INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, VOL. 25, 125?134 (1997) VOLTERRA SERIES ANALYSIS OF SPECTRAL COMPONENTS IN QUASI-RESONANT d.c.?d.c. CONVERTERS MAHINDA VILATHGAMUWA AND DENG JUNHONG School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798 SUMMARY The Volterra series method is used to predict the spectral components of a d.c.?d.c. quasi-resonant converter output wave-form. The Volterra kernels computed for quasi-resonant converters are based on their small-signal non-linear model. Illustrative analytic and simulation results are presented for several quasi-resonant converter topologies and are shown to match very closely. ? 1997 by John Wiley & Sons, Ltd. Int. J. Cir. Theor. Appl., vol. 25, 125?134 (1997) (No. of Figures: 2; No. of Tables: 2; No. of Refs: 6.) 1. INTRODUCTION Usually a quasi-resonant converter (QRC) circuit-averaged model exhibits non-linear characteristics. Therefore local stability conditions and transfer functions are obtained by perturbing the state space model around a steady state operating point and neglecting resulting higher-order components1 . However, this approximated smallsignal model falls short of providing the necessary information on output wave-form distortions. It can be seen that the inherent non-linear transfer characteristic of a QRC introduces distortions in the output wave-form which are larger than those introduced by switching actions. Therefore our motivation in this project is to determine these distortions accurately in the output wave-form of a QRC for given input conditions. We found that the classical Volterra series modelling approach will serve our purpose satisfactorily provided that signal levels are kept small. Once the Volterra kernels have been determined, the converter spectral characteristics containing higher-order harmonics, intermodulation and cross-modulation components can be identied. Distortion analysis in PWM-type d.c.?d.c. converters using the Volterra series method has been reported recently by Tymerski2 and Chan and Chau3 . The Volterra analysis of such circuits has been a manageable task because of the relative ease with which small-signal models are derived. However, for a QRC, the derivation of the small-signal model is exceedingly complex, let alone its Volterra analysis, owing to the dependence of its average control input () on other circuit variables. Tymerski2 has analysed only single-input d.c.?d.c. PWM converters. The eect of input voltage distortion has been neglected and the Volterra kernels obtained are found to be much simpler than those for multi-input systems. However, Chan and Chau3 have taken both input voltage and duty ratio distortions of d.c.?d.c. PWM converters into consideration in the process of developing Volterra kernels. The elaborate unied small-signal QRC model developed in this paper results in more complex but accurate expressions for Volterra kernels, especially in cross-modulations. We concentrate on multi-input systems, which should eventually help us to examine the output distortion due to both supply voltage and duty ratio disturbances. Moreover, we found that the contributions of the kernels diminish greatly with increasing order, so only up to third-order kernels are considered. The perturbed non-linear state space model of the quasi-resonant converter is analysed for multi-tone inputs. Usually this method is called the harmonic probing method, whereby higher-order Volterra kernels are determined by substituting the Volterra series expansion of the output and state variables with unknown coecients into the system model. As there exist several asymmetrical kernels for a given higher-order output, the symmetrical kernel is obtained by averaging the possible number of asymmetrical kernels. CCC 0098?9886/97/020125?10$17.50 ? 1997 by John Wiley & Sons, Ltd. Received 7 November 1995 Revised 10 May 1996 126 M. VILATHGAMUWA AND J. DENG We shall begin with the formulation of a unied small-signal model for both full-wave and half-wave quasi-resonant converters. Then the Volterra system equations for multi-input non-linear systems are derived and these equations are subsequently used to determine the higher-order Volterra kernels of QRCs. Finally, verication of this methodology for various quasi-resonant topologies is presented. 2. FORMULATION OF UNIFIED SMALL-SIGNAL MODEL OF QRCs Eorts made in the development of equivalent circuits and models for quasi-resonant converters for smallsignal a.c. analysis have been reported recently. The state space average modelling of PWM converters has been made easier by the justiable linear ripple approximation. However, the modelling of d.c.?d.c. quasiresonant converters has become complicated owing to the presence of high-frequency resonant elements. This obstacle can be overcome by circuit averaging in which the average value of the switch wave-form is derived from the average value of the lter network4 . This method provides a system model that is similar to the PWM state space average model and which is subsequently exploited to generate a unied small-signal model for QRCs in this section. The averaged state space equation for a quasi-resonant power converter operating in the continuous conduction mode can be expressed as4 , dx = [C1 + (1 ? )C2 ]x + [D1 + (1 ? )D2 ]vin dt y = Ex (1) (2) Perturbations in and vin cause perturbations in x and y. Thus = + ? (3) vin = vin + v?in (4) x = x + x? (5) y = y + y? (6) The state space equation describing its dynamical behaviour is d x? = k1 x? + k2 ? + k3 v?in + k4 x?? + k5 ?v?in dt y? = k6 x? (7) (8) where 1 + (1 ? )C 2; k1 = C k2 = (C1 ? C2 )x + (D1 ? D2 )vin ; k4 = C1 ? C2 ; k5 = D1 ? D2 ; k3 = D 1 + (1 ? )D 2; k6 = E The model given in equations (7) and (8) is similar to that of a PWM converter. However, the task of determining ? is normally tedious for quasi-resonant converters. Usually is given by1 , 1 ? FP for a zero-voltage-switching QRC = FP for a zero-current-switching QRC with F = fs =fn (9) where fs is the switching frequency and fn is the resonant frequency, and i p 1h 1 ?1 n n 2 1 ? (?1) (1 ? ) n ? (?1) sin + + P= 2 2 (10) where n = 1 for a half-wave QRC, n = 2 for a full-wave QRC and is a function of the characteristic impedance of the resonant circuit and the transconductance of the amplier. For a half-wave QRC, P is ? 1997 by John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl., Vol. 25, 125?134 (1997) VOLTERRA SERIES ANALYSIS OF SPECTRAL COMPONENTS 127 highly dependent on the switch parameter , but for a full-wave QRC the value of P is almost independent and can be approximated as unity. The switch average parameter can be expressed as a function of x; F and vin : @ @ @ @ @ F? + x? + v?in @F @ @x @ @vin ? = ? = G1 F? + G2 x? + G3 v?in where G1 = @ ; @F G2 = (11) @ @ ; @ @x G3 = @ @ @ @vin Note that @=@ ? 0 for a full-wave QRC. Substituting (11) into (7) and (8) yields d x? = N1 x? + N2 F? + N3 v?in + N4 x?F? + N5 F? v?in + N6 x?N7 x? + N8 x?v?in + N9 v?2in dt y? = N10 x? where (12) (13) N1 = k1 + k2 G2 ; N2 = k2 G1 ; N3 = k2 G3 + k3 ; N4 = k4 G1 ; N5 = k5 G1 ; N6 = k4 ; N8 = k4 G3 + k5 G2 ; N9 = k5 G3 ; N10 = k6 = E N7 = G2 ; Therefore equations (12) and (13) represent the unied small-signal non-linear model of the QRC. 3. VOLTERRA SERIES ANALYSIS OF NON-LINEAR SYSTEMS Volterra series are a generalization of power series and ideal for representing frequency-dependent small nonlinearities. Usually the rst three terms will be sucient to characterize the QRC used in this application. The response y(t) to an input u(t) of a QRC can be expressed by the Volterra series y(t) = ? P n=1 Z yn (t) = yn (t) (14) Z ? ?? иии ? ?? hn (1 ; : : : ; n ) n Q u(t ? r )dr (15) r=1 where hn (1 ; : : : ; n ) is the system non-linear impulse response of order n. The Laplace transform of hn (1 ; : : : ; n ), Z ? Z ? n Q Hn (s1 ; и и и ; sn ) = иии hn (1 ; : : : ; n ) e?jsr r dr (16) ?? r=1 ?? is the system non-linear transfer function of order n. Thus the use of Volterra series is a generalization of linear system theory. Alternatively the output yn (t) can be written as Z ? Z ? n Q иии Hn (s1 ; : : : ; sn ) u(sr )e jsr r dsr (17) yn (t) = ?? ?? r=1 The representation of the output by its Volterra expansion, equation (17), requires the determination of the non-linear transfer functions (or nth-order Volterra kernels) Hn (s1 ; : : : ; sn ). However, for a system with two ? 1997 by John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl., Vol. 25, 125?134 (1997) 128 M. VILATHGAMUWA AND J. DENG independent inputs a(t) and b(t) the output y(t) can be expressed as y= ? P n=1 Z y1 (t) = ? ?? Z y2 (t) = yn (t) ? ha1 ()a(t ? )d + Z ?? Z (18) ? ?? Z ? ?? ? Z ? ?? ? + Z y3 (t) = ?? Z ? ?? Z ?? ? ?? ? Z Z ? Z ? ?? ? + Z ?? ? Z ?? ? + ?? ?? hb1 ()b(t ? )d hab 2 (1 ; 2 ) a(t ? 1 )b(t ? 2 )d1 d2 hbb 2 (1 ; 2 ) b(t ? 1 )b(t ? 2 )d1 d2 haaa 3 (1 ; 2 ; 3 ) a(t ? 1 )a(t ? 2 )a(t ? 3 )d1 d2 d3 + ?? ? haa 2 (1 ; 2 ) a(t ? 1 )a(t ? 2 )d1 d2 + Z Z ?? haab 3 (1 ; 2 ; 3 ) a(t ? 1 )a(t ? 2 )b(t ? 3 )d1 d2 d3 habb 3 (1 ; 2 ; 3 ) a(t ? 1 )b(t ? 2 )b(t ? 3 )d1 d2 d3 hbbb 3 (1 ; 2 ; 3 ) b(t ? 1 )b(t ? 2 )b(t ? 3 )d1 d2 d3 ииииии The corresponding transfer functions are given by Z ? H1 (s) = h1 ()e?js d H2 (s1 ; s2 ) = H3 (s1 ; s2 ; s3 ) Z Z ?? ? ?? ? Z Z ? ?? ? ?j(s1 1 +s2 2 ) h d1 d2 2 (1 ; 2 ) e Z ? = ?? ?? ?? ?j(s1 1 +s2 2 +s3 3 ) h d1 d2 d3 3 (1 ; 2 ; 3 ) e ииииии It is important to note that the rth-order kernel hr (1 ; : : : ; r ) may have dierent values depending on the arrangement of 1 ; : : : ; r . In this case there are r! combinations of dierent asymmetrical kernels. Thus, by adding together all kernels for hr corresponding to all ways of rearranging the r s and dividing the sum by r!, we obtain the symmetrical kernel and its associated symmetrical transform: 1P hr (1 ; : : : ; r ) hr (1 ; : : : ; r ) = r! (и) 1P Hr (s1 ; : : : ; sr ) H r (s1 ; : : : ; sr ) = r! (и) (19) (20) where (и) represents all possible rearrangements of the s. ? 1997 by John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl., Vol. 25, 125?134 (1997) 129 VOLTERRA SERIES ANALYSIS OF SPECTRAL COMPONENTS 4. DETERMINATION OF VOLTERRA KERNELS (NON-LINEAR TRANSFER FUNCTIONS) OF QRCs A standard method of determining the Volterra kernels of a given non-linear system is the harmonic input probing method. It assumes the input to be given by sums of exponentials. Consider the two inputs, F? = a(t) = esa1 t + esa2 t + esa3 t + и и и (21) v?in = b(t) = esb1 t + esb2 t + esb3 t + и и и (22) These result in the state vector, ? ? ? P ? P P P x? = M1a (san )esan t + M1b (sbn )esbn t + M2aa (san1 ; san2 )e(san1 +san2 )t n=1 + n=1 ? P ? P n1 =1 n2 =1 M2bb (sbn1 ; sbn2 )e(sbn1 +sbn2 )t n1 =1 n2 =1 + ? P ? P n1 =1 n2 =1 M2ab (san1 ; sbn2 )e (san1 +sbn2 )t + M2ba (sbn1 ; san2 )e(sbn1 +san2 )t ? P ? P ? P + M3aaa (san1 ; san2 ; san3 )e(san1 +san2 +san3 )t n1 =1 n2 =1 n3 =1 + ? P ? P ? P M3bbb (sbn1 ; sbn2 ; sbn3 )e(sbn1 +sbn2 +sbn3 )t n1 =1 n2 =1 n3 =1 + ? P ? P ? P n1 =1 n2 =1 n3 =1 M3aab (san1 ; san2 ; sbn3 )e(san1 +san2 +sbn3 )t +M3aba (san1 ; sbn2 ; san3 )e(san1 +sbn2 +san3 )t + M3baa (sbn1 ; san2 ; san3 )e(sbn1 +san2 +san3 )t + ? P ? P ? P n1 =1 n2 =1 n3 =1 M3abb (san1 ; sbn2 ; sbn3 )e(san1 +sbn2 +sbn3 )t +M3bab (sbn1 ; san2 ; sbn3 )e(sbn1 +san2 +sbn3 )t +M3bba (sbn1 ; sbn2 ; san3 )e(sbn1 +sbn2 +san3 )t + и и и y? = N10 x? M1a (san ); (23) (24) M1b (sbn ); M2a a(san1 ; san2 ); The coecients etc. can be determined by substituting (21)?(23) into (12). These coecients are given in the Appendix. Then, assuming that the output is a linear relationship of x? as in (13), the following asymmetric self-kernels can be derived: H1a (s) = N10 M1a (s) (25) H1b (s) = (26) H2aa (s1 ; s2 ) H2bb (s1 ; s2 ) aaa H3 (s1 ; s2 ; s3 ) H3bbb (s1 ; s2 ; s3 ) = = = = N10 M1b (s) N10 M2aa (s1 ; s2 ) N10 M2bb (s1 ; s2 ) N10 M3aaa (s1 ; s2 ; s3 ) N10 M3bbb (s1 ; s2 ; s3 ) (27) (28) (29) (30) The symmetric forms can be obtained using equation (20); ? 1997 by John Wiley & Sons, Ltd. a H 1 (s) = H1a (s) (31) b H 1 (s) (32) = H1b (s) Int. J. Circ. Theor. Appl., Vol. 25, 125?134 (1997) 130 M. VILATHGAMUWA AND J. DENG 1 aa H 2 (s1 ; s2 ) = [H2aa (s1 ; s2 ) + H2aa (s2 ; s1 )] 2! 1 bb H 2 (s1 ; s2 ) = [H2bb (s1 ; s2 ) + H2bb (s2 ; s1 )] 2! 1 aaa H 3 (s1 ; s2 ; s3 ) = [H3aaa (s1 ; s2 ; s3 ) + H3aaa (s1 ; s3 ; s2 ) 3! +H3aaa (s2 ; s1 ; s3 ) + H3aaa (s2 ; s3 ; s1 ) (33) (34) +H3aaa (s3 ; s1 ; s2 ) + H3aaa (s3 ; s2 ; s1 )] 1 bbb H 3 (s1 ; s2 ; s3 ) = [H3bbb (s1 ; s2 ; s3 ) + H3bbb (s1 ; s3 ; s2 ) 3! +H3bbb (s2 ; s1 ; s3 ) + H3bbb (s2 ; s3 ; s1 ) (35) +H3bbb (s3 ; s1 ; s2 ) + H3bbb (s3 ; s2 ; s1 )] (36) Similarly the asymmetric forms of cross-kernels can be given as H2ab (s1 ; s2 ) = N10 M2ab (s1 ; s2 ) H2ba (s1 ; s2 ) H3aab (s1 ; s2 ; s3 ) H3aba (s1 ; s2 ; s3 ) H3baa (s1 ; s2 ; s3 ) H3abb (s1 ; s2 ; s3 ) H3bab (s1 ; s2 ; s3 ) H3bba (s1 ; s2 ; s3 ) = = = = = = = (37) N10 M2ba (s1 ; s2 ) N10 M3aab (s1 ; s2 ; s3 ) N10 M3aba (s1 ; s2 ; s3 ) N10 M3baa (s1 ; s2 ; s3 ) N10 M3abb (s1 ; s2 ; s3 ) N10 M3bab (s1 ; s2 ; s3 ) N10 M3bba (s1 ; s2 ; s3 ) (38) (39) (40) (41) (42) (43) (44) The symmetric forms are 1 ab H 2 (s1 ; s2 ) = [H2ab (s1 ; s2 ) + H2ba (s2 ; s1 )] 2! (45) 1 aab H 3 (s1 ; s2 ; s3 ) = [H3aab (s1 ; s2 ; s3 ) + H3aab (s2 ; s1 ; s3 ) 3! +H3aba (s1 ; s3 ; s2 ) + H3aba (s2 ; s3 ; s1 ) +H3baa (s3 ; s1 ; s2 ) + H3baa (s3 ; s2 ; s1 )] (46) 1 abb H 3 (s1 ; s2 ; s3 ) = [H3abb (s1 ; s2 ; s3 ) + H3abb (s1 ; s3 ; s2 ) 3! +H3bab (s3 ; s1 ; s2 ) + H3bab (s2 ; s1 ; s3 ) +H3bba (s3 ; s2 ; s1 ) + H3bba (s2 ; s3 ; s1 )] (47) 5. VERIFICATION Having characterized the non-linear system by a Volterra series, we can now determine the response of the system to any number of inputs. Owing to the non-linear nature of the system, the output response of the QRC contains numerous spectral components. Among other components, fundamentals, higher-order harmonics and intermodulations are predominant. A spectral component at a particular frequency is computed by adding the corresponding harmonic component and other contributing intermodulation and cross-modulation together. Although there exist a large number of intermodulation and cross-modulation components contributing to a certain frequency, only components with signicant contributions are taken into consideration. ? 1997 by John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl., Vol. 25, 125?134 (1997) 131 VOLTERRA SERIES ANALYSIS OF SPECTRAL COMPONENTS We consider two-tone inputs of the form F? = A1 cos(!a1 t) + A2 cos(!a2 t) (48) v?in = B1 cos(!b1 t) + B2 cos(!b2 t) (49) The spectral components of the output are determined using the Volterra kernels developed in Section 4. Alternatively the output behaviour of the QRC for these inputs can be analysed using the standard circuit simulation package PSPICE. Having obtained the steady state output wave-form of the QRC using PSPICE simulation, we then apply fast Fourier transform to get the harmonic spectrum. This spectrum provides us with the necessary information about harmonic components present in the output wave-form. 6. RESULTS As shown in Figure 1, the full-wave zero-current-switching quasi-resonant buck converter with Lr = 2и65 H; Cr = 0и106 F; L = 100 H; C = 10 F; R=5 is excited by the inputs F = 0и666 + 0и033 cos[2(2000)t] + 0и033 cos[2(3000)t] vin = 20 + cos[2(4000)t] + cos[2(5000)t] (50) (51) where F is the normalized switching frequency and vin is the input voltage. The actual switching frequency of the converter is 200 kHz. The converter system is modelled in PSPICE using ideal switching elements and linear magnetic elements. The steady state wave-form thus obtained is subjected to fast Fourier transform to evaluate the harmonic spectral components. The CPU time taken for this exercise is quite excessive (approximately 1?2h). In contrast, the CPU time required for Volterra kernel computations as given in Section 4 is insignicant. Therefore the Volterra kernel approach can be seen as a computationally ecient methodology in predicting the non-linear behaviour of the QRC in the frequency domain. Table I illustrates the harmonic, intermodulation and crossmodulation components in the output spectra evaluated using both software simulations and Volterra kernel computations. It can be seen that there is a close agreement between the two sets of results. The next example concerns the half-wave zero-voltage-switching boost converter shown in Figure 2. This circuit topology is specically chosen because of its complex small-signal circuit-averaged model. The coecients G2 and G3 in the small-signal model (equation (12)) are non-zero for the half-wave QRC while for the full-wave QRC they are nearly zero. Therefore additional non-linearities caused by the terms N6 x?N7 x? Figure 1. Zero-current-switching full-wave quasi-resonant buck converter ? 1997 by John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl., Vol. 25, 125?134 (1997) 132 M. VILATHGAMUWA AND J. DENG Table I Frequency (Hz) Harmonic amplitude obtained by Volterra series method Harmonic amplitude obtained by PSPICE simulation 1000 0и01622 0и01737 2000 0и77848 0и78730 3000 0и89128 0и89540 4000 5000 6000 7000 1и05353 1и04494 0и01865 0и02524 0и99310 0и95210 0и01547 0и02534 8000 0и00876 0и00904 9000 0и00018 0и00120 Major contributing component(s) !a1 ? !a2 !b1 ? !b2 !a2 ? !b1 !a1 !a1 ? !b1 !a2 ? !b2 !a2 !a1 ? !b2 !b1 !b2 !a1 + !b1 !a1 + !b2 !a2 + !b1 2!b1 !a2 + !b2 !b1 + !b2 Figure 2. Zero-voltage-switching half-wave quasi-resonant boost converter and N8 x?v?in need to be taken into account in determining the Volterra kernels for the half-wave QRC. The half-wave zero-voltage-switching QRC with the following circuit components is analysed in the frequency domain Lr = 2и65 H; Cr = 0и106 F; L = 200 H; C = 15 F; R=6 The circuit is excited by the inputs F = 0и666 + 0и033 cos[2(2000)t] + 0и033 cos[2(3000)t] vin = 20 + cos[2(4000)t] + cos[2(5000)t] (52) (53) where F is the normalized switching frequency and vin is the input voltage. The actual switching frequency of the converter is 200 kHz. As in the previous case, the PSPICE results obtained for the harmonic spectra are compared with the results from the Volterra kernel approach. The harmonic, intermodulation and cross-modulation components of the converter output for these two methods are illustrated in Table II. It can be seen that the results are in close agreement. ? 1997 by John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl., Vol. 25, 125?134 (1997) 133 VOLTERRA SERIES ANALYSIS OF SPECTRAL COMPONENTS Table II Frequency (Hz) Harmonic amplitude obtained by Volterra series method Harmonic amplitude obtained by PSPICE simulation 1000 0и02922 0и02390 2000 1и.37100 1и37670 3000 1и02900 1и04320 4000 0и41770 0и44160 5000 0и25470 0и26850 6000 0и00998 0и00990 7000 0и00418 0и00760 8000 0и00282 0и00370 9000 0и00282 0и00110 Major contributing component(s) !a1 ? !a2 !b1 ? !b2 !a2 ? !b1 !a1 !a1 ? !b1 !a2 ? !b2 !a2 !a1 ? !b2 !b1 2!a1 !b2 !a1 + !a2 2!a2 !a1 + !b1 !a1 + !b2 !a2 + !b1 2!b1 !a2 + !b2 !b1 + !b2 7. CONCLUSIONS In this paper, non-linear modelling of quasi-resonant converters using the Volterra series method is proposed. Distorted inputs of the switching frequency and supply voltage result in numerous harmonic, intermodulation and cross-modulation spectral components in the output of the QRC. These spectral components are predicted using Volterra kernels and are compared with results obtained by PSPICE simulation. The analytical and simulated results are shown to match very closely, thus verifying the Volterra series modelling approach. APPENDIX The coecients M1a (san ); M1b (sbn ); M2aa (san1 ; san2 ); etc. in equation (23) can be given as follows: M1a (s) = (sI ? N1 )?1 N2 M1b (s) M2aa (s1 ; s2 ) M2ba (s1 ; s2 ) M2ab (s1 ; s2 ) M2bb (s1 ; s2 ) M3aaa (s1 ; s2 ; s3 ) M3bbb (s1 ; s2 ; s3 ) M3aab (s1 ; s2 ; s3 ) ?1 = (sI ? N1 ) (54) N3 = [(s1 + s2 )I ? = [(s1 + s2 )I ? = [(s1 + s2 )I ? = [(s1 + s2 )I ? ?1 N1 ] N4 M1a (s1 ) N1 ]?1 N4 M1b (s1 ) ?1 N1 ] N8 M1a (s1 ) N1 ]?1 N8 M1b (s1 ) = [(s1 + s2 + s3 )I ? N1 ] ?1 + + + N6 M1a (s1 )N7 M1b (s2 ) N6 M1b (s1 )N7 M1b (s2 ) N4 M2aa (s1 ; s2 ) = + N6 M2aa (s2 ; s3 )N7 M1a (s1 ) [(s1 + s2 + s3 )I ? N1 ]?1 N8 M2bb (s1 ; s2 ) = + N6 M2bb (s2 ; s3 )N7 M1b (s1 ) [(s1 + s2 + s3 )I ? N1 ]?1 N8 M2aa (s1 ; s2 ) + N6 M2ab (s2 ; s3 )N7 M1a (s1 ) ? 1997 by John Wiley & Sons, Ltd. + (55) N6 M1a (s1 )N7 M1a (s2 ) N6 M1b (s1 )N7 M1a (s2 ) + + 1 2 N5 1 2 N5 + N9 + N6 M1a (s1 )N7 M2aa (s2 ; s3 ) + N6 M1b (s1 )N7 M2bb (s2 ; s3 ) + N6 M1a (s1 )N7 M2ab (s2 ; s3 ) (56) (57) (58) (59) (60) (61) (62) Int. J. Circ. Theor. Appl., Vol. 25, 125?134 (1997) 134 M. VILATHGAMUWA AND J. DENG M3aba (s1 ; s2 ; s3 ) = [(s1 + s2 + s3 )I ? N1 ]?1 N4 M2ab (s1 ; s2 ) + N6 M1a (s1 )N7 M2ba (s2 ; s3 ) + N6 M2ba (s2 ; s3 )N7 M1a (s1 ) M3baa (s1 ; s2 ; s3 ) = [(s1 + s2 + s3 )I ? N1 ]?1 N4 M2ba (s1 ; s2 ) + N6 M1b (s1 )N7 M2aa (s2 ; s3 ) + N6 M2aa (s2 ; s3 )N7 M1b (s1 ) M3abb (s1 ; s2 ; s3 ) = [(s1 + s2 + s3 )I ? N1 ]?1 N8 M2ab (s1 ; s2 ) + N6 M1a (s1 )N7 M2bb (s2 ; s3 ) + N6 M2bb (s2 ; s3 )N7 M1a (s1 ) M3bab (s1 ; s2 ; s3 ) = [(s1 + s2 + s3 )I ? N1 ]?1 N8 M2ba (s1 ; s2 ) + N6 M1b (s1 )N7 M2ab (s2 ; s3 ) + N6 M2ab (s2 ; s3 )N7 M1b (s1 ) M3bba (s1 ; s2 ; s3 ) = [(s1 + s2 + s3 )I ? N1 ]?1 N4 M2bb (s1 ; s2 ) + N6 M1b (s1 )N7 M2ba (s2 ; s3 ) + N6 M2ba (s2 ; s3 )N7 M1b (s1 ) (63) (64) (65) (66) (67) REFERENCES 1. M. Nakahara, T. Higashi, T. Ninomiya and K. Harada, ?Dynamic characteristics and stability analysis of resonant converter?, IEEE Power Electronic Specialist Conf. Rec., IEEE, New York, 1989, pp. 752?759. 2. R. Tymerski, ?Volterra series modelling of power conversion systems?, IEEE Trans. on Power Electron., PE-6, 712?718 (1991). 3. C. C. Chan and K. T. Chau, ?Spectral modeling of switched-mode power converters?, IEEE Trans. Ind. Electron., IE-41, 441?450 (1994). 4. A. F. Witulski and R. W. Erickson, ?Extension of state-space averaging to resonant switches and beyond?, IEEE Trans. Power Electron., PE-5, 98?109 (1990). 5. S. Narayanan ?Application of Volterra series to intermodulation distortion analysis of transistor feedback ampliers?, IEEE Trans. Circuit Theory, CT-17, 518?527 (1970). 6. H. M. Salgado and J. J. O?Reilly, ?Volterra series analysis of distortion in semiconductor laser diodes?, Proc. IEE , 138, Pt. J, 379?382 (1991). ? 1997 by John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl., Vol. 25, 125?134 (1997)

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