INTRODUCTION Simplified Analysis of PWM Converters Using Model of PWM Switch Part I: Continuous Conduction Mode VATCHEVORPERIAN Virginia Polytechnic Institute and State University A circuit-oriented approach to the analysis of pulsewidth modulation (PWM)converters is presented. This method relies on the identification of a three-terminal nonlinear device, called the PWM switch, which consists of only the active and passive switches in a PWM converter. Once the invariant properties of the PWM switch are determined, an average equivalent circuit model for it can be derived. This model is versatile enough that it can easily account for storage-time modulation of BJTs. The dc and small-signal characteristics of a large class of PWM converters can then be obtained by a simple substitution of the P W M switch with its equivalent circuit model. The methodology presented is very similar to linear amplifier circuit analysis .whereby the transistor is replaced by its equivalent circuit model. Consequently, for the novice, this method should serve as a very smooth introduction to the analysis of PWM converters. Manuscript received February 16, 1989; revised June 16, 1989. IEEE Log No. 33463A. Author’s address: Virginia Power Electronics Center, Dep’t. of Electrical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061. 0018-9251/90/0500-0490$1.00 @ 1990 IEEE 490 In this article the concept of the pulsewidth modulation (PWM) switch is introduced which leads to considerable simplification in the analysis (linear and nonlinear) and generation of dc-to-dc converters. To show the simplicity and elegance of the Pu.7M switch model the dc and small-signal analysis of basic dc-to-dc converters is given. Of course, the results of the dc and small-signal analysis of dc-to-dc PWM converters are well known today from the systematic method of state-space averaging and its canonical circuit model [l, 21. Hence, before going any further, a brief description of the two models mentioned above is given and the diflerence in the analytical approach between them is explained in order to justify the purpose of this article. First, the canonical circuit model completely represents the dc and small-signal characteristics of a PWM converter and is obtained after a considerable amount of matrix manipulations in order to single out the desirable input and output characteristics of the converter (input impedance, line-to-output transfer function, and output impedance) in addition to its control-to-output characteristics. The P W switch model, on the other hand, represents only the dc and small-signal characteristics of the nonlinear part of the converter, which consists of the active and passive switches (the PWM switch), and is obtained after a few lines of very simple algebra. The dc and small-signal characteristics of a PWM converter are then obtained by replacing the PWh4 switch with its equivalent circuit model in a manner similar to obtaining the small-signal characteristics of linear amplifiers whereby the transistor is replaced by its equivalent circuit model. There are two advantages in using the model of the PWM switch. First, the PWM switch model allows many PWM converters to be analyzed using simple linear electronic circuit analysis programs (P-SPICE, MICRO-CAP to name a few), which allow for user-defined models (macros), without recourse to special-purpose programs which manipulate state-space equations. The second and more important advantage is a pedagogical one. More and more universities are beginning to teach power electronics at the senior level in their undergraduate program. Although students at this level are familiar with some matrix algebra, it would be far easier for them to learn all about the dc and small-signal properties of PWM converters using the method of equivalent circuit model which they learned in their electronics courses earlier. Students spend quite a bit of time learning about the nonlinear characteristics of the transistor and its small-signal equivalent circuit model. They later use this model to analyze the small-signal characteristics of linear amplifiers simply by replacing the transistor with its equivalent circuit model. Why not d o the same in power electronics? Ultimately, whether or not the PWM switch model presented here is a useful IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 24, NO. 3 MAY 19N educational tool depends on how well it is received by the educators in the field of power electronics. The model of the PWM switch along with the model of the resonant switch were developed while work on the small-signal analysis of quasi-resonant With . a model converters was being conducted [MI of the PWM and resonant switches, the small-signal characteristics of quasi-resonant and PWM converters could be compared easily. The relationship of the PWM switch to the converter cell approach to the generation of converters has been discussed in [7], while a similar but more limited approach was given earlier in [ l l , 161. Subsequently, the nonlinear analysis of the PWM switch was performed [9, 101 which resulted in considerable simplification over previous attempts namely, the perturbation series method [SI, and the Volterra functional series method [14]. C PWM SWITCH A N D ITS INVARIANT PROPERTIES The four familiar PWM converters are shown in Fig. 1 where the active and passive switches are lumped together in a single functional block called the PWM switch. This functional block represents the total nonlinearity in these converters and is shown as a three-terminal nonlinear device in Fig. 2. For obvious reasons, the terminal designations a, p , c refer to active, passive, and common, respectively. The voltage and current port designations of this device are important if we are to think of the PWM switch as the basic building block of the converters in Fig. 1. It can be seen that in each converter the external circuit elements are connected to the switch in such a way as to provide the proper port conditions given in Fig. 2. Hence, we see that the converters of Fig. 1 are obtained by a simple permutation of the PWM switch. A similar cyclic permutation of the PWM switch to generate some basic converters was discussed in [ l l ] but its significance on the analysis of converters was never realized. Next we investigate the invariant relationships between the terminal currents and voltages of the PWM switch which are shown in Fig. 3. It can be easily seen that the instantaneous current in the active terminal is always the same as the current in the common terminal during the on-interval DT, no matter which configuration the switch is implemented in. Also, the instantaneous port voltages Gcp(t)and G a p ( t ) are always coincident during DT,. Hence, the invariant relations in the instantaneous terminal quantities are given by VORPmIAN: CONTINUOUS CONDUCTION MODE Fig. 1. Four basic PWM converters Fig. 2. PWM switch. Since in dc-to-dc converters the behavior of the average quantities is of greater interest in determining the dc and small-signal characteristics of these converters, we seek to determine the invariant relationships between the average terminal quantities. Hence, for the average terminal currents i, and i, we have from very simple considerations i, = di, (2) The instantaneous and average voltages across port a - p require special attention. Since this port is a voltage port, it is connected either across a voltage source or a capacitor which in general has an ESR. Hence, the voltage waveform across this port consists in general of a small square wave (with a tilted top) riding on top of a large average value as shown in Fig. 3. The source of the square wave is the ESR of the capacitor which absorbs a pulsating current of peak-to-peak amplitude equal to the maximum value of the current in the common terminal. Hence, if the ESR of this capacitor is equal to zero, then the instantaneous voltage Gap(t)is continuous and consists only of the capacitive ripple (which in the averaging process is neglected). Furthermore, this capacitor may be connected directly across terminals a - p as in the case of the Cuk, boost, and buck with input filter converters, or indirectly as in the case of the 491 7; + i -- I DT,+ * L- ’ n- DT$-u I 7; -4 U, , I I8 I 8 1 1 I FDT,? (1 , I1 ,1 vv- 3 5- - - - tGP(O 0 , I I , I I I I ‘ 0’7; A, , I ’ I Fig. 4. Terminal voltage C a P ( t ) in presence of ESR of capacitor which absorbs pulsating current in converter. DD‘rC I1 Fig. 3. Relation between terminal voItages and currents of PWM switch. buck-boost converter. If we neglect the ripple in the current and consider Only average value i,, then the peak-to-peak ripple voltage due to the ESR shown in Fig. 4 can be expressed as v, = icre (3) . . where re is in general a function of the ESR of the capacitor and the load resistor R. For example, in the boost and buck-boost converters the pulsating current of amplitude i, is absorbed by the outpucfilter capacitor which is in parallel with the output load resistor so that re is given by re = rc,IIR, boost, buck-boost. (4a) On the other hand, for the Cuk converter, we can see that re = rc,, Cuk (4b) + because the peak-to-peak pulsating current (i, = ii, io) is absorbed only by the energy transfer capacitor and the square ripple is only due to its ESR. Hence, depending upon the converter, one can figure out easily re. Referring to Fig. 4 we can now easily see that the relation between the average port voltages is given bY v,, = d(v,, - &red’), d’ = 1 - d . (5) The desired invariant relations of the PWM switch are given by (1) and (5) and are summarized below i, = di, v,, = d(v,, - icred’) of the PWM switch are perturbed because of some perturbation in either thk input voltage or the load of the converter. Perturbing (6a)and (6b) for a fixed duty ratio, we get basically the same thing 2, (6b) (7a) 2, (8b) ,D, + 1,; = D(P,, + I c r e i - icreD’)+ d(V,, = D;, - ZcreD’) (9a) which can be rearranged as Yap ci - + iCreD’- [V,, + 1,(D - D’)re]-. D Qc, = (9b) These equations correspond to the dc and small-signal model of the PWM switch shown in Fig. 6. Notice that if we neglect re, the model simplifies to the model shown in Fig. 7, which can also be obtained by perturbing (7a) and (7b).Equations (sa) and (9b) are summarized as follows: + Zd, = 0% ,D, DC AND SMALL-SIGNAL MODEL OF PWM SWITCH 492 Pa) These equations correspond to the model of the PWM switch for a fixed duty ratio as shown in Fig. 5 which is also valid all the way down to dc. Suppose we wanted to compute the open-loop line-to-output transfer function of a converter. The model to use then would be the one shown in Fig. 5 as is explained in the next section. If on the other hand, we would like to determine the response of the converter to perturbations in the duty ratio; we perturb (6a) and (6b) as follows v,, = dv,,. Let us first assume that the duty ratio is fixed at d = D and that the terminal currents and voltages = Di, 9,, = D(D,, - ;,reD’). A (6a) If the ESR of the capacitor, which absorbs the pulsating current, can be neglected then (6) reduce to i, = di, Fig. 5. Equivalent average circuit model of P W M switch for fixed duty ratio D . where = D 2 + ;,re D’ - VD D (lob) + ( W V, = Vap Zc(D- D’)re. IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 26, NO. 3 MAY 1990 ~ Fig. 6. Equivalent dc and small-signal model of PWM switch. Fig. 10. Boost converter to be analyzed for line-to-output and input impedance functions using model of PWM switch shown in Fig. 5. Open-Loop Line-to-Ouqut Danger Function: 6 P Fig. 7. Simplified dc and small-signal model of PWM switch. Under open-loop considerations the model in Fig. 5 still applies. Hence, to determine the line-to-output transfer the dc input voltage is replaced with a signal source 9, as shown in Fig. 10 whence we can immediately determine where Fig. 8. Boost converter. s,1 1 =‘Cf = Q Cf rLf + r,D‘ WO 1 , Fig. 9. Boost converter to be analyzed for dc characteristics using model of PWM switch shown in Fig. 5. ANALYSIS OF PWM CONVERTERS USING MODEL OF PWM SWITCH We now show how the model of the PWM switch is used to determine the dc as well as all the small-signal characteristics of PWM converters. The boost converter with all the parasitics shown in Fig. 8 is used as an example. DC [email protected]: Point-by-point substitution of the dc model of the PWM switch (Fig. 5 ) in the boost converter gives the circuit in Fig. 9 from which we can immediately determine the dc conversion ratio where re = rCf IIR as discussed earlier in (4a). Open-Loop Input Impedance: From Fig. 10 we can continue to determine the input impedance Zi, = R;, 1+ S/WOQ + s’/w,” 1 s/sp + where R;, = rLf i-r,DD’ + D”R Pa) The denominator is (of course) the same as in (13) and and Q are given by (14b) and (14c). Open-Loop Output Impedance: In this case we short the input to ground and connect a test voltage source at the output as shown in Fig. 11 and obtain WO where re = r C f IIR as explained in (4a). The two other dc quantities which determine the operating point of the switch are I, and Vap.These are easily determined from Fig. 9 where Hence VDcan be calculated from (1Oc). VORPkRIAN: CONTINUOUS CONDUCTION MODE 493 ~ ;*i.. - Fig. 11. Boost converter to be analyzed for output impedance function using model of PWM switch shown in Fig. 5. The quadratic in the denominator is the same as in (13) and szl is the same as in (14a). Control-to-&put Danger Function: In this case the input voltage source V, is shorted to ground and the P W M switch is replaced by its equivalent circuit model of Fig. 6 as shown in Fig. 12. Although one can directly determine the control-to-output transfer function in a straightforward manner from Fig. 12, we can take some short cuts and use some tricks of network analysis [15]. Hence, we expect the denominator of the control-to-output transfer function to be the same as that of the line-to-output transfer function as given in (13) so that we have where W O and Q are given by (14b) and (14c). The low frequency asymptote K d is simply given by as can be easily verified from (11). All that remains to be determined is the numerator in (19) which corresponds to the zeros or the nulls of the output voltage. To determine these zeros we simply consider the circuit of Fig. 12 under null conditions of the output, i.e., v0(sz2) = 0 and vo(-sZ1) = 0. The transform circuit under null conditions of the output is shown in Fig. 13. The first null is determined by the zero of the impedance of the output filter which is simply given by The second null in the output voltage is given by the null in the passive terminal current, i.e., i,(s,z) = 0. ForAthesecond null we see that the control generator VDd/D appears across the transformer whose secondary voltage appears across r ~ , r,DD' , and L f . Hence we have 1 Fig. 12. Boost converter to be analyzed for control-to-output transfer function using model of PWM switch shown in Fig. 6 . Fig. 13. Determination of zeros, or nulls, of control-toautput transfer function. Solving (22) and (23) we get Finally, use of the operating point, (L'D,&), in (1%) and (12b) in (24) gives determined Hence, the control-to-output transfer function has been completely determined. MODEL OF PWM SWITCH INCLUDING STORAGE-TIME M0DULATlON The model of the PWM switch is versatile enough to include parasitic effects such as storage-time modulation which Middlebrook has described in [12]. The effect of storage-time modulation in BJTs can be accounted for by a simple change in the perturbation of the duty ratio as follows where I,, is a modulation parameter which depends on the type of base drive (proportional, direct, etc.) as discussed in [13]. Substitution of the above in (1Oa)-(1Oc) gives (27) 2VD -D D = ic(rLf + r,DD' + s z 2 L f ) (22) Yap = - where re = rc, IIR as discussed earlier in (4a). Also, at the active terminal junction we have zc = I,d + i,D. * 494 * (23) D where VD r,,, = modulation resistance. (29) Zm e IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 26, NO. 3 MAY 1990 ’1 D a - + f l C Dr, C i + D’r, rd P Fig. 14. Equivalent circuit model of PWM switch including effects of storage-time modulation and resistances of diode and transistor branches. Under d c conditions, nondissipative modulation resistance rm disappears. Fig. 15. PWM switch showing resistances of transistor and diode branches. control-to-output transfer function was not observed, it was immediately suspected that the culprit was the numerator in the line-to-output transfer function. The According to the approximations explained in [12 converter was first modeled including all the parasitic and 131, D >> &/Ime so that (27) reduces to its elements of the passive components assuming ideal original form. With this approximation, the above switches, but no anomalous behavior in the phase equations correspond to a simple modification in the response was seen. Finally, when the modulation model of the P W M switch shown in Fig. 14 where resistance and the resistances of the diode and the a nondissipative modulation resistance r, has been transistor branches were included, a new zero was added in series with the common terminal. discovered which could move anywhere from the Other parasitic elements such as the lead left-half plane to the right-half plane depending resistances in the active and passive switches shown upon the values of rm (which can be made positive in Fig. 15, can be easily included in the switch model without much difficulty. Intuitively, it can be seen from or negative), rd, and r,. In order to appreciate the model of the P W M switch, one should go through Fig. 15 that the common terminal spends D-percent of the method of state-space averaging and its canonical its time in series with r, and D’-percent of its time in circuit model employed in [13] and compare it with series with rd so that the total effective resistance in the method given below using the model of the P W M the common terminal must be Dr, + D’rd as shown switch which yields the same desired results almost in Fig. 14. This can be verified formally if we write by inspection. Thus, the model of the PWM switch in the correct relationship between the average terminal Fig. 14, without the control sources, is substituted in voltages (as we did in (5)) in the presence of r, and rd the Cuk converter as shown in Fig. 16 to determine as vCp= d(vap- icred’ - ecrt)- d’icrd. (30) the open-loop line-to-output transfer function. Since we are interested here only in the numerator, we If this equation is perturbed, our intuitive derivation is study this circuit in the transform domain under null confirmed with a minor adjustment in VD conditions of the output. The first null is clearly given VD = V a p + (D - D’)lcrc zc(Td - r,). (31) by the null of the impedance of the output ffilter as in (21). The second null is given by the null in the Almost always one can use the approximation VD N s )0. For this second null we inductor current ? ~ ~ ( = Vapas can be seen from the above. The equations can immediately see from Fig. 16 that corresponding to the model of the PWM switch in Fig. 14 including all the parasitic effects, are summarized below + ia 21 D;, + Ice? (3h) which can be immediately solved for s to give the second zero 1 (34) where Recalling r, from (32c) and re = rc, from (4b) we get from (34) nn r UY s22 = As a final example we consider the line-to-output transfer function of the Cuk converter including all the parasitic elements as discussed in [13]. Researchers in the early days of this converter were puzzled with the peculiar behavior in the phase response of the line-to-output transfer function which at higher frequencies showed a variation from -90 to 90 degrees. Since a similar behavior in the VORP6RIAN: CONTINUOUS CONDUCTION MODE Cc(Dr, + D‘rd 4- r,) (35) where - VD - VQP = (VO+%) - - Vo (36) D I me where we have made use of the fact that for the Cuk converter Vo/V, = D/D’. Hence, the numerator of the rm Zme Ime I me 495 ~ Fig. 16. Cuk converter, including all parasitic elements, to be analyzed for audio. Null condition corresponding to second zero of audio transfer function is shown. line-toautput transfer function can now be written as where s1, and sZ2 are determined above in (21) and (35), respectively, and D ( s ) is a fourth-order polynomial. It is this second zero s,2 which is entirely dependent on the parasitic resistances of the switches that causes variations in the high end of the phase response. It has been shown in [13] that rm can be easily made sufficiently negative that the zero can migrate from the right-half plane to the left-half plane resulting in variations from -90 to 90 degrees at the high end of the phase response. CONCLUSIONS A new, simple, and circuit-oriented method of analysis of PWM converters, which uses the model of the PWM switch developed in this article, is presented. In this method, the model of the PWM switch is used in the same way as the model of the transistor is used in the analysis of electronic amplifier circuits. The PWM switch is presented as a three-terminal nonlinear device which represents the total nonlinearity in a PWM converter just as the transistor represents the total nonlinearity in an electronic amplifier. Hence, as one does not linearize the entire equations of an amplifier along with the Ebers-Moll equations of the transistor to determine its dc and small-signal characteristics, one does not need to linearize the entire equations of a PWM converter as is done in the case of state-space averaging, circuit averaging or hybrid modeling. Finally, with a model of the PWh4 switch PWM converters can be easily analyzed for dc and small-signal characteristics on standard electronic circuit analysis programs in closed-loop operation without the need for special-purpose programs. REFERENCES [l] 4% Cuk, S. (1976) Modelling, analysis and design of switching converters. Ph.D. dissertation, California Institute of Technology, Pasadena, Nov. 1976. Middlebmk, R. D., and Cuk, S. (1976) A general unified approach to modelling switching-converter power stages. In IEEE Power Electronics Specialists Conference Proceedings, 1976. VorptSrian, V. (1986) Average mcdelof the single-pole double throw switch in PWM converters. Internal correspondence, Feb. 11, 1986. VorptSrian, V., 'Qmerski, R., Liu, K., and Lee, E C. (1986) Generalized cesonant switches: part I. In 1986 W E C Conference Proceedings. [email protected], V., 'Qmerski, R., Liu, K., and Lee, E C. (1986) Generalized resonant switches: part 11. In 1986 W E C Conference Proceedings. VorptSrian, V. (1986) Equivalent circuit models for PWM and resonant switches. In Proceedings of 1986 Conference Internatwnal Symposium or Circuits and S y s t m , 3. qmerski, R., and Vorperian, V. (1986) Generation, classification and analysis of switched-mode dc-todc converters by the useof converter cells. INTELEC Conference Proceedings, 1986. Erickson, R. W. (1982) Large signals in switching converters. Ph.D. dissertation, California Institute of Technology, Pasadena, Nov. 1982. 'Qmerski, R., Vorperian, V., Baumann, W., and Lee, E C. Nonlinear modeling of the PWM switch. Accepted for publication, to appear in the IEEE Journal of Power Electronics. 'Qmerski, R. (1988) Topology and analysis in power conversion and inversion circuits. Ph.D. dissertation, Virginia Polytechnic Institute and State University, Blacksburg, VA, May 1988. Cuk, S. (1977) A new optimum topology switching dc-todc converter. In IEEE Power Electronics Specialists Conference Proceedings, 1977. Middlebmk, R. D. (1975) A continuous model of the tapped boost converter. In IEEE Power Electronics Specialists Conference Proceedings, 1975. Polivka, W., Chetty, P., and Middlebrook, R. (1980) State-space average modelling of converters with parasitics and storage-time modulation. In IEEE Power Electronics Specklists Conference Proceedings, 1980. 'Qmerski, R. (1985) The large signal forced response of switching amplifiers using the Volterra functional series expansion. Technical note T173, Power Electronics Group, Caltech, Pasadena, CA, Apr. 1985. VorptSrian, V. Circuitaiented analysis of PWM converters using the model of PWM switch. These are examples worked out in detail using many tricks of network analysis. Available from the author upon request. Landsman, E. E. (1979) A unifying derivation of switching dcdc converter topologies. In IEEE Power Electronics Specialists Conference Proceedings, 1979,239-243. lor's photograph and biography will be found on 505 of this issue following Part I1 of this article. IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 26, NO. 3 MAY 19W

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