RESONANT TECH PWE ELEC
RESONANT TECH PWE ELEC ECEN 5817
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CHAPTER 2 Sinusoidal Approximations n this chapter the properties of the series parallel and other resonant converters are investigated using the sinusoidal approximation Harmonics of the switching frequency are neglected and the tank waveforms are assumed to be purely sinusoidal This allows simple equivalent circuits to be derived for the bridge inverter tank recti er and output lter portions of the converter whose operation can be understood and solved using standard linear ac analysis This intuitive approach is quite accurate for operation in the continuous conduction mode with a highQ response but becomes less accurate when the tank is operated with a low Q factor or for operation in or near the discontinuous conduction mode The important result of this approach is that the dc voltage conversion ratio of a continuous conduction mode resonant converter is given approximately by the ac transfer function of the tank circuit evaluated at the switching frequency The tank is loaded by the effective output resistance nearly equal to the output voltage divided by the output current It is thus quite easy to determine how the tank components and circuit connections affect the converter behavior The in uence of tank component losses transformer nonidealities etc on the output voltage and converter ef ciency can also be found It is found that the series resonant converter operates with a stepdown voltage conversion ratio With a 11 transformer turns ratio the dc output voltage is ideally equal to the dc input voltage when the transistor switching frequency is equal to the tank resonant frequency The output voltage is reduced as the switching frequency is increased or decreased away from resonance On the other hand the parallel resonant converter is capable of both stepup and step down of voltage levels depending on the switching frequency and the effective tank Qfactor Switching loss mechanisms are also considered in this chapter Zero voltage switching is a property that can be obtained in resonant converters whenever the tank presents a lagging inductive load to the switch network This occurs for operation above resonance in the series Principles of Resonant Power Conversion resonant converter and it can lead to elimination of the switching loss which arises from the switch output capacitances Likewise zero current switching can be obtained when the tank presents a leading capacitive load to the switch network as in the series resonant converter operation below resonance This property allows natural commutation of thyristors and elimination of switching loss mechanisms associated with package and other parasitic inductances 2 1 First Order Network Models Consider the class of resonant converters which contain a controlled switch network NS and drive a linear resonant tank network NT The latter in turn is connected to an uncontrolled recti er NR lter N1 and load R which is illustrated in Fig 21 Many wellknown converters can be represented in this form including the series parallel LCC et al N s 1gt gt N T N R N F 1 8 II 1 K 1 LI Vg W n Vs VR v R 7 7 E 7 power controlled resonant uncontrolled low load input switch tank rectifier pass network network filter Fig 2 1 A class of resonant converters which consist of cascaded switch tank recti er anal lter networks A series resonant converter example is s own In the most common modes of operation the controlled switch network produces a square wave voltage output vst whose frequency fs is close to the tank network resonant frequency f0 In response the tank network rings with approximately sinusoidal waveforms of frequency fs The tank output waveform vR or iR is then recti ed by network NR and ltered by network NF to produce the dc load voltage V and current I By changing the switching frequency fs closer to or farther from resonance f0 the magnitude of the tank ringing response can be modi ed and hence the dc output voltage can be controlled A switch output v oltage spectrum 1 r5 3fs st re sonant tank response rg 3fs st tank current iS spectrum r5 3fg st Fig 22 The tank responds primarily to the fundamental component of the applied waveforms converter dc terminal quantities Vg V and I 1g 1 2 Vg 0 gt gt is gt Vs Fig 23 An ideal switch network Controlled Switch Network Chapter 2 Sinusoidal Approximations In the case where the resonant tank responds primarily to the fundamental r fs of the switch waveform vst and has negligible response at the harmonic frequencies nfs n 3 5 7 then the tank waveforms are well by their components As shown in Fig 22 this is indeed the case when the tank network approximated fundamental contains a highQ resonance at or near the switching frequency and a lowpass characteristic at higher frequencies Hence let us neglect harmonics and compute the relationships between the fundamental components of the tank terminal waveforms vst ist iRt and vRt and the 1 Vs1t vg A vsa Ts I Vg z SWIlCh position 1 2 Fig 2 4 IControlled switch network output voltage vst and its dndamental component vs1t If the switch of Fig 23 is controlled to produce a square wave of frequency fs as in Fig 24 then its output voltage waveform vst can be expressed in the Fourier series vsa 2 nl35 sin 2nTEfst 21 Principles of Resonant Power Conversion The fundamental component is 4V Vs1t g which has a peak amplitude of 471 times the dc input voltage Vg and is in phase with the original sin 27 fst 22 square wave vst Hence the switch network output terminal is modeled as a sinusoidal voltage generator vs1t It is interesting to model the converter dc input This requires computation of the dc component lg of the switch input current igt The switch input current igt is equal to the output current ist when the switches are in position 1 and its inverse ist when the switches are in position 2 Under the conditions described above the tank rings sinusoidally and ist is well approximated by a sinusoid of some peak amplitude 131 and phase pg ist 2 ISI Sin275fstPs 23 The input current waveform is shown in Fig 25 The dc component or average value of the input current can be found by averaging igt over one half switching period Ts2 ig TLS I igtdt 0 2 Ts2 isa 131 sin 27395st Pg dt 0 T Fig 25 Switch terminal current 2131 cos pg 24 waveforms igt and ist 75 Thus the dc component of the converter input current depends directly on the peak amplitude of the tank input current 131 and on the cosine of its phase shift pg An equivalent circuit for the switch is given in Fig 26 This circuit models the basic energy conversion properties of the switch the dc power supplied by the voltage source Vg is converted into ac power at the switch output Note that the dc power at the source is Vg times the dc component of igt and the ac power at the switch is the average of vst times ist Furthermore if the harmonics of vst are negligible then the switch output voltage can be represented by its fundamental a sinusoid vs1t of peak amplitude 4VgTE Chapter 2 Sinusoidal Approximations is1t E 1s1Sin27Efst PS 2 flswosaps V310 sin27cfst Fig 2 6 An equivalent circuit for the switch network which models the fundamental component of the output voltage waveform and the dc component of the input current waveform Uncontrolled rectifier with capacitive lter network In the series resonant converter the output recti er is driven by the nearly sinusoidal tank output current iRt and a large capacitor C1 is placed at the dc output so that the output voltage V contains negligible harmonics of the switching frequency fs as shown in Fig 28 The diode recti ers switch when iRt passes through zero as shown in Fig 27 and the recti er input voltage vRt is essentially a square wave equal to V when iRt is positive and V when iRt is negative Note that vRt is in phase with iRt 7s D V ma VRI D i D R 1 iRO T IliRt 1 I gt gt gt R1 v 7V uncontrolled low pass load rectifier filter Fig 28 Uncontrolled rectifier with F g39 239 7Recn er mput waveforms Rm capacitivefilter network as in the series and VRm resonant converter The diodes switch when iRt passes through zero If the tank output current iRt is a sinusoid with peak amplitude 1R1 and phase shift pR iRt 1R1 sin Z st PR 25 then the recti er input voltage may be expressed in the Fourier series vRt 2 sin mestiqu 26 n135 where pR is the phase shift of iRt This voltage waveform is impressed on the output terminal of the resonant tank network Again if the tank network responds primarily to the fundamental component of fs of vRt and has negligible response at the harmonic frequencies nfs n i 5 Principles of Resonant Power Conversion 357 then the harmonics of vRt can be ignored vRt is then well approximated by its fundamental component vRt lea V sin 2nfst4pR 27 The fundamental voltage component vR1t has a peak value of 4TE times the dc output voltage V and it is in phase with the current iRt The recti ed tank output current l iRt l is ltered by capacitor CF Since no dc current can pass through CF the dc component of l iRt l must be equal to the steadystate load current I Equating dc components yields Ts2 1 A 1Rl sin27cfstpR dt Ts fl 28 Therefore the load current and the tank output current amplitudes are directly related in steady state Since vR1t the fundamental component of vRt is in phase with iRt the recti er presents an effective resistive load Re to the tank circuit The value of Re is equal to the ratio of le t to iRt Division of Eq 27 by Eq 25 and elimination of 1R1 using Eq 28 yields 9 iRt n2 1 29 With a resistive load R VI this equation reduces to L Re n2 R 08106 R 210 Thus the tank network is damped by an effective load resistance Re equal to 81 of the actual load resistance R An equivalent circuit is given in Fig 29 i110 1R1 sin27cfst pT I gt gt 8 2 ma R87EZR E 1R1 v R Fig 2 9 An equivalent circuit for the uncontrolled recti er with capacitive lter network which models the funalamental components of the tank output waveforms iRt anal vR1t anal the alc components of the loaal waveforms I anal V The recti er presents an e ective resistive loaal Re to the tank network Chapter 2 Sinusoidal Approximations Resonant tank network We have postulated that the effects of harmonics can be neglected and we have consequently shown that the bridge behaves like a fundamental voltage source vs1t and that the recti er behaves like a resistor of value Re We can now solve the resonant tank network by standard linear analysis HS iRl As shown in Fig 210 the tank circuit is a of linear network with the voltage transfer function tank V R network R1 e VR1S ms Vs1S 211 Fig 210 The linear tank quotem0N6 696611861 Hence the ratio of the peak magnitudes of vR1t by an effective sinusoidal input source and driving an e ective resistive load and V310 ls glven by peak magnitude of vR1t 7 7 H 2 12 peak magnitude ofvs1t H S HSJ 2 fs In addition iRis given by ins Vs1S 213 So the peak magnitude of iR is H 1R1 W gt peak magnitude ofvs1t 214 e Solution of converter voltage conversion ratio VVg An equivalent circuit of a complete resonant converter is depicted in Fig 211 The complete voltage conversion ratio of the resonant converter can now be found VVg R g 39 le 39 Mensamg in H V v v X If i V7111 Ilvill I iR Vin HVs1ll 39 Vg Simpli cation by use of Eq 210 yields the nal result Principles of Resonant Power Conversion 1 H s 216 Vg Fjans Eq 216 is the desired result It states that the dc conversion ratio of the resonant converter is approximately the same as the ac transfer function of the resonant tank circuit evaluated at the switching frequency fs This intuitive result can be applied to converters with many different types of tank circuits However it should be reemphasized that Eq 216 is valid only if the response of the tank circuit to the harmonics of vst is negligible compared to the fundamental response an assumption which is not always valid In addition we have assumed that the switch network is controlled to produce a square wave and that the recti er network drives a capacitivetype network Finally the transfer function Hs is evaluated assuming that the load Re is effectively resistive and it is given by Eq 210 controlled rectif1er with dc input switch ac tank network capacitive filter dc output Hs A 1 iR1 t k an 2 v g 0 network VRI Re 7 HIRIH R 4V ig H1s1llcosltPs Vs1 Tgsmonfst Re R Fig 211 Steady state equivalent circuit which models the alc anal fundamental components of resonant converter waveforms Converter ef ciency The effects of tank component losses can also be easily estimated using the model of Fig 211 The converter input power is 2 17 Pm ngg Vg lllsllcows Note that HisH cos pg is equal to the real part of iss In addition iss is equal to the switch output voltage vs1s divided by the tank input impedance Zis Vs1S 218 1sS as 15 Vs1S Hence the real part of iss is Chapter 2 Sinusoidal Approximations 4V 219 Real1ss vs1s RealYis fr RealYis and the input power is Pin 12 ng RealYis 220 7 The converter output power is HVRIH2 Pout IV 27KB 221 But vR1s vs1s Hs and hence lllell2 vs1s2 Hs2 So H 2 n2V2 222 130 Vs1s2 8 Hltsgt2 Hence the converter ef ciency is 2 P HS 130 H H 223 In RerRealYis This expression models the losses associated with the tank network in a simple circuitoriented way Tank network ef ciency can be estimated by computing the tank transfer function Hs and the tank input admittance Yis and then evaluating Eq 223 An example is given in the following section in which the in uence of tank inductor core loss and capacitor esr on converter efficiency is determined 22 Series Resonant Converter Example The series resonant converter with switching frequency control is shown in Fig 21 For this circuit the tank network consists of a series LC circuit and Fig 211 can be redrawn as in Fig 212 The transfer function Hs is therefore Hs L L Z1S R9 sL L sC 5 1 Fig 212 Equivalent circuit which models Qewo the fundamental components of the tank 1 s if 23924 waveforms in the series resonant Qemo no converter Principles of Resonant Power Conversion 1 h 072nf were 0 m 0 L R0 6 QeR0Re The magnitude of Hi27cfs which coincides with the converter dc conversion ratio M VVg is M 111102751 le 2 if 2 225 1Qe F F where F fs f0 ZissLRe HZime at 300 L A 7 R 00 00C 0 SO 1 DO 7 VLC R0 V Re and ZiUwOjR039jR0Re Re 1 H HOltDH Qe Fig 213 Construction ofthe Bode diagrams onis and Hs for the series resonant converter The Bode diagrams of Zis and Hs are constructed in Fig 213 Equation 225 is plotted in Figs 214 and 215 for various values of Q9 and of switching frequency and the approximate results are compared with the exact results of chapter 3 It can be seen that the dc conversion ratio is unity when the converter is excited at resonance regardless of load This is true because the series LC circuit behaves as a short circuit at resonance the impedances of the Chapter 2 Sinusoidal Approximations tank inductor and capacitor are equal in magnitude but opposite in phase and their sum is zero The voltages vs and vR are therefore the same It can also be seen that a decrease in the load resistance R which increases the effective quality factor Qe causes a more peaked response in the vicinity of resonance M VVg 00 07 08 09 10 F 05 06 Fig 214 characteristics below resonance MVVg 1 2 3 4 F Comparison of approximate anal exact series resonant converter characteristics above resonance Fig 215 IHIHIH J exact M Q2 approx M Q2 exact M Q10 approx M Q10 exact M Q05 approx M Q05 Comparison of exact and approximate series resonant converter exact M Q05 approx M Q05 exact M Q10 approx M Q10 exact M Q2 approx M Q2 IHIHIMEI Over what range of switching frequencies is Eq 225 accurate The response of the tank to the fundamental component of vst must be suf ciently greater than the response to the 11 Principles of Resonant Power Conversion harmonics of vst This is certainly true for operation above resonance because Hs contains a bandpass characteristic which decreases with a single pole slope for fs gt f0 For the same reason Eq 225 is valid when the switching frequency is below but near resonance However for switching frequencies fs much less than the resonant frequency fo the sinusoidal approximation breaks down completely because the tank responds more strongly to the harmonics of vst than to its fundamental For example at fs f03 the third harmonic of vst is equal to f0 and directly excites the tank resonance Some other type of analysis must be used to understand what happens at these lower frequencies Also in the lowQ case the approximation is less accurate because the lter response is less peaked and hence does not favor the fundamental component as strongly As shown in a later chapter discontinuous conduction modes may then occur whose waveforms are highly non sinusoidal E f ci en c y A similar analysis can be used to compute the converter ef ciency Let us model the effects of tank inductor core loss by a resistance Rp and tank inductor winding resistance and tank capacitor equivalent series resistance esr by an effective resistance Rs as shown in Fig 216 Standard circuit analysis can be used to show that the tank transfer function Hs is given by H1 i Hs M 226 1 S i2 QemO 00 where 00 1 LC RPR sRe V RP l we ReC R 01 f1 L g Re RS Qe RP R0 R R RVL3941 S e 0 C Rp The ef ciency is found by evaluation of Eq 223 For the circuit of Fig 216 the ef ciency is 1 2 12 n Re 7 RP S 227 RsRe ZELY Rpmpw RsRe S 12 Chapter 2 Sinusoidal Approximations Hs A sketch of tank network ef ciency vs switching A frequency is given in Fig 217 To the extent that the sinusoidal approximation is valid the ef ciency RP at low frequency is asymptotic to the value Re Rs m1 Rs Re The tank esr R3 is effectively in series with M the load and leads to a power loss in direct L C z proportion to the load current Hence to obtain V 1 V SI gt R1 Re high ef ciency one should choose RsltltRe As the switching frequency is increased the inductor core loss begins to degrade the Fig 216 Equivalent circuit used to model ef ciency This occurs when the denominator of tank comPonem losses39 Eq 227 begins to increase To avoid this the switching frequency should be restricted to fs ltlt i VRMRP H RsRe 23928 Altematively given a maximum desired switching frequency Eq 228 can be used to determine a lower bound on Rp Re RPRSRe fs V RP RPM RsRe g 27cL Fig 21 7 Tank network ef ciency vs switching frequency Thus sinusoidal approximations give an effective means of estimating the losses and ef ciency degradations which arise owing to tank component nonidealities Principles of Resonant Power Conversion 23 switch output voltage spectrum fs resonant response fs tank A current is spectrum fs Fig 218 When the tank responds primarily 3 fs 5 fs 3 fs 5 fs 3fs sfs to the third harmonic of the switching frequency then frequency components other than the third harmonic may be neglected Vst E VSnO 4 Vg IITE Ts sin most Subharmonic Modes of the Series Resonant Converter If the nth ha1monic of the switch output waveform vst is close to the resonant tank frequency nfs N f0 and if the tank effective quality f factor Qe is suf ciently large then as shown in Fig 218 the tank responds primarily to harmonic n All other components of the tank waveforms can then be neglected and it is a good f approximation to replace vst with its nth harmonic component f 229 This differs from Eq 22 because the amplitude is reduced by a factor of ln and the frequency is nfs rather than fs Chapter 2 Sinusoidal Approximations U1quot cola fo f0 f0 fs Fig 219 The subharmonic modes of the series resonant converter These moales occur when the harmonics of the switching frequency excite the tank resonance The arguments used to model the tank and recti er lter networks are unchanged from section 21 The recti er presents an effective resistive load to the tank of value Re STUTEZ In consequence the converter dc conversion ratio is given by M L H UZTme 230 Vg n This is a good approximation provided that nfs is close to f0 ie n1fsltf0 ltn1fs 231 and Q is suf ciently large Typical characteristics are plotted in Fig 219 The series resonant converter is not generally designed to operate in a subhaimonic mode since the lndamental modes yield greater output voltage and power and hence higher ef ciency Nonetheless the system designer should be aware of their existence because inadvertent operation in these modes can lead to large signal instabilities 2 4 The Parallel Resonant Converter The parallel resonant converter is diagrammed in Fig 220 It differs from the series resonant converter in two ways First the tank capacitor appears in parallel with the recti er network rather than in series this causes the tank transfer function Hs to have a different form Second the recti er drives an inductiveinput lowpass lter In consequence the value of the effective resistance Re differs from that of the recti er with a capacitive lter Nonetheless sinusoidal approximations can be used to understand the operation of the parallel resonant converter Principles of Resonant Power Conversion iga NS N T N R N F ls iR I gt gt u gt Vg W y m I Vs VR V R 7 7 E 7 power controlled resonant uncontrolled low load input switch tank rectifier pass network network filter Fig 220 Block diagram of the parallel resonant converter As in the series resonant converter the switch network is controlled to produce a square wave vst If the tank network responds primarily to the fundamental component of vst then arguments identical to those of section 21 can be used to model the output fundamental components and input dc components of the switch waveforms The resulting equivalent circuit is identical to Fig 26 The uncontrolled recti er with inductive lter network can be described using the dual of the arguments of Section 21 In the parallel resonant converter the output recti ers are driven by the nearly sinusoidal tank capacitor voltage vRt and the diode recti ers switch when vRt passes through zero as in Fig 221 The recti er input current iRt is therefore a square wave of amplitude I and it is in phase with the tank capacitor voltage vRt The lndamental component of iRt is ima 471 sin2nfst p10 232 Hence the recti er again presents an effective resistive load to the tank circuit equal to R V1110 TEVR1 iR1t 4 1 233 Wquotquot p The ac components of the recti ed tank Fig 221 Parallel resonant converter capacnor VOItage l VRO l are removed by the waveforms vRt anal iRt output low pass lter In steady state the output voltage V is equal to the dc component of l vRt l Ts 2 vTL le sin27tfst pR dt 234 8 0 So the load voltage V and the tank capacitor voltage amplitude are directly related in steady state Substitution of Eq 227 and resistive load characteristics V IR into Eq 226 yields 16 Chapter 2 Sinusoidal Approximations 2 7 TE 7 Re 7 g R 7 12337R 235 An equivalent circuit for the uncontrolled recti er with inductive lter network is given in Fig 222 This model is similar to the one used for the series resonant converter Fig 29 except that the roles of the recti er input voltage vR and current iR are VRO VR1 Sinanfst pR interchanged and the effective resistance Re has a different value The model for the complete converter Fig 2 22 An equivalent circuit for the recti er with inductive output lter which models the funalamental Solution of Fig 223 yields the converter dc component of the ac siale voltage anal the alc component of the alc siale is given in Fig 223 conversion ratio voltage in the parallel resonant M Vlg L2 ll Hs llFJm converter 75 236 where Hs is the tank transfer function ZoS H S sL 237 and Z0S sL Hi llRe 238 controlled rectifier with dc input switch ac tank network inductive filter dc output 39 HS 39 H iR1t ig is1 t k z an 7 2 Vg V31 2 network VRO Re R l l 4Vg lg llis1ll005 s Vs1T sin27tfst V110 VR1Sin275fst 39 PR Fig 2 23 Equivalent circuit for the parallel resonant converter which moalels the funalamental components of the tank waveforms anal the alc components of the input current anal output voltage Principles of Resonant Power Conversion HZOOle HHOlt0H Fig 2 24 Construction ofBoale diagrams of Zis anal H s for the parallel resonant converter The Bode magnitude diagrams of Hs and Z0s are constructed in Fig 224 Z0s is the parallel combination of the impedance of the tank inductor L capacitor C and effective load Re The magnitude asymptote of the parallel combination of these components at a given frequency is equal to the smallest of the individual asymptotes 0L lDC and Re Hence at low frequency where the inductor impedance dominates the parallel combination H Z0s H 5 0L while at high frequency the capacitor dominates and Z0s H E lDC At resonance the impedances of the inductor and capacitor are equal in magnitude but opposite in phase so that their effects cancel H Z0 H is then equal to Re II M HF mo 1 1 o0CL JmoL J R h o 1 R were 0L DOC 0 The dc conversion ratio is therefore l Z0S SLHE Re at 0030 o L 71 7 R 0 DOC 0 SO 7 1 DO VLC L R0 7 HE anal Z0011 l 1 j 1 i f i i JRo R0 Re 1Qe Z S Hs 72L Chapter 2 Sinusoidal Approximations MAN 1 752 1 S L2 5l21f5 QewO 030 i 1 TC 4 1 F62 r 240 Equation 240 is compared with the exact converter solution in Fig 225 300 Fig 2 25 Comparison of exact parallel resonant converter characteristics solid lines vs the approximate solution Ea 2 40 shaded lines Principles of Resonant Power Conversion 25 Switching at Zero Current or Zero Voltage Operation of the F nll Bridge Below Resonance An of converters is their reduced switching loss advantage resonant When the series resonant converter is operated below resonance a phenomenon known as zero current switching can occur in which the transistors naturally switch off at zero current With a simple circuit modi cation the transistor tumon transition can also be caused to occur at zero current Let us consider the operation of the full bridge switch in more detail A full bridge circuit realized using power MOSFET s and antiparallel diodes is shown in Fig 226 The switch output voltage vst and its fundamental component vs1t as well as the approximately sinusoidal tank current waveform ist are plotted in Fig 227 At frequencies less than the tank resonant frequency the input impedance of the tank network Zis is dominated by the tank Hence the tank presents an effective capacitive load to the capacitor impedance bridge and switch current ist leads the switch voltage lndamental component vs1t in Fig 227 consequence the zero crossing of the as shown In current waveform ist occurs before the zero crossing of the voltage vst For the half cycle 0 St S Ts2 vs Vg For 0 S t S t5 the current ist is positive and transistors Q1 and Q4 conduct Then the diodes D1 and D4 conduct when quot 111 P3161 remainder of converter series resonant example Fig 226 Full bridge circuit implemented with powerMOSFETs and antiparallel diodes in a series resonant converter circuit Vs1t vg A Vs t V H Is11 Q1 ID1E 2 3D2 Q4 ID4 Q3 5D3 quothardquot 1 quotsoftquot I quothardquot quotsoftquot turnon turnoff tum0n turnof 0fQ1gtQ4 0fQ1gtQ4 0fQ2gtQ3 0fQ2gtQ3 Fig 227 Tank waveforms for the series resonant converter operatedbelow resonance ero current switching aids the transistor turn of transitions 20 VDs1 TS i TS rt 7 II II 2 II I i ll ist stored charge Is1 I I I soft tumoff I I I T T I lT T I 0 t I I73 ist I S B 2 2 B Q1 D1 Q2 iDzi 39 Q4 ID439 Q 39D3 Fig 2 28 Tank waveforms for the series resonant converter operated below resonance Zero current switching aids the transistor turn of transitions off while their respective antiparallel diodes conduct Chapter 2 Sinusoidal Approximations ist is negative t5 S t S TsZ The situation during Ts2 St S Ts is symmetrical Since i31 leads vs1 the transistors conduct before their respective antiparallel diodes Note that at any given time during the D1 conduction interval t5 S t S Ts2 transistor Q1 can be tumed off without incurring switching loss The circuit naturally causes the transistor tum off transition to be lossless and long turn off switching times can be tolerated In general zero current switching can occur when the resonant tank presents an effective capacitive load to the switches and the switch current zero crossings occur before In the bridge con guration zero current switching is the switch voltage zero crossings characterized by the conduction sequence Q1 D1Q2D2 such that the transistors are tumed It is possible if desired to replace the transistors with naturallycommutated thyristors whenever the zerocurrentswitching property II It rema1nder I I of converter Fig 229 Addition ofsmall inductors Lleg which effectively snub the transistor turn on transition and occurs The transistor turn on transition in Fig 228 is similar to that of a PWM switch and it is not lossless During the turn on transition of Q1 diode D2 must turn off Neither the transistor current nor the transistor voltage is zero Q1 passes through a period of high instantaneous power dissipation As in the PWM case the reverse recovery current and switching loss occurs of diode D2 ows through Q1 This current spike can be the largest component of switching loss In addition the energy stored in the drain tosource capacitances of Q1 and Q2 reduce turn on switching loss in the SRC operated below resonance Principles of Resonant Power Conversion Vs1t Vg Vst gt 3 TS t 2 Vg H V H l l 1s1t H H l l l l ta n I l t l I l I l l 1Q1t l H I H I n 39 n I l 39 l I l 39 l I I I l I I I II I M Humon n commutation I I interval 1D2t I ll I ltal I l I l l Fig 230 Waveforms for the circuit of Fig 229 During time t5 the tank current commutes from D2 to Q on time but is much shorter than normal Q1 D1 Q2 and D2 conduction intervals and in the depletion layer capacitance of D1 is lost when Q1 turns on To assist the transistor turn on process small inductors are often introduced into the legs of the bridge Fig 229 During the normal Q1 D1 Q2 and D2 conduction intervals these inductors appear in series with the tank inductor L and hence the effective total tank inductance is Leffective L l 2Lleg 23941 In addition these leg inductors introduce commutation intervals at transistor turn on At the instant when Q1 is tumed on the tank current owing through diode D2 begins to shift to Q1 at a rate determined by Vg and 2Lleg magnitude is therefore limited by Lleg The transistor current rather than being determined by the stored charge in diode D2 During time t5 2 Lleg iS0 Vg where is0 is the tank current at the t5 242 beginning of the commutation interval the current in diode D2 reaches zero and D2 turns off The leg inductance Lleg is chosen such that t5 is longer than the gatedriverlimited MOSFET tum Thus the MOSFET is switched fully on before the drain current rises signi cantly above zero and nearly lossless snubbing at tumon occurs This lossless snubbing of the diode D2 stored charge during the Q1 tumon transition is probably the most common reason to use zero current switching A nonideality not considered in the discussion above is the effect of other semiconductor device capacitances Transistor Q1 output capacitance diode D1 junction capacitance and diode D1 stored charge can be modeled as effective parallel capacitances which are shorted out whenever 22 Chapter 2 Sinusoidal Approximations transistor Q1 is turned on In consequence switching loss occurs equal to the total stored energy in these capacitances times the switching frequency Similar switching losses occur in the other three legs of the bridge This loss mechanism while not as great as the loss owing to the stored charge in D2 can nonetheless be quite signi cant in converters operating from high input voltages and at high switching frequencies It is often a significant disadvantage of zero current switching schemes Operation of the F nll Bridge Above Resonance When the series resonant converter is operated above resonance a dilTerent V810 phenomenon known as zero voltage Vg V80 switching can occur in which the transistors naturally switch on at zero voltage With a simple circuit modification the transistor tumoff transition can also be Vg caused to occur at zero voltage Ideally V this process is the dual of the zero current I switching process described in the I previous section i810 I For the full bridge circuit of Fig I 226 the switch output voltage vst and its fundamental component vs1t as well t as the approximately sinusoidal tank current waveform ist are plotted in Fig 231 At frequencies greater than the tank resonant ll 39 n D1 Q1 ID2I frequency the input impedance of the tank Q2 D4 Q4 D3 Q3 network Zis is dominated by the tank inductor impedance Hence the tank presents an effective inductive load to the I I I quotsoftquot quothardquot br1dge and the sw1tch current lst lags the tum0n tmmoff switch voltage fundamental component 0f Q1 gt Q4 0f Q1 gt Q4 Vsla as Shown m Flg39 23139 In Fig 231 Tankwaveforms for the series resonant Zero consequence the zero crossing of the converter operated above resonance voltage waveform vst occurs before the fragili osrxlmhmg aids the transistor tum 0n current waveform ist Principles of Resonant Power Conversion VDSl Vg nTS ITs t u 2 39 i A l I Q1 soft l hard 39 tumo tumo I I I Ts 2T3 t Fig 2 32 Detail of Q 1 drain current and drain source voltage waveforms SRC example operation above resonance For the half cycle 0 S t S Ts2 vs Vg For 0 S t S ta the current ist is positive and the transistors Q1 and Q4 conduct and the diodes D1 and D4 conduct when ist is negative ta S t S TsZ The situation during Ts2 S t S Ts is symmetrical Since vs1 leads igl the transistors conduct alter their respective antiparallel diodes Note that at any given time during the D1 conduction interval 0 S t S ta transistor Q1 can be turned on without The the transistor tumon incurring switching loss circuit naturally causes transition to be lossless and long tumon switching times can be tolerated In genera zero voltage switching can occur when the resonant tank presents an effective inductive load to the switches and hence the switch voltage zero crossings occur before the switch current zero crossings In the bridge con guration zero voltage switching is characterized by the conduction sequence DlQlDQQz such that the transistors are turned on while their respective antiparallel diodes conduct Since the transistor voltage is zero during the entire turn on transition switching loss due to slow tumon times or due to energy storage in any of the device capacitances does not occur at tumon The transistor tumoff transition in Fig 232 is similar to that of a PWM switch and is not lossless 4 l Q2 D2 Cleg CleglD4 Q4 remainder of 1 ism converter Fig 233 Introduction of small capacitors C leg which e ectively snub the transistor turn o transition and can reduce turn off switching loss in the SRC operated above resonance During the tumoff transition of Q1 diode D2 must turn on Neither the transistor current nor the transistor voltage is zero Q1 passes through a period of high instantaneous power dissipation and switching loss occurs To assist the transistor turn off process small capacitors may be introduced into the legs of the bridge as demonstrated in Fig 233 Alternatively the existing device capacitances can be used A delay is also introduced into the gate drive signals so that there is a short commutation interval when all four transistors are off During the normal Q1 D1 Q2 and D2 conduction intervals the leg capacitors appear Chapter 2 Sinusoidal Approximations in parallel with the semiconductor switches and have no effect on the converter operation However these capacitors introduce commutation intervals at transistor tumoff When Q1 is tumed off the tank current iLTs2 ows through the switch capacitances Cleg instead of Q1 and the voltage across Q1 and Cleg increases Eventually the voltage across Q1 reaches Vg diode D2 then becomes forwardbiased The length of this commutation interval is 2 Cleg Vg t i 5 1LTd2 where iLTs2 is the tank current at the beginning of the commutation interval the voltage reaches 2 43 zero and D2 becomes forwardbiased The leg capacitance Cleg is chosen such that t5 is longer than the gatedriverlimited MOSFET tumoff time but is much shorter than normal Q1 D1 Q2 and D2 conduction intervals Thus the MOSFET is switched fully off before the drain voltage rises signi cantly above zero and nearly lossless snubbing at tumoff occurs The fact that none of the semiconductor device capacitances or stored charges lead to switching loss is the major advantage of zerovoltage switching and is the most common motivation for its use An additional advantage is the reduction of EMI associated with device capacitances In conventional PWM converters and to a lesser extent in zero current switching converters signi cant high frequency ringing and current spikes are generated by the rapid charging and discharging of the semiconductor device capacitances at turn on andor turn off Converters in which all semiconductor devices switch at zero voltage inherently do not generate this type of EMI A nonideality not considered in the discussion above is the effect of conduwngu D1 Q1 I D1 t semiconductor package inductances These dewcesl D Q l XHD3I Q XI D 4 4 l I 3 39 4 1nductances are openc1rcu1ted whenever t t39 corm eurvzon the sem1conductor dev1ce 1s turned off In consequence switching loss occurs equal Fig 234 Waveformsfor the circuit ofFig 233 During the commutation interval tg all semiconductor devices are in the off state and inducmnces times the SWitChing frequency the tank circuit i S t charges or discharges the capacitors C leg to the total stored energy in these This loss mechanism can be signi cant in converters operating with high currents and low input voltages and at high switching frequencies 25 Principles of Resonant Power Conversion REFERENCE S 1 RL Steigerwald A Comparison of HalfBridge Resonant Converter Topologies IEEE Applied Power Electronics Conference 1987 Proceedings pp 135144 March 1987 2 R Severns Topologies for Three Element Resonant Converters IEEE Applied Power Electronics Conference 1990 Proceedings pp 712722 March 1990 PROBLEMS 1 Analysis of the LCC converter The LCC converter shown above contains both series and parallel tank capacitors 21 Using the sinusoidal approximation method find an expression for the dc conversion ratio M of this converter b For large C1 the circuit becomes essentially a parallel resonant converter with added blocking capacitor C1 Use the approximate factorization method to approximate the tank transfer function Hs for this case Show that your result of part a reduces to the parallel resonant converter M with an added rolloff inverted pole for low switching frequencies due to C1 Identify a resonant frequency and Q for this case and sketch typical curves ofM vs F for a few values of Qgt1 c Use the approximate factorization method to approximate the tank transfer function Hs for large C2 For this case the resonance occurs at approxim ately 1 f0 2139 V LC1 What is the Q d For the case when C1 C2 C sketch typical HHsH asymptotes for the high Q case ie Re gtgt R0 where R0 L 2C Identify salient features Chapter 2 Sinusoidal Approximations 2 Dual of the series resonant converter VCO LD 1 f 1 gt 0 27nch 1110 CF R I4 I LFI LFZ and CF are large filter elements whose switching ripples are small L and C are tank elements whose waveforms iL and vc are nearly sinusoidal 21 Using the sinusoidal approximation method develop equivalent circuit models for the switch network tank network and rectifier network b Sketch a Bode diagram of the parallel LC parallel tank impedance c Solve your model Find an analytical solution for the converter voltage conversion ratio M VVg as a function of the effective Qe and the normalized switching frequency F fsfo Sketch M vs F d What can you say about the validity of the sinusoidal approximation for this converter Which parts of your M vs F plot of part c are valid and accurate e Below resonance does the converter operate 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