RESONANT TECH PWE ELEC
RESONANT TECH PWE ELEC ECEN 5817
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Chapter 4 THE SERIES RESONANT CONVERTER T he objective of this chapter is to describe the operation of the series resonant converter in detail The concepts developed in chapter 3 are used to derive closedform solutions for the output characteristics and steadystate control characteristics to determine operating mode boundaries and to nd peak component stresses General results are presented for every continuous and discontinuous conduction mode using frequency control The origin of the discontinuous conduction modes is explained These results are used to consider three design problems First the variation of peak component stresses with the choice of worstcase operating point is investigated and some guidelines regarding the choice of transformer turns ratio and tank characteristic impedance are discussed Second the effects of variations in input line voltage and output load current are examined using the converter output characteristics Finally switching frequency variations are considered and the tradeoff between transformer size and tank capacitor voltage is exposed Q3I D3 D1 l Q1 VT D I ls 5 Vg ZSMZS t V L C D6 Q4 l IkD4 D21 l Qz VS vR FV R 28 I Fig 4 Series resonant converter schematic Principles of Resonant Power Conversion 4 1 Subintervals and Modes The series resonant converter Fig 21 is reproduced in Fig 41 It can be seen that the instantaneous voltage vTt applied across the tank circuit is equal to the difference between the switch voltage vst and the recti er voltage vRt VTO Vst V110 41 These voltages in turn depend on the conducting state of the controlled switch network and uncontrolled recti er network A subinterval is de ned as a length of time for which the conducting states of all of the semiconductor switches in the converter remain xed during each subinterval vst vRt and vTt are constant For example consider the case where transistors Q1 and Q4 conduct and iLt is positive so that diodes D5 and D8 also conduct as in Fig 42a In this case we have vS Vg vR V 42 vT VgiV a VT The applled tank voltage 1s therefore constant and VC equal to VgiV In normalized form one obtains VT 17M Vg 43 Vg iL gt 0 L C direction of CF V R current ow 39 state plane trajectory for this subinterval is a vT Vg V MT Hence according to section 33 the normalized circular arc centered at MT lM as shown in Fig 42b The radius depends on the initial conditions Note that since we have assumed that iLt is positive and diodes D5 and D8 conduct this particular switch conducting state can occur only in the upper halfplane 0L gt 0 For negative jL diodes D6 and D7 would conduct instead MT would be changed and an arc centered at a 39 t MT 1 M 1 m dlfferent locatlon would be obta1ned A sub1nterval utilizing the switch conduction state described Fig 42 Q conduction subinterval in which Q1 and Q4 conduct anal iL gt 0 so thatD 5 anang conduct a circuit b normalized state plane trajectory above and in Fig 42 is referred to in shorthand form as subinterval Q1 a VC iL lt 0 L C Vg C V R direction of F current ow 39 VT Vg V b JL lL lt 0 1 MT l M x InC N quota g Fig 43 D conduction subinterval in which iL lt 0 such that diodes D1 D4 D6 and D7 conduct a circuit b normalized state plane trajectory VT a VC L C Vg direction of CF V current ow VT Vg V b jL gt 0 K N MT l M l InC Fig 45 D2 conduction subinterval in which iL gt 0 such that diodes D2 D3 D5 and D8 conduct a circuit b normalized state plane trajectory Chapter 4 The Series Resonant Converter VT a VC iL lt 0 L C V g m CFV current ow 39 VT VgV b lL 1 MT 7 1 M F I11C jLlt0 Fig 44 Q2 conduction subinterval in which Q2 and Q3 conduct and iL lt 0 so that D6 and D7 conduct a circuit b normalized state plane trajectory Many other switch conduction states can occur Subinterval D1 is similar to subinterval Q1 except that the tank current iLt is negative The conducting devices are antiparallel diodes D1 and D4 and output recti er diodes D6 and D7 The applied tank voltage is therefore VT VgV or in normalized form MT 1M 44 The circuit and state plane trajectory for this subinterval are summarized in Fig 43 Note that this switch conduction state can only occur in the negative halfplane 0L lt 0 Symmetrical switch conduction states Q2 Fig 44 and D2 Fig 45 can also occur in which iL vs vR and vT have the opposite polarity from states Q1 and D1 respectively These correspond to MT 1M Q2 and MT 1M D2 Principles of Resonant Power Conversion a VT V iL gt 0 L C I g 9pm V c1rcu1t T b JL jL 0 InC mC does not change Fig 46 SubintervalX in which allfour rectifier diodes D5 D6 D7 and D8 are reverse biased The inductor current remains at zero and the tank capacitor voltage does not change a one possible circuit topology b normalized state plane trajectory Under certain conditions it is possible for all four uncontrolled recti er diodes D5 D6 D7 D8 to become simultaneously reverse biased When this occurs the circuit topology is as given in Fig 46 The tank inductor is then zero and the tank capacitor voltage remains at its initial value This switch conduction state is denoted X When phase control is used two other subintervals can occur iL gt 0 P1 which occurs for is summaiized in Fig 47 and P2 which occurs for iL lt 0 is summarized in Fig 48 An operating mode is de ned by a sequence of subintervals which combine to form a complete switching period Discontinuous iLgt0 L C direction of current ow 4 Fig 4 7 Subinterval P1 in which D2 and Q4 or Q andD3 conduct and iL gt 0 so that D5 and D8 also conduct a circuit b normalized state plane trajectory VT VC iLlt0 L C direction of C F V current ow vT V a Fig 48 Subinterval P2 in which D1 and Q3 or Q2 and D4 conduct and iL lt 0 so that D6 and D7 conduct a circuit b normalized state plane trajectory Chapter 4 The Series Resonant Converter conduction modes contain at least one X subinterval while continuous conduction modes contain no X subintervals As seen later in this chapter the different modes cause the series resonant converter to exhibit widely varying terminal characteristics Principles of Resonant Power Conversion 42 State Plane and Charge Arguments Mode State plane trajectory Typical inductor current iLt capacitor voltage vCt and applied tank voltage vTt waveforms are diagrammed in Fig 49 for the kl continuous conduction mode This mode is de ned by the subinterval sequence Q1D1Q2D2 In this mode the switching period begins when the control circuit switches transistors Q1 and Q4 on with the inductor current iLt positive The state plane trajectory for this subinterval is given in Fig 410a it begins at not 0 with some initial values of tank inductor current and capacitor voltage The tank rings with a circular state plane trajectory centered at MT 17M until at not B the inductor current rings negative and subinterval D1 begins The normalized state plane trajectory then follows a circular arc centered at MT 1M At time not B0L E 7 one half switching period the control circuit switches transistors Q1 and Q4 off and Q2 and Q3 are switched on Subinterval Q2 begins and as shown in Fig 410c the trajectory continues along a circular arc centered at MT 1M until the inductor current again reaches zero The output bridge recti ers then switch and subinterval D2 begins The trajectory follows an arc centered at MT lM for the remainder of the switching period The switching period ends when the control circuit switches Q2 and Q3 off and Q1 and Q4 on If the converter operates in equilibrium then the trajectory begins and ends at the same point in the state plane and the tank waveforms are 6 for the k1 Continuous Conduction iLt l Vct l VCl 506 I I subinterval Q1 D1 Q2 D2 Fig 49 Typical tank inductor current iL tank capacitor voltage vC and applied tank voltage vT waveforms for the k 1 continuous conduction mode Chapter 4 The Series Resonant Converter periodic Otherwise atransient occurs in which the trajectory for each switching period begins at a different point and follows a dilTerent path in the state plane If the circuit is stable then the trajectory eventually converges to a single closed path and the waveforms become periodic To nd the converter steadystate characteristics we need to solve the geometry of this closed path and to relate it to the load current using charge arguments Q1 subinterval c not 0 B rad1us MA I M wotB wotYB 1M l lM lMMA1 Inc c B radius MAl Q2 subinterval lL jL b d D2 subinterval I not y CC B 27 I 1 M at radius MA2 radiusMA2 I or I Dot0LB Y i wotB r I mC R 1M mc wotYB D1 subinterval Fig 410 Construction of the state plane trajectory for one complete switching period in the k1 CCM a subinterval Q1 b subintervalD 1 c subinterval Q2 61 subinterval D2 Capacitor charge arguments The inductor current waveform of Fig 49 contains only one positivegoing and one negativegoing zero crossing per switching period this is true throughout the kl continuous conduction mode In consequence the discussion of section 31 regarding tank capacitor charge variation applies directly to this case and leads to a result nearly identical to Eq 36 The tank Principles of Resonant Power Conversion inductor current coincides with the tank capacitor current in the series tank and hence the tank capacitor voltage vCt increases when the inductor current iLt is positive During the half switching period where iLt is positive the capacitor voltage increases from its negative peak VCl to its positive peak VCl The total change in vC is the peaktopeak value ZVCl This corresponds to a total increase in charge q on the capacitor given by the integral of the positive portion ofthe iL as shown in Fig 411 Hence we have q C 2Vc1 45 This q is also directly related to the dc load current I The load current is the dc component or average value of the recti ed tank inductor current l iL l1 I in i i i L07 l d1 iTs I ltliLlgt 46 since the integral in Eq 46 is equal to the charge q We can now eliminate q from Eqs 45 and 46 and solve for VCI IT Vc1 T5 47 or in normalized form J Mc1 g 48 where MCl VCl Vg areaq l iLt VC 0 VCI VC1 area q l l1Ltl Fig 41 1 Use of charge arguments to relate the peak capacitor voltage VC 1 load current and charge quantity q in the series resonant converter for k1 CCM This is a useful result because it allows us to relate the load current to the peak capacitor voltage In the normalized state plane we can now nd the circle radii directly in terms of the normalized load voltage and current M and J Chapter 4 The Series Resonant Converter 43 Solution of the k1 Continuous Conduction Mode Characteristics The radii MM and MA2 of the state plane trajectory redrawn in Fig 412 can now be found At time not B it can be seen that the radius M A1 of the Q1 subinterval is J MA1 MC1 1 M 731 M 49 and the radius M A2 of the D1 subinterval is W MA2 MCl71M 717M 410 So we know the radii and centers of the circular arcs in terms of the normalized output voltage M current J and control input 7 or switching frequency fS 7 f0 y D2 subinterval JL M Q1 subinterval not 27 not 0 39 N 06 i quot x J r 391 39 M x l M 1 Inc motYB V x I was radius MAl radius MA2 Q2 subinterval D1 subinterval Fig 412 Complete normalized state plane trajectory for steady state operation in the k1 CCM Finally it is desired to nd a closedform expression that relates the steadystate output voltage output current and the control input ie we want to directly relate M J and y In steadystate the endpoint of the state plane plot after one switching interval at time 00 2y DOTS coincides with the initial point at time not 0 and the trajectory is closed For a given M and J in the k1 CCM there is a unique set of values of y 0L 5 M A1 and MA2 which cause this Principles of Resonant Power Conversion to happen and which can be found the geometry of the state With these values the triangle of Fig 413 is formed whose by solving plane solution yields the converter steady state characteristics The lengths of the two radii are already known and the length of the triangle base is the distance between the Q1 and D2 subinterval centers or 2 Hence the jL xx I JTTETEOcTEBYTE i I x i 539 I I I 397 i K lt93 A g 5 V I I x s r s 47 5 067TE OL n B a a l M l M mc L 2 I lengths of the three sides of the triangle are known and are functions of only M J and y The included angles can be found using simple geometry The two angles adjacent to the base are T 7 0L and TE 7 B which are functions of the unknowns 0L and The remaining angle can be found knowing that the three angles of the triangle must sum to TB and is given by TPOOTEB Yin since 7 0L Note that this is a function of the control input 7 alone and does not depend separately on 0L or a2 b2 c2 2bc cos8 Fig 413 Magni cation of Fig 412 for the solution of steady state conditions a Fig 414 The Law ofCosines 4 11 The Law of Cosines Fig 414 can now be used to relate the top angle and the three sides and hence to find how M and J depend on y One obtains 22 JZ Y717M2 JZ Yi 1 M2 10 7 262171 7 1 M cosy4c 4 12 Chapter 4 The Series Resonant Converter Simpli cation yields 7 J Y 2 2 JY 2 2 4 7 2 71 2M 271 7M cosy 4 13 which can be rearranged to obtain J7 21cosy 2licosy 1 2 1 2 M 2 4 14 Trigonometric identities can now be used to obtain M2 sin2 17 12 cos2 1 4 15 This is the desired closedform solution for the series resonant converter operating in the k1 continuous conduction mode with frequency control Output characteristics At a given switching frequency f5 corresponding to a given 7 Tc f0 f5 Eq 415 shows that the relation between M and J is an ellipse centered at M 0 and J ZY A typical ellipse is plotted in Fig 415 The uncontrolled output 1 J no solutions I for M 2 1 39 DCM instead recti er diodes do not allow the load current to be negative so we must have J gt 0 Also with vahd solutions passive load requires I and V to have same polarity a passive load M must be positive when J is positive Hence the portions of the ellipse that lie in the second third and fourth quadrants are not valid physical solutions Also it is Shown in seem 44 Fig 415 Elliptical output characteristicM vs 1 Eq 4 15 that the solution is not valid for for a given 7 in the k 1 CCM M21 instead the k1 discontinuous conduction mode occurs for M1 Hence the solution is valid only for 0 S M lt l and J gt 0 Equation 415 is plotted in Fig 416 It can be seen that as the load current or J is increased the output voltage or M decreases Hence the output impedance of the openloop converter is substantial It is instructive to examine some limiting cases Principles of Resonant Power Conversion Cap k M Fig 41 6 Output characteristics of the series resonant converter operating in the k 1 continuous conduction mode Solutions occur over the range 0 SM lt1 2727 S J lt on solutions not shown here for gt 6 At F 05 half resonance Then fS 05 f0 and y TE 05 27 The output characteristic Eq 415 becomes 2 2 Mr07cJ71r171 M6 or J 2 7 417 which is independent of M The ellipse collapses to a horizontal line and the converter operates as a current source At F 10 resonance Then fS f0 andy TE 1 TE The output characteristic Eq 415 becomes J 2 M2gt1TTE71gt071 8 or M 1 419 which is independent of J The ellipse collapses to a vertical line and the converter operates as a voltage source For 05 lt F lt l the converter operates as neither a voltage source nor a current source Chapter 4 The Series Resonant Converter Value 0fJ when M 1 The critical minimum value of J occurs when M 1 for J less than this value the converter does not operate in k 1 CCM and Eq 415 is not valid Plugging M 1 into Eq 4 15 yields 00821 1 v n 420 y 7 2 M7121 s1n2 7 2 or J 47 4 21 which varies between 2 TE and 4 T for F between 05 and 1 Output short circuit current JSC When M 0 Eq 415 becomes 10 Jsc Y 2 2Y Ti 1 cos 5 7 1 4 22 8 Solve for JSC Jsc Z11 sec1l lt1lsecLl 6 y 2 7E 2F 2 1 4 423 Equation 423 is plotted in Fig 417 It can be 2 seen that the converter shortcircuit current is 0 inherently limited except at resonance 05 06 07 08 09 1 F The characteristics of a given load can be Fig 417 Normalized short circuit output current JSC vs switching frequency in superimposed on the converter output the k 1 CCM characteristics allowing graphical determination of output voltage vs switching frequency For example with a linear resistive load I V R 424 or in normalized form J M Q 425 with Q R0 R Equation 425 describes a line with slope Q as shown in Fig 418 The intersection of this load line with the converter elliptical output characteristic is the steadystate operating point for a given switching frequency As shown in the example of Fig 419 nonlinear load characteristics can also be superimposed and the operating point determined graphically Principles of Resonant Power Conversion 0 aha x M o IIM x M Fig 419 Nonlinear load characteristic superimposed over the converter output characteristics Chapter 4 The Series Resonant Converter Control plane characteristics It is also instructive to plot the voltage conversion ratio M vs normalized switching frequency F Doing so requires knowledge of the load characteristics so that J can be eliminated from Eq 415 In the case of a resistive load satisfying V I R Eq 425 can be substituted into Eq 415 yielding 2 Y MQY 2 Y 426 V s1n2 2 2 71 cos2 2 7 l Now solve for M M2sin2ltrm21Won 0 COSZ71 427 Use of the quadratic formula yields M l i 1 L2tan21 tan2z tanz Qy 2 2 2 428 Fig 420 Steady state control characteristicsM vs F for various values of Q RMR in the k1 continuous conduction mode Eq 4 28 Principles of Resonant Power Conversion To obtain the correct solution in which M gt 0 the plus sign should be used Equation 428 together with the identity 7 TE F 7 f0 f5 is a closedform representation of the control characteristics M vs F for the k1 CCM It is plotted in Fig 420 for various values of Q or load resistance R As R is decreased corresponding to heavy loading the Q is increased Loading the converter causes the output voltage to decrease and results in a peaked characteristic near resonance Control of diode conduction angle 06 Another popular scheme for controlling the series resonant converter when it operates below resonance is known as diode conduction angle control or 0L control Rather than using a voltage controlled oscillator to cause the switching frequency and y to be directly dependent on a control signal known as frequency control or Y control the OL controller causes the diode conduction time and angle cc to be directly dependent on the control signal The switching frequency varies indirectly and depends on both 0L and the load current This control scheme requires a current monitor circuit which senses the zero crossings of the tank current waveform A timing circuit then causes the transistors to switch on after a delay which is proportional to a control voltage The transistor off time which coincides with the diode conduction time is therefore proportional to the control voltage To understand the converter characteristics under 0L control we need to eliminate y from the k1 CCM solution Eq 415 in favor of 0c This can be done by again referring to Fig 413 Application of the Law of Cosines using the included angle at vertex C yields giHMr 2221717M2722le717McosTE706 429 This equation can be solved for y Paw J M 7 cosOL 43930 This describes how the switching frequency varies for a given range of 0c and load Equations 4 30 and 415 can now be used to eliminate y and to determine the OL control characteristics The result is 1 M 1 7 cos 0a sin0L 4 31 M 7 cos 06 75 t 171M 7 cos 06 where 0 S tan 1 S TE2 and M gt cosOL Equation 431 describes the output characteristics under 0L control and is plotted in Fig 421 It can be seen that the output characteristics resemble hyperbolae with vertical asymptotes 16 Chapter 4 The Series Resonant Converter M cos0L Also comparison with Fig 416 reveals that the switching frequency approaches resonance fS gtf0 as 0L gt0 and fs gt05f0 as OL gtTE Decreasing 0L causes M andor J to increase It can also be seen that for 0L lt TE2 the converter shortcircuit current is not inherently limited UI I L I Fig 421 k1 CCM output plane characteristics diode conduction angle control Mode boundaries k1 CCM So far we have studied only the kl continuous conduction mode characterized by the subinterval sequence Q17D17Q27D2 The state plane diagram of Fig 412 and the succeeding analysis are both based on the assumption that the transistors and diodes conduct in the order given by this sequence As stated previously the diode conduction angle 0L ie the lengths of the D1 and D2 subintervals vanishes as the switching frequency approaches resonance Increasing the switching frequency beyond resonance must therefore cause a different subinterval sequence to occur Likewise as fS approaches 05f0 the diode conduction angle 0L and transistor conduction angle 5 both approach TE or a complete resonant halfperiod For the series resonant converter no ringing subinterval can extend through an angle of more than T radians because the state plane centers and output diode switching boundary both lie on the jL 0 axis Therefore decreasing the switching Principles of Resonant Power Conversion frequency below fS 05f0 must cause new subintervals to occur Hence the kl CCM and Eq 415 are restricted in validity to the range 05f0 S fS S f0 By examination of Fig 416 it can be seen that the kl CCM solutions do not extend beyond the range J 2 2TE If it is desired to operate the converter at light loads corresponding to J lt 2TE then a different mode must be used most likely with a different range of switching frequencies In addition as shown in the next section the kl CCM is restricted to the range 0 S M lt 1 44 Discontinuous Conduction Modes At light loads all four bridge recti er diodes can become reversebiased during part of the switching period causing the converter to operate in a discontinuous conduction mode More that one discontinuous conduction mode is possible depending on the load current Each of these is characterized by a sequence of subintervals ending in subinterval X Fig 46 The k1 discontinuous conduction mode This mode is de ned by the subinterval sequence QlXQgX As shown in Fig 422 transistor Q1 conducts for a complete tank halfperiod The four bridge rectifier diodes then become reversebiased and subinterval X occurs for the remainder of the half switching period As in the kl continuous conduction mode the dc load current I is given by I ltl iL lgt 4 32 where q shown in Fig 422 is T52 q I iLtdt 433 0 The average input current is Ts2 ltig gt 1L iLt dt 434 sz 0 since ig iL when Q1 conducts Substitution of Eqs 432 and 433 into Eq 434 yields ltig gt 2i 1 TS 435 VC1 Chapter 4 The Series Resonant Converter So the converter dc input and output currents are equal If the converter is lossless and operates in equilibrium then the input and output powers must be equal this implies that the voltages are also equal Pin Vgltiggt Pout VI 436 Use of Eq 435 yields Vg V or M 1 437 Hence in the kl DCM the converter dc ratio M is unity and is independent of F ig 423 Tank inductor current anal capacitor voltage waveforms for the k 1 DCM JY Mai the values of load current and switching frequency The usual tank capacitor charge arguments can be used to complete the solution and compute the peak tank capacitor voltage VCl During the Q1 conduction subinterval the charge on the tank capacitor changes by an amount q corresponding to an increase in voltage of 2VCl see Fig 423 Hence q C Wei 438 Elimination of q using Eq 432 yields VC1 439 or in normalized form 440 Equation 440 happens to be identical to the result for the kl CCM Eq 48 Beware this does not occur for all other operating modes This mode cannot occur above resonance The switching period must be long enough that the tank can ring through one complete Q1 subinterval of length 7 during each half switching period of length 7 Hence a necessary condition for the occurrence of the kl DCM is Principles of Resonant Power Conversion Y gt 75 4 41 or in terms of F F gt 1 4 42 An additional necessary condition for occurrence of the kl DCM is given in the next subsection Reason for occurrence of the k1 DCM Why does the tank stop ringing at the end of the Q1 subinterval As suggested previously the reason is that all four bridge recti er diodes become reversebiased at this instant Physical arguments are used in this subsection to prove this assertion and to derive the conditions on load current and frequency which lead to operation in this mode These arguments also have a very simple stateplane interpretation As seen in Fig 41 the voltage applied to the tank inductor vL is diL vL 7 L 7 vsivcivR 443 dt During subinterval Q1 vs Vg and vR V We have already shown that V Vg in this mode Eq 437 and so vL becomes 7 VLLdt 7 VC 4 44 for subinterval Q1 vLt is plotted in Fig 423 At not 0 this is a positive quantity since as shown in Fig 422 vC0 VCl So initially diLdt is positive and iL increases At not TE2 vLt passes through zero and iL begins to decrease At not TE iL reaches zero Can the inductor current iL continue to decrease for not 2 TE This is possible only if the applied inductor voltage vL continues to be negative Note that if iL rings negative then the bridge recti er will switch from vR V to vR 7V and subinterval D1 will occur The applied tank inductor voltage Eq 441 would then become E VLLdt V V7 v 2V 7v g C g C 445 for subinterval D1 with vCTE VCl Note that it is possible for this voltage to be either positive or negative at not TE depending on whether or not VCl is greater than 2Vg In the kl discontinuous conduction mode 2Vg7VCl is a positive quantity As a result diodes D6 and D7 cannot turn on at not TE doing so would require that iL become negative which cannot occur if vL is positive since iLTE0 Instead all four bridge recti er diodes become reversebiased Chapter 4 The Series Resonant Converter 1L1 k1 DCM 1L1 k1 CCM I I gt I V I I V I I I I I I Dot I I I Dot I I I I I l I l I I I I I l I l I I I I Q1 I X I Q2 X Q1 ID1I Q2 ID2I VC I I I VC I I I I I l I l I I I VC139 I I I I I I l I I I I I I I I I I l I I I I I I I I I I I t I I I t I I I I I I I I I I l I l I I I I I l I l I I I VCI I I I I I I I I I l I l l l I l A I I A I I VL VgVVct VL VgVVct VgIVLVf VgVv ct 2V I VgVVct 2V i Fig 4 23 Comparison of tank waveforms of the k1 continuous and discontinuous conduction modes The inductor voltage and current remain at zero for the rest of the halfswitchingperiod Inductor voltage and current waveforms for the k1 DCM and k1 CCM are compared in Fig 423 Hence the requirement for the k1 DCM to occur is Vg V 7 VcTE 2Vg 7 VcTE gt 0 446 In normalized form this can be written Principles of Resonant Power Conversion 1M7MCI gt0 Substitution of Eqs 437 and 440 into this expression yields 4 lt JY 4 47 448 This is the basic condition for operation in this mode It can be seen that the kl DCM occurs at light load A simple state plane interpretation The above arguments can be given a simple As shown in chapter 3 for the series tank circuit the geometrical interpretation in the state plane state plane trajectories evolve in the clockwise direction about the applied tank voltage Consider the hypothetical state plane trajectories of Fig 424a At not TEjL reaches zero and mc MCl The gure is drawn for the case MCl lt 1M Note that a D1 subinterval cannot occur for not gt TE since such a subinterval would involve a trajectory centered at mc 1M and given either by hypothetical trajectory A or B Trajectory A is impossible because subinterval D1 cannot occur except for negative jL Trajectory B is also impossible because it does not travel clockwise about the center mc 1M Hence there can be no D1 subinterval and instead an X subinterval occurs as described in Fig 46 in which mc remains constant and equal to MCl as shown in Fig 424b In contrast a CCM trajectory is shown in Fig 424c In this case MCl gt 1M so that for negative jL the trajectory is able to evolve in the clockwise direction about the center mc 1M Thus the geometry of the state plane trajectory gives a simple interpretation of the boundary condition between the continuous and discontinuous conduction modes a lL Q1 D1 9 lQl M d 5 iDl lM Inc Inc MC1 at not 7 D1 b jL Q1 x lM MCl Inc 1 M1M MC1 lQ1 JDI Inc D1 Fig 4 24 Hypothetical state plane trajectories a forMC lt 1AI trajectoriesA anal B are impossible b actual k1 DCM trajectory withMC lt 1AI c actual k1 CCM trajectory withMC gt 1M Chapter 4 The Series Resonant Converter The k2 discontinuous conduction mode In the k2 discontinuous conduction mode the tank rings for two complete halfcycles during each halfperiod of length Ts2 A er the two complete halfcycles the output bridge recti er diodes become reversebiased Waveforms for this mode are given in Fig 425 For this mode to occur the switching period must be at least as long as twice the tank natural period To Hence 7 2 27 or F S i 449 This is a useful mode because the output is easily controllable the converter output behaves as a I Q1 D1 X I Q2 D2 X IL I I I I I I I I I I I I l l I I I I I I I I I I lt 7 gt 7 I I I a I I I I I I I Y 4 I I I I d I I 7 1L I I VLiLE VgVVC I I I WWW2539 I I I V I 2v vg v vc2n2g I I I I I VC I I I I I I I I Vc2 I I I I VCI I I I I I I I I I I I I I I I I I I I I I I I I I I 39VC1 39 I I I I Fig 4 25 Tank waveforms for the k2 discontinuous conduction mode 23 Principles of Resonant Power Conversion current source of value controllable by the switching frequency Also the switch tumon and tumoff transitions both occur at zero current so switching losses are low A disadvantage of this mode is its higher peak transistor currents than in the continuous conduction mode and hence higher conduction losses As is shown in this subsection the load current and switching frequency are directly related for the k2 DCM and hence a wide load current speci cation implies that the switching frequency must vary over a wide range This is less of a disadvantage than one might at first think because the transformer can be sized to the maximum switching frequency 05 f0 Operating point variations that cause the switching frequency to vary below this value do not necessitate use of a larger lower frequency transformer Analysis First capacitor charge arguments are used to relate the peak tank capacitor voltage to the dc load current The tank inductor current iLt which coincides with the tank capacitor current is re drawn in Fig 426 The total charge contained in the negative portion of the iLt waveform is de ned as iq The peaktopeak tank capacitor voltage is 2VC2 which represents a change in capacitor charge of q total charge q Fig 426 Tank inductor current waveform with emphasis on total charge owing through the tank capacitor Hence 2 vcz 450 The average output current I is 2 I ltl1Llgt q 4 51 S Elimination of q from Eqs 450 and 451 yields Chapter 4 The Series Resonant Converter 4CVC2 I Ts 452 Normalization of Eq 452 and solution for the normalized peak capacitor voltage MC2 yields J MC2 3y 4 53 This result is again similar to the kl DCM and kl CCM cases It states that the peak tank capacitor voltage is proportional to the load current lL MC2 1M Q2 Fig 427 State plane diagram for the k2 discontinuous conduction mode The state plane diagram for the k2 DCM is given in Fig 427 The switching period begins with mc jL 7MCl0 Subintervals Q1 and D1 are of length TE and are given by semicircles in the state plane Their radii are MC241M and MC271M respectively as indicated on the diagram Subinterval X ends the half switching period with mC jL MCl0 The converter output characteristics for this mode can be found in a manner similar to that used for the kl CCM If the converter operates in steady state then the state plane diagram is closed as indicated in Fig 427 The ending point of the D2 subinterval must therefore coincide with the beginning of the Q1 subinterval The portion of the state plane in the vicinity of this point is magni ed in Fig 428 It can be seen that in steady state the sum of the radii of the D2 and Q1 subintervals is equal to the distance between their centers or 2 McziliMMczilM 454 Principles of Resonant Power Conversion Simpli cation yields jL J MC2 2 Y 455 2 rad1us MC2 lM rad1us MC21M Now solve for the normalized load current J J i AF 456 lM Mc1 l IM mc Y 75 I I I I I4 gtI Hence the load current depends on switching 2 frequency but not 011 outpl voltage and the Fig 428 Illustration ofrelations between converter behaves as a current source in this mode M C M C2 and M As sketched in Fig 429 the k2 DCM output characteristics are straight horizontal lines With a resistive load we have J MQ F 457 with Q RRo Solution for M then yields the control plane characteristic M4F it Q 458 Thus the control plane characteristics M vs F for a linear resistive load vary linearly with F The som on can be completed by SOIVing Fig 429 Output plane characteristics for the normalized voltage MCl By inspection of emphasizing k2 DCM the state plane diagram of Fig 427 it can be seen that MCl is equal to MC2 minus twice the radius of M the D1 subinterval l quot MCl MCQ 7 2MCQ 7 l 7 M 2M 459 This result is used in the next section for kl derivation of the mode boundaries I CCM l u I I 3 I iquot l x I I k2 DCM boundaries i39 xix I 1 I z other modes I 1034 I 9 I I I I gt As noted prev1ously th1s mode 1s restr1cted 391 391 F to the frequency range 7 2 27 or F S 05 Since 2 E accordmg to Eq39 AI56 J and F are dlrealy Fig 430 Control plane characteristics related this restriction also places an upper limit on emphasizing k 2 DCM 26 Chapter 4 The Series Resonant Converter the load current J J F 3 460 Hence the k2 DCM is restricted to the portion of the output plane below J 2TE In addition for the k2 DCM to occur the output bridge recti er 1 must continue to conduct at not 7 between the Q1 and D1 subintervals and 2 must become reversebiased at not 27E after the D1 subinterval By use of the state plane it can be seen that requirement 2 leads to the constraint MCl gt 17M 4 61 and requirement 1 leads to MC2 gt 1 M 462 By substitution of Eqs 455 and 459 and after a small amount of algebra these two constraints become 1 gt M gt 463 To summarize the k2 DCM boundaries are 1 gt M gt L 3 l gt J gt 0 TE 464 L gt F gt 0 2 The k2 DCM output plane boundaries are given in Fig 429 and the control plane characteristics in Fig 430 As M gtl the converter enters the kl DCM while for M gtl3 the converter enters a higherorder k22 continuous or discontinuous conduction mode 45 A General Closed Form Solution The steadystate solutions for all frequencycontrolled continuous and discontinuous conduction modes are stated here Type k CCM The type k continuous conduction mode occurs over the frequency range f0 lt fs lt f0 k l k 465 Principles of Resonant Power Conversion CCM WAVEFORMS 1 type k CCM k odd I 1L I lt Y gt I I I Q1 I n Q1 I I n m D2 J J I D1 K TE H D1 H TE H Q2 Dot I symmetrical I4 kl complete halfcycles gt type k CCM k even I 1L1 Y gt I I I I D Q1 H Tc H Q1 I IH TE H Q1 2 J I V D1lt rc gt D1 K Tc H W I Q2 symmetrical I4 k complete halfcycles gtI I Fig 431 Tank inductor current waveforms for the general type k continuous conduction mode The output plane characteristics are elliptical and are described by the equation 2 2 Y 1 W k 2 Y 7 M E 91125 E7 1 cos2 7 1 466 where E is the subharmonic index k a Hg1 467 The voltage conversion ratio M is restricted to the range 0 s M s l 468 5 At light load the converter may enter a discontinuous conduction mode Typical tank current waveforms are shown in Fig 431 For k even diode D1 conducts rst for a fraction of a half resonant cycle If k is odd then transistor Q1 conducts rst for a time 28 Chapter 4 The Series Resonant Converter less than one complete halfcycle In either case this is followed by E 1 complete halfcycles of ringing The halfswitchingperiod is then concluded by a subinterval shorter than one complete resonant halfcycle in which the device that did not initially conduct is on The next half switching period then begins and is symmetrical The steadystate control plane characteristic can be found for a resistive load R by substituting J M Q into Eq 466 where Q R0 R Use of the quadratic formula and some algebraic manipulations yields M 2i 1k1 1W 469 54 tan2 coszg This is the closedform relationship between the switching frequency and the voltage conversion ratio for a resistive load It is valid for any continuous conduction mode k Type k DCM k add The type k discontinuous conduction modes for k odd occur over the frequency range f0 f lt S k 470 In these modes the output voltage is independent of both load current and switching frequency and is described by 471 M l k This mode occurs for the range of load currents 2k 1 gt J gt 2k 1 472 Y Y In the odd discontinuous conduction modes the tank current rings for k complete resonant halfcycles All four output bridge rectifier diodes then become reversebiased and the tank current remains at zero until the next switching halfperiod begins as illustrated in Fig 432 A dc equivalent circuit for the SRC operating in an odd discontinuous conduction mode is given in Fig 433 it is a dc transformer with effective turns ratio llk Converters are not usually designed to operate in these modes because the output voltage cannot be controlled by variation of the switching frequency Nonetheless all converters designed to perform below resonance can operate in one or more odd discontinuous conduction modes if the load current is sufficiently small and hence the converter designer should be aware of their existence 29 Principles of Resonant Power Conversion a I 1L X X Q1 E II Q1 D2 J lt TE gtI D1 H TE H Qz Dot I symmetrical I lt k complete halfcycles gt I A J h 75 10 i 3 k1 DCM k3 k5 DCM etc i l 1 M 5 3 Fig 432 General type k discontinuous conduction mode k odd a tank inductor current waveform b output characteristics 0 D 0 Fig 433 Steady state equivalent circuit model for an odd discontinuous conduction mode an e ective dc transformer W H Chapter 4 The Series Resonant Converter Type k DCM k even The type k discontinuous conduction modes for k even occur over the frequency range f0 f lt S k 473 These modes have current source characteristics in which the load current is a function of switching frequency and input voltage but not of the load voltage The output relation is J A 474 Y Operation in this mode occurs for 475 1 gtMgt l kil kl In the even discontinuous conduction modes the tank current rings for k complete resonant halfcycles All four output bridge rectifier diodes then become reversebiased and the tank current remains at zero until the next switching halfperiod begins as illustrated in Fig 434a The series resonant converter possesses some unusual properties when operated in an even discontinuous conduction mode A dc equivalent circuit is given in Fig 435 it is a gyrator with gyration conductance g ZldyRO The gyrator has the property of transforming circuits into their dual networks in this case the gyrator characteristic effectively turns the input voltage source Vg into its dual an output current source of value g Vg 9 Some very large converters have been designed to operate purposely in the k 2 DCM at light load ref Hughes and even at full load 10 Principles of Resonant Power Conversion a gt I 19 i D2 H 39 D1 W Q2 I symmetrical 4 k complete halfcycles gt b J A 10 l 39i3939V l n quot F 2 F 04 II I t 2 39 k2 39 F 7 025 I etc I n I DCM I I I F 01 I nF05I i l 1 M 5 3 Fig 434 General type k discontinuous conduction mode k even a tank inductor current waveform b output characteristics IggV I Vg I Fig 435 Steady state equivalent circuit model for an even discontinuous conduction mode an e ective gyrator The converter exhibits current source characteristics 32 Chapter 4 The Series Resonant Converter Q035 Q05 Q075 Q15 Q2 Q35 Q5 Q10 Q20 20 Fig 4 36 Complete control plane characteristics of the series resonant converter for the range 02 S F S 2 I M k1 DCM 1 I I I I I I I I I I I I I I I k2 DCM I I I I I I g I k1 CCM E k0 CCM k3 DCM 1 3 I I I I k 4 DCM I E I E I I l 1 0 l O I I 5 I I 0 I O I I k5 DCM E I E etc I I I I I I I I I I l l l l 1 F 5 4 3 2 Fig 4 3 7 Complete control plane characteristics for continuous anal discontinuous conduction mode boundaries Principles of Resonant Power Conversion Composite characteristics The complete control plane characteristics can now be plotted using Eqs 465 475 The result is shown in Fig 436 and the mode boundaries are explicitly diagrammed in Fig 437 It can be seen that for operation above resonance the only possible operating mode is the k0 CCM and that the output voltage decreases monotonically with increasing switching frequency Reduction in load current or increase in load resistance which decreases the Q causes the output voltage to increase A number of successful designs that operate above resonance and utilize zero voltage switching have been documented in the literature Operation below resonance is complicated by the presence of subharmonic and discontinuous conduction modes The kl CCM and k2 DCM are well behaved in that the output voltage increases monotonically with increasing switching frequency Increase in load current again causes the output voltage to decrease Successful designs which operate in these modes and employ zerocurrent switching are numerous However operation in the higherorder modes k2 CCM k4 DCM etc is normally avoided Given F and Q the operating mode can be evaluated directly using the following algorithm First the continuous conduction mode k corresponding to operation at frequency F with heavy loading is found i k INTlF 4 76 where INTx denotes the integer part of x Next the quantity k1 is determined r t 4 u E k1 INT2 4 2F 4 77 The converter operates in type k CCM provided that k1 gt k 478 Otherwise the converter operates in type k1 DCM A computer listing is given in Appendix 1 in which the conversion ratio M is computed for a given F and Q First the above algorithm is used to determine the operating mode Then the appropriate equation 469 471 or 474 is evaluated to find M The function works correctly for every frequencycontrolled mode Output plane characteristics for the k0 CCM plotted using Eq 466 are shown in Fig 438 The constantfrequency curves are elliptical and all pass through the point Ml J0 Output plane characteristics which combine the kl CCM kl DCM and k2 DCM are shown in Fig 439 These were plotted using Eqs 466 471 and 474 It can be seen that the constantfrequency curves are elliptical in the continuous conduction mode vertical voltage source characteristic in the kl DCM and horizontal current source characteristic in the k2 DCM Chapter 4 The Series Resonant Converter o o N 0 4 C Ox 0 oo Fig 438 Output characteristics in the k0 continuous conduction mode above resonance F 10 F9 2 F5 H i V 7Equot o F25 k2DCM Z 3 F 1 0 I I I I I I I I I I I I I I I I I I I I 0 02 04 06 08 1 M Fig 439 Composite output characteristics for the k1 CCM k1 DCM and k2DCM modes Mode boundaries are indicated by heavy lines 35 Principles of Resonant Power Conversion 4 6 Converter Losses If the transistors and diodes can be modeled as constant voltage drops while they conduct then there is a relatively easy way to adapt the ideal analysis of the previous sections to include these losses The transistor and diode forward voltage drops cause a dilTerent voltage to be applied to the tank circuit during each subinterval which can be accounted for by changing the effective Vg and V seen by the tank circuit Let us consider the halfbridge J V I example of Fig 440 The semiconductor Vg Dmx VT device forward voltage drops are de ned l m VDout as follows VT VT transistor onstate voltage Vg I I VDin antiparallel diode forward voltage T 39 39 VDOut output diode forward voltage V The applied tank voltage waveform vT is shown in Fig 441 for the kl CCM Since the applied tank voltage is by Fig 440 Half bridge circuitwith nonideal semiconductor devices assumptron constant dur1ng each subinterval the normalized state plane trajectories are again a series of circular arcs and the preceding analytical method can be used to solve this converter In fact it is not necessary to rederive the entire analysis because the converter circuit of Fig 442 in which the input and output voltages are modified but the converter is otherwise ideal exhibits exactly the same tank voltage waveform vT of Fig 441 In consequence the ideal kl CCM solution previously derived Eqs 415 and 428 also apply to this converter but with the input and output voltages replaced by the effective values Vgeff VDIJ1 7 VT actualVg 4 79 I VT Vg VDin V2VDout Vg 39VT 39V39ZVDout Q1 D1 I Q2 I D2 not Vg VT V2VDout l 39Vg 39VDin 39V39ZVDout Fig 441 Actual tank voltage waveforms for the half bridge circuit of F ig 440 36 Chapter 4 The Series Resonant Converter 1 EVDm ideal I switches VT 2 ideal diodes actual no forward Vg We drop actual Vg iv 2T 7 VDin Fig 442 H alf bridge circuit with explicit voltage drops due to non ideal diodes and transistors Veff ZVDout VDIJ1 iVT actualV 480 Note that these effective values must also be substituted into the normalization base quantities The result is given below V Meff V eff geff 481 I R Jeff V 0 geff 482 Mgff i2 sin2 L2 1k2 cos2 1 E 483 The effect of the semiconductor forward voltage drops is primarily to shift the output plane characteristics somewhat to the left and hence to lower the actual output voltage It is also possible but much more complicated to account for resistive losses such as tank inductor and capacitor equivalent series resistance esr shown in Fig 443 These losses effectively damp the tank circuit and cause the normalized state plane trajectories to become spirals of decreasing radius Closedform solutions are not known instead computer iteration can be used to evaluate the highly transcendental equations that are obtained The results show that the converter is very sensitive to small resistive loss elements when operated at large values of normalized current J This coincides with operation near resonance with a large Q JM The 37 Principles of Resonant Power Conversion same conclusions can be obtained more simply using the approximate sinusoidal methods of chapter 2 Rs RS Fig 443 H alf bridge circuit with explicit resistances due to non ideal diodes and switches 47 Design Considerations It may not be initially apparent how to best design a series resonant converter to meet given design objectives yet a suboptimal design may exhibit signi cantly higher tank current and voltage stresses than necessary andor may cause the switching frequency to vary over an unacceptably wide range In this section component stresses are analyzed and the results are overlaid on the output plane plots derived in the previous sections Also speci cations on the input voltage and output power ranges are translated into a region in the same output plane so that it can be seen how component stresses and switching frequency vary with operating point One can then choose the transformer turns ratio and tank characteristic impedance as well as tank inductance and capacitance to obtain a good converter design A 600W lll bridge example is given Approximate peak component stresses In the continuous conduction mode the tank inductor current and tank capacitor voltage peak magnitudes are both approximately proportional to the load current and nearly independent of the converter input and output voltages This is true because the dc load current I is equal to the average rectif1ed tank inductor current ltl iL lgt With a ln transformer turns ratio we have a iL 1711 lt l in gt b A l1Ltl 1L1 725 I ILP 39 39 I ltliLlgt c J A 3 Jun 4 2 JLP 3 JLP 2 1 gt l M Fig 444 Peak stress approximation where a series resonant converter has a 139n transformer turns ratio b dc load current equals average rectified tank inductor current c result shows that peak tank current is directly proportional to load current Chapter 4 The Series Resonant Converter L 39 I n ltl1Llgt 484 To the extent that the waveshape of iLt does not vary with converter operating point there is a direct relationship between the peak value of iL and Indeed this is the case near resonance where iL is nearly its average recti ed value ltliLlgt nI sinusoidal Equation 484 then becomes Ts 71 I 39 m dt lLILP 485 HTS LPlSln l n n 0 Hence I M1 L1 2 486 Or in normalized form one obtains J 2 EJ LP 487 Thus we expect the peak tank current ILp to be directly proportional to the load current I and We can plot contours of constant peak tank current in essentially independent of the load voltage the output plane to see how component stresses vary with operating point Eq 487 predicts that such contours should appear as in Fig 444c The approximation used above coincides with the sinusoidal approximation used in chapter 2 It is accurate for operation in the continuous conduction mode near resonance However in the discontinuous conduction mode or in the conduction mode near the DCM the tank highly nonsinusoidal and Eqs 484 487 become poor approximations continuous boundaries current is The tank capacitor voltage waveform is directly related to the tank inductor current since these components are connected in series Hence the peak tank capacitor voltage is also directly proportional to the load current and is essentially independent of the load voltage In the case of 39 Principles of Resonant Power Conversion sinusoidal tank waveforms the peak tank capacitor voltage ch is related to ILp through the characteristic impedance R0 V 21 R EMIR CP LP 0 2 0 488 Discussion Suppose that we have constructed a converter which is capable of producing the given rated output power Pmax at the normalized operating point M 05 J 5 This converter could also produce twice the rated power or 2Pmax by operating at the point M 10 J 5 ie by doubling the output voltage without changing the load current This is true because the peak component stresses are independent of M and depend only on J The component currents will be nearly the same in both cases and hence the losses will be almost the same also Furthermore a converter with lower peak currents could be constructed by halving the transformer turns ratio n so that the converter operates at rated output voltage and power with M 1 Equation 484 predicts that this would cause the peak tank current to be reduced by a factor of two Component stresses and losses would be reduced accordingly The conclusion is that the converter should be designed to operate with M as close to unity as other considerations will allow This implies that the transformer turns ratio should be minimized and it leads to low peak tank and transistor currents Converters designed to operate with lowerthannecessary values of M do not fully utilize the power components Exact peak component stresses Exact expressions for the peak tank current and voltage can be derived using the state plane diagram For example the state plane diagram for the kl CCM is reproduced in Fig 445 with the peak values of jL and mC identified It can be seen that the peak normalized tank inductor current is given by the circle radius during the Q1 conduction subinterval J Jun 7Y71M 489 The peak normalized tank capacitor voltage Mcp shown in Fig 445 was previously found It is J Y MCP MC1 7 490 We wish to plot these component stresses in the output plane to determine how stresses vary with operating point To do so we need to express JLp and Mg as functions of J and M with y eliminated The easiest way to do this is to solve for 72 in Eqs 489 and 490 40 Chapter 4 The Series Resonant Converter jL A peak value oij J jLPl1M circle radius duringQ1 conduction interval Q1 JV 2 InC peak value of mg J mCP MC1 Fig 445 State plane diagram for k1 CCM Y JLp l 7 M 491 2 J 7 492 2 J These expressions are then inserted into the converter output characteristic Eq 415 to eliminate y Solving for J one then obtains J JLP17M 493 TE 7tan 14 I JLP 7 My 7 1 17M2 J MCP 494 MCP 7 12 7 1 1 7 M2 Here it is necessary to use care to select the correct branch of the arc tangent function When the T 7 tan 1 denominator is written as shown then the correct answer will be obtained when the arc tangent function is de ned to lie in the domain 7 TE2 S tan 1 S TE2 Equations 493 and 494 can now be used to overlay contours of constant peak tank stress on the output plane characteristics The result is given in Fig 446 It can be seen that the contours are nearly horizontal lines and do not differ much from Fig 444 41 Principles of Resonant Power Conversion 0 025 05 075 1 M Fig 446 Superposition of peak tank current and voltage stress curves on the normalized output characteristics for the k1 continuous conduction mode below resonance Solid lines curves of constant switching frequency dashed lines contours of constant peak tank capacitor voltage shaded lines contours of constant peak tank inductor current Similar analysis can be used to derive the component stresses for above resonance operation The results for the k0 CCM are J MCP 495 mil MCP 12 71 17 M2 2 J JLP71M liMM ltJLP mil JLP M2 71 17 M2 JEP 496 71 1 7 2 7 2 J 1 M a 1 M gt JLP tan 1 7 17M2 Chapter 4 The Series Resonant Converter 0 025 05 075 1 M Fig 447 Superposition of peak stress curves on the normalized output plane above resonance k0 CCM Solid curves are contours of constant switchin frequency dashed curves are contours of constant peak tank capacitor voltage and shaded curves are contours of constant peak tank inductor current Two cases occur in Eq 496 depending on whether the peak occurs at the beginning of the switching period or at some time in the middle of the period Equations 495 and 496 can now be used to generate the plot of Fig 447 where peak tank stresses are overlaid on the converter output characteristics for the k0 CCM Use of the output plane In a typical voltage regulator design the output voltage is regulated to a given constant value V The input voltage Vg and output power P as well as the output current I P V vary over some speci ed range vgmax 2 vg 2 vgmin 4 97 F gtP2P 498 max mm Principles of Resonant Power Conversion where Pmax V Imax and Pmin V 1min To regulate the output voltage the controller will vary the switching frequency fS over some range and consequently the operating point will vary It is desired to choose the transformer turns ratio and the values of tank inductance and capacitance such that a good design is obtained in which the tank capacitor voltage and inductor current are low the range of switching frequency variations is small and the transformer and tank elements are small in size Note that the specifications 497 and 498 do not by themselves determine the operating region of the normalized operating plane By changing the tank characteristic impedance R0 and transformer turns ratio n the operating region can be moved to any arbitrary range of M and J This follows from the definitions of M and J M L n Vg 499 Here V and the range of Vg are specified but the transformer turns ratio n is not So we can choose n and hence also scale the range of M arbitrarily subject to M S 1 Also w J Vg 4100 The range of I and Vg are specified and n is given by the choice of M from Eq 499 above But we can still choose the tank characteristic impedance R0 arbitrarily and hence we can also scale the range of J arbitrarily So selection of an operating region or M and J is equivalent to choosing n and R0 Let us therefore map Eqs 497 and 498 into the normalized output plane for some given values of n and R0 Equations 497 and 499 imply that as the input voltage varies from its maximum to minimum values the voltage conversion ratio will also vary from Mmin to Mmax where M V mm H Mmax V 4101 n ngin The output power can be written PilV JVgV7JV2 4102 n R0 M R0 The expression on the right of Eq 4102 is the most useful for design because Vg has been eliminated in favor of the constant specified value of V Solution for J yields JM n2 R0 P 4103 v2 Chapter 4 The Series Resonant Converter For a given value of power P J is proportional to M if the input voltage varies causing M to vary as given by Eq 499 then J will also vary as given by Eq 4103 The peak J is given at maximum power and minimum input voltage n Pmax R0 J 4404 max V while the minimum J occurs at minimum power and maximum input voltage 4105 n Pmin R0 ngax V J min The region de ned by the speci cations can be plotted using Eqs 4 101 and 4103 4105 as in Fig 448 This region can be overlaid on the J converter output characteristics to see how max switching frequency and component stresses must vary as the operating point changes For the k0 CCM example shown the design Mmax Jmax 08 4 has been selected Given values for Pmax Pmin ngax and ngin the values of Mmin and J Jmin can be evaluated and the operating region is mm constructed It can be seen that the maximum switching frequency will occur at the point Mmin Jmin ie at maximum input voltage ngX and Fig39 43948 Valmge regulator Operating region plotted in the output plane m1n1mum power Pmin The component stresses are highest at maximum power Pmax 600W design example The preceding arguments are next applied to the example of a 500W offline lullbridge dc dc converter The converter speci cations are input voltage range 255 S Vg S 373 output power range 60 S Pout S 600 maximum switching frequency fsmax 1 MHz regulated output voltage V 24 Designs operating below resonance for various choices of Mmax and Jmax are compared in Table 41 Given the speci cations and the choice of Mmax and Jmax values of Mmin and Jmin can be computed from Eqs 4101 4 104 and 4105 The converter operating region can then be superimposed on the converter characteristics as in Fig 448 and the normalized peak stresses 45 Principles of Resonant Power Conversion and switching frequency variations can be determined graphically Altematively the exact equations can be evaluated to nd these quantities The tank resonant frequency f0 is then chosen such that the maximum switching frequency which occurs at the point Mmax Jmax for below resonance operation coincides with 1 MHz The required transformer tums ratio n is found by evaluation of Eq 4101 and the required tank characteristic impedance R0 can then be found using Eq 4104 Values of L C ILp and ch can then be evaluated below resonance CP Table 41 of nF AV All of the designs of Table 41 operate in the k2 discontinuous conduction mode at light load design C operates in DCM even at full load It is assumed that the transistors are allowed to conduct only once per halfswitchingperiod and hence the converter operates in k2 DCM for all switching frequencies below 05 f0 and for all values of J below 2TE To ensure that the rated output voltage can be obtained when semiconductor forward voltage drops and other losses are accounted for the maximum allowable value of M is taken to be 09 rather than 10 These designs all operate with zero current switching Point A Mmax Jmax 09 50 exhibits the lowest peak current 42A referred to the transformer primary However the peak tank capacitor voltage is very high 2080V referred to the primary side This voltage can be reduced by reducing the choice of Jmax for example to 15 as in point B This has little effect on the peak current but the peak capacitor voltage is reduced to 710V Further reductions in Jmax further reduce the peak capacitor voltage to some extent however the peak tank current is increased as the discontinuous conduction mode boundary is approached At point C where the converter operates in DCM for all load currents the peak tank current is approximately twice that of points A and B Point D illustrates the effect of a suboptimal choice of Mmax 045 rather than 09 this also approximately doubles the peak current Several designs for the same specifications are illustrated in Table 42 for the above resonance case No discontinuous conduction mode occurs in this case The design procedure is similar to the below resonance case except that the maximum switching frequency now occurs at minimum output power and maximum input voltage ie at the point Mmin Jmin These designs operate with zero voltage switching Chapter 4 The Series Resonant Converter Table 42 of above resonance nF A V Point A is again at Mmax 09 Jmax 50 The design is not signi cantly different from the corresponding design below resonance The peak current is again approximately 4A and the peak tank capacitor voltage is again very high approximately 2kV Reducing Jmax has the favorable effect of reducing this voltage In fact since there is no discontinuous conduction mode ch can be made arbitrarily small by reducing Jmax a sufficient amount The peak tank inductor current is slightly smaller than in the belowresonance case The problem with doing so is the large range of switching frequency variations For example point D illustrates a design with a peak capacitor voltage of 50 volts and peak current of 39A the problem with this design is the minimum switching frequency of 45kHz This design will require quite large magnetics A compromise is point C for which the peak capacitor voltage is 350V with low peak inductor current of 39A and a minimum switching frequency of l57kHz Point E illustrates again the effect of a suboptimal choice of Mmax this leads to increased peak tank current It is apparent that for operation above resonance large switching frequency variations can occur This is undesirable because it requires that the transformer and filter components be sized to the minimum switching frequency which can be quite low The benefits of high frequency operation are then lost Several control schemes have been described in the literature which circumvent this problem Constant frequency operation can be obtained by duty cycle 7 or phase control 1315 of the transistor bridge Large switching frequency variations can also occur below resonance but these do not lead to a large transformer The reason for this is the occurence of the k2 DCM at all switching frequencies below 05 f0 In this mode no voltage is applied to the transformer during the discontinuous X subintervals The transformer can therefore be designed as if its minimum frequency is 05 f0 and the low values of fsmin in Table 41 are not a problem Filter size is also not adversely affected by the wide range of switching frequency variations in this case because the low frequency operating points coincide with the low output current points where less ltering is needed The output capacitor value is therefore determined by the maximum power point which occurs at high frequency Principles of Resonant Power Conversion REFERENCE S FC Schwarz An Improved Method of Resonant Current Pulse Modulation for Power Converters IEEE PowerElectronics Specialists Conference 1975 Record pp 194204 June 1975 R King and T Stuart A Normalized Model for the Half Bridge Series Resonant Convener IEEE Transactions on Aerospace and Electronic Systems March 1981 pp 180193 V Vorperian and S Cuk A Complete DC Analysis of the Series Resonant Converter IEEE Power Electronics Specialists Conference 1982 Record pp 85100 June 1982 R King and TA Stuart Inherent Overload Protection for the Series Resonant Converter IEEE Transactions on Aerospace and Electronic Systems vol AES19 no 6 pp 820830 Nov 1983 R Oruganti and FC Lee Resonant Power Processors Part 1 State Plane Analysis IEEE Transactions on Industry Application vol 1A21 NovDec 1985 pp 14531460 A Witulski and R Erickson SteadyState Analysis of the Series Resonant Converter IEEE Transactions on Aerospace and Electronic Systems vol AES21 no 6 pp 791799 Nov 1985 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 IEEE Publication 87CH24596 RL Steigerwald High Frequency Resonant Transistor DcDc Converters IEEE Transactions on Industrial Electronics vol 1E31 no 2 pp 181191 May 1984 S Singer LossFree Gyrator Realization IEEE Transactions on Circuits and Systems no 2 pp 2634 January 1988 vol CAS28 Chambers 15kHz 35kW high voltage converter using 15 SCR s Y Cheron H Foch and J Salesses Study of a Resonant Converter Using Power Transistors in a 25kW Xray Tube Power Supply IEEE Power Electronics Specialists Conference Proceedings ESA Sessions pp 295306 June 1985 A Witulski and R Erickson Design of the Series Resonant Converter for Minimum Component Stress IEEE Transactions on Aerospace and Electronic Systems July 1986 SRC phase control reference Vandelac PESC 87 FS Tsai P Materu and FC Lee Constant Frequency Clamped Mode Resonant Converters IEEE PowerElectronics Specialists Conference 1987 Record pp 557566 June 1987 SRC phase control ref GE scheme high voltage converter PESC 90 KDT Ngo Analysis of a Series Resonant Converter PulsewidthModulated of CurrentControlled for Low Switching Loss IEEE Power Electronics Specialists Conference 1987 Record pp 527536 June 1987 Chapter 4 The Series Resonant Converter PROBLEMS 1 Using the phase plane derive the k0 continuous conduction mode characteristics Sketch your results in the output plane M vs I and control plane M vs F for resistive load 2 Analyze the k3 discontinuous mode a Draw the phase plane diagram b Draw waveforms for the inductor current capacitor voltage and inductor voltage39 c Use tank capacitor charge arguments to relate the normalized load current to normalized capacitor voltage boundary values d Solve for the output characteristics e Determine the complete set of conditions on normalized switching period and load current which guarantee operation in this mo e 3 It is desired to obtain a converter with current source characteristics Hence a series resonant converter is designed for operation in the k2 discontinuous mode The switching frequency is chosen to be fS 0225 f0 where fO is the tank resonant frequency consider only openloop operation The load R is a linear resistance whose value can change to any positive value a Plot the output characteristics M vs I for all values of R in the range 0ltgtltgt Label mode boundaries evaluate the shortcircuit current and give analytical expressions for the output characteristics b Over what range of R referred to the tank characteristic impedance R0 does the converter operate as intended in the k2 discontinuous mode 4 Derive the equations for the peak tank stresses in the k0 continuous conduction mode Eqs 495 and 496 5 Design of a high density series resonant converter Design a halfbridge series resonant converter as shown above to meet the following specifications Input voltage 1347176 volts Output voltage Vow 48 volts Output current I 171 0 amperes Maximum switching frequency 750 kHz Output voltage n39pple no greater than 1 volt peaktopeak Principles of Resonant Power Conversion Design the best converter that you can which combines high efficiency with small volume Use your engineering judgement to select the operating mode tank elements L and C and transformer turns ratio to attain what you consider to be the best combination of small transformer size small CF high minimum switching frequency low peak tank capacitor voltage and low peak transistor current The volume of a 50V lpF X7R ceramic chip capacitor is approximately 50 mm3 Capacitor volume scales as the product of capacitance and voltage rating so that the volume of a 100V 511E capacitor of the same dielectric material is 500 mm3 Capacitors are available with voltage ratings of 50V 75V 100V 200V 300V 400V and 500V and essentially any capacitance value You must choose a capacitor with a voltage rating at least 25 greater than the actual peak voltage applied by your design Use a ferrite EE core for the transformer Estimate the transformer size using the Kg method allowing a of 01T The core fill factor Ku of 05 total copper loss Pcu no greater than 05W and peak flux density Bmax geometrical constant Kg of the centertapped transformer is defined as follows 22 8 pk111 10 cm5 Kg 2 PCH Ku Bmax where p is the resistivity of copper 172410396Qcm k1 is the applied primary voltseconds and i1 is the applied primary rms current For this problem you may neglect core loss This is not valid in general especially for 750kHz 39 39 39f 39 39 for this problem The estimated volumes of transformers constructed 1 1 but 1s a J in this manner are as follows Core type vol mm3 g cm5 EE12 675 073103 EE16 1125 20103 EE19 1550 41103 EE22 2425 83103 EE30 8575 8610393 EE40 14700 021 EE50 31500 091 EE60 42250 14 Hence attempt to minimize the total volume Vtot of the transformer output filter capacitor and tank capacitor while keeping the peak currents reasonably low You may neglect the size of all other components Specify 1 your choices for L C and transformer turns ratio 2 the range of M J and fS over which your design will operate 3 the transformer core size required don t bother to compute number of turns or wire size 4 the value of CF required and 5 the total volume Vtot as defined above