ADVANCES IN AVERAGED SWITCH MODELING

Transcription

1 ADVANES IN AVEAGED SWITH MODEING obert W. Erickson Power Electronics Group Department of Electrical and omputer Engineering University of olorado Boulder, O USA rwe@boulder.colorado.edu Abstract - The averaged switch modeling technique unifies the modeling of PWM and resonant dc-dc converters, low-harmonic rectifiers, and inverters in a simple manner. This technique is briefly reviewed, and recent results in modeling current-programmed dc-dc converters, integrated single-phase low-harmonic rectifier-regulators, and single-switch three-phase lowharmonic rectifiers are described. A circuit element of importance in modeling these applications, the dependent power source, is also reviewed. I. INTODUTION Power v g (t) Time-invariant network containing converter reactive elements v (t) i 1 (t) port 1 Switch network ontrol Equivalent circuit modeling is an essential tool for design, worst-case analysis, and simulation of switching converters. The design engineer can obtain the insight needed to improve his or her power stage and controller design. onverter models are necessary in worst-case analysis, to prove that peak component stresses, transient response, regulation, controller stability, conducted emissions and conducted susceptibility all meet specifications. Equivalent circuits make the computer simulation of converter dynamic response a feasible undertaking. The use of averaging has been well accepted as a way to model the low-frequency components of the waveforms in a switching converter [1-5]. Averaging allows removal of the switching elements, and the harmonics they generate, from the converter model. The resulting equivalent circuit is much easier to solve, and allows insight to be gained into the important functional properties of the converter. The smallsignal transfer functions of the converter can also be derived. An important advance in averaged modeling of converters was the averaged switch approach [5-9]. This simple method is capable of modeling a wide variety of converters, including PWM and resonant converters operating in the continuous conduction mode, rectifiers, and inverters. In this paper, the averaged switch approach is utilized to represent the basic characteristics of several common example applications, including current-programmed converters, single-phase lowharmonic rectifiers, and three-phase low-harmonic rectifiers. Properties of the power source and dc transformer elements are also reviewed here; these elements represent the lowfrequency terminal characteristics of switch networks in a simple and clear manner. II. AVEAGED SWITH MODEING In development of the averaged switch approach [5-9], it was recognized that only the converter switching elements need to be averaged. The converter switches grouped into a switch network as in Fig. 1; the remainder of the converter is then a linear time-invariant network. The terminal waveforms of the switch network are averaged over one switching period to remove the switching harmonics. An equivalent circuit model for the switch network is then defined, based on the relations between the averaged terminal waveforms. Averaged switch models for two basic continuous conduction mode (M) PWM switch networks are illustrated in Fig. x; both large-signal dc and small-signal ac versions are shown. In Fig. 2, D is the switch duty cycle, angle brackets denote averaged quantities, and carats denote small-signal ac components. The ideal dc transformer symbol represents the basic properties of ideal PWM M switch networks: lossless conversion of power, with effective turns ratio equal to a function of the duty cycle [3]. When combined with the remainder of the converter, the averaged switch models of d(t) i (t) port 2 Fig. 1. Averaged switch modeling: a switch network, containing only the converter switching elements, is defined. The terminal waveforms of the switch network are then averaged over one switching period. oad

2 i 1 (t) i 1 (t) 1 : D I 1 i 1 1 : D I 2 i 2 d v 1 I 2 d V 2 v 2 i(t) i(t) i(t) = p(t) p(t) i 1 (t) i 1 (t) D' : D I 1 i 1 D' : D I 2 i 2 v 1 DD' d I 2 V 2 v 2 DD' d Fig. 2. Two simple PWM switch networks, and their averaged large-signal and small-signal models for continuous conduction mode operation. Fig. 2 can predict the dc and ac behavior of any PWM dutycycle-controlled converter operating in M. Other networks have been proposed to model the basic ideal proerties of switch networks operating in other modes [12]. The averaged switch modeling approach is quite general, and it effectively unifies the dc and ac modeling of PWM converters operating in both continuous and discontinuous conduction modes, resonant converters, low-harmonic rectifiers, and PWM inverters. It is able to predict the effect of every reactive element on the converter dynamic behavior, and consequently the averaged switching modeling approach has been widely accepted. Efficiency, losses, and dynamics of a wide variety of converters, operating modes, and control schemes can be represented using this approach. III. THE DEPENDENT POWE SOUE Several examples of extension of the averaged switch modeling approach to current programmed converters and to low harmonic rectifiers are listed in Section IV. In these examples, the basic low-frequency function of the switch network does not follow a dc transformer characteristic. ather, the averaged iðv characteristics of one port of the switch network follow constant power characteristics. Hence it is useful to model these networks using power source elements; properties of the dependent power source element are reviewed here. A. How the Power Source Arises in ossless Power Processing Networks Examples of power source characteristics in switching converters are widespread[10-12, 15-17]. onsider a lossless two-port network that contains no dynamics, such as the averaged switch models of Section II. Whenever such a network meets buffer conditions, then a dependent power source must arise [11]. When port 1 is buffered, then there is a functional relationship between its voltage and current that is independent of the voltage and current of port 2. The power flowing into port 1 is therefore independent of the waveforms at port 2. Since the two port network is lossless, all of the instantaneous power flowing into port 1 must flow out of port 2. Port 2 therefore exhibits a power source characteristic, equal to the power flowing into port 1. The Fig. 3. The dependent power source: (a) symbol, (b) iðv characteristic. voltage and current of port 2 are determined by the intersection of the port 2 power source characteristic with the iðv characteristic of the external circuit. Examples of classes of converters that exhibit power source characteristics at one or more of their terminals include low harmonic rectifiers, current-programmed dc-dc converters, open-loop dc-dc converters operating in the discontinuous conduction mode, and dc-dc converters having a regulated output voltage. In most of these cases, application of the averaged switch modeling technique leads to a switch model containing one or more dependent power sources. In view of the common occurrence of constant power characteristics in power electronics applications, it is useful to define a new circuit element, the dependent power source illustrated in Fig. 3(a). This element exhibits the iðv characteristic of Fig. 3(b). B. Properties of the Power Source The power source characteristic illustrated in Fig. 3(b) is symmetrical with respect to voltage and current; in consequence, the power source exhibits several unique properties. Similar to the voltage source, the ideal power source must not be short-circuited; otherwise, infinite current occurs. And similar to the current source, the ideal power P 2 P 3 n 1 : n 2 P 2 P 3 Fig. 4. ircuit properties of the dependent power source: (a) series and parallel combinations, (b) reflection through ideal transformer.

3 source must not be open-circuited, to avoid infinite terminal voltage. The power source must be connected to a load capable of absorbing the power p(t), and the operating point is defined by the intersection of the load and power source ið v characteristics. As illustrated in Fig. 4(a), series- and parallel-connected power sources can be combined into a single power source, equal to the sum of the powers of the individual sources. Fig. 4(b) illustrates how reflection of a power source through a transformer, having an arbitrary turns ratio, leaves the power source unchanged. Power sources are also invariant to duality transformations. v g i p Ts Averaged switch network Averaged switch network p Ts Ts IV. SEVEA APPIATIONS AND EXAMPES A. urrent Programmed onverters Substantial physical insight into the properties of current programmed (PM) converters can be obtained by use of the average switch modeling approach. The buck converter of Fig. 5 is used here as a simple example. We can define the terminal voltages and currents of the switch network as shown. When the buck converter operates in the continuous conduction mode, the switch network average terminal waveforms are related as follows: = d(t) i 1 (t) = d(t) (1) We now invoke the approximation in which the inductor current exactly follows the programmed control current i c [14]. In terms of the switch network terminal current i 2, we can therefore write (2) The duty cycle d(t) can now be eliminated from Eq. (1), as follows: i 1 (t) = d(t) = v 2(t) This equation can be written in the alternative form i 1 (t) = = p(t) (4) Equations (2) and (4) are the desired result, which describes v g v g (t) i 1 (t) Switch network i 1 i 2 p Ts Ts v 2 Averaged switch network i (t) i (3) Ts Fig. 5. Averaged switch modeling of a currentprogrammed buck converter, continuous conduction mode. Top: original converter. Bottom: averaged switch model. v g i the average terminal relations of the M currentprogrammed buck switch network. Equation (2) states that the average terminal current is equal to the control current. Equation (4) states that the port of the switch network consumes average power p(t) equal to the average power flowing out of the switch output port. The averaged equivalent circuit of Fig. 5 is obtained. The PM averaged switch model derived above can be inserted into any M converter that can be written as in Fig. 1 and that has a PM switch network with a transistor and diode connected in a manner similar to the buck converter. Figure 6 illustrates the resulting models of the M PM boost and buck-boost converters. A small-signal ac model can be constructed by linearization of the models of Figs. 5 and 6 about a quiescent Ts Fig. 6. Averaged switch models of the current programmed boost (top) and buck-boost (bottom) converters, continuous conduction mode. i 1 i 2 v v V g i 2 V I 1 1 c v c 2 i v V c 2 v 1 I 1 Switch network small-signal ac model Fig. 7. Small-signal linearization of the buck converter model of Fig. y leads to this ac model. i 1 I 1 Power source characteristic v 1 i 1 = p Quiescent operating point v 1 Fig. 8. Origin of the port negative incremental resistance r 1 : the slope of the power sink characteristic, evaluated at the quiescent operating point. 1 r 1 = I 1

4 v g (t) Ts v g (t) Ts i 1 (t) Ts Ts p(t) Ts p(t) Ts Ts Ts Ts Fig. 9. Averaged switch models of the currentprogrammed buck-boost (top) and buck (bottom) converters, operating in discontinuous conduction mode. v g i 1 v 1 r 1 f 1 i c g 1 v 2 operating point. For the buck converter, the ac model of Fig. 7 is obtained. The switch network output port is again a current source, of value. The switch network port model is obtained by linearization of the power sink characteristic. The port current i 1 (t) is composed of three terms. The effect of ac variations in is modeled by an independent current source, the influence of variations in output voltage is modeled by a dependent current source, and the dependence of current variations on voltage is modeled by an effective ac resistor having the negative value Ð /I 1. As illustrated in Fig. 8, this incremental resistance is determined by the slope of the power sink port characteristic, evaluated at the quiescent operating point. The power sink leads to a negative incremental resistance because an increase in causes a decrease in i 1 (t), such that constant p(t) is maintained. When the current-programmed converter enters the discontinuous conduction mode (DM), the i c current source becomes a power source. The resulting buck-boostand buck converter models are illustrated in Fig. 9. In the PM DM, the energy stored in the inductor during each switching period is fixed at 0.5i c 2, and all of this energy is transferred to the load during each switching period. Operation at constant switching frequency leads to constant power transfer from source to load, independent of the and output voltages. inearization of switch model leads to the small-signal ac model of Fig. 10 [1,12]. g 2 v 1 f 2 i c r 2 i 2 v 2 Ts Fig. 10. Small-signal averaged switch modeling of current-programmed discontinuous conduction mode converters, buck converter example. i v v ac (t) i ac (t) ac e (v control ) v control B. Single-Phase ow-harmonic ectifiers: BIFED and BIBED In recent years there has been much interest in lowharmonic single-phase rectifiers to meet the ac harmonic current limits requirements of IE-555. Such a rectifier is typically using a converter controlled to emulate the ideal rectifier equivalent circuit of Fig. 11. The port obeys OhmÕs law so that the current is sinusoidal and in pahse with the voltage. The instantaneous power is transferred to the output port. A typical single-phase power supply system therefore contains a low-harmonic rectifier, a bulk energy storage capacitor, and a dc-dc converter. Several schemes have been proposed for reduction of the cost of these systems. Two examples are the BIFED and BIBED of Fig. 12 [16,17]. These are single converters, containing a single switch, that simultaneously perform all three functions of low-harmonic rectification, internal bulk energy storage, and wide-bandwidth regulation of the dc i Ideal rectifier (F) 1 Q 1 Q 1 e p p(t) = v ac 2 / e D 1 D : n n : D 1 : n 2 m D 2 i(t) Fig. 12. Single-phase converters that simultaneously perform the functions of low-harmonic rectification, internal energy storage, and wide-bandwidth dc regulation. Top: BIFED. Bottom: BIBED. D dc output Fig. 11. Equivalent circuit that describes the functional properties of an ideal rectifier: resistor emulation, output power source characteristic, with controllable power flow. 3 (1D) : 1 2 v v n : 1 Fig. 13. Averaged switch model of the BIFED.

5 i 1 (t) i 3 (t) e oad e p tot = p a p b p c oad e Input filter Input filter r r r r r r D 1 D 2 D 3 D 4 D 5 D 6 D 1 D 2 D 3 Q 1 D 4 D 5 D 6 output voltage. The converters are designed such that their inductors operate in DM in conjunction with diode D 1, while the outputs operate in M. Internal capacitor 1 performs the function of low-frequency energy storage. Analysis of these converters is somewhat complex, and it is quite difficult simulate these converters using a program such as PSPIE (several ac line cycles are required to reach steady-state, and PSPIE typically diverges first). The averaged switch model of Fig. 13 (BIFED example shown) correctly predicts the low-frequency components of the converter waveforms (provided that the inductor operates in DM while the output inductor operates in M). These circuits can be solved or simulated by computer, leading to understanding of the converter characteristics.. Three-Phase ow Harmonic ectifiers Three-phase rectifiers that meet the line current harmonic limits of IE-555 or IEEE/ANSI STD-519 are required in applications such as ac motor drives, telecommunications power supplies, and electric vehicle battery chargers. A variety of converter circuits are again available. In the sixswitch boost topology of Fig. 14 (top), the IGBTs are controlled such that resistor emulation is obtained. Figure 14 (center) contains a single-switch buck-type converter whose quasi-resonant current waveforms naturally follow the respective ac voltage waveforms [18]. A transformer-isolated version based on the quasiresonant forward converter is illustrated in Fig. 14 (bottom). This converter is a simple and inexpensive way to obtain low-harmonic rectification in three-phase power supply applications. Quasi-resonant and multi-resonant topologies Q 1 D 7 D 8 D 7 r r Fig. 14. Examples of three-phase low-harmonic rectifiers: (top) six-switch boost topology, (center) single-switch quasi-resonant buck topology, (bottom) single-switch quasi-resonant forward topology. Input filter r r r e e p tot = p a p b p c e based on most of the well-known dc-dc converter circuits can be derived [18,19]. The active silicon utilization of these converters is quite low. Averaged switch models of the converters of Fig. 14 are listed in Fig. 15. esistor emulation at the ac terminals of the switch network is modeled by effective resistors e. The total three-phase instantaneous power p tot apparently consumed by these resistors is delivered to the dc load via the dependent power source. ow-frequency harmonics are neglected by this model, but could be included by refinement of the equivalent circuit. These models are suitable for system simulation, including modeling of rectifier and output impedances, and output voltage regulators. V. ONUSIONS The averaged switch modeling approach is a powerful method for representing the behavior of a wide variety of converters through equivalent circuits. In conventional PWM M dc-dc converters, the basic conversion properties are represented by the dc transformer symbol. osses and dynamics are represented by inclusion of additional circuit elements. Operation of PWM switches in DM dc-dc converters is modeled by replacing the dc transformer with a network containing an effective resistor e and a dependent power source element. In a similar manner, current-programmed switches can be modeled using a current source and dependent power source (M) or two power sources (DM). These models correctly predict the observed dc terminal characteristics, as well as the observed dynamics. inearization of these models yields a small-signal ac model suitable for controller design. The dependent power source element appears repeatedly in the averaged switch models of converters. This characteristic must arise whenever a lossless network meets buffer conditions. The power source exhibits the unique properties that it is invariant to reflection through a transformer, and any combination of series and parallel Fig. 15. Averaged switch models of the three-phase lowharmonic rectifier circuits of Fig. 14. r

6 sources can be simplified to a single source equal to the total power. It is also useful to model low-harmonic rectifiers using the averaged switch approach. In contrast to the ideal dc transformer model of the M dc-dc converter, the basic functions of the ideal rectifier are modeled by an effective resistor and dependent power source. As an example, the BIFED is modeled here. This converter integrates the basic functions of low harmonic rectification, bulk energy storage, isolation, and wide-bandwidth control of the output voltage into a single converter containing a single active switch. In the averaged switch model, the function of low harmonic rectification is represented by an effective resistor and dependent power source, while the output votlage control is represented by dc transformers. This averaged model makes simulation of this converter a feasible and reasonable undertaking. Two examples of averaged switch modeling of threephase low harmonic rectifiers are also given. The basic function of ideal three-phase rectification is represented by three effective resistors, one per phase. The total power apparently consumed by these resistors is transferred to an output dependent power source. EFEENES [1]. W. EIKSON, Fundamentals of Power Electronics, New York: hapman and Hall, [2] G. W. WESTE and. D. MIDDEBOOK, Òow-Frequency haracterization of Switched Dc-Dc onverters,ó IEEE Transactions an Aerospace and Electronic Systems, Vol. AES-9, pp , May [3]. D. MIDDEBOOK and SOBODAN UK, ÒA General Unified Approach to Modeling Switching-onverter Power Stages,Ó International Journal of Electronics, Vol. 42, No. 6, pp , June [4] P. T. KEIN, J. BENTSMAN,. M. BASS, and B.. ESIEUTE, ÒOn the Use of Averaging for the Analysis of Power Electronic Systems,Ó IEEE Transactions on Power Electronics, Vol. 5, No. 2, pp , April [5] S. FEEAND and. D. MIDDEBOOK, ÒA Unified Analysis of onverters with esonant Switches,Ó IEEE Power Electronics Specialists onference, 1987 ecord, pp [6] V. VOPEIAN,. TYMESKI, and F.. EE, ÒEquivalent ircuit Models for esonant and PWM Switches,Ó IEEE Transactions on Power Electronics, Vol. 4, No. 2, pp , April [7] V. VOPEIAN, ÒSimplified Analysis of PWM onverters Using the Model of the PWM Switch: Parts I and II,Ó IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-26, pp , May [8] A. WITUSKI and. EIKSON, "Extension of State-Space Averaging to esonant Switches Ñand Beyond," IEEE Transactions on Power Electronics, Vol. 5, No. 1, pp , January [9] D. MAKSIMOVI and S. UK, ÒA Unified Analysis of PWM onverters in Discontinuous Modes,Ó IEEE Transactions on Power Electronics, Vol. 6, No. 3, pp , July [10] S. SINGE, Òealization of oss-free esistive Elements,Ó IEEE Transactions on ircuits and Systems, Vol. AS-36, No. 12, January [11] S. SINGE and. W. EIKSON, ÒPower-Source Element and Its Properties,Ó IEE ProceedingsÑircuits Devices and Systems, Vol. 141, No. 3, pp , June [12] S. SINGE and. W. EIKSON, Òanonical Modeling of Power Processing ircuits Based on the POPI oncept,ó IEEE Transactions on Power Electronics, Vol. 7, No. 1, January [13] G. VEGHESE,. BUZOS, and K. MAHABI, ÒAveraged and Sampled-Data Models for urrent Mode ontrol: A eexamination,ó IEEE Power Electronics Specialists onference, 1989 ecord, pp [14]. D. MIDDEBOOK, ÒModeling urrent Programmed Buck and Boost egulators,ó IEEE Transactions on Power Electronics, Vol. 4, No. 1, January 1989, pp [15]. EIKSON, M. MADIGAN, and S. SINGE, ÒDesign of a Simple High Power Factor ectifier Based on the Flyback onverter,ó IEEE Applied Power Electronics onference, 1990 ecord, pp [16] M. MADIGAN,. EIKSON, AND E. ISMAI, ÒIntegrated High Quality ecitifier-egulators,ó IEEE Power Electronics Specialists onference, 1992 ecord, pp , June [17] M. MADIGAN, ÒSingle-Phase Integrated High Quality ectifier-egulators,ó Ph.D. thesis, University of olorado, [18] E. H. ISMAI and. W. EIKSON, ÒA Single Transistor Three-Phase esonant Switch for High Quality ectification,ó IEEE Power Electronics Specialists onference, 1992 ecord, pp [19] Y. JANG AND. EIKSON, ÒDesign and Experimental esults of a 6kW Single-Switch Three-Phase High Power Factor ectifier Using Multi-esonant Zero urrent Switching,Ó IEEE Applied Power Electronics onference, 1996 ecord, pp