Modeling and Simulation of Power Electronic
Converters
DRAGAN MAKSIMOVIĆ, MEMBER, IEEE, ALEKSANDAR M. STANKOVIĆ, MEMBER, IEEE,
V. JOSEPH THOTTUVELIL, MEMBER, IEEE, AND GEORGE C. VERGHESE, FELLOW, IEEE
Invited Paper
This paper reviews some of the major approaches to modeling
and simulation in power electronics, and provides references that
can serve as a starting point for the extensive literature on the subject. The major focus of the paper is on averaged models of various kinds, but sampled-data models are also introduced. The importance of hierarchical modeling and simulation is emphasized.
Keywords—Averaged models, boost converter, circuit averaging, dynamic phasors, hierarchical methods, modeling, power
electronics, power factor correction, sampled-data models, simulation, state-space averaging, switched models.
I. INTRODUCTION
A. Modeling and Simulation
Power electronic systems are widely used today to provide power processing for applications ranging from computing and communications to medical electronics, appliance
control, transportation, and high-power transmission. The associated power levels range from milliwatts to megawatts.
These systems typically involve switching circuits composed
of semiconductor switches such as thyristors, MOSFETs,
and diodes, along with passive elements such as inductors,
capacitors, and resistors, and integrated circuits for control.
Manuscript received November 28, 2000; revised February 1, 2001. The
work of D. Maksimović was supported by the National Science Foundation
under Grant ECS-9703449. The work of A. M. Stanković was supported
by the National Science Foundation under Grants ECS-9502636 and ECS9820977, and by the Office of Naval Research under Grant N14-97-1-0704.
D. Maksimović is with the University of Colorado, Boulder, CO 80302
USA.
A. M. Stanković is with Northeastern University, Boston, MA 02205
USA.
V. J. Thottuvelil is with Tyco Electronics Power Systems, Mesquite, TX
75149 USA.
G. C. Verghese is with the Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail: verghese@mit.edu).
Publisher Item Identifier S 0018-9219(01)04106-8.
The analysis and design of such systems presents significant
challenges.
Modeling and simulation are essential ingredients of the
analysis and design process in power electronics. They help
a design engineer gain an increased understanding of circuit
operation. With this knowledge the designer can, for a given
set of specifications, choose a topology, select appropriate
circuit component types and values, estimate circuit performance, and complete the design by ensuring—using Monte
Carlo simulation, worst case analysis, and other reliability
and production yield analyses—that the circuit performance
will meet specifications even with the anticipated variations
in operating conditions and circuit component values.
The increased availability of powerful computing has
made direct simulation widely accessible [1]–[12] and
has enlarged the set of tractable modeling and analysis
approaches. Simulation of a full production schematic still
remains an elusive goal; the obstacles include the need
for extensive model building, excessively long simulation
times, the challenges of automatically recognizing and
exploiting modular or hierarchical or time-scale structure
[13], the difficulties of coupling diverse modeling and
simulation modalities, and the effects of layout, packaging,
and parasitics. Even if it were possible to simulate a full
schematic with sufficient accuracy and efficiency, it is
doubtful whether this capability alone would provide the
basis for good design. Typically, crucial insight and understanding are provided by hierarchical modeling, analysis,
and simulation, rather than working directly with a detailed
schematic. The combination of these insights with hardware
prototyping and experiments constitutes a powerful and
effective approach to design.
Issues of modeling, simulation, and, more generally, computer-aided design in power electronics have been addressed
in this and other journals in past years. The papers [7], [14]
provide particularly valuable perspectives on these issues.
0018–9219/01$10.00 ©2001 IEEE
898
PROCEEDINGS OF THE IEEE, VOL. 89, NO. 6, JUNE 2001
Fig. 1.
Boost PFC converter circuit.
Our emphasis on modeling in this paper complements the
more detailed treatment of simulation in [7].
B. Example: The Boost PFC Rectifier
To illustrate the challenges in modeling and simulating
power electronic circuits, we build around the example
of a boost dc–dc converter in this paper. The basic boost
converter, intended to provide a voltage step-up function, is
embedded in applications ranging from power-factor-corrected (PFC) rectifier circuits (whose input current is made
to follow the waveshape of the input voltage) to circuits
that power the RF amplifiers in cell phones. This range of
applications illustrates an interesting characteristic of power
electronic circuits: depending on the application, the same
basic circuit can be embellished with additional elements or
used with different control methods that provide additional
functionality or work better at the power levels demanded
by the application. The particular details that are important
from a modeling and simulation perspective will vary
correspondingly, so this versatility presents a challenge.
The boost PFC rectifier circuit illustrates many of the challenges and opportunities to combine simulation with smart
analysis, as well as the need for a hierarchical approach [15].
Consider the particular version shown in Fig. 1. This circuit
is intended to provide a nominal 400 Vdc regulated output
from an ac voltage in the range of 85–264 Vac at the rectifier
input. The circuit functions as follows.
• The ac line input is fed through an electromagnetic interference (EMI) filter composed of , , and , to
a diode bridge that rectifies the input voltage (into pulsating dc).
• The rectified line voltage is applied to the boost converter, the basic elements of which are
, ,
,
serves to filter switching-freand . The capacitor
quency ripple current at the input of the boost converter
from the ac input. The circuit elements , , , ,
, and
form a turnoff snubber for diode
, to reduce the reverse recovery current.
The control loops described below are aimed at: 1)
making the input current of the boost converter closely
track the shape of its input voltage—i.e., track a multiple of the input voltage—over the duration of each
rectified ac cycle and 2) regulating the output voltage to
the desired value, by slowly adjusting this multiple over
several rectified ac cycles. The close current tracking at
the input causes the boost converter to appear resistive
to the ac line input, thereby resulting in a power factor
of (essentially) unity.
, ,
• The voltage-regulation loop is formed by ,
,
,
, and
, which derive an error voltage
from the fed back output voltage and a reference
serves as the “multiple”
voltage. The output of
referred to above, and is fed to a module in which it
multiplies (a signal proportional to) the rectified input
voltage, thereby creating the desired current reference
signal at the output of the multiplier block.
• The current-regulation loop is formed by current-sense
, and
,
,
,
,
, and
, creresistor
ating the current error signal at the positive input of
and deriving from it a modulating signal to be fed
MAKSIMOVIĆ et al.: MODELING AND SIMULATION OF POWER ELECTRONIC CONVERTERS
899
to a pulsewidth modulator (PWM) circuit. The PWM
circuit compares the modulating signal with a clocked
ramp voltage to create a duty cycle signal that is used
to drive the switch .
The basic analysis tasks for power electronic circuits can
be outlined as follows in the context of the boost PFC rectifier
of Fig. 1.
• Provide the steady-state relationships between input
and output voltages and currents, as a function of the
circuit and control parameters. From this operating
point information, one obtains peak and rms voltage
and current stresses and calculates element dissipation
due to losses in the key circuit elements (switch
, diode
, inductor
, and capacitor
). This
information can typically be obtained with sufficient
accuracy for a first-pass design through analytical
approaches employing linear circuit models for each
of the circuit configurations that arises through action
of the switches.
• Provide switching waveforms to design/select the components. This often requires a detailed determination
of the voltage and current waveforms associated with a
device. For example, the relevant stress levels used to
select a switch include peak switch current, peak switch
voltage, and power dissipation (which involves the average value of the product of the voltage and current
waveforms). While simplified analysis can predict idealized waveforms, real circuits typically have parasitic
elements and nonideal behavior that require detailed
circuit simulation, not only to predict stress levels but
also to take into account the sometimes complicated relationships among circuit elements.
• Obtain the dynamic characteristics of the circuit for two
key purposes: 1) to enable robust design of the control
loops that regulate the output voltage and shape the
waveform of the input current and 2) to confirm that the
circuit has acceptable transient response, with output
voltage and input current staying within specified limits
under changes of input voltage and load current as well
as during start-up and shutdown conditions.
• After the components have been selected and the
designer has gone through the physical realization
process, usually via a board layout, predict circuit
performance under abnormal conditions. Usually,
auxiliary circuits such as over-voltage, over-current,
and over-temperature detection and shutdown are
incorporated into the circuit. In addition, conducted
and radiated EMI performance also must be assessed
for conformance to regulatory requirements; this is
typically determined today by exhaustive experimental
testing.
C. Outline
Section II of the paper describes the process by which
simple analysis and simulations are performed to understand
the basic steady-state operation of a given circuit (in our case,
the boost converter embedded in a PFC rectifier), and how
900
Fig. 2. Simplified circuit with parameters V = 160 V, L = 0.32
mH, C = 22 F, R = 200 , f = 50 kHz, V
= 400 V.
these can be successively refined and extended to include
additional circuit details and gain more insight into circuit
operation. The methods used to systematically develop dynamic models (switched, averaged, or sampled) for power
converters are outlined in Section III, using the boost converter throughout as an example. Section IV describes how
to fold models such as those of Section III into simulation
tools that yield practically useful results in both the time
and frequency domains. Section V contains a concluding
discussion.
II. INITIAL ANALYSIS AND SIMULATION
The first step in the analysis of the circuit in Fig. 1 is to
obtain the voltage and current waveforms that describe basic
power-stage circuit operation. The input voltage to the boost
portion of the PFC rectifier is continuously varying at a frequency equal to twice the line frequency, since it is derived
as the rectified ac line voltage. However, we can make the
assumption that within a switching cycle (which is typically
25 s or smaller), and indeed over several switching cycles,
the input voltage is essentially constant.
With this assumption, the basic steady-state analysis can
be obtained using the simplified circuit shown in Fig. 2,
from Fig. 1,
denotes
from the
where denotes
denotes the load, modeled as being
earlier figure, and
purely resistive. Note that many of the circuit elements in
Fig. 1, such as those associated with the EMI filter, bridge
rectifier, snubber, and voltage- and current-feedback loops,
have been eliminated in Fig. 2, so as to focus on the basic
power processing function of the circuit. Note also that
has been replaced by a simple switch model
the switch
which is now controlled by a duty cycle pulse derived from
; this signal is compared with the
a modulating signal
sawtooth waveform at the input to the comparator IC in
order to establish the duty ratio of the converter (as discussed
in Section III).
PROCEEDINGS OF THE IEEE, VOL. 89, NO. 6, JUNE 2001
Fig. 3. Waveforms of simplified circuit. From top to bottom:
voltage that drives the switch, voltage across the switch, and
currents through the inductor, switch, and diode.
Fig. 3 shows key steady-state circuit waveforms obtained
from simulating the circuit in Fig. 2 using a simple switchedcircuit simulator of the general sort described in [1]–[12], for
example. The waveforms of the simplified power-stage circuit can also be obtained using straightforward circuit analysis techniques. Simulations or computations (such as determination of component stresses) at this modeling level
can frequently be implemented simply and conveniently in
a spreadsheet program such as Excel or using a general computational tool such as Mathcad or MATLAB.
The quasi-static analysis that describes boost circuit operation at any one input-voltage level can—with sufficient accuracy for most purposes—be extended to describe operation of
the full PFC rectifier over a rectified line cycle simply by setting up the analysis with the boost input voltage as a variable.
Key quantities such as rms currents through the switch and
, and ripple currents in
and
over a complete
diode
rectified line cycle, can be accurately estimated using such an
analysis. These results can be used to estimate dissipations
by calculating the conduction losses (which are products of
the rms currents and on-state resistances or forward drops),
core losses, and losses in the capacitor’s equivalent series resistance (ESR). Switching losses in switch and diode
can also be estimated by constructing analytical approximations for these quantities over a single switching cycle, and
accumulating them to compute the loss over a rectified line
cycle.
The next level of circuit elaboration would be to add a
snubber circuit. To analyze the more elaborate circuit, we
need a model that can replicate effects such as diode reverse recovery, as well as a circuit simulator that can deal
with such refinements. Although some models and simulators exist, in general, substantial investments in model construction and simulation time are needed to get useful results. In particular, the effects of temperature on device phenomena such as diode reverse recovery (and core losses in
the case of magnetic components) can be substantial and require a sophisticated simulation, with coupling between electrical and thermal simulators and an accurate representation
of the packaging of the devices, to yield even partially useful
results. Due to these constraints, such detailed circuit simulation is not commonly used in actual design (except when
undertaking failure analysis, particularly if experimental investigations have yielded little insight, or when ultrahigh reliability is needed, such as in space or military applications).
Rather, the practice is to simulate or analyze the effects of
diode turnoff with much simpler enhancements of the basic
diode model, e.g., having a switch in parallel with the basic
diode model, and closing it for a very short time in synchronism with the turning off of the diode, to allow a reverse current for a short duration. Using the results obtained with this
simple model, one can design the snubber circuit with significant margin, and finally carry out additional tests on a
prototype (especially at elevated temperatures), with further
adjustment of the snubber as needed.
III. LARGE-SIGNAL AVERAGED
MODELS
AND
SAMPLED-DATA
Elementary circuit modeling of a power converter
typically produces detailed continuous-time nonlinear
time-varying models in state-space form. These models
have rather low order, provided one makes approximations
that are reasonable from the viewpoint of control-oriented
modeling (as seen in the transition from Figs. 1 and 2):
neglecting dynamics that occur at much higher frequencies
than the switching frequency (for instance, dynamics due
to parasitics or snubber elements, whose time scales are
typically much shorter than the switching period), and
focusing instead on components that are central to the power
processing and control functions of the converter.
Such models capture essentially all the effects that are
likely to be significant for analysis of the basic power conversion function, but they are generally still too detailed and
awkward to work with. The first challenge, therefore, is to
extract from such a detailed model a simplified approximate
model, preferably time-invariant, that is well matched to
the particular analysis or control task for the converter
MAKSIMOVIĆ et al.: MODELING AND SIMULATION OF POWER ELECTRONIC CONVERTERS
901
being considered. There are systematic ways to obtain such
simplifications, notably through averaging, which blurs out
the detailed switching artifacts, and sampled-data modeling,
again to suppress the details internal to a switching cycle,
focusing instead on cycle-to-cycle behavior. Both methods
can produce time-invariant but still nonlinear models. In the
remainder of this section, and following the development
in [16], we illustrate the preceding comments through a
more detailed examination of the boost converter that was
introduced in the previous section. Extensions to other
converters can be made along similar lines.
Boost Converter Operation: In typical operation of the
boost converter under what may be called constant-frequency
PWM control, the switch in Fig. 2 is closed (or turned on)
seconds
every seconds, and opened (or turned off)
, so
represents the
later in the th cycle,
duty ratio in the th cycle. If we maintain a positive inductor
, then when the transistor is on, the diode
current,
is off, and vice versa. This is referred to as the continuous
conduction mode, and the waveforms in Fig. 3 correspond to
steady-state operation in this mode. In the discontinuous conduction mode, on the other hand, the inductor current drops
all the way to zero some time after the transistor is turned off,
and then remains at zero, with the transistor and diode both
off, until the transistor is turned on again. We focus on the
case of continuous conduction.
Let us mark the position of the switch using a switching
. When
, the switch is closed; when
function
, the switch is open. The switching function
may be thought of as (proportional to) the signal that has to be
applied to the gate of the MOSFET in Fig. 1 to turn it on and
off as desired. Under the constant-frequency PWM switching
jumps to 1 at the start of each
discipline described above,
later
cycle, every seconds, and falls to 0 an interval
in its th cycle, as reflected in the top waveform in Fig. 3.
over the th cycle is therefore ;
The average value of
, then
is
if the duty ratio is constant at the value
periodic, with average value .
corresponds to the signal at the output of the
In Fig. 2,
comparator. The input to the “ ” terminal of the comparator
is a sawtooth waveform of period that starts from 0 at the
beginning of every cycle, and ramps up linearly to by the
end of the cycle. At some instant in the th cycle, this ramp
at the “ ” tercrosses the level of the modulating signal
minal of the comparator. Hence, the output of the comparator
is set to 1 every seconds when the ramp restarts, and it resets to 0 later in the cycle, at time , when the ramp crosses
. (In practice, the output of the comparator would actually be used to trigger a latch, so that the switch does not
operate more than once each cycle.) The duty ratio of the
thus ends up being
in the corsignal
from cycle to
responding switching cycle. By varying
cycle, the duty ratio can be varied.
of
are what determine
Note that the samples
the duty ratios. We would therefore obtain the same seany signal
quence of duty ratios even if we added to
that stayed negative in the first part of each cycle and crossed
up through 0 in the th cycle at the instant . This fact
902
corresponds to the familiar aliasing effect associated with
sampling. Our assumption for the averaged models below
will be that
is not allowed to change significantly
is restricted to vary
within a single cycle, i.e., that
considerably more slowly than half the switching frequency.
in the th cycle, so
at
As a result,
any time yields the prevailing duty ratio [provided also that
, of course—outside this range, the duty ratio
is 0 or 1]. Generalizations to rapid small-signal variations in
can be found in [17]–[19], and a discussion of issues
of aliasing under such rapid variations may be found in [20]
is usually generated
and [21]. The modulating signal
by a feedback scheme, for instance, of the form shown by
the inputs to the PWM in Fig. 1.
A. Switched State-Space Models
and capacitor voltage
Choosing the inductor current
as natural state variables, picking the resistor voltage
, and using the notation in Fig. 2, it is
as the output
easy to see that the following state-space model describes the
idealized boost converter in that figure:
(1)
(where
Denoting the state vector by
the prime indicates the transpose), we can rewrite the above
equations as
(2)
where the definitions of the various (boldfaced) matrices and
vectors are obvious from (1). We refer to this model as the
switched or instantaneous model, to distinguish it from the
averaged and sampled-data models developed in later paragraphs. (Similar state-space models are not hard to obtain for
more elaborate, less idealized circuit models, for instance, including capacitor ESR.)
itself,
If our compensator were to directly determine
, then the
rather than determining the modulating signal
above model would be the one of interest. It is indeed possible to develop control schemes directly in the setting of the
switched model (2); see, for instance, [22]–[25], and references in those papers.
For the design of more conventional feedback control
compensation, we require a model describing the conor the duty
verter’s response to the modulating signal
, rather than the response to the switching
ratio
. Augmenting the model (2) to represent the
function
and
would introduce additional
relation between
nonlinearity and time-varying behavior, leading to a model
that is hard to work with. The averaged and sampled-data
models considered below are developed in response to this
difficulty.
PROCEEDINGS OF THE IEEE, VOL. 89, NO. 6, JUNE 2001
B. State-Space Averaged Models
To design an analog feedback control scheme, we seek a
or the
tractable model that relates the modulating signal
to the output voltage. In fact, since the
duty ratio
ripple in the instantaneous output voltage is made small by
design, and since the details of this small output ripple are not
of interest anyway in designing the feedback compensation,
what we really seek is a continuous-time dynamic model that
or
to the local average of the output
relates
voltage (where this average is computed over the switching
, the duty ratio, is the local
period). Also, recall that
in the corresponding switching cycle.
average value of
These facts suggest that we should look for a dynamic model
that relates the local average of the switching function
to that of the output voltage
.
to be
Specifically, let us define the local average of
the lagged running average
switching converter operating with low ripple in the state
variables. There are alternative sets of assumptions that lead
to the same approximations.
With the approximations in (6), the description (5) becomes
(7)
What has happened, in effect, is that all the variables in the
switched state-space model (1) have been replaced by their
average values. In terms of the matrix notation in (2), and
defined as the local average of
, we have
with
(8)
(3)
the continuous duty ratio. Note that
and call
, the actual duty ratio in the th cycle (defined as extending
to
). If
is periodic with period , then
from
, the steady-state duty ratio. Our objective is to
in (3) to the local average of the output voltage,
relate
defined similarly by
(4)
A natural approach to obtaining a model relating these averages is to take the local average of the state-space description in (1). The local average of the derivative of a signal
equals the derivative of its local average, because of the linear
time-invariant (LTI) nature of the local averaging operation
we have defined. The result of averaging the model (1) is
therefore the following set of equations:
(5)
where the overbars again denote local averages.
The terms that prevent the above description from being
and
; the average of a
a state-space model are
product is generally not the product of the averages. Under
reasonable assumptions, however, we can write
(6)
One set of assumptions leading to the above simplification
and
over the averaging interval
requires
to not deviate significantly from
and
, respectively. This condition is reasonable for a high-frequency
This continuous-time state-space model is referred to as
the state-space averaged model, [26], [27]. The model is
—with the
driven by the continuous-time control input
, and assuming continuous conducconstraint
. It is time-invariant
tion—and by the exogenous input
with respect to this pair of inputs, linear with respect to
, and bilinear with respect to
. [If
is fixed
at a constant value , then the model is LTI.] Note that,
, we can
under our assumption of a slowly varying
; with this substitution, (8) becomes
take
an averaged model whose control input is the modulating
, as desired. The use and interpretation of this
signal
model should be restricted to frequencies significantly
below half the switching frequency; converter dynamics up
to around one-tenth the switching frequency are generally
well captured by the averaged model.
The averaged model (8) can be used to solve for
steady-state or operating point relations obtained with conand
(by setting the derivatives
stant
to zero). It also leads to much more efficient simulations of
converter dynamic behavior than those obtained using the
switched model (2), provided only local averages of variables are of interest; the simulation can take larger time steps
because it no longer needs to track the switching-frequency
ripple. This averaged model also forms a convenient starting
point for various nonlinear control design approaches; see,
for instance, [28]–[30], and references in those papers. More
traditional small-signal control design to regulate operation
in the neighborhood of a fixed operating point can be based
on the corresponding LTI linearization of the averaged
model. Section IV has further discussion of these issues.
Current-Mode Control: The preceding averaged model
can also be easily modified to approximately represent the
dynamics of a high-frequency PWM converter operated
under so-called current-mode control [31]. The name comes
from the fact that a fast inner loop regulates the inductor
current to a reference value, while the slower outer loop
adjusts the current reference to correct for deviations of the
MAKSIMOVIĆ et al.: MODELING AND SIMULATION OF POWER ELECTRONIC CONVERTERS
903
Fig. 4. Nonlinear averaged circuit model of the boost converter.
output voltage from its desired value. Control of a PFC rectifier can be implemented on this basis as well. The current
monitoring and limiting that are intrinsic to current-mode
control are among its attractive features.
In constant-frequency peak-current-mode control, the
seconds, as before, but is
transistor is turned on every
turned off when the inductor current (or equivalently, the
transistor current) reaches a specified reference or peak
. The duty ratio, rather than being
level, denoted by
explicitly commanded via a modulating signal such as
in Fig. 2, is now implicitly determined by the inductor
. (Instead of constant-frequency
current’s relation to
control, one could use hysteretic or other schemes to confine
the inductor current to the vicinity of the reference current.)
A tractable and reasonably accurate continuous-time
model for the dynamics of the outer loop is obtained by
assuming that the average inductor current is approximately
equal to the reference current
(9)
.
and then making the substitution (9) in (7) to eliminate
The result is the following first-order nonlinear time-invariant model:
(10)
This model is simple enough that one can use it for simulations and to explore various nonlinear control possibilities
to control
or
; a linearized
for adjusting
version of this equation can be used to design small-signal
controllers for perturbations around a fixed operating point.
More continues to be written on averaged models in power
electronics (see, e.g., [32]–[36]) as well as further references
in Sections III-C, III-D, and IV.
C. Circuit Averaging and the Averaged Switch
Instead of averaging the converter state equations, we
could directly average the characteristics or waveforms
associated with each of the components in the converter
[37]. This circuit averaging approach is widely used (although sometimes in implicit rather than explicit ways).
Because manipulations are performed on the circuit diagram
instead of on its equations, the circuit averaging technique
often gives a more physical interpretation to the model.
The circuit averaging technique can be applied directly to a
904
number of different types of converters and switch elements,
including phase-controlled rectifiers, pulsewidth modulated
converters in continuous or discontinuous conduction mode,
resonant-switch converters, and so on. Because of its generality and the ease with which the resulting models are
simulated in standard circuit simulators such as SPICE or
SABER, there has been a recent resurgence of interest in
circuit averaging of switched networks; see [38]–[50] and
further references in Sections III-D and IV.
The first step in circuit averaging is to replace all voltages
and currents by their (running) averages. The resulting quantities still respect Kirchhoff’s laws, and therefore constitute
valid circuit variables. All LTI components of the original circuit impose the same constraints on the averaged quantities
as they do on the original instantaneous variables, and therefore remain the same in the averaged circuit. The switching
elements of the original circuit need to be handled differently, however. If we represent the switching elements in the
original circuit as appropriately controlled voltage or current
sources, then these can be circuit-averaged as well, but with
some approximations to convert the control relationships to
ones involving only averaged quantities.
As an example, consider the ideal boost dc–dc converter
of Fig. 2. The diode can be replaced by a controlled curand the switch by a controlled
rent source of value
, where
.
voltage source of value
By averaging these relationships and making the same approximations as in (6), we obtain
(11)
(12)
. These are the terminal relations that
where
characterize the averaged switching elements. The averaged
circuit model of Fig. 4 is the result of the above process. This
is a large-signal nonlinear, but time-invariant circuit model.
Not surprisingly, it is governed by the state-space averaged
model in (7), but we have not had to derive a state-space
model for each configuration in order to obtain the model.
Linearization of this circuit (as discussed in Section IV)
yields small-signal models that are suited to conventional
feedback control design.
It should be noted that the definition of the switch network and its dependent variables is not unique. Different definitions lead to equivalent but not identical averaged circuit
models; some choices may be better suited than others to any
particular analysis task.
PROCEEDINGS OF THE IEEE, VOL. 89, NO. 6, JUNE 2001
D. Generalized Averaging and Dynamic Phasors
The voltages and currents in power electronic converters
and electrical drives are typically periodic in steady state,
and often nonsinusoidal. The dynamics of interest for analysis and control are often those of deviations from periodic
behavior, for instance as manifested in deviations of the envelope of a quasi-sinusoidal waveform from its steady-state
value. For analysis of the steady state, one has familiar phasor
or harmonic or describing function methods [18], [51]–[53].
The analytical approach reviewed here is aimed at systematic derivation of phasor dynamics, from which the dynamic
behavior of the original waveform or its envelope can efficiently be deduced. (A distinct approach to the notion of envelope following, directly implemented in a simulation setting, may be found in [54].) The idea of deriving dynamical
models for Fourier coefficients goes back to classical averaging (see [32], [34] and references therein). The recent interest in these approaches for power electronics was sparked
by [49] and [55], which applied the approach to series resonant and switched mode dc–dc converters (also see [56]
for series resonant converters); the approach taken in [49]
involved direct circuit averaging. Some extensions may be
found in [57].
The generalized averaging that we perform to obtain our
models is based on the observation [55] that a (possibly comcan be represented on the
plex) time-domain waveform
using a Fourier series of the form
interval
(13)
and
are the complex Fourier cowhere
efficients, which we shall also refer to as phasors. These
Fourier coefficients are functions of time since the interval
under consideration slides as a function of time. We are interested in cases where a few coefficients suffice to provide
a good approximation of the original waveform, and where
those coefficients vary slowly with time.
The th coefficient (or -phasor) at time is determined
by the following averaging operation:
(14)
will be used to denote the averaging
The notation
operation in (14). Our analysis aims to provide a dynamic
model for the dominant Fourier series coefficients as the
window of length slides over the waveforms of interest.
More specifically, we aim to obtain a state-space model in
which the coefficients in (14) are the state variables.
are complex-valued, the
When the original waveforms
equals
(where is the complex
phasor
conjugate of ). Complex-valued waveforms arise, for instance, when using complex space vectors [58] in dynamical
descriptions of electrical drives. In the case of real-valued
and
, so
time-domain quantities,
(13) can be rewritten as a one-sided summation involving
for positive . If in additwice the real parts of
Fig. 5. Circuit schematic of a series resonant converter (typical
parameters: v = 3.3 V, I = 1 A, R = 5 , switching frequency
above 38 kHz).
tion
is time-invariant, the standard definition of phasors
from circuit theory is recovered. A key property is that the
derivative of the th Fourier coefficient is given by the following expression:
(15)
This formula is easily verified using (13) and (14), and integration by parts.
The definitions given in (13) and (14) can also be generalized for the analysis of polyphase systems, with the definition of dynamic positive-sequence, negative-sequence and
; see
zero-sequence symmetric components at frequency
[59].
The application of the above phasor calculus to obtaining
an averaged model proceeds just as with state-space averaging (and limiting attention to the zeroth-order phasor
actually recovers traditional state-space averaging). One begins with a standard state-space description of the instantaneous (switched) variables, then averages both sides, invoking the properties of dynamic phasors as needed. The next
step is to make approximations that allow the averaged model
itself to be written in state-space form, using the dynamic
phasors as state variables. The slow variation of the phasors
is usually one of the critical assumptions in making reasonable approximations.
1) Example: Resonant Converter: As an example of the
application of generalized averaging, consider the series resonant dc–dc converter shown in Fig. 5. Using the notation
given in the figure, a state-space model can be written as
(16)
denotes the switching frequency in rad/s, and
where
are the instantaneous resonant tank voltage and current reis the instantaneous output voltage, and the
spectively,
MAKSIMOVIĆ et al.: MODELING AND SIMULATION OF POWER ELECTRONIC CONVERTERS
905
load comprises a resistor in parallel with a current sink
(we have dropped the time argument from the variables ,
, and to avoid notational clutter). The “ ” in the above
equations is 1, the sign being that of its argument.
To derive a dynamical phasor model corresponding to
(16), it is assumed that both and are described with
)
sufficient accuracy by their respective fundamental (
components (with corresponding phasors , taken to have
, respectively), while is assumed
angle 0, and
to be slowly varying, hence well described by its
component, or local average. These assumptions are reasonable in well-designed dc–dc series resonant converters.
Then the following dynamic phasor model can be derived
from (16) using (15):
We illustrate how a sampled-data model may be obtained
for our boost converter example. The state evolution of (1),
(2) for each of the two possible values of
can be described very easily using the standard matrix exponential expressions for LTI systems, and the trajectories in each segment can then be pieced together by invoking the continuity
of the state variables. Recall that the matrix exponential can
be defined, just as in the scalar case, by the (very well behaved) infinite matrix series
(18)
from which it is evident that
(19)
Under the switching discipline of constant-frequency PWM,
for the initial fraction of the th switching
where
for the rest of the cycle, and assuming
cycle, and
, we find
the input voltage is constant at
(17)
(We have again dropped the time argument from ,
and .) This model can be written in the form of a fifthorder model involving real-valued quantities, for example,
by taking real and imaginary parts of the first two equations.
It turns out that the dynamic phasor model approximates the
switched model very closely, as shown in [55]. Control explorations using this model can be found in, e.g., [60] and
[61].
Dynamic phasors can be used to obtain models with
varying degrees of detail; for example, both the dc component and the fundamental switching-frequency component
were used to describe a boost converter in [57]. Dynamic
phasors have been used very naturally and effectively
for a variety of power electronic converters of interest
in high-power transmission systems. These flexible ac
transmission system (FACTS) applications include the
thyristor-controlled series capacitor (TCSC) described in
[62], [63], and an unbalanced unified power flow controller
(UPFC) that utilizes polyphase dynamic phasors, treated in
[64]. Application to unbalanced three-phase machines can
be found in [59]. The notion of a dynamic phasor can be
of use in power systems even when no power electronics is
involved; see [65] and [66].
E. Sampled-Data Models
Sampled-data models are naturally matched to power electronic converters, firstly because of the cyclic way in which
power converters are operated and controlled, and secondly
because such models are well suited to the design of digital controllers, which are increasingly used in power electronics. Like averaged models, sampled-data models allow us
to focus on cycle-to-cycle behavior, ignoring details of the intracycle behavior. This makes them effective in studying and
controlling ripple instabilities (i.e., instabilities at half the
switching frequency), and also in general simulation, analysis, and design.
906
(20)
where
(21)
For a well-designed high-frequency PWM dc–dc converter in continuous conduction, the state trajectories in
each switch configuration are close to linear, because the
switching frequency is much higher than the filter cutoff
frequency. What this implies is that the matrix exponentials
in (20) are well approximated by just the first two terms in
their Taylor series expansions
(22)
If we use these approximations in (20) and neglect terms
in , the result is the following approximate sampled-data
model:
(23)
This model is easily recognized as the usual forward-Euler
approximation of the continuous-time model in (8), obtained
by replacing the derivative there by a forward difference.
leads to more refined, but still very
(Retaining the terms in
simple, sampled-data models.) For an example of the use
in simulation of sampled-data and continuous-time models
based on this sort of approximation, see [67] and [68].
The sampled-data models in (20) and (23) were derived
from (1), (2), and therefore used samples of the natural state
and
, as state variables. However, other
variables,
choices are certainly possible, and may be more appropriate
PROCEEDINGS OF THE IEEE, VOL. 89, NO. 6, JUNE 2001
for a particular implementation. For instance, we could reby
, i.e., the sampled local average of
place
the capacitor voltage.
An early reference on sampled-data models in power electronics is [69]. For more on sampled-data models, see [45]
and references there, and also, e.g., [70]–[72]. In particular,
[45] derives a sampled-data model for the boost PFC (but
sampling at the period of the rectified ac voltage rather than
the switching period of the boost converter), and uses it to design a discrete-time feedback controller (with time constant
on the order of the period of the ac input).
IV. SIMULATION OF SWITCHED AND AVERAGED DYNAMIC
MODELS
In the design verification of power electronic systems by
simulation, it is often necessary to use component and system
models of various levels of complexity. This section, which
elaborates on some of the issues raised in Sections I and II,
is focused on switched and averaged models, although sampled-data models have their particular role as well, especially
in careful stability studies and in control design for digital
controllers.
• Detailed, complex models that attempt to accurately
represent the physical behavior of devices are necessary for tasks that involve finding switching times,
details of switching transitions and switching loss
mechanisms, or instantaneous voltage and current
stresses. Component vendors often provide libraries
of such device models for use with general-purpose
circuit simulators such as SPICE or SABER. To complete a detailed circuit model, one must also carefully
examine effects of packaging and board interconnects.
With fast-switching power semiconductors, simulation
time steps corresponding to a few nanoseconds or less
may be required, especially during ON–OFF switching
transitions. Because of the complexity of detailed
device models and the fine time resolution, the simulation tasks can be very time consuming. In practice,
time-domain simulations using detailed device models
are usually performed only on selected parts of the
system, and over short time intervals involving a few
switching cycles.
• Since an ON–OFF switching transition usually takes
only a small fraction of a switching cycle, the basic
operation of switching power converters can be explained using simplified, idealized device models. For
example, a MOSFET can be modeled as a switch with
a small (ideally zero) resistance when on, and a very
large resistance (ideally an open circuit) when off.
Such simplified models yield physical insight into the
basic operation of switching power converters, and
provide the starting point for the development of the
analytical models described earlier. Simplified device
models are also useful for time-domain simulations
aimed at determining or verifying converter and
controller operation, switching ripples, current and
voltage stresses, responses to load or input transients,
and small-signal frequency-response characteristics.
With simple device models, and ignoring details of
switching transitions, simulations over many switching
cycles can be completed efficiently, using general-purpose circuit simulators or specialized simulators that
are developed to support fast transient simulation
based on idealized, piecewise-linear device models,
or based on a combination of piecewise-linear and
nonlinear models (see [1]–[12]).
• Averaged models are well suited for prediction of
converter steady-state and dynamic responses. These
models are essential design tools because they provide
physical insight and lead to analytical results that can
be used in the design process to select component and
controller parameter values for a given set of specifications. A large-signal averaged circuit model, such as
the model in Fig. 4, is very convenient for application
with general-purpose circuit simulators such as SPICE
or SABER. Simulations of averaged circuit models
can be performed to test for losses (apart from those
due to switching) and efficiency, steady-state voltages
and currents, stability, and large-signal transient responses. Since switching transitions and ripples are
removed by averaging, simulations over long time
intervals and over many sets of parameter values can
be completed efficiently. Therefore, averaged models
are also well suited for simulations of large electronic
systems that include multiple switching converters
[73]. Furthermore, although large-signal averaged
models are nonlinear, they are time-invariant and can
be linearized about any constant operating condition
to produce LTI small-signal models, from which one
can generate various frequency responses of interest
(see Section IV-B). References on averaged converter
modeling for simulation include [74]–[81].
A. Transient Response Analysis
In the design of control loops around converters, it is
often necessary to perform transient simulations over many
switching cycles. For example, in dc voltage regulator
designs, it is necessary to verify whether the output voltage
remains within specified limits when the load current takes
a step change. In the boost PFC rectifier of Fig. 1, transient
simulations can be used to determine current harmonic
distortion, component stresses during start-up or load transients, and so on. Such simulations can be performed on a
switching circuit model using a switched-circuit simulator
or a general-purpose simulator, or on the converter averaged
model, or using a sampled-data model.
As an example, let us apply the first two approaches
to investigate a transient response of the boost converter
shown in Fig. 2 due to a step change in the switch duty
cycle. Fig. 6 shows the inductor current and the capacitor
voltage waveforms during the transient. The waveforms
obtained by switched-circuit transient simulation are shown
together with the waveforms obtained by simulation of the
averaged circuit model in Fig. 4. The converter transient
response is governed by the natural time constants of the
MAKSIMOVIĆ et al.: MODELING AND SIMULATION OF POWER ELECTRONIC CONVERTERS
907
(a)
(b)
Fig. 6. Transient waveforms in the boost converter example, for
i (t) and v (t). The duty cycle is increased from d = 0.55 to
d = 0.6 at t = 0.5 ms.
converter. Since these time constants are much longer than
the switching period, the converter transient responses take
many switching cycles to reach a new steady state. In the
results obtained by simulation of the averaged circuit model,
the switching ripples are removed, but the low-frequency
portions of the converter transient responses match very
closely the responses obtained by switched-circuit simulation. (Note that the converter goes through an interval in the
discontinuous conduction mode, from around 1.2 to 2 ms.
An appropriate averaged-switch model can be derived to
handle this transition to discontinuous conduction and back;
see [50].)
B. Steady-State and Small-Signal Analysis
There are many numerical/simulation approaches to determining the steady state of a switched model (see [82]–[86]
and references therein, for example). Small-signal models,
and particularly sampled-data small-signal models, can now
be constructed to represent small deviations from this steady
state.
A designer is also often interested in determining the
boundaries in the space of parameters (such as input voltage
amplitude, frequency, load resistance or current, and so on)
that mark transitions from one steady-state operating mode
to another in a switched circuit. Of complementary interest is
the determination of stability domains in the state space for
particular operating modes, i.e., the sets of initial conditions
that respectively converge to these operating modes. Models
908
and numerical approaches for such problems may be found
in [87], which also examines the modeling and simulation of
more exotic phenomena such as chaos in power electronics.
Circuit averaging leads to a nonlinear, time-invariant
circuit model, as illustrated by the example shown in Fig. 4.
Both steady-state computations and the construction of
small-signal models are easily carried out with averaged
circuits. As an example, Fig. 7 shows the steady-state dc and
small-signal ac circuit model obtained by standard linearization of the nonlinear controlled sources in Fig. 4 around a
steady-state operating point. This circuit model includes
an ideal transformer that explicitly illustrates the major
features of the boost dc–dc converter, namely a
dc
), small-signal natural
conversion ratio (where
time constants determined by energy storage components, as
well as effects of duty cycle variations through the sources
and , where
, the duty-ratio deviation from
steady state. This circuit model can be easily solved for
transfer functions of interest for classical controller design
based on the LTI model, including control-to-output and
line-to-output responses, as well as the output impedance.
Linearized averaged models are also the starting point for
the modeling and stability analysis of paralleled converters
(see [46], [88], and [89]).
Frequency responses of interest can alternatively be obtained by appropriate time-domain simulations of switchedcircuit models (see [90]–[93]), or by ac simulations of nonlinear averaged circuit models (see [50], [74]–[80]). As an
example, Fig. 8 shows magnitude and phase responses of
the boost control-to-output transfer function
(where is the perturbation in output voltage), obtained by
ac simulation of the model in Fig. 4.
V. CONCLUDING DISCUSSION
First, an important disclaimer. Although we have cited several relevant references, there are at least as many other ones
that we have not. The references listed here are intended to
serve as pointers for the interested reader, and will quickly
lead to much more that is likely to be useful.
Hierarchical approaches, using a variety of layered models
and simulations, form the basic strategy used today to analyze and design power electronic circuits. Proceeding up
the hierarchy typically involves modeling individual modules or portions of the circuit in a more aggregated or abstracted form, allowing larger portions of the circuit or of a
system with multiple circuits to be simulated in reasonable
times with adequate accuracy.
Switched-circuit and averaged simulators have also
proven to be very valuable in the synthesis of new power
electronic circuits. Generally, there are large numbers of possible combinations of switches and passive elements that can
be combined to create new circuit topologies. Simulation of
these topologies remains a key tool in comparing topologies
for an application, discovering problems in a new circuit
or control approach, trying out variations to overcome each
successively discovered hurdle, and then refining the circuit
or controller to meet performance requirements. The ability
PROCEEDINGS OF THE IEEE, VOL. 89, NO. 6, JUNE 2001
Fig. 7.
DC and small-signal ac averaged circuit model of the boost converter.
Fig. 8. Magnitude and phase responses of the control-to-output
transfer function G (s) = v^=d^ of the boost dc–dc converter.
in a simulator to build models of increasing complexity,
starting from very idealized models, provides a strong tool
for the power electronics design engineer exploring a new
design concept, in essence by using the computer tool to help
understand how a new circuit works (or does not work). The
development of approaches to rapidly determining the cyclic
steady-state sequence of a switching circuit has meant that
long simulations to reach steady state are avoided, leading
to significant increases in design engineer productivity.
Although general-purpose circuit simulators such as
SPICE are increasingly being used for power electronics
simulation, they are still beset with problems when used
to simulated detailed device behavior. These include the
following.
• Paucity of accurate device models for generally
available commercial devices, especially for power
switching devices such as diodes, MOSFETs, thyristors, and IGBTs. Although there have been numerous
papers on modeling power semiconductor devices [94],
[95], the models remain difficult to develop and involve
somewhat large investments in time and equipment
to build and validate. Also, power electronic design
engineers, who are generally focused on circuit-level
design, rarely have a deep enough understanding of
device behavior to understand the structure of the
models or how to change model parameters to reflect
different devices. These barriers remain as significant
obstacles to the use of detailed circuit simulations in
power electronics for the future.
• Inadequate understanding and modeling of the role that
thermal effects play in changing electrical character-
istics. Although a few papers and simulators have reported some work in this area [96]–[98], they still remain highly specialized activities requiring major investments in time to set up the models and simulations,
leading to practical use only in a few cases.
• Difficulties in getting reliable convergence of simulators also remains an ongoing problem and a source of
frustration for power electronic circuit designers.
In addition to circuit simulators and analytical methods
such as averaging, control-system-oriented tools such as
MATLAB are also used today. They are used in much the
same way as for other control-system analysis problems,
with the exception that an appropriate circuit-averaged or
sampled-data model is used for the switching power stage
and portions of the control circuitry that are not continuous.
Another area with increased activity recently is in deriving models that are related to the physical geometry for
such components as magnetics and printed wiring boards,
[99]–[101]. Results from these analyses may be used to
derive circuit models that can then be combined with other
elements in a circuit simulator.
Many of the methods described above can be folded into
a framework that assesses circuit performance with variation
of circuit parameters or operating conditions. These include
worst case analysis (see [102] and references therein), and
yield analysis using Monte Carlo or similar techniques. Most
commercially available simulators used in power electronics
include such capabilities.
In summary, analysis of power electronic systems requires
multiple methods and tools to understand circuit operation
and obtain enough information to achieve a robust design.
This is no different than in other fields of electronics such
as digital systems where hierarchical approaches and multiple tools are routinely used. A key difference, valid even
today, is that experimental methods are still practical, effective, and heavily relied upon in power electronics. Nevertheless, the widespread availability of inexpensive computing
and the refinement of simulation tools and techniques over
the last decade have allowed us to come closer to the day
when a complete power electronic circuit can be simulated
and studied in software, then built with high confidence that
it will work right the first time.
ACKNOWLEDGMENT
The authors are grateful to an anonymous reviewer, and to
V. Caliskan, G. Escobar, S. Leeb, and D. Perreault for comments that helped improve the paper.
MAKSIMOVIĆ et al.: MODELING AND SIMULATION OF POWER ELECTRONIC CONVERTERS
909
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Dragan Maksimović (Member, IEEE) was born
in Belgrade, Yugoslavia, on July 15, 1961. He
received the B.S. and M.S. degrees in electrical
engineering from the University of Belgrade and
the Ph.D. degree from the California Institute of
Technology, Pasadena, in 1984, 1986, and 1989,
respectively.
From 1989 to 1992, he was with the University
of Belgrade. Since 1992, he has been with the Department of Electrical and Computer Engineering
at the University of Colorado, Boulder, where he
is currently an Associate Professor and Co-Director of the Colorado Power
Electronics Center (CoPEC). His current research interests include simulation and control techniques, low-harmonic rectifiers, and power electronics
for low-power, portable systems.
In 1997 he received the NSF CAREER Award, and a Power Electronics
Society Transactions Prize Paper Award.
912
Aleksandar M. Stanković (Member, IEEE) received the Dipl. Ing. degree
in 1982 and the M.S. degree in 1986, both from the University of Belgrade,
Yugoslavia, and the Ph.D. degree from the Massachusetts Institute of Technology, Cambridge, in 1993, all in electrical engineering.
He has been with the Department of Electrical and Computer Engineering
at Northeastern University, Boston, since 1993, presently as an Associate
Professor.
Dr. Stanković is a member of IEEE Power Engineering, Power Electronics, Control Systems, Circuits and Systems, Industrial Electronics, and
Industry Applications Societies. He serves as an Associate Editor for the
IEEE TRANSACTIONS ON CONTROL SYSTEM TECHNOLOGY, and as chair of
the technical committee on Power Electronics and Power Systems of the
IEEE Circuits and Systems Society.
V. Joseph Thottuvelil (Member, IEEE) received the B.S. degree from the
Indian Institute of Technology, Madras, and the M.S. and Ph.D. degrees from
Duke University, Durham, NC, all in electrical engineering, in 1978, 1980,
and 1984, respectively.
He was with Digital Equipment Corporation from 1984 to 1993.
Between 1993 and 2000, he was with Bell Laboratories, Lucent Technologies, working on telecommunications and computer power systems and
components. He is now with Tyco Electronics Power Systems as Technical
Manager of the Applications Engineering Group.
Dr. Thottuvelil was Secretary of the IEEE Power Electronics Society from
1995–1998, and is currently Associate Editor for the IEEE TRANSACTIONS
ON POWER ELECTRONICS, covering the area of telecommunications.
George C. Verghese (Fellow, IEEE) received the
B.Tech. degree from the Indian Institute of Technology at Madras in 1974, the M.S. degree from
the State University of New York at Stony Brook
in 1975, and the Ph.D. degree from Stanford University, Stanford, CA, in 1979, all in electrical engineering.
Since 1979, he has been at the Massachusetts
Institute of Technology, Cambridge, where
he is Professor of Electrical Engineering in
the Department of Electrical Engineering and
Computer Science and a member of the Laboratory for Electromagnetic
and Electronic Systems. His research interests and publications are in the
areas of systems, control, estimation, and signal processing, with a focus on
applications in power electronics, power systems, and electrical machines.
He is co-author (with J. G. Kassakian and M. F. Schlecht) of Principles
of Power Electronics (Addison-Wesley, 1991), and co-editor (with S.
Banerjee) of Nonlinear Phenomena in Power Electronics: Attractors,
Bifurcations, Chaos, and Nonlinear Control (IEEE Press, 2001). He has
served on the AdCom and other committees of the IEEE Power Electronics
Society, and also as founding co-chair (with V. J. Thottuvelil) of its
technical committee and workshop on Computers in Power Electronics.
Dr. Verghese has served as an Associate Editor of Automatica, the IEEE
TRANSACTIONS ON AUTOMATIC CONTROL, and the IEEE TRANSACTIONS ON
CONTROL SYSTEMS TECHNOLOGY.
PROCEEDINGS OF THE IEEE, VOL. 89, NO. 6, JUNE 2001