METHODS FOR MODELING AND SIMULATION
OF POWER ELECTRONICS AND DRIVES
P.J. van Duijsen
Simulation Research
P.O.Box 397, 2400 AJ, Alphen aan den Rijn
The Netherlands, Tel/Fax +31 172 492353
Abstract
The availability of personal computers to
electronic engineers created a wide range of
simulation programs. In general these
programs were designed for modeling and
simulation of analog circuits. For power
electronics various modeling methods were
developed.
In this paper an overview is presented of the
various methods for modeling and simulation
of power electronics and electrical drives
hereafter referred to as Power Conversion
Systems; (PCS). Mathematical modeling
methods such as state space equations,
modified nodal analysis, differential algebraic
equations and transmission lines are discussed
and compared. The minimum requirements for
modeling (circuit, block-diagram, behavioral
equations) and the types of analysis (transient,
steady-state, small-signal) are discussed.
1
Introduction
Computer Aided Modeling and Simulation of
electric circuits started when the first
computers in large research centers and
universities became available. One of the first
circuit simulation program which became quite
famous was ECAP, developed at an IBM
research laboratory. Although very simple, it
was one of the first general programs for
solving time varying circuit equations.
Different disciplines in electrical engineering
required different methods for modeling and
simulation. In some disciplines the need for
modeling and simulation became more urgent
than in other disciplines. For example, the
development of integrated circuits, stimulated
the design of SPICE, (Simulation Program
with Integrated Circuit Emphasizes) [Nagel,
1975].
With the availability of mathematical equations
solving programs, which could handle blockdiagram models or modeling languages, for
example CSMP, [Korn, 1978], it was possible
to build models of power electronic systems or
drive systems with the use of Ordinary
Differential Equations: (ODE). The use of
modeling and simulation methods for power
electronics and drive systems was concentrated
mainly towards the analysis of dynamical
effects in the mechanical part of a drive
system. The main problem when defining
ODE's were caused by the switches in the
electronic
power
conversion
circuit,
introducing an acausal non-linear relation
[Nelms, 1988].
Next to the growing number of modeling and
simulation programs, the number of methods
performing a specific analysis upon a PCS was
growing. State space averaging [Middlebrook,
1976] is a good example of a modeling method
which serves as a mathematical method for
deriving insight in the dynamic behavior of
switched-mode power supplies.
Recently a large number of methods became
available for the modeling and simulation of a
PCS [Revankar, 1973], [Sankara, 1975],
[Kelkar, 1986]. Most of these methods are
especially designed for one class of converters,
for example DC-to-DC converters with a fixed
mode of operation. The problem with these
methods is, that they are limited to the
application they were intended for. The state
space averaging method was originally
developed for hard-switched DC-to-DC
converters and Switched Mode Power Supplies
(SMPS). It has a limited applicability to
resonant converters [Yang, 1993].
In this chapter the existing methods for
simulation, cyclic-steady-state and small-signal
analyses are discussed. The advantages and
disadvantages of each method will be
highlighted.
Simulation
Simulation is performed in various ways, but
they are all based on numerically solving of
non-linear state equations, where independent
storage elements like inductors and capacitors,
are described by differential equations.
Because of the differences between the various
models for circuits, digital controllers, analog
controllers and components, a multilevel
approach is introduced which combines the
various models like a circuit model, a blockdiagram and even computer program
instructions. The combination of these models
is called a multilevel model and is translated
into one mathematical model. Numerical
solving of this mathematical model reveals the
time responses.
Cyclic-steady-state analysis
Existing methods for cyclic-steady-state
analysis are based on the assumption that a set
of state equations is defined from which a
periodic response can be calculated. This is
achieved by setting up non-linear state
equations where the state variables at the
beginning of the period have to be equal to the
state variables at the end of the period of a
cyclically switching PCS. The resulting set of
equations is solved numerically [Aprille,
1971]. For piece-wise linear circuits direct
calculation of the state variables is possible
[Lavers, 1986].
Small-signal analysis
Small-signal analysis is important for the
design of the control of a PCS. Extensive
research has been done for DC-to-DC switched
mode supplies and numerous methods have
been developed [Kassakian, 1991]. The main
disadvantage of these existing methods is that
they are nearly only applicable to the class of
converters the analysing method was originally
designed for. For example, state space
averaging was originally designed for DC-toDC converters with a fixed switching
frequency larger than the bandwidth of the
converter. For resonant converters the
switching frequency lies within the bandwidth
of the internal waveforms of the converter, so
state-space averaging can not be applied.
2
Modeling the PCS
If non-linear mathematical relations are
included, the formulation of a mathematical
model of a PCS is limited to the time-domain.
With numerical methods, time responses can
be calculated. If a mathematical model can
describe the behavior with linear mathematical
relations and constant parameters, also the
frequency-domain can be used. Using
numerical methods, a frequency response can
be calculated.
For mathematical models with linear
mathematical
relations
and
constant
parameters, with the use of the Discrete
Fourier Transformation (DFT) [Papoulis,
1980], a time response can be transformed into
a frequency response. Inverse Fourier
transformation can be used to transform a
frequency response into a time response.
Figure 1 :
Circuit
with
components,
constant
parameters
and
switches.
Figure 2 :
Time intervals. a)Time interval
fixed. b)Time interval variable.
•
2.1
Time-domain
A PCS can be described by Differential
Algebraic Equations; (DAE):
f (x (t), x(t), y(t) , u(t), t) = 0
(1)
The DAE describes the non-linear, possible
acausal, relations among the time-varying state
variables x(t), their time-derivative x(t), the
variables y(t) and the input variables u(t) of a
PCS.
If there are no acausal relations in the
mathematical model, the DAE can be
simplified to an Ordinary Differential
Equation (ODE):
x (t) = f (x(t), u(t) )
(2)
Switches
Semiconductor switches are the main problem
in modeling PCSs. There are two possibilities
for the operation of the switches
•
defined with known on and off times
(ton and toff).
defined by the value of the state
variables.
In the first case ton and toff are independent of
the value of the state variables. An example is
a DC converter with continuous conduction
mode without control, see figure 1a.
In the second case ton and toff are defined both
by the control of the PCS and the value of the
state variables of the PCS. An example is a DC
converter with discontinuous conduction
mode. There ton of the freewheeling diode is
dependent on the zero crossing of the inductor
current, see figure 1b. The mathematical model
contains an implicit relation describing the
dependency between the inductor current and
ton of the diode.
A piece-wise linear relation in the
mathematical model, consisting of two linear
relations, can describe ideal switches:
On : u s = 0
Off : i s = 0
(3)
If the mathematical model of the PCS does not
contain any non-linear relations but only linear
and piece-wise linear relations, as given by (3),
the mathematical model can be simplified. If
the operation of the switches is known in
advance and the mathematical model consists
of linear ODEs, a simplification can be made.
In this case the mathematical model is piecewise linear, which means that the
mathematical model can be replaced by a finite
number of sets of linear ODEs.
The PCS with switches as indicated by figure 2
is replaced by a set of sub-circuits without
switches. Each sub-circuit is solely described
by linear mathematical relations and the state
of the switches defines the connections
between the components in the sub-circuit, see
figure 3.
Figure 3 :
Circuit with components,
constant
parameters
and
without switches.
n
T per =  T i
(4)
i=1
A cycle starts at t=t0 with an initial value y(t0)
and ends at t=tn with y(tn). For a cyclic-steadystate y(t0) = y(tn), as shown in figure 5.
Figure 5 :
Time intervals for a piece-wise
linear circuit.
Models of components
Modeling components of the PCS is not
unique. Depending on the need of the user a
model can be either simple, detailed, or can
contain just enough details to model the timedomain behavior satisfactory.
The piece-wise linear circuit changes its
topology each time the status of the switches
changes. The operation of the switches is
modeled by selecting, for a specific time
interval Ti( = ti-ti-1), the sub-circuit with the
valid switch-configuration, (see figure 4).
Figure 6 :
Figure 4 :
Piece-wise linear circuit.
For each time interval Ti only one sub-circuit is
valid. For cyclic-steady-state analysis the cyclic
time interval Tper is equal to the sum of the
time intervals per sub-circuit. If the cyclic time
interval of the cyclic-steady-state equals n subcircuits, the cyclic time interval equals:
Complexity of models.
The complexity of the mathematical model is
not necessarily related to the complexity of the
model of the component. As shown in figure 6
a simple model can contain a (non-linear)
acausal mathematical relation and therefore a
DAE has to be used for the mathematical
model. On the other side a more detailed model
can be described by ODEs if it doesn't contain
any acausal relations.
All the non-linear mathematical relations
describing the components in a PCS are
functions of time and/or functions of variables.
These mathematical relations can be
formulated as a DAE, making the description
by DAEs more general than any other
mathematical modeling approach. In using
DAEs, the user has more freedom to set-up a
mathematical model than with other modeling
approaches, such as block-diagrams, where
acausal relations are not allowed.
2.2
2
If one harmonic influences another harmonic,
the calculation of the frequency response can
not be carried out for each harmonic separately.
In this case a Harmonic Balance technique
[Nakhla, 1976] is required. The mathematical
model has to contain all harmonics that are of
interest. The solution for all harmonics is
calculated at the same time by solving:
g HB ( y1 , ..., y n , p1 (  1 , ..., n ), ...,
pn (  1 , ..., n ) , 1 , ..., n ) = 0
Frequency-domain
In the frequency-domain the frequency
response is defined for a dynamic system. The
frequency response is given by the gain and
phase difference between the frequency
components at the input and output of the
system with equal frequency. When describing
a mathematical model in the frequency
domain, there are two possibilities:
1
p(i) denotes the parameters of the
mathematical model which are dependent of
i.
linear mathematical relations with
constant parameters and no dependency
between the different harmonics.
non-linear mathematical relations with
time-varying
parameters
and
dependency
between
different
harmonics
If there is no dependency between the
harmonics, the frequency response can be
calculated for each harmonic separately. This
means that for each harmonic i the following
equation has to be solved:
g( yi , p(  i ) , i ) = 0
(5)
where yi are the variables of the model and
(6)
Here n denotes the number of harmonics.
Compared to (5), the mathematical model (6)
is considerable more complex, because of the
relations among the harmonics.
The size of the mathematical model (6) is one
drawback of modeling in the frequencydomain. Relations between harmonics exist in
nearly all elements of the PCS. The main
contributions are caused by
•
switches in the electronic power
converter.
•
saturation of components. (For
example magnetic components like
inductances and electrical machines).
•
limiters in controllers.
If all the mathematical relations can be
described by only using (6) without any extra
ODEs, the cyclic-steady-state can be calculated
directly. This approach is used for
telecommunication systems [Nakhla, 1976].
Also for electric machines, Harmonic Balance
can be used to describe the influence of
harmonics in the machine.
Another problem is the definition of the
models. The formulation of a mathematical
model describing ton and toff of a switch in the
PCS is more understandable than a description
of the mathematical relations of the switch in
the frequency-domain. In this frequency domain description the harmonics in the
voltage and current representations, which are
caused by the cyclic operation of the switches,
are approximated and used in (6).
Example
Figure 7 shows schematically a control
algorithm for a resonant converter. The switch
S is turned on if a specific set-signal equals 1
and the output voltage uo is below the reference
voltage uref. It turns off at the zero crossing of
the switch current is. The set-signal can be
defined in a time table in the control algorithm
of the PCS.
mathematical relations in the frequencydomain, because of the time events taking
place in the control of the resonant converter as
modeled in figure 7.
A concluding remark is that from the frequency
response only the cyclic-steady-state in the
time-domain can be calculated. Therefore the
frequency-domain is not well suited for a
general approach of the analysis of a PCS,
which has to include the transient behavior, for
example the start-up of a PCS.
3
Simulation
The number of algorithms for simulation is
large. They all require time-domain models.
The majority of simulation algorithms is based
on state space equations:
x (t) = A(x, t) x(t) + B(x, t) u(t)
(7)
Piece-wise linear model
For switched mode power supplies the piecewise linear circuit description is applied to
model the switches in the circuit. The matrices
A(x,t) and B(x,t) are considered to have
constant parameters. Doing so (7) is replaced
by:
x = Ai x + Bi u
Figure 7 :
Control of a resonant converter.
This is a simple example of a model for a
component that includes:
1
function of time
2
function of variables : uo, is
This
model
cannot
: set-signal
be
described
by
i = 1,..., n
(8)
where i denotes the sub-circuit of the piecewise linear circuit. The state space approach
and the use of piece-wise linearity is used by
many authors. In [Kassakian, 1991] state space
equations for simulation is discussed for
general use. The piece-wise linear description
was introduced for sampled data modeling of
PCS [Verghese, 1986], [Elbuluk, 1988] and
[Kelkar, 1986]. The obtained sampled data
models are used to derive transfer functions
between the input and output variables of a
PCS, through the z-transform [Huliehel,1991].
Problems during switching from one subcircuit to another sub-circuit in a piece-wise
linear circuit is explained in [Dirkman, 1987].
Here models are derived which allow a sudden
parallel connection of capacitors and secure the
continuous current through a series connection
of two inductors, which can occur because of
the closing or opening of switches. This is
assured by inserting current or voltage sources,
which cancel the current or voltage spike
occurring because of the parallel or series
connection. The problem with this method is
that the exact value of the current or voltage
source value is dependent of the circuit and
therefore extra calculation work is needed to
define these values.
For predefined time intervals transition
matrices are calculated which give the solution
of the state space equations over a certain
interval [t0, t1], [Hsiao, 1987]:
t1
x( t1) = x( t0) eA1( t1 - t0 ) +  eA1( t1 - t0 -  ) B1 u( ) d
t0
(9)
If the input u(t) is not taken into consideration,
the transition matrix i(t) is defined as:
x( ti + Ti) = i (Ti) x( ti)
(10)
and calculated for a fixed time interval Ti.
An efficient method to calculate the transition
matrix i(Ti) for varying time intervals Ti can
be found in [Wong, 1987] where transition
matrices with a fixed time interval are
precalculated and stored. The lengths of the
different time intervals are related to a power
of two. A simulation is performed and with the
use of a binary search method the transition
matrices are obtained for a variable time
interval. This final time interval has to be an
integer multiple of the smallest precalculated
time interval.
The general problem with transition matrices is
that they are calculated for a fixed time
interval. This time interval is dependent on
events occurring in the circuit or on control
actions. Therefore in [Luciano, 1990] an
attempt is made to make the transition matrices
independent of the time interval. This
approximation is only valid for small variations
of the fixed predefined time interval:
x( t 0 + T1 +  T1) = x( t 0) eA1( T1 + T1) +
(11)
t 0 + T1 +  T1

(
eA1 T1
+  T1 -  )
B1 u( ) d
t0
where Ti<<Ti. A transition matrix has to be
evaluated analytical from (11) and has to
include Ti. This is considerable more
complex than (10).
Recently Transmission Line Modeling; (TLM)
is proposed for modeling switching power
converters, [Hui, 1991]. For transmission line
modeling a matrix can be defined which is
independent of the status of the switches.
Therefore a single system matrix models the
piece-wise linear circuit. This is achieved by
replacing the switch by a transmission line.
The transmission line has either a small
inductance or small capacitance. This
inductance or capacitance models the parasitics
of the switch. The resulting mathematical
model consists of a square matrix ATLM with
constant entries, the vector x(t) contains the
state variables and the vector bTLM(t) includes
the time-varying variables like the independent
sources and also a variable indicating the status
of the switch:
ATLM x(t) = bTLM (t)
(12)
The solution is obtained from:
1
x(t) = A-TLM
bTLM (t)
(13)
Changing the status of a switch only affects the
entries of bTLM(t). Since the inversion of ATLM
has to be carried out only once, this method
seems to have certain advantages over other
methods, where the matrix A has to be inverted
each time step, [Hui, 1991].
The drawback of the method is, that in order to
keep the parasitic inductance and capacitance
of the switch low, the time step of the
simulation has to be smaller than in the case of
the simulation of an equivalent piece-wise
linear state space equation. As a result applying
transmission line modeling, compared to
simulation with state space equations does not
reduce the simulation time.
In general, the various algorithms and methods
have different drawbacks. For a general
simulation of a mathematical model as given
by (7), numerical integration is applied. The
parameters of the mathematical model can
remain non-constant.
3.1
Circuit simulation
A popular program for the simulation of
electric circuits is the SPICE "family" of circuit
simulators. The most common is the SPICE2
circuit simulator which, is originally developed
at the University of California, Berkeley,
during the mid-1970s. SPICE2 [Nagel, 1975]
evolved from the original SPICE program,
which evolved from another circuit simulator
called CANCER that was developed in the
early 1970s. SPICE2 became an industry
Figure 8 :
SPICE algorithm.
standard tool. U.C. Berkeley does not support
SPICE like commercial software, nor does
U.C. Berkeley provide consulting services for
these programs. These lacks of support led to
commercial versions of SPICE that have the
kind of support industrial customers require.
Also, many companies have an in-house
version of SPICE that has modifications to suit
particular needs.
The SPICE program is based on the Modified
Nodal Analysis (MNA) method, [Ho, 1975].
Suppose a model can be formulated as:
A(t, x (t), x(t), h) x(t) = b(t)
(14)
The parameters of the matrix A are dependent
on the variables and state variables in the
vector x(t). Vector b(t) stores the values of the
independent sources. The time derivative of
x(t) is replaced by a numerical integration
approximation, where the parameter h is the
step size of the numerical integration. The
algorithm for SPICE is shown in figure 8.
There are three loops inside the algorithm.
Loop number 1 exists because of the recursive
Newton-Raphson method [Burden, 1985].
Here x(t) is solved from (14) for the time t. If
the convergence of the Newton-Raphson
method fails, the step size h is reduced and (14)
is solved again. This is indicated by loop
number 2. If convergence of the NewtonRaphson is reached, the next point in time can
be calculated. This is achieved by increasing
the time with the step size h as indicated by
loop number 3.
The main problem of SPICE for the simulation
of PCSs is the divergence of Newton-Raphson
in loop number 1, which occurs during the
zero-crossing
of
currents
through
semiconductors. As a result the step size is
decreased, which can lead to many cycles
through loop 1 and loop 2. In figure 9 the
points in time calculated by a SPICE
simulation are shown.
and to solve x(t) from (15) using the NewtonRaphson method. Mathematical non-linear and
acausal relations describing components and
their interconnections in the PCS can directly
be incorporated in (15). Solving x(t) from (15)
using Newton-Raphson implies that the
Jacobian matrix has to be set-up:
Jacobian =
For a general solution to the MNA approach,
the inverse of the matrix A(t,x(t),h) has to be
calculated inside loop number 1. Another
approach is to rewrite (14) to:
(15)
The evaluation of the Jacobian matrix is time
consuming and therefore the main drawback of
the use of (15) as a mathematical model. The
evaluation can be simplified if only the most
important entries of the Jacobian matrix are
used. The greatest simplification can be
reached if the Jacobian matrix is equal to a
block-diagonal matrix, which is the case for
MNA in SPICE [Schwarz, 1987].
The modern versions of SPICE are based on
(15), where Newton-Raphson is used to solve
x(t). During loop 1 in figure 8 the parameters
of the matrix A are only dependent of the
Jacobian matrix (16).
3.2
f(t, x (t), x(t), h) =
Figure 9 :
Zero-crossing of a current
(16)
through a semiconductor in
A(t, x (t), x(t), h) x(t) - b(t) = 0
Spice.
f
x
Multilevel modeling and simulation.
Multilevel modeling [Duijsen, 1994] is a
technique where different model descriptions
can be combined. A multilevel simulation is
done with the mathematical model of the
multilevel model. A multilevel model can
contain different types of models. For the
modeling of a PCS the models in table 1 are
selected. The ODE is combined with the DAE
and during the simulation the computer code
which is compiled from the Computer program
instructions is executed.
Model
Mathematical
model
Circuit
DAE, MNA
Block-diagram
ODE
Computer
program
instructions
Computer code
for the simulation
Table 1 :
Multilevel
model
mathematical model
and
The multilevel modeling technique is
advantageous compared to other modeling
techniques for two major reasons.
•
•
Each part of the implementation like
the power electronic circuit, the load or
the control, is described by its most
efficient modeling technique. This
allows a straightforward description
without, for example, modeling an
inverter in a block diagram or modeling
a control algorithm by lumped circuit
elements.
Secondly, the simulation time can be
decreased because each part of the
implementation can be solved more
efficiently.
In [Duijsen, 1992] a comparison is made
between SPICE and a multilevel simulation of
a buck converter with PID controller. For
SPICE the PID control is modeled by lumped
circuit components. A switch model with the
same parameters as used in the multilevel
model replaced the semiconductor model in
SPICE. The model for the diode in SPICE was
simplified such, that it took only one or two
recursive steps to reach convergence. In the
multilevel approach the PID controller is
described by an ODE. The multilevel
simulation used less simulation time compared
to the SPICE simulation. A reduction of the
simulation time of two orders of magnitude
was achieved in [Duijsen, 1992], because the
simulation of the ODE requires less time than
the simulation using the MNA method in
SPICE.
4
Cyclic-steady-state analysis
The known cyclic-steady-state methods are
based on a state space formulation as given by
(7). Instead of performing a simulation until all
transients are damped, (7) is reformulated as a
set of algebraic equations, from which the
cyclic-steady-state solution is calculated
directly.
Cyclic-steady-state analysis of circuits
containing periodically interrupted switches
was first presented in [Liou, 1972]. In this
method one transition matrix is derived for one
time interval. The state vector at the end of this
time interval should equal the state vector at
the beginning of the time interval. From this
equation the state vector is solved as function
of the input vector. In [Lavers, 1986], [Cheung,
1986] and [Cheung, 1987] this method is
extended to AC-AC switching power
converters. A Basis Transformed State Space
(BTTS) formulation is applied to describe time
varying sources by a transition matrix as
introduced in [Balbania, 1969]. In [Cheung,
1987] the cyclic-steady-state calculation is
performed when the time intervals are varying
due to state events occurring in the variables
and events caused by the control. Here the
Newton-Raphson algorithm is used to calculate
the time interval for each transition matrix of
the piece-wise linear circuit.
For non-linear circuits a steady-state
calculation may be performed as described in
[Aprille, 1972] and [Colon, 1973] where the
cyclic-steady-state solution is calculated using
the Newton Raphson algorithm. The
mathematical model has to have the following
form:
xend = f ( x begin)
(17)
where x contains the state variables. To
calculate the cyclic-steady-state (17) is
extended with:
xend = x begin
(18)
xbegin - f ( xbegin) = 0
(19)
transfer functions. However the averaging
process assumes a piece-wise linear system and
constant time intervals. Furthermore, the length
of the time intervals has to be shorter than the
time constants of the system.
Averaging is applicable if the magnitude of the
basic harmonic is much higher than the other
harmonics of the waveform that has to be
averaged. This is true for hard-switched DC
converters where the basic harmonic is the DC
component. For resonant converters the
resonant waveforms consists of more than one
harmonic with equal magnitude. Therefore
averaging can not be applied to resonant
converters.
From the relation:
xbegin is solved iteratively. In [Wong, 1987] the
cyclic-steady-state of a closed loop regulated
switched mode power supply is calculated. The
mathematical relation for the closed loop is
included in the total mathematical model.
5
Small-signal analysis
In literature small-signal analysis is realized by
two different approaches:
1.
2.
averaging techniques,
sampled data modeling.
Averaging
Averaging is studied extensively in literature
[Middlebrook, 1976]. The motivation for this
method is that it provides equations that can be
solved analytically, providing expressions for
Figure 10 :
Approximation of a switched
current (straight line), by its
average (dashed line) and
sampled data calculation (dots).
An exception can be made for soft-switched
converters [Lee, 1993], where the resonant
waveform is only of interest during switching.
The basic harmonic is still a DC component, so
averaging can be applied, although the
dynamics of the resonant circuit are not
included in the averaged mathematical
model.In figure 10 a waveform, being an
inductor current in a chopper, is shown in
combination with its averaged value, indicated
by the dashed line. The dynamic behavior is
indicated by the averaged value. The frequency
spectrum of the original waveform differs from
the frequency spectrum of the average
waveform. The original waveform shows a
high-frequency component, which could for
example be caused by a periodically operating
switch. The averaging process filters this
frequency.
Sampled data modeling
The dots on the original waveform in figure 10,
indicate the points which are calculated when
using sampled data modeling. Sampled data
modeling describes the propagation per cyclic
time interval of the state variables of the PCS.
The method is based on transition matrices
with fixed-time intervals.
A method based on a transient response
obtained by simulation, is presented by
[Maranesi, 1990]. The transient response is
used to identify the parameters of a discrete
state space model. A perturbation is imposed in
the cyclic-steady-state and the deviation from
the cyclic-steady-state after one cyclic time
interval Tper is measured in a simulation:
x([k + 1] Tper ) = A x(k Tper ) + B u(k Tper )
(20)
In [Tymerski, 1991], fourier analysis of a
piece-wise linear circuit is presented. The
piece-wise linear circuit is described by a set of
state equations, which are transformed to the
frequency-domain. Although the derivation is
complex, for simple circuit models the result
can be presented in symbolic expressions. The
method is based on a piecewise-linear state
equation:
x i (t) = Ai xi (t) + Bi ui (t)
(21)
i = 1, ..., n
For each time interval Ti the solution x(t) is
expressed as function of Ai, Bi and u(t):
xi (t) = (t, Ai , Bi , xi (0), ui (t) )
Xi (j ) =  xi (t) e - jt dt
-
ti - 1  t  ti
This expression is transformed to the
frequency-domain by using the Fourier
transform of (22):
The result is an expression for Xi(j) in the
frequency-domain for each time interval. The
time-varying system (21) is approximated by
[Tymerski, 1991]:
X̂(j ) =
y(k Tper ) = C x(k Tper ) + D u(k Tper )
For each simulation over a cyclic time interval
of Tper, one of the entries of x or u is changed
and the deviation of all other entries of x and y
are measured. From the input u and the
deviation of the state variables x and output y,
the parameters of (20) are calculated. The
number of simulations is equal to the sum of
the number of state variables x and input
variables u. The method is only applicable for
the approximation of linear systems.
(23)
(22)
1
Tper
 x (t) e  dt + ... +

 
t 0 + T1
tn -1 + Tn
1
-j t
t0
tn -1
x n (t) e - jt dt


n
Tper = Ti
i =1
(24)
where Ti are the time intervals of the piece-
wise linear state equation (21). From the
resulting expressionX (j), the transfer
function between X(j) and U(j) is
approximated via a kind of averaging given by
the separate integrals for each time interval.
The mathematical model (21) has constant
parameters.
6
Conclusion
Modeling
The PCS contains components that can be
modeled by different approaches, depending on
the mathematical relations describing the
component and the interconnection between
Circuit
Non-linear
acausal
mathematical
relations : DAE
Linear acausal
mathematical
relations : MNA
Block diagram
Non-linear causal
mathematical
relations : ODE
Computer
instructions
Program language
:
Pascal, C++,
4GL, ASM, etc.
Table 2 :
Multilevel
model
with
mathematical relations.
incorporates the models as indicated in table 2.
Simulation
Simulation is performed with the mathematical
model containing DAEs, MNA matrices,
ODEs and computer code. It gives the time
responses of all the time-varying variables in
the mathematical model.
Time-domain versus Frequency-domain
The frequency-domain is limited in application
for the modeling of PCSs, because of the timedomain dependency of the parameters and
mathematical relations of the mathematical
model. The frequency response only yields the
steady state in the time-domain. Therefore the
frequency-domain is not suited for a general
analysis of a PCS.
Cyclic-steady-state analysis
Instead of performing the simulation until a
steady state is reached, the cyclic-steady-state
can be calculated directly. It requires however
the knowledge of the periodicity of the PCS.
Small-signal analysis
Small-signal analysis is generally based upon
averaging or sampled data modeling. The
small-signal analysis is valid until half the
switching frequency because of alaising
effects.
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