Analysis of Resonant Converters Citation Vorpérian, Vatché (1984) Analysis of Resonant Converters. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/NWMH-AP27. https://resolver.caltech.edu/CaltechETD:etd-04082005-150525 Abstract The load parameter, Q, and the ratio of the switching frequency to the resonant frequency, F s /F o , characterize the operation of resonant converters. An accurate do analysis of the series and parallel resonant converters is given in terms of these parameters whereby the conversion ratio, peak stresses, and diode conduction time are determined. An exact and systematic method of small-signal analysis is given whereby the control-to-output transfer function, audio susceptibility, and input impedance are determined at a given operating point. In addition, simple and approximate transfer functions are obtained under high-Q assumption. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Electrical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Cuk, Slobodan Thesis Committee: Cuk, Slobodan (chair) Caughey, Thomas Kirk McGill, Thomas C. Martel, Hardy Cross Papas, Charles Herach Defense Date: 21 May 1984 Funders: Funding Agency Grant Number Caltech UNSPECIFIED Office of Naval Research (ONR) UNSPECIFIED Naval Ocean Systems Center UNSPECIFIED IBM UNSPECIFIED Record Number: CaltechETD:etd-04082005-150525 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-04082005-150525 DOI: 10.7907/NWMH-AP27 ORCID: Author ORCID Vorpérian, Vatché 0000-0002-8534-187X Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1308 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 08 Apr 2005 Last Modified: 29 Oct 2025 22:42 Thesis Files Preview PDF (Vorperian_v_1984.pdf) - Final Version See Usage Policy. 5MB Repository Staff Only: item control page

ANALYSIS OF RESONANT CONVERTERS Thesis by Vatche Vorperian In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1984 (Submitted May 21, 1984) ii to my parents 1ii ACKNOWLEDGMENTS I wish to thank my advisor, Professor Slobodan Cuk, for accepting me into the Power Electronics Group at Caltech and for the kind support that he and Professor R.D. Middlebrook provided during the course of this work. I appreciate the financial support of the California Institute of Technology by way of Graduate Teaching Assistantship, the Office of Naval Research and the Naval Ocean System Center by way of Graduate Research Assistantship. Also, I thank the International Business Machines Corporation for their generous support during my last year at Caltech by way of their Graduate Fellowship Program. Finally, I wish to thank my friends who made my stay an enjoyable and productive experience. iv ABSTRACT The load parameter, @, and the ratio of the switching frequency to the resonant frequency, f3//>, characterize the operation of resonant converters. An accurate de analysis of the series and parallel resonant converters is given in terms of these parameters whereby the conversion ratio, peak stresses, and diode conduction time are determined. An exact and systematic method of small-signal analysis is given whereby the control-to-output transfer function, audio susceptibility, and input impedance are determined at a given operating point. In addition, simple and approximate transfer functions are obtained under high-Q@ assumption. TABLE OF CONTENTS | ACKNOWLEDGEMENTS ABSTRACT CHAPTER 1 INTRODUCTION CHAPTER 2 DC ANALYSIS OF THE PARALLEL RESONANT CONVERTER gl Introduction 2.2 Operation 2.3 Analysis 2.4 Nonidealities 2.5 Experimental Results 2.6 Conclusion CHAPTER 3 DC ANALYSIS OF THE SERIES RESONANT wonuwa Ooh wWnore CHAPTER 4 4.1 4.2 4,3 44 CONVERTER Introduction Operation Analysis: Fy /f 21/2 General Analysis Nonidealities Experimental Results SMALL-SIGNAL ANALYSIS OF RESONANT CONVERTERS Introduction Analysis in Continuous Conduction Mode and in the Range #4, /Fo2=i/2 Experimental Results Conclusion iv 48 49 53 79 112 118 127 127 128 161 170 vi Fr CHAPTER 5 HIGH-Q APPROXIMATION 5.1 Introduction 5.2 Approximate Analysis 5.3 Conclusion CHAPTER 6 CONCLUSIONS APPENDICES APPENDIX A APPENDIX B APPENDIX C REFERENCES 172 172 173 190 191 193 194 197 218 2el CHAPTER 1 INTRODUCTION In the past two decades resonant converters have gained importance owing to certain advantages of switching sinusoidal voltages and currents. The principal motivation behind the development of these converters has been their ability, when operating properly, to provide natural turn-off for the switching devices. This minimizes turn-off losses and facilitates the implementation of thyristor switches which are indispensable at power levels greater than 5OkW. Operation at higher frequencies is another desirable feature in power-processing circuits because smaller filters can be used and consequently higher power densities can be attained. When field-effect or bipolar transistors are used at lower or moderate power levels, resonant converters can operate at higher frequencies with greater ease than PWM square-wave converters. Aithough, undoubtedly, the turn-on and turn-off switching stresses are far less in resonant converters than in PWM converters, the total switching losses may mot necessarily be less in comparison, because the ratio of peak resonant current to the average input current is generally higher in resonant converters than in PWM converters. It is nof possible to compare the efficiencies of resonant and PWM converters in general, and if such comparisons are made, they should be interpreted with caution. Therefore, to help the engineer understand and design these converters, an accurate analysis of their operation must be given. Whereas for PWM converters numerous methods of analysis exist, to date only limited dc analysis of resonant converters has been given while their small-signal analysis has not even been touched. The purpose of this thesis is to provide an accurate analysis of resonant converters and characterize their operation in terms of two simple and relevant parameters: the load parameter @ and the control parameter F,//Fy5, which is the ratio of switching frequency to the resonant frequency. A block diagram that follows from this characterization is shown in Fig. 1.1a. In Chapters 2 and 3 the de analysis of the series and the parallel resonant converters is given. The conversion ratio characteristics, peak stress levels, output-network switching time, and diode conduction time are determined in terms of @ and F,/ Fy. A systematic method of small-signal analysis of resonant converters is outlined in detail in Chapter 4. In this analysis the system equations are linearized at the operating point, determined from de analysis, assuming small-signal perturbations in the switching frequency and the input voltage. A block diagram representing the small-signal perturbations is shown in Fig. 1.1b. The results of the analysis are determined numerically because of the occurrence of exponential matrices which are difficult to determine in expression form. In Chapter 5 simple and approximate transfer functions are obtained for high-@ approximation. + Vg “== Resonant Converter | F5/Fo (a) A Vg Resonant Converter V + g T _ AAA The: Fo tf, (b) A Vo + <y> Fig. 1.1 a) Block diagram of resonant converters and b) perturbations in input voltage and switching frequency. CHAPTER 2 DC ANALYSIS OF THE PARALLEL RESONANT CONVERTER 2.1 Introduction In this chapter the steady-state operation and the de analysis of the parallel resonant converter are discussed. The basic operation is reviewed in Sec. 2.2. The de analysis is presented in Sec. 2.3 where first a thorough treatment of the conversion ratio characteristics is given in terms of the normalized load and control parameters. Other steady-state quantities such as diode conduction time, output-network switching time, and the initial state variables at the beginning of the switching interval are determined. Maximum stress levels such as the peak capacitor voltage and the peak inductor current are determined as well. Proper selection of the the range of the normalized load and control parameters for optimum operation is discussed. In Sec. 2.4, the efficiency and the conversion ratio are discussed in the presence of such nonidealities as parasitic resistances and semiconductor voltage drops. Experimental results given in Sec, 2.5 are in good agreement with the theoretical predictions. 2.2 Operation The parallel resonant converter with two different methods of voltage excitation is shown in Fig. 2.1a and b. In Fig. 2.1a the capacitors C., serve as voltage dividers and are assumed to be much larger than the resonant capacitor, Cp, ie, C.2>Cp. The parallel combination of the transistors and the diodes forms bidirectional switches which operate at fifty percent duty ratio to generate a symmetrical square voltage waveform Vr across the resonant circuit. For the converter in Fig. 2.1a the peak value, ¥,, of the square wave is equal to half the input voltage whereas for the converter in Fig. 2.1b the peak value is equal to the input voltage. The ideal circuit is shown in Fig. 2.1c. The switching times 7,' and 7, are shown in Fig. 2.1 and are defined as follows T,' = Switching interval F, = 1/T,' = Switching frequency TT, = T,'/2 The resonant frequency is defined as 1 Fo = —— © BrV/Tolo (2.2) The voltage across the resonant capacitor is rectified and applied to a low-pass filter from which the output voltage is obtained. By analogy to linear circuits it is clear that the output voltage can be controlled by changing the ratio of the switching frequency to the resonant frequency. This is shown in Fig. 2.2 where the converter is represented by a block a) b) + | [+ Vr Oe {I 0 + - * Ip — Ve $2 Vg T- V1.1) ; _ “\, c) na ia 'Le l+-——T , —o Ly + — Vy SR Fig, 2.1 The parallel resonant converter with two different methods of excitation a) and b), and ideal equivalent circuit c). diagram with an input voltage VJ, an output voltage Vp across a load PR, and a control parameter F,/Fy. In the following section the conversion ratio characteristics, M=V,/V,, will be determined for continuous conduction mode operation and for the range of the control parameter F./Fo> 5. if Vy Parallel - M= Vo _|_ > Mz a Resonant Converter _ Vg _ R Qp= Wolg Fig. 2.2 Block diagram of the parallel resonant converter, 2.3 Analysis The basic operation of the parallel resonant converter is discussed in [1,2,3]. The analysis given in [3] is strictly an iterative computational method whereby the circuit equations are solved numerically cycle by cycle until the initial conditions converge. No attempt is made in [3] to determine neither the initial conditions nor the conversion ratio for a given load F and control parameter F,//F>. The analysis given in this section begins with the determination of the conversion ratio characteristics and the initial conditions for a given F,//'y) and normalized load parameter, Gp. defined by _ _R Gp = ole (2.2) The waveforms during a complete switching interval are shown in Fig. 2.3. Since the operation of the circuit is symmetrical over an entire switching interval 7;,', the analysis is performed over half the switching interval 7;, It is assumed that the output filter inductor, Ly, is much larger than the resonant inductor, Lp, so that the output filter section can be represented by a current source Jp=Vo/FR. Because of the output rectifiers, as the voltage across the resonant capacitor changes sign at t=7,, the output network, or the current source Jp, switches polarity, This results in two switched networks over half the switching interval, 7,, as shown in Fig. 2.4. According to Fig. 2.4a, the equations for the resonant capacitor voltage, Vp(t), and the resonant inductor current, /e(t), for the time interval O<!< 7, are given by 4 Vi COSWpt + noo [% — Ve(0)]sinwot - a (23a) sinwot + Vy { Vo Va(t) = -|% — Vp(0) Jooswot + volo| +Ip(0) (2.3b) According to Fig. 2.4b, the equations for 7, <f<T, are t Ip(Ta (t-7,) + 2 (t-7,) + 2 Ir )= R\ )-3 COSHg a Wolo SINWp a R (2.4a) Vp ( i \ ‘a Vg V V TL Vv R(Ts) Ip(t) \ LI Ty —— 0 -I, 7 —>f | 1 Vp(0)| ! I ! I I i _ I Vp ' 1 I Te ! ! 1 ; 1 | 1 11 1 1 it ! ! rt | ! Mh) \ wor I I i a | 1 rt | ! 14 <-Ip-! 1 1 ; 1 a) Lo J \ I 1 1 Ip(Ts) 1” <———Q ] ——>- D1 Fig. 2.3 Voltage and current waveforms of the parallel resonant converter, 10 Ip(t) (a) ' {| * I (b) Fig. 2.4 The two switched networks of the parallel resonant converter during half the switching interval T,. a) O<t<T, and b) Ty <t <T, 11 V Ve(t) = —Vygcoswo(t—-T,) + wolo In( Ta) — G-|sinwo(t ~Ta) + Vy (2.4b) When Eqs. (2.3) and Eqs. (2.4) are matched at t=7, and cyclic stability required, ie., Jp(7,)=—Jp(0) and Ve(7,)=—Vp(0), the following two equations are obtained | Ip(0)R % sinwp 7, cos(y—wg Tq) GV, 1 + COSWy Tz. co8(¥—t9 Ta ) - a [1 + cos(y¥—W 7, ) (cost Tq -2)| =sintwg 7, cos(y—wy 7, ) + sin(y—w9 To) dy (2.5a) Ip(0)R Vq Vp(0) COSWy Ty sin(y¥—Wp Ty) Vg sinwg Tysin(y—w9 7a) — 1| - ~ [coswx Ty - ] sin(y—W Ty) = 1 +sinwy Tysin(y—wo Ty) — cos(y—wo Ty) Qp (2.5b) The requirement Vp(7,)=0 in Eq. (2.3a) gives Vp(0) Ip(O)R M COSWpT, + sinwo7T, + ——sinwpT, = cosug7Ty — 1 where - fo yore (2.6) Eqs. (2.5) are three equations in which the three unknowns J/p(0), Vp(0), and M can be solved for : 12 1 + cosy —2cos(y—tw9 Ty) + 2coswo Ty [cos(y—wo Te) - 1] V Ip(0) = woLpsiny sin(y—2w9 Ty) + sinwy Tz —sin(y—w Tr) VR (0) ~ siny cos(y—wWo Tz) + Cost Ty — 1 —cosy Mo siny P Vy g (2.7a) (2.7%) (2.7c) The unknown time 7, in Eqs. (2.7) is determined by equating input and output powers. The input power is given by - tf” Pry = To ViIp(t) dt and the output power is given by Pag = Af" Vp(t)Ig(t) at ot = 7 Jp p(t )Ip(t) where (2.8) (2.9) (2.10) When the integration is performed and Fy, is set equal to Fay an implicit equation of the form Gp(¥.Q@p Ta) = 0 is obtained from which the time 7, can be determined for a given @ and y: 13 Gp(¥,@p.Ta) = cos(y-woTa)| sin®aoTa + cos2p Ta | +sinwg7,|siny — sin(y—2u9 Ta) | +(y¥ —2t29 Tq )siny —2coswg Ty + Gy + COSY ~ COSWy Ty —cos(7—wp Ta) | =0 (2.11) Once 7, is determined, the conversion ratio M and the initial conditions are determined from Eqs. (2.7). A plot of M vs. F,/Fo for different values of @, is shown in Fig. 2.5. It can be seen from Fig. 2.5 that for a given @ the maximum value of the conversion ratio Mma, and the control parameter (F,/Fo)mex at which the maximum occurs depend on the value of @. A plot of (Fs/Fo)max and Mmax VS. Q is shown in Fig, 2.6. The maximum value of M for large @ is approximately given by Mmax ™ Qp (2.12) Figure 2.5 reveals that in order to achieve significant control over the conversion ratio M with a reasonable change in the control parameter, @, should be selected larger than two or three. For instance, if @=1, then even a large variation of F;/Fo produces very little change in M, which clearly is not suitable for control purposes. Also, according to Fig. 2.5, when @, is selected large and the converter is operated away from the resonant peaks, the conversion ratio characteristics are less sensitive to changes in the load #. Therefore, only when operating away from the resonant peaks with @, larger than two or three , can the parallel resonant converter be somewhat considered as a voltage-fed converter. il 10 Fig. 2.5 Conversion ratio characteristics of the parallel resonant converter 14 5 in continuous conduction mode. 15 For a given design application, the range of the control parameter F,/Fo can be selected either above or below (F3/Fo)mx. It is advantageous, however, to operate this converter below (F,/F'o)max because the transistor stops conducting as the resonant current, or the collector current, passes through zero. This eliminates problems of turn-off and makes it very attractive for SCR applications. Actually, for a given Q,, the critical value of the control parameter (F,/Fo)z- below which this natural turn-off occurs is slightly less than (/,/Fp)max and is plotted in Fig. 2.6. This is illustrated in Fig. 2.7 where it is shown how the shape of the resonant current changes as F,//y is swept through (F;/Fp)zc- When the converter is operating below (F,//)z,, as shown in Fig 2.7a, the transistor conducts first because the resonant current at the beginning of the switching interval is positive, ie, Jp(0)>0. As the collector current passes through zero the parallel diode begins to conduct for a duration 7p. When using SCRs, it is important to know this diode conduction time 7p in order to allow sufficient turn-off time for the SCR. When operating exactly at the critical value (F,/Fp)z-, the transistors turn on and off at zero current and the parallel diodes never conduct. When operating above (F;/F)z¢, as shown in Fig. 2.7c, the diodes conduct first because /p(0) is negative. As the diode current passes through zero, the transistor in parallel with it starts to conduct and is switched off while still in conduction. Therefore, this range of operation is not suitable for SCR applications. (F,/ Fp). is determined numerically by requiring /p(0) = 0 for a given &- The diode conduction time 7p, shown in Fig. 2.7a, is now determined. The resonant current passes through zero at t=T7, in the interval 7, <t<T, as shown in Fig. 2.7a.. Therefore, Eq. (2.4a) is set equal 1.0 16 . Joon 49 } (Fs/Fo) zc 48 . 47 L 46 Kd . 45 L 44 . 43 a “ 2 } al 1 2 3 4 5 7 8 9 10 421i Qp Fig. 2.6 The critical control parameters (F,/Fo)mex and (F,/Fo)z, and the maximum value of M vs. @. Note that Mme, is almost linearly related ta @,. . a) 17 Ve ; i<— 02 ——>- D2> I, (t) Tp(0)” yoR Th 0 i<—— T, a -V, —_— Q1 —#D1> Ip (t) R b) Vv a“ g ee lk Q2. —____-» 0) b~e—_—_—— Q] —__—_> ne, c) V __ & ke D2. Q2-— I, (t) N\ 0 Tp (0) “— g F,/ Fo< (F,/ Fo) F./F, = (Fe/ Fo) Fo/Fo> (Fs/ Fo) zc ZC ZC fig. 2.7 Explanation of the change in the shape of the resonant current as F,/F, is swept through (F5/Fo)zc- FR/ Fo=(Fa/Folec &) Fe/ Fo>(Fs/ Fo)ac- a) Fy/ Fo <(F5/ Fo)zc b) 18 to zero for t=7, and is written here as Eq. (2.13) after multiplying it out | rol Ts) Wolo _ M_ costig(T, ~T,) + sinwg(T, -—T,) + Me 0 q &p Qp (2.13) The angle won is defined as follows - wolo M_ which when manipulated further gives _ L+cosy—Rcoswg Ty tan(woa) = siny (2.15) Substitution of Eq. (2.15) in Eq. (2.13) gives M = Q, cos(woa) + sinwy(T, —Tg ta) = 0 (2.16) From Eq. (2.16) the diode conduction time Tp =7, —T, is obtained woTp = y¥ — 1 + to(a-Ty) — sin™}( HM cos(aga)) * @ (2.17) Therefore, for a given @ and y, the time 7p is determined from Eq. (2.17) where 7,, M and a are determined from Eqs. (2.11), (2.%7c) and (2.15) respectively. The normalized conduction time or the conduction angle vs. F;/ Fo is plotted in Fig 2.8 for different values of @. It can be seen from Figs. 2.6 and 2.8 that if @ is selected small, the useful range of the control parameter 5<F,/Fo<(Fs/Fo)e gets smaller. For all practical purposes then it is desirable to design the parallel converter with a minimum Q@, no 19 WoT p/T Fig. 2.8 Diode conduction angle characteristics of the parallel resonant converter, 20 less than two or three. The condition at (F,/Fp)z, corresponds to wpTp=0. A review of the linear parallel resonant circuit shown in Fig. 2.9a will now be given to show its similarity to the corresponding resonant converter. The transfer function characteristics of the linear circuit for different values of @ are shown in Fig. 2.9b. A comparison of these transfer function characteristics and the conversion ratio characteristics of the parallel converter shown in Fig. 2.5 shows that the peak of the characteristics and its position in both cases depend on @. Furthermore, for large @, the peak is approximately equal to @, and occurs approximately at resonance. For the parallel converter this is shown in Fig. 2.6 where it can be seen that (F,/Fo)max approaches unity for large Gp and Mmex is approximately equal to 4. The critical value of the control parameter (F,/Fo)z, for the parallel converter can be likened to the critical value w,/ w= V1-17 Gy” of the linear resonant circuit for which the input impedance is real. For the linear circuit when the input impedance is real, the input voltage and the input current are in phase. Figures 2.9c and d show the similarity of the voltage and current relationship of the linear circuit at w,/w) and the converter at (F,/Fp)e; in both cases the voltage and current are of the Same sign. For large values of @, both w/w, of the linear circuit and (F5/Fo)zc of the parallel converter tend to unity. For proper component selection, the peak stress levels must be determined. First, the peak capacitor voltage and its dependence on and F,/Fp is determined. The variation of the capacitor voltage with F,/Fo is shown in Fig. 2.10. For a given 4, as the control parameter is 21 lly 1 Qp™lo + Vo/Vin (a) 5 6 7 8 9 1.0 1.1 1.2 1.3 1.4 1.5 w, / W, (b) [NO Van Vg —— “e U (c) (d) Fig. 2.9 a) The linear parallel resonant circuit and b) the magnitude of its transfer function characteristics. c) Input voltage and current relationships for the linear circuit at w./wg=V1-1/ Qn” and da) for the converter at F,/Fo=(Fe/ Fo)ac- 22 swept through (F3/Fo)max, the peak of the capacitor voltage shifts from the interval 7, <f<7, to the interval 0<t<7,. Therefore, depending on which range F,//f'p falls in, the peak capacitor voltage is determined either from Eq. (2.38b) or Eq. (2.4b). First, the range Fy/Fo<(F5/Fo)max is considered where the peak capacitor voltage occurs in the interval 7,<t<T7,. The capacitor voltage in the interval 7,<f<T, is given by Eq. (2.4b) which after some manipulation is rewritten here as Eq. (2.18) Vat) [ Wolo MI. = —coswg(t -T,) + p(T, — —|sinwy(t—-7,) +1 Substitution of Eq. (2.14) in Eq. (2.18) gives Va(t) _ 1 ~sec(tpa) coswo(t —Ty +a) (2.19) From Eq. (2.19) it is clear that the peak capacitor voltage Voeux is given by Voeak —“—— = 14+ lseclona F. / Fo<tf J~ A | ( ‘0 )| s 0 ( s o) max (2.20) When the converter is operating in the range of the control parameter F,/ Fo >(Fs/ Fo)max. the peak occurs in the interval 0<t<T, in which case the capacitor voltage is given by Eq. (2.3b) which, after multiplying out by Wplo/ V4, is rewritten here as Eq. (2.21) Ve(t) _ | Vp(0) l Ip(0) | = —llcoswpt + |—— + 0.L,—— f£+1 Vg Vg nono a Wore Vg sw (2.21) The angle w 8 is defined as 23 Voeak V Vp (t) Ty R 0 i Fg/ Fy <(Fs/ Fo) max 1 > «Ta t < Ts Vg —! b) Vp —_—— Vr ' ~|+—- Tg ——>| t ' 0 Fo/ Fo >(Fs/ Fo) max Vp (t) _— ~Vpeak Fig. 2.10 Explanation of the shift in the peak capacitor voltage fram a) the interval T,<t<T, to 6b) to the interval O<t<T, as the control parameter is swept through (F5/ Fo) max- 24 Ve(0)/ Vy — 1 (2.22) tan(w»f) = where Jp(0) and Vp(0) are given by Eqs. (2.7a) and (2.7b) respectively. When Eq. (2.22) is substituted in Eq. (2.21), the following equation is obtained [ ee = sec(of)| ae -1 cosw,(t—-B) + 1 (2.23) Since the capacitor voltage in this interval is negative, it can be seen from Eq. (2.23) that Voe is given by Vv, Vp(0) peak e R = -i + lseclw —1 Fe/ Fo > (EF V, ( of) Vy | s Q ( s 0) max (2.24) The results are summarized in the following equation 1+ |sec(wpa) | > FAA Fo <(FS/ Fo) max Vreak - Y, Vp (0 g —1+ lsec(wof) x ) -| > F/ Fo>(Fs/ Fo) mex g (2,25) A plot of the normalized peak resonant capacitor voltage vs. F;/ Fp for different values of @ is shown in Fig. 2.11, These characteristics are very similar to the conversion ratio characteristics because the output voltage is equal to the average rectified resonant capacitor voltage. The variation of the resonant inductor current, /p(t), with the control parameter is slightly different from that of the capacitor voltage, Ve(t). The peak of /p(t), shown in Fig. 2.12a, occurs in the time interval Ts <t <T, and is given by the maximum value of Eq. (2.4a). As the control 25 lo em e b+ oe) oO te -~ Vpeak / Vg ON nN Fig. 2.11 Normalized peak resonant capacitor voltage characteristics of the parallel resonant converter. a) 26 [ F,/Fo< (Fs /Fo) 0X (Fg/F 0 oic -Vp —— Toeak b) Ve — Ip (t) Vp— “S 0 (Fg/Fo) 2c<Fg/Fo< (Fg /Fo)ic Va (t) -V, —_3 - Ts —eo c) Ve ena Ip (T.)=1 Ip(t) R\*s peak ; \ Te 0 VA aN F/Fo? (Fs/Foic Vp (t) Fig. 2.12 Explanation of the change in the peak resonant current as Fo/Fo is swept through (F3/ Foie a) and 6) For Fy/fo<(UG/Fo)iz the peak is given by the maximum value of the resonant current in the interval T,<t<T, and c) for Fy/Fo>(F/ Foc the peak is given by Ip(T,). 27 parameter is swept through some critical value (F,/Fp5),,, the maximum value of Eq. (2.4a) does not occur because the time interval 7, <t<T, is not long enough. Consequently, when the converter is operating above (F,/Fo)ic. AS Shown in Fig. 2.12b, the peak is given by the value of the resonant current at the end of the switching time 7%, ie., Jpeak =Ip(Ts)=—Ip(0). First, operation in the range Fy/Fo<(Fi/ Fo) is considered for which the peak inductor current is determined from Eq. (2.4a) which, after multiplying out by wl o/Vj, is rewritten here as Eq. (2.28) wolo _ | Wolo MM M Iplt = [J p(T. — — |cosw (ft -T,) +sinw,(t—-T,) + — Substitution of Eq. (2.14) in Eq. (2.28) gives Wolo M . Ip({t) % @, sec(woa)sinwg(t -T, +a) (2.29) The maximum value of Jp(t) in Eq. (2.29) is I peak Wolo = + |secaa | Fy/ Fo<(Fs/ Fodic Vy (2.30) As explained earlier, for F,/Fo>(Fj/Fo)ic. the peak is given by Ip(T,)=—ip(0). The normalized peak resonant current can be summarized by the following equation 28 M —— + |seclwga abe _ | Beetoee) g Fy/ Fo <(F5/ Fo)ic —Ip(0) ( Eq. (2.7a)) Fe/ Fo >(F/ Foc (2.31) The normalized peak resonant inductor current vs. F,/ Fo for different values of @, is plotted in Fig. 2.13. According to Fig. 2.13, an interesting feature of the peak resonant current is that, in a certain range below resonance depending on 4, it decreases with increasing 9. This of course is expected because, as explained earlier, in this range of operation the converter behaves like a voltage-fed converter. Consequently, as the output power decreases the peak current decreases while on the other hand the peak capacitor voltage, like the output voltage, remains mostly unaffected. Also, a comparison of the peak current below and above resonance shows that, for the same conversion ratio, the peak current below resonance is less than the peak current above resonance. The critical control parameter (/,/Fo)z is determined numerically and is plotted in Fig. 2.14. A comparison of Figs, 2.14 and 2.6 reveals the following important relation (Fs/ Fo)ec < (F3/ Fo) max < (Fs/ Fo)ic (2.32) According to the inequality in Eq. (2.32), when the converter is operating in the more useful range of the control parameter, 5<F,/Fo<(F3/Fo)ec, where natural turn-off of the transistors occur, the peak capacitor voltage and peak inductor current can be summarized by the following equation 29 16+ - 14+ ye 12} \ > ~~ 10+ 9 =] L f ! s ag SF Ge) wv L ou 4h 2k = a) 6 7 8 9 1.0 1.1 1.2 1.3 1.4 1.5 Fg/Fo Fig. 2.13 Normalized peak resonant inductor current characteristics of the parallel resonant converter, 30 1.24 1.1 (Fs/Fodic 1.04 Fig. 2.14 Critical value of the control parameter (F;/Fo)ie us. @ ic US. Qpy. ( Vv, Perk = 1+ |sec(wa) | Vg 1/2 Fy/ Fos (Fs/ Foe woh, M J peak 7 a= QQ |sec(woa) | g P (2.33a-b) 2.4 Nonidealities In Sec. 2.3 the analysis was carried out with the assumption that all the active and passive devices were ideal. Nonidealities, such as parasitic resistances of the resonant inductor and capacitor and the output filter inductor, as well as the voltage drops in the active components, are considered in this section. These parasitics introduce losses which deteriorate the efficiency of the converter. The losses due to parasitic resistances and voltage drops in the semiconductors are considered separately to show the contribution of each loss mechanism to the efficiency. Since these parasitics are small and do not affect the main operation of the converter, the results of the analysis given in Sec. 2.3 are modified slightly in this section by introduction of fudge factors. The first nonideality to be considered is the voltage drop Vgp in the output rectifiers. The two different methods of output rectification are shown in Fig. 2.15 where the isolation transformer is assumed to be ideal and of unity turns ratio. In Fig. 2.16a the output network is reflected to the primary side of the transformer and the voltage drop in the rectifiers is separated and represented by a voltage source Vgp in series with an ideal diode. The voltage source, Vgp, is either equal to one diode voltage drop, Vp, or two diode voltage drops, 2Vp, depending on the ‘method of 32 — STOO ® a Fig. 2,15 Two methods of implementing the isolation transformer in the parallel resonant converter. a) center-tapped and 6) full- bridge, Pout — Plost Va(t) > Cc All diodes are idea! Fig. 2.16 a) Nonideal circuit with voltage drop Vpp in the output rectifiers and b) ideal circuit that delivers the same power P,,' as the nonideal circuit in a). 33 rectification Vp center-tapped Vap = 2V¥p full-bridge The power flow is shown in Fig. 2.16a where P;, represents losses in the diode voltage drops and P,,, represents the power delivered to the load. The input power is given by P| is given by Eq. (2.9) and rewritten here as Eq. (2.35) ee t) dt Pp! = TA Va(t) Ip(t) (2.35) It is clear that in the absence of losses Py,'=Pyy. Equation (2.11) was obtained by equating input and output powers. In the absence of rectifier losses, the power delivered at the terminals of the resonant capacitor is equal to the output power. In the presence of these losses the power, Pin’, delivered at the terminals of the resonant capacitor must be correctly accounted for. In what follows, an ideal circuit, such as the one shown in Fig. 2.16b, will be determined which has the same power, Fi,', delivered at the terminals of the resonant capacitor as the nonideal circuit in Fig. 2.16a, and to which the same analysis given in Sec. 2.3 can be applied. First, the power Fj,' in Fig. 2.16a is considered which is equal to the sum of the power lost in the rectifiers and the power delivered to the load 34 Pin'= Venlo + Volo 3 Lo = (2.36) According to Fig. 2.16a the output voltage Vp is equal to the average rectified capacitor voltage <|Vp(t)|> less Vgp Vo = <|Ve(t)|> — Veo (2.37) where <|Vp(t)|> = af" Vp(t)| dt (2.38) substitution of Eq. (2.37) in Eq. (2.36) gives P,' = <|Va(t) |>? | Ve | ™ R <|Ve(t)|> | (2.39) Now let V' = <| Vp(t)|> (2.40) ro Vv = Vg (2.41) Ri= — BD MY, (2.42) Equations (2.40) through (2.43) are satisfied by the ideal circuit shown in 35 Fig. 2.16b which has a conversion ratio M' and a load Ff’. According to Eq. (2.42) the normalized load parameter, @,', of this ideal circuit is related to the original @, of the nonideal circuit by _& _ _Vep_ M'V, (2.44) G' = 1 The results of the analysis given in Sec. 2.3 will now be applied to the ideal circuit in Fig. 2.16b. First, the conversion ratio M' is considered which according to Eq. (2.7c) is given by ue cos(y—wo7a) tcoswg7g —1—cosy _ , ~ siny P (2.45) Substitution of Eq. (2.44) in Eq. (2.45) gives cos(y—Wo Ty) + coswgTg — 1 —cosy Vp sin G * y Vy (2.46) M' = Next, the implicit Equation (2.11), from which wo7, is determined, is modified. The last line in Eq. (2.11) can be seen to be equal to —Mysiny. Now, when M is substituted by M' a new implicit equation G'(y,Q@),7u) is obtained which is related to G(7.@),7,) by the following equation __ Vep ' iL = a ’ ~ > = G (7, @ a) Gp (y Gp Ta) —ysiny V, 0 (2.47) Once wo7, is determined from Eq. (2.47) for a given @ and y, the conversion ratio M' can be determined from Eq. (2.46). The actual conversion ratio M, of the nonideal circuit can now be obtained from Eq. (2.37) as follows S ie) %, V, (2.48) According to Eqs. (2.46) and (2.48), the conversion ratio of the nonideal circuit M, is still given by Eq. (2.7c), but the angle wo7, is now determined from the modified implicit equation given by Eq. (2.47). The efficiency of the converter due to losses in the output rectifiers is considered next. This efficiency 7p, due to rectifier losses is easily seen to be given by ~ Fy; Vo *! (2.49) The first thing to note from Eq. (2.49) is that, just as in the case of PWM converters, the efficiency deteriorates for lower output voltages. According to Eq. (2.48), substitution of Vp in Eq. (2.49) gives another expression for the efficiency np TR MV, (2.50) The actual gain can now be written as Ma = NpM' (2.51) The efficiency Np vs. F,/Fp is plotted in Fig. 2.17 for two different values of Vgp/ Vg. The efficiency is seen to be higher near resonance where the conversion ratio is higher. These curves do not suggest that the converter should be operated near the resonant peak; all they show is the behavior of 7p in open loop operation. For a closed loop regulator with a 37 100 } Qp=10 eh 3 80 ON Oo T NR &% wn ° 30+ Ven/Vor.l 20 Bp/Vg 10+ es) 6 7 8 9 1.0 1.1 21.2 1.3 1.4 1.5 100 20 | Vpp/Vg=.2 i 1 n L aL J. 1 1 n ae) 6 7 8 -9 1.0 12.1 ° 1.2 1.3 1.4 1.5 Fig. 2.17 a-b) Efficiency of the parallel resonant converter due to Losses in the output rectifiers for two different ratios of Vap/ Vy. 38 fixed output voltage, np is given by Eq. (2.49). The losses in the switches are now considered briefly. These switches can be realized by implementing BJTs, FETs [1,2], or SCRs [3]. In all these cases the voltage drop during forward conduction, Vzg, is different from the voltage drop during reverse conduction, Vps. The losses in the switches are given by 1 qT, T, . Prost = tts f |Ip(t)| at + Ves Sr, [Ip(t) | ae (2.52) where 7, is the zero-crossing time of Jp(f). In Eq. (2.52), the resonant current, Ip(t), must be determined taking into account the voltage drops Vrs and Vrs. Such an analysis would be rather complicated, however, if Vrs and Vps are small compared to Vj, which is generally the case for most applications, Jp(t) in Eq. (2.52) can be determined assuming ideal switches. Therefore, for thermal design, the losses can be estimated according to Eq. (2.52), in which Ip(t) is determined assuming ideal switches. Experimental results given in Sec. 2.5 are in good agreement with the predicted results assuming ideal switches. The losses in the parasitic resistances are considered next. The losses in the esr of the output filter capacitor are not considered because in the de analysis given in Sec. 2.3 the current in the output filter inductor was assumed to be free of ripple. The input power is given by Where Ji, is the average input current shown in Fig. 2.1a and b. The 39 output power, in the presence of parasitics, is given by According to Eq. (2.53) and (2.34), the efficiency due to the parasities is given by _ Vo fo 1p To Tn (2,55) It will be assumed now that in the presence of these parasitics, the current conversion ratio Ji, / Jp remains unaffected and is still given by the ideal voltage conversion ratio. Therefore, the actual conversion ratio, Ma =Vo/ Vin, of the converter in Fig. 2.1a, according to Eq. (2.55) is now given by _, M Ma = Np 5 (2.56a) and for the converter in Fig. 2.1b, M, is given by Ma = 7,M (2.56b) The results given in Eqs. (2.56) hold for PWM converters as well. The actual conversion ratio for PWM converters in the presence of parasitics is derived by the method of state-space averaging [4]. It was shown in [4] that the current conversion ratio remains unaffected in the presence of parasitic resistances and that the actual conversion ratio is given by the product of the efficiency and the ideal conversion ratio. Since for resonant converters the steady-state results are not as easily obtained, rather than deriving it, the current conversion ratio is assumed to remain unchanged. The power 40 lost in the parasitic resistances is given by Prost = F,*ry, + 1,?r¢, + Lo"r1, (2.57) where TL, TC, and TL, are the parasitic resistances of the resonant inductor, resonant capacitor and output filter inductor respectively. The currents /, and J, are the rms values of the resonant inductor and capacitor currents determined in Appendix A. The efficiency is now given by 1 I,” TL, te z °C, TL, + + Ig Rk Ig RR (2.58) Np = 1+ As before, it will be assumed that the ratios 4/J) and J,/ Jp remain unchanged in the presence of parasitics. Consequently, the rms values are determined from the expressions derived in Sec. 2.3. The following normalized loss parameters are defined _ Wgho Q, = \ g, = doko : g, = Poko TL, TC, Ly (2 5 9) The efficiency due to the parasitics can be written now as 1 Np = @{e Be), 1, The constants B, and B, are determined in Appendix A. According to Eq. (2.60) it can be seen that the efficiency due to the parasitics in the 41 100 80+ 70 + 60 + a) £50} 30 + 20 + 10} 75 1005 90} 80} 70+ b) 504 30} 20+ + a Q1= 175 Qo = 437 1 Fig. 2.18 Efficiency of the parallel resonant converter due to losses in the parasitic Q=175 @,=437. 9 1.0 Fs/Fo resistances, 1 1 a) 1. 2 1.3 a = a. =B75 1.4 1.5 and b) 42 resonant capacitor and inductor depend on the operating point whereas the efficiency due to the parasitic resistance in the output filter inductor does not. In Fig. 2.18 the efficiency vs. F,/F is plotted for two different values of &, @, and @ assuming ry, =0. It can be seen that the efficiency deteriorates for increasing @ and F,/Fpo. Clearly the efficiency deteriorates for small @, and large Q as can be seen by comparing Fig. g.19a and b. It can also be seen that the efficiency is higher in the important range of the control parameter 5<F,/Fo<(F/Fo)sec determined earlier in Sec. 2.3. 2.5 Experimental Results The experimental circuit used to verify the conversion ratio characteristics is shown in Fig. 2.19. The circuit values are Lo =49ynH Co =.1uF Vg =12V (2.61) The experimental and predicted results are shown in Fig. 2.20. The predicted characteristics are obtained considering only the diode voltage drops using Eqs. (2.46) through (2.48). The experimental and the predicted results are in good agreement except near the peaks. This discrepancy owes to the fact that the analysis for the nonideal circuit is not exact. An exact analysis with all the parasitic losses (including core losses) would result in better agreement near the peaks, but since it is desirable to operate the converter away from the peaks, such a tedious analysis would be unnecessary. 43 D44H10 42uF ent => -—— F oie: Ss 42uF quam 1 é D45H10 1.35mH All diodes UES1303 32uF au» Vo vco Buffer Fy ——e— Intersil > -——>— 8038 DS0026 Fig. 2.19 Experimental circuit for the parallel resonant _converter 44 Aet=% att =99 Hey = 97 ‘£3}.L8NU09 JUDUOS3L 727 ]7DLDa ay} fo SOUJSLLAJODLDYI 01704 ‘worszanuoa 8yz fo sjjnsas pajoipasd pup youaumadry oo2 ‘bw GT 7° T eT ami TT O°T 6" 8° Z° 9° T Lf T T T 7 T T T W 45 ae / Lov N\ 5A _— 2usec Fig. 2.21 Experimentally measured waveforms, left, and predicted waveforms, right, of the parallel resonant converter, a) Resonant capacitor voltage Vp(t) and excitation voltage Vr, and b) resonant inductor current Ip(t). Lo=49uH, Co=.1yF, Q@ =2.95, Fy/Fo=.75, and Vz =15V. 46 Figure 2.21 shows experimental and predicted voltage and current waveforms for the following circuit values Gp =2.95 F/Fo=.75 Co=.1uF Ly=49uH Vv, =15V The following table summarizes the results relevant to Fig. 2.23 Predicted Measured Vocak 41.4V 40V peak 1.57A 1.6A Tp 2.5Busec 2,5usec Th l.l4ysec 1.3usec 2.6 Conclusion In this chapter the parallel resonant converter is first described as a black box with a control signal F,/ Fy, input voltage source Vy, and an output connected across a load FR. The output voltage is controlled by changing the ratio of switching frequency to the resonant frequency, F/ Fo. The conversion ratio characteristics are determined for continuous conduction mode in terms of the normalized load parameter, &, and control parameter, F,/Fp. Only the range of the control parameter F./Fy>5 is considered. The peak resonant voltage and current -characteristics are determined likewise in terms of @ and Fy/ Fo. The critical value of the control parameter (/,/F)z,, below which natural turn-off of the switches occurs, is determined in terms of &. It is Shown that (f,/F)z, tends to unity for large &. The diode conduction time, Tp, in the range .5<F,/Fo<(F./Fo)ec. Which is an important design 47 parameter when SCRs are used, is determined in terms of @ and F,/ Fo. The efficiency in the presence of parasitic resistances is discussed and it is shown that the efficiency decreases for increasing F,/ Fp. According to the conversion ratio, diode conduction time and efficiency characteristics it is shown that it is best to operate this converter below resonance and away from the peak with a @ no less than two or three. This will insure higher efficiency and minimum sensitivity of the conversion ratio to variations in the load. 48 CHAPTER 3 DC ANALYSIS OF THE SERIES RESONANT CONVERTER 3.1 Introduction This chapter discusses the steady-state operation and the dc analysis of the series resonant converter. After a brief review of the basic operation in Sec, 3.2, the analysis is presented in Sec. 3.3 and Sec. 3.4. In Sec. 3.3 the analysis in the range *;/Fp=1/2 is given, and in Sec. 34 a general analysis is given in the entire range 0<f,;/Fp<™, The various modes of operation in discontinuous and continuous conduction are identified, and the conversion ratio for each mode of operation is determined in terms of the normalized load parameter, Q,, and the control parameter F;/Fre. The boundaries between the various modes of operation are determined in terms of Q@ and F,/Fp as well. It is shown that certain types of discontinuous conduction modes should be avoided because the conversion ratio for such modes is insensitive to the control parameter. For example, in the range 1/2<F,/Fy<1 a simple equation is given for the proper selection of @ in order to avoid the occurrence of dem. Other Steady-state quantities such as peak stress levels and diode conduction time are determined as well. Nonidealities due to the parasitic resistances and rectifier voltage drops are considered in Sec. 3.5. The experimental results given in Sec. 3.6 are in good agreement with the predictions. 49 3.2 Operation The series resonant converter was first introduced by Schwarz [5]. The major drawback of the converter described in [5] is that its conversion ratio is fixed. The converter shown in Fig. 3.1a, which has an adjustable conversion ratio, was introduced later by Schwarz [6]. The reason for the drawback of the converter described in [5] is the absence of the diodes D1 and D2 shown in Fig. 3.1a. It is shown in the next section that in the absence of D1 and D2, the converter is forced to operate in a certain discontinuous conduction mode for which the conversion ratio is uncontrollable and fixed at unity. Other methods of exciting this converter are shown in Fig. 3.1b-d. All these converters are equivalent to the ideal circuit shown in Fig. 3.1e. Switches S1 and S2 operate at fifty percent duty ratio and generate a symmetrical square waveform, Vr, of amplitude Vz across the resonant circuit. For the converters in Fig. 3.1acc, Vy, is g equal to V,/2, whereas for the converter in Fig, 3.1d Va is equal to Vin. In Fig, 3.1a the capacitors Cg/2 serve as resonant capacitors as well as voltage dividers whereas in Fig. 3.1b the capacitors C,, serve only as voltage dividers and are much larger than the Cp, ie, C,2>Cp.' The switching times 7, and 7,', shown Figs. 3.1, are defined as before T,' = Switching interval ay fy = 7 = Switching frequency s 50 51 — QQ, - © (d) Vi 1 “r | S| V, g = } '+— Vae—*| O- — 041 (t) V TL Valt) ‘ $2 - -V, 7 | ht To 8 PTS + (e) Fig. 3.1 a-d) The series resonant converter with various methods of excitation and e) the ideal equivalent circuit. 52 T. ' T, = 5 The resonant frequency is defined as Fox 1 += end Lolo (3.1) The normalized load parameter is given by _ Wolo & = —p (3.2) The constant K is defined as _ BY K= 5 (3.3) where __fo YOne (3.4) The resonant current, Jp(t), is rectified and applied to a low-pass filter from which the output voltage is obtained. It is then clear, by analogy to linear circuits, that the output voltage can be regulated by controlling the ratio of the switching frequency to the resonant frequency, F,/ Fo. The same block diagram given for the parallel converter is repeated here for the series resonant converter in Fig. 3.2. 53 V + Series + 6 — RS Vo Me— q - Resonant Converter = ° Vg WoL Q,= “o=9 TF /F Fig. 3.2 Block diagram of the series resonant converter. 3.3 Analysis: F,/ Fo = 1/2 The analysis of the series resonant converter for the range of the control parameter F,/Fp=1/2 is divided into two parts; below resonance, 1/2<F,/fo=1, and above resonance, F,/Fo>=1. First the range 1/2<F,/Fo<1 is discussed. 3.3.1 1/2<F,/Fo<1 According to Fig. 3.2 the analysis in this section begins by the determination of the conversion ratio, M, in terms of @, and F,/Fp in continuous and discontinuous conduction modes. The analyses given in [6] and [7] have considered the operation of this converter in terms of the diode conduction angle, w»7Tp and have not explicitly determined the conversion ratio nor the occurrence of discontinuous conduction mode (dem) in the range 1/2<F,/ Fo<1. 54 The voltage and current waveforms in continuous conduction mode (ccm) are shown in Fig. 3.3. In this range, the resonant current at the beginning of the switching interval is always positive, ie., Ip(0)=0. This means that the transistor always conducts first and thereafter, as the collector current passes through zero, the parallel diode begins to conduct for a duration wp7p. This natural turn-off is very attractive for SCR applications. For the parallel converter it was shown that the range of the control parameter in which natural turn-off occurs is load dependent. In the case of the series resonant converter this range is given by 1/2<F,/Fo<1 and is independent of the load. As in the case of the parallel converter, owing to the symmetry of operation over an entire switching interval, the analysis is carried out over half the switching interval, 7,. It is assumed that the output filter capacitor, C;, is much larger than Cg so that the output section can be represented by a voltage source Vp, As the resonant current changes sign at £=7,, the output section, or the voltage source Vp, switches polarity because of the output rectifier bridge. This results in two switched networks over the interval 7, as shown in Fig. 3.4. According to Fig. 3.4a, the equations for Jp(t) and Vp(t) in the interval O<f<T, are given by Ip{t) = Ip(O)coswpt + wl [% — Vo — Ve(0)) sinwot (3.5a) _ _f _ve t+ Ip(0) : _ V(t) = ~[Vq— Vo Ye(0)|cosagt + Soc -sinwot + Vy - Vo 1g According to Fig. 3.4b, the equations in the interval 7,<t<T7, are 55 0 g ' j—Ta —te-Th 1 l-———T ! \ \ I { I I | \ I Vp (t) R ; ‘ ; 1 I 0 I ! I Vp (0)! R60) A Fig. 3.3 Voltage and current waveforms of the series resonant converter when operating in continuous conduction mode and in the range l/e <F,/ Fo 1. 56 + Vg —==— - 4 L. 7 F I ,(t) a) | V(t) R 1 ~ ly Vo + Vg" if Lo - r Ip(t) b) |, Valt) R | Vo ig. 3.4 The two surtched networks of the series resonant converter during half the switching interval T, when operating in the 1/2 <Fo/Fos1. a) O<t<T, and b) Ti<t<T,. For range the sequence is switched operation in the range F,/Fo2=1, around, 57 Vp(t) = ~ [V+ Yo —Ve(Ta)] cosie(t — Ta) ++ Vo (3.6b) For the series resonant converter, the initial conditions Vp(0) and Ip(0) can be easily determined by the following simple approach which is also given in [6]. The input power to the the circuit is given by (A, — Ae) Pe |x Pay = fl Sf, Inlt)at + Sp tet) (3.7) where A; and Ag are the areas under the resonant current waveform shown in Fig. 3.3. The output power is given by f Pop = Te Ay telat ~ Stata | = Fo (A +42) (3.8) The requirement Pi, =Pou gives Vg A, —Ag _~ M=— Vy, A, +Ag (3.9) Since A, and Ag are positive, it follows immediately that M cannot exceed unity M<i (3.10) According to Fig. 3.3, the peak-to-peak capacitor voltage is given by 2 peat = 2Ve(Ta) = Z— (Ai +As) 3.11) Also, the output voltage is given by the average rectified resonant current 58 multiplied by the load FR 1 ris Vo= Raf "lip(t)|at = © (4, +A4,) Ts ~0 Ts (3.12) Substitution of Eq. (3.12) in Eq. (3.11) gives - - Js%o _ Ve(To) = Vooux = Bae = MAY, (3.13) where K is given by Eq. (3.3). The initial capacitor voltage Vp(0) can be expressed as ¥e(0) = Vooat 2 fe "Fel Ta)at = Vea — 7 (3.14) which after some manipulation, gives Ve(0) = —M?KV, (3.15) The requirement Jp(7,)=0 in Eq. (3.5a) gives Ip(0) = —t2pCoVe 1 —M+MPK |tanwy7, (3.16) When cyclic stability conditions are required, i, Jp(0)=—Jp(T;) and Vp(0) =—Vp(7;), the following two equations are obtained 1+M—-MK tanwole = +e sin(y ~ 97a) (3.17) 1+M —M®K cos(y—WoTa) = Tar | (3.18) In Appendix B.1 Egs. (3.17) and (3.18) are solved simultaneously and an 59 implicit equation of the form G,(M,Q,,7y)=0 is obtained from which M can be determined for a given @ and F,/ Fp. This is given by _ 1-M+MPK G(M 7) = M(K+1)-1 “°S” V(1—M*)(MK—-2)MK M(K+1)-1 siny + 14+M—-M?K _ 0 1+M(1—-K) (3.19) Also, the angle wo7, is shown to be given by _ 1-M+MPK cosmo la = THK +1) (3.20) A plot of the conversion ratio characteristics in cem is shown in Fig. 3.5. These characteristics are obtained by solving G@(M,Q,,7)=0 numerically for each @, and for values of F,/Fy) ranging from 1/2 to 1. As expected, according to (3.10), the maximum value of M for any 4, does not exceed unity. The characteristics for Q,=4/m are seen to extend over the entire range 1/2<F,/F)<1 while those corresponding to 4,<4/m are restricted to a narrower range of F,/Fpo. For example, for @,=3/m the characteristics for ccm are restricted to the range 1/2<F,/F 9<3/4. This means that for Q,=3/n the real solutions of Eq. (3.19) are restricted to this range and that no real roots exist in the range 3/4</,/Fpo<1. This restricted range of F,/ Fp will now be determined analytically and it will be shown that outside this range the converter operates in dem. 60 | Fs/Fo=3/4 Fig. 3.5 Conversion ratio characteristics of the series resonant converter when operating in continuous conduction mode and in the range 1/2 <F,/ Fo 1. 61 Since M <1, an examination of the radical in Eq. (3.19) reveals that MK >2 (3.21) which can be written as Fs - MQ,0 Fo” 4 (3.22) According to (3.22), the range of F,/Fp in which Eq. (3.19) admits a real solution is given by MQ;T 4 (3.23) 1,-k. 2 Fy Since the maximum value of M is unity, the upper bound in (3.23) can be written as @, 71 4 (3.24) < oa IA Do A comparison of (3.24) to 1/2<F,/Fo<i shows that for Q>=4/m Eq. (3.19) has a real root M in the entire range 1/2< F,/Fo<1, whereas for Q@<4/n this range is restricted to (3.24). This then explains the characteristics shown in Fig. 3.5. For the example of Q=3/ 1, (3.24) gives 1/22 F,/ Fy S3/4. In order to complete Fig. 5, the characteristics for @,<4/7 outside the range given in (3.24) is now determined. From the preceding discussion and Fig 3.5, M reaches unity for 4,<4/7 below resonance at F,/Fo=@.n/4. According to Eq. (3.9) unity gain implies Ag=0. The only 62 Ip (t) a wVorVg _ Va 0 i 4 I | \ | -V | qT 1 g —©O ___»! 1 , 2 | { bee I | ) 1! I ! { i t it leVa(T,) | \ | { Vp (t) ; i \ I \ { ; I } \ i I Vp (0) Fig. 3.6 Voltage and current waveforms of series resonant converter when operating in discontinuous conduction mode and in the range 1/2 <F,/Fo<1. Note that during T,-nT)/2, the voltage at the input side of the bridge, Vy, is less. than the output voltage. 63 / CCM ~«<—}—» dem 1.0} Bt Fg/Fo= Qs™/4 5 6 7 8.9 1.0 Fg/Fo Fig. 3.7 A typical conversion ratio characteristics curve of the series resonant converter for 2/n=4,<4/nm when operating in the range 1/2 <F,/Fo<1. For such values of @,, the operating mode changes from continuous to discontinuous at Fo/ Fo = 91/4. 64 way this can happen below resonance is for the current in the interval T,<£<T, to be zero as shown in Fig. 3.6. This means that for such values of @ the converter enters dem at F,/F)=@,n/4. As F,/Fo is increased beyond 4.7/4, the converter remains in dem with M fixed at unity. This is illustrated in Fig 3.7. The combined characteristics in continuous and discontinuous conduction modes are shown in Fig. 3.8. These characteristics show that this converter is not well suited for applications that require very large load changes. This has been a known fact since the introduction of this converter, because of its apparent current fed nature. It is shown now that this is true only under certain conditions. A plot of the normalized output current, which is given by Ipwolio/ Vy, vs. Fs/Fo for different values of @ is shown in Fig. 3.9. These characteristics are obtained by multiplying M by @. Now, it can be seen from Fig. 3.9 that only for @ greater than three or four and away from resonance does the converter behave like a current fed converter. For a given @ the modes of operation in the range 1/2<F,/Fp)<1 can be summarized as follows 4 1. Fs >— =—< < @= 7 comin 3-7 <} (3,25a) ; 1 Fs Q 7 —s= < ; . A ecm in 2 Fo 4 s So F. 7 | dem in an < <1 8 (3.25b-c) Q<7 demin 3 7 <1 (3.254) Condition (3.25d) follows immediately from (3.24) whence it can be seen 65 o> 4 OK) oo Qs y) 3F- eK) 5 ~ oy 09 2% o2+- ~LF a a 1 1 L L L lL i 5 6 7 8 9 1.0 Fig. 3.8 Combined conversion ratio characteristics of the series resonant converter in continuous and discontinuous conduction modes and in the range 1/2 <F,/Fo<1. 11 10 66 29,4/T Q,=3.5/7 Q5=3/t Qo=2.5/T 6 1 0 Fig. 3.9 Normalized average output current characteristics of the series resonant converter in continuous and discontinuous conduction modes and in the range 1/2 <F,/Fo<1. 67 that for @,=2/n the upper and lower bounds of cem coincide. Alternately, the boundary between dem and ccm as a function of @; for a given F,/F in the range 1/2<F,/Fo<1, can be given as follows Fy dem Q = 1/2<F./Fo<1 Q, = ak 1 a ccm rit (3.26a-b) It is quite clear that dem should be avoided if a controllable output is desired. This completes the discussion on the determination of M@ in both modes of operation. In what follows a physical reason for the occurrence of dem is given. According to Fig. 3.6 and Eqs. (3.5) and (3.6), the voltages and the currents in dem are given by Ip(t) =—woCg Vp(0) sinwot Va(t)=Ve(O)coswpt O<t<T, (3.27) Ip(t)=0 Vp{t) =—Vp(0) Tyst<T, (3.28) The output voltage, which in this case is equal to the input voltage, is given by = = i Ty = a Fs Yo= Vy = Ray le(t)idt = --5-e-Va(0) (3.29) In the time interval 7,<t<T7, the net voltage at the input side of the bridge, Vp, is given by 68 ar Fo _, 2 F, (3.30) Ve = Ve(Ts)—Vy = vl The range of F,/Fy in which dem occurs is given by (3.25c) from which it can be easily shown that 2 (3.31) which according to Eq. (3.30), implies that the voltage at the input side of the bridge is less than the output voltage. This means that in the interval T,<t=T,; the bridge becomes reverse biased and the diodes Di and D2 do not conduct. The occurrence of this discontinuous conduction was never explained in [6] or [7] where it was assumed that the diodes D1 and D2 always remain in conduction. It should be clear now that if these diodes are removed the converter will always be forced to operate in dem with M fixed at unity as in the case of the converter described in [5]. For proper component selection, the peak stress levels must be determined. According to Fig. 3.3 the peak of the resonant current occurs in the interval O<f<7, during which the resonant current is given by Eq. (3.5a). When Vp(0) and Jp(0), given by Eqs. (3.15) and (3.16) respectively, are substituted in Eq. (35a), the following is obtained 1-M+M?K , a coswyTs sinwe(t — T,) Substitution of cosw97, from Eq. (3.20) in Eq. (3.32) gives (after taking the positive value |1-M(K+1)|=M(K+1)-1) 69 16 F Q,=10 14 + Q,=8 12 + 10 + Q,=6 wo CE > ~— < 8b rf > * Q5=4 6 a L 093 44 5 Qg=2 Q_=4/T 2 0835 Q,=3/1 L r Q,=2.5/t 5 6 7 8 9 1.0 F./Fo Fig. 3.10 Normalized peak resonant capacitor voltage characteristics of the series resonant converter in continuous and discontinuous conduction modes and in the range 1/2 <F,/Fo<1. 70 16+ Q,=10 14+ Qg=8 124+ 10+ Qg=6 oO s rh ee - 3 = st oO jor eI A 6 F os# . Q,=3 4} L Qg=2 2F — a" Q,=3.5/T \Q,=2.5/1 5 6 7 8 9 1.0 Fig. 3.11 Normalized peak resonant inductor current characteristics of the series resonant converter in continuous and discontinuous conduction modes and in the range 1/2 <F,/Fo<1. 71 Ipeak = CoV, [M(K +1) —1] (3.33) Vocar = MKV, (3.34) where the peak capacitor voltage, given earlier in Eq. (3.13), is repeated here in Eq. (3.34). For completeness, the peak stress levels in dem are given now. From Eqs. (3.27) and (3.29) these are easily given by Vpeak = KV, (3.36) It can be seen that the peak stress levels in dem can be obtained from their corresponding expressions in ccm by letting M=1 in Eqs. (3.33) and (3.34). This, of course, is expected because at the boundary between dem and ccm both expressions should match. Plots of Vpeok ANd Jpeqe in continuous and discontinuous conduction mode are shown in Figs. 3.10 and 3.11 respectively. When SCRs are used, it is important to know the diode conduction time, 7p, shown in Fig. 3.3, in order to allow sufficient time for the SCR to turn off. This angle according to Eq. (3.18) is given by if 1+M—MeK woTp = y-WoTg = cos” +m —K) (3.37) This is plotted in Fig 3.12. The characteristics for Q@<4/m terminate at F5/Fo=@,7/4 because, as explained earlier, for such values of @ the converter operates in dem in the range Q.n/4<F,/Fo<1. 72 Fe/Bo Fig. 3.12 Diode conduction angle characteristics of the series resonant converter in the range 1/2 <F,/Fo<1. Note that for Q,<4/n the curves terminate at F,/Fo=@Q,n/4 as the converter enters discontinuous conduction mode, 73 3.3.2 F./Fo=1 The voltage and current waveforms in the range F,/Fp=1 are shown in Fig. 3.13. In this range the diode Di conducts first because Ip(0) is always negative. As the resonant current passes through zero at t=7, the transistor enters conduction and is switched off at t=T7, while still in conduction. Therefore, this range of operation is not very suitable for SCR applications since natural turn-off does not occur. It is clear that discontinuous conduction cannot occur in this range because 7, <7 /2. The analysis is similar to the analysis given earlier for the range 1/2<sF,/Fo=1. In this section only the final results are given. The details of the analysis are given in Appendix B.2. The implicit equation from which M is obtained is given by 14+M+MP?K G, (MQ. 7) = W(k+ iar CO V(1 —M*)(MK + 2)MK + M(K+i)t1 siny _1sM=MPK _ 1-M-K) (3.38) The conversion ratio characteristics are shown in Fig. 3.14 along with the characteristics in the range 1/2<F,/Fo<1. It was shown earlier for Q,<4/mn that the converter operates in dem in the range @,n/4=<F,/Fo<1. For such values of @, the converter once again enters ccm when excited above resonance. 74 VR (0); Fig. 3.13 Voltage and current waveforms of the series resonant converter when operating in the range F,/Fo2=1. 75 oF » 5.6 «67 68 9 1.0 1.2 1.2 1.3 1.4 1.5 F5/Fo Fig. 3.14 Conversion ratio characteristics of the series resonant converter in the range Fy/Fo2=1/2. 76 The output-network switching time, or the diode conduction time, is given by COSsO T. = 1+M+MPK O'e ~ M(K+1)41 (3.39) The voltages and currents for O<f£<T7, are given by Ip(t) = CoVq [1+ M(K+1)] sine t ~ 7a) (3.40a) Vat) = -V [: +M(K+1)] coswo(t Tu) + Vy(M +1) (3.40b) and for 7,<t<T, are given by Ip(t) = wolyoVy [2 -M + UK| sinag(t —T,) (3.41a) Ve(t) = -Vy [ -M +UK | cosao(t —T,) + %j(1-M) (3.41b) The initial values Jp(0) and Vp(0) are given by Vp(0) = —M® KV, (3.42) Ip(0) = —woCyV¥yV(1 —M*)MK(MK +2) (3.43) The peak stress levels are given by Vpcak = MKV, (3.44) tooCoVg| —-M+ MK ; 1-M+M®K <0 peak = woCoVyV(1—-M*)MK(MK +2) ; 1-M+M?K = 0 (3.45a-b) The peak value value of the resonant current is either given by the maximum value of /p{f) or by /r(7;) depending on whether this maximum 77 Ip (T,) <Tpeak= Ip(Ts) < I E ie) | p ~-~-t--4j-__-~ Fig. 3.15 Explanation of the change in the peak resonant current when the series resonant converter is operating in the range Fe/Fo=1. a) when y-woTy>=n/2, Ipex iS given by the maximum value of Ip(t) in the interval T, <t <T,. 0b) When Y¥~-WpTgsn/2, the maximum value of Ip(t) cannot occur and Teak = Ir Ts). 5 -6 7 8 9 1.0 1.1 1.2 1.3 1.4 1.5 Fs/Fo Fig. 3.16 Normalized peak resonant inductor current characteristics of the series resonant converter in the range Fy/ Foz 172. 78 16+ 144 12+ 10+ Vpeak/Ve oo Fig. 3.17 Normalized peak resonant capacitor voltage characteristics of the series resonant converter in the range F,y/Fo=1/2. 79 can occur or not in the interval 7,<t<7, as shown in Fig. 3.15. The boundary between these two cases, according to Fig. 3.15, is determined by y—wWpT, =7/2. From Appendix B.2 we have _ 1-M-MPK cos(y wT) 7 1-M(1-K) (3.46) If y-Wo7,=7/2, then the numerator in Eq. (3.46) is negative (since the denominator is always positive) and the peak is given by Eq. (3.45a), otherwise it is given by Eq. (3.45b). The peak current and voltage characteristics are plotted in Figs. 3.16 and 3.17 respectively. 3.4 General Analysis In this section the operation of the series resonant converter is given in the entire range O0<F,/Fo<~. The general discontinuous and continuous conduction modes are identified, and the conversion ratio for each mode of operation is determined. The various modes of conduction are discussed first before the details of the analysis are given. In Fig. 3.1e the switches Si and S2 operate alternately at fifty percent duty ratio and for a duration 7,;. In the following analysis, first, it is assumed that at no time S1 and S2 open simultaneously. Later, this restriction is removed in Sec. 3.4.7. Assume then that a turn-on pulse of duration 7, is applied to @1 and simultaneously a similar turn-off pulse is applied to @2 as shown in Fig. 3.1. Depending upon the duration of 7, and load conditions, Qi and D1 may conduct alternately for the entire duration of the turn-on pulse, 7, as may be the case for cem, or only during a Portion of 7;, as may be the case for dem. For example, in Sec. 3.3.1 it 80 was shown that when the converter operates in ccm in the range 1/2e<Fs/Fo=1, @1 and Di each conduct once in sequence during 7,, but in dem only @1 conducts for a duration 7)/2 while during the remaining time, 7,—T7)/2, D1 does not conduct. In the range F,/Fo<1/2 since T, may be many times larger than 7)/2, @1 and D1 may conduct several times during 7,. This will be investigated presently. There are two general types of dem. These are shown in Fig. 3.1Ba and 3.1Bb whence it can be seen that the resonant current, Jp{t), ceases to flow after completing n complete half-cycles each of duration T7)/2. During the remaining time, 7,-nT)/2, neither D1 nor @1 conduct even though @1 is still on. The reason for this discontinuity in the resonant current is that, under certain load conditions, the output bridge becomes reversed biased during the time interval 7,-n7)/2. In the previous section this was explained for the particular type of dem that occurs in the range l/2sF/Fo¥ il. The two different types of dem are shown in Figs. 3.1B8a and 3.18b. Each type corresponds to whether the number of conduction cycles, n, is even or odd, and is appropriately called odd or even type n dem. The reason for this distinction becomes clear when the conversion ratio for each type of dem is determined in the next section. Clearly, when n is Specified explicitly, the words even or odd become redundant, ie., type 3 dem instead of odd type 3 dem. From Fig. 3.1Ba or b it can be easily seen that if the converter is operating in type n dem (odd or even), then 81 s2 on 30 $4 ON S2 ON S41 OFF S2 OFF S41 OFF ‘I 14 (t) I, (t)4 , inna (t) A, \Lve [> | An-4 Ap A WZ 7 T, | dytt)! | Pan te ef l | | eo 7.— I at | D4 | a4 sj at; o4 | — | - = | Yo, kV peak Vt | ] boy bo 1Vp (t) | | | | | 1 oN. ; | 1 by rot | | | | | | Wog | ] | t=0 Fig. 3.180 Voltage and current waveforms of the series resonant converter in the general even type n discontinuous conduction mode, 82 s2 on 9 S1 ON S2 ON S4 OFF S2 OFF “) S4 OFF Vr in (t u 4 {t) Ty (t)4 , ip (t) A, \ Ve 7 Ap An-4 Wi 1 3 | | ing (t)! x To 7 ip (t) ; ' | a4 7. | | at | pa | as Sj; Of; Oty -\ = I | | _— Vo, IVneak Vea I | | | | | | | ! | 1 ; 1 | ; | | I | | J Von-4 | | 1 | ! Ven+4 | vy ; Weg cn ! ; \ \ \ | 4p (0) =Vo4 t=0 Fig. 3.18b Voltage and current waveforms of the series resonant converter in the general odd type n discontinuous conduction mode, 83 p oF ov ei n rN) o (3.47) The range of the control parameter given in (3.47) is only a sufficient condition for the occurrence of type n dem, For a given 9, a necessary and sufficient condition for the occurrence of type n dem is given by 0 < C2(Q.n) < Pee C1(Q,.n) < Fo 5 | (3.48) where C,(@,n) and C2(Q,n) are determined in Sec. 3.4.3. In continuous conduction mode the resonant current never ceases to flow. According to Fig. 3.19a and b there are two different types of ccm depending on whether Jp(0) is positive or negative at the beginning of the switching cycle when S1 is closed. In Fig. 3.19a since /p({0)>0, @1 conducts first for a duration 7, after which D1 and @1 conduct alternately for n complete half-cycles, each of duration 79/2, and after which D1 conducts for a duration 7p=T,-n7T)/2—-T, until Si is opened and S2 is closed. Cyclic stability requires that if @1 conducts first then Di conducts last, which implies that the number of half-cycles n, denoted by i,(t) through i,(f) in Fig. 3.19, must either be even or zero. This type of ccm is therefore appropriately called +fype nm ccm, where the positive sign refers to the fact that Jp(0) is positive, and n refers to the conducted number of complete half-cycles. It can be seen from Fig. 3.19a that n=0 corresponds to the case of cem in the range 1/2<F,/Fo<1 discussed in Sec. 3.3.1. According to Fig. 3.20a if the converter is operating in +type n ccm, then 84 V s2 on 9 Si ON $2 ON Si OFF~ S2 OFF S41 OFF VF ig, (t) As M in (t) i 2 ipa (t) ol t28 i, (t) Xa ¥ Ago |r. eB Aa An T — = ng! WY | | fj ts -a 14 (t} | { ; ; r 1 | T 1 s > pe} at ,; DA | ot a1 'p4| a2 — vel | & “Vg peak Vc4 I (LLL Ld t {ot eee ee 2 —| Z en Fig. 3.194 Voltage and current waveforms of the series resonant converter in the general +type n continuous conduction mode. Fig, 3.200 Expanded view of the currents ig,(t) and ige(t) in Fig. 3.190. 85 Vv s2 on 9 S1 ON S2 ON S1 OFF S2 OFF Si OFF om Vv ! T T x 2 te! Ty (t)] te) | Hos ipa (t) —_ INES Mosc A<Ago ¢ NJ A, 1 4, (t)! | } ! | ! I ! a2] | D1 1G4;y 92 “Vi ' 11 pe 1 Pony I 1 nTg ly T.— —S ta me s— 5 >" — | I leVon ly 1 J Ii 1 4 t |} j Py 4 ! ry t ! >A fa ty t=0 ‘ Fig. 3.196 Voltage and current waveforms of the _ series resonant converter in the general —fype n continuous conduction mode, Fig. 3.20b Expanded view of the currents ig,(t) and ige(t) in Fig. 3.19b. 86 To nfo 1 Fs 1 <1, <To or oye * F * nal (3.49) The range of the control parameter given in (3.49) is only a sufficient condition for the occurrence of +type n ccm. For a given @ a necessary and sufficient condition for the occurrence of this type of cem is given by 1 Fs 1 nee < Fy FG@n) = (3.50) where 8,(@,n) is determined in Sec. 3.4.6. The other type of cem, which is called —type n ccm, is shown in Fig. 3.19b. In this case it can be seen that /p(0) is negative so that D1 conducts first for a duration 7, after which @1 and D1 conduct alternately for n complete half-cycles and after which 91 conducts for a duration Tp=7T,—-nTp/2—-T, until Si is opened and S2 is closed. At ft =7,, when @1 is switched off and @2 is switched on, D2 conducts first and the current is diverted from @1 to D2. As before, n can only be even or zero. It can be easily seen that n=0 corresponds to the case of cem in the range F;/Fo=1 discussed previously. According to Fig. 3.20b, it can be seen that if the converter is operating in this type of ccm then xy nTo _ To or 1 <- 2S 2. 2 nt+1 1 <n (3.51) a As before, the range of the control parameter in (3.51) is only a sufficient condition for the occurrence for this type of cem. A necessary and sufficient condition for the occurrence of —type n ccm is given by 87 1 Fs < < B2(Q,.n) < “ (3.52) where #2(Q,n) is determined in Sec. 3.4.6. The results of the previous discussion are summarized in the following table. Mode Conduction Sequence Frequency Interval even type n dcm odd type n dem Q1 D1... Qt D1 ee” n complete half-cycles Fee Q1 D1....Q1 D1 Qi F, 0 <Co(Q,.n) < =< C,(Q.n) < 1 neven Fo n 0 <Co(@.n) < “<0,(Q.n)<+ nodd Fo n +type n ccm ~type ncem Q1 D1 Q1.... D1 Q1 D1 —- n complete half-cycles Paneer wae D1 Q1 D1... Q1 D1 Q1 F. s— <B,(Q.n)< 1 n+2 Fo nt+1 n=0,2,4,...even 1 fs 1 < <a att Fo <82(Q@.n)< a 3.4.1 Even Type n dcm The voltage and current waveforms for even type n dem are shown in Fig. 3.18a. The current i,,(t) is easily determined im(t) = cagCo | Vg + (—1)" Vo — Vor sinWof (3.53) In Eq. (3.53) Vim is the value of Vp(t) at the beginning of each complete half-eycle, and the time reference is taken at the beginning of each cycle, 88 im(t). The area, Ay, under the current i,,(f) is given by Tf? Am = fy” lim(t)|dt = 2Cp|¥y + {—1)"Vo— Vera] (1) The average output current can be expressed in terms of A, _ Vv _ 1 fis 1 Io = Rp To |Ip(t} | dt = Te Am 3 m= which gives n m=1 The input and output powers are given by 1 Ts _ V, Te Ip(t)V,dt = a) Am(—1)"*! S m=] 1 Ts _ Vo n Ty Uelt)| Vodt = 7 Am Substitution of Eq. (3.56) in (3.58) and the requirement Pz, = Png gives SY Ag(—1)™*1 Me m=) 2CpVoK (3.54) (3.55) (3.56) (3.57) (3.58) (3.59) According to Fig. 3.18a the peak-to-peak capacitor voltage, 2Vpenx, is given by 89 = 3 An(-1)* 24, m=1 (3.60) and Vp(0), or Vegi, is given by Ver = Veo, - A= - BOp Am 1)™*1 = —MKYV, or Peak Co aR m ° (3.61) In Eq. (3.61) the last equality is obtained by using Eq. (3.59). In Eq. (3.54) Vem is unknown except for m=1, however, by successive application of Am-1 = —j})}m a recursive expression for Am is obtained in Appendix B.3 given by Am = 2CoV, (1 —(2m-1)M +MPK (3.63) When Eq. (3,63) is substituted in (3.59), the summation for even n gives (Appendix B.3) Mz ne en Fs K Qn Fo (3,64) which is a linear function of the load Af and the control parameter. Equation (3.64) also gives for the normalized average output current T Moho _ on Fs, o'Vy n Fo (3.65) which is independent of the load #. Therefore, when the series resonant converter is operating in an even type dem, it acts as a true current—fed 90 converter. The peak stress levels, determined in Appendix B.3, are given by - _2n_, ne Vpoak wle KK (3.68) [ 2 nh n Ipeak = woCoVs |! K K (3.67) 3.4.2 Odd type n dcm The only difference here is that n is odd in Eq. (3.59). In Appendix B.3 it is shown that for n odd M is given by -i Meo (3.68) which implies that when the converter is operating in an odd type dem the conversion ratio is insensitive to F;/Fo and FR. The special case for n=1 was discussed earlier. Clearly this mode of operation should be avoided since the output in this mode is uncontrollable. The peak stress levels are given by 2 K peak Tn ne? (3,69) f CoV, , + K =o -a2 4 peak 00 a | ne (3.70) 91 3.4.3 Boundary between two Discontinuous Conduction Modes Assume that the converter is operating in even type mdcm. The conversion ratio according to Eq. (3.64) is given by = 71Q; Fo (3.71) and is plotted in Fig. 3.21 for three different cases. These are straight lines through the origin with slope 2m/7Q@,. The conversion ratios for odd type (m—1) and odd type (m+1) dem, given by Eq. (3.68), are also plotted in Fig. 3.21. Since a sufficient condition for type m dem to occur is that F,/Fo<i/m, Eq. (3.71) is valid in a region to the left of F,/Fp=1/m on the M vs. F,/Fo plot. In case I, as the switching frequency is increased, M increases linearly until the converter enters odd type (m-1) dem at (F,/Fo),. If Fs/f'p is increased further , say up to (F3/Fo)g, M will stay at a fixed value of 1/(m—1) because of operation in odd type (m—-1) dem. (To complete the discussion for all F,/Fo=(Fs/Fo)c. we need to know results from operation in cem given in the next section, but for the present discussion we need only consider the results obtained for dem). If F3/Fo on the other hand is decreased, M will decrease linearly until the converter enters odd type (m+1) dem at (F,/F po)». If F;//p is decreased further, say down to (F,/Fo)a, M will remain at a fixed value of 1/(m+1). Therefore, Case I represents an even type m dem that is bounded between odd type (m~—1) and type (m+1) dem. Case II on the other hand, which corresponds to a larger value of @ than case I, represents an even type m dem that is bounded between odd type (m+1) dem and +ttype (m-2) ccm. To understand this, we simply determine which of the two general types of 92 +type (m-2) ccm odd type (m-l)~ dem 1 >» m=-1 1 -- odd ml type (m+1) 1 dem - 1 1 Nt -type m ccm | i | ! m Fig. 3.21 Boundaries of even type n discontinuous conduction mode. even type (mtl) dem even type (m-1) dem ~otype (m-1) ccm +type m ccm |- Jee F,/Fo GE) am =] Fig, 3.22 Boundaries of odd type n discontinuous conduction mode. 93 cem has a lower bound given by the reciprocal of an even integer. An inspection of (3.50) and (3.52), in both of which n can only be even or zero, reveals that only the lower bound of +type n ccm, given by 1/(n+2), can correspond to i/m. This implies that n=m-~-2 and the continuous conduction mode in question is +type (m—2). The third case represents an even type m dem that is bounded between —type m ccm and +type (m-2) com. In this case as F,/Fp) is decreased, the converter enters —type m ccm in the range 1/(m+1)<F,/Fo<1/m when M=1/(n+1). The reason how this happens is given in Sec. 3.4.6 where the boundaries between dem and cem are determined. The lower bound C2(4,.n) in (3.48) can now be easily determined. From Fig. 3.21 it can be seen that at the lower bound M=1/(m+1) which according to Eq. (3.71) gives Q C2(@,.m) = ome Fi) (3.72) Also, according to Fig. 3.21 a necessary condition for this type of dem to occur is that C2(Q,n)<1/m which gives @ = (3.73) This last condition covers all three cases in Fig. 3.21. The upper bound in case I can be easily determined by setting M=1/(m-—1) which according to Eq. (3.71) gives 94 Q, 7 (F3/ Foe = Sm(m-—1) (3.74) The condition on Q@ for case I to occur is obtained by requiring (F,/Fo)-¢<1/m which from (3.74) gives 2(m—1 &=— (3.75) According to Eqs. (3.73), (3.74), and (3.75) the upper bound C,(@,n) in (3.4B) is given by Q37 . em(m—-1) g, < ma) T 2(m—1) <Qg,< 2imt1) T ~ se 7 C,(@,m) = 1 m (3.76a-b) A complete description of the conditions on Q for this mode to occur and the range of F;//'p in which it can occur can be summarized by the following ( a fs _ &r . Qs < 2(m—-1) em(m+i) Fp 2@m(m-—1) 1 type m dem Qs Fs Ji 2(m—1 2(m+1 < <— ; <@,< 2m(m + 1) o m 1 7 { (3.77a-b) Alternately for a given F3/Fo<1/m the range of Q in which even type m dem can occur is easily determined from Eqs. (3.72) and (3.74) 95 @m(m—1) fs — —em(mti) ¥s | Fs 21 7 Fy 7 ~ om Fo ' Fo (3.78) In odd type m dem the conversion ratio according to Eq. (3.68) is constant. This is shown in Fig. 3.22 by a straight line parallel to the F,/ axis and extending to F,/Fy=1/m. There are three cases to be considered as shown in Fig. 3.22. In the first case the value of @ is such that as F,/Fo is increased the converter enters even type (m~1) dem at (F5/Fo)c. and when decreased the converter enters even type (m+1) dem at (F,/ Fo)e. For the second and third cases, as F,/F is increased the converter enters -type (m-1) ccm at F,/Fo=1/m, and when decreased, the converter enters even type (m+1) dem in the second case and +type (m—-1) ccm in the third case. The boundary between odd type m dem and +type (m-1) ccm in the third case is discussed in Sec. 3.4.6 where it is shown that at this boundary K is equal to m(m+1) just like at the boundary between even type (m+1) dem and odd type m dem in the first two cases. Proceeding exactly as before, we can determine the conditions on @, for the occurrence of odd type m dem. The results obtained are identical to the results obtained for even type m dem given by Eqs. (3.77) and (3.78), Therefore, these equations give a complete analytic description of odd and even types of discontinuous conduction mode. The boundaries between even and odd type discontinuous conduction modes can be put together as shown in Fig. 3.23. The conversion ratio characteristics for even type type m dem, which are linear functions of F,/Fpo, are confined to the shaded areas, and _ the characteristics for odd type m dem, which are insensitive to F,/Fp, are 96 type 1 dcm: = +type 0 ccm -type 0 type 3 dem; ecm A 3 5 5 5 lf o Y 5 f eI N N type 5 demf 2! y v 7UA: Teal &] & type 7 dem}+~ : BRI 2 ” 417 lia 1 1 cee u OF z Fe/Fo * n=4 =2 n=0 Fig. 3.23 Boundaries of odd and even type n discontinuous conduction modes in the entire range O<F,/Fo <=. The characteristics for even type discontinuous conduction modes are confined to the shaded areas, whereas the characteristics for odd type discontinuous conduction mode are confined to straight lines parallel to the Fo/Fo axis. The characteristics for continuous conduction mode are confined to the unshaded areas, 97 confined to lines parallel to the F,/Fp axis. The areas, to which the conversion ratio characteristics of the various continuous conduction modes are confined, can be easily identified by using (3.50) and (3.52). In Fig. 3.23 examples are given for n=0,2 and 4 which correspond to 1/2sF5/Fos1 and Fy/Fo=1, 1/4$F,/Fo<1/3 and 1/3<F,/Fo<1/2, and 1/6<F,/Fo<1/5 and 1/5<F,/Fo<1/4 respectively. Note that the subdivision of F,/Fy axis according to (3.50) and (3.52) is only with respect to the continuous conduction modes. For example, the proper way to refer to the range 1/4<F,/Fy)<1/3 with respect to cem is ; n=2 and not 1/(n+1)<F,/Fo<1/n in which n=8, Therefore, investigation of the possibility of occurrence of ccm in an arbitrary range requires proper splitting of the arbitrary range. For example, a range such as ASF,/Fo<.75 is split into 4<F,/Fpo<i/2 and 1/2<F,/F os.75. The first range belongs to 1/(n+1)<F,/Fo<1/n for n=2, and consequently only —type 2 ccm can occur in this range, whereas the second range belongs to 1/(n+2)<F,/Fo<1/(nt+1) for n=0, and only +type 0 com can occur in the Second range. This completes the discussion on the boundaries between two general discontinuous conduction modes. The boundaries between discontinuous and continuous conduction in the ranges 1/(n+2)<F,/Fosi/(nt1) and 1/(nt+1)<F,/Fo<1/n are determined in Sec. 3.4.6. 98 3.4.4 +type n cem In this section the most important steps for the determination of M are given. The details of the analysis are given in Appendix B.4. The yoltage and current waveforms are shown in Fig. 3.19a. As before, the expression of the output voltage in terms of the areas under the resonant current is given by R n =J,>R = =~ + +5 A Vo = Ig T, {Aoi + Aoz >») m) (3.79) When the input and output powers are set equal, an expression for M in terms of these areas is obtained which is given by Ao: ~ Aoz + 3 An(-1)™ M = m=] Aoi tApet+ >, A Q1 02 > m (3.80) The current i,,(¢) is given by im(t) = woCo| Vy + Vo(—1)™*! — Vim|sinwot (3.81) In Eq. (3.81) Vim is the initial capacitor voltage at the beginning of each complete half-cycle, i,{¢), and the time origin is taken at the beginning of each cycle. The recursive relation for A, is determined to be An = 2C5 Vey (2m —1)Vo-V (3.B2) and the summation in Eq. (3.80) is obtained for even n 99 n > Am(-1)™ = -2nCpVo m=1 (3.83) The currents io,(t) and igg(t), which are shown in detail in Fig. 3.20a, are given by igg(t) = woCaV, HRB sin(wot —c0Te +7) (3.85) The corresponding areas Ap, and Apg are given by Ao = CoV, [1-H + HK |[1 ~secwTa (3.86) Aoz = CoV, [mex —m -1] [sec(y—wo7a) —1] (3.87) Substitution of Eqs. (3.83), (3.86) and (3.87) in (3.80) and the requirement ip,(0) =to2(0) give two simultaneous trigonometric equations in w 97, which when solved together give 1-M+M?K 1—-M(n+1)"-MK G,(M.Q5.y.n) = (n +1) cosy _ [MK +n][M?(n+1)? -1][n+2—-MK] 1-M(n+1)?-MK siny M®K-—M-—-1 1+M(n+1)?-MK (n+) =0 (3.88) 100 Equation (3.88) is the implicit equation, which when solved numerically, gives the value of M for a given @ and F,/Fp. The peak stress levels are given by | MK-1 = + ‘peak Vy L+nM+ ntl (3.89) Ho C' V, Teak = <i [M(n+1)*°+MK—1] (3,90) 3.4.5 -type n ccm ‘A similar analysis, given in Appendix B.5, for this mode of operation gives 1+M+M?K 1+M(n+1)?+MK G,-(M,Q,.7n) = (n+1) cosy V[MK-n][ 1 —M?(n+1)?][n+2+MK] 1+M(n+1)?+MK + siny 1-M—M?K _ i-Min+ife+mR ot) = 9 (3.91) which is another implicit equation which must be solved numerically. It can be seen that Gi(M,Q..y.n) = G(—M,—Q,—y,n) (3.92) The peak stress levels for n#0 are given by 101 _ _ 1+M(n+1)?+M@K - Vpcok = Vg j1-3M + nal n=2,4,6..... (3.93) _ trp Cy Vg 2 —_ Ipoak = —— [1+ M(n? -1+K)] n=2,4.6..... (3.94) For n=0 the peak stress levels are given in Sec. 3.3.2. The reason the results for n=0 are different from the results for n#0 can be seen from Fig. 3.19b. For -type n com Vg, occurs at the end of i,(#) which for n=0 does not exist and consequently Vigg, is given by Eq. (3.44) of Sec. 3.3.2. The same reason applies for Jpeq,, and consequently for n=0 Jpeak iS given by Eq. (3.45a-b). Before the conversion ratio characteristics are plotted for the two general types of ccm, the boundaries between continuous and discontinuous conductions mode will be determined. 3.4.6 Boundaries between Discontinuous and Continuous Conduction Mode For —type n ccm to occur in the interval 1/(n+1)<F,/F)<1i/n, an examination of the radical in Eq. (3.91) reveals that 1 MK2>n and M<« ne (3.95) Therefore, the maximum value for —type n ccm is 1/(n+1). This explains the boundary line between -type n cem and even type n dem in the interval 1/(n+1)<F,/F)<1/n which in Fig. 3.23 is shown by dashed lines for n=2 and 6. As explained earlier, in the range F,/Fp=1, which corresponds to n=0, dem cannot occur. This is consistent with the 102 nomenclature, since such a dem would have to be called type 0 dem which implies that the resonant current is zero all the time which corresponds to no load conditions. According to (3,95) it can be seen that K =n(n+1) (3.96) which implies Q, > Fs en(n +1) or Fs < QT The first inequality in (3.97) gives the condition on 4 for —type n ccm to occur which can be rewritten as g, < fs 2n(nti) dem 1 he Fo on nti My nigs =e Bota t)) -type n com 0 Tr (3.98a-b) For example, F,/Fp=.4 falls in the interval 1/3<F,/Fo<1/2, which corresponds to n=2e. Therefore, according to (3.98) for Q>=(.4)12/n, the converter operates in -type 2 ccm at F,/Fy)=.4, otherwise it operates in dem. A comparison of (3.97) and the necessary and sufficient condition for the occurrence of —type n ccm given by (3.52) shows that 2n(n+1) 7 1 Be Qn) = i . Q> 2(n +1) n ty er (3.99a-b) 103 An examination of the radical in Eq. (3.88) reveals that, for +type n cem to occur in the interval 1/(n+2)<F,/F)<1i/(nt1), the following must be true 1 n+l (3.100) MK >nt+2 and Ms Therefore, as in the case of —type n ccm, the maximum value of M for +type n ccm is 1/(n+1). This explains the boundary line between +type n ecm and odd type (n+1) dem in the range 1/(n+2)<F,/Fo<1/(nt1) as shown in Fig. 3.23. According to (3.100) it can be seen that K => (nti)(n+2) (3.101) which implies that Q. > Fy e{n+1)(n+2) or Fy < Q, 7 Ss rT The first inequality in (3.102) gives the condition on Q, for +type n cem to occur in the range 1/(n+2)<F,/ Fo <1/(n+1), which can be rewritten as 1 Fs < 1 8 ” a F. nt+e Fg n+1 Q, = 2(n+1}(n+2) Z +type n com 7 0 (3.103a-b) For example, F,/Fo=.3 falls in the interval 1/4<F,/Fo<1/3, which corresponds to n=2e in the range 1/(n+2)<F,/Fo<1i/(nt+1). Therefore, according to (3.103) for @,=(.3)24/m the converter operates in +type 2 ccm at F,/F,=.3, otherwise it operates in dem. 104 A comparison of (3.103a-b) and the necessary and sufficient condition for the occurrence of +type n ccm given in (3.50) shows that @, . B(nt1) 2(n+2) Q(nt+1)(n+2) 7 << T By(@,.n) = 1 ; Q> e(n+e2) nt+1 , cot (3.104a-b) For a given f,/Fyo the boundary between discontinuous and conduction modes can be determined as a function of Q, either from (3.98) or (3.103) depending on which range F,/F falls in. To complete this discussion, the boundary between dem and ecem for a given @ as a function of F3/F9 is determined next. Consider now two adjacent cases of ccm corresponding to n and n-2. According to 8)(@n-2) in (3.104b) if @2>2n/m the converter operates in +type (n—2) ccm in the entire range 1/n<F,/Fo<1/(n-1). Also, according to Bz(@.n—-2) in (3.99b) the converter operates in -type (n—2) com in the range 1/(n-1)<F,/Fo<1/(n—2). In fact, for Q,=2n/7 ecm occurs for all /;/F9=1/n, because the conditions for cem according to B,(@,n-m) in Eq. (3.104b) and B2(Q@,n—-m) in Eq. (3.99b) — for m even -— is met by @2=2n/mn. Consider now two adjacent cases of ccm corresponding to n and n+e. If Q<2(n+1)/m then according to B,(Q@,.n) in (3.104b), the converter operates in dem in the range 1/(nt2)<F,/Fo<1/(nt1). Also, according to Be(Q.n+2) in Eq. (3.99a) the converter operates in dem in the range 1/(n+3)<F,/Fy<1/(n+2) since Q@<2(nt+1)/m satisfies the condition @<2(n+2)/7m. In fact, the converter will be in dem for all F,/Fo<1/(n+1), since the condition for dem 105 according to B,(@,n+m) in (3.104a) and B2(@,n+m) in Eq. (3.99a) — for m even — is satisfied by Q,€2(nt+1)/ nr. summarized by the following Fs -_t Fo ~n+i 1 fs .__&7 2n << 2(nt1) n+l Fo en(n+1) no Sg Qn Fs 1 gn{(n+1)° Fp on fy Fo — n A similar discussion gives Root Fo nti — %T OA T Sot 1 <fs Qs nt+2 Fo e(nt+1)(nt2) fs Fo n+2 Therefore, for a given determined as a function The preceding discussion can be dem —type n ccm even type n dem com (3.105a-d) ecm odd type (n+1) dem +fype n ccm dem (3.106a-d) @ the boundaries between dem and ccm are of F;/ Fo either from (3.105) or (3.106) depending on which range 4; falls in. For example if Q,=4 then of the following two ranges n+2> ar nti Ss (n even) (3.107) Qsé | 2(nt1) T > 2(ns2) | 106 oO xv) usae n+l > 2(n+1) T n+2 ntl n Fig. 3.24 Conversion ratio characteristics of +type mn continuous conduction mode in the range 1/(n+2)<F,/Fo<1/(n+1) and -type mn continuous conduction mode in the range i/(nt+1)<F,/Fosi/n. The boundary between +type n com and odd type (n+1) dem is at the line M=1i/(n+1), and the boundary between —type n com and even type n dem is at the dotted line M=1/(n+1). The shaded regions [ and If correspond to values of Q; in (3.105) and (3.106) respectively. 107 ‘os l7/S75Q abun. aayua ay] UL 137.48NU0I JUDUOSAL Sauias ayy fo SojSWaJDDLDYS OFDL UOWSAINUOD 7DLIUaD Ge'E 25/85 L ; jn g's 108 =n (n even) (3.108) only (3.108) is satisfied for n=6, and the boundaries as a function of F/ Fo are given by (3.101). On the other hand if @=2, then only (3.107) is satisfied for n=2, and the boundaries as a function of F,/Fp are given by (3.106). The general boundaries between dcm and ccm for a given even n are shown in Fig. 3.24. The characteristics corresponding to the two ranges of @, given by (3.105) and (3.106) are shown in the shaded regions IJ and II respectively. The general conversion ratio characteristics are plotted in Fig. 3.25 for n=0, 2, and 4. It can be seen that these characteristics admit multiple resonant peaks at F,/Fp=1,1/3,1/5...1/(n+1) with corresponding values of M=1,1/3,1/5... at the peaks. 3.4.7 Operation in the Special Case When Qi and Q2 Are Switched Off Simultaneously Until now the switches S1 and S2 were allowed to conduct for the entire duration of 7;. If conduction is inhibited by switching off Q@1 and @2 before the end of 7, then the converter can operate only in a finite number of continuous and discontinuous conduction modes. To illustrate this, consider the operation of this circuit when 5CRs are used which are fired only once during 7;. In this case, when the circuit is excited in the range F,/Fo<1/2, the switches Si and S2 conduct only once during the forward and reverse directions. Consequently, the only allowed modes of operation in the range F,/Fo<1/2 are type 1 dem and type 2 dem. In the 109 1.0 a) = 35 + F./Fy 1 1.0 b) = 5 4 Y/) 1] 1 w 3 2 F,/Fo , Fig. 3.26 a) Conversion ratio characteristics of the series resonant converter when SCRs are used or when @1 and @2 are allowed to conduct only once during T,. 6) The same characteristics when @1 and @@ are allowed to conduct at most twice during T,. in the first case only one resonant peak at F,/Fo=1 accures whereas in the second case only two resonant peaks, at Fy/Fo=1 and F,/Fo=1/3, occur. Im general if Q@1 and Q2 are allowed to conduct only j number of times, only j resonant — peaks occur. 110 range F,/Fo=1/2 the operation remains unaffected since, as before, in this range Si and S2 conduct only once as restricted by the switching time 7,. The conversion ratio characteristics for this case are shown in Fig. 3.26a where it can be seen that in the range F,/Fo<1/2 only those characteristics corresponding to type 1 and type 2 dem are present. In the range F,/fo9=1/e the characteristics remain unchanged. Also, since no other mode occurs in the range F,/Fp<1/2, the lower bound of type 2 dem is now zero instead of C2(Q,,2)=@m/12 as required by (3.48) and (3.72). In general, @1 and @2 can be made to conduct at most 7 number of times. In Fig 3.26b the characteristics for j7=2 are shown. In this case the maximum allowed discontinuity is type 4 dem and consequently C2(Q,,4)=0. The M for type 2 dem is now confined, as in the general case, between M=1 and M=1/3 because of the occurrence of type 3 dem and —type 2 ccm. Also, in addition to the first resonant peak at F,/F)=1, the second resonant peak at F,//)>=1/3 occurs because of the occurrence of ttype 2 ccm. In general, if @1 and @2 are allowed to conduct at most j number of times then only jg resonant peaks occur. at Fy/Fo = 1, 173, 1/5,..1/(27-1) and C2(Q,,27) =0. This completes the de analysis of the series resonant converter in the general continuous and discontinuous conduction modes. lil 3.5 Nonidealities As in the case of the parallel resonant converter, the only two nonidealities considered in this section are the voltage drop in the output rectifier bridge and the parasitic resistances in the tank and the output filter circuits. The voltage drop in the output rectifiers is considered first. The two methods of output rectification are shown in Fig. 3.27a and b where the isolation transformer is assumed to be ideal and of unity turns ratio. The nonideal circuit is shown in Fig. 3.28a where the output network is shown reflected to the primary side. All the diodes in this circuit are assumed to be ideal. The voltage drop in the output rectifiers is separated and is represented by a voltage source Vgp in series with an ideal diode. For the center-tapped circuit Vgp=Vp and for the full-bridge circuit Vep=2Vp. As in the oase of the parallel resonant converter, an ideal circuit, shown in Fig, 3.28b, is determined which delivers the same power f,' at the input side of the bridge as the nonideal circuit. The conversion ratio, M'=V'/¥V,, and the load parameter, Q,'=wpLo/F', of the ideal circuit are related to My =Vo/V, and @ =Wolio/F of the nonideal circuit by M' = M+ Vy V, (3.109) [ Vap R ts 1- or R= — The derivation of Eqs. (3.109) and (3.110) is the same as the derivation given in Sec. 2.4 for the parallel resonant converter and will not be 112 | | Fig. 3.27 Two methods of implementing the isolation transformer in the series resonant converter. a) Center-tapped and b) full-bridge. Pin Ip(t) Lo Co Pin Ip(t) Lo - Pout Cy All diodes Cs ore ideol tly tk wy wy - Rt - Ri+ Vo V Fig. 3.28 a) Nonideal circuit with voltage drop Vpn in the output rectifiers and b) ideal circuit that delivers the same power Fi,,' as the nonideal circuit in a). 113 repeated here. According to Eq. (3.109) M' can be measured directly by measuring the voltage Vg at the input side of the bridge which is a square wave of amplitude Vg+Vgp. For the parallel converter M', given by Eq. (2.41), cannot be measured directly. M' is determined from Eqs. (3.64), (3.88), and (3.91) in which K is modified to K" by Oy 2 (3.111) [ Ven g M K'= Kp _ i Vap Pas Since for odd type n dem the conversion ratio is insensitive to the load and control parameters, Eq. (3.68) remains unchanged, and we have ‘= L = iL —_—_ M n ° Ma n Vg (3,112) Substitution of Eq. (3.111) in (3.64) gives for even type n dem V; 1D BD - mo WaK ty oF Ma= (3.113) Therefore, the actual conversion ratio for even type n dem remains unchanged. In Appendix B.6 a numerical method for determining M' and M for continuous conduction mode is discussed. The modification in the boundaries between dem and cem are discussed by way of an example in the section on experimental results. The efficiency due to the losses in the output rectifiers is the same as for all other converters and is given by 114 _ 1 TR ~ TE Vpp/ Vo (3.114) According to Eqs. (3.109) and (3.114) the 7p can be written as np = 17 Vap 2 H'V, (3.115) The actual conversion ratio can now be written as Ma = NpM' (3.118) Equation (3.115) is plotted in Fig. 3.29 for two different cases of Vgp/ V, and @, in the range F,/Fo=1/2. The efficiency near the resonant peak is higher since M is larger near the peak. In a closed loop regulator, for a given output voltage requirement, the efficiency 7p is fixed and is given by Eq. (3.114). The curves shown in Fig. 3.29 do not suggest the converter should be operated near the resonant peak; all they show is the behavior of Np for an open loop converter. The values of Vgp/V, chosen for the series resonant converter are less than those of the parallel converter (Fig. 2.17) because in the useful range of the operation the conversion ratio of the parallel converter is greater than unity while that of the series resonant converter is always less than unity. Therefore, for a given Vp/V, the series resonant converter requires a step-up transformer to match the conversion ratio of the parallel converter. Consequently, the reflected value of Vgp/ V, will be reduced by the step-up turns ratio. Since M is smaller for larger values of @, the values of Vgp/ V, chosen for @,=10 are smaller than those chosen for Q=3. It should be noted that when the parallel resonant converter is excited well beyond resonance, or if @ is selected very small, 100 90 20 10 100 90 80 + 70 ~ 60 40 fF 30 20 10 50 F 115 oe r Vpp/Vg = +04 5 6 ./ 8 9 1.0 1.1 1.2 1.3 1.4 1.5 Fs/Fo L Q, = 10 a) -6 | 8 9 1.0 1.1 1.2 1.3 1.4 1.5 Fs/Fo Fig. 3.29 a-b) Efficiency of the series resonant converter due to losses in the output rectifiers for two different values of Vpp/V, and 43. 116 = 3 100k {2s - oe 90 Q, = 10 80} 70} xe 60F a) 50F 40} 30 Q, = 400 20k Q- = 100 10} 5 6 7 8 9 1.0 1.1 1.2 1.3 1.4 1.5 F/Fo 100 fs" 3 —_— 90 Na, 10 80 + 70 f+ 60 F b) ®&® 50 F 40 F 30 + Q,. = 200 20 t Qe = 100 10 F 5 6 7 8 9 1.0 1.1 1.2 1.3 21.4 1.5 Fs/Fo Fig. 3.80 Efficiency of the series resonant converter due to losses in the parasitic resistances, and Qs; = 100. a) @=400 and @,=100 and b) @,=200 117 M will be less than unity as in the case of the series resonant converter, but since this type of operation is not useful, the choices of Vgp/V, made were relevant. The efficiency due to the losses in the parasitic resistances in the tank circuit and the output filter capacitor is given by 1 n = ? Lr, 1,2 Te, 1+ 7 24 - Ip? Ro Ip? FR (3.117) where 7, is the total parasitic resistance in the tank circuit, TG is esr of the output filter capacitor Cy, and J, and J, are the rms tank current and output capacitor current respectively. As explained earlier in Sec. 2.4, in the presence of the parasitic resistances, the ratios J,//Jp and J,/Jp are assumed to remain unchanged from the ideal conditions. In Appendix B.7 these rms values are determined and 7, is given by 1 Np = i G&D ii. i , eG a1e& (3.118) where D, is given in Appendix B.7 and @, and Q@, are the normalized loss parameters defined as WoL, i Qa and Gao Equation (3.118) is plotted in Fig. 3.30 for two different cases of Q . Q, and Q,. 118 3.6 Experimental Results The experimental circuit used is shown in Fig. 3.31. The isolation transformer is eliminated in order to avoid additional losses. The circuit parameters used for the verification of the conversion ratio characteristics are Vy=19.5V Lo=.19%mH Co=.051pF Fy =50.21kHz The conversion ratio for the various modes of operation is verified in the range f,/Fo21/4 as shown in Fig. 3.32a and b, The agreement between the predicted and measured results is generally good except near the resonant peaks at F,/F')=1/3 and F,/Fy)=1 for high Q where the effect of the parasitics is dominant. The only nonideality considered in obtaining the predicted characteristics is the voltage drop in the output rectifiers, which was taken to be .7V, ie, Vep=1.4. Certainly, an analysis with all the parasitics considered (including core losses) will result in better agreement near the resonant peaks, however, such an analysis would be complicated and unnecessary because it is undesirable to operate this converter near the resonant peak. Figure 3.33 shows predicted and measured voltage and current waveforms with the following circuit conditions W,=19.5V F./Fo=.75 @,=1.94 The following table summarizes the results relevant to Fig. 3.33 119 f D44H10 42uF 1.35mH , en Vin bn —_—e Ss 42uF .051pF | D45H10 f All diodes 32uF $ UES1303 Ve VCO Buffer Fs ——e— Intersil > i 8038 DS0026 Fig. 3.31 Experimental circuit for the series resonunt converter. 120 b) =° 1/4 1/3 Fs/Fo 1/2 Fig. 3.32 a) Measured and predicted conversion ratio characteristics in the range 1/2<F,/Fo<1.5 and 6) in the range 1/4<F,/Fosi/2 5V/div. .2A/div. 20V/div. .2A/div. 121 VAS c \L A Vi >| |e 5usec Fig, 3.33 Experimentally measured waveforms, left, and predicted waveforms, right, of the series resonant converter. a) The resonant current, Ip(t), and the voltage, Vp, at the input side of the bridge. 6b) The resonant capacitor voltage and the resonant current. The circuit parameters are given in the text, 122 Predicted Measured Fpeak 72A TA Vocak 50.33V 49V Vp 13.7BV 138V To B.?usec B.9usec Figure 3.34 shows how the operation changes from one mode to another for a fixed value of F,/ Fp) as @ is varied. The boundaries between the various modes of operation are verified now by way of this example. The circuit parameters for the results in. Fig. 3.34 are V,=15V F/Fo=.4e @; = .49,1.94, and 5.18 First, F,/F'p = .42 falls in the range ot IA IA B |e =) I rs) pB + be =] a and consequently, the only allowed modes of operation are —type 2 ccm, type 2 dem, and type 1 dem. The occurrence of these modes is verified in Figs. 3.34 a-c. Now we will verify the conditions on Q@ for each of these modes to occur. According to Eqs. (3.98a-b), -type 2 ccm occurs if Ge Py Ah | (3,120) Fig. 3.34 123 . 1A/div. a) . 5usec/div. .1A/div. b) . 5usec/div. .lA/div. c) 4 . H Susec/div. Experimental verification of the change of operation of the series resonant converter from a) type 1 dem to 6) type 2 dem to c) —type 2 ccm as Q changes from 49 to 1.94 to 5.18 respectively af Fy / Fy =.42. 124 If the voltage drop in the output rectifiers is to be considered, then Q must be modified according to Eq. (3.110) 1. Vp Q, _ Qs ee (3.121) The value of M' that must be used here is that of the boundary between —type 2 ccm and type 2 dem which is simply M'=1/3 and we have f 2x.7 @' = G5 |1 = Q,(.72) ~ (473)(15) (3.122) and the condition for —type 2 ccm to occur is modified to ' 16 _ Bince @ =5.18 satisfies this condition, the converter operates in -type 2 ecm for this value of @ as shown in Fig.3.34c. Since 4, =1.94 does not satisfy the condition in (3.123), for this the converter operates discontinuous conduction mode. But which type of dem?, From the value of F,/Fo=.42 the only two types of dem possible are type 2 dem and type 1 dem. To find out which occurs for a given Q we refer to the boundaries between two discontinuous conduction modes discussed earlier in Sec. 3.4.3. According to (3.78) type 2 dem occurs if x 42g<Bh . & ry nm Fo Fo 1 <3 3 ; (3.124) The upper bound of @ in (3.124) defines the boundary between -type 2 ccm and type 2 dem which is in agreement with the condition in (3.120). 125 The lower bound in (3.124) gives the boundary between type 1 dem and type 2 dem. Substitution of F,/Fg = .42 in (3.124) gives 535 < Q < 16 (3.125) If the voltage drop in the output rectifiers is taken into account, then (3.125) must be modified. The upper bound, which is the boundary between —-type 2 ccm and type 2 dem was modified earlier in (3.123). At the lower bound @ is modified according to Eq. (3.121), in which this time M'=1 because at the boundary between type 2 dem and type 1 dem M'=1. We then have 1.4 "= 1-—-—| = 91 & @| (1)(15) @. (81) (3.126) and consequently, the condition in (3.125) is modified to 535 1.6 “91 = a; < 72. or 9 < Qs <= 2.22 (3.127) @, = 1.94 satisfies the condition in (3.127) for type 2 dem to occur which is verified in Fig. 3.34b. Finally, since @=.49<.59 type 1 dem occurs for this value of @ as shown in Fig. 3.34a. 3.7 Conclusion In this chapter a thorough dc analysis of the series resonant converter is given in terms of the normalized load parameter, @;, and the control parameter, F;/Fo. The various modes of continuous and discontinuous conduction are identified and the conversion ratio in each mode of operation is determined. The boundaries between the various 126 modes of operation are determined by simple equations. There are two general types of continuous conduction mode, namely, +type n ccm and —type n ccm for which the conversion ratio is determined numerically. There are, also, two general types of discontinuous conduction mode defined as even type n dem and odd type mn dem. For the even type n dem the conversion ratio is given by a simple equation which is a linear function of F,/Fp and FA. Therefore, it is immediately seen that when operating in an even type n dem the converter behaves as a frue current—fed converter because the average output current is independent of the load A. The conversion ratio for the odd type n dem is simply M=1i/n which is insensitive to variations in the load and the control parameter. The most useful range of continuous conduction mode is 1/2<F,/Fo<1 because in this range the switches turn off naturally and the maximum value of M is unity. It is shown that in this range the converter behaves as a current fed converter only away from resonance and for values of @ larger than three or four. It is also shown that, in this range type 1 dem can occur for which M becomes uncontrollable. Therefore, in order to avoid this type of discontinuous conduction mode, a simple boundary equation is given from which Q, can be chosen properly. According to the conversion ratio characteristics, this converter is not very well suited for applications that require very large load variations. Other steady-state quantities such as peak stress levels and diode conduction time are determined as well. 127 CHAPTER 4 SMALL-SIGNAL ANALYSIS OF RESONANT CONVERTERS 4.1 Introduction This chapter discusses the small-signal response of resonant converters to perturbations in the switching frequency and the input voltage at a given operating point. For the series and parallel resonant converters, the operating point is determined from the de analysis given in Chapters 2 and 3. In Sec. 4.2 the small-signal analysis of these converters is given only for operation in continuous conduction mode and in the range Fy/Fo=1/2. For the series resonant converter, it was shown if discontinuous conduction mode (type 1 dem) occurs in the range 1/2<F,/Fo<1, the output becomes uncontrollabe and hence a small-signal analysis would be useless. It was also shown that this type of discontinuity can be avoided in the entire range 1/2<F,/Fo<1 if Q,=4/mn (Eq. (3.25a)). For the parallel resonant converter the occurrence of dem was not analyzed although this mode can occur if @ is large enough because Ly/Lo is finite. For the parallel converter dem is ignored because it corresponds to very light load conditions. The various modes of operation that occur in the range F;/Fo<1/2 are ignored as well because the most important range of operation of these converters is 1/2<F,/Fo=l, however, the analysis given is also valid for Fy/Fo=1. The control-to- output transfer function is given in Sec. 4.2.1, the audio susceptibility is 128 given in Sec. 4.2.2, and the input impedance is given in Sec. 4.2.3. The experimental results given in Sec. 4.3 are in good agreement with the predictions of the analysis. A block diagram of resonant converters for small-signal perturbations is shown below in Fig. 4.1. ' Resonant Converter S Vo + <v> V, — 9 T - A Tee Fs +f, Fig. 4.1 Block diagram of a resonant converter with perturbations in the switching frequency and in the input voltage. 4.2 Analysis in Continuous Conduction Mode and in the Range F,/ Fo=1/2 In this section a systematic method of small-signal analysis of resonant converters in frequency control is outlined in detail, This method employs state-space analysis without the linear ripple approximation, since it is clear that such an approximation is not valid for resonant converters. In this analysis steady-state terms of the form e4? are retained while perturbation terms of the form eft which arise from perturbations in the Switching times, are linearized under small-signal assumption. This is Summarized by the following equation 129 ett =@A(T +t) = gAT el AT | 7 + At] ; t <« ALT (4.1) where A, are the eigenvalues of A and 7 is the steady-state switching time. The justification of the linearization of the perturbation term in the last step, explained in Appendix C.1, requires that f be much smaller than 7 and the time constants of the circuit A;!. Since for resonant converters 7 cannot be assumed to be much smaller than the circuit time constants \,), as in the case of PWM converters (buck, boost, Cuk,...), the linear approximation of e4? is not performed here [8]. Because of the symmetric operation of the circuit over an entire switching interval, as in the case of the dc analysis, the small-signal analysis is carried out over half the switching interval, 7,. 4.2.1 Control-to-output Transfer Function In this section, the small-signal response of the series and parallel resonant converters to perturbations in the switching frequency is determined. If the switching frequency is perturbed by a small amount F,(t), by injecting a control signal, v,(t), in the voltage-controlled oscillator (VCO), then the switching time f,(f) will be modulated by Km -(t) ~ 1 os 1 t,(t) = ~ ape st) = ~ BRE (4.2) The switching interval, however, can only be considered on a discrete basis. This does not present a problem, and the connection between the continuous injected signal and the discrete perturbation in the switching interval will be explained presently. Consider the arbitrary switching interval shown in Fig. 4.2a 130 Lg Ty, (nT) —»| i i a) -- - modulated nTs ts( nT.) — b) ——--—1 ! -- unmodulated nTs (n+! Ts i Ts+ 10) = n=O c) ~~ 0 t.(0) modulated Fig. 4.2 a) An arbitrary modulated switching interval. 6) An arbitrary unmodulated sutching interval, c) A modulated interval for n=0. 131 nT, <t < t,(nT,) (4.3) In the absence of any modulation, as shown in Fig. 4.2b, this interval is given by nT, <t < (n+1)7, (4.4) In the presence of modulation, the perturbation in the duration of this interval is denoted by #,"(nT,) and (4.3) is written as nT, <t <(n+1)T, +t, "(nTp) (4.5) The time t;(nT,) can be summarized by the following equation (n +1)T7; unmodulated t,(nT,) = (n+1)% +f, "(n7,) modulated (4.6a-b) The discrete perturbation t,"(nT,) in the switching time is now related to the continuous control signal by #5 "(nTz) = t,(t) 6(t-nTs) (4.7a) __ f(t) =~ ope o(t —nT,) (4.7b) In addition to the switching interval, the output-network switching time, 7,, is modulated as well. In the parallel resonant converter ft, is determined by the zero-crossing of the resonant capacitor voltage while in the series resonant converter fg is determined by the zero crossing of the resonant current. Since in the presence of modulation these states are 132 modulated, it follows that the zero-crossing times determined by these states are modulated as well. The perturbation in the zero-crossing time is denoted by £4 °"(nT.) and is considered to be a sampled point of a continuous function f,(t) tq"(nTs) = tq (t) 6(t-nT.) (4.8) The switching time tg(n7,), inside the interval in (4.3), is given by t,(nT,) = nT, +7, +tq"(nT) (4.9) Whereas #,(f) is a continuous function that can be related directly to the measurable quantity u,(t), #,(t) is just a continuous function with no such physical significance. In step v, the unknown modulation #4 (t) is determined in terms of the perturbations in the state vector from a certain constraint. For conciseness and without loss of generality we let n =O in (4.3) through (4.9). The resonant current waveform for the series resonant converter and the perturbations in the switching times #, and f, in the interval O<t<t,(0) are shown in Fig. 4.3. A similar figure for the parallel resonant converter applies with the resonant current replaced by the resonant capacitor voltage since in this case it is the capacitor voltage that determines the output-network switching time f,. To simplify the notation further, whenever possible, ¢, and £, will be used instead of f£,(0) and ¢,(0). In what follows the steps of the small-signal analysis are outlined in detail. 133 ig(t) a) ip(O)—" ° 1 Te tt X(0) 2 \ arta tN | an (te) t,(0) \ { a 1 T,+t3*(0) ___, | ! ; t,(0) =t, b) j t2(0) | txcT.) oS | 1 | c) je2(0) tT.) o— a dq) frelt)! A ee <i te) | | e) B(t T 5b ott) f 8-1.) @) T, Fig, 4.3 a) The resonant current ip(t) in ah modulated interval O<t<#,(0), 06) The perturbations 3 (0) in the switching interval and the perturbation tf, *(0) in the zero-crossing time, c) The continuous functions f,(t) and f,(t) which when sampled at t=nT, give the perturbations in the switching interval nTI,<t<t,(nT,;) and the zero-crossing time ty(nT,) in that interval respectively. 134 i) State Equations The two switched networks of the series and parallel resonant converters over half the switching interval are shown in Fig, 4.4. For the series resonant converter the sequence of switched networks shown is for the range of operation 1/2<F,/Fo<1, and is reversed in the range F;/Fo2=1, but for the parallel resonant converter the sequence remains the same for all f,/Fo2=1/2. According to Fig. 4.4a the state equations in the interval O<f<f, are a(t) = Ayz(t) + diy (4.10) u(t) = cf x(t) (4.11) where for the series resonant converter the matrix A, and the vectors x(t), b; and cf are given by x(t) = [ig(t) up(t) vo,(t)] _ re ee ee (Rllre, +70) Lo Lig Lio it+re,/f 1 A, = = 0 0 1 Co i se Cy 1+7¢,/R Cr Rtro | (4.12) bf =[1/Ly 0 Oj (4.18) cf =[Rlirg, 0 1/(1+7¢,/R)] (4.14) For the parallel resonant converter these are given by 135 Fig. 4.4 The two switched networks for the series and the parallel resonant converters a) during O<t<t, and b) during ty<t<t,. 136 x? = [ip(t) up(t) 4, (t) ue, (t) | _ 19+7 ep) 1 Te, Lo “Lo Lo ° 1 1 A Co ° Co ° i= Co 1 1 1 oF -“]Z ip Ar, +o +o, ||") 7 ly Bre, Roi -_——_L 0 0 R+r¢, Cy (R+1¢,)Cy LL J bf =[1/Lp 0 0 O] (4.16) ef = [00 Rilrc, 1/(1+7¢,/R)] (4.17) The solution of Eq. (4.10), with the assumption that V, is constant, is given from linear system theory by z(t) = el 2(0) + By(t)b,V% (4.18) where the matrix 8,(t) is given by Bilt) = fem ar = Ay eA —7] (4.19) According to Fig. 4.4b, the corresponding expressions for the series resonant converter in the interval {,<f<f, are 137 fa - 2 wi 121 (Filre, +f) Lo Lo Lo 1 +7¢,/R - A Az = C 0 9) Cy itre/k C} R+To, | (4.20) bg =, ; of =[-Rl|ro, 0 1/(1+r¢,/R)] (4.21) For the parallel resonant converter these are given by [ (r1.+7e,) a Tey Lo Lo 0 ° 1 1 Co Q - 0 Az = Te 1 1 1 oR iG iy "1, +85 *7o,|18) 7 “ly Bera Rk 1 1 0 0 R +TC Cy (Rk +7¢,)C bg =b, ; cf =a? (4.23) The solution in this interval is given by w(t) = eM#@ ‘oo (t,) + Bolt —t)bVy (4.24) where the matrix B2(t) is given by t _ - Bolt ~ta) = f 282 ae = Age Aalt fe) _ 7] (4.25) a The evolution of the state vector z(¢) over the interval 0<f<t¢, is obtained by matching the solutions given by Eqs. (4.18) and (4.24) at t =¢,. This gives 138 x(t,) = Ax(0) + BbvV, (4.26) where the matrices A and B are given by A = efelteta) Arte (4.27) B = efetsta) Bt) + Bolt,—te) (4.28) ii) State-—Space Averaging When the system is not perturbed or modulated, the output voltage consists of a de component and a ripple with a fundamental at twice the switching frequency. Let the steady-state output voltage be denoted by V(t) and defined as follows V(t) = Steady-state or unmodulated output voltage This is a periodic waveform of period 7, and is shown in Fig. 4.5a in a thin line. The spectrum of V(t), which consists of a de component and other harmonics at 2n¥F;, is shown in Fig. 4.6a. The de component, which is the average output voltage, is determined from the de analysis discussed in Chapters 2 and 3, and is simply given by Vo = MV, (4.29) This can also be written in terms of V(f) as 1 (rn +1)7, == dt Yo Ts Son, Mt) (4.30) Vp is a constant throughout all the intervals and is shown in Fig. 4.5a with ig, 4.5 139 <V> (0) v(t) Zw, oN yey “N Kd < 0 ! 1 tg=t, (0) =T,+t (0) | } n~ n~ * % P \<v> (0) p<" (76) | I I I yt | 0 re ry £(0) ' t*(T.) w<V> (t)t d) ' 0 ! i ie ! 1 I | e) 0 ; i en een tet) | F) 5(0 t—T, 5 18) {8ct-r,) 0 T. a) The steady-state or unmodulated output voltage, V(t), and its average value, Vp, are shown in thin lines. The modulated output voltage, u(t), and its average, <u>(nT,), in each interval in the presence of modulation are shown in thick lines. b) Perturbations in the average output voltage, <u>"(nT,), im each interval are shown as sampled points of a continuous function d) <u>(t) which is the component in the output voltage that has the same frequency as the modulating signal. 140 Vo a) I \ fe) 2F, OF. Vo b) <¥>(t) =A (wy) sin (wmt—@ (wm)) [Alem i, a | ry A 4 O f, 2F, 4F, Fig. 4.6 a) The spectrum of the unmodulated or steady-state output voltage, V(t), and b) the spectrum of the modulated output voltage, u(t). Vot<t>tt HP3570A ir one Wave Analyzer f_ LI Ch. A g 9 ChB 4 rY A F, +f, (t) =z tg | Computer_| fg (t) = Kmn¥e (t) vco A HP3330B tn Synthesizer = FM input -—<# me eed Fig. 4.7 Experimental method of determining the control-to-output transfer function. 141 a thin straight line. Let the switching frequency be modulated now by a small amount f.(t)=esinant fot) = Fe + Fe(t) (4.31) The output voltage, which is a modulated signal now, is denoted by w(t) and defined as follows u(t) = Modulated output voltage This is shown in Fig. 4.5a with a thick line, and is no longer a periodic waveform. The spectrum of u(t) is shown in Fig. 4.6b and it consists of, in addition to the de component and the harmonics of twice the switching frequency, a sideband accompanying the zero frequency component and sidebands around each harmonic. Strictly speaking, u(t) is a demodulated signal since the harmonics of the carrier and their sidebands are well Suppressed because of the small ripple requirement on the output. The only component in this spectrum that is of interest to us is the sideband accompanying the zero frequency component because it is the only component which has the same frequency, w,, as the modulating signal. This sideband is denoted by <u>(t) and is defined as follows <u >(t) = Sideband accompanying the zero frequency component of the output voltage which has the same frequency as the modulating signal. 142 The experimental method used to verify the small-signal transfer functions [9] is shown in Fig. 4.7, At the output, the analyzer searches for that signal which has the same frequency as the injected signal. In other words the transfer function that can be verified experimentally is <u>(s) - <ur(s) Fs(s) ve(s) f(s) U(s) (4.32a) K <u>(s) ” Fels) (4.32b) H Our goal, then, is to determine <u>(t). To do this, first, the average output voltage in each interval is determined in the presence of modulation t,(n7,) <u>(nT,) = iatjcate Ln uv (t)dt (4.33) In steady state, this reduces to Eq. (4.30). In the presence of modulation, in each interval, <v>(n7,) differs from Vp by an amount <u>"(n7,) <u>(nT,) = Vo + <u>"(nT,) (4.34) In Fig. 4.5a, <u>(n7,) is shown in each interval (n=-1,0,1) by a thick straight line, and the perturbation <u>"(nT,) in each interval is shown in Fig. 4,5b as a sampled data point of the continuous function <u>(t) <u>"(nT,) = <u>(t) 6(t -nT,) (4.35) It is, therefore, proposed that <uv>(t¢), the component of interest to us, be determined from a knowledge of the perturbation in the average output voltage, <u>"(nT,), in each interval. Since the sampled points <u>"(nT,) 143 are spaced at intervals corresponding to twice the switching frequency, <u>(t) can be determined for modulating frequencies reaching near the switching frequency. Our first task, then, is to determine <v>(n7,) in Eq. (4.33). Again, without loss of generality, we let n =0, so that Eq. (4.33) becomes <u>(0) = - fcfa(t)at + f'cka(t)at (4.36) This integration is carried out by substituting z(f) from Eqs. (4.18) and (4.24) in the proper appropriate integrands. The result is t,<u>(0) = c7x(0) + d7by, (4.37) where the vectors c” and d? are given by cfBy(ta) + cSBo(t,—t,)e** (4.38) of a” = c{By(tg)A,;* + cfBo(ts—ta)[Bi(te) + Ae] —cfA,"'t, — cf427\(t,—-ta) (4.39) Equation (4.37) is the statement of state-space averaging, according to which the average output voltage in each interval is expressed as a linear combination of the states at the beginning of that interval. wii) Perturbations in <v>(0), (0), and x(T7;) In the presence of modulation, the switching times ¢,(0) and t,(0) (or ft, and ¢,) are perturbed according to Eqs. (4.5) and (4.9) 144 ts = Ts +#,"(0) =T; +t, (4.40a) ta =Tyt+b_'(0)=Tatte (4.40) When these are substituted in Eq. (4.37), in which the average output voltage in each interval is perturbed by <u>"(nT,), the following is obtained [Vor <u>*(0) ][ % +7, "(0) ]=[07 +87 J[X(0) +27(0) ] + [a7 +d" ]bv, (4.41) The linearization of this equation will be discussed shortly. It is clear from Eq. (4.41) that the dynamics of <u>"(nT,) depends on the dynamics of z"(nT,), which is pursued next. The state vector at f =O is perturbed as follows z(0) = X(0) + 2"(0) (4.42) At f=7,, since ip(7,)=—ip(0) and we(T,)=—-vp(0), these states will be perturbed as follows in(Ts) = In(Ts) —te"(Ts) (4.43a) up(Ts) = Vp(Ts) — Up"(Te) (4.43b) The output capacitor and inductor states, on the other hand, are perturbed as usual in, (Ts) = In,(Ts) + 71, "(Ts) - (4.48e) 145 UG(Ts) = Vo,(Ts) + Ue, “(Ts) (4.434) Equations (4.43) can be summarized by x(T.) = X(T.) + Mz "(T.) (4.44) where the matrix M, for the series resonant converter is given by -1 oOo Oo M,=|0 -1 0 09 oO 1 (4.45) 0 0 0 -1 0 0 Mr=\o g 1 0 Q Qo oO i (4.45b) It should be noted that, instead of writing z(7,) and 2°(T7,) in Eq. (4.44), one should actually write z(t,) and 2*(t,), but expanding around 7, we get x (ts) = x(T, +25) = 2(T;) + Gobet s (4.46) Since the second term represents a product of two perturbations, it is ignored and thus a(ts)*2(Ts) and 2 (t,) ¥2"(7,) (4.47) Va (T,) ~ my ValT.) =Val(T,) “Va (T,) Fig. 4.8 The modulated components of the state vector x(t) of the series rasonant converter, The steady-state components are shown in dotted lines inside the envelope of modulation. a) The resonant current, b) the resonant capacitor voltage, and c) the output capacitor voltage. 147 Fig. 4.9 The envelope of the resonant current of the series resonant converter due to modulations in the switching frequency. The triggering level is set such that all the zero-crossings at ft=t, are coincident. 148 The modulated components of the state vector x(t) for the series resonant converter are shown in Fig. 4.8. The shaded areas represent the envelope of modulation in which the steady-state or unmodulated waveforms are shown in dashed lines. An oscillogram of the envelope of modulation of the resonant current of the series converter, with the trigger adjusted such that all the zero-crossings are coincident, is shown in Fig. 4.9, Now, when the perturbations in f,, #,, ~(0) and x(t,) are substituted in the equation of the evolution of the state vector given by Eq. (4.26), the following is obtained M,&"(Ts)+X(Ts) = [A+A][X(0) +2 °(0)] + [B+B ov, (4.4B) iv) Linearization Equations (4.41) and (4.48) are now linearized under small-signal assumption using Eq. (4.1). As an example of this linearization, consider the matrix [A +A] in Eq. (4.48) At A = otelTeTa) , Aalts —ty) eftta Ate (4.49) en core” linearize The perturbation terms are linearized according to Eq. (4.1) as follows eAdle ta) oAtta ws LF + Ao(Eg—ta) JL + Arta] RI + Alts-ta) + Aiba (4.51) where the second line is obtained by neglecting products of two perturbation terms. When Eq. (4.51) is substituted in the Eq. (4.49), the 149 following is obtained A = ApAt, + (AA\—ApA) Eg (4.52) where the matrix A is given by A = eh2la-Ta) 941% (4.53) Following the same procedure on the remaining terms in Eq. (4.48), cancelling de terms, and substituting #,°(0)=-f, "(0)/2F2 as given by Eq. (4.7b), we get the following linear discrete equation ax “kk ~ * ~ * x (Ts) = M, Ax (0) + dat, (0) + ds fs (0) (4.54) The vectors dg and d, are given in Appendix C.2. Equation (4.54) is equivalent to the following linear difference equation a(t +75) = M, Az (t) + dat ,(t) + df ,(t) (4.55) where the continuous perturbation vector is defined as follows z(t) = The continuous perturbation vector which hos the same frequency as the modulating frequency, Om, and which when sampled at t=nT,, gives the perturbation z*(nT,) on the state vector x(nT;). When sampled at t=(n+1)Ts, M,x(t) gives the perturbation M,z"((n+1)T,) on the state vector X((n +1)T;). It should be clear that 150 a(t) # X(t) + x(t) (4.58) because a modulated signal, in this case z(f), cannot be simply represented by the sum of the unmodulated signal and a signal that has the same frequency as the modulating signal. Following the same linearization procedure, we obtain from Eq. (4.41) <u>(t) = c?z(t) + kglg(t) + kefe(t) (4,57) where the vector c? is given by Eq. (4.38), and the constants k, and k, are given in Appendix C.2. From this equation it is clear that the dynamics of <u>(t) depends linearly on the dynamics of the components of z(t). Equations (4.55) and (4.57) have an unknown modulation term, 7#,(f), in them which needs to be resolved. This is explained in the following step. vu) Determination of t,(t) in Terms of x(t) Since the time f, is determined by the zero-crossing of either the resonant inductor current (for the series resonant converter) or the resonant capacitor voltage (parallel resonant converter), the perturbations on these states at f =f, is zero as well. Let ¢ =f, in Eq. (4.18) x(tz) = e“"8z(0) + Bi(ta)bV, (4.58) When this equation is perturbed and linearized as explained earlier, the following is obtained 151 3°(1q) = of%3%(0) + ef" [4,X(0) + bY Bo"(0) (4.59) : > * : : The constraint iz (t,)=0 gives for the series resonant converter > * Qi7 * Qig-~ * Qig~ * t =— - _ a (0)=- Fr (0) —FvR (0)- Fo Ve, (0) (4.60) For the parallel resonant converter, the constraint v p(t.) =0 gives oF Ay G21 en! ~ Geg7 *,.. Geaan~ * fa (O)=- Fo tr *(0) - n (0) Fg 1 O- ZV (0) (4.61) The constants a, are the elements of the matrix eft" and the constants k; are the elements of the vector e417 4,X(0) + Vy] [a] = e417 and [ki] = e417 [4,X(0)+bV,] (4.62 a-b) Equations (4.60) and (4.61) can be put in continuous form and summarized as follows ta(t) = Ve'x(t) (4.83) where, for the series resonant converter, the vector V,’ is given by Ve =[-ay1/k, —Gy2/ky —043/k, | (4.64a) and for the parallel resonant converter VJ is given by Val =[-Gg1/ke —Gee/kq —Ggs/kg —Aa4/ke | (4.64b) | When Eq. (4.63) is substituted in Eq. (4.55), the final desired difference equation is obtained for x(t) 152 z(t+ 7.) = M,z(t) + df ,(t) (4.65) where the matrix M, is given in Appendix C.2. The equation for <u >(t) follows after substitution of Eq. (4.63) in Eq. (4.57) <u>(t) = aATZ(t) +k F,(t) (4.68) where the constant k, and the vector A’ are given in Appendix C.2. vi) The Control—-to—Output Transfer Function The control-to-output transfer function is now easily determined by taking the Laplace transform of Eqs. (4.65) and (4.66) su>(s) _ yr 2(s) yy F s(s) f(s) (4.67a) z s = T st, _ M. ag Fey UE = Me, (4.87) Substitution of Eq. (4.67b) in (4.67a) gives SPS) = ntie®® — Mid, + bk f(s) (4.68) This is the control-to-output transfer function. As mentioned earlier, f s(t) is generated via a control signal u,(t) that is fed into the FM input of the VCO. This control-to-output transfer function is given <u>(s) _ <u>r(s) fs(s) u_(s) f(s) @,(s) (4.69) The particular VCO used for the experimental verification of the control-to- output transfer function was the Intersil 8038 waveform generator for 153 which f,(s)/u,(s) is frequency independent so that <u>(s) _ , <u>(s) aes)” Pals) (4.70) where A, is a constant. Consequently, no other dynamics was observed in the experimental results other than that given by Eq. (4.68). 4.2.2 Audio Susceptibility In this section the response to small-signal variations in the input voltage is determined. Let the perturbation in % be denoted by v(t) ug(t) = Vy + v(t) (4.71) If V(t) varies slowly during 7,, then it can be assumed to be constant during this interval and the results in Eqs. (4.18) and (4.24) are still valid. Therefore, unlike the control-to-output transfer function, the line-to-output transfer function determined is valid only for low frequencies. The equations for the evolution of the state vector and the average output voltage in each interval is now perturbed for Vg. First, consider Eq. (4.37) in which in addition to z(0) and V,, the vectors c’ and d? are perturbed because they are functions of ft, T,<u>"(0) = c72"(0) + 67X(0) + d7bV, + d7bu,"(0) (4.72) let c7=c'Tf, and d’=d'"t, T,<u>*(0) = c72*(0) + [c'?X(0) + a'7bV, Jf, "(0) + dTov,*(0) (4.73) The equation for <ur(t) follows from the continuous form of this equation 154 T,<u>(t) = c7B(t) + [o'TX(0) + d'7bV, Jeg (t) + a7bU,(t) (4.74) Similarly, for the perturbation vector z(t) the following difference equation is obtained M,x(t+T;) = Ax(t) + [A'X(0)+B'bV, ]fg(t) + Bou, (t) (4.75) The unknown modulation ta can now be determined as before from the requirement that the perturbation on the state that determines f, be zero at f=T,. Equation (4.58) is repeated here as Eq. (4.76) x(ty) = e4tex(0) + By(t,)bV, (4.76) Perturbation of V, in this equation gives @*(Ty) = e41792*(0) +6417 (4,X(0) + bVg lea "(0) +B, (Ta)bvg"(0) (4.77) The constraint ip (ta) =0 gives for the series resonant converter *(9) = — 2G - =a ~ Rp -z0 -t1y ~ * For the parallel resonant converter, the constraint uv Rp (ta) =0 gives 7 * Gai <2 ~ * 2234 ~ * q"(0)=— FE in"(0) - FAO n"(0) - Fan, (0) — Fave, (0) — 2 ¥4"(0) ks” (4.79) In Eqs. (4.7B) and (4.79) aj and k; are given by Eqs. (4.62a-b) and the constants p, and pz are the elements of the vector B,(t,)b 155 [pi] = By(Tq)b (4.80) Equations (4.77) and (4.78) can be put in continuous form and summarized as follows tg(t) = Vj'a(t) + kv, (t) (4.B1) where V," is given by Eqs. (4.64a-b) as before and k, is given by Pk (4.B2) where ~=1 corresponds to the series resonant converter and i=2 corresponds to the parallel converter. Substitution of Eq. (4.81) in Eqs. (4.74) and (4.75) gives <u>(t) = ATE(t) + kydg(t) (4.83) a(t+T,) = Me&(t) + dyvg(t) (4.84) where ky and dg are given in Appendix C.2 and where h? and M, are the same as in Eqs. (4.65) and (4.66) and are also given in Appendix C.2. The audio susceptibility is now determined by taking the Laplace transform of Eqs. (4.83) and (4.84) Svs) = atte — Med, + ky Vg(s) (4.85) 156 4.2.3 Input Impedance The input impedance of a converter is determined by the small- signal response of the average input current to perturbations in the input voltage. Because of the various methods of excitation of these converters as shown in Figs. 2.1 and 3.1, the input impedance seen by the source, Vin, in each of these converters is different. The ideal equivalent circuits of these converters, shown in Fig. 4.10, relate all these different input impedances to a single input impedance given by _ _v,(s) eT Sig >(S) (4.86) where <i,> is the average branch current in Fig. 4.10. It can be easily seen that the input impedance, Z,'(s), seen by the source, Vm, for the converters in Figs. 2.1a and 3.1b is related to Z,(s) of Eq. (4.86) by Zq'(8) = 2 =| Zn(s) m st, 0 ™ (4.B7) For the converters in Figs. 2.1b and 3.1id, which implement four switches, the input impedance Z;,,,'(s) is given by Zin'(S) ~ 2 (4.88) Finally, for the converters in Figs. 3.1a and c, Z'(s) is given by Zin'(8) = 2 Zin(S) | (4.89) 157 Zin(S) | M $j» ely “7 : $2 Ce Z,,(s) AG Wh R (a) Zin(S) . 'b LE _~ il + va L000 ® if ‘se ‘sr. R CSE MSR (b) Fig. 4.10 The input impedance, Z,,(s), of the ideal equivalent circuit of a) the series resonant converter and b) of the parallel resonant converter. 158 The branch current % is the same as the resonant current ip(t) during the interval 0 <f <7, and is zero during the interval 7, <t <27%. The steady-state and modulated waveforms of i are shown in Fig. 4.11. Since the first component of the state vector z(t) is the resonant current, the average branch current <i,>(0) in each interval is given by the first component of the average state vector <z>(0) <z>(0) = atl f*x(t)at + f'n(t)at 2Te (4.90) In Eq. (4.90), the averaging interval is taken as 27, while the integral is taken from 0 to 7, because the branch current is zero during 7, <t <2T,. Equation (4.90) gives <z>(0) = Agz(0) + Bd (4.91) where the matrices Ag and By are given by Ae = apt Bilta) + Bol To —ty)e" (4,92) Ba = gp lArBilte) + Bel Te ~ta) Fi (ta) +A, "BT, —ty) —( Ts —ta)Ag™ ~tgAy" | (4.93) The steady-state or unmodulated current, /,(¢), is shown in Fig. 4.11a in a thin line and the modulated current, i,(f), is shown in a thick line. The steady-state average current, </,>, shown in a thin straight line throughout all the intervals, can be expressed in terms of (tf) and the 159 I, (t) a) i,(t) <i, >(0) ~ l |<ap>” (0) \<in>” (2T,) — Wt \\E Aca,(27,) r— Ts ——my b) | t | 0 ~ = (0) ! <p>” (2T.) c) | V(2T5) O—--: : 1 V*(0) | | ! { I “ d) 1 <ap> (thy, e) f) Fig. 4.11 Vq (t) a O 1 i 4 1 T 1 J I { | { 810) { 6(t-2T,) fe) 6) eT. The branch current i, of Fig. 4.10. a) The unmodulated branch current, I(t), and its steady-state average value, </,>, are shown in thin lines, The modulated branch current, i,(t), and its average, <i,>(nT,), in each interval are shown in thick lines. 6) The perturbations in the average branch current, > A : . ~ <ip> (nTs), in each interval in the presence of Ug are shown as sampled data points of a continuous function d) <i,>(t) from which the input impedance, given by Eq. (4.86), is determined. 160 conversion ratio M as _ 1. rls _ MV, <b> = or J, I(t)dt = (4.94) where the last equality is simply obtained from the following dc consideration Mo MPV, _ 2 From the last equality, it can be seen that the de input impedance is given by | _ 2h lim Zin(s) = (4.98) s>0 The modulated branch current i,(f), in the presence of modulation of the input voltage, is shown in a thick line and the average branch current in each interval is shown in a thick straight line in that interval. In each interval the modulated average branch current deviates from the steady- state average by <i, >*(0) Sip>"(0) = <i,>(0)-<hy> (4.97) To determine the small-signal perturbations in the average branch current, Eq. (4.91) is perturbed for ¥. This perturbation gives <z>(t) = Agt(t) + [Aq'X(O) + Ba'b ]fg(t) + Babu g(t) (4.98) The perturbation in the evolution of the state vector and the unknown modulation #,(f) are given by Eqs. (4.81) and (4.84) as before. Substituting 161 Eq. (4.81) in Eq. (4.98) and rewriting Eq. (4.84) as Eq. (4.100) we get <z>(t) = A,Z(t) + byvy(t) (4.99) a(t+T,) = Mpz(t) + dyv,(t) (4.100) where the matrix A, and the vector 6b, are given in Appendix C.2. Equations (4.99) and (4.100) are solved by taking their Laplace transform SB2(8) 2 4 fye®™ —y,J1d, +b, Vy(s) (4.101) The first component in Eq. (4.101) is the input admittance. 4.3 Experimental Results In this section the experimental and predicted results of the small-signal response are given with special emphasis on the control-to- output transfer function. The circuit parameters of the series resonant converter are fo=50.2kHz Lo=.19%7mH Co=.051uF C, =32uF For the parallel resonant converter, the circuit parameters are £9 =88.7KHZ Lo =364H Co=.47uF Ly =1.85mH Cy =32uF The control-to-output transfer function for the series resonant converter is shown in Figs. 4.12a-d where it can be seen that predicted and experimental results are in good agreement. In Figs: 4.12a and b the operating point is below resonance and in Figs. 4.12c and d the operating point is above resonance. Comparing the transfer functions below and 162 above resonance we can see that there is a 1B0° phase shift in the phase response owing to the reversal of the slope of the conversion ratio characteristics as the point F,//'p=1 is traversed. An interesting point in these transfer functions is the following: in Fig. 4,12d, for the operating point 9,=3 (or R=20.70) and F,/F)=1.3, the low-frequency behavior is essentially that of a single pole ( f,=240Hz ) at RC;, which is in agreement with our intuition since the output low-pass network is current fed by the resonant branch. This intuition fails completely for the operating conditions in Figs, 4.12b and c, but the results of the exact analysis are in good agreement with the measurements. If the modulation of the zero-crossing time f,, determined in step vu, is neglected, then this change in the dominant behavior will not be predicted properly. Therefore, the dominant behavior of the transfer functions depends on the modulation of the zero—crossing time tg. An example of a transfer function, shown in Fig. 4.12a, is given for the operating point Q=4 and F,/F )=.8 <u>(s) _ , ltay2 +a22* +a328 u(s) 1+b,z +bpz* + bgz8 where the coefficients in the numerator and the denominator are given by a,=-23.813 a2,=-19.9 a3g=—2.086 b,=-2.Bi11 62,=3,382 b6,=-1.593 and the constant K depends on the input voltage, the gain of the VCO, and values of the circuit components. 163 20dB r —— prediction measurement -90° Operating point Q,= 4.0 ~40- F, /F920.65 ~180° -50- -60 boil po tl po ~270° 20Hz lOO Hz Ik Hz 1OkHz 25kHz ome prediction 10dB measurement Pea 0° Operating point ~40r Qy71.5 ' F,/Fo=0.8 -50- -~180° “6OF a tl po ta po ta __} -270° ISHz 1OOHz tkHz lOkHz 30OkHz 164 ome prediction 20dBF ens measurement . 180° 0 1. 90° -20k c) L 0° ~40F Operating point Q,=3.0 r Fy /Fo=l.l ° ~6ok 90 -80 rn po tl 2 tail 1 1 boiil 4 : -!80 1OHz 1OOHz IkHz lOkHz 40kHz 30dB r we prediction 20 measurement 10 + 180° (@) + -10 4 90° d) ~20 4 -30 + 0° Operating point -40 Q,= 3.0 7 Fe/Fo=t.3 3 -50- -~90 - ~60b 4 —-70L. 1 aii il A atuiiil po a J -180 ISHz |OOHz 'kHz lOkHz SOkHz Fig. 4.12 a-d) Experimental transfer functions and pradicted results of the control-to-output of the series resonant converter. a) 165 90° 20dBF —— prediction —— measurement lOr O° mag OF a ~!Ob a -90° -20- phase ~30 -180° -40ob Operating point r Qp= 5 -50 Fy /Fo=0.6 -270 “60+ L -70 ee areas | 1 i rn ~360 20Hz 1OOHz tkHz lOkHz =2tkHz 20dB a mag ome prediction Te) measurement | 0° 0 -10 -90° ~-20 -30 -180° ~40 Operating point -270° -50- Qp=2.22 L F, /F,=0.85 -60- b ~360° ~ TOF 4 -80 F toil 4 r iil a iil ‘ 20Hz lOOHz tkHz lOkHz 22kHz 166 20dB i a=emme prediction 10 measurement L 4 0° OF r ang: 7 -10h | -90° i phase c) -20+ : ; | -180 -30r L Operating point | Qp=l79 -40- Fy /F,=0.65 L se -|-270 -50- -~60 tet a il 1 re rarer | L to ta J -360 20Hz lOOHz [kHz 10 kHz 23kHz 20dB E —— prediction ioL —— measurement 0 es 180° -I0 -20 90° d) -30 -40r ; ; 0° i Operating point “en Qp= 1.79 507 Fy / Fo =1.05 -60F -90 -80 re reer | 1 arn arererere | 1 re rarer | 1 -180 20Hz 100 Hz I kHz lOkHz 3OkHz Fig. 4.18 a-d) Experimental and predicted results of the control-to-output transfer functions of the parallel resonant converter. 167 OdBF Audio Susceptibility prediction mecsurement -20 ~4O0 a) -60 - 80 - Operating point L Q,=4 F,/ F206 ry - 180° -100 F se ~“l20 + sisal 1 Fr rererere | to tI A -270° 90Hz lOOHz tkHz lOkHz 28kHz : speeds mamma Orediction oda Audio Susceptibility meosurement mag a -40 b) -60 90° -80 Operating point r Os*4 180° Fs/Fo=0.9 ~ -100 + “e -1!20F rn ptaaiil i toa tirial i n pisiiyl -270° 20Hz 100Hz IkHz {OkHz 2OkHz Fig. 4.14 a-b) Experimental and predicted results of the audio susceptibility of the series resonant converter, 168 ~20dB vgs Audio Susceptibility prediction meosurement er ~60r a) gol ~100 F Operating point ] ad Qp = 3 ° ~Rol Fy/ Fo =0.6 ~'80 -~140 il 1 1 i 1 oa l n nN 1 1 a -270° 80Hz IOOHz {kHz lOkHz Audio Susceptibility == prediction —40dBE —— measurement mag =~ 0 -60 L | -80- + -90° b) r -l100F L 4 -180° -120b Operating point | Fg /Fo = 0.83 4 -270° -140- -|I60 1 { i 1 1 1 coe oe il i 1 n js re ee l -360° 80Hz 100Hz tkHz lOkHz I2kHz Fig. 4.15 a-b) Experimental and predicted results of the audio susceptibility of the parallel resonant converter. 169 60dB Input Impedance prediction 50F measurement 180° . Operating point 90° Of Qs =4 =0.92 fe) 4 0° -I0 ; -20F- 7 -30 Lairiil n ao tisiil 1 n ro ee | 1 1 -90° 40Hz I00Hz ikHz !OkHz 40kHz Fig. 4.16 Experimental and predicted results of the input impedance of the series resonant converter, 170 The control-to-output transfer function for the parallel converter is given in Figs. 4.13a-d. The audio susceptibility, or the line-to-output transfer function, is shown in Figs. 4.15 and 4.16 and the input impedance is shown in Fig. 4.17. Because of low frequency limitation in the analysis of the response to perturbations in input voltage, there is some discrepancy at high frequencies between predicted and measured results. 4.4 Conclusion In Chapters 2 and 3 an accurate de analysis of the series and the parallel resonant converters is given whereby the steady-state vector is determined in terms of @ and F,/Fpy at a given operating point. In this chapter, a systematic method of small-signal analysis is given whereby the response of resonant converters to perturbations in switching frequency and input voltage is determined. Experimental and predicted results for the parallel and series resonant converters are in good agreement. In this analysis the discrete and average small-signal responses are determined. The average response is important because in the experimental verification of the small-signal response the network analyzer searches for that component in the output voltage or the input current which has the same frequency as the modulating signal. This component which the analyzer searches for is determined from a knowledge of the perturbations in the average output voltage or the average input current in each interval. The average state in each switching interval is determined as a linear combination of the initial states at the beginning of that switching interval. Consequently, the average response is determined in 171 terms of the discrete response of the initial states. The control-to-output transfer function is valid for modulating frequencies approaching near the switching frequency because the discrete analysis is carried out over half the switching interval which corresponds to a sampling rate of twice the switching frequency. The line-to-output transfer function and the input impedance are valid only for low-frequency variations in the input voltage because the solutions of the differential equations, during each output-network switching time, are obtained assuming slow variations in the input voltage. The results are determined numerically because of the occurrence of functions of matrices that are difficult to determine in expression form. 172 CHAPTER 5 HIGH-Q APPROXIMATION 5.1 Introduction The results of the analysis in the previous chapters were obtained numerically. To gain further insight, an approximate small-signal analysis is given in this chapter assuming high Q and operation away from the resonant peak. It was shown in Chapter 2 that it is desirable to operate the parallel resonant converter with a high @ (greater than 3) and below resonance because in this range it behaves like a voltage source (Fig. 2.5). Although this high Q approximation is relevant for the series resonant converter as well, it is slightly restrictive because it is not always necessary to design this converter for Q,>3 provided of course Q; is not small enough for discontinuous conduction mode to occur in the range 1/2<F,/Fo<1 (Q,=4/m as given by Eq. (3.25a)). Therefore, improved approximate results, valid over a wider range of Q, are given for the series resonant converter. The results of the approximate and exact analysis are in good agreement over the range of interest. 173 5.2 Approximate Analysis In this section, an intuitive rather a formal approximation of the exact analysis is given. An inspection of the Bode plots of the control-to- output transfer functions given in the previous chapter shows that for the series resonant converter the dominant behavior is given by a single-pole and the high-frequency behavior is given by a quadratic <u>(s) _ A Fs(s) 1+7—|J1+—2—+ sé] Ors sos Wn” | (5.1) For the parallel resonant converter a similar observation leads to <u >(s) _ A fF ~ 2 ) 2 Fe(s) 1+ A+ S14 SG W rp Os Wp J WsoQs Wso (5.2) The low-frequency gain, A, in both of these transfer functions is proportional to the slope of the conversion ratio characteristics, Ap, which is determined numerically and plotted in Fig. 5.1a and b. A is given by _~ 4." A= Ao Fo (5.3) where Ap is obtained from Fig. 5.1 and is equal to the slope of M aM Ao = aR /Fo) (5.4) Next, the dominant behavior of these transfer functions is determined. For the series resonant converter, it can be seen from Fig. 3.9 that the average output current is almost independent of the load for high 174 Qgrl0 7 10 Ne Ao 9.8 8 Qo=8 Q,.=10 -10+ ~*~ Fig. 5.1a Slope of the canversion ratio characteristics of the series resonant converter in continuous conduction mode. 175 10 1.1 1.2 421.3. 2.4 -8 Q5=3 ~12t ~16 - J Sp=4 -20 -24 Qp=5 Q,=10 |e Q.=10 - P 28 we Qp=7 Ps Fig. 5.16 Slope of the conversion ratio characteristics of the parallel resonant converter in continuous conduction mode, 176 @, and away from the resonant peak. A circuit that represents this approximation is shown in Fig. 5.2b where the average rectified current [Jp| is represented by a stiff current source. It is clear from this circuit and Fig. 3.9 that small variations in F, will cause the output voltage to vary with a single-pole frequency dependence. Therefore, Wr, in Eq. (5.1) is given by oye = RC} (5.5) For the parallel resonant converter, the dominant behavior is determined by referring to the conversion ratio characteristics given in Fig. 2.5 whence it can be seen that for high Q and away from the resonant peak — specially below resonance — the output voltage is almost independent of the load. This is shown in Fig. 5.8b where the rectified capacitor voltage |Vp| is represented by a stiff voltage source. From this circuit and the conversion ratio characteristics it is clear that the dominant behavior of the transfer function in Eq. (5.2) is given by the output low-pass filter and hence = —L_ and = —# 7) Q, = The high-frequency quadratic behavior of these transfer functions is determined next. In Figs. 5.2a and 5.3a, the resonant converter is viewed as a circuit driven by a square voltage voltage, %, with Fourier components at Fs, 2F3,...nP,; as shown in Fig. 54a. When the frequency, or the amplitude, of V, is small-signal modulated, its spectrum will contain sidebands around each Fourier component. The most important sidebands 177 Fig. 5.2 a) The series resonant converter viewed as a circuit driven by a square voltage source, V,, and b) an approximate circuit valid for high Q and away from the resonant penok, Ht + LVV Olvel c= vsR b) Fig. 5.3 a) The parallel resonant converter viewed as a circuit driven by a square voltage source, V,, and b) an approximate circuit valid for high @ and away from the resonant peak. , 178 a) oF, b) Fig. 5.4 The spectrum of the voltage source, V,, in Fig. 6.3, a) when unmodulated and 6b) when small-signal amplitude or frequency modulated. 179 are Fy+fm and F,—-fm. When either circuit is excited below resonance, the response of the resonant branch to the modulation is strongest when the frequency of the sideband at F,+/,, is equal to the resonant frequency Fe+fm=Fo or Iim=Fo-Fs (5.7) and when excited above resonance, the response is strongest when the sideband at #,—f,, is equal to the resonant frequency Fy—-fm=Fo or Sm=Fs—Fo (5.B) Since the voltage and current sources, shown in Figs. 5.2b and 5.3b, are determined by the resonant branch, their response to the modulating signal is strongest when the frequency of the sideband is equal to the resonant frequency. This explains the high-frequency quadratic behavior in these transfer functions. The modulating frequency for which Eqs. (5.7) and (5.8) are satisfied is denoted by w,) and is the frequency in the quadratic factor in Eqs (5.1) and (5.2). This is summarized as follows Wg — Ws Fy< Fo Weg = The @ in this quadratic factor is the same as the original resonant @ _ Wolo _ Rk 180 Equations (5.1) and (5.2) are now compared to the results of the exact analysis in Figs. 5.5 and 5.6. These results are in good agreement for high @ and operation away from the resonant peak. The values of the circuit components, used in determining these transfer functions, for the series resonant converter are Lo=.197mMH Co=.05iuF Cy, =32uF For the parallel resonant converter, the values of these components are Lo = 36H Co= .47mF Ly =1.35mH = C, = 32 uF These are the same values used in the experimental circuits in the previous chapter, Equations (5.1) and (5.2) break down as the operating point gets closer to the resonant peak as shown in Fig. 5.7a and b. These figures are the same as Figs. 4.12b and 4.13b where the results of the exact analysis and measurements are given. The degree of closeness to the peak is dependent on the @; the higher the @ the closer the operating point can get to the peak before the approximate results begin to deteriorate. For the parallel converter this does not present a serious problem since as explained earlier it is desirable to operate this converter for @>3 and below resonance. For the series resonant converter, this high @ assumption must be relaxed since there is no apparent advantage in designing this converter for high @ provided of course, that the @ is not selected low enough for discontinuous conduction mode to occur (Eq. (3.25a)). For example, according to the conversion ratio characteristics in Fig. 3.8, @,=1.5 and F,/F)=.B8 is a reasonable point to operate at, and it 181 OdB - —_—mee approximate exoct -10 ma ~_— 9 -20 °° -30 ~40 -30° -50 Operating point ° -60 Q,26 4 -180 F,/F,=0.6 -70 ~80 rtiiiil po te tI ~270° OHz lOOHz ikHz lOkHz 28kHz — opproximote OdB exact ~10 mog O° -20 -30 phose—~ -90° -40 -50 Operating point Q,=4 - 180° -60 Fg/Fo= 0.65 -70 fan rer ere | 1 rn ptiiril 4 rt ap trriil -270° 20Hz tOOHz ikHz tOkHz Hz Fig. 5.5 Comparison of approximate and exact control-to-output transfer functions of the series resonant converter for high Q and operation away from the resonant peak. 182 2048 i — opproximote exact (OF oF 0° -10e -20- -90° -30b L -180° aol 80 -sobk Operating point L Qp=3 -270° -60 - F,/Fy=0.6 -70F- a rare | m" pa tl 1 re eres | ~360° 20Hz 100Hz tkHz lOkHz 2O0kHz — = approximate 20dB F exoct ~-20 -40 Operating point -60 i Qp=6 | -270° Fy /Fot0.6 “B80 - po td 1 oo td 1 po tol -360° 20Hz |1OOHz tk Hz lOkHz 2OkHz Fig. 5.6 Comparison of approximate and exact control-to-output transfer functions of the parallel resonant converter for high @ and operation away from the resonant peak. 183 —= opproximote OdB - exoct -10 — O° -20 -30 a) -+ -90° -40 phase -5O0F . Operating point a - o 60 Q521.5 '80 F, /Fp=0.8 -70b re rere | po a a a tl 1 -270° ISHz 100Hz IkHz lOkHz 30kHz L — opproximate 20dB - : oO IOoF 7 0G OF L + -90° -lOk “20 F 4 -180° b) _ | -30 4 - f -270° 40 Operating point -50- Qp= 2.22 F F,/Fp= 0.85 -360° -60 “70K 1. 1 pil rs aera e | a ta I -450° 20Hz 100Hz tkHz lOkHz 20kHz Fig. 5.7 Breakdown of approximate control-to-output transfer functions for operating points close fo the resonant peak and low @: a) for the series resonant converter and b) for the parallel resonant converter. 184 would be nice if the results of the approximation could be improved for this converter. According to Fig, 5.7b the discrepancy between the exact and the approximate analysis is in the position of the dominant pole. As the operating point moves towards the peak, the dominant pole begins to move forward and is no longer given by Eq. (5.5). For @=1.5 and F,/Fpo=.8 the dominant pole is already at twice the pole given by Eq. (5.5). This new pole that must be determined is given by i Wrs = KW fs } Wrs = RC} (5.11) The modification factor « is shown in Fig. 5.8 and is determined by a low- frequency numerical approximation of the exact transfer functions. The form of the transfer function given by Eq. (4.68) is “ y" a,e7"s <u>(s) _ n=0 r 1 ~ — , s FT s(s) y be”! ef n=0 (5.12) 8 If we let ee tsl, in} b,e 8 then the real root of n=0 b,(1+sT,)" = 0 corresponds to the dominant pole o,,' in Eq. (5.11) from which « is determined and plotted in Fig. 5.8. It should be pointed out that the quadratic factor in Eq. (5.13) does not correspond to the quadratic factor in Eqs. (5.1) and (5.2) determined earlier. Also, a heuristic modification of 185 4.0 Fig. 5.8 The dominant pole modification factor, x, of the series resonant converter, 186 ame improved approximation OdBr- exact -10 O° -20 -30 -90° -40 -~50 . : Operating point o Qg= 1.5 ~!80 -60f-r- = Fs /F20.8 -7TOF 1 1 ptrirril 1. po tirrel rn po ta | 4 -270° ISHz 100Hz tkHz lOkHz 3OkHz Fig, 5.9 Improved approximate control-to-output transfer function for the series resonant converter, given by Eq, (5.15), after modification of the dominant pole and the @ by x. 187 Q@, is given by _. a (5.14) where « is the same pole modification factor shown in Fig. 5.8. The modified transfer function for the series resonant converter can now be written as >(s) A < ¢ ~ 2 Fs(s 14 Ji+4- 4,44, KW 5 Wso Q Oso (5.15) This improved result is shown in Fig. 5.9 where it can be seen that the magnitude now is excellent agreement but there is still some discrepancy in the phase. The approximate line-to-output transfer function is obtained by an entirely similar argument. These approximate transfer functions are the same as the control-to-output transfer functions except for the low frequency gain, which in this case is given by the conversion ratio M. For the parallel resonant converter the approximate line-to-ouput transfer function is given by <u>(s) _ M _ 2 Ug(s) 14—S, 8 | ppQ Wy” |} s s® + Ws Qs Wg o” it (5.16) For the series resonant converter the approximate line-to-output transfer function is given by b) OdB -20 -40 -60 -100 -120 OdB -80 -100 -120 -140 188 = . epege — approximote Audio Susceptibility exact o° L - XN -90° ~ Operating point Q,=4 = -180° - F,/Fo=0.6 are | A a arrears | rtirial A -270° 40Hz lOOHz Ik Hz lOkHz 28kHz Audio Susceptibility =“ approximate exact O° L ~ -90° i Opercting point | L 0,24 n 1 aaa | 4 i a taryiil 1 ature, | -270° 20Hz 100 Hz IkHz {OkHz 20kHz Fig. 5.10a-b Approzimate and exact line-to-output transfer functions of the series resonant converter. 189 2048 | Audio Susceptibility ommm== approximate exact OF r mag -20F 0° ~40b a) i + -90° -60 -B80- Operating point 4 -180° 5 Qp=4 -100 + F,/Fo=0.7 ~270° -l20- roi! 1 1 oe rere | rn Ll pote pial -360° S5OHz I00OHz tkHz lIOkHz I6kHz 20d8 Audio Susceptibility opproximation exact re) = mag -20- 0° by Laok r + -90° ~60F Operating point 4 L Q,=3 . -B0 F,/Fo =0.6 ~180 -{00 Lit 1 1 a. posi if 2 rn 1 {4123 /-270° 80Hz !00Hz IkHz : tOkHz Fig. 5.11a-b Approximate and ezact line-to-output transfer functions of the parallel resonant converter. 190 <u >(s) M Ug(s) [+ s |p S_ «st Kiss Ws0 @s Weg” (5.17) The approximate and exact line-to-output transfer functions are compared in Figs. 5.10 and 5,11 using the same circuit values as before. Note that for the series resonant converter, the pole position in Fig. 5.10b has been modified by the factor «=1.73, 5.3 Conclusion In this short chapter approximate transfer functions of the series and parallel resonant converters are given for high @ and operating points away from the resonant peak. It is shown that the dominant behavior of these transfer functions is determined by the output low-pass filter while the high-frequency behavior is given by a second order response in the denominator whose frequency is at the difference between the switching and the resonant frequency and whose @ is the same as the original resonant @. These approximations break down for lower @ and operating points near the resonant peak. For the parallel resonant converter this breakdown of the approximate results can be ignored since this converter operates satisfactorily for @,= 3 and below resonance (F,/Fo<.8). For the series resonant converter the results are improved for lower @ since it may still be desirable to operate this converter for Q in the range 1.5 to 2.5. The results of the approximate and the exact analyses are in good agreement. 191 CHAPTER 6 CONCLUSIONS Resonant converters belong to a class of dc-to-de switching converters in which the circuit time constants are comparable to the switching time. For this reason their analysis is tedious and has often been neglected or given in limited and nonilluminating form. In this thesis, the load parameter, @, and the ratio of switching frequency to the resonant frequency, F;/Fpo, are considered as the only two parameters required to characterize the operation of these converters. The de and small-signal ac analyses are given. In the de analysis, the conversion ratio characteristics, peak stresses, and diode conduction time are determined in terms of Q@ and F,/Fp for the series and parallel resonant converters. In the small-signal analysis, the control-to-output transfer function, audio susceptibility, and input impedance are determined at a given operating point (Q, F,;/p)) determined from the dc analysis. Most of the results are obtained numerically. For the de analysis, the only numerical routine required is a root finder, which today can be found in hand-held calculators. The results obtained are plotted in numerous useful graphs to help the engineer design as well as evaluate the performance of these converters for any specific application. In the small- signal analysis, the coefficients in the numerator and the denominator of the transfer functions are determined numerically using a root finder and a matrix multiplication routine. 192 To gain further insight simple and approximate transfer functions are obtained under high Q assumption, 193 APPENDICES 194 APPENDIX A A.1 Determination of The RMS Values of the Resonant Inductor and Capacitor Currents of the Parallel Resonant Converter The square of the rms value of the resonant current is given by [or 7, _ 1 a ° B= Te p(t )dt + fr Tet )at (A.1) The resonant current in the interval 0< f< T, is given by Eq. (2.3a) which can be rewritten as L [ Vp(O an e(t) = at sec(wf) [t~ a sinwp(t — 8) (A.2) where f is given by Eq. (2.29). In the interval 7,< t< T,, /p(t) is given by Eq. (2.29) Wolo __M , _ “y, 7Rtt) = a + sec(woa)sinwy(t —T, +a) (4.3) where a is given by Eq. (2.14). Integration in Eq. (A.1) gives Vy" RB iF = —$+5, | where 5, is given by 195 09 To 2 B, = AP + = 2 + + (cea) ee +24lc (coswp( 7, —8) —coswe8) — B(cos(y—wwp Tq + wa) - costipa) — ZF [er(sin2wo Ty — 8) + sin2wof) + B?(sin2(y — wo Ty + pa) — sin2w9x) | (A.5) where = ML 4" @ (a6) B = secwpa (A.7) _ _ Vr(0) C = secwB |1 V, (4.8) The current through the resonant capacitor, J,({t), is equal to the difference between the resonant inductor current and the reflected load current. In the interval O< ¢< 7,, J,(t) is given by [ — Vp(0) I(t) = See aof) [1 — V. sinw(t —p) (A9) and in the interval 7,< t< 7,, J,(t) is given by I-(t) = sec(woa)sinwo(t — Ty + 2) (A.10) The rms square value is then given by 196 pe We B ne ae (A.11) where &, is given by to Te B, = By (C? — B?) ——— Jy |CMlsinzen( To —B) + sin2wf) + B*(sin2(y —ap To + wa) —sin29a) | (A.12) 197 APPENDIX B B.1 Determination of G(M,Q,,y)=0 in the Range 1/2<F,/Fy<1 Expansion of Eq. (3.18) gives _ ye Cosy coswp7q + sinysinwoT, = ene (B.1) The resonant current, Jp(t), in the interval O<t<T, is given by Eq. (3.5a) which, after substitution of Vp(0) and Jp(0) given by Eqs. (3.15) and (3.16) respectively, is rewritten here as 1-M+MPeK In{t) = Wolo Vy COSW Ty sinwo{t —7,) (B.2) Similarly, the current in the interval T,<t< 7, given by Eq. (3.6a), is rewritten after substitution of Vp(7,), given by Eq. (3.13), as Ip{t) = woCoVy [1+M@ —MK]sinwo(t — Tz) (B.3) The positive areas A, and Ag, shown in Fig. 3.3, are determined by integration of Eqs. (B.2) and (B.3), Ty A = fy Ip(t)dt = CoVy(1-M +MPK)(1 —secw Te) (B.4) T, Az = fo In(t)dt = CoV, (1+.M —MK)(cos(y —woTq) - 1) (B.5) Substitution of cos(y—-wp7_) from Eq. (3.18) in Eq. (B.5) gives 198 Ag = CyV,MK(1—M) (B.6) From Eqs. (3.11) and (3.13) we have A, + Ag = CoVj2MK (B.7) Substitution of Eqs. (B.4), (B.6), and (B.7) gives 1-M+M?K cosiols = TK (B.B) The expression for sinw 7, follows from Eq. (B.8) _ V(1—M*)(MK —2)MK - MK +M -1 (B.9) sinw 7, Substitution of Eqs. (B.8) and (B.9) in Eq. (B.1) gives the implicit equation G,(M,@,,.7) =O in Eq. 3.19 from which the conversion ratio is determined for a given @ and F,/ Fp. In deriving Eq. (B.9) from Eq. (B.8), we made sure that the sign of the denominator was positive because sinwy7,>0. For operation in the range 1/25F,/ Fo¥1, it is important to note that the range of wo7, is given by 1 which implies COswy7Ty <0 and = sinwyT, = 0 (B.11) In [6] and [7] it is assumed that the range of wT, is given by 0<wy7% <7, which does not restrict the range of woT, properly. It should be clear from 199 Fig. Bi, that if wo9%, <m/2 then A,—Ag <0 which implies negative conversion ratio. Hence the proper range of wo7,, when the converter is operating in the range 1/2<F,/Fo<1, is given by Eq. (B.10). Fig. B.1 The resonant current of the series resonant converter in the range 1/2<F,/Fo<1. WoT, is always larger than n/2 and less than 1, otherwise if woT,< 1/2 then A,—Ag <0 and the conversion ratio will be negative which is impossible. B.2 Determination of G(M,Q,,y)=0 in the Range F,/F,>1 It can be seen that in this range of operation, M, Voeak aNd Vp(0) are still given by Eqs. (3.9), (3.11), (3.13), and (3.15) as in the case of operation in the range 1/2<F,/F <1. A, —Ap M = tA (B.12) 1 A, tA Voeak = Ve(Ta) = G2 2s MKYV, (B.13) 200 Vp(0) = —MP®KV, (B.14) The resonant current in the interval 0 <t <7, is given by Ip{t) = wpCo[ Vg + Vo—Ve(0) ]sinwgt + Ip(0)coswot (B.15) and in the interval 7, <t <7,, Jp(t) is given by Ip(t) = 9Co[ Vy —Vo + Ve( Tq) | sinwo(t —T.) (B.16) Substitution of Vp(0) and Vp(7,) gives for Jp(t) in the interval 0 <t <7, Ip(t) = wpCo¥y [1+ M+M*K ]sinwot + Ip(0)coswot (B.17) and for the interval 7, <t <7, In(t) = apCo Vy [1 —M + MK] sinwo(t —T,) (B.18) The requirement /p(7,) =0 in Eq. (B.17) gives Tp(0) = —woCoV, [1+M +MPK ]tanwoTy (B.19) which when substituted in Eq. (B.17) gives (B.20) Similarly the equations for Vp(t), given by Eqs. (3.40b) and (3.41b), are obtained. The requirements Vp(0)=—Vp(T,) and Jp(0)=—Jp(T,) gives 1+M+MPK sin(y —wp7.) = Toman taneoTe : (B.21) 201 1—-M -—M?®K cos(y—2oTa) = a aK (B.22) Expansion of Eq. (B.22) gives 1—M —M*®K COSY COSWy7, + sinysinwp7, = T-M+MK (B.23) Next, the angle wo7, is determined. The positive areas A, and Ag, shown in Fig. 3.13, are determined by integration of /p(t). 7, Ae = -f "Ip(t)dt = CoV, [1+M +M®K ][secwpT, -1] (B.24) tr, Ay = fy In(t)dt = CoV, [1—M + MK ][1-cos(y-woTs)] (B.25) Substitution of Eg. (B.22) in Eq. (B.25) gives A, = CoV, MK([1+M] (B.26) Substitution of Eqs. (B.13), (B.24), and (B.26) in Eq. (B.12) gives 14+M+MPK cosola = (K+ 1) +1 (B27) The expression for sinwo7, follows immediately from Eq. (B.27) . _ V(t — M*)(MK + 2)MK sind Ta = 1+M(K+1) (B.28) Substitution of Eqs. (B27) and (B.28) in Eq. (B.23) gives the implicit Equation (3.38), G,(M,Q,.y)=0, from which the conversion ratio M is determined for a given 9, and F,/F». 202 When the converter is operating in the range F,/Fo>1, it is important to note that the range of wo7, is given by 7 O< wola = 5 (B.29) which implies that Sinwp7, =O and cosuo7, = 0 (B.30) The range given in Eq. (B.29) ensures that A,—Ap>=0 so that the conversion ratio M is always positive. It is clear from Fig. B.2 that if woT,=n/2, then A,;—Ag<0 which implies negative conversion ratio which of course is impossible. Fig. B.2 The resonant current of the series resonant converter in the range F,/Fo=1. In this case woT, is always less than n/2 , otherwise if woT,>n/2 then A,—Ag <0 and the conversion ratio will be negative which is impossible, 203 B.3 Derivation of the Equations in Discontinuous Conduction Mode Equations (3.54), (3.61), and (3.62) are repeated here Am = 2Col Vy +(-1)®Vo— Vem (1) (B.31) Am-1 = —} na Ver = —MKVq = —M®KV, (B.33) A recurrence expression for Ay is obtained now by writing the first few terms of Am. For m=1, A, is given by A, = 20% -Vo-Ker] (B.34) Substitution of Eq. (B.33) in Eq. (B.34) gives A, = 2CoVgl[1-M+M?K] (B.35) From Eq. (B.32) Ve is determined next Vea = Ver + Ges eyg(i— + EA (3.36) From Eq. (B.31) Ag is given by Ag = -2C | Vy + Vo- Ve] (B.37) Substitution of Eq, (B.36) in Eq. (B.37) gives Ag = 2C [1-3 + PK } (B.3B) When Ves, determined by substitution of Eqs. (B.36) and (B.38) in Eq. (B.32), 204 is substituted in Eq. (B.31) the expression for As is obtained As = 2CoV¥[1-5M + MPK ] (B.39) From Eqs. (B.35), (B.38), and (B.39) the recurrence relation for A, given by Eq. (3.63), becomes clear Am = 2CoVj[1—(2m—1)M + M2K] (B.40) The summation in Eq. (3.59) is written as Sp = YS Ap(—1)™?} 1 (B.41) aM The sequence of partial sums Sy, for even n are computed as Se = Ay—Ag = 2Cy¥,(2M) (B.42a) S4 = Sg+Ag—Ag = 2CoVy (4M) (B.42b) Sy = Sy-2+An-1-An = 2CoV,(n¥) (B.42c) Substitution of Eq. (B.42c) in Eq. (3.59) gives the conversion ratio of even type n dem (B.43) >| To determine the conversion ratio for odd type n dem, the summation in Eq. (3.59) is computed for odd n. The sequence of partial sums, Sy, for odd n is given by 205 S, = A, = 2C9V,[1-M + MK] (B.44a) Sg = A,-AgtAg = 2CyV,[1-3M + MK] (B.44b) Sy = Sy-27An-1+4n = 2CoV,[1—-nM + M?K] (B.44c) Substitution of Eq. (B.44c) in Eq. (3.59) gives the conversion ratio for odd type n dem 1 Ma (B.45) The peak stresses are now easily determined. Since Vogge =Vzg, substitution of Eq. (B.43) in Eq. (B.36) gives Eq. (3.66) and substitution of Eq. (B.45) in Eq. (B.36) gives Eq, (3.69). The peak resonant current is given by the amplitude of 4,(f) in Eq. (3.53) Voi _ 2 in which if M is substituted for odd and even type n dem the corresponding expressions for Jpegx are obtained. B.4 Derivation of the Equations for +type n ccm The area under each complete half-cycle of the resonant current, im(t), is given by 206 Tf? Am = fo" lim(t)|dt = 2Co[ Vy + Vol—1)™*? — Vem |(—1)™ (B.47) The recursive relation for Am, given by Eq. (3.82), is determined next. From Eq. (B.47) we have for m=1 and 2 Ay = -2Co[ Vy +Vo-Vei] Ag = 20 Vy —Vo- Mee] Voi is related to Ye by Substitution of Eqs. (B.48) and (B.50) in Eq. (B.49) gives Ag = 2C)[%1-3Vp-% | Following the same procedure we get for m=3 Ag = 201 Vii -5Vo— Vy | (B.48) (B.49) (B.50) (B.51) (B.52) From Eqs. (B.48), (B.51) and (B.52), the recursive relation of Ap, given by Eq. (3.82), becomes apparent (B.53) The summation in the numerator of Eq. (3.80) is determined next. This summation is denoted by S, 207 S, = Bits Am(-1)™ (B.54) The recursive expression for S,, given in Eq. (3.83), follows from the expressions of the first few terms of Eq. (B.54) given by Sg =-A, +Ap = —20p(2Vo) (B.55a) 4 = Se-Agt+Aq = —2C9(4Vo) (B.55b) Sp = Sy-g—-Ap-1+An = —2Co(nVo) (B.55c) The currents io,(f) and igg(t), shown in Figs. 3.19a and 3.20a are given by toa{t) = woCol—Vy —Vo—- Ve(—Th) ] sinwo(t + Ty) (B.56b) The initial capacitor voltage, Ve(0), is determined as follows [ ValTe) = Ya(0)+ [Aon + & Al -1)— Ace ce) Since Vp(0)=—Vp(7,), we get from Eq. (B.57) —2Vp(0) = 2+] 3° Ap(—-1)" + Aoi —Ape Co | ™ ° (B.58) Substitution of Eq. (8.79) in Eq. (3.80) gives 208 nh Am(—-1)™+Ap;—Ape = 2MCoVoK = MK yy m(—1) 01 ~ Agog oVoK = 2CoV, (B.59) Substitution of Eq. (B.59) in (B.58) gives Vp(0) = —M®KV, (B.60) Substitution of Eq. (B.60) in (B.56a) and the requirement i9;(7,) =O gives Ip(0) = —wpCoV[1—-M + MPK ]tanwp Ty (B.61) Substitution of Eqs. (B.61) and (B.60) in Eq. (B.56a) gives . 1-M+M?K #)= — _ toi(t) = cogCoVy cose Ts sin(wot —w Tq) (B.62) which is Eq. (3.84). The area, Ao), is given by T Aor = fy = CoVgli-M + MPK ][1 —secwy Ts | (B.63) The area, App, is given by integrating Eq. (B.56b) 0 . Ag = fp toe(t)dt = Col Vy + Vot Ve(—Ty) ]lcoswo Ty -1] (B.64) Vp(-Ty) is related to Vp(0) by _ Apg _ 2 Age Substitution of Eq. (B.65) in Eq. (B.64) gives for Ago Apg = CoV,[M®K-M-1][secwoT, -1] — (B.68) n7o a since 7, = 7, - —T, we have 209 COSWyT, = cos(y—nT—wWoT,) = cos(y—wpT,) n=0,2,4,6,... (B67) The area, Agg, is now rewritten as Agog = CoV,[M*®K —M —1 ][sec(y—wpT%)—-1] (B.68) Substitution of Eq. (B.68) in (B.65) gives Va(-T)) = —MV, —V, —V,[M?K -M -1 ]sec(y—wyT,) (B.69) Substitution of Eq, (B.69) in Eq. (B.56b) gives Eq. (3.85) for ige(t) M?K-M-1 sin(cot — wT. +7) (B.70) Substitution of Eq. (3.83), or Eq. (B.55c), and Eq. (3.79) in Eq. (3.80) gives —2nCy Vo +Ao —Age M = 2VeCpK (B.71) Substitution of Ap, and Apg from Eqs. (B.63) and (B.68) in Eq. (B.71) gives 2M(n+1) = [1+M —MP?K ]sec(y—wo7,) —[1 —M + MK ]secwyTy (B.72) Also the requirement 19;(0) =i2(0) gives 1-M+M*®K tan(y—wW pT.) = an(y~@oTa) = sa eK tanwyT, (B.73) Equations (B.72) and (B.73) are solved simultaneously as follows. In Egs. (B.72) and (B.73) let 210 C=1+M-M’K and D=1-M+M?K (B.74) Equations (B.72) and (B.73) can be written as D z + secly—Wo7,) = G secwnTa + Hine) (B.75) D tan(y—woT,) = o taneoTa (B.76) Squaring both sides of Eq. (B.76) and using the identity tn?=sce®—-1 we get sec*(y—woT,) = 2” (seek Ty-1)+1 Squaring both sides of Eq. (B.75) 2 2 ; _ pe. 2{nt1 n+1 sec®(y— wT.) = 5 secu Ty +4M {oe + 4MD ce Secu Tg (B.78) Equating Eqs. (B.78) and (B.77) we get - _ 1-M+MPK COSWypT, = 1-M(n+1)°-MK (n+1) (3.79) sinw7, follows from Eq. (B.79) sino, T, = SK +n [MP (n+ 1)?-1 in +2 -MK] we MK+M(nt+1)2-1 (B.80) In obtaining Eq. (B.B0) from Eq. (B.79) we made sure the sign of the denominator was positive by realizing that M>1/ (n+1) because sinwo7, must be positive. Substitution of Eq. (B.79) in Eq. (B.72) gives 211 1+M—-MPK — = + cos(y— Ta) i+M(n+i1—MK Ot) (B.B1) which upon expansion gives 1+M—-M*®K cosy cosw97, + sinysinwy7, = 1+M(nti)°-MK (n+1) (B.82) Substitution of Eqs. (B.79) and (B.80) in Eq. (B.82) gives the implicit equation G.(M,Q,,7,.n) =0 in Eq. (3.88). The peak stresses are easily determined now. According to Fig. 3.19a Voeae is given by Substitution of Eqs. (B.60), (B.63) and (B.79) in Eq. (B.83) gives Eq. (3.89). The peak resonant current, Jpeqx, according to Fig. 3.19a, is given by the amplitude of i9;(f) which from Eq. (B.62) is given by 1—-M+MPK L, = wl peak or0"T) costg Fa (B.B4) Substitution of Eq. (B.79) in Eq. (B.84) gives Eq. (3.90). B.5 Derivation of The Equations For -type n ccm Since the analysis of —type n ccm is very similar to the analysis of +type n cem, given in the previous section, only the key equations are given in this section. Referring to Fig. 3.19b we proceed as before by equating input and output powers and obtain 212 n — } Am(-1)™ + Aoi —Aog m=1 AmtApitA y mt 4191 T Ag (B.85) M= The output voltage is given by the product of the average rectified resonant current and the load # R Vo = <|Ip(t)|>R = = AmtAoi +A 0 = <[Inlt)I>R = F—(¥ Am +Aor + Aoe) (86) The recursive relation for areas, Aj, under each complete half- cycle is determined exactly as before and is given by Am = 2Col Vy —(2m—-1)Vo~ V1] (B.B7) where V,;, shown in Fig. 3.19b, is the resonant capacitor voltage at the beginning of the first complete half-cycle. The summation in the numerator of Eq. (B.85) is determined using Eq. (B.87) which for even n gives nh Am(—1)™ = -—2encoV, =0,2,4,6,.... Ant) mavens * (B.88) Substitution of Eqs. (B.88) and (B.86) in Eq. (B.85) gives Me eng Vo + Ag ~—Age 2CyVoK (B.B9) To complete the analysis, the currents i9;(f) and Zipg(f) must be determined. These are easily given by 213 ; 1-M—-M’K toi (t) = woCoV, cos(y—woTn) sinwot (B 90) a . . 1+M+M*K ipe(t) = tg Cg Vg COSW)T. sin(y — wot) (B 91) a . where the time origin for these equations has been shifted to the beginning of to,(é) as shown in Fig. 3.20b. The areas Ap; and App are given by Ap, = CoV,[1-M —M?K ][sec(y —wo7,) —1 | (B.92) Aoz = CoVg[1+M +M?K ][secwpT, —1 ] (B.93) Substitution of Ap, and Ape in Eq. (B.B9) gives 2M(n+1) = (1+M+MPK)sectipT, —(1-M—M?K)secly—woT,) (B94) T, The requirement %9;(7,) =to2(7,) and use of 7, =T7,— = —T, give 1+M+M?K _ Tom oe tanwo7, = tan(y—wgTy) (B.95) Equations (B.94) and (B.95) are solved simultaneously to eliminate wT. We then have 1+M+MPK (n+1) 1+M(n+1)°+MK (B.96) COSHyT, = V[MK -—n][1—M?(n+1)*][n+2+MK] 1+M(n+1)®?+MK (B.97) SinwyT, = 1-M—M?K 1~M(n+1)°+MK cos(y—wWpTa) = (n+1) (B.98) Substitution of Eqs. (B.96) and (B.97) in the expansion of Eq. (B.98) gives 214 the implicit equation G_(M,.Q,,7.n)=0 in Eq. (3.91). B.6 Numerical Methods of Solving G,.(M,Q,,y.n)=0 Equations (3.88) and (3.91) are solved numerically for a given Q,, y and n. These equations have only one root that lies in the range O0<M<i/(n+1). There are several numerical routines available for solving such implicit equations, even some that can be found on hand-held calculators. The particular numerical routine used in this work was the "BISECT’ method available on Hewlett-Packard System 45 [10]. These equations can also be solved using a polynomial root-finder routine (subroutine "SILJACK" [10]) if they are put in polynomial form. The polynomial that correspond to these implicit equations is given by 6 dai) P(Q.y.n) = 0 (B.99) where a=1 corresponds to G4 (M,Q.yn)=0 and a=~-1i corresponds G_-(M,Q,,7,n)=0. The coefficients P, are given by P,=2m*(1—-cosy) -sin®y P,=4K[sin®y+m*(cosy—1) ] P,=2(m*—3K*)sin®y —2(K*® —m*— 1)m®cosy + 2m®(K* + m* —4m? + 1) P,=4K(K* -m‘)sin®?y -8Km®cosy + BKm*(2m? — 1) P,=- (K*® —m*)*sin®y + (6.K* —m*)2m*cosy + 2m*(6K* +m*—4k*m*) 215 Ps=—4Km"[2k? +m* + (2K*—m*)cosy] P,=2kK*m*| kK? + mt +(K®—m*)cosy] (B,100a-b) where m=n+i (B.101) Of all six roots only that which satisfies the original equation, Eq. (3.88) or Eq. (3.91), is accepted. In the presence of diode voltage drops, Q, is modified according to Eq. (3.110). If Eqs. (3.88) and (3.91) are modified with this change in , they can still be solved numerically using the subroutine "BISECT” to obtain the value of M’. The polynomial method in this case must also be modified. One method is to find a new polynomial which corresponds to either Eq. (3.88) or Eq. (3.91) in which 4 has been modified, or an iterative method can be employed without changing the polynomial in Eq. (B.99) as shown in the diagram. Normally, five to six iterations are sufficient. Of course, if in the range of interest Van/ MV, <<1, this correction for Q, is not necessary. ¥ (an)! P, = 0 1=0 | G, (4. Q,.y.R) = 0 } No Has the value of M’converged | ves M’ . Fig. B3 An iterative method for the determination of M’ of the series resonant converter when taking into account the voltage drop in the output rectifiers. This refinement can be ignored if Vap HV, Q," = afi 216 B.7 Determination of the RMS Values of the Resonant Current and the Output Filter Capacitor Current of the Series Resonant Converter in the Range F,/Fo=1i/2 In the range 1/2<F,/Fy<1 the resonant current in the interval O<t<T, is given by Eq. (B.2) in which substitution of Eq. (B.8) gives Ip(t) = weCoVg(1 —M —MK)sinw(t — 7.) (B.102) In the interval 7,<t<T,, Ip(t) is given by Eq. (B.3). The rms square value of Ip(t) is given by 2_ 1 Ta rg "72 Pe a SJ, TRE )dt + Sr, Lilt at (B,103) Integration in Eq. (B.103) gives I? r= w09"Co” Vy *D, (B.104) where r D 1 ee UKY’ (aM URS Fe — 2- sintoota| 4 (Lt - MK)” ; _ Feb smely oot) 2 T, 2y (B.105) A similar computation gives for D, in the range Fy /Fo>1 2] D _ (+M+MK) a 1 sin2uT, Te ey r 2 217 _ ef +i +UK [i- Te By sine —o0Te) (B.108) Note that eq. (B.106) can be obtained form Eq. (B.105) by changing M to —M. The current through the output filter capacitor is given by I, = |In(t)| - Io (B.107) The rms square value of J, is then given by IP = <({Ip(t)| -Io)®> = 1? - 18 (B.108) c where J” is given by Eq. (B.104). Substitution of Eqs. (B.108) and (B.104) in Eq. (3.117) gives Eq. (3.118). 218 APPENDIX C C.1 Small-Signal Approximation of eAt If A is diagonalizable (A has distinct eigenvalues or is Hermitian) then f (A) is given by J(A) = TH(N)T = EF Oder? (C.1) where x; and ; are the eigenvectors and the eigenvalues of A, and the matrix T is given by T= [zy \ze| +> [ap] T= [ry|re| °° Ira]? Let f (A) =e? and g(A)=1+At and assume that \,f «1 so that Fm) = ew 14AP = g(r) From Eq. (C.1) we have F(A) = US Adan? ® Nig dan” = 9 (A) which gives f(A) g9(A) or ett a s+at (C.2) (C.3) (C.4) (C.5) (C.6) 219 A similar argument applies for an arbitrary square matrix A for which f(A) = 7f(J)T~! where J is the Jordan form A. In this case, in addition to f (A,), the derivatives of f(A,) are considered while still (C.4) is satisfied. C.2 Matrices, Vectors, and Constants Occurring in Chapter 4 Some of the vectors and matrices in Chapter 4 are redefined here for conciseness. These are g, = efit vo = e42Ts~Ta) (c.7) B, = Bi(T%) Be = B2(T, — Ty) (C.B) The vectors in Eqs. (4.54) and (4.55) are given by dy = M,[(AA;—AgA)X(0) + g2(y, —AgB, —1)bV, | (C.9) d, = —M,|AzAX(0) + ye(AgB, +/)bV, 1/2FP) (C.10) where A=go¢, as given by Eq. (4.27). The constants k, and k, in Eq. (4.57) are given by 1 kg = q leie + c (Bog1A,—A) ]X(0) s 1 + pe lerBy ted (Beds —¥2) Fi JOY, (C.11) k, = —cf[AX(0) + (ye(B, +Ag!)~Ag)bVy J27, + 2Ts Vo (C.12) 220 The matrix M, in Eq. (4.65) is given by where Vj is the vector given by Eqs. (4.64). The vector A? in Eq. (4.66) is given by where the vector c’ is given by Eq. (4.38). M, = M,A+d,V0 hRTac™ +k, VE The constant k, in Eq. (4.83) is given by - T ky = kpkg + d7 where k, is given by Eq. (4.82) and d” is given by Eq. (4.39). The vector d, in Eq. (4.84) is given by d, = kd, + Bb where B=928,+ Be as given by Eq. (4.28). The matrix A, and the vector 6, in Eq. (4.99) are given by where the matrices A, and BH, A, = Ay + 9a Ve by = kp9a + Bad respectively and the vector g, is given by Ja = 1 RT. [1+ BA, —¢2]e,X(0) + 1 eTs are given by Eqs. (4.92) and [7 + BeAi ~¢2|BibV, (C.13) (C.14) (C.15) (C.16) (C.1'7) (C.18) (4.93) (C.19) [1] 221 REFERENCES Dilip A. Amin, "Applying Sinewave Power Switching Techniques to the Design of High-Frequency Off-line Converters," Proc. of the Seventh National Solid-State Power Conversion Conference (Powercon 7), March 1980, pp. Al.1-A1.7 Richard Myers and Robert D. Peck, "200 khz Power Technology in New Modular Power Supplies," Hewlett-Packard Journal, Aug. 1981, pp. 3-10 V.T. Ranganathan, Phoivos D. Ziogas, Victor Stefanovic, " A Regulated De-De Voltage Source Inverter Converter Using a High Frequency Link," IEEE Transactions on Industry and Applications, vol. IA-18, No. 3, May/June 19B2, pp. 279-287, Slobodan Cuk, "Modeling, Analysis and Design of Switching Converters," Ph.D. Thesis, California Institute of Technology, pp. 49-44, Nov. 1976. F.C. Schwarz, "A Method of Resonant Current Pulse Modulation for Power Converters," IEEE Transactions on Industrial Flectronics and Control Instrumentation, vol. IECI-17, No. 3, pp.209-221, May 1970. [6] [9] [10] 222 F.C, Schwarz, "An Improved Method of Resonant Current Pulse Modulation for Power Converters,” IEEE Power Elecronics Specialists Conference, 1975 Record, pp. 194-204 (IEEE Publication 75CH0965-4 AES); also IEEE Transactions on Industrial Electronics and Instrumentation, vol. JECI-28, pp. 134-141. May 1976. R. King and T.A. Stuart, "A Normalized Model for the Half-Bridge Series Resonant Converter,” IEEE Transactions on Aerospace and Electronic Systems, vol. AES-17, No. 2, pp. 190-198. March 1981. R.D. Middlebrook and Slobodan Cuk, "A General Unified Approach to Modelling Switching-Converter Power Stages," IEEE Power Electronics Specialists Conference, 1976 Record, pp. 18-34, (IEEE Publication 76CH1084-3 AES); also International J. of Electronics, vol. 40, no. 1, pp. 1-4, Jan. 1976. Farhad Barzegar, "Problems in Switched-mode DC and AC Power Conversion,” Ph.D Thesis, California Institute of Technology, pp. 38-49, May 1983. Hewlett-Packard System 45 General Utility Routines.