Modeling and Simulation of Power Electronic Converters

Transcription

1 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, The work of D. Maksimović was supported by the National Science Foundation under Grant ECS The work of A. M. Stanković was supported by the National Science Foundation under Grants ECS and ECS , and by the Office of Naval Research under Grant N D. Maksimović is with the University of Colorado, Boulder, CO USA. A. M. Stanković is with Northeastern University, Boston, MA USA. V. J. Thottuvelil is with Tyco Electronics Power Systems, Mesquite, TX USA. G. C. Verghese is with the Massachusetts Institute of Technology, Cambridge, MA USA ( verghese@mit.edu). Publisher Item Identifier S (01) 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 /01$ IEEE 898 PROCEEDINGS OF THE IEEE, VOL. 89, NO. 6, JUNE 2001

2 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 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,,, and. The capacitor serves to filter switching-frequency 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 voltage. The output of serves as the multiple 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 resistor, and,,,,, and, creating 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

3 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 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, where denotes from Fig. 1, denotes from the earlier figure, and denotes the load, modeled as being 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 the switch has been replaced by a simple switch model which is now controlled by a duty cycle pulse derived from a modulating signal ; this signal is compared with the sawtooth waveform at the input to the comparator IC in order to establish the duty ratio of the converter (as discussed in Section III). 900 PROCEEDINGS OF THE IEEE, VOL. 89, NO. 6, JUNE 2001

4 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. 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. 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 diode, and ripple currents in and over a complete 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), III. LARGE-SIGNAL AVERAGED AND SAMPLED-DATA MODELS 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

5 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) every seconds, and opened (or turned off) seconds later in the th cycle,,so represents the duty ratio in the th cycle. If we maintain a positive inductor current,, then when the transistor is on, the diode 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 function. When, the switch is closed; when, 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 discipline described above, jumps to 1 at the start of each cycle, every seconds, and falls to 0 an interval later in its th cycle, as reflected in the top waveform in Fig. 3. The average value of over the th cycle is therefore ; if the duty ratio is constant at the value, then is periodic, with average value. In Fig. 2, corresponds to the signal at the output of the 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 crosses the level of the modulating signal at the terminal 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 signal thus ends up being in the corresponding switching cycle. By varying from cycle to cycle, the duty ratio can be varied. Note that the samples of are what determine the duty ratios. We would therefore obtain the same sequence of duty ratios even if we added to any signal that stayed negative in the first part of each cycle and crossed up through 0 in the th cycle at the instant. This fact 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 within a single cycle, i.e., that is restricted to vary considerably more slowly than half the switching frequency. As a result, in the th cycle, so at 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] and [21]. The modulating signal is usually generated by a feedback scheme, for instance, of the form shown by the inputs to the PWM in Fig. 1. A. Switched State-Space Models Choosing the inductor current and capacitor voltage as natural state variables, picking the resistor voltage as the output, and using the notation in Fig. 2, it is easy to see that the following state-space model describes the idealized boost converter in that figure: Denoting the state vector by (where the prime indicates the transpose), we can rewrite the above equations as 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.) If our compensator were to directly determine itself, rather than determining the modulating signal, then the 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 converter s response to the modulating signal or the duty ratio, rather than the response to the switching function. Augmenting the model (2) to represent the relation between and would introduce additional 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. (1) (2) 902 PROCEEDINGS OF THE IEEE, VOL. 89, NO. 6, JUNE 2001

6 B. State-Space Averaged Models To design an analog feedback control scheme, we seek a tractable model that relates the modulating signal or the duty ratio to the output voltage. In fact, since the 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 relates or to the local average of the output voltage (where this average is computed over the switching period). Also, recall that, the duty ratio, is the local average value of in the corresponding switching cycle. 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. Specifically, let us define the local average of to be the lagged running average and call the continuous duty ratio. Note that, the actual duty ratio in the th cycle (defined as extending from to ). If is periodic with period, then, the steady-state duty ratio. Our objective is to relate in (3) to the local average of the output voltage, defined similarly by 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: where the overbars again denote local averages. The terms that prevent the above description from being a state-space model are and ; the average of a product is generally not the product of the averages. Under reasonable assumptions, however, we can write One set of assumptions leading to the above simplification requires and over the averaging interval to not deviate significantly from and, respectively. This condition is reasonable for a high-frequency (3) (4) (5) (6) 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 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 with defined as the local average of,wehave This continuous-time state-space model is referred to as the state-space averaged model, [26], [27]. The model is driven by the continuous-time control input with the constraint, and assuming continuous conduction and by the exogenous input. It is time-invariant 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, under our assumption of a slowly varying, we can take ; with this substitution, (8) becomes an averaged model whose control input is the modulating signal, as desired. The use and interpretation of this 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 constant and (by setting the derivatives 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 (7) (8) MAKSIMOVIĆ et al.: MODELING AND SIMULATION OF POWER ELECTRONIC CONVERTERS 903

7 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 transistor is turned on every seconds, as before, but is turned off when the inductor current (or equivalently, the transistor current) reaches a specified reference or peak level, denoted by. The duty ratio, rather than being explicitly commanded via a modulating signal such as in Fig. 2, is now implicitly determined by the inductor current s relation to. (Instead of constant-frequency 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 and then making the substitution (9) in (7) to eliminate. The result is the following first-order nonlinear time-invariant model: (9) (10) This model is simple enough that one can use it for simulations and to explore various nonlinear control possibilities for adjusting to control or ; a linearized 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 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 current source of value and the switch by a controlled voltage source of value, where. By averaging these relationships and making the same approximations as in (6), we obtain (11) (12) where. These are the terminal relations that 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. 904 PROCEEDINGS OF THE IEEE, VOL. 89, NO. 6, JUNE 2001

8 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 complex) time-domain waveform can be represented on the interval using a Fourier series of the form (13) where and are the complex Fourier coefficients, 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) The notation will be used to denote the averaging 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. When the original waveforms are complex-valued, the phasor equals (where is the complex 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 time-domain quantities, and,so (13) can be rewritten as a one-sided summation involving twice the real parts of for positive. If in addi- Fig. 5. Circuit schematic of a series resonant converter (typical parameters: v = 3.3 V, I = 1A, 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 zero-sequence symmetric components at frequency ; see [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) where denotes the switching frequency in rad/s, and are the instantaneous resonant tank voltage and current respectively, is the instantaneous output voltage, and the MAKSIMOVIĆ et al.: MODELING AND SIMULATION OF POWER ELECTRONIC CONVERTERS 905

9 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 angle 0, and, respectively), while is assumed 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): (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. 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 from which it is evident that (18) (19) Under the switching discipline of constant-frequency PWM, where for the initial fraction of the th switching cycle, and for the rest of the cycle, and assuming the input voltage is constant at, we find where (20) (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. (Retaining the terms in leads to more refined, but still very 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 variables, and, as state variables. However, other choices are certainly possible, and may be more appropriate 906 PROCEEDINGS OF THE IEEE, VOL. 89, NO. 6, JUNE 2001

10 for a particular implementation. For instance, we could replace by, i.e., the sampled local average of 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

11 Fig. 6. (a) (b) 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 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 conversion ratio (where ), small-signal natural 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 908 PROCEEDINGS OF THE IEEE, VOL. 89, NO. 6, JUNE 2001