EQUIVALENT CIRCUIT MODELLING OF BIPOLAR TRANSISTORS
by
Monica
A thesis
for
submitted
the degree
Dowson
to the University
of Doctor
March 1982
of Leicester
of Philosophy
ACKNOWLEDGMENTS
The author gratefully
given
with
by her
supervisor,
and the provision
of Philips
who provided
Research
Dr.
acknowledges the advice and encouragement
O. P. D. Cutteridge;
of data
Laboratories;
encouragement,
by Mr.
P. J.
the helpful
Rankin
and Mr.
and the understanding
food and drink
at all
the
discussions
K. W. Moulding
of her husband,
appropriate
times.
EQUIVALENT CIRCUIT
MODELLING OF BIPOLAR TRANSISTORS
Monica
Dowson
ABSTRACT
Existing
transistors
equivalent
circuit
are
models of bipolar
for the evaluation
reviewed together
with techniques
values
of suitable
A method enabling the optimisation
of the model elements.
of the element
values of any particular
model in order to match the measured S parameThis method uses
ters of a device that is to be modelled is described.
function
Gauss Newton algorithm
a modified
to minimise an objective
defined as the sum of the squares of the weighted errors between the reS parameters and those of the model.
quired
for the
Details
algorithm
are then given of a new modelling
development of accurate equivalent
circuit
models which was developed
from this original
The new modelling
algorithm
optimisation
method.
be
device
S
to
the
modelled'
of
requires
some
parameter measurements
together
with a potentially
over an appropriate
range of frequencies,
The initial
and then,
model elements are optimised
suitable
model.
involving
the addition
or
if necessary,
topological
changes,
suitable
having
the.
deletion
both
model
a
of
elements and nodes, are made until
is obtained.
required
accuracy or complexity
circuit
equivalent
A number of examples are given of small-signal
These
developed
the
bipolar
algorithm.
transistors
using
models of
lGHz
frequencies
to
up
transistors
at
particular
were operating
.A
in
the
further
the
is
algorithm
modelling
the
of
use
example
given of
bipolar
two
similar
development of a bias dependent small-signal
model of
S
Additional
para2GHz
frequencies
to
transistors
up
operating
at
.
that
demonstrate
is
to
data
for
the same two transistors
also used
meter
for the development of non-linear
the algorithm
can be used successfully
models.
CONTENTS
Page
CHAPTER 1
INTRODUCTION
1
1.1
Background
2
1.2
Aims of the Research
3
1.3
Achievements
4
1.4
Computing
CHAPTER 2
Facilities
The Bipolar
2.2
Hybrid
Pi Model
2.2.1.
Basic
2.2.2.
Modified
Ebers Moll
Pi Model
Hybrid
13
Pi Model
14
Model
2.3.2.
Extended
2.4
Other
Non-Linear
2.5
Choice
16
Ebers Moll
Model
Ebers Moll
EMI
-
16
Model - EM2
18
20
Models
of Models
21
TRANSISTOR PARAMETERMEASUREMENTS
AND ANALYSIS OF MODELS
23
y Parameters
25
3.1.1.
Nodal Analysis
26
3.1.2.
Generation
of the Network
from the Nodal Admittance
3.1.3.
3.2
10
13
Hybrid
Basic
3.1
7
Transistor
2.3.1.
CHAPTER 3:
5
MODELLING THE BIPOLAR TRANSISTOR
2.1
2.3
Used
Calculation
Condensation
3.1.4.
Accuracy
3.1.5.
Further
Polynomial
Matrix
of the y Parameters
Using
Coefficients
28
Pivotal
30
of Calculations
Considerations
32
on the
s Parameters
Choice
of Algorithm
39
39
3.2.1.
Measurement
of the s Parameters
41
3.2.2.
Calculation
of the
s Parameters
42
43
NUMERICAL OPTIMISATION
CHAPTER4:
4.2
Values
4,1.1.
Algorithms
Using
Function
4.1.2.
Algorithms
Using
Derivatives
4.1.3.
Designed
Algorithms
Squares Functions
Some Details
CHAPTER5
46
Methods
Available
4.1
for
46
Only
47
Sum of
Hinimising
50
of the Optimisation
Chosen
Algorithm
54
DEVELOPMENTOF THE MODELLING ALGORITHM
5.1
Details
5.2
Construction
Values
First
of the
to Optimise
of an Algorithm
54
Exercise
Modelling
51
Model Element
57
5.2.1.
Construction
5.2.2.
Generation
of the Jacobian
5.2.3.
Application
of Constraints
5.2.4.
Exit
of the Error
Function
57
Matrix
59
60
61
Criteria
5.3
Example of Results
61
S. 4
Improvement
63
5.5
Example of Results
5.6
Description
5.7
of Models
65
of the
Final
Modelling
5.6.1.
Stage
1-
Obtaining
5.6.2.
Stage
2-
Element
5.6.3.
Stage
3-
Node Addition
Checking
the
Final
the First
Local
Addition
Modelling
Common Collector
6.2
Common Emitter
6.3
6.4
.
Configuration
Configuration
6.2.1.
First
6.2.2.
Second Attempt
Attempt
Minimum
71
76
81
Algorithm
CHAPTER 6 : MODELLING A VERTICAL N-P-N TRANSISTOR
6.1
70
Algorithm
87
92
96
100
100
102
Common Base Configuration
107
General
112
Model
CHAPTER 7:
First
6.4.2.
Second Attempt
6.4.3.
Suggested
Optimisation
7.2
Examination
7.3
Resultant
7.4
Models
APPENDIX 1:
APPENDIX 2:
REFERENCES
Attempt
116
Alternative
Approaches
123
A BIAS DEPENDENTMODEL FOR TWO SIMILAR BIPOLAR TRANSISTORS 125
7.1
CHAPTER 8:
112
6.4.1.
of the
Bias
for
127
of the Model
Bias
Dependent
High Emitter
Dependent
Elements
133
Model
141
Currents
147
'ORK
CONCLUSIONSAND SUGGESTIONSFOR FURTHERV,
151
S PARMIETER DATA PROVIDED BY PHILIPS RESEARCH
LABORATORIES
156
RELEVANT PUBLICATIONS BY THE AUTHOR
176
194
I
CHAPTER 1
INTRODUCTION
Since
sought
the invention
of the
simulate
the behaviour
which
by the author
bipolar
modelling
are discussed
Models
in Chapter
at ease with
a circuit
that
the
reflects
analysis
packages
sistors
diagram
rather
devices
from Philips
ested in obtaining
the electronics
for
small-signal
equivalent
industry
direction
generation
of
the techniques
linear
linear)
models
of bipolar
to other
here,
was a bias
dependent
non-linear
operation
of a bipolar
models
of tran-
to be analysed.
circuits
for which they were interIn addition
that
to proving
of transistors,
this
The requirement
was
designed
transistors
0.5 MHz and 2 GHz.
to
MacleanI
has
type of problem.
described
applicable
which
between
in this
are equally
are many circuit
in the modelling
for
to the
data on a number of bipolar
models.
models
more
a model
modifications
for the research.
in ranges
also expressed an interest
Although
circuit
to feel
prefer
circuit
larger
in receiving
is interested
(implying
at frequencies
operate
enabling
use equivalent
in
circuit
tend
designers
and there
to incorporate
rela-
of numbers or mathematical
circuit
the model;
that
designers
a set
Research Laboratories
a definite
also provided
than
of the device
construction
The author has been fortunate
transistors
of which
of mathematical
Equivalent
circuit
and integrated
now available
and other
terms
circuits.
reasons:
to be mapped. onto
construction
in
can be either
several
transistor
relationships;
with
entirely
and applications
construction
form of equivalent
for
The work undertaken
2.
of a transistor
models are popular
of the device.
has been concerned
the
transistors,
or in the
tionships
here,
and described
1948, models have been
in
transistor
here were originally
modelling
model,
the
transistors,
problems.
the
transistor.
author
devised for the
techniques
In the
obtained
final
models
Mathews and Ajose2
involved
problem
for
the
have
2
to model both
technique
used one optimisation
and field
bipolar
effect
transistors.
1.1.
Background
In recent
years,
in network synthesis
at Leicester
network synthesis
7
Hegazi
3,4
by di Mambro . Wrights,
more concerned
has been of direct
the optimisation
with
benefit
The techniques
9,
Dowson Henderson"
whilst
Theses on passive
and optimisation.
have been presented
8,
and Savage
ted theses
interest
there has been a great deal of research
to the
Krzeczkowski6.,
have presen-
and Dimmer"
This
processes.
expertise
author.
used for the analysis
particularly
networks,
of passive
12
3,4
conthose developed by di Mambro
were
and di Mambro and Cutteridge
,
sidered
however,
the
for the author's
only
used for
also
author,
coefficient
used was not
also
to the local
made some contributions,
involved
difficult
optimisation
problems
with
of the optimisation
developed
to
suit
author
the problem
problem
and the
It
algorithm
type,
This
the
that
for
author9
optimisation
and was designed
the
to solve
type
of
the Gauss Newton section,
was an ideal
techniques
there
available.
to which
was discovered
one part,
by the author.
of networks,
importance.
of
programs
studied.
techniques
by Henderson",
were of great
by the
being
in the analysis
the use of a two-part
to be solved
adaptations
the problems
The technique
synthesis
network
earlier
by
written
routine
3, was developed.
experience
problems.
in Chapter 3; these were,
analysis
knowledge of optimisation
the technique
technique
for
suitable
was also a considerable
In particular
these
all
the
until
in Chapter
which
considered
In addition
time
a short
described
matching
as described
application
technique.
used are given
Details
in Chapter
4.
3
1.2.
Aims of the
As already
Laboratories
circuit
The first
set
an integrated
difficult
was that
was to develop
research
and if
verified,
be produced
could
original
If
models.
to optimise
techniques
the
topology
the required
is
this
These aims necessitated
techniques
which
were then
The development
strategy.
is
transistor
from this
described
experiment
is
This new technique
by Philips
circuit
each of the
Finally,
still
in
a set
not
is
of an improved
Chapter
5
-
time a vertical
the
then
it
the
be
could
These improved
of the
values
becomes necessary
element
values
obtained.
analysis
and optimisation
model for
the
and the modelling
modelling
lateral
p-n-p
technique
devised
5.6.
on the second set of data provided
This data was for
another
n-p-n transistor,
common base and common collector
of data'5'16
aim of this
to form an overall
in Section
14
correlation
of devices
with
of
together
combined
Laboratories
common.
adequate
Thus,
adequate
together
was then tested
common emitter,
give
quite
the element
the development
detailed
this
is
be produced.
models
or complexity
Research Laboratories
transistor,
do not
of the models
accuracy
Research
2.5,
by optimising
merely
model of
stage
be developed.
whereby models
improved
necessary,
a first
of the devices.
characteristics
was
to us [Philips
interest
in Section
models of devices
standard
This
as "a notoriously
by Philips
model of the device
stages.
transistor.
p-n-p
to the data,
The requirement
as discussed
in obtaining
in three
and was described13
In addition
requirement,
the measured
a lateral
and "one of current
to model"
a more accurate
Frequently,
until
was for
was provided.
This
models
of data13
transistor
Research
were interested
they
which
The data was provided
Laboratories]".
the device
for
by Philips
was provided
models.
circuit
device
Research
some data
mentioned,
on some devices
equivalent
with
Research
was provided
for
two similar
integrated
with data for
configurations.
n-p-n
tran-
4
at different
sistors
two voltages
bias
had published
a model for
devices.
of these
taining
as few bias
in Chapter
bias
elements
models
was available
the first
the
type
Slatter17
second type.
improve
as possible
for those elements.
bias
a wider
range
the use
with
consistent
This
aspects
model by ob-
on this
covering
at
and at two
based on physical
range
aim was to
dependent
dependent
of simple 'relationships
in detail
a limited
bias
for
currents
The author's
more accurate
for
currents
and at seven different
voltages
using
different
and at six
Measured data
conditions.
described
is
problem
7.
Achievements
As will
valent
be seen later
the author has produced a method by which equi-
models of bipolar
circuit
can be verified
transistors
of the model can be modified
In order to achieve this,
to analyse
a network
an optimisation
Using
model for
this
accuracy or
of its
p-n-p
transistor
which
model were 0.1 dB in
r. m. s.
(root
mean square)
errors
nodes with
12
of
in the
and 1.2*
0.04 dB and 0.4*.
elements
using
data
of these
in the
operated
the modulus
with
strategy.
The first
errors
this
and combined
a new modelling
The maximum absolute
final
6
parameters
Laboratories.
of the
of
with
s
a number of models have been produced
Research
10 MHz.
the author has produced a computer program
together
technique
a lateral
0.5 MHz to
in terms
routine
by Philips
consisted
necessary,
is obtained.
complexity
supplied
Then..' if
the required
until
a set
they are inadequate
of s parameter measurements of the devices and if
.
the element values of the* models can be optimised.
the topology
against
and included
was a
frequency
s
range
parameters
in the phase with
This
final
one current
model
source
only.
The next
device
modelled
was a vertical
n-p-n
transistor
model which
5
operated
in the
produced
for
frequency
each of the common emitter,
and r. m. s. errors
0.3 dB
of
became a task
of tremendous
r. m. s. errors
of
this
type
1.3 dB
is possible,
of task
extremely
transistors
dependent
a bias
for
of the devices
both
in the range
currents
8 mA for
the
dependent
the bias
Thus,
these
tests
author
is
electronic
1.4.
developed
this
have been with
of the opinion
Computing
at the University
easily
is
s
that
to
2 GHz
for
.
linear
oper-
emitter
0.5 mA to
and
and good agreement
for
the techniques
models
of the transistors.
has been extensively
data
the
with
Accurate
was achieved.
operation
parameter
bipolar
The same
the
and for
models.
for
could
tested
bipolar
and although
the
transistors,
be applied
to many
problems.
Facilities-Used.
The majority
Manchester
technique
with
Simple-expressions
two transistors.
the non-linear
modelling
two similar
3V
that
accurate
the first
4 mA for
of the transistors
for
modelling
0.5 V and
of
computer
for
and was valid
giving
believes
sufficiently
0.1 GHz
of
were developed
elements
measured characteristics
were also
to generate
transistors
second of the
fast
a large,
requires
0.5 mA to
This
successful
the author
model was produced
at voltages
was made to produce
partially
Whilst
.
at frequencies
operating
model was produced
ation
it
code in order
efficient
Finally,
12.9*
4"
and
any of the configurations.
and was only
volume
and
An attempt
for
model applicable
general
1.5-dB
of
errors
models were
and common base
common collector
3*
and
Separate
1 GHz.
These gave maximum absolute
configurations.
a single
0.1 GHz to
range
of the computing
of Leicester
(UMRCC).
transportable
the word length
work has been done on the
and the CDC 7600 at the University
These machines
between
which
is
CDC Cyber 73
the two.
60 bits.
are compatible
and programs
The main feature
This
means that
of these
high
of
are
machines
precision
can
6
be easily
'powerful'
Leicester
The UMRCC CDC 7600 is
obtained.
the Leicester
than
time
limit
of
the UMRCCfacilities
s
The p-n-p
on the Cyber 73 but
of c. p. u.
the n-p-n
the bias
model in Chapter
was generally
the GHOST library
with
were produced
layout
circuit
library.
the
output
on the University
program
written
Flow charts
interactive
All
of graphical
program
the computer
an encumbrance,
depends
complexity
of the
in Chapter
5 was
in
of the computing
for
73.
Cyber
done
the
on
was
obtained
subroutines.
by the author
produced
the Cyber 73 together
using
The model circuit
PDP-11/44
using
were written
diagrams
an interactive
and which used the GINO
using
the PDP-11/44,
FLOW18.
programs
where the
models described
the majority
of Leicester
were also
and the
more
problem
the time
transistor
the CDC 7600, whilst
Graphical
proved
model described
transistor
7
jobs
larger
time
available
Chapter 6 required
dependent
for
In general,
data
parameter
involved.
produced
33 minutes
were used.
on the amount of
models
Cyber 73, thus
12 times
approximately
in FORTRANIV.
using
7
CHAPTER2
MODELLING THE BIPOLAR TRANSISTOR
A special
leading
stages
Shocklaythat
they had invented
was a point
junction
in
device.
contact
their
Since
behaviour
will
hand,
equivalent
interested
less
applications.
beginning
cated
GetreU20 lists
with
models
the
starts
are
the effects
of
can be predicted.
with
the
accuracy
in the model have
can be used in conjunction
in the design and analysis
to assist
without
a wide
device.
the time and expense of manua number of such uses of models,
the performance of an integrated
This chapter
and its
circuits
need
models which
that
the elements
whether
analyses of waveforms and frequency
the parasitics
without
the device
of
over
require
in order
models of transistors
circuit
the circuits.
to predicting
of the transistor
in the manufactured
or integrated
from performing
Prize
their
designers
are more concerned
with computer aided design techniques
facturing
and device
of the device
designers
describing
models
generally
or geometry
circuit
of the model and are
of discrete
transistor
were awarded the Nobel
inventors
designers
structure
changes in the construction
Equivalent
went on to develop the
Both circuit
Device
based on the physical
a physical
and
Shockleythen
the performance
simulate
of operation.
On the other
Brattain
This first
of the transistor,
have been sought.
models that
1948 by Bardeen,
work.
invention
the
in
the
the transistor.
These three
transistor.
1956 for
range
to the announcement
describes
in Electronics19
on the transistor
report
responses of circuits
circuit
at high frequencies
a breadboard would introduce.
with
a brief
description
Then some of the popular
classical
hybrid
such as the Ebers Moll
pi
of the bipolar
models
model and going
model and its
variants.
transistor
are described
on to more compli-
8
2.1.
The Bipolar
A bipolar
Transistor
transistor
may be either
in Figure
given
defines
2.1 together
with
Active
mode
the
-
2.
Saturated
mode
both
-
the emitter
are forward
3.
Cutoff
both
mode
Inverse
the
mode
mode the
active
of the active
schematic
circuit
by using
a single
sisting
battery
R3
of resistors
resistance
and
Navon2l
signal
rapid
frequencies
is
the
that
load
In the diagram
R,
amplifier
as stated
achieved
network
is
the
con-
source
fast,
of
the order
in
of
signal
In addition
amplifier
10-
9
can also
the most used
it
at
common base
amplifiers
(22),
the
seconds,
amplification
to the
is probably
Reference
such as this,
circuit
and common collector
The common emitter
is
where the biasing
divider
gigahertz.
common emitter
2.3 shows an
Figure
2.2.
the voltage
can be used to achieve
response
because,
(22)
transistor
an n-p-n
resistance.
in a linear
can be quite
up to several
configuration,
practice
states
transmission
and this
used.
R2
in Figure
with
R4
for
arrangements
in Reference
and
A
may be used as an amplifier.
given
together
and
is reverse biased.
biasing
is
described
amplifier
are
biased
forward
is
junction
transistor
in common base configuration
junctions
and collector
collector
region
junctions
and collector
biased.
the emitter
In the
and the
biased.
the emitter
reverse
4.
are
biased.
reverse
collector
biased
forward
is
types
as follows.
transistor
junction
emitter
and
Navon2l
symbols.
circuit
of the bipolar
the modes of operation
1.
their
of both
diagrams
Block
type.
or p-n-p
an n-p-n
base and collector
of an emitter,
consists
is
be
in
9
EC
emitter
IpIn
L
71
pI
IB
Ec
npn
0
collector emitter
'o"
coltector
B
base
base
EC
EC
B
p-n-p
Figure
2.1.
type
n-p-n
type
The Structure
and Circuit
Symbols for
r
Bi polar
Transistors
T_
L
B
Ve
Figure
2.2.
Typical
Bias Arran
the Active Rer,,ion
Ib
ent
for
vc
an N-P-N Transistor
Biased
in
10
V
it
Vout
Vin
0-
Figure
2.3.
%%ý
A Simple
0-
wo
The
u
Ttme
Common Base Amplifier
Ic
F
out
C
Vid
ýý, f
F]
1ime
Figure
2.4.
An N-P-N Transistor
4
Vout
Used in Switching
Mode
Time
11
the only
In the
saturated
in common emitter
connected
The two parts
and a small
normally
this
V
in
but with the collector
than that
than that
can cause the
to the conducting
RL*
defined
which
terminals
and collector
Navon2l
this
causes
thus
state
states
of the
as the ratio
in
that
collector
can be
increase,
designed then the operation
described
of
with the active
interchanged.
However, in practice
doped and the collector
cross-sectional
Thus the p-n-p transistor
mode
area
should
as a p+-n-p type.
diagram of the construction
of a planar
n-p-n trans-
given in Figure 2.1 is now shown in Figure 2.5 . This diagram
aid the identification
described
load
of the emitter.
A more accurate
will
shown in Figure
switch
mode would be identical
and emitter
is more heavily
be more accurately
istor
the
was symmetrically
in the inverse
the transistor
is greater
emitter
to convert
to the base current
the transistor
the emitter
simple
to the base terminal
gain,
is normally
50.
than
If
the
are the
across
the current
ificrease
higher
V
out
a voltage
configuration
current
mode as in
transistor
can be used
transistor
mode, the transistor
switching
applied
'
non-conducting
producing
modes the bipolar
of the switch
signal,
and power gain.
voltage
current,
and cutoff
When used in
as a switch.
2.4.
to provide
configuration
of some of the physical
in the following
of the integrated
to the modelling
of this
sections
circuit
chapter
elements in the models
and is also relevant
described
transistors
in later
chapters.
A full
description
in the construction
Briefly,
of these devices
an integrated
Figure 2-5, is built
layer
of the theory
circuit
up layer
by layer
collector
the same single
on a p-type
collector
substrate.
resistance
structure
as the substrate.
involved
and Howard23.
such as that
is grown by means of epitaxial
crystal
processes
is given by Hamilton
n-p-n transistor
serves to reduce the series
The n-type
and manufacturing
shown in
The n+ buried
of the transistor.
growth and is part
The p+ channels
of
12
i
n+ emittercontact
p base contacts
n+ coltectorcontact
P+substrate contact
isolation
channel
p+
AP (base) I/
Ln/epitaxiat
layer
p
collector)
n+
buried[aye
ýýL4
r
substrate
Figure
2.5
Construction
of an Integrated
Circuit
N-P-N Transistor
13
the device
isolate
base and n-type
followed
2.2.
from the rest
?i
The hybrid
Its
range
Basic
Hybrid
model is the most popular
24
it
sistor,
is possible
its
Pi
linear
is
model of a bipolar
its
simplicity,
and the ease of parameter
accuracy
over
a
determination.
Model
and Brown
the model with
in
lies
seven element,
by Chua and Lin
and this
process
of the contacts.
appeal
wide frequency
The basic
The p-type
circuit.
Model
pi
transistor.
paring
are formed by a diffusion
emitter
by metalisation
Hybrid
integrated
of the
25.126
Figure
hybrid
common emitter
and is
2.5
which
shown here
model is
pi
in Figure 2.6.
showed the construction
to see the relevance
described
By comof a tran-
of most of the elements.
Wc
b
C
rce
e
Figure
2.6
Common Emitter
Ilybrid
Pi
Transistor
Model
14
Of particular
spreading
interest
resistance,
Searle27 describe
how this
majority
flowing
carriers
the
into
5Q
the
active
100 Q.
and
Gray and
of base region
between the emitter
C
ble
is
capacitances
a
to allow for the transverse
region
capacitors
for
constant
caused by the drift
of
layer
space charge
is practically
between
term the base
Chua and Lin24
element is inserted
The inclusion
collector.
which
this
typically
drops in the base region
voltage
for
r bbt"
that
stating
and is
transistor
given
is
C
b1c
and
described
also
and
to account
by Gray and
Searle27.
Brown26 describes
circuit
admittance
how measurements
particular
linear
model for
2.2.2.
Modified
frequencies
Brayden28
collector-base
of
conductance
Pi
Hybrid
also
while
increases,
phase shift
.A
gm and,
as at higher
around
agreed
them,
that
the interlead
than for
hybrid
is
term
these
C
be
capacitance.
capacitances
taking
split,
is
model in which
pi
also
frequencies
interlead
The new element
some interlead
model is a good
deteriorates.
of the transistor
capacitance
bI
pi
the agreement between the basic hybrid
depletion
generally
include
smaller
any
Model
a modified
significant.
also
the basic hybrid
describes
by the capacitance
is
selected
up to a few megahertz.
model and the performance
instead
with other
the values of the elements in the model for
agreed that
As the frequency
pi
y
case.
is generally
It
(short-
parameters
can be used together
parameters)
measurements to determine
of the
rbIc
of it
part
included
in the mutual
and
r
and case capacitances
by Brayden2B
Gray and Searle27
in an integrated
the same component in discrete
b
to
are swamped
ce
from the model.
are eliminated
inserted
the
circuit
form,
it
are also
will
state
thus
that
component are
the junction
15
bc
Cce
Cb
Figure
2.7
Modified
Hybrid_
Pi
Model
16
capacitances
The above thus
Figure
2.7 which
higher
than
for.
be accounted
must still
leads
hybrid
to the modified
25,26
Brown
states
should
if
300 MHz, especially
model shown in
pi
be accurate'at
is
inductance
lead
some emitter
frequencies
also
included.
As with
hybrid
the basic
pi
model,
which must be made to determine
measurements
describes
Brown26 also
the
the
for
values
element
this
model.
2.3.
Ebers Moll
Model
The Ebers Moll
for bipolar
M01127 in
1954 and there
though
the work in
linear
models,
Moll
the
this
published
involved
the non-linear
important
mainly
to be ignored.
GetreU20 is used to differentiate
original
here because,
with
a transistor
modelling
by Ebers and
on this
included
was concerned
problem
Also,
region.
model are too
thesis
final
model is
This
model
non-linear
have been numerous variations
then.
since
the most popular
This model was first
transistors.
model published
non-linear
model is probably
even
small-signal
which
problem
entered
the
and the Ebers
Here the notation
used by
between the stages in the development of
the models.
2.3.1.
Basic
A diagram
basic
model is
in all
Ebers Moll
of the basic
IF
describes
this
Ebers Moll
a d. c. model that
modes of its
currents
Model - EMI
and
operat -ion.
IR'
the
describes
through
given
the behaviour
The model is
currents
model as follows.
is
model,
described
the diodes.
in Figure
2.8.
The
of the
transistor
by its
reference
GetreU20
17
I
C
b
e
Figure
2.8
Basic
Ebers Moll
The diodes represent
I
of the transistor.
base-emitter
junction
Model
EM1
-
the base-emitter
junctions
and base-collector
is the current that would flow across the
F*
collector
Vbe ' if the
region were replaced by an ohmic contact without disturbing
base.
I
es
given value of
for a given base-emitter
is the saturation
current
voltage
of this
junction.
the
Thus, for
V
be
q Vbe
(2.1)
kT
es
where
q=
electron
charge
Boltzmann's
constant
A6soluke. +etr,?
er&ture
Similarly,
disturbing
if
the emitter
the base and
base junction
Ics
were replaced
is
the
saturation
by an ohmic contact
current
of the
without
collector-
a
1
18
IR=I
Coupling
base region
kT
e-
cs
(2.2)
-11
the junctions
between
and is modelled
of
is provided
the transistor
dependent
by the two current
current
by the
sources.
Thus
(2.3)
Ic=FIF-I
where
aF
is
the
and is
transistor
of a common emitter
large
signal
forward
current
large
signal
to the
related
by the
transistor
gain
of a common base
forward
current
transistor
current
Ib ý' (1 -F
(i
The derivation
)I
F+
signal
reverse
signal
2.3.2.
hybrid
pi
Extended
current
gain of a commonbase
(2.4)
(2.6)
of these equations
active
(2. S)
ol
R
1- aR
GetreU20 shows how the linearised
in the forward
can be written
)I
-aR
R
to equation
and similarly
R
region
is given by Ebers and M01129.
version
model for operation
of this
reduces to the d. c. portion
of the small-
model.
Ebers Moll
The EM1 model is
a basic
Model - EM2
d. c. model of
a bipolar
transistor.
next stage in the development of the model is to take into
charge storage
F)
(2.4)
the base terminal
aR is the large
where
(a
expression
aF
F1aF
Similarly
gain
effects
The
account the
and to improve the d. c. representation.
19
In
currents
r
tors
tor
are added to improve
eel
its
base,
storage effects
(i)
are modelled by
for incremental
two diffusion
circuit
the substrate
,C
Dc
capacitor
,c
sub -'
rc-
of the transisThe charge
capacitors
and
CJc
,
junction
CDe
-,
which model
in the transistor,
and
in integrated
which is important
C
-I-
-
rcc
sub
Tc
S
III
b
r bb' ý
CDC
T
3T,
J12
Cj
C
Jljý
1
CDe-T
T,
C
el
e
2.9
Ebers Moll
(transformed)
i
Je
ree/
Figure
I EM1 model
113
Model - EM2
i
and
in the transistor's
transistors.
r--
cl,
These resis-
changes in the associated
capacitors
the
011 model a
r bb"
region
charges stored
with the mobile carriers
the charge associated
(iii)
active
respectively.
fixed
for
Expressions
resistors
two junction
in
shown here
characterisation.
of the
and emitter
collector
layers
(ii)
voltages,
and
Around this
The three
the d. c.
incremental
C
the
model
which
Je '
space-charge
by Getreu.
the ohmic resistance
represent
to
are derived
have been added.
number of elements
and
13
12 and
I,,,
by GetreU20
EMILmodel has been transformed.
2.9, the basic
Figure
fully
described
EM2 model
the
20
As with
to the
2.4.
hybrid
linear
Other
model reduces
region.
active
to the Ebers Moll models described
a further
and a junction
capacitor
tion
gain
S
in Section
model includes
a further
and models
the base-width
modulation
with
current
2.3,
of the Ebers Moll
refinement
This
the EM3 model.
of current
this
Models
goes on to describe
which he calls
that
forward
model in the
pi
Non-Linear
In addition
GetreU20
GetreU20demonstrates
the E?I1 model,
model
two diodes
and varia-
and some temperature
and voltage,
effects.
Another
model described
is almost entirely
states
by GetreU20
is
the Gummel Poon model which
basically
still
he
improvements to the d. c. charact-
concerned with
of the EM3 model but
eristics
is
to the 013
equivalent
model.
Other popular
to the physics
closely
adhering
of
parameters
the Linvill30
models*include
namely
elements
Hamilton
Ebers Moll,
degree
a list
Linvill
and Beaufoy
of references
presents
a total
device modelling
transistors.
Sparkes
and give
are many variations
controlled
comparison
similar
current
models
results
although
not all
are all
for
model as lying
generators.
conclude
similar
transient
described
concerned with high-injection
SparkeS31
two new network
of models
on the models
of 486 references
this
only
with
the new network
and the Beaufoy
He describes
models
and charge
32
in their
et al.
of approximation
There
and Linvill
storance
introduces
and combinance,
model which is favoured by Brown26.
between the Ebers Moll
device,
of the
diffusance
storance,
lumped mpdel which, while
here,
effects
in his bibliography
that
the
in their
problems.
Ruch33 gives
and Agajanian34
on semiconductor
of these are concerned with
bipolar
21
2.5.
Choice
of Models
described
For each of the models
measurements
enabling
For example,
Orlik35
made to determine
the
element
for
first
description
is
It
fit
model is
the bipolar
of physical
effects
have been incorporated
but while
time and effort
modelling,
an opinion
Similarly,
there
if
is
that
Thus,
could
the
complexity
knowledge
produce
acceptable
the simplest
existing
should be used, Ruch33
even a complex model has
models.
is
and one which
as to what action
aspect of
often
overlooked.
be taken
should
inadequate.
aim was to present
and if
and accuracy
of the
that
is an important
by Ruch33,
no guidance
A number
saves the wasted time and
model verification
virtually
author's
be verified
it
that
a
models with varying
spent on modelling
confirmed
the model chosen is
the
is not
there
transistor.
many models or of using unsuitable
The author feels
Probably
2.4 that
to be undertaken
in particular
expense of evaluating
into
Getreu2O states
model adequate for the analysis
advantages,
the
of the
the values
to a set of measured data.
model for
accepted single
definite
to optimise
in Section
universally
suggests that
model may be determethod of determining
Another
from the discussion
degrees of complexity
can be
that
measurements
method was by Bassett37.
of this
obvious
is published.
values
of a similar
data.
any particular
the best
to give
elements
element
one set of
of a Gummel Poon model while
values
how the parameters
from fabrication
at least
chapter,
a set of electrical
the parameter
mined directly
values
of the
evaluation
describes
describes
Roulston36
in this
better
necessary,
It
required.
further
models
effects
could
that
here
a method by which
models
was also
could
be extended.
be evolved
hoped that
must be incorporated
models
to the
by so doing,
in order
to
the
22
The
indicated
techniques
described
work by Bassett37
on optimising
a method by which models
developed
in the later
by the
author
chapters
could
to
of this
the
element
be verified.
achieve
thesis.
the
values
of models
The optimisation
aims stated
are
23
CHAPTER 3
TRANSISTOR PARWETER MEASUREMENTS
AND ANALYSIS OF MODELS
In order
their
to obtain
element
modelled must be available.
there are prescribed
Similarly,
can be measured
for
parameters
are described
by a set
of equations
to generate passive
was represented
investigated
by Luca138,
and the
in
later
also
the
are
measurements
chapter
later.
or
s
and are
The
y
para-
as shown in Section
has used coefficient
at Leicester
In these cases the network
networks.
by network polynomials
by Ilegazi7
for
scattering,
this
given
by nodal analysis
work on network synthesis
to be synthesized
Suitable
func-
that
be calculated
can also
comparisons.
y,
meters of a network can be obtained
matching techniques
that
chapter
of a model require
the transistor
or
These parameters
to each other
to be
transistor
may be made to enable the calculation
and optimisation
admittance,
parameters.
and to check
models.
to make the necessary
the short-circuit
Earlier
the
was mentioned in the previous
measurements that
the validation
model in order
related
It
for specific
of element values
that
of transistors
of models
a set of measurements characterising
validity,
tions
values
such as those given
and Savage8 and shown in equations
(3.1).
36p4 + 2o58p3_+ 6552p2 + 4638p + 36
3+
2+
36p
216p
396p + 216
-Y 12
36P4 + 36p3_+ 72p2 + 36p + 36
3+
36p
216p + 396P + 216
(3.1)
364 +
+ 1572p2 + 1183P + 36
_53323
3+
2+
36p
216p
396p + 216
22
where
y
and
p
11 ,y 12
is
and
y
the complex
22
are the
frequency
short
variable.
circuit
admittance
parameters
24
Cutteridge
of
coefficients
thus
matrix
formed.
and di Dlambro12 developed
for
p
enabling
network
an analysis
ients
networks
for
As it
frequency
of the network elements with
the derivatives.
Also,
as stated
network
by Savage, it
it
,
be
therefore
would
model.
were first
sugges-
for many years.
method include
good convergence
functions
are multi-linear
the consequent easy and accurate
formulation
of
matching the minimum size of the
using coefficient
synthesize
is clearly
in the
parameters
synthesis
and derivatives
to effectively
coeffic-
example in a similar
for
approach has been favoured
since the coefficients
network required
for
the advantage's of this
that
y
or
on a transistor
matching
techniques
ted by Calahan4l and this
di Mambr039 later
polynomial
s
measured
by Enden and Groenendaal40
matching
SavageB states
the network
admittance
to be per-
matching
networks,
functions,
rational
coefficient
the nodal
using
elements.
to convert
complex
suggested
Coefficient
properties
active
containing
to perform
possible
to calculate
would be possible
manner to that
passive
work with
routine
domain into
network
by coefficient
synthesis
to his
In addition
developed
two-port
a passive
the
a method to calculate
the functions
is well
defined
and,
has been
apparent when a realization
achieved.
However, there
functions
in
is the disadvantage
known as the approximation
p,
errors
before
decide
somehow on the orders
any matching
to choose orders
the formulation
that
giving
is
even begun.
of
less
stage,
It
the polynomials
than
a specified
is
would
in
error
of the rational
bound to introduce
also
be necessary
to
One way would be
p.
at the approximation
stage.
This
rejected
seems to be over-complicated
coefficient
choosing direct
matching as a suitable
precedent:
methods for transistor
Bassett37,
modelling,
thus
technique for this
matching of the measur,ed parameters
This is not without
tion
and e:ýror-prone,
the
author
application,
ýn the frequency
one of the
first
chose to match the
domain.
to use optimisay
parameters
25
directly
3.1.
have followed
and most others
this
example.
Y Parameters
The
y
means of describing
whose interior
small-signal
the complex
is
a two-port
Y11
1
12
By examination
and taking
Y12
1+
Y21V1
bias
at any particular
point
(3.2)
yIl
12
v2
when V,
Y21
! Zvi
when
V
when
V
.1=0
as
it
can be seen that
the input
(3.3)
the
20
short-circuit
admittance
parameters.
TWO-POTv
BLACKBOX
=T
2
I
11and12 are the port small signal currents
Two-Port
'Black
port
0
V, and V2are the port smaLL
signal voltages
3.1
and are
the parameters
or output
2
Figure
3.1,
where
121v
the port
thus
vi
description
of
(3.2)
admittance,
= VZ
2
in terms
2
when V0
2
Y22
i. e. one
network,
and frequency.
!
their
Y22
and measured by short-circuiting
the relevant
box'
are shown in Figure
and
are a
parameters,
+ Y22 v2
of equations
y1
hence
which
y 112 Y122 Y2 ,
Ivv
may be defined
They are defined
and currents,
functions
as a 'black
network
inaccessible.
voltages
admittance
or short-circuit
parameters,
Box'
26
3.1.1.
Nodal Analysis
Nodal
network
is
analysis
a method by which
2
m
input
are the
as a reference
Current
of a two-port
parameters
y
can be calculated.
Consider an m-terminal
and
the
passive
and output
and denote,
nodes respectively,
Spence 42 describes
node.
Law to each node in turn
nodes
network where, conventionally,
we can obtain
how applying
the
say,
1
node
Kirchhoff's
equations
in matrix
which
form give
(Y
I=
y
where
I
and V
entering
is the vector
of voltages
at each node with reference
network,
thus
i
j,
and
by the
are non-zero.
ind
the
'
where in general
there
incident
m; iý
connected
terms the admittance
j)
may be
matrix,
each diagonal
2 l, m; iýl,
the admittances
sum of all
admittance
the admittances
(i
Y
element
at node
is
directly
i
formed
between
11
and
by the
nodes
Y between any two nodes
is given by
Y=
where
p
is the complex frequency
C is the capacitance
and
PC +G+1
G
is
L
is the inductance
the
conductance
(3.5)
pL
variable
in Farads between the two nodes
in Siemens between
in Henrys
between
m.
nodes from external
indefinite
Thus,
sources
to node
nodes are inaccessible,
12
sum of
Yij
element
of the
Y
each node from external
these internal
of the network.
formed
each off-diagonal
into
and
shows how
formed by inspection
= I, m) is
since the internal
injected
I,
only
Spence also
negative
matrix
of currents
cannot be any current
(i
admittance
is the vector
For a two-port
sources,
ind3V-
is the indefinite
ind
(3.4)
the
the two nodes
two nodes.
27
can be seen that
It
matrix
This
the
terms
in
having
the
mutual
indefinite
and
node
i
(voltage
Z
into
node
j
If
at
gM to element
Y
a time constant
Section
2.2.2,
deletion
reference
p,
tions
admittance
nodal
of the
mth
the nodal
An alternative
Y
jk
of
Y
Y
jq.
of
Y
matrix
admittance
admittance
matrix
by Aitken 43 to obtain
This
procedure
approach
is
developed
earth
flows
Y
ind
are
ind
Y
PT
to in
Thus we are able to form
network.
ind
node.
y
from
ind
where
described
is
n=m-1,
where node
m
using
parameters
in detail
polynomials
obtained
is
pivotal
is
described
value
condensation
as defined
in Section
by
the
At any particular
by di Hambro 4 and Cutteridge
di Mambro12 making use of the network
3.1.2.
between nodes
is
(3.6)
may be reduced
the
source
controlled
ind
of an active
matrix,
the
to
as in the model referred
row and column of
node and is usually
(3.2).
Y
is also required,
T
n xn
as described
it
of
42
Spence,
ind
then the terms become ±gme
the indefinite
The
to element
ik
Y
the
current
terms in
of
controlled
referring
voltage
controlled
then the additional
Y
+gM
controlling
and the
gM to element
due to
matrix
the
k)
gM to element
If
admittance
a voltage
Again
gm
admittance
high
include
to
conductance
can be formed by inspection.
k
indefinite
the
network
can now be extended
analysis
source
current
a passive
be symmetric.
always
will
for
in equa3.1.3.
and
in Section
of
28
3.1.2.
of
Admittance
Matrix
in the previous
As described
a total
of
are the input
2
and
is
the
n xn
nodal
and output
the reference
node,
(3.7)
n
I,
and
matrix
admittance
and di flambro12 describe
Cutteridge
i=1,
= Ii
yijvj
[Y]
containing
network
may be written
equations
n
where
1
a two-port
designated
is
the
which
earth
node
plus
respectively,
the nodal
for
section,
n+I nodes, in which nodes
from the Nodal
Coefficients
Polynomial
the Network
Generation
how the
y
i=3,
=0,
parameters
n .
can be
written
A22
2'' 61122
Yll
A21
Y12
y
61122
(3.8)
A12
61122
21
All
Y22
A
where
Al,
and
They then
using
by the product
more work
A
Extending
be seen that
frequency
division
a purely
Gaussian
and
the
calculation
the
analyses
all
variable,
elimination
of
to
the determinants
p,
by some power of
622
and if
p.
produced
can be obtained
parameters
y
in
with
3
rows
only
is
61,22
since
to
n
slightly
alone.
1122
networks
that
and state
elements
A
the
network,
resistive
of the diagonal
'ý12P
'ý2 1
113
than
(YI
of
are the unsigned minors of
how, for
describe
inclusive,
as the determinant
A
1133
and
obtained
can-be
given
is defined
1ý1122
containing
involved
inductances
Cutteridge
reactive
are polynomials
are present
elements,
in
these
it
can
the complex
include
and di Mambro state
a
the upper
29
limits
of the powers of
by multiplying
p,
by
through
of the polynomials
order
m
,i=O,
method any inverse
In their
and RLC networks.
m +1 values
produced,
m
is
are assigned
to
if
Then,
RC, RL
are removed
p
powers of
as appropriate.
p
for
and denominator
in the numerator
p
the highest
say
p,
giving
a
pp2pm
000
P,
2m
P,
P,
A(p
0
0
A(pl)
a
(3.9)
pp2pm
mmm
is the value of a cofactor
A(p)
where
The (m +1)
x (m +1)
the Vandermonde matrix
algorithm
and this
hand side
enabling
value of
polynomial.
(3.9)
of equation
and di Mambro invert
Cutteridge
by Traub 44 thus
given
of the corresponding
left
on the
matrix
m
at a particular
evaluated
ail, i = Om , are the coefficients
and
A(p
a
m
is
an
using
of the polynomial
calculation
coefficients.
12,,
45
and di Mambro
also
Cutteridge
of the polynomial
efficiently
coefficients
elements
constants,
matrix.
sources
T,
developed
a time
without
program
of a computer
current
to the network
respect
derivatives
how the partial
elements
may be
computed.
di Mambro4 further
of active
with
describe
with
written
time
introduce
this
by di Mambr039 which
was received
exponential
this
terms
problem
for
to allow
A later,
constant.
constants
di Mambro overcame
technique
development,
allowed
by the
PT
te
the presence
into
by approximating
voltage
version
controlled
author.
Time
the nodal
admittance
these
terms
using
p
30
the first
of the
two terms
xx
PT
thus replacing
The author
calculating
the network
polynomials.
there
were some errors
were included.
it
was felt
Calculation
If
the nodal
frequency
In particular,
approximation
method using pivotal
procedure
a systematic
condensation,
pT
for
Also,
as des-
method and the method described
di Mambro's
using
Parameters
Y
matrix
the
y
described
procedure
1+
would be more suitable.
of the
admittance
version
the method was
too inaccurate.
was considered
are compared in Section
at which
p
elimination
in
this
3.1.4.
together
in a network are inserted
the value
with
parameters
Pivotal_Condensation
using
the values of the elements present
all
fault,
even on correcting
time constants
from
when inductors
occurring
the
stages
frequency
a development
being
this
one of these
present,
J-n the early
program
at any particular
Unfortunately,
obtained
in the next section
of di Hambro's
parameters
a more direct
that
The results
3.1.3.
.
y
in the next section,
cribed
into
the
However,
the terms involving
PT
version
satisfactory.
not
still
1+
used this
of research,
X3
X2
-5-1-+ 31
l+T
by
e
series
are required,
in the previous
known as pivotal
section
condensation
of
the complex
then
the Gaussian
can be performed
which
is
described
by Aitken43.
In pivotal
two-port
reduced
'black
to an
condensation,
box' are zero,
(n -1)
x (n -1)
since the currents
the
1- to
3
n xn nodal admittance
matrix
using
the
formula
In+j of the n+InOAe
matrix
[y]
is
31
Figure
3.2
Flow Chart
for
Pivotal
Condensation
Procedure
32
Y
[YI]
where
This
matrix
is
chart
of the
calculation
nn
of
the
giving
in
Figure
y
parameters
y
1, n-1
(n -1)
size
x (n -1).
(YIj
on
,
x2
a2
and so on until
parameters.
3.2 gives
the
computational
from the nodal
for
procedure
matrix
admittance
the
using
technique.
this
In the FORTRANsubroutine
for
j
be repeated
can then
thus
obtained
The flow
ij
in
the new matrix
procedure
is
nj
y
1,
was decided
to set up two-dimensional
of conductance,
contributions
the nodal
admittance
the nodal admittance
9e
m
PT
[Y]
, I", ij
G, L
and
and capacitance
parameters
y
C
it
to contain
the
to
respectively,
frequency
an element
in
was given by
PT
+1+9me
ýLij -
term due to a current
was the
the
since
at a number of frequencies,
Then at any given
matrix
ij
author,
arrays
inductance
matrix.
yG
where
would be required
network
any particular
by the
written
source,
(3.12)
which
was included
as
appropriate.
As complex
is
arithmetic
in the
no difficulty
available
in
FORTRANon modern computersthere
due to
arithmetic
complex
p
which
is
given
was
by
j2Trf
where
f=
frequency
3.1.4.
Accuracy
in Hertz.
of Calculations
When calculating
possible,
in order
just
(3.13)
the
as with
to preserve
Spence 42, the nodal
y
parameters
Gaussian
using
elimination,
the maximum accuracy.
admittance
matrix
should
pivotal
condensation
to choose the pivotal
However,
always
as pointed
it
is
element
out
by
have non-Azero elements
33
on the diagonal,
therefore
anyway,
eliminated
a node is
unless
They observe
the calculation.
frequencies
or at resonant
become very
large
this
occurrence
In the
written
frequency
polynomial
coefficients,
it
inaccurate
in the Mz
frequency
it
9
by 10- and the reactive
in order to preserve
factors
of
method,
partly
very
and
for
large
10-9
and very
small
values
Checks were then made using
it
the
but
in the
circuit
of 0.5 MHz, 10 ?fflz and 100 MHz and with
generator
0, -0.01 ns
of
to be prone
The notation
delay.
to
loss
values
and
of accuracy,
used here
is
that
by 109.
that
condensation,
that
Remembering
it
this,
+0.01 ns .t
was analysed
a negative
values
of all,
was felt,
should
was decided to apply the
same way as was necessary
of consistency
reasons
To overcome
arithmetic.
word length
Additionally,
in the
were grossly
the frequency
by multiplying
constant
method
of the network
calculation
the author decided first
the computers in use had a 60 bit
even
in-built
by the author.
required
should be performed in double precision
109
of the
the calculations
and time
that
this
precautions.
the maximum accuracy in the pivotal
adequate protection.
cies
that
can
and that
felt
they
no special
used in the
the values
component
provide
both
range
in mind the above points
Bearinj
section
was found
the matrix
in the matrix
However,
and took
components
to scale
was necessary
in
on
low or high
at very
some elements
elements
be
should
of the frequency
by di Mambr039, because
up the
that
other
case it
well-conditioned.
the effect
of calculation.
of setting
this
generally
in particular,
frequencies,
was an infrequent
subroutine
is
consider
that,
compared with
to inaccuracies
can lead
also
in which
connected
the matrix
and Basset46
Fairbrother
not
also
to avoid
with
di Mambrols
the presence
of
arithmetic.
shown in Figure
3.3 at frequen-
time
T
constantS
This circuit,
manually
value
Of
current
which appeared
to obtain
T
on the
indicates
as accurate
a time
34
cl
C-'
Ar
2
1
0
Node 1-
Input
Node 2-
Output
Node 0-
Ground
L1L
(1-4)
= 4.15E+00 nH
L2L
(2-6)
= 2.38E+01 nH
L3L
(3-0)
= 5.71E+00 nH
C1C
(4-5)
= 3.16E-02
nF
C2C
(5-3)
= 8.89E-02
nF
G1G
(4-5)
= 2.8SE-02
S
G2G
(5-3)
= 5.96E-03
S
g (6-3)
= 6.84E-01
S
g,
T0
ns , +0.01 ns
S.
nodes 3
-0.01
V across
j2rrf
p=
f=0.5
Figure
3.3
MHz
Circuit
J.
10 MHz
for
.
100 flHz
Testing
Analysis
Methods
35
of the
values
y
of this
matrix
using
circuit,
27f
w=
00
-110
JwLj
The definite
as possible.
parameters
is
,
in equation
given
admittance
nodal
(3.14).
0
jwL,
01000 jwL.
jwL
jWT
0
0+G
0-
+jwc, +gle
3WL
3
+jwC
--!
jwL -+G
1111
00
jWT
Using
2
of Section
the notation
3.1.2
0
0
jw (C +C? )+GlG2
1
i-I
JWT
0
-gle
jwc
-G
-G, -jwC,
-G 2-jWC2
-1jwL
w-r)0
(3.14)
100
-..JWL
0
(G2 +jwc +g
2 1ej
2
gle
jwL
terms
and cancelling
2
where appropriate
we get
0
All
CI
CiR
A12
w l -LlL2L
3
+GI+jGR+
A21
0
ý122
(GIC2 + G-,Cj)
w,' LlL2L3
1&1122
-
L3Gl(G, ) +R)
(G, +G2)
4 LIL2
j
CJC?
w LlL2L3
+ C, +Cq +GIG2L,
ý'
w -L,
-
(GlC2
+CI(G2+R))
w
where
and
R=
Real
part
I=
Imaginary
w=
21rf
f=
frequency.
of
part
g1e
of g1
JWT
W2 (Ll+
(3.15)
L2
CII
L3)CIC2
L2L3
+jw
jWT
GG
w LlL2
+
(C2G, +CIG2)
L3
GI
36
Thus the
y,
of this
parameters
Agg
Yll
1ý1122
A21
61122
Y12
y
by
are given
circuit
0
(3.16)
A12
61122
21
Ali
A1122
The values
double
using
of
f
Table
and
y 11
precision
and
on the
figures
significant
Starting
3.2.
21
to
figures
14 significant
were then made between
the values
described.
analysis
corresponded
with
expressions
on the Cyber 73 at the required
two methods of
that
from these
were evaluated
The results
Comparisons
the variations
y
accuracy
T.
3.1.
in Table
of
with
those
of
T=0
a value
are given
obtained
using
3.1 are given
the methods
compared were:
di Mambrols method
P9
in
as
-
but with
P
frequencies,
factors
reactive
of
9
10
component
9
and 10applied
values
and time
to
constants
as appropriate
pivotal
M9
in
as
-
N19S- as in
condensation
NI but with
1.19 but using
using double precision
factors
single
applied
precision
in
The number of
in Table
only,
values
arithmetic
as in P9
arithmetic.
37
t-1
00
(M
CD \O
rm
Glý
"0
%0
lqt
CD
-1
ýi
00
rt4)
1-1
4.
r-
cli
cq
le
M
"l
\o
r-ý
ý
0
C%
%0
4
tn
"
t41
P-4
\C
Ln
\C
CD
rq
00
Itt
%0
nt
er)
00
00
tn
m
m
le
Lr)
e
ý4
P-4
r-1 m
t'n
1--4 (N
o
r-i
r--1
0
rt-
r, 1
Ln
M
CD
CD
-4
1-4
00
P-4
Lt-i
r*I
e
00
tn
-1
t41
r4
rq
CD
r-
ci
t-
CO
Ln
00
-4
Co
--. i
-zr
'Z
L',
Ln
cý
ýo
r-
CD
LA
M
tn
-4
r, 4
ýI
e
00
r, 1
(N
ul
1,0
C
0
"
00
CD
CD
le
rlý
00
fli
00
ý4
%0
C)
CD
M
LA
r--
CD
r_
cý
oo
-4
Ln
%0
c71
r41
00
r-CD
00
P-4
(N
-t
CD
LI-
m
%0
tn
cn
C%
CD
t41
f.-4
%0
00
qt
Lr)
(M
m
0
it,r_
CD
r_
Ln
CD
1-4
(2)
m
Z (ý "ý
';
\o
-e
m
ei
(111
tCý
te)
«e
Cý
C)
-4
Ln
r, 4
tn
"q
00
00
t-'
e
m
e
Ln
Ln
CD
CD
r-i
M
--1
c;
Ltli
\JD
\O
c;
lý
-, -4
4-4
u
., 4
4.4
. r4
0
m
rlet
rq e
rq
00
cq
t4ý
le
CD
-i
CD
rq
00
Izt
le
Vý
Co
00
00
CD
r00
cý
(M
CD
CD
(71
tn
Izt
le
e
Ln
en
Ln
tn
Cä
Lt')
M-4
Ln
e
rq
00
r-4
P-4
tn
ýr
rM
tn
tl
CD
e
cý
-4
NI
00
"4
rq
Co
-4
CD
c)
e
-e
Ln
CD
ul
Ilt
CD
rP-4
Ul
C%
\o
00
e
r, 1
Co
(N
tn
1-t
%0
1gt
tP-4
Ln
Izt
CD
P-4
N
"
le
ul
Ln
re-i
cý
rbi
00
0
t-ý
rn
r,
In
In
r-,
rn
r. '
r-,
+
+
+
+
+
+
+
+
+
00
Ln
rq
Ln
IKT,
tn
Lri
M
CD
10
nt
et
\JD
00
00
tn
P-4
Lt)
%M
Ln
r-
Lti
le
(14
r-I
CD
rn
00
r-4
Lr)
-4
\Z
\o
Co
vj
bl
00
(N
le
F-4
P--1
(ýq
le
"-1
CD
CD
tn
00
CN
vj
0)
%0
t4ý
\Z
00
Ln
00
CD
Ln
\JD
00
rý
cý
P-4
(N
rq
Ln
te
gt
cý
00
rý
(-4
tn
e
ei
(Z
tn
t-e
tn
00
0
c
CD
(N
oh
00
cl-
Lf)
00
r--
44
0
cz
0
.0
zi
Ln
;
c
(Z
;
c
CD
;
c
LA
;
c
C
c
CD
7
(Z
Ln
CD
(Z
C
38
0.5 ISIZ
Yll
Method
T ns
0
0.01
0.01
I
The results
Yll
Y21
Y21
R
I
R
I
R
I
R
I
R
I
P9
13
14
13
14
14
14
14
14
14
14
14
14
M
14
11
14
14
14
12
14
14
14
14
13
14
M9
14
11
14
14
14
12
14
14
14
14
14
14
m9s
13
9
13
14
13
11
13
14
14
13
13
14
P9
13
10
9
10
8
7
7
7
6
4
5
5
M9
14
10
14
14
14
11
14
14
14
13
14
14
P9
13
10
9
10
8
7
7
7
5
4
5
5
M9
14
11
14
14
14
11
14
14 ý14
1
13
14
14
using
were so large
ficant
figure
Significant
P
method
that
in Computed Values
Figures
shown in Table
are not
none of the values
3.2 because
the
corresponded even to one signi-
and many values were an order of magnitude in error.
As can be seen from Table
followed
closely
had a major effect
and double
3.2,
P,
M9
M.
arithmetic
M9S might
T=0
P9
method
Whilst
M.
and
was the most
improvement
However, the differences
versions
be considered
still
M9
of methods
factors
109
the
there was only a very slight
to method
length
more marked although
with
by methods
on method
when they were applied
single
Yll
I
I
errors
accurate
[
R
Number of Correct
Table 3.2
I
Y21
100 Mllz
10 MHz
between the
M9S were
and
enough for
accurate
most purposes.
Taking
time
sons with
from Table
P9
-0.01
ns
and
at 0.5 MHz method
P9
has lost
constants
3.2 that
T
be expected,
this
by a frequency
of
as might
so that
methods,
the two most accurate
M9
of
loss
of accuracy
100 NIHz the
has suffered
Meanwhile,
method
unaffected
by the magnitude
of the
frequency.
M9
,
for
further
+0.01 ns , it
rises
results
negligible
and
can be seen
and that,
some accuracy
as the
frequency
are becoming
loss
of
Thus,
compari-
increases
unacceptable.
accuracy
although
and is
method
P9
39
is slightly
more accurate
generators,
method
when no time constants
M9 is significantly
to any current
are applied
when time constants
superior
are
applied.
In all
of less than
to the
consideration
should
P9
The storage
M9
in
calculation
savings
calculation
P9
time
.
P9
,
.
from accuracy
in terms of speed and storage
M9S is considered
time and storage
M9
r-almead. by
to being more suitable
than
by
taken
4"fiadt
M9
-by
space make it
space.
then the obvious
sufficient
approach the author elected
Taking a cautious
time
70t
N19 was
in addition
,
the accuracy of method
the best choice.
to use method
M9
.
S Parameters
3.1 it
In Section
network
turn
50% of the
took
by
was also better
speed of
algorithm.
BOX of that-raquitred
M9S required
considerations,
3.2.
M9S
to the
M9 took 90% of the computation
method
space required
Thus, method
If
while
,
a computer
when choosing
be given
also
by the
space required
storage
by method
while
errors
of Algorithm
of calculations,
accuracy
For the given analyses,
taken
on the Choice
Considerations
In addition
and the
y 22 had absolute
y 12 and
20.
10-
Further
algorithm,
for
cases the computed values
could
above 100 NIHz it
circuits
The
the
appropriate
s,
y
parameters
the
in obtaining
difficult
input
to measure
good short
of a two-port
and output
However,
admittances.
becomes increasingly
ok the difficulty
often
how the
be measured by short-circuiting
and measuring
because
was described
circuits
in
ports
at frequencies
the
y
parameters
and because
short
cause oscillation.
or scattering,
parameters
are
similar
to the
y
parameters
in
40
that
they describe
the inputs
47
describes
Weinert
incident
in Figure
measured up to the GHz frequency
they are easily
the advantage that
how the
and reflected
s
parameters
(a,,
parameters
3.4 and are defined
1
black box, but have
of a two-port
and outputs
bl)
in equations
are defined
and
(a.,
range.
in terms of the
b. )
are shown
which
(3.17).
[a
0
rzvrz,
I,
+
0
2
0
rz
0
12
/Z-0
b2 'ý
The s
12rz,
0
parameters
b,
(3.17)
+YZ0
12
r'(Z
12
0
are then given by
= s1la,
+ Sl2a2
(3.18)
s
21
a1S
22
a2'
12
vi
bl
Fig. 3.4
Incident
and Reflected
TWO-PORT
BLACKBOX
Parameters
CL2
V2
2
z0
b2
Used to Define
the
s
Parameters
'
41
Measurement
3.2.1.
Just
as the
measurement,
incident
the
s
Therefore
ratio.
may be defined
parameters
a,
parameters
or
we can
the
and taking
port
or output
so the
and measured by short-
can be defined
parameters
y
input
the
circuiting
Parameters
s
of the
to
a2
S
the
express
and measured by setting
and taking
zero
admittance
appropriate
the
parameters
appropriate
as:
b,
s
when
a2ý0
when
a=0
b
s
-L
a21
12
(3.19)
s221
a12
s222
The
s
parameters
and incident
a2
of
zero
or
output
port
of
measurement
with
forward
cies for
the
SOQ
s
11
8405A
to
a1
set
is
test
jig
and
vector
s
21
voltmeter
to
is
The
jig
which
thereby
in
is
is
by Hewlett
SOQ loads
and
s 12
device
characteristic
described
reversed
to
the
connect
in its
lines
and
how the matching
respectively
a2
line
easily
s 22
or
lines
transmission
is matched.
when one port
is explained
A measurement
500
of the reflected
(48) it
transmission
parameters
s
ports
the transmission
under
reverse
Packard
voltages
Thus the
the
the device
parameters
a Hewlett
required
necessary
3.4.
49,
incorporating
Packard
such a way that
and output
of reference
in Figure
Z0
0
be expressed as ratios
and then terminating
impedance,
the
a,
by using
may be achieved
under test
when
at the input
voltages
input
a0
can therefore
In the introduction
the
when
the
used
used
the
enabling
same
in
to
in
set-up
as
conjunction
measure
the
and phase angles.
parameters
any particular
may be easily
device.
measured
over
a range
of frequen-
42
Calculation
3.2.2.
The
s
s
of the
Parameters
of a model may be obtained
parameters
of the model using
y
parameters
this
chapter
(49)
and reproduced
and then
(3.20)
as equations
given
earlier
in
in reference
below.
*d Y22) + Y12Y21
Yll)(1
sil
formulae
the
calculating
described
one of the methods
the conversion
applying
by first
I
12
dy
(3.20)
2 y2l
dy
21
=
22
sll)(l-+
parameters
to the
y
parameters
equations
(i
The
y
to
Z,
0
conversion
(3.21)
2 IZ1
dt,
(I
ds =
+ S12S21
ds
Y21
Y22
S22)
ds
2 sig
Y12-
lised
s
may be used.
Yll
where
+ Y12Y21
+ Y22) - Y12Y21
from the
For conversion
(3.21)
-d Y22)
y
(1 + Yll)(1
cý =
where
(1 + Yll)(1
ý-
+
S22)
ds
+ S22)
+s 11)(1
parameters
given
if
therefore
is performed
+ Si2s2i
- S12S21
in equations
y!
ij
is
(3.20)
and (3.21)
one of the actual
are all
norma-
parameters,
using
ij
yii
z01,2;
j=1,2
(3.22)
43
CHAPTER 4
NUMERICAL OPTIMISATION
Numerical optimisation
is a widely
not just
range of applications,
in electronics
most basic,
but usually
some error
between the ideal
difference
defined*
is
most problems there
depends on the
of 'best'
which indicates
the
value of a parameter which is dependent on one
parameter
currently
problem thus becomes one of minimising
optimisation
At its
as a means by which the best value
function
and the value of that
or more variables
but encompassing
modelling
The definition
may be obtained.
of some variable
application
can be described
optimisation
area with a vast
subject
problems in many disciplines.
design and manufacturing
modelling,
studied
the error
therefore
are a number of criteria,
The
available.
function.
In
the sum of squares
function
F(x)
where
fi (x),
vector
of variables
i=1,
m, are the individual
is often
x,
If
equations.
can be seen that
the equations
in the solution
indicator.
error
of nonlinear
sim-
are
(4.2)
i'm
a zero minimum of the sum of squares function
There are numerous other
error
overall
dependent on a
would
of the equations.
give a solution
all
functions
used as a single
fi (x) =
then it
error
is also useful
The sum of squares function
ultaneous
(4.1)
=
function.
formulations
Another popular
that
are suitable
for
one is the minimax function
an overwhich
takes the value
F(x) = maxIf (X)l
i
where
f1 (x),
for the single
i=l,
m are individual
function
i=
error
to be minimised
i'm
functions.
is the objective
(4.3)
The general
function.
term
44
The optimum value
equation
real
not
individual
a zero minimum of
x
of values
all
In any particular
F(x)
the
However,
.
for
whether
or
to be at
defined
minimum is
global
=0
be negative
cannot
fi (X)
functions
F(x)
such that
where
of
values
other
found
is
since
exists,
F (x )
for
a zero minimum may not necess-
x
of
error
F(x)
squares
although
=0,
of values
any set
of the
the vector
sum of
minimum has been found
a global
values
all
F(x)
when
If
exist.
arily
then
is
(4.1)
of
in
as defined
problems
ý<
(4.4)
F (x)
x.
R
region
there
be a non-zero
might
minimum at
where
x1
F
for
all
x
of
values
R
within
< F(xl)
F(x*)
The vector
xI
found so far
in their
survey of global
ing points,
50.
et al.
misation
values
optimisation
global
and those
methods observe that
At Leicester
techniques
with
optimisation.
may occur
sat ion problems,
University,
the situation
problems is very poor.
optimisation
based on Branin's
Gomulka and Szeg650
Dixon,
multistart
is performed many times from different
The distinction
strained
of knowing whether the lowest minimum
methods are random methods, including
minimisation
minimum.
and time consuming operation,
is the lowest minimum overall.
regard to solving
popular
.
is a very difficult
because of the uncertainty
not least
(4.5)
case is termed a local
in this
Global optimisation
with
but where
methods where local
start-
randomly-selected
trajectory
method,
*52 has worked
Price5l,
The most
by Dixon
reviewed
opti-
on global
some success.
should also be made between constrained
Unconstrained
at any values
however, there
between
is
optimisation
-to
and
are restrictions
+-
.
when the
and unconsolution
In many real
on the acceptable
optimi-
solution
45
to the variables
transformations
transformations,
the simplest
Lootsma54 has reviewed
It
can be seen,
local
to take
definition
of the
therefore,
minimisation
current
A reasonable
approximation
terms of the series,
where
Also,
only.
constraints
function
to which
be applied.
of a bipolar
fact
are in
problems
to
n
in terms of the
transistor
and inductance
together
with
minimum could be likened
the global
sources,
X6
6!
Cos x
(x <
(4.6)
at least
requires
the first
n
is given by
(4.7)
xm<M!
where
m= 2(n - 1) .
Thereafter
it
to
series
X4
ý12 ý,
2!
41
Cos x-+
of the variables,
more complex
objective
capacitance
cos x by the
to
the app.roximation
work as des-
problems.
standard elements of resistance,
controlled
values
most design
that
In the case of the modelling
voltage
logarithms
positive
methods may then
optimisation
constrained
domain of the
of
One of
by BoX53.
the author's
a number of methods whereby
imposed
by
be
suitable
can
unconstrained
described
is used in
many constrained
by the application
as unconstrained
which
the variables
constraining
In practice
such as those
to work in the
is
later,
cribed
can be treated
problems
optimisation
thus
must be imposed.
and so constraints
values
each further
could be argued that
term improves the accuracy of the approximation
only an infinite
number of terms will
but
give the true
value.
Similarly,
example a voltage
reasonable
to obtain
in a model of a bipolar
controlled
model whilst
higher
the inclusion
accuracy,
to get a true match.
current
but it
Therefore,
transistor,
source,
are necessary
of other
would require
in this
some elements,
to obtain
elements is desirable
an infinite
context
for
any
in order
number of elements
a good model should contain
46
only
those
minimum is
the model a new local
The modelling
Chapter S.
if
a model
element
is
added to
developed by the author is dealt
strategy
method actually
Numerical methods of optimisation
can be divided
Available
with optimi-
to those of the
applications
the optimisation
in
with
is concerned firstly
chapter
have been used in similar
author and secondly with
make a
Thus,
the model.
of
that
found.
The remainder of this
methods that
4.1.
those
with
by element, as each new significant
up element
sation
to the accuracy
contribution
significant
is built
h&ve a, majewafficttogether
that
elements
used by the author.
Methods
into
three main
categories.
Brief
Ci)
Methods using function
(ii)
Methods using derivatives.
(iii)
Methods designed for minimising
details
of
some of the algorithms
indication
an
with
4.1.1.
of applications
Algorithms
In this
Using
first
Function
category
discussed
were
methods
briefly
to determine model parameters
method are its
The disadvantage,
narrow,
Values
Only
and its
assisted
in n-dimensional
searches.
of this
for MOStransistor
in the function.
(or simplex)
together
whe-re they have been found useful.
A simple pattern
by Price57
follow,
each category
chapter.
Random
Pattern
search was used by KnudsenS5
models.
suitability
as noted by Knudsen, is its
Mead56, also studied
sum of squares functions.
at the beginning
simplicity
curved valleys
polyhedron
in
only.
are random and pattern
searches are more systematic.
of this
values
for
The attractions
small computers.
tendency to get trapped
in
The simplex method of Nelder and
by the author58,
space.
forms a regular
On examination
of the
47
function
minimum is located.
obtain
A simplex algorithm
by matching
transistors
s
searches together
trical
with a simplex algorithm
they noted that
although
circuits,
function
to be evaluated
best points
N random points
at
described
in the next chapter.
starting
The classic
51#52
evaluating
selecting
each new
from the
simplex of points
N
was used by the author6O
model of a p-n-p
and second derivatives
descent,
In the method of steepest
objective
by initially
methods are those of steepest
optimisation
Newton Raphson which use first
function
of Price
search algorithm
thereafter
a trial
due to
transistor
as
Derivatives
Using
Algorithms
for
values
suitable
of elec-
method was limited
their
This algorithm
available.
to obtain
4.1.2.
in the d. c. optimisation
using a randomly selected
currently
Durbin,
and random
of simple pattern
a simplex technique
combines a random search with
the objective
data of the devices.
The global
the large computing time required.
point
models of microwave FETs and
parameter
Montaron and Heydemann59 used a combination
a suitable
was used by Mathews and Ajose2 to
in
particular
of
elements
values
optimum
bipolar
is then expanded, contrac-
the process being repeated until
about a vertex,
ted or reflected
it
of the simplex,
values at the vertices
respectively.
the gradient
is used as a search direction,
descent and
vector
g
of the
hence from any position
gk, an improved
given by the vector' xk with associated gradient vector
k+1
k
the
+o
inclicate
by
iteraficmý
be
should
obtainedusing
x
position
x
Xk
where
It
tion
of
agreed that
a variation
filter
k9k
= xk
is chosen by linear
is generally
however,
k+l
this
(4.8)
search to minimise
the objective
method has bad convergence properties,
on the method was used by Agnew"
circuits,
to generate the smallest
function.
used only
active
step length
residuals
necessary
at
who,
for
the
any particular
to produce a specified
optimisastage
48
"reasonable"
in a minimax
reduction
descent
be made here of the gradient
by Henderson"
improvements
of,
Algorithms
gradients
using conjugate
methods using first
derivatives.
number and
sequence of points
in the iterative
k+ 1k
for
the solution
problem encountered
transistor.
are more popular
and effective
of these is by
method, using the superscripts
denote the iteration
T
matrix
descent",
of steepest
The most well-known
In this
and ReeveS63.
also
later
the Hessian
difficult
a particularly
should
(with
algorithm
a d. c. Ebers Moll model of a bipolar
in producing
Fletcher
stage in a two-part
applications,
amongst other
included
a "curve
to generate
initial
the
at
was
used
which
Mention
method of Cutteridge62
and Dowson9), which
derivatives
of second partial
function.
error
to denote the transpose
k
to
the
of a vector,
scheme is given by
k+l
Xky
x_
(4.9)
k+l
y9y
where
and
kkk
Xk
is the linear
gk
"
ly
is the gradient vector
is the Vector of Conýuq&tdAirer-tions
ak
is given by
k
a0
search parameter minimising
k=O
for
,
(n+l),
function
the objective
2(n+l)
(4.10)
(ýk) T (ýk)
(ak-1)
ýk
function
On a quadratic
tions,
where
n
this
method locates
is the number of variables
by both Krzeczkowski6
of lumped linear
otherwise
and Savage8
three-terminal
for
where
the gradient
vector
the initial
itera-
The method was used
x.
stages in the synthesis
based on the Taylor
F (X + 6) =FW+6Tg+...
is
in
n
networks.
The Newton Raphson method is
g
the minimum in at most
of
F.
series
expansion
(4.11)
49
higher
Ignoring
to zero,
tives
differentiating
terms,
order
where
the Hessian
is
the direction
along
given
by
k+l
Xk
where
very
successful
=xk-Xk
tive
definite
the initial
where
hk
thus
giving
A variable
k+l
x-x
k
k+l
k
and Powell65.
as quasi-Newton
if
deriva-
Davidon 64 and the later,
are those of
Variable
they satisfy
to the inverse
k+l
metric
the criteria
Hessian matrix,
6khk
is posi-
(4.14)
the Newton Raphson iteration
metric
algorithm
by
Orlik-35
was used
Poon models
of algor-
and that
of networks to time-domain
Powell
are a class
C4.13) using the first
equation
approximation
of the Hessian
evaluation
algorithms
metric
by Fletcher
s
k
becomes
function.
the
requires
search
(4.13)
the objective
thus
may be classified
a linear
k) -1
(H
g
Variable
algorithm
it
F
of
the Newton Raphson iteration
thus
Among these algorithms
only.
algorithms
0
S
that
to include
to approximate
ithms which attempt
tives
descent algorithm,
6,
The Newton Raphson algorithm
matrix
(4.12)
derivatives
is chosen to minimise
at each iteration.
gives
of second partial
matrix
as with the steepest
is usual,
F,
a minimum of
H6
0g+
H
for
i. e. the equation
deriva-
first
the
and equating
of integrated
for
a quadratic
was used by Lanca and Nichols"
specifications
to obtain
circuit
function.
for
the
and the method of Fletcher
some of the
transistors.
element
values
design
and
in Gummel
so
for
Designed
Algorithms
4.1.3.
Sum of Squares
Minimising
Functions
sum of squares functions
When minimising
i-i
functions
the
component
are
m,
fi Cx) ,i=l,
where
n
variables
tion
and its
x,
then,
derivatives,
ent functions
and their
If we say that
then taking
in addition
the Taylor
there
which,
higher
where
Equating
we are attempting
to obtain
f1()
m
series
=0I=1,
(x)
+
m=n
and J
(x)
If
m>n
squares
derivatives.
(4.16)
we get
.
then
J-1 f (x)
(4.20)
.
then we can use
-1
J)
JTf (x)
6=-QT
DimmerIl
(4.17)
+ j6
from equation
is non-singular
6=-
i'm
gives
of first
to zero as required
6. +
3
axj
order terms,
gives
afi(x)
n
16
If
func-
(4.16)
expansion of each component function
is the Jacobian matrix
this
of the objective
the values of the compon-
are also available
f (X + 6) =f
J
to the values
describes
solution
of
derivatives.
(X + 6) =f
fi-i-j11
ignoring
dependent on the vector
how equation
of equation
(4.21)
(4.19).
is
.
derived
by finding
the
least
51
introducing
Thus,
a linear
search
we get the Gauss
parameter
Newton iteration
kT
Xk (P )i
k
k+1
xx_
kkTk
(i
(4.22)
f (x
k)
k)=0
(J
f(x
at
that
point
which
a
provided
shows
k)ik
is non-singular,
then there will
been reached and (J
Dimmer"
X>0
which will
the differences
function.
reduce the objective
has not
be a value of
Dimmer also emphasises
between the Gauss Newton and Newton Raphson methods and
of performance by the Gauss Newton method over the Newton
notes a superiority
Raphson method on sum of squares problems.
The Gauss Newton method has been used extensively
University
from the synthesis
problems arising
on optimisation
at Leicester
of electrical
networks.
Most other methods designed for minimising
sum of squares functions
Newton
Gauss
based
the
method and are concerned either
on
are
of the matrix
singularity
to be inverted
for
the calculation
requiring
method
by Broyden
transistor
bipolar
of the Optimisation
It
however it
of
misation.
was the author's
Cutteridge
At this
favour,
4.1.1
the starting
intention
values
thus
only as in the
to,
stage
the reason
Chosen
as no single
so no single
applications
that
values
62A
10
for
and Henderson
early
just
that
even for
in
section
mentioned
was
Price
to obtain
of
search
Algorithm
4.1
has found universal
has become the standard
algorithm
ithm
of the component function
from
be
section
seen
can
It
the Jacobian matrix
6 9.
Some Details
4.2.
avoiding
8
Levenberý57
Marquardt6
the
as with
the
use of approximations
or
with
method
with
optimisation
of a similar
nature.
the author used the global
for
one of the trial
initially,
the major
for
model of the
this
problems;
use the two-part
part
choice
algor-
of the model optiwas largely
52
because of familiarity
tion
of its
of this
soon discovered
her
in
particular
needed
this
adequate
for
this
Therefore,
that
observa-
involved.
from the modelling
point
Gauss Newton algorithm
several
in one iteration
was the major
a preliminary
kkkkk
Xo , 2XO , 3X
series
0,
The aim of this
Should this
values
Sx
0 ...
where
not be achieved,
optimisation
method used
x
This was achieved in the
limit.
k
X
of
search included
The linear
Xk
from the Fibonacci
calculated
Xo was a given starting
search was to attempt
preliminary
became apparent.
also
the change in any value
was to limit
search used to choose the value of
of
due to Henderson".
modifications
to some predetermined
search using trial
of view
proved
benefits
Furthermore,
fact
in
was not
section
Gauss Newton section
the
since
descent
Gauss Newton section.
descent
the gradient
One of the modifications
set
first-hand
of a gradient
and a modified
application
and included
by the author
4.1.2,
the optimisation
second section
linear
with
consisted
algorithm
in section
mentioned
The author
quite
together
algorithm
efficacy.
The two parts
section,
the
with
the preliminary
to bracket
search continued
value.
a minimum.
the pre-
until
in
in
value
reached
was
each variable.
change
maximum
If
was located
a bracket
was obtained
the occurrence
of non-unimodal
the
through
minimum
onto
convergence
of
slowness
uation
too close to the current
search then the minimum
interpolation
using a quadratic
more accurately
was safeguarded against
in the preliminary
taking
algorithm
functions
and against
for
new points
used in the quadratic
three points
which
eval-
interpola-
tion.
Henderson also categorised
The first
of these,
concerned conditions
which was not really
when it
values of the correction
category
criteria
terms
for
termination
6
calculated
failure
terminations
relevant
was not advisable
The second concerned successful
described
the possible
of the algorithm.
in the author's
to attempt
using
application,
the algorithm.
which was based on the absolute
by Gauss Newton.
due to the JTJ matrix
The third
being
or
53
almost
failure
(i)
if
being
The fourth
singular.
and final
There were three
of the algorithm.
the number of iterations
category
predicted
criteria
in this
exceeded a certain
number, (ii)
161
k, and (iii)
if
6k
60,
100
at iteration
where
max
max
max
k
k-1
increase
(6
iterations
the
10
with
successive
on
-6
max
max
increase
i.
iteration,
the
rate
of
accelerating.
e.
successive
The author,
diction,
especially
misation
algorithm.
models it
Dimmer1l,
of ultimate
that
as possible.
use of this
is
of
failure,
the
algorithm.
opinion
that
is an important
When the optimisation
is most important
carded as early
author's
like
unsuccessful
ultimate
category:
if
6k>
max
6k>
6k-1
max max
larger at each
termination
pre-
aspect of any opti-
problem is one of developing
models be detected
We thus have a further
justification
and disfor
the
54
CHAPTER S
DEVELOPMENTOF THE MODELLING ALGORITHM
specified
s
that
these were the quantities
the equivalent
sets of
formulae would alter
formation
models which they thought
author
a suitable
the
request
were
operation
on
the standard
applying
of the fit.
the weightings
the
trans-
PRL also provided
starting
for
point
a model
exercise.
optimisation
Thus, this
initially,
fit
to
model
first
chapter
chapters
three previous
igned,
this
measured, and (ii)
actually
would provide
for
One of these was that
models.
parameters obtained
y
data
The reasons given for
directly.
parameters be fitted
(i)
for their
conditions
certain
(PRL) who provided
Laboratories
Research
Philips
describes
were combined together
to calculate
the
to the conditions
s
how the component parts
to form a single
the optimum element values
for
parameter measurements of any particular
set by PRL.
Details
of the
method desany particular
device
subject
are then given of the possible
ways
in which models may be improved and of the methods chosen by the author.
At each stage examples are given of the results
model developed for PRL, that
first
results
It
of this
exercise
should be noted that
and confirming
some suggestions
5.1.
Details
The first
original
s
further
of the First
in
p-n-p
this
first
common emitter
the period
included
in Appendix
modelling
of research
in Chapter 8.
Exercise
by PRL was for
configuration
parameter measurements given in Table 5.1.
form is
and Dowson60.
the process of refining
improvements are included
Modelling
The
for the final
exercise,
throughout
continued
the
transistor.
by Cutteridge
the major criteria
set of data provided
connected
transistor
set of
although
the techniques
for
jointly
were published
were developed during
algorithm
of a lateral
for
obtained
1.
a lateral
p-n-p
and consisted
of
The data in its
the
and
55
S
Zo -, 5006
11
Phase Modulus
dB
degrees
Modulus
dR
Frequency
Miz
0.5
2.0
-5.40
-0.20
5.0
-11.20
-0.40
10.0
-19.40
-1.20
A measurement
It
Measured
5.1
Table
can be seen that
cies
the modulus
trolled
take
It
the diagram.
three
Price
cribed
elements
51,52
with
zero time
was felt
5.2.1.
values.
init-ial
the
as
used
-0.50
-14.00
132.90
0.00
-0.90
available
for
P-N-P. Transistor
s
s
for
12
of these
and "phase
trial
for
point
are in
parameters
which
frequen-
a measurement
measurements
angles
were
0"
near
are
a simple model which
the derivation
of a model.
of four nodes (including
consisted
that
model was too basic
this
The global
author.
the
single
in order to obtain
This
model,
model for
these generators
These values
constants.
was then used to minimise
unknown element
0.00
It was suggested that
were added by the
in Section
-12.80
155.00
elements of which two elements were voltage
sources.
values
specific
-0.30
to these measurements, PRL provided
and five
current
0.00
and near 180* to about ±2*11.
This model, which is shown in Figure 5.1,
the
-12.50
169.90
The accuracies
would be a good starting
0)
node
-0.20
each of the
10 MHz except
available.
to ± 0.50,
suggested
-12,50
0.00
of a Lateral
for
Modulus
Phase
dB
degrees
178.10
0.5 NHz was not
Parameters
s
22
dB and phase in degrees and are given at four
In addition
they
-52.00
76.30
t O. S dBII for
as "about
certain
-57.20
84.00
0.5 DIHz to
at 0.5 MHz was not
given
-64.70
the measurements
terms of modulus in
in the range
Modulus
Phase
dB
degrees
Phase
degrees
88.70
S12 at
of
S
21
---
-1.40
-0.10
S
S12
which
suitable
is
the lateral
p-n-p
in
are given
technique
function
error
starting
shown in
should
and so a further
search
overall
con-
values
Figure
transistor.
5.2,
of
des-
for
the
was then
S6
2
c
1
No. oF nodes =i
No. oF elements
(2-3) = 3.53E-03
(T :- 0)
(V across nodes
(0-3) = 2.07E-05
(T = 0)
(V across nodes
S
P
S
31)
S
9ase
Collector
Substrate and Emitter
Node I
Node 2
Node 0
Figure
5.1
Simple
Model of Lateral
P-N-P Transistor
2
1
b
C.
S
g (2-3) = 3ME-03
(T = 0)
1)
(V across nodes 3
(0-3) = 2.0? E-05 S
(T = 0)
(U across nodes 31)
G (1-3)
G. 12P--10 S
C (1-3)
B. GSE-02 nF
S. 22E-OS S
G (2-3)
C (2-3)
S. OGP--04nF
C (0-3)
G. 77E-03 S
C (0-3)
3.71E-06 nF
Node INode 2Node 0-
Figure
No. oF nodes =i
No. oF elements
5.2
Initial
Base
rollector
Substrote and Ernitter
Model of
Lateral
P-N-P Transistor
57
5.2.
Given a set
was for
function
algorithm
tion
of how good the model was.
function
the objective
were obviously
were also
obtained
to these.
the exit
Therefore,
Finally,
version
of the
The calculation
tal
condensation
following
PRL had requested
was decided
the
given
choice
real
frequency
former
for
and imaginary
parts
was that
by weighting
s
are also
parameters
discussed.
of a model were
used by the author,
3.1.2.
the
y
and used for
parameters
using
and to then use the conparameters.
s
done by the method of pivo-
3.1.3,
the examples in the
and all
s
s
the
or whether
seen in
errors
in
favour
the
at each of
be in terms
should
and it
There was a
parameters.
functions
parameters
directly
be calculated
should
the
One advantage
equally
be fitted
parameters
functions
error
of the
be matched.
s
was later
each of
the individual
of whether
phase should
the
error
values
algorithm.
method.
that
individual
that
model would be
to obtain*the
in Section
used that
chapters
3.2.2.
yý parameters
as described
any con-
been stated
by the optimisation
of the
in Section
given in Section
formulae
stage
the model
of
whether
has already
was to calculate
chapter,
the method of dillambro described
to decide
of
Function
The method first
the examples given in this
values
Gauss Newton algorithm
of the
in Chapter 3.
derivatives
partial
it
first
and a
an indica-
to give
The element
minimum had been found
criteria
first
was necessary
Techniques for the calculation
discussed
the
an acceptable
view,
of the Error
Construction
5.2.1.
it
be applied
when a local
4.2,
5.1
had chosen the Gauss
author
required.
but
in the author's
that,
in Section
the
Values
by the optimisa-
requirement
to be constructed
Since
the variables,
should
straints
an error
described
first
the
Model Element
as shown in Table
parameters
5.2,
tion
Newton algorithm
s
of measured
model as shown in Figure
trial
to Optimise
of an Algorithm
Construction
of
the modulus
of matching
real
the
and
the
and imaginary
58
for
s
of the
parts
in
errors
the modulus
each of the
s
it
parameters
would
important
the
most
emphasise
that
phase.
tive
errors,
that
exists
3600
that
absolute
a similar
while
different
gives a totally
absolute
there is negligible
in a modulus of
0*
rela-
gives a
different
A slightly
in decibels.
would feel
Since the
value means that
that
than an error
an error
errors
in the measurements by PRL, it
the weighted absolute
errors
would be suitable
ldB
of
in
ldB
of
In view of the stated
error
than
where the paradox
a large negative
was less significant
in the
error
in a phase angle close
error.
gain and many engineers
-5dB
10"
to a
error
is stated
unit,
to
PRL, were such
with
were more appropriate
relative
a logarithmic
and for
of the model in question.
in a phase angle close to
error
the
the weightings
the opportunity
based on discussions
errors
be
could
frequencies
an engineer
give
of operation
author,
a small. absolute
is basically
different
weights
to consider
by allowing
and that
the
for
the modulus when it
occurs
problem
decibel
be preferable
would
in the case of the phase errors
particularly
error
the phase in degrees
in the modulus was equivalent
was felt
large relative
to
1 dB
of
an error
It
region
chosen by the
The weightings
it
the appropriate
choosing
and in
for
individually
to be stated
on these
that
of
and degrees
of decibels
in terms
errors
in. decibels
was felt
it
However,
avoided.
the problems
parameters
SdB
.
was decided that
for both types of individual
functions.
The overall
error
function
to be minimised
individual
the
the
of
weighted
squares
of
sum
of each of the
s
parameters
was then made up of the
in the modulus and phase
errors
at each of the frequencies
for which measure-
ments were given.
In the early
and
89.600
stages,
were given for
for
ease of programming,
the modulus and phase of
these values being chosen by eye from plots
the initial
of
32
problem was one of minimising
individual
functions
in
8
of
values of
s 12
S12 against
at
-77.40
0.5 MHz
frequency.
a sum of squares function
unknowns
(the
elements
in
dB
Figure
,
Thus
made up
5.2. ).
59
of the Jacobian
Generation
5.2.2.
first
When analytic
here,
then numerical
author first
derivatives
of the Jacobian
estimates
displacement
k ýj,
d
where
(S. 1)
during
the value
of the value
af
1
(X)/ax
prohibitive
10-
3=
was based on the reasoning
7
iI
large
to be of the order
jxjj
of
values
One further
2.6x
will
i
be 10-
be elements
which
Figure 5.2,
this
applied
of Gill
raised
have values
provides
values
reduces
should
the
forward
already
accuracy
recommended
I
,
a fixed
who suggest
of the computer in use, assuming
by this
smaller
a further
differences
of the Jacobian
if
Sx
-
(5.2)
and Murray70,
also
choice
in many models
Since
The use of central
can be reduced
However,
for
allowed
the possibility
of
the minimum value
of
.
to the element and frequency
each element
To do this
times.
the value of
(1 + Ix 1)
i
but which
unity,
point
7.
computing
equation
This value was given by
small value governed by the word length
Ix
using
can be checked.
3
accuracy.
gave sufficient
2 6x
which
(5.1)
and recalculating
which proved that
were obtained
by Henderson"
then
(X - ax)
d
of
of
run produces
an optimisation
some results
dxk = d. x k=j
y
2 6x.
reducing
the
accuracy
,
The
to evaluate
such that
is some small value,
3x.
By progressively
so that
is defined
fý (X + ax) -fi
af. (x)
1--1----.
can be used.
matrix
differences
dx
vector
as was the case
are not available
experimented using central
afi (ý) / axia
and 6xk0O,
Matrix
than
is
that
of bipolar
this
reason for
as seen in
The number of
differences
are used since
of the estimates
Although
the
will
example
in
in Chapter 3.
two function
matrix.
be available,
there
the use of the factors
values described
requires
transistors
function
the
the use of
of the Jacobian
evaluations
evaluations
current
forward
matrix,
for
function
differences
Brown and
60
them to be adequate
DenniS71 found
in methods
for
sum of
squares
function
necessary
for
model elements
to only
minimisation.
of Constraints
Application
5.2.3.
it
Although
feasible
take physically
if
they
is not
initial
one of the
model elements
tive-valued
It
model elements of capacitance,
g Vejwr where
is
V
then A, negative
the
seemed reasonable
resistance
and inductance
controlled
current,
direction
to take both negative
constant
gressed.
So that
the value
should
Of
always
T
that
merely
be positive.
However,
never
that
The time
value
indi-
there were many instances
compensated
for
the time constant
could be that
for
stage in
a particular
elements
a positive
which
were not
time
were cases where a positive
became negative*as
the value
was
correct
values only.
since it
there
by
constant,
value.
decided to allow
values
same constraints
should
time
the current
a positive
was found that
Certainly
stage
the
that
with
the
that
to positive
The author
stage
in the model.
was decided
to the author
the model is sufficiently
and positive
at any particular
at an early
it
a nega-
should be constrained
is
T
and
value was the optimum value
the development of a model.
yet
included
be expected to take a small negative
However it
a time delay.
present
to that
could be restricted
g
where a small positive
value
that
sources represented
would merely indicate
g.
T would normally
constant
voltage
be assumed that
could generally
cating
controlling
value of
in the opposite
the value of
which
therefore
With regard to voltage
it
significance,
was
values on.ly.
to take positive
flowing
exception
by PRL, who were most insistent
have some physical
capacitor.
models more acceptable
An interesting
elements.
component
models provided
should
find
most engineers
values,
the
can visualise
strictly
could
to be optimised
be more negative
the model development
be applied
should
than
to
all
the variables,
be (1 + T)nS.
-I
ns
then
pro-
Since
(1 + T)
61
Thus,
a logarithmic
only,
values
transformations,
other
situation,
success
at Leicester
scaling
exhibited
4.2
some details
6
corrections
part
of the present model and that
for that particular
Id
was set at when iI<
model.
10-8
the
convergence
of
i=1,
one of the failure
the model was not
then
become necessary
failure
should
termination
termination
Obviously
by all
characterised
becoming negligible.
the elements were all
The successful
6
if
necessary as
been ob-
the optimum element values-had
n , where
the
termination
criterion
is the vector
of correc-
domain.
in the logarithmic
that
termination
suitable
and that
to study
the model,
of the optimisation
then it
modes occurred
it
should
the
algorithm
could be
be modified.
element
values
in order
It
and the
to determine
would
reas-
what
be taken.
Example of Results
5.3.
Applying
in
aided
some
the natural
by Henderson.
categorised
then one could say that
occurred
action
that
of the various
were given
to the element values
this
ons for
found
transformation
the one aimed for was the successful
If
be applied
could
algorithms.
modes of the Gauss Newton algorithm
tions
Although
has been used with
and Krzeczkowski6
University
In Section
tained
transformation,
positive
Criteria
Exit
individual
be applied.
could
transformation
by the use of this
optimisation
5.2.4.
transformation
logarithmic
to take
were constrained
such as a square
the
in this
his
the variables
all
since
Figure
was that
5.2,
there
the algorithm
the
algorithm
described
failed
to converge.
were too many successive
the maximum modulus
correction.
in Section
This
increases
failure
5.2 to the model shown
The reason
in the
rate
mode, which
for
of
failure
increase
we will
refer
of
to
62
At this
examples.
ations
fact
mode 1, was in
as failure
before
the most common mode encountered
failure
stage,
the optimisation
mode 1 had to occur
was being
maximum modulus correction
to
applied
this
between nodes 1 and 3. , At termination
11
10
S while the
S to 9.91 x 106.12 x 10from
had
domain
changed
mic
the other
to its
The author
the optimisation
model and restarted
in value.
which was increasing
current
failure
mode 1.
On restarting
The value of 9
tending
was
0-3
the model invalid.
was felt
that
However, there
On attempting
tion.
its
reversed,
the model.
to optimise
was still
tended to zero,
on applying
the optimisation
model is
shown in Figure
failed
the algorithm
this
was C
3-0
element should
one of the
caused a further
to vary,
As there were two
to zero.
the question
flowing
the element values
value still
_3
from the
removing one of them would not make
should be connected with the current
generator
progression
element
the algorithm,
both of which had been allowed
in the model it
generators
Again
was decided that
It
also be removed open-circuit.
generators,
removed that
algorithm.
all
would make G,
this
This time the element causing the failure
mode 1.
in failure
this
from
logarith-
Meanwhile,
Taking
transformation
in value
in the
term
unchanged.
therefore
the conductor
x 105 .
-4.95
iter-
example the
had reduced
correction
to
logarithmic
because of the
limit,
i. e. open-circuit.
zero,
-8-01
were virtually
values
element
X 104
this
G
1-3'
element
element
the
on 10 successive
and in
terminated
algorithm
in all
it
therefore
of whether the
in the opposite
with the generator
too was removed from
to this
algorithm
direc-
model, convergence
was obtained.
The new
values of the remaining
optimised
values,
overall
errors
ute
error
change
function
are shown later
errors
11.02*
largest
the
of
about
20% being
reduced
in Table
5.2 and it
in the phase.
It
elements are quite
had been
have been reduced
to 6.52*
5.3.
from
from
in
384.2
C
can be seen that
similar
At
2-3*
to
114.9.
can be seen that
0.55 dB to
the
to their
this
original
stage
The
the
individual
the maximum absol-
0.43 dB in the modulus
and from
63
I
1. ISE+02
b
No. oF nodes =4
No. oF elements
g (2-3) = 3.94E-03 S
(T = 0)
C
G
C
G
(V across nodes 1)
(1-3) = 1. OSE-01 nF
(2-3) = i. 67E-OS S
(2-3) = 3.90E-Oi nF
(0-3) = 6.63E-03 S
S
Base
Collector
Su6strake and Emitter
Node INode 2Node 0-
Figure
Model at a Local
First
S. 3
The question
better
tion
Minimum
of why a model containing
fewer elements should be
than one with more elements could be asked.
of this
would be that
better
be
only
firstly,
a model containing
fewer elements if
than one with
types and values and in the correct
the right
element values
the original
in this
The author's
more elements would
the extra
positions,
elements were of
and secondly,
example had been optimised
the elements had optimum values of zero,
while
explana-
a third
and two of
had an optimum value
of infinity.
5.4.
Improvement
of Models
Having found a model which gave a minimum value of the objective
functionp
it
improved.
a model giving
improvement,
then became necessary to consider
It
would not be expected that
a local
therefore
'
the removal of any elements from
minimum of the objective
the addition
how the model could be
function
would make an
of elements must be the only course of
64
There are many strategies
action.
If
tor
that
insert
that
not
should
we
and
the most likely
inductors
of all
to replace
any current
any additional
generators,
reason for the presence of inductance
physical
genera-
than at the terminals
other
con-
Other restric-
structure.
has been removed, we should not insert
then
as possible
we should first
the existing
we could attempt
although
the addition
now,
Therefore
nodes,
on adding elements within
might be that
tions
for
we want the model to be as simple
we assume that
we should avoid adding extra
centrate
be employed
could
be considered
some of these will
of elements,
that
since
would be
lead inductance.
It
might
be that
the
engineer
those elements for which some physical
be given.
Certainly,
to as "intuitive
the engineer's
preconceived
could preclude
the development of another
This factor
model.
For the
first
modelling
function
was generally
caused by the corrections
or if
one of the other
element
and more accurate
the
by Lucal given in Chapter 3,
the
involved
If
result.
was reduced,
the
of the
elements
algorithm
If
and the
converged
the algorithm
mode 1 occurred.
modes occurred
addition.
value.
it
same type
the optimisation
restarting
to the new element
would not make a suitable
of
then the new element was left
because failure
failure
used an intuitive
author
values
model and another new element was tried.
verge it
a model
to add one element at a time to the
based on the
Each attempt
and checking
error
overall
for
that
by HegaZi7 to discover
the
exercise,
by attempting
value
model at an arbitrary
algorithm
is possible
topology
acceptable
to the set of equations
approach and experimented
already
presence could
and one which Nabawi and
concepts of a suitable
to the failure
only
found by SavageB.
which was later
present.
their
However, it
modelling".
contributed
six node realisation
for
strategy
NicolS72 refer
to be present
there
explanation
is a popular
this
like
would
failed
Usually,
If
this
to conthis
was the
was assumed that
There
in the
was one case,
was
case
the new
which
can
65
be seen in the example in the next
by one of the existing
in and the
optimisation
algorithm
The entire
mode 1 was caused
In this
case the new elem-
failure
the
causing
element
terminated
then
where failure
in the model,
elements
ent was left
section,
successfully,
to the first
additions
sequence of successful
model are
in the next section.
described
Example of Results
S. S.
failing
After
to replace
between nodes 1
capacitance
from the
apart
successful
each of the elements removed from the
singly
was decided to attempt
model, it
initial
0
and
conductor
to add elements of conductance and
lpF
at values of
too,
These values
to converge
F
is
the
onto
given
pointed
-4
10
and
local
minima.
at each stage
and 0
The order
can be seen that
errors
0.5 MHz.
The function
against
values
later
and it
three
adjusted
13
of the overall
it
at this
these
from
for
be
should
stage included
values
include
func-
error
It
cal-
iterations
was reduced
new elements.
extrapolated
do not
quoted
to optimisa-
prior
11 and
can be seen that
the
and conductance
to the optimum values
close
The value.
the
small amount
respectively
being minimised
calculated
individual
S
took between
of these
the function
as the weightingswere
It
5.4.
were reasonably
114.9 to 56. SO by the addition
out that
2
between nodes
Newton
Gauss
the
algorithm
and
culated
These were all
The new elements of capacitance
between each addition.
tion.
0.
and
elements changed by only a relatively
values of the existing
were inserted
2
and nodes
in which they were added is shown in Figure
tion
was removed and the
S
the
12
individual
at
errors
to exclude them from the objective
function.
Next,
external
first
it
nodes.
of these
was decided
to attempt
The two successful
was originally
inserted
to
insert
attempts
inductors
are shown in
at a value
of
4nH
at
Figure
each of the
5.5.
The
and was optimised
66
2F=i.
C.
iSE+02
No. oF nodes
No. nF elements
(2-3) = 3.91E-03 5
(T = 2)
(V across nodes 13)
C (1-3)
i. OSE-01 np
C (2-3)
4.67r--QS S
2.90e-gi nF
C (2-3)
G.G'jE--03 F
G (0-3)
(S
(I
Q
F
b
No. nF nodes ai
No. oF elements =6
o (2-3) = 6.35S-03 5
(T = 01
(V a,:ro.; s nodes 13)
(1-3) =I. 99p-ol np
G (2-3)
4..jgE-GSS
C (2-3)
nP
r1 (0-3)
4,91E-03 S
C (1-0)
2. ýSF-92 nF
f5
F-S.
SGE+gl
No. nP nodes m4
No. oF etements z7
=i . 36F-02 S
(U across nodes 1)
C (1-3) 21.47E Ol nP
G (2-3) = 4.29F--OS S
c (2-3) - 4.02E-04 nF
G (2-3) - 3.23E-03 S
C (1-0) a 3.20E-02 mp
G (1-9) = 2.1! E-94 S
9 (2-3)
F=S.
GSE+0I
No. nF nodes =I
No. nF elements a8
C
G
r.
ri
(2-3) - &. SSE-03 S
(T = e)
(U across noues
(1-3)
(2-3)
Q-3)
(0-2)
1.2SE-01
4.36p-os
4.10C-94
3.48F-03
nr3
nP
S
3.03S-02 np
VE-91
S
, Q--O) = i. 9se-33 )F-
os
a
Made iliode 2Node
Figure
5.4
Addition
a-
Base
CollectmSubstrate
of Elements
*n4
Emitter-
Between Nodes 1 and 0 and Nodes 2 and 0
b7
2
F=i.
95E+gl
C
No. oF modes =S
No. oF elements
I. IGE-02 S
9 (2-3)
(T=
(V across nodes 13)
C (1-3) = 1.53F-01 np
G (2-3) = 1.3sr--OS S
C (2-3) = 4-19r--04 nF
G (4-3)
3.14E-.-03 S
C (1-4)
3.42E-02 nF
G (1-4)
2.27E-04 S
C (2-4)
i-SOE-03 nF
L (4-0)
1.27E+01 nH
0
s
I
b
2
c
F :-2.2SE+01
No. oF nodes =6
No. aF elements
9 (2-3) = 8,79E-03 S
(T = 0)
0 So
Node INode 2Node 0-
Figure
5.5
Base
Collector
Substrate and EmIttex
Addition
of
Inductors
(V across nodes sl)
C (S-3)
1.31E-01 nF
C (2-3)
I. ISP-OS S
C (2-3)
4.04E-Oi nP
G (4-3)
3.33C-03 S
C (5-4)
2. G?E-02 nF
G (S-4)
1.2se-04 S
C (2-1)
2-IOE-03 nF
L (4-0)
G. 4SE+Oj nH
L (5-1)
3. S2E+02 nH
tJote: The values given here
,
were later Found to be in error
due to a FaýAt in the analysis
routine
in use at the Lime
68
'first
inductor
these
cases converged
The overall
in dillambro's
and in addition
this
C
(where,
3-4 ,G 2-4
between nodes
the
I
for
element
to
additions
example,
3)
and
added the
was
both
of
Gauss Newton algorithm.
22.45.
to
At this
in Figure
point,
was detected.
to earlier,
5.5 was rý-optimised
the value
values,
model was reduced
and
changed by
However,
referred
second model
in the
different
Then several
inductor
the
of
50 nR
of
elements
five,
of over
routine,
analysis
to changes
for
function
existing
had now been reduced
of the error
correction
at a value
second
11 iterations
within
function
error
the error
error
by a factor
changed
the
but when the
two,
of up to about
factors
In eachcase,
to 352 nH.
was optimised
After
The second was inserted
of 12.7 nH,
to a value
of the overall
12.27.
including
were attempted
G
signifies
1-3
an element
but each of these failed.
G
1-3
of conductance
On attempting
to add
to element C
C
the failure
mode 1 was due to the corrections
5-4 '
1-4
C5-4, was therefore removed and on convergence the overall error function
had been reduced to 9.70.
have been considered
one last
that
current
The values
of many of the elements
without
5.6.
However,
difficulty
The individual
of these
in the opposite
was flowing
seen from Figure
achieved
the current
This last
model.
are considerably
also
smaller
than
attempt
to that
of
had been
that
in
the
although
initial
the Gauss Newton algorithm
error
function
stated
was reduced
measurement
model.
as can be
considerably
model are shown in Table
the
might
However,
was successful
changed quite
convergence
of this
generator
direction
and the overall
errors
S. 6 and it
improvements had been made.
sufficient
attempt was made to replace
initial
from
the
removed
the
This model is shown in Figure
was
to 6.13.
5.2 and most
errors.
69
S
1
b
No. oF nodes =6
No. oF elemenLs = 10
9 (2-3) - 7.9SE-03 S
(T = 0)
(V across nodes sP
C
0
C
G
G
C
L
L
C
(S-3)
(2-3)
(2-3)
(4-3)
(5-4)
(2-4)
(1-0)
(S-1)
(1-4)
=
=
=
=
=
=
=
=
=
I. IBE--Gl np
i. 17F-05 S
4.20E-gi nF
3.48G-03 S
1.29E-04 S
2. SGE--03 nP
8. SSE+01 nH
S. 99e+02 nH
2. SOE--92 nF
5
1
G. 13E+00
No. oF nodes =6
No. oF elements
9 (2-3) -- I n; --04 S
(T = 0)
0
C
G
C
G
G
C
L
L
C
9
(U across nodes '3)
(S-3) = i. 9GE-03 mp
(2-3) = 4.43E-OS S
(2-3) = 8.19E-gi nF
(4-3) = 2.? iE-03 S
(5-1)
I. ISE-0i S
(2-4)
2. Gie-03 np
(4-0)
I. SSE+02 mH
(S-1)
1.9"E+03 nH
(1-i)
3.23e-02 rip
(3-4)
2,72ý-03 S
(T =V
(V across noces 13)
Node INode 2Node 0-
Figure
5.6
Further
Base
Collector
Substrate and Emitter
Improvements
70
sss
Frequency
Model
MHz
IFig,
S, 2
'Fig.
5 3
.
Phase
degrees
Modulus
dR
10
_Q,
O's
2,0
5,0
10ýO
-0,14
-0,02
O. OS
i
O. S
2.0
5.0
10.0
0.09
_
ll
_O,
0.10
0.38
!Fig.
6.6
l(Final
Model:
Table
-0.05
0.04
-
-1.48
0.00
0.10
0.00
1.03
0.04
-0.13
0.02
in the S Parameters
Final
of the
Description
5.6.
0.00
2,19
1.97
-0.19
of Models
Modelling
a more automatic
was felt,
it
gave,
or not,
was suitable
puter
run for
ately,
but it
indication
was extremely
each possible
ones.
successful
should
cover
rejecting
those
In addition
it
stage,
as many possible
felt
was
However, it
possible.
so that
algorithm
that
was felt
placements
in
be
batch
have
to
run
would
program
it
was possible
prejudices,
-0.14
-0.07
0.07
0 24
.
-0.05
0.01
0.07
0.03
.
0.00
0.00
0.00
0. 00
available
of
a Lateral
P-N-P Transistor
was felt
manner it
that
that
The Gauss Newton algorithm
of whether any particular
model
time consuming to do one batch com-
was required
to
were unsuitable
continuously
and retaining
As the
as was practicable.
mode, user
interaction
each intermediate
attempt
the
improvements
the attempted
that
Unfortun-
execution.
on-line
would
not
computer
be
stage should be stored
before manually
restarting
the modelling
This way, the model development would not be inhibited
although
-0.19
-0.27
-0.42
-0.74
0.67
0.13
0.06
5
-0.0
0.52
for the user to choose a model from any particular
it,
modify
perhaps
and
process.
-0.19
-0.26
-0.41
-0.71
stage in the development of a model.
modelling
improvements,
model
-1,89
12
-S,
-11.02
Phase
degrees-
0.03
0.03
0.03
0.03
-0.49
-2.00
-6.52
the program was too large and slow to permit
The author's
0,17
-0.30
-0.19.
0.07
0.43
and error
could be developed.
a very reliable
Modulus
dB
Algorithm
Having developed a model in a trial
algorithm
Phase
degrees
0.03
0.03
0.03
0.03
of S12 at 0.5 NSIz was not
A measurement
Errors
0,27
0,34
0,46
0.46
-0,17
-3.41
-9.66
0,00
0.08
-0.21
-0.66
0.09
O. OS
Modulus
dB
0.00
-0,36
-0.21
-0.71
-0.36
-2.21
-0.04
0.04
0.00
5.2
0ý00
O, SS
0,20
-0,37
-1.33
-1.60
-3,31
0.00
0.5
2.0
5.0
10.0
Phase
degrees
Modulus
dB
22
21
12
the user could still
exercise
some control
by the user's
over the
71
were introduced
New criteria
process.
model would be suitable
developed.
elling
The final
The first
stages.
ting
to find
a local
the addition
we assume that
ately
Obtaining
Stage 1-
is
algorithm
to the
applied
is identical
termination
initial
sooner.
increase
in the maximum modulus
and the mode we could
by the
the number of iterations
obtaining
of corrections
earlier
The third
is
stage
can be repeated
is obtained.
for
altern-
These three
model.
described
local
minimum, the Gauss Newton
for
The criterion
in Chapter 4.
than those described
are more stringent
correction
now call
successful
for
has increased
failure
if
the
are
rate
of
on 5 successive
mode 2, is
Gauss Newton algorithm
failure
deemed fatal
exceeds
if
50 without
convergence.
Whichever failure
elements
stage
nu! nherpf, nodes. since
mode 1 is now deemed to be fatal
Failure
remov-
necessary
The second
the two most common reasons
detected
iterations
if
Local Minimum
the first
failure
4, in particular,
separate
model and attemp-
existing
two and three
the First
to that
for
The criteria
in Chapter
the
mod-
in more detail.
to find
In attempting
three
obtained.
accuracy or complexity
now be described
stages will
5.6.1.
Stages
into
function,
we want as few nodes as possible.
the required
until
within
finaI
1981.
initial
the
a minimum is
of new elements
of a new node.
the addition
of taking
minimum of the objective
from the model until
ing elements
involves
consists
stage
in
a
of elements
of the
can be divided
algorithm
modelling
the addition
Some details
by the author73
were published
whether
more quickly
for
and techniques
or not
and nodes to a model were also
algorithm
indicate
to
it
might
6
mode occurs the next step is to examine the vector
(which is in the logarithmic
be best
experimentation
it
domain) to determine
to remove from the model.
was found
that
removal
Throughout
of the
elements
all
which
the
having
the
72
maximum modulus corrections
the Gauss Newton algorithm,
generally
If the element being removed is a
tion)
(or similarly
is positive
is assumed that
then it
having.. a- negative
ection)
of this
will
the deletion
the removal of any other
element.
of a node although
provided
element in parallel
was set
arbitrarily
quite
new elements
is
with
a
by a resistor.
replaced
discussed
Again this
is less likely
than
at
capacitor.
The initial
1K 0.
The setting
in more detail
that
with
there
a resistor.
of premature
should be removed open-circuit
element is in parallel
one other
as any
nodes, elements
in order to reduce the possibility
that
at least
it
an inductor
with
collapse
then
was decided that,
It
of a model, elements of capacitance
is
the
elements which are
occur only at one of the external
elements in parallel
It was also decided that,
is no other
corr-
case necessitates
type should always be removed as a short-circuit.
be other
it
the removal of a reshaving a positive
conductor
is not so straightforward.
would. normally
necessitates
and
with regard to the removal of elements of inductance
The situation
inductor
correc-
to an open-circuit
This latter
with the offending
corresponding
having a negative
Conversely,
(or
correction
involves
node
and
a
of
removal
connected in parallel
and its
the element is tending
is achieved by a short-circuit.
and capacitance
and
from the
continuing
resistor
conductor
a
is
removed open-circuit.
the
element
so
istor
to
at the time of the failure.
element values current
correction
aim is
the maximum modulus correction
from
the
the
element giving
model
remove
to then restart
Thus the basic
convergence.
aided
that
If
there
is being removed
value of this
of the
in Section
it.
with
initial
resistor
values
5.6.2.
There are a number of cases when short-circuiting
an element is
These are
deemed to have failed.
if
any of the
Cii)
if
any current
(iii)
if
the
external
controlling
short-circuited.
nodes are connected
generator
voltage
is
together
short-circuited
of
any current
generator
is
of
73
it
Furthermore,
generator
current
uation
where removal of a current
then it
should be done manually.
ised
flow
in the
scheme is
Stage
to remove only
when there
were a number of elements
restart
of several
elements
removal
cycles.
The criterion
removal
domain),
this
element
following
If,
prescribed
5.8.
devised
Although
the maximum modulus
it
modulus
was found
for
multiple
16ij
powers of 10 of all
If
that
the
corrections,
failure/
save on the number of
could
element
removal
was
j+2
> 10
where
,
the corrections
j
is
(in the
6
none of the corrections
then only the element having the maximum modulus
the removal scheme already
elements are successfully
then the remaining
time of entry
to Stage 1 and the process restarted,
abandoned.
If
the user to first
is made to
If
elements are given their
values
otherwise
model for
model, perhaps based on the initial
errors
this
at the
Stage 1 is
should happen then the best course of action
check the initial
elements.
none of the
modulus correction.
is successful
this
described,
removed, then an attempt
having
highest
the
the
second
element
remove
additional
summar-
is removed.
correction
different
Figure
causing
large
causing
should be removed.
criterion
option,
is
of elements
shown in
those elements whose modulus correction
logarithmic
satisfy
1 is
at one time
removal
the average of the positive
there was a sit-
the Gauss Newton algorithm,
correction
all
if
that
was the only remaining
the
the
and to then
that
as most models
5.7.
scheme for
The overall
the basic
in Figure
chart
particularly
generator
above for
The method described
a
since the presence of current
was felt
It
generator.
for
not be permitted
would
to a model,
critical
only one current
contain
it
that
to be removed automatically
is normally
generators
was decided
and to then try
model but containing
is for
a
some
74
Figure
5.7
Scheme for
Removal of Elements
75
STAGE I
lGauss
Newton
I
Yes
SuccessPuL
ocaL minimum
i
No
exit
J=
Average power oF positive
power corrections
CLIM = 10.0*m((J+2)
CMAX = Max moduLus correction
> CLIM
Yes
N-o
Remove eLement
giving CMAX
Y
Remove aLL eLements with
> CLIM
moduLus correction
At Least one
cessFuL removaL
No
Remove onLy the eLement
having the second highest
moduLus correction
-essPu Lýýo
Yes
I Reset remaining eLements
to vaLues at entry
to STAGEII
Figure
5.8
Scheme for
Stage
1
FaiLure
exit
76
Stage 2-
5.6.2.
Having
obtained
F
min'
function,
the
parallel
Elements of inductance
stage.
inductors
The author also felt
model, and their
positions
that
The user always has the ability
and then restart
the modelling
The problems
elements of capacitance
had developed
expressions
tual
which
elements
matching
felt
techniques
Savage8 however,
,
model.
In this
are systeriatically
scheme is
had been utilised
shoum in
made to
nodes where an element of that
Figure
the
of
Cutteridge74
using
of vir-
coefficient
by Krzeczkowski6.
of virtual
scheme for
been
placements
the placement
synthesis
the use of this
each type
insert
possible
for
values
to network
rejected
scheme, taking
stage
have already
should be attempted.
the optimum
developed a similar
This
in a
sources to
of current
modelling
in Stage 2, all
Gauss
Newton
based
the
on
corrections
one
independently
should be present
voltages
intuitive
with
were applicable
eventually
ideas about the
they should not be added automatically.
and resistance
and which
was
process.
that
for
it
to modify a model at any particular
associated
The author
discussed.
in
coupled with the problems of
and controlling
led to the decision
Therefore,
to have strong
This factor,
inserted
a new node at Stage 3.
sources that
current
positions.
choosing suitable
be inserted,
users were likely
controlled
number of voltage
are expected to occur
node.
should only be added with
that
and resistance
to be successfully
modes and are unlikely
stage
the number of
only elements of capacitance
with another element even at an external
decided that
The aim of this
increasing
the model without
The author decided that
only at the external
minimum of the objective
may now be entered.
second stage
should be added at this
a local
a model giving
elements into
is to add extra
nodes.
Addition
Element
in
technique
of
The author
elements.
the addition
favour
of elements to a
5.9.
of new element
new element
in turn,
between
type does not already
exist.
attempts
each pair
If
of
the addition
Figure
5.9
Scheme for
Stage
2
78
is successful,
for which will
the criteria
remains in place,
until
was found that
it
it
to be added at one stage,
even if
might be successful
strategy
might
more laborious
stage
in Section
5.7 -
from a model at one stage
have been to evaluate
of the best until
and addition
felt
However, the author
be added.
at a later
Indeed,
placements and to only add the best of these,
cycles of evaluation
element has
later.
An alternative
possible
nor a res-
a capacitor
a particular
there is an example where an element was deleted
and then re-inserted
These
The reason for these repea--
has changed slightly.
when the model topology
then the element
former topology.
neither
has been added in one complete cycle.
ted attempts. is that
failed
its
the model retains
otherwise
placements are attempted repeatedly
istor
be given later,
and that
that
the same final
this
topology
each of the
followed
by further
no more elements could
latter
would be
option
would be obtained
in most
cases.
Obviously,
with
so many possible
to detect
of the algorithm
efficiency
placements being attempted,
the success or failure
the
of each
addition
is paramount.
It was observed that,
inserted
into
model, the moduli of the Gauss Newton corrections
an existing
to be applied
to the existing
elements were usually
to be applied
modulus of the correction
much smaller
to the new element.
upon by the scheme devised
was capatilized
when a new element was
for
than the
This fact
element addition
summarised
in Figure 5.10.
Thus,
corrections
when attempting
to add a new element,
If
are calculated.
the maximum modulus correction,
normally.
However, if
new element,
then this
in the logarithmic
first
Gauss Newton
the new element does not give rise
then the Gauss Newton algorithm
the maximum modulus correction
correction,
the
directly
proceeds
does apply to the
up to a maximum absolute
domain, is applied
to
value of ten
to the new element.
The
79
ELEMENT ADDITION
lCaLcuLate
0No
No
Gauss Newton corrections
I
L42
have
Does new eeLement
correction
xmmod col
max
m
Yy
ess
CCC Correction
Orr
10
Ensure
sm
mod C4
AppLyjC C to new eLement
RecaLcuLate
Gauss Newton corrections
a--No
Does new eLement have
max mod correction
again
Yes
Cthse
Is'0 the new correction
1
9same sign & Larger
than bePore
No
YY
ee
ss
ILaR
s it aR
tending 3 to zero
No
Yes
Yes
Remove
appropriate
eLement
Continue
Gauss Newton
No
iLed du
ýFaiLed
duee to
tc0
.,:,.
new eLement
-wL
eLem
ment
No
Yes
Y s
: onverged
Yes
2 eLements
aLready removed
and F> FMIN
->-<
N
No
Is iaR
it aL
to zerro
tending
z
Yes
No
FaiLure
exit
< O. SGxFMIN
I-I
ýSuccessPuL
Figure
5.10
exit
Scheme for
Attempted
es
Element
Additions
at
Stage
2
80
are then
Gauss Newton corrections
is no longer
ulus
is
is
is of the opposite
its
and
sign
same
value is greater
assumed that
the new element is tending
infinity.
If
Newton algorithm
is
the author
both
attempting
by ensuring
that
the starting
has been found
It
even when the starting
with this
that
into
addition
addition
of element
and resistance
was felt
that
is
very
reliable
inaccurate.
the values
Thus,
at which new
arbi-
the new elements should be of such values
on the existing
Alternative
resistance
was 10
the actual
values of each type of element already
the time of the addition,
reasonable.
is
model.
value chosen for new elements of capacitance
.
more
are added, were chosen quite
they would not have too great an effect
the initial
way,
the Gauss Newton algorithm
Gauss Newton algorithm
convergence,
but it
choices
was felt
that
is
it
the model.
failure
of the new element
the
aid to rapid
additional
It
Otherwise,
in this
of
to zero,
and the Gauss
to the new element
convergence
value
is
is tending
values of new elements are quite
elements of capacitance
trarily.
for
is of the
zero or
suggests is permitted
to detect
and to aid more rapid
and it
to the reduced model.
the Gauss Newton correction
By applying
the
then it
to a value of either
the new element is unsuitable
assumed that
quickly,
this
is then applied
or if
than before,
the new element is a resistor
then the removal of the node that
the mod-
however, the new correction
If,
absolute
if
than before,
smaller
the
then
correction
then again the Gauss Newton algor-
sign,
ithm is allowed to proceed.
the new element
Similarly,
normally.
continued
to the new element
correction
correction
the maximum modulus
the one generating
0
Gauss Newton algorithm
If
re-calculated.
that
Therefore,
was lpF
and of
could have been based on
present
this
in the model at
would not offer
any
improvement.
Those cases where the Gauss Newton algorithm
now continue
in a similar
is
manner to Stage 1 except that
allowed
to proceed
in certain
cases
81
is deemed to have failed
the element addition
Gauss Newton algorithm
a local
onto
before
convergence of the
minimum has been obtained.
These
cases are;
(i)
if
of the Gauss Newton algorithm
failure
new element and it
if
element removals are indicated
further
elements have already
of the objective
value for
If
if
is
only
accepted
model
Fmin'
that
the model contains
bution
value
function
F
is
still
value
than
greater
the
accepted model, F
min*
at a new local
converges
the currently
accepted
minimum,
function
the value of the objective
for
than
when two or more
been removed and the current
the currently
the Gauss Newton algorithm
the
to short
tending
a pair- of noJe3
circuit
Cii)
is not a resistor
is due to the
only those elements that
is 4% lower
This
model.
the new
is
to ensure
make a significant
to the model accuracy and to avoid adding an excessive
contri-
number of
elements to the model.
Thus, Stage 2 ends when no further
If
the model is still
in
to add elements
S. 6.3.
Stage
3-
not accurate
such
a way that
successful
additions
are possible.
enough then the next step is to attempt
a new node is
generated.
Node Addition
Savage8 in his
work on network
synthesis
be
introduced
by
nodes
could
which
methods
into
discussed
an existing
a number of
network.
Among these methods are to:
(i)
split
an existing
attempted
element to form a new node to which
element additions
should be made
82
duplicate
Cii)_
T-network
an existing
perform a delta-wye
(iv)
transformation
in place
a T-network
-substitute
of
a negative
virtual
element.
Savage found that
if
indicated
element addition
a potential
ponding proposed negative
the author
felt
that
element type and its
each of these methods which,
two extra
of these,
where
the element should take a
should be inserted
value,
of at least
the generation
that
based on that
value then a T-network,
negative
was the last
technique
the most effective
in its
However,
from (iii),
apart
ignored
elements,
place.
corres-
caused
the most likely
method for the improvement of a model.
If
rather
the current
we assume that
the internal
than disturb
developed at the external
If
involves
again
element,
Thus,
5.11,
is
the addition
of a different
the
author's
the evaluation
be seen in Figure
1.
in series
of only
a model should
and include
alieady
to
the
this
Figure
involves
the best
select
on the placements
elements.
we assume that
to a minimum,
in order
a new
shown in
Since
new
of new elements
can
following.
node is only permitted
at that
elements of inductance,
be kept
is
which
one element.
placement
Expansion at an external
inductor
with one of the existing
method of node addition,
The reSt-Lictions
5.11
is to insert
of only one new element,
of each possible
node and element.
of a new node
of only one new element as in the examples
addition
the number of nodes in
then,
nodes are to be added then a method which
type,
based on the
This way, the addition
nodes.
internal
correct
of the model, new nodes should be
structure
by
be
the addition
achieved
can
in Figure S. S.
model is basically
external
resistance
node.
if
there
Then the
or capacitance
is not an
addition
is permitted.
of
83
STAGE 3
FOR I=192
Inductor
and
at externaL
I
node
Yes
No
Attempt
between
oP each oP
addition
R and C
L,
node
node I and new externaL
NEXT I
FOR I=19
no.
JELement
I oP type
Attempt to add L
iP at externaL node
lAttempt
to add CI
I
IC
LI
6
R
oF eLements
g
Attempt to add C
at the
node
non-externaL
Attempt
to add R
Attempt to add R
at the
node
non-externaL
NEXT I
y successFuL
additions
Yes
best
SeLect
ýSuccessFuL
Figure
5.11
Scheme for
Stage
3
addition
exit
FaiLure
No
exit
84
2.
An element
of resistance
added in series
3.
only
to an external
connected
tor
it
with
may have an inductor
or conductance
if
by so doing
the
does not
node which
:Lrductor
already
be
will
have an induc-
connected.
Elements
may have a capacitor
of resistance
or conductance
of inductance
may have an element
added
in series.
4.
Elements
in series
tance added
S.
An element
6.
A voltage
at the non-external
may have a
of capacitance
current
controlled
of resistance
or capaci-
node.
added in
resistor
series.
may not have any element
source
added
in series.
Further
added is
1.
given
some of these
elements
are
below.
The addition
already
of the way in which
clarification
an element
is not attempted
element
of a particular
type
of that
in
in that
series
if
is
there
branch
of the
model.
2.
The new elements
ment, usually
3.
are only
on the
When an element
is
side
added on one side
having
connected
generator
taken
the new element
Thus, although
most topologies
Stages 2 and 3, some could
take
is
shown in Figure
5.12.
As with
a controlling
then the controlling
voltage
voltage
is
also.
are possible
an inordinate
The method used to'determine
ele-
node number.
at a node at which
is taken,
of a current
across
the highest
of any particular
whether
time
with
repeated
to develop.
a node addition
the element
to
entries
addition
is
successful
at Stage 2, it
is
1
85
to add
Attempt
eLement B in series
with eLement A
there aLready an eLement oF
type B in series with
eLement A>
t
No
F-Inuert
eLement and
ca LC Late Gauss Newton
corrections
ýs the moduLus correction
to either
eLement A or
> 100
Yes
FaiLure
exit
'it
B
Yes
No
Continue
Gauss Newton
aLgorithm
FNýo
onverged
Yes
Have any eLements
Lready been removed Z>
t
r e0s
No
43
Was PaiLure
due to
eLements A or 8
No
-<
--Lý
< FMI ý,
Yes
Yes
No
Remove appropriate
eLement
IF
CFaiLure 7e)
(Su
ccessPuL
exit
exit
Figure
5.12
Scheme for
Attempted
Node Addition
at Stage 3
86
The initial
based on the Gauss Newton algorithm.
one order
of inductance
stage
and the only
(in
correction
the
to the new node,
prediction
logarithmic
domain)
elements
connected
entered
the
provided
finds
and the best
recommendation
It
of these
at which
is that
it
be suggested
further
element
new node.
affect
element
that
additions
However
is
lower
is
stage
be after
be gained
1000.
should
Otherwise,
the
author
"
said
saved for
element
therefore,
rather
using
to
of the
removed and
again
then
minimum obtained
to be successful.
future
reference
Although
the
of Stage 2, in order
number of nodes.
in the vicinity
the addition
is better
Stage 2 again.
by the
the author's
than re-entering
only
it
is
to converge
be accepted,
completion
has found that
fails
the
the Gauss Newton algorithm
is
be attempted
connected
the algorithm
from any particular
to save time,
are attempted
fails
the model should
should
at
the absolute
best
F
the
min ,
than
effort
was not due to either
it
If
if
then used as the new model.
the whole model and that,
additions
If
If
node addition
the maximum benefit
might
for
value
of the elements
the appropriate
to Stage 3, then the addition
entry
user may decide
that
is
is
exceeds
to converge
restarted.
minimum that
Each successful
user
failure
is deemed to have failed.
a new local
before
to either
normally.
to the new node,
the Gauss Newton algorithm
the addition
The initial
of failure
by Gauss Newton,
is
Gauss Newton algorithm
then,
respectively,
to reduce the computational
early
calculated
converge
2.
0.1 pF
new elements
was set at 10 nH .
It was found more difficult
this
100 0 and
at
at Stage
than
smaller
of magnitude
elements
were set
and capacitance
of resistance
for
values
Stage 2,
of the
of a new node can
if
all
permitted
87
the Final
Checking
5.7
As a final
after
section,
the first
instead
time
using
of the trial
routine
C3_,, and then
G
and
1-3
to remove elements
should
be removed.
also
5.3.
C2-0 and R1_0.
in Figure
the element
in
model shown in Figure
quite
different
from those
manually
the
S. S.
given
of
and re-
added the elements
same as the
model in
last
order
in the analysis
value
function,
F,
had produced
in Figure
to that
routines,
were slightly
differ-
was 50.97 compared
Stage 3 and the new element
The values
F,
that
the model shown in Figure
was in a different
error
then entered
and the value
and
the same model as the
of the elements
were,
S. S due to the errors
here,
in the
which had not been recalculated
in the
was found to be 13.50.
trials,
From this
point,
thus immediately
that
this
differently
the model developed
in which r-&3&. the next
was thought
element
function
Thus the algorithm
first
routine
this
addition
error
proceeded
an indication
model was thus
of the overall
The algorithm
the algorithm
with
Owing to the differences
node added was L 4-0*
analysis
5.2,
Stage 2, the algorithm
and the overall
values
56.50.
into
the element
5.4.
ent and the new value
it
by the author,
written
then went on to obtain
This
5.4 even though
Figure
earlier
the models in Figures
stopped
On removing
from there
Continuing
with
,
it
the algorithm,
starting
given
condensation
model shown in Figure
From the initial
Cl-01
algorithm
of the models were calculated
parameters
pivotal
using
was re-
of di Mambro's routine.
place
gO-3
transistor,
p-n-p
of the modelling
version
s
the
in the previous
6 and 7 had been developed,
method used to produce
and error
In addition,
by the analysis
in Chapters
the final
described
algorithm
to model a lateral
exercise,
modelling
5.3 to S. 6.
Algorithm
check on the modelling
the models described
but this
peated
Modelling
generating
from the
step was to add an inductor
an additional
made a significant
node.
improvement,
Wode'l
at
,
Although
it
v%oaalln
at the time
can now be seen
88
P=3.84E+02
No. oF nodes =4
No. oF etements =8
9 (2-3) = -; IE-03 3
0),
,T=
(V across nades 13)
(0-3) = 2.07E--95 S
(T = 9)
S
(I
C(2-9)
Delete G( 1-3)
Add
C(1-0)
C(2-9)
Add new node with L(4-0)
e
(V across nodes ',)
G (1-3) = S. 12E-ig S
C (1-3) - 9. GSE-02nF
G '2-3) = S. 22E-OSS
C (2-2) 2 S. OGE-Oinp
G (9-3) = 6.77E-03 S
C (0-3) = 3.71E-06 np
9(0-3)
R(1-0)
F=1.3SE+91
No. oF nodes =S
No. oF elements =9
C( 1-3)
6 (2-3)
C (2-3)
9 (2-3)
(T =
G
C
C
R
L
=I . 2GE--9l
= i. 64E-03
x 3.62E-01
= 9.23E-03
0)
(V across nodes
(4-3)
= 3.27E-03
(1-4)
w 2.97E-02
(2-4)
!. 39E--9i
7.76E+93
(1-4)
(0-4)
i. 28E+02
nF
S
np
S
30
S
nF
nF
ohm
nH
Add C(I-2) i Delete C(2-3)
Add C(2-0)
Add new node with C(Q-3)
F=7.3GE-+Q9
No. oF nodes -6
Mo. oF elements = 11
C (1-3)
I. ISE-91 nF
6 (2-S)
8.28E-96 S
9.14E-03 S
9 (2-3)
(T = 9)
G
C
C
R
L
C
C
C
e
Add C(2-3)
Add R(4-5)
Node iNode 2Made
Figure
5.13
9-S.
i Delete
C(I-2)
ease
Collector
i:,atrQte,
Modelling
"a
Emmet
the Lateral
P-N-P Transistor
(V across nodes 1)
(4-3) = 3.42E-03 S
(1-4) = 2.8GE-02 nF
(2-1) = I. G?E-94 nF
(1-1) = 9.29E+03 ohm
(0-4) = i. iSE+02 nH
(1-2) = 3.94E--13i np
(2-0) = 2.21F-93 nF
(S-3) = 3.20G-03 nF
89
the change in the value
that
In the final
In this
deleting
and C
2-0 ,
F
algorithm,
modelling
is re-entered.
of
section
C
while
2-3
after
one successful
the algorithm
adding
C
to
obtain
1-2 ,
in the value of
gave a very small reduction
was formed by the addition
the algorithm
F
5.13 together
These additions
of R4-5*
tion
mutually
its
attempted
a value
the additions
of F=
12.26.
at the input node only
and the new node selected
in
This sequence is shown in
made in the final
entry
to Stage 2.
However,
deletion
element
serves
the addition
to confirm
should
additions
at this
of C
2-3
the point
be repeated
late
stage
after
made in Section
5.6.2
that
at each stage
of a model's
development.
V, I
S.??E+00
No. oF nodes =6
No. oF elements = 12
C (1-3) = 1.19e-01 nF
G (2-S) = 4.63E-05 S
9 (2-3) = B-47E-03 S
(T = 0)
(U across nodes 13)
G (4-3)
2.98E--03 S
C (1-4)
2.99E-02 nF
C (2-4)
1.18E-04 nF
R (1-4)
B. iOE+03 ohm
L (0-1)
4. IOE+02 nH
C (2-0)
2. iGF:-03 nF
C (S-3)
2. S7E-01 nF
C (2-3)
3. S3C--Oi nF
R (4-S)
2.44E+03 ohm
'e
Node INode 2-
Base
Coliector
Node 0-
Substrate
Figure
5.14
by
C
followed
by the addiwhile deleting
were to add C
2-3
1-2 ,
It is interesting
C
that elements C
to be
and
appear
1-2
2-3
exclusive.
earlier
with
C
1-2
of an element of capacitance
series with G2-3 , and gave a value of F=7.36.
Figure
Stage 2
node addition,
added two new elements
inductor
of an
to Stage 3, the addition
On re-entry
from 13.50 to 12.27.
was actually
Final
and Emitter
Model
of
the
Lateral
P-N-P
Transistor
90
This'final
giving
which included
The values
of the
and the
parameters.
s
S
0.5
2.0
5.0
10.0
11
S
S
21
22
0.5
2.0
5.0
10.0
-12.50
-12.50
-12.80
-14.00
0.5
2.0
5.0
10.0
0.00
0.00
0.00
0.00
Table
5.3
Although
final
intuitive
compared with
the errors
-0.40
1.24
178.28
169.81
154.72
133.11
-0.18
0.09
0.28
-0.02
-0.19
-0.49
-0.97
-0.18
-0.11
-0.01
0.07
-0.01
-0.01
-0.01
-0.01
0.01
0.01
0.01
0.01
-0.20
-0.30
-0.50
-0.90
MHz was not
is
F
of
these
errors
0.13 dB and 1.03*.
for
smaller
5.6,
for
-0.21
available
this
this
for
model than
being
values
Model
and Final
P-N-P Transistor
of Lateral
-0.55
5.77
the
and 6.13 res-
model are 0.08 dB and 1.24*
Thus the errors
are again
comparable
with
in the measurements.
would suggest
of a lateral
-0.08
0.01
0.00
88.70
84.00
76.30
99.54
89.10
82.76
76.85
-0.03
model in Figure
In the author's
controlled
0.00
0.07
-0.26
-0.88
-0.31
0.35
-0.06
-0.00
0.08
the maximum absolute
pectively,
-0.04
0.03
0.00
Error
-1.14
-4.52
-10.89
-19.75
-12.44
-12.50
-12.88
-13.97
the value
5.3.
-1.40
-5.40
-11.20
-19.40
178.10
169.90
155.00
132.90
s Parameters
in Table
j Eodel
Measured
0.01
-76.89
-64.77
-57.12
-52.01
Of S 12 -at 0-5
A measurement
Error
-0.11
-0.16
-0.43
-1.20
-64.70
-57.20
-52.00
are given
generator.
Phase (degrees)
Model
-0.10
-0.20
-0.40
-1.20
0.5
2.0
5.0
10.0
12
S
Measure dI
errors
current
(dB)
Modulus
Frequency
MHz
controlled
one voltage
only
of F=5.77,
of 6 nodes and 12
of 0.04 dB and ý0.4" and consisted
r. m. s. errors
elements
had a value
5.14,
shown in Figure
model,
that
opinion
the original
current*generators
p-n-p
However, Callahan's
transistor
both
of these
model proposed
in an attempt
models are equally
by PRL contained
the
lateral
p-n-p
and she
two voltage
to model the substrate
in the manner suggested
model was for
valid
interaction
by Callahan75.
transistor
operating
in
91
the saturated
region
data provided
by PRL.
in the final
it
5.6,
taken
should
sidered
for
means of reducing
ter
computers
priority
if
it
neglects
final
being
taken.
to
The author
the computing
but with
produce
little
a good madel.
time
the present
that
was not
point
trend
opinion
produces
the
an effective
8, some suggestions
in the author's
of an algorithm
There is
50 min-
However,
was to produce
In Chapter
developed,
in
more than that
CDC Cyber 73.
intention
time,
com-
would suggest
model was approximately
of models and that
be the development
of the time
regardless
algorithm
the author's
a supplementary
of the two models.
University
the computing
constantly
should
this
generator
a model of much greater
5.14 has one element
importance.
to be of prime
one,
is produced.
on the Leicester
the development
method for
second current
By adding
only
to the set of
applicable
why the
the more simple
to produce
be made that
not
reversed.
effectiveness,
actually
of c. p. u. time
point
model is
to a model needing
is
The time
utes
explain
could
the model in Figure
although
Figure
This
and-potential
plexity,
that
probably
intuitive
generator
current
and therefore,
con-
are made
of fas-
the main
results
in developing
,
a fast
1
92
CHAPTER6
MODELLINGA VERTICAL N-P, N TRANSISTOR
The second set of data provided
the lateral
transistor.
Like
an integrated
circuit
provided
is reproduced
original
form.
refers
in the integrated
circuit.
in Appendix
the data
between
ments at 12 frequencies
on input
the original
example,
(included
figuration
s
parameter
phase. readings
each of the common
The original
Similarly,
program.
for
in an erratic
s1l
and 336.1*
manner,
in the common base conof 0.6 GHz and
0.9 GHz are 355.3*,
-11.70,
Therefore,
the data was read in to the computer program
as it
although
was given,
although
example,
As the calculated
in the same range,
were produced
the algorithm
this
In this
the graphs that
the frequency
error
are shown later,
range,
therefore
of the models
phase angles
made the calculation
absolute
possible
of the errors
cases where,
is
error
the absolute
values
is
for
phase angle
6* and not
in a phase angle
the phase angles
exactly
to values
was -1780 and the calculated
case the correct
thus the maximum absolute
readings.
converted
had to check for
still
the measured phase angle
was +176*.
on consecutive
were immediately
the phase angles
between -1800 and t1800.
easier
-17.6*
to
the measurements
1) between the frequencies
in Appendix
measure-
and were converted
were discontinuous
in degrees
an example of the data's
configurations.
the magnitude
to the computer
of the phase angles
The data
0.1 GHz and 1.0 GHz for
modulus measurements were simply
decibels
of
a component of
to the way in which
with
consists
common base and common collector
emitter,
for
1, together
n-p-n
a vertical
is
this
'vertical'
can be seen that
It
transistor
p-n-p
and the term
is produced
the transistor
by PRL was for
3540.,
1800.
are made continuous
In
over
of the phase angles
exceed 1300 in some cases.
PRL also provided
a general
model which they
had developed.
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to obtain
any particular
a low frequency
give
good agreement
the measurements
in the range 0.1 GHz to 0.3 GHz.
become very
In each of these
shown later.
model and is used for
original
One interesting
for
problem
modelling
configuration
an increase
The first
individual
and common collector
as the PRL general
to ground.
Of these
PRL
although
realistic
model contains
ele-
a negative-
by attempting
general
common collector
model for
rapidly
to give
model,
satisfactory
paper73 describing
in the model.
equally,
with
a
ldB
in the phase.
the common emitter
model
the appropriate
connected
model produced
one node and several
therefore
connected
terminal
a
few changes in the topology
case,
decided
appropriately,
to try
This model developed
result,
which was included
algorithm.
elements
using
as another
case.
the modelling
too
In each case the initial
relatively
the common emitter
another
were for
the common collector
The author
The aim
model.
as a 10* error
models attempted
and with
in Stage l..
was weighted
the same weighting
model with
to model each
range of the model without
frequency
In the common emitter
initial
author's
frequency
two models,
quickly
very
of the model.
the final
the
configurations.
was taken
were deleted
this
them,
that,
of physically
was approached
at every
in the modulus having
good result
model is
this
in the numbers of nodes and elements
Each of the parameters
error
to this
graph refers
purposes.
and to produc. e a single
separately
in each case was to improve
great
the black
the presence
for
at higher
However,
as can be seen in the graphs
graphs,
about
with
C3-6.
capacitor
This
large
comparison
in models produced
ments only
valued
point
preference
a strong
stated
node has
model was primarily
at frequencies
the errors
ground.
This
s
frequencies
the base,
the appropriate
model and its
parameters
node 1 is
and node 0 is
the collector
configuration,
to node 0.
to be connected
diagram,
In this
node 3 is
node 2 is the emitter,
Thus,
6.1.
shown in Figure
model is
more
in the
Following
this,
95
the common base case was also
model.
to develop
a single
the
In attempting
the major
configurations,
problem
initial
model and this
such a level
that
of complexity
firstly,
complex model would be. inappropriate,
second general
bination
model that
of those
function
Full
tions
value
of this
of the development
details
the same absolute
the same ratio
of the results
scales
ranges
for
factors
Each of the models developed
PRL model.
The major
feature
are more interconnections
probably
tical
this
be expected
aspects
of the device
approximation
characteristics
since
The
initial
model,
were common
from this
contained
and its
elements,
errors
in order
with
in the following
to aid
of the
comparison
applied
to these.
bears
still
some resemblance
of each of the new models is
nodes.
that
This
being
modelled.
elements
that
to the
there
would
the PRL model was based on the primary
of the device.
sec-
of the modulus and phase are in
between the internal
by including
over-
were smaller.
models together
are given
a com-
each graph of modulus and of phase cover
and the ranges
as the weighting
of computing
models that
of these
In the graphs,
chapter.
the vertical
errors,
individual
but had fewer
model,
a more
required.
as the
used
and maximum individual
showing the accuracy
graphs
problems
The model developed
models.
that
was considered
and secondly,
of the first
one node. more than the first
error
it
was produced
elements
to two or more of those
all
used the PRL model as the
caused by the amount of calculation
time were being
of this,
when the model had reached
was terminated
attempt
three
all
In spite
size.
of these
common collector
model for
general
was one of
The first
two models were produced.
the final
using
initial
for
model as the basis
attempted
theore-
The new models improve
model less
significant
on
96
CoiAfiguration
Comnon Collector
model for
The initial
Since the author's
6.2.
to take positive
values
reduced the overall
actually
After
Figure
This
was 1.12E + 03,
F
of
the second model shown in
is
6.2.
and the details
were several
of these
model thus
obtained,
7 nodes and 15 elements
a further
factor
of
of ten to 1.61E + 02.
would find
extremely
the simpler
6.2,
The r. m. s. errors
model obtained
consisted
of
in Stage 2 by
of this
final
were 0.32 dB
errors
and it
accurate
in
The
entered.
had been reduced
F
and the maximum absolute
This model is
of elements
The model obtained
model shown in Figure
and the value
model were 0.13dB and 1.3"
and 1.74*.
last
the
6.2.
Stage 3 was not
that
accurate
and deletions
additions
in Figure
are given
Stage 2 was sufficiently
people
from 7 nodes and 15
The new value
reduction.
In Stage 2, there
final
+ 04 were
were 3.6 dB and 85*, these
errors
Stage 1, the model had been reduced
a significant
already
F, from 5.84E + 04 to
function,
to 7 nodes and 12 elements.
elements
change in value
This
pF.
C4-7'
capacitor,
frequency.
at the highest
occurring
the element
constrained
-0.8
The maximumabsolute
2.4 dB and 240.
model shown in
first
given by the value of F=5.54E
The r. m. s. errors
5.54E + 04.
of
error
configuration
the negative-valued
only,
values
the
algorithm
of 0.1 pF instead
a value
was given
6.1 is
model in Figure
from the PRL general
Figure
the common collector
derived
could
be that
many
at the end of Stage 1 quite
acceptable.
The graphs
final
Y
in Figures
As has already
model in blue.
axes of all
comparison
the graphs
of the errors.
30 dH and the
phase
6.3a and 6.3b show the
scales
in this
been mentioned,
chapter
a total
parameters
the
scales
have been standardised
Thus the modulus scales
cover
s
range
of
cover
3000.
a total
The black
of the
of the
to aid
range of
points
97
r- a S.SiE+94
No. (nF nodes w7
No. oF elements = IS
ý
9C-
Delete R(4-G) i R(4-9)
Replace C(1-0) bg R(1-9)
Delete C(2-9)
R
R
R
R
L
L
C
C
C
C
C
C
C
C
9
(3-6) a 1,12E+92 ahm
(4-6) 2 Z. SDE+93 ohm
(4-9) = 9.43E+04 ohm
(4-5) - 4.99E+99 ohm
(1-3) = 4.79E+Iag nH
(S-2) = 4. igE-+OgM
S. OOE-OSnF
(1-9)
U-9)
I. GOE-04 np
(3-4)
i. 74E-93 nP
(4-6)
1,99F-02 nF
9.20E-OS nF
(6-9)
IME-95
(4-0)
nF
S. OGE-93 Mp
(2-9)
S. OOE-OSnF
(1-2)
3.90E-02 S
(0-4)
(T u-9.29E-93 ns
(V across nodes
r- a
1.12E+03
No. aF nodes -7
No. oF elements = 12
R (3-6) - 9.92E491 ohm
R (4-S) w 2. L2E+gl ohm
L (1-3) x G. S2StG9 nH
L (S-2) a 3.99E+09 nH
R U-9) x 3.29E+93 ohm
C8 9) x 9.49F-91 nF
C (3: 1) 2 I, GSE-93 nF
C (i-G) -t 2. OOE-92 nF
C ý6-9)
i. 02E-ei nF
C ý4-9)
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6.2iE-gi
nF
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9 (9-1)
Ua3.
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(V across nodes 'I)
0
C.
Add C( 1-4)
Replace C(I-2) by R(I-2)
Add C( 1-6)
Delete R(I-2)
Add C(2-3)
Add C(2-9)
Delete C(I-S)
Add R(I-G)
Add R(3-0)
Add C(S-6)
Delete
R(1-0)
Add C(S-0)
Delete
C(4-9)
i C(2-3)
F=I.
GIE+02
No. oF nodes =7
No. oF elements = IS
R (3-G) - 2. QIE+02ohm
R ýi-S) 2 2.22E+91ohm
L ý1-3) a 1.986+91nH
L (S-V = 7.70E+99nH
C 0-9) = 1.41E-03np
C (3-4) = 2.12E-gi nF
C (i-G) = 1.9iE-92 nF
C (6-9)
1.8CE-04np
5.71E-92 S
9 (0-1)
(T = ?. DOE-92ns
(V across nodes
C (1-4) = 2. S6E-03nF
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R (3-9) = 1.48r:+24 ohm
C (S-G) = 9.9SE-04nF
C (5-9) a 1.93E-94nF
e1
0C.
tiode IMode 2Node 0-
Figure
6.2
F-miLter
Base
C0114ct*r
Modelling
the Common Collector
Configuration
98
MEASUREMENTS
PRL MODEL
,EWML
MODOB
PHASE DEG
511
811
50
5
FREQ GHZ
)0
0
REG (GHZ
0
--50
-100
-150
-200
-20
moo DB
PHASE DEG
521
10
100
5
50
FREQ GHZ
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FREQ GHZ
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-15
-150
Figure
521
6.3a
s
in
S
and
21
11
Common Collector
-
-
Configuration
() ()
MEASUREMENTS
PRL MODEL
MODOB
PHASEDFG
512
512
iso
00
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0
REQ GHZ
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522
150
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100
150
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Figure
6.3b
s
in
S.
and
12
21
Common Collector
Configuration
100
and 6.3b
6.3a
in Figures
in
model connected
general
are the
negative).
The red points
aimed for.
The graphs
the
frequency
entire
cases the red and blue
seconds
time
Two attempts
the commonemitter
to generate
The first
configuration.
In many
this
model was approximately
in
transistor
n-p-n
of these used an initial
model and the second used an initial
model produced by the author
for
the common
Attempt
caused a reduction
The r. m. s. errors
a value
in the value
given
by this
of
0.1 pF instead
from
F
of
occurred
at the
model is the first
common collector
8.60E
of
+ 04
the negative-
case,
-0.8
to
pF.
This
7.72E
+ 04.
model were 3.0 dB and 30.1-* while
were 10.3 dB and 134.40.
errors
errors
As in the
6.4.
was given
capacitor
from the PRL general
model derived
model shown in Figure
largest
taken
were made to model the vertical
The initial
absolute
frequencies.
configuration.
First
valued
over
Configuration
final
based
the
on
model
collector
at higher
obtained
are indistinguishable.
from the PRL general
model derived
show the values
improvement
the
C4-7
on the UMRCCCDC 7600.
Common Emitter
6.2.
and thus
PRL
tile
(With
configuration
clearly
and especially
for
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points
The computing
600 c. p. u.
common coi.'lector
show quite
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s
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higher
As will
frequencies,
be seen later,
those
on
S
12
the maximum
the
being
the most obvious.
After
ments giving
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Stage 1, the model had been reduced to 6 nodes and 10 elea value
is
minimum
the
of F=1.45E
+ 04.
second model in Figure
The model giving
6.4.
this
first
The maximum errors
in
101
40.
4c.
12
R (3-1)
R (ý-S)
C
f
C
C
C
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DejeLe '-ý1-3)
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Combine
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R (3-4) - 3.46E+02 ohm
R (5-2) - 2,93G+gi oh-ft
L ý4-0) = 7. ilE+99 nq
C (1-4) = 2XF-93
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C 13-2) = i. 31F-gi nF
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Figure
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capacitor)
Figure
and the graphs
6.4
for
in red and the values
in blue,
together
PRL general
the
frequency
the entire
Although
it
reduced model, this
can be seen that
while
the PRL model gives
Thus, it
the model is very
but the reduced model gives similar
errors
over
range.
should be feasible
to build
the commonemitter
At the same time as this
using an initial
known that
first
attempt
was
model based on the final
case, was also being made.
the commoncollector
was already
up a model from this
to produce a model for
attempt
being made, the second attempt,
for
developed
model
model
the negative-valued
was abandoned.
configuration
with
(with
frequencies
low
the
at
measurements
with
good agreement
poor at high frequencies,
in Figure
configuration
in the common emitter
connected
sj,
were on
configuration
the second attempt had produced,
after
Stage 1, a model with more nodes and elements which gave a considerably
lower value of
It
F-
decided that
was therefore
showed more promise and should be continued
the second attempt
in preference
to the first
attempt.
6.2.2.
Second Atteýt
The initial
6.6 and it
produced
can be seen that
for
this
the common collector
interchanged.
C
model used in the
This
model,
and R7-0 in Figure
3-7
6.1,
F=4.95E
of
value
a
gave
and
values of the individual
However,
capacitor
being
in Stage
replaced
model is
identical
lacked
in
were present
+ 05 with
shown in Figure
to the
but
configuration
in particular,
that
is
second attempt
with
the
the
final
nodes
model
2 and 0
substrate
elements,
PRL general
model,
high absolute
correspondingly
errors.
1 only
one element
by a resistor,
was deleted
resulting
in
the
together
with
second model
one
103
P-4.9SE+9S
No. oF modes w7
No. oF elements = IS
R
R
L
L
C
C
C
C
9
bI
0
(3-6)
(4-S)
(1-3)
(S-9)
(3-2)
(3-4)
(4-G)
(6-2)
(2-4)
=
ft
x
=
=
2
x
2.21E+02
2.22E+gl
i, 28F491
7.72E+09
1.41E-93
2, k2E-04
1.94E-92
L. HE-ai
S. 71E-93
ohm
ohm
rJ4
nH
nF
nP
nF
nF
S
ýT w 7.29E-92 ns;
(V across nodes
C (1-4) it 2. SGE-03rF
C (2-9) - 8.25F-gi rF
R (I-G) x i. 41E+02ohm
R (3-2) xI 19F494ohm
.
C (S-G) x 8.9SE-94 Ap
C (S-2) x i. 93E-94 nr-
4L
DeleLe C(3-i)
Replace C(S-G) by R(S-G)
r -, 1.77E#93
No. oF modesjr 7
No. oF elements a 14
bi
c-
0e
R (3-G) x 9, ISE+92 ohm
R (4-S) a 2.22E+gl ohm
L (1-3) w 1.17E+02nH
L (S-0) x 3.33E+09 nH
C (3-2) N 2. i2E-94 Mp
C (4-6) a 1.23E-92 nF
C (6-2) x 3. GBE-95nP
g (2-4) x 4.31E-92 S
(T a-S. 7GE-92ns
(U across nodes
C (1-i) - 3.3iE-23 nF
C (9-2) - 9.90E-94 nF
R (1-6) x 8.2, E+91 ohm
R (3-2) x 3. G3E+94ohm
R (S-6) a 1.916+93 ohm
C ýS-2) ar i. 19E-93 nP
Delete R(S-6)
Add R(I-i)
Delete R(I-i)
Add R(I-S)
Delete R(I-S)
Add R(1-0)
Delete C(6-2)
FIdd R(2-i)
.Add R(2-0)
Add new node with L(2-7)
r -r 3.94E+22
tic. oF modesw9
No. oF elements x IS
(3-6)
1.21E+03 Ohio
R (4-S)
3.32E+91 ohm
L (1-3)
92 rJ4
1ASE-o,
L (S-9)
2.37E*29 nH
C (3-7)
1.97E-94 np
C (4-G)
2.94E-02 np
7.72E--02 S
9 (7-1)
(T -LUE-92
ns
(U across nodes
cR
C
R
R
C
Q
R
R
L
o AAdd R(3--S)
Node INode 2Made 0-
Figure
6.6
Modelling
(1-4)
(i-6)
(3-2)
(S-7)
(1-0)
(7-1)
(7-9)
(2-7)
4.29E-93 rF
a. ! OF-Ql ohm
!. SGE+gs ohm
I. G3F--93 nF
8.9SE+92 ohm
2. SGE+93 ohm
- 1.18e+9i ohm
a 3.49G+93 nH ,
Base
C-illector
Elnitter
the
Common Emitter
Configuration
(Second
Attempt
104
6.6 which had a value
shown in Figure
of F=1.77E
+ 03.
and deletions
In Stage 2a number of additions
of 7 nodes and 15 elements
in a model consisting
were made resulting
giving
a value
of
1.45E + 03.
On entry
at node 2, the
since
collector
PRL had stated
of F=9.84E
they
was an interesting
development
an inductor
collector
had wanted
had been unable
last
an inductor
to
model shown in
at the
insert
one successfully.
Figure
6.6 which
to Stage
2 resulted
model shown in Figure
6.7.
in one further
This final
element
0.30 dB and 3.0* and the maximum absolute
addition
model consisted
and 16 elements and gave a value of*F = 8.81E + 02.
R (3-6) - i. M-493 ohm
R (i-S) a 3.22E+01 ohm
L (1-3) a 1.39E.+22 r.H
L (S-9) - 2.3SE400 nH
C (3-7) - 2. OSE-04 nF
C (i-S)
3.11E-02 np
(7-1)
8.29E--92 S
(T =--3.OGE--92ns )
(V across nodes 'I)
Node INode 2Node 9-
Figure
Final
C
R
Q
C
R
R
Q
L
p
(1-i)
(1-6)
(3-7)
(S-7)
(1-9)
(7-4)
(2-0)
(2-7)
(3-S)
=
=
=
=
-
Base
Collector
EmMar
Model for
Common Emitter
of 8 nodes
were 1.56 dB and 4.1*.
errors
No. oF nodes -, 8
No. oF elements a iG
0e
giving
The r. m. s. errors
r=9.916+02
6.7
had a value
+ 02.
Re-entry
the final
that
gave the
new addition
This
terminal.
model but
of their
terminal
This
to Stage 3a new node was added by inserting
Configuration
4.41E-03 np
8.19E+gl ohm
7.! SE-+OSohm
i. GSE-33 nF
i. 41E+23 ohm
3.23E+93 ohm
S. 99ý+33 ohm
3. S3E+22 MH
3.32E+93 ohm
were
los
*
MI-ASUREMENTS
ODFL
*PRLM
NEW MODIL
*
C
iiI
Li j0
Mr-in
5
50
J,]riz
GHZ
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G
r
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-5
-100
2,0
-
C.
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PHASE'Df-ýG 521
2 00
5
FREQ GHZ
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i5o
i 1-110
r
,-o
n
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E!ý Gri Z
-15
r
-50
-20
Figure
6.8a
s
Final
S
of
and
21
11
Model for
Common Emitter
Configuration
106
MEA5UREMEN15
PRL M0Fj L
t'
INL-ý MODEL
PHA'ýE DýG
'i i2
MSlri
ý '12
20-5
20
i 50
-Pli
i fl%
r,
ju
-30
Griz
--40
b9
tl Z
10,
10
D, OlI.,
IF
rý LL5G
PWA. DFG
322
ý22
11,00
10
0
F-rlIE5 Griz
GHZ
I
0
1r1lo
-10
1 r5c
-13
Figure
6.8b
s
S
of
and
22
12
Final
Model
for
Common Emitter
Configuration
107
The absolute
of the modulus
where the value
appears
great
significance.
-11.54
The
s
dB in
and 6.8b.
6.8a
graphs
erable
improvement
might
6.3.
Common Base Configuration
the negative-valued
Changing the value
value
F
of
of this
model are shown in blue in
can be seen that
by the
given
to 2.12E + 05.
17 elements
as an initial
The attempt
was terminated
trolled
current
final
based on the author's
was then used.
of
This
is
7 nodes and 15 elements
initial
tion,
the
on the UMRCCCDC 7600.
initial
in the PRL general
In Stage
based on the PRL general
F=1.67E
of
0.1 pF,
pF to
to
that
find
+ 05.
changed the.
model consisting
1 failed
described
first
giving
for
of
8 nodes and
a local
minimum.
the voltage
either
in Section
6.2,
con-
which
an initial
the common collector
model shown in
a value
in
the second attempt
model used
this
model in the second
the
be made short-circuit.
should
model
the
model.
be removed or the nodes across
should
In view of the results
-0.8
indications
after
was taken
this
Stage
model,
generator
voltage
controlling
Using
were
are a consid-
PRL general
gave a value
from
they
the final
seconds
capacitor
capacitor
to be of
in the moduli
errors
The model for the commonbase configuration
model with
this
although
dB.
-13.23
1900 c. p. u.
was approximately
0.9 GHz
of
be considered
not
The computing time taken to generate
attempt
Thus,
absolute
It
the values
over
dB.
final
of this
parameters
in Figure
it
highest
The next
dB and 0.73
s 2-1 at a frequency
on
was -17.52
large,
comparatively
error
0.9 dH in
1. S6 dR occurred
of
error
Figure
of F=8.65E
for
model lacked the substrate
model
configuration
6.9
and consisted
+ 05.
As with
the commonemitter
elements that
the
configura-
were present
model.
1, three
in
Figure
shoArn
model
6.9,
elements
which
were removed,
gave a value
resulting
of F=3.09E
in
the
+ 03.
second
108
EMU
No. oF nodes -7
No. oF elements = iS
R (3-6) u 2. OIE+02 ohm
R (i-S) x 2.22E+01 ohm
L (0-3) 2 1.99E+91 nH
L (5-1) a 7.70G+29 MH
C (3-2) a 1.41E-03 nF
C (3-4) 2 2.12E-gi nF
C (1-6) 321.9iE-02 nF
1.8GE-8i nF
,C (6-2)
5.71E-02 S
9 (2-1)
(T = 7. DQE-02 ns )
W across nodes '4)
C (0-i)
2. SGE-03 np
C (1-2)
p
9.2SE-04 1-,
R (0-6)
1.41E+02 ohm
R (3-2)
1.49e+04 ohm
C (S-G) z 8.95E-94 nP
C (S-2) J: 1.83E-94 nF
c
e1
1)
DeleLe RO-G) i C(4-6) t C(S-2)
Replace C(I-2) by R(I-2)
3.29E+03
No. oF nodes -7
No. oF elements a 12
R (4-S) x 4.9ge+20 ohm
L (0-3) x B. SOE+20nH
L (S-1) a 4.24E-+Q9nH
C 0-2) u I. SSE-03 np
C (3-1) x 4.48F-01 nF
C (6-2) a 1.92E-04 nP
(2-1) S, 2.90E-02 S
(T a-1.39E-01 ns
(V across nodes
C (0-4)
8.7? E-Di nF
R (1-2)
4.29E+04 ohm
R (0-G)
7. G4E+gl ohm
R (3-2)
1.3sE+23 ohm
C (S-G)
1.36E-02 nF
t
e1
b
Add C(1-3)
Add C(2-0)
M-0)
Delete Rý1-2) P Wi-S)
(deleting
node)
r- = I. SSE+93
No. oF nodes c6
No. oF eleitemts = 13
i
L (0-3) - 1.78E+21 nH
L (4-1) - 1.31E+31 nH
C (3--2)
7.3SE-91 np
(3-1)
1.49E+03
ohm
IR
C (S-2)
8.31E-OS nP
2. SIE-22 S
9 (2-1)
(T a-9.2? E-03 ns
(V across nodes
C (9-1)
1.77E-03 nF
R (0-S)
9.3SE+Ial ohm
R (3-2)
1.81E-+03 Ohm
C (4-S)
8. GiE-93 np
C (1-3)
-1.32E-Oi Mp
C (1-12)
;., K-03 np
C (2-2)
!. ZGF--03nF
e1
b0
Add C( 1-2 )i
C( 5-9 )
Oelete C(O-1)
Add R(I-1)
C(4-0)
Add C(1-2)
Add new node with R(G-2)
Made 1Mode 2Mode a-
Figure
6.9
9(3-4)
EmlLter
Collector
Base
Modelling
the Common Base Configuration
1,09
2, duting
In Stage
and deletion
the addition
The model produced after
also deleted.
Continuing
2, further
in Stage
deletion
elements
Although
giving
sufficient
substrate
a value
of F=6.92E
accuracy
had already
in the addition
Stage 3 resulted
into
elements in a similar
The value of
of this
+ 03.
C
1-2
+ 02 was produced.
been obtained,
an entry
of a new node which created
form to those in the PRL general
6.10
in
Figure
is
gave r. m. s. errors
shown
which
errors
node is the
a model having 6
until
This final
F was also reduced to 3.63E + 02.
The maximum absolute
one node was
were added and deleted,
being added, then removed_and then added again,
nodes and 15 elements
elements,
This model gave a value of F=1.88E
model in Figure 6.9.
third
of
the
model.
model,
of 0.2 dB and 1.9*.
were 0.7 dB and 3.40.
F=3.
G3E+02
No. oF nodes a7
No. oF elements = IG
c.
e'
6
0b
Node INode 2Node 9-
L (9-3)
1.99S+91nH
L (i-I ) 3.03F+91nH
C (3-2? G.77E-gi rfc (S-2)
1.99E-04 nF
2. G2E--02S
9 (2-1)
(T -1.98F-01 ns
(U across nodes
R (9-S) a 3.40E+01 ohm
R (3-2) a !. i7E-+9I ohm
C (4-S) - IME-02 nrC (1-3) = 3.91E-gi np
C (G-0) a 1.42E-03 nF
C (S-9) = Lile-92 nF
R (1-4)
9. nE+gl aN,a
R (3-S)
9.99F:492 nhm
C (1-2)
S. 97E--GS
nP
C (4-3)
i. 2Sý-03 nFR (G-2)
1.9sr-+91 ohth
GniLter
Collector
Sme
I
Figure
6.10
Final
Model for
Common Base Configuration
110
-Is
VEýSupEwllýý
PRL MOD,
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F
ML
'ý
'N
morl, 5 EI
nFilýC*)Eri lu
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-
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-
-20
-1%J
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'15
r0k,
'121
PHASE DEG
5
521
0
HZ
0
-i,,,
-15
-20
Figure
6.11a
S
S
in
and
21
11
Common Base Configuration
FQEIIJGriz
0s
MEA5UREMENTS
PRLM CCU'
L
N
m0'n-.5EI'
Fill-G
PLIA'JE-
'i j2
12
0
1OL
2ri
25
ýjrjz
35
-31i
-40
-15C
-45
ýýHZ
10
nki
nj !-jb,
'J22
rj,,.
-G
322
10
FRE5 CjHZ
r"
D
Griz
0
IC'
-I
-13
-200
Figure
6.11b
S
S2
and
12
in
Common Base Configuration
112
As can be seen from the
final
improvement
a considerable
The computing time taken to generate
The total
configurations
3620 c. p. u.
was approximately
to be used in an attempt
that
expected
at
least
of the
as long
times
errors
as for
considerable
putation
for
the
for
the
nodes
of
number
of
general
PRL general
initial
an
produced
author
6.4.1.
value
be expected
could
to
to a
contribute
The amount of com-
by the order
of
the
square
the available
data.
to develop an improved
attempt
In the
model.
However, the
second attempt,
common emitter
the
models generated
for
and common base configurations.
AtteýTt
The PRL general
function
model would
model based on the final
common collector,
First
take
would
node.
model used in the first
model was the
each of the
stage
way to reduce the computing time would be to fit
author decided to proceed using all
The initial
to be
was only
Furthermore,
parameter data at fewer frequencies.
S
the measured
each additional
it
was
models.
computing'time.
of a model increases
analysis
The simplest
this
data
available
any particular
a general
transistor
n-p-n
model, then
individual
models,
in the required
increase
in the
individual
at
the
in
the
all
a general
as the number of nodes and elements
than
If
individual
the three
the vertical
of
seconds.
to produce
the analysis
three
be greater
model.
model was approximately
computing time taken to generate
the different
for
parameters
s
Model
General
models
this
this
seconds on the UMRCCCDC 7600.
1120 c. p. u.
6.4.
and 6.11b,
PRL general
the
over
6.11a
the measured
agreement with
model gives excellent
and is
in Figures
graphs
model shown in
of F=3.11E
+ 05.
Figure
6.1
gave an overall
The r. m. s. errors
error
were 3.3 dB and
113
Fa3.
IIE+05
No. oF nodes -9
No. oF elements a 22
5
b
R (4-S) z 1.13G+92 ohm
R (S-G) a 2. SIE+03 ohm
R (3-6) a 9.43E+94 ohm
R (6-8) - 1.99[+99 ohm
R (7-0) - 6.23E499 ohm
1. ( 1-0 it 4.72E+09 nH
L (9-2)
4. i GE+9QnH
C (4-G)
I. VIG-03 np
C (5-6) x I. OSE-02 nP
C (S-3) u 8.23G-95 nF
C (3-6)
i. 20E-OS np
C (3-7)
1.31E-93 mrC (4-3)
I. G2E.
-94 nF
C (1-3)
SME-as nF
(1-0)
1.23E-03 np
S. SQF-94 np
(3-0)
3.92E-02 S
(2-G)
(T w-L. DOE-92 ns )
(U across nodes 5g)
C (2-9) x 1.22E-93 nF
C (1-2) a S. WE-QS nF
C (2-3) x SME-gs nF
3
7
jE
2
I0
-
No. oF nodes w7
No. oF elements a IS
C.
___
___
e2
R (1-4)
I. O?E+92 ohm
G (4-S)
1.38E-93 S
R (6-0)
i. 9SE+QIohm
L (5-2) x 1.48E+21nH
2.! IF--03 nF
C (1-5)
7.8ae-93 MF
C (4-5)
7. ise-gs np
C (1-3)
R (3-S)
1.47E+03 ohm
2.43E-03 nF.
C (2-6)
C (1-3)
1.62e-04 nF
C (1-0) - 1.83E-93 nF
R (3-0) 0 3.63E-+03ohm
9 (3-S) a 2.97E-92 S
(T 2 1.7SE-92 ns )
(U across nodes 34)
C (2-0) wI G7E--93nF
C (1-2) u SAIE-94 nF-
0
x
G.Vt+24
No. OF modes a8
No. OF elements a 21
3
Figure
6.12
General
i230-
Base
Emitter
Collector
Ground
Model - First
x 9.3SC-+03ohm
a 1.31E+91 ohm
2.32E-+9l nH
2.3SE-03 nF
3.32E-03 nF
S. ISEAS MF
2.2SE--93nP
a 8.47E-+23ohm
& 2. SK-02 S
(T w-4. S.)G-92 ns )
(V Qcross nodes S)
C (I-G)
2.33E-93 Mp
C (2-G)
i . 48ri-.23 nrR (1-2)
2.91E+02 ohm
R (1-3)
!. D! E+OS ohm
R (2-S)
G. 20E+92 ohn
C (1-4)
?. S7E--94 np
L U-1)
I. I7E+02 nH
C (3-7)
I. SGE-04 nF:
R (S-7)
1.71E+97 ohm
C (S-9)
2.92E-OG nF
R (7-9) = 2.99E+93 ohm
R (1-4) 2 1.9? E+92 ohm
0
Made
Node
hoe
Node
R (7-4)
R (G-2)
L (S-2)
C (I-S)
C (4-S)
C (4-3)
C (3-G)
R (3-9)
9 ý3-9)
Attempt
114
N
As can be seen
32.8".
from
shown
graphs
the
earlier,
gave
model
a
at higher
but there were gross errors
at low frequencies
good fit
the
frequencies.
on using this
model as the initial
These are the first
to 7 nodes and 15 elements.
The two node deletions
inductor
of the base terminal
in series with the emitter
Stage
distributed
although
case,
two models shown in
were attained
(ii)
and
The individual
they were still
by (i)
the deletion
inductor.
terminal
1 was 1.67E + 05.
the general
of the model from 9 nodes and 20 elements
Stage I caused the reduction
Figure 6.12.
model for
the deletion
of the resistor
The value of
F
after
now became more evenly
errors
for
smallest
the common collector
configuration.
After
one entry
each to Stages 2 and 3, during
the base terminal
inductor
Stage 2, the final
model shown in Figure
the errors
CDC 7600 to develop.
element
addition
therefore
a different
were still
model had taken
Because of the
was now taking
model.
and 8.0 dB and 59* in
a large
6000 c. p. u.
size
of
an excessive
attempt
configuration,
improvement had
large.
rather
approximately
decided to make another
initial
configuration
Thus, although
the commonbase configuration.
This
This
but the maximum
were 1.5 dB and 380 in the common collector
6.3 dB and 530 in the commonemitter
been achieved,
to
entry
6.12 had been produced.
model were 1.45 dB and 14.50,
of this
The r. m. s. errors
absolute
and one further
was re-inserted,
of which
of 8 nodes and 21 elements and gave a value of F =6.07E +-04.
model consisted
errors
the latter
seconds
on the UNIRCC
the model each further
amount of
time.
to produce a general
It
was
model using
lis
r- = 9.78F+GS
No. oF nodes -9
No. oF elements
L (1-4) a 3.20E421 nw
q (4-S) x 2.30E+92 Ohm
R (I-S) a 8.29E+91 ohm
C (1-3) x 7.90E-94 nF
R (4-3) it I. OQE+giohm
C (S-6) x 2.30E-92 nr9 (3-G) 2 SME-92 S
(T a-1.00E-92 ns )
(V across nodes 11)
L (7-2) a S.02EQ9 01
C (1-6) N 3.20E-03 nF
C (S-3) - 2.20E-94 rF
R (6-7) a 3. ME-+Ql ohm
C (3-7) a S. 09E-04 nrC (3-2) a L.QOE-OS
Mr.
I
lo
i3
r- -
3.0SE+0S
No. OF nodes -7
No. OF elements a9
L (1-4) 2 S.SPE+99nH
R (4-S) a 9.96E+91 ohm
C (4-3) - 2. S7E-94 nF
C (S-6) a i. ISE-92 rf
9 (3-6) a 2.37E-02 S
(T. -SAGE-02 ns
(U across nodes
L (G-2) x S. 9GE+99nH
C (1-6) a 2.25E-93 nP
C (3-6) u 4.37F-OGnp
R (3-2) u S.?SF+03ohm
b
r- a 4.78F+94
No. OF modes-9
No. OF elements
L( 1-4)
C (4-3)
b1
= 19
I. M+01 rAi
G. GIF-OS nP
U. -G) 2.026-02 S
(T --L. GOE-91ns
(U across nodes
3.23E-+QlnH
L (6-2)
C (1-6)
3.97E-93 np
C (1-9)
i. 3SE-03 MF
C (2-1)
1.21E-03 nF
C (2-9)
i. i2F-03 np
9 ý4-5) x G.Sir=+33ohm
R (2-4) - 2.; ýIZE+02ohm
C U-3)
2.23E..93 pip
R
C
R
L
C
0
Mode I
Mode 2
Mode 3
Mode 9
Figure
6.13
General
ease
Emitter
Collector
Ground
Model
Second
AtteTpt
-
(S-? )
(7-9)
(1-7)
(6-7)
ý2-3)
(9-9)
(1-3)
1.49R+gi
3. ýDE401
1.3ýE+93
9.22E-24
2.33E-+24
2.17E+90
i. ý2E-gi
ohm
ohm
ohm
np
ohm
nH
Mp
116
Second AttepRt
6.4.2.
For the second attempt,
initial
the
used-as
This model was derived
model.
elements of the three
of 8 nodes and 13
This model consisted
+ 05.
F=9.78E
of
a
value
elements and gave
was reduced
by coinbining those
models developed by the author that
individual
in
two
models.
any
appeared
7 nodes and 9 elements
to
model shown in Figure 6.13 was
the first
Stage 1 the model
After
giving
This simple model, which is the second model in Figure
of
errors
3.3 dB and 32.5*
After
to each of Stages
two entries
this
seen to be merely splitting
However, this
7.
additions
S.
to
node
connected
L
a late
each of the
emitter,
and common collector
formed
in the region
element
of the
elements had been
This
obtained
ground
node.
model therefore
in the first
gives
and 12.9* and the maximum absolute
commoncollector
configuration
configuration,
but
in
their
inclusion
a better
overall
The r. m. s. errors
attempt.
errors
is thus one inductor
However,
reduction
to
prior
to Stage 3 to generate
in
the
at
common emitter
complicated
a rather
and node caused a significant
7.23E + 04 immediately
there
and base terminals,
configurations,
of
series
to be made concerns the inductor
The second point
collector
can be
between nodes-1 and
stage other
In commonbase configuration,
a new node.
about
node 5, at
was developed by a complicated
and until
model
is taken,
generator
This element was added in the second entry
0-8*
from
is that
an element of resistance
situation
and deletions
point
of the current
voltage
which the controlling
final
2 and 3, the
The first
provoke comment.
gave r. m. s.
There are two points
been
had
6.13
in
Figure
obtained.
shown
model that
6.13,
11.0 dB and 1520.
of
and maximum errors
+ 05.
F-3.05E
of
a value
is
structure
addition
result
of this
this
function
the objective
to 4.78E
of
+ 04.
than
that
model are 1.3 dB
are 4.8 dB and 28.1* in the
2.3 dB and 26.2* in the common emitter
and 6.2 dB and 36.50 in the commonbase configuration.
Iýt
117
MEASUREMENTý
PRL MODý-L
NLW MODF-L
MODD6
311
PHASE DEG
311
50
FýEG GHZ
)0
0
-,\EG GHZ
-1ý0
-100
-150
-20
-200
S21
MOD DB
PHASE DEG
10
IN
5
50
FREQ GHZ
)0
0
0
-5
-ýo
-10
-100
-15
-150
Figure
6.14a
s
S
and
1 1-ý----21
for
General
S21
Model
GHZ
in
Common Collector
Configuration
118
*
w
MEASUREMENTS
PRL MODEL
*
NEL-W
MODEL
PHýSF.DEG
S12
MOD DB
S12
150
5
ý-REQGHZ
0
0
1100
50
0
EQ GHZ
10-1
-10
-50
-
-15
'10
-1 Ili
-20
522
MOD DB
PHýSF
-DFG
S22
150
0
FREQ GHZ
)0
100
50
GHZ
0
-50
-20
-100
-25
Figure
6.14b
s
and S 22 for
12
General
Model
in
Common Collector
Configuration
119
PRL MODEL
511
moo IDB
PH,
A3E- rj'EG
311
5
FREU GHZ
0
FREQ GH7
)0
1;
-10
-15
-20
-200
521
MODOB
PHASE DEG
5
321
200
FREQ GHZ
0
150
D0
5)
100
-10
50
-15
0
-20
-50
Figure
6.15a
s
for
S
and
21
11-
General
Model
GHZ
in
Common Emitter
Configuration
120
*
MEASUREMENTS
PRL MODEL
L
PHASE DEG
S12
MODDB
-15
S12
200
-20
150
-25
100
-30
50
-35
FREQ GHZ
D0
-40
",--REGGHZ
0
10
1
10'
MOD DB
-130
322
PHAK DEG
S22
100
10
50
5
0
10-
Q GHZ
0
-50
4,4
.
FREQ GHZ
)0
-
-5
-100
-10
-150
-15
Figure
6. lSb
s
and S 22 for
12
General
Model
in
Common Emitter
Configurat
ion
_
121
MEASUREMENTS
P.RL M'uDf-L.
NEWMODEL
FIHýCJF-
li jI
MOD
FF-fý,
EQ GrIz
1K
r
oro
0
-ý'EQGHZ
rý0
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25
menD
L)
OL"
PHASEý"L-ýG
J21
0
rý
F
L-ý
Gmz
.
'IHZ
50
1fil0
150
-10
20 ri
? 50
Figure
6.16a
S
General
S2
and
11
___for
Model
in
Common Base Configuration
122
MEASUREMENTS
'--L
PR'
-0
ý41EW mn DL
512
PHýSE DEG
MO DE
,0 lý
50
-REQ GHZ
C)
10
30
-50
-35
-100
40
-150
45 IKI
Griz
-7S9
10-,'
MOD DB
PHASE DEG
522
10
522
50
5
0
FREQ GHZ
)0
ýE:Q GHZ
0
-50
-5
-100
-10
-150
-15
Figure
-200
6.16b
S
for
S
and
22
12
General
Model
in
Common Base Configuration
123
can be seen that
the final
errors
as the
initial
The
S
that
of this
6.14,6.15
general
not
of the
the model chosen for
model if
use
and that
a satisfactory
the PRL model exhibited
shown for
As expected,
these
as the
each configgraphs
individual
frequencies
that
that
errors
has lost
some of
the
low frequencies.
at
The time taken to develop this
it
CDC
7600
UMRCC
the
and
on
onds
but
show
models
model has removed the gross
in the ýRL model at high
were present
distribution
model are
as accurate
The new general
were developed.
for
those
on the results
effect
and 6.16.
model is
the new general
than
one attempt.
after
parameters
in Figures
accuracy
demonstrates
This
the
to use more than one initial
model is not obtained
that
but
attempt
Model can have a significant
can be worthwhile
uration
first
in-the
different.
also
lower
are the maximum errors
only
model obtained
is
it
not
model was 8300 c. p. u. sec-
general
was decided that
no further
improvements
be
attempted.
should
Alternative
Suggested
6.4.3.
Approaches
of hindsight,
With the benefit
the author now feels
the
two general
to
produce
used
approach
both
Knowing that
required
would
be large,
the
size
first
the
priority
the algorithm
operated
that
some accuracy
was lost.
Therefore,
could
have been used and,
precision
this
for
would
the
5.2.2,
have achieved
analysis
the
calculation
have employed
although
this,
as efficiently
forward
too,
in the
of the
approximation
differences
involves
rather
a loss
of
even if
analysis
this
meant
single
as shown in Section
3.1.5,
computing
as mentioned
to the Jacobian
than
time
routine
in the
Furthermore,
computing
have been to ensure
as possible
a 50% reduction
of the models.
and the
should
that
arithmetic
the
models was too ambitious.
the problem
of
that
central
accuracy,
it
time
in Section
matrix
differences,
would
required
could
and
have halved
the
124
function
of
number
Further
savings
data
s parameter
the measured
s
that
attempt.
results
was provided.
frequency
given.
approach using the initial
at low frequencies,
it
and the maximum frequency
instead
given,
of the
of using
they could have
This certainly
might
model used in the second
might have been better
matching
increasing
at which the
less
since the PRL model gave good
attempt,
to improve the model in stages by initially
and gradually
calculation.
For example,
parameters at every frequency
However, in the first
at low frequencies
that
have been made by using
also
could
been taken at every alternate
have been a better
in
required
evaluations
to have attempted
the
S parameters
the number of frequencies
S parameters
were matched.
125
CHAPTER 7
A BIAS DEPENDENTMODEL FOR TWO SIMILAR BIPOLAR TRANSISTORS
The third
s
set
of data provided
by PRL15 consisted
in
the appendix,
given
measurements,
parameter
transistors
in commonemitter
in silicon
with implanted
The JE2M had one emitter
stripe
The data provided
the
two devices
collector-emitter
arsenic
and the 3E2M had three
of the
consisted
at 16 frequencies
voltages,
currents,
Ie,
in the ranges
27 mA for
the
3E2M.
V
ce,
in
of
0.5 V and
in
lOmA
Figure
emitter
for
7.1
and with
the
stripes.
measurements
0.1 GHz to
3V
made
baron bases.
parameter
the range
0.5 mA to
The graphs
s
had the
n-p-n transistors
and diffused
emitters
2GHz
with
emitter
fT
and
the
modulus
I fT
1TM
2-0-
: ýýI
1M
E2
.01
Mod
S, I
a?lGHZ
(dB)
le(mA)
Figure
7.1
Transistor
Characteristics
for
Vce ý0*5V
for
lE2M and 0.5 mA to
showing
(GHz)
3.0-
of
bipolar
two similar
These transistors
configuration.
numbers lE2M and 3E2M and were isolated
reference
for
the sets
of
126
of
s2l
I GHz plotted
at
of 0.5 V,
voltage
against
for
8mA
and 0.6 mA to
that
the
small-signal
Slatter
fT
such that
currents
of
up to a frequency
dependent
bias
Some of
elements.
the values
of
a good fit
general
as a guide,
emitter
this
parameters
1GHz with the bias variation
of up to
currents
s
for
2 mA for
had been obtained
quency
the
limits
errors
for
be
1GHz for
3E2M.
the
the
s1
that
for the lE2M
the higher
fre-
lE2M and 1.6 GlIz for
3E2M.
The author
apply
the models might
4 mA for
the measured s para-
He also states
than expected.
for
for
simulated
for
lE2M
than
that
the
accurately
and that
more
meters
shows larger
in
model show that
well
for
the 3E2Mfits
the
that
model
states
Slatter16
for
considerations.
lE2M and up to
the
while
The relationships
techniques.
by Slatter17
to the measured
up to
frequencies
published
a number
the model elements
based
dependent
bias
were
elements
on theotetical
the
The results
The model
and contained
of the devices
were chosen using optimisation
emitter
peak value.
of the device
structure
were chosen using the construction
others
lE2M
the
the small signal
1GHz for
of
less than its
was still
was based on the physical
for
ImA
had been developed by
This model was designed to represent
of the transistors
behaviour
operating
3E2M.
the
A bias dependent model for these devices
16,17
a collector-emitter
were 0.5 mA to
IGHz
up to
for
current
lead to the assumption
of. the transistors
ranges
emitter
the modelling
producing
optimised
method of studying
achieved.
decided
an interesting
that
algorithm
described
models for
to this
each of the bias conditions,
would
be to
problem.
By
an alternative
the bias dependence of the model elements could be
Some of the results
76 in 1981.
author
earlier,
experiment
of this
experiment
were published
by the
127
of the Model
Optimisation
7.1.
The author
the individual
a model
earlier
and using
8 nodes and 12 elements,
7.2.
IMI
initial
and the
the frequency
over
This
Ie `2 4 mA.
overall
region
model for
initial
the
used as
Ie = 0.5 mA, further
the measured
s
of F=4.35E
individual
7.4 show the
s
This
model,
from the initial
shown, were
V
= 0.5 V and
ce
this
and gave an
model was at
of the lE2M and when it
with V
and
=3V
ce
in the less complicated
+ 02 for
the
of 6 nodes and 9
lE2M with
v
ce
= 3V
model were 0.2 dB and 1.8"
of this
were 0.7 dB and 3.8".
errors
was
of this
parameters
The
model compared with
The scales of the graphs cover the same
parameters.
time
The
chapter.
values
2GHz for
This model consisted
developed
in
approximately
was
model
CDC 7600.
on
However,
of operation
ranges as those in the previous
absolute
same weightings
of 6 nodes and 14 elements
The r. m. s. errors
and the maximum absolute
in
Figure
graphs
0.1 GHz to
the same transistor
7.3.
and gave a value
0.
S
Ie
mA.
`ý
and
the
A new model was produced for
changes were made resulting
in
Figure
shown
model
elements
the
element
of 5.15E + 02.
F,
of the linear
the limit
range
model consisted
function,
error
lE2M using
This model, which consisted
based on the PRL bias dependent model.
the
the
in the previous
described
n-p-n transistor
model used is shown in Figure
initial
for
in the modulus and phase as were used when modelling
errors
the vertical
of
described
algorithm
modelling
by producing
started
includes
the
chapter.
This very accurate
2200 c. p. u. seconds on the UMRCC
in the
two stages
model in Figure
7.2,
through
development
the
first
of the
model
final
V,
for V
0.5
to
this
3V
model for V
=
=
.
ce
ce
Now, using this latest model as the initial
model for the lE2M
developed
with
Vce 'ý 0.5 V and Ie=0.5
were obtained
found
that
this
without
model,
mA, it
any further
with
was found
topology
the element
that
changes.
values
excellent
results
Furthermore,
optimised,
also
it
was
gave good
128
No. oF nodes =8
No. oF elements = 12
c.
bI
Mode INode 2Node 0-
Figure
7.2
R (1-3) = I. OOE+01ohm
R (3-4) = 2. OOE+01ohm
C (4-S) = 3ME-05 nF
R (5-2) = I. OOE+01ohm
C (3-6) = IME-03 nF
R (1-6) = S. OOE+01ohm
C (4-6) = IME-0i
nF
g (S-G) = I. OOE-02S
(T =-2 SOE-03 ns )
.
(V across nodes 4)
R (6-0) = I. OOE+00ohm
C (2-7) = 1,00E-03 nF
R (7-0)
2. OOE+02ohm
C (1-2)
I. OOE-OSnF
Base
Collector
Emitter
Model Based on PRL Bias Dependent Model
Initial
P=
V30+02
No. oF nodes =6
No. oF etements
bI
C (1-2) = 1.33E-Oi nF
9 (2-5) = 2.5SE-02 S
(T =-2. GGE-02ns
(V across nodes s
C (i-S)
8. ISE--Oinp
C (3-5)
I. We-03 nF
R (3-1)
i. giE+01 ohm
R (1-2)
1.9! E+02 ohm
R (1-5)
I. Sep+03ohm
R (O-S)
1. 90"E+01oh-n
C (2-3)
9.07E--OSnP
0
Figure
7.3
Node INode 2Node 0-
Base
Collector
Emitter
Final
Model for
1E211 for
V
ý3V
ce
and I.
= 0.5 mA
129
*
MEA5UREMENTS
,k
NEW MODEL
PH
1ýiI
DF
L-'
F-E
: - -,
ýEI-,
I, -- -jFlZ
riZ
i 5c
'121
MOD DE
PýAl-j ýjHlý
15
'121
25ý
1I"
IL,
-, ,
r,
5
ý- F\ t ýJ 'i Fl Z
0
I
'\'
It
ý rýll-ý li'OZ
Figure
7.4a
s
S
of
and
21
11
Optimised
Model
for
V
ce
=3V
and
10=0.5
mA
130
MEASUREMENTS
NEWMODEL
ti
-12
2CIG
-25
r
--3,
IE IJ-1
'I--1ý
-
39
Iýriz
10
-40
EQ 1ý z
10
MC,DD In'
22
PýW)f: D,-,G
ý22
I (\(
1C,
I JLI
5
C
-1Ei. JdZ
F-1ý1ý0GHZ
-
I 'r
-15C
Figure
7.4b
S
Optimised
S
of
and
22
12
Model for
V
=3V
ce
and 10=O.
S mA
131
for
results
the
MM
Following
with
this,
the
each of
four
Excellent
Ie*
higher
of
value
next
of V
of the values
was used as the
new models
were again obtained without
results
in
in
topology
the
and
every case the optimisation
changes
further
ce
and the samevalue of Vce with the
for
the same transistor
model
initial
both
Ie=0.. 5 mA and with
of the
Leicester
200
the
less
than
on
seconds
took
c.
p.
u.
model element values
Therefore the process was repeated for still
CDCCyber 73.
University
higher values of Ie , until
the results
was necessary to achieve similar
The values of the objective
accuracy.
7.1.
in Table
7.3 are given
Objective
e
3E2?4
ce
V
= 0. sV
ce
= 3V
V
V
ce
= 3V
3.33E + 02
4.35E
+ 02
4.12E
+ 02
S. 82E + 02
1.0
6.62E + 02
8.94E
+ 02
4.04E
+ 02
6.96E + 02
2.0
6.80E + 02
1.22E + 03
3.7SE + 02
3.99E + 02
4.0
3.80E + 03
1.6SE + 03
3.94E
6.27E
was felt
4mA
Of Ie up to
that
for
of F=3.80E
occurred
ranged
lE2M and up to
the
1EN with
8mA for
V
ce
-8.9
dB to
frequency,
dB.
28
-IS.
Models
had been obtained
the
3E2M.
errors
Although
high,
the
mA,
it
was still
of 1.7 dB and 15.2*
where the moduli
The effective
of the
at values
= 0.5 V and Ie=4.0
since the maximum absolute
2GHz, the highest
from
accuracy
of 0.5 dB and S. 4% was relatively
acceptable
at
the
+ 03 for
r. m. s. errors
considered
satisfactory
+ 02
1.11E + 03--
for Optimised
Function
of the Objective
lE2M and 3EN Transistors
Values
7.1.
giving
+ 02
1.46E + 03
It
meters
ce
= O. SV
O. S
8.0
value
F
Function,
lE2M
mA
V
Table
that a topology change
for each of the models having the same topology as that shown in
function
Figure
indicated
of the
ranges
of
s
para-
Ie of
132
this
model for the two types of transistor
operating
signal
ranges of the devices
for
the small-signal
model
accurate
transistors
the suggested smallThus a single
given earlier.
bipolar
of the two similar
operation
the frequency
covering
agree with
range 0.1 GHz to 2.0 GHz had been
developed.
process had gone through a number of diff-
Although the modelling
stages,
erent
this
new model was quite
to the PRL bias dependent
similar
final
in
Looking
the
the
elements
at
-
model.
model in Figure
in the PRL model in Figure
which have obvious equivalents
7.3 those
7.2 are C
1-2"
having
Those
C
a slightly
weaker resemblance to elements
and g2-5'
3-2
C4-, and R
The remaining elements: R
in the PRL model are R,
5-6'
1-32
-4.
form a 11 arrangement where in the PRL model there
to
R
seem
and C
,
3-5
1-5
T-Hor
The
transformation
well-known
star-delta
arrangement.
aT
was
is described
bility
the
that
by Weinberg77 and the author
the
arrangement in the final
n
arrangement in the PRL model, with
T
would be present
by this
would be equal to R,
inductor.
series
resistor,
the
some of the extra
is equivalent
of
elements that
rendered unnecessary by the optimi-
equal to Rl_, in the PRL model, in series
3
and
Thus it
in series.
is the possi-
in the new model
R
1-3
+R 3-4 in the PRL model and should also have a
_,
The 11 arm between nodes 1 and 5 should consist of a
n arm between nodes
capacitor
Rl_,
transformation
there
model is a transformation
Thus, by the T-11 transformation,
process.
sation
suggests that
to R,
+R
5
should consist
could be conjectured
with
a capacitor,
of a resistor
that
and
and a
in the new model
in the PRL model, R1_5 is equivalent
3-4
_,
in the new model is due to C4-6 plus
in the PRL model and C
R
3-5
1-3
capacitance from the T-H transformation.
additional
to
some
Having developed the new model, the next step was to examine the
variation
in the values of the elements in the model with
tions.
The results
of this
At values of Ie greater
exercise
than
the bias condi-
are given in the next section.
4mA for
the lE2M transistor
and
133
8mA for
3E2M, topology
the
for
models developed
Examination
7.2.
these
changes were indicated.
in
cases are given
Details
Section
graphs were drawn of
Taking each of the model elements in turn,
7.5a and 7.5b.
in Figures
by straight
lines.
lines
to the
blue
the emitter
the functions
that
These graphs are shown
current.
joined
In each of these graphs the points-are
The red
3E2M.
lines
refer
to the
The solid-colour
lines
are for
lines
the dotted-colour
7.4.
Elements
of the Bias Dependent
the element values against
some
of
were fitted
lE2M transistor
are for
The black
V
= 3v .
ce
to these points
V
and the
= 0.5 V and
ce
dotted
lines
are
be described
and will
later.
These graphs were extremely
showed a distinct
pattern
collector-emitter
voltage.
as most of the elements
encouraging
of variation
the emitter
with
Although the model was different
current
from the PRL bias dependent model,
in
be
two
the
the
models
could
compared.
elements
of
some
relationships
in the PRL model involved
being based on theoretical
curve fitting
obtained
only
functions
I
were studied, allowing
of
eeee
Then the effect of Vce was taken into
thereof.
with
dependence of each element is discussed
the model shown in Figure
7.2 for
the
I,
these
e
decided to use a
E02ADF78, to examine the fits
First,
to each of the sets of points.
The bias
and
VVce ,V ce
The author
considerations.
from the NAG library,
routine
and the
only the relationships
VT
12
,I,
and reciprocals
The bias
consideration.
below with reference
comparisons
with
the
being made to
PRL bias
depen-
dent model.
C1-2 - This
they
the
suggested
element
had a direct
was not
current
lE2M and 0.621VV
ce
pF for
equivalent
dependent,
the
3E2DI.
with
in the
PRL model which
values
of
In the author's
0.33/VV-
ce
pF for
model the
134
C' (1
-2)
GM
N'
"3
ý
30,39
---:
X
0, -ýý
311)
ýý61
[iJrN
NU-C
M
c (4-51 N,
/
/
/
/
0.
-- -- -----------
Li
I
',
_)
c_)
C kj Q
t?,, _-1, j -ý !JA
C (3 -T
-
lE2M
Oý.5V
1E2M 3, OV
3lý2M
OAV
3
ýUM
jý 3v
=
-w=.
1.0'1
ý-L1
iNk'7f C,1,-)
CURQt:NT,
Figure
7.5a
Variation
of
Model
NA
Element
Values
with
Emitter
Current
13.5
(3 -1) 0 1-11ý
R (4- 2) C,
20
--
40
3 ri
--------
40
2O
.3r
"r
0-
2'7e
MIA
R (i -ý'jý 0`11'ý
!IA
R"0, -,J:ý 011N
22'3
IA
14
12
140
203
!JA
ý
C (2--3) N'
E2M 0.5V
-1
1E2M 3. OV
-
ýE2M 0.5V
7 ?m
11 ýv
,
----------
- --
F-u INý' -1 cl,Ni1ý
11
-----------
5' 6
Figure
7.5b
Variation
of
Model
Element
Values
with
Emitter
Current
(j ý r, ý- KI -, !I
136
limit
C
to
the maximum values obtained
reach
a
where
appeared
of
value
1-2
were
in the ranges of le und.er consideration
and
for
IE2M at 0.5 V
giving
0.33/YV
0.24 pF
for
lE2M at 3V
giving
0.42/VV
0.99 pF
for
3E2M at 0.5 V
giving
0.701VrV
0.45 pF
for
3E2M at 0.5 V
giving
0.78/Y'V
the value
However,
at
0.47 pF
lower
values
of C1-2 was definitely
of
a 5% change in
change in the value
significant
of the
ThL- black dotted
in Table 7.2.
7.5a show the function
PRL
suggested
-
that
ce
ce
PF
PF
pF
the value
with
sufficiently
of both
Ie
of
Ie
to
sensitive
Of CI-2
function,
objective
lines
made a
F.
Thus the
Vce
and
as shown
on the graphs of C1-2 in Figure
at the appropriate
evaluated
ce
the optimum value
chosen for C1-2 were functions
functions
g2-5
The model was also
Ie.
of C1-2 that
the value
varying
ce
PF
the value
of the
of V and Ie.
ce
values
current
generator
be
should
A further
relationship
stated by PRL was that
proportional
1
where hfe re was element R4-6 in Figure 7.2 and h fe 0 so
r.
e
It is possible that R4-6 in Figure 7.2 could be equivalent
to element
to Ie
R
1-5
in Figure
If
7.3.
Checking
the optimised
although
this
taking
However,
the values
Ie=8
were so then
of g
for
g2-,
values
were 44.4
with
were not
e*
2-5
to the
for
accurate.
linear
of this
function
element for
mA with V
= 0.5 V was taken as a sign that
ce
non-linear.
This effect
and in each case it
can be seen in several
occurs at the highest
values
SO/R
1-5,
that
50 was not
of
lE2M and 46.3
sufficiently
to
conclusion
the value
the optimised
chosen for 92-5 was a piecewise
The sudden change in the value
be equal
could
was possible,
values
average
g
R
led
and
2-5
1-5
of relationship
these
calculated
relationship
values
The average
right.
quite
on I
type
this
for
3E2M.
of Rl_,,,
Thus the
dependent only
the 3EN at
the device was becoming
of the model elements
value of Ie for
which an acceptable
137
value
had been obtained
F
of
model this
fits
and the
effect
-r - The time
can be seen that
It
model vary with the emitter
on the
to the
for
in mind the PRL values
it
T,
The values
of operation.
in the PRL model was
of
the
However, as there
was decided that
constant
at low emitter
values
currents
since
in the linear
for
ps
-28
chosen were therefore
the higher
and bearing
T
were most certainly
for
for
T
in the values of
JUDI, the emphasis being on the values
the
in the author's
T
the values
should be chosen based on the optimum values
region
Of
voltage.
these were the values when the devices
7.5a.
the lE2M and -26 ps for
and that
in the variations
was no apparent pattern
chosen
in Figure
of g2-5
generator
values
lE2M also vary with the collector-emitter
was made to
functions
linear
current
the optimised
current
No attempt
graphs
for
ps
-25
given the independent values of
3E2M.
ce
lines
applied
constant
"" O'S V-
of the piecewise
dotted
black
be
the
seen
as
can
V
with
voltage,
the
and -32 ps
the MM.
for
function
The
-
C4-5
A/F
was
1.16 PF. V1 for
V
voltage
A
where
c-sub
the
while
current
Therefore,
inverse
C
large
it
at
F
F=
were not modelled.
of
1.12 pF. Vi
of the
+ 0.75.
element
ignoring
values
values
PRL model
for
the
lE2M and
collector-substrate
This
element
at high
values
was one
of emitter
were almost
the high current
of
of
this
element,
emitter
7.2 perhaps
discussed
as with
current
C
be considered
could
3-5
due to the T-TItransformation
the
constant.
effects,
an
of V
as given in Table 7.2 was appropriate.
ce
values
in the PRL model in Figure
in
element
was a function
the
values
function
higher
value
YV
c-sub
was decided that,
at
equivalent
changed drastically
values
The
optimised
-
changes
the
a constant
3E2M and
lower
square root
3-5
took
such that
where the optimised
for
given
with
and
C
4-5'
again,
underwent
these
effects
equivalent
to element C4-6
some additional
capacitance
earlier.
C
in the PRL model
4-6
138
was dependent
on the value
of the current
possibilities
of this
of relationship
type
for
were obtained
curves
of C3-51g2-5
of
value
g
against
both
emitter
Investigating
generator.
showed that
transistors
smooth
the optimised
on plotting
The function
current.
similar
the
dependent
values
on the
.
2-5'
was found
in
Table
7.2,
element
could
be taken
given
to
fit
a best
give
for
this
element.
R3-1 - This
in the PRL model in Figure
and the combined
the transistors
in the
PRL model were 66 Q for
R
model,
author's
from
These resistors
7.2 were used by PRL. to model
Figure
of
7.2.
3-1
in Figure
the graphs
some of
constant
values
of these
7.5b.
the
C
with
frequency
lE2M and 23 Q for
the
R3-4
and
1-3
together
the high
to depend on both
appeared
to R
as equivalent
in
3-6
effects
two resistors
3MI.
In the
Ie and V
as can be seen
ce
The maximum values
this
element
attained
for the
were 54SI with V
`ý 0.5 V and 56 0 with V
and
=3V
ce
ce
3EMI were 22 SI with V
3V
These
=O. SV and 26a withV
=
values
.
ce
ce
However
were quite similar to the constant values used in the PRL model.
for
the UZI
of the model to the value of R3-1 meant that
the sensitivity
changing current
and voltage
should Ve included.
effects
increase
be
to
to
appeared
range
of
Ie plotted
element
is
given
entire
this
of
values
1890
for
the
the
in Figuie
of this
element
at
Ie=0.5
voltage
these
mA.
7.5b.
The function
for
7.2.
element
that
of increasing
developed
lE2M and 185sl
7.5b
The effect
in Figure
lE2M and 14S Q for
graphs
it was decided that both the current
amount over the
were the most appropriate
for
of
by a constant
the value of R
3-1
in Table
The
R
equivalent
4-2
209LI
Therefore,
was significant.
the effect
in the
for
the
PRL model was given
3EN.
It
MM.
values
was felt
chosen by the
and the values
the
constant
It
that
constant
author
can be seen from
were based on the
values
optimised
were
the
values
139
R1_5 - This
g2-5 .
No relationship
was also
decided
Functions
of
that
in Table
This
R
0-5
element
for
arly
as 18 Q,
the
this
a function
element
The functions
Figure
did
not
for
l. lSI
of
agree
it
by R
for
values
giving
the best
7.5b.
in the
to R
0-8
values
of R0-5 at
the value
which
PRL model in
the
lE2M and
the values
with
of R
0-5
Ie=2
Ie00.5
Functions
mA.
of
mA was as high
Ie were therefore
element.
of the voltage
was calculated
no element
was discovered.
average
in
of
This element appeared to depend on Ie particul-
PRL model,
In
C
the
2-3
is
to the
be equivalent
to 4SI at
reducing
the discussion
on Vce exhibited
each transistor.
could
during
elements
7.2 and are plotted
model.
lE2M for
developed for
voltage
fitted
These values
3E2M.
the
in the author's
for
two
these
PRL gave R
constant
0-8
7.2.
0.8SI
been mentioned
was no dependence
there
for
of V
ce
are given
Figure
between
Ie were therefore
the two values
fit
has already
element
the equivalent
C
4-5'
between nodes 5 and 8 in Figure
as (V
equivalent
element,
-I
ce
e*
R2-5)*.
In the
to PRL's R
2-5*
However,
was defined
7.2.
This
model there
author's
the
as
optimised
values
both
depend
Ie
functions to the
Fitting
C
to
and V
on
seemed
of 2-3
ce'
because
difficult
the element values at V
proved
=3V
element values
ce
became zero.
values
element
necessary
for
The final
for
V
= 0.5 V first
ce
the higher
7.2 is not
given
in Table
using
the function
was to
compromise
value
of V
entirely
fit
and then
ce'
to modify
The final
satisfactory
are shown in Figure
a function
7.5b.
this
complicated
and the
It
of
fits
Ie to the
function
as
function
obtained
can be seen that
the
0.5
V
V
are considerably
`2
with
more accurate than those with
values
ce
Fortunately,
the model was not highly sensitive
V=3V.
to these
ce
low values of C2-3'
140
Element
Constants
for
lE2M
Bias Dependent
Element
Value
of
in V)
in mA, V
(I
ce
e
(A
C
+IB
1-2
e,
10
nF
A=
B=
2
(A + B. I ). 10e
2-5
S
A=
B=
T
I-
3-5
nF
R
B=-1.81
1s1
e
1>1
e
A= - 0.47
mA B
= 4 . 64
A= 1.5
mA B= 3.0
0.028
A=0.024
e
B = 0.018
B=0.055
B
+I+C.
e
A = 62.9
B = 43.1
C = 1.7
A= 22.9
B= 17.2
C=1.2
A = 189
= 145
B
V
ce
0
A+
Q
R
6-5
SI
B. I . .
+BIeV
nF
Table
7.2.
105
e
A+B
C
2-3
>1
92-5
A
1-5
e
mA
A=-0.032
A = 0.024
4-2
R
1s1
e
V17ce
+ I
A
1.12
7 21
.
3.4
3.0
A=0.34
[A
3-1
B =-1.12
A = 0.54
i+
nF
R
3
10-
A+
C
A=6.22
A =
A
ns
C4-5
A = 2.84
ce
L
9
Constants
for
3E2M
Bias
C
5
loce
Dependent
A = 8.385
A=7.371
B = 108.32
B= 76.19
A = 1.0
B = 8.0
A=0.1
A = 3.27
B = 5.95
A=8.2
Relationships
B=0.8
for
B=9.7
the Model
in Figure
6.3
mA
141
of the different
The effect
for
the determination
for
using
optimised
ent
the number of values
exercise
of the
effects
of different
Although
on the model element
values
of collector-emitter
voltage,
bias
the
conditions
of differ-
effects
could
be seen,
generally
for
V
ce ,
data
which
in the proposed functions
2mA for
than
the lE2M and greater
lGHz against
they
As expected
up to its
and measured
show that
model which,
in the paper
3E211 before
the peak value
S parameters
calculated
the measured
7.7 and 7.8 for
the
=3V
of V
ce
has some doubts
s
is
appears
by Slatter17
accurate
is
,
s
shown to
is
21
using the author's
the
As
s
of
at
21
against
accuracy
of
values
of modulus
better
than
lose
the
S21
PRL
for
accuracy
the
Graphs of the
reached.
frequency
are shown in Figures
at the
Looking
mA .
7.6.
bias dependent model, together
lE2M and 3E2M respectively,
and Ie=2
for
to be slightly
of modulus
parameters,
about
the 3E2M.
of the modulus
values
the model
This
peak value.
was
These graphs are shown in Figure
current.
emitter
4mA for
than
Ie
of the new bias dependent model, graphs were
a check on the effectiveness
drawn of th6 calculated
would not give the
0.5 V when
elements for V
0
ce
values of certain
optimised
greater
some of the functions
was known that
It
at these
the measurements
arbitrary
graphs
of
s
12
bias
the
author
for
the
However, the graphs show good agreement between the measured and
IEV.
calculated
values of all
the
s
parameters
improvement on those shown by Slatter17
tinues
were developed
However, a bias dependent model had been developed.
of Vce'
values
of the
be seen as a feasibility
could
was too small to give confidence
was provided
with
7.2 and functions
models of transistors.
currents
emitter
on the values
conditions
This
each of the elements.
study
bias
in Section
was discussed
model elements
Model
Bias Dependent
Resultant
7.3.
up to a frequency
of
2GHz while
and would appear to offer
especially
,
Slatter's
an
as the agreement conmodel was only valid
142
MOD'321
lE2M
0
VALUES
o MEASURED
,, MODELVALUES
0.5V
CURRENTMA
10 Je
246
ýD
-10
-15 -0
MODS21
lE2M 3. OV
10
5
n
CURRENT
MA
To
-5
-10
-15
MOD521
3E2M 0.5V (Vre)
lo
5
0-
CURRENT
MA
ýQ
0m5---
20
----
-5
25
-15 -
MOD321
-0
3E2M 3. OV (Vet)
10 -
5
0
5
10
Is
20
-5
25
CURRENT
MA
le
-10
L
-15
Figure
7.6
Modulus
S21 at
1GHz for
the
Bias
Dependent
Model
14
MEASUREMENT5
NEWMODEL
DHPE: iE
MIU
C:
5
iI I
'1
HZ
_rd
"I
-
I "
c,21
0DD bi
PLI'Vi.
ý .1
1-
ýI
J2
20
IC
1,r3
1c
1 (r
rz
HZ
5C
r
_________________________
-;
'ilL
-'
-\
10'
Figure
7.7a
s
for
S
and
21
11
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S
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21
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147
1GHZ.
up to
The conclusions
drawn from
were that
this
developed
be
the
author's
using
could
models
it
was possible
modelling
simple functions
to develop quite
of likely
retical
absolutely
functions
different
of
number
choice of the most suitable
7.4.
Models
-
After
developed,
at the highest
forms the
the possible
technique
could assist
with
a
in the
function.
the bias dependent, model for
the transistors
they were not
Currents
High Emitter
had been successfully
some theo-
of the model element
of the bias variables
final
for
of the variation
Then the use of a curve fitting
could take.
function
Although
could indicate
bias conditions
at different
values
Examination
necessary.
and that
the depen-
could be useful,
relationships
dependent
algorithm
describing
dence of the model elements on the bias conditions.
indications
bias
accurate
the linear
of operation
were made to develop models fot
attempts
of emitter
values
region
current
for
which data
was provided.
For each transistor,
values
of
Ie
the
using
s
both
values
with
and
given
the
appropriate
was
used
model
from the earlier
optimised
Applying
work.
different
four
cases produced
V
ce
model for
of
the modelling
each of the four
be the most effective
with
given,
the initial
Ie=4
mA or Iea8
algorithm
another
to these four
each of
10 mA and 28 m.A for
function
individual
in each case was less
errors
than
the
for
model was the most effective
highest values
and their respective
=3V
with V
ce
The value of the objective
These models are shown in Figure 7.9.
both transistors
of I
4.3-E + 03 and the maximum absolute
were 1.5 dB and 12*.
mA
cases, one of the models proved to
V= '2 0.5 V and I
ce
e
while
data at the highest
However, re-optimising
models.
these four models for
lE2M and 3E2M respectively,
parameter
The models were again effective
e
148
b
No. oF nodes =2
No. oF elemenLs
Node INode 2Node 0-
Base
Collector
ErMter
0
IE2M O.SV lOmFl
26E+03
F=i.
9
C
R
R
L
C
C
C
C
R
R
3E2M O.SV 27mA
F --- I SSE-+03
=. 1.80E-01 S
=-I. O?E-01 ns
(V across nodes s
(3-5) = G. iOE-02 nF
(3-1) = 3.61E+01 ohm
(4-6) = 1.92E+01 ohm
(0-S) = 1.13E+00 nH
1,1-4) = 1.18e-03 nF
(3-4) = i. SOE--03 nF
(1-5) = 6. SGE-04 nF
(6-0) = 1.30C--03 nF
(1-5) = 1.33C-+02 ohm
(2-6) = ]. ý3E+01 ohm
C
R
R
L
(6-S) = i. 97E.-Ol S
(T =--3.09E-02 ns )
(V across nodes 3)
(3--S)
(3-1. )
(1-6)
(0-5)
C (3-4)
C (I-S)
(6-0)
R (I-S)
R (2-6)
1.20E-01 nP
1.36E+gl ohm
2.36E+00 ohm
S. iOE--Ol nH
3.06E-03 nF
9.04F.-03 nF
4.03e.-03 nF
1.90E-03 nF
ý. GH+01 ohm
J. ISE+01ohm
bi
c
lio. oF modes m6
No. oF eletents a 12
Node INode 2Node 0-
0
IE211 3.9V
F=i.
3E2M 3.2V
lOmA
P=1.42E+93
2SE+23
(2-0) = ý. S4e-gl S
(T ::-3.2iE--02 ns
(V acrass nodes 3)
9)
(i-0) = 3.16E-03 np
C (3--0) = 3.97ý-02 nF
q (3---) = 2. Z8F+31 ohm
R (1-2) = 2JOE+02 ohm
p (S-0) - G.S7E+el ohm
SSS-03 nP
C=2.
= !. ý7E-101ohm
i. s3e-oi n"(2-0)
MH-01
(1-5)
nH
C (1-2)
S. iSE-Oi nFC (1-3)
9. ýH-04 nF
SOE--g
IS
(2-9)
=I
9
ýT =-i. 3SE-22 ns
(v across nodes 2
(1-0) = i. 90E-03 nF
C (3-0) = !. SýE-02 nF
q O-S) = S. 93E+91 ohm
R (1-2) = 2.91E+02 ohm
9 (5-0) = j. S@E+02ohm
C (5-3ý = 9.42E-04 nF
1.98E+Oi ohm
1.39E-01 nP
(2-0)
1.9ýE+00 nH
L (I-S)
2,72F-V nP
C (1-2)
i. 02E-04 nP
C (i-3)
Figure
7.9
2?mA
Models
for
High Emitter
Currents
Base
Collectir
ErAitter
149
MEASUREMENTS
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Figure
7.10
Modulus
S
21
Against
Frequency
for
the
High
Current
Models
iso
up to
are
2GHz and graphs
shmm in Figure
The author
the
operate
in a similar
demonstrates
of
operation
found
this
interesting
result
3E2M would
manner to the
of collector-emitter
same values
models
that
frequency
for
each case
7.10.
in the
emitter
different
were
21
against
there
at the
of the transistors.
appear
to
enable
lE2M at higher
voltages.
two values
are major
as it
The effect
Of the two transistors.
similarity
stripes
the
s
of modulus
effects
of
seems to confirm
the extra
of
the
transistor
emitter
The fact
collector-emitter
on changing
to
currents
that
for
there
vol. tage
in this
V
ce
region
151
CHAPTER 8
CONCLUSIONSAND SUGGESTIONSFOR FURTHERWORK
In Chapter 2, the standard models for
described
with methods by which the values
together
chosen gives adequate accuracy there
there was insufficient
author
native
determination
the
on
and
models
Chapters 3 and 4 provided
the
author's
where a suitable
how models
device
for
described
could
transistor
which parameter
device
in order
measurements
The algorithm
of particular
started
From this,
details
transistor
p-n-p transistor
Then, after
vertical
measurcments
to
using the final
in Chapter
those
readily
available.
and gave details
performance
were available.
Chapter
a
4 then
optimisation.
modelling
algorithm
as one to optimise
and the
s
was described
the element
between the
necessary,
s
of the model.
parameters
the topology
The development of a model for
a complete description
of the modelling
6 details
of
with
the development of the
version
to
cases
was used to illustrate
demonstration
n-p-n transistor
relevant
compare their
were given of the development of a model for
As a further
algorithm,
was not
were developed whereby, if
techniques
be
optimised.
the
also
could
model
of
algorithm.
for
models in order to reduce the errors
parameter measurements of a transistor
lateral
was developed
which
a number of methods of numerical
in Chapter S.
element values.
some background information
The development of the author's
values
of their
parameter
be analysed
However, the
guidance on the choice of alter-
model of a particular
3 described
Chapter
algorithm
modelling
Where the model
are no problems.
felt
were
of the elements in a
device could be determined.
model of any particular
that
bipolar"transistors
of the modelling
of the efficacy
were given
the lateral
a
algorithm,
p-n-p
algorithm.
of the modelling
of the models
produced
in each of commonbase, common emitter
for
and
a
152
common collector
but
accurate,
to produce
attempts
model
bipolar
the
dependent model of
The modelling
was first
algorithm
model and then to optimise
of the model for different
bias conditions
the element values
Using
of the transistors.
element values as a guide,
model elements as functions
of
was in
algorithm
used to develop a suitable
these optimised
three
all
required.
the development of a'bias
transistors.
for
because
was mainly
example of use of the modelling
Chapter 7, which described
two similar
time
and the computing
of the problem
The final
This
were highly
produced
general
a single
so successful.
was not
configurations
size
The models
configurations.
suitable
expressions
of the bias conditions
for
the
were suggested by the
author.
The modelling
producing
small,
was developed with
was on the achievement of results
fast
computer program.
faster
becoming
ever
and larger;
should be written
efficiently,
cause the failure
to find
that,
with
a solution
to a particular
the facilities
the vertical
available
models for
some large problems.
tests
although
algorithm
of the models,
precision
could
However,
problem.
transistor
n-p-n
to the author,
of the modelling
could be made faster.
the use of single
that
in order to achieve
could be conducted to check the effect
the modelling
Computer programs
However, these sacrifices
on the efficacy
a
computers are
they should not take short-cuts
sacrifices
effect
than on producing
view is that
therefore,
of accuracy might be necessary
have a detrimental
The emphasis,
rather
The author's
the development of the general model for
demonstrated
the main aim of
transistors.
models of particular
accurate
therefore,
algorithm
some
satisfactory
might not
algorithm
and
of a number of ways in which
These include
arithmetic
in the analysis
153
the Jacobian
loss
the
increase
any
cause
improve
could
matrix
offset
could
in the
increment
The use of a smaller
the
of
of accuracy
due to
(ii)
this
calculation.
above,
and would not
time.
computing
the author decided that
algorithm,
should remain in the model.
addition
One effect
of this
was that
sometimes elements were added and then removed again during
addition
of-a different
cular
native
The first
additions
evaluate
all
the possible
former suggestion
for
additions,
further
each possible
Then a suitable
one at a time,
elements,
successful
would be worth evaluating.
of these would be to test
in Stage 2 could be, either,
in order
A second,
each possible
element addition
together
re-optimised
at their
in a similar
The
of only one element
be inserted.
for
Stage
2 could
all
the successful
The model could
optimum values.
manner to Stage 1 so that
In this
be to evaluate
then
be
any elements that
were
way a number of element
be
in
achieved
one operation.
might
For large
transistor,
to re-
or,
might seem to be the most effic-
and then to insert
be
removed.
could
not necessary
additions
to add the remaining
elements which could be successfully
technique
alternative
further
only the best again.
taking
to the change, could no longer
added prior
additions
that
for
of effectiveness,
by
the
but
addition
a
model
changing
method,
differences
such
cause
could
and to
addition
technique
to attempt
additions
There are two alter-
times.
the author considers
only the best of these.
select
ient
that
the
Cases have been seen where a parti-
element.
element is added and removed several
techniques
to
of the approximation
calculation
accuracy
in the required
of the
calculation
matrix.
the
In Stage 2 of the modelling
each successful
in
to the Jacobian
approximation
This
differences
the use of forward
(ii)
problems
even these
such as the
suggestions
might
general
not
model of
reduce
the
the
vertical
required
n-p-n
computing
154
time
sufficiently.
for
available
might
in view
Therefore,
that
be feasible
to use only
of the
part
in Chapter
as suggested
problem,
particular
amount of data
large
the
of
data.
available
In the development of the bias dependent model described
Chapter 7, the modelling
was used to calculate
algorithm
different
the
of
each
at
element values
choosing the constant
might be possible
it
author
n-p-n
transistor,
create
technique
to use an optimisation
like
This would,
optimum values.
a very large
both to
their
model of the vertical
problem and it
scale optimisation
might be necessary to use only a selection
of
and to ascertain
values
the general
Instead
manner used by the
arbitrary
take
constant
should
elements
which
choose
in
the optimum
bias conditions.
in the rather
values
it
6,
S parameter
of the available
data.
After
the constant
functions
accurate
have been ascertained,
element values
for the bias dependent elements might be obtained if
the models are then re-optimised
the constant
with
region
could be possible
approach to this
The author's
algorithm.
using
higher
for
progressively
models
values
of
Ie
perhaps,
be incorporated
which would allow,
to zero,
resistor
ternatively,
after
V
ce *
Hopefully
for
which it
and these changes could,
example, the gradual
reduction
Introduction
of a
Al-
would remain as a short-circuit.
bias conditions
is the author's
the
the models by the use of mathematical
by the use of diodes
described
Moll
Ebers
in
the
models
way as
in the
and
elements or groups of elements could be made part
model under certain
It
modelling
problem would be to develop new
of the models would only change slowly
functions
beyond the linear
the author's
topology
into
fixed,
element values
The development of bias dependent models valid
of operation
more
opinion
that
have been achieved.
of the.
in much the same
in Chapter 2.
the aims of the research
A modelling
algorithm
stated
has been
iss
developed
which will
to produce
order
model element
optimise
measurements
algorithm
has been demonstrated
these
Although
hybrid
pi
modelling
of most bipolar
transistors.
transistors
using
was hoped that
models of bipolar
2.2.2
would
here,
region
different
for
were provided
be suitable
of bias
operation
for
for
has also
development
of
functions
the bias
the
shown that
dependent
two similar
of
regearch might indicate
this
transistors
dependent
elements
the author is of the opinion
applicability
accurate
that
any conclusions
of the non-standard
development of the new modelling
means by which this
how the standard
could be improved especially
before
be
devices
modelled
must
of
the general
several
The author
Although a number of different
quencies.
described
simple
of this
by the author.
developed
It
linear
the
s
model based on the modified
can be used in the
algorithm
A model for
topologies
an initial
that
in Section
models.
was
feels
model described
the modelling
bipolar
model
of
the
using
-The effectiveness
by the modelling
suggested
the author
cases,
transistors
of the devices.
parameter
transistors.
of bipolar
models
accurate
in
and topologies
values
algorithm,
can now be achieved.
at high fre-
models have been
a much larger
selection
can be drawn concerning
elements.
However, by the
the author has provided
a
156
APPENDIX 1
S PAWIETER DATA PROVIDED BY PHILIPS RESEARCHLABORATORIES
157
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APPENDIX 2
RELEVANT PUBLICATIONS BY THE AUTHOR
177
International Conference on
ComputerAided Design
Pilanulacture.
ElectLrnnic
of
and
Components,Circuits and Systems
3-6July 1979
Organisedbythe
Electronics Division of the
Institution of Electrical Engineers
in association with the
British Computer Society
Institute of Electrical and Electronics Engineers
(Circuits and Systems Society)
Institute of Electrical and Electronics Engineers
(Region 8)
Institute of Mathematics and its Applications
Institution of Electronic and Radio Engineers
with the support of the
Convention of National Societies of Electrical Engineers of Western Europe (EUREL)
Venue
Universityof Sussex
178
Cutteridge
O. P. D.
INTEGRATED
IN MODELLING
EXPERIENCE
University
U. K.
Leicester,
of
TRANSISTORS
Dowson
Monica
and
CIRCUIT
did
INTRODUCTION
than
increasing
is
Industry
to
attention
paying
integrated
linear
circuit
modelling
of
the
The
have
fortun.
been
authors
transistors.
from
data
Philips
in
Research
receiving
ate
(1)
Javices,
firstly,
Laboratories
on two
such
transistor,
secondly,
and
a
pnp
a lateral
The
data
transistor.
npn
vertical
first
is
in
in
Table
the
case
shown
provided
I and
the
of measurements
of
consisted
lateral
S-parameters
the
transistor
of
pnp
O. S MHz
frequencies
in
four
the
to
range
at
MHz.
10.0
RESULTS OBTAINED
A brief
of the method
used by the
account
devices
for
linear
Jelling
active
authors
....
Figures
in the following
is given
section.
for
1.2
the
obtained
and 3 show the models
different
lateral
transistor
at three
pnp
in the modelling
the
procedure.
with
stages
in Table
2.
tabulated
errors
corresponding
METHOD
mODELLING
Construction
Function
Error
of
S-parameter
of
each
and phase
modulus
for
frequency
at
every
real
matched
were
Every
values
were
given.
measured
which
the
error
was
given
same weighting.
modulus
these
every
phase
error,
although
as was
different.
An overall
error
were
weightings
by taking
function
the
sum of
constructed
was
individual
the
errors.
of
weighted
the
squares
The
Strategy
General
the
original.
The S-parameter
to the lateral
Table
earlier.
S-parameters
of
1,2
and 3.
data
I refers
shown in Table
pnp
transistor
mentioned
2 contains
in the
the errors
the models
shown in Figures
The element
for
initial
values
the
model'
I were those
shown In Figure
obtained
using
the
(21).
global
search
The value
method
of
the overall
function
error
at this
stage
was
384.2.
After
Gauss Newton
optiaksation
of
during
this
model
three
which
elements
ware
improved
removed,
the
model
shown in Figurg
2. which
gave an overall
function
error
value
114.9
of
Elements
was
obtained.
ware
,
then
added until
the final
in
model
shown
Figure
3, having
function
an overall
error
6.1
value
of
was produced.
As can be
.
2, the errors
seen from Table
were
reduced
from maximum errors
O-S do
of
It
and
degrees
in the initial
0.1 do
model
to
and
I
degree
in the
final
model.
The order
of element
addition
was found
to be
quite
critical,
especially
that
of thv
second
Although
generator
g2 .
this
element
w4s
in the originally
present
suggested
modoi,
the optimisatLon
ttchniquo
would
not
tolurato
its
insertion
final
the
un-til
It
stage.
be noted
might
that
the direction
of the
is reversed
current
generator
in the
ftnal
model
as compared
initial
the
with
model.
ACK%OWLEDCEMENT
The method
rea uires
an initial
model with
* to be
Although
values
provided.
a
element
by the donors
model was suggested
topological
were given.
no element
values
of the data.
to obtain
in order
suitable
starting
values
the g! obal
for
the element'.
search
method
by Price
12) was used.
With
described
these
the tain
values
optimisatLon
proce.
starting
Gauss Newton
dure,
the modified
section
of a
dge (3),
descrioed
by Cutter.
program
computer
utilised.
was then
this
optimisation
logarithms
the
(repre5entznq
By running
domain
of
variables
it
is
the
ef
(; auts
possL 'ble
on
error
improvement
such
to
a
function.
addition
of
elements
Une
immcJi ate!
I"IMn
a*
ro
are
rejucing
VIn'!
the
generated
at
successi.
ons
be
the
which,
From
OnIf
new
altoments.
a time
-&I.
nvcra!
- rj
liaving
Obt3LrkCd
3, Y
Lt
it I, not
I, -
- ith
'. u-)r
a
10,182-L84.
ovural'.
ftirther
point
bV
tht)
thev
FIR
I,..
now
no%LtIvu
ý'un,:?
P#% P. -II
Aru efPvvt
-C4
K. W. gn J Rankin,
communication.
P. J.,
19? 7.
Price.
W. L.,
1977.
"A Contrall0#1
Random
Search
Procedure
for
Global
OPtimL4atavn".
COmn- J.,
20,367-370.
Simultaneous
conver.
3d'JLnq
hle
no-4%.
is
Moulding,
Privato
Cutteridge,
O. P. D..
1974,
Program
I.. part
for
Solution
the
such
`)C
Can
ve
climina-
3tgartthm
of
REFERENCES
vaiiies
any,
After
the
elements
Loca!
minimum
-it
if
rLmuved.
e tý.,! r
je w:,
Ph..
, ),
: len-ifici
re taIAFt,
later
iivUcts
from
iterat.
should
of
the
determine
ions
corru,;,
Newton
elements
ting
all
ges
to
in
algorithm
the
Independent
the
element
ealue. s)
The authors
like
would
to express
their
gratitude
to K. W. 4oulding
and P. J. Rankin
of
Philips
Research
Laboratories
for
providing
them with
the accurately
S-paramoter
moasured
data
for
tl%e two tYpes
of transistor.
t Fe
ý
vuL, ji:, i ,k
ios
ar,,
it
It -S fn
P. 'ý'i ts
trý
-nu-ý')er
oi
noL! cs
4 greater
Equations...
"Poarejal
()f
C. tact
.
sunlinvar
I. Ott
179
TARLr
1.
S-parameters
Measured
a lateral
of
S12
S1
Frequ 0 ncy
MHI
0.5
2.0
5.0
10.0
modulus
dB
-0.10
. 0.20
-0.40
-1 . 20
A measurement
TABLE
2.
Errors
in
of
Phase
degrees
Modu I us
dB
-t. 40
. 5.40
--11.20
-19.40
---64.70
-S7.20
-52.00
S12
at
the_S-parameters
of
Improved
Model
Final
model
Pha s 0
degri ýcs
--88.70
94.00
76.3n
not
Modulus
da
179.10
169.00
ISS. 00
t32.90
O. S
2.0
S. 0
10.0
-0.10
-0.14
-0.02
OAS
-0.37
-1.33
-1.60
. 3.31
O. S
2.0
S. 0
10.0
-0.09
. 0.11
0.10
0.38
-0.21
-0.71
-0.36
-2.21
0.00
0.08
-O. OS
O. S
2.0
S. 0
10.0
0.00
-0.21
-0.66
0.09
0.08
0.00
0.10
-0.13
0.02
-0.19
O. S MHz was
not
S
12 at
0.00
O. SS
0.1.0
-0.36
-0.04
0.00
0.00
0.00
0.00
0.00
-0.17
-3.41
-9.66
0.00
2.19
1.97
. 1.43
0.0(1
1.03
0.04
-0.2 0
. 0.30
-0.5 0
. 0.90
avaLlable.
Phas:
deg re s
Phase
degrees
of
-12. SO
-12. SO
-12.80
-14.00
Phu::
egr
Modulus
45
onp
transistor.
S21
modulus
dB
A measurement
Ph*% :
dog re '
S12
Fr*qu e ncy
MH
-0.04
0.04
0.00
%todulus
(III
a lateral
of
models
S.
.2
S21
O. S M11: was
S
initial
model
tratishttor.
pnp
modblus
dB
0.27
0.34
0.46
0.46
-0.30
-0.19
0.07
0.43
-O. OS
0.01
(). a1
-0.03
avaklable.
3
Pha : :
degr
,
0.17
-1.89
-S. 12
-11.02
0.52
-0.49
-2.00
-6. S2
0.67
9.13
0.06
-0.05
Modulus
dB
22
Phase
degrees
0.03
0.03
0.03
0.03
-0.19
-0.26
-0.41
. 0.71
0.03
0.03
0.03
0.03
-0.19
. 0.27
-0.42
-0.74
0.00
0.00
0.00
0.00
-0.14
-0.07
0.07
0.24
,
180
G,a 6.12xlO-IOS
G2a 5.22x 10-SS
G] a 6.77x 10-3S
gIz 3.53x 10-3s
g2x 2.07x 10-5S
C, a 8.68 x 10-S PF
C22 5.06 X 10,7 ILF
C] z 3.71 x 10-9 gF
C,,.
1.1S
A 110"S
223 2.72 x 10-'S
C2z6.19x 10 gF
2 C4 xV
0uF
Uj-;
C4,-3 3.7 3xV'gF
Figure 3 Final modei
C, al. OSX10-4 ILF
I
CZa3.90xlOpF
Figure 2 Improved model
Flgxe 1 Initica model
01 a 4.43 x 10- 5
G3s 2. X x 10-Is
G2 z 4.67 x 10' SS
o3a 6.63 x 10'3s
g, a 3.94 x 10'3 s
LZz 0.16gH
181
Algorithms
supplement
riots bYthe EdRor
AlSonthm No. 107 is publisW i
out AjFCCI t with the
Institute of Mathemtscs &M IU Appliations. and is snociated
with the authon' article in the JIMA. Volum 24. Number 1.
Aug= 1979.
AlPrithm 107
A WEJG117ZD S1.04PLEXPROCEDURE FOR THE SOLLMON
OFS114ULTANEOUSNONLLVEAX EQUA71ONS
W. I- Prke and. W. Dowsm
Audo",
Not"
The weighted simplex (WS) procedux provides an altaustive to die
t4ewwn-Raphsoo
CNR) procedure for the solution of Y simui.
tancous nonlinear equations in N real vwubk-& rapid convergence
being obtained from a suffiactitty good itutuil appro,, unution. in
NR the US
contrast on "
peac ure does not involve the computation Of dcrivatnres. The principle of dw %%Smethod. totether
VMh the rewilts o( comparative performance two. are Publisheil
elsewhm (Price. 191,n
Given the set of equationsfe (zj...
0.1 m I.... JV. the
xx)
=
,
Procedure OPCI tes on the data -esoci-ted with a S1161PLEX of
N+I
Points in N-sp- cc Prior to each iteration the co-ordinates;
of dme Points. So (%hem I-1.
V + 1). and the
j-1.
-,
..
-V;
..
COr. CSPOndinji function values. f, #. are held in store. For each point I
of the simpkx a WEIGHT. wj, is computed try "ving
iby Gauss
CliwilutiOU) the set Of. V +I linear equations Zwj ft 1; Zwjfj m 0.
j
The
Xwjýzq
JV.
co-ordinates
x,
.
of t'he weighted
have three solutions, the
The pair of equations A-0
and A-0
arproximate locations at (0,0), (1.5,3.1) and (1.5, -2.5) having
been obtained by a global search procedure. The WS procedure
was used to obtain a refinement of the solution for which the initial
was chosen
approximation is xt - 1.5. xt - 3.5. A zone size z-I
so as to exclude the neighbourhoods of the other two solution
points. The precision was specified by ACC - 10-4. The WS
procedure achieves the exact solution at (Z4) in six iterations. the
same number as is required by the NR procedure from the same
starting point and with the sameprecision.
Rtferewe
Nim W. L (1979).A Weighted Simplex Procedure for the Solution
of Simultaneous Nonlinear Equations, JIMA, Vol. 24 no. 1,
pp. 1-9).
bEIGNTED SlkPLEI PROGRAM
hNITTIN BY W.L. PRICS
INTO FORTRANBY h. 00b.SON
TUhSLAM
DIMMUN
V(3.4),
SM-RICUS
baM
U(3.4). 1(2), F(2)
UZ*UT
&.3 MLLOb3
&. NO. Or VAAIABLL3 AbU nhCTIONS
U-T 1/0 MEAKS
1101
I&PUt N "D
centroid.
STIATING VALUES Or Vk&lkblAS
X. of dw simpleit are then dctermijxý
wid each Of tiw v functions
is evaluated
at X. If. at X. ttw magiutude
is less
of every function
ACCý & user-supplied
thin
mejoun
tlien
of tbc mquimd
ccurscy.
the Procedure
Otheraise
UMMIes.
one of ON* simplex
Points is
discarded.
its place being taken by X and tise
moddied
amplex
Which MUlts form
the haw of the next ites tion. The discs
point
is chosen to be 1, that Point
Of the ample% vnth the least positive
(MIDU nCg1tivC) %Vghl4
unless L harrew
to be the X mnt
of the
in ii hich can the discard
Previous iteration
point is chosen randomly
ftm
the simplex
but excluding
w, that Point with the
the point
onost Positive "ght
(Prwlý 1979 for full explanation).
The procedure
is initialised
by lim"atIng
a ample% o( A-I
points randomly
Positioned
'kidlin
A hYPercL; be of linear 4imenston
zt the zom an)
the hypercube
bang
entred on 11w initial
4rMximabon
supplied
by tbC user- The chOICT
but ideally the hvDa=be
Of: a not critical.
obould be IUP
enough to embrwe
the exact solution
sought yet
small tmugh
90 gIckids
other rosw bit solutions
to the given system
0(equations.
ILArmth. loom
ULC(lb. 1001)(X(I). I. I'M)
hPl. "I
T'le Procedure has been
PCOVInuned to ANM FORTRAN so as to
be gimerally applicable
and to require the mmunum of um coding
10 run a sPCc& rroWelit. All am%s
used " the program am
'kriared in the "win MOVIns so that the uw tias only to change
the DIMENSION
sute'llent to &Jopi t1w program kir use vnth any
Particular maximum %Cut of V. Tim us" must surply & submtne
FUNCT(-r., F. N) aluch
c&kutstc, the values *(his set o(functions
F- corresponding to the
Jr and F are "th
variables X -two
of
dimension V. lie
must 11-) Suvrly. as data. the value of N. the
C40"Ord'notts Of the U-sal at)rroxtmanon.
the zone size -, and the
required 4ccUrsC-y ACC. T,-w
user must code a random number
llenctatOr apPrOOnAtt to %is
is a standard
coir.
puter-RANF
function On the CYbcf 72
Used bv the authors.
%&lT6(IOt, T. 2OC3)
As an example Of the
OMralien of the proV= a printout is
surP"ed (Or a run 1`0311119
to it,* sgm=fk mo-vanable problem in
%hich tM (", -txws am
F& - ZZ, 3 So
#2
A641 - ills + X1
l(k)
LP2. b. 2
42-02
CALCULAIR IPUhCTIONVALLE3 AND FRUT ThEh
CALL FIACT(I. P. L)
2000)
kalml"T.
I-AM6(louT. 2001) Q.
JbPVT ZOAL SIZI AND ACCURACI ILQUIkED
AUZ(lb, lool)Z. Acc
kAlT&%la%lT.230a)Z. ACc
CALL SD%pLn KUTIhs
I. I. V. U. bPl. bP2. N2, Z. ACC)
CALL bS"U(I.
CJJ'tft? 3LLUIXON VALUES
a T-4
c
fOrASTC12)
1000
10&1
rchhWatio.
3)
SIMPLEX PhCGAAh///16H STAAIIKG VALUES//
I
51. ýLVAIIAbLES, ISX. 9hFLhC-,I,; t4S/)
2001 POPPATO-h X(. 12.4h)
IPEIO-3)
(,FACT hLQ41RED
Z002 iýW. ATG'/13t. =14. SIZE IeLIO. 3121h AC&.
I
-,FEIO. 3)
2003 f"ihAT(//16h
50"TION VALLLS//54,9hVARIABLI: S. 191.9hiUNCTIOr. S/)
ZKD
zcoo
SMPLI(A.
SUMOINIL
F. A. ý. 6. NPI, NP2,42.
Z. ACC)
SMOLEX SUEROUTILE
*91"10
IAITIEk 81 I.. L. PRICE
MANSLATED Ib-O rOATRAb DI H. C44SCh
VALUES
Ob ENTRY . SIARTING
CK LILT
. SOLUTION VALUES
10 1
VALUES CURALSeChOlhG
Cf ktkCTIQN
ANL, VAIlAbLý
Of K64.11CLS
16.AARAT Of
#*A*mAl
N. &UkW
VAKALL&S.
Th*ComputerJournal
Volumt22
Nurnber3
182
cV
c
c
G
F.
c
c
c
c
AnD U ARE ARMS USLO 61 6SMPLX
hpl. h. 1
NP2. N. 2
N2. b*2
ZxZOhE SIZE
ACC. RLGUIREO ACCURACI
00 70 12I. N
IPI*I*l
KKKaNPI+lPi
DO 50
K-Lkk-kk
It
E* 40 J-1, N?2
b*U(9. J)
DikENSION V(A.Pl, h2), U(NPl. NP2). X(N)d(Ns
FWPI. FLCAI(NPI)
GENLRATL IhITIAL
SlhPLEX
40 cohilfuh
so CONTINUE
00 60 Kal, N
KPInK*l
Do 60 J. I. NPI
JPIOJ+l
CALL FUNCUM, k)
00 10 W, N
10 CONTIMA
DO 40 Jal. b
jpl. j*l
40 20 K. I, N
IS
AW
NANKING
RMON
hUht: 01 UFMMICR
%4&hLF,. Wtu
(A-IrChOU
Mhhl
hIlk
VAL6LS IN
AKG4hihT
hAhQ& 0.0
V(Jpl, K)21(K)
20 COI-TlhtV
CALL fbhCI(X, F,; -)
DO 30 Ksl, V
KfhýK+N
VCJPI, LFN)SI(k)
30 CONTINTL
#;L*;? t;Tb, WEIGhIED MUM,
I
50 CALL
00 60 ksl, N
l(K)SO-0
Cri 60 JOI, Npl
X(K)@l(K)+b(jdlp2)bv(j,
60 COMILUE
k6VALUAIL &I X AND ILbl
Ull
It
TO
1.0
60 CONIII#UE
70 CONTME
Npl)
l; p2)/U(Npl,
U(NPI. k? 2)su(bpl,
L-kpl
ti. Np,r
DO*90 Ktal, N
90API-Kk
1.0.0
KFI. K*l
DO 90 JoKpl. tpt
fiRB*U(K. J)*U(J, NFf)
so CONTME
K)
VK, I.P2)s(U(L, NP2)-l0/Q(I.
FIbD k AND L
40 CONTINUE
IDII
CkLL FUCI(l.
00 TO L. I, h
If (AUMM.
TO CChTIt. UL
GO TO 50
K)
It (U(e. K?2). LT. U(L, NP2)) LsK
If (U(K, KP2). GT. U(h, N?2)) K-K
go CON11hum
RETURN
END
3UBhOUTINE fUhCT(X. F. N)
MIkSION
X(N). F(h)
l0)s2-Q*l(l)'*3*X(2)-X(2)*113
F(2)-6.0*X0)-1(2)0*2+1(2)
MORN
ED
tch
COt.VLhGbJ.CL
t. 0
Algoridim 108
M. ACE) Q
IG 80
C&hTLj%GEC
EFFICIENTSOLU77ONOF TRIDIAGONAL LINEAR S Y=SfS
Ole Osterby
ComputerScienceDepartment
AarhusUniversity
RiT4R%
Author'sNote
The solution of a tridiagonalsystemof linear equationsof the form
ChOGSL DISCARD POINT
60 It (L. NL. 10 GO TO 90
LaUINTUNP16RANEW. 0))
go IWL
RLPLACL DISCARD POW
bi
at
x,
XI
c3
di
dt
61 1
00 100till'k
an b*
IGO LONlIbUL
-COTO 50
LND
SUEbOt,Tlbl&
#A*.PUTrb 11LIGMS AhL FIND MUST POSITIVi 6EIGNI, k
AWLoLEAST P03111VE VALIGH1, L
i; ImLNSIC#; V(NPl, h2), U(hPl, kP2)
IkITIALISik
ct
bi
U ARNAI
00 10 tal, h?2
u0,10.1.0
10 cohTINUE
DO 20 K-2, týPl
u(g. 6P2). 0 a
Ii2
20 CONT111.
DO 10 J. I. t.
jvl. j*l
jph. J. h
to
0 Kal, Npl
30
Ck)h*lijlk. ý.ElGhTS
The Computer Journal Volume22 Number3
6xsý
dm
is carried out efficiently by meansor Gaussianelimination. In the
following we assumethat pivoting is not necessary.
The idea behind
the algorithm is not new ýSprague,1960;Leavenworth. 1960).but
wehavesuppliedthe procedurefor easyreference.
The operation count is: 2n -I divisions, 3n -3 multiplications
and 3n -3 additions. The only new feature in our procedureis a
slightly more efficient use or simply subscripted variables. The
number of such referencesis lOn -5 compared to 13n -9 in
Leavenworth(1960)and Sprague(1960).The solution is returned
in array D and array B is destroyed.
The reasonfor this very detailedoperation count is that on most
computersdivision is slowerthan multiplication. and floating point
addition is not so much faster that it makesthe time ;ns:gnif cant.
Subscriptedvariablesare countednot only becauseof the subscnpt
In cor.trast,
handlingbut alsobecausetheyimply memoryreferences.
the simple variables can (and -should, if possible) stay in Ust
registers, of which most modern computers have sufficiently
many.
If severalsystemswith the samecoefficientmatrix are to besolved
subsequently,we can store the intermediateresults from the LUdecomr)ositionin array A for later use when the right hand sides
become available. The operat-on count for subsequcntsystems
283 -
183
IEE
DIVISION
ELECTRONICS
COLLOQUIUMON
FORCIRCUIT
PARAMETERS
'MODEL
IS
ANALYS
ORGANISED BY
GROUPE10
PROFESSIONAL
(CIRCUIT THEORYAND DESIGN)
E3
GROUP
PROFESSIONAL
AND
(MICROELECTRONICS
AND SEMICONDUCTOR
DEVICES)
FRIDAY 27 MARCH1981
DIGEST
"
""-" . "- r
'
I
No: 1981/25
_.
" "-
"
184
ON MDEL MODIFICATION
CONSIDERATIONS
Monica Dowson
1.
Introduction
devices
A number of equivalent
circuit
models of electronic
exist
to match the measured characteristics
of an actual
which can be optimised
Vher; the model does not give a su-E-ficiently
device.
accurate
have
to
then
the
measured
characteristics
mod4fications
with
agreement
This paper discusses
be made to the model.
some aspects
of model
in order to match the S-parameter
of high
measurements
modification
integrited
frequency
transistors.
circuit
Determination
of original
model
For any given device a model based on
device is normally available
where perhaps
given fixed values but most of the element
be optimised
element values must therefore
required characteristics.
the physical
aspects of the
some of the elements are
These
values are unhnown.
to provide a fit to the
Given a set of S-parameter measurements of a device at a range of
frequencies
together with a suggested model, the optimisation
method,
favoured by the author involves setting up an eiýror function
given by the
between the measured S-parameters
sum of the weighted absolute differences
This error function
for the model.
can then be
and those calculated
By running this optimisation
minimised using the Gauss Newton method.
(i. e. the
algorithm in the domain of the logarithms of the variables
to be positive.
the element values are constrained
element values),
Vhen convergence of the algorithm onto a minimum is obtained all the
to the element values are very close to zero and this is a
corrections
indication
that the model is a suitable
one.
positive
3.
Modifications
to the model
If the convergence of the gauss Newton algorithm
is not obtained this
By looking at the
that the model is not suitable.
can be an indication
to the individual
values oi the corrections
elements it is possible to
determine what action should be taken.
3.1.
Removal of elements from the model
Since the algorithm is operating in the logaritimic
domain it is
correction
to an element could
possible to say that a large negitive
indicate
If, after applying that
that the element should be : ero.
for that same element is still
to the element, the correction
correction
large and negative,
Thus, if for example,
this acts as confi=ation.
A
then this should be short circuited.
the element was a resistance
lttnica Dowson is with
of Leicester.
the Department of Engineering,
.:).. I.
University
iss
that are
similar
argument may be used for corrections
low frequency
Fig. 1 shows a suggested
physical
model
0.1
1.0
to
of
transistor
at frequencies
operating
here the model given
described
tion of the technique
in Guass Newton over the entire
range of frequencies
S
improvement
in the high frequency
of
response
12
in Fig. 3.
3.2.
Addition
large and positive.
for a vertical
npn
CH.,
After
applica.
in Fig. 2 gave coývergence
with an attendant
as shown
n particular,
of elements
Just as this technique can be used to remove unwanted elements, so it can
be used to determine whether a particular
element could be added to the model.
The placing of additional
elements is the cause of some disagreement.
One view is that all elements in a model must have a physical meaning within
To comply with this requirement,
attempts
the construction
of the device.
in
should only be made to place elements in certain predetermined positions
However, as sho%mby Cutteridge
the model.
and Dowsonl, it should not be
once, it
of a particular
element has failed
a*ssumedthat because the addition
will not be acceptable at a later stage.
Where a model is required that gives close agreement with the measured
based elements can
the requirement of adding only physically
characteristics,
if the view is taken that any model
be an unnecessary restriction
particularly
In order to
is only an approximation
to an infinite
connection of elements.
be certain that each element added is the best one at that point in the model
This can be most time
growth all possible placements must be attempted.
Thus, to add elements
are usually necessary.
consuming and some restrictions
without creating a new node one can either attempt to add elements in parallel
with elements already present in the model Can approach most physicists
would
not find too unorthodox) or attempt to add elements between each combination
To add a new node, the simplest way is to attempt to add one
of nodes.
element although,
of course, this does not
element in series with an existing
cover all possibilities.
Results
Fig. 4 shows an example of a model grown from the reduced model in Fig. 2.
for this model is shown in Fij. 5 and
The frequency response of modulus S.
1
in Fig. 6. - As can be seen this model now gives
of modulus and phase S,,
'required
good agreement with the
characteristics.
References
Cutteridge,
O. P. D. and Dawson 1!. "Experience in Modelling Integrated
,
Circuit
Transistors",
I. E. E. Conierence on Computer Aided Design and
Components, Circuits
Manufacture of Electronic
and Systems, July 1979.
2. D-
186
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NC0MFECE
EUR0PE ko'-A..
DESIGN
ELEC-7FO'MIC
on
AUTO'wý"IQATIOM,
1-4
September 1981
Organisedbythe
Electronics and Management and Design Divisions
Engineers
Institution
Electrical
the
of
of
in association with the
British Computer Society (BCS)
Convention of National Societies of Electrical
Engineers of Western Europe
Institute of Electrical and Electronics Engineers Inc
(Circuits and Systems Society) (IEEE)
Institute of Electrical and Electronics Engineers Inc
(Region 8) (IEEE)
Institution of Electrical Enginers (IEE)
(Southern Centre)
Institution of Electronic and Radio Engineers (IERE)
Venue
University of Sussex,
Brighton, United Kingdom
189
24
EQUIVALENT CIRCUIT
BIAS CCNDITIONS
Monica
MCDELLING
or
A SMALL SIGNAL
HIGH
FREQUENCY BIPOLAR
TRANSISTOR
AT DIFFERENT
Dowson
University
of
Leicester,
UK
Details
technique
of this
modelling
been given
by Cutteridge
previously
(3) and Dawson (4).
INTRODUCTION
is to describe
the
The purpose
of this
paper
circuit
made in the equivalent
progress
high
small
signal
of two similar
modelling
different
bipolar
frequency
transistors
uitder
bias
conditions.
The optimised
model
for
tial
type
model
any further
without
being
The transistors
modelled
were bipolar
designed
to be used in beam lead
transistors
data
for
S-parameter
integrated
circuits.
in the fretransistors
the
was available
2 GHz
for
100 MHz to
several
range
quency
different
currents.
emitter
description
A brief
techniques
Mi33tiOn
tained
are illustrated
together
with
model
compared
with
model
transistors.
*
DESCRIPTION
here of the optiis given
obused and the results
by diagrams
of the
S parameters
the
of the
for
the
those
measured
being
The transistors
modelled
were isolated
imwith
transistors
made in silicon
n-p-n
boron
and diffused
arsenic
emitters
planted
(A) had
(1)).
The first
(Slatter
type
bases
(a)
type
stripe
and the second
one emitter
for
The data
had three
stripes.
emitter
by Philips
transistors
these
was supplied
(2))
(Slatter
Laboratories
Research
and conS parameters
of the transisof the
sisted
12
in common emitter
at
configuration
tors
; GHz
100 MHz to
frequencies
in the range
10 MA for
O. S MA to
of
currents
at emitter
27 mA for
O. S mA to
type B.
A and of
type
MODELLING
DEPENDENT MODEL
to
OF THE TRANSISTORS
From the graphs
the normal
that
I CHz
was
to
I mA
B was
was then used as the ini.
B and gave good results
topology
changes.
Thus.
taking
the model as the basis
of a bias
dependent
the optimisation
model,
algorithm
described
above was then used to obtain
suitable
element
at different
values
values
of
current.
emitter
BIAS
the aim was
Using
techniques,
optimisation
the
simulating
models
accurate
produce
S parameters
of the transistors.
measured
dependent
as
elements
as few bias
with
possible.
have
and Dawson
in Fig. 1 it
can be deduced
A up
range
of type
operating
2 mA and of type
O. S mA to
8 mA
to
.
TECHNTQUE
by working
The modelling
process
was started
Using
A only.
an initial
model as
on type
model was obshown in Fig. 2. an optimised
for
the whole
of the given
range
of
tained
frequencies
of emitter
value
at a single
the normal
operating
region
of
within
current
This
model,
shown in
optimised
the device.
by using
Fig. 3, was obtained
successive
applithe
to minimise
of Gauss Newton
cations
both
between
the calculated
errors
weighted
S par&and phase of each of the
modLius
measurements
of the model and the giYen
meters
The correction
terms
calcuof the device.
by Gauss Newton were used to indicate
lated
were required,
changes
conver.
where topology
in Gauss Newton being
at each
required
gence
in the development
of the model.
stage
for
The model
shown in Fig. 3 gave good results
B transistors
both
A and type
type
over
the entire
of frequencies
range
at emitter
the normal
operating
currents
regions
within
described
However,
in
at currents
above.
further
in
excess
of the maxima stated
changes
the topology
of the model were required.
The original
hypothesis
there
was that
would
be a small
number
of linearly
varying
elements
in the model
describe
that
the bias
would
Within
the normal
variation.
operating.
this
to be a valid
regions
was found
assump.
for
For example.
this
tion
particular
model.
in Fij. 4,
by the graphs
R
as is suggested
is inversely
to the emitter
proportional
C,
by a
current
and
can be approximated
function
of the form
b
where
a
and
b
are constants.
Taking the model beyond the normal operating
regions.
evidence of the type of topology
is given by C,
in
changes required
which
,
Fig. 4 shows a rapid increase
in'valut
as the
emitter
current
rites
and which the optimisadeletes
tion algorithm
open circuit
at higher
values of current.
and by R2 which is later
deleted
short circuit.
RESULTS
Examples
S parameters
for
of the
calculated
the model compared
the measured
with
parameA and type
B transistors
ters
of type
are
in Fils.
5 and 6.
As can be seen these
given
frequency
the entire
give
good agreement
over
100 Milz
2 Gil:
is a
to
range
and there
in cases
there
smoothing
effect
where
appear
to be measurement
errors.
CONCLUSIONS
dependent
The bias
model obtained
shows severby
to the one described
al similarities
(1) which
Slatter
by a different
was produced
to the problem.
approach
The model described
here
simulates
istors
being
modelled
over
a wide
the trans.
of
range
190
25
frequencies
and emitter
currents
changes
gives
good results
pology
larger
of emitter
current.
values
and
at
with
even
to-
ACKNOWLEDGEMENT
The author
like
to
would
and P. J. Rankin
of Philips
for providing
tories
the
K. W. Moulding
thank
Research
Laboradata.
necessary
REFERENCES
1.
Slatter,
1977,
"Optimisation
of a Small
Signal
High Frequency
Bipolar
Transistor
9
Model Using
Parameter
Measurements
at
Different
Btas Conditions".
Conference
on
Modelling
Semiconductor
Devices,
Ecole
Polytechnic
do Lausanne,
October
1977.
2.
Slatter,
Philips
3.
Cutteridge,
O. P. D. and Dowson.
M.,
in Modelling
"Experience
Integrated
Circuit
Transistors".
I. E. E. Conference
kided
Design
on Computer
and Manufacture
Components,
Circuits
of Electronic
and
Systems.
July
1979.
4.
Dowson,
M., "Considerations
on model
Modification",
I. E. E. Colloquium
on Model
Parameters
for
Circuit
Analysis,
March
1981.
J. A. G..
Research
FT(MHz)
3000
2000
Internal
Publication,
Laboratories.
1977.
Hill.
1000
niiii ---Fi
1IV. O-sv
-7-
2-0-
Mod
S21
dB at
-2,
1 GHz
Type b
\yp eA
L
-T,
ilu
Lig
1
10.0
T-cL.
nsis2r
chcrc: "Mr-,:tics --t, 1GH
1000.0
le(mA)
191
26
Cl
I
2
Ri
RI
2
1
C4
fjg..] QptimisedModel
fig. 2 Initiat Model
Rin
2400
C,pF
2000
1600
1200
800
4w
4-
i-1
.
.00
0-70.5
0-3
02468
le mA
R20
02468
lemA
C2nF
0.04.
so
0-03-
40
30
0,02-
20.
0.01
10
0
0-00
lemA
lemA
-Type
Erg,.
L E?
etementvalues
ýcmoesof bias de eriddent
_p_
A
Type 8
192
27
ModdB
lu ,
ModdB
Phase11
Sil
lu
Phaseo
S12
Freq GHz
Freq GHz
Phase*
S21
ModdB
ModdB
Phase*
S22
-1
-I
lu
FreqGHz
ModulusSmodelx measured
Phase S --model
o measured
Lg, 5
lu-
JU
lez 1mA
v
0.5v
cez
Measured and modet S parameters-TypeA
FreqGHz
193
28
ModdB
Pnasel
S11
10 ,
ModdB
I-
lu
Freq GHz
Phase*
S21
s
ModdB
Phase
12
lu,
lu
S22
ModdB
FreqGHz
Phase*
160
140
120
-100
80
rveq unz
rreq unz
Modulus Smodel x measured
Phase S ----model
o measured
_fg,
I, = 2mA
Vce=O'5V
6 Measured and model S parameters -Tyoe B
194
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