RESONANT TECH PWE ELEC
RESONANT TECH PWE ELEC ECEN 5817
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State Plane Analysis of Basic Zero Voltage Transition ZVT Full Bridge Converter Schematic and waveforms igl n DS I Vg V00 V V2 0 v4 I V20 Vg J 0 0 u 7 1 V40 Vg 3 3 quot1 C12g3C12g4 quot1 C12g3C12g4 0 0 gt gt v50 J V3 0 O 0 u 1 7 Vg 13950 nI u 7 I in n Subinterval 0 4 5 6 11 0 Q2 Q1 Q1 Q1 Q2 Q2 P7A Conducting A7 devices D4 94 X D3 X D4 D5 D5 D5 D5 X D5 Secondary D6 X X D6 D6 D6 diodes EmbwaQwiL l 3v6 LOLW Q quotfume a DLD D3 D6 CD AAUJD we 4w vow D5 Wt D6 mu Com on M HiL WsJVDYW Mag 5 Skovr rthnlej L A h to 0 Jvu a 0210 C CHLLUL loecotwi T L NONka za 5 73 V Plane X VLAW J b6L 06 3 gt K 1 LC I R0 D Cl51Clefgti wo v s n 1 j LG clay Cle zgt RC h1 VV 32 J 3 F 239 c RawL 39Llnsz Vase V dcquot quot 3 Mvzo kw Kw any c S Ek flame Wei39l b rj 120003 C39 cuar 66Mqu 0 Orlljllm was Y 3 IVA PWOJ W5 ham V7 V3 quotgmaszLMNb D 39 H ad aquot Mzzl aw Jajm SchRONA 5g Flame 320 wail quot t 1 Am F j 339 quot l ax W QWA X boo t 1 am JC lt 1 gt or m vounag 9396 we vaZuV 323 14 341 Pwvx ma waver rmLzs ij auA w 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pb39xvuw a 39 5 quotQckemg 9129343 M h vwxs var Xums ra x y Sme ll 03 01M Sam 49 I V 5 6W5 955 thn j Sm yaksz 5Q M 0 lo WW VLCT USEA CDAAuc J 39m 035 Dun W46 WO CC MT U0 o 15gt 1ij ugtmq WqS 7 1 0 am 0 SukAPWA5 D 9 3 39 cow ek 54 Flame n edmb quotquot2 6 JQ o on k V O wmevar 1 MA D Mk V0 5 h AVD CDMAUL4 O Wig 1 H75quot I t erANA Dab C99 I quotI I l 2 I A L Vogt T5 39VNchLri Zv b fg uh L 2L kPWWSLv as megTL LNOSINJ 36 8737 230 Q fL m msrvm Qm 35w WP 3L0 0 5 L319 OULW H O m W P 4v mf lint ff illii k a I 4 v 765 nlw WV 9 lt4 9 HS n16me 026 351 Cof vm NW m gg 86 x 1 P 4N AEL lt4 91m 0 0 iv vcnf vly Cw W skefts mgb N H m mam a 50 mg T fir i 4 tz t3 Ham e 4607 5 h 5 39tl fvth V 0 i L M f 3 M quot 2 E I kbt5 L ES V3 AV5 L 4m L0 Zl we 7w 21 may Avg 3 6539 J i ijg Clej fB t4 ub93 0mg AOMK ZJL b 104 HA F 39L J v w 32 2 M 2 whisk Cs 0 quotat QM my 32 1 M r F Paur33 39 39 W Mk F 0 PW i 3 oisgt Qlt3W 2w Milq5 95 PPM 32 F 3 9 wm E ZWT 43 31 05 15 EWCD 2 25 A W5 Ew EVTC 5 we q h ver 6m w 39 vex maxi v A130 w w mm m a smugch 06 m W Azmpg fean m P441 new FFBW3gt 4 1 Amok Mi m QKQOAY M 5 Ma226 y qum FN39 MIA PM L fin z C st Md app255 5 l i o 45 5u 1 L 9 4Q amp tui LL 4 9 may 0 x ganv w v u I ME I 0 rm Ohm mi Er r fr 35 etc prowful ii ME isn m Tu V a bmlwnv 2 U xlkqwljoml qml1tzw ltP Q M 4 A N O 925 PM TL Erma muva min 3 4 94 u 9 SE y 6 05 ob wiryHP 91F 93 wk Ow HS Frugal A nV h Altle V 1 WWRQU dvnw u m WGU Ceinvu Tramway Crv 1 C H 3M Q mrirvw m0 va III wringrd I mu ASIVV crn 42 9L 2 m Q 6 Left w piJ cgt 9 Mr W w 503 WrL W waltz up zu ckg 4 a v1 u4v 39 wag 6 quot39 J gt RN lb F F vugt4gt Ka 11quot cam M5 ave nag MM 5133 SflCE sf D39Rmf gxwlibf Lax e LJLB LL5 SLA C C C gt ofevx 2 L IL14 P Q K RAuc 0 u 39 d Ad ANHIHVM LHK H4 73L 33 3 i p A g 3 3 3 2 4 2w 733 4amp2 1 me a 3 2 1 37 i JSHK a CTQJ 0 x n36 6 271de EHHJH 7 6 L n l I U I quotAU L 3K 13 9 I gt K 9 37K U AKIQ 69 W pd7UAV K quotN7 11 A0 fame n7 25 1 v M H 5 n9 Pduv quotAdJV AIV CHAPTER 3 STATE PLANE ANALYSIS AVERAGING AND OTHER ANALYTICAL TOOLS T he sinusoidal approximations used in the previous chapter break down when the effects of harmonics are signi cant This is a particular problem in the case of discontinuous conduction modes where harmonics cannot be ignored To obtain a complete understanding of the behavior of resonant converters another approach is needed In this chapter the fundamental principles necessary for an exact time domain analysis of resonant converters are explained These principles are used in later chapters to examine not only the series and parallel resonant converters but also quasiresonant converters The state plane can be used to reduce the complicated tank waveforms of resonant converters to geometry When properly normalized the tank waveforms are described by segments of circles lines andor other simple gures in the state plane Determination of converter steadystate characteristics is often a matter of piecing together these segments then solving a few triangles or other gures Equally important is the use of averaging in which the dc and lowfrequency ac components of the converter terminal waveforms are found while neglecting high frequency switching harmonics The average output current of the series resonant converter is related to the charge that ows through its output terminals per switching period This charge also ows through the tank capacitor where it excites an ac voltage The load current and tank capacitor voltage magnitude are therefore closely related and considerable insight can be gained by use of some simple arguments regarding the ow of charge during a switching period Similarly the average output voltage of the parallel resonant converter is the voltseconds ux linkages applied to the tank inductor per switching period and therefore it is also related to the peak tank current Thus some simple charge and uxlinkage arguments are discussed in this chapter and are used in later chapters to easily relate the tank waveforms to the dc terminal voltages and currents The various lndamental principles which describe the ow of charge and ux linkages in a resonant circuit and their relations to the average terminal waveforms are collected in section 31 and are illustrated using the series and parallel resonant converters as examples Systems of Principles ofResonani Power Conversion notation and normalization a perennial source of confusion in any discussion of resonant converters are described in section 32 In sections 33 and 34 the ringing responses of series and of parallel resonant tank circuits are derived and they are plotted in the state plane It is apparent that an exact time domain analysis of resonant converters is more complex than the use of the sinusoidal approximations and frequency domain methods of chapter 2 Nonetheless simple and exact closedform solutions can be obtained for the many continuous and discontinuous conduction modes of the series resonant converter as well as for the parallel resonant and many quasiresonant converters These ideas are also useful in modeling the dynamics of these converters and the basic ideas developed in this chapter are used throughout the remainder of this monograph 31 Averaging and Related Concepts The signals in a power electronics system generally contain substantial switching harmonics By speci cation and design the magnitude of these harmonics must be negligible at the converter output Hence when analyzing the behavior of a converter we usually neglect the switching harmonic components of the converter terminal waveforms and model only their dominant dc and lowfrequency ac components This simpli es the analysis considerably and allows a much better understanding of the converter properties The basic arguments used to average the converter waveforms were described by Wester and Middlebrook 1 Although these arguments were originally developed for modeling PWM converters they also assist the analysis of resonant converters Averaging the system signals over a period does not signi cantly alter the waveforms so long as the period is short compared to the system s natural response times This is similar to passing the waveforms through a lowpass lter if the lter comer frequency is suf ciently high then the important dc and lowfrequency ac components are not affected In particular it is useful to average the terminal waveforms over one switching period This effectively removes the switching and ringing harmonics without modifying the desired dc and low frequency ac response and signi cantly simpli es the analysis This approximation is justi ed because it is normally required that switching harmonics be negligibly small at the load and therefore suf cient lowpass ltering is incorporated into any welldesigned converter The implication is that the converter output current can be adequately represented if we simply nd the total charge which ows out of the output port during one switching period Dividing this charge by the switching period yields the average output current Dual arguments allow representation of the output voltage knowing the total uxlinkages or voltseconds which the converter applies to the output during one switching period In this section some basic Chapter 3 State Plane Analysis principles are discussed which allow the average terminal voltages and currents to be directly related to the tank ringing waveforms Averaging charge arguments Let us consider how to average a dependent terminal current iN of a switch network as illustrated in Fig 31 IfiN is periodic with period T then the average value can be written T ltmgtLimom 31 al zquot T where qN I iNt dt is the net charge transferred at the port over period T 0 The form ltiNgt qN T is useful because as shown later qN can be related to other salient features of the resonant network waveforms In particular qN is a function of the change in tank capacitor charge and hence also the change in tank capacitor voltage over a portion of the switching period CONVERTER i1 iN v1 Port 1 P011 N vN 7 qN ma areaqN lt1Ngt T b t 0 T 2T Fig 3 1 Arbitrary output of switch network average output current computed from charge transfer Principles ofResonant Power Conversion Averaging uxlinkage arguments Dependent terminal voltages can be averaged using dual arguments Consider a dependent terminal voltage vN of a switch network as illustrated in Fig 32 If vN is periodic with period T then the average value can be written ltvN gt vNtdt 0 32 alzy T where XN J vNt dt is the net voltseconds applied at the port over period T 0 The formlt vN gt 1N T is useful because as shown later 1N can be related to other salient features of the resonant network waveforms In particular IN is a function of the change in tank inductor ux linkages and hence also the change in tank inductor current over a portion of the switching period CONVERTER l1 N V1 Port 1 P011 N vN 7v 7 N area 7 IN lt VNgt T VNO I I I t 0 T 2T Fig 32 Computation of average terminal voltage ltvNgt using flux linkages 1N Chapter 3 State Plane Analysis Tank capacitor charge variation Over one switching cycle charge is transferred from the switch power input through the tank capacitor to the output The amount of charge transfer can be directly related to the capacitor voltage waveform In particular over a given interval tatB if a given amount of charge q is deposited on the tank capacitor then we know that the capacitor voltage changes from vCta to vCtB where q C VCOb VCOa 33 Hence the capacitor voltage initial and nal values vCta and vCtB are related to the charge transfer and therefore also to the switch average terminal current by Eq 31 For example consider the circuit of Fig 33 It is desired to compute the average or dc component of the bridge recti er output current lti2gt and to relate it to the capacitor voltage waveform vCt Typical waveforms are sketched in Fig 34 The average value of i2t is given by T lti2gt LJ i2tdt T 0 Fig 3 3 Demonstration of direct relation Zq between alc component of load current 34 anal peak to peak capacitor voltage T 2 where q I i2t dt 0 During the interval 0 St S T2 the capacitor current ict is identical to the bridge recti er output current i2t and hence the same net charge q is deposited on the capacitor The maxima and minima of the capacitor voltage waveform vct coincide with the zero crossings of the current ict and hence the capacitor voltage changes from its minimum value chp to its maximum value ch during this interval The capacitor charge relation is therefore q C VCP VCP 2CVCP 35 Elimination of q from Eqs 34 and 35 yields lti2gt 4cvcp T 36 Principles ofResonant Power Conversion A area q area q i2 licl 7 V C A VCP T T VCP Fig 34 Waveformsfor the circuit ofFig 33 Hence the component of the resistor current lti2gt and average value or dc the peak capacitor voltage ch are directly related These arguments are used in chapter 4 to derive a nearly identical relation between the peak tank capacitor voltage and the load current of the series resonant converter Tank inductor ux linkage variation The dual of the tank capacitor charge relation follows from the de nition 7 L i Inductor uxlinkage 7 has the dimensions of voltseconds and is the integral of the applied voltage as de ned in Eq 32 voltseconds are transferred from the During one switching cycle switch power input through the tank inductor to the output So over a given interval twtg if a given amount of ux linkages 7 are stored in the tank inductor then the inductor current changes from iLta to iLtB where 7 LiLtb iLta Hence the inductor current boundary values iLta and iLtB are related to the voltsecond transfer and hence also to the converter average terminal voltage by Eq 32 Chapter 3 State Plane Analysis For example consider the circuit of Fig 35 We wish to compute the average or dc component of the bridge rectifier output voltage ltv2gt and to relate it to the inductor current waveform iLt Typical waveforms are sketched in Fig 36 The average value of Fig 35 Sinusoidal current source driving V t is iven b an inductor in parallel with bridge 2 g y rectifier and resistor gr ltv2gt v2tdt ET 0 H a gt 38 1T 2 where 7 I v2t dt 0 During the interval 0 St S T2 the inductor voltage vLt is identical to the bridge recti er output voltage v2t and hence the same net ux linkages 7 are stored in the inductor The maXima and minima of the inductor current waveform iLt coincide with the zero crossings of the voltage vLt and hence the inductor current changes from its minimum value 71141 to its maximum value ILp during this interval The inductor ux linkage relation is therefore 7V L ILP ILP 2LILP 39 Elimination of 7 from Eqs 38 and 39 yields ltV2gt 4LILp T 3 10 Principles ofResonant Power Conversion L ILP T T 391 LP Fig 36 Waveformsfor the circuit ofFig 35 Hence the average value or dc component of the resistor voltage ltv2gt and the peak inductor current ILp are directly related Similar arguments are used in chapter 5 to derive a relation between the peak tank inductor current and the load voltage of the parallel resonant converter Kirch off s laws in integral form We know from Kirchoff s Current Law KCL that the total currents owing into a given node must be zero ik 0 k 3 l l The net charge which enters the node over a given interval ta t5 must also be zero 20 k 312 tb where qk I ika dt ta The integral form of Kirchoff s Current Law is use ll for relating terminal charge quantities to the change in tank capacitor charge For example consider the circuit of Fig 37 This circuit is similar to the tank circuits of both the parallel resonant converter and the zero current resonant switch during one ringing subinterval In conjunction with the determination of the average input current of this network we wish to compute the charge contained in i1 during the given ringing subinterval tut5 115 I i1tdt tac By KCL we know that i1 iC i2 Hence this qCB 125 tu l t where lCB J iCt dt tun 313 314 Chapter 3 State Plane Analysis qlB qCB l qu area q1B area q2B ta tOLtB Fig 3 7 Illustration of use of integral form of K CL for a typical tank network tut5 and 12B I i2tdt tac Therefore the ringing interval input charge qlg is related to qCB the change in tank capacitor charge over the ringing interval and to qgg the charge transferred to the output during the ringing interval Some of the input charge is stored on the tank capacitor while the remainder ows to the output Principles ofResonani Power Conversion An integral form of Kirchoff s Voltage Law KVL is also useful network loop is zero vk 0 k 315 The total voltseconds applied over a given interval The total voltage around a tat3 across the elements of this loop must also be Elk0 k zero 316 tb where kk J Vka dt ta When element k is an inductor or transformer kk has the physical interpretation of winding ux linkages The integral form of KVL relates terminal voltsecond quantities to the change in tank inductor ux linkages For example in conjunction with the determination of the average output voltages of the parallel resonant converter and the zero current resonant switch we wish to compute the voltseconds contained in v2 during the ringing interval see Fig 38 irts 12B J V20 11 ton By KVL we know that v2 v1 7 vL Therefore 12B kl ikLB 33918 317 irts where 11B J v1t dt ton irts XLB J VLt dt tun V2612 7MB XLBt MB V1 A areaklg gt ta tOLFtB t H tB gt1 1 VL areakLB ta tatB t areakgg V2 ta t Fig 38 llustration of the use of the integral form of K VL Therefore the ringing interval output voltseconds 125 is related to 1L5 the change in tank inductor ux linkages over the ringing interval and to MB the voltseconds applied to the input during the ringing interval Some of the input voltseconds are stored in the tank inductor while the remainder are applied to the switch output Chapter 3 State Plane Analysis Steadystate capacitor charge balance When a capacitor operates with periodic steadystate waveforms then the initial and final values of the capacitor voltage waveform are identical In consequence no net charge is deposited in the capacitor and the integral of the capacitor current waveform over one complete cycle is zero The dc component of capacitor current is zero Formally this follows from the definition ict C t 319 Integration over one complete period T yields T VCTVC0 ictdt 320 In periodic steady state vCT vc0 Hence T 0 iCt dt 321 0 Division by the period Tthen shows that the average value lticgt or dc component must also be zero T J iCtdt ltiCgt 0 322 0 This is the wellknown principle of steadystate capacitor ampsecond or charge balance It is true for any capacitor which operates with periodic steadystate waveforms i For example consider the capacitive output lter circuit of Fig 39 In steady state no net charge is deposited on capacitor C1 during a switching period and hence the average recti er output current i2 is equal to the dc component I of the load current it By Ohm s law the Fig 39 Capacitive lter circuit dc component V of the load voltage vt is V IR Hence we have V lti2gt R 323 For this example the input current i2t is a rectified sinusoid i2t 1212 l Sinlt0st l 324 whose average is Ts 7 l 7 2 lt gt 7 7 12 TS I 120 ll TE I21 0 325 Substitution of this expression into Eq 323 yields v lm R 326 Principles ofResonant Power Conversion iCFO l area qC M w 39395 140W V 0st area q C WV V Dst Fig 310 Typical waveforms for the circuit ofFig 39 Eq 327 over this interval 140 46 1c 39 54 I2P l SinUDSt l Evaluation of the integral yields qc 00671212 Ts d0st TE Os Output voltage ripple can also be estimated using charge arguments 310 sketched for the case in which CF is large In Fig the capacitor current waveform is enough that its voltage ripple induced by the switching harmonics in i2t is small compared to the dc component V In this case the voltage harmonics applied to resistor R are also small and hence by Ohm s law the current through R is essentially dc Therefore the dc of i2t ows exclusively through R while the switching harmonic of i2t flows overwhelmingly through CF The capacitor current waveform is then given by mmwwmemmmmv w 327 The capacitor current is positive over the interval 3954 lt Dst lt 14046 During this interval the capacitor voltage increases by an amount 2Avc from its minimum value to its maximum value This corresponds to an increase in charge qc given by the integral of 328 329 The peaktoaverage capacitor voltage ripple is therefore Avc 0067 2C Or in terms of the load current 1 Ts A 7 00526 VC C 330 331 This gives a simple estimate which is useful for choosing the output filter capacitance However it does not include the effects of capacitor esr equivalent series resistance which can cause the voltage ripple to be significantly larger than that predicted by Eq 331 Chapter 3 State Plane Analysis Steadystate inductor uxlinkage balance When an inductor operates with periodic steadystate waveforms then the initial and final values of the inductor current waveform are identical In consequence no net ux linkage is induced in the inductor and the integral of the inductor voltage waveform over one complete cycle is zero The dc componentof inductor voltage is zero Formally this follows from the definition vLt Ldllig 332 Integration over one complete period T yields T amino ma dt 333 In periodic steady state iLT iL0 Hence T 0 iCt dt 334 0 Division by the period Tthen shows that the average value ltvLgt or dc component must also be zero T J vLtdt ltVLgt 0 335 0 This is the wellknown principle of steadystate inductor voltsecond or uxlinkage balance It is true for any inductor which operates with periodic steadystate waveforms 32 Normalization and Notation The geometries of the state plane plots of the next sections are simplified considerably when the waveforms are normalized using a base impedance Rbase equal to the tank characteristic impedance R0 The normalizing base voltage Vbase can be chosen arbitrarily and is usually chosen to be equal to the power input voltage Vg Other normalizing base quantities can then be derived base impedance Rbase R0 W base voltage Vbase Vg base current Ibase Vbase Rbase Vg R0 base power Phase Vbase Ibase ng R0 336 In the system of notation developed at the University of Colorado CU the symbol for a normalized voltage contains the same subscripts and case as the original unnormalized voltage but the character V is replaced by M For example M V Vbase normalized loaal voltage mct vct Vbase normalized tank capacitor voltage 337 Principles ofResonant Power Conversion For currents I is replaced by J J I Ibase normalized load current jLt iLt Ibase normalized tank inductor current 338 When a converter contains a transformer the base quantities should be referred to the proper side of the transformer by multiplication by the appropriate function of the transformer turns ratio Some other authors use the same normalizing base quantities but denote normalized variables using the subscript n The symbol q is also sometimes used to denote the normalized output voltage Neither of these conventions is used here It is convenient to normalize frequency using the tank resonant frequency f0 and to convert time to angular form fbase f0 l 27E m base frequency 00 l V LC tank resonant angular frequency F fs f0 normalized switching frequency 0L 00 tx angular length of interval tX 339 where fs is the switching frequency and Ts lfs is the switching period The following notation is also traditional in the series resonant converter literature 7 00 T3 2 TE F angular length of one half switching period 0L 00 ta diode conduction angle 5 00 t5 transistor conduction angle 340 When performing exact timedomain or stateplane analysis Q is defined using the actual load resistance R as opposed to the effective resistance R9 of chapter 2 Q R0 R for the series resonant converter Q R R0 for the parallel resonant converter 341 4 EJ 3 gt 4 g 1 lt etc l k 2gtlt k1 gtlt k 0 gt f S ifo H0 f0 Fig 311 Switching frequency ranges over which various mode indices k and subharmonic numbers 50ccur Final definitions for the series resonant converter are the mode indeX k and subharmonic number E as follows The continuous conduction mode k occurs over the frequency range fio lt f lt fio L lt F lt L k1 S k or k1 k 342 for integer k The subharmonic number is then Chapter 3 State Plane Analysis 1391k Ev k 2 343 For example continuous conduction mode operation at switching frequency fs 04 f0 would imply that k 2 and E 3 33 State Plane Trajectory of a Series Tank Circuit Let us next examine the timedomain response of a series resonant c1rcu1t A ser1es tank c1rcu1t exc1ted by a constant L voltage VT is shown in Fig 312 As shown in the next chapter VT the series resonant converter can be reduced to a circuit of this V form dur1ng each sub1nterval The state equatrons of th1s c1rcu1t C are L d1Lt VT iVCa Fig 312 Series tank circuit dt 344 excited by constant dVCt voltage VT C dt 1L0 Let us normalize the state equations according to the conventions of section 32 Note that R L 0 and C 345 00 00 R0 where 00 is the tank angular resonant frequency and R0 is the tank characteristic impedance as defined in Eqs 336 and 339 Division of Eqs 345 by Vg and use of the identities 346 and 336 339 yields MN m0 dt MT Inca 3 46 1dmclttgt t 39 m0 dt JLO where MT VT Vg The solution of this secondorder system of linear differential equations is mCt A cos00t p MT jLt iAsin00t p 347 where the constants A and p depend on boundary conditions It can be seen that the solution contains a dc term mc MT or vc VT which represents the dc solution of the circuit plus a sinusoidal term which represents the ac ringing response of the resonant tank Principles ofResonant Power Conversion yl Y0 p 0 X X X0 A cos9 y Y0 7 A sin9 Fig 314 Normalized state plane trajectory for the circuit of F ig 312 corresponding Fig 313 Parametric representation of a czrcle to Eq 347 The normalized state plane The normalized state plane is a plot of mCt vs jLt with t as an implicit parameter As shown in Fig 314 the solution 347 above describes a circle in the normalized state plane of radius A If we let the radius go to zero we can see that the circle is centered at mc MT jL 0 which coincides with the dc solution of the circuit As time increases the solution moves in the clockwise direction around the center this must be true because the capacitor is in series with the inductor and if the normalized inductor current is positive then the capacitor charges and mc increases In general the normalized state plane trajectory of an undamped twoelement resonant circuit is circular and is centered at the dc solution of the circuit The radius depends on the initial values of jL and mg and remains constant It can also be seen from Eq 347 and Fig 314 that time is related to the angle through which the trajectory moves During an interval of time t1 the trajectory moves through an arc of angle 00t1 So the length of ringing intervals and their angles in the normalized state plane can be easily related Note this is not necessarily true for systems other than the resonant tank circuit considered here The tank circuit of the parallel resonant converter The name of the parallel resonant converter can present some confusion because although its load is connected in parallel with the tank capacitor the tank capacitor and inductor are effectively in series In consequence the time domain response and normalized state plane trajectory are quite similar to that of the series resonant converter tank circuit Fig 315 Tank circuit driven by constant voltage source VT and constant current source I T The state equattions of the circuit are L it vT 7 vclttz C T 1Lt 7 IT In normalized form the state equations become DI 0 Mr emc ml Odngf now where JT IT Ro Vg The solution is Inca MT mc0 MT 005mot P f 110 JT Sin10t P 111 JTJr110 JT 3OS10139 Fig 3 1 6 Normalized state plane trajectory for the circuit of F ig 315 Chapter 3 State Plane Analysis As shown in chapter 5 during each subinterval of the operation of the parallel resonant converter its tank circuit can be reduced to a con guration of the form shown in Fig 315 This differs from the circuit of Fig 312 only by The effect of this extra source is to shift the dc solution the addition of constant current source IT of the circuit and hence also the center of the circular trajectory 348 3 49 ltp7mc0iMT sinrmot ltp 33950 As shown in Fig 316 this represents a circular arc centered at the dc solution mc MT jL JT whose radius r depends on the initial conditions and is given by r 4 mC0 MT2UL0 JT2 351 As in the case ofthe circuit of Fig 312 the length of ringing intervals and their angles in the normalized state plane can be easily related During an interval of time t1 the trajectory moves through an arc of angle motl All of the fundamental concepts necessary for an exact analysis of the series parallel and other resonant converters have now been discussed Various arguments involving the ow of charge and uxlinkages can be used to relate the tank waveforms to the average terminal voltages and currents of the converter The waveforms can be normalized which causes the state plane trajectories to assume circular paths As seen in the next two chapters these concepts allow Principles ofResonant Power Conversion closedform analytical solution of the characteristics of the series and parallel resonant converters in a direct manner They also aid in the understanding of resonant switch converters REFERENCE S 1 GW Wester and RD Mddlebrook Low Frequency Characterization of Switched DcDc Converters IEEE Transactions on Aerospace andElectronic Systems vol AES9 May 1973 pp 376385 2 Steven G Trabert and Robert W Erickson quotSteadyState Analysis of the Duty Cycle Controlled Series Resonant Converter IEEE Power Electronics Specialists Conference 1987 Record pp 545556 3 R Oruganti and FC Lee Resonant Power Processors Part 1 State Plane Analysis IEEE Transactions on Industry Applications vol 1A21 NovDec 1985 pp 14531460 4 R Oruganti and FC Lee State Plane Analysis of the Parallel Resonant Converter IEEE Power Electronics Specialists Conference 1985 Record pp 5673 June 1985 5 CQ Lee and K Siri Analysis and Design of Series Resonant Converter by State Plane Diagram IEEE Transactions on Aerospace andElectronic Systems vol AES22 no 6 pp 757763 November 1986 APPENDIX 1 Computer Listings A11 Series Resonant Converter MQF The following routine correctly computes the voltage conversion ratio M as a function of Q and F It is assumed that the converter operates with linear resistive load R and QR0 R The algorithm and equations used are described in section 45 function MQF computation of series resonant converter M as a function of load Q and frequency F define kksigqgtcgttgtmrkl gPIF gamma kINTlF check for mode klINTO5sqrtO25QPI2F if klgtk type k CCM ksikl 1Ak2 subharmonic number qgtQg2 intermediate variable cgtcosg2A2 intermediate variable tgttang2A2 intermediate variable mr contains the result M if Flksi mr tangent function may give unpredictable results at resonance and subharmonics else mrqgtkSiA4tgtqgtA2lAklSQRTlksiA2 cgtksiA4tgtqgtA2qgtA2cgt end if else type kl DCM if kl2INTkl2 even DCM mr2klFPIQ else odd DCM mrlkl end if end if return mr end function A12 Parallel Resonant Converter MJF The following routine computes the operating point of the parallel resonant converter It works in both the continuous conduction mode and the discontinuous conduction mode provided that F205 The CCM solution is a straightforward application of Eqs 531 and 535 In the case of the discontinuous conduction mode Eqs 545 must be solved iteratively Doing so is Principles of Resonant Power Conversion not straightforward and can become an exercise in numerical analysis methods The routine below first nds the solution for E MB This is done by combining the third and fourth lines of Eq 545 to obtain GfEcosE05 2J77205 EHey52J7Ysin sin 1 This equation is solved iteratively Once E is known the other angles 0LB and 5 can be evaluated directly and then M can be found The routine rst searches for the neighborhood of the root by simply evaluating Gf at regular intervals note that E can take on values between 0 and Y beginning at E y Once Gf changes polarity the routine uses Newton s method to converge quickly to the root function MJF computation of parallel resonant converter M as a function of load current J and frequency F equations are valid only for Fgt05 define phi g qgt jcrit mr phicrit ksil ksi2 ksi3 G1 G2 G3 convg epsln a b d dc ic kc Gc dGl jsc gPIF gamma check for mode if Flt05 return quotinvalid Fquot end if jcrit Jcrg if j lt jcrit CCM phithiJg mr 2gphi sinphicosg2 else DCVM jsc05g test whether J39 exceeds the short circuit current exit if it mrquoterror Igtshort ckt currentquot return mr exit function end if find neighborhood of root where convergence is likely dc0l Gcl while Gcgt0 kcg GcGfkcJg dcdc0l icdc while Gcgt0 and kcgt0 kckc ic GcGfkcJg end while end while ksi2kc ksilkcic iterationz Newton s method convg000000l criterion to test convergence Appendix 1 Computer Listings epsln0l factor to reduce gain of iteration algorithm GlGfksilJg dGldeksilJg while absGlgtconvg iteration loop ksi alpha beta ksi3ksil epslnGldGl GlGfksi3Jg dGlde ksi3 J g ksilksi3 end while a acos05lcosksi3 b ksi3 a d g b mr l2gJ d end function Other functions called by the function MJF function GfkJg function required for iteration in DCVM de ine ru u2J g rcoskksink05uA2kA2uksink l return r end function function dekJg derivative of Gf used for Newton iteration method define r rkcoskk2J glcosk return r end function function Jcrg evaluation of Jcrit the critical value of J CCM for JltJcrit DCVM for JgtJcrit define r r O5sing sqrtsing05A2O25singA2 return r end function function thiJg evaluation of the angle phi needed for CCM solution define r r acoscosg2Jsing2 if gltpi r r r end function A13 Other functions for evaluation of stresses and other quantities for the parallel resonant converter The functions below evaluate the boundary values MCO and JLO Also listed are routines for determination of peak stresses Mg and Jun as well as functions which indicate the operating Principles of Resonant Power Conversion mode and whether the converter operates with zero current or zero voltage switching at the given operating point These functions require that M J and 7 all be speci ed it is intended that MJF be used rst to solve iteratively for the operating point and then the results used in the routines below All functions operate correctly for both CCM and DCM They use the relevant equations of Chapter 5 Boundary values May and JLg function McoMJg evaluation of tank capacitor initial voltage Mco given the steady state solution MJ and gamma define dbrjcritphi jcritJcrg if Jgtjcrit DCM soln dJg2Ml else CCM soln phithiJg r Jsinphicosg05 end if return r end function function JLoMJg evaluation of tank inductor initial current JLo given the steady state solution MJ and gamma define dbrjcritphi jcritJcrg if Jgtjcrit DCM soln dJg2Ml bg d rJsinb else CCM soln phithiJg r J 2 1 tang05 end if return r end function Peak stresses JLP and MCP function JkaMJg evaluation of peak tank inductor current Jka given the steady state solution MJ and gamma define dbrjcritphijlle jcritJcrg if Jgtjcrit DCM soln dJ g2M l Appendix 1 Computer Listings else CCM soln phithiJg leJLoMJg if McoMJgltl and legtO rle else jl sinphicos05g rJsqrtjl JA2l end i end if return r end function function MCpkMJg evaluation of peak tank capacitor voltage MCpk given the steady state solution MJ and gamma define dbrjcritphimc0lejl jcritJcrg leJLoMJg if Jgtjcrit DCM soln dJg2Ml else r2 end if else CCM soln phithiJg if legtJ mc0McoMJg rsqrtchlA2J leA2 l else jl sinphicos05g rlsqrtjl JA2l end if end if return r end function Operating mode function ZCVSMJg determination of whether the converter operates with zero current or zero voltage switching at the designated operating point define rle Principles of Resonant Power Conversion else rquotZVSquot end if return r end function function PRCmodeJg determination of operating mode define jcritjscr if ggt2pi return quotmode unknown Flt05quot exit function return quotinvalid JltOquot exit function end if qsc05g else if Jgtjsc return quotno soln Jgtjscquot else return quotDCVMquot end if en if end function For an LCQ 0123 4435 4 Bale Ax mm 639 wk in USquot fW39rQAW Gl 9 La can 934 Huge lf i v We OH AM MD U HW st HM sLoFF mar 090 MW me 223003 1 m LWLB 2415 we 3 2WD LVYQA SMHMQ 3T 4KC 00 0 0P an b FM 458 4amp1 QWTIQ 3834 mm M 5ka ma sz were C5 r X52835 quotLquot CP X o quot447 51 a Cgfwz H3 gee d xs J 7335L whgk 5 MJQ QKJ 290 ZJ OCS fXF 2 quot1374103 395 64044 one Comm af rm 4 mi 12 w a Evs 3 gunma an wads of an 1 Mb 5 opemcucu f7 s 391 Wu 4 shut Mui sr 4 fo eile e M l nml so Hm f We nd 22 is More U39AJJud39kl rQa Far examska aw AM lvxoLuc l of39 quotGWM M45ka 45 4mm Ih MfeJamz becomes 290 W 37 Inc Mn 44 AX7M H 32636 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