]:= solDC = NDAESolve[rectifierMNA /. Vin −> Sin[10^6 t],
{t, 0., 10^−5}, AnalysisMode −> DC]
Out[9]=
V$1 InterpolatingFunction0, 0.00001, ,
V$2 InterpolatingFunction0, 0.00001, ,
I$V0 InterpolatingFunction0, 0.00001, ,
I$AC$D1 InterpolatingFunction0, 0.00001, 2. Tutorial
108
The influence of the capacitor C1 can be clearly recognized from the plot of the node voltages V$1
and V$2.
In[10]:= TransientPlot[solDC, {V$1[t], V$2[t]}, {t, 0., 10^−5}]
1
0.5
V$1[t]
-6
2. 10
-6
4. 10
-6
6. 10
-6
8. 10
t
0.00001
V$2[t]
-0.5
-1
Out[10]= Graphics 2.7.2 Analysis of a Differential Amplifier
Transient Analysis
In this section, we want to examine a more complicated circuit to demonstrate the features and capabilities of the DAE solver. Consider the differential amplifier circuit shown in Figure 7.2 consisting
of four bipolar junction transistors. We shall compute the transient response Vout t to a rectangular
voltage pulse applied at node 1.
2.7 Time-Domain Analysis of Nonlinear Dynamic Circuits
109
8
VCC
RC1
5
4
Cload
1
RBIAS
RC2
RS1
Q1
Q2
Vout
3
2
V1
RS2
6
7
Q3
Q4
9
VEE
Figure 7.2: Differential amplifier circuit
Again we start by writing a netlist (of course we could have used ReadNetlist as well). Let the
values of the circuit elements be given as RS1 RS2 k , RC1 RC2 k , RBIAS k , and
CLOAD pF. The supply voltages are VCC V and VEE V. For all transistors Q1, , Q4,
we use the full Gummel-Poon transistor model implemented in the Analog Insydes model library
(see Section 4.3.2). We assume the following model parameter values: transport saturation current
IS A, large-signal forward and reverse current gains BF and BR , and
the global temperature TEMP K.
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In[11]:= diffAmp =
Circuit[
Netlist[
{RS1, {1, 2}, Symbolic −> RS1, Value −> 1.*10^3},
{RS2, {3, 0}, Symbolic −> RS2, Value −> 1.*10^3},
{RC1, {4, 8}, Symbolic −> RC1, Value −> 1.*10^4},
{RC2, {5, 8}, Symbolic −> RC2, Value −> 1.*10^4},
{RBIAS, {7, 8}, Symbolic −> RBIAS, Value −> 2.*10^4},
{CLOAD, {4, 5},
Symbolic −> CLOAD, Value −> 5.*10^−12},
{Q1, {4 −> C, 2 −> B, 6 −> E}, Model −> BJT,
Selector −> GummelPoon, Parameters −> BJTNPN},
{Q2, {5 −> C, 3 −> B, 6 −> E}, Model −> BJT,
Selector −> GummelPoon, Parameters −> BJTNPN},
{Q3, {6 −> C, 7 −> B, 9 −> E}, Model −> BJT,
Selector −> GummelPoon, Parameters −> BJTNPN},
{Q4, {7 −> C, 7 −> B, 9 −> E}, Model −> BJT,
Selector −> GummelPoon, Parameters −> BJTNPN},
{VCC, {8, 0}, Type −> VoltageSource,
Symbolic −> VCC, Value −> 12.},
{VEE, {9, 0}, Symbolic −> VEE, Value −> −12.},
{V1, {1, 0}, V1}
],
ModelParameters[Name −> BJTNPN, Type −> "NPN",
BF −> 255.9, BR −> 6.092, IS −> 14.34*10^−15],
GlobalParameters[TEMP −> 300.15]
]
Out[11]= Circuit Setting up the circuit equations in the time-domain yields the following DAE system of 32 equations
in 32 variables.
In[12]:= diffAmpMNA = CircuitEquations[diffAmp,
AnalysisMode −> Transient]
Out[12]= DAETransient, 32 32 Next, we define the input signal V t. We choose a rectangular voltage pulse with an amplitude
of V, a duration of Μs and a delay of Μs. This waveform can be set by the Analog Insydes
command PulseWave (Section 4.1.3).
In[13]:= Vpulse[t_] :=
PulseWave[t, 0, 2, 2.5*10^−6, 0, 0, 4.*10^−6, 10^−5]
A graphical representation of the stimulus signal is shown below.
2.7 Time-Domain Analysis of Nonlinear Dynamic Circuits
111
In[14]:= TransientPlot[Vpulse[t], {t, 0., 10^−5},
PlotRange −> {−1, 3}]
3
2.5
2
1.5
Vpulse[t]
1
0.5
-6
2. 10
-6
4. 10
-6
6. 10
-6
8. 10
t
0.00001
-0.5
-1
Out[14]= Graphics Then, we call
NDAESolve to simulate the transient behavior of the differential amplifier for t " Μs. The
NonlinearMethod option allows for choosing between different algorithms for solving systems of
nonlinear algebraic equations numerically. By default, Mathematica’s numerical solver FindRoot is
applied, whereas in this example we want to make use of another implementation of Newton’s
method called NewtonIteration .
In[15]:= transient = NDAESolve[diffAmpMNA /. V1 −> Vpulse[t],
{t, 0., 10^−5}, NonlinearMethod −> NewtonIteration];
Now, we use TransientPlot to plot the waveforms of the input and output voltages V$1 and V$5.
In[16]:= TransientPlot[transient, {V$1[t], V$5[t]},
{t, 0., 10^−5}]
12
10
8
V$1[t]
6
V$5[t]
4
2
-6
2. 10
-6
4. 10
-6
6. 10
-6
8. 10
t
0.00001
Out[16]= Graphics DC-Transfer Analysis
Next, we shall compute the DC-transfer characteristic Vout Vin V$5[V1] of the differential amplifier circuit as the input voltage is swept from V to V. Therefore, we first set up the
static circuit equations by calling CircuitEquations with the option setting AnalysisMode −> DC.
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Additionally, we formulate symbolic equations by setting ElementValues −> Symbolic. The option
setting Symbolic −> None causes all device model parameters to be replaced by their numerical
values.
In[17]:= staticequations = CircuitEquations[diffAmp,
AnalysisMode −> DC, ElementValues −> Symbolic,
Symbolic −> None];
DisplayForm[staticequations]
Out[18]//DisplayForm=
V$1 V$2
V$1 V$2
I$V1 0, I$BS$Q1 0,
RS1
RS1
V$3
V$4 V$8
I$BS$Q2 0, I$CS$Q1 0,
RS2
RC1
V$5 V$8
I$CS$Q2 0, I$CS$Q3 I$ES$Q1 I$ES$Q2 0,
RC2
V$7 V$8
I$BS$Q3 I$BS$Q4 I$CS$Q4 0,
RBIAS
V$4 V$8
V$5 V$8
V$7 V$8
I$VCC 0,
RC1
RC2
RBIAS
I$ES$Q3 I$ES$Q4 I$VEE 0, I$BS$Q1 I$CS$Q1 I$ES$Q1 0,
I$BS$Q1 ib$Q1, I$BS$Q1 I$ES$Q1 ic$Q1,
ic$Q1 1.434 1014 1. E38.6635 V$2V$6 1.16415
1.434 1014 1. E38.6635 V$2V$4 1. 1012 V$2 V$4 1. 1012 V$2 V$6, ib$Q1 0.16415
1.434 1014 1. E38.6635 V$2V$4 1. 1012 V$2 V$4 0.00390778
1.434 1014 1. E38.6635 V$2V$6 1. 1012 V$2 V$6,
I$BS$Q2 I$CS$Q2 I$ES$Q2 0,
I$BS$Q2 ib$Q2, I$BS$Q2 I$ES$Q2 ic$Q2,
ic$Q2 1.434 1014 1. E38.6635 V$3V$6 1.16415
1.434 1014 1. E38.6635 V$3V$5 1. 1012 V$3 V$5 1. 1012 V$3 V$6, ib$Q2 0.16415
1.434 1014 1. E38.6635 V$3V$5 1. 1012 V$3 V$5 0.00390778
1.434 1014 1. E38.6635 V$3V$6 1. 1012 V$3 V$6,
I$BS$Q3 I$CS$Q3 I$ES$Q3 0,
I$BS$Q3 ib$Q3, I$BS$Q3 I$ES$Q3 ic$Q3,
ic$Q3 1.434 1014 1. E38.6635 V$7V$9 1.16415
1.434 1014 1. E38.6635 V$6V$7 1. 1012 V$6 V$7 1. 1012 V$7 V$9, ib$Q3 0.16415
1.434 1014 1. E38.6635 V$6V$7 1. 1012 V$6 V$7 0.00390778
1.434 1014 1. E38.6635 V$7V$9 1. 1012 V$7 V$9,
I$BS$Q4 I$CS$Q4 I$ES$Q4 0,
I$BS$Q4 ib$Q4, I$BS$Q4 I$ES$Q4 ic$Q4, ic$Q4 0. 1.434 1014 1. E38.6635 V$7V$9 1. 1012 V$7 V$9,
ib$Q4 0. 0.00390778
1.434 1014 1. E38.6635 V$7V$9 1. 1012 V$7 V$9,
V$8 VCC, V$9 VEE, V$1 V1, 1,
DesignPoint 1
Then, we run NDAESolve sweeping the parameter V1 in the interval V V. Note that the
sweep variable can be any network parameter, for instance a voltage or current source parameter, a
model parameter, or even any unknown in the system of equations.
2.7 Time-Domain Analysis of Nonlinear Dynamic Circuits
113
In[19]:= dctransfer = NDAESolve[staticequations, {V1, −0.5, 0.5}];
To display the DC-transfer curve graphically we again apply the function TransientPlot . For our
DC-transfer analysis we want to plot the output voltage V$5 versus the input voltage V1:
In[20]:= TransientPlot[dctransfer, {V$5[V1]}, {V1, −0.5, 0.5}]
12
10
8
V$5[V1]
6
4
2
-0.4
-0.2
0.2
0.4
V1
Out[20]= Graphics The resulting plot shows the typical tanh-shaped transfer characteristic of this type of differential
amplifier.
Parametric Analyses
Finally, we demonstrate the usage of NDAESolve for carrying out parametric analyses. As an example, we perform a parametric analysis of the above computed DC-transfer characteristic V$5[V1] . As
additional sweep parameter we choose the positive reference voltage VCC which shall take the values
VCC " V V V. Note that for parametric analyses NDAESolve returns a multi-dimensional
data object (see Section 3.7.1).
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114
In[21]:= paramdctransfer = NDAESolve[staticequations,
{V1, −0.5, 0.5}, {VCC, {10, 11, 12}}]
Out[21]=
V$1 InterpolatingFunction0.5, 0.5, ,
V$2 InterpolatingFunction0.5, 0.5, ,
V$3 InterpolatingFunction0.5, 0.5, ,
V$4 InterpolatingFunction0.5, 0.5, ,
V$5 InterpolatingFunction0.5, 0.5, ,
V$6 InterpolatingFunction0.5, 0.5, ,
V$7 InterpolatingFunction0.5, 0.5, ,
V$8 InterpolatingFunction0.5, 0.5, ,
V$9 InterpolatingFunction0.5, 0.5, ,
I$BS$Q1 InterpolatingFunction0.5, 0.5, ,
I$CS$Q1 InterpolatingFunction0.5, 0.5, ,
I$ES$Q1 InterpolatingFunction0.5, 0.5, ,
ib$Q1 InterpolatingFunction0.5, 0.5, ,
8, I$ES$Q3 InterpolatingFunction0.5, 0.5, ,
ib$Q3 InterpolatingFunction0.5, 0.5, ,
ic$Q3 InterpolatingFunction0.5, 0.5, ,
I$BS$Q4 InterpolatingFunction0.5, 0.5, ,
I$CS$Q4 InterpolatingFunction0.5, 0.5, ,
I$ES$Q4 InterpolatingFunction0.5, 0.5, ,
ib$Q4 InterpolatingFunction0.5, 0.5, ,
ic$Q4 InterpolatingFunction0.5, 0.5, ,
I$VCC InterpolatingFunction0.5, 0.5, ,
I$VEE InterpolatingFunction0.5, 0.5, ,
I$V1 InterpolatingFunction0.5, 0.5, ,
SweepParameters VCC 10., 1,
1
The above generated multi-dimensional data format is supported by TransientPlot . Thus, we can
display all sweep traces of the DC-transfer characteristic in a single plot.
In[22]:= TransientPlot[paramdctransfer, {V$5[V1]},
{V1, −0.5, 0.5}]
12
10
8
V$5[V1]
6
4
2
-0.4
-0.2
Out[22]= Graphics 0.2
0.4
V1
2.7 Time-Domain Analysis of Nonlinear Dynamic Circuits
115
2.7.3 Flow of Transient Circuit Analysis
A Summary of the Transient Analysis Procedure
In this subsection, we summarize the steps which are necessary to perform a transient analysis of a
dynamic circuit with Analog Insydes.
1) Write a netlist of the circuit to be simulated, specifying numerical values for all circuit elements or
use ReadNetlist (Section 3.10.1) for automatically importing netlists from external simulator files.
circuit =
Circuit[
Netlist[ ],
Model[ ],
]
2) Set up a system of time-domain circuit equations using CircuitEquations with the option setting
AnalysisMode −> Transient . For numerical circuit simulation you should use modified nodal analysis, which is the default formulation. Modified nodal equations are smaller in size and therefore
require less simulation time.
mnaequations =
CircuitEquations[circuit, AnalysisMode −> Transient]
3) Apply the command NDAESolve to the circuit equations to calculate the transient solution in the
time interval t " t t , where t is usually zero.
transient = NDAESolve[mnaequations, {t, t , t }]
4) Use TransientPlot for transient waveform display. Select the variables to be plotted by means
of the list vars and specify the display time interval ta tb . The latter need not be the same as
the simulation time interval t t . You can use any other values for ta and tb to zoom in onto a
particular section of a trace as long as the condition t & ta tb & t holds.
TransientPlot[transient, vars, {t, ta , tb }]
NDAESolve Program Flow
Now, let’s take a closer look at the strategy NDAESolve employs to integrate a system of differential
and algebraic equations. The main idea behind the algorithm is to transform the problem of solving
a nonlinear system of differential and algebraic equations into a sequence of linear and purely algebraic
problems which can be solved rather easily. The transformation is carried out in two stages. In
the first step, the differential equations are discretized by replacing all time derivatives with a finite
difference approximation. The differential equations are thus evaluated and solved at discrete time
steps only. The finite difference approximation scheme used by NDAESolve is an implicit integration
method known as the trapezoidal rule. In a second step the resulting nonlinear algebraic system of
116
2. Tutorial
equations is then solved for the voltages and currents using the multi-dimensional Newton-Raphson
method.
Circuit equations are usually stiff, which means that their solutions involve both very rapid and very
slow transients. To avoid loss of accuracy and convergence problems during rapid transients as well
as excessive computation time requirements for slow transients the step size is chosen adaptively at
each time step according to the curvature of the waveform. Whenever a waveform changes quickly
the step size is cut back whereas it is increased during periods of latency. The behavior of the
step size control algorithm depends on the values of five parameters which will be discussed in the
following section. The associated NDAESolve options by which the default parameter settings can be
changed are StartingStepSize , MaxStepSize , MinStepSize , StepSizeFactor , and MaxDelta .
Figure 7.3 shows a flow graph of the algorithm implemented in NDAESolve .
2.7 Time-Domain Analysis of Nonlinear Dynamic Circuits
117
input of circuit netlist
set up timedomain
circuit equations
compute initial transient solution
DC operating point
substitute time derivatives
at time step t using trapezoidal method
solve nonlinear system of
equations at time step t
using Newton’s method
iteration
i
no
convergence
AND iimax
Y
N
iimax
step size control
Y
N
store solution at time step t
ttmax
time step
th
Y
N
interpolation of data
output of solutions as
interpolated functions
Figure 7.3: NDAESolve program flow
h StepSizeFactor
or
h StepSizeFactor
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2.7.4 NDAESolve Options
Options[NDAESolve]
Several options are available for NDAESolve most of which need not be changed unless there are
problems with the default settings, such as convergence problems. To list all options associated with
NDAESolve inspect the value of Options[NDAESolve] :
In[23]:= Options[NDAESolve]
Out[23]=
AccuracyGoal 6, AnalysisMode Transient,
BreakOnError False, Compiled True, CompressEquations False,
DesignPoint Automatic, InitialConditions Automatic,
InitialGuess Automatic, InitialSolution Automatic,
InterpolationOrder 1, MaxDelta 1., MaxIterations 100, 20,
MaxSteps Automatic, MaxStepSize Automatic,
MinStepSize Automatic, NonlinearMethod FindRoot,
OutputVariables All, Protocol InheritedAnalogInsydes,
RandomFunction RandomReal, 0., 0.1&,
ReturnDerivatives False, Simulator InheritedAnalogInsydes,
SimulatorExecutable Automatic, StartingStepSize Automatic,
StepSizeFactor 1.5, WorkingPrecision 16
Below, a set of NDAESolve options is explained in more detail:
InitialGuess
With the default setting InitialGuess −> Automatic , zero is used as initial guess for all variables
when solving a system of nonlinear equations by means of Newton iterations. Whenever this setting causes numerical problems, such as division-by-zero errors or failure to converge, you can use
InitialGuess to specify your own initial guess as a list of rules of the form:
InitialGuess −> {var −> value , , varn −> valuen }
Note that the variables vari are specified without the time variable tvar.
InitialSolution
With the option InitialSolution we can force NDAESolve to use a certain DC solution as initial
operating point for a transient analysis. A set of initial values must be provided as a list of rules
which associate a numerical value with a variable from the system of equations. Missing variables
are then computed consistently.
InitialSolution −> {var −> value , , varn −> valuen }
Note that the variables vari are specified without the time variable tvar.
MaxDelta
One of the conditions that are checked by the integration step size control mechanism is whether the
values of some variables change abruptly from one time step to the next. The maximum allowed
2.7 Time-Domain Analysis of Nonlinear Dynamic Circuits
119
difference between two successive values is controlled by the option MaxDelta. The default setting
is MaxDelta −> 1.0. This value is large enough to allow sudden steps of input voltages which are
quite usual in the case of pulse excitations.
MaxIterations
The option MaxIterations specifies the maximum number of steps that the nonlinear equation
solver should use in attempting to find a solution. The option setting can either be an integer int or
a list of two integers int int . If it is specified as a single integer int then it is equivalent to the
list int int. The first integer value defines the iteration limit for finding a DC operating point and
the second integer value the iteration limit for transient computations, respectively. If the number
of iterations for the operating-point computation exceeds the limit, an error message is generated
and the computation is interrupted. If the iteration limit for transient computations is exceeded,
the step size is reduced by the factor given by the option StepSizeFactor . The default setting is
MaxIterations −> {100, 20}.
MaxSteps
The option MaxSteps limits the total number of integration steps. The simulation is stopped immediately if the limit is exceeded. The default setting is MaxSteps −> Automatic , which means
MaxSteps /t t / StartingStepSize .
MaxStepSize
It is sometimes possible to speed up simulations by allowing a larger maximum integration step size.
With the option MaxStepSize the present step size limit can be extended or reduced. The default
setting is MaxStepSize −> Automatic , which means MaxStepSize /t t / .
MinStepSize
The lower limit of the integration step size can be specified by the option MinStepSize . NDAESolve
stops a simulation immediately if the integration step size falls below this limit. The default setting
is MinStepSize −> Automatic , which means MinStepSize /t t / .
NonlinearMethod
The option NonlinearMethod chooses among different numerical algorithms for solving the nonlinear algebraic systems of equations which arise due to the discretization of DAEs. By default,
Mathematica’s numerical solver FindRoot is used for this purpose. Alternatively, you can employ
Analog Insydes’ own implementation of the multi-dimensional Newton-Raphson method by specifying NonlinearMethod −> NewtonIteration .
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Protocol
With the option setting Protocol −> Notebook, NDAESolve can be instructed to print protocol information to your notebook as the computation proceeds. The protocol includes an output of the
initial guess, the initial transient solution, shows the percentage of the computation completed, and
the number of time steps which were necessary to carry out the computation. The default setting is
Protocol −> Inherited[AnalogInsydes] which means that NDAESolve inherits the option setting
from the global setting specified in Options[AnalogInsydes] . See also Section 3.14.5.
StartingStepSize
The option StartingStepSize helps to overcome convergence problems at the beginning of a simulation. If NDAESolve fails to converge during the first time step you can use StartingStepSize to
specify a smaller value for the initial integration step size. The default setting is
StartingStepSize −> Automatic , which means StartingStepSize /t t / .
StepSizeFactor
To reduce computation time and increase simulation accuracy NDAESolve employs an automatic step
size control scheme. If the estimated simulation error in between two integration time steps is very
low then the step size is increased. On the other hand, if the estimated error is too large the step
size is cut back. Increasing and reducing the step size is performed by multiplying or dividing the
current step size by a factor given by the value of the option StepSizeFactor . The default setting
is StepSizeFactor −> 1.5.
2.7.5 Transient Analysis with Initial Conditions
Circuits with Initial Conditions
The handling of initial conditions was rather complicated with version 1 of Analog Insydes. Therefore, the netlist format was extended to treat initial conditions of dynamic circuit elements such as
capacitors or inductors. This is done by specifying the setting InitialCondition −> value in the
option field of the particular netlist entry (see Section 3.1.4).
Consider the RLC circuit shown in Figure 7.4. The initial voltage VC1 across the capacitor C1 at t is given as VC1 Vic . Assume that the following numerical values are given for the circuit elements:
R1 , L1 ΜH, C1 ΜF, and Vic mV.
2.7 Time-Domain Analysis of Nonlinear Dynamic Circuits
121
Vic
2
1
L1
C1
R1
Iout
Figure 7.4: Oscillator circuit with initial condition
We start with writing a circuit netlist which includes the initial condition of the capacitor C1.
In[24]:= oscillator =
Netlist[
{L1, {0, 1}, Symbolic
{C1, {1, 2}, Symbolic
InitialCondition −>
{R1, {2, 0}, Symbolic
]
−> L1, Value −> 1.*10^−5},
−> C1, Value −> 3.*10^−7,
0.002},
−> R1, Value −> 1.}
Out[24]= NetlistRaw, 3 Following, we distinguish two alternatives for considering initial conditions of dynamic circuit
elements.
Initial Conditions for some Dynamic Elements
With the option setting InitialConditions −> Automatic the function CircuitEquations can be
instructed to use initial conditions for those elements for which such a condition has been specified
in the netlist entry. All others are then computed consistently by NDAESolve . For our example, we
set up a system of modified nodal equations from the circuit configuration at the time t .
In[25]:= oscillatorMNA1 = CircuitEquations[oscillator,
AnalysisMode −> Transient, ElementValues −> Symbolic,
InitialConditions −> Automatic];
DisplayForm[oscillatorMNA1]
Out[26]//DisplayForm=
I$L1t C1 V$1 t V$2 t 0,
V$2t
C1 V$1 t V$2 t 0, V$1t L1 I$L1 t 0,
R1
V$10 V$20 0.002, V$1t, V$2t, I$L1t,
DesignPoint L1 0.00001, C1 3. 107 , R1 1.
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Note that the initial condition for the capacitor C1 is given by the node voltage difference
V$1[0] V$2[0] and is included in the set of time-domain equations. Applying NDAESolve yields
the following DC operating-point values for the node voltages and the initial inductor current I$L1:
In[27]:= NDAESolve[oscillatorMNA1, {t, 0.}]
Out[27]= V$1 0., V$2 0.002, I$L1 0.002
For the DC analysis the inductor was replaced by a short circuit, therefore V$1 is zero. I$L1 has
the same numerical value as V$2 because the resistor has the unit value . Now, we compute the
transient response:
In[28]:= transient1 = NDAESolve[oscillatorMNA1, {t, 0., 10^−4}]
Out[28]=
V$1 InterpolatingFunction0, 0.0001, ,
V$2 InterpolatingFunction0, 0.0001, ,
I$L1 InterpolatingFunction0, 0.0001, Finally, we plot the transient waveform of the output current Iout (given by the inductor current I$L1)
in the simulated time interval.
In[29]:= TransientPlot[transient1, {I$L1[t]}, {t, 0., 10^−4},
PlotRange −> All]
0.0015
0.001
0.0005
0.00002
0.00004
0.00006
0.00008
t
0.0001
I$L1[t]
-0.0005
-0.001
-0.0015
-0.002
Out[29]= Graphics Initial Conditions for all Dynamic Elements
Alternatively, the function CircuitEquations can be instructed to use initial conditions for all dynamic elements by applying the option setting InitialConditions −> All. Initial conditions which
are not explicitly specified in the netlist are then assumed to be zero. For our example, we again set
up a system of modified nodal equations from the circuit configuration at the time t .
2.7 Time-Domain Analysis of Nonlinear Dynamic Circuits
123
In[30]:= oscillatorMNA2 = CircuitEquations[oscillator,
AnalysisMode −> Transient, ElementValues −> Symbolic,
InitialConditions −> All];
DisplayForm[oscillatorMNA2]
Out[31]//DisplayForm=
I$L1t C1 V$1 t V$2 t 0,
V$2t
C1 V$1 t V$2 t 0,
R1
V$1t L1 I$L1 t 0, I$L10 0,
V$10 V$20 0.002, V$1t, V$2t, I$L1t,
DesignPoint L1 0.00001, C1 3. 107 , R1 1.
Note that this time the time-domain equations contain not only the initial condition for the capacitor C1 (given by the node voltage difference V$1[0] V$2[0]) but also an initial condition for the
inductor current I$L1[0] . Now, we compute the DC operating point and the transient response by
applying NDAESolve .
In[32]:= NDAESolve[oscillatorMNA2, {t, 0.}]
Out[32]= V$1 0.002, V$2 0., I$L1 0.
In[33]:= transient2 = NDAESolve[oscillatorMNA2, {t, 0., 10^−4}]
Out[33]=
V$1 InterpolatingFunction0, 0.0001, ,
V$2 InterpolatingFunction0, 0.0001, ,
I$L1 InterpolatingFunction0, 0.0001, Finally, we plot the output current.
In[34]:= TransientPlot[transient2, {I$L1[t]}, {t, 0., 10^−4},
PlotRange −> All]
0.0002
0.0001
0.00002
0.00004
0.00006
0.00008
t
0.0001
I$L1[t]
-0.0001
-0.0002
-0.0003
Out[34]= Graphics Please note the different transient waveforms dependent on the usage of initial conditions.
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2.7.6 TransientPlot Options
Options[TransientPlot]
As was shown previously, the function TransientPlot provides a convenient way for displaying
results of NDAESolve or NDSolve computations. The appearance of TransientPlot graphics can be
changed with the following set of options:
In[35]:= Options[TransientPlot]
Out[35]=
1
AspectRatio , Axes True,
GoldenRatio
AxesLabel None, AxesOrigin Automatic, AxesStyle Automatic,
Background GrayLevel0.92, ColorOutput Automatic,
Compiled True, DefaultColor Automatic,
Epilog , Frame False, FrameLabel None,
FrameStyle Automatic, FrameTicks Automatic, GridLines None,
ImageSize Automatic, MaxBend 10., PlotDivision 30.,
PlotLabel None, PlotPoints 30, PlotRange Automatic,
PlotRegion Automatic, PlotStyle RGBColor1., 0., 0.,
AbsoluteThickness1., AbsolutePointSize3.,
RGBColor0., 0., 1., AbsoluteThickness1.,
AbsoluteDashing10., 8., AbsolutePointSize3.,
RGBColor0., 0.8, 0., AbsoluteThickness1.,
AbsoluteDashing4., 8., AbsolutePointSize3.,
RGBColor0.7, 0., 0.9, AbsoluteThickness1.,
AbsoluteDashing10., 8., 4., 8., AbsolutePointSize3.,
Prolog , RotateLabel True,
Ticks Automatic, DefaultFont $DefaultFont,
DisplayFunction $DisplayFunction, FormatType $FormatType,
TextStyle $TextStyle, GraphicsSpacing 0.1,
Parametric False, ShowLegend True, ShowSamplePoints False,
SingleDiagram True, SweepParameters Automatic
Most options are standard options of Graphics objects. Therefore, we will restrict ourselves to the
discussion of the remaining keywords Parametric , ShowSamplePoints , and SingleDiagram .
Parametric
With Parametric −> True parametric plots of two interpolating functions can be displayed. In this
case, TransientPlot requires a list of two display variables xvar and yvar, where xvar denotes
the x-axis variable and yvar the y-axis variable. TransientPlot then shows a trace of the points
{xvar[par], yvar[par]} as the parameter par is swept from par to par .
TransientPlot[numericaldata, {xvar[par], yvar[par]},
{par, par , par }, Parametric −> True]
The axes are automatically labeled by the specified variable names. As an example we perform a
parametric plot of the transient response of the diode rectifier circuit introduced in Section 2.7.1 (see
Figure 7.1).
2.7 Time-Domain Analysis of Nonlinear Dynamic Circuits
125
In[36]:= TransientPlot[solutions, {V$1[t], V$2[t]},
{t, 10^−6, 10^−5}, Parametric −> True]
V$2[t]
0.4
0.35
0.3
0.25
-1
-0.5
0.5
1
V$1[t]
Out[36]= Graphics ShowSamplePoints
Changing the default setting of the option ShowSamplePoints from False to True allows for visualizing the sample points stored in InterpolatingFunction objects. By this, the variation of the
step size in a computed interval can be observed. The simulation data is displayed via Mathematica’s
ListPlot .
In[37]:= TransientPlot[solutions, {V$1[t], V$2[t]},
{t, 0., 2.*10^−5}, ShowSamplePoints −> True]
1
0.5
V$1[t]
-6
5. 10
0.00001
0.000015
t
0.00002
V$2[t]
-0.5
-1
Out[37]= Graphics SingleDiagram
With the option SingleDiagram you can choose whether to combine the traces of several interpolating functions in a single diagram or display them in a separate plot each. The default SingleDiagram −> True causes all traces to be combined. To display an array of plots use
SingleDiagram −> False. Note that it is possible to generate not only plots of circuit variables, but
also plots of expressions involving these variables. For example, the following graphics array contains
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a plot of the voltage across the capacitor Cload of the differential amplifier circuit from Section 2.7.2
(see Figure 7.2). The capacitor voltage is given by the difference of the node voltages V$5 and V$4.
In[38]:= TransientPlot[transient, {V$4[t], V$5[t] − V$4[t]},
{t, 0., 10^−5}, SingleDiagram −> False]
6
5
V$4[
4
3
-6
2. 10
-6
4. 10
-6
6. 10
-6
t
0.00001
8. 10
10
8
6
V$5[
4
2
-6
2. 10
-6
4. 10
-6
6. 10
Out[38]= GraphicsArray -6
8. 10
t
0.00001
2.8 Linear Symbolic Approximation
127
2.8 Linear Symbolic Approximation
2.8.1 Introduction to Symbolic Approximation
The Complexity Problem
If you have previous experiences with symbolic circuit analysis programs or if you have done some
experiments with Analog Insydes before reading this chapter you may have already become aware
of a major obstacle which is inherent to symbolic computations. While the carefully selected example
circuits analyzed in the preceding chapters all yield transfer functions of no more than a few lines
in length minor modifications such as adding an element or two to the circuit or choosing a slightly
more complex transistor model may already lead to expressions of incredible size. In fact, expression
complexity increases exponentially with the number of symbols in your circuit descriptions, allowing
for full symbolic analysis of very small circuits only.
6
RC
2.2k
R1
100k
C2
3
5
C1
1u
2
1
RL
4
100n
V1
VCC
Q1
R2
RE
47k
1k
47k
Vout
Figure 8.1: Common-emitter amplifier with coupling capacitors and resistive load
Let us demonstrate this effect by computing the symbolic voltage transfer function of the commonemitter amplifier displayed in Figure 8.1. This circuit is essentially the same as the common-emitter
amplifier from Section 2.3.1 (see Figure 3.1) except that coupling capacitors and a resistive load have
been added.
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Following, the netlist description of the circuit is imported via the Analog Insydes command
ReadNetlist (Section 3.10.1) from already existing simulator files. For more details refer to Section 2.9.2.
In[1]:= <<AnalogInsydes‘
In[2]:= ceamp = ReadNetlist[
"AnalogInsydes/DemoFiles/emitter.cir",
"AnalogInsydes/DemoFiles/emitter.out",
KeepPrefix −> False, Simulator −> "PSpice"]
Out[2]= Circuit From the netlist description, we set up a system of symbolic sparse tableau equations. We
use a complexity-reduced transistor model from the Analog Insydes model library by specifying the
option ModelLibrary (Section 3.14.4).
In[3]:= ceampsta = CircuitEquations[ceamp,
ElementValues −> Symbolic, Formulation −> SparseTableau,
ModelLibrary −> "BasicModels‘"]
Out[3]= DAEAC, 32 32 Then we solve the equations for the output voltage V$RL and extract the expression for V$RL from
the result.
In[4]:= ceampvout = Together[
V$RL /. First[Solve[ceampsta, V$RL]]]
Out[4]=
C1 R1 R2 s2 C2 RC RE RL C2 gm$Q1 RC RL Ro$Q1 Rpi$Q1 C2 Cbc$Q1 RC RE RL Ro$Q1 s C2 Cbx$Q1 RC RE RL Ro$Q1 s C2 Cbc$Q1 RC RE RL Rpi$Q1 s C2 Cbe$Q1 RC RE RL Rpi$Q1 s C2 Cbx$Q1 RC RE RL Rpi$Q1 s C2 Cbc$Q1 RC RL Ro$Q1 Rpi$Q1 s C2 Cbx$Q1 RC RL Ro$Q1 Rpi$Q1 s C2 Cbc$Q1 gm$Q1 RC RE RL Ro$Q1 Rpi$Q1 s C2 Cbx$Q1 gm$Q1 RC RE RL Ro$Q1 Rpi$Q1 s C2 Cbc$Q1 Cbe$Q1 RC RE RL Ro$Q1 Rpi$Q1 s2 C2 Cbe$Q1 Cbx$Q1 RC RE RL Ro$Q1 Rpi$Q1 s2 V1 R1 R2 RC R1 R2 RE R1 RC RE R2 RC RE R1 R2 Ro$Q1 R1 RE Ro$Q1 R2 RE Ro$Q1 R1 RC Rpi$Q1 R2 RC Rpi$Q1 R1 RE Rpi$Q1 R2 RE Rpi$Q1 R1 Ro$Q1 Rpi$Q1 R2 Ro$Q1 Rpi$Q1 gm$Q1 R1 RE Ro$Q1 Rpi$Q1 gm$Q1 R2 RE Ro$Q1 Rpi$Q1 C1 R1 R2 RC RE s 153 C2 Cbe$Q1 Cbx$Q1 R1 R2
RC RE Ro$Q1 Rpi$Q1 s3 C1 C2 Cbc$Q1 R1 R2 RC RL Ro$Q1 Rpi$Q1 s3 C2 Cbc$Q1 Cbe$Q1 R1 R2 RC RL Ro$Q1 Rpi$Q1 s3 C1 C2 Cbx$Q1 R1 R2 RC RL Ro$Q1 Rpi$Q1 s3 C2 Cbe$Q1 Cbx$Q1 R1 R2
RC RL Ro$Q1 Rpi$Q1 s3 C1 C2 Cbe$Q1 R1 R2 RE RL Ro$Q1 Rpi$Q1 s3 C2 Cbc$Q1 Cbe$Q1 R1 R2 RE RL Ro$Q1 Rpi$Q1 s3 C2 Cbe$Q1 Cbx$Q1 R1 R2 RE RL Ro$Q1 Rpi$Q1 s3 C2 Cbc$Q1 Cbe$Q1 R1 RC RE RL Ro$Q1 Rpi$Q1 s3 C2 Cbe$Q1 Cbx$Q1 R1 RC RE RL Ro$Q1 Rpi$Q1 s3 C2 Cbc$Q1 Cbe$Q1 R2 RC RE RL Ro$Q1 Rpi$Q1 s3 C2 Cbe$Q1 Cbx$Q1 R2 RC RE RL Ro$Q1 Rpi$Q1 s3 C1 C2 Cbc$Q1 gm$Q1 R1 R2 RC RE RL Ro$Q1 Rpi$Q1 s3 C1 C2 Cbx$Q1 gm$Q1 R1 R2 RC RE RL Ro$Q1 Rpi$Q1 s3 C1 C2 Cbc$Q1 Cbe$Q1 R1 R2 RC RE RL Ro$Q1 Rpi$Q1 s4 C1 C2 Cbe$Q1 Cbx$Q1 R1 R2 RC RE RL Ro$Q1 Rpi$Q1 s4 2.8 Linear Symbolic Approximation
129
Even for this very small circuit we obtain a transfer function which contains more than product
terms. If you are not impressed yet try calculating the symbolic transfer function of two amplifier
stages connected in series, or rather, use ComplexityEstimate (Section 3.11.1) to get an estimate of
the expression size.
Analog Insydes provides the command ComplexityEstimate for computing an estimate of the number of product terms in a transfer function computed from a symbolic matrix equation A x b
(note that the equations have to be given as a system of AC circuit equations). More precisely, the
function computes a lower bound for the number of product terms in the determinant of A. For our
common-emitter amplifier this yields the following estimate:
In[5]:= ComplexityEstimate[ceampsta]
Out[5]= 185
A primary goal of symbolic circuit analysis is to give us qualitative insight into circuit behavior.
From a symbolic formula we can read off which circuit elements have an influence on particular
circuit characteristics, such as voltage gains or poles and zeros, and we can draw conclusions on
how to modify element values in order to make a circuit meet some performance specifications. It
is quite obvious, though, that this requires the formulas to be rather small – ideally no more than
one line on the screen – because an expression as large as the one above hardly yields any insight.
Unfortunately, there is little we can do to reduce the size of symbolic analysis results in a purely
algebraic way. Algebraic simplifications such as factoring and cancelling common terms seldom have
a sufficiently large impact on expression complexity because transfer functions are usually composed
of irreducible polynomials. In addition, factored or partially factored expressions are not necessarily
easier to understand than their expanded forms. Exact symbolic analysis is therefore infeasible for
most circuits of practical size.
Approximating Symbolic Expressions
There are two ways to cope with the complexity problem. The first one is to keep circuit and device
models always as simple as possible. Carefully balancing model accuracy and simplicity holds the
key to obtaining meaningful results from symbolic circuit analysis programs, so any knowledge about
a particular analysis task should be exploited before running the analyzer. For instance, if our task
was to find a symbolic expression describing the voltage gain of the common-emitter amplifier in the
passband frequency range for a high-impedance load we should short-circuit the coupling capacitors,
neglect the load resistor, and use a transistor model which just meets our accuracy requirements.
In other words, to obtain compact results we should model and analyze the circuit exactly like we
already did in Chapter 2.3. The major disadvantage of such a-priori simplifications on circuit level
is, however, that we cannot always control the final error in our computations which is invariably
introduced by neglecting some elements.
The second approach to the generation of interpretable formulas is known as symbolic approximation
or symbolic simplification, which refers to a whole family of hybrid symbolic/numeric algorithms for
expression simplification. Symbolic approximation techniques require more numerical knowledge
about the circuit under examination than manual simplifications but yield compact expressions with
predictable error in a fully automatic way. The basic idea behind all these algorithms is to compare
the magnitudes of symbolic terms in a transfer function or a system of equations on the basis of
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numerical reference values and then to discard all those terms which have negligible influence on
the result. In most cases, this leads to a dramatic reduction of expression complexity, thus allowing
for approximate symbolic analysis of reasonably large circuits.
1
R1
R3
V0
3
2
R2
R4
Figure 8.2: Double voltage divider
To understand how symbolic approximation works let’s look at a simple example – the transfer
function of the double voltage divider shown in Figure 8.2:
In[6]:= doubleVoltageDivider =
Netlist[
{V0, {1, 0}, V0},
{R1, {1, 2}, R1},
{R2, {2, 0}, R2},
{R3, {2, 3}, R3},
{R4, {3, 0}, R4}
]
Out[6]= NetlistRaw, 5 In[7]:= dvdmna = CircuitEquations[doubleVoltageDivider];
DisplayForm[dvdmna]
Out[8]//DisplayForm=
1
1
R1
R1
1
1
1
1
R1
R1
R2
R3
1
0
R3
0
1
0
1
R3
1
1
R3
R4
0
1
V$1 0 0 V$2
0
.
0
V$3 0
I$V0 V0 0
In[9]:= tfdvd = (V$3 / V0) /. First[Solve[dvdmna, V$3]]
R2 R4
Out[9]= R1 R2 R1 R3 R2 R3 R1 R4 R2 R4
Without any additional knowledge about the circuit this transfer function cannot be simplified much
further algebraically. Now, let’s assume that in this particular circuit the values of R1 and R2 are
approximately equal, as well as those of R3 and R4, but that R1 and R2 have much smaller values
2.8 Linear Symbolic Approximation
131
than R3 and R4: R1 R2 R3 R4. Under these conditions we can discard the product term R1 R2 in
the denominator of the transfer function because it is the product of two small quantities and thus
contributes little to the total value of the expression as compared to the other terms:
In[10]:= simptfdvd = tfdvd /. R1*R2 −> 0
R2 R4
Out[10]= R1 R3 R2 R3 R1 R4 R2 R4
The resulting function can now be factored and simplifies to a more meaningful form.
In[11]:= Factor[simptfdvd]
R2 R4
Out[11]= R1 R2 R3 R4
This approximate formula holds for all sets of resistor values for which the condition R1 R2 R3 R4
is satisfied, but no longer in the general case. Symbolic approximation thus always implies a trade-off
between low expression complexity on the one hand and generality and precision on the other.
Numerical Reference Values: Design Points
Deciding on which terms to discard from a symbolic expression on the basis of vague information
such as "X is much larger than Y" may be feasible for humans but is virtually impossible to do for a
computer. This is particularly true when an expression contains a large number of symbols in nontrivial combinations. To enable a computer to reduce a symbolic expression to its dominant content
we must provide input which allows for clear yes-or-no decisions on whether a term is important
or negligible. This input can be given as a set of accompanying numerical reference values for the
symbols, known as a design point, based on which the contributions of compound symbolic terms
can be compared numerically. Design-point values need not always be the exact element values for
which a circuit works within specifications. However, the reference values should reflect the relative
magnitude relations among the symbols in a realistic way.
In the case of the double voltage divider we could express the conditions on the relative magnitudes
of the resistors by an (arbitrary) numerical assignment of design-point values such as R1 R2 and R3 R4 . We rewrite the netlist of the double voltage divider keeping both a symbolic
element value and an associated design-point value together in each netlist entry.
In[12]:= doubleVoltageDivider =
Netlist[
{V0, {1, 0}, Symbolic
{R1, {1, 2}, Symbolic
{R2, {2, 0}, Symbolic
{R3, {2, 3}, Symbolic
{R4, {3, 0}, Symbolic
]
−>
−>
−>
−>
−>
V0,
R1,
R2,
R3,
R4,
Value
Value
Value
Value
Value
−>
−>
−>
−>
−>
1.},
10.},
10.},
1000.},
1000.}
Out[12]= NetlistRaw, 5 Now, we set up the corresponding equations using the option setting ElementValues −> Symbolic.
The design-point values are then automatically stored in the DAEObject and can be extracted via
GetDesignPoint (Section 3.6.12).
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In[13]:= dvdmna = CircuitEquations[doubleVoltageDivider,
ElementValues −> Symbolic]
Out[13]= DAEAC, 4 4 In[14]:= designpoint = GetDesignPoint[dvdmna]
Out[14]=
V0 1., R1 10., R2 10., R3 1000., R4 1000.
With these reference values we can evaluate the symbolic expression numerically. Below, we apply
the function HoldForm to the Plus operation which prevents the denominator from being evaluated
to a single number.
In[15]:= tfdvd /. Plus −> HoldForm[Plus] /. designpoint
10000.
Out[15]= Plus100., 10000., 10000., 10000., 10000.
By comparing the individual numerical values we, as well as a computer, can now clearly see that
the first argument of the Plus expression, corresponding to the term R1 R2, contributes only to
the total value of the denominator. The term can thus be safely removed from the transfer function
if we allow for an error of, say, or less in the design point.
2.8.2 Solution-Based Symbolic Approximation
Groups of Symbolic Approximation Techniques
Symbolic approximation techniques can be divided into three different groups of methods which
perform Simplifications Before, During, or After the Generation of symbolic formulas (SBG, SDG,
and SAG methods). The approach discussed above is thus an SAG, or solution-based, method because we computed the complete transfer function first and then simplified it. While the principle
behind this SAG technique was demonstrated on the transfer function of a non-dynamic system it
can be applied to general transfer functions containing powers of the frequency variable s as well.
In this case, all symbolic coefficients of the different powers of s are simplified separately up to a
given maximum error.
Approximating Transfer Functions
Analog Insydes provides the function ApproximateTransferFunction (Section 3.11.2) for solutionbased symbolic approximation. The argument sequence is
ApproximateTransferFunction[tfunc, fvar, dp, maxerr]
where tfunc is a rational transfer function in the frequency variable fvar, dp is a design-point specification, and maxerr is the maximum relative error of the simplified coefficients of fvar in the design
point. The design point must be specified as a list of rules which associate numerical values with
the symbols in tfunc, and maxerr is a real number between and .
2.8 Linear Symbolic Approximation
133
Let’s apply solution-based approximation to the transfer function of the common-emitter amplifier calculated in the preceding section. First, we extract the list of design-point values from the
DAEObject using GetDesignPoint .
In[16]:= dpceamp = GetDesignPoint[ceampsta]
Out[16]=
C1 1. 107 , C2 1. 106 , R1 100000., RC 2200.,
R2 47000., Rpi$Q1 2200., Ro$Q1 36400., Cbc$Q1 4.39 1012 ,
Cbe$Q1 7.01 1011 , gm$Q1 0.0808, Cbx$Q1 0., RE 1000.,
V1 1., RL 47000., Simulator PSpice
Then we let ApproximateTransferFunction remove all numerical insignificant terms from the coefficients of the frequency variable s, thereby allowing for a maximum design-point error of in
each coefficient. Note that a coefficient error does not imply error on the magnitude of
the transfer function in the design point. Ideally, the coefficient error should be a common factor
and thus cancel from numerator and denominator, yielding zero overall error in the design point.
In practice, however, there is no direct correlation between the overall error and the maximum
coefficient error. The former is usually lower than the latter but may, in some cases, even be larger.
In[17]:= voutsimp = ApproximateTransferFunction[ceampvout,
s, dpceamp, 0.1] // Together
Out[17]=
C1 C2 gm$Q1 R1 R2 RC RL Rpi$Q1 s2 V1 C1 C2 Cbc$Q1 gm$Q1 R1 R2 RC RE RL Rpi$Q1 s3 V1 C1 C2 Cbc$Q1 Cbe$Q1 R1 R2 RC RE RL Rpi$Q1 s4 V1 R1 R2 gm$Q1 R1 RE Rpi$Q1 gm$Q1 R2 RE Rpi$Q1 C2 R1 R2 RL s C1 gm$Q1 R1 R2 RE Rpi$Q1 s C2 gm$Q1 R1 RE RL Rpi$Q1 s C2 gm$Q1 R2 RE RL Rpi$Q1 s C1
C2 gm$Q1 R1 R2 RE RL Rpi$Q1 s2 C1 C2 Cbe$Q1 R1 R2 RE RL Rpi$Q1 s3 C1 C2 Cbc$Q1 gm$Q1 R1 R2 RC RE RL Rpi$Q1 s3 C1 C2 Cbc$Q1 Cbe$Q1 R1 R2 RC RE RL Rpi$Q1 s4 Here, the approximation process reduces the original transfer function to only significant terms.
We can check the quality of the approximation by evaluating it with the design-point values and
comparing it graphically with the original function.
In[18]:= voutexactn[s_] = Together[ceampvout /. dpceamp]
Out[18]=
470. s2 6.68943 108 3.00693 s 2.54816 109 s2 1.15576 1015 5.9939 1013 s 1.52545 1011 s2 1543.08 s3 1.19764 106 s4 In[19]:= voutsimpn[s_] = voutsimp /. dpceamp
Out[19]=
8.63878 106 s2 0.0379242 s3 3.29021 1011 s4 3.08307 1010 1.53259 109 s 3.92672 106 s2 0.041331 s3 3.29021 1011 s4 For the following BodePlot we choose a linear scaling of the magnitude values to get a better
impression of the true magnitude error. It turns out that the simplified expression (dashed line)
approximates the original transfer function (solid line) quite well, although it comprises only about
one tenth of the terms of the latter.
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In[20]:= BodePlot[{voutexactn[2. Pi I f], voutsimpn[2. Pi I f]},
{f, 1., 1.*10^9}, MagnitudeDisplay −> Linear]
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Out[20]= Graphics 2.8.3 Equation-Based Symbolic Approximation
Coping with the Limit of Solution-Based Approximation Techniques
Solution-based expression approximation techniques have one obvious drawback: they can only be
applied to symbolic transfer functions after these have been computed from systems of circuit equations. This precludes the use of SAG techniques in those cases where calculating an unsimplified
symbolic transfer function is technically impossible. In practice, the limit is reached very quickly,
because even for small analog building blocks with ten or less active devices the term count of the
transfer functions can be expected to grow into the millions, if not into billions.
To cope with these cases we must resort to simplification-before-generation techniques which simplify the problem beforehand, i.e. even before a transfer function is computed symbolically. Such
simplifications can be done manually on circuit level, for instance by replacing negligible components
with zero-admittance or zero-impedance elements. However, as controlling the error introduced by
manual simplifications is usually difficult, Analog Insydes provides an automatic and more reliable
SBG algorithm which operates on systems of symbolic circuit equations. Hence, it belongs to the
class of equation-based approximation techniques.
Given a system of symbolic linear circuit equations in matrix form A x b and a design point,
the equation-based approximation algorithm implemented in Analog Insydes finds and removes all
symbolic matrix entries whose numerical contribution to the design-point value of the transfer function is negligible, thereby keeping track of the accumulated approximation error. As the number of
removed matrix entries is usually quite large the solution of the resulting simplified matrix equation
will be much less complex than the solution of the unsimplified system. Therefore, we can calculate
simplified symbolic transfer functions without ever having to compute the full expressions at all.
2.8 Linear Symbolic Approximation
135
This will be illustrated by an example right after the following explanations on how to specify design
points for the matrix approximation function.
Design Points for Matrix Approximation
Since the matrix approximation algorithm can work with multiple design points with individual
error bounds, the design points must be specified in a slightly different form as compared to those
for solution-based approximation:
designpoint = {
{symbol −> value, , MaxError −> maxerr1 },
{symbol −> value, , MaxError −> maxerr2 },
}
Note that for symbols which are not specified here the corresponding design-point value is automatically extracted from the design point stored in the DAEObject. Each set of design-point values
must be accompanied by a rule with the keyword MaxError on its left-hand side. This rule specifies
the maximum relative error by which the magnitude of the solution of the approximated system of
equations may deviate from the value of the magnitude of the transfer function in the corresponding
design point. The error bound maxerr must be a real number greater than zero. Note that the error
definition used here is different from the one used with ApproximateTransferFunction in that it
pertains to the total error on the magnitude of the entire transfer function at certain frequencies and
not to the error on individual coefficients. It is thus a more reliable error measure than the coefficient
error.
Let’s define a design point for applying matrix approximation to our double-voltage-divider example
(see Figure 8.2 in Section 2.8.1). We can use the same numerical values for the resistors and the
voltage source as we did for solution-based approximation. If we allow for up to magnitude
error of the simplified result with respect to the unsimplified solution we must write the error bound
specification as MaxError −> 0.05. Note that although we have only one single design point here
we must wrap the list of rules into an additional level of list braces.
In[21]:= dpdvd = {{MaxError −> 0.05}};
Approximating Symbolic Matrix Equations
Symbolic matrix equations can be approximated by means of the command
ApproximateMatrixEquation (Section 3.11.3), which has the following command syntax:
ApproximateMatrixEquation[dae, var, dp, opts]
The first argument dae represents a DAEObject containing small-signal circuit equations in matrix
form as returned by CircuitEquations , var specifies the variable for which the equations should be
solved after approximation, and dp denotes a list of design points as discussed above. The remaining
argument opts is a sequence of options which may also be empty.
We are now ready to try out equation-based approximation on the MNA equations of the voltage
divider. For comparison, here are the original unsimplified equations again.
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In[22]:= DisplayForm[dvdmna]
Out[22]//DisplayForm=
1
1
R1
R1
1
1
1
1
R1
R1
R2
R3
1
0
R3
1
0
0
1
R3
1
1
R3
R4
0
1
0 V$1 0 V$2
0
.
0 V$3 0
I$V0 V0 0
Since we are interested in a simplified solution for the voltage at the output node 3, we specify V$3
as the variable for which the equations are to be approximated.
In[23]:= approxdvd = ApproximateMatrixEquation[dvdmna,
V$3, dpdvd];
DisplayForm[approxdvd]
Out[24]//DisplayForm=
0
0
1
1
1
R1
R1
R2
1
1
0
R3
R3
0
1
0
0
1
R4
0
1
0 V$1 0
0 V$2
.
0
V$3
0
V0
I$V0
0 The result is a matrix equation from which all numerical irrelevant symbolic terms have been removed. We can now compute a simplified transfer function directly by solving the approximated
equations.
In[25]:= Solve[approxdvd, V$3]
R2 R4 V0
Out[25]= V$3 R1 R2 R3 R4
In this case, we obtain identical results from both equation-based and solution-based approximation.
However, as opposed to the latter, equation-based approximation does not require an exact symbolic
solution to be computed first. In such a small example this advantage makes nearly no difference,
but it will turn out to be a key factor in larger applications.
Approximating the Equations of the Amplifier
As an example for a slightly larger application let’s continue with our well-known common-emitter
amplifier once more. Again, we start by setting up a design point for
ApproximateMatrixEquation , reusing the numerical values we defined for simplifying the exact
solution with ApproximateTransferFunction . For matrix approximation, however, we still need
some more numerical information. While the SAG algorithm approximates the coefficients of a transfer function independently, the SBG method uses the total value of a transfer function in a design
point as reference quantity. Therefore, we must specify one or more particular frequency points at
which ApproximateMatrixEquation should monitor the change in the value of the transfer function
caused by the removal of terms.
Here, we might be interested in computing a simplified expression describing the passband voltage
gain of the amplifier. The Bode plot above shows that the passband extends from about Hz to
several MHz, so we choose a design-point value for the operating frequency in the middle of the
passband, e.g. at f kHz. The corresponding value for the Laplace frequency s is then given by
2.8 Linear Symbolic Approximation
137
s Π I f . Finally, we limit the approximation error in this frequency point to of the original
magnitude value.
In[26]:= dpceamp2 = {{s −> 2. Pi I 10^4, MaxError −> 0.1}};
In the next step, we approximate the sparse tableau equations of the amplifier for the given designpoint data with respect to the output voltage V$RL.
In[27]:= approxceamp = ApproximateMatrixEquation[ceampsta,
V$RL, dpceamp2]
Out[27]= DAEAC, 32 32 Then we solve the approximated system for V$RL to obtain a simplified voltage transfer function.
In[28]:= voutsimp2 = (V$RL / V1) /. First[Solve[approxceamp, V$RL]]
RC
Out[28]= RE
By equation-based approximation we have reduced the symbolic transfer function to the simple
quotient of two resistor values. In comparison to the original transfer function, which is composed
of more than terms, this may seem to be a rather surprising result. From a technical point of
view, however, the extreme reduction of complexity is not surprising at all because the amplifier
was designed to have a voltage gain of approximately −RC/RE. Therefore, our approximate symbolic
analysis just extracted knowledge which was put into this circuit structure at the time of its design.
Again, we evaluate the simplified expression with the design-point values and compare the result
with the original transfer function.
In[29]:= voutsimp2n[s_] = voutsimp2 /. dpceamp
Out[29]= 2.2
In[30]:= BodePlot[{voutexactn[2. Pi I f], voutsimp2n[2. Pi I f]},
{f, 1., 1.*10^9}, MagnitudeDisplay −> Linear,
PlotRange −> {{0, 2.5}, Automatic}]
1.0E0
2.5
1.0E2
1.0E4
1.0E6
1.0E8
1.0E2
1.0E4
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2
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Out[30]= Graphics It becomes immediately apparent from the plots that the result obtained by means of SBG does not
constitute a global description of the frequency response of the amplifier. As opposed to the SAG
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solution, the SBG solution is only valid in the passband region and does not reflect the behavior the
amplifier exhibits for very low or very high frequencies. But remember that such a local approximation was precisely what we asked for because we specified only one reference point on the frequency
axis at kHz.
Equation-based approximation thus allows us to compute reduced-order circuit models for limited
frequency ranges. Of course, we can also define several frequency points at which
ApproximateMatrixEquation monitors the approximation error in order to extend range of validity
of a simplified expression. However, best results in terms of expression complexity are usually
obtained through local approximations.
2.8.4 Combining SBG and SAG
While equation-based approximation can reduce the complexity of a symbolic expression greatly the
result may still contain some insignificant terms. So it is generally a good idea to apply solutionbased simplification to such a transfer function as well in order to remove the remaining irrelevant
contributions. The following example demonstrates the combination of SBG and SAG.
First, we use matrix approximation to compute a reduced-order transfer function of the amplifier
from Section 2.8.1 (see Figure 8.1) which is valid for the passband and also for low frequencies.
Therefore, we add a second reference point at Hz to the list of design points. We allow for an error
of up to in that point because we do not need high precision there.
In[31]:= dpceamp3 = {{s −> 2. Pi I 10^4, MaxError −> 0.1},
{s −> 2. Pi I 1, MaxError −> 0.2}};
In the subsequent call to ApproximateMatrixEquation we specify the option
CompressEquations −> True. The effect of this option is that all rows and columns of the matrix
equation which are not needed to compute the variable of interest will be deleted after approximation. Unlike approximation this is a mathematically exact operation, so no further information will
be lost. The benefit gained from compressing a matrix equation is a speed-up in solving the system.
In[32]:= approxceamp3 = ApproximateMatrixEquation[ceampsta,
V$RL, dpceamp3, CompressEquations −> True]
Out[32]= DAEAC, 25 25 With the CompressEquations option the approximated equations are reduced from a to a
system without changing the symbolic solution for V$RL. The compression factor usually
increases with larger matrix sizes and error bounds. Solving the approximated system yields the
following expression.
2.8 Linear Symbolic Approximation
139
In[33]:= voutsimp3 = Together[
V$RL /. First[Solve[approxceamp3, V$RL]]]
Out[33]=
C1 C2 R1 R2 RL
RC RE s2 RC Rpi$Q1 s2 gm$Q1 RC Ro$Q1 Rpi$Q1 s2 V1 R1 RC RE R2 RC RE R1 RE Ro$Q1 R2 RE Ro$Q1 R1 RC Rpi$Q1 R2 RC Rpi$Q1 R1 Ro$Q1 Rpi$Q1 R2 Ro$Q1 Rpi$Q1 gm$Q1 R1 RE Ro$Q1 Rpi$Q1 gm$Q1 R2 RE Ro$Q1 Rpi$Q1 C1 R1 R2 RC RE s C2 R1 RC RE RL s C2 R2 RC RE RL s C1 R1 R2 RE Ro$Q1 s Cbc$Q1 R1 R2 RE Ro$Q1 s C2 R1 RC RE Ro$Q1 s C2 R2 RC RE Ro$Q1 s C2 R1 RE RL Ro$Q1 s 22 C2 Cbc$Q1 R1 R2 RC RE Ro$Q1 s2 C1 C2 R1 R2 RE RL Ro$Q1 s2 C2 Cbc$Q1 R1 R2 RE RL Ro$Q1 s2 C2 Cbe$Q1 R1 R2 RC RE Rpi$Q1 s2 C1 C2 R1 R2 RC RL Rpi$Q1 s2 C2 Cbe$Q1 R1 R2 RC RL Rpi$Q1 s2 C2 Cbe$Q1 R1 R2 RE RL Rpi$Q1 s2 C1 C2 R1 R2 RC Ro$Q1 Rpi$Q1 s2 C2 Cbc$Q1 R1 R2 RC Ro$Q1 Rpi$Q1 s2 C2 Cbe$Q1 R1
R2 RC Ro$Q1 Rpi$Q1 s2 C1 C2 gm$Q1 R1 R2 RC RE Ro$Q1 Rpi$Q1 s2 C2 Cbc$Q1 gm$Q1 R1 R2 RC RE Ro$Q1 Rpi$Q1 s2 C1
C2 R1 R2 RL Ro$Q1 Rpi$Q1 s2 C2 Cbc$Q1 R1 R2 RL Ro$Q1 Rpi$Q1 s2 C2 Cbe$Q1 R1 R2 RL Ro$Q1 Rpi$Q1 s2 C2 Cbc$Q1 gm$Q1 R1 R2
RC RL Ro$Q1 Rpi$Q1 s2 C1 C2 gm$Q1 R1 R2 RE RL Ro$Q1 Rpi$Q1 s2 C2 Cbc$Q1 gm$Q1 R1 R2 RE RL Ro$Q1 Rpi$Q1 s2 As compared to the result of our previous equation-based simplification this transfer function is quite
lengthy again. Still, approximating the matrix has reduced the degree of the solution. Therefore,
we can apply solution-based approximation to compute a simplified transfer function which is more
compact than our SAG-only solution and valid over a larger frequency range than our first SBG
result.
In[34]:= voutSBGSAG = ApproximateTransferFunction[voutsimp3,
s, dpceamp, 0.1]
Out[34]=
C1 C2 gm$Q1 R1 R2 RC RL Ro$Q1 Rpi$Q1 s2 V1 gm$Q1 R1 RE Ro$Q1 Rpi$Q1 gm$Q1 R2 RE Ro$Q1 Rpi$Q1 C1 gm$Q1 R1 R2 RE Ro$Q1 Rpi$Q1 C2 gm$Q1 R1 RE RL Ro$Q1 Rpi$Q1 C2 gm$Q1 R2 RE RL Ro$Q1 Rpi$Q1 s C1 C2 gm$Q1 R1 R2 RE RL Ro$Q1 Rpi$Q1 s2 The common factor which appears in the coefficients of the numerator and the denominator of this
transfer function can be cancelled by algebraic simplification:
In[35]:= voutsimp3s = Simplify[voutSBGSAG]
C1 C2 R1 R2 RC RL s2 V1
Out[35]= RE R1 R2 C1 R1 R2 s 1 C2 RL s
The above result is very useful because it allows for reading off approximate symbolic expressions
for the poles. To calculate these explicitly we must solve the denominator of the transfer function
for s:
In[36]:= poles = Solve[Denominator[voutsimp3s] == 0, s]
R1 R2
1
Out[36]= s , s C1 R1 R2
C2 RL
From our SAG-only approximation (voutsimp ) alone we cannot determine the poles so easily because
the denominator is a polynomial of degree . On the other hand, the SBG-only approximation is
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too large to yield meaningful expressions for the poles. This shows that combining SBG with SAG
techniques can also be a powerful approach for computing other circuit characteristics than only
transfer functions.
In[37]:= voutsimp3n[s_] = voutsimp3s /. dpceamp
48.598 s2
Out[37]= 1 0.047 s 147000. 470. s
Finally, we plot this simplified transfer function together with the original function to visualize the
effects of our low-and-mid-frequency approximation. The Bode plot shows that the approximation
is valid from zero frequency to the upper end of the passband at about MHz. High-frequency
effects are not described because we did not specify a design point in the frequency region beyond
the passband.
In[38]:= BodePlot[{voutexactn[2. Pi I f], voutsimp3n[2. Pi I f]},
{f, 1., 1.*10^9}, MagnitudeDisplay −> Linear,
PlotRange −> {{0, 2.5}, Automatic}]
1.0E0
2.5
1.0E2
1.0E4
1.0E6
1.0E8
1.0E2
1.0E4
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1.0E6
1.0E8
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2
1.5
1
0.5
Phase (deg)
1.0E0
1.0E0
0
-50
-100
-150
-200
-250
-300
-350
1.0E0
1.0E2
1.0E4
1.0E6
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1.0E2
1.0E4
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1.0E6
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Out[38]= Graphics 2.8.5 Options for ApproximateMatrixEquation
ApproximateMatrixEquation can be called with a number of options, most of which are for very
advanced usage only and need not be changed unless there appear to be problems with the default settings. Since all options from Options[ApproximateMatrixEquation] are documented in
Section 3.11.3, we will restrict ourselves to the discussion of the option SortingMethod . Note that
the option CompressEquations has already been introduced in Section 2.8.4.
SortingMethod
ApproximateMatrixEquation removes negligible contributions from a matrix equation term by term
following a ranking scheme which causes the term with the smallest influence on the solution to be
removed first. The term ranking is obtained by sorting the list of the individual numerical influences
2.8 Linear Symbolic Approximation
141
of all terms by least influence. If only one design point is present, the sorting order is obvious, but
it is not obvious for two or more design points. For example, if removing matrix entry X causes a
magnitude error of in design point and an error of in design point while the error
values for matrix entry Y are and respectively, which entry should be removed first?
Removing X first would introduce a very small error in design point but an excessively large one
in design point . On the other hand, Y has a much larger influence in design point than X but
the total error in both design points would be smaller if Y is removed first.
SortingMethod is an option which selects the sorting strategy that is applied to the influence list
to obtain the term rankings. By default, terms are ranked by least influence on the solution in
the primary design point, that is design point . In the above example, this strategy would give X
precedence over Y. The corresponding option setting in Options[ApproximateMatrixEquation] is
SortingMethod −> PrimaryDesignPoint
The other available strategy ranks terms by least arithmetic mean influence in all design points,
which would give Y precedence over X:
SortingMethod −> LeastMeanInfluence
Usually, LeastMeanInfluence is the better choice if the calculations involve more than one design
point. To examine the effect of selecting a different term ranking method we repeat the previous
approximation run with SortingMethod −> LeastMeanInfluence .
In[39]:= approxceamp4 = ApproximateMatrixEquation[ceampsta,
V$RL, dpceamp3, CompressEquations −> True,
SortingMethod −> LeastMeanInfluence]
Out[39]= DAEAC, 19 19 Solving these approximated equations now yields a much simpler expression.
In[40]:= voutsimp4 = V$RL /. First[Solve[approxceamp4, V$RL]]
Out[40]=
C1 C2 gm$Q1 R1 R2 RC RL Rpi$Q1 s2 V1 1 C2 RL s R1 R2 gm$Q1 R1 RE Rpi$Q1 gm$Q1 R2 RE Rpi$Q1 C1 gm$Q1 R1 R2 RE Rpi$Q1 s
Here, we determined the approximation sequence by using the average of the design-point errors as
sorting criterion. In our first approximation run with two design points the term influences on the
second design point at Hz were not taken into consideration when the ranking was computed. The
error bound at design point was reached quickly because some terms were removed first which
had low influence in design point but large influence in design point .
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2.9 Frequency-Domain Analysis of Linear Circuits
In the preceding chapters, many aspects of the circuit modeling and analysis functionality provided
by Analog Insydes have been introduced in a rather loosely connected fashion. In this chapter we
will now discuss how these techniques and functions can be applied to small-signal analysis (or AC
analysis) of linear or linearized circuits in the frequency domain. Primarily, this includes classical
symbolic analysis tasks such as computing transfer functions, input and output impedances, or calculating two-port parameters. However, we will see that by making use of Mathematica’s vast symbolic
computation and expression manipulation capabilities it is easy to extend the standard techniques
for calculating nominal small-signal characteristics to the analysis of many other quantities which
are of interest to circuit designers. Such quantities are, for example, sensitivities or effects caused by
nonidealities of circuit elements due to process tolerances or temperature dependence.
2.9.1 Transistor Models
MOS Transistors
We will show how Analog Insydes can be used to solve practical AC circuit analysis problems by
demonstrating the above-mentioned analysis tasks on a number of basic transistor circuits. Although
Analog Insydes has built-in device models (see Chapter 4.3), we start with implementing a set of
suitable transistor models in order to better understand the modeling language of Analog Insydes.
Since bipolar transistor models have already been dealt with to some extent, we begin with a
discussion of MOS field-effect transistor models.
D
G
D
B
S
G
B
S
Figure 9.1: NMOS (left) and PMOS (right) transistor
Figure 9.1 shows the schematic representations of an n-channel type (NMOS) and a p-channel type
(PMOS) MOSFET on the left-hand side and right-hand side, respectively. The letters D, G, B, and
S denote the drain, gate, bulk (substrate), and source connections of the devices. For both NMOS
and PMOS transistors, the small-signal operation at DC or low frequencies is characterized by the
small-signal equivalent circuit shown in Figure 9.2. The symbol VGS represents the gate-source voltage, which is the main controlling factor for the drain current of the MOSFET, and VBS denotes the
bulk-source, or back-gate, voltage. The model parameters of the small-signal equivalent circuits are
the transconductance gm, the drain-source conductance GDS and the back-gate transconductance gmb,
which is usually, but not always, considerably smaller than gm.
2.9 Frequency-Domain Analysis of Linear Circuits
143
D
G
B
GDS
VGS
VBS
gm*VGS
gmb*VBS
S
S
Figure 9.2: Low-frequency small-signal model for MOS transistors
To describe the MOSFET AC operation at higher frequencies more accurately a number of parasitic
capacitances are added to the low-frequency model, namely the gate-source and gate-drain capacitances CGS and CGD, and the bulk-source and bulk-drain capacitances CBS and CBD. The resulting
high-frequency AC model is displayed in Figure 9.3. For simplicity, we have neglected the ohmic
resistances in the drain and source regions which are usually accounted for by two additional resistors in between the physical drain and source connections and the corresponding terminals of the
internal MOSFET device.
D
CBD
CGD
B
G
GDS
VGS
CGS
gm*VGS
S
CBS
VBS
gmb*VBS
S
Figure 9.3: High-frequency small-signal model for MOS transistors
Implementation of Small-Signal MOS Transistor Models
For the following calculations we do not need the high-frequency MOSFET equivalent circuit. Therefore, we only implement the low-frequency model at this point. The former will be dealt with later
when you have learned more about advanced techniques for associating numerical design-point
values with symbolic subcircuit parameters.
In[1]:= <<AnalogInsydes‘
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In[2]:= lfMOSmodel =
Circuit[
Model[
Name −> MOSFET,
Selector −> LowFrequency,
Scope −> Global,
Ports −> {D, G, S, B},
Parameters −> {gm, gmb, Gds},
Definition −>
Netlist[
{VCG, {G, S, D, S}, gm},
{VCB, {B, S, D, S}, gmb},
{GDS, {D, S}, Gds}
]
]
]
Out[2]= Circuit To store the model definition in the global subcircuit database we expand the Circuit object using
the function ExpandSubcircuits (Section 3.4.1). The contents of the database can be inspected with
the command GlobalSubcircuits (Section 3.3.4).
In[3]:= ExpandSubcircuits[lfMOSmodel];
GlobalSubcircuits[]
Out[4]= MOSFET, LowFrequency
2.9.2 Transfer Functions
A CMOS Differential Amplifier
The most basic application of linear symbolic circuit analysis is to compute transfer functions as
analytic expressions of the circuit parameters and the Laplace frequency s. For instance, consider the
single-ended CMOS differential amplifier stage shown in Figure 9.4 where we might be interested in
computing the AC transfer function from the input voltage at node 1 to the output voltage across
the load capacitor CL.
2.9 Frequency-Domain Analysis of Linear Circuits
145
VDD
6
M4
M3
5
4
1
M1
2
M2
3
CL
V2
V1
IBIAS
Figure 9.4: Single-ended CMOS differential stage
Following, the netlist description of the circuit is not written by hand, but imported via the Analog
Insydes command ReadNetlist (Section 3.10.1) from already existing simulator files. This function supports the standard application of symbolic analysis which is the extension of a simulation
performed with a numerical circuit simulator. By this, a time-consuming and error-prone data conversion by hand is ruled out. In our case we start from a PSpice simulation. Thus, we need to
specify the PSpice netlist and output file in the ReadNetlist call, as well as the option setting
Simulator −> "PSpice" for applying the simulator-specific interface functionality.
In[5]:= cmosdiffamp = ReadNetlist[
"AnalogInsydes/DemoFiles/CMOSdiffamp.cir",
"AnalogInsydes/DemoFiles/CMOSdiffamp.out",
KeepPrefix −> False, Simulator −> "PSpice"]
Out[5]= Circuit In[6]:= Statistics[cmosdiffamp]
Number of Capacitors
: 1
Number of CurrentSources
: 1
Number of VoltageSources
: 3
Number of models MOSFET/MODN : 2
Number of models MOSFET/MODP : 2
Total number of elements
: 9
The automatically generated netlist description of the amplifier contains generic model references of
the form
2. Tutorial
146
{refdes, {connections},
Model −> Model[devtype, parset, refdes],
Selector −> Selector[devtype, parset, refdes],
Parameters −> Parameters[devtype, parset, refdes],
}
This allows for an easy exchange of device models and parameter sets for each device or group
of devices by means of pattern matching and rewriting. The command CircuitEquations (Section 3.5.1) makes use of the above syntax when automatically selecting device models for a specific
analysis mode.
Computing the Single-Input to Single-Ended Voltage Gain
Now, we are ready to analyze the circuit. In order to compute the voltage transfer function from
node 1 to node 5 we must first set up a system of circuit equations for the small-signal configuration
of the amplifier. This means that all DC sources, namely VDD and IBIAS, are set to zero. Moreover,
all signal sources whose contribution is not of interest, i.e. V2, are set to zero as well. The resulting
equivalent circuit is shown in Figure 9.5.
VDD=0
6
M4
M3
5
4
1
2
M1
M2
3
CL
V2=0
V1=1
IBIAS=0
Figure 9.5: Differential stage, configuration for small-signal transfer function analysis
In the frequency domain, the relation between a response Ys, an excitation Xs, and the corresponding transfer function Hs of a linear system is given by the equation
2.9 Frequency-Domain Analysis of Linear Circuits
147
Ys Hs Xs
Hence, the Laplace transform of the output signal, Ys, is identical to the transfer function if we let
Xs :
Xs $
Ys Hs
Therefore, we can obtain the transfer function of the differential amplifier by applying a unit voltage
to the input and computing the output voltage V$5[s] at node 5 because V$5[s] H[s] for V1 .
In the following command line we select the low-frequency small-signal MOSFET model by application of the CircuitEquations option setting DefaultSelector −> LowFrequency (see Section 3.5.1).
Additionally, we set up symbolic circuit equations in modified nodal formulation (which is the
default for CircuitEquations ):
In[7]:= mnacmos = CircuitEquations[cmosdiffamp,
ElementValues −> Symbolic,
DefaultSelector −> LowFrequency]
Out[7]= DAEAC, 9 9 Next, we solve the MNA equations for V$5 using the function Solve (Section 3.5.4):
In[8]:= solcmos = V$5 /. First[Solve[mnacmos, V$5]]
Out[8]=
Gds$M1 Gds$M2 gm$M2 Gds$M1 Gds$M2 gm$M1 gm$M2 gm$M4
gm$M1 Gds$M1 Gds$M2 gm$M1 gm$M2 V1 Gds$M1 gm$M1 gm$M1 V1 gm$M2 V2 Gds$M1 Gds$M1 gm$M1 Gds$M1 Gds$M2 gm$M1 gm$M2 Gds$M1 Gds$M3 gm$M3
gm$M2 Gds$M1 Gds$M2 gm$M1 gm$M2 V2 Gds$M2 gm$M2 gm$M1 V1 gm$M2 V2 Gds$M2 Gds$M1 gm$M1 Gds$M1 Gds$M2 gm$M2 Gds$M1 Gds$M2 gm$M1 gm$M2 gm$M4 Gds$M1 Gds$M1 gm$M1 Gds$M1 Gds$M2 gm$M1 gm$M2 Gds$M1 Gds$M3 gm$M3
Gds$M2 Gds$M2 gm$M2 Gds$M1 Gds$M2 gm$M1 gm$M2 Gds$M2 Gds$M4 CL s
Finally, we apply a list of replacement rules to replace the values of the DC sources and V2 by zero,
set the input voltage V1 to , and rewrite the desired transfer function into canonical sum-of-products
form using the Mathematica function Together :
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In[9]:= tfcmos = Together[
solcmos /. {V1 −> 1, V2 −> 0, IBIAS −> 0, VDD −> 0}]
Out[9]=
Gds$M2 Gds$M3 gm$M1 Gds$M3 gm$M1 gm$M2 Gds$M2 gm$M1 gm$M3 gm$M1 gm$M2 gm$M3 Gds$M2 gm$M1 gm$M4 gm$M1 gm$M2 gm$M4 Gds$M1 Gds$M2 Gds$M3 Gds$M1 Gds$M2 Gds$M4 Gds$M1 Gds$M3 Gds$M4 Gds$M2 Gds$M3 Gds$M4 Gds$M2 Gds$M3 gm$M1 Gds$M3 Gds$M4 gm$M1 Gds$M1 Gds$M4 gm$M2 Gds$M3 Gds$M4 gm$M2 Gds$M1 Gds$M2 gm$M3 Gds$M1 Gds$M4 gm$M3 Gds$M2 Gds$M4 gm$M3 Gds$M2 gm$M1 gm$M3 Gds$M4 gm$M1 gm$M3 Gds$M4 gm$M2 gm$M3 Gds$M1 Gds$M2 gm$M4 Gds$M2 gm$M1 gm$M4 CL Gds$M1 Gds$M2 s CL Gds$M1 Gds$M3 s CL Gds$M2 Gds$M3 s CL Gds$M3 gm$M1 s CL Gds$M1 gm$M2 s CL Gds$M3 gm$M2 s CL Gds$M1 gm$M3 s CL Gds$M2 gm$M3 s CL gm$M1 gm$M3 s CL gm$M2 gm$M3 s
Note that alternatively one could use the functions UpdateDesignPoint (Section 3.6.14) and
ApplyDesignPoint (Section 3.6.13) to modify or insert design-point values.
Computing Differential Gains
Similarly, we can compute the differential gain of the amplifier by splitting up the input signal into
two equal parts with opposite phase, i.e. V1 and V2 .
In[10]:= diffgain = Together[
solcmos /. {V1 −> 1/2, V2 −> −1/2, IBIAS −> 0, VDD −> 0}]
Out[10]=
Gds$M2 Gds$M3 gm$M1 Gds$M1 Gds$M3 gm$M2 2 Gds$M3 gm$M1 gm$M2 Gds$M2 gm$M1 gm$M3 Gds$M1 gm$M2 gm$M3 2 gm$M1 gm$M2 gm$M3 Gds$M2 gm$M1 gm$M4 Gds$M1 gm$M2 gm$M4 2 gm$M1 gm$M2 gm$M4 2 Gds$M1 Gds$M2 Gds$M3 Gds$M1 Gds$M2 Gds$M4 Gds$M1 Gds$M3 Gds$M4 Gds$M2 Gds$M3 Gds$M4 Gds$M2 Gds$M3 gm$M1 Gds$M3 Gds$M4 gm$M1 Gds$M1 Gds$M4 gm$M2 Gds$M3 Gds$M4 gm$M2 Gds$M1 Gds$M2 gm$M3 Gds$M1 Gds$M4 gm$M3 Gds$M2 Gds$M4 gm$M3 Gds$M2 gm$M1 gm$M3 Gds$M4 gm$M1 gm$M3 Gds$M4 gm$M2 gm$M3 Gds$M1 Gds$M2 gm$M4 Gds$M2 gm$M1 gm$M4 CL Gds$M1 Gds$M2 s CL Gds$M1 Gds$M3 s CL Gds$M2 Gds$M3 s CL Gds$M3 gm$M1 s CL Gds$M1 gm$M2 s CL Gds$M3 gm$M2 s CL Gds$M1 gm$M3 s CL Gds$M2 gm$M3 s CL gm$M1 gm$M3 s CL gm$M2 gm$M3 s
From this formula, the DC gain can be obtained by setting the complex Laplace frequency s :
In[11]:= diffgainDC = diffgain /. s −> 0
Out[11]=
Gds$M2 Gds$M3 gm$M1 Gds$M1 Gds$M3 gm$M2 2 Gds$M3 gm$M1 gm$M2 Gds$M2 gm$M1 gm$M3 Gds$M1 gm$M2 gm$M3 2 gm$M1 gm$M2 gm$M3 Gds$M2 gm$M1 gm$M4 Gds$M1 gm$M2 gm$M4 2 gm$M1 gm$M2 gm$M4 2 Gds$M1 Gds$M2 Gds$M3 Gds$M1 Gds$M2 Gds$M4 Gds$M1 Gds$M3 Gds$M4 Gds$M2 Gds$M3 Gds$M4 Gds$M2 Gds$M3 gm$M1 Gds$M3 Gds$M4 gm$M1 Gds$M1 Gds$M4 gm$M2 Gds$M3 Gds$M4 gm$M2 Gds$M1 Gds$M2 gm$M3 Gds$M1 Gds$M4 gm$M3 Gds$M2 Gds$M4 gm$M3 Gds$M2 gm$M1 gm$M3 Gds$M4 gm$M1 gm$M3 Gds$M4 gm$M2 gm$M3 Gds$M1 Gds$M2 gm$M4 Gds$M2 gm$M1 gm$M4
2.9 Frequency-Domain Analysis of Linear Circuits
149
2.9.3 Device Mismatch
Applying Matching Information
Many analog circuit designs, including our CMOS differential amplifier, rely upon the theoretical
assumption that some semiconductor devices are perfectly matched. For instance, the transistors M1
and M2, as well as M3 and M4, must have equal small-signal characteristics to make the differential
stage function ideally, i.e. with zero offset voltage and infinite common-mode rejection ratio (CMRR).
For applying matching information Analog Insydes provides the function MatchSymbols (Section 3.6.15). All symbols of a matching group are replaced by a common symbol. In our example,
the matching group specification {"$M1", "$M2", "12"} causes all symbols which end with "$M1"
or "$M2" to be replaced by a symbol which ends with "12". Thus, gm$M1 and gm$M2 are replaced
by gm12, as well as Gds$M1 and Gds$M2 are replaced by Gds12. For the amplifier stage this yields
an enormous reduction of the expression size for the differential gain:
In[12]:= dgmatch = Simplify @ MatchSymbols[diffgainDC,
{{"$M1", "$M2", "12"}, {"$M3", "$M4", "34"}}]
gm12 Gds34 2 gm34
Out[12]= 2 Gds12 Gds34 Gds34 gm34
Computing Common-Mode Gains with Mismatch
In practice, however, the matching condition cannot always be fulfilled completely because process
tolerances or temperature gradients can cause slight differences in transistor parameters that were
originally meant to be identical. The result is a deviation from the nominal circuit behavior which
may even be unacceptably large. Therefore, it is necessary to take possible performance degradations
due to device mismatch into account during circuit design.
With a symbolic analyzer we can derive formulas which express circuit characteristics in terms of
nominal parameter values plus mismatch contributions. In Analog Insydes, accounting for mismatch
is simply achieved by assigning the same symbolic element value to two nominally equal components and adding a "delta term" to one of the element values. Two resistors R1 and R2 with the same
nominal value R would thus be assigned the values R and R + dR, respectively, where dR represents
the mismatch contribution.
To demonstrate this procedure let’s examine the influence of device mismatch on the common-mode
gain of the CMOS amplifier from Section 2.9.2 (see Figure 9.4). To consider the corresponding
mismatch information, we set up a list of replacement rules. Since M1 and M2 should match, both
transconductances gm$M1 and gm$M2 are assigned the value gm12. Mismatch is then accounted
by adding the delta term dgm12 to the transconductance of gm$M2. Similarly, all other transistor
parameter values are expressed in terms of a nominal value plus a mismatch term.
In[13]:= mismatchparams = {
gm$M1 −> gm12, Gds$M1 −> Gds12,
gm$M2 −> gm12 + dgm12, Gds$M2 −> Gds12 + dGds12,
gm$M3 −> gm34, Gds$M3 −> Gds34,
gm$M4 −> gm34 + dgm34, Gds$M4 −> Gds34 + dGds34};
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First, we can compute the common-mode gain taking mismatch into account by applying the same
unit signal to both input nodes simultaneously, i.e. V1 V2 and by applying the above defined
replacement rules describing the device mismatch:
In[14]:= cmgmismatch = Together[solcmos /. mismatchparams
/. {V1 −> 1, V2 −> 1, IBIAS −> 0, VDD −> 0}]
Out[14]=
dgm12 dgm34 Gds12 dgm12 Gds12 Gds34 dGds12 dgm34 gm12 dGds12 Gds34 gm12 2 dgm12 Gds12 gm34 2 dGds12 gm12 gm34 dGds12 dGds34 Gds12 dGds34 dgm12 Gds12 dGds12 dgm34 Gds12 dGds34 Gds122 dgm34 Gds122 dGds12 dGds34 Gds34 dGds34 dgm12 Gds34 2 dGds12 Gds12 Gds34 2 dGds34 Gds12 Gds34 dgm12 Gds12 Gds34 2 Gds122 Gds34 dGds12 Gds342 dgm12 Gds342 2 Gds12 Gds342 dGds12 dgm34 gm12 dGds34 Gds12 gm12 dgm34 Gds12 gm12 dGds12 Gds34 gm12 2 dGds34 Gds34 gm12 2 Gds12 Gds34 gm12 2 Gds342 gm12 dGds12 dGds34 gm34 dGds34 dgm12 gm34 2 dGds12 Gds12 gm34 2 dGds34 Gds12 gm34 2 Gds122 gm34 dGds12 Gds34 gm34 dgm12 Gds34 gm34 2 Gds12 Gds34 gm34 2 dGds12 gm12 gm34 2 dGds34 gm12 gm34 2 Gds12 gm12 gm34 2 Gds34 gm12 gm34 CL dGds12 Gds12 s CL dgm12 Gds12 s CL Gds122 s CL dGds12 Gds34 s CL dgm12 Gds34 s 2 CL Gds12 Gds34 s CL Gds12 gm12 s 2 CL Gds34 gm12 s CL dGds12 gm34 s CL dgm12 gm34 s 2 CL Gds12 gm34 s 2 CL gm12 gm34 s
Now, we compute the common-mode gain in the ideal case where all mismatch terms are zero:
In[15]:= Simplify[cmgmismatch
/. {dgm12 −> 0, dGds12 −> 0, dgm34 −> 0, dGds34 −> 0}]
Out[15]= 0
Just as expected, for perfectly matching devices the common-mode gain is zero.
In the presence of mismatch terms the common-mode gain is a rather lengthy transfer function
whose size makes it hard to read off how the tolerances of individual circuit components contribute
to the deviation from the nominal behavior. However, we can find out which mismatch terms have
dominant influence by employing symbolic approximation methods (see Chapter 2.8). Below, we
choose the design-point values for the MOSFET small-signal parameters gm and Gds taken from the
PSpice operating-point simulation. Note that the corresponding data is already included in the above
imported Circuit object cmosdiffamp .
Assuming a transistor matching precision of the delta terms are given by multiplying the nominal
values with .
In[16]:= dpcmos = {CL −> 1.*10^−13,
gm12 −> 6.*10^−5, dgm12 −> 6.*10^−7,
Gds12 −> 3.*10^−7, dGds12 −> 3.*10^−9,
gm34 −> 6.*10^−5, dgm34 −> 6.*10^−7,
Gds34 −> 8.*10^−7, dGds34 −> 8.*10^−9};
With these design-point values we approximate the expression for the common-mode gain to a
coefficient error of .
2.9 Frequency-Domain Analysis of Linear Circuits
151
In[17]:= cmgSAG = ApproximateTransferFunction[
cmgmismatch, s, dpcmos, 0.05] // Simplify
dgm12 Gds12 dGds12 gm12
Out[17]= gm12 Gds12 Gds34 CL s
The numerator of the result indicates that the nonideal behavior is largely due to mismatch between
M1 and M2. On the other hand, tolerances between M3 and M4 have little impact on common-mode
operation.
Computing CMRR
Using the same design point we can also simplify the differential DC gain a bit further, which yields
the known approximate gain formula for this amplifier topology:
In[18]:= dgSAG = ApproximateTransferFunction[
dgmatch, s, dpcmos, 0.05] // Simplify
gm12
Out[18]= Gds12 Gds34
Having derived expressions for both the differential and the common-mode gain we can now calculate the common-mode rejection ratio which is defined as the quotient of both quantities. We
let s to focus on the DC component only.
In[19]:= CMRR = (dgSAG / cmgSAG) /. s −> 0
gm122
Out[19]= dgm12 Gds12 dGds12 gm12
The conclusion that can be drawn from this result is that increasing CMRR requires making the
transconductances gm12 of M1 and M2 larger.
2.9.4 Input and Output Impedances
Circuit Configuration for Impedance Calculations
Since input and output impedances are transfer functions just like voltage gains we can compute
the former using the same circuit analysis procedures as for any other transfer function. An input
impedance is the transfer function from the current flowing into a port to the voltage across the
same port (see Figure 9.6).
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V$in
Iin=1
circuit
Figure 9.6: Unit current source excitation for input impedance calculations
Likewise, an input admittance is the transfer function from a voltage across the port to the current
into the port. The same applies to output impedances and admittances which are, in fact, just the
input impedances and admittances at a port which is designated as an output. Consequently, to
calculate an output impedance we must connect a unit current source to the port, set up a system of
circuit equations, and solve for the port voltage.
Computing the Output Impedance of the CMOS Amplifier
To determine the output impedance of the differential amplifier (see Figure 9.4 in Section 2.9.2) we
replace the load capacitor by a stimulus current source as displayed in Figure 9.7.
VDD=0
6
M4
M3
5
4
1
M1
2
M2
IZOUT=1
3
V2=0
V1=0
IBIAS=0
Figure 9.7: Differential amplifier, small-signal configuration for computing the output impedance
2.9 Frequency-Domain Analysis of Linear Circuits
153
Replacing the load capacitor is done by editing our via ReadNetlist imported netlist. Therefore,
we first remove the netlist entry for CL using the Analog Insydes command DeleteElements (Section 3.6.2) and in a second step we add an independent current source IZ using the command
AddElements (Section 3.6.1).
In[20]:= cmosdiffamp2 = DeleteElements[cmosdiffamp, CL];
cmosdiffamp2 = AddElements[cmosdiffamp2,
{IZ, {0, 5}, IZout}];
After the modification of the netlist we set up the modified nodal equations and solve for the voltage
at the output port, i.e. V$5.
In[22]:= mnaZout = CircuitEquations[cmosdiffamp2,
ElementValues −> Symbolic,
DefaultSelector −> LowFrequency]
Out[22]= DAEAC, 9 9 In[23]:= zout = Together[V$5 /. First[Solve[mnaZout, V$5]]
/. {IBIAS −> 0, VDD −> 0, V1 −> 0, V2 −> 0, IZout −> 1}]
Out[23]=
Gds$M1 Gds$M2 Gds$M1 Gds$M3 Gds$M2 Gds$M3 Gds$M3 gm$M1 Gds$M1 gm$M2 Gds$M3 gm$M2 Gds$M1 gm$M3 Gds$M2 gm$M3 gm$M1 gm$M3 gm$M2 gm$M3 Gds$M1 Gds$M2 Gds$M3 Gds$M1 Gds$M2 Gds$M4 Gds$M1 Gds$M3 Gds$M4 Gds$M2 Gds$M3 Gds$M4 Gds$M2 Gds$M3 gm$M1 Gds$M3 Gds$M4 gm$M1 Gds$M1 Gds$M4 gm$M2 Gds$M3 Gds$M4 gm$M2 Gds$M1 Gds$M2 gm$M3 Gds$M1 Gds$M4 gm$M3 Gds$M2 Gds$M4 gm$M3 Gds$M2 gm$M1 gm$M3 Gds$M4 gm$M1 gm$M3 Gds$M4 gm$M2 gm$M3 Gds$M1 Gds$M2 gm$M4 Gds$M2 gm$M1 gm$M4
For matching the MOS transistors forming the differential pair and the current-mirror load we again
apply the command MatchSymbols :
In[24]:= zoutmatch = Together @ MatchSymbols[zout,
{{"$M1", "$M2", "12"}, {"$M3", "$M4", "34"}}]
Gds12 2 Gds34 2 gm34
Out[24]= 2 Gds12 Gds34 Gds34 gm34
As usual, this expression contains many numerically small terms which are identified and discarded
by ApproximateTransferFunction . The output impedance is then given by the following simple
formula in terms of the drain-source conductances Gds12 and Gds34.
In[25]:= zoutSAG = ApproximateTransferFunction[
zoutmatch, s, dpcmos, 0.05] // Simplify
1
Out[25]= Gds12 Gds34
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2.9.5 Two-Port Parameters
Two-Port Representations
It is easy to extend the analysis procedures for computing transfer functions to two-port parameter
analysis because two-port or, more generally, n-port matrices can be regarded as transfer functions
in several dimensions. Indeed, every single coefficient in a two-port matrix is a transfer function
from one port current or voltage to another, so we could obtain the four coefficients of a two-port
matrix by means of four separate transfer function analyses. However, we will apply a more efficient
strategy which allows for calculating complete n-port matrices in only one step.
I1
I2
V1
V2
Figure 9.8: Two-port current and voltage reference directions
Figure 9.8 shows the voltages and currents at the terminals of a two port. The relation between the
port quantities V , V , I , and I is generally given as a linear mapping of two equations in four
variables:
p p I q q V p p I q q V Solving these equations for any combination of two-port voltages or currents yields one of six possible two-port representations known as the admittance, impedance, hybrid I and II, and cascade I
and II forms. For example, solving for I and I in terms of V and V results in the admittance, or
Y-matrix, representation
I y y V I y y V whose coefficients y y are known as the Y-parameters of the two port. Similarly, solving for
V and V in terms of I and I yields the impedance, or Z-matrix, representation:
V z z I V z z I 2.9 Frequency-Domain Analysis of Linear Circuits
155
Stimulus Sources for Two-Port Calculations
A particular two-port matrix can be computed by connecting appropriate stimulus sources to the
ports and solving for the opposite port quantities. For instance, if we want to compute a Z-matrix we
must connect current sources to the ports, set up circuit equations and solve for the corresponding
port voltages. Likewise, a Y-matrix is obtained by connecting voltage sources V1 and V2 to the ports
and determining the port currents I$V1 and I$V2 in terms of these voltages.
I$V2
I$V1
Y
V1
V2
Figure 9.9: Voltage source excitation for Y-matrix calculation
Note that a straightforward combination of the two-port circuit and the excitation sources as shown
in Figure 9.9 will not yield an entirely correct result. The resulting Y-parameters would have a wrong
sign because the positive reference directions for the branch currents of the voltage sources are not
oriented according to the reference directions for two-port terminal currents.
Remember that the positive reference direction for the current and voltage in a circuit branch is always
determined by the order of the nodes in the corresponding netlist entry (see Chapter 2.2).
The reference direction for the port currents can be aligned with the two-port terminal current
directions by reversing the order of the nodes in the netlist entries belonging to the stimulus sources.
However, the voltage reference directions, whose initial settings were correct, are changed by this
operation as well. Hence, to compensate this unwanted effect, the voltages must be given negative
values in the netlist (see Figure 9.10).
I$V2
I$V1
-V1
Y
-V2
Figure 9.10: Correct voltage source excitation for computing Y-parameters
Computing Y-Parameters
Let’s demonstrate the procedure for two-port analysis by calculating the Y-matrix of the circuit shown
in Figure 9.11.
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CM
I1
I2
V1
RO
RB
V2
gm*V1
Figure 9.11: Two-port circuit
As discussed in the preceding subsection we must connect stimulus voltage sources to the circuit as
shown in Figure 9.12 and write the corresponding netlist. If you want to calculate two-port parameters of a circuit whose netlist was read by ReadNetlist , you can use DeleteElements (Section 3.6.2)
and AddElements (Section 3.6.1) to edit the netlist.
CM
I$V11
-V1
2
I$V2
RO
RB
-V2
gm*V1
0
Figure 9.12: Correct voltage source excitation for computing Y-parameters
In[26]:= twoport =
Netlist[
{V1, {0, 1},
{V2, {0, 2},
{RB, {1, 0},
{CM, {1, 2},
{VC1, {1, 0,
{RO, {2, 0},
]
−V1},
−V2},
RB},
CM},
2, 0}, gm},
RO}
Out[26]= NetlistRaw, 6 Next, we set up a system of circuit equations and compute the port currents I$V1 and I$V2 as
functions of the stimulus voltages V1 and V2.
2.9 Frequency-Domain Analysis of Linear Circuits
157
In[27]:= twoportmna = CircuitEquations[twoport];
DisplayForm[twoportmna]
Out[28]//DisplayForm=
1
CM s
1 0 V$1 0 RB
CM s
1
0 V$2
gm CM s CM
s
0
1
RO
.
V1
I$V1
1
0
0
0
V2
I$V2
0
1
0
0
In[29]:= ypar = Solve[twoportmna, {I$V1, I$V2}]
V1 CM RB s V1 CM RB s V2
Out[29]= I$V1 ,
RB
1
I$V2 gm CM s V1 CM s V2
RO
The Y-parameters of the two-port circuit are given by the coefficients of V1 and V2 on the right-hand
sides of the solutions.
In[30]:= yrhs = {I$V1, I$V2} /. First[ypar];
MatrixForm[yrhs]
V1CM RB s V1CM RB s V2
RB
Out[31]//MatrixForm= 1
CM
s
V1
CM
s
V2
gm
RO
To set up the Y-matrix we must extract the coefficient matrix from these two linear expressions, for
example by computing the Jacobian with respect to V1 and V2.
In[32]:= JacobianMatrix[eqs_, vars_] := Outer[D, eqs, vars]
In[33]:= ymatrix = JacobianMatrix[yrhs, {V1, V2}];
MatrixForm[ymatrix]
1CM RB s
RB
Out[34]//MatrixForm= gm
CM s
CM s 1
CM
s
RO
2.9.6 Advanced Transistor Modeling
Implementation of MOS Models with Inline Design-Point Information
By using the value-field keywords Value and Symbolic we can provide design-point information
in subcircuit or model definitions. However, directly specifying numerical values in the netlist definition of a model object hardly makes sense because, in general, different model instances have
different sets of design-point values. Therefore, instead of treating these values as global quantities
we must pass individual parameter sets to every model instance.
The techniques for passing parameters to subcircuit instances have already been dealt with in Chapter 2.3, and the same approaches can be used to handle instance-specific design-point information.
Here, we just have to observe the additional requirement that every element in a model netlist now
needs two symbolic parameters: one that represents the symbolic value of the element and one that
is replaced by the associated numerical value during subcircuit expansion.
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For standard applications, you do not need to take care of these things as on the one hand Analog Insydes
provides a library with built-in device models and on the other hand there is no need to write netlists by hand
as this is automatically done by ReadNetlist. Following, we explain the internal model mechanism which
is implemented in the Analog Insydes model library.
Consider the following circuit description which contains a model definition for the high-frequency
MOSFET model introduced in Section 2.9.1 (see Figure 9.3). The symbolic element values are given
as arguments to the Symbolic options while the design-point values, denoted by the names with
a trailing $ac, are given as arguments to the Value options. Both sets of symbols must appear in
the parameter lists of the model definitions to allow for passing design-point values from a model
reference to a model instance and to ensure that symbolic parameters are correctly instantiated.
In[35]:= hfMOSmodel =
Circuit[
Model[
Name −> MOSFET,
Selector −> HighFrequency,
Scope −> Global,
Ports −> {D, G, S, B},
Parameters −> {gm, gmb, Gds, Cgs, Cgd, Cbs, Cbd,
GM$ac, GMB$ac, GDS$ac, CGS$ac, CGD$ac, CBS$ac,
CBD$ac},
Definition −> Netlist[
{CGS, {G, S}, Value −> CGS$ac, Symbolic −> Cgs},
{CGD, {G, D}, Value −> CGD$ac, Symbolic −> Cgd},
{VCG, {G, S, D, S}, Value −> GM$ac,
Symbolic −> gm},
{GDS, {D, S}, Value −> GDS$ac, Symbolic −> Gds},
{VCB, {B, S, D, S}, Value −> GMB$ac,
Symbolic −> gmb},
{CBD, {B, D}, Value −> CBD$ac, Symbolic −> Cbd},
{CBS, {B, S}, Value −> CBS$ac, Symbolic −> Cbs}
]
]
]
Out[35]= Circuit In[36]:= ExpandSubcircuits[hfMOSmodel];
GlobalSubcircuits[]
Out[37]=
MOSFET, HighFrequency, MOSFET, LowFrequency
Generating Design Points
Please consider the circuit description of the differential amplifier circuit from Section 2.9.2 (see Figure 9.4). From the netlist entries for M1 to M4 it becomes apparent how the design-point information is
specified for each transistor model instance. Values are assigned only to parameters which represent
numerical information. No assignments are made to the corresponding symbolic model parameters
to let them be instantiated automatically. Note again, that this is all taken care of automatically by
the command ReadNetlist .
2.9 Frequency-Domain Analysis of Linear Circuits
159
Let’s examine the effects of the preceding parameter definitions and assignments by recalculating the
single-input-to-single-ended voltage transfer function of the differential amplifier, this time using the
high-frequency MOS model we just defined. First, we set up symbolic circuit equations using the
option setting DefaultSelector −> HighFrequency .
In[38]:= eqscmoshf = CircuitEquations[cmosdiffamp,
ElementValues −> Symbolic,
DefaultSelector −> HighFrequency]
Out[38]= DAEAC, 9 9 As a result of default parameter instantiation (see Chapter 2.3) unique symbolic values have been
generated for all instances of the transistor model elements. In addition, the arguments of the Value
keywords have been replaced by the numerical values specified in the model references. Therefore,
we can now use GetDesignPoint to extract the design-point list from the DAEObject.
In[39]:= GetDesignPoint[eqscmoshf]
Out[39]=
V2 0.0001, V1 1., CL 1. 1013 ,
Cgs$M3 2.38 1013 , Cgd$M3 0., gm$M3 0.0000654,
Gds$M3 7.89 107 , gmb$M3 0.00002169999999999999, Cbd$M3 0.,
Cbs$M3 0., Cgs$M4 2.38 1013 , Cgd$M4 0., gm$M4 0.0000654,
Gds$M4 7.89 107 , gmb$M4 0.00002169999999999999,
Cbd$M4 0., Cbs$M4 0., Cgs$M1 2.33 1013 ,
Cgd$M1 0., gm$M1 0.00005959999999999999, Gds$M1 3.01 107 ,
gmb$M1 0.0000441, Cbd$M1 0., Cbs$M1 0., Cgs$M2 2.33 1013 ,
Cgd$M2 0., gm$M2 0.00005959999999999999, Gds$M2 3.01 107 ,
gmb$M2 0.0000441, Cbd$M2 0., Cbs$M2 0., Simulator PSpice
2.9.7 AC Analysis of the CMOS Amplifier
To conclude this chapter let’s carry out an AC analysis of the CMOS amplifier. Our task shall be
to determine a symbolic formula which approximates the frequency response of the voltage gain
to first order. We begin with importing the numerical data from the PSpice small-signal simulation
applying the Analog Insydes command ReadSimulationData (Section 3.10.3). Note that we have to
specify the simulator-specific option setting Simulator −> "PSpice".
In[40]:= data = ReadSimulationData[
"AnalogInsydes/DemoFiles/CMOSdiffamp.csd",
Simulator −> "PSpice"]
Out[40]=
V5 InterpolatingFunction1., 1. 109 , In[41]:= tfcmosPSpice := "V(5)" /. First[data]
The Bode diagram of the simulation result shows that the transfer function has a dominant pole
which is responsible for the corner in the magnitude response at MHz.
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Magnitude (dB)
In[42]:= BodePlot[tfcmosPSpice[f], {f, 1., 1.*10^9}]
1.0E0
1.0E2
1.0E4
1.0E6
1.0E8
30
20
10
0
-10
-20
-30
1.0E0
1.0E2
1.0E4
Frequency
1.0E6
1.0E8
Phase (deg)
1.0E0
0
1.0E2
1.0E4
1.0E6
1.0E8
1.0E2
1.0E4
Frequency
1.0E6
1.0E8
-20
-40
-60
-80
-100
1.0E0
Out[42]= Graphics Next, we set up a system of circuit equations in symbolic form. Remember that this requires specifying the option
ElementValues −> Symbolic in the call to CircuitEquations . Furthermore, we use a complexityreduced device model for the transistors by the option setting ModelLibrary −> "BasicModels‘"
(see Section 3.14.4) as we shall not be interested in high-frequency parasitic effects.
In[43]:= stacmoshf = CircuitEquations[cmosdiffamp,
ElementValues −> Symbolic, Formulation −> SparseTableau,
ModelLibrary −> "BasicModels‘"]
Out[43]= DAEAC, 50 50 We can now perform a numerical reference simulation within Analog Insydes to ensure that the
imported data is correct and that the employed transistor models are sufficient for computing the
small-signal voltage gain and the subsequent symbolic approximations. The corresponding Analog
Insydes command for carrying out a numerical small-signal analysis is ACAnalysis (Section 3.7.3).
In[44]:= reference = ACAnalysis[stacmoshf, {V$CL}, {f, 1., 1.*^9}]
Out[44]=
V$CL InterpolatingFunction1., 1. 109 , Within the Bode diagram we compare the results of the PSpice simulation and the Analog Insydes
simulation. One can see that the curves match perfectly in the computed frequency range.
2.9 Frequency-Domain Analysis of Linear Circuits
161
Magnitude (dB)
In[45]:= BodePlot[reference, {tfcmosPSpice[f], V$CL[f]},
{f, 1., 1.*10^9}, ShowLegend −> False]
1.0E0
1.0E2
1.0E4
1.0E6
1.0E8
30
20
10
0
-10
-20
-30
1.0E0
1.0E2
1.0E4
Frequency
1.0E6
1.0E8
Phase (deg)
1.0E0
0
1.0E2
1.0E4
1.0E6
1.0E8
1.0E2
1.0E4
Frequency
1.0E6
1.0E8
-20
-40
-60
-80
-100
1.0E0
Out[45]= Graphics Following, we can proceed with carrying out the symbolic approximation. Note that statistical information concerning DAEObjects can be displayed via the Analog Insydes command Statistics
(Section 3.6.17):
In[46]:= Statistics[stacmoshf]
Number of equations
: 50
Number of variables
: 50
Number of entries
: 2500
Number of non−zero entries : 139
Complexity estimate
: 184
You may ask yourself why we set up the sparse tableau above instead of modified nodal equations which would
be considerably smaller. The reason for this is that, in most cases, equation-based symbolic approximation
tends to produce better results when applied to tableau equations.
To capture the corner in the magnitude response we select two representative design points on the
frequency axis, one to the left of the corner at kHz and the other to the right of the corner at
MHz (see Chapter 2.8).
In[47]:= dpSBG = {{s −> 2. Pi 10^4 I, MaxError −> 0.1},
{s −> 2. Pi 10^6 I, MaxError −> 0.1}};
Then, we approximate the tableau matrix with respect to the voltage across the load capacitor CL:
In[48]:= stacmoshfSBG = ApproximateMatrixEquation[
stacmoshf, V$CL, dpSBG]
Out[48]= DAEAC, 50 50 Please note the reduced complexity of the returned DAEObject inspected via Statistics :
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In[49]:= Statistics[stacmoshfSBG]
Number of equations
: 50
Number of variables
: 50
Number of entries
: 2500
Number of non−zero entries : 87
Complexity estimate
: 48
Next, we solve for the capacitor voltage V$CL to obtain a simplified transfer function:
In[50]:= tfcmoshf = Together[V$CL
/. First[Solve[stacmoshfSBG, V$CL]]]
Out[50]=
Gds$M2 Gds$M3 gm$M1 V1 Gds$M3 gm$M1 gm$M2 V1 Gds$M2 gm$M1 gm$M3 V1 gm$M1 gm$M2 gm$M3 V1 Gds$M2 gm$M1 gm$M4 V1 gm$M1 gm$M2 gm$M4 V1 Gds$M1 Gds$M3 gm$M2 V2 Gds$M3 gm$M1 gm$M2 V2 Gds$M1 gm$M2 gm$M3 V2 gm$M1 gm$M2 gm$M3 V2 Gds$M1 gm$M2 gm$M4 V2 gm$M1 gm$M2 gm$M4 V2 Gds$M1 Gds$M2 Gds$M3 Gds$M1 Gds$M2 Gds$M4 Gds$M1 Gds$M3 Gds$M4 Gds$M2 Gds$M3 Gds$M4 Gds$M2 Gds$M3 gm$M1 Gds$M3 Gds$M4 gm$M1 Gds$M1 Gds$M4 gm$M2 Gds$M3 Gds$M4 gm$M2 Gds$M1 Gds$M2 gm$M3 Gds$M1 Gds$M4 gm$M3 Gds$M2 Gds$M4 gm$M3 Gds$M2 gm$M1 gm$M3 Gds$M4 gm$M1 gm$M3 Gds$M4 gm$M2 gm$M3 Gds$M1 Gds$M2 gm$M4 Gds$M2 gm$M1 gm$M4 Cgd$M2 Gds$M1 Gds$M2 s CL Gds$M1 Gds$M2 s Cgd$M2 Gds$M1 Gds$M3 s Cgd$M4 Gds$M1 Gds$M3 s CL Gds$M1 Gds$M3 s Cgd$M2 Gds$M2 Gds$M3 s Cgd$M4 Gds$M2 Gds$M3 s CL Gds$M2 Gds$M3 s Cgd$M2 Gds$M3 gm$M1 s Cgd$M4 Gds$M3 gm$M1 s CL Gds$M3 gm$M1 s Cgd$M2 Gds$M1 gm$M2 s CL Gds$M1 gm$M2 s Cgd$M2 Gds$M3 gm$M2 s Cgd$M4 Gds$M3 gm$M2 s CL Gds$M3 gm$M2 s Cgd$M2 Gds$M1 gm$M3 s Cgd$M4 Gds$M1 gm$M3 s CL Gds$M1 gm$M3 s Cgd$M2 Gds$M2 gm$M3 s Cgd$M4 Gds$M2 gm$M3 s CL Gds$M2 gm$M3 s Cgd$M2 gm$M1 gm$M3 s Cgd$M4 gm$M1 gm$M3 s CL gm$M1 gm$M3 s Cgd$M2 gm$M2 gm$M3 s Cgd$M4 gm$M2 gm$M3 s CL gm$M2 gm$M3 s Cgd$M4 Gds$M1 gm$M4 s Cgd$M4 Gds$M2 gm$M4 s Cgd$M4 gm$M1 gm$M4 s Cgd$M4 gm$M2 gm$M4 s
Approximating the tableau equations has reduced the transfer function, which was initially of order
four, to the first-order formula we were looking for. In a last postprocessing step, we can remove all
remaining numerically irrelevant terms by solution-based approximation.
In[51]:= dpcmos2 = GetDesignPoint[stacmoshf];
In[52]:= tfcmoshfSAG = Simplify @
ApproximateTransferFunction[tfcmoshf, s, dpcmos2, 0.1]
Out[52]=
gm$M1 gm$M2 gm$M3 gm$M4 V1 Gds$M2 gm$M1 gm$M3 gm$M4 gm$M1 gm$M2 Gds$M4 gm$M3 Cgd$M2 gm$M3 Cgd$M4 gm$M3 CL gm$M3 Cgd$M4 gm$M4 s
From the above result we can now easily compute a formula for the pole of the transfer function by
solving the denominator of the expression for s:
2.9 Frequency-Domain Analysis of Linear Circuits
163
In[53]:= Solve[Denominator[tfcmoshfSAG] == 0, s] // Simplify
Out[53]=
s Gds$M4 gm$M1 gm$M2 gm$M3 Gds$M2 gm$M1 gm$M3 gm$M4 gm$M1 gm$M2
Cgd$M2 gm$M3 CL gm$M3 Cgd$M4 gm$M3 gm$M4
Finally, we verify the result by evaluating the approximated formula with the design-point values
and comparing its frequency response to that of the original PSpice simulation. The simplified
formula turns out to be a good approximation from DC up to a frequency of about MHz.
In[54]:= tfcmoshf2 = Together[tfcmoshfSAG
/. dpcmos2 /. s −> 2 Pi I f]
4.64623 1013
Out[54]= 8.49729 1015 1.0776 1020 I f
Magnitude (dB)
In[55]:= BodePlot[reference, {tfcmosPSpice[f], tfcmoshf2},
{f, 1., 1.*10^9}, ShowLegend −> False]
1.0E0
1.0E2
1.0E4
1.0E6
1.0E8
30
20
10
0
-10
-20
-30
1.0E0
1.0E2
1.0E4
Frequency
1.0E6
1.0E8
Phase (deg)
1.0E0
0
1.0E2
1.0E4
1.0E6
1.0E8
1.0E2
1.0E4
Frequency
1.0E6
1.0E8
-20
-40
-60
-80
-100
1.0E0
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2.10 Nonlinear Symbolic Approximation
The linear simplification techniques implemented in Analog Insydes proved to be very effective
and powerful. As of version 2, simplification routines for nonlinear circuits have been added to
Analog Insydes. They are able to reduce the complexity of symbolic nonlinear differential-algebraic
equations systems (DAE) with automated error control and can be used, for example, for behavioral
model generation.
Section 2.10.1 describes the main principles of the nonlinear simplification techniques in Analog
Insydes. Section 2.10.2 shows their application on an example.
2.10.1 Introduction to Nonlinear Approximation
As opposed to the linear case, for nonlinear approximation techniques there is no distinction between
simplification-before-generation and simplification-after-generation methods. All nonlinear simplification commands expect a symbolic DAEObject as input and return an approximated symbolic
DAEObject. This simplified system can then be approximated further until the desired complexity
reduction is reached. During simplification, the error is measured by numerical simulations and the
simplification is stopped as soon as the error exceeds a user-given error bound. Different user-defined
numerical analyses can be performed for error calculation.
Before the nonlinear simplification techniques can be applied, the equation system has to be prepared
using the command NonlinearSetup (Section 3.12.1). The command syntax is as follows:
NonlinearSetup[dae, inputs, outputs, mode1 −> {settings1 }, ]
The first argument of NonlinearSetup is the DAEObject to be simplified. It has to be set up in
symbolic formulation for DC or transient analysis mode. See the command CircuitEquations (Section 3.5.1), especially the options ElementValues and AnalysisMode , for setting up an appropriate
equation system.
Then you have to specify the input and output symbols of the equation system. During simplification, the input symbols are swept in a user-given range, whereas the value of the output symbols is
used to calculate the numerical error. All parameters of the DAEObject can be used as input symbols.
To get a list of all valid input symbols, use the command GetParameters (Section 3.6.9). On the
other hand, all variables of the DAEObject can be used as output symbols. To get a list of all valid
output symbols, use the command GetVariables (Section 3.6.7). Both arguments inputs and outputs
can either be a single symbol or a list of symbols. Since several input and output symbols can be
specified, the nonlinear simplification techniques are able to approximate multi-input/multi-output
(MIMO) systems.
Finally, the analysis modes used for error calculation have to be given as rules of the form
modei −> {settingsi }, where modei is either DT (for DC-transfer analysis) or AC (for AC analysis).
The argument settingsi is a sequence of rules for specifying the sweep range (using the symbol
Range), the simulation function (using the symbol SimulationFunction ), and the error function
(using the symbol ErrorFunction ). The sweep range is a required argument and determines for
2.10 Nonlinear Symbolic Approximation
165
each input symbol (as specified by inputs) the numerical range in which to sweep the symbol during
numerical simulation. The sweep range is given by a standard Analog Insydes parameter sweep
specification as described in Section 3.7.2. Both the simulation function and error function arguments
are optional and for advanced usage only. For usual applications you do not need to specify them.
For example, the argument
DT −> {Range −> {VIN, {0., 5., 1.}}}
specifies that a DC-transfer analysis has to be performed for error calculation. For this, the input
symbol VIN has to be swept from 0. to 5. in steps of 1. and the default simulation and error
functions have to be used.
NonlinearSetup returns a new DAEObject which is prepared for nonlinear approximation. All data
given as arguments to NonlinearSetup are stored in the returned DAEObject and automatically
extracted by subsequent commands, such that there is no need to specify them again. Additionally,
NonlinearSetup performs numerical simulations and stores the result as reference values used for
error calculations. You can use the command NonlinearSettings (Section 3.12.5) to inspect which
data has been added to the DAEObject for nonlinear simplifications.
Once the DAEObject is prepared, there are two major commands for nonlinear simplifications:
CompressNonlinearEquations and CancelTerms .
CompressNonlinearEquations
CompressNonlinearEquations (Section 3.12.2) removes equations and eliminates variables from the
DAEObject which are irrelevant for solving for the output variables. The general syntax is as follows:
CompressNonlinearEquations[dae, vars]
If the given DAEObject is prepared via NonlinearSetup , the output variables specified in the call
to NonlinearSetup are added to the given list vars of variables to hold in the equation system.
Note that compressing equations is a mathematically exact operation, there is no need to perform
numerical error calculations.
Since CompressNonlinearEquations reduces the number of equations in a DAEObject, it is recommended as the first step in a nonlinear simplification procedure.
CancelTerms
CancelTerms (Section 3.12.3) simplifies a DAEObject by successive removal of terms in the equation
system. The general syntax is as follows:
CancelTerms[dae, maxerrors]
After each each term removal, CancelTerms performs a numeric simulation (as set by
NonlinearSetup ), calculates the error, and compares it to the user-given error bound maxerrors. If
the error caused by a simplification exceeds this error bound, the simplification is undone.
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CancelTerms returns the simplified DAEObject. All nonlinear options which are stored in dae are
copied to the new system. Thus, the returned system can be used as input for subsequent simplifications.
By default, CancelTerms removes terms in top-level sums of the equation system. Using the option
Level, removal of terms can be performed in deeper levels, too. The value of this option is either
a non-negative integer, a list of non-negative integers, or the symbol All. For the latter, removal of
terms is performed in all levels, starting on top level.
2.10.2 Analyzing a Square Root Function Block
In this section the application of the nonlinear simplification functions is demonstrated on an example. Figure 10.1 shows the schematic of a nonlinear square root function block consisting of four
bipolar transistors.
VLOAD
VCC
I0
Q3
Q2
Q1
5
Q4
II
Figure 10.1: Square root function block
The input is given by the current source II and the voltage source VLOAD, the output is given by the
current through the voltage source VLOAD. Using the nonlinear approximation routines we want to
generate a symbolic expression for the static input output behavior of the circuit.
Importing Numerical Data
In PSpice the numerical simulation is performed as a DC sweep on ii and vload. The corresponding
part of the .cir-file looks as follows:
.op
.dc ii 0.0 0.001 0.00001 vload 0 3.5 0.875
.probe/csdf
.options numdgt=8
2.10 Nonlinear Symbolic Approximation
167
As a first step we import the numerical simulation data calculated with PSpice into Mathematica
using ReadSimulationData (Section 3.10.3).
In[1]:= <<AnalogInsydes‘
In[2]:= data = ReadSimulationData[
"AnalogInsydes/DemoFiles/Sqrt.txt",
Simulator −> "PSpice"]
Out[2]=
V3 InterpolatingFunction0., 0.001, ,
V5 InterpolatingFunction0., 0.001, ,
V1 InterpolatingFunction0., 0.001, ,
Vout InterpolatingFunction0., 0.001, ,
V4 InterpolatingFunction0., 0.001, ,
III InterpolatingFunction0., 0.001, ,
IIB InterpolatingFunction0., 0.001, ,
IVLOAD InterpolatingFunction0., 0.001, ,
Ivdc InterpolatingFunction0., 0.001, ,
ICQ1 InterpolatingFunction0., 0.001, ,
IBQ1 InterpolatingFunction0., 0.001, ,
IEQ1 InterpolatingFunction0., 0.001, ,
ISQ1 InterpolatingFunction0., 0.001, ,
ICQ2 InterpolatingFunction0., 0.001, ,
IBQ2 InterpolatingFunction0., 0.001, ,
IEQ2 InterpolatingFunction0., 0.001, ,
ISQ2 InterpolatingFunction0., 0.001, ,
ICQ3 InterpolatingFunction0., 0.001, ,
IBQ3 InterpolatingFunction0., 0.001, ,
IEQ3 InterpolatingFunction0., 0.001, ,
ISQ3 InterpolatingFunction0., 0.001, ,
ICQ4 InterpolatingFunction0., 0.001, ,
IBQ4 InterpolatingFunction0., 0.001, ,
IEQ4 InterpolatingFunction0., 0.001, ,
ISQ4 InterpolatingFunction0., 0.001, ,
SweepParameters VLOAD 0., 1,
1, 1, V3 InterpolatingFunction0., 0.001, ,
V5 InterpolatingFunction0., 0.001, ,
V1 InterpolatingFunction0., 0.001, ,
Vout InterpolatingFunction0., 0.001, ,
V4 InterpolatingFunction0., 0.001, ,
III InterpolatingFunction0., 0.001, ,
IIB InterpolatingFunction0., 0.001, ,
IVLOAD InterpolatingFunction0., 0.001, ,
10, IBQ3 InterpolatingFunction0., 0.001, ,
IEQ3 InterpolatingFunction0., 0.001, ,
ISQ3 InterpolatingFunction0., 0.001, ,
ICQ4 InterpolatingFunction0., 0.001, ,
IBQ4 InterpolatingFunction0., 0.001, ,
IEQ4 InterpolatingFunction0., 0.001, ,
ISQ4 InterpolatingFunction0., 0.001, ,
SweepParameters VLOAD 3.5
ReadSimulationData returns the two-dimensional sweep data as a list of lists of one-dimensional
interpolating functions as described in Section 3.7.1. To transform the data into a list of twodimensional interpolating functions we can use the command ConvertDataTo2D (Section 3.13.1). In
the following step we extract the interpolating function corresponding to the output value I(VLOAD)
and assign it to the function ivloadPSpice .
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In[3]:= data2d = ConvertDataTo2D[data]
Out[3]=
V3 InterpolatingFunction0., 3.5, 0., 0.001, ,
V5 InterpolatingFunction0., 3.5, 0., 0.001, ,
V1 InterpolatingFunction0., 3.5, 0., 0.001, ,
Vout InterpolatingFunction0., 3.5, 0., 0.001, ,
V4 InterpolatingFunction0., 3.5, 0., 0.001, ,
III InterpolatingFunction0., 3.5, 0., 0.001, ,
IIB InterpolatingFunction0., 3.5, 0., 0.001, ,
IVLOAD InterpolatingFunction0., 3.5, 0., 0.001, ,
Ivdc InterpolatingFunction0., 3.5, 0., 0.001, ,
ICQ1 InterpolatingFunction0., 3.5, 0., 0.001, ,
IBQ1 InterpolatingFunction0., 3.5, 0., 0.001, ,
IEQ1 InterpolatingFunction0., 3.5, 0., 0.001, ,
ISQ1 InterpolatingFunction0., 3.5, 0., 0.001, ,
ICQ2 InterpolatingFunction0., 3.5, 0., 0.001, ,
IBQ2 InterpolatingFunction0., 3.5, 0., 0.001, ,
IEQ2 InterpolatingFunction0., 3.5, 0., 0.001, ,
ISQ2 InterpolatingFunction0., 3.5, 0., 0.001, ,
ICQ3 InterpolatingFunction0., 3.5, 0., 0.001, ,
IBQ3 InterpolatingFunction0., 3.5, 0., 0.001, ,
IEQ3 InterpolatingFunction0., 3.5, 0., 0.001, ,
ISQ3 InterpolatingFunction0., 3.5, 0., 0.001, ,
ICQ4 InterpolatingFunction0., 3.5, 0., 0.001, ,
IBQ4 InterpolatingFunction0., 3.5, 0., 0.001, ,
IEQ4 InterpolatingFunction0., 3.5, 0., 0.001, ,
ISQ4 InterpolatingFunction0., 3.5, 0., 0.001, In[4]:= ivloadPSpice[VLOAD_, II_] =
"I(VLOAD)"[VLOAD, II] /. data2d
Out[4]=
InterpolatingFunction0., 3.5, 0., 0.001, VLOAD, II
The interpolating function ivloadPSpice can now be plotted in a three-dimensional plot:
In[5]:= Plot3D[ivloadPSpice[VLOAD, II],
{VLOAD, 0., 3.5}, {II, 0., 0.001},
AxesLabel −> {"VLOAD", "II", ""},
PlotLabel −> "IVLOAD (PSpice)"]
IVLOAD (PSpice)
0.0003
0.0002
0.0001
0
0
0.001
0.0008
0.0006
II
0.0004
1
VLOAD
0.0002
2
3
Out[5]= SurfaceGraphics 0
2.10 Nonlinear Symbolic Approximation
169
Equation Setup
In the next step the equation system has to be set up. For this we import the PSpice .cir-file and
convert it into an Analog Insydes Circuit object using the command ReadNetlist (Section 3.10.1).
In[6]:= netlist = ReadNetlist[
"AnalogInsydes/DemoFiles/Sqrt.cir",
Simulator −> "PSpice"];
DisplayForm[netlist]
Out[7]//DisplayForm=
Circuit:
Netlist Raw, 8 Entries:
Q1, 3 C, 5 B, 0 E, Model ModelBJT, BJTNPN, Q1, Selecto
Q2, 1 C, 3 B, 5 E, Model ModelBJT, BJTNPN, Q2, Selecto
Q3, OUT C, 3 B, 4 E, Model ModelBJT, BJTNPN, Q3, Selec
Q4, 4 C, 4 B, 0 E, Model ModelBJT, BJTNPN, Q4, Selecto
VLOAD, 1, OUT, Type VoltageSource, Value AC 0, DC Tran
II, 5, 0, Type CurrentSource, Value AC 0, DC Transient
IB, 1, 3, Type CurrentSource, Value AC 0, DC Transient
VDC, 1, 0, Type VoltageSource, Value AC 0, DC Transien
ModelParametersName BJTNPN, Type NPN
GlobalParametersSimulator PSpice
Afterwards, we set up a symbolic DC equation system using the modified nodal formulation. Since
we are not interested in temperature effects, we instruct CircuitEquations (Section 3.5.1) to replace
TEMP, TNOM, the elementary charge $q, and Boltzmann’s constant $k by their numerical values.
In[8]:= mnaFull = CircuitEquations[netlist,
AnalysisMode −> DC, ElementValues −> Symbolic,
Value −> {"*" −> {TEMP, TNOM, $q, $k}}]
Out[8]= DAEDC, 27 27 In[9]:= GetVariables[mnaFull]
Out[9]=
V$1, V$3, V$4, V$5, V$OUT, I$BS$Q1,
I$CS$Q1, I$ES$Q1, ib$Q1, ic$Q1, I$BS$Q2, I$CS$Q2, I$ES$Q2,
ib$Q2, ic$Q2, I$BS$Q3, I$CS$Q3, I$ES$Q3, ib$Q3, ic$Q3, I$BS$Q4,
I$CS$Q4, I$ES$Q4, ib$Q4, ic$Q4, I$VLOAD, I$VDC
In[10]:= GetParameters[mnaFull]
Out[10]=
AREA$Q1, AREA$Q2, AREA$Q3, AREA$Q4, BF$Q1, BF$Q2, BF$Q3, BF$Q4,
BR$Q1, BR$Q2, BR$Q3, BR$Q4, GMIN, IB, II, IS$Q1, IS$Q2, IS$Q3,
IS$Q4, VDC, VLOAD
Numerical Reference Calculation
Using NDAESolve (Section 3.7.5) we now perform the same numerical simulation as in PSpice, transform the result again into a two-dimensional interpolating function, and plot it in a three-dimensional
plot.
In[11]:= data = NDAESolve[mnaFull,
{II, 0.0, 0.001}, {VLOAD, 0., 3.5, 0.5}];
2. Tutorial
170
In[12]:= data2d = ConvertDataTo2D[data];
In[13]:= ivload[VLOAD_, II_] = I$VLOAD[VLOAD, II] /. data2d
Out[13]=
InterpolatingFunction0., 3.5, 0, 0.001, VLOAD, II
In[14]:= Plot3D[ivload[VLOAD, II],
{VLOAD, 0., 3.5}, {II, 0., 0.001},
AxesLabel −> {"VLOAD", "II", ""},
PlotLabel −> "IVLOAD (Analog Insydes)"]
IVLOAD (Analog Insydes)
0.0003
0.0002
0.0001
0
0
0.001
0.0008
0.0006
II
0.0004
1
VLOAD
0.0002
2
3
0
Out[14]= SurfaceGraphics Model Validation
One key step is now to check if the numerical simulation in Analog Insydes is identical to the result
of the PSpice simulation, i.e. to check if we have chosen appropriate transistor models. This can for
example be done by comparing the graphical output. We choose some arbitrary value for VLOAD and
plot both output values in one graph sweeping II. The plot shows that both curves are identical
(for this value of VLOAD). Alternatively, we can calculate the maximum difference of both output
values evaluated on a uniform grid. The result shows that the deviation is of order ΜA.
2.10 Nonlinear Symbolic Approximation
171
In[15]:= TransientPlot[{ivloadPSpice[2., II], ivload[2., II]},
{II, 0., 0.001}]
0.0003
0.00025
ivloadPSp
0.0002
0.00015
ivload[2.
0.0001
0.00005
II
0.00020.00040.00060.0008 0.001
Out[15]= Graphics In[16]:= Max @ Map[Abs[Apply[ivload,#] − Apply[ivloadPSpice,#]] &,
Flatten[Table[{VLOAD, II},
{VLOAD, 0., 3.5, 0.1}, {II,0., 0.001, 0.000025}], 1]]
Out[16]= 2.62136 107
Starting Nonlinear Simplifications
Now that we have ensured to use the same equation system as PSpice does, we can proceed with the
nonlinear simplification routines. The task is to generate a simplified static equation system which
has the same input-output behavior as the original system mnaFull . To get an impression of the
complexity of the original system, we use the command Statistics (Section 3.6.17).
In[17]:= Statistics[mnaFull]
Number of equations
: 27
Number of variables
: 27
Length of equations
: {4, 4, 3, 3, 2, 3, 2, 3, 8, 8,
3, 2, 3, 9, 9, 3, 2, 3, 9, 9, 3, 2, 3, 4, 4, 3, 2}
Terms with derivatives : 0
Sums in levels
: {113, 20}
Compressing Equations
In a first step we eliminate irrelevant variables using the function CompressNonlinearEquations
(Section 3.12.2). The argument {I$VLOAD} to CompressNonlinearEquations assures that our variable of interest is not eliminated from the equation system. Eliminating variables is a mathematically
exact operation, thus no error calculation is needed. As in our example, this usually reduces the number of equations drastically (but not necessarily the number of terms) and is therefore recommended
as the first step in the simplification procedure.
2. Tutorial
172
In[18]:= step1 = CompressNonlinearEquations[mnaFull, {I$VLOAD}]
Out[18]= DAEDC, 6 6 In[19]:= Statistics[step1]
Number of equations
: 6
Number of variables
: 6
Length of equations
: {16, 24, 9, 23, 3, 2}
Terms with derivatives : 0
Sums in levels
: {77, 42}
Preparing Approximation Routines
To further reduce the complexity of the DAE system we now cancel terms in the equations. Since
this operation alters the output of the system, numerical control simulations have to be applied to
check the current error. We use NonlinearSetup (Section 3.12.1) to prepare the DAEObject for the
following simplification steps. Since we are dealing with a static equation system, we choose the
DT analysis mode for error checking, specifying sweep ranges for both input values VLOAD and II.
NonlinearSetup performs a numerical reference simulation and stores the result in the returned
DAEObject. These numerical values are used automatically to calculate the error in subsequent
simplification steps.
In[20]:= step2 = NonlinearSetup[step1, {II, VLOAD}, {I$VLOAD},
DT −> {Range −>
{{VLOAD, 0., 3.5, 0.5}, {II, 0., 0.001, 0.0002}} }]
Out[20]= DAEDC, 6 6 Cancelling Terms
Once we have set up the DAE system we call CancelTerms (Section 3.12.3) where we specify an
error bound for the output variable. CancelTerms then deletes terms in the equation system using
this error bound. Afterwards we again use Statistics to inspect the complexity reduction achieved
by CancelTerms .
In[21]:= step3 = CancelTerms[step2, {DT −> {I$VLOAD −> 25.*^−6}}]
Out[21]= DAEDC, 6 6 In[22]:= Statistics[step3]
Number of equations
: 6
Number of variables
: 6
Length of equations
: {2, 2, 2, 4, 2, 1}
Terms with derivatives : 0
Sums in levels
: {12, 4}
Note that CancelTerms reduces the number of terms drastically.
2.10 Nonlinear Symbolic Approximation
173
Compressing Equations again
Deletion of terms changes the equations in such a way that it is often possible to further solve
equations for some variables. Thus, we can again try to eliminate variables using
CompressNonlinearEquations . Here, two more equations can be removed from the system. Note
that CompressNonlinearEquations automatically retrieves the settings made by NonlinearSetup ,
so there is no need to specify the variable of interest I$VLOAD as a second argument. Afterwards we
drop those parameters from the design point which no longer appear in the equation system using
UpdateDesignPoint (Section 3.6.14).
In[23]:= step4 = CompressNonlinearEquations[step3]
Out[23]= DAEDC, 4 4 In[24]:= Statistics[step4]
Number of equations
: 4
Number of variables
: 4
Length of equations
: {2, 2, 2, 4}
Terms with derivatives : 0
Sums in levels
: {10, 4}
In[25]:= step5 = UpdateDesignPoint[step4]
Out[25]= DAEDC, 4 4 In[26]:= step5 // DisplayForm
Out[26]//DisplayForm=
II 1. AREA$Q2 E38.6635 V$3V$5 IS$Q2 0,
1. IB 1. AREA$Q1 E38.6635 V$5 IS$Q1 0,
1. AREA$Q3 E38.6635 V$3V$4 IS$Q3 1. I$VLOAD 0,
1. IB 1. AREA$Q1 E38.6635 V$5 IS$Q1 1. AREA$Q4 E38.6635 V$4 IS$Q4 1. I$VLOAD 0,
V$3, V$4, V$5, I$VLOAD, DesignPoint II 0.0001, IB 0.0001,
AREA$Q1 1., IS$Q1 1. 1016 , AREA$Q2 1., IS$Q2 1. 1016 ,
AREA$Q3 1., IS$Q3 1. 1016 , AREA$Q4 1., IS$Q4 1. 1016 Looking at the simplified DAE system one can see that the parameter VLOAD disappeared from the
equations. The three-dimensional plot of the output from the beginning shows that the output does
not depend on VLOAD, thus the simplification procedure automatically removes this parameter from
the equation system.
Validating the Result
Now we can compare the output of the simplified system step5 with the output of the original system. Since VLOAD no longer appears in the DAE system, we no longer need to perform a
multi-dimensional parametric sweep of NDAESolve . A single sweep on II suffices, resulting in a
one-dimensional interpolating function.
In[27]:= data = NDAESolve[step5, {II, 0., 0.001}];
2. Tutorial
174
In[28]:= ivloadSimp[II_] = I$VLOAD[II] /. First @ data
Out[28]=
InterpolatingFunction0, 0.001, II
In[29]:= Plot3D[ivloadSimp[II],
{VLOAD, 0., 3.5}, {II, 0., 0.001},
AxesLabel −> {"VLOAD", "II", ""},
PlotLabel −> "IVLOAD (simplified system)"]
IVLOAD (simplified system)
0.0003
0.0002
0.0001
0
0
0.001
0.0008
0.0006
II
0.0004
1
VLOAD
0.0002
2
3
0
Out[29]= SurfaceGraphics To show that the requested error bound is met, we plot the deviation of the output comparing the
original and the simplified system (note the plot range).
In[30]:= Plot3D[Abs[ivloadSimp[II]−ivload[VLOAD, II]],
{VLOAD, 0., 3.5}, {II, 0., 0.001},
PlotRange −> {0, 25.*^−6},
AxesLabel −> {"VLOAD", "II", ""},
PlotLabel −> "absolute error"]
absolute error
0.00002
0.001
0.0008
0.0006
II
0.0004
0.00001
0
0
1
VLOAD
0.0002
2
3
Out[30]= SurfaceGraphics 0
2.10 Nonlinear Symbolic Approximation
175
Further Postprocessing
In the final step we further reduce the complexity of the equation system by applying some standard
Mathematica functions to manipulate the equations symbolically. In this example it is possible to solve
the equations for the output variable I$VLOAD symbolically yielding an explicit formula describing
the input-output behavior.
In[31]:= eqs = GetEquations[step5];
In[32]:= eqs2 = Eliminate[eqs, {V$3, V$4, V$5}]
Eliminate::ifun:
Inverse functions are being used by Eliminate, so some
solutions may not be found.
Out[32]=
I$VLOAD
Log AREA$Q3 IS$Q3
IB
II
I$VLOAD
Log Log 1. Log AREA$Q1 IS$Q1
AREA$Q2 IS$Q2
AREA$Q4 IS$Q4
In[33]:= Solve[Map[Exp, eqs2], I$VLOAD]
Out[33]=
1. AREA$Q3 AREA$Q4 IB II IS$Q3 IS$Q4
I$VLOAD ,
AREA$Q1 AREA$Q2 IS$Q1 IS$Q2
1. AREA$Q3 AREA$Q4 IB II IS$Q3 IS$Q4
I$VLOAD AREA$Q1 AREA$Q2 IS$Q1 IS$Q2
From the result it can be seen that the output current I$VLOAD depends on the square root of the
input current II and that it is independent of the input voltage VLOAD.
Reference Manual
3.1
Netlist Format
. . . . . . . . . . . . . . . . . . . . . . . 178
3.2
Subcircuit and Device Model Definition . . . . . . . . . . 190
3.3
Model Library Management . . . . . . . . . . . . . . . . 211
3.4
Expanding Subcircuits and Device Models . . . . . . . . 221
3.5
Setting Up and Solving Circuit Equations . . . . . . . . . 227
3.6
Circuit and DAEObject Manipulation . . . . . . . . . . . 255
3.7
Numerical Analyses . . . . . . . . . . . . . . . . . . . . 280
3.8
Pole/Zero Analysis . . . . . . . . . . . . . . . . . . . . . 304
3.9
Graphics Functions . . . . . . . . . . . . . . . . . . . . . 330
3.10 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 365
3.11 Linear Simplification Techniques . . . . . . . . . . . . . . 385
3.12 Nonlinear Simplification Techniques . . . . . . . . . . . . 401
3.13 Miscellaneous Functions . . . . . . . . . . . . . . . . . . 418
3.14 Global Options . . . . . . . . . . . . . . . . . . . . . . . 425
3.15 The Analog Insydes Environment . . . . . . . . . . . . . 432
3. Reference Manual
178
3.1 Netlist Format
The following chapter explains the circuit-description language used by Analog Insydes. In Analog
Insydes, circuits are described in terms of Netlist and Circuit objects. Section 3.1.2 shows the
general syntax for Circuit objects, Section 3.1.1 for Netlist objects. Sections 3.1.3–3.1.8 describe the
syntax of the Netlist object entries, Sections 3.1.9–3.1.11 describe the syntax of the Circuit object
entries.
3.1.1 Netlist
Netlist[args]
defines a circuit in terms of basic components and model
references where args may be any sequence of netlist
entries
Command structure of Netlist.
A flat netlist is described in Analog Insydes by means of a Netlist object. A Netlist consists of
a sequence of netlist entries which can be either basic components or model references. Each netlist
entry is a list. The general syntax for a Netlist object is as follows:
netlistname =
Netlist[
netlist entry 1,
netlist entry 2,
]
The format of a netlist entry is described in Section 3.1.3.
Usually, a Netlist object is contained in a Circuit object together with model and parameter
definitions.
See also: Circuit (Section 3.1.2), CircuitEquations (Section 3.5.1), Sections 3.1.3–3.1.7.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
A simple voltage divider
netlist.
In[2]:= voldiv =
Netlist[
{V0, {1, 0}, V0},
{R1, {1, 2}, R1},
{R2, {2, 0}, R2}
]
Out[2]= NetlistRaw, 3 3.1 Netlist Format
179
3.1.2 Circuit
Circuit[args]
defines a circuit in terms of top-level netlists and models
where args may be any sequence of Netlist, Subcircuit ,
Model, ModelParameters , or GlobalParameters objects
Command structure of Circuit.
With Analog Insydes you can describe electrical circuits and control systems hierarchically in terms
of netlists, subcircuit or device model definitions, and model parameter sets. Circuits are defined
with the commands Circuit or Netlist. The general syntax for a Circuit object is as follows:
Circuit[
Netlist[top-level netlist with subcircuit references],
Model[subcircuit/model definition],
ModelParameters[model parameters specification],
GlobalParameters[global parameters specification],
]
Multiple Netlist sections in a Circuit object will be concatenated in order of appearance
upon subcircuit expansion.
See also:
Netlist (Section 3.1.1), ModelParameters (Section 3.1.10), GlobalParameters (Section 3.1.11), Model
(Section 3.2.1), CircuitEquations (Section 3.5.1), ReadNetlist (Section 3.10.1).
Examples
The following is an example for a complete Analog Insydes circuit description including device
model and parameter definitions.
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
3. Reference Manual
180
A complex circuit
description.
In[2]:= amplifier = Circuit[
Netlist[
{V0, {VCC, 0}, Symbolic −> {AC−>0, _ −> VCC},
Value −> {DC | Transient −> 12}},
{V1, {1, 0}, Symbolic −> V1, Value −> {AC −> 1,
DC | Transient −> SinWave[Time, 0.8, 0.01, 1000]}},
{Q1, {2 −> C, 1 −> B, 0 −> E},
Model −> Model[BJT, DEFNPN, Q1],
Selector −> Selector[BJT, DEFNPN, Q1],
Parameters −> Parameters[BJT, DEFNPN, Q1],
GM$ac −> 0.02, RPI$ac −> 5.0*^3, RO$ac −> 1.0*^4},
{RC, {VCC, 2}, Value −> 500, Symbolic −> RC}
],
Model[Name −> BJT, Selector −> ACsimple,
Scope −> Global, Ports −> {C, B, E},
Parameters −> {gm, Rpi, Ro, GM$ac, RPI$ac, RO$ac},
Definition −> Netlist[
{RPI, {B, E}, Value −> RPI$ac, Symbolic −> Rpi},
{VC, {B, E, C, E}, Value −> GM$ac, Symbolic −> gm},
{RO, {C, E}, Value −> RO$ac, Symbolic −> Ro}
]
],
ModelParameters[Name −> DEFNPN, Type −> "NPN",
IS −> 1.0*^−16, BF −> 100, BR −> 1, VAF −> 150],
GlobalParameters[TEMP −> 300.15, GMIN −> 1.0*^−12]
]
Out[2]= Circuit 3.1.3 Netlist Entries
Netlist entries must be written in one of the following ways:
{refdes, {nodes}, value}
simple netlist entry format
{refdes, {nodes}, Value−>value, opts}
extended value field format for netlist entries
{refdes, {node −>port , † † † }, Model−>name, Selector−>sel, par −>val , † † † }
netlist entry format for model references
Netlist entry formats.
The reference designator (refdes) is a unique name (a symbol or a string) by which a circuit component
can be identified, such as R1 and CL for a resistor R and a load capacitor CL . For linear netlist
elements, the first one, two, or three letters of the reference designator (the type tag) determine the
type of the element (see Chapter 4.2 for a list of available linear element types). In the following,
the simple and extended netlist entry format is described. Section 3.1.8 describes the netlist format
for model references in more detail.
3.1 Netlist Format
181
Reference designators must be symbols or strings that can be converted to symbols.
Reference designators are case sensitive. Therefore, R1 and r1 denote different elements.
All reference designators are converted internally to strings. Therefore, R1 and "R1" refer to
the same element.
The second argument of a netlist entry (nodes) specifies the nodes to which the element is connected.
The number of nodes that have to be specified depends on the element type. You may use nonnegative integers, symbols, and strings as node identifiers. The number 0, or equivalently the string
"0", designates the ground node.
Numeric node identifiers are not required to be consecutive.
Node identifiers are case sensitive. Therefore, N1 and n1 denote different nodes.
All node identifiers are converted internally to strings. Therefore, N1 and "N1" refer to the
same node.
The value of a circuit element may be a symbol, a number, or any Mathematica expression involving
both numeric and symbolic quantities.
Whenever the extended value field format is used for a netlist entry, the value of the
corresponding element must be specified with the keyword Value; this keyword may not be
omitted.
See also: Section 3.1.4.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
This defines an RLC
lowpass filter circuit.
In[2]:= rlcf =
{V1,
{R1,
{L1,
{C1,
]
Netlist[
{1, 0}, V},
{1, 2}, R},
{2, 3}, L},
{3, 0}, C}
Out[2]= NetlistRaw, 4 3. Reference Manual
182
This netlist is semantically
identical to the previous
one.
In[3]:= rlcf2 = Netlist[
{"V1", {1, 0}, Value
{"R1", {1, 2}, Value
{"L1", {2, 3}, Value
{"C1", {3, 0}, Value
]
−>
−>
−>
−>
V},
R},
L},
C}
Out[3]= NetlistRaw, 4 Set up a system of circuit
equations.
In[4]:= CircuitEquations[rlcf] // DisplayForm
Out[4]//DisplayForm=
1
1
0
R
R
1
1
1
1
R
R L s
L s
1
1
0
L s
L s
C s
0
0
1
1
V$1 0
0
V$2
0
.
0
V$3
0
I$V1 V
0
The example below shows a valid mixture of data types and formats for reference designators, node
identifiers, and value specifications.
This defines a voltage
divider circuit.
In[5]:= divider = Netlist[
{V1, {"in", 0}, Value −> 5},
{"R1", {in, out}, R},
{R2, {"out", 0}, 2*R}
]
Out[5]= NetlistRaw, 3 3.1.4 Netlist Entry Options
The extended value field format allows you to specify additional information related to a circuit
element expressed in terms of rules with the following keywords:
option name
default value
InitialCondition
Automatic
the initial condition for a dynamic element
(L or C)
Pattern
Automatic
the MNA pattern for a two-terminal
impedance or admittance element (R, G, C,
L, Z, Y)
Symbolic
(no default)
the symbolic value of a circuit element
Type
Automatic
explicit specification of the element type
Netlist entry options.
A netlist entry in extended value field format looks like
3.1 Netlist Format
183
{refdes, {nodes}, Value −> value,
Symbolic −> symbolicvalue,
Pattern −> pattern,
Type −> type,
InitialCondition −> initialcondition}
where parameters in slanted typewriter font are optional.
InitialCondition
With InitialCondition −> ic, you can specify initial currents and voltages for inductors and capacitors, respectively. If no initial condition is given explicitly, it will be assumed to be zero (AC
analysis) or will be calculated automatically from operating-point data (transient analysis).
Pattern
The option Pattern allows you to select the matrix fill-in patterns for two-terminal admittances and
impedances (immittances) to be used in modified nodal analysis (MNA). By default, all impedances Z
are converted to equivalent admittances Y Z to keep the dimensions of the circuit equations
small. The following values are allowed:
Automatic
convert impedances to admittances for MNA
Admittance
use admittance pattern for immittances
Impedance
use impedance pattern for immittances
Values for the Pattern option.
An explicit Pattern specification overrides the setting of the option ConvertImmittances
locally.
Symbolic
With Symbolic −> symbol, you can specify an alternative symbolic element value in addition to a
numerical value given by Value −> value. This feature is used to associate design-point data with
symbolic element values. Note that symbol is not strictly required to be a symbol; you may also
specify any Mathematica expression. However, design-point management will work as intended only
if the value of the Symbolic option is a symbol.
3. Reference Manual
184
You may not use the Symbolic option without the Value option. If the extended value field
format is used, the Value option must always be present (except in model references).
Type
The Type option allows you to override the automatic element type detection mechanism. With the
default setting Type −> Automatic , the type of an element is determined from the leading characters
of its reference designator. With Type −> typename, the element is assumed to be of type typename regardless of what the reference designator implies. The argument typename must be a full type name.
To get a list of all available type names, inspect the contents of the global variable ElementTypes .
See also: ElementTypes (Section 3.1.5), Section 3.1.3, Section 3.5.1.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
The following netlist illustrates the use of the netlist entry options described above. The option
Pattern −> Impedance , prevents the resistance R1 from being converted to an equivalent admittance 1/R1 when setting up modified nodal equations. The Type option in the value field of the
component Load tells Analog Insydes to treat the element as a resistor although the type tag is L
(inductor). Finally, the InitialCondition option in the netlist entry for C1 sets an initial capacitor
voltage of V.
This illustrates the use of
the netlist entry options.
In[2]:= filter = Netlist[
{V1, {1, 0}, Value −> 5, Symbolic −> V1},
{R1, {1, 2}, Value −> 1000., Symbolic −> R1,
Pattern −> Impedance},
{Load, {2, 3}, Value −> 10., Symbolic −> RL,
Type −> Resistor},
{C1, {3, 0}, Value −> 1.*^−6, Symbolic −> C,
InitialCondition −> 2.5}
]
Out[2]= NetlistRaw, 4 Set up a system of
symbolic circuit equations.
In[3]:= CircuitEquations[filter, ElementValues −> Symbolic,
InitialConditions −> Automatic] // DisplayForm
Out[3]//DisplayForm=
0
0
0
1
1
0
RL
RL
1
1
0 RL
RL
C s
1
0
0
1
0
1
1 1 0
V$1 0 1 0
V$2 2.5
C
.
V$3
0 0 V1
I$V1
0 0 0
I$R1
0 R1 3.1 Netlist Format
185
3.1.5 ElementTypes
ElementTypes
contains the names of all primitive circuit element types
Global variable ElementTypes.
The global variable ElementTypes contains a list of the internal names of all primitive circuit element
types supported by Analog Insydes. Any name from this list can be used as value of the option
Type in the value fields of netlist entries.
See also: Section 3.1.4, Chapter 4.2.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Get the list of element
type names.
In[2]:= ElementTypes
Out[2]=
Resistor, Conductance, Admittance, Impedance, Capacitor,
Inductor, CurrentSource, VoltageSource, CCCSource, CCVSource,
VCCSource, VCVSource, Nullator, Norator, Fixator, OpAmp, OTAmp,
ABModel, OpenCircuit, ShortCircuit, SignalSource, Amplifier,
Integrator, Differentiator, TransmissionLine, TransferFunction
3.1.6 Value Specifications for Independent Sources
Independent sources may require multiple value specifications for different analysis modes. For
example, a supply voltage source may have a value of V for DC and transient analysis and an
AC value of V (e.g. for PSRR calculation). Multiple source values for both numerical and symbolic
values can be specified in a netlist entry using a special form of the extended value field syntax (see
Section 3.1.3).
Value | Symbolic−>{modespec−>value, † † † }
mode-specific source value specification
AC
mode key for AC analysis
DC
mode key for DC analysis
Transient
mode key for transient analysis
Mode specific source value specifications.
A mode specification (modespec) is a Mathematica pattern which is compared against the value of
the option AnalysisMode given to CircuitEquations (Section 3.5.1) during equation setup. Mode
specifications are evaluated from left to right. The first matching pattern yields the element value
that will be used.
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If no value is specified for a particular analysis mode, the value is assumed to be zero.
Multiple element values may be given in both Value and Symbolic specifications.
The value specification in the following netlist entry will yield the source value for the option
settings AnalysisMode −> DC and AnalysisMode −> Transient . For AC analysis the value will be
used.
{VSUP, {VCC, 0}, Value −> {DC | Transient −> 12, AC −> 1}}
The value specification in the following netlist entry sets both the numeric and symbolic AC values
of V1 to zero whereas the values Vin and 12 are used for all other analysis modes (DC and Transient).
{V1, {1, 0}, Symbolic −> {AC −> 0, _ −> Vin},
Value −> {AC −> 0, _ −> 12}}
Note that the above approach allows you to keep symbols that represent pure DC values out of AC
equations (see example in Section 3.1.2):
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define filter circuit.
In[2]:= filter = Netlist[
{V0, {1, 0}, Value −> {DC|Transient −> 5, AC −> 0},
Symbolic −> {AC −> 0, _ −> V0}},
{V1, {1, 2}, Value −> {AC −> 1, DC −> 0,
Transient −> SinWave[Time, 0.8, 0.01, 1000]},
Symbolic −> V1},
{R1, {2, 3}, Value −> 1000., Symbolic −> R1},
{C1, {3, 0}, Value −> 1.*^−6, Symbolic −> C}
]
Out[2]= NetlistRaw, 4 Set up AC equation
system. Note that
symbol V0 does not appear
in the AC equations.
In[3]:= CircuitEquations[
filter, AnalysisMode −> AC,
ElementValues −> Symbolic] // DisplayForm
Out[3]//DisplayForm=
0
0
0
1
1
0 R1
R1
1
1
0 R1
R1
C s
1
0
0
0
1 1
1 1 0 V$1 0 1 0 V$2
0
.
V$3
0 0 0
I$V0
0 0 V1
I$V1
0 0 3.1.7 Time and Frequency
The keywords Time and Frequency can be used in the value fields of netlist entries to refer to time
and the complex Laplace frequency, respectively. For example, the netlist entry
3.1 Netlist Format
187
{L1, {2, 3}, L[Time]}
defines a time-variant inductance, and
{VC2, {1, 2, 3, 4}, Frequency*C}
defines a transcapacitance, i.e. a voltage-controlled current source with a transconductance that
depends linearly on frequency.
Upon setting up circuit equations, the keywords Time and Frequency are replaced by the symbols
that designate time and frequency as set in Options[CircuitEquations] (see also Section 3.5.1).
The following analysis mode specific settings are performed:
For DC analysis, Time is replaced by the value of InitialTime and Frequency is set to zero.
For AC analysis, Time is replaced by the value of InitialTime .
For transient analysis, Frequency is set to zero.
Mode-specific replacement of Time and Frequency.
3.1.8 Model References
Subcircuits and device models are referenced via special netlist entries of the following form. A
model reference is recognized by the presence of the keyword Model in the value field.
{refdes, {node −>port , † † † }, Model−>name, Selector−>sel, par −>val , † † † }
netlist entry format for model references
Referencing models.
The following netlist entry references a transistor model NPN/DC and specifies numerical values for
the transistor parameters IS, BF, and BR:
{Q1, {1 −> C, 2 −> B, 3 −> E}, Model −> NPN, Selector −> DC,
IS −> 1.*^−16, BF −> 100, BR −> 1}
To reference a parameter set, use the value-field option Parameters −> name, where name denotes
the name of the parameter set defined with ModelParameters (see Section 3.1.10).
The following netlist entry includes a reference to the model parameter set DEFNPN.
{Q1, {1 −> C, 2 −> B, 3 −> E}, Model −> NPN, Selector −> DC,
Parameters −> DEFNPN}
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A model reference may contain multiple parameter set references.
References to parameter sets are replaced by the corresponding model parameters in the
given order.
When you read in circuit descriptions using the command ReadNetlist (Section 3.10.1), the Model,
Selector , and Parameters keys in the value fields of model references are returned in the generic
form key −> key[devtype, parset, refdes], for example
{Q1, {1 −> C, 2 −> B, 3 −> E},
Model −> Model[BJT, DEFNPN, Q1],
Selector −> Selector[BJT, DEFNPN, Q1],
Parameters −> Parameters[BJT, DEFNPN, Q1]}
This approach permits an easy exchange of device models and parameter sets for each device or
group of devices by means of pattern matching and rewriting (see also Section 3.4.1). In the default
case, i.e. when no replacement rules are applied to these keys, the model name, the selector, and the
parameter set key are rewritten as follows upon circuit equation setup:
Model −> devtype, Selector −> mode, Parameters −> parset
For the following model reference Q1, the generic right-hand sides of these rules are transformed as
shown below if AnalysisMode −> AC.
{Q1,{2 −> C, 1 −> B, 0 −> E}
Model −> BJT, Selector −> AC, Parameters −> DEFNPN,
GM$ac −> 0.02, RPI$ac −> 5.0*^3, RO$ac −> 1.0*^4}
See also: Model (Section 3.2.1), ModelParameters (Section 3.1.10), Section 3.1.3.
3.1.9 Model and Subcircuit
In Analog Insydes, Subcircuit is a synonym for Model. You may use either function name as you
like.
For a detailed description of the Model command, see Section 3.2.1.
3.1 Netlist Format
189
3.1.10 ModelParameters
ModelParameters[Name−>name, par −>val , † † † ]
defines a local model parameter set
ModelParameters[Name−>name, Scope−>Global, par −>val , † † † ]
stores a model parameter set in the global subcircuit
database
Command structure of ModelParameters .
In addition to subcircuit and device models, you can also define parameter sets (model cards) that
can be used in conjunction with model references in a circuit description.
The following statement defines a model parameter set for the device type DEFNPN.
ModelParameters[Name −> DEFNPN, IS −> 1.0*^−16, BF −> 100,
BR −> 1, VAF −> 150]
These model parameter sets can be used in model references by means of the Parameters keyword.
See also: Section 3.1.8.
3.1.11 GlobalParameters
GlobalParameters[par −>val , † † † ]
defines global parameter settings
Command structure of GlobalParameters .
With GlobalParameters you can specify settings for global parameters used in a Circuit object, as
for instance TEMP or GMIN.
Global parameter definitions may be self-referencing, i.e. a global parameter may be
specified as a function of one or several others (make sure that there are no circular
references).
The list of global parameters is implicitly appended to the value fields of all model
references in a circuit. Therefore, you can use global parameter settings in a model without
having to pass these settings explicitly in each model reference.
The following examples are valid declarations for global parameters:
GlobalParameters[TEMP −> 300.15, GMIN −> 1.0*^−12]
GlobalParameters[TEMP −> TNOM, TNOM −> 300.15,
GMIN −> 1.0*^−12]
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3.2 Subcircuit and Device Model Definition
This chapter describes the usage of the command Model (Section 3.2.1), which enables you to describe
your circuits hierarchically in terms of subcircuits and device models. In the following sections, Analog Insydes’ subcircuit and device modeling functionality is presented mainly from the perspective
of language syntax. Therefore, it is recommended that you also read Chapter 2.3 and Chapter 2.6 of
the Analog Insydes Tutorial to understand the philosophy behind the modeling language and learn
more about methodologies for modeling linear and nonlinear circuits and devices.
Analog Insydes provides a unified modeling scheme that lets you implement and instantiate both
subcircuits and device models in exactly the same way. Therefore, the terms model and subcircuit shall
be treated as synonyms throughout this chapter. Whenever the word model is used in the following
text, it may refer to a component of a hierarchically structured circuit as well as to an equivalent circuit or set of equations that describes a physical device. Consequently, the Analog Insydes command
Subcircuit is just an alias for Model, and you may use both names interchangeably.
3.2.1 Model
Model[keyword −>value , keyword −>value , † † † ]
defines a subcircuit or device model in terms of a netlist or
a system of behavioral equations
Command structure of Model.
Subcircuits and device models can be implemented either as netlists, i.e. as equivalent circuits composed of electrical primitives and model references (see Section 3.1.3), or as behavioral descriptions
in terms of linear and nonlinear algebraic and differential equations. Both equivalent circuits and
analog behavioral models (ABMs) can be defined with the Model command. Model takes a sequence of
named arguments, four of which are required while the remaining arguments are optional.
Name
Selector
Ports
Definition
Required arguments of the Model command.
the model class name
the selector for a particular member from a model class
the list of model port nodes
the model definition (a netlist or a system of equations)
3.2 Subcircuit and Device Model Definition
Scope
Parameters
Defaults
Translation
Symbolic
Variables
InitialGuess
InitialConditions
191
the scope of the model definition (local or global)
the list of model parameter names
default values for model parameters
translation rules for parameter conversion
the list of ABM parameters to be kept in symbolic form
the variables in a system of ABM equations
initial guess for the numerical values of ABM variables
initial conditions for dynamic ABM variables
Optional arguments of the Model command.
A valid Model definition has the following general form:
Model[
Name −> name,
Selector −> selector
Ports −> {port nodes},
optarg −> value ,
Definition −> Netlist[ ] | Circuit[ ] | Equations[ ]
]
Arguments Description
The arguments of the Model command are explained in detail below.
Name
Name−>"string"
Name−>symbol
Name−>{"string ", "string ", † † † }
the name of the model class to which the model definition
is assigned
equivalent to Name−>ToString[symbol]
assign the model definition to several model classes
Format of the Name argument.
The value of the argument Name identifies a class of model definitions for a non-primitive circuit
object or a device type, such as current mirror, operational amplifier, bipolar junction transistor, or
MOSFET. The class name must be a symbol or a string. For example, the class names of BJT and
MOSFET models are defined in the Analog Insydes model library as:
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Name −> "BJT"
Name −> "MOSFET"
You may assign one or more alias names to a model class by specifying a list of strings:
Name −> {"BJT", "Q"}
Internally, all model names are converted to strings. Therefore, Name−>symbol and
Name−>ToString[symbol] are equivalent name specifications.
Likewise, you can specify the model class name in a model reference (see Section 3.1.8)
either as a symbol or a string regardless of the data type used in the model definition.
Selector
Selector−>"string"
Selector−>symbol
the key by which the model can be identified among the
members of the model class
equivalent to Selector−>ToString[symbol]
Selector−>{"string ", "string ", † † † }
assign several alias names to the model definition
Format of the Selector argument.
The value of the argument Selector selects a particular member from the class of models identified
with Name. For instance, the AC, DC, and dynamic large-signal models of the device type MOSFET
can be defined and distinguished with the following Name/Selector pairs:
Model[Name −> "MOSFET", Selector −> "AC", ],
Model[Name −> "MOSFET", Selector −> "DC", ],
Model[Name −> "MOSFET", Selector −> "Transient", ]
By assigning several selector keys to a model definition, you can set up a list of alias names for
the model. This feature allows you to configure models for use with several closely related analysis
modes, such as AC and noise analysis (see DefaultSelector in Section 3.5.1). To prepare a model
for use with both AC and noise analysis, your selector specification should be written as follows:
Model[Name −> "MOSFET", Selector −> {"AC", "Noise"}, ]
3.2 Subcircuit and Device Model Definition
193
Internally, all model selectors are converted to strings. Therefore, Selector−>symbol and
Selector−>ToString[symbol] are equivalent selector specifications.
Likewise, you can specify the selector in a model reference (see Section 3.1.8) either as a
symbol or a string regardless of the data type used in the model definition.
Scope
Scope−>Local
Scope−>Global
restrict the visibility of the model definition to the
surrounding Circuit object
store the model definition in the global subcircuit database
Format of the Scope argument.
The optional argument Scope determines the scope of the model definition. With the default setting
Scope −> Local, the model can only be referenced from Netlist objects inside the Circuit object
that contains the corresponding Model statement. With Scope −> Global, the model is stored in the
global subcircuit database and can be accessed by other Circuit objects.
You can use the Scope mechanism to implement a library of frequently used device models separately
from any top-level circuit description and load the models into Analog Insydes’ global subcircuit
database. This can be accomplished by implementing a set of models in a Mathematica *.m file as
illustrated below.
(* Begin MyModels.m *)
Circuit[
Model[Name −> "BJT", Selector −> "AC", Scope −> Global, Definition −> ],
Model[Name −> "MOSFET", Selector −> "AC", Scope −> Global, Definition −> ],
] // ExpandSubcircuits;
(* End MyModels.m *)
The library file "MyModels.m" can then be loaded using the command Get["MyModels.m"] .
See also: GlobalSubcircuits (Section 3.3.4), RemoveSubcircuit (Section 3.3.8).
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Ports
Ports−>{portspec , portspec , † † † }
specify the nodes in the model definition which serve as
external connection points
Format of the Ports argument.
With the Ports keyword, you can define the port nodes of a model. The value of the argument
must be a list of the identifiers of the model nodes which serve as external connection points. Nodes
in a model definition which are not listed in the Ports statement (internal nodes) are not accessible
from the outside.
"string" or symbol
declare a required port node
Optional[port]
declare port as optional port node; if port remains
unconnected in a model reference, treat it as if it were an
internal node of the model
Optional[port, default]
declare port as optional port node; if port remains
unconnected in a model reference, connect it to the node
default instead
Values for the port specification portspec.
For a three-pin BJT model with collector (C), base (B), and emitter (E) terminals, the list of port nodes
can be specified as a list of strings
Ports −> {"C", "B", "E"}
or, equivalently, as a list of symbols
Ports −> {C, B, E}
Although there is no semantic difference between these two Ports declarations, it is
recommended that you always use strings as port node identifiers. This avoids potential
errors caused by accidentally assigning a value to a Mathematica symbol which is used as a
port node identifier in some model definitions. For example, the assignment C = 1 + x will
corrupt the latter of the above Ports declarations whereas the version that uses strings is
not affected.
The global ground node 0 (or "0") may not be used as a port identifier.
In a model reference, all required ports listed in the Ports argument of the corresponding model
definition must be connected to an external node exactly once; otherwise, subcircuit expansion fails.
3.2 Subcircuit and Device Model Definition
195
By declaring a port as Optional , you can tell Analog Insydes to ignore any missing connections to
this port and perform some default action instead, such as connecting the port to the ground node.
This is useful for modeling integrated semiconductor devices with substrate pins. If the substrate
pin of a BJT or MOSFET model is an optional port, you can use the same model definition for
both three-pin and four-pin devices. If you write the Ports specifications for these device types as
shown below, then the substrate pin is connected implicitly to the ground node and source terminal,
respectively, whenever a BJT or MOSFET model is instantiated as a three-pin device.
The following Ports declaration causes the substrate pin S of a BJT device to be connected to the
ground node if S is left unconnected:
Ports −> {"C", "B", "E", Optional["S", 0]}
This causes the bulk node B of a MOSFET device to be connected to the source terminal if the port
node B is left unconnected:
Ports −> {"D", "G", "S", Optional["B", "S"]}
You can also tell Analog Insydes to leave an unconnected node floating by using Optional without
a second argument. The following Ports declaration treats S as if it were an internal node of the
model definition if S is left unconnected.
Ports −> {"C", "B", "E", Optional["S"]}
The second node given in an Optional declaration must be a required port, an internal
node of the model definition, or the global ground node.
Parameters
Parameters−>{parspec , parspec , † † † }
specify the symbols which represent parameters of a model
definition
Format of the Parameters argument.
With the Parameters keyword, you can specify the set of symbols which represent the parameters
of a model definition. Possible values are:
symbol
Global[symbol]
Values for the parameter specification parspec.
declare symbol as an instance-specific model parameter
declare symbol as a global quantity but allow
instance-specific value assignments
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Any symbol which appears in the value field of a model netlist entry or in a set of behavioral equations can be designated as a parameter. The difference between parameters and other symbols is that
you can assign instance-specific values only to parameters. Other symbols are regarded as global
quantities which cannot be bound to instance-specific values in a model reference. In addition, if a
model parameter p is left unbound in a model reference, it will be replaced by a uniquely instantiated
symbol p$instance upon subcircuit expansion whereas symbols representing global quantities are not
changed. The following line shows a valid example for the Parameters declaration of a BJT model:
Parameters −> {IS, N, BF, BR}
In some cases it is necessary to override the value of a global circuit parameter locally for a particular
model instance. For example, you may have set the environmental temperature TEMP in a Circuit
object with the GlobalParameters statement
Circuit[
Netlist[ ],
GlobalParameters[TEMP −> 300.15]
]
but you want to specify a different operating temperature for an individual device, such as a temperature sensor or the output transistor of a power amplifier. In order to override the global value
of TEMP for this device, the local temperature value must be specified explicitly in the corresponding
model reference:
{QOUT, {VCC −> C, 5 −> B, out −> E},
Model −> BJT, Selector −> DC,
IS −> 1.*^−16, N −> 1., BF −> 100., BR −> 1., TEMP −> 340.}
However, TEMP is a global quantity and not an instance parameter of the model BJT/DC. Therefore,
the local setting TEMP −> 340. in the model reference QOUT has no effect unless the Parameters
specification in the model definition is modified as follows:
Parameters −> {IS, N, BF, BR, Global[TEMP]}
Now the setting Global[TEMP] allows you to change the value of TEMP individually for any instance
of BJT/DC while the setting from the GlobalParameters list will be used for all instances for which
no local temperature value is given.
Defaults
Defaults−>{param −>value , param −>value , † † † }
specify default values for model parameters
Format of the Defaults argument.
With the Defaults keyword, you can specify default values for model parameters which have been
left unbound in a model reference. For example, with the settings
3.2 Subcircuit and Device Model Definition
197
Parameters −> {IS, N, BF, BR, Global[TEMP]},
Defaults −> {IS −> 1.*^−16, N −> 1., BF −> 100.,
BR −> 1., TEMP −> 300.15},
you can instantiate a default BJT device without having to specify values for the parameters in the
model reference:
{QDEF, {nout −> C, 2 −> B, 3 −> E}, Model −> BJT, Selector −> DC}
During subcircuit expansion the numerical default values specified by the Defaults argument are
used as model parameter values. The above model reference is thus processed as if its value field
had been specified as shown below.
{QDEF, {nout −> C, 2 −> B, 3 −> E}, Model −> BJT, Selector −> DC,
IS −> 1.*^−16, N −> 1., BF −> 100., BR −> 1., TEMP −> 300.15}
Translation
Translation−>{param −>expr , param −>expr , † † † }
specify model parameter translation rules
Format of the Translation argument.
The optional argument Translation allows you to specify conversion rules for calculating values of
internal model parameters from external parameters declared with the Parameters argument.
For example, the following code defines a simple BJT small-signal model with the external parameters RB (base resistance) and beta (current gain). Internally, the model is implemented using a
VCCS whose transconductance gm is calculated automatically from RB and beta by means of the
Translation rule gm −> RB*beta.
Model[
Name −> "BJT", Selector −> "SimpleACgm",
Ports −> {"C", "B", "E"},
Parameters −> {RB, beta},
Translation −> {gm −> RB*beta},
Definition −> Netlist[
{"RB", {"B", "E"}, RB},
{"VC", {"B", "E", "C", "E"}, gm}
]
]
Translation rules may also be specified using the RuleDelayed operator (:>). This enables you to
defer the evaluation of function calls on the right-hand sides of translation rules until model expansion. In the following list of rules, the thermal voltage Vt of a semiconductor device is calculated by
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a delayed rule to prevent the function ThermalVoltage to be evaluated before the value of TEMP is
known for an individual device.
Translation −> {Is −> Js*Area, Vt :> ThermalVoltage[TEMP]}
If Vt were calculated using a simple Rule (−>), ThermalVoltage would already be called at the time
of model definition, i.e. execution of the Model command.
Symbolic
Symbolic−>{symbol , symbol , † † † }
Symbolic−>{symbol −>alias , † † † }
declare behavioral model parameters as symbolic quantities
declare behavioral model parameters as symbolic
quantities and specify alias names to be used in symbolic
circuit equations
Format of the Symbolic argument.
The optional argument Symbolic is used only in conjunction with analog behavioral model definitions. It lets you specify the subset of the model parameters declared in the Parameters section
that will be treated as symbolic quantities in a set of behavioral model equations. Model parameters
which are not listed in the Symbolic section are always replaced by their numerical values when
you set up circuit equations. This feature allows you to restrict the parameters that will appear in a
system of symbolic circuit equations to the ones you really wish to see.
The following code defines a DC model for a temperature-dependent linear resistor. The model
parameters are the nominal resistance RES, the first-order temperature coefficient TC1, the ambient
temperature TEMP, and the reference temperature TNOM. The Symbolic argument declares RES as a
model parameter which will appear in symbolic circuit equations. The other parameters will always
be evaluated numerically and will not appear explicitly in circuit equations.
Model[
Name −> "RES", Selector −> "DC",
Ports −> {"P", "N"},
Parameters −> {RES, TC1, Global[TEMP, TNOM]},
Symbolic −> {RES},
Variables −> {Voltage["P", "N"], Current["P", "N"]},
Definition −> Equations[
Voltage["P", "N"] ==
RES*(1 + TC1*(TEMP − TNOM))*Current["P", "N"]
]
]
Names of device model or instance parameters in netlists imported from external circuit simulators
with ReadNetlist sometimes have lengthy or ugly names. The alternative format of the Symbolic
3.2 Subcircuit and Device Model Definition
199
argument allows you to assign alias names to model parameters which will appear in symbolic
circuit equations in place of the original names.
For example, the following declaration causes the symbol R to be used instead of the original
model parameter name RES whenever you set up symbolic circuit equations with the above model
definition.
Symbolic −> {RES −> R}
Variables
Variables−>{varspec , varspec , † † † }
specify the identifiers which represent unknowns in a set
of model equations
Format of the Variables argument.
The Variables argument is used only in conjunction with analog behavioral model definitions. It
serves to declare the identifiers in a set of model equations which represent the unknowns. The
identifiers have to be of the following form:
symbol
declare symbol as an internal model variable
Voltage[port , port ] or Current[port , port ]
declare a model variable as a branch voltage or branch
current
Values for the variable specification varspec.
For each port branch you must declare both the branch voltage and the branch current as
model variables, even in cases where one of the two does not appear explicitly in the
equations.
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Definition
Definition−>Netlist[† † †]|Circuit[† † †]
define a model in terms of an equivalent Netlist or
Circuit object
Definition−>Equations[† † †]
define a behavioral model in terms of a system of
equations
Format of the Definition argument.
With the Definition argument, you specify the implementation of a model. A model can be
implemented either as an equivalent circuit or as a set of linear or nonlinear differential-algebraic
equations.
You may use a Netlist or a Circuit object to implement an equivalent circuit. The netlist that
represents the equivalent circuit may contain primitive devices as well as further model references.
There is no fixed limit of the subcircuit nesting level.
The following example shows a model implementation for a simple RC tank.
Definition −> Netlist[
{R, {in, out}, R},
{C, {out, ref}, C}
]
A Circuit object may contain local ModelParameters sections as well as local Model definitions.
Local model definitions and parameter sets should have Scope −> Local to prevent interference
with the global subcircuit and parameter databases. Using a Circuit instead of a simple Netlist
feature allows you to define models with instance-specific model parameter sets, i.e. device model
parameters whose values are calculated from model instance parameters.
For example, the following code implements a device-level description of a bipolar differential pair
with emitter degeneration resistors. The model definition contains a local ModelParameters statement in which the ohmic collector and emitter resistances RC and RE are calculated as functions of
the BJT area factor Area for each instance of the subcircuit DifferentialPair/DeviceLevel .
3.2 Subcircuit and Device Model Definition
201
Subcircuit[
Name −> "DifferentialPair", Selector −> "DeviceLevel",
Ports −> {"H1", "H2", "IN1", "IN2", "L"},
Parameters −> {Area, RED},
Definition −> Circuit[
Netlist[
{"Q1", {"H1" −> "C", "IN1" −> "B", "ED1" −> "E"},
Model −> "BJT", Selector −> Selector["BJT", "Q1"],
Parameters −> "LocalNPN"},
{"RED1", {"ED1", "L"}, RED},
{"Q2", {"H2" −> "C", "IN2" −> "B", "ED2" −> "E"},
Model −> "BJT", Selector −> Selector["BJT", "Q2"],
Parameters −> "LocalNPN"},
{"RED2", {"ED2", "L"}, RED}
],
ModelParameters[Name −> "LocalNPN", Type −> "NPN",
IS −> 2.*^−15, BF −> 200., RC −> 50./Area, RE −> 2./Area]
]
]
Analog Insydes allows you to implement arbitrary multi-dimensional voltage-current relations as
analog behavioral models. ABMs are implemented using sets of Equations instead of Netlist or
Circuit objects. Each model equation must be written in the form lhs == rhs, where lhs and rhs
denote arbitrary Mathematica expressions.
If you implement a behavioral model using the Equations keyword, you must declare the
unknowns of the model equations in the Variables section.
The following code implements a nonlinear DC model for a bipolar junction diode with series
resistance Rs. The first equation serves to compute the effective diode voltage vdeff. It is written in
implicit notation to demonstrate that Analog Insydes does not require model equations to be written
in explicit form var == expr, where var denotes a single symbol or branch quantity identifier.
Model[
Name −> "Diode", Selector −> "DC",
Ports −> {"A", "C"}, Parameters −> {Rs, Is, Global[Vt]},
Variables −> {Voltage["A", "C"], Current["A", "C"], vdeff},
Definition −> Equations[
Voltage["A", "C"] − Rs*Current["A", "C"] − vdeff == 0,
Current["A", "C"] == Is*(Exp[vdeff/Vt] − 1)
]
]
In addition, the Model command lets you implement behavioral models involving ordinary differential equations (ODEs). In general, the independent variable of such equations is the time variable,
3. Reference Manual
202
denoted by the symbol Time, but other quantities than time may be used as independent variable as
well (see Section 3.5.1).
You can attach a trailing tick mark (’) to a variable in a set of model equations to denote a time
derivative. The following model definition implements the voltage-current relation of a linear, timeinvariant capacitor in the time domain. The time derivative of the capacitor voltage is denoted by
Voltage["P", "N"]’.
Variables −> {Current["P", "N"], Voltage["P", "N"]},
Definition −> Equations[
Current["P", "N"] == C*Voltage["P", "N"]’
],
To denote a derivative of a variable var with respect to some other quantity q, you must use
Mathematica’s full derivative notation
Derivative[n][var][q]
where n is the order of the differentiation. Of course, you can also use this notation for time derivatives. For example, the first derivative of the variable vdeff with respect to time can be written
as:
Derivative[1][vdeff][Time]
Since Time is the default quantity with respect to which derivatives are computed, the above line
can be abbreviated as follows.
Derivative[1][vdeff]
Any derivative term lacking an explicit specification of the independent variables is
assumed to be a derivative with respect to Time.
Your equations may involve second and higher-order derivatives.
Your equations may not involve partial derivatives.
See also: IndependentVariable (Section 3.5.1), Section 3.1.7.
InitialGuess
InitialGuess−>{var −>value , var −>value , † † † }
specify initial guesses for the values of behavioral model
variables
Format of the InitialGuess argument.
3.2 Subcircuit and Device Model Definition
203
With the InitialGuess argument, you can specify a set of numerical values that will be used
by NDAESolve (Section 3.7.5) as initial guess for the solution of behavioral model variables. Initial
guesses may be specified for some or all of the variables listed in the corresponding Variables
declaration.
The default value for initial guesses is 0.
When you analyze strongly nonlinear circuits, an educated initial guess helps to speed up
DC calculations with NDAESolve considerably.
The effect of initial guess specifications on the convergence behavior of NDAESolve depends
on the type of circuit equations.
In the example shown below, initial guesses of V and V are specified for the model variables vdeff and Voltage["A", "C"], respectively. No initial guess is given for Current["A", "C"],
therefore, NDAESolve uses 0 as initial guess for this current.
Model[
Name −> "Diode", Selector −> "DC",
Ports −> {"A", "C"}, Parameters −> {Rs, Is, Global[Vt]},
Variables −> {Voltage["A", "C"], Current["A", "C"], vdeff},
Definition −> Equations[
Voltage["A", "C"] − Rs*Current["A", "C"] − vdeff == 0,
Current["A", "C"] == Is*(Exp[vdeff/Vt] − 1)
],
InitialGuess −> {vdeff −> 0.7, Voltage["A", "C"] −> 0.8}
]
InitialConditions
InitialConditions−>{var −>value , var −>value , † † † }
specify initial conditions for dynamic model variables
Format of the InitialConditions argument.
With the InitialConditions argument, you can specify initial conditions for dynamic variables in
behavioral model equations. Initial conditions can be assigned to any quantity x whose derivative x’
appears in the equations. This applies to internal model variables as well as to branch voltages and
currents.
The following example shows how to specify an initial condition for the branch quantity
Voltage["P", "N"]. Note that it is not important whether this quantity appears directly in the
equations or not. However, its derivative Voltage["P", "N"]’ must be present as a prerequisite for
the assignment of an initial condition.
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Model[
Name −> "Capacitor", Selector −> "Behavioral",
Ports −> {"P", "N"}, Parameters −> {C},
Variables −> {Current["P", "N"], Voltage["P", "N"]},
Definition −> Equations[
Current["P", "N"] == C*Voltage["P", "N"]’
],
InitialConditions −> {Voltage["P", "N"] −> 1.52}
]
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
The following model commands implement a CCCS-based model for bipolar transistors and an
equivalent VCCS-based model. The transconductance of the VCCS in BJT/SimpleACgm is calculated
automatically from the parameters RB and beta by means of a translation rule.
CCCS and VCCS-based
BJT small-signal models.
In[2]:= Circuit[
Model[
Name −> "BJT", Selector −> "SimpleACbeta",
Scope −> Global, Ports −> {"C", "B", "E"},
Parameters −> {RB, beta},
Definition −> Netlist[
{"RB", {"B", "X"}, RB},
{"CC", {"X", "E", "C", "E"}, beta}
]
],
Model[
Name −> "BJT", Selector −> "SimpleACgm",
Scope −> Global, Ports −> {"C", "B", "E"},
Parameters −> {RB, beta},
Translation −> {gm −> RB*beta},
Definition −> Netlist[
{"RB", {"B", "E"}, RB},
{"VC", {"B", "E", "C", "E"}, gm}
]
]
] // ExpandSubcircuits;
List the contents of the
global subcircuit database.
In[3]:= GlobalSubcircuits[]
Out[3]= BJT, SimpleACbeta, BJT, SimpleACgm
The Circuit object shown below describes a common-source amplifier with a MOSFET device. In
the model definition for MOSFET, the bulk terminal B is defined as an optional port. Since this node
is not connected in the top-level netlist, the bulk terminal is connected implicitly to the source node.
3.2 Subcircuit and Device Model Definition
This netlist describes a
common-source amplifier.
205
In[4]:= csamp = Circuit[
Netlist[
{V1, {1, 0}, Value −> {DC −> 1.5, AC −> 1}},
{M1, {"out" −> "D", 1 −> "G", 3 −> "S"},
Model −> "MOSFET", Selector −> "DCSmallSignal",
GM −> 2.*^−4, GMB −> 3.*^−5, GDS −> 1.*^−6},
{RD, {"VDD", "out"}, Symbolic −> Rd, Value −> 1.*^6},
{RS, {3, 0}, Symbolic −> Rs, Value −> 5000.},
{VDD, {"VDD", 0}, Value −> {DC −> 3.3, _ −> 0}}
],
Model[
Name −> "MOSFET", Selector −> "DCSmallSignal",
Ports −> {"D", "G", "S", Optional["B", "S"]},
Parameters −> {gm, gmb, Gds, GM, GMB, GDS},
Definition −> Netlist[
{VCG, {"G", "S", "D", "S"}, Symbolic −> gm,
Value −> GM},
{VCB, {"B", "S", "D", "S"}, Symbolic −> gmb,
Value −> GMB},
{GDS, {"D", "S"}, Symbolic −> Gds,
Value −> GDS}
]
]
]
Out[4]= Circuit Expand the MOSFET
model.
In[5]:= ExpandSubcircuits[csamp] // DisplayForm
Out[5]//DisplayForm=
Netlist Flat, 7 Entries:
V1, 1, 0, Value DC 1.5, AC 1
VCG$M1, 1, 3, out, 3, Symbolic gm$M1, Value 0.0002
VCB$M1, 3, 3, out, 3, Symbolic gmb$M1, Value 0.00003
GDS$M1, out, 3, Symbolic Gds$M1, Value 1. 106 RD, VDD, out, Symbolic Rd, Value 1. 106 RS, 3, 0, Symbolic Rs, Value 5000.
VDD, VDD, 0, Value DC 3.3, _ 0
This example shows the effect of the argument Symbolic in a Model statement. The model equations
describe the DC behavior of a linear resistor including first-order temperature dependence. Note that
although there are four model parameters, only the resistance appears explicitly in symbolic circuit
equations. Note also that the alias name R is used instead of RES to denote the resistance.
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206
A circuit with a resistor
model.
In[6]:= resckt = Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0, Value −> 1},
{R1, {1 −> "P", 0 −> "N"},
Model −> "RES", Selector −> "DC",
RES −> 1000., TC1 −> 10^−3}
],
Model[
Name −> "RES", Selector −> "DC",
Ports −> {"P", "N"},
Parameters −> {RES, TC1, Global[TEMP, TNOM]},
Defaults −> {TC1 −> 0, TNOM −> 300.15, TEMP −> 300.15},
Symbolic −> {RES −> R},
Variables −> {Voltage["P", "N"], Current["P", "N"]},
Definition −> Equations[
Voltage["P", "N"] ==
RES*(1 + TC1*(TEMP − TNOM))*Current["P", "N"]
]
],
GlobalParameters[TEMP −> 310.]
]
Out[6]= Circuit Note that only R$R1 is
kept in symbolic form.
In[7]:= CircuitEquations[resckt, AnalysisMode −> DC,
ElementValues −> Symbolic] // DisplayForm
Out[7]//DisplayForm=
I$PN$R1 I$V0 0,
V$1 V0, V$1 1.00985 I$PN$R1 R$R1, V$1, I$V0, I$PN$R1,
DesignPoint V0 1, R$R1 1000., TEMP 310.
The following example demonstrates the use of local model parameter sets in a subcircuit definition.
The netlist shown below represents a bipolar differential pair with emitter degeneration resistors.
The Circuit object contains a local ModelParameters statement in which the values of the BJT’s
ohmic collector and emitter resistances are calculated as a function of the area factor.
3.2 Subcircuit and Device Model Definition
A differential pair with
emitter degeneration
resistors.
207
In[8]:= dpwedr = Subcircuit[
Name −> "DifferentialPair", Selector −> "DeviceLevel",
Ports −> {"H1", "H2", "IN1", "IN2", "L"},
Parameters −> {Area, RED},
Definition −> Circuit[
Netlist[
{"Q1", {"H1" −> "C", "IN1" −> "B", "ED1" −> "E"},
Model −> "BJT", Selector −> Selector["BJT", "Q1"],
Parameters −> "LocalNPN"},
{"RED1", {"ED1", "L"}, RED},
{"Q2", {"H2" −> "C", "IN2" −> "B", "ED2" −> "E"},
Model −> "BJT", Selector −> Selector["BJT", "Q2"],
Parameters −> "LocalNPN"},
{"RED2", {"ED2", "L"}, RED}
],
ModelParameters[Name −> "LocalNPN", Type −> "NPN",
IS −> 2.*^−15, BF −> 200., RC −> 50./Area,
RE −> 2./Area]
]
]
Out[8]=
SubcircuitDifferentialPair, DeviceLevel, 5 This netlist describes a
differential amplifier on
the basis of the subcircuit
implemented above.
In[9]:= diffamp = Circuit[
Netlist[
{V0, {"VCC", 0}, Value −> {DC −> 9., _ −> 0}},
{V1, {2, 0}, Value −> {DC −> 4.5, AC −> 1}},
{V2, {3, 0}, Value −> {DC −> 4.5, AC −> 0}},
{R1, {"VCC", 4}, 5000.},
{R2, {"VCC", 5}, 5000.},
{DP1, {4 −> "H1", 5 −> "H2", 2 −> "IN1",
3 −> "IN2", 6 −> "L"},
Model −> "DifferentialPair",
Selector −> "DeviceLevel",
RED −> 100., Area −> 2.},
{IBIAS, {6, 0}, Value −> {DC −> 0.001, _ −> 0}}
],
dpwedr
]
Out[9]= Circuit Set up circuit equations.
In[10]:= diffampeqs = CircuitEquations[diffamp,
AnalysisMode −> DC, ElementValues −> Symbolic]
Out[10]= DAEDC, 25 25 The nominal collector and
emitter resistances have
been divided by the area
factor.
In[11]:= {RC$Q1$DP1, RE$Q1$DP1} /. GetDesignPoint[diffampeqs]
Out[11]= 25., 1.
The following code is a slightly more complex example for a behavioral model definition. The
Model statement implements a nonlinear DC model for a bipolar junction diode, taking into account
junction voltage reduction due to series resistance (Rs) and DC convergence issues (GMIN).
3. Reference Manual
208
This code defines a diode
model and stores it in the
global subcircuit database.
In[12]:= Circuit[
Model[
Name −> "Diode", Selector −> "DC",
Scope −> Global, Ports −> {"A", "C"},
Parameters −> {Is, Rs, Global[TEMP, GMIN]},
Defaults −> {Is −> 1.*^−15, Rs −> 0, TEMP −> 300.,
GMIN −> 1.*^−12, $q −> 1.6021918*^−19,
$k −> 1.3806226*^−23},
Translation −> {Vt −> $k*TEMP/$q},
Symbolic −> {Is, Vt, Rs, GMIN},
Variables −> {Voltage["A", "C"],
Current["A", "C"], vdeff},
Definition −> Equations[
Current["A", "C"] ==
Is*(Exp[vdeff/Vt] − 1) + vdeff*GMIN,
vdeff == Voltage["A", "C"] − Rs*Current["A", "C"]
],
InitialGuess −> {vdeff −> 0.7}
]
] // ExpandSubcircuits;
This defines a circuit with
a diode in series
connection with a resistor.
In[13]:= diodeckt = Circuit[
Netlist[
{V1, {1, 0}, Vin},
{R1, {1, 2}, Symbolic −> R1, Value −> 1000.},
{D1, {2 −> "A", 0 −> "C"},
Model −> "Diode", Selector −> "DC", Rs −> 10.}
],
GlobalParameters[TEMP −> 330.]
]
Out[13]= Circuit Set up modified nodal
equations.
In[14]:= diodeeqs = CircuitEquations[diodeckt,
AnalysisMode −> DC, ElementValues −> Symbolic]
Out[14]= DAEDC, 5 5 Display the equations.
In[15]:= diodeeqs // DisplayForm
Out[15]//DisplayForm=
V$1 V$2
V$1 V$2
I$V1 0, I$AC$D1 0,
R1
R1
V$1 Vin, I$AC$D1 1 Evdeff$D1Vt$D1 Is$D1 GMIN vdeff$D1,
vdeff$D1 I$AC$D1 Rs$D1 V$2,
V$1, V$2, I$V1, I$AC$D1, vdeff$D1,
DesignPoint Vin Vin, R1 1000., Is$D1 1. 1015 ,
Vt$D1 0.0284364, Rs$D1 10., GMIN 1. 1012 , TEMP 330.
Calculate the DC response
while sweeping Rs$D1
from to .
In[16]:= diodedc = NDAESolve[diodeeqs,
{Vin, −1., 5.}, {Rs$D1, 0., 200., 50.}];
3.2 Subcircuit and Device Model Definition
Plot the diode voltage.
209
In[17]:= TransientPlot[diodedc, V$2[Vin], {Vin, −1., 5.}]
1.5
1
0.5
-1
V$2[Vin]
1
2
3
4
5
Vin
-0.5
-1
Out[17]= Graphics This example demonstrates how to specify initial conditions. The netlist describes an RLC series
resonator. The behavior of the LC subcircuit is modeled by a system of differential equations.
An RLC circuit with a
behavioral description for
the LC resonator.
In[18]:= rlcfilter = Circuit[
Netlist[
{R, {1, 0}, 100.},
{LC, {1 −> "P", 0 −> "N"},
Model −> "LCSeriesResonator",
Selector −> "Behavioral", L −> 0.02, C −> 5.*^−7}
],
Model[
Name −> "LCSeriesResonator",
Selector −> "Behavioral",
Ports −> {"P", "N"}, Parameters −> {L, C},
Variables −> {Current["P", "N"], Voltage["P", "N"],
vind, vcap},
Definition −> Equations[
vind == L*Current["P", "N"]’,
Voltage["P", "N"] − vcap − vind == 0,
Current["P", "N"] == C*vcap’
],
InitialConditions −> {vcap −> 5.,
Current["P", "N"] −> −0.02}
]
]
Out[18]= Circuit Set up time-domain circuit
equations.
In[19]:= rlcfeqs = CircuitEquations[rlcfilter,
AnalysisMode −> Transient,
InitialConditions −> Automatic]
Out[19]= DAETransient, 6 4 Display the circuit
equations.
In[20]:= rlcfeqs // DisplayForm
Out[20]//DisplayForm=
I$PN$LCt 0.01 V$1t 0, vind$LCt 0.02 I$PN$LC t,
vcap$LCt vind$LCt V$1t 0, I$PN$LCt 5. 107 vcap$LC t, vcap$LC0 5., I$PN$LC0 0.02,
V$1t, I$PN$LCt, vind$LCt, vcap$LCt, DesignPoint 3. Reference Manual
210
Simulate the circuit.
In[21]:= rlcftran = NDAESolve[rlcfeqs, {t, 0., 0.002}]
Out[21]=
I$PN$LC InterpolatingFunction0, 0.002, ,
vcap$LC InterpolatingFunction0, 0.002, ,
V$1 InterpolatingFunction0, 0.002, ,
vind$LC InterpolatingFunction0, 0.002, Display the capacitor
voltage and the inductor
current.
In[22]:= TransientPlot[rlcftran, {vcap$LC[t], I$PN$LC[t]},
{t, 0., 0.002}, SingleDiagram −> False, PlotRange −> All]
4
2
vcap
0.0005
0.001
0.0015
t
0.002
-2
0.01
0.005
t
0.0005 0.001 0.0015 0.002
-0.005
-0.01
-0.015
-0.02
-0.025
I$PN$
Out[22]= GraphicsArray 3.2.2 Subcircuit
The command Subcircuit is an alias for Model (Section 3.2.1). You may use either of the two
function names to define a subcircuit or device model.
3.3 Model Library Management
211
3.3 Model Library Management
Analog Insydes provides a number of functions for inspecting and managing the contents of its global
subcircuit database and external model libraries. These functions are described in detail below.
ListLibrary (Section 3.3.1)
FindModel (Section 3.3.2)
listing contents of a library
locating models
FindModelParameters (Section 3.3.3)
locating model parameters
GlobalSubcircuits (Section 3.3.4)
inspecting global subcircuit database
GlobalModelParameters (Section 3.3.5)
inspecting global model-parameters database
LoadModel (Section 3.3.6)
loading models
LoadModelParameters (Section 3.3.7)
loading model parameters
RemoveSubcircuits (Section 3.3.8)
removing subcircuits from global database
RemoveModelParameters (Section 3.3.9)
removing model parameters from global database
3.3.1 ListLibrary
ListLibrary[]
lists the contents of the default model library
ListLibrary["libfile"]
lists the contents of the model library "libfile"
Command structure of ListLibrary.
With ListLibrary you can list the contents of one or several model library files. ListLibrary
returns the model definitions and model parameter sets stored in the specified files.
ListLibrary has the following options:
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212
option name
default value
LibraryPath
Inherited[AnalogInsydes]
the search path for device model libraries
(see Section 3.14.3)
ModelLibrary
Inherited[AnalogInsydes]
the name of the library to be searched for
device models (see Section 3.14.4)
Options for ListLibrary.
See also: FindModel (Section 3.3.2), FindModelParameters (Section 3.3.3), LoadModel (Section 3.3.6),
LoadModelParameters (Section 3.3.7).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
List the contents of the
default model library.
In[2]:= ListLibrary[]
List the contents of the
specified libraries.
In[3]:= ListLibrary[ModelLibrary −> {"BasicResistorModels‘",
"FullDiodeModels‘"}]
Out[2]=
FullModels‘ ModelBJT, DC,
SpiceDC, GummelPoonDC, Transient, Spice, GummelPoon ,
ModelBJT, AC, SpiceAC, Noise, SpiceNoise ,
ModelMOSFET, DC, SpiceDC, Transient, Spice ,
ModelMOSFET, AC, SpiceAC, Noise, SpiceNoise ,
ModelDiode, DC, SpiceDC, Transient, Spice ,
ModelDiode, AC, SpiceAC, Noise, SpiceNoise ,
ModelJFET, DC, SpiceDC, Transient, Spice ,
ModelJFET, AC, SpiceAC, Noise, SpiceNoise ,
ModelRES, DC, Transient , ModelRES, AC ,
ModelRES, Noise , ModelCAP, DC, Transient ,
ModelCAP, AC, Noise , ModelIND, DC, Transient ,
ModelIND, AC, Noise Out[3]=
BasicResistorModels‘ ModelRES, AC, BasicSpiceAC,
DC, BasicSpiceDC, Transient, BasicSpice ,
ModelRES, Noise, SpiceNoise ,
FullDiodeModels‘ ModelDiode, DC, SpiceDC, Transient, Spice ,
ModelDiode, AC, SpiceAC, Noise, SpiceNoise 3.3 Model Library Management
213
3.3.2 FindModel
FindModel[name, selector]
searches the default library for the model definition
name/selector
Command structure of FindModel.
FindModel searches a list of model library files for a specified model definition and returns the
name of the first library in which the model was found. FindModel returns the symbol $Failed if
it cannot find the model definition in any of the library files.
FindModel has the following options:
option name
default value
LibraryPath
Inherited[AnalogInsydes]
the search path for device model libraries
(see Section 3.14.3)
ModelLibrary
Inherited[AnalogInsydes]
the name of the library to be searched for
device models (see Section 3.14.4)
Options for FindModel.
See also: FindModelParameters (Section 3.3.3), ListLibrary (Section 3.3.1), LoadModel (Section 3.3.6).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Determine the name of the
library file which contains
the model
"BJT/GummelPoon" .
In[2]:= FindModel["BJT", "GummelPoon"] // InputForm
Return the name of the
first library from the given
list which contains the
model "BJT/AC" .
In[3]:= FindModel["BJT", "AC",
ModelLibrary −> {"FullDiodeModels‘", "BasicBJTModels‘",
"SimplifiedModels‘"}
] // InputForm
Out[2]//InputForm= "FullModels‘"
Out[3]//InputForm= "BasicBJTModels‘"
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214
3.3.3 FindModelParameters
FindModelParameters[name]
searches the default library for the parameter set name
Command structure of FindModelParameters .
FindModelParameters searches a list of model library files for a specified model parameter set and
returns the name of the first library in which the parameter set was found. FindModelParameters
returns the symbol $Failed if it cannot find the parameter set in any of the library files.
FindModelParameters has the following options:
option name
default value
LibraryPath
Inherited[AnalogInsydes]
the search path for device model libraries
(see Section 3.14.3)
ModelLibrary
Inherited[AnalogInsydes]
the name of the library to be searched for
device models (see Section 3.14.4)
Options for FindModelParameters .
See also: FindModel (Section 3.3.2), ListLibrary (Section 3.3.1), LoadModelParameters (Section 3.3.7).
3.3.4 GlobalSubcircuits
GlobalSubcircuits[]
lists the names and selectors of the subcircuit definitions
stored in the global subcircuit database
Command structure of GlobalSubcircuits .
With GlobalSubcircuits you can inspect the contents of the global subcircuit database.
GlobalSubcircuits returns a list of the names and selectors of the subcircuits stored in the database.
GlobalSubcircuits not only returns the name under which a model was loaded but also all
alias names assigned to the model definition. For example, "Diode", "DC" | "Transient" denotes
a model definition which can be accessed through Name −> "Diode", Selector −> "DC" as well
through Name −> "Diode" , Selector −> "Transient" .
See also: FindModel (Section 3.3.2), GlobalModelParameters (Section 3.3.5).
3.3 Model Library Management
215
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Load two model
definitions from the model
library and store them in
the global subcircuit
database.
In[2]:= Circuit[
LoadModel["Diode", "Transient", Scope −> Global],
LoadModel["JFET", "AC", Scope −> Global]
] // ExpandSubcircuits;
List the contents of the
global subcircuit database.
In[3]:= GlobalSubcircuits[]
Out[3]= Diode, DC SpiceDC Transient Spice,
JFET, AC SpiceAC Noise SpiceNoise
3.3.5 GlobalModelParameters
GlobalModelParameters[]
lists the names of the model parameter sets stored in the
global model parameter database
Command structure of GlobalModelParameters .
With GlobalModelParameters you can inspect the contents of the global model parameter database.
GlobalModelParameters returns a list of the names of the parameter sets stored in the database.
See also: FindModelParameters (Section 3.3.3), GlobalSubcircuits (Section 3.3.4).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define two diode model
parameter sets.
In[2]:= Circuit[
ModelParameters[Name −> "Diode1", Scope −> Global,
IS −> 1.2*^−12, N −> 1.1],
ModelParameters[Name −> "Diode2", Scope −> Global,
IS −> 1.0*^−14, N −> 1.5]
] // ExpandSubcircuits;
List the contents of the
global model parameter
database.
In[3]:= GlobalModelParameters[] // InputForm
Out[3]//InputForm= {{"Diode1"}, {"Diode2"}}
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3.3.6 LoadModel
LoadModel[name, selector]
loads the model definition name/selector from the default
model library
LoadModel["libfile", name, selector]
loads the model definition name/selector from the model
library "libfile"
LoadModel[{{name , selector }, † † † }]
loads the models name /selector , † † †
Command structure of LoadModel.
With LoadModel you can load a model definition from a library file. LoadModel searches a list of
model library files and returns the first model definition that matches the specified name and selector.
You can use LoadModel within a Circuit object to load a model definition at run time and assign
global scope to the model.
LoadModel has the following options:
option name
default value
Scope
Local
LibraryPath
Inherited[AnalogInsydes]
the search path for device model libraries
(see Section 3.14.3)
ModelLibrary
Inherited[AnalogInsydes]
the name of the library to be searched for
device models (see Section 3.14.4)
the designated scope of the loaded model
definition
Options for LoadModel.
LoadModel only loads Model records from library files; it does not store the models in the
database.
To store a loaded model in the database, you must use Circuit and ExpandSubcircuits .
See also: Circuit (Section 3.1.2), ExpandSubcircuits (Section 3.4.1), LoadModelParameters (Section 3.3.7).
3.3 Model Library Management
217
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Load two model
definitions from the model
library and store them in
the global subcircuit
database.
In[2]:= Circuit[
LoadModel["Diode", "Transient", Scope −> Global],
LoadModel["JFET", "AC", Scope −> Global]
] // ExpandSubcircuits;
List the contents of the
global subcircuit database.
In[3]:= GlobalSubcircuits[]
Out[3]= Diode, DC SpiceDC Transient Spice,
JFET, AC SpiceAC Noise SpiceNoise
3.3.7 LoadModelParameters
LoadModelParameters[name]
loads the model parameter set definition name from the
default model library
LoadModelParameters["libfile", name]
loads the model parameter set definition name from "libfile"
LoadModelParameters[{name , † † †}]
loads the model parameter set definitions name , † † †
Command structure of LoadModelParameters .
With LoadModelParameters you can load a model parameter set definition from a library file.
LoadModelParameters searches a list of model library files and returns the first model parameter
set definition that matches the specified name.
You can use LoadModelParameters within a Circuit object to load a model parameter set definition
at run time and assign global scope to it.
LoadModelParameters has the following options:
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option name
default value
Scope
Local
LibraryPath
Inherited[AnalogInsydes]
the search path for device model libraries
(see Section 3.14.3)
ModelLibrary
Inherited[AnalogInsydes]
the name of the library to be searched for
model parameter set definitions (see
Section 3.14.4)
the designated scope of the loaded model
parameter set definition
Options for LoadModelParameters .
LoadModelParameters only loads ModelParameters records from library files; it does not
store the model parameter sets in the database.
To store a loaded model parameter set in the database, you must use Circuit and
ExpandSubcircuits .
See also: Circuit (Section 3.1.2), ExpandSubcircuits (Section 3.4.1), LoadModel (Section 3.3.6).
3.3.8 RemoveSubcircuit
RemoveSubcircuit[name, selector]
removes the subcircuit definition name/selector from the
global database
RemoveSubcircuit[All]
clears the global subcircuit database
Command structure of RemoveSubcircuit .
RemoveSubcircuit allows you to remove individual subcircuit definitions from the global subcircuit
database or to clear the database completely. RemoveSubcircuit returns the list of the names and
selectors of the remaining subcircuit definitions.
See also: GlobalSubcircuits (Section 3.3.4), RemoveModelParameters (Section 3.3.9).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
3.3 Model Library Management
219
Load two model
definitions from the model
library and store them in
the global subcircuit
database.
In[2]:= Circuit[
LoadModel["Diode", "Transient", Scope −> Global],
LoadModel["JFET", "AC", Scope −> Global]
] // ExpandSubcircuits;
List the contents of the
global subcircuit database.
In[3]:= GlobalSubcircuits[]
Out[3]= Diode, DC SpiceDC Transient Spice,
JFET, AC SpiceAC Noise SpiceNoise
Removing the model definition automatically clears all alias names for the model. Thus, if you
remove the model "JFET/AC" , the aliases "JFET/SpiceAC" , "JFET/Noise" etc. will be deleted as
well.
Remove the subcircuit
definition "JFET/AC" .
In[4]:= RemoveSubcircuit["JFET", "AC"]
Delete all remaining
subcircuit definitions.
In[5]:= RemoveSubcircuit[All]
Out[4]= Diode, DC SpiceDC Transient Spice
Out[5]= 3.3.9 RemoveModelParameters
RemoveModelParameters[name]
RemoveModelParameters[All]
removes the model parameter set name from the global
database
clears the global model parameter database
Command structure of RemoveModelParameters .
RemoveModelParameters allows you to remove individual model parameter sets from the global
parameter database or to clear the database completely. RemoveModelParameters returns the list of
the names of the remaining model parameter sets.
See also: GlobalModelParameters (Section 3.3.5), RemoveSubcircuit (Section 3.3.8).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define two diode model
parameter sets.
In[2]:= Circuit[
ModelParameters[Name −> "Diode1", Scope −> Global,
IS −> 1.2*^−12, N −> 1.1],
ModelParameters[Name −> "Diode2", Scope −> Global,
IS −> 1.0*^−14, N −> 1.5]
] // ExpandSubcircuits;
List the contents of the
global parameter database.
In[3]:= GlobalModelParameters[]
Out[3]= Diode1, Diode2
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Remove the parameter set
"Diode1" .
In[4]:= RemoveModelParameters["Diode1"]
Delete all remaining global
parameter sets.
In[5]:= RemoveModelParameters[All]
Out[4]= Diode2
Out[5]= 3.4 Expanding Subcircuits and Device Models
221
3.4 Expanding Subcircuits and Device Models
Before Analog Insydes can set up a system of circuit equations from a netlist, all subcircuits and
model references in the netlist must be expanded and instantiated with ExpandSubcircuits . Subcircuit expansion is taken care of automatically by CircuitEquations , so that, except for debugging
purposes, there is usually no need to call ExpandSubcircuits explicitly.
3.4.1 ExpandSubcircuits
ExpandSubcircuits[netlist, opts]
expands and instantiates all subcircuit and model
references in netlist
Command structure of ExpandSubcircuits .
Given a Netlist or Circuit object, ExpandSubcircuits expands all model references and stores
model definitions to the global model data base. It returns a Netlist object. Expanding subcircuits is taken care of automatically by CircuitEquations , so that there is usually no need to call
ExpandSubcircuits explicitly nor to take care of its options.
ExpandSubcircuits has the following options:
option name
default value
AutoloadModels
True
whether to load device models
automatically from the model library
DefaultSelector
None
the default value to be used to replace
generic subcircuit selectors
HoldModels
None
a list of patterns matching the names and
selectors of model references that shall
remain unexpanded
InstanceNameSeparator
Inherited[AnalogInsydes]
the separator string for name components
of model variables and reference
designators (see Section 3.14.2)
KeepLocalModels
False
Options for ExpandSubcircuits , Part I.
whether to keep local model definitions in a
netlist after subcircuit expansion
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option name
default value
LibraryPath
Inherited[AnalogInsydes]
the search path for device model libraries
(see Section 3.14.3)
ModelLibrary
Inherited[AnalogInsydes]
the name of the library to be searched for
device models (see Section 3.14.4)
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
Symbolic
All
the parameters of behavioral models to be
kept symbolic when setting up symbolic
circuit equations (see Section 3.5.1)
Value
None
the parameters of behavioral models to be
kept numeric when setting up symbolic
circuit equations (see Section 3.5.1)
Options for ExpandSubcircuits , Part II.
Options Description
A detailed description of all ExpandSubcircuits options is given below in alphabetical order:
AutoloadModels
With the option AutoloadModels , you can specify how ExpandSubcircuits handles references to
device models which have not been loaded into the global subcircuit database or defined locally in
the netlist. The following values are possible:
True
False
Values for the AutoloadModels option.
search model libraries for device models and parameter
sets (model cards).
do not load models from the model library automatically;
abort subcircuit expansion when a reference to an
unavailable model is found
3.4 Expanding Subcircuits and Device Models
223
DefaultSelector
With the option DefaultSelector , you can specify the key used by ExpandSubcircuits to replace the right-hand sides of generic device model selector specifications in netlists generated with
ReadNetlist (Section 3.10.1). The setting DefaultSelector −> selector causes all generic selectors
to be replaced by the key selector. Possible values are:
None
"string"
symbol
do not use a default value for generic model selectors
use "string" as default model selector
use SymbolName[symbol] as default model selector
Values for the DefaultSelector option.
With the setting DefaultSelector −> None, ExpandSubcircuits will print an error
message and abort upon encountering a generic Selector specification.
Usually ExpandSubcircuits is automatically called by CircuitEquations which sets the
DefaultSelector option to an appropriate value based on the chosen analysis mode.
HoldModels
With HoldModels −> patterns, you can specify a list of Mathematica patterns matching the names and
selectors of model references ExpandSubcircuits should not expand. Note that the application of
the HoldModels option is only necessary for advanced tasks. Usually there is no need to use this
option.
A practical application of the HoldModels option is to stop expansion of a hierarchical circuit description at the device level. The function ReadNetlist (Section 3.10.1) makes use of this feature to
assign small-signal data read from a simulator output file to individual devices in a circuit which
contain subcircuit definitions. The following values are allowed:
pattern
do not expand model references whose name specification
matches pattern
{pattern , pattern }
do not expand model references whose name specification
matches pattern and whose selector specification matches
pattern
{{pattern , pattern }, † † † }
Values for the HoldModels option.
do not expand model references that match at least one of
the pattern pairs
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When HoldModels has a value other than None, ExpandSubcircuits returns a checked
Netlist or a Circuit instead of a flat Netlist regardless of whether any model references
were kept unresolved.
If the netlist contains references to local model definitions or model parameter sets, you
should use the setting KeepLocalModels −> True with the HoldModels option to prevent
local model data from being discarded prematurely.
In the above cases, ExpandSubcircuits must be called at least one more time to produce a
flat netlist.
KeepLocalModels
When you expand the model references in a Circuit only partially using the HoldModels option,
then the held model references may still contain unresolved links to local model definitions and model
parameter sets. By default, local model data is discarded after subcircuit expansion, so that essential
information for expanding the netlist completely may be lost. The setting KeepLocalModels −> True
tells ExpandSubcircuits to keep local model data for future reference. Note that for usual applications there is no need to change the default value of this option. Possible values are:
False
True
discard local model definitions and parameter sets from
the netlist after subcircuit expansion
return local model definitions and parameter sets together
with the expanded netlist in a Circuit object
Values for the KeepLocalModels option.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Import PSpice netlist.
In[2]:= buffer = ReadNetlist[
"AnalogInsydes/DemoFiles/Buffer.cir",
Simulator −> "PSpice"]
Out[2]= Circuit The following call to ExpandSubcircuits results in an error because the model "BJT/AC" needs to
be loaded from the model library.
Turn off model
autoloading.
In[3]:= ExpandSubcircuits[buffer, DefaultSelector −> AC,
AutoloadModels −> False]
ExpandSubcircuits::undefsc:
Subcircuit "BJT/AC" is undefined.
Out[3]= $Failed
3.4 Expanding Subcircuits and Device Models
225
With AutoloadModels −> True, ExpandSubcircuits searches the model library for undefined device
models.
Load undefined models
from the library.
In[4]:= ExpandSubcircuits[buffer, DefaultSelector −> AC,
AutoloadModels −> True, Protocol −> Notebook]
Searching model libraries for model "BJT/AC".
Loading model "BJT/AC" from library FullModels‘.
Instantiating model "BJT/AC" for reference Q$T1.
Instantiating model "BJT/AC" for reference Q$T3.
Instantiating model "BJT/AC" for reference Q$T4.
Instantiating model "BJT/AC" for reference Q$T5.
Instantiating model "BJT/AC" for reference Q$T2.
Out[4]= NetlistFlat, 55 The following command tells Analog Insydes to load a model from the library into memory and store
it in the global subcircuit database for future reference. Further requests for the model "BJT/AC"
will not invoke a library search.
Load the model "BJT/AC"
and store it in the global
subcircuit database.
In[5]:= Circuit[
LoadModel["BJT", "AC", Scope −> Global]
] // ExpandSubcircuits;
List the contents of the
global subcircuit database.
In[6]:= GlobalSubcircuits[]
Out[6]= BJT, AC SpiceAC Noise SpiceNoise
Now that the model "BJT/AC" has been loaded into memory, turning off the autoloading mechanism
no longer results in an error when expanding the netlist buffer.
Use only model definitions
from the subcircuit
database; do not search
the library.
In[7]:= ExpandSubcircuits[buffer, DefaultSelector −> AC,
AutoloadModels −> False, Protocol −> Notebook]
Instantiating model "BJT/AC" for reference Q$T1.
Instantiating model "BJT/AC" for reference Q$T3.
Instantiating model "BJT/AC" for reference Q$T4.
Instantiating model "BJT/AC" for reference Q$T5.
Instantiating model "BJT/AC" for reference Q$T2.
Out[7]= NetlistFlat, 55 Clear the subcircuit
database.
In[8]:= RemoveSubcircuit[All]
Out[8]= To illustrate how to write patterns for HoldModels , we make a list off all Model and Selector
specifications in the netlist buffer.
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226
List the Model and
Selector specifications.
In[9]:= Cases[buffer, _Model | _Selector, Infinity] // InputForm
Out[9]//InputForm= {Model["BJT", "Q2N2222", "Q$T1"],
Selector["BJT", "Q2N2222", "Q$T1"],
Model["BJT", "Q2N2907A", "Q$T3"],
Selector["BJT", "Q2N2907A", "Q$T3"],
Model["BJT", "Q2N2907A", "Q$T4"],
Selector["BJT", "Q2N2907A", "Q$T4"],
Model["BJT", "Q2N2907A", "Q$T5"],
Selector["BJT", "Q2N2907A", "Q$T5"],
Model["BJT", "Q2N2222", "Q$T2"],
Selector["BJT", "Q2N2222", "Q$T2"]}
You can use HoldModels to prevent expansion of a particular device type or model instance.
Do not expand references
to "Q2N2222" devices.
In[10]:= ExpandSubcircuits[buffer,
DefaultSelector −> AC, Protocol −> Notebook,
HoldModels −> Model[_, "Q2N2222", _]]
Searching model libraries for model "BJT/AC".
Loading model "BJT/AC" from library FullModels‘.
Instantiating model "BJT/AC" for reference Q$T3.
Instantiating model "BJT/AC" for reference Q$T4.
Instantiating model "BJT/AC" for reference Q$T5.
Out[10]= NetlistChecked, 37 If you want to flatten a netlist completely after partial expansion, you must specify
KeepLocalModels −> True to prevent local model definitions and parameter sets from being discarded by ExpandSubcircuits .
Keep the transistor
instance "Q$T3" and
references to "Q2N2222"
devices; do not discard
local model card data for
these devices.
In[11]:= xbuffer = ExpandSubcircuits[buffer,
DefaultSelector −> AC, Protocol −> Notebook,
HoldModels −> {{Model[_, "Q2N2222", _], _},
{Model[__, "Q$T3"], _}},
KeepLocalModels −> True]
Searching model libraries for model "BJT/AC".
Loading model "BJT/AC" from library FullModels‘.
Instantiating model "BJT/AC" for reference Q$T4.
Instantiating model "BJT/AC" for reference Q$T5.
Out[11]= Circuit Expand the held model
references.
In[12]:= ExpandSubcircuits[xbuffer,
DefaultSelector −> AC, Protocol −> Notebook]
Instantiating model "BJT/AC" for reference Q$T1.
Instantiating model "BJT/AC" for reference Q$T3.
Instantiating model "BJT/AC" for reference Q$T2.
Out[12]= NetlistFlat, 55 3.5 Setting Up and Solving Circuit Equations
227
3.5 Setting Up and Solving Circuit Equations
Setting up a system of equations from a netlist description of a circuit is the central step of most
circuit analysis problems you will solve with Analog Insydes. All types of equations can be set
up using the command CircuitEquations (Section 3.5.1). Additionally, there are the commands
ACEquations (Section 3.5.2) and DCEquations (Section 3.5.3) which transform a nonlinear dynamic
equation system into a linear and static equation system, respectively. In order to solve systems of
equations symbolically, Mathematica’s built-in function Solve has been extended to handle DAEObjects as well (see Section 3.5.4).
3.5.1 CircuitEquations
CircuitEquations[netlist, opts]
sets up a system of circuit equations from a netlist
description
CircuitEquations[circuit, opts]
sets up a system of circuit equations from a circuit
description
Command structure of CircuitEquations .
The first argument of CircuitEquations must be a Circuit or Netlist object. If it is not a flat
netlist, then ExpandSubcircuits will be called first automatically to expand any subcircuit or device
model references. CircuitEquations sets up modified nodal, sparse tableau, or extended sparse
tableau equations for AC, DC, or transient analysis. CircuitEquations returns a DAEObject.
The type and appearance of equations generated with CircuitEquations can be changed with the
following options:
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option name
default value
BranchCurrentPrefix
"I$"
the prefix used for branch current identifiers
BranchVoltagePrefix
"V$"
the prefix used for branch voltage identifiers
ControllingCurrentPrefix
"IC$"
the prefix used for controlling current
identifiers
ControllingVoltagePrefix
"VC$"
the prefix used for controlling voltage
identifiers
InstanceNameSeparator
Inherited[AnalogInsydes]
the separator string for name components
of model variables and reference
designators (see Section 3.14.2)
NodeVoltagePrefix
"V$"
the prefix used for node voltage identifiers
Naming convention options for CircuitEquations .
option name
default value
AnalysisMode
AC
the circuit analysis mode
FrequencyVariable
s
the symbol which denotes the complex
Laplace frequency
IndependentVariable
Automatic
the independent variable for transient
analysis
InitialGuess
Automatic
the initial guess for the DC solver
InitialTime
0
the initial time for transient analysis
TimeVariable
t
the symbol which denotes time in transient
analysis
Analysis mode specific options for CircuitEquations .
3.5 Setting Up and Solving Circuit Equations
229
option name
default value
ConvertImmittances
True
whether to convert impedances to
admittances for nodal analysis
ElementValues
Value
whether to set up numeric or symbolic
equations
Formulation
ModifiedNodal
the circuit analysis formulation
InitialConditions
None
whether to use and how to handle initial
conditions for dynamic quantities
MatrixEquation
True
whether to represent a system of AC
equations as matrix equation
Symbolic
All
the parameters of behavioral models to be
kept symbolic when setting up symbolic
circuit equations
Value
None
the parameters of behavioral models to be
kept numeric when setting up symbolic
circuit equations
Equation-formulation options for CircuitEquations .
option name
default value
DefaultSelector
Automatic
LibraryPath
Inherited[AnalogInsydes]
the search path for device model libraries
(see Section 3.14.3)
ModelLibrary
Inherited[AnalogInsydes]
the name of the library to be searched for
device models (see Section 3.14.4)
Model library options for CircuitEquations .
the default value to be used by
ExpandSubcircuits to replace generic
model selectors
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option name
default value
DesignPoint
Automatic
the design point for a symbolic DAEObject
IgnoreMissingGround
False
whether to ignore a missing analog ground
node
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
Miscellaneous options for CircuitEquations .
See also: Netlist (Section 3.1.1), Circuit (Section 3.1.2), ExpandSubcircuits (Section 3.4.1), Chapter 4.3.
Options Description
A detailed description of all CircuitEquations options is given below in alphabetical order:
AnalysisMode
The option AnalysisMode −> mode specifies the circuit analysis mode for which CircuitEquations
sets up a system of equations. Possible values are:
AC
small-signal frequency-domain (AC) analysis
DC
operating-point (DC) or DC-transfer (DT) analysis
Transient
large-signal time-domain (transient) analysis
Values for the AnalysisMode option.
With the setting AnalysisMode −> AC, CircuitEquations sets up a system of small-signal Laplacedomain equations. CircuitEquations does not linearize nonlinear circuits. Therefore, the first
argument of CircuitEquations must represent a linear circuit. If it contains references to device models, the AC equivalent circuits of the referenced device are used to set up the equations.
By default, AC equations are set up in matrix formulation, but this can be changed using the
MatrixEquation option (see below).
With the setting AnalysisMode −> DC, CircuitEquations yields a system of (usually nonlinear) DC
equations. Any time derivatives resulting from the constitutive equations of dynamic elements as
well as derivatives with respect to other independent quantities are set to zero.
3.5 Setting Up and Solving Circuit Equations
231
With AnalysisMode −> Transient , you can set up a system of differential-algebraic equations (DAE)
for nonlinear dynamic circuits. The independent variable of transient analysis is given by the value
of the option IndependentVariable , which is not necessarily time.
BranchCurrentPrefix
With
BranchCurrentPrefix −> "string", you can specify the prefix string CircuitEquations prepends to
an element’s reference designator to generate a unique branch current identifier.
For example, with BranchCurrentPrefix −> "I$", the current through a resistor R1 will be named
I$R1.
Symbol["string" <> "refdes"] must yield a valid symbol. Therefore, "string" must not
contain characters which have a special meaning in Mathematica.
BranchVoltagePrefix
With
BranchVoltagePrefix −> "string", you can specify the prefix string CircuitEquations prepends to
an element’s reference designator to generate a unique branch voltage identifier.
For example, with BranchVoltagePrefix −> "V$", the voltage across a resistor R1 will be named
V$R1.
Symbol["string" <> "refdes"] must yield a valid symbol. Therefore, "string" must not
contain characters which have a special meaning in Mathematica.
ControllingCurrentPrefix
With ControllingCurrentPrefix −> "string", you can specify the prefix string CircuitEquations
prepends to the reference designator of a current-controlled source to generate a unique identifier
for the controlling current.
For example, with ControllingCurrentPrefix −> "IC$", the controlling current of a currentcontrolled voltage source CV1 will be named IC$CV1.
Symbol["string" <> "refdes"] must yield a valid symbol. Therefore, "string" must not
contain characters which have a special meaning in Mathematica.
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ControllingVoltagePrefix
With ControllingVoltagePrefix −> "string", you can specify the prefix string CircuitEquations
prepends to the reference designator of a voltage-controlled source to generate a unique identifier
for the controlling voltage.
For example, with ControllingVoltagePrefix −> "VC$", the controlling voltage of a voltagecontrolled voltage source VV1 will be named VC$VV1.
Symbol["string" <> "refdes"] must yield a valid symbol. Therefore, "string" must not
contain characters which have a special meaning in Mathematica.
ConvertImmittances
With the option ConvertImmittances , you can control the way in which CircuitEquations handles
impedances when setting up modified nodal equations. The following values are allowed:
True
convert impedances Z to equivalent admittances Z
False
do not convert impedances to equivalent admittances
Values for the ConvertImmittances option.
For example, with ConvertImmittances −> True, a resistance R1 will appear as an admittance 1/R1
in a system of modified nodal equations. With ConvertImmittances −> False, the resistance will
not be converted to an admittance, and the MNA equations will be augmented by the branch
current I$R1.
The setting of ConvertImmittances has no influence on sparse tableau equations.
An explicit Pattern specification in a netlist entry for an impedance or admittance
overrides the setting of ConvertImmittances for this element.
DefaultSelector
With the option DefaultSelector , you can specify the key used by ExpandSubcircuits to replace the right-hand sides of generic device model selector specifications in netlists generated with
ReadNetlist (Section 3.10.1). The setting DefaultSelector −> selector causes all generic selectors
to be replaced by the key selector. Generic selector specifications have the form
Selector −> Selector["device", "type", "instance"]
for example,
3.5 Setting Up and Solving Circuit Equations
233
Selector −> Selector["BJT", "Q2N2222", "Q1"]
Possible values are:
Automatic
"string"
symbol
use value of option AnalysisMode as default model
selector for ExpandSubcircuits
use "string" as default model selector for
ExpandSubcircuits
use SymbolName[symbol] as default model selector for
ExpandSubcircuits
Values for the DefaultSelector option.
The standard setting DefaultSelector −> Automatic causes the default selector to be determined
from the value of the option AnalysisMode . This enables ExpandSubcircuits to load semiconductor
device models from the model library which are compatible with the current analysis mode.
DesignPoint
The DesignPoint option allows you to specify a set of numerical reference values for the circuit
parameters in a symbolic system of circuit equations. It allows the following values:
Automatic
extract design-point information from Value and Symbolic
specifications in the netlist
{symbol −>value , symbol −>value , † † † }
specify numerical values for symbolic circuit parameters
explicitly
None
do not extract design-point information from the netlist
Values for the DesignPoint option.
The design point is stored in the DAEObject returned by CircuitEquations . If available, designpoint information is applied automatically whenever numerical calculations are performed with a
system of equations which has been set up with ElementValues −> Symbolic . The design point can
be extracted from the DAEObject with the function GetDesignPoint (Section 3.6.12).
ElementValues
With ElementValues −> value, you can decide whether CircuitEquations sets up a system of circuit
equations symbolically or numerically. Possible choices are:
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Value
Symbolic
use Value specifications from the value fields of netlist
entries as element values
use Symbolic specifications from the value fields of netlist
entries as element values
Values for the ElementValues option.
With the settings ElementValues −> Symbolic and DesignPoint −> Automatic , CircuitEquations
will extract design-point information from a netlist automatically. The design point is stored in the
DAEObject returned by CircuitEquations . It can be retrieved with GetDesignPoint (Section 3.6.12).
Formulation
The option Formulation −> type selects the type of circuit equations set up by CircuitEquations
and allows the following values:
ModifiedNodal
modified nodal analysis (MNA)
SparseTableau
sparse tableau analysis (STA)
ExtendedTableau
extended sparse tableau analysis (ESTA)
Values for the Formulation option.
Modified nodal analysis expresses systems of circuit equations in terms of node voltages and currents
through impedance branches.
Sparse tableau analysis yields systems of circuit equations in terms of all branch currents and branch
voltages.
In addition to the branch currents and voltages, extended sparse tableau analysis also includes all
node voltages of a circuit.
To set up equations for circuits including control network elements, you must use
Formulation −> ModifiedNodal .
FrequencyVariable
With FrequencyVariable −> symbol, you can change the symbol CircuitEquations uses to denote
the complex frequency in AC equations. The default setting is FrequencyVariable −> s.
3.5 Setting Up and Solving Circuit Equations
235
IgnoreMissingGround
IgnoreMissingGround is an option needed for analyzing pure control circuits (also known as block
diagrams). Equation setup for control circuits is based on the modified nodal analysis functionality
for electrical circuits, which are expected to contain a ground node. CircuitEquations displays an
error message when you try to set up equations from a netlist in which no reference to the ground
node is present. As pure control networks do not have ground nodes, these error messages must be
suppressed by means of the IgnoreMissingGround option, which allows the following values:
False
display a warning when no ground node is present
True
ignore missing ground node when analyzing control
circuits
Values for the IgnoreMissingGround option.
You do not need to set IgnoreMissingGround −> True when a ground node is present in a
circuit containing both electrical and control components.
IndependentVariable
With IndependentVariable −> var, you can specify a variable other than time as independent
variable for transient analysis.
Note that in Analog Insydes, transient analysis does not necessarily mean analysis in the time domain. You can perform transient analysis, i.e. solving differential-algebraic equations, with respect
to any other independent variable, for instance, temperature. This feature is useful for tasks such
as parametric analyses or sizing problems with differential-algebraic constraints. Possible values for
the IndependentVariable option are:
Automatic
symbol
use value of option TimeVariable as independent variable
for transient analysis
use symbol as independent variable for transient analysis
Values for the IndependentVariable option.
If AnalysisMode −> Transient , then time-dependent variables or derivatives of circuit variables
with respect to time or other quantities are handled as follows.
3. Reference Manual
236
If the IndependentVariable is not identical to the TimeVariable , then all time derivatives
in a system of circuit equations are set to zero and all explicit references to Time are
replaced by the value of InitialTime .
If the IndependentVariable is identical to the TimeVariable , then any derivatives with
respect to other variables, such as temperature, are set to zero.
InitialConditions
The option InitialConditions allows you to specify how CircuitEquations handles initial conditions for dynamic quantities, such as capacitor voltages and inductor currents. The following values
are allowed:
Automatic
use initial conditions where given in the value field of
netlist entries
All
use initial conditions where given explicitly, set initial
conditions of all other dynamic quantities to zero
None
do not use any given initial conditions
Values for the InitialConditions option.
InitialTime
With InitialTime −> t , you can specify the point of time for which initial conditions are given. By
default, t , but you may choose any other value.
NodeVoltagePrefix
With NodeVoltagePrefix −> "string" you can specify the prefix string CircuitEquations prepends
to a node name to generate a node voltage identifier.
For example, with NodeVoltagePrefix −> "V$", the node voltage at node 1 will be named V$1.
Symbol["string" <> "node"] must yield a valid symbol. Therefore, "string" must not contain
characters which have a special meaning in Mathematica.
3.5 Setting Up and Solving Circuit Equations
237
Protocol
This option describes the standard Analog Insydes protocol specification (see Section 3.14.5). The
top-level function for the Protocol option is CircuitEquations . The second-level function is
ExpandSubcircuits . Protocol output generated by models is printed as third level.
Symbolic
The Symbolic option gives you control over the set of symbolic parameters of behavioral models
which are kept as symbolic quantities in a system of circuit equations. The Symbolic option is the
complement of the Value option and expects the following values:
All
None
{instances −> params, † † † }
keep all symbolic parameters of behavioral models in
symbolic form
replace all symbolic model parameters by numerical values
keep only the specified parameters from the given
instances in symbolic form
Values for the Symbolic option.
If the value form {instances −> params, † † † } is chosen, both instances and params can be a string
pattern or a list of string patterns. The instances specification may have the following format:
"pattern"
{"pattern ", "pattern ", † † † }
matches any reference designator "refdes" for which
StringMatchQ["refdes", "pattern"] yields True
matches any reference designator that matches at least one
of the patterns; patterns are evaluated in the given order
Format specifications for instances.
The params specification may have the following format:
symbol
"pattern"
{"pattern" | symbol, † † † }
Format specifications for params.
matches symbol literally
matches any symbol arg for which
StringMatchQ[ToString[arg], "pattern"] yields True
matches any symbol that matches at least one of the
patterns; patterns are evaluated in the given order
3. Reference Manual
238
Value takes precedence over Symbolic . Thus, a model parameter listed in the arguments of
both options will be converted to a numerical value.
Symbolic−>None is equivalent to Value−>All .
TimeVariable
With the option TimeVariable −> var, you can change the symbol which CircuitEquations uses to
denote time. By default, this symbol is used to denote the independent variable in transient analysis.
In addition, all explicit references to the identifier Time are replaced by the time variable.
Value
The Value option gives you control over the set of symbolic parameters of behavioral models which
are kept as numeric quantities in a system of circuit equations. The Value option is the complement
of the Symbolic option.
All
None
{instances −> params, † † † }
replace all symbolic parameters of behavioral models by
numerical values
keep all symbolic parameters of behavioral models in
symbolic form
replace only the specified parameters from the given
instances by numerical values
Values for the Value option.
If the value form {instances −> params, † † † } is chosen, both instances and params can be a string
pattern or a list of string patterns. The instances specification may have the following format:
"pattern"
{"pattern ", "pattern ", † † † }
matches any reference designator "refdes" for which
StringMatchQ["refdes", "pattern"] yields True
matches any reference designator that matches at least one
of the patterns; patterns are evaluated in the given order
Format specifications for instances.
The params specification may have the following format:
3.5 Setting Up and Solving Circuit Equations
symbol
"pattern"
{"pattern" | symbol, † † † }
239
matches symbol literally
matches any symbol arg for which
StringMatchQ[ToString[arg], "pattern"] yields True
matches any symbol that matches at least one of the
patterns; patterns are evaluated in the given order
Format specifications for params.
Value takes precedence over Symbolic . Thus, a model parameter listed in the arguments of
both options will be converted to a numerical value.
Value−>All is equivalent to Symbolic−>None .
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
This defines an RLC
lowpass filter circuit.
In[2]:= rlcf =
{V1,
{R1,
{L1,
{C1,
]
Netlist[
{1, 0}, V},
{1, 2}, R},
{2, 3}, L},
{3, 0}, C}
Out[2]= NetlistRaw, 4 Set up a system of AC
equations.
Set up a system of DC
equations.
In[3]:= CircuitEquations[rlcf, AnalysisMode −> AC] // DisplayForm
Out[3]//DisplayForm=
1
1
0
R
R
1
1
1
1
R
R
Ls
L s
1
1
0
L s
L s
C s
1
0
0
1
0
V$1 0
V$2
0
.
0
V$3
0
I$V1 V
0
In[4]:= CircuitEquations[rlcf, AnalysisMode −> DC] // DisplayForm
Out[4]//DisplayForm=
V$1 V$2
V$1 V$2
I$V1 0, I$L1 0, I$L1 0,
R
R
V$1 V, V$2 V$3 0, V$1, V$2, V$3, I$V1, I$L1,
DesignPoint 3. Reference Manual
240
Set up a system of
time-domain equations.
In[5]:= CircuitEquations[rlcf,
AnalysisMode −> Transient] // DisplayForm
Out[5]//DisplayForm=
V$1t V$2t
I$V1t 0,
R
V$1t V$2t
I$L1t 0, I$L1t C V$3 t 0,
R
V$1t V, V$2t V$3t L I$L1 t 0,
V$1t, V$2t, V$3t, I$V1t, I$L1t, DesignPoint Change the branch current
prefix to "ib$".
In[6]:= CircuitEquations[rlcf,
BranchCurrentPrefix −> "ib$"] // DisplayForm
Out[6]//DisplayForm=
1
1
0
R
R
1
1
1
1
R
R L s
L s
1
1
0
L s
L s
C s
0
0
1
Change the branch voltage
prefix to "ub$".
In[7]:= CircuitEquations[rlcf, Formulation −> SparseTableau,
BranchVoltagePrefix −> "ub$"] // DisplayForm
Out[7]//DisplayForm=
1 1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0 1 0
0
0
0 1 0
0
0 Cs
0
Do not convert R and L to
equivalent admittances.
1
0
V$1 0
V$2 0
.
0
V$3
0
ib$V1 V
0
0
0 0
0
0 ub$V1 0
1 1
0
0 ub$R1
0
0 1 1
0 ub$L1 0
0 0 1 1 ub$C1
.
V
0 0
0
0 I$V1
0
0 R
0
0 I$R1 0
0 0 Ls 0 I$L1
0 0
0 1 I$C1 0
In[8]:= CircuitEquations[rlcf,
ConvertImmittances −> False] // DisplayForm
Out[8]//DisplayForm=
0
0
0 1
0
0
0 0
0
0
C
s 0
1
0
0 0
0 0
1 1
0 1 1 0
1
0 V$1 0
1 1 0
V$2 0 1 0
V$3
.
0
0 V
I$V1 R
0 0
I$R1
0 L s I$L1 0
The DefaultSelector option allows you to choose a different set of device models than would be
normally used for the selected analysis mode. You can use this feature to set up circuit equations
for noise analysis. Noise equations are essentially AC equations with some extensions, that can be
included by selecting noise models instead of plain AC models using DefaultSelector −> Noise.
Read in a PSpice netlist
and small-signal data.
In[9]:= buffer = ReadNetlist[
"AnalogInsydes/DemoFiles/Buffer.cir",
"AnalogInsydes/DemoFiles/Buffer.out",
Simulator −> "PSpice"]
Out[9]= Circuit 3.5 Setting Up and Solving Circuit Equations
Set up a system of AC
equations.
241
In[10]:= CircuitEquations[buffer, AnalysisMode −> AC,
Protocol −> {None, Notebook}]
> ExpandSubcircuits...
> Searching model libraries for model "BJT/AC".
> Loading model "BJT/AC" from library FullModels‘.
> Instantiating model "BJT/AC" for reference Q$T1.
> Instantiating model "BJT/AC" for reference Q$T3.
> Instantiating model "BJT/AC" for reference Q$T4.
> Instantiating model "BJT/AC" for reference Q$T5.
> Instantiating model "BJT/AC" for reference Q$T2.
Out[10]= DAEAC, 18 18 Set up a system of AC
equations with noise
extensions.
In[11]:= CircuitEquations[buffer, AnalysisMode −> AC,
DefaultSelector −> Noise, Protocol −> {None, Notebook}]
> ExpandSubcircuits...
> Searching model libraries for model "BJT/Noise".
> Loading model "BJT/Noise" from library FullModels‘.
> Instantiating model "BJT/Noise" for reference Q$T1.
> Instantiating model "BJT/Noise" for reference Q$T3.
> Instantiating model "BJT/Noise" for reference Q$T4.
> Instantiating model "BJT/Noise" for reference Q$T5.
> Instantiating model "BJT/Noise" for reference Q$T2.
Out[11]= DAEAC, 18 18 Specify numerical
reference values for the
circuit parameters.
In[12]:= eqs = CircuitEquations[rlcf, DesignPoint −> {V −> 1.,
R −> 1000., L −> 0.001, C −> 2.2*10^−6}]
Retrieve design point from
DAEObject.
In[13]:= GetDesignPoint[eqs]
Define an RLC lowpass
filter circuit with
design-point information.
In[14]:= rlcf2 = Netlist[
{V1, {1, 0}, Symbolic
{R1, {1, 2}, Symbolic
{L1, {2, 3}, Symbolic
{C1, {3, 0}, Symbolic
]
Out[12]= DAEAC, 4 4 Out[13]= V 1., R 1000., L 0.001, C 2.2 106 −>
−>
−>
−>
V,
R,
L,
C,
Value
Value
Value
Value
−>
−>
−>
−>
1.},
1000.},
0.001},
2.2*10^−6}
Out[14]= NetlistRaw, 4 Set up a system of AC
equations with numerical
coefficients.
In[15]:= CircuitEquations[rlcf2,
ElementValues −> Value] // DisplayForm
Out[15]//DisplayForm=
0.001
0.001
1000.
0.001 0.001 s
1000.
0
s
1
0
0
1000.
s
1000.
2.2 106
s
0
1
0 V$1 0
0 V$2
.
0
V$3
s 0
I$V1 1.
0
3. Reference Manual
242
Set up a symbolic system
of AC equations.
In[16]:= eqssym = CircuitEquations[rlcf2,
ElementValues −> Symbolic]
Out[16]= DAEAC, 4 4 Show the equations.
In[17]:= eqssym // DisplayForm
Out[17]//DisplayForm=
1
1
0
R
R
1
1
1
1
R
R
Ls
L s
1
1
0
L s
L s
C s
1
0
0
1
0
V$1 0
V$2
0
.
0
V$3 0
I$V1 V
0
Get the design point from
the DAEObject.
In[18]:= GetDesignPoint[eqssym]
Set up a system of sparse
tableau equations.
In[19]:= CircuitEquations[rlcf,
Formulation −> SparseTableau] // DisplayForm
Out[18]= V 1., R 1000., L 0.001, C 2.2 106 Out[19]//DisplayForm=
1 1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0 1 0
0
0 Cs
0
Set up a system of
extended sparse tableau
equations.
0
0 0
0
0 V$V1 0
1 1
0
0 V$R1 0
0 1 1
0 V$L1
0
0 0 1 1 V$C1 .
V
0 0
0
0 I$V1
0
0 R
0
0 I$R1
0
0 0 Ls 0 I$L1 0
0 0
0 1 I$C1 In[20]:= CircuitEquations[rlcf,
Formulation −> ExtendedTableau] // DisplayForm
Out[20]//DisplayForm=
0
0 0 0
0
0 1 0
0 V$V1 0
1 0
0
0 1
0
0 0 0
0
0 1 1
0 V$R1 0
0
0
1
0
0
0
0
0
0
1
1
V$L1
0
0
0
0
1
0
0
0
0
0
0
1
V$C1
0
0 0
0
0 1 1
0
0
0
0
0 I$V1 0
.
0
0
0
0
0
1
1
0
0
0
0
I$R1
0
0
0
0
0
0
0
1
1
0
0
0
I$L1
V
1
0
0
0
0
0
0
0
0
0
0
I$C1 0
0
1
0
0
0
R
0
0
0
0
0
V$1
0
0
0
1
0
0
0
L
s
0
0
0
0
V$2
0
0
0
0
C
s
0
0
0
1
0
0
0
V$3
The default setting for the complex frequency is FrequencyVariable −> s, which is the most widely
used symbol. However, some people prefer to use the setting FrequencyVariable −> p.
3.5 Setting Up and Solving Circuit Equations
Use the symbol p to
denote the complex
frequency.
This defines a simple
control circuit with a
signal source, a forward
signal path, and a
feedback path.
243
In[21]:= CircuitEquations[rlcf,
FrequencyVariable −> p] // DisplayForm
Out[21]//DisplayForm=
1
1
0
R
R
1
1
1
1
R
L p
R
L p
1
1
0
L p
L p
C p
0
0
1
1
0
V$1 0
0
V$2
.
0
V$3
0
I$V1 V
0
In[22]:= fbloop = Netlist[
{HS1, {in}, X[Frequency]},
{HG1, {in, out}, Hf[Frequency]},
{HG2, {out, in}, Hfb[Frequency]}
]
Out[22]= NetlistRaw, 3 By default, CircuitEquations prints a warning because the netlist does not contain a reference to
the electrical ground node 0. To turn off the warning, you must set IgnoreMissingGround −> True.
Set up modified nodal
equations for the control
circuit.
In[23]:= CircuitEquations[fbloop, NodeVoltagePrefix −> "x$"]
Netlist::gndnode: Found no connections to the ground node.
Netlist::lttc: Less than two connections at node(s) {0}.
Out[23]= DAEAC, 2 2 This tells Analog Insydes
to ignore the missing
reference to ground.
In[24]:= fbleqs = CircuitEquations[fbloop,
IgnoreMissingGround −> True, NodeVoltagePrefix −> "x$"];
fbleqs // DisplayForm
Out[25]//DisplayForm=
1
Xs
Hfbs
x$in
.
1
0
Hfs
x$out
This defines a circuit with
two time-varying and
temperature-dependent
resistors.
In[26]:= tempckt = Netlist[
{V1, {1, 0}, V1},
{R1, {1, 2}, R1[Time]*(1 + TC11*TEMP)},
{R2, {2, 0}, R2[Time]*(1 + TC12*TEMP)},
{C1, {2, 0}, C1}
]
Out[26]= NetlistRaw, 4 By default, TEMP is simply regarded as a global parameter. Equations for transient analysis are
formulated with time as independent variable.
Set up circuit equations
for time-domain analysis.
In[27]:= CircuitEquations[tempckt,
AnalysisMode −> Transient] // DisplayForm
Out[27]//DisplayForm=
V$1t V$2t
I$V1t 0,
1 TC11 TEMP R1t
V$2t
V$1t V$2t
C1 V$2 t 0,
1 TC12 TEMP R2t
1 TC11 TEMP R1t
V$1t V1, V$1t, V$2t, I$V1t,
DesignPoint 3. Reference Manual
244
Now we use TEMP as independent variable. This causes the time derivative due to C1 to be set to
zero and Time to be replaced by the value of InitialTime .
Designate temperature as
independent variable.
In[28]:= CircuitEquations[tempckt, AnalysisMode −> Transient,
IndependentVariable −> TEMP] // DisplayForm
Out[28]//DisplayForm=
V$1TEMP V$2TEMP
I$V1TEMP 0,
1 TC11 TEMP R10
V$2TEMP
V$1TEMP V$2TEMP
0,
1 TC12 TEMP R20
1 TC11 TEMP R10
V$1TEMP V1, V$1TEMP, V$2TEMP, I$V1TEMP,
DesignPoint This assigns an initial
condition to the inductor
L1.
In[29]:= rlcfic = Netlist[
{V1, {1, 0}, V},
{R1, {1, 2}, R},
{L1, {2, 3}, Value −> L, InitialCondition −> iL0},
{C1, {3, 0}, C}
]
Out[29]= NetlistRaw, 4 By default, initial conditions are ignored by CircuitEquations . To use the given initial conditions,
you must specify InitialConditions −> Automatic or InitialConditions −> All.
Set up circuit equations
for transient analysis.
In[30]:= CircuitEquations[rlcfic,
AnalysisMode −> Transient] // DisplayForm
Out[30]//DisplayForm=
V$1t V$2t
I$V1t 0,
R
V$1t V$2t
I$L1t 0, I$L1t C V$3 t 0,
R
V$1t V, V$2t V$3t L I$L1 t 0,
V$1t, V$2t, V$3t, I$V1t, I$L1t, DesignPoint Use initial conditions only
where specified explicitly.
In[31]:= CircuitEquations[rlcfic, AnalysisMode −> Transient,
InitialConditions −> Automatic] // DisplayForm
Out[31]//DisplayForm=
V$1t V$2t
I$V1t 0,
R
V$1t V$2t
I$L1t 0, I$L1t C V$3 t 0,
R
V$1t V, V$2t V$3t L I$L1 t 0, I$L10 iL0,
V$1t, V$2t, V$3t, I$V1t, I$L1t, DesignPoint 3.5 Setting Up and Solving Circuit Equations
Use given initial
conditions, set all other
initial conditions to zero.
245
In[32]:= CircuitEquations[rlcfic, AnalysisMode −> Transient,
InitialConditions −> All] // DisplayForm
Out[32]//DisplayForm=
V$1t V$2t
V$1t V$2t
I$V1t 0, I$L1t 0,
R
R
I$L1t C V$3 t 0, V$1t V,
V$2t V$3t L I$L1 t 0, I$L10 iL0, V$30 0,
V$1t, V$2t, V$3t, I$V1t, I$L1t, DesignPoint Use the symbol t0 to
denote the initial time.
In[33]:= CircuitEquations[rlcfic, AnalysisMode −> Transient,
InitialConditions −> All, InitialTime −> t0
] // DisplayForm
Out[33]//DisplayForm=
V$1t V$2t
V$1t V$2t
I$V1t 0, I$L1t 0,
R
R
I$L1t C V$3 t 0, V$1t V,
V$2t V$3t L I$L1 t 0, I$L1t0 iL0, V$3t0 0,
V$1t, V$2t, V$3t, I$V1t, I$L1t, DesignPoint Change the node voltage
prefix to "vn$".
In[34]:= CircuitEquations[rlcf,
NodeVoltagePrefix −> "vn$"] // DisplayForm
Out[34]//DisplayForm=
1
1
0
R
R
1
1
1
1
R
R L s
L s
1
1
0
L s
L s
C s
0
0
1
1
vn$1 0
0
vn$2
0
.
0
vn$3
0
I$V1 V
0
With the default settings
Symbolic −> All and Value −> None, all device parameters of the diode appear in symbolic form
when you set up a system of circuit equations with ElementValues −> Symbolic.
This defines a simple
half-wave diode rectifier
circuit.
In[35]:= rectifier = Netlist[
{V1, {1, 0}, Symbolic −> Vin,
Value −> 5*Sin[2*Pi*50*Time]},
{D1, {1 −> A, 2 −> C}, Model −> "Diode",
Selector −> Selector["Diode", "Spice", "D1"]},
{RL, {2, 0}, Symbolic −> RL, Value −> 1000.0},
{CL, {2, 0}, Symbolic −> CL, Value −> 2.2*^−5}
]
Out[35]= NetlistRaw, 4 3. Reference Manual
246
Set up symbolic circuit
equations for transient
analysis.
In[36]:= CircuitEquations[rectifier, AnalysisMode −> Transient,
ElementValues −> Symbolic] // DisplayForm
Out[36]//DisplayForm=
V$2t
I$AC$D1t I$V1t 0, I$AC$D1t CL V$2 t 0,
RL
TEMP
1.11 1. TNOM
$q
TEMP $k
V$1t Vin, I$AC$D1t AREA$D1 E 3.
1. $q V$1tV$2t
TEMP
TEMP $k
1 E IS$D1 GMIN V$1t V$2t,
TNOM
V$1t, V$2t, I$V1t, I$AC$D1t,
DesignPoint Vin 5 Sin100 Π t, RL 1000., CL 0.000022,
AREA$D1 1., BV$D1 , CJO$D1 0, GMIN 1. 1012 ,
IBV$D1 0.001, IS$D1 1. 1014 , RS$D1 0, TEMP 300.15,
TNOM 300.15, $k 1.38062 1023 , $q 1.60219 1019 You can also tell CircuitEquations to treat all device parameters as numeric quantities.
Replace all parameters in
device model equations by
numerical values.
In[37]:= CircuitEquations[rectifier, AnalysisMode −> Transient,
ElementValues −> Symbolic, Symbolic −> None
] // DisplayForm
Out[37]//DisplayForm=
V$2t
I$AC$D1t I$V1t 0, I$AC$D1t CL V$2 t 0,
RL
V$1t Vin, I$AC$D1t 1. 1014 1 E38.6635 V$1tV$2t 1. 1012 V$1t V$2t,
V$1t, V$2t, I$V1t, I$AC$D1t,
DesignPoint Vin 5 Sin100 Π t, RL 1000., CL 0.000022
Finally, the Symbolic option allows you to specify for each device or group of devices the parameters
which CircuitEquations should keep in symbolic form.
Keep only the parameters
AREA and TEMP of all
diode instances "D*" in
symbolic form, replace all
other model parameters by
numerical values.
In[38]:= CircuitEquations[rectifier,
AnalysisMode −> Transient, ElementValues −> Symbolic,
Symbolic −> {"D*" −> {AREA, TEMP}}] // DisplayForm
Out[38]//DisplayForm=
I$AC$D1t I$V1t 0,
V$2t
I$AC$D1t CL V$2 t 0, V$1t Vin,
RL
12881.4 1.0.00333167 TEMP
TEMP
I$AC$D1t 3.69815 1022 AREA$D1 E 11604.8 V$1tV$2t
TEMP
1 E TEMP3. 1. 1012 V$1t V$2t,
V$1t, V$2t, I$V1t, I$AC$D1t,
DesignPoint Vin 5 Sin100 Π t, RL 1000., CL 0.000022,
AREA$D1 1., TEMP 300.15
3.5 Setting Up and Solving Circuit Equations
Use the symbol Τ to
denote time.
247
In[39]:= CircuitEquations[tempckt, AnalysisMode −> Transient,
TimeVariable −> \[Tau]] // DisplayForm
Out[39]//DisplayForm=
V$1Τ V$2Τ
I$V1Τ 0,
1 TC11 TEMP R1Τ
V$2Τ
V$1Τ V$2Τ
C1 V$2 Τ 0,
1 TC12 TEMP R2Τ
1 TC11 TEMP R1Τ
V$1Τ V1, V$1Τ, V$2Τ, I$V1Τ,
DesignPoint With the default settings
Symbolic −> All and Value −> None, all device parameters of the diode appear in symbolic form
when you set up a system of circuit equations with ElementValues −> Symbolic.
Set up a system of
symbolic circuit equations
for transient analysis.
In[40]:= CircuitEquations[rectifier, AnalysisMode −> Transient,
ElementValues −> Symbolic] // DisplayForm
Out[40]//DisplayForm=
V$2t
I$AC$D1t I$V1t 0, I$AC$D1t CL V$2 t 0,
RL
TEMP
1.11 1. TNOM
$q
TEMP $k
V$1t Vin, I$AC$D1t AREA$D1 E 3.
1. $q V$1tV$2t
TEMP
TEMP $k
1 E IS$D1 GMIN V$1t V$2t,
TNOM
V$1t, V$2t, I$V1t, I$AC$D1t,
DesignPoint Vin 5 Sin100 Π t, RL 1000., CL 0.000022,
AREA$D1 1., BV$D1 , CJO$D1 0, GMIN 1. 1012 ,
IBV$D1 0.001, IS$D1 1. 1014 , RS$D1 0, TEMP 300.15,
TNOM 300.15, $k 1.38062 1023 , $q 1.60219 1019 You can also tell CircuitEquations to treat all device parameters as numeric quantities.
Replace all parameters in
device model equations by
numerical values.
In[41]:= CircuitEquations[rectifier, AnalysisMode −> Transient,
ElementValues −> Symbolic, Value −> All] // DisplayForm
Out[41]//DisplayForm=
V$2t
I$AC$D1t I$V1t 0, I$AC$D1t CL V$2 t 0,
RL
V$1t Vin, I$AC$D1t 1. 1014 1 E38.6635 V$1tV$2t 1. 1012 V$1t V$2t,
V$1t, V$2t, I$V1t, I$AC$D1t,
DesignPoint Vin 5 Sin100 Π t, RL 1000., CL 0.000022
Finally, the Value option allows you to specify for each device or group of devices the parameters
which CircuitEquations should treat as numeric quantities.
3. Reference Manual
248
Replace the parameters
TEMP, TNOM, $k, and $q of
the model instance D1 by
their numerical values.
In[42]:= CircuitEquations[rectifier,
AnalysisMode −> Transient, ElementValues −> Symbolic,
Value −> {"D1" −> {TEMP, TNOM, $k, $q}}] // DisplayForm
Out[42]//DisplayForm=
V$2t
I$AC$D1t I$V1t 0, I$AC$D1t CL V$2 t 0,
RL
V$1t Vin, I$AC$D1t 1. AREA$D1
1 E38.6635 V$1tV$2t IS$D1 GMIN V$1t V$2t,
V$1t, V$2t, I$V1t, I$AC$D1t,
DesignPoint Vin 5 Sin100 Π t,
RL 1000., CL 0.000022, AREA$D1 1., BV$D1 , CJO$D1 0,
GMIN 1. 1012 , IBV$D1 0.001, IS$D1 1. 1014 , RS$D1 0
3.5.2 ACEquations
ACEquations[dae, oppoint, sources]
linearizes equations about oppoint, replaces sources by
values (specified via sources), and returns an AC DAEObject
ACEquations[dae, oppoint]
linearizes equations using the AC source specifiations
given by the ModeValues option stored in dae
Command structure of ACEquations.
Given a Transient or DC DAEObject, ACEquations linearizes the equation system about the given
operating point oppoint and returns an AC DAEObject. The operating point is given by a list of rules
of the form var −> value for each variable of the equation system. It is possible to assign values to
time derivatives of variables, too. The list of value mappings is added to the DesignPoint option
of the returned DAEObject. During linearization, the values of AC sources are replaced by their AC
value as given by sources, where sources is a list of rules of the form acsource −> val for each parameter representing an AC source. If this third argument is omitted, the AC values are automatically
extracted from the ModeValues option stored in the DAEObject.
Note that, given a Netlist or Circuit object, you can set up AC equations directly using
CircuitEquations (Section 3.5.1).
ACEquations provides the following options:
3.5 Setting Up and Solving Circuit Equations
249
option name
default value
FrequencyVariable
Automatic
specifies the symbol denoting the complex
Laplace frequency
OperatingPointPostfix
"$op"
string to append to variable names
denoting symbolic operating points
DerivativePostfix
"d"
string to append to variable names
denoting derivatives
Options for ACEquations.
See also: DCEquations (Section 3.5.3), CircuitEquations (Section 3.5.1).
Options Description
A detailed description of all ACEquations options is given below in alphabetical order:
DerivativePostfix
If symbol names denoting derivates of varibles have to be created during linearization, the value
of the DerivativePostfix option is appended to the variable name. Thus, the value of the
DerivativePostfix option has to be a string which can be converted to a valid Mathematica
symbol. The default setting is DerivativePostfix −> "d".
FrequencyVariable
The option FrequencyVariable specifies the symbol denoting the complex Laplace frequency. If set
to Automatic , the value of the option FrequencyVariable as stored in the DAEObject dae is used.
The default setting is FrequencyVariable −> Automatic .
OperatingPointPostfix
If the given operating point list oppoint contains a rule of the form var −> value, the rule opvar −> value
is added to the DesignPoint option of the returned DAEObject, where opvar is given by appending
the value of the OperatingPointPostfix option to the variable name var. Thus, the value of the
OperatingPointPostfix option has to be a string which can be converted to a valid Mathematica
symbol. The default setting is OperatingPointPostfix −> "$op".
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
3. Reference Manual
250
Define netlist description
of a simple diode rectifier
circuit.
In[2]:= rectifier = Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −>2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C},
Model −> "Diode", Selector −> "Spice",
GMIN −> 0, AREA −> 1}
]
]
Out[2]= Circuit Set up transient equation
system.
In[3]:= tran = CircuitEquations[rectifier,
ElementValues −> Symbolic,
AnalysisMode −> Transient,
Value −> {"*" −> {TEMP, TNOM, $q, $k, AREA, GMIN}}]
Out[3]= DAETransient, 4 4 Show equation system.
In[4]:= tran // DisplayForm
Out[4]//DisplayForm=
V$2t
I$AC$D1t I$V0t 0, I$AC$D1t C1 V$2 t 0,
R1
V$1t V0, I$AC$D1t 1. 1 E38.6635 V$1tV$2t IS$D1,
V$1t, V$2t, I$V0t, I$AC$D1t, DesignPoint V0 2. Sin1000000 t, R1 100., C1 1. 107 , BV$D1 ,
CJO$D1 0, IBV$D1 0.001, IS$D1 1. 1014 , RS$D1 0
Calculate operating point.
In[5]:= op = First[NDAESolve[tran, {t, 0.}]]
Out[5]= V$2 0., V$1 0., I$V0 0., I$AC$D1 0.
Linearize equations about
the operating point.
In[6]:= ac = ACEquations[tran, op, {V0 −> 1}]
Extract equation system.
In[7]:= GetEquations[ac]
Out[6]= DAEAC, 4 4 Out[7]=
1
I$AC$D1 I$V0 0, I$AC$D1 C1 s V$2 0,
R1
V$1 V0, I$AC$D1 38.6635 E38.6635 V$1$opV$2$op IS$D1 V$1 38.6635 E38.6635 V$1$opV$2$op IS$D1 V$2 0
Display design point
stored in the DAEObject.
Note that new parameters
have been generated.
In[8]:= GetDesignPoint[ac]
Out[8]=
R1 100., C1 1. 107 , BV$D1 , CJO$D1 0, IBV$D1 0.001,
IS$D1 1. 1014 , RS$D1 0, V0 1, V$1$op 0., V$2$op 0.,
I$V0$op 0., I$AC$D1$op 0., V0$op 2. Sin1000000 t$op
3.5 Setting Up and Solving Circuit Equations
251
3.5.3 DCEquations
DCEquations[dae]
transforms a dynamic Transient DAEObject into a static
DC DAEObject
Command structure of DCEquations.
Given a Transient DAEObject, DCEquations transforms the dynamic equations into a static system
and returns a DC DAEObject. All derivatives with respect to the IndependentVariable (as stored
in the DAEObject) are replaced by zero. The initial time is inserted into both the equations and the
design point. Depending on the value of the ModeValues option, the values of independent sources
stored in the DesginPoint option of the DAEObject are replaced by the corresponding DC values
stored in the ModeValues option of the DAEObject.
Note that, given a Netlist or Circuit object, you can set up DC equations directly using
CircuitEquations (Section 3.5.1).
DCEquations provides the following options:
option name
default value
InitialTime
Automatic
initial time to insert into the equation and
the design point
ModeValues
Automatic
whether to extract independent source
values from the ModeValues option of the
DAEObject
Options for DCEquations.
See also: ACEquations (Section 3.5.2), CircuitEquations (Section 3.5.1).
Options Description
InitialTime
The option InitialTime specifies the value (which can be any Mathematica expression) to be substituted for the independent variable in the equation system and the design point. If the value of the
InitialTime option is set to Automatic , the initial time is extracted from the InitialTime option
stored in the DAEObject. The default setting is InitialTime −> Automatic .
ModeValues
The option ModeValues specifies whether values of independent sources should be updated in the
DesignPoint option of the returned DAEObject. If the value of the ModeValues option is set to
3. Reference Manual
252
Automatic , DC values of independent sources are extracted from the ModeValues option stored in the
DAEObject. If the value of the ModeValues option is set to None, the values of independent sources
are used as-is. Note that in both cases the initial time is inserted into the design-point values. The
default setting is ModeValues −> Automatic .
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define netlist description
of a simple diode rectifier
circuit.
In[2]:= rectifier = Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −>2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C},
Model −> "Diode", Selector −> "Spice",
GMIN −> 0, AREA −> 1}
]
]
Out[2]= Circuit Set up transient equation
system.
In[3]:= tran = CircuitEquations[rectifier,
ElementValues −> Symbolic,
AnalysisMode −> Transient,
Value −> {"*" −> {TEMP, TNOM, $q, $k, AREA, GMIN}}]
Out[3]= DAETransient, 4 4 Show equation system.
In[4]:= tran // DisplayForm
Out[4]//DisplayForm=
V$2t
I$AC$D1t I$V0t 0, I$AC$D1t C1 V$2 t 0,
R1
V$1t V0, I$AC$D1t 1. 1 E38.6635 V$1tV$2t IS$D1,
V$1t, V$2t, I$V0t, I$AC$D1t, DesignPoint V0 2. Sin1000000 t, R1 100., C1 1. 107 , BV$D1 ,
CJO$D1 0, IBV$D1 0.001, IS$D1 1. 1014 , RS$D1 0
Transform dynamic
equations into a static
system.
Show equation system.
In[5]:= dc = DCEquations[tran]
Out[5]= DAEDC, 4 4 In[6]:= dc // DisplayForm
Out[6]//DisplayForm=
V$2
I$AC$D1 I$V0 0, I$AC$D1 0, V$1 V0, I$AC$D1 R1
1. 1 E38.6635 V$1V$2 IS$D1, V$1, V$2, I$V0, I$AC$D1,
DesignPoint V0 0, R1 100., C1 1. 107 , BV$D1 ,
CJO$D1 0, IBV$D1 0.001, IS$D1 1. 1014 , RS$D1 0
3.5 Setting Up and Solving Circuit Equations
253
3.5.4 Solve
Solve[dae]
Solve[dae, var]
Solve[dae, {var , var , † † † }]
solves a system of equations given by dae for all variables
solves for one variable
solves for several variables
Command structure of Solve.
In order to symbolically solve systems of equations formulated by the function CircuitEquations ,
Mathematica’s built-in function Solve has been extended to handle DAEObjects as well. Solve returns its result as lists of rules of the form {{var −> expr }, † † † }, where the variables var are assigned
symbolic expressions expr.
See also: CircuitEquations (Section 3.5.1), ACAnalysis (Section 3.7.3), NDAESolve (Section 3.7.5).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define netlist description.
In[2]:= vdiv =
{V0,
{R1,
{R2,
]
Netlist[
{1, 0}, V0},
{1, 2}, R1},
{2, 0}, R2}
Out[2]= NetlistRaw, 3 Set up a system of
small-signal equations.
In[3]:= vdiveqs = CircuitEquations[vdiv]
Display equations.
In[4]:= DisplayForm[vdiveqs]
Out[3]= DAEAC, 3 3 Out[4]//DisplayForm=
1
1
1
V$1 R1
R1
0 1
1
1
0 .
V$2
0
R1
R2
R1
V0
I$V0
0
0
1
Solve only for the variable
V$2.
In[5]:= Solve[vdiveqs, V$2]
Solve for the variables V$1
and V$2.
In[6]:= Solve[vdiveqs, {V$1, V$2}]
Solve for all variables.
In[7]:= solution = Solve[vdiveqs]
R2 V0
Out[5]= V$2 R1 R2
R2 V0
Out[6]= V$2 , V$1 V0
R1 R2
V0
R2 V0
Out[7]= I$V0 , V$2 , V$1 V0
R1 R2
R1 R2
3. Reference Manual
254
Extract value of V$2 from
the list of solutions.
In[8]:= V$2 /. First[solution]
R2 V0
Out[8]= R1 R2
3.6 Circuit and DAEObject Manipulation
255
3.6 Circuit and DAEObject Manipulation
This chapter describes a number of functions which simplify the usage of the key objects existing
in Analog Insydes, the Netlist object, the Circuit object, and the DAEObject. These functions
allow you to manipulate the objects without knowledge of the internal data format. The following
functions are available:
AddElements (Section 3.6.1)
DeleteElements (Section 3.6.2)
inserting netlist elements
deleting netlist elements
GetElements (Section 3.6.3)
extracting netlist elements
RenameNodes (Section 3.6.4)
renaming node identifiers
Statistics (Section 3.6.17)
showing contents of a Circuit object
Functions dealing with Netlist and Circuit objects.
GetEquations (Section 3.6.5)
GetMatrix (Section 3.6.6)
GetVariables (Section 3.6.7)
GetRHS (Section 3.6.8)
GetParameters (Section 3.6.9)
extracting equations
extracting matrix
extracting variables
extracting right-hand side vector
extracting parameters
GetDAEOptions (Section 3.6.10)
extracting options
SetDAEOptions (Section 3.6.11)
setting and modifying options
GetDesignPoint (Section 3.6.12)
extracting design point
ApplyDesignPoint (Section 3.6.13)
inserting design point
UpdateDesignPoint (Section 3.6.14)
modifying design point
MatchSymbols (Section 3.6.15)
ShortenSymbols (Section 3.6.16)
Statistics (Section 3.6.17)
Functions dealing with DAEObjects.
matching symbol names
shortening symbol names
showing contents of a DAEObject
3. Reference Manual
256
3.6.1 AddElements
AddElements[netlist, element]
adds netlist entry element to netlist
AddElements[circuit, element]
adds netlist entry element to the top-level netlist of circuit
AddElements[circuit, element, subcirspec]
adds netlist entry to subcircuits specified by subcirspec
Command structure of AddElements.
Given a Netlist or Circuit object, AddElements adds a new netlist element to the top-level netlist
or to subcircuits defined in the circuit and returns the new Netlist or Circuit object, respectively.
The subcircuits are given by means of the subcircuit specification subcirspec which is a sequence of
rules of the following form:
sequence
description
Name−>stringpattern , Selector−>stringpattern
adds netlist entry to subcircuits whose Name matches
stringpattern and whose Selector matches stringpattern
Name−>stringpattern
equivalent to Name−>stringpattern, Selector−>"*"
Selector−>stringpattern
equivalent to Name−>"*", Selector−>stringpattern
Possible subcircuit specifications.
The new netlist element is added to all subcircuits, whose Name and Selector fields match the string
pattern given by the subcircuit specification.
Note that there is no check whether the returned circuit defines a valid netlist.
See also: DeleteElements (Section 3.6.2), GetElements (Section 3.6.3).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define a simple voltage
divider circuit.
In[2]:= net =
Netlist[
{V0, {1, 0}, V0},
{R1, {1, 2}, R1},
{R2, {2, 0}, R2}
]
Out[2]= NetlistRaw, 3 Add a load resistance to
the voltage divider.
In[3]:= netload = AddElements[net, {Rload, {2, 0}, Rload}]
Out[3]= NetlistRaw, 4 3.6 Circuit and DAEObject Manipulation
Display new netlist.
257
In[4]:= DisplayForm[netload]
Out[4]//DisplayForm= Netlist Raw, 4 Entries:
V0, 1, 0, V0
R1, 1, 2, R1
R2, 2, 0, R2
Rload, 2, 0, Rload
3.6.2 DeleteElements
DeleteElements[netlist, stringpattern]
removes those netlist entries from netlist whose reference
designator matches stringpattern
DeleteElements[circuit, stringpattern]
removes those netlist entries from the top-level netlist of
circuit whose reference designator matches stringpattern
DeleteElements[circuit, stringpattern, subcirspec]
removes netlist entries in subcircuits specified by subcirspec
Command structure of DeleteElements .
Given a Netlist or Circuit object, DeleteElements removes netlist elements from the top-level
netlist or from subcircuits defined in the circuit and returns the new Netlist or Circuit object,
respectively. The argument stringpattern is either a single string pattern or a list of string patterns. Netlist elements whose reference designator matches any of the string patterns are removed.
To delete elements in subcircuits, referencing to certain subcircuits is performed by means of the
subcircuit specification subcirspec which is a sequence of rules of the following form:
sequence
description
Name−>stringpattern , Selector−>stringpattern
deletes netlist entries in subcircuits whose Name matches
stringpattern and whose Selector matches stringpattern
Name−>stringpattern
equivalent to Name−>stringpattern, Selector−>"*"
Selector−>stringpattern
equivalent to Name−>"*", Selector−>stringpattern
Possible subcircuit specifications.
Note that there is no check whether the returned Circuit object defines a valid circuit.
See also: AddElements (Section 3.6.1), GetElements (Section 3.6.3).
3. Reference Manual
258
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define a simple voltage
divider with load
resistance.
In[2]:= netload =
Netlist[
{V0, {1, 0}, V0},
{R1, {1, 2}, R1},
{R2, {2, 0}, R2},
{Rload, {2, 0}, Rload}
]
Out[2]= NetlistRaw, 4 Remove all elements
whose reference designator
match the string pattern
"*load" .
Display new netlist.
In[3]:= net = DeleteElements[netload, "*load"]
Out[3]= NetlistRaw, 3 In[4]:= DisplayForm[net]
Out[4]//DisplayForm= Netlist Raw, 3 Entries:
V0, 1, 0, V0
R1, 1, 2, R1
R2, 2, 0, R2
3.6.3 GetElements
GetElements[netlist, stringpattern]
extracts those netlist entries from netlist whose reference
designator matches stringpattern
GetElements[circuit, stringpattern]
extracts those netlist entries from the top-level netlist of
circuit whose reference designator matches stringpattern
Command structure of GetElements.
Given a Netlist or Circuit object, GetElements extracts netlist elements from the top-level netlist
and returns a list of these elements. The argument stringpattern is either a single string pattern or a
list of string patterns. Netlist elements whose reference designator matches any of the string patterns
are extracted.
See also: AddElements (Section 3.6.1), DeleteElements (Section 3.6.2).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
3.6 Circuit and DAEObject Manipulation
Define a simple voltage
divider.
259
In[2]:= netload =
Netlist[
{V0, {1, 0}, V0},
{R1, {1, 2}, R1},
{R2, {2, 0}, R2},
{Rload, {2, 0}, Rload}
]
Out[2]= NetlistRaw, 4 Extract all elements whose
reference designator match
the string pattern "R*".
In[3]:= GetElements[netload, "R*"]
Out[3]=
R1, 1, 2, R1, R2, 2, 0, R2, Rload, 2, 0, Rload
3.6.4 RenameNodes
RenameNodes[netlist, repl]
renames nodes in netlist
RenameNodes[circuit, repl]
renames nodes in the top-level netlist of circuit
Command structure of RenameNodes.
Given a Netlist or Circuit object, RenameNodes changes node names in the top-level netlist and
returns the new Netlist or Circuit object, respectively. The replacements repl are given as a single
rule, a sequence of rules, or as a list of replacement rules of the form oldnodename −> newnodename.
The node names can be given as symbols or as strings.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define a simple voltage
divider.
In[2]:= net =
Netlist[
{V0, {1, 0}, V0},
{R1, {1, 2}, R1},
{R2, {2, 0}, R2}
]
Out[2]= NetlistRaw, 3 Show variable names.
In[3]:= GetVariables[CircuitEquations[net]]
Out[3]= V$1, V$2, I$V0
The variable of interest – the voltage at node 2 – is called V$2. To mark this variable as the output
variable we rename node 2 to OUT. Then the output variable is called V$OUT.
Rename node 2 to OUT.
In[4]:= net2 = RenameNodes[net, {2 −> OUT}]
Out[4]= NetlistRaw, 3 3. Reference Manual
260
Display netlist.
In[5]:= DisplayForm[net2]
Out[5]//DisplayForm= Netlist Raw, 3 Entries:
V0, 1, 0, V0
R1, 1, OUT, R1
R2, OUT, 0, R2
Show variable names.
Now the variable of
interest is called V$OUT.
In[6]:= GetVariables[CircuitEquations[net2]]
Out[6]= V$1, V$OUT, I$V0
3.6.5 GetEquations
GetEquations[dae]
returns the list of equations stored in dae
Command structure of GetEquations.
Given a DAEObject dae, GetEquations returns the equation system stored in the object as a list of
equations. In case of an AC DAEObject the internal matrix representation is converted to a list of
linear equations.
See also: GetMatrix (Section 3.6.6), GetVariables (Section 3.6.7), GetRHS (Section 3.6.8),
GetParameters (Section 3.6.9), GetDAEOptions (Section 3.6.10), GetDesignPoint (Section 3.6.12).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define netlist description
of a simple diode rectifier
circuit.
In[2]:= cir =
Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −> 2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C}, Model −> "Diode"}
]
]
Out[2]= Circuit Set up Transient DAE
system.
In[3]:= dae = CircuitEquations[cir, AnalysisMode −> Transient,
DefaultSelector −> "Spice", ElementValues −> Symbolic]
Out[3]= DAETransient, 4 4 3.6 Circuit and DAEObject Manipulation
Return list of equations.
261
In[4]:= GetEquations[dae]
Out[4]=
I$AC$D1t I$V0t 0,
V$2t
I$AC$D1t C1 V$2 t 0, V$1t V0, I$AC$D1t R1
TEMP
1.11 1. 1. $q V$1tV$2t
TEMP 3.
TNOM
$q
TEMP $k
TEMP $k
1 E IS$D1 AREA$D1 E TNOM
GMIN V$1t V$2t
Set up AC DAE system.
In[5]:= daeac = CircuitEquations[cir, AnalysisMode −> AC,
DefaultSelector −> "SpiceAC", ElementValues −> Symbolic]
Out[5]= DAEAC, 3 3 The system is set up as a
matrix equation.
Return value is again a
list of equations.
In[6]:= DisplayForm[daeac]
Out[6]//DisplayForm=
1
Rd$D1 Cd$D1 s
1
Cd$D1 s
Rd$D1
1
1
Rd$D1 Cd$D1 s
1
0 V$1 1
1
0 .
V$2
C1
s
Cd$D1
s
0
R1
Rd$D1
V0
I$V0
0
0
In[7]:= GetEquations[daeac]
Out[7]=
1
1
I$V0 Cd$D1 s V$1 Cd$D1 s V$2 0,
Rd$D1
Rd$D1
1
1
1
Cd$D1 s V$1 C1 s Cd$D1 s V$2 0,
Rd$D1
R1
Rd$D1
V$1 V0
3.6.6 GetMatrix
GetMatrix[dae]
returns the matrix equations stored in dae
Command structure of GetMatrix.
Given a DAEObject dae, GetMatrix returns the matrix A of the linear equation system A x b
stored in the object.
Please note that GetMatrix is only valid for AC DAEObjects.
See also: GetEquations (Section 3.6.5), GetVariables (Section 3.6.7), GetRHS (Section 3.6.8),
GetParameters (Section 3.6.9), GetDAEOptions (Section 3.6.10), GetDesignPoint (Section 3.6.12).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
3. Reference Manual
262
Define a simple voltage
divider netlist.
In[2]:= net =
Netlist[
{V0, {1, 0}, V0},
{R1, {1, 2}, R1},
{R2, {2, 0}, R2}
]
Out[2]= NetlistRaw, 3 Set up AC DAE system.
In[3]:= dae = CircuitEquations[net]
Out[3]= DAEAC, 3 3 Return matrix A.
In[4]:= GetMatrix[dae] // MatrixForm
1
R1
1
Out[4]//MatrixForm= R1
1
1
R1
1
R1
0
1
R2
1
0
0
3.6.7 GetVariables
GetVariables[dae]
returns the list of variables stored in dae
Command structure of GetVariables.
Given a DAEObject dae, GetVariables returns the list of variables stored in the object. For AC
and DC DAEObjects this is a list of symbols, for Transient DAEObjects this is a list of the form
symbt where symb is a symbol and t is the independent variable.
See also: GetEquations (Section 3.6.5), GetMatrix (Section 3.6.6), GetRHS (Section 3.6.8),
GetParameters (Section 3.6.9), GetDAEOptions (Section 3.6.10), GetDesignPoint (Section 3.6.12).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define netlist description
of a simple diode rectifier
circuit.
In[2]:= cir =
Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −> 2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C}, Model −> "Diode"}
]
]
Out[2]= Circuit Set up AC DAE system.
In[3]:= dae = CircuitEquations[cir, AnalysisMode −> AC,
DefaultSelector −> "SpiceAC"]
Out[3]= DAEAC, 3 3 3.6 Circuit and DAEObject Manipulation
Return list of variables
stored in dae.
In[4]:= GetVariables[dae]
Set up Transient DAE
system.
In[5]:= daetran = CircuitEquations[cir,
AnalysisMode −> Transient, DefaultSelector −> "Spice"]
263
Out[4]= V$1, V$2, I$V0
Out[5]= DAETransient, 4 4 Return list of variables
stored in daetran.
In[6]:= GetVariables[daetran]
Out[6]= V$1t, V$2t, I$V0t, I$AC$D1t
3.6.8 GetRHS
GetRHS[dae]
returns the right-hand side source vector stored in dae
Command structure of GetRHS.
Given a DAEObject dae, GetRHS returns the right-hand side source vector b of the linear equation
system A x b stored in the object.
Please note that GetRHS is only valid for AC DAEObjects.
See also: GetEquations (Section 3.6.5), GetMatrix (Section 3.6.6), GetVariables (Section 3.6.7),
GetParameters (Section 3.6.9), GetDAEOptions (Section 3.6.10), GetDesignPoint (Section 3.6.12).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define a simple voltage
divider netlist.
In[2]:= net =
Netlist[
{V0, {1, 0}, V0},
{R1, {1, 2}, R1},
{R2, {2, 0}, R2}
]
Out[2]= NetlistRaw, 3 Set up AC DAE system.
In[3]:= dae = CircuitEquations[net]
Out[3]= DAEAC, 3 3 Return vector b.
In[4]:= GetRHS[dae] // MatrixForm
0 Out[4]//MatrixForm= 0 V0 3. Reference Manual
264
3.6.9 GetParameters
GetParameters[dae]
returns the list of parameters stored in dae
Command structure of GetParameters.
Given a DAEObject dae, GetParameters returns the list of parameters occuring in the equation
system of the object.
Note that GetParameters differs from GetDesignPoint in that it returns only those parameters
which appear in the equations of dae. Moreover, it returns a list of symbols, not a list of replacement
rules.
See also: GetEquations (Section 3.6.5), GetMatrix (Section 3.6.6), GetVariables (Section 3.6.7),
GetRHS (Section 3.6.8), GetDAEOptions (Section 3.6.10), GetDesignPoint (Section 3.6.12).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define netlist description
of a simple diode rectifier
circuit.
In[2]:= cir =
Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −> 2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C},
Model −> "Diode", Selector −> "Spice"}
]
]
Out[2]= Circuit Set up DAE system.
In[3]:= dae = CircuitEquations[cir,
AnalysisMode −> Transient, ElementValues −> Symbolic]
Out[3]= DAETransient, 4 4 Return list of parameters
occuring in the equations
of dae.
In[4]:= GetParameters[dae]
Out[4]=
AREA$D1, C1, GMIN, IS$D1, R1, TEMP, TNOM, V0, $k, $q
3.6 Circuit and DAEObject Manipulation
265
3.6.10 GetDAEOptions
GetDAEOptions[dae]
GetDAEOptions[dae, symb]
returns the list of options stored in the DAEObject dae
returns the value of the option symb stored in dae
GetDAEOptions[dae, {symb , symb , † † † }]
returns a list of values of the options symbi stored in dae
Command structure of GetDAEOptions .
Besides the equation system and the list of variables, a DAEObject contains a list of arbitrary options.
Use GetDAEOptions to extract these options: GetDAEOptions[dae] returns a list of all options stored
in the DAEObject dae. If a second argument is specified and is an option symbol, the value of this
option stored in the DAEObject is returned. If it is a list of option symbols, a list of the option
values is returned.
The function CircuitEquations (Section 3.5.1) automatically adds a number of options to the
DAEObject which have been used during equation setup.
To extract the DesignPoint option of a DAEObject it is recommended to use the function
GetDesignPoint .
See also: SetDAEOptions (Section 3.6.11), GetEquations (Section 3.6.5), GetMatrix (Section 3.6.6),
GetVariables (Section 3.6.7), GetRHS (Section 3.6.8), GetDesignPoint (Section 3.6.12).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define netlist description
of a simple diode rectifier
circuit.
In[2]:= cir =
Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −> 2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C},
Model −> "Diode", Selector −> "Spice"}
]
]
Out[2]= Circuit Set up DAE system.
In[3]:= dae = CircuitEquations[cir,
AnalysisMode −> Transient, ElementValues −> Symbolic]
Out[3]= DAETransient, 4 4 3. Reference Manual
266
Return the list of all
options stored in dae.
In[4]:= GetDAEOptions[dae]
Return special options
stored in dae.
In[5]:= GetDAEOptions[dae, {IndependentVariable, InitialTime}]
Out[4]=
AnalysisMode Transient, BranchCurrentPrefix I$,
BranchVoltagePrefix V$, ControllingCurrentPrefix IC$,
ControllingVoltagePrefix VC$, ConvertImmittances True,
DesignPoint V0 2. Sin1000000 t, R1 100., C1 1. 107 ,
AREA$D1 1., BV$D1 , CJO$D1 0, GMIN 1. 1012 ,
IBV$D1 0.001, IS$D1 1. 1014 , RS$D1 0, TEMP 300.15,
TNOM 300.15, $k 1.38062 1023 , $q 1.60219 1019 ,
ElementValues Symbolic, Formulation ModifiedNodal,
FrequencyVariable s, IgnoreMissingGround False,
IndependentVariable t, InitialConditions None,
InitialGuess I$AC$D1 0, V$1 V$2 0,
InitialTime 0, InstanceNameSeparator $,
MatrixEquation True, ModeValues , NodeVoltagePrefix V$,
Symbolic All, TimeVariable t, Value None
Out[5]= t, 0
3.6.11 SetDAEOptions
SetDAEOptions[dae, name−>val]
stores the value val of the option name in dae, removing
any previously stored value for this option
SetDAEOptions[dae, {name −>val , name −>val , † † † }]
stores the values vali for the options namei in dae
Command structure of SetDAEOptions.
Besides the equation system and the list of variables, a DAEObject contains a list of arbitrary options.
Use SetDAEOptions to store or to update options in a DAEObject: SetDAEOptions[dae, name −> val]
stores the value val of the option name in dae and returns the new list of options. If the second
argument is a list of option rules, the value for each option rule is stored. Previously stored values
for the given options are replaced by the new values and new options are appended to the list of
options. Note that SetDAEOptions alters the given DAEObject rather than returning a new object.
Instead it returns the updated option list.
In contrast to the Mathematica function SetOptions , the function SetDAEOptions does not check
whether a given option is a valid DAEObject option. Any option is a valid DAEObject option and
can be stored in the object (there is no Options[DAEObject] ).
To alter the DesignPoint option of a DAEObject it is recommended to use the function
UpdateDesignPoint .
See also: GetDAEOptions (Section 3.6.10), GetEquations (Section 3.6.5), GetMatrix (Section 3.6.6),
GetVariables (Section 3.6.7), GetRHS (Section 3.6.8), GetDesignPoint (Section 3.6.12),
UpdateDesignPoint (Section 3.6.14).
3.6 Circuit and DAEObject Manipulation
267
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define netlist description
of a simple diode rectifier
circuit.
In[2]:= cir =
Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −> 2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C},
Model −> "Diode", Selector −> "Spice"}
]
]
Out[2]= Circuit Set up DAE system.
In[3]:= dae = CircuitEquations[cir,
AnalysisMode −> Transient, ElementValues −> Symbolic]
Out[3]= DAETransient, 4 4 Inspect the value of the
InitialTime option in
dae.
In[4]:= GetDAEOptions[dae, InitialTime]
Alter the value of the
InitialTime option.
In[5]:= SetDAEOptions[dae, InitialTime −> 0.5]
Note that dae was altered
by SetDAEOptions .
In[6]:= GetDAEOptions[dae, InitialTime]
Out[4]= 0
Out[5]=
AnalysisMode Transient, BranchCurrentPrefix I$,
BranchVoltagePrefix V$, ControllingCurrentPrefix IC$,
ControllingVoltagePrefix VC$, ConvertImmittances True,
DesignPoint V0 2. Sin1000000 t, R1 100., C1 1. 107 ,
AREA$D1 1., BV$D1 , CJO$D1 0, GMIN 1. 1012 ,
IBV$D1 0.001, IS$D1 1. 1014 , RS$D1 0, TEMP 300.15,
TNOM 300.15, $k 1.38062 1023 , $q 1.60219 1019 ,
ElementValues Symbolic,
Formulation ModifiedNodal, FrequencyVariable s,
IgnoreMissingGround False, IndependentVariable t,
InitialConditions None, InitialGuess I$AC$D1 0, V$1 V$2 0, InstanceNameSeparator $,
MatrixEquation True, ModeValues , NodeVoltagePrefix V$,
Symbolic All, TimeVariable t, Value None, InitialTime 0.5
Out[6]= 0.5
3. Reference Manual
268
You can set arbitrary
options with arbitrary
values.
In[7]:= SetDAEOptions[dae,
Comment −> "This object describes a rectifier circuit"]
Out[7]=
AnalysisMode Transient, BranchCurrentPrefix I$,
BranchVoltagePrefix V$, ControllingCurrentPrefix IC$,
ControllingVoltagePrefix VC$, ConvertImmittances True,
DesignPoint V0 2. Sin1000000 t, R1 100., C1 1. 107 ,
AREA$D1 1., BV$D1 , CJO$D1 0, GMIN 1. 1012 ,
IBV$D1 0.001, IS$D1 1. 1014 , RS$D1 0, TEMP 300.15,
TNOM 300.15, $k 1.38062 1023 , $q 1.60219 1019 ,
ElementValues Symbolic,
Formulation ModifiedNodal, FrequencyVariable s,
IgnoreMissingGround False, IndependentVariable t,
InitialConditions None, InitialGuess I$AC$D1 0, V$1 V$2 0, InstanceNameSeparator $,
MatrixEquation True, ModeValues , NodeVoltagePrefix V$,
Symbolic All, TimeVariable t, Value None, InitialTime 0.5,
Comment This object describes a rectifier circuit
3.6.12 GetDesignPoint
GetDesignPoint[dae]
GetDesignPoint[dae, param]
returns the design point stored in dae
returns the value of the parameter param from the design
point stored in dae
GetDesignPoint[dae, {param , param , † † † }]
returns the value of each parameter parami
Command structure of GetDesignPoint .
Given a DAEObject dae, GetDesignPoint returns the design point list stored in the object. If a
symbol is given as a second argument, the value of the corresponding parameter stored in the design
point is returned. If the second argument is a list of symbols, a list of the corresponding parameter
values is returned.
Note that GetDesignPoint differs from GetParameters in that it returns the complete design point
stored in dae and not only those parameters which appear in the equations of dae.
See also: GetEquations (Section 3.6.5), GetMatrix (Section 3.6.6), GetVariables (Section 3.6.7),
GetRHS (Section 3.6.8), GetParameters (Section 3.6.9), GetDAEOptions (Section 3.6.10).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
3.6 Circuit and DAEObject Manipulation
Define netlist description
of a simple diode rectifier
circuit.
269
In[2]:= cir =
Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −> 2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C},
Model −> "Diode", Selector −> "Spice"}
]
]
Out[2]= Circuit Set up DAE system.
In[3]:= dae = CircuitEquations[cir,
AnalysisMode −> Transient, ElementValues −> Symbolic]
Out[3]= DAETransient, 4 4 Return design point stored
in dae.
In[4]:= GetDesignPoint[dae]
Return the values of the
parameters R1 and C1.
In[5]:= GetDesignPoint[dae, {R1, C1}]
Out[4]=
V0 2. Sin1000000 t, R1 100.,
C1 1. 107 , AREA$D1 1., BV$D1 , CJO$D1 0, GMIN 1. 1012 ,
IBV$D1 0.001, IS$D1 1. 1014 , RS$D1 0, TEMP 300.15,
TNOM 300.15, $k 1.38062 1023 , $q 1.60219 1019 Out[5]= 100., 1. 107 3.6.13 ApplyDesignPoint
ApplyDesignPoint[dae, stringpattern]
replaces all parameters of dae matching stringpattern with
their numerical values
ApplyDesignPoint[dae, {stringpattern , stringpattern , † † † }]
replaces all parameters matching a list of string patterns
with their numerical values
ApplyDesignPoint[dae]
replaces all parameters of the design point with their
numerical values
Command structure of ApplyDesignPoint .
Given a DAEObject, ApplyDesignPoint returns a new DAEObject where all parameters in the
equation system which match a string pattern stringpattern are replaced with their numerical values
stored in the DesignPoint option of dae. If a list of string patterns is given, all parameters which
match at least one of the patterns are replaced. By default, all replaced parameters are removed from
the DesignPoint option in the returned DAEObject. You can use the option Cleanup to change this
behavior (see below). Note that matching is performed using string patterns.
3. Reference Manual
270
It is possible to specify a symbol symb instead of a string pattern as parameter to
ApplyDesignPoint , which is equivalent to specifying the string pattern "symb". This is also valid
for the option KeepSymbolic (see below).
ApplyDesignPoint has the following options:
option name
default value
Cleanup
True
removes unused parameters from the
design point
KeepSymbolic
{}
specifies parameters to be kept symbolic
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
Options for ApplyDesignPoint.
See also: UpdateDesignPoint (Section 3.6.14).
Options Description
Cleanup
This option allows for removing parameters from the design point of a DAEObject that do not occur
in the corresponding equations.
The default setting is Cleanup −> True, which means that at least all those parameters which have
been inserted numerically are removed from the design point.
KeepSymbolic
By default, all parameters which match one of the string patterns stringpatterni are replaced by their
numerical value. The option KeepSymbolic allows for filtering this list of replaced parameters. The
option value must be a list of string patterns.
All parameters which match at least one of these string patterns are kept symbolic even if they
match the string patterns given as arguments to ApplyDesignPoint . This option is especially useful
to insert the numerical value for a model parameter keeping the parameter of a specific instance
symbolic. For example,
ApplyDesignPoint[dae, "AREA*", KeepSymbolic −> {"AREA$Q1"}]
inserts the numerical value of the model parameter AREA for each instance except for the instance Q1
for which the parameter is kept symbolic.
3.6 Circuit and DAEObject Manipulation
271
The default setting is KeepSymbolic −> {}, which means not to filter the parameters to be inserted
numerically.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define netlist.
In[2]:= cir =
Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −> 2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C},
Model −> "Diode", Selector −> "Spice"}
]
]
Out[2]= Circuit Set up DAE system.
In[3]:= dae = CircuitEquations[cir, AnalysisMode −> Transient,
ElementValues −> Symbolic]
Out[3]= DAETransient, 4 4 Show DAE system.
In[4]:= DisplayForm[dae]
Out[4]//DisplayForm=
V$2t
I$AC$D1t I$V0t 0, I$AC$D1t C1 V$2 t 0,
R1
TEMP
1.11 1. TNOM
$q
TEMP $k
V$1t V0, I$AC$D1t AREA$D1 E 3.
1. $q V$1tV$2t
TEMP
TEMP $k
1 E IS$D1 GMIN V$1t V$2t,
TNOM
V$1t, V$2t, I$V0t, I$AC$D1t,
DesignPoint V0 2. Sin1000000 t, R1 100., C1 1. 107 ,
AREA$D1 1., BV$D1 , CJO$D1 0, GMIN 1. 1012 ,
IBV$D1 0.001, IS$D1 1. 1014 , RS$D1 0, TEMP 300.15,
TNOM 300.15, $k 1.38062 1023 , $q 1.60219 1019 Insert numerical values for
all parameters except
AREA$D1 .
In[5]:= daenew = ApplyDesignPoint[dae, "*",
KeepSymbolic −> {AREA$D1}]
Show new DAE system.
Note that parameters are
removed from the
DesignPoint option of
daenew .
In[6]:= DisplayForm[daenew]
Out[5]= DAETransient, 4 4 Out[6]//DisplayForm=
I$AC$D1t I$V0t 0,
I$AC$D1t 0.01 V$2t 1. 107 V$2 t 0,
V$1t 2. Sin1000000 t, I$AC$D1t 1. 1014 AREA$D1
1 E38.6635 V$1tV$2t 1. 1012 V$1t V$2t,
V$1t, V$2t, I$V0t, I$AC$D1t,
DesignPoint AREA$D1 1.
3. Reference Manual
272
3.6.14 UpdateDesignPoint
UpdateDesignPoint[dae, repl]
allows for modifying the DesignPoint by replacement
rules
Command structure of UpdateDesignPoint .
Given a DAEObject dae, UpdateDesignPoint replaces all parameters in the DesignPoint option
stored in dae according to the replacement rules repl and returns the new DAEObject. The replacements repl are given as a list of rules of the form symbol −> newvalue which means to replace the value
of the parameter symbol by the new numerical value newvalue in the design point of the DAEObject.
If a replacement rule is of the form stringpattern −> newvalue then the value of each parameter which
matches the string pattern is replaced by newvalue. This can for example be used to set a model
parameter to a common value for each instance of this model.
If UpdateDesignPoint is called without a second argument, the values of the design point are not
changed. However, if the option Cleanup is set to True (which is the default), unused parameters
are removed from the design point. Thus, this pattern is useful for stripping the design point to the
relevant data.
UpdateDesignPoint has the following options:
option name
default value
Cleanup
True
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
removes unused parameters from the
design point
Options for UpdateDesignPoint .
See also: ApplyDesignPoint (Section 3.6.13).
Options Description
Cleanup
This option allows for removing parameters from the design point of a DAEObject that do not occur
in the corresponding equations.
The default setting is Cleanup −> True, which means to remove all those parameters from the design
point which do not occur in the equation system.
3.6 Circuit and DAEObject Manipulation
273
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define netlist.
In[2]:= cir =
Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −> 2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C},
Model −> "Diode", Selector −> "Spice"}
]
]
Out[2]= Circuit Set up DAE system.
In[3]:= dae = CircuitEquations[cir, AnalysisMode −> Transient,
ElementValues −> Symbolic]
Out[3]= DAETransient, 4 4 Show DAE system.
In[4]:= DisplayForm[dae]
Out[4]//DisplayForm=
V$2t
I$AC$D1t I$V0t 0, I$AC$D1t C1 V$2 t 0,
R1
TEMP
1.11 1. TNOM
$q
TEMP $k
V$1t V0, I$AC$D1t AREA$D1 E 3.
1. $q V$1tV$2t
TEMP
TEMP $k
1 E IS$D1 GMIN V$1t V$2t,
TNOM
V$1t, V$2t, I$V0t, I$AC$D1t,
DesignPoint V0 2. Sin1000000 t, R1 100., C1 1. 107 ,
AREA$D1 1., BV$D1 , CJO$D1 0, GMIN 1. 1012 ,
IBV$D1 0.001, IS$D1 1. 1014 , RS$D1 0, TEMP 300.15,
TNOM 300.15, $k 1.38062 1023 , $q 1.60219 1019 Store new values for the
parameters R1, C1, and V0
in the design point.
In[5]:= daenew = UpdateDesignPoint[dae,
{R1 −> 10., C1 −> 1.*^−6, V0 −> 1.}]
Show updated design
point of daenew. Note
that unused parameters
are removed from the
design point.
In[6]:= GetDesignPoint[daenew]
Out[5]= DAETransient, 4 4 Out[6]=
V0 1., R1 10., C1 1. 106 , AREA$D1 1.,
GMIN 1. 1012 , IS$D1 1. 1014 , TEMP 300.15, TNOM 300.15,
$k 1.38062 1023 , $q 1.60219 1019 3. Reference Manual
274
3.6.15 MatchSymbols
MatchSymbols[expr, matchgroups]
matches symbols in an expression expr
MatchSymbols[dae, matchgroups]
matches symbols in a system of equations dae
Command structure of MatchSymbols.
Given an expression expr or a DAEObject dae, MatchSymbols replaces all symbols which belong to a given matching group matchgroups by a common symbol and returns a new expression or DAEObject, respectively. The matching groups must be specified as lists of the form:
{suffix , suffix , † † † , matchsuffix}.
Please note that the function pattern for equations uses the design-point information stored in the
DAEObject to validate the matching specifications.
MatchSymbols has the following options:
option name
default value
DesignPoint
Automatic
list of numerical reference values for
symbolic element values in a set of circuit
equations
Tolerance
0.1
tolerance specification for matching the
symbols in a matching group
Options for MatchSymbols.
See also: ShortenSymbols (Section 3.6.16).
Options Description
DesignPoint
With
DesignPoint −> {symbol −> value , † † † } you can overwrite the design point given in the DAEObject
(see also CircuitEquations in Section 3.5.1). The default setting is DesignPoint −> Automatic ,
which means to use the design point given in the DAEObject.
3.6 Circuit and DAEObject Manipulation
275
Tolerance
Tolerance −> tol specifies the maximum relative deviation of the values of the symbols in a matching
group from the group’s mean value.
Please note that the option Tolerance is neglected if the DAEObject contains no design-point
information.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define netlist.
In[2]:= net =
Netlist[
{V0, {1, 0}, 1},
{R1, {1, 2}, Symbolic −> R1, Value −> 10.},
{R2, {2, 3}, Symbolic −> R2, Value −> 10.},
{R3, {3, out}, Symbolic −> R3, Value −> 10.},
{RL, {out, 0}, Load}
]
Out[2]= NetlistRaw, 5 Set up equations.
In[3]:= dae = CircuitEquations[net, ElementValues −> Symbolic];
DisplayForm[dae]
Out[4]//DisplayForm=
1
1
0
R1
R1
1
1
1
1
R1
R1
R2
R2
1
1
1
0
R2
R2
R3
1
0
0
R3
0
0
1
0
0
1
R3
1
Load
1
R3
0
1
0
V$1 0
0
V$2 0
.
V$3
0
0
V$out
0
I$V0 1
0
Compute voltage transfer
function.
In[5]:= vout = V$out /. First[Solve[dae]]
Match symbols of resistors
R1, R2, and R3.
In[6]:= MatchSymbols[vout, {"R*", R}]
Match symbols of resistors
in the DAEObject.
In[7]:= daematch = MatchSymbols[dae, {"R*", R}];
DisplayForm[daematch]
Load
Out[5]= Load R1 R2 R3
Load
Out[6]= Load 3 R
Out[8]//DisplayForm=
1
1
0
0
R
R
1
2
1
0
R
R
R
2
1
1
0
R
R
R
1
1
1
0
0
R
Load R
0
0
0
1
1
0
V$1 0
0
V$2 0
.
V$3
0
0
V$out
0
I$V0 1
0
3. Reference Manual
276
Compute matched voltage
transfer function.
In[9]:= V$out /. First[Solve[daematch]]
Modify values of resistors
R2 and R3.
In[10]:= daenew = UpdateDesignPoint[dae, {R2 −> 15., R3 −> 20.}]
Match symbols of resistors
of the modified
DAEObject.
In[11]:= MatchSymbols[daenew, {"R*", R}] // DisplayForm
Load
Out[9]= Load 3 R
Out[10]= DAEAC, 5 5 MatchSymbols::mtol:
Design−point values of matching group {R1, R2, R3}
do not meet the tolerance specification.
Out[11]//DisplayForm=
1
1
0
R1
R1
1
1
1
1
R1
R1
R2
R2
1
1
1
0
R2
R2
R3
1
0
0
R3
0
0
1
0
0
1
R3
1
Load
1
R3
0
1
0
V$1 0
0
V$2 0
.
V$3
0
0
V$out
0
I$V0 1
0
3.6.16 ShortenSymbols
ShortenSymbols[dae]
replaces symbols in dae by a set of symbols with shorter
symbol names
Command structure of ShortenSymbols .
Given a DAEObject dae, ShortenSymbols replaces all symbols (variables and parameters) by a set of
new symbols with shorter symbol names and returns a new DAEObject. Please note that the new
symbols are generated at the expense of instance information as name components enclosed by the
InstanceNameSeparator are dropped.
ShortenSymbols has the following options:
option name
default value
InstanceNameSeparator
Inherited[AnalogInsydes]
specifies the character(s) used to separate
name components of variables and
parameters, see InstanceNameSeparator
(Section 3.14.2)
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
Options for ShortenSymbols.
3.6 Circuit and DAEObject Manipulation
277
See also: MatchSymbols (Section 3.6.15).
3.6.17 Statistics
Statistics[netlist]
prints type and number of netlist elements in a Netlist
object netlist
Statistics[circuit]
prints type and number of netlist elements in a Circuit
object circuit
Statistics[dae]
prints complexity information of a DAEObject dae
Command structure of Statistics.
The command Statistics can be used to display both type and number of the netlist elements in
a circuit or a netlist. Additionally, you can obtain detailed information concerning the complexity of
a DAEObject.
Statistics[circuit] and Statistics[netlist] provide the following information:
type and number of occurrence of each netlist element
type and number of occurrence of each model/selector call
total number of entries
Statistics[dae] for DC and Transient DAEObjects provides the following information:
number of equations
number of variables
number of top-level terms for each equation
number of terms which involve derivatives
number of terms for each level
Statistics[dae] for AC DAEObjects provides the following information:
3. Reference Manual
278
number of equations
number of variables
total number of entries
number of non-zero entries
complexity estimate (only if the DAEObject option ElementValues is set to Symbolic , and
if the option value for the time constraint is chosen large enough)
Statistics has the following options:
option name
default value
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
TimeConstraint
10
specifies the maximum number of seconds
for the complexity estimate computation for
AC DAEObjects
Options for Statistics.
TimeConstraint
TimeConstraint specifies the number of seconds after which the computation of the complexity
estimate for an AC DAEObject is aborted. The option value may be any positive numeric value or
Infinity . The default setting is TimeConstraint −> 10.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define netlist.
In[2]:= cir =
Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −> 2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C},
Model −> "Diode", Selector −> "Spice"}
]
]
Out[2]= Circuit 3.6 Circuit and DAEObject Manipulation
Display netlist information
using Statistics .
279
In[3]:= Statistics[cir]
Number of Resistors
: 1
Number of Capacitors
: 1
Number of VoltageSources
: 1
Number of models Diode/Spice : 1
Total number of elements
Set up DAE system.
: 4
In[4]:= dae = CircuitEquations[cir, AnalysisMode −> Transient];
dae // DisplayForm
Out[5]//DisplayForm=
I$AC$D1t I$V0t 0,
I$AC$D1t 0.01 V$2t 1. 107 V$2 t 0,
V$1t 2. Sin1000000 t, I$AC$D1t 1. 1014 1 E38.6635 V$1tV$2t 1. 1012 V$1t V$2t,
V$1t, V$2t, I$V0t, I$AC$D1t, DesignPoint Display complexity
information using
Statistics .
In[6]:= Statistics[dae]
Number of equations
: 4
Number of variables
: 4
Length of equations
: {2, 3, 2, 5}
Terms with derivatives : 1
Sums in levels
: {12, 2}
3. Reference Manual
280
3.7 Numerical Analyses
This chapter describes the standard numerical analyses types you can perform with most numerical
circuit simulators. The following table shows the supported analyses and their corresponding Analog
Insydes functions:
ACAnalysis (Section 3.7.3)
NoiseAnalysis (Section 3.7.4)
small-signal (AC) analysis
noise analysis
NDAESolve (Section 3.7.5)
operating-point (DC) analysis
NDAESolve (Section 3.7.5)
DC-transfer (DT) analysis
NDAESolve (Section 3.7.5)
large-signal (transient) analysis
Section 3.7.1 starts with an introduction of the Analog Insydes numerical data format, which all of
the above mentioned functions rely on. Section 3.7.2 describes the format which is used in Analog
Insydes to perform parameter sweeps.
3.7.1 Analog Insydes Numerical Data Format
This section introduces the simulation data format used in Analog Insydes. Some functions like
ACAnalysis (Section 3.7.3), NoiseAnalysis (Section 3.7.4), NDAESolve (Section 3.7.5), or
ReadSimulationData (Section 3.10.3) need to return numerical data which represents zero, one, or
multi-dimensional numerical data. This data will be returned in the following form:
{{l1 −> v1 , l2 −> v2 , ,
SweepParameters −> {p1 −> s1 , p2 −> s2 , }}, }
The variable labels l and the parameter labels p can be of type String or Symbol. For operating-point
data the value vi is a single numerical value, for sweep data it is an InterpolatingFunction . The
parameter value si is always a single numerical value.
SweepParameters denotes the parameters which have been swept. It can be omitted if only one
data set is present.
3.7.2 Parameter Sweeps
A few Analog Insydes functions make use of parameter sweeping: Besides the integration variable,
NDAESolve (Section 3.7.5) allows for varying several parameters performing a multi-dimensional
parameter sweep. For preparation of the nonlinear simplification methods the range of the input
variables has to be specified as a parameter sweep using the command NonlinearSetup (Section 3.12.1).
3.7 Numerical Analyses
281
Analog Insydes supports the following parameter sweep formats:
format
{symbol, {f , † † † , fm }}
{symbol, start, stop, incr}
description
symbol has values f , † † † , fm
equivalent to {symbol, Range[start, stop, incr]}
{{symb −>f , † † † , symbn −>fn }, † † † , {symb −>fm , † † † , symbn −>fmn }}
First, symb has value f , , and symbn has value fn , and
so forth. In the last step, symb has value fm , , and symbn
has value fmn
{sweep , † † † , sweepm }
combinations of the above sweep formats
Valid parameter sweep formats.
The following example shows valid parameter sweeps:
{{R1, {100, 1000, 10000}},
{V1, 1, 5, 1},
{{P1 −> 10, P2 −> 20}, {P1 −> 20, P2 −> 10}}}
If several sweep formats are combined in a list (as shown in the example above) the outer product
of all sweep specifications is computed.
You can use ParamSweepQ to check whether a given sweep specification is valid in the above sense.
Examples
To demonstrate the parameter sweep format, in this example section the internal command
ResolveParamSweep is used. This function returns a list where the first element is the canonical
representation of the sweep specification (i.e. a flat list of rules). The second element of the result is
for internal purposes only.
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
A one-dimensional sweep
with values given
explicitly.
In[2]:= First @ ResolveParamSweep[{R1, {100, 1000, 10000}}]
A one-dimensional sweep
with values given by an
interval.
In[3]:= First @ ResolveParamSweep[{V1, 1, 5, 1}]
Out[2]= R1 100., R1 1000., R1 10000.
Out[3]=
V1 1., V1 2., V1 3., V1 4., V1 5.
3. Reference Manual
282
The combination of both
sweeps yields a
two-dimensional sweep.
In[4]:= First @ ResolveParamSweep[
{{R1, {100, 1000, 10000}}, {V1, 1, 4, 1}}]
You can mix any sweep
format to build valid
multi-dimensional sweeps.
In[5]:= ParamSweepQ[{{R1, {100, 1000}}, {V1, 1, 5, 1},
{{P1 −> 10, P2 −> 20}, {P1 −> 20, P2 −> 10}}}]
Out[4]=
R1 100., V1 1., R1 100., V1 2.,
R1 100., V1 3., R1 100., V1 4., R1 1000., V1 1.,
R1 1000., V1 2., R1 1000., V1 3., R1 1000., V1 4.,
R1 10000., V1 1., R1 10000., V1 2.,
R1 10000., V1 3., R1 10000., V1 4.
Out[5]= True
3.7.3 ACAnalysis
ACAnalysis[dae, {fvar, fstart, fend}]
computes the small-signal solution of a linear DAEObject
dae within a specified frequency range fstart fend as
function of the frequency variable fvar for all variables
ACAnalysis[dae, var, {fvar, fstart, fend}]
solves for the variable var
ACAnalysis[dae, {var , var , † † † }, {fvar, fstart, fend}]
solves for several variables
Command structure of ACAnalysis.
This functions allows for carrying out a numerical small-signal analysis. Given an AC DAEObject,
ACAnalysis computes the small-signal solution in the specified frequency range and returns the
result according to the Analog Insydes numerical data format described in Section 3.7.1. It consists
of lists of rules, where the variables are assigned InterpolatingFunction objects.
The result of the small-signal analysis can be displayed with Analog Insydes graphics functions such
as BodePlot (Section 3.9.1).
ACAnalysis has the following options:
3.7 Numerical Analyses
283
option name
default value
FrequencyVariable
Automatic
specifies the symbol that denotes the
complex Laplace frequency
InterpolateResult
True
specifies the return format of the result
InterpolationOrder
1
specifies the interpolation order of the
resulting interpolating functions
PointsPerDecade
10
specifies the number of sampling points per
frequency decade
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
SweepPath
(2. Pi I #1 &)
specifies a mapping from a frequency to a
point in the complex plane
Options for ACAnalysis.
See also: BodePlot (Section 3.9.1), NoiseAnalysis (Section 3.7.4), Section 3.7.1.
Options Description
A detailed description of all ACAnalysis options is given below in alphabetical order:
FrequencyVariable
This option specifies the symbol that denotes the complex Laplace frequency. See CircuitEquations
in Section 3.5.1.
The default setting is FrequencyVariable −> Automatic , which means to use the symbol given in
the DAEObject.
InterpolateResult
If InterpolateResult is set to True, the result is returned as an interpolating function for each
variable. If the option is set to False, the result is returned as a list of {frequency, value} pairs.
InterpolationOrder
With InterpolationOrder −> integer you can change the interpolation order of the resulting interpolating functions (see the Mathematica object InterpolatingFunction ).
The default setting is InterpolationOrder −> 1, which means linear interpolation.
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284
PointsPerDecade
PointsPerDecade specifies the number of sampling points per frequency decade.
The default setting is PointsPerDecade −> 10.
SweepPath
SweepPath specifies a mapping from a frequency to a point in the complex plane. The option value
has to be a pure function with one argument which returns a complex number.
The default setting is SweepPath −> (2. Pi I #1 &), which specifies a sweep along the IΩ axis.
Examples
Below we compute the frequency response of a RC lowpass filter circuit.
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define netlist description.
In[2]:= lowpass =
Circuit[
Netlist[
{V1, {1,
{R1, {1,
{C1, {2,
{R2, {2,
{C2, {3,
{RL, {3,
]
]
0},
2},
0},
3},
0},
0},
1},
100.},
1.*^−6},
100.},
1.*^−6},
1000.}
Out[2]= Circuit Set up system of
small-signal equations.
In[3]:= lowpasseqs = CircuitEquations[lowpass]
Display equations.
In[4]:= DisplayForm[lowpasseqs]
Out[3]= DAEAC, 4 4 Out[4]//DisplayForm=
0.01
0.01
0
1
0
V$1 6
0
0.01
0.02
1.
10
s
0.01
0
V$2
.
6
0
V$3 0
0.01
0.011
1.
10
s
0
I$V1 1
0
0
0
1
Perform numerical
small-signal analysis with
ACAnalysis .
In[5]:= acsweep = ACAnalysis[lowpasseqs, {f, 1., 1.*^6}]
Out[5]=
V$1 InterpolatingFunction1., 1. 106 , ,
V$2 InterpolatingFunction1., 1. 106 , ,
V$3 InterpolatingFunction1., 1. 106 , ,
I$V1 InterpolatingFunction1., 1. 106 , 3.7 Numerical Analyses
285
To solve only for one
variable, specify the
variable as second
argument.
In[6]:= ACAnalysis[lowpasseqs, V$3, {f, 1., 1.*^6}]
Display the simulation
result with BodePlot .
In[7]:= BodePlot[acsweep, {V$2[f]}, {f, 1., 1.*^6}]
Phase (deg)
Magnitude (dB)
Out[6]=
V$3 InterpolatingFunction1., 1. 106 , 1.0E0
0
-10
-20
-30
-40
-50
1.0E1
1.0E2
1.0E3
1.0E4
1.0E5
1.0E6
1.0E0
1.0E1
1.0E2 1.0E3 1.0E4
Frequency
1.0E5
1.0E6
1.0E0
0
1.0E1
1.0E2
1.0E4
1.0E5
1.0E6
1.0E1
1.0E2 1.0E3 1.0E4
Frequency
1.0E5
1.0E6
1.0E3
-20
-40
-60
-80
1.0E0
V$2[f]
Out[7]= Graphics 3.7.4 NoiseAnalysis
NoiseAnalysis[dae, invar, outvar, {f, fstart, fstop}]
computes the output noise and the equivalent input noise
for a linear DAEObject dae where the input signal is
specified by invar and the output by outvar
Command structure of NoiseAnalysis .
This functions allows for carrying out a numerical noise analysis. The output noise and the equivalent input noise are computed from a linear DAEObject dae, where the parameter invar denotes the
input and the variable outvar denotes the output, respectively. Use GetParameters to get a valid list
of input symbols and GetVariables for a list of valid output symbols.
NoiseAnalysis returns the result according to the Analog Insydes numerical data format (see Section 3.7.1) for the output noise (denoted by the symbol OutputNoise ), the equivalent input noise
(denoted by the symbol InputNoise ), the output variable outvar, and each noise source. The computed values for the noise sources are given in A Hz and V Hz, input and output noise in A Hz
and V Hz, respectively.
The result of the numerical noise analysis can be displayed with Analog Insydes graphics functions
such as BodePlot (Section 3.9.1).
The DAEObject dae must be set up symbolically (use ElementValues −> Symbolic during equation
setup). Note that at least all sources (noise sources and the one small-signal input source invar)
3. Reference Manual
286
have to be accessible symbolically. When setting up equations for noise analysis, specify the option
settings AnalysisMode −> AC and DefaultSelector −> "Noise" for CircuitEquations in order
to use the noise models. See CircuitEquations (Section 3.5.1) for a description of the options
DefaultSelector and ElementValues .
Note that the noise sources are expressed in the following units:
source type
unit
example: thermal noise
CurrentSource
A Hz
Sqrt[4 k T / R]
VoltageSource
V Hz
Sqrt[4 k T R]
Units of noise sources.
While specifying the value of a noise source it is also assumed that the bandwidth is Hz.
NoiseAnalysis has the following options:
option name
default value
DesignPoint
Automatic
specifies a list of numerical reference values
for symbolic element values in a set of
circuit equations
FrequencyVariable
Automatic
specifies the symbol that denotes the
complex Laplace frequency
InterpolationOrder
1
specifies the interpolation order of the
resulting interpolating functions
PointsPerDecade
10
specifies the number of sampling points per
frequency decade
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
SweepPath
(2. Pi I #1 &)
specifies a mapping from a frequency to a
point in the complex plane
Options for NoiseAnalysis
See also: BodePlot (Section 3.9.1), ACAnalysis (Section 3.7.3), GetVariables (Section 3.6.7),
GetParameters (Section 3.6.9), Section 3.7.1.
3.7 Numerical Analyses
287
Options Description
A detailed description of all NoiseAnalysis options is given below in alphabetical order:
DesignPoint
With DesignPoint −> {symbol −> value, † † † }, you can specify the design point. See also
CircuitEquations (Section 3.5.1).
The default setting is DesignPoint −> Automatic , which means to use the design point given in the
DAEObject.
FrequencyVariable
This option specifies the symbol that denotes the complex Laplace frequency. See CircuitEquations
in Section 3.5.1.
The default setting is FrequencyVariable −> Automatic , which means to use the symbol given in
the DAEObject.
InterpolationOrder
With InterpolationOrder −> integer, you can change the interpolation order of the resulting interpolating functions. See the Mathematica object InterpolatingFunction .
The default setting is InterpolationOrder −> 1, which is linear interpolation.
PointsPerDecade
PointsPerDecade specifies the number of sampling points per frequency decade.
The default setting is PointsPerDecade −> 10.
SweepPath
SweepPath specifies a mapping from a frequency to a point in the complex plane. The option value
has to be a pure function with one argument which returns a complex number.
The default setting is SweepPath −> (2. Pi I #1 &), which specifies a sweep along the IΩ axis.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
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288
Read the netlist and the
operating-point
information.
In[2]:= bufcir = ReadNetlist[
"AnalogInsydes/DemoFiles/Buffer.cir",
"AnalogInsydes/DemoFiles/Buffer.out",
Simulator −> "PSpice"]
Out[2]= Circuit Choose noise models and
set up system of equations
symbolically.
In[3]:= bufeq = CircuitEquations[bufcir, AnalysisMode −> AC,
DefaultSelector −> "Noise", ElementValues −> Symbolic]
View circuit variables. The
variable V$99 is chosen as
output symbol.
In[4]:= GetVariables[bufeq]
View circuit parameters.
The parameter V$VIN is
chosen as input symbol.
In[5]:= GetParameters[bufeq]
Out[3]= DAEAC, 18 18 Out[4]=
V$1, V$2, V$3, V$4, V$5, V$99, V$BS$Q$T1, V$BS$Q$T2, V$BS$Q$T3,
V$BS$Q$T4, V$BS$Q$T5, V$CS$Q$T1, V$CS$Q$T2, V$CS$Q$T3,
V$CS$Q$T4, V$CS$Q$T5, I$V$VDD, I$V$VIN
Out[5]=
Cbc$Q$T1, Cbc$Q$T2, Cbc$Q$T3,
Cbc$Q$T4, Cbc$Q$T5, Cbe$Q$T1, Cbe$Q$T2, Cbe$Q$T3, Cbe$Q$T4,
Cbe$Q$T5, Cbx$Q$T1, Cbx$Q$T2, Cbx$Q$T3, Cbx$Q$T4, Cbx$Q$T5,
Gmu$Q$T1, Gmu$Q$T2, Gmu$Q$T3, Gmu$Q$T4, Gmu$Q$T5, gm$Q$T1,
gm$Q$T2, gm$Q$T3, gm$Q$T4, gm$Q$T5, Ib$Q$T1, Ib$Q$T2, Ib$Q$T3,
Ib$Q$T4, Ib$Q$T5, Ic$Q$T1, Ic$Q$T2, Ic$Q$T3, Ic$Q$T4, Ic$Q$T5,
Irc$Q$T1, Irc$Q$T2, Irc$Q$T3, Irc$Q$T4, Irc$Q$T5, Irx$Q$T1,
Irx$Q$T2, Irx$Q$T3, Irx$Q$T4, Irx$Q$T5, Rc$Q$T1, Rc$Q$T2,
Rc$Q$T3, Rc$Q$T4, Rc$Q$T5, Ro$Q$T1, Ro$Q$T2, Ro$Q$T3, Ro$Q$T4,
Ro$Q$T5, Rpi$Q$T1, Rpi$Q$T2, Rpi$Q$T3, Rpi$Q$T4, Rpi$Q$T5,
Rx$Q$T1, Rx$Q$T2, Rx$Q$T3, Rx$Q$T4, Rx$Q$T5, V$VIN
3.7 Numerical Analyses
289
Perform numerical noise
analysis with
NoiseAnalysis . Note the
additional symbols
OutputNoise and
InputNoise in the result.
In[6]:= bufnoi = NoiseAnalysis[bufeq, V$VIN, V$99,
{f, 10., 1.*^8}]
Display the simulation
result.
In[7]:= BodePlot[bufnoi, {OutputNoise[f], InputNoise[f]},
{f, 10., 1.*^8}, PhaseDisplay −> None,
TraceNames −> {"noise(out)", "noise(in)"}]
Out[6]=
OutputNoise InterpolatingFunction10., 1. 108 , ,
InputNoise InterpolatingFunction10., 1. 108 , ,
V$99 InterpolatingFunction10., 1. 108 , ,
Ib$Q$T1 InterpolatingFunction10., 1. 108 , ,
Ib$Q$T2 InterpolatingFunction10., 1. 108 , ,
Ib$Q$T3 InterpolatingFunction10., 1. 108 , ,
Ib$Q$T4 InterpolatingFunction10., 1. 108 , ,
Ib$Q$T5 InterpolatingFunction10., 1. 108 , ,
Ic$Q$T1 InterpolatingFunction10., 1. 108 , ,
Ic$Q$T2 InterpolatingFunction10., 1. 108 , ,
Ic$Q$T3 InterpolatingFunction10., 1. 108 , ,
Ic$Q$T4 InterpolatingFunction10., 1. 108 , ,
Ic$Q$T5 InterpolatingFunction10., 1. 108 , ,
Irc$Q$T1 InterpolatingFunction10., 1. 108 , ,
Irc$Q$T2 InterpolatingFunction10., 1. 108 , ,
Irc$Q$T3 InterpolatingFunction10., 1. 108 , ,
Irc$Q$T4 InterpolatingFunction10., 1. 108 , ,
Irc$Q$T5 InterpolatingFunction10., 1. 108 , ,
Irx$Q$T1 InterpolatingFunction10., 1. 108 , ,
Irx$Q$T2 InterpolatingFunction10., 1. 108 , ,
Irx$Q$T3 InterpolatingFunction10., 1. 108 , ,
Irx$Q$T4 InterpolatingFunction10., 1. 108 , ,
Irx$Q$T5 InterpolatingFunction10., 1. 108 , 1.0E0
1.0E2
1.0E4
1.0E6
1.0E8
1.0E2
1.0E4
Frequency
1.0E6
1.0E8
-140
Magnitude (dB)
-150
-160
-170
-180
1.0E0
noise(out) noise(in)
Out[7]= Graphics 3. Reference Manual
290
3.7.5 NDAESolve
NDAESolve[dae, {tvar, t }]
computes the DC solution of a nonlinear DAEObject dae at
the time point tvar t
NDAESolve[dae, {tvar, t }, params]
carries out a parametric DC analysis
NDAESolve[dae, {tvar, tstart, tend}]
computes the transient or the DC-transfer solution of a
nonlinear DAEObject dae within a specified integration
interval tstart tend as function of the independent
variable tvar
NDAESolve[dae, {tvar, tstart, tend}, params]
carries out a parametric transient or a parametric
DC-transfer analysis
Command structure of NDAESolve.
The function NDAESolve allows for carrying out several numerical analyses, such as an operatingpoint (DC) analysis, a DC-transfer (DT) analysis, or a large-signal (transient) analysis. NDAESolve
returns the result according to the Analog Insydes numerical data format described in Section 3.7.1.
It consists of lists of rules, where the variables are assigned InterpolatingFunction objects in case
of a transient analysis or numerical values in case of a DC analysis. Both analysis modes can also
be carried out as parametric analyses. For this, specify a parameter sweep params according to the
Analog Insydes parameter sweep format described in Section 3.7.2.
The result of the numerical transient analysis can be displayed with Analog Insydes graphics functions such as TransientPlot (Section 3.9.6).
Note that NDAESolve still supports the command structure from Release 1 where the first argument
is given as {eqns, vars} instead of a DAEObject:
NDAESolve[{eqns, vars}, {tvar, t }]
carries out a DC analysis
NDAESolve[{eqns, vars}, {tvar, t }, params]
carries out a parametric DC analysis
NDAESolve[{eqns, vars}, {tvar, tstart, tend}]
carries out a transient or a DC-transfer analysis
NDAESolve[{eqns, vars}, {tvar, tstart, tend}, params]
carries out a parametric transient or a parametric
DC-transfer analysis
Additional patterns of NDAESolve.
3.7 Numerical Analyses
291
NDAESolve has the following options:
option name
default value
AccuracyGoal
6
number of digits of accuracy to seek in the
result (see also FindRoot)
Compiled
True
whether to compile the equations (see also
FindRoot )
MaxIterations
{100, 20}
maximum number of iterations of the
nonlinear solver
WorkingPrecision
16
number of digits of precision to keep in
internal computations (see also FindRoot )
NonlinearMethod
FindRoot
allows for switching the nonlinear equation
solver
Nonlinear solver options of NDAESolve.
option name
default value
MaxDelta
1.
limit for the deviation of variable values
during the automatic step size control
MaxSteps
Automatic
maximum number of integration steps
MaxStepSize
Automatic
upper limit of the integration step size
MinStepSize
Automatic
lower limit of the integration step size
StartingStepSize
Automatic
initial integration step size
StepSizeFactor
1.5
modification factor of the integration step
size
Step size control options of NDAESolve.
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option name
default value
InitialConditions
Automatic
whether to consider initial conditions
InitialGuess
Automatic
initial guess for the operating-point
computation
InitialSolution
Automatic
initial solution for the operating-point
computation
RandomFunction
(Random[Real, {0., 0.1}]&)
random function for perturbing the initial
guess
Operating-point computation options of NDAESolve.
option name
default value
InterpolationOrder
1
interpolation order of the resulting
interpolating functions
OutputVariables
All
variables to appear in the solution vector
ReturnDerivatives
False
whether to include solutions for the
derivatives in the solution vector
Return format options of NDAESolve.
3.7 Numerical Analyses
293
option name
default value
AnalysisMode
Transient
circuit analysis mode
BreakOnError
False
whether to interrupt a parametric DC or
transient analysis in case of an error
CompressEquations
False
whether to carry out a symbolic
pre-processing of the equation system
DesignPoint
Automatic
list of numerical reference values for
symbolic element values in a set of circuit
equations
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
Simulator
Inherited[AnalogInsydes]
standard Analog Insydes simulator
specification (see Section 3.14.6)
SimulatorExecutable
Automatic
name of the external simulator executable
Miscellaneous options of NDAESolve.
See also: TransientPlot (Section 3.9.6), Section 3.7.1, Section 3.7.2.
Options Description
A detailed description of all NDAESolve options is given below in alphabetical order:
AnalysisMode
The option AnalysisMode specifies the circuit analysis mode, where the default setting is
AnalysisMode −> Transient . Valid option values are as follows:
DC
Transient
carries out a DC-transfer analysis
carries out a large-signal time-domain analysis
Values for the AnalysisMode option.
Note that this option is only valid for the NDAESolve pattern where you have to specify a start value
tstart and an end value tend. In this case NDAESolve is able to carry out a DC-transfer analysis by
setting the option to AnalysisMode −> DC, although the input is a Transient DAEObject.
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Note that the DC-transfer analysis method replaces all derivatives in the circuit equations by zero.
Thus, inductors are replaced by short circuits because all inductor voltages are set to zero:
Inductor:
VL t L
#IL t
#t
DC
$
VL t %
Short Circuit
and capacitors are replaced by open circuits because all capacitor currents are set to zero:
Capacitor:
IC t C
#UC t
#t
DC
$
IC t %
Open Circuit
BreakOnError
The option BreakOnError allows for interrupting a parametric DC or transient analysis in case of an
error. If set to True and the computation for one parameter sweep fails, the simulation is interrupted
and $Failed is returned. The default setting is BreakOnError −> False.
CompressEquations
The option CompressEquations allows for carrying out a symbolic pre-processing of the equation
system before it is solved numerically. This means that equations and variables which are not relevant for computing the variables of interest (given by the option OutputVariables ) are removed.
Note that this option is only effective if the option OutputVariables is not set to All. The option
is realized by an internal call of the function CompressNonlinearEquations (Section 3.12.2). The
default setting is CompressEquations −> False.
DesignPoint
With
DesignPoint −> {symbol −> value , † † † } you can overwrite the design point given in the DAEObject
(see also CircuitEquations in Section 3.5.1). The default setting is DesignPoint −> Automatic ,
which means to use the design point given in the DAEObject.
InitialConditions
The option InitialConditions allows for considering initial conditions (see also CircuitEquations
in Section 3.5.1). The default setting is InitialConditions −> Automatic . Valid option values are
as follows:
3.7 Numerical Analyses
295
None
Automatic
All
neglects initial conditions
considers initial conditions
same as Automatic
Values for the InitialConditions option.
InitialGuess
The option InitialGuess allows for explicitly specifying an initial guess for the Newton iteration
when computing the operating point. Note that missing variables are set to zero. The default setting
is InitialGuess −> Automatic, which means that all variables have the initial value zero. Possible
option values are as follows:
Automatic
{var −> value , † † † }
uses zero values as initial guess
uses specified values as initial guess and sets values not
specified to zero
Values for the InitialGuess option.
InitialSolution
The option InitialSolution allows for specifying an initial solution for certain variables of the
operating point. Note that missing variables are computed consistently. The default setting is
InitialSolution −> Automatic , which automatically computes the operating point. Possible option values are as follows:
Automatic
{var −> value , † † † }
computes operating point automatically
uses specified values as initial solution
Values for the InitialSolution option.
InterpolationOrder
With InterpolationOrder −> integer you can change the interpolation order of the resulting interpolating functions (see the Mathematica object InterpolatingFunction ). The default setting is
InterpolationOrder −> 1, which means linear interpolation.
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MaxDelta
One of the conditions included in the automatic step size control is a check for rapidly changing
variable values from a certain point in time to its succeeding time instant. The maximum allowed
deviation is controlled via the option MaxDelta. The default setting is MaxDelta −> 1.
MaxIterations
MaxIterations specifies the maximum number of iterations the nonlinear equation solver should use
attempting to find a solution. The option setting can either be an integer int or a list of two integers
{int , int }. If it is specified as a single integer int then it is equivalent to the list {int, int}. The
first integer value defines the iteration limit for finding a DC operating-point and the second integer
value the iteration limit for transient computations, respectively. If the number of iterations for the
operating-point computation exceeds the limit, an error message is generated and the computation is
interrupted. If the iteration limit for transient computations is exceeded, the step size is reduced by
the factor given by the option StepSizeFactor . The default setting is MaxIterations −> {100, 20}.
MaxSteps
The option MaxSteps limits the number of integration steps. The computation will be stopped
immediately if the limiting value is exceeded. The default setting is MaxSteps −> Automatic , which
means MaxSteps tend tstart StartingStepSize .
MaxStepSize
The option MaxStepSize specifies the upper limit of the integration step size. The default setting is
MaxStepSize −> Automatic , which means MaxStepSize tend tstart .
MinStepSize
The option MinStepSize specifies the lower limit of the integration step size. The computation will
be stopped immediately if the integration step size falls below this limit. The default setting is
MinStepSize −> Automatic , which means MinStepSize tend tstart .
NonlinearMethod
The option NonlinearMethod allows for choosing between different algorithms for numerically solving nonlinear equation systems. The default setting is NonlinearMethod −> FindRoot. Valid option
values are as follows:
3.7 Numerical Analyses
297
FindRoot
NewtonIteration
uses Mathematica’s numerical equation solver FindRoot
uses multi-dimensional Newton-Raphson implementation
Values for the NonlinearMethod option.
OutputVariables
The option OutputVariables specifies those variables which should be included in the solution
vector. The default setting is OutputVariables −> All. Valid option values are as follows:
All
{var , † † † }
returns solutions for all variables
returns solutions for the specified variables only
Values for the OutputVariables option.
RandomFunction
The option RandomFunction specifies a function used for generating random numbers for perturbing
the initial guess during the operating-point computation. Note that the generated random numbers
are added to the initial guess. The return value of the function should be a real number. The default
setting is RandomFunction −> (Random[Real, {0., 0.1}]&).
ReturnDerivatives
The option ReturnDerivatives allows for including solutions of the derivatives in the solution
vector. The default setting is ReturnDerivatives −> False.
Simulator
The option Simulator specifies the simulator which should be applied for numerically solving the
system of equations. The default setting is Simulator −> Inherited[AnalogInsydes] . Valid option
values are as follows:
"AnalogInsydes"
applies internal Analog Insydes solver
"PSpice"
applies PSpice kernel
"Saber"
applies Saber kernel
Values for the Simulator option.
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In case of an external simulator kernel a behavioral model in the simulator-specific language is generated with the function WriteModel (Section 3.10.5). Then the respective simulator kernel given by
the option SimulatorExecutable is applied, and finally the generated simulation data is imported
via the function ReadSimulationData (Section 3.10.3).
Please note that this option is in an experimental state. Additionally, it is extremely dependent on
the applied simulator, its underlying operating system, and even the available simulator version.
SimulatorExecutable
The option SimulatorExecutable specifies the name of the external simulator executable. Note that
this option is only effective if an external simulator is applied via the option Simulator . The default setting is SimulatorExecutable −> Automatic , which means "pspice" in case of PSpice and
"saber" in case of Saber, respectively. If set to a string make sure that the simulator call corresponds
to the setting of the Simulator option. Otherwise the internal calls to the functions WriteModel and
ReadSimulationData fail. Valid option values are as follows:
Automatic
"string"
generates simulator call according to the Simulator option
allows user specific simulator call
Values for the SimulatorExecutable option.
StartingStepSize
The option StartingStepSize specifies the initial integration step size. The default setting is
StartingStepSize −> Automatic , which means StartingStepSize tend tstart .
StepSizeFactor
To speed up the computation time NDAESolve includes an automatic step size control. If the deviations of the variable values from one time point to its succeeding time instant fulfill certain conditions
the actual step size will be multiplied or divided by the value given by the option StepSizeFactor .
The default setting is StepSizeFactor −> 1.5.
Examples
Consider the following diode rectifier circuit with a sinusoidal input voltage.
3.7 Numerical Analyses
299
D1
1
2
V0
C1
R1
Vout
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define netlist description
of a simple diode rectifier
circuit.
In[2]:= rectifier =
Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −> 2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C},
Model −> "Diode", Selector −> "Spice"}
]
]
Out[2]= Circuit Set up system of symbolic
time-domain equations.
In[3]:= rectifiereqs = CircuitEquations[rectifier,
AnalysisMode −> Transient, ElementValues −> Symbolic]
Out[3]= DAETransient, 4 4 Display equations.
In[4]:= DisplayForm[rectifiereqs]
Out[4]//DisplayForm=
V$2t
I$AC$D1t I$V0t 0, I$AC$D1t C1 V$2 t 0,
R1
TEMP
1.11 1. TNOM
$q
TEMP $k
V$1t V0, I$AC$D1t AREA$D1 E 3.
1. $q V$1tV$2t
TEMP
TEMP $k
1 E IS$D1 GMIN V$1t V$2t,
TNOM
V$1t, V$2t, I$V0t, I$AC$D1t,
DesignPoint V0 2. Sin1000000 t, R1 100., C1 1. 107 ,
AREA$D1 1., BV$D1 , CJO$D1 0, GMIN 1. 1012 ,
IBV$D1 0.001, IS$D1 1. 1014 , RS$D1 0, TEMP 300.15,
TNOM 300.15, $k 1.38062 1023 , $q 1.60219 1019 Perform numerical DC
analysis with NDAESolve .
In[5]:= NDAESolve[rectifiereqs, {t, 0.}]
Out[5]=
V$2 0., V$1 0., I$V0 0., I$AC$D1 0.
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300
Perform numerical
transient analysis with
NDAESolve .
In[6]:= tran = NDAESolve[rectifiereqs, {t, 0., 2.*^−5}]
Display the simulation
result.
In[7]:= TransientPlot[tran, {V$1[t], V$2[t]}, {t, 0., 2.*^−5}]
Out[6]=
V$2 InterpolatingFunction0, 0.00002, ,
V$1 InterpolatingFunction0, 0.00002, ,
I$V0 InterpolatingFunction0, 0.00002, ,
I$AC$D1 InterpolatingFunction0, 0.00002, 2
1
V$1[t]
-6
t
0.00001 0.000015 0.00002
5. 10
V$2[t]
-1
-2
Out[7]= Graphics Perform numerical
DC-transfer analysis with
NDAESolve .
In[8]:= dctran = NDAESolve[rectifiereqs, {t, 0., 1.*^−5},
AnalysisMode −> DC]
Display the simulation
result.
In[9]:= TransientPlot[dctran, {V$1[t], V$2[t]}, {t, 0., 1.*^−5}]
Out[8]=
V$1 InterpolatingFunction0, 0.00001, ,
V$2 InterpolatingFunction0, 0.00001, ,
I$V0 InterpolatingFunction0, 0.00001, ,
I$AC$D1 InterpolatingFunction0, 0.00001, 2
1
V$1[t]
-6
2. 10
-6
4. 10
-1
-2
Out[9]= Graphics t
-60.00001
-6
6. 10
8. 10
V$2[t]
3.7 Numerical Analyses
Perform parametric
transient analysis with
NDAESolve .
301
In[10]:= paramtran = NDAESolve[rectifiereqs, {t, 0., 2.*^−5},
{R1, Table[10^i, {i, 0, 3}]}]
Out[10]=
V$2 InterpolatingFunction0, 0.00002, ,
V$1 InterpolatingFunction0, 0.00002, ,
I$V0 InterpolatingFunction0, 0.00002, ,
I$AC$D1 InterpolatingFunction0, 0.00002, ,
SweepParameters R1 1.,
V$2 InterpolatingFunction0, 0.00002, ,
V$1 InterpolatingFunction0, 0.00002, ,
I$V0 InterpolatingFunction0, 0.00002, ,
I$AC$D1 InterpolatingFunction0, 0.00002, ,
SweepParameters R1 10.,
V$2 InterpolatingFunction0, 0.00002, ,
V$1 InterpolatingFunction0, 0.00002, ,
I$V0 InterpolatingFunction0, 0.00002, ,
I$AC$D1 InterpolatingFunction0, 0.00002, ,
SweepParameters R1 100.,
V$2 InterpolatingFunction0, 0.00002, ,
V$1 InterpolatingFunction0, 0.00002, ,
I$V0 InterpolatingFunction0, 0.00002, ,
I$AC$D1 InterpolatingFunction0, 0.00002, ,
SweepParameters R1 1000.
By default, TransientPlot displays the solutions for all parameter sweep values within a single
plot.
Display the simulation
result.
In[11]:= TransientPlot[paramtran, {V$2[t]}, {t, 0., 2.*^−5}]
1.2
1
0.8
0.6
V$2[t]
0.4
0.2
t
-6 0.00001 0.000015 0.00002
5. 10
Out[11]= Graphics Consider the following oscillator circuit with initial conditions at the time t .
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302
Vic
2
1
L1
C1
R1
Iout
Define netlist description
of a simple oscillator
circuit.
In[12]:= oscillator =
Netlist[
{L1, {0, 1}, Symbolic
{C1, {1, 2}, Symbolic
InitialCondition −>
{R1, {2, 0}, Symbolic
]
−> L1, Value −> 1.*^−5},
−> C1, Value −> 3.*^−7,
2.*^−3},
−> R1, Value −> 1.}
Out[12]= NetlistRaw, 3 Set up system of symbolic
time-domain equations
with initial conditions
where given in the netlist.
In[13]:= oscillatoreqs1 = CircuitEquations[oscillator,
AnalysisMode −> Transient, ElementValues −> Symbolic,
InitialConditions −> Automatic];
DisplayForm[oscillatoreqs1]
Out[14]//DisplayForm=
I$L1t C1 V$1 t V$2 t 0,
V$2t
C1 V$1 t V$2 t 0, V$1t L1 I$L1 t 0,
R1
V$10 V$20 0.002, V$1t, V$2t, I$L1t,
DesignPoint L1 0.00001, C1 3. 107 , R1 1.
Set up system of symbolic
time-domain equations
with initial conditions for
all dynamic netlist
elements.
In[15]:= oscillatoreqs2 = CircuitEquations[oscillator,
AnalysisMode −> Transient, ElementValues −> Symbolic,
InitialConditions −> All];
DisplayForm[oscillatoreqs2]
Out[16]//DisplayForm=
I$L1t C1 V$1 t V$2 t 0,
V$2t
C1 V$1 t V$2 t 0,
R1
V$1t L1 I$L1 t 0, I$L10 0,
V$10 V$20 0.002, V$1t, V$2t, I$L1t,
DesignPoint L1 0.00001, C1 3. 107 , R1 1.
Perform numerical
transient analysis for both
equation systems.
In[17]:= tran1 = NDAESolve[oscillatoreqs1, {t, 0., 1.*^−4}];
tran2 = NDAESolve[oscillatoreqs2, {t, 0., 1.*^−4}];
3.7 Numerical Analyses
303
With InitialConditions −> Automatic , initial conditions are applied only where specified in the
netlist, all others are computed consistently. With InitialConditions −> All, initial conditions are
set to zero for all elements for which there is no condition specified in the netlist. You can see the
difference between both possibilities in the following two plots.
Display the first
simulation result.
In[19]:= TransientPlot[tran1, {I$L1[t]}, {t, 0., 1.*^−4},
PlotRange −> All]
0.0015
0.001
0.0005
0.00002
0.00004
0.00006
0.00008
t
0.0001
I$L1[t]
-0.0005
-0.001
-0.0015
-0.002
Out[19]= Graphics Display the second
simulation result.
In[20]:= TransientPlot[tran2, {I$L1[t]}, {t, 0., 1.*^−4}]
0.0002
0.0001
0.00002
0.00004
-0.0001
-0.0002
-0.0003
Out[20]= Graphics 0.00006
0.00008
t
0.0001
I$L1[t]
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3.8 Pole/Zero Analysis
This chapter describes the Analog Insydes functions for numerical and symbolic pole/zero analysis
by solving generalized eigenvalue problems (GEPs). The GEPs considered here have the form
A ΛBu v A ΛB H
where A and B are real-valued square matrices that arise from decomposing the coefficient matrix
of a system of circuit equations Ts x b into the contributions from the static and the dynamic
elements.
Ts A sB
In the above GEP, Λ is an eigenvalue, and u and v denote the right and left eigenvectors corresponding to Λ (vH denotes the Hermitian conjugate of v). In the following, the pairs Λ u and
Λ v are referred to as (left and right) eigenpairs of the matrix pencil A B. Analog Insydes includes
two different GEP solvers: an enhanced version of the QZ algorithm and a variant of the Jacobi
orthogonal correction method (JOCM).
The following table shows the available Analog Insydes functions for solving GEPs:
GeneralizedEigensystem (Section 3.8.1)
computing eigenvalues and eigenvectors by QZ
GeneralizedEigenvalues (Section 3.8.2)
computing eigenvalues by QZ
PolesAndZerosByQZ (Section 3.8.3)
computing poles and zeros by QZ
PolesByQZ (Section 3.8.4)
computing poles by QZ
ZerosByQZ (Section 3.8.5)
computing zeros by QZ
RootLocusByQZ (Section 3.8.6)
LREigenpair (Section 3.8.7)
computing root locus by QZ
computing eigenvalues and eigenvectors by JOCM
3.8 Pole/Zero Analysis
305
3.8.1 GeneralizedEigensystem
GeneralizedEigensystem[A, B]
computes the eigenvalues and the corresponding left and
right eigenvectors of the matrix pencil A B
Command structure of GeneralizedEigensystem .
GeneralizedEigensytem computes the eigenvalues and the corresponding left and right eigenvectors of a matrix pencil A B by the QZ algorithm. The function returns a list of lists of the form
{{lambda, v, u, iter}, † † † }, where lambda denotes a finite eigenvalue of A B, v and u the corresponding left and right eigenvectors, and iter the number of iterations the QZ algorithm needed to
compute lambda.
Note that this function is accessible only if the global Analog Insydes option UseExternals is set to
True and if a native version of the external MathLink module QZ.exe is available for your machine
(see Section 3.13.4).
GeneralizedEigensystem has the following option:
option name
default value
Normalize
True
whether to normalize the eigenvectors
Option for GeneralizedEigensystem .
Unless you specify Normalize −> False, the eigenvectors are scaled such that vi ui .
See also: GeneralizedEigenvalues (Section 3.8.2), UseExternals (Section 3.14.7).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define two square
real-valued matrices.
In[2]:= A = {{1, 2}, {−3, 1}};
B = {{1, −1}, {0, 0}};
Compute the eigenvalues
and eigenvectors of A B.
In[4]:= eigsys = GeneralizedEigensystem[A, B]
Get the eigenvalue and
corresponding
eigenvectors.
In[5]:= {lambda, v, u, iter} = First[eigsys];
Out[4]=
3.5, 0.5547, 0.83205, 0.316228, 0.948683, 0
To show that lambda, v, and u are indeed solutions of the GEP within machine precision, we compute
the L norms of the residual vectors rv and ru :
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306
rv vH A ΛB
ru A ΛBu
Compute the residual
norms.
In[6]:= {L2Norm[Conjugate[v].(A − lambda*B)],
L2Norm[(A − lambda*B).u]}
Out[6]= 4.57757 1016 , 3.29562 1016 3.8.2 GeneralizedEigenvalues
GeneralizedEigenvalues[A, B]
computes the eigenvalues of the matrix pencil A B where
A and B must be real-valued square matrices
Command structure of GeneralizedEigenvalues .
GeneralizedEigenvalues computes the eigenvalues of a matrix pencil A B by the QZ algorithm.
The function returns a list of the finite Λi that satisfy the GEP defined by A B.
Note that this function is accessible only if the global Analog Insydes option UseExternals is set to
True and if a native version of the external MathLink module QZ.exe is available for your machine
(see Section 3.13.4).
See also: GeneralizedEigensystem (Section 3.8.1), UseExternals (Section 3.14.7).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define two square
real-valued matrices.
In[2]:= A = {{1, 2}, {−3, 1}};
B = {{1, 4}, { 2, 1}};
Compute the eigenvalues
of A B.
In[4]:= GeneralizedEigenvalues[A, B]
Out[4]= 0.514618, 1.94319
GeneralizedEigenvalues also works if some or all of the eigenvalues are complex and in the case
where A or B are singular.
Define two further B
matrices.
In[5]:= B2 = {{1, −5}, {2, 1}};
B3 = {{1, −1}, {0, 0}};
Compute the eigenvalues
of A B .
In[7]:= GeneralizedEigenvalues[A, B2]
Compute the eigenvalues
of A B .
In[8]:= GeneralizedEigenvalues[A, B3]
Out[7]=
0.772727 0.198132 I, 0.772727 0.198132 I
Out[8]= 3.5
3.8 Pole/Zero Analysis
307
3.8.3 PolesAndZerosByQZ
PolesAndZerosByQZ[dae, var]
calculates the poles of the system described by the AC
DAEObject dae and the zeros with respect to var
Command structure of PolesAndZerosByQZ .
To calculate the poles and zeros of a linear circuit, the coefficient matrix of the corresponding system
of circuit equations must be decomposed into a matrix pencil A B. PolesAndZerosByQZ performs
this decomposition and calls GeneralizedEigenvalues to compute the poles and zeros numerically.
The argument dae must be an AC DAEObject written in matrix form (see CircuitEquations in
Section 3.5.1).
PolesAndZerosByQZ returns a list of rules of the form:
{Poles −> {poles}, Zeros −> {zeros}}
The return value can directly be used as input for RootLocusPlot for visualizing the pole/zero
distribution.
Calculating zeros requires the circuit to be a single-input system, i.e. the circuit must be
excited by a single AC input source.
Note that this function is accessible only if the global Analog Insydes option UseExternals is set to
True and if a native version of the external MathLink module QZ.exe is available for your machine
(see Section 3.13.4).
See also: PolesByQZ (Section 3.8.4), ZerosByQZ (Section 3.8.5), RootLocusPlot (Section 3.9.5),
UseExternals (Section 3.14.7).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Read in a PSpice netlist
and small-signal data.
In[2]:= buffer = ReadNetlist[
"AnalogInsydes/DemoFiles/Buffer.cir",
"AnalogInsydes/DemoFiles/Buffer.out",
Simulator −> "PSpice"]
Out[2]= Circuit Set up a system of AC
equations.
In[3]:= mnabuffer = CircuitEquations[buffer, AnalysisMode −> AC]
Out[3]= DAEAC, 18 18 3. Reference Manual
308
Calculate both poles and
zeros with respect to the
output voltage V$99 of the
circuit.
In[4]:= pz = PolesAndZerosByQZ[mnabuffer, V$99]
Show the pole/zero
distribution.
In[5]:= RootLocusPlot[pz, LinearRegionLimit −> Infinity,
PlotRange −> 5.0*^6]
Out[4]=
Poles 1.93945 1011 , 3.51956 1010 , 279586. 2.89305 106 I,
279586. 2.89305 106 I, 3.20148 106 , 3.22422 106 ,
2.26133 109 , 1.31955 1010 , 6.86928 109 , 8.83669 109 ,
Zeros 1.31966 1010 , 7.47584 109 , 2.26016 109 ,
1.52864 106 4.19572 106 I, 1.52864 106 4.19572 106 I,
2.98706 106 , 2.40024 106 , 1.59415 1011 ,
2.1154 1011 4.95934 109 I, 2.1154 1011 4.95934 109 I
Im s
6
4. 10
6
2. 10
6
-4. 10
6
6
-2. 10
2. 10
6
Re s
4. 10
6
-2. 10
6
-4. 10
Out[5]= Graphics 3.8.4 PolesByQZ
PolesByQZ[dae]
calculates the poles of the dynamic system described by
the AC DAEObject dae
Command structure of PolesByQZ.
To calculate the poles of a linear circuit, the coefficient matrix of the corresponding system of circuit
equations must be decomposed into a matrix pencil A B. PolesByQZ performs this decomposition
and calls GeneralizedEigenvalues to compute the poles numerically. The argument dae must be
an AC DAEObject written in matrix form (see CircuitEquations in Section 3.5.1).
Note that this function is accessible only if the global Analog Insydes option UseExternals is set to
True and if a native version of the external MathLink module QZ.exe is available for your machine
(see Section 3.13.4).
See also: PolesAndZerosByQZ (Section 3.8.3), ZerosByQZ (Section 3.8.5),
UseExternals (Section 3.14.7).
3.8 Pole/Zero Analysis
309
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Read in a PSpice netlist
and small-signal data.
In[2]:= buffer = ReadNetlist[
"AnalogInsydes/DemoFiles/Buffer.cir",
"AnalogInsydes/DemoFiles/Buffer.out",
Simulator −> "PSpice"]
Out[2]= Circuit Set up a system of AC
equations.
In[3]:= mnabuffer = CircuitEquations[buffer, AnalysisMode −> AC]
Calculate the poles of the
circuit.
In[4]:= PolesByQZ[mnabuffer]
Out[3]= DAEAC, 18 18 Out[4]=
1.93945 1011 , 3.51956 1010 , 279586. 2.89305 106 I,
279586. 2.89305 106 I, 3.20148 106 , 3.22422 106 ,
2.26133 109 , 1.31955 1010 , 6.86928 109 , 8.83669 109 3.8.5 ZerosByQZ
ZerosByQZ[dae, var]
calculates the zeros of the transfer function from the AC
DAEObject dae to the output variable var
Command structure of ZerosByQZ.
To calculate the zeros of a linear circuit, the coefficient matrix of the corresponding system of circuit
equations must be decomposed into a matrix pencil A B. ZerosByQZ performs this decomposition
and calls GeneralizedEigenvalues to compute the zeros numerically. The argument dae must be
an AC DAEObject written in matrix form (see CircuitEquations in Section 3.5.1).
Calculating zeros requires the circuit to be a single-input system, i.e. the circuit must be
excited by a single AC input source.
Note that this function is accessible only if the global Analog Insydes option UseExternals is set to
True and if a native version of the external MathLink module QZ.exe is available for your machine
(see Section 3.13.4).
See also: PolesAndZerosByQZ (Section 3.8.3), PolesByQZ (Section 3.8.4),
UseExternals (Section 3.14.7).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
3. Reference Manual
310
Read in a PSpice netlist
and small-signal data.
In[2]:= buffer = ReadNetlist[
"AnalogInsydes/DemoFiles/Buffer.cir",
"AnalogInsydes/DemoFiles/Buffer.out",
Simulator −> "PSpice"]
Out[2]= Circuit Set up a system of AC
equations.
In[3]:= mnabuffer = CircuitEquations[buffer, AnalysisMode −> AC]
Calculate the zeros of the
circuit with respect to the
output voltage V$99.
In[4]:= ZerosByQZ[mnabuffer, V$99]
Out[3]= DAEAC, 18 18 Out[4]=
1.31966 1010 , 7.47584 109 , 2.26016 109 ,
1.52864 106 4.19572 106 I, 1.52864 106 4.19572 106 I,
2.98706 106 , 2.40024 106 , 1.59415 1011 ,
2.1154 1011 4.95934 109 I, 2.1154 1011 4.95934 109 I
3.8.6 RootLocusByQZ
RootLocusByQZ[dae, var, {k, k , k , incr}]
calculates the root locus of the AC DAEObject dae as the
parameter k is swept from k to k in steps of incr; zeros
are calculated with respect to the output variable var
Command structure of RootLocusByQZ.
With RootLocusByQZ you can sweep a parameter of a linear system over an interval and calculate the
root locus of the system directly from the corresponding circuit equations using the QZ algorithm.
The argument dae has to be an AC DAEObject written in matrix form (see CircuitEquations in
Section 3.5.1).
RootLocusByQZ returns data in the following form:
{{Poles −> {poles}, Zeros −> {zeros},
SweepParameters −> {k −> k0 }}, }
The return value can directly be used as input for RootLocusPlot for visualizing the root locus.
Note that this function is accessible only if the global Analog Insydes option UseExternals is set to
True and if a native version of the external MathLink module QZ.exe is available for your machine
(see Section 3.13.4).
RootLocusByQZ has the following option:
3.8 Pole/Zero Analysis
311
option name
default value
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
Option for RootLocusByQZ .
See also: PolesAndZerosByQZ (Section 3.8.3), RootLocusPlot (Section 3.9.5), UseExternals (Section 3.14.7).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Read in a PSpice netlist
and small-signal data.
In[2]:= buffer = ReadNetlist[
"AnalogInsydes/DemoFiles/Buffer.cir",
"AnalogInsydes/DemoFiles/Buffer.out",
Simulator −> "PSpice"]
Out[2]= Circuit Set up system of symbolic
AC equations.
In[3]:= mnabuffersym = CircuitEquations[buffer,
AnalysisMode −> AC, ElementValues −> Symbolic]
Out[3]= DAEAC, 18 18 Get the design-point value
of Cbc$Q$T5 .
In[4]:= Cbc$Q$T5 /. GetDesignPoint[mnabuffersym]
Calculate root locus of the
circuit as Cbc$Q$T5 is
varied from pF to pF
in steps of pF.
In[5]:= rloc = RootLocusByQZ[mnabuffersym, V$99,
{Cbc$Q$T5, 5.0*^−12, 50.0*^−12, 5.0*^−12}];
Out[4]= 8.39 1012
3. Reference Manual
312
Show the root locus.
In[6]:= RootLocusPlot[rloc, LinearRegionLimit −> Infinity,
PlotRange −> 5.0*^6]
Cbc$Q$T5 = 5.000e-12 .. 5.000e-11 (0% .. 100%)
Im s
Poles
0%
6
4. 10
6
2. 10
100
6
6
-4. 10 -2. 10
6
2. 10
6
4. 10
Re s
Zeros
0%
6
-2. 10
6
-4. 10
100
Out[6]= Graphics 3.8.7 LREigenpair
LREigenpair[A, B, theta0]
LREigenpair[dae, theta0]
solves the GEP described by the matrix pencil A B for an
eigenvalue Λ near the initial guess theta0 where A and B
must be real-valued square matrices of equal dimensions
solves the GEP defined by the coefficient matrix of the AC
DAEObject dae for an eigenvalue Λ near theta0
Command structure of LREigenpair.
As an alternative to the QZ algorithm, Analog Insydes provides the function LREigenpair for solving
GEPs iteratively using a variant of the Jacobi orthogonal correction method. This approach is more efficient and reliable than the QZ algorithm for large GEPs when only one single eigenvalue is sought.
In addition to the eigenvalue, LREigenpair also yields the corresponding left and right eigenvectors.
The argument dae has to be an AC DAEObject written in matrix form (see CircuitEquations in
Section 3.5.1).
If the target eigenvalue is complex, the initial guess theta0 must also be a complex number.
LREigenpair returns a list of the form {{lambda, v, u, rv , ru , iter}}, where lambda, v, and u denote
the calculated eigenvalue and eigenvectors, rv and ru the corresponding L norms of the residual
vectors
3.8 Pole/Zero Analysis
313
rv vH A ΛB ru A ΛBu The last return value, iter, denotes the number of iterations needed to compute lambda with the
desired accuracy.
LREigenpair has the following options:
option name
default value
AccuracyGoal
Round[0.5*$MachinePrecision]
the accuracy goal for the sought eigenvalue
DesignPoint
Automatic
the design-point values for the coefficients
of dae
FrequencyVariable
Automatic
the name of the variable that represents the
complex frequency in dae
InitialGuess
Automatic
initial guesses for the left and right
eigenvectors
MaxIterations
60
the maximum number of iterations for the
Jacobi orthogonal correction method
Options for LREigenpair, Part I.
3. Reference Manual
314
option name
default value
MaxResidual
Automatic
the maximum L norm of the residual
vectors rv and ru
Prescaling
True
whether to prescale the rows of the matrix
pencil
ProjectionVectors
{LeftEigenvector, RightEigenvector}
the projection vectors used by the Jacobi
orthogonal correction method
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
Tolerance
Infinity
the maximum logarithmic distance between
the initial guess theta0 and the calculated
eigenvalue Λ
Options for LREigenpair, Part II.
See also: GeneralizedEigensystem (Section 3.8.1).
Options Description
A detailed description of all LREigenpair options is given below in alphabetical order:
AccuracyGoal
AccuracyGoal −> n specifies the desired accuracy of the sought eigenvalue in terms of the number
of accurate significant digits. The value of AccuracyGoal is used to determine an appropriate setting
for the MaxResidual option if MaxResidual −> Automatic .
DesignPoint
With DesignPoint −> dp, you can specify a list of design-point values for the coefficients of
a symbolic system of circuit equations. The default setting DesignPoint −> Automatic causes
LREigenpair to use the design-point information stored in dae. You can use the DesignPoint
option to override design-point data stored in dae.
FrequencyVariable
LREigenpair needs to know the symbol which represents the complex frequency in order to
be able to decompose the system of circuit equations dae into the matrices A and B. With
3.8 Pole/Zero Analysis
315
FrequencyVariable −> Automatic, the symbol is determined automatically from the status information stored in dae. You can specify the frequency variable manually with FrequencyVariable −> var.
InitialGuess
With InitialGuess −> {v0, u0}, you can specify initial guesses for the sought eigenvectors v and u.
Both v0 and u0 must be real or complex-valued numeric vectors whose dimensions are compatible
with dae.
MaxIterations
MaxIterations −> n specifies the maximum number of orthogonal correction iterations LREigenpair
should use in order to find an eigenvalue and the corresponding eigenvectors. If LREigenpair fails
to find solutions which satisfy the MaxResidual specification within n iterations, a warning is
generated and the most recent iterates are returned.
MaxResidual
With MaxResidual −> maxres, you can set the convergence criterion for LREigenpair . The value
maxres denotes the maximum L norm the residual vectors rv and ru may have such that the current
set of iterates can be regarded as valid approximations of the sought eigenpairs. LREigenpair stops
iterating as soon as both rv and ru fall below maxres.
With the default setting MaxResidual −> Automatic , the maximum residual norm is computed as
maxres AccuracyGoal /* / B A Note that the value of maxres depends on the setting of the Prescaling option because the norms
of A and B are computed after prescaling.
Prescaling
With Prescaling −> True, LREigenpair scales the rows of the matrix pencil A B such that the absolute value of the largest coefficient in each row is 1. In general, prescaling improves the numerical
properties of ill-conditioned GEPs and helps to reduce the number of iterations. Prescaling does not
change the eigenvalues and eigenvectors of the matrix pencil. You can turn off prescaling by setting
Prescaling −> False.
ProjectionVectors
Starting with a given initial guess * v u for the sought eigenvalue and eigenvectors, LREigenpair
repeatedly solves the correction equation
3. Reference Manual
316
A *i B w̃ z r ỹH
for updates z of u and v. The symbols w̃ and ỹ denote two arbitrary projection vectors, which
must be chosen appropriately in each step to ensure convergence of the iterates. Suitable choices
for w̃ and ỹ include combinations of the initial guesses and the most recent approximations of the
eigenvectors v and u.
With the option ProjectionVectors −> {wt, yt}, you can choose the vectors LREigenpair uses as
projection vectors. Possible values are:
LeftEigenvector
RightEigenvector
InitialLeftEigenvector
InitialRightEigenvector
the most recent approximation of the left eigenvector v
the most recent approximation of the right eigenvector u
the initial guess v for the left eigenvector
the initial guess u for the right eigenvector
Values for the ProjectionVectors option.
In most cases, the default setting ProjectionVectors −> {LeftEigenvector, RightEigenvector}
constitutes an appropriate choice. You will need to change the setting of this option only if you
encounter convergence problems.
Tolerance
Tolerance −> tol specifies the radius of the tolerance region around the initial guess theta0 in which
the sought eigenvalue Λ should lie. LREigenpair generates a warning if it converges to an eigenvalue that lies outside the tolerance region. The Tolerance option allows you to check whether
LREigenpair has found the eigenvalue you wished to compute or whether the iterations have
converged to a completely different solution.
Tolerance uses a logarithmic measure for distances. For example, Tolerance −> 1.0 allows the
value of the solution Λ to lie within and times the value of the initial guess theta0.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define three square
real-valued matrices.
In[2]:= A = {{1, 2}, {−3, 1}};
B = {{1, 4}, { 2, 1}};
B2 = {{1, −1}, {0, 0}};
Solve the GEP A B iteratively using the initial
guess theta .
In[5]:= LREigenpair[A, B2, −3.0]
Out[5]=
3.5, 0.5547, 0.83205, 0.316228, 0.948683, 9.48575 1016 ,
1.17657 1015 , 5
3.8 Pole/Zero Analysis
317
Solve the GEP A B.
Specify initial guesses for
the eigenvectors.
In[6]:= LREigenpair[A, B, −1., InitialGuess −> {{0, 1}, {1, 0}}]
Use the default projection
vectors.
In[7]:= LREigenpair[A, B, 1., MaxResidual −> 1.*^−15]
Out[6]=
1.94319, 0.288369, 0.957519, 0.957519, 0.288369,
5.05207 1010 , 8.70217 1010 , 4
Out[7]=
0.514618, 0.992822, 0.1196, 0.1196, 0.992822,
2.23773 1016 , 2.48253 1016 , 4
Note that the following setting for ProjectionVectors increases the number of iterations.
Use right eigenvectors as
projection vectors.
In[8]:= LREigenpair[A, B, 1., MaxResidual −> 1.*^−15,
ProjectionVectors −>
{RightEigenvector, RightEigenvector}]
Out[8]=
0.514618, 0.992822, 0.1196, 0.1196, 0.992822, 0.,
4.04127 1016 , 5
The setting Tolerance −> 1.0 allows the value of the solution Λ to lie within and times the
value of the initial guess theta0.
Search for an eigenvalue
in the interval In[9]:= LREigenpair[A, B, −100., Tolerance −> 1.]
LREigenpair::tolx:
Eigenvalue lies outside the specified tolerance region.
Out[9]=
1.94319, 0.288369, 0.957519, 0.957519, 0.288369,
2.70895 1011 , 4.66522 1011 , 6
With the default setting Tolerance −> Infinity , any solution of the GEP is accepted without a
warning.
Do not specify a tolerance
region.
In[10]:= LREigenpair[A, B, −100., Tolerance −> Infinity]
Read in a PSpice netlist
and small-signal data.
In[11]:= buffer = ReadNetlist[
"AnalogInsydes/DemoFiles/Buffer.cir",
"AnalogInsydes/DemoFiles/Buffer.out",
Simulator −> "PSpice"]
Out[10]=
1.94319, 0.288369, 0.957519, 0.957519, 0.288369,
2.70895 1011 , 4.66522 1011 , 6
Out[11]= Circuit Set up a system of
symbolic AC equations.
In[12]:= mnabuffersym = CircuitEquations[buffer,
AnalysisMode −> AC, ElementValues −> Symbolic]
Out[12]= DAEAC, 18 18 3. Reference Manual
318
Find the pole of the buffer
circuit near
theta i and the
corresponding left and
right eigenvectors.
In[13]:= lreigenpairs = LREigenpair[mnabuffersym, 3.0*^6 I]
Show the eigenvalue.
In[14]:= lreigenpairs[[1, 1]]
Out[13]=
279586. 2.89305 106 I,
1.70056 1018 2.18582 1018 I, 0.101704 0.218124 I,
0.0281952 0.414952 I, 0.175266 0.0104981 I,
1.70111 1018 2.18582 1018 I, 0.177763 0.0195432 I,
0.000145936 0.000121065 I, 0.177854 0.0196025 I,
0.175419 0.0103065 I, 0.175107 0.0106381 I,
0.0284306 0.414988 I, 0.0281891 0.414953 I,
0.175267 0.0105033 I, 0.0281827 0.414949 I,
0.175266 0.0104982 I, 0.177756 0.019539 I,
0.0000145936 0.0000121065 I, 0.0000145936 0.0000121065 I,
9.51237 1019 4.50366 1019 I, 0.055128 0.256781 I,
0.108991 0.0369544 I, 0.0952012 0.0116717 I,
9.51493 1019 4.50349 1019 I, 0.109187 0.515199 I,
0.000239817 0.0000548798 I, 0.108876 0.515278 I,
0.0952466 0.0115085 I, 0.0952183 0.0115916 I,
0.10892 0.0367968 I, 0.108987 0.0369774 I,
0.0952038 0.0116972 I, 0.109 0.0369492 I,
0.0951948 0.011671 I, 0.109209 0.515193 I,
0.0000239817 5.48798 106 I, 0.0000239817 5.48798 106 I,
3.61927 1013 , 6.03057 1012 , 4
Out[14]= 279586. 2.89305 106 I
Compare the eigenvalue
with the result of the QZ
algorithm.
In[15]:= Cases[PolesByQZ[mnabuffersym], _Complex]
Out[15]=
279586. 2.89305 106 I, 279586. 2.89305 106 I
3.8.8 ApproximateDeterminant
ApproximateDeterminant[dae, lambda, error]
approximates the equations dae with respect to the pole
closest to the initial guess lambda where dae has to be an
AC DAEObject and error has to be a positive real value
ApproximateDeterminant[dae, zvar, lambda, error]
approximates the equations dae with respect to the zero of
the transfer function from the input signal to the output
zvar
Command structure of ApproximateDeterminant .
With ApproximateDeterminant you can approximate a linear symbolic matrix equation Ts x b
directly with respect to a particular eigenvalue Λ (a pole or a zero). By discarding all terms which
have little or no influence on Λ, ApproximateDeterminant reduces both the complexity and the
degree of the characteristic polynomial Ps det Ts. Provided that the eigenvalue of interest is
located sufficiently far apart from other eigenvalues, the polynomial degree can be reduced to if Λ
3.8 Pole/Zero Analysis
319
is real or if Λ is complex. The argument dae must be an AC DAEObject written in matrix form (see
CircuitEquations in Section 3.5.1).
The following options can be used with ApproximateDeterminant :
option name
default value
AccuracyGoal
Round[0.5*$MachinePrecision]
the desired numerical accuracy of numerical
computations
DesignPoint
Automatic
the design-point values for the coefficients
of dae
ErrorFunction
Automatic
the function to be used for calculating the
approximation error
FrequencyVariable
Automatic
the symbol which denotes the complex
Laplace frequency
GEPSolver
LREigenpair
the GEP solver to be used for calculating
the initial reference solution
GEPSolverOptions
Automatic
options which are passed to the GEP solver
Options for ApproximateDeterminant , Part I.
3. Reference Manual
320
option name
default value
InitialSolution
Automatic
the initial reference solution
MaxDivergentSteps
1
the maximum number of divergent
iterations allowed in the error tracking
process
MaxIterations
3
the maximum number of eigenvalue
correction iterations allowed in each error
tracking step
MaxResidual
Automatic
the convergence criterion for the error
tracking iterations
MaxShift
1.0
the maximum relative eigenvalue sensitivity
a matrix entry may have to be considered
as a candidate for removal
MinMAC
0.95
the minimum MAC value between the
original eigenvector and the corresponding
eigenvector of the approximated system
Options for ApproximateDeterminant , Part II.
3.8 Pole/Zero Analysis
321
option name
default value
Prescaling
True
ProjectionVectors
{LeftEigenvector, RightEigenvector}
the projection vectors used by the Jacobi
orthogonal correction method
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
SingularityTest
SingularityTestByLU
whether to prescale the rows of the matrix
pencil
the function used to determine whether an
approximated matrix is singular
TestFrequency
1.
the frequency or point in the complex plane
at which singularity tests are performed
Tolerance
0.05
the maximum logarithmic distance between
the initial guess and the calculated
eigenvalue Λ
Options for ApproximateDeterminant , Part III.
See also: GeneralizedEigensystem (Section 3.8.1), LREigenpair (Section 3.8.7).
Options Description
A detailed description of all ApproximateDeterminant options is given below in alphabetical order:
AccuracyGoal
AccuracyGoal −> n specifies the desired accuracy of the sought eigenvalue in terms of the number
of accurate significant digits. The value of AccuracyGoal is used to determine an appropriate setting
for the MaxResidual option if MaxResidual −> Automatic . See also LREigenpair (Section 3.8.7).
DesignPoint
With DesignPoint −> dp, you can specify a list of design-point values for the coefficients of
a symbolic system of circuit equations. The default setting DesignPoint −> Automatic causes
ApproximateDeterminant to use the design-point information stored in dae. You can use the
DesignPoint option to override design-point data stored in dae.
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322
ErrorFunction
ErrorFunction −> func specifies the function to be used for calculating the error of an approximated
eigenvalue with respect to the reference solution. The following values are allowed:
Automatic
select the error function according to the properties of the
eigenvalue of interest: use LogError for real eigenvalues
and DirectedLogError for complex ones
LogError
yields the logarithmic difference of the magnitudes of two
eigenvalues; most suitable for real eigenvalues
DirectedLogError
similar to LogError , but takes into account the phases of
complex numbers; most suitable for complex eigenvalues
Values for the ErrorFunction option.
FrequencyVariable
ApproximateDeterminant needs to know the symbol which represents the complex frequency in
order to be able to decompose the system of circuit equations dae into the matrices A and B. With
FrequencyVariable −> Automatic , the symbol is determined automatically from the status information stored in dae. You can specify the frequency variable manually with FrequencyVariable −> var.
GEPSolver
With GEPSolver −> func, you can select the GEP solver ApproximateDeterminant uses to calculate
the initial reference solution. Possible option values are:
GeneralizedEigensystem
use the QZ algorithm for computing the reference solution
LREigenpair
use the Jacobi orthogonal correction method for computing
the reference solution
Values for the GEPSolver option.
GEPSolverOptions
With GEPSolverOptions −> opts, you can specify a list of option settings that will be passed to the
selected GEP solver, for example:
GEPSolverOptions −> {MaxResidual −> 1.*^−7,
MaxIterations −> 100}
3.8 Pole/Zero Analysis
323
Note that you may also change the option settings of the selected GEP solver directly with
SetOptions[gepsolver, opts]. However, with GEPSolverOptions , you can specify private option
settings which will only be used in conjunction with ApproximateDeterminant .
InitialSolution
With InitialSolution −> initsol, you can specify the initial reference solution for the GEP to be
approximated. The value of InitialSolution must be given in the same format as the return value
of LREigenpair (Section 3.8.7). Possible values are:
Automatic
{lambda , v , u , † † † }
compute the initial reference solution using the GEP solver
specified with GEPSolver
use the given initial reference solution
Values for the InitialSolution option.
MaxDivergentSteps
MaxDivergentSteps −> n specifies the maximum number of divergent iterations allowed in the error tracking step following the elimination of a matrix entry. Iterates are considered divergent if
the residual of the numerical solution of the GEP becomes larger between two consecutive steps.
If the number of divergent steps exceeds the specified maximum in the error tracking process,
ApproximateDeterminant aborts the iterations, reinserts the current term into the matrix, and
continues with the next term.
MaxIterations
MaxIterations −> n specifies the maximum number of error tracking iterations performed after
removing a matrix entry. An approximation is considered valid if the iterations converge within
n steps, and if both the MaxResidual and MinMAC specifications are satisfied. See also LREigenpair
(Section 3.8.7).
If you set GEPSolver −> LREigenpair , note that the MaxIterations setting given for
ApproximateDeterminant is not passed to LREigenpair . To specify the maximum number
of iterations for the GEP solver, change the value of GEPSolverOptions .
MaxResidual
MaxResidual −> posreal specifies the convergence criterion for the error tracking iterations.
MaxResidual is one of the key options you should play with in order to obtain good results from
ApproximateDeterminant . You should choose the value as large as possible to allow for a reason-
3. Reference Manual
324
able error but small enough to prevent the iterations from “converging” to an arbitrary value. This
may require some experimentation. See also LREigenpair (Section 3.8.7).
The value of MaxResidual depends on the setting of the Prescaling option.
If you set GEPSolver −> LREigenpair , note that the MaxIterations setting given for
ApproximateDeterminant is not passed to LREigenpair . To specify the maximum residual
for the GEP solver, change the value of GEPSolverOptions .
MaxShift
MaxShift −> posreal restricts the list of matrix entries which are candidates for removal to those
entries with a eigenvalue sensitivity figure less than posreal. With MaxShift −> 1.0, a term will be
considered for removal if the eigenvalue shift caused by removing the term has been predicted to be
less than . To restrict the list of matrix entries to terms with small eigenvalue sensitivities, use
a setting such as MaxShift −> 0.1.
MinMAC
MinMAC −> posreal specifies the minimum allowed value of the modal assurance criterion between the
right eigenvector of the original GEP, u, and the right eigenvector of the approximated GEP, u+ .
The MAC between u and u+ constitutes a measure for the correlation of the two eigenvectors. It is
defined as
+
MAC u u uH u+ H
+H
u u u u+ The value of the MAC ranges from to . A value of means that one eigenvector is a multiple of
the other. A value of means that the two vectors are completely uncorrelated.
The use of the MAC helps to prevent ApproximateDeterminant from accidentally converging to
some other eigenvalue than the one of interest. For corresponding eigenpairs of the original and
approximated GEP, the value of the MAC should be close to , say MAC , whereas the MAC
between the eigenvector of the original GEP and some other eigenvector of the approximated system
should be small. ApproximateDeterminant considers an approximation step as invalid if the MAC
between the two eigenvectors falls below the value of MinMAC.
Prescaling
With Prescaling −> True, ApproximateDeterminant scales the rows of the matrix pencil A B
such that the absolute value of the largest coefficient in each row is 1. In general, prescaling improves the numerical properties of ill-conditioned GEPs and helps to reduce the number of iterations.
3.8 Pole/Zero Analysis
325
Prescaling does not change the eigenvalues and eigenvectors of the matrix pencil. You can turn off
prescaling by setting Prescaling −> False.
The value of Prescaling has an influence on the required setting of the MaxResidual
option.
If you set GEPSolver −> LREigenpair , note that the Prescaling setting given for
ApproximateDeterminant is not passed to LREigenpair . To turn on prescaling for the GEP
solver, change the value of GEPSolverOptions .
ProjectionVectors
With the option ProjectionVectors −> {wt, yt}, you can choose the vectors
ApproximateDeterminant uses as projection vectors for the Jacobi orthogonal correction method
(JOCM) during error tracking operations:
Starting with a given initial guess * v u for the sought eigenvalue and eigenvectors, the JOCM
repeatedly solves the correction equation
A *i B w̃ z r ỹH
for updates z of u and v. The symbols w̃ and ỹ denote two arbitrary projection vectors, which
must be chosen appropriately in each step to ensure convergence of the iterates. Suitable choices
for w̃ and ỹ include combinations of the initial guesses and the most recent approximations of the
eigenvectors v and u.
Possible settings for the ProjectionVectors option are:
LeftEigenvector
RightEigenvector
InitialLeftEigenvector
InitialRightEigenvector
the most recent approximation of the left eigenvector v
the most recent approximation of the right eigenvector u
the initial guess v for the left eigenvector
the initial guess u for the right eigenvector
Values for the ProjectionVectors option.
In most cases, the default setting ProjectionVectors −> {LeftEigenvector, RightEigenvector}
constitutes an appropriate choice. You will need to change the setting of this option only if you
encounter convergence problems. See also Section 3.8.7.
3. Reference Manual
326
If you set GEPSolver −> LREigenpair , note that the ProjectionVectors setting given for
ApproximateDeterminant is not passed to LREigenpair . To choose the projection vectors
for the GEP solver, change the value of GEPSolverOptions .
SingularityTest
After each approximation step, ApproximateDeterminant applies a singularity test to the matrix
A s B to ensure that removing a matrix entry has not rendered the GEP singular. The singularity
test is performed by numerical computation of the rank of A s B for some s " C which is not
an eigenvalue of the GEP. However, there is no guarantee that numerical rank computation always
yields a mathematically correct result. This may cause singularity to remain undetected in some
situations, particularly when the GEP is ill-conditioned. With the option SingularityTest , you
can select the function ApproximateDeterminant uses to determine whether the GEP is singular. If
you encounter singularity problems, i.e. if detA s B after approximating a system of circuit
equations, then you should change the value of SingularityTest and rerun the approximation.
The possible values for SingularityTest are:
SingularityTestByLU
perform singularity tests by rank computation using
LUDecomposition
SingularityTestByQR
perform singularity tests by rank computation using
QRDecomposition
Function[{M}, expr]
specify a user-defined function which returns True if the
complex-valued floating-point matrix M is singular
Values for the SingularityTest option.
TestFrequency
With the option TestFrequency , you can specify the value of the complex frequency variable s which
is used for testing whether the numerical matrix A s B is singular. For best numerical accuracy and
computing performance, it is recommended that you choose a real value of the same order of magnitude as the modulus of the target eigenvalue. For example, if you wish to approximate a GEP with
respect to an eigenvalue s j , then the option setting TestFrequency −> 1.0*^8
constitutes an appropriate choice.
The point in the complex plane represented by the value of TestFrequency should not lie
in the neighborhood of any eigenvalue of the GEP to be approximated. An inappropriate
choice of the test frequency may result in ill-conditioned rank computation problems.
3.8 Pole/Zero Analysis
327
Tolerance
Tolerance −> tol specifies the radius of the tolerance region around the initial guess theta0 in which
the sought eigenvalue Λ should lie. ApproximateDeterminant generates a warning if it converges
to an eigenvalue that lies outside the tolerance region. The Tolerance option allows you to check
whether ApproximateDeterminant has found the eigenvalue you wished to compute or whether
the iterations have converged to a completely different solution.
Tolerance uses a logarithmic measure for distances. For example, Tolerance −> 1.0 allows the
value of the solution Λ to lie within and times the value of the initial guess theta0.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define an AC model for
BJTs.
In[2]:= Circuit[
Model[
Name −> BJT,
Selector −> AC,
Scope −> Global,
Ports −> {B, C, E},
Parameters −> {Rbe, Cbc, Cbe, beta, Ro,
RBE, CBC, CBE, BETA, RO},
Definition −> Netlist[
{CBE, {B, E}, Symbolic −> Cbe, Value −> CBE},
{RBE, {X, E}, Symbolic −> Rbe, Value −> RBE},
{CBC, {B, C}, Symbolic −> Cbc, Value −> CBC},
{CC, {B, X, C, E},
Symbolic −> beta, Value −> BETA},
{RO, {C, E}, Symbolic −> Ro, Value −> RO}
]
]
] // ExpandSubcircuits;
This netlist describes a
common-emitter amplifier.
In[3]:= ceamplifier =
Circuit[
Netlist[
{V1, {1, 0}, 1},
{V0, {6, 0}, 0},
{C1, {1, 2}, Symbolic −> C1, Value
{R1, {2, 6}, Symbolic −> R1, Value
{R2, {2, 0}, Symbolic −> R2, Value
{RC, {6, 3}, Symbolic −> RC, Value
{RE, {4, 0}, Symbolic −> RE, Value
{C2, {3, 5}, Symbolic −> C2, Value
{RL, {5, 0}, Symbolic −> RL, Value
{Q1, {2 −> B, 3 −> C, 4 −> E},
Model −> BJT, Selector −> AC,
CBE −> 30.*^−12, CBC −> 5.*^−12,
RO −> 10000., BETA −> 200.}
]
]
Out[3]= Circuit −>
−>
−>
−>
−>
−>
−>
1.0*^−7},
1.0*^5},
47000.},
2200.},
1000.},
1.0*^−6},
47000.},
RBE −> 1000.,
3. Reference Manual
328
Set up system of symbolic
AC equations.
In[4]:= eqs = CircuitEquations[ceamplifier,
ElementValues −> Symbolic]
Out[4]= DAEAC, 10 10 Calculate poles and zeros
of the voltage transfer
function.
In[5]:= pz = PolesAndZerosByQZ[eqs, V$5]
Show the pole/zero
distribution.
In[6]:= RootLocusPlot[pz, LinearRegionLimit −> 100,
PlotRange −> 5.0*^10]
Out[5]=
Poles 20.4778, 375.176, 9.50936 107 , 6.75675 109 ,
Zeros 0., 1.35569 1027 , 1.91793 108 , 6.94846 109 Im s
1.0E8
1.0E5
100.
-1.0E8-1.0E5 -100.
100. 1.0E5 1.0E8
Re s
-100.
-1.0E5
-1.0E8
Out[6]= Graphics The numerical pole/zero analysis shows that the characteristic polynomial of the equations has a
degree of four because the amplifier has four poles. Although polynomial equations with a degree
up to four can be solved analytically, the resulting expressions are usually very complex if the degree
is greater than two. Therefore, we use ApproximateDeterminant to compute simplified formulas
for the poles.
We begin by computing an approximate expression for the pole near s . The logarithmic error
bound specification corresponds to a tolerance region from to of the
design-point value p .
Approximate the equations
with respect to p using
the QZ algorithm for
computing the reference
solution.
In[7]:= sbgp2 = ApproximateDeterminant[eqs, −400, 0.05,
MaxIterations −> 4, GEPSolver −> GeneralizedEigensystem]
Out[7]= DAEAC, 10 10 To check if the algorithm has successfully isolated the pole of interest, we compute the poles of the
approximated system.
Compute the remaining
poles numerically.
In[8]:= PolesByQZ[sbgp2]
Estimate the number of
remaining terms.
In[9]:= ComplexityEstimate[sbgp2]
Out[8]= 0., 362.766
Out[9]= 4
3.8 Pole/Zero Analysis
Show complexity of
original system.
329
In[10]:= ComplexityEstimate[eqs]
Out[10]= 132
The result shows that the algorithm has eliminated the two high-frequency poles and has approximated the low-frequency pole p near s by zero. On the contrary, the pole p has been
preserved, and its numerical value lies in the specified tolerance region. In addition, the complexity
of the characteristic polynomial has been greatly reduced. Therefore, we can expect to obtain an
interpretable formula for p .
Compute the
approximated
characteristic polynomial.
In[11]:= detp2 = Det[GetMatrix[sbgp2]] // Together
Solve for the poles.
In[12]:= sbgpoles = Solve[detp2 == 0, s]
Out[11]=
C2 R1 R2 s beta$Q1 C2 R1 RE s beta$Q1 C2 R2 RE s beta$Q1 C1 C2 R1 R2 RE s2 R1 R2 Rbe$Q1 RC RE
Out[12]=
R1 R2 beta$Q1 R1 RE beta$Q1 R2 RE
s 0, s beta$Q1 C1 R1 R2 RE
Extract the pole of interest.
In[13]:= p2 = s /. Last[sbgpoles] // Simplify
1
1
1
R1
R2
beta$Q1 RE
Out[13]= C1
Next, we try to extract the right-half plane zero z near s .
Approximate the equations
with respect to z .
In[14]:= sbgz3 = ApproximateDeterminant[eqs, V$5, 2.0*^8, 0.05,
GEPSolver −> GeneralizedEigensystem]
Out[14]= DAEAC, 10 10 Compute the remaining
zeros numerically.
In[15]:= ZerosByQZ[sbgz3, V$5]
Compute the
approximated polynomial.
In[16]:= detz3 = Det[GetMatrix[sbgz3]] // Simplify
Solve for the zeros.
In[17]:= sbgzeros = Solve[detz3 == 0, s]
Out[15]= 0., 0., 2. 108 beta$Q1 C1 C2 s2 1 Cbc$Q1 RE s
Out[16]= Rbe$Q1 RE
1
Out[17]= s 0, s 0, s Cbc$Q1 RE
Extract the zero of interest.
In[18]:= z3 = s /. Last[sbgzeros]
1
Out[18]= Cbc$Q1 RE
3. Reference Manual
330
3.9 Graphics Functions
Analog Insydes extends Mathematica’s graphics functionality by several custom functions for producing frequently used graphs in circuit design. The package provides special functionality for
displaying the following types of plots:
BodePlot (Section 3.9.1)
FourierPlot (Section 3.9.2)
NicholPlot (Section 3.9.3)
NyquistPlot (Section 3.9.4)
Bode plot (frequency response: magnitude and phase vs.
frequency)
Fourier spectrum plot (frequency spectrum: spectral
magnitude vs. frequency)
Nichol plot (frequency response: gain vs. phase)
Nyquist plot (frequency response: imaginary vs. real part)
RootLocusPlot (Section 3.9.5)
pole/zero and root locus plot (parameter response:
imaginary vs. real part)
TransientPlot (Section 3.9.6)
transient waveform plot (signal value vs. time)
3.9.1 BodePlot
BodePlot[func, {var, f , f }]
displays a Bode plot of the transfer function func[var] as
the independent variable var is swept from f to f
BodePlot[{func , func , † † † }, {var, f , f }]
displays the transfer functions func , func , † † †
simultaneously in a Bode plot
Command structure of BodePlot.
BodePlot displays one or several transfer functions in a Bode plot. A Bode plot consists of two
separate charts in which the magnitude and phase of a transfer function are plotted vs. frequency.
Magnitude and phase are displayed, respectively, on logarithmic and linear scales. The frequency
axis is scaled logarithmically for both charts.
Note that BodePlot has attribute HoldFirst .
BodePlot supports additional patterns for displaying numerical data generated with the functions ACAnalysis (Section 3.7.3), NoiseAnalysis (Section 3.7.4), or ReadSimulationData (Section 3.10.3):
3.9 Graphics Functions
331
BodePlot[traces, {var, f , f }]
displays a list of AC traces computed with e.g.
ACAnalysis in a Bode plot
BodePlot[traces, expr, {var, f , f }]
displays the value of an expression in terms of AC traces
BodePlot[traces, {expr , expr , † † † }, {var, f , f }]
displays the values of several expressions
Displaying numerical data with BodePlot.
You can customize the appearance of Bode plots with the options listed below. In addition, BodePlot
inherits many options from LogLinearListPlot and Legend. Both the options which are specific to
BodePlot as well as the inherited options can be set with SetOptions[BodePlot, opts].
option name
default value
AspectRatio
0.75
the aspect ratio of the whole plot
AspectFactor
0.8
a correction factor by which the aspect
ratios of the magnitude and phase plots are
multiplied so as to fill the individual plot
areas optimally
FrequencyScaling
Exponential
whether to distribute sampling points
exponentially or linearly along the
frequency axis
GraphicsSpacing
0.05
the distance between the magnitude and the
phase plot in percent of the total plot height
MagnitudeDisplay
Decibels
how to scale and display magnitude values
Options for BodePlot, Part I.
3. Reference Manual
332
option name
default value
PhaseDisplay
Degrees
whether to display phase traces and which
unit to use for phase values
PlotPoints
25
the number of frequency sampling points
PlotRange
Automatic
the plot ranges for the magnitude and the
phase plot
ShowLegend
True
whether to display a plot legend
TraceNames
Automatic
the trace names displayed in the plot legend
UnwrapPhase
True
whether to unwrap phase traces
UnwrapTolerance
0.2
the maximum relative difference between
two successive phase values that triggers
phase unwrapping
Options for BodePlot, Part II.
See also: ACAnalysis (Section 3.7.3), NoiseAnalysis (Section 3.7.4), ReadSimulationData (Section 3.10.3), NicholPlot (Section 3.9.3), NyquistPlot (Section 3.9.4).
Options Description
A detailed description of all options that are specific to BodePlot or are used in a non-standard way
is given below in alphabetical order:
FrequencyScaling
The option FrequencyScaling determines how sampling points are distributed over the frequency
axis. Possible values are:
Exponential
Linear
use exponentially spaced sampling points
use linearly spaced sampling points
Values for the FrequencyScaling option.
MagnitudeDisplay
With the option MagnitudeDisplay , you can specify how magnitude values are displayed in a Bode
plot. The following values are allowed:
3.9 Graphics Functions
333
Decibels
AbsoluteValues
Linear
convert magnitude values to decibels
show absolute magnitude values on a logarithmic scale
show magnitude values on a linear scale
Values for the MagnitudeDisplay option.
PhaseDisplay
The option PhaseDisplay specifies whether and how to display phase values. Possible choices are:
Degrees
display phase values in degrees
Radians
display phase values in radians
None
suppress the phase plot
Values for the PhaseDisplay option.
PlotRange
With the PlotRange option, you can set the plot ranges for the y-axes of the magnitude and phase
plots. Possible values are:
Automatic
{magrng, phsrng}
determine the plot ranges automatically
specify plot ranges for magnitude and phase plot; the
values of magrng and phsrng can be Automatic , All, or
{ymin, ymax}
Values for the PlotRange option.
TraceNames
The option TraceNames allows you to specify the labels that are shown in the plot legend for the
displayed traces if ShowLegend −> True. The following values are possible:
3. Reference Manual
334
Automatic
string or expr
{string, † † † } or {expr, † † † }
use trace names from AC sweeps computed with e.g.
ACAnalysis
the name for a single trace
names for all traces; the number of names must equal the
number of traces
Values for the TraceNames option.
UnwrapPhase
With the option UnwrapPhase , you can specify whether BodePlot should restrict phase values to
the interval or try to unwrap the phase values of a frequency response trace that wraps
around the origin of the complex plane more than once.
UnwrapTolerance
With UnwrapTolerance −> tol, you can specify the maximum percentage of a full revolution by
which two successive phase values may differ such that phase unwrapping is triggered.
Examples
In[1]:= <<AnalogInsydes‘
Define two transfer
functions.
In[2]:= H1[s_] := 1/(s + 1);
H2[s_] := 10/(s^2 + s + 10)
This produces a Bode plot
of H jΩ.
In[4]:= BodePlot[H1[I w], {w, 0.01, 100}]
Phase (deg)
Magnitude (dB)
Load Analog Insydes.
1.0E-2
0
1.0E-1
1.0E0
1.0E1
1.0E2
-40
1.0E-2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
1.0E-2
0
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
-10
-20
-30
-20
-40
-60
-80
1.0E-2
Out[4]= Graphics 3.9 Graphics Functions
In[5]:= BodePlot[{H1[I w], H2[I w]}, {w, 0.01, 100}]
Phase (deg)
Magnitude (dB)
This produces a Bode plot
of H jΩ and H jΩ.
335
1.0E-2
10
0
-10
-20
-30
-40
-50
-60
1.0E-2
1.0E-2
0
-25
-50
-75
-100
-125
-150
-175
1.0E-2
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
Out[5]= Graphics This defines an RLC filter
circuit.
In[6]:= rlcf =
{V1,
{R1,
{L1,
{C1,
{R2,
];
Set up modified nodal
equations.
In[7]:= eqsrlcf = CircuitEquations[rlcf]
Do an AC analysis from
Hz to MHz.
In[8]:= acsweep = ACAnalysis[eqsrlcf, {f, 10, 10^6}]
Increase the number of
plot points.
In[9]:= SetOptions[BodePlot, PlotPoints −> 200];
Netlist[
{1, 0}, Symbolic
{1, 2}, Symbolic
{2, 3}, Symbolic
{3, 0}, Symbolic
{3, 0}, Symbolic
−>
−>
−>
−>
−>
V,
R1,
L,
C,
R2,
Value
Value
Value
Value
Value
−>
−>
−>
−>
−>
1},
1000.},
0.001},
2.2*10^−6},
1000.}
Out[7]= DAEAC, 4 4 Out[8]=
V$1 InterpolatingFunction10., 1. 106 , ,
V$2 InterpolatingFunction10., 1. 106 , ,
V$3 InterpolatingFunction10., 1. 106 , ,
I$V1 InterpolatingFunction10., 1. 106 , 3. Reference Manual
336
Magnitude (dB)
In[10]:= BodePlot[acsweep, {f, 10, 10^6}]
1.0E1
0
-20
-40
-60
-80
-100
1.0E1
1.0E2
1.0E3
1.0E4
1.0E5
1.0E6
1.0E2
1.0E3
1.0E4
Frequency
1.0E5
1.0E6
1.0E1
0
-50
-100
-150
-200
-250
-300
-350
1.0E1
1.0E2
1.0E3
1.0E4
1.0E5
1.0E6
Phase (deg)
Display all traces in a
Bode plot.
1.0E2
1.0E3
1.0E4
Frequency
1.0E5
1.0E6
V$1
V$2
V$3
I$V1
Out[10]= Graphics Magnitude (dB)
In[11]:= BodePlot[acsweep, {V$2[f], V$3[f]}, {f, 10, 10^6}]
1.0E1
0
-20
-40
-60
-80
-100
1.0E1
1.0E2
1.0E3
1.0E4
1.0E5
1.0E6
1.0E2
1.0E3
1.0E4
Frequency
1.0E5
1.0E6
1.0E1
0
-50
-100
-150
-200
-250
-300
-350
1.0E1
1.0E2
1.0E3
1.0E4
1.0E5
1.0E6
Phase (deg)
Display only the traces
V$2[f] and V$3[f].
1.0E2
1.0E3
1.0E4
Frequency
1.0E5
1.0E6
V$2[f]
Out[11]= Graphics V$3[f]
3.9 Graphics Functions
In[12]:= BodePlot[acsweep, {V$2[f], V$2[f]−V$3[f]}, {f, 10, 10^6}]
Phase (deg)
Magnitude (dB)
Display the traces V$2[f]
and V$2[f]−V$3[f] .
337
1.0E1
0
1.0E2
1.0E3
1.0E4
1.0E5
1.0E6
1.0E2
1.0E3
1.0E4
Frequency
1.0E5
1.0E6
-20
-40
-60
-80
1.0E1
1.0E1
0
-50
-100
-150
-200
-250
-300
-350
1.0E1
1.0E2
1.0E3
1.0E4
1.0E5
1.0E6
1.0E2
1.0E3
1.0E4
Frequency
1.0E5
1.0E6
V$2[f] V$2[f] - V$3[f
Out[12]= Graphics In[13]:= BodePlot[H1[I w], {w, 0.01, 100},
MagnitudeDisplay −> AbsoluteValues]
Magnitude
1.0E-2
1.0E0
5.0E-1
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
2.0E-1
1.0E-1
5.0E-2
2.0E-2
1.0E-2
1.0E-2
1.0E-2
0
Phase (deg)
Show absolute magnitude
values instead of decibels.
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
-20
-40
-60
-80
1.0E-2
Out[13]= Graphics 3. Reference Manual
338
Show magnitude values
on a linear scale.
In[14]:= BodePlot[H1[I w], {w, 0.01, 100},
MagnitudeDisplay −> Linear]
Phase (deg)
Magnitude
1.0E-2
1
1.0E-1
1.0E0
1.0E1
1.0E2
0
1.0E-2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
1.0E-2
0
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
0.8
0.6
0.4
0.2
-20
-40
-60
-80
1.0E-2
Out[14]= Graphics In[15]:= BodePlot[acsweep, {V$2[f], V$2[f]−V$3[f]},
{f, 100, 10^5}, PhaseDisplay −> None]
1.0E2
0
5.0E21.0E3
5.0E31.0E4
5.0E41.0E5
5.0E21.0E3
5.0E31.0E4
Frequency
5.0E41.0E5
-10
Magnitude (dB)
Do not show the phase
plot.
-20
-30
-40
-50
-60
1.0E2
V$2[f] V$2[f] - V$3[f
Out[15]= Graphics 3.9 Graphics Functions
In[16]:= BodePlot[H1[I w]^5, {w, 0.01, 100}, UnwrapPhase −> False]
Magnitude (dB)
Restrict the phase to the
interval .
339
1.0E-2
0
1.0E0
1.0E1
1.0E2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
-50
-100
-150
-200
1.0E-2
Phase (deg)
1.0E-1
1.0E-2
0
-50
-100
-150
-200
-250
-300
-350
1.0E-2
Out[16]= Graphics In[17]:= BodePlot[H1[I w]^5, {w, 0.01, 100}, UnwrapPhase −> True]
Phase (deg)
Magnitude (dB)
Unwrap the phase trace.
1.0E-2
0
1.0E-1
1.0E0
1.0E1
1.0E2
-200
1.0E-2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
1.0E-2
0
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
-50
-100
-150
-100
-200
-300
-400
1.0E-2
Out[17]= Graphics In the following example, the number of plot points is reduced so that the differences between
successive phase values become too large for phase unwrapping.
3. Reference Manual
340
In[18]:= BodePlot[H1[I w]^5, {w, 0.01, 100},
PlotPoints −> 10, PlotRange −> {Automatic, {−360, 0}}]
Magnitude (dB)
Reduce the number of plot
points.
1.0E-2
0
1.0E0
1.0E1
1.0E2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
-50
-100
-150
-200
1.0E-2
Phase (deg)
1.0E-1
1.0E-2
0
-50
-100
-150
-200
-250
-300
-350
1.0E-2
Out[18]= Graphics To unwrap the phase trace, you can specify a larger number of plot points or increase the value of
UnwrapTolerance .
Magnitude (dB)
In[19]:= BodePlot[H1[I w]^5, {w, 0.01, 100},
PlotPoints −> 10, UnwrapTolerance −> 0.3]
Phase (deg)
Increase the unwrap
tolerance to 30% of .
1.0E-2
0
1.0E-1
1.0E0
1.0E1
1.0E2
-200
1.0E-2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
1.0E-2
0
1.0E-1
1.0E0
1.0E1
1.0E2
1.0E-1
1.0E0
Frequency
1.0E1
1.0E2
-50
-100
-150
-100
-200
-300
-400
1.0E-2
Out[19]= Graphics 3.9 Graphics Functions
341
3.9.2 FourierPlot
FourierPlot[func, {t, t , t }]
plots the frequency spectrum of the function func where
the independent variable t is swept from t to t
FourierPlot[{func , func , † † † }, {t, t , t }]
superimposes the Fourier plots of several functions
Command structure of FourierPlot.
FourierPlot performs a discrete fourier analysis by applying Fourier on the given functions.
FourierPlot inherits all options from ListPlot . Additionally the following option is available:
option name
default value
PlotPoints
250
specifies the number of sample points
Options for FourierPlot.
PlotPoints
The option PlotPoints −> integer specifies the number of sample points. The maximum displayed
frequency is given by integert t .
See also: Fourier.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Plot a spectrum of two
signals with Hz and
Hz.
In[2]:= FourierPlot[{0.5 Sin[2. Pi 20. t], 2. Cos[2. Pi 80. t]},
{t, 0, 1.}, PlotStyle −> {{Hue[0]}, {Hue[0.4]}}]
Amplitude
1.5
1.25
1
0.75
0.5
0.25
0
0
Out[2]= Graphics 20
40
60
80
Frequency
100
120
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342
3.9.3 NicholPlot
NicholPlot[func, {var, f , f }]
displays a Nichol plot of the transfer function func[var] as
the independent variable var is swept from f to f
NicholPlot[{func , func , † † † }, {var, f , f }]
displays the transfer functions func , func , † † †
simultaneously in a Nichol plot
Command structure of NicholPlot.
NicholPlot displays one or several transfer functions in a Nichol plot. A Nichol plot is similar to a
Nyquist plot but shows magnitude (in dB) vs. phase with the axis origin at the point dB .
Note that NicholPlot has attribute HoldFirst .
NicholPlot supports additional patterns for displaying numerical data generated with the functions ACAnalysis (Section 3.7.3), NoiseAnalysis (Section 3.7.4), or ReadSimulationData (Section 3.10.3):
NicholPlot[traces, {var, f , f }]
displays a list of AC traces computed with e.g.
ACAnalysis in a Nichol plot
NicholPlot[traces, expr, {var, f , f }]
displays the value of an expression in terms of AC traces
NicholPlot[traces, {expr , expr , † † † }, {var, f , f }]
displays the values of several expressions
Displaying numerical data with NicholPlot.
You can customize the appearance of Nichol plots with the following options. In addition,
NicholPlot inherits many options from ListPlot and Legend. Both the options which are specific
to NicholPlot as well as the inherited options can be set with SetOptions[NicholPlot, opts].
3.9 Graphics Functions
343
option name
default value
FrequencyScaling
Exponential
whether to distribute sampling points
exponentially or linearly along the
frequency axis
PhaseDisplay
Degrees
the unit for phase values
ShowLegend
True
whether to display trace names
TraceNames
Automatic
the trace names displayed in the plot legend
Options for NicholPlot.
See also: ACAnalysis (Section 3.7.3), NoiseAnalysis (Section 3.7.4), ReadSimulationData (Section 3.10.3), BodePlot (Section 3.9.1), NyquistPlot (Section 3.9.4).
Options Description
A detailed description of all options that are specific to NicholPlot or are used in a non-standard
way is given below in alphabetical order:
FrequencyScaling
The option FrequencyScaling determines how sampling points are distributed over the frequency
axis. Possible values are:
Exponential
Linear
use exponentially spaced sampling points
use linearly spaced sampling points
Values for the FrequencyScaling option.
PhaseDisplay
The option PhaseDisplay specifies the unit for phase values. Possible values are:
Degrees
display phase values in degrees
Radians
display phase values in radians
Values for the PhaseDisplay option.
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344
TraceNames
The option TraceNames allows you to specify the labels that are shown in the plot legend for the
displayed traces if ShowLegend −> True. The following values are possible:
Automatic
string or expr
{string, † † † } or {expr, † † † }
use trace names from AC sweeps computed with e.g.
ACAnalysis
the name for a single trace
names for all traces; the number of names must equal the
number of traces
Values for the TraceNames option.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define two transfer
functions.
In[2]:= H1[s_] := (60 + 20*s)/(100*s + 45*s^2 + 15*s^3 + 2*s^4)
H2[s_] := 10/(s^2 + s + 10)
Draw a Nichol plot of
H jΩ.
In[4]:= NicholPlot[H1[I w], {w, 0.1, 5}, AspectRatio −> 0.8]
dB
15
10
5
-360 -300 -240
-120 -60
-5
-10
-15
-20
Out[4]= Graphics 0
deg
3.9 Graphics Functions
Display two transfer
functions in a Nichol plot.
345
In[5]:= NicholPlot[{H1[I w], H2[I w]}, {w, 0.1, 100},
AspectRatio −> 0.7, PlotRange −> {{−300,0}, {−60,15}},
PlotPoints −> 100]
dB
10
-300
-240
-120
-60
0
deg
-10
-20
-30
-40
-50
-60
Out[5]= Graphics This defines an RLC filter
circuit.
In[6]:= rlcf =
{V1,
{R1,
{L1,
{C1,
{R2,
];
Set up modified nodal
equations.
In[7]:= eqsrlcf = CircuitEquations[rlcf]
Do an AC analysis from
Hz to MHz.
In[8]:= acsweep = ACAnalysis[eqsrlcf, {f, 10, 10^6}]
Netlist[
{1, 0}, Symbolic
{1, 2}, Symbolic
{2, 3}, Symbolic
{3, 0}, Symbolic
{3, 0}, Symbolic
−>
−>
−>
−>
−>
V,
R1,
L,
C,
R2,
Value
Value
Value
Value
Value
−>
−>
−>
−>
−>
1},
1000.},
0.001},
2.2*10^−6},
1000.}
Out[7]= DAEAC, 4 4 Out[8]=
V$1 InterpolatingFunction10., 1. 106 , ,
V$2 InterpolatingFunction10., 1. 106 , ,
V$3 InterpolatingFunction10., 1. 106 , ,
I$V1 InterpolatingFunction10., 1. 106 , 3. Reference Manual
346
Display several
expressions in a Nichol
plot.
In[9]:= NicholPlot[acsweep, {V$2[f], V$3[f]}, {f, 10, 10^6},
AspectRatio −> 0.8]
dB
-360-300-240
deg
-120 -60 0
-20
-40
-60
-80
-100
V$2[f]
V$3[f]
Out[9]= Graphics 3.9.4 NyquistPlot
NyquistPlot[func, {var, f , f }]
displays a Nyquist plot of the transfer function func[var]
as the independent variable var is swept from f to f
NyquistPlot[{func , func , † † † }, {var, f , f }]
displays the transfer functions func , func , † † †
simultaneously in a Nyquist plot
Command structure of NyquistPlot.
NyquistPlot displays one or several transfer functions in a Nyquist plot. A Nyquist plot is a
parametric plot of the imaginary part vs. the real part of a frequency response as the frequency is
swept from f to f .
Note that NyquistPlot has attribute HoldFirst .
NyquistPlot supports additional patterns for displaying numerical data generated with the functions ACAnalysis (Section 3.7.3), NoiseAnalysis (Section 3.7.4), or ReadSimulationData (Section 3.10.3):
3.9 Graphics Functions
347
NyquistPlot[traces, {var, f , f }]
displays a list of AC traces computed with e.g.
ACAnalysis in a Nyquist plot
NyquistPlot[traces, expr, {var, f , f }]
displays the value of an expression in terms of AC traces
NyquistPlot[traces, {expr , expr , † † † }, {var, f , f }]
displays the values of several expressions
Displaying numerical data with NyquistPlot.
You can customize the appearance of Nyquist plots with the following options. In addition,
NyquistPlot inherits many options from ListPlot and Legend. Both the options which are specific
to NyquistPlot as well as the inherited options can be set with SetOptions[NyquistPlot, opts].
option name
default value
FrequencyScaling
Exponential
whether to distribute sampling points
exponentially or linearly along the
frequency axis
ShowLegend
True
whether to display trace names
ShowUnitCircle
False
whether to draw the unit circle
TraceNames
Automatic
the trace names displayed in the plot legend
UnitCircleStyle
RGBColor[0., 0., .5]
the plot style for the unit circle
Options for NyquistPlot.
See also: ACAnalysis (Section 3.7.3), NoiseAnalysis (Section 3.7.4), ReadSimulationData (Section 3.10.3), BodePlot (Section 3.9.1), NicholPlot (Section 3.9.3).
Options Description
A detailed description of all options that are specific to NyquistPlot or are used in a non-standard
way is given below in alphabetical order:
FrequencyScaling
The option FrequencyScaling determines how sampling points are distributed over the frequency
axis. Possible values are:
3. Reference Manual
348
Exponential
Linear
use exponentially spaced sampling points
use linearly spaced sampling points
Values for the FrequencyScaling option.
ShowUnitCircle
With ShowUnitCircle −> True, NyquistPlot adds the unit circle to a plot.
TraceNames
The option TraceNames allows you to specify the labels that are shown in the plot legend for the
displayed traces if ShowLegend −> True. The following values are possible:
Automatic
string or expr
{string, † † † } or {expr, † † † }
use trace names from AC sweeps computed with e.g.
ACAnalysis
the name for a single trace
names for all traces; the number of names must equal the
number of traces
Values for the TraceNames option.
UnitCircleStyle
With UnitCircleStyle −> style, you can change the plot style for the unit circle.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define two transfer
functions.
In[2]:= H1[s_] := 1/(s + 1);
H2[s_] := 10/(s^2 + s + 10)
Increase the number of
plot points.
In[4]:= SetOptions[NyquistPlot, PlotPoints −> 200];
3.9 Graphics Functions
Draw a Nyquist plot of
H jΩ.
349
In[5]:= NyquistPlot[H2[I w], {w, 0.01, 100}, PlotRange −> All]
Im
-1 -0.5
0.5
1
Re
1.5
-0.5
-1
-1.5
-2
-2.5
-3
Out[5]= Graphics This produces a Nyquist
plot of H jΩ and H jΩ.
In[6]:= NyquistPlot[{H1[I w], H2[I w]}, {w, 0.01, 100},
PlotRange −> All]
Im
-1 -0.5
0.5
1
Re
1.5
-0.5
-1
-1.5
-2
-2.5
-3
Out[6]= Graphics This defines an RLC filter
circuit.
In[7]:= rlcf =
{V1,
{R1,
{L1,
{C1,
{R2,
];
Set up modified nodal
equations.
In[8]:= eqsrlcf = CircuitEquations[rlcf]
Netlist[
{1, 0}, Symbolic
{1, 2}, Symbolic
{2, 3}, Symbolic
{3, 0}, Symbolic
{3, 0}, Symbolic
Out[8]= DAEAC, 4 4 −>
−>
−>
−>
−>
V,
R1,
L,
C,
R2,
Value
Value
Value
Value
Value
−>
−>
−>
−>
−>
1},
1000.},
0.001},
2.2*10^−6},
1000.}
3. Reference Manual
350
Do an AC analysis from
Hz to MHz.
In[9]:= acsweep = ACAnalysis[eqsrlcf, {f, 10, 10^6}]
Display the trace of
V$2[f].
In[10]:= NyquistPlot[acsweep, {V$2[f]}, {f, 10, 10^6}]
Out[9]=
V$1 InterpolatingFunction10., 1. 106 , ,
V$2 InterpolatingFunction10., 1. 106 , ,
V$3 InterpolatingFunction10., 1. 106 , ,
I$V1 InterpolatingFunction10., 1. 106 , Im
0.5
0.4
0.3
0.2
0.1
0.2 0.4 0.6 0.8
Re
-0.1
-0.2
V$2[f]
Out[10]= Graphics Display several
expressions in a Nyquist
plot.
In[11]:= NyquistPlot[acsweep, {V$2[f]−0.5, 0.5−V$2[f]},
{f, 10, 10^6}, Frame −> True, ShowLegend −> False]
Im
0.4
0.2
Re
0
-0.2
-0.4
-0.4 -0.2
Out[11]= Graphics 0
0.2 0.4
3.9 Graphics Functions
Display the unit circle.
351
In[12]:= NyquistPlot[H2[I w], {w, 0.01, 100},
PlotRange −> {{−3, 3}, {−3.5, 1.5}},
ShowUnitCircle −> True]
Im
1
-3
-2
-1
1
2
3
Re
-1
-2
-3
Out[12]= Graphics Display the unit circle
using a different style.
In[13]:= NyquistPlot[H2[I w], {w, 0.01, 100},
PlotRange −> {{−3, 3}, {−3.5, 1.5}},
ShowUnitCircle −> True,
UnitCircleStyle −>
{GrayLevel[0], Dashing[{0.04, 0.03}]}]
Im
1
-3
-2
-1
1
-1
-2
-3
Out[13]= Graphics 2
3
Re
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352
3.9.5 RootLocusPlot
RootLocusPlot[func, {k, k , k }]
displays the root locus of func as k is swept from k to k
RootLocusPlot[func]
RootLocusPlot[rootloc]
displays a pole/zero diagram of func
displays a root locus calculated with RootLocusByQZ
Command structure of RootLocusPlot.
RootLocusPlot displays a root locus plot of a transfer function Hs k, where k denotes a realvalued parameter. A root locus plot shows the trajectories of the poles and zeros of Hs k in the
complex plane as k varies from k to k . RootLocusPlot can also be used to produce a pole/zero
diagram of transfer function Hs without parameters as well as to display root loci calculated with
RootLocusByQZ .
By default, RootLocusPlot chooses a logarithmic representation of the complex plane for better
visualization of widely separated poles and zeros. However, as logarithmic scaling is inappropriate
for coordinates near the axes, the regions around the axes are scaled semi-logarithmically, and the
region around the origin is scaled linearly. RootLocusPlot marks the linearly scaled region with a
gray background. The semi-logarithmic regions are the regions above and below as well as to the
left and the right of the linear region.
You can customize the appearance of root locus plots with the following options. In addition,
RootLocusPlot inherits many options from Graphics . Both the options which are specific to
RootLocusPlot as well as the inherited options can be set with SetOptions[RootLocusPlot, opts].
3.9 Graphics Functions
353
option name
default value
LinearRegionLimit
1.0
the extent of the linearly scaled plot region
LinearRegionSize
0.3
the relative size of the linearly scaled plot
region
LinearRegionStyle
GrayLevel[0.9]
the fill color used to mark the linear region
PlotPoints
5
the number of points between k and k at
which the poles and zeros are calculated
PlotRange
Automatic
the plot range
PoleStyle
CrossMark[0.02, Hue[0.15 (1. − #1)] & , Thickness[0.007]]
the mark and plot style for poles
ShowLegend
True
ZeroStyle
CircleMark[0.02, Hue[0.25 #1 + 0.3] & , Thickness[0.007]]
the mark and plot style for zeros
whether to display a plot legend
Options for RootLocusPlot.
See also: PolesAndZerosByQZ (Section 3.8.3), RootLocusByQZ (Section 3.8.6).
Options Description
A detailed description of all options that are specific to RootLocusPlot or are used in a non-standard
way is given below in alphabetical order:
LinearRegionLimit
With LinearRegionLimit −> value, you can specify the extent of the linearly scaled plot region
around the origin. For example, the setting LinearRegionLimit −> 100 causes the region where
x and y to be scaled linearly. LinearRegionLimit −> Infinity turns
off logarithmic scaling.
LinearRegionSize
LinearRegionSize −> size specifies the percentage of the total plot area occupied by the linear
region. The option value size must be a real number between and .
LinearRegionStyle
With LinearRegionStyle −> style, you can specify the fill color RootLocusPlot uses for the linearly
scaled region.
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354
PlotRange
The meaning of the PlotRange option is the same as for other Mathematica graphics functions, but
the option has slightly different formats.
{{xmin, xmax}, {ymin, ymax}}
{{xmin, xmax}}
value
Automatic or All
specify individual limits for all axes directions
specify plot range for the x-axis.
equivalent to {{−value, value}, {−value, value}}
determine plot range automatically
Values for the PlotRange option.
PoleStyle
With the option PoleStyle −> style, you can change the appearance of the symbol RootLocusPlot
uses to mark poles. The following styles are available:
CrossMark[size, colorfunc, grstyle]
marks a root with a cross
PlusMark[size, colorfunc, grstyle]
marks a root with a plus
CircleMark[size, colorfunc, grstyle]
marks a root with a circle
SquareMark[size, colorfunc, grstyle]
marks a root with a square
PointMark[size, colorfunc, grstyle]
marks a root with a point
Styles for the PoleStyle and ZeroStyle options.
In the above table, size denotes the size of a root marker in scaled coordinates, colorfunc a pure
function that returns a color value for an argument between 0 and 1, and grstyle an additional list of
graphics styles for the root marker. The color function is used to visualize the parameter dependency
of the root.
ZeroStyle
With the option ZeroStyle −> style, you can change the appearance of the symbol RootLocusPlot
uses to mark zeros. For a list of available styles see the description of the PoleStyle option.
3.9 Graphics Functions
355
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define a transfer function
Ha s a.
In[2]:= Ha[s_, a_] := (a + 2*s + s^2)/(10 + 3*a*s + 4*s^2 + s^3)
Display a root locus plot
of Ha s a as a is swept
from to .
In[3]:= RootLocusPlot[Ha[s, a], {a, −4, 10}, PlotPoints −> 10]
a = -4.000e0 .. 1.000e1 (0% .. 100%)
Im s
5.0E0
Poles
0%
2.0E0
1.
-5.0E0
-2.0E0
100
-1.
1.
Re s
Zeros
0%
-1.
-2.0E0
100
-5.0E0
Out[3]= Graphics Display the pole/zero
distribution of Ha s a for
a .
In[4]:= RootLocusPlot[Ha[s, 0], PlotRange −> 6]
Im s
5.0E0
2.0E0
1.
-5.0E0
-2.0E0 -1.
1.
-1.
-2.0E0
-5.0E0
Out[4]= Graphics 2.0E0
5.0E0
Re s
3. Reference Manual
356
Change boundaries, plot
size, and color of the
linear region.
In[5]:= RootLocusPlot[Ha[s, a], {a, −4, 10},
PlotPoints −> 10, LinearRegionLimit −> 4.0,
LinearRegionSize −> 0.8,
LinearRegionStyle −> GrayLevel[0.95]]
a = -4.000e0 .. 1.000e1 (0% .. 100%)
Im s
Poles
0%
4.
2.
100
-4.
-3.
-2.
-1.
1.
Re s
Zeros
0%
-2.
-4.
100
Out[5]= Graphics Set equal plot limits for all
axes directions.
In[6]:= RootLocusPlot[Ha[s, a], {a, −4, 10},
PlotPoints −> 10, PlotRange −> 100, Frame −> True]
a = -4.000e0 .. 1.000e1 (0% .. 100%)
Im s
Poles
0%
100
Re s
Zeros
0%
100
Out[6]= Graphics 3.9 Graphics Functions
Change the style for pole
and zero markers.
357
In[7]:= RootLocusPlot[Ha[s, a], {a, −4, 10},
PlotPoints −> 10,
PoleStyle −> PlusMark[0.02,
GrayLevel[1−0.8*#]&, Thickness[0.005]],
ZeroStyle −> SquareMark[0.02,
GrayLevel[1−0.8*#]&, Thickness[0.005]]]
a = -4.000e0 .. 1.000e1 (0% .. 100%)
Im s
5.0E0
Poles
0%
2.0E0
1.
-5.0E0
-2.0E0
-1.
100
1.
Re s
Zeros
0%
-1.
-2.0E0
-5.0E0
100
Out[7]= Graphics 3.9.6 TransientPlot
TransientPlot[func, {tvar, t , t }]
displays the transient waveform func[tvar] as the
independent variable tvar is swept from t to t
TransientPlot[{func , func , † † † }, {tvar, t , t }]
displays the transient waveforms func , func , † † †
simultaneously in a single plot
Command structure of TransientPlot .
TransientPlot displays one or several transient waveforms in a single plot, where the signal values
are usually plotted versus time.
Note that TransientPlot has attribute HoldAll.
Additionally,
TransientPlot supports the multi-dimensional data format of Analog Insydes described in Section 3.7.1, which consists of lists of rules, where the variables are assigned InterpolatingFunction
objects. Therefore, call TransientPlot where the first argument is compatible to the return value of
the numerical solver functions NDAESolve (Section 3.7.5) and NDSolve , or the data interface function
ReadSimulationData (Section 3.10.3):
3. Reference Manual
358
TransientPlot[traces, expr, {tvar, t , t }]
displays the transient waveform of an expression in terms
of the computed traces as the independent variable tvar is
swept from t to t
TransientPlot[traces, {expr , expr , † † † }, {tvar, t , t }]
displays the transient waveforms of several expressions
expr , expr , † † † simultaneously in a single plot
Displaying numerical data with TransientPlot.
TransientPlot inherits most of its options from Plot and GraphicsArray . Both the options which
are specific to TransientPlot as well as the inherited options can be set with
SetOptions[TransientPlot, opts].
option name
default value
Parametric
False
whether to carry out a parametric plot
PlotRange
Automatic
the plot ranges for the different transient
waveforms
ShowLegend
True
whether to display a plot legend
ShowSamplePoints
False
generates a ListPlot of the simulation data
SingleDiagram
True
combines plots of several functions in a
single diagram
SweepParameters
Automatic
allows for filtering the multi-dimensional
data
Options for TransientPlot.
See also: NDSolve, NDAESolve (Section 3.7.5), ReadSimulationData (Section 3.10.3).
Options Description
A detailed description of all options that are specific to TransientPlot or are used in a non-standard
way is given below in alphabetical order:
3.9 Graphics Functions
359
Parametric
Changing the default setting from False to True allows for carrying out parametric plots of transient
waveforms. Therefore, specify a list of two expressions, where the first one refers to the x-axis and
the second to the y-axis. The expressions are automatically added as labels to the x and y-axes.
Please note that this option also supports the multi-dimensional data format.
PlotRange
The usage of the option PlotRange is different as compared to Plot. The PlotRange of each transient waveform can be separately modified for the setting SingleDiagram −> False. The default
setting is PlotRange −> Automatic . Possible option values are as follows:
Automatic or All
{ymin, ymax}
determine PlotRange automatically
specify same PlotRange for all waveforms
{Automatic | All | {ymin, ymax}, † † † }
specify individual PlotRange for each waveform
Values for the PlotRange option.
ShowLegend
The option ShowLegend allows for displaying a plot legend. The default setting is
ShowLegend −> True.
ShowSamplePoints
Changing the default setting from False to True allows for generating a ListPlot of the simulation
data.
Please note that this option also supports the multi-dimensional data format.
SingleDiagram
The option SingleDiagram combines plots of several transient waveforms in a single diagram.
Changing the default setting from True to False produces a GraphicsArray of separately plotted
transient waveforms.
Please note that this option also supports the multi-dimensional data format.
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SweepParameters
The option SweepParameters allows for filtering the multi-dimensional data. The default setting is
SweepParameters −> Automatic . Possible option values are as follows:
Automatic
leave the multi-dimensional data unchanged
{param −> value | param −> _, † † † }
specify explicit value or Blank[] for each valid sweep
parameter
Values for the SweepParameters option.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Plot two sinusoidal
functions.
In[2]:= S[T_] := Sin[10.T];
TransientPlot[{S[T], 2 S[T−1]}, {T, 0, 1}]
2
1
S[T]
0.2
0.4
0.6
0.8
1
T
2 S[T - 1
-1
-2
Out[3]= Graphics In the following, the transient solution of the diode rectifier circuit shown below is plotted using TransientPlot .
D1
1
V0
2
C1
R1
Vout
3.9 Graphics Functions
Define netlist description
of a simple diode rectifier
circuit.
361
In[4]:= rectifier =
Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −> 2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C},
Model −> "Diode", Selector −> "Spice"}
]
]
Out[4]= Circuit Set up system of symbolic
time-domain equations.
In[5]:= rectifiereqs = CircuitEquations[rectifier,
AnalysisMode −> Transient, ElementValues −> Symbolic]
Out[5]= DAETransient, 4 4 Perform numerical
transient analysis.
In[6]:= tran = NDAESolve[rectifiereqs, {t, 0., 2.*^−5}]
Display the simulation
result with TransientPlot
for the variables V$1[t]
and V$2[t].
In[7]:= TransientPlot[tran, {V$1[t], V$2[t]}, {t, 0., 2.*^−5},
Axes −> False, Frame −> True, GridLines −> Automatic]
Out[6]=
V$2 InterpolatingFunction0, 0.00002, ,
V$1 InterpolatingFunction0, 0.00002, ,
I$V0 InterpolatingFunction0, 0.00002, ,
I$AC$D1 InterpolatingFunction0, 0.00002, 2
1
V$1[t]
0
V$2[t]
-1
-2
0
-6 0.00001 0.000015 0.00002
5. 10
t
Out[7]= Graphics 3. Reference Manual
362
View signals separately.
In[8]:= Vd[t_] := V$1[t] − V$2[t];
TransientPlot[tran, {Vd[t], V$2[t]}, {t, 0., 2.*^−5},
PlotRange −> {{−4., 2.}, {0., 1.5}},
SingleDiagram −> False]
2
1
-6
0.00001
0.000015
t
0.00002
5. 10
Vd[t
-1
-2
-3
-4
1.4
1.2
1
0.8
V$2[
0.6
0.4
0.2
-6
0.00001
0.000015
t
0.00002
5. 10
Out[9]= GraphicsArray Display the simulation
data.
In[10]:= TransientPlot[tran, {V$1[t], V$2[t]}, {t, 0., 2.*^−5},
ShowSamplePoints −> True]
2
1
V$1[t]
-6
t
0.00001 0.000015 0.00002
5. 10
-1
-2
Out[10]= Graphics V$2[t]
3.9 Graphics Functions
Generate a parametric
plot.
363
In[11]:= TransientPlot[tran, {V$1[t], V$2[t]},
{t, 2.*^−6, 2.*^−5}, Parametric −> True]
V$2[t]
1.2
1.1
-2
-1
1
2
V$1[t]
0.9
0.8
Out[11]= Graphics Perform parametric
transient analysis.
In[12]:= paramtran = NDAESolve[rectifiereqs, {t, 0., 2.*^−5},
{R1, Table[10^i, {i, 0, 3}]}]
Out[12]=
V$2 InterpolatingFunction0, 0.00002, ,
V$1 InterpolatingFunction0, 0.00002, ,
I$V0 InterpolatingFunction0, 0.00002, ,
I$AC$D1 InterpolatingFunction0, 0.00002, ,
SweepParameters R1 1.,
V$2 InterpolatingFunction0, 0.00002, ,
V$1 InterpolatingFunction0, 0.00002, ,
I$V0 InterpolatingFunction0, 0.00002, ,
I$AC$D1 InterpolatingFunction0, 0.00002, ,
SweepParameters R1 10.,
V$2 InterpolatingFunction0, 0.00002, ,
V$1 InterpolatingFunction0, 0.00002, ,
I$V0 InterpolatingFunction0, 0.00002, ,
I$AC$D1 InterpolatingFunction0, 0.00002, ,
SweepParameters R1 100.,
V$2 InterpolatingFunction0, 0.00002, ,
V$1 InterpolatingFunction0, 0.00002, ,
I$V0 InterpolatingFunction0, 0.00002, ,
I$AC$D1 InterpolatingFunction0, 0.00002, ,
SweepParameters R1 1000.
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364
Display the
multi-dimensional
simulation result: show all
traces of the parameter
sweep.
In[13]:= TransientPlot[paramtran, {V$2[t]}, {t, 0., 2.*^−5}]
1.2
1
0.8
0.6
V$2[t]
0.4
0.2
t
-6 0.00001 0.000015 0.00002
5. 10
Out[13]= Graphics Filter the
multi-dimensional data:
display only the trace
corresponding to the
parameter value R1=1.
In[14]:= TransientPlot[paramtran, {V$2[t]}, {t, 0., 2.*^−5},
SweepParameters −> {R1 −> 1.}]
1
0.8
0.6
V$2[t]
0.4
0.2
t
-6 0.00001 0.000015 0.00002
5. 10
Out[14]= Graphics 3.10 Interfaces
365
3.10 Interfaces
In the following chapter the different import and export features of Analog Insydes are discussed.
Section 3.10.1 describes how to translate netlists from external simulators to the internal Analog
Insydes netlist format using ReadNetlist . The commands ReadSimulationData (Section 3.10.3)
and WriteSimulationData (Section 3.10.4) can be used to import or export numerical simulation
data. For behavioral model generation the command WriteModel (Section 3.10.5) can be used, which
translates a symbolic DAEObject to an external behavioral model description language such as
Saber MAST. For importing schematics stored in DXF format (Drawing Interchange File) as native
Mathematica graphic objects see DXFGraphics (Section 3.13.2).
3.10.1 ReadNetlist
ReadNetlist["netfile", Simulator−>sim]
reads a netlist file "netfile" from a simulator sim and
converts it into an Analog Insydes netlist
ReadNetlist["netfile", "outfile", Simulator−>sim]
additionally adds the operating-point information from the
simulator output file "outfile"
Command structure of ReadNetlist.
This function converts a simulator-specific netlist file netfile into an Analog Insydes netlist description.
Additionally, the operating-point information can be extracted from a given simulator-specific output
file outfile. ReadNetlist then returns a Circuit object which can be used for setting up circuit
equations by applying the function CircuitEquations (Section 3.5.1). For more information on the
simulator-specific features see Section 3.10.2.
ReadNetlist has the following options:
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option name
default value
CharacterMapping
{"_" −> "$"}
InstanceNameSeparator
Inherited[AnalogInsydes]
specifies the character or string Analog
Insydes uses to separate name components
of reference designators and node
identifiers generated for instantiated
subcircuit objects (see Section 3.14.2)
KeepPrefix
True
if set to False the prefix of the reference
designator introduced by the PSpice
Schematics Editor and the Saber Designer
will be omitted
LevelSeparator
"/"
specifies the character or string Analog
Insydes uses to separate name components
of local models, parameters, and subcircuits
generated by ReadNetlist
LibraryPath
{"."}
specifies the directories which contain
device model libraries
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
Simulator
Inherited[AnalogInsydes]
standard Analog Insydes simulator
specification (see Section 3.14.6)
list of translation rules for character
mapping used while transforming simulator
parameters to Mathematica symbols
Options for ReadNetlist.
See also: ReadSimulationData (Section 3.10.3), Simulator (Section 3.14.6), Section 3.10.2.
Options Description
A detailed description of all ReadNetlist options is given below in alphabetical order:
CharacterMapping
With CharacterMapping −> {charactera −> characterb } the internal character mapping scheme can
be modifed. Use this option with caution, because this rule will be applied to each symbol and
expression in the netlist. For example CharacterMapping −> {"." −> "$"} will replace all dots
3.10 Interfaces
367
(".") into dollar signs ("$"), even if it is a decimal dot like in "1.234" , which will lead to a
syntax error ("1$234"). This option is useful if your netlist contains e.g. element names including "_" and "$" and which will be ambiguous if the "_" is mapped to a "$". You can use
CharacterMapping −> {"_" −> "x"} to avoid this ambiguity.
KeepPrefix
Some schematic capture tools add a prefix to all element names to make sure that the element type
is correct and independent from the actual element name. If KeepPrefix −> False is set, the prefix
will be removed. If a Saber netlist is read, the template name is removed.
Use this option with caution. It might cause ambiguous element names.
LevelSeparator
Locally defined subcircuits must be transformed to a top-level subcircuit with unique names. To
generate unique names the value of LevelSeparator −> string is used to separate the different levels
of subcircuits. Because these names are never converted to a Mathematica symbol it is possible to
specifiy a non-alphanumeric character.
The default is LevelSeparator −> "/".
LibraryPath
A netlist can contain calls to further files and libraries. With LibraryPath the search path for these
files can be specified.
The default is LibraryPath −> {"."}. The current working directory is set to the directory from the
file name specification for netfile (see also the Mathematica command DirectoryName ).
Protocol
This option describes the standard Analog Insydes protocol specification (see Section 3.14.5). The
top-level function for the Protocol option is ReadNetlist . The second-level function is
ExpandSubcircuits .
Simulator
ReadNetlist supports the following simulators and file types:
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simulator
netlist file type
output file type
"AnalogInsydes"
*.m file
"Eldo"
*.cir file
*.chi file
"PSpice"
*.cir file
*.out file
"Saber"
*.sin file
saved ssparam output
Supported simulators and file types by ReadNetlist.
ReadNetlist appends Simulator −> simulator to the GlobalParameters field of the returned
Circuit object. This enables the Analog Insydes models to determine which simulator-specific
properties of the models have to be chosen. For more information on the simulator-specific features
see Section 3.10.2.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Read and convert a PSpice
netlist.
In[2]:= netdc = ReadNetlist[
"AnalogInsydes/DemoFiles/Multivibrator.cir",
Simulator −> "PSpice"]
Out[2]= Circuit Display the netlist.
In[3]:= DisplayForm[netdc]
Out[3]//DisplayForm=
Circuit:
Netlist Raw, 9 Entries:
R1, 1, 2, Type Resistor, Value 4700., Symbolic R1
R2, 2, 0, Type Resistor, Value 4700., Symbolic R2
R3, 4, 0, Type Resistor, Value 39000., Symbolic R3
R4, 5, 0, Type Resistor, Value 82000., Symbolic R4
R5, 1, 3, Type Resistor, Value 18000., Symbolic R5
IC, 4, 5, Type CurrentSource, Value AC 0, DC Transient
Q1, 3 C, 2 B, 4 E, Model ModelBJT, BC182, Q1, Selector
Q2, 1 C, 3 B, 5 E, Model ModelBJT, BC182, Q2, Selector
VCC, 1, 0, Type VoltageSource, Value AC 0, DC Transien
ModelParametersName BC182, Type NPN, CJC 1. 1012 GlobalParametersSimulator PSpice
Read a PSpice netlist
including operating-point
information.
In[4]:= buffer = ReadNetlist[
"AnalogInsydes/DemoFiles/Buffer.cir",
"AnalogInsydes/DemoFiles/Buffer.out",
Simulator −> "PSpice"]
Out[4]= Circuit 3.10 Interfaces
369
3.10.2 Simulator-Specific Remarks on ReadNetlist
The command ReadNetlist translates netlists from several external simulators to the Analog Insydes
netlist format. The following section describes the different features supported by ReadNetlist for
each simulator.
AnalogInsydes
ReadNetlist["netfile", Simulator −> "AnalogInsydes"] is basically only a wrapper function for
the Mathematica command Get. Additionally, ReadNetlist checks if the netfile loaded contains a
Circuit object and appends Simulator −> "AnalogInsydes" to the GlobalParameters field.
PSpice
ReadNetlist["netfile", Simulator −> "PSpice"] supports the following devices:
PSpice reference designator
Analog Insydes type
C
Capacitor or model call
D
model call
E
VCVSource or model template
F
CCCSource
G
VCCSource or model template
H
CCVSource
I
CurrentSource
J
model call
L
Inductor or model call
M
model call
Q
model call
R
Resistor or model call
V
VoltageSource
X
subcircuit call
PSpice reference designators supported by ReadNetlist.
All model calls have functions as references for Model, Selector, and Parameters were applicable:
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{rd, {n1 −> p1 , },
Model −> Model[type, model, rd],
Selector −> Selector[type, model, rd],
Parameters −> Parameters[type, model, rd], }
where type is one of "BJT", "CAP", "Diode", "IND", "JFET", "MOSFET" , or "RES". The value model
is the model name, rd the reference designator of the device.
Linear resistors (R), capacitors (C), and inductors (L) are converted to their generic counterpart in
Analog Insydes. Model calls are generated if temperature dependencies are given or a model is
defined.
Linear controlled sources (E, F, G, H) are converted to their generic counterpart in Analog Insydes. In
case of current controlled sources (F, H) additional nodes are introduced. In case of nonlinear voltage
controlled sources (E, G) a behavioral model is generated and appended to the netlist. Supported
types are POLY, VALUE, and TABLE.
Independent sources (I, V) of type EXP, PULSE, PWL, SFFM, and SIN are supported (see Chapter 4.1).
Parameterized subcircuits without optional nodes are also supported and may contain local model
cards.
The following cards are supported:
PSpice card
.INCLUDE
.LIB
action
include the given file
search for missing models and subcircuits
.MODEL
convert to ModelParameters
.PARAM
convert to GlobalParameters
.OPTIONS
.SUBCKT
scan for GMIN, DEFL, DEFW, DEFAD, DEFAS, DEFPD, DEFPS,
DEFNRD, DEFNRS, TNOM and convert to GlobalParameters
convert to Subcircuit
.TEMP
convert to GlobalParameters
.TRAN
scan for tstep and tstop and convert to GlobalParameters
.TITLE
PSpice cards supported by ReadNetlist.
The following cards are ignored:
print title according the setting of the Protocol option
3.10 Interfaces
371
.AC
.ALIASES
.DC
.NOISE
.PLOT
.PRINT
.PROBE
.OP
.WIDTH
PSpice cards ignored by ReadNetlist.
Note that all temperatures like TEMP are converted from C to K.
ReadNetlist["netfile", "outfile", Simulator −> "PSpice"] additionally scans the operating-point
information section in the output file outfile. Afterwards the circuit is flattened via an internal call
to ExpandSubcircuits and the small-signal parameters are appended to the corresponding model
instance. The parameter names are postfixed with "$ac" to avoid ambiguities.
Eldo
ReadNetlist["netfile", Simulator −> "Eldo"] supports the following devices:
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Eldo reference designator
Analog Insydes type
C
Capacitor or model call
D
model call
E
VCVSource or model template
F
CCCSource or model template
G
VCCSource
H
CCVSource
I
CurrentSource
J
model call
L
Inductor or model call
M
model call
Q
model call
R
Resistor or model call
V
VoltageSource
X
subcircuit call
Y
model call
Eldo reference designators supported by ReadNetlist.
All model calls have functions as references for Model, Selector , and Parameters were applicable:
{rd, {n1 −> p1 , },
Model −> Model[type, model, rd],
Selector −> Selector[type, model, rd],
Parameters −> Parameters[type, model, rd], }
where type is one of "BJT", "CAP", "Diode", "IND", "JFET", "MOSFET" , or "RES". The value model
is the model name, rd the reference designator of the device.
Linear resistors (R), capacitors (C), and inductors (L) are converted to their generic counterpart in
Analog Insydes. Model calls are generated if temperature dependencies are given or a model is
defined.
Linear controlled sources (E, F, G, H) are converted to their generic counterpart in Analog Insydes. In
case of current controlled sources (F, H) additional nodes are introduced. In case of nonlinear voltage
controlled sources (E, G) a behavioral model is generated and appended to the netlist. Supported
types are POLY, VALUE, and TABLE.
Independent sources (I, V) of type EXP, PULSE, PWL, SFFM, and SIN are supported (see Chapter 4.1).
3.10 Interfaces
373
Parameterized subcircuits are also supported and may contain local model cards and local subcircuit
definitions.
For calls to HDLA models (Y), ReadNetlist reads the file "hdlaInfo" and scans the section
[HdlaPins] to set up the correct node to port mapping:
{rd, {n1 −> p1 , },
Model −> Model[entity, architecture, rd],
Selector −> architecture, }
with the HDLA entity name entity and the HDLA architecture name architecture. If the file cannot
be found a generic port list is generated:
{rd, {n1 −> 1, n2 −> 2, }
Model −> Model[entity, architecture, rd],
Selector −> architecture, }
The environment variables HDLALIBPATH and HDLAWORKPATH are taken into account while searching
for the file "hdlaInfo" .
The following cards are supported:
Eldo card
action
.ADDLIB
search for missing models and subcircuits
.DC
scan for TEMP and convert to GlobalParameters
.HDLALIB
search for macro interface definitions of HDLA models
.INCLUDE
include the given file
.LIB
search for missing models and subcircuits
.MODEL
convert to ModelParameters
.PARAM
convert to GlobalParameters or subcircuit parameters
.OPTIONS
.SUBCKT
scan for GMIN, DEFL, DEFW, DEFAD, DEFAS, DEFPD, DEFPS,
DEFNRD, DEFNRS, TNOM and convert to GlobalParameters
convert to Subcircuit
.TEMP
convert to GlobalParameters
.TRAN
scan for tstep and tstop and convert to GlobalParameters
.TITLE
Eldo cards supported by ReadNetlist.
The following cards are ignored:
print according the setting of the Protocol option
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.AC
.ALIASES
.NOISE
.PLOT
.PRINT
.PROBE
.OP
.RAMP
.VIEW
.WIDTH
Eldo cards ignored by ReadNetlist.
Note that all temperatures like TEMP are converted from C to K.
ReadNetlist["netfile", "outfile", Simulator −> "Eldo"] additionally scans the operating-point information section in the output file outfile. Afterwards the circuit is flattened via an internal call
to ExpandSubcircuits and the small-signal parameters are appended to the corresponding model
instance. The parameter names are postfixed with "$ac" to avoid ambiguities.
Saber
ReadNetlist["netfile", Simulator −> "Saber"] reads in Saber netlists. Due to the fact that all elements in Saber MAST are calls to MAST templates, ReadNetlist replaces some template calls with
generic Analog Insydes elements:
3.10 Interfaces
375
MAST template name
Analog Insydes type
C
Capacitor
CL
Capacitor
CAP
Capacitor
L
Inductor
R
Resistor
RES
Resistor
SPI_DC
CurrentSource
I_DC
CurrentSource
I_PULSE
CurrentSource
I_SIN
CurrentSource
I
CurrentSource
SPV_DC
VoltageSource
V_DC
VoltageSource
V_PULSE
VoltageSource
V_SIN
VoltageSource
V
VoltageSource
MAST templates replaced with generic Analog Insydes elements.
All other elements are converted to
{name, {n1 −> p1 , },
Model −> Model[model, model, name],
Selector −> Selector[model, model, name],
Parameters −> Parameters[model, model, name], }
where name is the name of the element and model is the template name.
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3.10.3 ReadSimulationData
ReadSimulationData["file", Simulator−>sim]
reads a data file "file" from a simulator sim and converts
its content to the Analog Insydes data format
ReadSimulationData["file", key, Simulator−>sim]
reads data sets marked with key
Command structure of ReadSimulationData .
ReadSimulationData returns the result according to the Analog Insydes numerical data format described in Section 3.7.1. It consists of lists of rules, where the variables are assigned
InterpolatingFunction objects. Note that Analog Insydes supports a multi-dimensional data
format.
The return value of ReadSimulationData can be used as first argument to most Analog Insydes
graphics functions such as BodePlot (Section 3.9.1) or TransientPlot (Section 3.9.6) for visualizing
the simulation data.
ReadSimulationData has the following options:
option name
default value
InterpolationOrder
1
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
Simulator
Inherited[AnalogInsydes]
standard Analog Insydes simulator
specification (see Section 3.14.6)
specifies the interpolation order of the
resulting InterpolingFunction
Options for ReadSimulationData .
See also: WriteSimulationData (Section 3.10.4), ReadNetlist (Section 3.10.1), BodePlot (Section 3.9.1), TransientPlot (Section 3.9.6), Simulator (Section 3.14.6), Section 3.7.1.
Options Description
InterpolationOrder
With InterpolationOrder −> integer, you can change the interpolation order of the resulting interpolating functions. See the Mathematica object InterpolatingFunction .
3.10 Interfaces
377
The default setting is InterpolationOrder −> 1, which is linear interpolation.
Simulator
To distinguish between different file formats used by different simulators it is necessary to specify
the simulator which generated the data file. Possible values for Simulator are:
simulator
"AnalogInsydes"
"Eldo"
"PSpice"
"Saber"
file type
Mathematica *.m file
Binary *.cou file
CSDF file, *.csd or *.txt
SaberScope file
Supported simulators and file types by ReadSimulationData .
If file contains multiple data sets like DC and AC analysis results, a key can be given as a second
argument to ReadSimulationData to distinguish between these sets ("Eldo" and "PSpice" only):
simulator
"Eldo"
"PSpice"
possible keys
"HERTZ", "VOLT", "SEC"
"AC Sweep", "DC Sweep", "Transient"
Simulator-dependent keys for ReadSimulationData .
Please note that the keys may depend on the used simulator version.
The return value has the data format as described in Section 3.7.1. The labels are of type String.
Reading a CSDF file generated with PSpice which contains simulation data from various simulation
runs, results in misleading values for SweepParameters .
An Eldo "*.cou" file generated with the Eldo command line option "−dbp" (the default) contains
the complex data expressed as magnitude and phase, with "−ri" as real and imaginary values.
ReadSimulationData automatically adds the corresponding complex waveforms.
Only Eldo "*.cou" binary files written with big endian machines are supported.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
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Read simulation data from
PSpice data file.
In[2]:= data = ReadSimulationData[
"AnalogInsydes/DemoFiles/Multivibrator.txt",
Simulator −> "PSpice"]
Out[2]=
V4 InterpolatingFunction0.00018,
V5 InterpolatingFunction0.00018,
SweepParameters VCC 0.,
V4 InterpolatingFunction0.00018,
V5 InterpolatingFunction0.00018,
SweepParameters VCC 6.,
V4 InterpolatingFunction0.00018,
V5 InterpolatingFunction0.00018,
SweepParameters VCC 12.
The data can easily be
plotted with
TransientPlot .
0.00012, ,
0.00012, ,
0.00012, ,
0.00012, ,
0.00012, ,
0.00012, ,
In[3]:= TransientPlot[data, {"V(4)"[IC], "V(5)"[IC]},
{IC, −0.00018, 0.00012}]
10
8
V(4)[IC]
6
V(5)[IC]
4
2
-0.00015
-0.0001
-0.00005
0.000050.0001
IC
Out[3]= Graphics 3.10.4 WriteSimulationData
WriteSimulationData["file", exprs, {x, xmin, xmax}, Simulator−>sim]
writes sampled data of exprs to file for usage with
simulator sim
WriteSimulationData["file", traces, vars, {x, xmin, xmax}, Simulator−>sim]
writes sampled data of the variables vars as a function of x
from xmin to xmax where traces corresponds to the return
value of numerical equation solvers like NDSolve or
NDAESolve
Command structure of WriteSimulationData .
WriteSimulationData can be used for displaying Analog Insydes results in a post-processor of
an external design framework. Therefore WriteSimulationData writes different file formats which
are addressed by the option Simulator . WriteSimulationData does not support multi-dimensional
data.
3.10 Interfaces
379
WriteSimulationData has the following options:
option name
default value
ComplexValues
False
whether the data should be written as
complex numbers
Compiled
True
if exprs should be compiled
DataLabels
Automatic
list of strings to give the traces a more
convenient name
InterpolationOrder
1
specifies the interpolation order of the
resulting interpolating functions
PlotPoints
100
number of sample points
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
Sampling
Linear
Simulator
Inherited[AnalogInsydes]
standard Analog Insydes simulator
specification (see Section 3.14.6)
whether to sample Linear or Exponential
Options for WriteSimulationData .
See also: ReadSimulationData (Section 3.10.3), Simulator (Section 3.14.6).
Options Description
A detailed description of all WriteSimulationData options is given below in alphabetical order:
ComplexValues
To generate complex-valued data from a small-signal analysis set ComplexValues −> True to write
the data as complex numbers.
The default ComplexValues −> False writes real numbers which can be used for DC and transient
analyses.
Compiled
With Compiled −> True the expression exprs is compiled with the Mathematica command Compile
before evaluation.
3. Reference Manual
380
DataLabels
With DataLabels −> {string , string , † † † } you can specifiy new labels for the return value of exprs
or for each variable in vars. This is necessary because post-processors often rely on a special name
scheme for trace names. Using a Mathematica expression may irritate the post-processor.
InterpolationOrder
With InterpolationOrder −> integer, you can change the interpolation order of the resulting interpolating functions which are generated if the option Simulator is set to "AnalogInsydes" . See the
Mathematica object InterpolatingFunction .
The default setting is InterpolationOrder −> 1, which is linear interpolation.
PlotPoints
The option PlotPoints −> integer specifies the number of sample points. The default is 100.
Sampling
With Sampling −> Exponential the functions are sampled exponentially. The default is
Sampling −> Linear.
Simulator
The following file formats are supported:
simulator
"AnalogInsydes"
"PSpice"
"Saber"
file type
Mathematica *.m file
CSDF file, *.csd or *.txt
ASCII SaberScope file
Simulators and file types supported by WriteSimulationData .
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Write a SaberScope file.
In[2]:= WriteSimulationData["Test.p1.dat",
{(1. − Exp[−t])*Cos[t]}, {t, 0., 10.},
DataLabels −> {"F"}, Simulator −> "Saber"]
3.10 Interfaces
381
View the file
"Test.p1.dat" with
SaberScope. Alternatively,
read it again into
Mathematica with
ReadSimulationData .
In[3]:= data = ReadSimulationData["Test.p1.dat",
Simulator −> "Saber"]
Plot the data with
TransientPlot .
In[4]:= TransientPlot[data, "F"[t], {t, 0., 10.}]
Out[3]=
F InterpolatingFunction0., 10., 1
0.5
2
4
6
8
10
t
F[t]
-0.5
-1
Out[4]= Graphics 3.10.5 WriteModel
WriteModel["file", modelname, dae, ports, connections, Simulator−>sim]
converts a DAEObject dae into a simulator-specific model
with respect to the ports and the defined circuit interface
given by connections
Command structure of WriteModel.
The function WriteModel converts a DAEObject dae into a behavioral model for a given simulator.
An interface can be defined which connects the model with a circuit. The connections are then given
by a list of rules of the form:
{param1 −> Voltage[port1 , port2 ],
param2 −> Current[port3 , port4 ],
var1 −> Voltage[port5 , port6 ],
var2 −> Current[port7 , port8 ]}
This interface definition will generate two input port branches from port to port (voltage) and from
port to port (current) and two output branches from port to port (voltage) and from port to port
(current).
WriteModel has the following options:
3. Reference Manual
382
option name
default value
DesignPoint
Automatic
specifies a list of numerical reference values
for symbolic element values in a set of
circuit equations
InitialConditions
Automatic
specifies alternative initial conditions
InitialGuess
Automatic
specifies an alternative initial guess
Simulator
Inherited[AnalogInsydes]
standard Analog Insydes simulator
specification (see Section 3.14.6)
Options for WriteModel.
Options Description
A detailed description of all WriteModel options is given below in alphabetical order:
DesignPoint
With
DesignPoint −> {symbol −> value , † † † } you can overwrite the design point given in the DAEObject
(see also CircuitEquations in Section 3.5.1). The default setting is DesignPoint −> Automatic ,
which means to use the design point given in the DAEObject.
InitialConditions
The option InitialConditions allows for explicitly specifying initial conditions. The default setting is InitialConditions −> Automatic , which means to use the initial conditions given in the
DAEObject. Valid option values are as follows:
Automatic
{var == value , † † † }
uses initial conditions given in the DAEObject
uses specified equations as initial conditions
Values for the InitialConditions option.
InitialGuess
The option InitialGuess allows for explicitly specifying an initial guess for the Newton iteration
when computing the operating point. The default setting is InitialGuess −> Automatic , which
means to use the initial guess given in the DAEObject. Possible option values are as follows:
3.10 Interfaces
383
Automatic
{var −> value , † † † }
uses initial guess given in the DAEObject
uses specified values as initial guess
Values for the InitialGuess option.
Simulator
The following simulator-dependent modeling languages are supported:
simulator
"AnalogInsydes"
"PSpice"
"Saber"
model type
Analog Insydes Model object
netlist model consisting of controlled sources
Saber MAST model
Simulators and model types supported by WriteModel.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Read a netlist.
In[2]:= net = ReadNetlist[
"AnalogInsydes/DemoFiles/Multivibrator.cir",
Simulator −> "PSpice"]
Out[2]= Circuit Set up a system of
equations.
In[3]:= dae = CircuitEquations[net,
ElementValues −> Symbolic, AnalysisMode −> DC]
Out[3]= DAEDC, 16 16 Display the variables of
dae. These variables can
be used as outputs.
In[4]:= GetVariables[dae]
Display the parameters of
dae. These parameters can
be used as inputs.
In[5]:= GetParameters[dae]
Out[4]=
V$1, V$2, V$3, V$4, V$5, I$BS$Q1, I$CS$Q1, I$ES$Q1, ib$Q1, ic$Q1,
I$BS$Q2, I$CS$Q2, I$ES$Q2, ib$Q2, ic$Q2, I$VCC
Out[5]=
AREA$Q1, AREA$Q2, BF$Q1, BF$Q2, BR$Q1, BR$Q2, GMIN, IS$Q1, IS$Q2,
R1, R2, R3, R4, R5, TEMP, TNOM, VCC, $k, $q
384
Generate the Saber MAST
model test. The first
branch ("IP" −> "IN")
defines IC as the
corresponding input
current. The second
branch ("VP" −> "VN")
defines an input voltage
VCC and an output current
I$VCC.
3. Reference Manual
In[6]:= WriteModel["test.sin", "test", dae,
{"IP", "IN", "VP", "VN"},
{IC −> Current["IP", "IN"], VCC −> Voltage["VP", "VN"],
I$VCC −> Current["VP", "VN"]},
Simulator −> "Saber"]
3.11 Linear Simplification Techniques
385
3.11 Linear Simplification Techniques
A special feature of Analog Insydes is its capability of analyzing large analog circuits symbolically
by means of symbolic approximation techniques. Analog Insydes provides two different methods
for computing approximated transfer functions, namely a simplification-before-generation (SBG, see
function ApproximateMatrixEquation in Section 3.11.3) and a simplification-after-generation method
(SAG, see function ApproximateTransferFunction in Section 3.11.2). SBG refers to methods that
simplify a circuit analysis problem before a symbolic transfer function is generated from a system
of equations or a network graph. SAG algorithms approximate a transfer function by discarding
insignificant terms after the exact expression has been computed.
Furthermore, this section contains the function ComplexityEstimate (Section 3.11.1) for estimating
the number of product terms in a symbolic transfer function, and the function
CompressMatrixEquation (Section 3.11.4) for compressing a system of matrix equations.
3.11.1 ComplexityEstimate
ComplexityEstimate[dae]
ComplexityEstimate[netlist]
estimates the number of product terms in the expanded
determinant of dae, which must be a system of AC circuit
equations in matrix form
estimates the complexity of any transfer function calculated
from netlist, which must be a flat netlist
Command structure of ComplexityEstimate .
Before you try to solve a system of circuit equations symbolically with Solve (Section 3.5.4), you
should use the function ComplexityEstimate to check whether the computation is feasible at all.
ComplexityEstimate computes an estimate of the number of product terms in a transfer function
computed from a symbolic matrix equation A x b. More precisely, the function computes a lower
bound for the number of product terms in the determinant of A.
As a rule of thumb, solving a system of circuit equations symbolically is technically feasible if the
number returned by ComplexityEstimate is less than . If you expect a symbolic expression
to be interpretable, its complexity should be in the range from to .
See also: Statistics (Section 3.6.17).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
3. Reference Manual
386
Define an AC model for
BJTs.
In[2]:= Circuit[
Model[
Name −> BJT,
Selector −> AC,
Scope −> Global,
Ports −> {B, C, E},
Parameters −> {Rbe, Cbc, Cbe, beta, Ro,
RBE, CBC, CBE, BETA, RO},
Definition −> Netlist[
{CBE, {B, E}, Symbolic −> Cbe, Value −> CBE},
{RBE, {X, E}, Symbolic −> Rbe, Value −> RBE},
{CBC, {B, C}, Symbolic −> Cbc, Value −> CBC},
{CC, {B, X, C, E},
Symbolic −> beta, Value −> BETA},
{RO, {C, E}, Symbolic −> Ro, Value −> RO}
]
]
] // ExpandSubcircuits;
This netlist describes a
common-emitter amplifier.
In[3]:= ceamplifier =
Circuit[
Netlist[
{V1, {1, 0}, 1},
{V0, {6, 0}, 0},
{C1, {1, 2}, Symbolic −> C1, Value
{R1, {2, 6}, Symbolic −> R1, Value
{R2, {2, 0}, Symbolic −> R2, Value
{RC, {6, 3}, Symbolic −> RC, Value
{RE, {4, 0}, Symbolic −> RE, Value
{C2, {3, 5}, Symbolic −> C2, Value
{RL, {5, 0}, Symbolic −> RL, Value
{Q1, {2 −> B, 3 −> C, 4 −> E},
Model −> BJT, Selector −> AC,
CBE −> 30.*^−12, CBC −> 5.*^−12,
RO −> 10000., BETA −> 200.}
]
]
−>
−>
−>
−>
−>
−>
−>
1.0*^−7},
1.0*^5},
47000.},
2200.},
1000.},
1.0*^−6},
47000.},
RBE −> 1000.,
Out[3]= Circuit Set up a system of
symbolic AC equations.
In[4]:= eqs = CircuitEquations[ceamplifier,
ElementValues −> Symbolic]
Out[4]= DAEAC, 10 10 Compute the complexity
estimate.
In[5]:= ComplexityEstimate[eqs]
Out[5]= 132
The number returned by ComplexityEstimate indicates that solving the equations for any transfer
function is feasible, but the result cannot be expected to yield much insight into circuit behavior.
3.11 Linear Simplification Techniques
Compute the voltage
transfer function
symbolically and expand
the result.
387
In[6]:= solution = Solve[eqs, V$5];
v5 = Together[V$5 /. First[solution]]
Out[7]=
C2 R1 R2 RC RL s C1 RE s beta$Q1 C1 Ro$Q1 s C1 Cbc$Q1 Rbe$Q1 RE s2 C1 Cbe$Q1 Rbe$Q1 RE s2 C1 Cbc$Q1 Rbe$Q1 Ro$Q1 s2 C1 Cbc$Q1 RE Ro$Q1 s2 beta$Q1 C1 Cbc$Q1 RE Ro$Q1 s2 C1 Cbc$Q1 Cbe$Q1 Rbe$Q1 RE Ro$Q1 s3 R1 R2 RC R1 Rbe$Q1 RC R2 Rbe$Q1 RC R1 R2 RE R1 Rbe$Q1 RE R2 Rbe$Q1 RE R1 RC RE R2 RC RE R1 R2 Ro$Q1 R1 Rbe$Q1 Ro$Q1 R2 Rbe$Q1 Ro$Q1 R1 RE Ro$Q1 beta$Q1 R1 RE Ro$Q1 R2 RE Ro$Q1 beta$Q1 R2 RE Ro$Q1 C1 R1 R2 Rbe$Q1 RC s Cbc$Q1 R1 R2 Rbe$Q1 RC s Cbe$Q1 R1 R2 Rbe$Q1 RC s C1 R1 R2 Rbe$Q1 RE s 94 C2 Cbe$Q1 R2 Rbe$Q1 RE RL Ro$Q1 s2 C2 Cbc$Q1 R1 RC RE RL Ro$Q1 s2 beta$Q1
C2 Cbc$Q1 R1 RC RE RL Ro$Q1 s2 C2 Cbc$Q1 R2 RC RE RL Ro$Q1 s2 beta$Q1 C2 Cbc$Q1 R2 RC RE RL Ro$Q1 s2 C1 C2 Cbc$Q1
R1 R2 Rbe$Q1 RC RE RL s3 C1 C2 Cbe$Q1 R1 R2 Rbe$Q1 RC RE RL s3 C1 C2 Cbe$Q1 R1 R2 Rbe$Q1 RC RE Ro$Q1 s3 C1
Cbc$Q1 Cbe$Q1 R1 R2 Rbe$Q1 RC RE Ro$Q1 s3 C2 Cbc$Q1 Cbe$Q1 R1 R2
Rbe$Q1 RC RE Ro$Q1 s3 C1 C2 Cbc$Q1 R1 R2 Rbe$Q1 RC RL Ro$Q1 s3 C2 Cbc$Q1 Cbe$Q1 R1 R2 Rbe$Q1 RC RL Ro$Q1 s3 C1 C2 Cbe$Q1 R1 R2 Rbe$Q1 RE RL Ro$Q1 s3 C2 Cbc$Q1 Cbe$Q1
R1 R2 Rbe$Q1 RE RL Ro$Q1 s3 C1 C2 Cbc$Q1 R1 R2 RC RE RL Ro$Q1 s3 beta$Q1 C1 C2 Cbc$Q1 R1 R2 RC RE RL Ro$Q1 s3 C2 Cbc$Q1 Cbe$Q1 R1 Rbe$Q1 RC RE RL Ro$Q1 s3 C2 Cbc$Q1 Cbe$Q1 R2 Rbe$Q1 RC RE RL Ro$Q1 s3 C1 C2 Cbc$Q1 Cbe$Q1 R1 R2 Rbe$Q1 RC RE RL Ro$Q1 s4 Here, the complexity estimate is identical to the true number of terms in the denominator of the
symbolic transfer function. In the general case, the estimate yields a lower bound for this number.
Determine number of
terms in the denominator
of the transfer function.
In[8]:= Length[Denominator[v5]]
Out[8]= 132
3.11.2 ApproximateTransferFunction
ApproximateTransferFunction[expr, fvar, dp, error]
approximates the transfer function expr by discarding
insignificant terms where fvar, dp, and error denote the
complex frequency variable, the design point, and the
bound for the coefficient error
Command structure of ApproximateTransferFunction .
ApproximateTransferFunction approximates a symbolic transfer function by discarding insignificant terms from its coefficients (simplification after generation, SAG). The significance or insignificance of a term is evaluated on the basis of numerical reference values for the symbols (the design
point). For each coefficient, the algorithm removes the numerically least significant terms until the
maximum coefficient error is reached.
3. Reference Manual
388
The fourth argument of ApproximateTransferFunction denotes the maximum coefficient
error. It does not constitute a bound on the absolute error of a transfer function. The
absolute error may be larger or lower than the maximum coefficient error.
See also: ApproximateMatrixEquation (Section 3.11.3).
Examples
Below, we approximate the transfer function of the common-emitter amplifier. We allow for a
maximum error of in all coefficients of powers of the complex frequency s.
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define an AC model for
BJTs.
In[2]:= Circuit[
Model[
Name −> BJT,
Selector −> AC,
Scope −> Global,
Ports −> {B, C, E},
Parameters −> {Rbe, Cbc, Cbe, beta, Ro,
RBE, CBC, CBE, BETA, RO},
Definition −> Netlist[
{CBE, {B, E}, Symbolic −> Cbe, Value −> CBE},
{RBE, {X, E}, Symbolic −> Rbe, Value −> RBE},
{CBC, {B, C}, Symbolic −> Cbc, Value −> CBC},
{CC, {B, X, C, E},
Symbolic −> beta, Value −> BETA},
{RO, {C, E}, Symbolic −> Ro, Value −> RO}
]
]
] // ExpandSubcircuits;
This netlist describes a
common-emitter amplifier.
In[3]:= ceamplifier =
Circuit[
Netlist[
{V1, {1, 0}, 1},
{V0, {6, 0}, 0},
{C1, {1, 2}, Symbolic −> C1, Value
{R1, {2, 6}, Symbolic −> R1, Value
{R2, {2, 0}, Symbolic −> R2, Value
{RC, {6, 3}, Symbolic −> RC, Value
{RE, {4, 0}, Symbolic −> RE, Value
{C2, {3, 5}, Symbolic −> C2, Value
{RL, {5, 0}, Symbolic −> RL, Value
{Q1, {2 −> B, 3 −> C, 4 −> E},
Model −> BJT, Selector −> AC,
CBE −> 30.*^−12, CBC −> 5.*^−12,
RO −> 10000., BETA −> 200.}
]
]
Out[3]= Circuit −>
−>
−>
−>
−>
−>
−>
1.0*^−7},
1.0*^5},
47000.},
2200.},
1000.},
1.0*^−6},
47000.},
RBE −> 1000.,
3.11 Linear Simplification Techniques
Set up a system of
symbolic AC equations.
389
In[4]:= eqs = CircuitEquations[ceamplifier,
ElementValues −> Symbolic]
Out[4]= DAEAC, 10 10 Compute the complexity
estimate.
In[5]:= ComplexityEstimate[eqs]
Compute the voltage
transfer function
symbolically and extract
the result from the
solution vector.
In[6]:= solution = Solve[eqs, V$5];
v5 = Together[V$5 /. First[solution]]
Get the design-point
information from the
DAEObject.
In[8]:= dp = GetDesignPoint[eqs]
Discard insignificant terms
in the transfer function.
In[9]:= sag = ApproximateTransferFunction[v5, s, dp, 0.2]
Out[5]= 132
Out[7]=
C2 R1 R2 RC RL s C1 RE s beta$Q1 C1 Ro$Q1 s C1 Cbc$Q1 Rbe$Q1 RE s2 C1 Cbe$Q1 Rbe$Q1 RE s2 C1 Cbc$Q1 Rbe$Q1 Ro$Q1 s2 C1 Cbc$Q1 RE Ro$Q1 s2 beta$Q1 C1 Cbc$Q1 RE Ro$Q1 s2 C1 Cbc$Q1 Cbe$Q1 Rbe$Q1 RE Ro$Q1 s3 R1 R2 RC R1 Rbe$Q1 RC R2 Rbe$Q1 RC R1 R2 RE R1 Rbe$Q1 RE R2 Rbe$Q1 RE R1 RC RE R2 RC RE R1 R2 Ro$Q1 R1 Rbe$Q1 Ro$Q1 R2 Rbe$Q1 Ro$Q1 R1 RE Ro$Q1 beta$Q1 R1 RE Ro$Q1 R2 RE Ro$Q1 beta$Q1 R2 RE Ro$Q1 C1 R1 R2 Rbe$Q1 RC s Cbc$Q1 R1 R2 Rbe$Q1 RC s Cbe$Q1 R1 R2 Rbe$Q1 RC s C1 R1 R2 Rbe$Q1 RE s 94 C2 Cbe$Q1 R2 Rbe$Q1 RE RL Ro$Q1 s2 C2 Cbc$Q1 R1 RC RE RL Ro$Q1 s2 beta$Q1
C2 Cbc$Q1 R1 RC RE RL Ro$Q1 s2 C2 Cbc$Q1 R2 RC RE RL Ro$Q1 s2 beta$Q1 C2 Cbc$Q1 R2 RC RE RL Ro$Q1 s2 C1 C2 Cbc$Q1
R1 R2 Rbe$Q1 RC RE RL s3 C1 C2 Cbe$Q1 R1 R2 Rbe$Q1 RC RE RL s3 C1 C2 Cbe$Q1 R1 R2 Rbe$Q1 RC RE Ro$Q1 s3 C1
Cbc$Q1 Cbe$Q1 R1 R2 Rbe$Q1 RC RE Ro$Q1 s3 C2 Cbc$Q1 Cbe$Q1 R1 R2
Rbe$Q1 RC RE Ro$Q1 s3 C1 C2 Cbc$Q1 R1 R2 Rbe$Q1 RC RL Ro$Q1 s3 C2 Cbc$Q1 Cbe$Q1 R1 R2 Rbe$Q1 RC RL Ro$Q1 s3 C1 C2 Cbe$Q1 R1 R2 Rbe$Q1 RE RL Ro$Q1 s3 C2 Cbc$Q1 Cbe$Q1
R1 R2 Rbe$Q1 RE RL Ro$Q1 s3 C1 C2 Cbc$Q1 R1 R2 RC RE RL Ro$Q1 s3 beta$Q1 C1 C2 Cbc$Q1 R1 R2 RC RE RL Ro$Q1 s3 C2 Cbc$Q1 Cbe$Q1 R1 Rbe$Q1 RC RE RL Ro$Q1 s3 C2 Cbc$Q1 Cbe$Q1 R2 Rbe$Q1 RC RE RL Ro$Q1 s3 C1 C2 Cbc$Q1 Cbe$Q1 R1 R2 Rbe$Q1 RC RE RL Ro$Q1 s4 Out[8]=
C1 1. 107 , R1 100000., R2 47000., RC 2200., RE 1000.,
C2 1. 106 , RL 47000., Cbe$Q1 3. 1011 , Rbe$Q1 1000.,
Cbc$Q1 5. 1012 , beta$Q1 200., Ro$Q1 10000.
Out[9]=
beta$Q1 C1 C2 R1 R2 RC RL Ro$Q1 s2 beta$Q1 C1 C2 Cbc$Q1 R1 R2 RC RE RL Ro$Q1 s3 C1 C2 Cbc$Q1 Cbe$Q1 R1 R2 Rbe$Q1 RC RE RL Ro$Q1 s4 R1 R2 Ro$Q1 beta$Q1 R1 RE Ro$Q1 beta$Q1 R2 RE Ro$Q1 C2 R1 R2 RL Ro$Q1 beta$Q1 C2 R1 RE RL Ro$Q1 beta$Q1 C2 R2 RE RL Ro$Q1 s beta$Q1 C1 C2 R1 R2 RE RL Ro$Q1 s2 beta$Q1 C1 C2 Cbc$Q1 R1 R2 RC RE RL Ro$Q1 s3 C1 C2 Cbc$Q1 Cbe$Q1 R1 R2 Rbe$Q1 RC RE RL Ro$Q1 s4 3. Reference Manual
390
Determine the complexity
of the approximated
expression.
Simplify the result.
In[10]:= Length[Denominator[sag]]
Out[10]= 7
In[11]:= Simplify[sag]
Out[11]=
C1 C2 R1 R2 RC RL s2
Cbc$Q1 Cbe$Q1 Rbe$Q1 RE s2 beta$Q1 1 Cbc$Q1 RE s beta$Q1 R2 RE 1 C2 RL s R1 beta$Q1 RE 1 C2 RL s R2 1 C2 RL s beta$Q1 C1 C2 RE RL s2 beta$Q1 C1 C2 Cbc$Q1 RC RE RL s3 C1 C2 Cbc$Q1 Cbe$Q1 Rbe$Q1 RC RE RL s4 To validate the result, we compute the frequency response of the original system numerically and
compare it graphically with the approximated result.
Compute the frequency
response numerically.
In[12]:= acsweep = ACAnalysis[eqs, V$5, {f, 1, 1.0*^9}]
Evaluate the approximated
expression numerically.
In[13]:= sagn[s_] = sag /. dp
Plot the exact and
approximated functions
together.
In[14]:= BodePlot[acsweep, {V$5[f], sagn[2 Pi I f]},
{f, 1, 1.0*^9}, TraceNames −> {"Exact", "Approximated"}]
Out[12]=
V$5 InterpolatingFunction1., 1. 109 , Phase (deg)
Magnitude (dB)
Out[13]=
9.7196 1010 s2 485.98 s3 7.2897 108 s4 3.41 1014 1.6027 1013 s 4.418 1010 s2 485.98 s3 7.2897 108 s4 1.0E0
1.0E2
1.0E4
1.0E6
1.0E8
5
2.5
0
-2.5
-5
-7.5
-10
1.0E0
1.0E2
1.0E4
1.0E6
Frequency
1.0E8
1.0E2
1.0E4
1.0E6
1.0E8
1.0E2
1.0E4
1.0E6
Frequency
1.0E8
1.0E0
0
-50
-100
-150
-200
-250
-300
-350
1.0E0
Exact
Out[14]= Graphics Approximated
3.11 Linear Simplification Techniques
391
3.11.3 ApproximateMatrixEquation
ApproximateMatrixEquation[dae, var, errspec]
approximates a symbolic matrix equation dae with respect
to the output variable var and the error specification errspec
Command structure of ApproximateMatrixEquation .
The function ApproximateMatrixEquation approximates a linear symbolic matrix equation by discarding insignificant matrix entries before the system is solved (simplification before generation,
SBG). The significance of a symbolic matrix entry is determined by calculating its numerical influence on the magnitude of the transfer function in one or several frequency sampling points. The
approximation process follows a term ranking obtained by computing the large-change sensitivities of
the output variable with respect to all matrix entries and sorting the resulting list by least influence.
The algorithm deletes all matrix entries whose cumulative influence is less than the user-specified
error bound. The error specification errspec has to be given in the following form:
{fvar−>freq, MaxError−>maxerr}
error specification for a single frequency sampling point
{{fvar−>freq , MaxError−>maxerr }, {fvar−>freq , MaxError−>maxerr }, † † † }
error specification for several frequency sampling points
Format for sampling point and error specifications.
For a given errspec, the algorithm assures that the relative magnitude deviation of the output
variable var measured at the frequency points fvar = freqi is less than maxerri .
To approximate the frequency response of a dynamic circuit, sampling points must be
placed on the imaginary axis where s jΩ, or equivalently s Πj f . Therefore, you should
specify frequency sampling points in the form s −> 2. Pi I f.
By placing sampling points to the left and right of a corner in a frequency response, you can
use ApproximateMatrixEquation to extract poles and zeros symbolically.
For optimal approximation results, you should set up circuit equations in Sparse Tableau
formulation (Formulation −> SparseTableau , see Section 3.5.1).
ApproximateMatrixEquation has the following options:
3. Reference Manual
392
option name
default value
AbsolutePivotThreshold
1.0*10^−8
the absolute pivot threshold for the sparse
matrix solver used in the MathLink binary
MSBG.exe
CompressEquations
False
whether to compress the matrix equation
after approximation
DesignPoint
Automatic
the numerical reference data for the symbols
NumericalAccuracy
1.0*10^−8
the numerical accuracy threshold for
ranking recomputations
PivotThreshold
0.01
the relative pivot threshold for the sparse
matrix solver used in the MathLink binary
MSBG.exe
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
Options for ApproximateMatrixEquation , Part I.
option name
default value
QuasiSingularity
1.0*10^−15
threshold for detecting numerical
singularities
2.0
threshold for triggering a ranking
recomputation
RecomputationThreshold
SortingMethod
PrimaryDesignPoint
how to sort the list of matrix entry
influences to obtain a term ranking
StripIndependentBlocks
False
UseExternals
whether to search for and discard
algebraically independent blocks from a
matrix equation after approximation
Inherited[AnalogInsydes]
whether to use the MathLink binary
MSBG.exe (see Section 3.14.7)
Options for ApproximateMatrixEquation , Part II.
3.11 Linear Simplification Techniques
393
See also: ApproximateTransferFunction (Section 3.11.2).
Options Description
A detailed description of all ApproximateMatrixEquation options is given below in alphabetical
order:
AbsolutePivotThreshold
With AbsolutePivotThreshold −> posreal you can specify the minimum absolute value a matrix
entry must have to be considered as a pivot candidate during LU decomposition. There is no
need to change the value of this option unless ApproximateMatrixEquation reports problems with
ill-conditioned equations.
CompressEquations
The setting CompressEquations −> True causes the matrix equation to be compressed after approximation. The option is realized by an internal call of the function CompressMatrixEquation
(Section 3.11.4). You should turn matrix compression on whenever you are working with large
matrices, i.e. when the matrix dimensions are significantly larger than .
DesignPoint
With DesignPoint −> {symbol −> value , † † † } you can overwrite the design point given in the
DAEObject. The default setting is DesignPoint −> Automatic , which means to use the design
point stored in the DAEObject.
NumericalAccuracy
The option NumericalAccuracy −> posreal specifies the minimum influence a matrix entry must have
to trigger a recomputation of the term ranking. This threshold is used to prevent frequent ranking
recomputations due to numerical inaccuracies in very small influence values.
If the ranking is recomputed too often at the beginning of the approximation process, you should
increase the value of NumericalAccuracy by an order of magnitude. Note that information about
ranking recomputations is available only if UseExternals −> False and Protocol −> Notebook.
See also the description of RecomputationThreshold .
PivotThreshold
With PivotThreshold −> posreal you can specify the minimum relative magnitude a matrix entry
must have to be considered as a pivot candidate during LU decomposition. The value posreal must
be a number between and . If it is , then the pivoting method becomes complete pivoting,
3. Reference Manual
394
which is very slow and tends to fill up the matrix. If it is set close to zero, the pivoting method
becomes strict Markowitz with no threshold. There is no need to change the value of this option
unless ApproximateMatrixEquation reports problems with ill-conditioned equations.
Protocol
This option describes the standard Analog Insydes protocol specification (see Section 3.14.5). Note
that detailed protocol information is not available if UseExternals −> True.
QuasiSingularity
QuasiSingularity −> posreal specifies a threshold value for determining whether a matrix is numerically singular. In case the matrix approximation function returns a singular matrix, increase the
value of QuasiSingularity by one or several orders of magnitude.
RecomputationThreshold
RecomputationThreshold −> posreal specifies a factor for determining whether the current term
ranking should be recomputed. For example, a value of means that a ranking recomputation is
triggered if the true influence of a matrix entry at the current approximation step turns out to be two
times larger than predicted initially. Ranking recomputations are triggered only by influence values
greater than the value of NumericalAccuracy .
SortingMethod
The option SortingMethod specifies how the list of matrix entry influences is sorted to obtain a term
ranking.
PrimaryDesignPoint
sort matrix entries by least influence on the primary
design-point solution
LeastMeanInfluence
sort matrix entries by mean influence on all design-point
solutions
Values for the SortingMethod option.
With the default setting SortingMethod −> PrimaryDesignPoint , the influence list is sorted by
influence on the solution of the matrix equation at the first sampling point specified in errspec. The
setting SortingMethod −> LeastMeanInfluence causes the least to be sorted by average influence
on the solutions at all sampling points.
The setting of the SortingMethod option is relevant only if you specify more than one sampling
point. In general, SortingMethod −> LeastMeanInfluence is the preferred setting for multiple
sampling points.
3.11 Linear Simplification Techniques
395
StripIndependentBlocks
With StripIndependentBlocks −> True, clusters of equations which are decoupled from the variable of interest, are searched for and discarded after approximation. This option is effective only in
conjunction with the setting CompressEquations −> True.
UseExternals
The setting UseExternals −> True causes the MathLink application MSBG.exe to be used for matrix
approximation. This is the recommended setting for all platforms for which native Analog Insydes
versions exist. You can set UseExternals −> False to run Analog Insydes on other platforms or
for debugging purposes in conjunction with the option Protocol . Alternatively, you can also load
and unload the module manually as described in Section 3.13.3. For more information refer to
UseExternals (Section 3.14.7).
Examples
In the following we approximate the system of nodal equations representing the small-signal equivalent circuit of the common-emitter amplifier with respect to the output variable V$5. To obtain an
approximation of the amplifier’s passband voltage gain, we select a sampling frequency of kHz.
At the sampling point we allow the magnitude of the solution of the approximated equations to
deviate from the original design-point value by .
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define an AC model for
BJTs.
In[2]:= Circuit[
Model[
Name −> BJT,
Selector −> AC,
Scope −> Global,
Ports −> {B, C, E},
Parameters −> {Rbe, Cbc, Cbe, beta, Ro,
RBE, CBC, CBE, BETA, RO},
Definition −> Netlist[
{CBE, {B, E}, Symbolic −> Cbe, Value −> CBE},
{RBE, {X, E}, Symbolic −> Rbe, Value −> RBE},
{CBC, {B, C}, Symbolic −> Cbc, Value −> CBC},
{CC, {B, X, C, E},
Symbolic −> beta, Value −> BETA},
{RO, {C, E}, Symbolic −> Ro, Value −> RO}
]
]
] // ExpandSubcircuits;
3. Reference Manual
396
This netlist describes a
common-emitter amplifier.
In[3]:= ceamplifier =
Circuit[
Netlist[
{V1, {1, 0}, 1},
{V0, {6, 0}, 0},
{C1, {1, 2}, Symbolic −> C1, Value
{R1, {2, 6}, Symbolic −> R1, Value
{R2, {2, 0}, Symbolic −> R2, Value
{RC, {6, 3}, Symbolic −> RC, Value
{RE, {4, 0}, Symbolic −> RE, Value
{C2, {3, 5}, Symbolic −> C2, Value
{RL, {5, 0}, Symbolic −> RL, Value
{Q1, {2 −> B, 3 −> C, 4 −> E},
Model −> BJT, Selector −> AC,
CBE −> 30.*^−12, CBC −> 5.*^−12,
RO −> 10000., BETA −> 200.}
]
]
−>
−>
−>
−>
−>
−>
−>
1.0*^−7},
1.0*^5},
47000.},
2200.},
1000.},
1.0*^−6},
47000.},
RBE −> 1000.,
Out[3]= Circuit Set up a system of
symbolic AC equations.
In[4]:= eqs = CircuitEquations[ceamplifier,
ElementValues −> Symbolic]
Out[4]= DAEAC, 10 10 Compute the complexity
estimate.
In[5]:= ComplexityEstimate[eqs]
Allow for magnitude
error at kHz.
In[6]:= errspec = {s −> 2. Pi I 10^4, MaxError −> 0.1}
Approximate the system of
circuit equations.
In[7]:= sbg = ApproximateMatrixEquation[eqs, V$5, errspec]
Estimate the number of
terms after approximation.
In[8]:= ComplexityEstimate[sbg]
Compute the solution of
the approximated
equations.
In[9]:= sbgresult = Solve[sbg, V$5]
Out[5]= 132
Out[6]= s 62831.9 I, MaxError 0.1
Out[7]= DAEAC, 10 10 Out[8]= 1
RC
Out[9]= V$5 RE
To validate the result, we compute the frequency response of the original system numerically and
compare it graphically with the approximated result.
Compute the frequency
response numerically.
In[10]:= acsweep = ACAnalysis[eqs, V$5, {f, 1, 1.0*^9}]
Evaluate the approximated
expression numerically.
In[11]:= sbgn = V$5 /. First[sbgresult] /. GetDesignPoint[eqs]
Out[10]=
V$5 InterpolatingFunction1., 1. 109 , Out[11]= 2.2
Note that the approximated expression is valid only within the passband because we did not specify
sampling points at low or high frequencies.
3.11 Linear Simplification Techniques
In[12]:= BodePlot[acsweep, {V$5[f], sbgn},
{f, 1, 1.0*^9}, TraceNames −> {"Exact", "Approximated"}]
Phase (deg)
Magnitude (dB)
Plot the exact and
approximated functions
together.
397
1.0E0
1.0E2
1.0E4
1.0E6
1.0E8
10
5
0
-5
-10
1.0E0
1.0E2
1.0E0
0
-50
-100
-150
-200
-250
-300
-350
1.0E0
1.0E4
1.0E6
Frequency
1.0E8
1.0E2
1.0E4
1.0E6
1.0E8
1.0E2
1.0E4
1.0E6
Frequency
1.0E8
Exact
Approximated
Out[12]= Graphics Below, we specify a second sampling point at Hz to extend the validity region of the approximation
to low frequencies. Note that you may use a different error bound for each sampling point.
Add a second sampling
point at Hz.
In[13]:= errspec2 = {errspec, {s −> 2. Pi I, MaxError −> 0.3}}
Approximate the matrix
equation.
In[14]:= sbg2 = ApproximateMatrixEquation[eqs, V$5, errspec2,
SortingMethod −> LeastMeanInfluence]
Out[13]=
s 62831.9 I, MaxError 0.1, s 6.28319 I, MaxError 0.3
Out[14]= DAEAC, 10 10 Estimate the number of
terms after approximation.
In[15]:= ComplexityEstimate[sbg2]
Solve the approximated
equations.
In[16]:= sbgresult2 = Solve[sbg2, V$5] // Simplify
Out[15]= 9
Out[16]=
C1 C2 R1 R2 RC RL s2
V$5 RE R1 R2 C1 R1 R2 s 1 C2 RC RL s
Extract the solution for
V$5.
In[17]:= v52 = V$5 /. First[sbgresult2]
Evaluate the approximated
expression numerically.
In[18]:= sbgn2[s_] = v52 /. GetDesignPoint[eqs]
C1 C2 R1 R2 RC RL s2
Out[17]= RE R1 R2 C1 R1 R2 s 1 C2 RC RL s
48.598 s2
Out[18]= 1 0.0492 s 147000. 470. s
3. Reference Manual
398
Magnitude (dB)
In[19]:= BodePlot[acsweep, {V$5[f], sbgn2[2. Pi I f]},
{f, 1, 1.0*^9}, TraceNames −> {"Exact", "Approximated"}]
1.0E0
5
2.5
0
-2.5
-5
-7.5
-10
1.0E0
1.0E2
1.0E4
1.0E6
1.0E8
1.0E2
1.0E4
1.0E6
Frequency
1.0E8
1.0E0
0
-50
-100
-150
-200
-250
-300
-350
1.0E0
1.0E2
1.0E4
1.0E6
1.0E8
Phase (deg)
Plot the exact and
approximated functions
together.
1.0E2
1.0E4
1.0E6
Frequency
1.0E8
Exact
Approximated
Out[19]= Graphics Note that matrix approximation has reduced the polynomial order of the transfer function from four
to two. Therefore, we can now solve the transfer function for the low-frequency poles.
Solve for the
low-frequency poles.
In[20]:= Solve[Denominator[v52] == 0, s]
R1 R2
1
Out[20]= s , s C1 R1 R2
C2 RC RL
3.11.4 CompressMatrixEquation
CompressMatrixEquation[dae, var]
compresses the matrix equation dae with respect to the
variable of interest var
Command structure of CompressMatrixEquation .
Computing a transfer function from a linear system of circuit equations A x b requires solving
the matrix equation for one single variable xk . The values of all other variables xj , j , k, are of
no interest in this context. Systems of circuit equations usually contain a significant amount of
information which is only needed to determine the irrelevant variables, and the proportion of such
information increases during the approximation process as equations are algebraically decoupled by
removing negligible terms.
To reduce the computational effort needed for solving an approximated matrix, all unnecessary
information should be discarded from the equations prior to calling Solve (Section 3.5.4), using the
function CompressMatrixEquation . This function performs a recursive search on a matrix equation
to find and delete all rows and columns which are not needed to compute the variable of interest.
Note that compressing a matrix equation is a mathematically exact operation which does not change
the value of the output variable.
3.11 Linear Simplification Techniques
399
CompressMatrixEquation has the following option:
option name
default value
StripIndependentBlocks
False
whether to search for and discard
algebraically independent blocks which are
decoupled from the variable of interest
Option for CompressMatrixEquation .
See also: CompressNonlinearEquations (Section 3.12.2),
ApproximateMatrixEquation (Section 3.11.3).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define an AC model for
BJTs.
In[2]:= Circuit[
Model[
Name −> BJT,
Selector −> AC,
Scope −> Global,
Ports −> {B, C, E},
Parameters −> {Rbe, Cbc, Cbe, beta, Ro,
RBE, CBC, CBE, BETA, RO},
Definition −> Netlist[
{CBE, {B, E}, Symbolic −> Cbe, Value −> CBE},
{RBE, {X, E}, Symbolic −> Rbe, Value −> RBE},
{CBC, {B, C}, Symbolic −> Cbc, Value −> CBC},
{CC, {B, X, C, E},
Symbolic −> beta, Value −> BETA},
{RO, {C, E}, Symbolic −> Ro, Value −> RO}
]
]
] // ExpandSubcircuits;
3. Reference Manual
400
This netlist describes a
common-emitter amplifier.
In[3]:= ceamplifier =
Circuit[
Netlist[
{V1, {1, 0}, 1},
{V0, {6, 0}, 0},
{C1, {1, 2}, Symbolic −> C1, Value
{R1, {2, 6}, Symbolic −> R1, Value
{R2, {2, 0}, Symbolic −> R2, Value
{RC, {6, 3}, Symbolic −> RC, Value
{RE, {4, 0}, Symbolic −> RE, Value
{C2, {3, 5}, Symbolic −> C2, Value
{RL, {5, 0}, Symbolic −> RL, Value
{Q1, {2 −> B, 3 −> C, 4 −> E},
Model −> BJT, Selector −> AC,
CBE −> 30.*^−12, CBC −> 5.*^−12,
RO −> 10000., BETA −> 200.}
]
]
−>
−>
−>
−>
−>
−>
−>
1.0*^−7},
1.0*^5},
47000.},
2200.},
1000.},
1.0*^−6},
47000.},
RBE −> 1000.,
Out[3]= Circuit Set up a system of
symbolic AC equations.
In[4]:= eqs = CircuitEquations[ceamplifier,
ElementValues −> Symbolic]
Out[4]= DAEAC, 10 10 Approximate the system of
circuit equations.
In[5]:= sbg = ApproximateMatrixEquation[eqs, V$5,
{s −> 2. Pi I 10^4, MaxError −> 0.1}]
Out[5]= DAEAC, 10 10 Inspect the list of
unknowns in the matrix
equation.
In[6]:= GetVariables[sbg]
Solve the equations for
V$5.
In[7]:= Solve[sbg, V$5]
Discard rows and columns
which are not needed to
compute V$5.
In[8]:= cmat = CompressMatrixEquation[sbg, V$5]
Inspect the list of
remaining unknowns.
In[9]:= GetVariables[cmat]
Out[6]=
V$1, V$2, V$3, V$4, V$5, V$6, V$X$Q1, I$V1, I$V0, IC$CC$Q1
RC
Out[7]= V$5 RE
Out[8]= DAEAC, 7 7 Out[9]= V$1, V$2, V$3, V$4, V$5, V$X$Q1, IC$CC$Q1
Note that matrix compression does not change the solution of the equations.
Solve the compressed
equations for V$5.
In[10]:= Solve[cmat, V$5]
RC
Out[10]= V$5 RE
3.12 Nonlinear Simplification Techniques
401
3.12 Nonlinear Simplification Techniques
As of version 2, Analog Insydes provides completely new functionality for the simplification of nonlinear circuits: Besides the main functions CompressNonlinearEquations and CancelTerms , there
are a number of supporting procedures. The following table shows all commands which build the
simplification functionality for nonlinear circuits:
NonlinearSetup (Section 3.12.1)
setting up the nonlinear approximation procedure
CompressNonlinearEquations (Section 3.12.2)
eliminating variables in a DAEObject
CancelTerms (Section 3.12.3)
removing terms in different levels in a DAEObject
SimplifySamplePoints (Section 3.12.4)
generating sparsed sample-point grids
NonlinearSettings (Section 3.12.5)
showing nonlinear settings in a DAEObject
ShowLevel (Section 3.12.6)
showing terms of a DAEObject in different levels
ShowCancellations (Section 3.12.7)
showing terms of a DAEObject which have been removed
3.12.1 NonlinearSetup
NonlinearSetup[dae, inputs, outputs, mode −>{settings }, † † † ]
initializes the nonlinear approximation routines
Command structure of NonlinearSetup .
Before applying the nonlinear simplification routines the DAEObject has to be set up using
NonlinearSetup . This function returns a DAEObject containing additional information which is
automatically used later on by the simplification functions. Furthermore, NonlinearSetup performs
the numerical reference simulations which are used for error calculation. Besides the input and output
variables you have to specify which analysis modes to consider during subsequent simplifications.
The DAEObject dae has to be a DC or Transient DAEObject. The argument inputs is a symbol or
a list of symbols denoting the input parameters to be swept. In this context input parameters are
parameters of the DAEObject (not variables). Use GetParameters (Section 3.6.9) to obtain a list of
valid parameters. Further, outputs is a symbol or a list of symbols denoting the output variables
which are used for error control. In this context output variables are variables of the DAEObject (not
parameters). Use GetVariables (Section 3.6.7) to obtain a list of valid variables.
3. Reference Manual
402
The rules modei −> {settingsi } specify the analysis modes to be used during error control and some
mode-specific settings for the simplification routines. Here, modei can be DT (DC-transfer analysis) or
AC. If the AC analysis is chosen, DT settings have to be specified, too. The mode settings settingsi
are given by a sequence of rules. The format is described below.
Additionally, NonlinearSetup provides the following option:
option name
default value
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
Options for NonlinearSetup.
You can use NonlinearSettings (Section 3.12.5) to inspect the settings of the nonlinear simplifications currently stored in a DAEObject. Use GetDAEOptions[dae, NonlinearOptions] to get a list of
all nonlinear simplification settings stored in the DAEObject.
See also: GetVariables (Section 3.6.7), GetParameters (Section 3.6.9), NonlinearSettings (Section 3.12.5), Section 3.7.2.
DT mode specifications
Range−>paramsweep
SimulationFunction−>simfct
ErrorFunction−>errfct
MaxError−>errlist
parameter sweep specification for all input variables
simulation function used for DT analysis; default value:
DefaultDTSimulation
error function used for DT analysis; default value:
DefaultDTError
Maximum error specification for the output variables;
default value: {}
Settings for DT mode.
The range in which to sweep the input parameters during simplification is specified by the paramsweep
value of the Range option. The range specification has to be given in the standard format for Analog
Insydes parameter sweeps which is described in Section 3.7.2. Note that you have to specify a range
for each input parameter.
SimulationFunction , ErrorFunction , and MaxError are optional, whereas Range is a required
argument.
If no SimulationFunction option is specified, the default function is used. The default DT simulation function DefaultDTSimulation calls NDAESolve (Section 3.7.5) to calculate the DC operating
3.12 Nonlinear Simplification Techniques
403
points. DefaultDTSimulation inherits almost all options from NDAESolve . The following options
have a special meaning when used with DefaultDTSimulation :
option name
default value
MaxIterations
Inherited[NDAESolve]
maximum number of iterations for
NDAESolve (only used if option
InitialGuess is set to Automatic )
InitialGuess
None
if set to Automatic , the values of the
reference simulation are used as initial
guess for NDAESolve
Options for DefaultDTSimulation .
Besides the default simulation function you can define your own DT simulation function and specify
it as SimulationFunction in the DT mode settings for NonlinearSetup . During the setup and the
nonlinear simplifications this function will be called with the following arguments:
simfct[dae_, refsim_List:{}, opts___]
Pattern for a user-defined DT simulation function.
Here, dae is a DC or Transient DAEObject, refsim is the list of numerical reference values and opts
are additional options.
The simulation function must return the DC operating points in the same format as NDAESolve
returns DC operating points of a parameter sweep (see Section 3.7.2).
Similarly, you can define your own DT error function instead of the default error function and
specify it as ErrorFunction in the DT mode settings for NonlinearSetup . During the simplification
routines this function is used to calculate the error with the following arguments:
errfct[newval_List, oldval_List, outputsymbols_List]
Pattern for a user-defined DT error function.
Here, newval and oldval are the numerical solutions of the simplified and the original DAE system,
outputsymbols is a list of all output symbols.
The error function must return a list of rules {symb −> err , † † † }, which specifies the numerical error
for each output symbol. The default DT error function calculates the maximum absolute error.
An example for a user-defined simulation and error function and of the corresponding call to
NonlinearSetup is given in the example section below.
3. Reference Manual
404
Instead of specifying the maximum allowed error for output variables for each call to the nonlinear
simplifications routines separately, you can specify global maximum errors for output variables using
the MaxError argument. These values are used if no error value is given in the call to the nonlinear
simplification routine. The value of this argument has to be a list of rules of the form outvar −> error.
AC mode specifications
Range−>{paramsweep, freqlist, sourceval}
paramsweep must be a subset of the sweep given in the DT
settings, freqlist is a list of frequency points and sourceval is
a list of replacement rules for all AC sources
SimulationFunction−>simfct
ErrorFunction−>errfct
MaxError−>errlist
simulation function used for AC analysis; default value:
DefaultACSimulation
error function used for AC analysis; default value:
DefaultACError
Maximum error specification for the output variables;
default value: {}
Settings for AC mode.
SimulationFunction , ErrorFunction , and MaxError are optional, whereas Range is a required
argument.
If you specifiy AC settings you have to specify DT settings, too. The parameter sweep settings
paramsweep must be a subset of the parameter sweep given in the DT settings.
The default AC simulation function is called DefaultACSimulation . This function calls ACAnalysis
(Section 3.7.3) to perform the numerical calculation. DefaultACSimulation inherits almost all options from ACAnalysis .
Besides the default simulation function you can define your own AC simulation function and specify
it as SimulationFunction in the AC mode settings for NonlinearSetup . During the setup and the
nonlinear simplifications this function will be called with the following arguments:
simfct[dae_, refsim_List:{}, opts___]
Pattern for a user-defined AC simulation function.
Here, dae is a Transient DAEObject, refsim is the list of numerical reference values and opts are
additional options.
The simulation function must return a list of the AC solutions in the same format as ACAnalysis
returns the result (with the option setting InterpolateResult −> False).
3.12 Nonlinear Simplification Techniques
405
Similarly, you can define your own AC error function instead of the default error function and
specify it as ErrorFunction in the AC mode settings for NonlinearSetup . During the simplification
routines this function is used to calculate the error with the following arguments:
errfct[newval_List, oldval_List, outputsymbols_List]
Pattern for a user-defined AC error function.
Here, newval and oldval are the numerical solutions of the simplified and the original dae system,
outputsymbols is a list of all output symbols.
The error function must return a list of rules {symb −> err , † † † } which specifies the numerical error
for each output symbol. The default AC error function calculates the maximum relative error.
Instead of specifying the maximum allowed error for output variables for each call to the nonlinear
simplifications routines separately, you can specify global maximum errors for output variables using
the MaxError argument. These values are used if no error value is given in the call to the nonlinear
simplification routine. The value of this argument has to be a list of rules of the form outvar −> error.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Read PSpice netlist.
In[2]:= net = ReadNetlist["AnalogInsydes/DemoFiles/Sqrt.cir",
Simulator −> "PSpice"]
Out[2]= Circuit Set up DAE system.
In[3]:= dae = CircuitEquations[net,
AnalysisMode −> DC, ElementValues −> Symbolic]
Out[3]= DAEDC, 27 27 Get all valid symbols that
can be used as inputs for
NonlinearSetup .
In[4]:= GetParameters[dae]
Get all valid symbols that
can be used as outputs for
NonlinearSetup .
In[5]:= GetVariables[dae]
Out[4]=
AREA$Q1, AREA$Q2, AREA$Q3, AREA$Q4, BF$Q1, BF$Q2, BF$Q3, BF$Q4,
BR$Q1, BR$Q2, BR$Q3, BR$Q4, GMIN, IB, II, IS$Q1, IS$Q2, IS$Q3,
IS$Q4, TEMP, TNOM, VDC, VLOAD, $k, $q
Out[5]=
V$1, V$3, V$4, V$5, V$OUT, I$BS$Q1,
I$CS$Q1, I$ES$Q1, ib$Q1, ic$Q1, I$BS$Q2, I$CS$Q2, I$ES$Q2,
ib$Q2, ic$Q2, I$BS$Q3, I$CS$Q3, I$ES$Q3, ib$Q3, ic$Q3, I$BS$Q4,
I$CS$Q4, I$ES$Q4, ib$Q4, ic$Q4, I$VLOAD, I$VDC
In the following we choose VLOAD and II as inputs and V$5 and I$VLOAD as outputs, treating dae as
a multi-input/multi-output system. We want to control the DC behavior sweeping VLOAD between
V and V and II between A and A on a uniform two-dimensional grid with points.
The following call to NonlinearSetup prepares a DAEObject with these settings.
3. Reference Manual
406
Specify settings for
nonlinear simplifications.
In[6]:= step1=NonlinearSetup[dae, {VLOAD, II}, {V$5, I$VLOAD},
DT −> {Range −>
{{VLOAD, 0, 3, 1}, {II, 0, 0.003, 0.001}} }]
Out[6]= DAEDC, 27 27 Display settings stored in
step1.
In[7]:= NonlinearSettings[step1]
Input symbols
: {II, VLOAD}
Output symbols
: {I$VLOAD, V$5}
Ranking−Cache
: −not set−
DT range
: {{VLOAD −> 0., II −> 0.}, <<15>>}
DTSimulation
: DefaultDTSimulation
DTError
: DefaultDTError
DTReference
: {<<16>>}
OperatingPoints : {<<16>>}
The following code illustrates how to define your own simulation and error function and how to
call NonlinearSetup such that these functions are used during nonlinear simplifications.
Define a DT simulation
function.
In[8]:= MyDTSimulationFct[dae_, refsim_List:{}, opts___] :=
Block[
{nonlinopts, indepvar, inittime},
{indepvar, inittime, nonlinopts} = GetDAEOptions[
dae,
{IndependentVariable, InitialTime, NonlinearOptions}
];
NDAESolve[
dae,
{indepvar, inittime},
Range /. (DT /. nonlinopts),
opts
]
]
Define a DT error
function.
In[9]:= MyDTErrorFct[newval_, oldval_, outputsymbols_] :=
Map[
Rule[#, Max[Abs[(# /. newval) − (# /. oldval)]]] &,
outputsymbols
]
Set up nonlinear
simplifications using new
simulation and error
function.
In[10]:= step1 = NonlinearSetup[dae, {VLOAD, II}, {V$5, I$VLOAD},
DT −> {
Range −> {{VLOAD, 0, 3, 1}, {II, 0, 0.003, 0.001}},
SimulationFunction −> MyDTSimulationFct,
ErrorFunction −> MyDTErrorFct
}
]
Out[10]= DAEDC, 27 27 3.12 Nonlinear Simplification Techniques
Display settings stored in
step1.
407
In[11]:= NonlinearSettings[step1]
Input symbols
: {II, VLOAD}
Output symbols
: {I$VLOAD, V$5}
Ranking−Cache
: −not set−
DT range
: {{VLOAD −> 0., II −> 0.}, <<15>>}
DTSimulation
: MyDTSimulationFct
DTError
: MyDTErrorFct
DTReference
: {<<16>>}
OperatingPoints : {<<16>>}
3.12.2 CompressNonlinearEquations
CompressNonlinearEquations[dae, outvars]
removes equations and eliminates variables from a
nonlinear DAEObject dae which are irrelevant for solving
for the variables outvars
CompressNonlinearEquations[dae]
removes as many equations and variables as possible
Command structure of CompressNonlinearEquations .
This function contains a compression algorithm which allows for identification and elimination of
irrelevant equations and variables and the removal of decoupled clusters of equations and variables
in a system of symbolic nonlinear differential and algebraic circuit equations. The return value is
again a DAEObject.
Note that the argument outvars also supports string patterns. Additionally,
CompressNonlinearEquations automatically retrieves the settings made by NonlinearSetup , so
there is no need to specify the variables of interest as a second argument.
Note that CompressNonlinearEquations also supports the data format from Release 1, where
equations are given as {eqns, vars}, instead of a DAEObject:
CompressNonlinearEquations[{eqns, vars}, outvars]
removes equations and eliminates variables from a set of
nonlinear equations eqns formulated in the variables vars
which are irrelevant for solving for the variables outvars
CompressNonlinearEquations[{eqns, vars}]
removes as many equations and variables as possible
Additional patterns of CompressNonlinearEquations .
CompressNonlinearEquations has the following options:
3. Reference Manual
408
option name
default value
EliminateVariables
Automatic
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
eliminates irrelevant variables
StripIndependentBlocks
removes irrelevant independent subsystems
of equations
True
Options for CompressNonlinearEquations .
See also: CompressMatrixEquation (Section 3.11.4).
Options Description
EliminateVariables
EliminateVariables allows for eliminating variables from a set of equations. Settings are
Automatic for eliminating variables occuring linear everywhere, All for eliminating variables occuring linear somewhere, and None for no variable elimination, respectively. The default setting is
EliminateVariables −> Automatic .
StripIndependentBlocks
With StripIndependentBlocks −> True irrelevant independent subsystems of equations will be
searched for and removed. The default setting is StripIndependentBlocks −> True.
Examples
D1
1
V0
Load Analog Insydes.
2
C1
In[1]:= <<AnalogInsydes‘
R1
Vout
3.12 Nonlinear Simplification Techniques
Define netlist description
of a simple diode rectifier
circuit.
409
In[2]:= rectifier =
Circuit[
Netlist[
{V0, {1, 0}, Symbolic −> V0,
Value −> 2. Sin[10^6 Time]},
{R1, {2, 0}, Symbolic −> R1, Value −> 100.},
{C1, {2, 0}, Symbolic −> C1, Value −> 1.*^−7},
{D1, {1 −> A, 2 −> C},
Model −> "Diode", Selector −> "Spice"}
]
]
Out[2]= Circuit Set up the system of
symbolic time-domain
equations.
In[3]:= rectifiereqs = CircuitEquations[rectifier,
AnalysisMode −> Transient,
ElementValues −> Symbolic] // UpdateDesignPoint
Out[3]= DAETransient, 4 4 Show equation system.
In[4]:= DisplayForm[rectifiereqs]
Out[4]//DisplayForm=
V$2t
I$AC$D1t I$V0t 0, I$AC$D1t C1 V$2 t 0,
R1
TEMP
1.11 1. TNOM
$q
TEMP $k
V$1t V0, I$AC$D1t AREA$D1 E 3.
1. $q V$1tV$2t
TEMP
TEMP $k
1 E IS$D1 GMIN V$1t V$2t,
TNOM
V$1t, V$2t, I$V0t, I$AC$D1t,
DesignPoint V0 2. Sin1000000 t, R1 100., C1 1. 107 ,
AREA$D1 1., GMIN 1. 1012 , IS$D1 1. 1014 , TEMP 300.15,
TNOM 300.15, $k 1.38062 1023 , $q 1.60219 1019 Eliminate variables which
occur linear everywhere.
In[5]:= rectifiercomp = CompressNonlinearEquations[rectifiereqs,
V$2[t], EliminateVariables −> Automatic]
Out[5]= DAETransient, 2 2 Show compressed equation
system.
In[6]:= DisplayForm[rectifiercomp]
Out[6]//DisplayForm=
TEMP
1.11 1. 1. $q V$1tV$2t
TEMP 3.
TNOM
$q
TEMP $k
TEMP $k
1. AREA$D1 E 1. E IS$D1 TNOM
V$2t
1. GMIN V$1t 1. V$2t C1 V$2 t 0,
R1
V$1t V0, V$1t, V$2t,
DesignPoint V0 2. Sin1000000 t, R1 100., C1 1. 107 ,
AREA$D1 1., GMIN 1. 1012 , IS$D1 1. 1014 , TEMP 300.15,
TNOM 300.15, $k 1.38062 1023 , $q 1.60219 1019 Eliminate variables which
occur linear somewhere.
In[7]:= rectifiercomp = CompressNonlinearEquations[rectifiereqs,
V$2[t], EliminateVariables −> All]
Out[7]= DAETransient, 1 1 3. Reference Manual
410
Show compressed equation
system.
In[8]:= DisplayForm[rectifiercomp]
Out[8]//DisplayForm=
TEMP
1.11 1. 1. $q V0V$2t
TEMP 3.
TNOM
$q
TEMP $k
TEMP $k
1. AREA$D1 E 1. E IS$D1 TNOM
V$2t
1. GMIN V0 1. V$2t C1 V$2 t 0, V$2t,
R1
DesignPoint V0 2. Sin1000000 t, R1 100., C1 1. 107 ,
AREA$D1 1., GMIN 1. 1012 , IS$D1 1. 1014 , TEMP 300.15,
TNOM 300.15, $k 1.38062 1023 , $q 1.60219 1019 Perform numerical
transient analyses with
NDAESolve .
In[9]:= orig = NDAESolve[rectifiereqs, {t, 0., 2.*^−5}];
comp = NDAESolve[rectifiercomp, {t, 0., 2.*^−5}];
Compare the simulation
results with
TransientPlot .
In[11]:= V$2Orig := V$2 /. First[orig];
TransientPlot[comp, {V$2[t], V$2Orig[t]},
{t, 0., 2.*^−5}]
1.2
1
V$2[t]
0.8
0.6
0.4
V$2Orig[t
0.2
t
-6 0.00001 0.000015 0.00002
5. 10
Out[12]= Graphics 3.12.3 CancelTerms
CancelTerms[dae, maxerrors]
CancelTerms[dae]
applies cancellation of terms in dae for a given error
specification maxerrors
applies cancellation of terms using error specification given
by NonlinearSetup
Command structure of CancelTerms.
CancelTerms simplifies the given DAEObject by successive removal of terms in the equation system.
It assures that the behavior of the simplified system differs not more than maxerrors from the behavior
of the original system. Before using CancelTerms , the DAEObject has to be prepared for nonlinear
simplification using NonlinearSetup (Section 3.12.1). CancelTerms performs the numerical analyses set by the call to NonlinearSetup to calculate the error a simplification causes. The error
3.12 Nonlinear Simplification Techniques
411
list maxerrors must contain an error bound value for each analysis mode and each output variable,
i.e. it has to be a list of the form
{mode1 −> {var1 −> bound1 , }, }
Variables for which an error value has been specified in the call to NonlinearSetup can be omitted.
If error specifications have been given for each analysis mode and each output variable in the call
to NonlinearSetup , the maxerrors specification can be omitted completely. If the error caused by a
simplification exceeds any of the maximum errors given in the error list, the simplification is undone.
Note that the error is measured by the error function specified in the call to NonlinearSetup .
Removal of terms is done in the level given by the option Level (see below).
CancelTerms returns the simplified DAEObject. All settings for nonlinear simplifications stored in
the original system are added to the returned DAEObject, too. This means that you do not have to
call NonlinearSetup again on the new system in order to apply subsequent simplifications.
Note that CancelTerms reduces the number of terms only. Use CompressNonlinearEquations
(Section 3.12.2) to reduce the number of equations.
CancelTerms provides the following options:
option name
default value
Clusterbound
−1
logarithmic bound for cluster calculation
Errorbound
3
number of successive error-bound violations
before the algorithm terminates
Level
0
level in which to remove terms
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
Ranking
Automatic
method to use for ranking
RecordCancellations
True
whether to record term cancellations
Options for CancelTerms.
After removal of one or more terms a numerical simulation is performed. Comparing the result to
a reference calculation yields the error on the output variables.
See also: NonlinearSetup (Section 3.12.1), ShowCancellations (Section 3.12.7), ShowLevel (Section 3.12.6), Statistics (Section 3.6.17).
Options Description
A detailed description of all CancelTerms options is given below in alphabetical order:
3. Reference Manual
412
Clusterbound
Usually, CancelTerms deletes several terms at once before performing a numerical error calculation.
The computation of suitable term clusters for this is done automatically by CancelTerms . If the
precasted influence of a term exceeds a given threshold, calculation of clusters is stopped and cancellation is done term by term. This threshold value is specified by the option Clusterbound . The
following values are allowed:
integer
stop calculation of clusters if the logarithm (to base ) of
the precasted term influence is bigger than integer
−Infinity
do not cancel terms in clusters
Infinity
always cancel terms in clusters
Values for the Clusterbound option.
The default setting is Clusterbound −> −1. Note that deletion of terms using clusters speeds up the
computation significantly.
Errorbound
If the computed error exceeds the given error bound maxerrors, then the term or terms are re-inserted.
The option Errorbound specifies the number of successive re-insertions before the algorithm terminates. Use Errorbound −> Infinity to check all terms for cancellation. The default setting is
Errorbound −> 3.
Level
By default, CancelTerms removes terms from top-level sums (level 0). It is also possible to simplify
sums in deeper levels, i.e. sums occuring inside of expressions. The option Level determines which
level is affected by the simplification. Possible values are:
n
{n , n , † † † }
All
apply cancellation in level n
apply cancellation in levels n , n , apply cancellation in all levels, starting on top level
Level specifications for CancelTerms.
If given explicitly, level specifications have to be non-negative integers. The default setting is
Level −> 0.
3.12 Nonlinear Simplification Techniques
413
The level specification of CancelTerms does not correspond to the level of the internal
Mathematica representation of dae. Use ShowLevel (Section 3.12.6) or Statistics
(Section 3.6.17) to see which levels can be influenced by CancelTerms .
Ranking
Cancellation of terms is performed in a pre-calculated order which sorts the terms by increasing
influence. A method which estimates the influence of a cancellation on the output behavior is
called ranking. CancelTerms lets you choose which ranking method to use for each analysis mode
separately by means of the Ranking option. Possible values are:
{mode−>Automatic|Simulation, † † † }
set mode dependent ranking functions
Automatic
equivalent to {_−>Automatic}
Simulation
equivalent to {_−>Simulation}
Values for the Ranking option.
The ranking method can either be Automatic , which means to use the internal implemented ranking
function, or Simulation , which means to use the corresponding simulation function to calculate
the influence. The former is much more efficient while the latter is more accurate. If the option
value is the symbol Automatic or Simulation the corresponding ranking function is set for all
analysis modes simultaneously. If you want to set the ranking function for each mode separately,
you have to give a list of rules. The current analysis mode is then matched against the left hand
side of the rules and the corresponding right hand sides are chosen as ranking functions. Thus, mode
can either be one of the symbols DT or AC, or a pattern specification like _. The default setting is
Ranking −> Automatic .
RecordCancellations
The value of the RecordCancellations option determines whether term cancellations are recorded
during computation. If set to True, a cancellation history is written to the options section of the
DAEObject given as argument to CancelTerms . Note that it is not written to the returned DAEObject. The command ShowCancellations (Section 3.12.7) can then be used to visualize the history list.
If set to False, no history list is recorded. The default setting is RecordCancellations −> True.
If you cancel terms in different levels, the history list of the first level will be recorded only.
3. Reference Manual
414
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Read PSpice netlist.
In[2]:= netlist = ReadNetlist[
"AnalogInsydes/DemoFiles/Sqrt.cir",
Simulator −> "PSpice"]
Out[2]= Circuit Set up symbolic circuit
equations.
In[3]:= mnaFull = CircuitEquations[netlist,
AnalysisMode −> DC, ElementValues −> Symbolic,
Value −> {"*" −> {TEMP, TNOM, $q, $k}}]
Out[3]= DAEDC, 27 27 Prepare DAE system for
nonlinear simplifications.
In[4]:= step1 = NonlinearSetup[mnaFull, {II, VLOAD}, {I$VLOAD},
DT −> {Range −>
{{VLOAD, 0., 3.5, 0.5}, {II, 0., 0.001, 0.0002}} }]
Out[4]= DAEDC, 27 27 Display complexity
information of the original
system.
In[5]:= Statistics[step1]
Number of equations
: 27
Number of variables
: 27
Length of equations
: {4, 4, 3, 3, 2, 3, 2, 3, 8, 8,
3, 2, 3, 9, 9, 3, 2, 3, 9, 9, 3, 2, 3, 4, 4, 3, 2}
Terms with derivatives : 0
Sums in levels
: {113, 20}
Cancel top-level terms in
the equations.
In[6]:= step2 = CancelTerms[step1, {DT −> {I$VLOAD −> 25.*^−6}}]
Display complexity
information of the
simplified system. Note
that the number of terms
decreased, while the
number of equations
stayed the same.
In[7]:= Statistics[step2]
Out[6]= DAEDC, 27 27 Number of equations
: 27
Number of variables
: 27
Length of equations
: {1, 2, 2, 2, 2, 2, 1, 2, 2, 1,
1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1}
Terms with derivatives : 0
Sums in levels
: {32, 4}
3.12.4 SimplifySamplePoints
SimplifySamplePoints[func, range, tolerance]
reduces the number of points in range for a given tolerance
specification tolerance
Command structure of SimplifySamplePoints .
3.12 Nonlinear Simplification Techniques
415
SimplifySamplePoints returns a sparsed list of sample points such that the function values, evaluated on adjacent points, do not exceed the given tolerance. The argument range must be a valid
parameter sweep specification as described in Section 3.7.2. If range is a n-dimensional sweep range
then func must be a function of n arguments which returns a single real number.
SimplifySamplePoints provides the following options:
option name
default value
ReturnSymbols
True
whether to return a list of rules or a list of
numerical values
SearchDirections
All
specifies the direction in which to search for
adjacent points
Options for SimplifySamplePoints .
See also: Section 3.7.2.
Options Description
ReturnSymbols
If ReturnSymbols is set to True, the sparsed list is returned as a flat list of rules. If it is set
to False, the sparsed list is returned as a flat list of numerical values. In the former case the return
value is a valid parameter sweep specification and can be used as argument to NonlinearSetup
(Section 3.12.1). The default setting is ReturnSymbols −> True.
SearchDirections
The option SearchDirections specifies the direction in which to search for adjacent points. The
value can be All or a list of symbols which appear in the given parameter sweep. If the value is
a list of symbols, adjacent points are only searched for the given variables. The default setting is
SearchDirections −> All which means to search in all directions.
3.12.5 NonlinearSettings
NonlinearSettings[dae]
Command structure of NonlinearSettings .
prints all settings for nonlinear simplifications stored in dae
3. Reference Manual
416
As mentioned in the description of NonlinearSetup (Section 3.12.1) additional data has to be
stored in a DAE system in order to apply the nonlinear simplification routines. The command
NonlinearSettings prints the according data currently stored in a DAEObject.
NonlinearSettings provides the following information:
list of input symbols
list of output symbols
value of the ranking cache
mode-specific settings (range, simulation function, error function, and reference values)
See also: NonlinearSetup (Section 3.12.1).
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Read PSpice netlist.
In[2]:= net = ReadNetlist["AnalogInsydes/DemoFiles/Sqrt.cir",
Simulator −> "PSpice"]
Out[2]= Circuit Set up DAE system.
In[3]:= dae = CircuitEquations[net,
AnalysisMode −> DC, ElementValues −> Symbolic]
Out[3]= DAEDC, 27 27 Specify settings for
nonlinear simplifications.
In[4]:= setup = NonlinearSetup[dae, {VLOAD, II}, {V$5, I$VLOAD},
DT −> {Range −>
{{VLOAD, 0, 3, 1}, {II, 0, 0.003, 0.001}} }]
Out[4]= DAEDC, 27 27 Display settings stored in
the DAEObject.
In[5]:= NonlinearSettings[setup]
Input symbols
: {II, VLOAD}
Output symbols
: {I$VLOAD, V$5}
Ranking−Cache
: −not set−
DT range
: {{VLOAD −> 0., II −> 0.}, <<15>>}
DTSimulation
: DefaultDTSimulation
DTError
: DefaultDTError
DTReference
: {<<16>>}
OperatingPoints : {<<16>>}
3.12 Nonlinear Simplification Techniques
417
3.12.6 ShowLevel
ShowLevel[dae]
ShowLevel[dae, level]
highlights parts of dae influenced by nonlinear
simplification routines for all levels
highlights parts in levels of dae given by level
Command structure of ShowLevel.
The command ShowLevel displays the equation system dae and highlights those parts of the equations
which are influenced by nonlinear simplification functions such as CancelTerms (Section 3.12.3).
The second argument level can either be a number or a list of numbers. If it is a number, then exactly
this level is highlighted. If it is a list of numbers {l , † † †, ln } then the levels l , , ln are highlighted.
The level argument corresponds to the level argument of CancelTerms and to the output of
Statistics (Section 3.6.17).
The level specification of ShowLevel does not correspond to the level of the internal
Mathematica representation of dae.
See also: ShowCancellations (Section 3.12.7), Statistics (Section 3.6.17), CancelTerms (Section 3.12.3).
3.12.7 ShowCancellations
ShowCancellations[dae]
highlights those parts of dae which have been removed by
nonlinear simplification routines
Command structure of ShowCancellations .
The command ShowCancellations displays the equation system dae and highlights those parts of
the equations which have been removed by nonlinear simplification functions such as CancelTerms
(Section 3.12.3). You have to set the value of the CancelTerms option RecordCancellations to True
to activate the history recording feature.
Note that dae is not the return value of a nonlinear simplification routine but its argument.
See also: ShowLevel (Section 3.12.6), CancelTerms (Section 3.12.3).
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418
3.13 Miscellaneous Functions
3.13.1 ConvertDataTo2D
ConvertDataTo2D[data]
converts numerical simulation data into two-dimensional
interpolating functions
Command structure of ConvertDataTo2D .
All Analog Insydes functions which return numerical data, e. g. ACAnalysis (Section 3.7.3),
NoiseAnalysis (Section 3.7.4), NDAESolve (Section 3.7.5), or ReadSimulationData (Section 3.10.3)
use the data format described in Section 3.7.1. This format is suitable for representing one, two, and
multi-dimensional numerical data. As standard Mathematica graphic functions do not support the
multi-dimensional Analog Insydes data format, ConvertDataTo2D converts the data such that it can
be plotted with standard Mathematica graphic functions such as Plot3D.
The input data must be two-dimensional data as described in Section 3.7.1, for example:
Data format for
two-dimensional numerical
simulation data.
{
{
V$1 −> InterpolatingFunction[{{t1, t2}}, <>],
,
SweepParameters −> {VIN −> v1}
},
,
{
V$1 −> InterpolatingFunction[{{t1, t2}}, <>],
,
SweepParameters −> {VIN −> v2}
}
}
The data will then be converted to a set of two-dimensional interpolating functions:
Converted data format.
{
V$1 −> InterpolatingFunction[{{v1, v2}, {t1, t2}}, <>],
,
}
Please note the order of the parameters for the two-dimensional interpolating function. The interpolation order is set to {1, 1}.
See also: Section 3.7.1, Section 3.7.2.
3.13 Miscellaneous Functions
419
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Read the simulation data
with ReadSimulationData .
In[2]:= data = ReadSimulationData[
"AnalogInsydes/DemoFiles/Multivibrator.txt",
Simulator −> "PSpice"]
Out[2]=
V4 InterpolatingFunction0.00018,
V5 InterpolatingFunction0.00018,
SweepParameters VCC 0.,
V4 InterpolatingFunction0.00018,
V5 InterpolatingFunction0.00018,
SweepParameters VCC 6.,
V4 InterpolatingFunction0.00018,
V5 InterpolatingFunction0.00018,
SweepParameters VCC 12.
0.00012, ,
0.00012, ,
0.00012, ,
0.00012, ,
0.00012, ,
0.00012, ,
Convert it into
two-dimensional
interpolating functions.
In[3]:= data2d = ConvertDataTo2D[data]
Display simulation result
with Plot3D.
In[4]:= Plot3D[
Evaluate[("V(4)"[VCC, IC]−"V(5)"[VCC, IC]) /. data2d],
{IC, −0.00018, 0.00011}, {VCC, 0., 12.},
BoxRatios −> {1., 1., 1.},
AxesLabel −> {"IC", "VCC", "V(4)−V(5)"}]
Out[3]=
V4 InterpolatingFunction
0., 12., 0.00018, 0.00012, , V5 InterpolatingFunction0., 12., 0.00018, 0.00012, VCC 10
7.5
5
2.5
0
5
V(4)-V(5) 0
-5
-10
-0.0001
0.0001
IC 0
0.0001
Out[4]= SurfaceGraphics 3.13.2 DXFGraphics
DXFGraphics["file"]
Command structure of DXFGraphics.
displays a DXF file as a native Mathematica graphic object
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420
The main purpose of importing DXF (Drawing Interchange File) is to include a schematics into a
Mathematica notebook for a better understanding and documentation of circuit analysis tasks. If your
schematic editor does not support the generation of DXF files it might be possible to use tools like
pstoedit and hpgl2ps to convert the graphic data to DXF.
The DXF file should only contain two-dimensional graphic objects. The following DXF entities are
supported by DXFGraphics :
ARC
CIRCLE
LINE
POINT
POLYLINE
SHAPE
SOLID
TEXT
DXF entities supported by DXFGraphics.
DXFGraphics inherits all options from Graphics. Additional options for DXFGraphics are:
option name
default value
FillColors
None
specifies the colors to be filled
LayerColor
False
whether the layer color should be used
LineDashingScale
5000.
the line dashing scale
Protocol
Inherited[AnalogInsydes]
standard Analog Insydes protocol
specification (see Section 3.14.5)
ShowColors
False
whether to display the legend of used colors
TextSizing
True
whether to size the text
TextStyle
{FontSize −> 12}
the used TextStyle
ThicknessFunction
AbsoluteThickness
function translating the DXF line thickness
into a graphics primitive
ColorList
{† † †}
list which translates DXF colors into
Mathematica colors
Options for DXFGraphics.
3.13 Miscellaneous Functions
421
Options Description
A detailed description of all DXFGraphics options is given below in alphabetical order:
FillColors
Most DXF generators are not able to convert filled graphic objects into DXF correctly. The option
FillColors can be used to fill POLYLINE and CIRCLE entities by specifying a list of pairs of DXF
colors and Mathematica graphic primitives. For example, FillColor −> {1, RGBColor[0, 0, 0]}
will fill all converted POLYLINE and CIRCLE entities of color 1 with the Mathematica color directive
RGBColor[0, 0, 0]. More fill colors can be set as lists of pairs of a DXF color and a Mathematica
graphic primitive or a DXF color. If only a DXF color number is given the entity will be filled with
its original color.
LayerColor
Each DXF entity belongs to a layer having its own color property. DXFGraphics takes the color of
the entity if LayerColor −> False and the color of the layer otherwise.
LineDashingScale
The value given by the option LineDashingScale is directly multiplied with the DXF dashing value.
The result is the argument of AbsoluteDashing used by DXFGraphics .
ShowColors
ShowColors −> True displays a legend of used colors which allows for identifying the wanted DXF
colors easily.
TextSizing
If TextSizing is set to True, the text will be scaled according to the DXF data. The average size
of all text primitives in the resulting Mathematica graphics object is calculated. Afterwards the text
sizes are fitted such that the average equals the value of FontSize given by the option TextStyle
(see below).
TextStyle
The option TextStyle specifies the default style and font options used for displaying text. The
format is {opt −> val , † † † }. If TextStyle contains the rule TextStyle −> val, the value val denotes
the average size of the font sizes if TextSizing is True.
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ThicknessFunction
This option can be used for setting the line thickness. To generate a graphic using a ten times larger
line thickness and a minimum line thickness of five specify
ThicknessFunction −> (AbsoluteThickness[10.*#+5.]&) .
ColorList
The option ColorList can be used to modify the color translation table. For example, to set the
DXF color 7 to the Mathematica color white use the option
ColorList −> ReplacePart[Options[DXFColor, ColorList][[1,2]], GrayLevel[1], 7].
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Import a DXF file into
Mathematica.
In[2]:= DXFGraphics["AnalogInsydes/DemoFiles/Buffer.dxf"]
Q2N2907a5
Q2N2907a
T4
T3
Q2N2907a
T5
4
3
VDB
1Q2N2222
T1
+
T2
Q2N2222
2 VOUTVDD
VIN
+
I1
I2
+
+
0
Out[2]= Graphics 3.13 Miscellaneous Functions
Use the option TextStyle
to fit the font size.
423
In[3]:= DXFGraphics["AnalogInsydes/DemoFiles/Buffer.dxf",
TextStyle −> {FontSize −> 4}]
5
Q2N2907a
Q2N2907a
T4
T3
Q2N2907a
T5
4
3
1
Q2N2222
T1T2
2Q2N2222
VIN
+
-
VOUTVDB
I1+
I2+
-
-
VDD
+
-
0
Out[3]= Graphics With the option
FillColors certain picture
elements can be filled.
In[4]:= DXFGraphics["AnalogInsydes/DemoFiles/colorcirc.dxf",
FillColors −> {{0}, {1}, {2}, {3}, {4}, {5}, {6},
{7, −1}, {8}, {9}, {20}, {30}, {33}, {40}, {60},
{70}, {80}, {100}, {110}, {120}, {140}, {150},
{160}, {180}, {190}, {200}, {220}, {230}, {240},
{250}, {251}, {252}, {253}, {254}},
TextStyle −> {FontSize −> 4}]
Red GreenBlue Yellow
Pink Cyan ’White’
Out[4]= Graphics 3. Reference Manual
424
3.13.3 MathLink Module MSBG
LoadMSBG[] or LoadMSBG[False]
loads and installs the MathLink module if it is not installed
yet
LoadMSBG[True]
UnloadMSBG[]
loads and installs the MathLink module; forces unloading
and reinstallation if the module was already installed
uninstalls the MathLink module
Manually loading and unloading of the MathLink module MSBG.exe.
Analog Insydes loads the MathLink SBG module automatically when it is needed, but you can
also load and unload the module manually by applying one of the above functions. Note that
since the SBG module is a MathLink application, the global Analog Insydes option UseExternals
(Section 3.14.7) must be set to True to enable these functions.
3.13.4 MathLink Module QZ
LoadQZ[] or LoadQZ[False]
LoadQZ[True]
UnloadQZ[]
loads and installs the MathLink module if it is not installed
yet
loads and installs the MathLink module; forces unloading
and reinstallation if the module was already installed
uninstalls the MathLink module
Manually loading and unloading of the MathLink module QZ.exe.
Analog Insydes loads the MathLink QZ module automatically when it is needed, but you can also load
and unload the module manually by applying one of the above functions. Note that since the QZ
module is a MathLink application, the global Analog Insydes option UseExternals (Section 3.14.7)
must be set to True to enable these functions.
3.14 Global Options
425
3.14 Global Options
Analog Insydes provides a number of global options for controlling some fundamental settings
shared by all program modules. An overview of the global options and their default settings (Section 3.14.1) as well as a detailed description of each option (Section 3.14.2 – Section 3.14.7) are given
below. Section 3.14.8 describes the option inheritance scheme implemented in Analog Insydes.
3.14.1 Options[AnalogInsydes]
The current settings of the global Analog Insydes options can be inspected with
Options[AnalogInsydes] . You can change the value of a global option in a Mathematica session
with SetOptions[AnalogInsydes, optname −> value].
There are the following global Analog Insydes options available:
option name
default value
InstanceNameSeparator
"$"
LibraryPath
Prepend[$Path, "aidir/AnalogInsydes/ModelLibrary"]
the search path for device model libraries
ModelLibrary
"FullModels‘"
the default device model library searched
by ExpandSubcircuits
Protocol
StatusLine
the amount and location of protocol
information generated by Analog Insydes
commands
Simulator
"AnalogInsydes"
the source/target simulator for data
import/export and circuit simulation
functions
UseExternals
True
whether to use external MathLink modules
Global options for Analog Insydes.
See also: Options, SetOptions .
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
separator string for name components of
model variables and reference designators
3. Reference Manual
426
View global Analog
Insydes option settings.
In[2]:= Options[AnalogInsydes]
Out[2]=
InstanceNameSeparator $,
LibraryPath Prepend$Path, aidirAnalogInsydesModelLibrary,
ModelLibrary FullModels‘, Protocol StatusLine,
Simulator AnalogInsydes, UseExternals True
3.14.2 InstanceNameSeparator
The option InstanceNameSeparator −> "string" specifies the character or string Analog Insydes
uses to separate name components of variables, node identifiers, and reference designators generated
for device model or subcircuit instances.
For example, the default setting InstanceNameSeparator −> "$" causes the transconductance gm of
a transistor Q1 to be named gm$Q1 after model instantiation.
The value of InstanceNameSeparator must be a valid symbol name.
The given "string" may not contain characters which have a special meaning in Mathematica,
such as _ or ‘.
See also: CircuitEquations (Section 3.5.1).
3.14.3 LibraryPath
With the option LibraryPath , you can specify the paths to be searched for device model library
files. Possible values are:
"path"
{"path ", "path ", † † † }
search for model libraries in "path"
search paths "path ", "path ", † † † in the given order for
model libraries
Values for the LibraryPath option.
The default setting is LibraryPath :> Prepend[$Path, "aidir/AnalogInsydes/ModelLibrary"] ,
which adds the installation path of your Analog Insydes model library to the Mathematica variable $Path.
See also: Info[AnalogInsydes] (Section 3.15.6), Chapter 3.3.
3.14 Global Options
427
3.14.4 ModelLibrary
With the option ModelLibrary , you can specify the names of the files or contexts searched by
ExpandSubcircuits for device model definitions during netlist expansion. Possible values are:
"lib"
{"lib ", "lib ", † † † }
select file or context "lib" as default device model library
select files or contexts "lib ", "lib ", † † † as default device
model libraries (files are searched in the given order)
Values for the ModelLibrary option.
Analog Insydes provides the following semiconductor device model libraries:
"FullModels‘"
"SimplifiedModels‘"
"BasicModels‘"
full-precision SPICE device models
medium-precision device models (some parasitic effects
excluded)
low-precision device models (without parasitic effects)
Analog Insydes device model libraries.
The default setting is ModelLibrary −> "FullModels‘" , which applies the full-precision SPICE
device models.
See also: LibraryPath (Section 3.14.3), CircuitEquations (Section 3.5.1), Chapter 3.3, Chapter 4.3.
Examples
Load Analog Insydes.
In[3]:= <<AnalogInsydes‘
Use basic models for all
semiconductor devices.
In[4]:= SetOptions[AnalogInsydes,
ModelLibrary −> "BasicModels‘"]
Out[4]=
InstanceNameSeparator $,
LibraryPath Prepend$Path, aidirAnalogInsydesModelLibrary,
ModelLibrary BasicModels‘, Protocol StatusLine,
Simulator AnalogInsydes, UseExternals True
The fact that library files are searched in the given order and the way in which the Analog Insydes
model library is organized allow you to change the default model complexity setting for a particular
device type by using the ModelLibrary option as follows.
3. Reference Manual
428
Use full-precision models
for diodes and basic
models for all other
devices.
In[5]:= SetOptions[AnalogInsydes,
ModelLibrary −> {"FullDiodeModels‘", "BasicModels‘"}]
Out[5]=
InstanceNameSeparator $, LibraryPath Prepend
$Path, aidirAnalogInsydesModelLibrary, ModelLibrary FullDiodeModels‘, BasicModels‘, Protocol StatusLine,
Simulator AnalogInsydes, UseExternals True
3.14.5 Protocol
With the option setting Protocol −> levelspecs, you can control the amount of protocol information
generated by Analog Insydes commands and specify the location on the screen where protocol
messages are displayed. You can distinguish between top-level functions and lower-level functions
(i.e. functions called by top-level functions). Possible values are:
leveltop
specify destination for protocol messages generated by
top-level function; do not show protocol messages from
lower levels
{leveltop , level , † † † }
specify destinations for protocol messages generated by
top-level function and lower-level functions called by
top-level function
The format of levelspecs.
Protocol information can be printed in a notebook or displayed in a notebook’s status line. The
following values may be specified as destinations level for Protocol , where the default setting is
Protocol −> StatusLine .
Notebook
StatusLine
All
None
display protocol information in notebook
display protocol information in status line
display protocol information in both notebook and status
line
do not print protocol information
Destinations for protocol messages.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
3.14 Global Options
429
Turn off protocol display.
In[2]:= SetOptions[AnalogInsydes, Protocol −> None]
Out[2]=
InstanceNameSeparator $,
LibraryPath Prepend$Path, aidirAnalogInsydesModelLibrary,
ModelLibrary FullModels‘, Protocol None,
Simulator AnalogInsydes, UseExternals True
Display top-level
information in status line,
send messages from
second level to notebook.
In[3]:= SetOptions[AnalogInsydes,
Protocol −> {StatusLine, Notebook}]
Out[3]=
InstanceNameSeparator $,
LibraryPath Prepend$Path, aidirAnalogInsydesModelLibrary,
ModelLibrary FullModels‘, Protocol StatusLine, Notebook,
Simulator AnalogInsydes, UseExternals True
3.14.6 Simulator
The value of the option Simulator −> "string" determines the source or target simulator for Analog
Insydes’ simulator interfaces and circuit analysis functions. The following simulators are supported:
"AnalogInsydes"
"PSpice"
Analog Insydes (Fraunhofer-ITWM)
PSpice (MicroSim, OrCAD)
"Eldo"
Eldo (Anacad)
"Saber"
Saber (Avant!)
Supported circuit simulators.
The default setting is Simulator −> "AnalogInsydes" , which disables the usage of the external
simulator interface.
Although the Simulator option of ReadNetlist , ReadSimulationData , etc. is inherited from the
global Analog Insydes option Simulator , it is recommended to specify the Simulator option in
each call of the above mentioned commands rather than altering the global value.
3.14.7 UseExternals
The option UseExternals enables or disables the use of the external MathLink applications of Analog
Insydes. You can set UseExternals −> False if you prefer not to use MathLink programs or if you
wish to run Analog Insydes on a platform for which no native support is provided. Possible values
are:
3. Reference Manual
430
True
False
use the MathLink extensions of Analog Insydes
do not use the MathLink extensions; use internal algorithms
if available
Values for the UseExternals option.
The default setting is UseExternals −> True, which enables the usage of the MathLink applications.
If you set UseExternals −> False, some functions may not be available or run with
reduced performance.
Note that the external routines are called transparently for the user. There is no need to perform
any additional setup.
3.14.8 Option Inheritance
Analog Insydes employs an option inheritance scheme that allows you to change the values of shared
options both globally and locally in a convenient way. If the value of an option optname is specified
as Inherited[symbol] , then the actual value of optname will be retrieved from Options[symbol] as
soon as it is needed.
For example, the Analog Insydes functions ExpandSubcircuits and ReadNetlist both inherit the
options InstanceNameSeparator and Protocol from Options[AnalogInsydes] . You can change
the settings for both functions simultaneously by specifying new values for
Options[AnalogInsydes] . You can also change any of these options locally for either function in
Options[ExpandSubcircuits] or Options[ReadNetlist] . See example below.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Display the options of
ExpandSubcircuits .
In[2]:= Options[ExpandSubcircuits]
Out[2]=
AutoloadModels True, DefaultSelector None, HoldModels None,
InstanceNameSeparator InheritedAnalogInsydes,
KeepLocalModels False, LibraryPath InheritedAnalogInsydes,
ModelLibrary InheritedAnalogInsydes,
Protocol InheritedAnalogInsydes, Symbolic All,
Value None
Note that some of the option values are marked as Inherited[AnalogInsydes] . The values of these
options will be retrieved from Options[AnalogInsydes] when they are neeeded.
3.14 Global Options
431
Display the options of
ReadNetlist .
In[3]:= Options[ReadNetlist]
Change the value of
InstanceNameSeparator
globally.
In[4]:= SetOptions[AnalogInsydes, InstanceNameSeparator −> "$$"]
Change the value of
Protocol locally for
ExpandSubcircuits .
In[5]:= SetOptions[ExpandSubcircuits, Protocol −> None]
Out[3]=
CharacterMapping _ $,
InstanceNameSeparator InheritedAnalogInsydes,
KeepPrefix True, LevelSeparator ,
LibraryPath ., Protocol InheritedAnalogInsydes,
Simulator InheritedAnalogInsydes
Out[4]=
InstanceNameSeparator $$,
LibraryPath Prepend$Path, aidirAnalogInsydesModelLibrary,
ModelLibrary FullModels‘, Protocol StatusLine,
Simulator AnalogInsydes, UseExternals True
Out[5]=
AutoloadModels True, DefaultSelector None, HoldModels None,
InstanceNameSeparator InheritedAnalogInsydes,
KeepLocalModels False, LibraryPath InheritedAnalogInsydes,
ModelLibrary InheritedAnalogInsydes, Protocol None,
Symbolic All, Value None
Internally, Analog Insydes resolves inherited options by means of the function ResolveOptions .
We use ResolveOptions here for documentation purposes only. Option inheritance is taken care of
automatically by Analog Insydes. Below, ResolveOptions is used to determine the final values of
the options for ExpandSubcircuits . Note that the local value of the Protocol option overrides the
value from Options[AnalogInsydes] .
Get option settings for
ExpandSubcircuits .
In[6]:= ResolveOptions[{}, ExpandSubcircuits,
{InstanceNameSeparator, Protocol}] // InputForm
Out[6]//InputForm= {"$$", None}
For ReadNetlist , the values of both options are retrieved from Options[AnalogInsydes] .
Get option settings for
ReadNetlist .
In[7]:= ResolveOptions[{}, ReadNetlist,
{InstanceNameSeparator, Protocol}] // InputForm
Out[7]//InputForm= {"$$", StatusLine}
3. Reference Manual
432
3.15 The Analog Insydes Environment
The Analog Insydes loading procedure is completed by execution of several init files which allow for
adapting Analog Insydes to your local needs. Section 3.15.1 explains the details of the initialization
procedure.
Once Analog Insydes is started, there are several commands for obtaining detailed information
concerning your local Analog Insydes installation:
ReleaseInfo[AnalogInsydes] (Section 3.15.2)
Analog Insydes license type information
ReleaseNumber[AnalogInsydes] (Section 3.15.3)
Analog Insydes release number
License[AnalogInsydes] (Section 3.15.4)
Analog Insydes license number
Version[AnalogInsydes] (Section 3.15.5)
Analog Insydes version information
Info[AnalogInsydes] (Section 3.15.6)
Analog Insydes installation information
Obtaining Analog Insydes release information.
3.15.1 The Initialization Procedure
At the end of the Analog Insydes loading procedure, after setting up autoload declarations for all
Analog Insydes functions, three configurable init files are loaded (via the Mathematica function Get)
in the following order:
If a file called aiinit.m exists in the Analog Insydes installation directory, this file is loaded first.
If a file called .airc exists in the directory given by the Mathematica variable $HomeDirectory ,
this file is loaded next.
If a file called .airc exists in the current working directory (i.e. the directory returned by
Directory[] ; see the Mathematica command Directory ), this file is loaded last.
These files allow for adapting Analog Insydes to your local needs: Simply add the corresponding
Mathematica commands to one of the init files (see example below).
By setting the following environment variables you can suppress loading of init files:
3.15 The Analog Insydes Environment
433
AI_NO_SITE_INIT
suppress loading of init file in Analog Insydes installation
directory
AI_NO_USER_INIT
suppress loading of init file in home directory
AI_NO_WORKDIR_INIT
suppress loading of init file in current working directory
Environment variables for suppressing init file loading.
Since the init files are loaded after installing the package autoload declarations, you can access all
Analog Insydes functions and options in the init files. For example, if you always want to produce
protocol output in your notebook, you can set the global Analog Insydes option Protocol by adding
the following line to your user init file:
Setting the default
protocol.
SetOptions[AnalogInsydes, Protocol −> Notebook];
If you want to configure the location of an init file by an environment variable, add the following to
your user init file:
Loading an init file given
by an environment
variable.
If[ (file = Environment["MY_AI_INIT_FILE"]) =!= $Failed,
Get[file]];
This loads the init file given by the environment variable MY_AI_INIT_FILE if the variable is set.
3.15.2 ReleaseInfo
ReleaseInfo[AnalogInsydes]
prints information on your local Analog Insydes
installation
Command structure of ReleaseInfo.
The command ReleaseInfo[AnalogInsydes] prints general information concerning your local Analog Insydes installation such as the release number and a copyright notice.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Print information on
Analog Insydes release.
In[2]:= ReleaseInfo[AnalogInsydes]
Analog Insydes for Mathematica, Release 2.0
Copyright (c) 1996−2001 by Fraunhofer−ITWM
3. Reference Manual
434
3.15.3 ReleaseNumber
ReleaseNumber[AnalogInsydes]
returns the release number of your local Analog Insydes
installation
Command structure of ReleaseNumber.
The command ReleaseNumber[AnalogInsydes] returns the release number of your Analog Insydes
installation as a real number.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Get Analog Insydes
release number.
In[2]:= ReleaseNumber[AnalogInsydes]
Out[2]= 2.
3.15.4 License
License[AnalogInsydes]
returns the Analog Insydes license number currently used
Command structure of License.
The command License[AnalogInsydes] returns the license string used by your Analog Insydes
installation. As your license file might contain a list of licenses, License[AnalogInsydes] displays
the license actually used.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Get Analog Insydes license
number.
In[2]:= License[AnalogInsydes]
Out[2]=
AIM1200R0100014238592786992445254318
3.15.5 Version
Version[AnalogInsydes]
Command structure of Version.
shows detailed information on the version of your local
Analog Insydes installation
3.15 The Analog Insydes Environment
435
The command Version[AnalogInsydes] returns a list of version strings of your Analog Insydes
installation. Each loaded Analog Insydes package contributes a separate version string to this list,
thus the return value of this commands differs depending on the commands used during the current
session. Please always state Version[AnalogInsydes] when reporting problems or bugs.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Print detailed Analog
Insydes version
information.
In[2]:= Version[AnalogInsydes]
Out[2]=
Common: 2.157, Main: 2.96, Master: 2.45, Miscellaneous: 2.70
3.15.6 Info
Info[AnalogInsydes]
prints general information of your running Analog Insydes
installation
Command structure of Info.
The command Info[AnalogInsydes] provides general information of the running Analog Insydes
installation. It prints:
path to the Analog Insydes installation directory
list of all init files loaded during startup
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Print general information
on Analog Insydes
installation.
In[2]:= Info[AnalogInsydes]
Path to Analog Insydes:
/usr/local/Mathematica/3.0/AddOns/Applications/AnalogIns\
ydes
Init files loaded: {/usr/local/Mathematica/3.0/AddOns/App\
lications/AnalogInsydes/aiinit.m, /home/aiuser/.airc}
Appendix
4.1
Stimuli Sources . . . . . . . . . . . . . . . . . . . . . . . 438
4.2
Netlist Elements
4.3
Model Library . . . . . . . . . . . . . . . . . . . . . . . 460
4.3
Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
. . . . . . . . . . . . . . . . . . . . . . 446
4. Appendix
438
4.1 Stimuli Sources
This chapter describes time-dependent (HSPICE-compatible) sources which can be used for numerical
circuit simulations. The following table shows the supported time-dependent stimuli sources and
their corresponding Analog Insydes functions:
AMWave (Section 4.1.1)
ExpWave (Section 4.1.2)
PulseWave (Section 4.1.3)
PWLWave (Section 4.1.4)
SFFMWave (Section 4.1.5)
SinWave (Section 4.1.6)
amplitude-modulated source
exponential source
pulse source
piecewise linear source
frequency-modulated source
sinusoidal source
4.1.1 AMWave
AMWave[time, u , u , fm, fc, td]
describes a time-dependent amplitude modulation source
Command structure of AMWave.
AMWave returns a numerical value in case the first argument is numeric, otherwise a CheckedAMWave
is returned. CheckedAMWave is a data type which marks a AMWave description as being syntactically
correct. CheckedAMWave should not be defined directly.
A CheckedAMWave can be displayed with standard Mathematica graphic functions such as Plot or
with Analog Insydes graphic functions such as TransientPlot .
AMWave has the following optional arguments:
4.1 Stimuli Sources
439
argument name
default value
u
0 V or A
voltage or current amplitude
u
0
offset constant
fm
1 Hz
modulation frequency
fc
1 Hz
carrier frequency
td
0 s
delay time
Arguments of AMWave.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define time-dependent
amplitude modulated
source.
In[2]:= Vam[t_] = AMWave[t, 10, 1, 100, 1.*^3, 1.*^−3]
Display the signal.
In[3]:= TransientPlot[Vam[t], {t, 0, 5.*^−3}]
Out[2]=
CheckedAMWavet, 10., 1., 100., 1000., 0.001
20
10
0.001
0.002
0.003
0.004
t
0.005
Vam[t]
-10
-20
Out[3]= Graphics 4.1.2 ExpWave
ExpWave[time, u , u , tdr, taur, tdf, tauf]
describes a time-dependent exponential source
Command structure of ExpWave.
ExpWave returns a numerical value in case the first argument is numeric, otherwise a CheckedExpWave
is returned. CheckedExpWave is a data type which marks a ExpWave description as being syntactically
correct. CheckedExpWave should not be defined directly.
4. Appendix
440
A CheckedExpWave can be displayed with standard Mathematica graphic functions such as Plot or
with Analog Insydes graphic functions such as TransientPlot .
ExpWave has the following optional arguments:
argument name
default value
u
0 V or A
initial value of voltage or current
u
0 V or A
pulsed value of voltage or current
tdr
0 s
rise delay time
taur
1 s
rise time constant
tdf
0 s
fall delay time
tauf
1 s
fall time constant
Arguments of ExpWave.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define time-dependent
exponential source.
In[2]:= Vexp[t_] = ExpWave[t, 1, −1, 1, 0.1, 2]
Display the signal.
In[3]:= TransientPlot[Vexp[t], {t, 0, 8}]
Out[2]= CheckedExpWavet, 1., 1., 1., 0.1, 2., 1.
1
0.5
2
-0.5
-1
Out[3]= Graphics 4
6
8
t
Vexp[t]
4.1 Stimuli Sources
441
4.1.3 PulseWave
PulseWave[time, u , u , td, tr, tf, pw, per]
describes a time-dependent pulse source
Command structure of PulseWave.
PulseWave returns a numerical value in case the first argument is numeric, otherwise a
CheckedPulseWave is returned. CheckedPulseWave is a data type which marks a PulseWave description as being syntactically correct. CheckedPulseWave should not be defined directly.
A CheckedPulseWave can be displayed with standard Mathematica graphic functions such as Plot
or with Analog Insydes graphic functions such as TransientPlot .
PulseWave has the following optional arguments:
argument name
default value
u
0 V or A
initial value for voltage or current pulse
u
0 V or A
pulse value for voltage or current pulse
td
0 s
delay time
tr
0 s
rise time
tf
0 s
fall time
pw
0 s
pulse width
per
$MaxMachineNumber
s
pulse period
Arguments of PulseWave.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define time-dependent
pulse source.
In[2]:= Vpulse[t_] =
PulseWave[t, 0.5, 2, 0.2, 0.2, 0.3, 0.3, 0.9]
Out[2]=
CheckedPulseWavet, 0.5, 2., 0.2, 0.2, 0.3, 0.3, 0.9
4. Appendix
442
Display the signal.
In[3]:= TransientPlot[Vpulse[t], {t, 0, 2}]
2
1.8
1.6
1.4
Vpulse[t]
1.2
0.5
1
1.5
2
t
0.8
0.6
Out[3]= Graphics 4.1.4 PWLWave
PWLWave[time, data, delay, repeat]
describes a time-dependent piecewise linear source
Command structure of PWLWave.
PWLWave returns a numerical value in case the first argument is numeric, otherwise a CheckedPWLWave
is returned. If the last argument is not specified PWLWave returns an interpolating function object.
CheckedPWLWave is a data type which marks a PWLWave description as being syntactically correct.
CheckedPWLWave should not be defined directly.
A CheckedPWLWave can be displayed with standard Mathematica graphic functions such as Plot or
with Analog Insydes graphic functions such as TransientPlot .
PWLWave has the following optional arguments:
argument name
default value
data
{{0, 0}} s V or s A
data pairs {{t , v }, † † † } of time values
and voltage or current values
delay
0 s
delay time
repeat
Null s
time point at which the waveform is to
be repeated
Arguments of PWLWave.
4.1 Stimuli Sources
443
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Select data pairs.
In[2]:= data = {{0.1, 0}, {0.5, 0}, {1, 1}, {1.5, 1},
{2, 0.5}, {2.5, 0.5}, {3, 0}};
Define piecewise linear
source.
In[3]:= Vpwl1[t_] = PWLWave[t, data]
Define repeated piecewise
linear source.
In[4]:= Vpwl2[t_] = PWLWave[t, data, 0.5, 1]
Display the signals.
In[5]:= TransientPlot[{Vpwl1[t], Vpwl2[t]}, {t, −1, 7}]
Out[3]=
InterpolatingFunction1.79769 10308 , 1.79769 10308 , t
Out[4]=
CheckedPWLWavet,
InterpolatingFunction1.79769 10308 , 1.79769 10308 , ,
0.5, 1., 3.
1
0.8
Vpwl1[t]
0.6
0.4
Vpwl2[t]
0.2
2
4
6
t
Out[5]= Graphics 4.1.5 SFFMWave
SFFMWave[time, u , u , fc, mi, fs]
describes a time-dependent single-frequency frequency
modulation source
Command structure of SFFMWave.
SFFMWave returns a numerical value in case the first argument is numeric, otherwise a
CheckedSFFMWave is returned. CheckedSFFMWave is a data type which marks a SFFMWave description
as being syntactically correct. CheckedSFFMWave should not be defined directly.
A CheckedSFFMWave can be displayed with standard Mathematica graphic functions such as Plot or
with Analog Insydes graphic functions such as TransientPlot .
SFFMWave has the following optional arguments:
4. Appendix
444
argument name
default value
u
0 V or A
voltage or current offset
u
0 V or A
voltage or current amplitude
fc
1 Hz
carrier frequency
mi
0
modulation index
fs
1 Hz
signal frequency
Arguments of SFFMWave.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define time-dependent
frequency modulated
source.
In[2]:= Vfm[t_] = SFFMWave[t, 0, 1.*^6, 2.*^4, 10, 5.*^3]
Display the signal.
In[3]:= TransientPlot[Vfm[t], {t, 0, 2.*^−4}]
Out[2]=
CheckedSFFMWavet, 0., 1. 106 , 20000., 10., 5000.
6
1. 10
500000
t
0.00005 0.0001 0.00015 0.0002
Vfm[t]
-500000
6
-1. 10
Out[3]= Graphics 4.1.6 SinWave
SinWave[time, u , u , freq, td, alpha, phi]
describes a time-dependent sinusoidal source
Command structure of SinWave.
SinWave returns a numerical value in case the first argument is numeric, otherwise a CheckedSinWave
is returned. CheckedSinWave is a data type which marks a SinWave description as being syntactically
correct. CheckedSinWave should not be defined directly.
4.1 Stimuli Sources
445
A CheckedSinWave can be displayed with standard Mathematica graphic functions such as Plot or
with Analog Insydes graphic functions such as TransientPlot .
SinWave has the following optional arguments:
argument name
default value
u
0 V or A
voltage or current offset
u
0 V or A
voltage or current amplitude
freq
1 Hz
frequency
td
0 s
delay time
alpha
0 s damping factor
phi
0 phase delay
Arguments of SinWave.
Examples
Load Analog Insydes.
In[1]:= <<AnalogInsydes‘
Define time-dependent
sinusoidal source.
In[2]:= Vsin[t_] = SinWave[t, 0.5, 1, 2, 0.2, 1.5, 180]
Display the signal.
In[3]:= TransientPlot[Vsin[t], {t, 0, 3}]
Out[2]=
CheckedSinWavet, 0.5, 1., 2., 0.2, 1.5, 180.
1
0.8
0.6
Vsin[t]
0.4
0.2
0.5
1
-0.2
Out[3]= Graphics 1.5
2
2.5
3
t
4. Appendix
446
4.2 Netlist Elements
Analog Insydes supports the following primitive electrical circuit and control network elements:
element type
type tag
Resistor
R
Section 4.2.1
Conductance
G
Section 4.2.2
Admittance
Y
Section 4.2.3
Impedance
Z
Section 4.2.4
Capacitor
C
Section 4.2.5
Inductor
L
Section 4.2.6
OpenCircuit
OC
Section 4.2.7
ShortCircuit
SC
Section 4.2.8
Linear two-terminal elements.
element type
type tag
CurrentSource
I
Section 4.2.9
VoltageSource
V
Section 4.2.10
Independent sources.
element type
type tag
CCCSource
CC
Section 4.2.11
CCVSource
CV
Section 4.2.12
VCCSource
VC
Section 4.2.13
VCVSource
VV
Section 4.2.14
OPAmp
OP
Section 4.2.15
OTAmp
OT
Section 4.2.16
Controlled sources.
4.2 Netlist Elements
447
element type
type tag
Nullator
NUL
Section 4.2.17
Norator
NOR
Section 4.2.18
Fixator
FIX
Section 4.2.19
Singular network elements.
element type
type tag
SignalSource
HS
Section 4.2.20
Amplifier
HP
Section 4.2.21
Integrator
HI
Section 4.2.22
Differentiator
HD
Section 4.2.23
TransmissionLine
HT
Section 4.2.24
TransferFunction
H
Section 4.2.25
Control network elements.
4.2.1 Resistor
type name
syntax
Resistor
{Rname, {node1, node2}, valueR }
denotes a linear resistor
Circuit element type Resistor.
R
node1
node2
Modified nodal equation setup for admittances is influenced by the value-field option Pattern. Note
that the symbol Admittance is also a possible option value for the value-field option Pattern.
4. Appendix
448
4.2.2 Conductance
type name
syntax
Conductance
{Gname, {node1, node2}, valueG }
denotes a linear conductance
Circuit element type Conductance.
G
node1
node2
Modified nodal equation setup for conductances is influenced by the value-field option Pattern.
4.2.3 Admittance
type name
syntax
Admittance
{Yname, {node1, node2}, valueY }
denotes a linear (complex) admittance
Circuit element type Admittance.
Y
node1
node2
Modified nodal equation setup for admittances is influenced by the value-field option Pattern. Note
that the symbol Admittance is also a possible option value for the value-field option Pattern .
4.2.4 Impedance
type name
syntax
Impedance
{Zname, {node1, node2}, valueZ }
denotes a linear (complex) impedance
Circuit element type Impedance.
Z
node1
node2
4.2 Netlist Elements
449
Modified nodal equation setup for impedances is influenced by the value-field option Pattern . Note
that the symbol Impedance is also a possible option value for the value-field option Pattern .
4.2.5 Capacitor
type name
syntax
Capacitor
{Cname, {node1, node2}, valueC }
denotes a linear capacitor
Circuit element type Capacitor.
C
node1
node2
Modified nodal equation setup for capacitors is influenced by the value-field option Pattern .
4.2.6 Inductor
type name
syntax
Inductor
{Lname, {node1, node2}, valueL }
denotes a linear inductor
Circuit element type Inductor.
L
node1
node2
Modified nodal equation setup for inductors is influenced by the value-field option Pattern.
4.2.7 OpenCircuit
type name
syntax
OpenCircuit
{OCname, {node1, node2}, 0}
denotes an open-circuit branch
Circuit element type OpenCircuit.
4. Appendix
450
node1
V1
node2
An open-circuit branch is essentially the same as an independent current source with zero source
current. The value specification is ignored. However, it should be set to 0 (zero) to reflect the fact
that the source current is zero. Note that open circuits are mostly used implicitly as controlling
branches of voltage-controlled sources but they may also be used explicitly in a netlist when a
voltage meter is needed. In the latter case, modified nodal analysis employs a special matrix fill-in
pattern which augments the vector of unknown node voltages by the branch voltage across the open
circuit. This allows for directly computing the differential voltage between two nodes neither of
which is ground. Without the voltage sensor, two node voltages would have to be solved for and
then subtracted from one another, which would be computationally more expensive.
4.2.8 ShortCircuit
type name
syntax
ShortCircuit
{SCname, {node1, node2}, 0}
denotes a short-circuit branch
Circuit element type ShortCircuit .
node1
I1
node2
A short-circuit branch is essentially the same as a voltage source with zero source voltage. The
value specification is ignored. However, it should be set to 0 (zero) to reflect the fact that the
source voltage is zero. Note that short circuits are mostly used implicitly as controlling branches of
current-controlled sources. They may also be used explicitly as current meters in combination with
modified nodal analysis.
4.2 Netlist Elements
451
4.2.9 CurrentSource
type name
syntax
CurrentSource
{Iname, {node+, node−}, valueI0 }
denotes an independent current source
Circuit element type CurrentSource.
node+
I0
nodeThe positive reference direction for the source current is from node+ to node−.
4.2.10 VoltageSource
type name
syntax
VoltageSource
{Vname, {node+, node−}, valueV0 }
denotes an independent voltage source
Circuit element type VoltageSource.
node+
V0
nodeThe positive reference direction for the source voltage as well as the current through the source is
from node+ to node−.
4. Appendix
452
4.2.11 CCCSource
type name
syntax
CCCSource
{CCname, {cnode+, cnode−, node+, node−}, valuebeta }
denotes a linear current-controlled current
source (CCCS) with gain valuebeta
Circuit element type CCCSource.
node+
cnode+
I1
beta*I1
node-
cnode-
The terminals of the controlling branch are denoted by cnode+ and cnode−. The positive reference
direction for the sensor current and the source current is from cnode+ to cnode− and from node+ to
node−, respectively. Note that Analog Insydes automatically inserts a short circuit in between cnode+
and cnode− as a current meter. As opposed to the SPICE netlist format there is no need to add a
sensor voltage source to the netlist.
4.2.12 CCVSource
type name
syntax
CCVSource
{CVname, {cnode+, cnode−, node+, node−}, valuer }
denotes a linear current-controlled voltage
source (CCVS) with transresistance valuer
Circuit element type CCVSource.
cnode+
node+
I1
cnode-
r*I1
node-
The terminals of the controlling branch are denoted by cnode+ and cnode−. The positive reference
direction for the sensor current and the source voltage is from cnode+ to cnode− and from node+ to
4.2 Netlist Elements
453
node−, respectively. Note that Analog Insydes automatically inserts a short circuit in between cnode+
and cnode− as a current meter. As opposed to the SPICE netlist format there is no need to add a
sensor voltage source to the netlist.
4.2.13 VCCSource
type name
syntax
VCCSource
{VCname, {cnode+, cnode−, node+, node−}, valuegm }
denotes a linear voltage-controlled current
source (VCCS) with transconductance
valuegm
Circuit element type VCCSource.
cnode+
node+
V1
gm*V1
cnode-
node-
The terminals of the controlling branch are denoted by cnode+ and cnode−. Note that as opposed to
the SPICE netlist format the controlling nodes are listed before the terminals of the controlled branch.
4.2.14 VCVSource
type name
syntax
VCVSource
{VVname, {cnode+, cnode−, node+, node−}, valuev }
denotes a linear voltage-controlled voltage
source (VCVS) with voltage gain valuev
Circuit element type VCVSource.
cnode+
V1
cnode-
node+
v*V1
node-
4. Appendix
454
The terminals of the controlling branch are denoted by cnode+ and cnode−. Note that as opposed to
the SPICE netlist format the controlling nodes are listed before the terminals of the controlled branch.
4.2.15 OPAmp
type name
syntax
OPAmp
{OPname, {cnode+, cnode−, node+, node−}, valuev }
denotes an ideal operational amplifier with
finite or infinite gain valuev
Circuit element type OPAmp.
cnode+
V1
node+
v*V1
cnode-
node-
For infinite gain, valuev must be Infinity . Note that, functionally, an OpAmp is the same as a
voltage-controlled voltage source. The difference to a VCVSource is that during circuit equation
setup another matrix fill-in pattern is used which makes limit calculations for large OpAmp gains
easier.
4.2.16 OTAmp
type name
syntax
OTAmp
{OTname, {cnode+, cnode−, node+, node−}, valuegm }
denotes an ideal operational
transconductance amplifier with finite or
infinite gain valuegm
Circuit element type OTAmp.
cnode+
gm*V1
V1
node+
cnode-
node-
4.2 Netlist Elements
455
For infinite gain, valuegm must be Infinity . Note that, functionally, an OTAmp is the same as a
voltage-controlled current source. The difference to a VCCSource is that during circuit equation
setup another matrix fill-in pattern is used which makes limit calculations for large OTAmp gains
easier.
4.2.17 Nullator
type name
syntax
Nullator
{NULname, {node1, node2}, 0}
denotes a nullator
Circuit element type Nullator.
node1
V=0
I=0
node2
A nullator is an ideal two-terminal circuit element which enforces both a zero voltage and a zero
current between its terminals. It is thus neither a voltage nor a current source but rather both at
the same time. The value specification in the netlist entry of a nullator is irrelevant and should be
set to 0 (zero). Note that an element with a arbitrary but fixed branch current and branch voltage
is also known as a fixator. With a zero current and voltage, a nullator is a special case of a fixator.
A nullator usually appears paired with a norator, thus forming a nullor. A nullor represents a
controlled source of arbitrary type with infinite gain, such as an ideal operational amplifier.
4.2.18 Norator
type name
syntax
Norator
{NORname, {node1, node2}, 0}
denotes a norator
Circuit element type Norator.
4. Appendix
456
node1
node2
A Norator is an ideal two-terminal circuit element which imposes no constraints on its branch current
and voltage. The value specification is ignored and should be set to 0 (zero). Note that a norator
usually appears paired with a nullator, thus forming a nullor. A nullor represents a controlled source
of arbitrary type with infinite gain, such as an ideal operational amplifier.
4.2.19 Fixator
type name
syntax
Fixator
{FIXname, {node+, node−}, {valueI0 , valueV0 }}
denotes a fixator
Circuit element type Fixator.
node+
V0
I0
nodeA fixator is an ideal two-terminal circuit element which enforces both a fixed voltage and a fixed
current between its terminals. It is thus neither a voltage nor a current source but rather both at the
same time. Note that the netlist entry for fixators has a special form because two element values,
a source current and a source voltage, must be specified. Fixators are hardly ever used for circuit
analysis purposes. Their main application field is circuit sizing where they can be used to model
operating points of nonlinear devices such as transistors.
4.2 Netlist Elements
457
4.2.20 SignalSource
type name
syntax
SignalSource
{HSname, {output}, valueX }
denotes a signal source in a control network
Circuit element type SignalSource.
output
X(s)
Signal sources do not have an input node. Therefore, the connectivity field contains only the node
identifier of the output node.
4.2.21 Amplifier
type name
syntax
Amplifier
{HPname, {input, output}, valueA }
denotes a proportional amplifier with
amplification valueA
Circuit element type Amplifier.
input
A
output
4. Appendix
458
4.2.22 Integrator
type name
syntax
Integrator
{HIname, {input, output}, valueTi }
denotes an integrator with an integration
time constant valueTi
Circuit element type Integrator.
input
Ti
output
4.2.23 Differentiator
type name
syntax
Differentiator
{HDname, {input, output}, valueTd }
denotes a differentiator with a
differentiation time constant valueTd
Circuit element type Differentiator .
input
Td
output
4.2.24 TransmissionLine
type name
syntax
TransmissionLine
{HTname, {input, output}, valueT }
denotes an ideal lossless transmission line
with a delay time constant valueT
Circuit element type TransmissionLine .
4.2 Netlist Elements
459
input
T
output
4.2.25 TransferFunction
type name
syntax
TransferFunction
{Hname, {input, output}, valuetfunc }
denotes a generic transfer function block
Circuit element type TransferFunction .
input
H(s)
output
The transfer function valuetfunc may be any function of the Laplace frequency variable s. To avoid
confusion with the other control network element types the block name must not begin with the
letters S, P, I, D, or T.
4. Appendix
460
4.3 Model Library
The Analog Insydes model library provides full-precision SPICE-compatible models for the most
important devices such as Diodes (Section 4.3.1), BJTs (Section 4.3.2), MOSFETs (Section 4.3.3), and
JFETs (Section 4.3.4).
Each section contains two subsections describing the large-signal as well as the small-signal model.
Following an equivalent circuit schematic the particular model parameters are introduced including their default values and a short description. Note that the default values of the small-signal
parameters are given symbolically with the suffix "$ac" as they are automatically generated by
ReadNetlist (Section 3.10.1). Each model section concludes with a description of those parameters
which are influenced by the automatic model-complexity reduction controlled via the global Analog
Insydes option ModelLibrary (Section 3.14.4).
4.3.1 Diode
syntax
{Dname, {n −>A, n −>C}, Model−>Diode, Selector−>selector, parameters}
denotes a subcircuit reference for a Diode
Subcircuit reference for Diode.
A
C
Valid values for selector are: Spice, SpiceDC , SpiceAC, and SpiceNoise . Possible parameters are
described below.
Large-Signal Model
The analog behavioral model included in the Analog Insydes model library has the following port
characteristics:
Voltage[A,C]
C
A
Current[A,C]
where A denotes the Anode and C the Cathode, respectively. The port currents can be expressed in
the port variables as follows:
4.3 Model Library
461
iA t Current[A,C]t
Anode:
iC t Current[A,C]t
Cathode:
A
RS
CD
id
C
Equivalent schematic for the DC and transient analysis
The large-signal model includes the following model parameters:
parameter name
default value
BV
- V
IBV
IS
N
emission coefficient
NBV
reverse breakdown coefficient
reverse breakdown voltage
A A reverse breakdown current
saturation current
DC related model parameters.
parameter name
default value
RS
Parasitic resistance related model parameters.
ohmic resistance
4. Appendix
462
parameter name
default value
CJO
F
zero-bias junction capacitance
FC
forward-bias depletion capacitance
coefficient
M
grading coefficient
TT
s
transit time
VJ
V
junction potential
Capacitance related model parameters.
parameter name
default value
EG
eV
energy gap
TEMP
K
model temperature
TNOM
K
nominal temperature
XTI
saturation current temperature exponent
Temperature related model parameters.
parameter name
default value
AF
flicker noise exponent
KF
flicker noise coefficient
Noise related model parameters.
parameter name
default value
AREA
GMIN
LEVEL
relative device area
minimum conductance
model index
Miscellaneous model parameters.
The automatic reduction of the model complexity controlled via the global Analog Insydes option
ModelLibrary influences the following parameters:
4.3 Model Library
463
parameter name
FullModels
SimplifiedModels
BasicModels
BV
used
used
default
RS
used
default
default
TT
used
default
default
Simplification related model parameters.
Small-Signal Model
A
Rs
Rd
Cd
C
Equivalent schematic for the AC analysis
The small-signal model includes the following model parameters:
parameter name
default value
Cd
CAP$ac
junction and diffusion capacitance
Rd
REQ$ac
small-signal resistance
Rs
RS/AREA
resistance
Small-signal model parameters.
The automatic reduction of the model complexity controlled via the global Analog Insydes option
ModelLibrary influences the following parameters:
4. Appendix
464
parameter name
FullModels
SimplifiedModels
BasicModels
Rs
used
ignored
ignored
Simplification related model parameters.
4.3.2 Bipolar Junction Transistor
syntax
{Qname, {n −>C, n −>B, n −>E, n −>S}, Model−>BJT, Selector−>selector, parameters}
denotes a subcircuit reference for a BJT
Subcircuit reference for BJT.
C
C
B
B
E
E
Valid values for selector are: GummelPoon , GummelPoonDC , Spice, SpiceDC, SpiceAC, and SpiceNoise .
Possible parameters are described below. Please note that the substrate port S is optional.
Large-Signal Model
The analog behavioral model included in the Analog Insydes model library has the following port
characteristics:
C
Voltage[C,S]
Current[C,S]
Voltage[B,S]
S
B
Current[B,S]
Voltage[E,S]
Current[E,S]
E
4.3 Model Library
465
where B denotes the Base, C the Collector, E the Emitter, and S the Substrate, respectively. The port
currents can be expressed in the port variables as follows:
Base:
iB t Current[B,S]t
Collector:
iC t Current[C,S]t
Emitter:
iE t Current[E,S]t
Substrate:
iS t Current[B,S]t Current[C,S]t Current[E,S]t
C
RC
ics
CBX
CBC
CJS
RB
S
B
CBE
ic
ib
RE
E
Equivalent schematic for the DC and transient analysis for the vertical bipolar transistor (NPN, PNP)
4. Appendix
466
C
RC
CBC
CBX
RB
B
CXS
CBE
CJS
ibs
ic
ib
RE
S
E
Equivalent schematic for the DC and transient analysis for the lateral bipolar transistor (LPNP)
The large-signal model includes the following model parameters:
4.3 Model Library
467
parameter name
default value
BF
ideal maximum forward current gain
BR
ideal maximum reverse current gain
IKF (IK)
- A corner current for forward-beta high-current
roll-off
IKR
- A corner current for reverse-beta high-current
roll-off
IS
ISC (C4)
A base-collector leakage saturation current
ISE (C2)
A base-emitter leakage saturation current
ISS
A substrate junction saturation current
NC
base-collector leakage emission coefficient
NE
base-emitter leakage emission coefficient
NF
forward current emission coefficient
NK
high-current roll-off coefficient
NR
reverse current emission coefficient
NS (NSS)
substrate junction emission coefficient
VAF (VA)
- V
forward early voltage
VAR (VB, VBAR)
- V
reverse early voltage
A transport saturation current
DC related model parameters.
parameter name
default value
IRB
- A current at which RB falls halfway to its
minimum value
RB
zero-bias base ohmic resistance
RBM
RB
RC
collector ohmic resistance
RE
emitter ohmic resistance
Parasitic resistance related model parameters.
minimum base ohmic resistance
4. Appendix
468
parameter name
default value
CJC
F
zero-bias base-collector depletion
capacitance
CJE
F
zero-bias base-emitter depletion capacitance
CJS (CCS)
F
zero-bias substrate capacitance
FC
forward-bias depletion capacitance
coefficient
MJC (MC)
base-collector junction grading factor
MJE (ME)
base-emitter junction grading factor
MJS (MS)
substrate junction grading factor
VJC (PC)
V
base-collector built-in potential
VJE (PE)
V
base-emitter built-in potential
VJS (PS)
V
substrate junction built-in potential
XCJC
fraction of base-collector capacitance
connected to internal base node
XJBS
fraction of base-substrate capacitance
connected to internal base node (for lateral
devices only)
Junction capacitance related model parameters.
parameter name
default value
ITF
A transit time dependency on collector current
PTF
excess phase at f Π TF Hz
TF
s
ideal forward transit time
TR
s
ideal reverse transit time
VTF
- V
transit time dependency on base-collector
voltage
XTF
transit time bias dependence coefficient
Dynamic model parameters.
4.3 Model Library
469
parameter name
default value
EG
eV
GAP1
GAP2
K
energy gap
eV K
first bandgap correction factor
second bandgap correction factor
TEMP
K
model temperature
TNOM
K
nominal temperature
TRB1
K linear temperature coefficient for RB
TRB2
K quadratic temperature coefficient for RB
TRC1 (TRC)
K linear temperature coefficient for RC
TRC2
K quadratic temperature coefficient for RC
TRE1 (TRE)
K linear temperature coefficient for RE
TRE2
K quadratic temperature coefficient for RE
TRM1 (TRBM1)
K linear temperature coefficient for RBM
TRM2 (TRBM2)
K quadratic temperature coefficient for RBM
XTB (TB, TCB)
forward and reverse beta temperature
exponent
XTI (PT)
IS temperature effect exponent
Temperature related model parameters.
parameter name
default value
AF
flicker noise exponent
KF
flicker noise coefficient
Noise related model parameters.
4. Appendix
470
parameter name
default value
AREA
GMIN
LEVEL
relative device area
minimum conductance
model index
Miscellaneous model parameters.
The automatic reduction of the model complexity controlled via the global Analog Insydes option
ModelLibrary influences the following parameters:
4.3 Model Library
471
parameter name
FullModels
SimplifiedModels
BasicModels
CJS
used
default
default
IKF
used
default
default
IKR
used
default
default
IRB
used
default
default
ISC
used
default
default
ISE
used
used
default
ISS
used
default
default
ITF
used
default
default
PTF
used
default
default
RB
used
used
default
RBM
used
used
default
RC
used
default
default
RE
used
default
default
TF
used
default
default
TR
used
default
default
VAF
used
used
default
VAR
used
used
default
VTF
used
default
default
XCJC
used
used
default
XJBS
used
used
default
XTF
used
default
default
Simplification related model parameters.
4. Appendix
472
Small-Signal Model
C
Rc
Cjs
S
Gmu
Cbx
Cbc
Rx
Ro
B
vpi
Rpi
gm*vpi
Cbe
Re
E
Equivalent schematic for the AC analysis for the vertical bipolar transistor (NPN, PNP)
C
Rc
Gmu
Cbx
Cbc
Rx
Ro
B
Cxs
Cjs
vpi
Rpi
Cbe
gm*vpi
Re
S
E
Equivalent schematic for the AC analysis for the lateral bipolar transistor (LPNP)
The small-signal model includes the following model parameters:
4.3 Model Library
473
parameter name
default value
Cbc
CBC$ac (CMU$ac)
base collector capacitance
Cbe
CBE$ac (CPI$ac)
base emitter capacitance
Cbx
CBX$ac
external base collector capacitance
Cjs
CJS$ac (CBS$ac, CCS$ac)
substrate capacitance
Cxs
CXS$ac
external base substrate capacitance (for
lateral devices only)
gm
GM$ac
transconductance
Gmu
GMU$ac
base collector conductance
Rc
RC/AREA
collector resistance
Re
RE/AREA
emitter resistance
Ro
RO$ac
collector emitter resistance
Rpi
RPI$ac
base emitter resistance
Rx
RX$ac
base resistance
Small-signal model parameters.
The automatic reduction of the model complexity controlled via the global Analog Insydes option
ModelLibrary influences the following parameters:
parameter name
FullModels
SimplifiedModels
BasicModels
Cjs
used
used
ignored
Cxs
used
used
ignored
Gmu
used
ignored
ignored
Rc
used
ignored
ignored
Re
used
ignored
ignored
Rx
used
ignored
ignored
Simplification related model parameters.
4. Appendix
474
4.3.3 MOS Field-Effect Transistor
syntax
{Mname, {n −>D, n −>G, n −>S, n −>B}, Model−>MOSFET, Selector−>selector, parameters}
denotes a subcircuit reference for a MOSFET
Subcircuit reference for MOSFET.
D
G
D
G
B
B
S
S
Valid values for selector are: Spice, SpiceDC , SpiceAC, and SpiceNoise . Possible parameters are
described below.
Large-Signal Model
The analog behavioral model included in the Analog Insydes model library has the following port
characteristics:
D
Voltage[D,B]
Current[D,B]
Voltage[G,B]
B
G
Current[G,B]
Voltage[S,B]
Current[S,B]
S
where G denotes the Gate, D the Drain, S the Source, and B the Bulk, respectively. The port currents
can be expressed in the port variables as follows:
Gate:
iG t Current[G,B]t
Drain:
iD t Current[D,B]t
Source:
iS t Current[S,B]t
Bulk:
iB t Current[G,B]t Current[D,B]t Current[S,B]t
4.3 Model Library
475
D
RD
CGB
ibd
CGD
CBD
G
B
ids
CGS
CBS
ibs
RS
S
Equivalent schematic for the DC and transient analysis
The large-signal model includes the following model parameters:
parameter name
default value
GAMMA
V
IS
JS
Am2 bottom bulk junction saturation current
density
JSSW
Am
sidewall bulk junction saturation current
density
KP
LAMBDA
V channel length modulation
N
bulk junction emission coefficient
PHI
V
surface potential
VTO
V
zero-bias threshold voltage
DC related model parameters.
bulk threshold parameter
A AV2 bulk junction saturation current
transconductance parameter
4. Appendix
476
parameter name
default value
L
DEFL m
channel length
LD
m
lateral diffusion length
W
DEFW m
channel width
WD
m
lateral diffusion width
Area related model parameters.
parameter name
default value
NRD
DEFNRD square
number of squares for drain resistance
NRS
DEFNRS square
number of squares for source resistance
RD
drain ohmic resistance
RDS
- drain source shunt resistance
RS
source ohmic resistance
RSH
square
drain and source diffusion sheet resistance
Parasitic resistance related model parameters.
parameter name
default value
CBD
F
zero-bias bulk drain capacitance
CBS
F
zero-bias bulk source capacitance
CJ
Fm2 zero-bias bottom bulk capacitance
CJSW
Fm
zero-bias sidewall bulk capacitance
FC
forward-bias bulk junction capacitance
coefficient
MJ
bulk junction bottom grading coefficient
MJSW
bulk junction sidewall grading coefficient
PB
V
bulk bottom junction potential
PBSW
V
bulk sidewall junction potential
Junction capacitance related model parameters.
4.3 Model Library
477
parameter name
default value
CGBO
Fm
gate-bulk overlap capacitance
CGDO
Fm
gate-drain overlap capacitance
CGSO
Fm
gate-source overlap capacitance
TT
s
transit time
Dynamic model parameters.
parameter name
default value
TEMP
K
TNOM
model temperature
K
nominal temperature
Temperature related model parameters.
parameter name
default value
AF
flicker noise exponent
KF
flicker noise coefficient
Noise related model parameters.
parameter name
default value
AD
DEFAD m2 drain diffusion area
AS
DEFAS m2 source diffusion area
GMIN
LEVEL
model index
M
DEFM
device multiplier
PD
DEFPD m
drain diffusion perimeter
PS
DEFPS m
source diffusion perimeter
Miscellaneous model parameters.
minimum conductance
4. Appendix
478
parameter name
default value
DEFAD
m2 default value for AD
DEFAS
m2 default value for AS
DEFL
DEFM
default value for M
DEFNRD
square
default value for NRD
DEFNRS
square
default value for NRS
DEFPD
m
default value for PD
DEFPS
m
default value for PS
DEFW
m
default value for L
m
default value for W
Parameter defaults set by global options.
The automatic reduction of the model complexity controlled via the global Analog Insydes option
ModelLibrary influences the following parameters:
parameter name
FullModels
SimplifiedModels
BasicModels
CGBO
used
used
default
CGDO
used
used
default
CGSO
used
used
default
CJSW
used
default
default
GAMMA
used
used
default
RD
used
default
default
RDS
used
default
default
RS
used
default
default
RSH
used
default
default
TT
used
default
default
Simplification related model parameters.
4.3 Model Library
479
Small-Signal Model
D
Cgb
Rd
Cgd
Gbd
Cbd
Gds
G
B
gm*vgs
Cgs
gmb*vbs
Rs
vgs
Gbs
Cbs
vbs
S
Equivalent schematic for the AC analysis for the level 1-3 model
4. Appendix
480
s*DqdDvgb*vgb
s*DqdDvdb*vdb
s*DqdDvsb*vsb
D
Cgb
vgd
Rd
vbd
Cgd
Gbd
Cbd
s*DqbDvgb*vgb
s*DqbDvdb*vdb
s*DqbDvsb*vsb
Gds
G
B
gm*vgs
s*DqgDvgb*vgb
s*DqgDvdb*vdb
s*DqgDvsb*vsb
Cgs
gmb*vbs
vgs
Rs
Gbs
Cbs
vbs
S
s*DqsDvgb*vgb
s*DqsDvdb*vdb
s*DqsDvsb*vsb
Equivalent schematic for the AC analysis for the BSIM model (simulator PSpice)
4.3 Model Library
481
s*Cdd*vd
s*Cdg*vg
s*Cds*vs
s*Cdb*vb
D
Rd
vgd
Gbd
vbd
Gds
G
gm*vgs
gmb*vbs
vgs
Gbs
s*Cbd*vd
s*Cbg*vg
B s*Cbs*vs
s*Cbb*vb
vbs
Rs
s*Cgd*vd
s*Cgg*vg
s*Cgs*vs
s*Cgb*vb
S
s*Csd*vd
s*Csg*vg
s*Css*vs
s*Csb*vb
Equivalent schematic for the AC analysis for the BSIM model (simulator Eldo)
The small-signal model includes the following model parameters:
4. Appendix
482
parameter name
default value
Cbd
CBD$ac
bulk drain capacitance
Cbs
CBS$ac
bulk source capacitance
Cgb
CGB$ac+CGBOV$ac
gate bulk capacitance
Cgd
CGD$ac+CGDOV$ac
gate drain capacitance
Cgs
CGS$ac+CGSOV$ac
gate source capacitance
Gbd
GBD$ac
bulk drain conductance
Gbs
GBS$ac
bulk source conductance
Gds
GDS$ac
drain source conductance
gm
GM$ac
transconductance
gmb
GMB$ac
bulk transconductance
Rd
RD
drain resistance
Rs
RS
source resistance
Level 1-3 small-signal model parameters.
4.3 Model Library
483
parameter name
default value
Cbdb
DQBDVDB$ac
derivative of bulk charge w.r.t. drain bulk
voltage
Cbgb
DQBDVGB$ac
derivative of bulk charge w.r.t. gate bulk
voltage
Cbsb
DQBDVSB$ac
derivative of bulk charge w.r.t. source bulk
voltage
Cddb
DQDDVDB$ac
derivative of drain charge w.r.t. drain bulk
voltage
Cdgb
DQDDVGB$ac
derivative of drain charge w.r.t. gate bulk
voltage
Cdsb
DQDDVSB$ac
derivative of drain charge w.r.t. source bulk
voltage
Cgdb
DQGDVDB$ac
derivative of gate charge w.r.t. drain bulk
voltage
Cggb
DQGDVGB$ac
derivative of gate charge w.r.t. gate bulk
voltage
Cgsb
DQGDVSB$ac
derivative of gate charge w.r.t. source bulk
voltage
Csdb
DQSDVDB$ac
derivative of source charge w.r.t. drain bulk
voltage
Csgb
DQSDVGB$ac
derivative of source charge w.r.t. gate bulk
voltage
Cssb
DQSDVSB$ac
derivative of source charge w.r.t. source
bulk voltage
Additional small-signal model parameters for BSIM PSpice.
4. Appendix
484
parameter name
default value
Cb
Cbb$ac
derivative of bulk charge w.r.t. bulk
potential
Cbd0
Cbd$ac
derivative of bulk charge w.r.t. drain
potential
Cbg0
Cbg$ac
derivative of bulk charge w.r.t. gate
potential
Cbs0
Cbs$ac
derivative of bulk charge w.r.t. source
potential
Cdb0
Cdb$ac
derivative of drain charge w.r.t. bulk
potential
Cd
Cdd$ac
derivative of drain charge w.r.t. drain
potential
Cdg0
Cdg$ac
derivative of drain charge w.r.t. gate
potential
Cds0
Cds$ac
derivative of drain charge w.r.t. source
potential
Cgb0
Cgb$ac
derivative of gate charge w.r.t. bulk
potential
Cgd0
Cgd$ac
derivative of gate charge w.r.t. drain
potential
Cg
Cgg$ac
derivative of gate charge w.r.t. gate potential
Cgs0
Cgs$ac
derivative of gate charge w.r.t. source
potential
Csb0
Csb$ac
derivative of source charge w.r.t. bulk
potential
Csd0
Csd$ac
derivative of source charge w.r.t. drain
potential
Csg0
Csg$ac
derivative of source charge w.r.t. gate
potential
Cs
Css$ac
derivative of source charge w.r.t. source
potential
Additional small-signal model parameters for BSIM Eldo.
The automatic reduction of the model complexity controlled via the global Analog Insydes option
ModelLibrary influences the following parameters:
4.3 Model Library
485
parameter name
FullModels
SimplifiedModels
BasicModels
Cgb
used
ignored
ignored
Gbd
used
used
ignored
Gbs
used
used
ignored
gmb
used
used
ignored
Rd
used
ignored
ignored
Rs
used
ignored
ignored
Simplification related level 1-3 model parameters.
parameter name
FullModels
SimplifiedModels
BasicModels
Cbd0
used
used
ignored
Cbg0
used
used
ignored
Cbs0
used
used
ignored
Cdb0
used
used
ignored
Cdg0
used
used
ignored
Cds0
used
used
ignored
Cgb0
used
used
ignored
Cgd0
used
used
ignored
Cgs0
used
used
ignored
Csb0
used
used
ignored
Csd0
used
used
ignored
Csg0
used
used
ignored
Simplification related BSIM Eldo model parameters.
4.3.4 Junction Field-Effect Transistor
syntax
{Jname, {n −>D, n −>G, n −>S}, Model−>JFET, Selector−>selector, parameters}
denotes a subcircuit reference for a JFET
Subcircuit reference for JFET.
4. Appendix
486
D
D
G
G
S
S
Valid values for selector are: Spice, SpiceDC , SpiceAC, and SpiceNoise . Possible parameters are
described below.
Large-Signal Model
The analog behavioral model included in the Analog Insydes model library has the following port
characteristics:
D
Voltage[D,G]
Current[D,G]
G
Voltage[S,G]
Current[S,G]
S
where G denotes the Gate, D the Drain, and S the Source, respectively. The port currents can be
expressed in the port variables as follows:
Drain:
iD t Current[D,G]t
Source:
iS t Current[S,G]t
Gate:
iG t Current[D,G]t Current[S,G]t
4.3 Model Library
487
D
RD
igd
CGD
G
CGS
ids
igs
RS
S
Equivalent schematic for the DC and transient analysis
The large-signal model includes the following model parameters:
parameter name
default value
ALPHA
V BETA
IS
ISR
A gate junction recombination current
parameter
LAMBDA
V channel length modulation
N
gate junction emission coefficient
NR
emission coefficient for ISR
VK
V
ionization knee voltage
VTO
V
threshold voltage
DC related model parameters.
ionization coefficient
AV2 A transconductance coefficient
gate junction saturation current
4. Appendix
488
parameter name
default value
RD
drain ohmic resistance
RS
source ohmic resistance
Parasitic resistance related model parameters.
parameter name
default value
CGD
F
zero-bias gate drain junction capacitance
CGS
F
zero-bias gate source junction capacitance
FC
forward depletion capacitance coefficient
M
gate junction grading coefficient
PB
V
gate junction potential
Capacitance related model parameters.
parameter name
default value
BETATCE (BETATC)
K BETA exponential temperature coefficient
EG
eV
energy gap
TEMP
K
model temperature
TNOM
K
nominal temperature
TRD1
K RD temperature coefficient
TRS1
K RS temperature coefficient
VTOTC
V K
VTO temperature coefficient
XTI
saturation current temperature exponent
Temperature related model parameters.
4.3 Model Library
489
parameter name
default value
AF
flicker noise exponent
KF
flicker noise coefficient
Noise related model parameters.
parameter name
default value
AREA
GMIN
LEVEL
relative device area
minimum conductance
model index
Miscellaneous model parameters.
The automatic reduction of the model complexity controlled via the global Analog Insydes option
ModelLibrary influences the following parameters:
parameter name
FullModels
SimplifiedModels
BasicModels
ALPHA
used
default
default
ISR
used
used
default
RD
used
default
default
RS
used
default
default
VK
used
default
default
Simplification related model parameters.
4. Appendix
490
Small-Signal Model
D
Rd
Ggd
Cgd
Gds
G
vgs
Ggs
Cgs
gm*vgs
Rs
S
Equivalent schematic for the AC analysis
The small-signal model includes the following model parameters:
parameter name
default value
Cgd
CGD$ac
gate drain capacitance
Cgs
CGS$ac
gate source capacitance
Gds
GDS$ac
drain source conductance
Ggd
GGD$ac
gate drain conductance
Ggs
GGS$ac
gate source conductance
gm
GM$ac
transconductance
Rd
RD/AREA
drain resistance
Rs
RS/AREA
source resistance
Small-signal model parameters.
The automatic reduction of the model complexity controlled via the global Analog Insydes option
ModelLibrary influences the following parameters:
4.3 Model Library
491
parameter name
FullModels
SimplifiedModels
BasicModels
Ggd
used
used
ignored
Ggs
used
used
ignored
Rd
used
ignored
ignored
Rs
used
ignored
ignored
Simplification related model parameters.
Credits
Developers
Dipl.-Ing. Thomas Halfmann
Dr.-Ing. Eckhard Hennig
Dipl.-Ing. Jutta Praetorius
Dr.-Ing. Ralf Sommer
Dr.-Ing. Manfred Thole
Dipl.-Math. Tim Wichmann
Contributions
Michael Brickenstein
Dipl.-Math. Yves Drexlmeier
Stefan Feitig
Dipl.-Math. Jan Mark Tweer
Dipl.-Math. Michael Wiese
Anton Winterfeld
Design and Cover Art
Juliette Armbrecht
Dipl.-Math. Steffen Grützner
Dipl. BK Gertrud Schrenk
Scientific Advisor
Prof. Dr. Gert-Martin Greuel
Director Fraunhofer-ITWM
Prof. Dr. Dieter Prätzel-Wolters
Index
ABM (analog behavioral model), 90
AbsolutePivotThreshold, option for
ApproximateMatrixEquation, 393
AbsoluteValues, option value for MagnitudeDisplay,
82, 333
AC, mode option for source value specification, 185
mode specification for NonlinearSetup, 404
option value for AnalysisMode, 230
AC analysis, 282
ACAnalysis, 282
example application, 159
examples for, 284
options description for, 283
AccuracyGoal, option for ApproximateDeterminant, 321
option for LREigenpair, 314
option for NDAESolve, 291
ACEquations, 248
examples for, 250
options description for, 249
AddElements, 256
examples for, 256
Admittance, 448
option value for Pattern, 183
option value for Type, 43
Admittance elements, 43
AI_NO_SITE_INIT, 433
AI_NO_USER_INIT, 433
AI_NO_WORKDIR_INIT, 433
aiinit.m, 432
airc, 432
Amplifier, 457
AMWave, 438
examples for, 439
Analog behavioral models (ABM), 90
AnalogInsydes, option value for Simulator, 429
AnalysisMode, option for CircuitEquations, 230
option for NDAESolve, 293
ApplyDesignPoint, 269
examples for, 271
options description for, 270
ApproximateDeterminant, 318
examples for, 327
options description for, 321
ApproximateMatrixEquation, 134, 135, 391
examples for, 395
options description for, 393
ApproximateTransferFunction, 132, 387
examples for, 388
Approximation, design point specification, 135
methods, 132
of expressions, 132
of linear circuits, 127, 385
of linear equations, ApproximateMatrixEquation,
134, 391
of nonlinear circuits, 164, 401
of nonlinear equations, CancelTerms, 165, 410
of transfer functions,
ApproximateTransferFunction, 387
AspectFactor, option for BodePlot, 331
AspectRatio, option for BodePlot, 331
AutoloadModels, option for ExpandSubcircuits, 222
BasicModels, option value for ModelLibrary, 427
Behavioral models, 90
defining, 90
referencing, 94
Bias point calculation, example for, 100
Bipolar junction transistor (BJT), 464
BJT, 464
example implementation, 98
large-signal model, 464
small-signal model, 472
Bode diagram, 81, 330
BodePlot, 81, 330
examples for, 334
options description for, 332
Branch currents, naming scheme for, 70
Branch voltages, naming scheme for, 70
BranchCurrentPrefix, option for CircuitEquations,
70, 231
Branches, of controlled sources, 35, 38
BranchVoltagePrefix, option for CircuitEquations,
70, 231
BreakOnError, option for NDAESolve, 294
Bug reports, 12
494
CancelTerms — Design points
CancelTerms, 165, 410
example application, 172
examples for, 414
options description for, 411
Capacitor, 449
Capacitors, initial conditions for, 44
CCCSource, 452
CCVSource, 452
CharacterMapping, option for ReadNetlist, 366
CheckedAMWave, data type returned by AMWave, 438
CheckedExpWave, data type returned by ExpWave, 440
CheckedPulseWave, data type returned by PulseWave,
441
CheckedPWLWave, data type returned by PWLWave, 442
CheckedSFFMWave, data type returned by SFFMWave, 443
CheckedSinWave, data type returned by SinWave, 445
CircleMark, marker style for options PoleStyle and
ZeroStyle, 87, 354
Circuit, 46, 179
examples for, 179
Circuit equations, setting up and solving, 227
Circuit object, 178
information on, Statistics, 277
manipulating, 255
showing contents, DisplayForm, 47
Circuit-description language, 178
CircuitEquations, 71, 227, 231, 234, 235, 236
examples for, 239
options description for, 230
Cleanup, option for ApplyDesignPoint, 270
option for UpdateDesignPoint, 272
Clusterbound, option for CancelTerms, 412
CMRR, 151
ColorList, option for DXFGraphics, 422
Common-mode gain, 149
Common-mode rejection ratio, 151
Compatibility to Version 1, 29
Compiled, option for NDAESolve, 291
option for WriteSimulationData, 379
Complex frequency, 79
Complexity problem, 127
ComplexityEstimate, 129, 385
examples for, 386
ComplexValues, option for WriteSimulationData, 379
CompressEquations, option for
ApproximateMatrixEquation, 393
option for NDAESolve, 294
CompressMatrixEquation, 398
examples for, 399
CompressNonlinearEquations, 95, 165, 407
example application, 171
examples for, 409
options description for, 408
Conductance, 448
Connectivity field, format for referencing subcircuits, 51
Connectivity list, 34
Index
Control system description, Circuit, 179
Controlled sources, 35, 38
Controlling branch, of controlled sources, 38
Controlling currents, naming scheme for, 70
Controlling voltages, naming scheme for, 70
ControllingCurrentPrefix, option for
CircuitEquations, 70, 231
ControllingVoltagePrefix, option for
CircuitEquations, 70, 232
ConvertDataTo2D, 418
examples for, 419
ConvertImmittances, option for CircuitEquations, 74,
232
CrossMark, marker style for options PoleStyle and
ZeroStyle, 87, 354
Current, into pin, 93
naming scheme for brach currents, 70
naming scheme for controlling currents, 70
reference direction, 155
Current, 92
Current meter, 450
Current-controlled sources, connection, 38
CurrentSource, 451
DAE (differential-algebraic equations), 104
DAEObject, information on, Statistics, 277
manipulating, 255
showing contents, DisplayForm, 40
Data format, 280
Database, for subcircuits, 62
DataLabels, option for WriteSimulationData, 380
DC, mode option for source value specification, 185
option value for AnalysisMode, 230, 293
DC analysis, 290
DC-transfer analysis, 107, 111, 290
DCEquations, 251
examples for, 252
options description for, 251
Decibels, option value for MagnitudeDisplay, 333
DefaultACError, option value for ErrorFunction, 404
DefaultACSimulation, option value for
SimulationFunction, 404
DefaultDTError, option value for ErrorFunction, 402
DefaultDTSimulation, option value for
SimulationFunction, 402
Defaults, parameter for Model, 196
DefaultSelector, option for CircuitEquations, 232
option for ExpandSubcircuits, 223
Definition, parameter for Model, 48, 90, 92, 200
Degrees, option value for PhaseDisplay, 333, 343
DeleteElements, 257
examples for, 258
Demo notebooks, 11
DerivativePostfix, option for ACEquations, 249
Design points, applying, ApplyDesignPoint, 269
extracting, GetDesignPoint, 268
Index
Design points — GetDAEOptions
for matrix approximation, 135
generation of, 158
numerical reference values, 131
updating, UpdateDesignPoint, 272
DesignPoint, option for ApproximateDeterminant, 321
option for ApproximateMatrixEquation, 393
option for CircuitEquations, 233
option for LREigenpair, 314
option for MatchSymbols, 274
option for NDAESolve, 294
option for NoiseAnalysis, 287
option for WriteModel, 382
Device mismatch, 149
Differential gain, 148
Differential-algebraic equations (DAE), 104
Differentiator, 458
Diode, 460
example model implementation, 91
large-signal model, 460
small-signal model, 463
DirectedLogError, option value for ErrorFunction, 322
DisplayForm, 34, 40, 47
Dominant pole, 159
DT analysis, 107, 111, 290
DXF, 419
DXFGraphics, 419
examples for, 422
options description for, 421
Ebers-Moll transistor model, example for, 96
Eldo, option value for Simulator, 429
Element types, 37
Element values, 36
numerical values, 41
symbolic values, 41
Elements, admittance elements, 43
impedance elements, 43
ElementTypes, 37, 185
examples for, 185
ElementValues, option for CircuitEquations, 79, 233
EliminateVariables, option for
CompressNonlinearEquations, 408
Email address, 6
Environment variables, 433
Equations, compressing nonlinear equations,
CompressNonlinearEquations, 165, 407
extracting, GetEquations, 260
for behavioral models, 92
setting up, CircuitEquations, 227
solving nonlinear equations, 95
solving of, 72
Equations, parameter value for Definition, 90, 92
Error, specification for equations, MaxError, 135
specification for expressions, 133
Errorbound, option for CancelTerms, 412
495
ErrorFunction, AC mode option value for
NonlinearSetup, 404
DT mode option value for NonlinearSetup, 402
option for ApproximateDeterminant, 322
ESTA (extended sparse tableau analysis), 76
ExpandSubcircuits, 53, 221
examples for, 224
options description for, 222
Expansion, of subcircuits, 53
Exponential, option value for FrequencyScaling, 85,
332, 343, 348
option value for Sampling, 380
Exporting, behavioral models, WriteModel, 381
simulation data, WriteSimulationData, 378
ExpWave, 439
examples for, 440
Extended sparse tableau analysis (ESTA), 76
ExtendedTableau, option value for Formulation, 76, 234
Features, detailed description of new features, 26
overview of new features, 25
Fill-in pattern, 43
FillColors, option for DXFGraphics, 421
FindModel, 213
examples for, 213
FindModelParameters, 214
FindRoot, option value for NonlinearMethod, 297
Fixator, 456
Flat netlist, 53
Formulation, option for CircuitEquations, 74, 234
Four-terminal elements, 35
FourierPlot, 341
examples for, 341
Frequency, 186
Frequency-domain analysis, 142
FrequencyScaling, option for BodePlot, 332
option for NicholPlot, 86, 343
option for NyquistPlot, 85, 347
FrequencyVariable, option for ACAnalysis, 283
option for ACEquations, 249
option for ApproximateDeterminant, 322
option for CircuitEquations, 79, 234
option for LREigenpair, 314
option for NoiseAnalysis, 287
FullModels, option value for ModelLibrary, 427
GeneralizedEigensystem, 305
examples for, 305
option value for GEPSolver, 322
GeneralizedEigenvalues, 306
examples for, 306
GEPSolver, option for ApproximateDeterminant, 322
GEPSolverOptions, option for
ApproximateDeterminant, 322
GetDAEOptions, 265
examples for, 265
496
GetDesignPoint — KeepLocalModels
GetDesignPoint, 268
examples for, 269
GetElements, 258
examples for, 259
GetEquations, 260
examples for, 260
GetMatrix, 261
examples for, 262
GetParameters, 264
examples for, 264
GetRHS, 263
examples for, 263
GetVariables, 262
examples for, 262
Global, parameter specification, 196
parameter value for Scope, 62, 91, 193
Global subcircuit database, 62
examining content, GlobalSubcircuits, 62
Global subcircuits, 61
GlobalModelParameters, 215
examples for, 215
GlobalParameters, 189
GlobalSubcircuits, 62, 214
examples for, 215
Graphics, Bode plot, 81, 330
Nichol plot, 85, 342
Nyquist plot, 83, 346
root locus plot, 86, 352
transient waveform plot, 89, 106, 357
GraphicsSpacing, option for BodePlot, 332
Ground node, 34, 50, 181
missing, CircuitEquations, 235
Hierarchical circuit, keep local model data,
KeepLocalModels, 224
stop expansion, HoldModels, 223
History list, recording, 413
visualization of, ShowCancellations, 417
HoldModels, 223
option for ExpandSubcircuits, 223
Home page, 6
IgnoreMissingGround, option for CircuitEquations,
235
Impedance, input, 151
output, 151
Impedance, 448
option value for Pattern, 183
option value for Type, 43
Impedance elements, 43
Importing, netlists, ReadNetlist, 365
schematics, DXFGraphics, 419
simulation data, ReadSimulationData, 376
Independent sources, examples for value specification,
186
value specification, 185
Index
IndependentVariable, option for CircuitEquations,
235
Inductor, 449
Inductors, initial conditions for, 44
Info, 435
examples for, 435
Init files, 432
list of loaded init files, 435
Initial conditions, for capacitors and inductors, 44
transient analysis, 120
InitialCondition, option for netlist entries, 44, 183
InitialConditions, example application, 121, 122
option for CircuitEquations, 236
option for NDAESolve, 294
option for WriteModel, 382
parameter for Model, 203
InitialGuess, option for CircuitEquations, 229
option for DefaultDTSimulation, 403
option for LREigenpair, 315
option for NDAESolve, 118, 295
option for WriteModel, 382
parameter for Model, 202
Initialization procedure, 432
InitialLeftEigenvector, option value for
ProjectionVectors, 316, 325
InitialRightEigenvector, option value for
ProjectionVectors, 316, 325
InitialSolution, option for ApproximateDeterminant,
323
option for NDAESolve, 118, 295
InitialTime, option for CircuitEquations, 236
option for DCEquations, 251
Input impedance, 151
InputNoise, return value of NoiseAnalysis, 285
InstanceNameSeparator, global Analog Insydes option,
426
option for CircuitEquations, 71, 228
option for ExpandSubcircuits, 222
option for ReadNetlist, 366
option for ShortenSymbols, 277
Integrator, 458
Interfaces, 365
InterpolateResult, option for ACAnalysis, 283
InterpolationOrder, option for ACAnalysis, 283
option for NDAESolve, 295
option for NoiseAnalysis, 287
option for ReadSimulationData, 376
option for WriteSimulationData, 380
Jacobian matrix, 157
JFET, 485
large-signal model, 486
small-signal model, 490
Junction field-effect transistor (JFET), 485
KeepLocalModels, 224
Index
KeepLocalModels — Model
option for ExpandSubcircuits, 224
KeepPrefix, option for ReadNetlist, 367
KeepSymbolic, option for ApplyDesignPoint, 270
Laplace frequency, 79
Laplace-domain, AC equation setup, 230
Large-signal analysis, 290
introduction, 104
Large-signal model, BJT, 464
Diode, 460
JFET, 486
MOSFET, 474
LayerColor, option for DXFGraphics, 421
LeastMeanInfluence, option value for SortingMethod,
394
LeftEigenvector, option value for ProjectionVectors,
316, 325
Level, option for CancelTerms, 412
Levels, for nonlinear approximations, ShowLevel, 417
LevelSeparator, option for ReadNetlist, 367
Library, list contents, ListLibrary, 211
searching models, FindModel, 213
LibraryPath, global Analog Insydes option, 426
option for CircuitEquations, 229
option for ExpandSubcircuits, 222
option for FindModel, 213
option for FindModelParameters, 214
option for ListLibrary, 212
option for LoadModel, 216
option for LoadModelParameters, 218
option for ReadNetlist, 367
License, 434
examples for, 434
Linear, option value for FrequencyScaling, 85, 332,
343, 348
option value for MagnitudeDisplay, 82, 333
option value for Sampling, 380
Linearization, of nonlinear DAEObjects, 248
LinearRegionLimit, option for RootLocusPlot, 353
LinearRegionSize, option for RootLocusPlot, 353
LinearRegionStyle, option for RootLocusPlot, 353
LineDashingScale, option for DXFGraphics, 421
ListLibrary, 211
examples for, 212
Loading Analog Insydes, 11
Loading procedure, 432
LoadModel, 216
examples for, 217
LoadModelParameters, 217
LoadMSBG, 424
LoadQZ, 424
Local, parameter value for Scope, 62, 193
Local subcircuits, 61, 63
LogError, option value for ErrorFunction, 322
LPNP, 464
LREigenpair, 312
497
examples for, 316
option value for GEPSolver, 322
options description for, 314
MagnitudeDisplay, option for BodePlot, 82, 332
MAST, modeling language supported by WriteModel,
383
Matching, of devices, 149
precision, 150
MatchSymbols, 149, 274
example application, 149
examples for, 275
options description for, 274
Mathlink applications, MSBG.exe, 424
QZ.exe, 424
Matrix formulation, CircuitEquations, 234
MatrixEquation, option for CircuitEquations, 78, 229
MaxDelta, option for NDAESolve, 118, 296
MaxDivergentSteps, option for
ApproximateDeterminant, 323
MaxError, 135
AC mode option value for NonlinearSetup, 404
DT mode option value for NonlinearSetup, 402
named parameter for error specifications in design
points, 391
MaxIterations, 119
option for ApproximateDeterminant, 323
option for DefaultDTSimulation, 403
option for LREigenpair, 315
option for NDAESolve, 119, 296
MaxResidual, option for ApproximateDeterminant, 323
option for LREigenpair, 315
MaxShift, option for ApproximateDeterminant, 324
MaxSteps, 119
option for NDAESolve, 119, 296
MaxStepSize, option for NDAESolve, 119, 296
MinMAC, option for ApproximateDeterminant, 324
MinStepSize, option for NDAESolve, 119, 296
Mismatch of devices, 149
MNA (modified nodal analysis), 74
Model, behavioral models, 90
definition of, 190
exchanging device models, 187
expanding, 221
exporting, WriteModel, 381
global model library, 62
loading from library, LoadModel, 216
model library, 460
parameter cards, 189
referencing, 51, 187
variables of behavioral models, 93
Model, 47, 188, 190
argument description, 191
defining behavioral models, 90
examples for model implementation, 204
option for netlist entries, 45
498
Model library — Options
Model library, 460
defining global model library, 62
library management, 211
ModelLibrary, examples for, 427
global Analog Insydes option, 427
option for CircuitEquations, 230
option for ExpandSubcircuits, 222
option for FindModel, 213
option for FindModelParameters, 214
option for ListLibrary, 212
option for LoadModel, 216
option for LoadModelParameters, 218
ModelParameters, 189
ModeValues, option for DCEquations, 251
Modified nodal analysis (MNA), 74
ModifiedNodal, option value for Formulation, 74, 234
MOS field-effect transistor, 474
MOSFET, 474
advanced modeling, 157
large-signal model, 474
modeling examples, 143
small-signal model, 142, 479
MSBG.exe, 424
Name, parameter for Model, 48, 191
Name/selector specification, 49
Naming conventions, 70
NDAESolve, 105, 115, 290
examples for, 299
implemented algorithms, 115
options description for, 118, 293
Netlist, format of, 33, 178
importing, ReadNetlist, 365
Netlist, 178, 185
examples for, 178
Netlist entries, examples for, 181
extended format, 41
format, 34, 180
format for subcircuit references, 51, 60
option examples for, 184
options, 182
Netlist object, 178
information on, Statistics, 277
manipulating, 255
showing contents, DisplayForm, 34
New features in Analog Insydes Version 2, 25
Newsletter, 6
Newton-Raphson algorithm, 116
NewtonIteration, example application, 111
option value for NonlinearMethod, 297
Nichol diagram, 85, 342
NicholPlot, 85, 342
examples for, 344
options description for, 343
NMOS, 474
small-signal model, 142
Index
Node identifier, 34
Node voltages, naming scheme for, 70
Nodes, 34, 181, 194
NodeVoltagePrefix, option for CircuitEquations, 70,
236
Noise analysis, 285
NoiseAnalysis, 285
examples for, 288
options description for, 287
Nonlinear approximation, CancelTerms, 165, 410
example application for, 166
setting up, NonlinearSetup, 401
status of, NonlinearSettings, 415
term levels, ShowLevel, 417
Nonlinear equation solving, NDAESolve, 115
number of integration steps, MaxSteps, 119
number of Newton iterations, MaxIterations, 119
Nonlinear equations, compressing,
CompressNonlinearEquations, 165, 407
setting up, 94
solving, 95
NonlinearMethod, option for NDAESolve, 119, 296
NonlinearSettings, 415
examples for, 416
NonlinearSetup, 401
AC mode specifications for, 404
DT mode specifications for, 402
example application, 172
examples for, 405
Norator, 455
Normalize, option for GeneralizedEigensystem, 305
Notebook, option value for Protocol, 428
NPN, 464
Nullator, 455
Numerical solving, ACAnalysis, 282
NDAESolve, 290
NoiseAnalysis, 285
NumericalAccuracy, option for
ApproximateMatrixEquation, 393
Nyquist diagram, 83, 346
NyquistPlot, 83, 346
examples for, 348
options description for, 347
Online help system, 23
OP analysis, 290
OPAmp, 454
OpenCircuit, 449
Operating point, 118
Operating-point analysis, 290
OperatingPointPostfix, option for ACEquations, 249
Option inheritance, 430
examples for, 430
Optional, model port specification, 194
Options, global Analog Insydes options, 425
of netlist entries, 41
Index
Options — Referencing
retrieving DAEObject options, GetDAEOptions, 265
setting DAEObject options, SetDAEOptions, 266
setting default options on startup, 433
Options[AnalogInsydes], 425
examples for, 426
OTAmp, 454
Output impedance, 151
example application, 152
OutputNoise, return value of NoiseAnalysis, 285
OutputVariables, option for NDAESolve, 297
Palette, 23
Parameter sweep, 280
examples for, 281
Parameters, declaration for subcircuits, 57
implicit translation, 65
inspecting parameter database,
GlobalModelParameters, 215
loading from library, LoadModelParameters, 217
of subcircuits, 56
removing, RemoveModelParameters, 219
Parameters, parameter for Model, 57, 60, 195
Parametric, option for TransientPlot, 124, 359
Parametric analysis, 290
example circuit, 113
Pattern, option for netlist entries, 43, 183
PhaseDisplay, option for BodePlot, 82, 333
option for NicholPlot, 86, 343
PivotThreshold, option for
ApproximateMatrixEquation, 393
PlotPoints, option for BodePlot, 332
option for FourierPlot, 341
option for RootLocusPlot, 353
option for WriteSimulationData, 380
PlotRange, option for BodePlot, 82, 333
option for RootLocusPlot, 354
option for TransientPlot, 359
Plots, Bode plot, 81, 330
Nichol plot, 85, 342
Nyquist plot, 83, 346
root locus plot, 86, 352
transient waveform plot, 89, 106, 357
PlotStyle, option for RootLocusPlot, 354
PlusMark, marker style for options PoleStyle and
ZeroStyle, 87, 354
PMOS, 474
small-signal model, 142
PNP, 464
PointMark, marker style for options PoleStyle and
ZeroStyle, 87, 354
PointsPerDecade, option for ACAnalysis, 284
option for NoiseAnalysis, 287
Pole/zero analysis, 304
PolesAndZerosByQZ, 307
examples for, 307
PolesByQZ, 308
499
examples for, 309
PoleStyle, option for RootLocusPlot, 87
Ports, of behavioral models, 92
optional port nodes, 194
port currents, 92
port voltages, 92
Ports, parameter for Model, 48, 194
Prefix, customizing prefix settings, 70
for branch voltages and currents, 70
for controlling voltages and currents, 70
for node voltages, 70
specification, CircuitEquations, 231, 236
Prescaling, option for ApproximateDeterminant, 324
option for LREigenpair, 315
PrimaryDesignPoint, option value for SortingMethod,
394
ProjectionVectors, option for
ApproximateDeterminant, 325
option for LREigenpair, 315
Protocol, examples for, 429
global Analog Insydes option, 428
PSpice, option value for Simulator, 429
PulseWave, 441
examples for, 441
PWLWave, 442
examples for, 443
QuasiSingularity, option for
ApproximateMatrixEquation, 394
QZ.exe, 424
Radians, option value for PhaseDisplay, 333, 343
RandomFunction, option for NDAESolve, 297
Range, AC mode option value for NonlinearSetup, 404
DT mode option value for NonlinearSetup, 402
Ranking, least mean influence, 141
of terms, 140
Ranking, option for CancelTerms, 413
ReadNetlist, 365
example application, 145
examples for, 368
options description for, 366
supported features for AnalogInsydes, 369
supported features for Eldo, 371
supported features for PSpice, 369
supported features for Saber, 374
ReadSimulationData, 376
examples for, 378
options description for, 376
RecomputationThreshold, option for
ApproximateMatrixEquation, 394
RecordCancellations, option for CancelTerms, 413
Reference designator, 34, 71, 180
Reference directions, for currents and voltages, 35, 155
for port branches, 92
Referencing, behavioral models, 94
500
Referencing — SortingMethod
subcircuits, 51
ReleaseInfo, 433
examples for, 433
ReleaseNumber, 434
examples for, 434
RemoveModelParameters, 219
examples for, 219
RemoveSubcircuit, 63, 218
examples for, 219
RemoveSubcircuits, 63, 218
RenameNodes, 259
examples for, 259
Resistor, 447
ReturnDerivatives, option for NDAESolve, 297
ReturnSymbols, option for SimplifySamplePoints, 415
RightEigenvector, option value for
ProjectionVectors, 316, 325
Root locus diagram, 86, 352
animated, 89
of transfer functions without parameters, 88
RootLocusByQZ, 310
examples for, 311
RootLocusPlot, 86, 352
examples for, 355
options description for, 353
Saber, option value for Simulator, 429
SAG (simplification after generation), 132, 385
ApproximateTransferFunction, 387
combination with SBG, 138
Sampling, option for WriteSimulationData, 380
SBG (simplification before generation), 132, 134, 385
ApproximateMatrixEquation, 391
combination with SAG, 138
Schematic pictures, 419
Scope, of subcircuit definitions, 61
Scope, option for LoadModel, 216
option for LoadModelParameters, 218
parameter for Model, 62, 63, 193
SDG (simplification during generation), 132
SearchDirections, option for SimplifySamplePoints,
415
Searching, model parameters, FindModelParameters, 214
models, FindModel, 213
Selector, parameter for Model, 48, 192
SetDAEOptions, 266
examples for, 267
SFFMWave, 443
examples for, 444
ShortCircuit, 450
ShortenSymbols, 276
options for, 276
ShowCancellations, 417
ShowColors, option for DXFGraphics, 421
ShowLegend, option for BodePlot, 332
option for NicholPlot, 343
Index
option for NyquistPlot, 347
option for RootLocusPlot, 353
option for TransientPlot, 359
ShowLevel, 417
ShowSamplePoints, option for TransientPlot, 125, 359
ShowUnitCircle, option for NyquistPlot, 348
SignalSource, 457
Simplification, design point specification, 135
methods, 132
of expressions, 132
of linear circuits, 127, 385
of linear equations, ApproximateMatrixEquation,
134
of nonlinear circuits, 164, 401
of nonlinear equations, CancelTerms, 165, 410
Simplification after generation (SAG), 132, 385
combination with SBG, 138
Simplification before generation (SBG), 132, 134, 385
combination with SAG, 138
Simplification during generation (SDG), 132
SimplifiedModels, option value for ModelLibrary, 427
SimplifySamplePoints, 414
Simulation data, exporting, WriteSimulationData, 378
importing, ReadSimulationData, 376
SimulationFunction, AC mode option value for
NonlinearSetup, 404
DT mode option value for NonlinearSetup, 402
Simulator, global Analog Insydes option, 429
option for NDAESolve, 297
option for ReadNetlist, 367
option for ReadSimulationData, 377
option for WriteModel, 383
option for WriteSimulationData, 380
SimulatorExecutable, option for NDAESolve, 298
SingleDiagram, option for TransientPlot, 125, 359
SingularityTest, option for ApproximateDeterminant,
326
SingularityTestByLU, option value for
SingularityTest, 326
SingularityTestByQR, option value for
SingularityTest, 326
SinWave, 444
examples for, 445
Small-signal analysis, 282
Small-signal model, BJT, 472
Diode, 463
JFET, 490
MOSFET, 142, 479
Solve, 72, 253
examples for, 253
Solving, ACAnalysis, 282
NDAESolve, 290
NoiseAnalysis, 285
Solve, 253
SortingMethod, option for
ApproximateMatrixEquation, 140, 394
Index
Sources — Type tag
Sources, stimuli for two-port calculations, 155
Sparse tableau analysis (STA), 75
extended sparse tableau analysis (ESTA), 76
SparseTableau, option value for Formulation, 75, 234
SquareMark, marker style for options PoleStyle and
ZeroStyle, 87, 354
STA (sparse tableau analysis), 75
Stability analysis, 83
Starting Analog Insydes, 432
StartingStepSize, option for NDAESolve, 120, 298
Statistics, 277
examples for, 278
StatusLine, option value for Protocol, 428
StepSizeFactor, option for NDAESolve, 120, 298
Stimulus sources, 155
StripIndependentBlocks, option for
ApproximateMatrixEquation, 395
option for CompressMatrixEquation, 399
option for CompressNonlinearEquations, 408
Subcircuit, definition of, 190
expanding, 53, 221
generic selectors, 52
global and local definitions, 61
global database for, 62
inspecting subcircuit database, GlobalSubcircuits,
214
local overrides, 63
parameters, 56, 57
passing parameter values to, 60
redefining global subcircuits, 63
referencing, 51
removing, RemoveSubcircuits, 63, 218
translation of parameters, 65
Subcircuit, 47, 188, 210
option for netlist entries, 45
SweepParameters, Analog Insydes data format, 280
option for TransientPlot, 360
SweepPath, option for ACAnalysis, 284
option for NoiseAnalysis, 287
Symbolic, option for CircuitEquations, 237
option for ExpandSubcircuits, 222
option for netlist entries, 41, 183
option for source value specification, 185
option value for ElementValues, 79, 234
parameter for Model, 198
Symbolic approximation, complexity problem, 127
design point specification, 135
introduction, 127
methods, 132
of expressions, 132
of linear circuits, 127, 385
of linear equations, ApproximateMatrixEquation,
134
of nonlinear circuits, 401
of nonlinear equations, CancelTerms, 410
Symbolic expressions, approximation of, 129
501
Symbolic solving, Solve, 253
TestFrequency, option for ApproximateDeterminant,
326
TextSizing, option for DXFGraphics, 421
TextStyle, option for DXFGraphics, 421
ThicknessFunction, option for DXFGraphics, 422
Time, 186
Time variable, 105
Time-domain analysis, 104
TimeConstraint, option for Statistics, 278
TimeVariable, option for CircuitEquations, 238
Tolerance, option for ApproximateDeterminant, 327
option for LREigenpair, 316
option for MatchSymbols, 275
TraceNames, option for BodePlot, 333
option for NicholPlot, 344
option for NyquistPlot, 348
Transfer function, approximation,
ApproximateTransferFunction, 132
complexity, ComplexityEstimate, 129
example application, 162
plotting gain vs. phase, NicholPlot, 85, 342
plotting magnitude and phase, BodePlot, 81, 330
plotting real vs. imaginary part, NyquistPlot, 83,
346
TransferFunction, 459
Transient, mode option for source value specification,
185
option value for AnalysisMode, 230, 293
Transient analysis, 290
example circuit, 106, 108
initial conditions, 120
introduction, 104
procedure, 115
Transient waveforms, 89, 357
TransientPlot, 106, 357
examples for, 360
options description for, 124, 358
Transistor models, 142
advanced modeling, 157
Translation, of subcircuit parameters, 65
Translation, parameter for Model, 65, 197
TransmissionLine, 458
Trapezoidal rule, 115
Two-port parameters, 154
Two-port representation, admittance form, 154
cascade form, 154
hybrid form, 154
impedance form, 154
Two-terminal elements, 35
Type, option for netlist entries, 42, 184
Type tag, 34
for controlled sources, 38
overriding default type, 42
502
UnitCircleStyle — ZeroStyle
UnitCircleStyle, option for NyquistPlot, 348
Units, 36
UnloadMSBG, 424
UnloadQZ, 424
UnwrapPhase, option for BodePlot, 334
UnwrapTolerance, option for BodePlot, 334
UpdateDesignPoint, 272
examples for, 273
options description for, 272
UseExternals, global Analog Insydes option, 429
option for ApproximateMatrixEquation, 395
Value, option for CircuitEquations, 238
option for ExpandSubcircuits, 222
option for netlist entries, 41, 181
option for source value specification, 185
option value for ElementValues, 79, 234
Value field, 41
Value specification, for different analysis modes,
Netlist, 185
Variables, independent, CircuitEquations, 235
of behavioral model equations, 93
Variables, parameter for Model, 93, 199
VCCSource, 453
VCVSource, 453
Version, 434
examples for, 435
Voltage, naming scheme for branch voltages, 70
naming scheme for controlling voltages, 70
naming scheme for node voltages, 70
reference direction, 155
Voltage, 92
Voltage gain, 146
Voltage meter, 450
VoltageSource, 451
Web page, 6
WorkingPrecision, option for NDAESolve, 291
WriteModel, 381
examples for, 383
options description for, 382
WriteSimulationData, 378
examples for, 380
options description for, 379
Y-parameters, 155
Zero node, 34, 50
Zero-state response, 41
ZerosByQZ, 309
examples for, 310
ZeroStyle, option for RootLocusPlot, 87, 354
Index