A generalized steady-state analysis of resonant converters using two-port model and Fourier-series...

142 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 13, NO. 1, JANUARY 1998

A Generalized Steady-State Analysis ofResonant Converters Using Two-PortModel and Fourier-Series Approach

Ashoka K. S. Bhat,Fellow, IEEE

Abstract—A generalized steady-state analysis of resonant con-verters using a two-port model and Fourier-series approach ispresented. Analysis is presented for both voltage-source (VS) andcurrent-source (CS)-type loads. Analysis can also be used eitherfor variable-frequency (half- or full-bridge) or fixed-frequency(phase-shift control for full-bridge) operation. Steady-state so-lutions have been obtained. Particular cases are considered toshow the method of application in analyzing different schemes.A simple design procedure is given for two particular cases toillustrate the use of analysis in obtaining design curves and indesigning the converters. Experimental results obtained froma MOSFET-based 500-W fixed-frequency LCL-type resonantconverter are presented to verify the analysis.

Index Terms—Fixed frequency, Fourier series, generalizedanalysis, resonant converters, two-port model, variablefrequency.

I. INTRODUCTION

DURING the last decade, many authors have presented[1]–[15] analyses and designs of different resonant con-

verter configurations. This increased interest in the area ofresonant converters is due to the availability of high-frequency(HF) switches and HF magnetics, coupled with the advan-tages of resonant switching techniques. Analysis of resonantconverters is more difficult compared to the pulse-width-modulation (PWM) converters. This is due to the nonlinearloading [viz. voltage- or current-source (CS)-type loading]and due to a number of operating modes entered by theseconverters. Basically, three analysis methods are available foranalyzing resonant converters in the steady state.

1) Approximate analysis using complex ac-circuit analysis[1]–[4]: in this method, fundamental components of thewaveforms are used for voltages and currents. Thismethod cannot predict the various voltage and currentwaveforms appropriately, and the accuracy reduces asthe switching frequency is away from resonance fre-quency (in variable-frequency-controlled converters) oras the duty ratio decreases (in fixed-frequency-controlledconverters).

Manuscript received April 16, 1996; revised May 6, 1997. This workwas supported in part by a grant from the Natural Science and EngineeringResearch Council of Canada. Recommended by Associate Editor, L. Xu.

The author is with the Department of Electrical and Computer Engineering,University of Victoria, Victoria, B.C., V8W 3P6, Canada.

Publisher Item Identifier S 0885-8993(98)00482-7.

2) State-space or differential equations approach (e.g., see5): although this method is accurate, analysis is verycumbersome and usually very difficult (especially forhigher order converters) to use.

3) Fourier-series method or frequency domain approach[7]–[14]: in this method, all the predominant harmonicsare taken into account, and classical ac-circuit analysistechniques are used to analyze the converter. Therefore,this method is simple and gives good results.

There are a large number of resonant converter configu-rations available for the power electronics design engineer.Selection of a particular configuration depends on the par-ticular application and requirements. A proper selection ofthe configuration for given constraints is possible only ifall the configurations are analyzed and then compared. Thiscan be simplified using a generalized analysis approach. Thefirst attempt to give a generalized approach for resonantinverters using the Fourier-series approach was presentedin [16]. However, in the case of resonant inverters, loadpresented is usually the R-L type, and this simplifies theanalysis. In the case of resonant converters, load presentedis either the CS or voltage-source (VS) type [1]–[5]. A simplegeneralized analysis of resonant converters using complex ac-circuit analysis [1] was presented in [3] using fundamentalwaveforms approximation. Disadvantages of this method arealready explained above.

The Fourier-series approach using the superposition princi-ple has been used to analyze parallel resonant [8], [9], seriesresonant [10]–[12], (LC)(L)-type [14], and double-tuned [13]resonant converters. In this paper, a generalized steady-stateanalysis of resonant converters is presented using a two-port model [2], [17] and the Fourier-series approach. Thismethod is novel and simple and can be used to analyze allthe configurations reported in the literature and also to analyzenew possible configurations [3], [15] without analyzing eachconfiguration separately. Some of the features of the proposedmethod are as follows.

1) Results for different resonant converters can be simplyobtained by opening or shorting particular elements inthe “generalized tank circuit” and eliminates the needto analyze each converter separately. In a computerprogram written using the generalized equations derivedin this paper, each configuration can be treated as aparticular case of the generalized scheme [3], [16].

0885–8993/98$10.00 1998 IEEE

BHAT: GENERALIZED STEADY-STATE ANALYSIS OF RESONANT CONVERTERS 143

Fig. 1. Basic circuit diagram of an HF resonant converter employing a generalized tank circuit. Parasitics of the HF transformer are taken care of inthe tank circuit. Load can be VS or CS types.

2) The method presented can be used for resonant con-verters employing either variable-frequency control orfixed-frequency (phase-shift) control.

3) Both types of loads (VS or CS) can be modeled easily.4) This method gives accurate results by considering first

few harmonics and, therefore, it gives results very fast.5) Parasitics of the HF transformer and losses in the tank

circuit can be included in the analysis.6) All the design curves required to design and optimize a

resonant converter can be obtained easily.

The layout of this paper is summarized below. SectionII briefly reviews operation and modeling of the resonantconverters for both variable-frequency and fixed-frequencycontrol. The generalized analysis of resonant converters ispresented using the two-port model and Fourier-series ap-proach in Section III. An analysis is presented for VS- aswell as CS-type loads and is focused for lagging powerfactor (pf) (above resonance) mode of operation due to itsadvantages compared to leading pf (below resonance) modeof operation [1]. However, analysis is equally valid for lead-ing pf mode. Particular cases are presented in Section IVto illustrate the application of the generalized analysis tosome resonant converter configurations. A design method ispresented together with design examples in Section V for(LC)(LC)-type and LCL-type resonant converters to prove theusefulness of the analysis presented. Theoretically predictedwaveforms for the designed converters are also presentedin this section and verified using SPICE simulation for the(LC)(LC)-type converter. Experimental results obtained froma fixed-frequency LCL-type experimental converter rated at500-W output are presented in Section VI to verify the theory.

II. OPERATING PRINCIPLE AND MODELING

Fig. 1 shows the basic circuit diagram of the full-bridgeversion of the HF resonant converter employing a “generalizedtank circuit” (Fig. 2). Power control is achieved by usingeither variable-frequency control [1]–[5] or fixed frequency

Fig. 2. Generalized tank circuit that can be used in Fig. 1.

with phase-shift control [6], [13], [14]. For variable-frequencyoperation, will be a square wave of amplitude . Inthe case of fixed-frequency (phase-shift) control, switchingfrequency is kept constant while the power control is obtainedby changing the phase shift angle between the gatingsignals to vary the pulse width of waveform . Itmust be noted that the HF transformer shown is ideal with itsparasitics absorbed in the “generalized tank circuit” (Fig. 2).The voltage (current) waveform at the input of the rectifieris a square-wave voltage (for VS load) [10]–[12], [14] or asquare-wave current (for CS load) [8], [9]. The models canalso be used for half-bridge converters [1]–[5] operating withvariable-frequency control noting that the square-wave voltageapplied to the tank circuit has an amplitude of 2, where

is the supply voltage.The voltage waveform across and the output square-

wave voltage (or current) can be represented by their Fourier-series. The phasor-equivalent circuit model at the output ofthe inverter of Fig. 1 for the th harmonic is shown in Fig. 3.The two-port model for the two cases for theth harmonic isshown in Fig. 4. The parameters and all the relevant equationsare derived in the next section.

III. STEADY-STATE ANALYSIS

The converter shown in Fig. 1 is analyzed using the Fourier-series approach for theth harmonic phasor-equivalent circuit


Fig. 3. Phasor-equivalent circuit for thenth harmonic for the generalizedtank circuit shown in Fig. 2.

(a) (b)

Fig. 4. Two-port models used for (a) VS-type load and (b) CS-type load.

model shown in Fig. 3. Fig. 3 can be represented by two-port models as shown in Fig. 4. The superposition principle isapplied to Fig. 3 (and Fig. 4) to analyze the converter. Fig. 5illustrates this for a VS-type load. In Fig. 5(a), voltage source

is shorted, whereas is shorted in Fig. 5(b). Variousvoltages and currents for theth harmonic are obtained usingthe superposition theorem to Fig. 5. Then, the effect of allthe harmonics is obtained using them in the Fourier-seriesexpressions. All the solutions are given in closed form (exceptfor , the phase angle between the voltages and ),allowing the designer to design the converters easily.

A. Assumptions Used in the Analysis

The following assumptions are made in the analysis pre-sented.

1) All the switches and diodes are ideal.2) The effect of snubber capacitors is neglected.3) Load voltage and/or load current is assumed to be

constant so that the constant VS model (for capacitiveoutput filter) or constant CS model (for inductive outputfilter) can be used.

B. Base Values

All the equations presented in this paper are normalizedusing the following base quantities:

(1)

The base value chosen for is for converters having aseries (LC) circuit as part of its tank (e.g., Figs. 6 and 7).isdifferent for other types of converters. However, the equationsderived in the generalized form will not change—only variousimpedances in per-unit (p.u.) form will be different.

C. Notations Used

All the th harmonic quantities are represented by anadditional subscript “,” and the normalized quantities aredenoted by an extra subscript “ ” The normalized reactances

Fig. 5. Superposition principle applied for VS-type load model shown inFig. 4(a).

and impedances (neglecting losses) for theth harmonic (e.g.,for Fig. 6) are

(2)

and the ratio of switching frequency to series resonancefrequency is

(3)

All the normalized instantaneous voltages and currentsare represented by and , respectively. Converter gain

and normalized load current. Phasors are represented by bold italic letters,

e.g., represents the inverter output phasor current for theth harmonic.

D. Case I: VS-Type Load

1) Two-Port Model for VS-Type Load:The two-port model[17] (for the th harmonic) for the generalized tank circuitwith VS-type load [Fig. 4(a)] using parameters (short-circuitadmittance parameters) is

and

(4)

where

(4a)

The square-wave voltages across (for fixed-frequencyoperation) and the output (: instantaneous voltage at theinput of the rectifier bridge referred to the primary side)are represented by the following normalized Fourier-seriesexpressions (note: is the phase angle between the voltages

and , and is the pulse width of when fixed-frequency control is used):

p.u. (5)


(a) (b) (c)

Fig. 6. (a) Equivalent circuit of a fourth-order resonant converter employing the (LC)(LC)-type tank circuit. (b) Phasor-equivalent circuit of (a)for thenthharmonic. (c) Simplified phasor model of (b) for comparison with the generalized tank circuit.

(a) (b)

Fig. 7. (a) Equivalent circuit of a third-order resonant converter employing the (LC)(L)-type tank circuit. (b) Phasor-equivalent circuit of (a) forthe nth harmonic.

p.u. (6)

The th harmonic components of and can berepresented in phasor form as given in

(7)

(8)

For variable-frequency operation, in the above expres-sions.

2) Expressions for Voltages and Currents:The normalizedinstantaneous current at the output of the inverter is givenby [using (4)–(6)]

(9)

p.u. (10)

where

and(10a)

(10b)

(10c)

The normalized output (i.e., rectifier input) current reflectedto the primary side can be written as [using (4)–(6)]

p.u.

(11)

Since the power is delivered to the load at the output, theactual polarity of is opposite [refer to Figs. 4(a) and 5]to that of (11). Similarly, expressions for the voltages andcurrents across other elements can be derived. These equationsare used to obtain the voltage and current stresses on differentcomponents. They are also used to calculate the kilovoltampererating of the tank circuit.

In order to evaluate the above equations, the value of(phase angle between the voltages and ) must beknown. At , the voltage (also rectifier input current)changes polarity. Therefore, the value ofis obtained bysolving [using (11) with ]

(12)

The phase angle between the fundamental componentsof the voltage sources and is given by

(13)


(a) (b)

Fig. 8. (a) Equivalent circuit of a third-order resonant converter employingan (LC)(C)-type tank circuit with CS-type load. (b) Phasor-equivalent circuitof (a) for thenth harmonic.

Using as the initial guess value, (12) has to be solvednumerically (e.g., using the Newton–Raphson technique) forthe value of . The normalized load current is given by

p.u.

(14)

Therefore, for each value of (converter gain), can beevaluated to get a family of curves of versus in theoutput plane.

E. Case II: CS-Type Load

1) Two-Port Model for CS-Type Load:The two-port model[17] (for the th harmonic) for the generalized tank circuit withCS-type load [Fig. 4(b)] using (inverse-hybrid) parameters is

and

(15)

where

(15a)The phasor expression for is the same as (7), and the

normalized th harmonic component of in phasor form is

p.u. (16)

2) Expressions for Voltages and Currents:The normalizedinstantaneous current at the output of the inverter is givenby (10), and (9) and (10a)–(10c) are also valid for CSload with replaced by normalized load current andparameters replaced byparameters. The normalized value ofinstantaneous output current is given by

p.u. (17)

Since power is delivered to the load, has opposite signto that of (17). The equation for the instantaneous voltage atthe output (i.e., at the rectifier input) is given by

p.u.

(18)

In order to evaluate the above equations, the value of(phase angle between the voltage and current ) must beknown. The value of is obtained by equating (18) to zero(since becomes zero when changes polarity) at .The phase angle between the fundamental components ofsources and is given by

(19)

Using as the initial guess value, (18) has to be solvednumerically for the value of .

Similarly, expressions for the voltage and current stressesacross other elements can be derived. The normalized con-verter gain is given by

p.u.

(20)

Therefore, for each value of, can be evaluated to get afamily of curves of versus in the output plane.

Comparing the equations for VS- and CS-type loads, it canbe observed that all the equations of the second case can beobtained by replacing by (or by ) and parametersby parameters in the equations of the first case.

IV. EXAMPLES (PARTICULAR CASES)

Four particular cases are presented in this section.

A. Example 1, (LC)(LC)-Type SeriesResonant Converter, VS Load

Fig. 6 shows the scheme. In this case,and . Then, using (4a)

(21)


(a) (b) (c)

Fig. 9. Typical design curves obtained for (LC)(LC)-type resonant converter: (a) normalized load current(J), (b) inverter rms output current (A) for the500-W converter designed, and (c) total kilovoltampere rating of tank circuit per kilowatt of output power versus converter gain(M). Here, � = �;Ls=Lp = 0:1; and Cs=Cp = 2.

Using the analysis of Section III-D, all the equations as aparticular case can be obtained. For variable-frequency control,

. For example, (9) and (12)–(14) become (22)–(25)

p.u. (22)

This inverter output current flows through the switches,,and

(23)

(24)

p.u. (25)

The normalized instantaneous current throughis given by

p.u.

(26)

B. Example 2, (LC)(L)-Type Series ResonantConverter, VS Load

Fig. 7 shows the scheme [15]. In this case, all the equationsderived in Example 1 are valid (taking into account) with

. It can be easily verified that all theequations derived in [15] for fixed-frequency operation areobtained.

C. Example 3, (LC)(C)-Type ParallelResonant Converter, CS Load

This is a standard configuration (Fig. 8) [1]–[3] and usingthe equations of Section III-E and noting that and

, all the relevant equations for the designcan be obtained. Using the same notations as in Case I, with

, equations given for CS load can berewritten as follows:

(27)

p.u. (28)

(29)

p.u. (30)

D. Example 4, Parallel Resonant Converter

This type of converter has been analyzed using Fourierseries in [8] and [9]. It can be verified that all the equationsgiven in [8] and [9] can be obtained as a particular case givenfor CS-type load in Section III.


(a) (b)

Fig. 10. (a) Theoretically predicted waveforms for the (LC)(LC)-type converter designed in Section V, operation at full load. (b) Corresponding waveformsobtained from SPICE simulation. All the waveforms are referred to the primary side of the HF transformer.

V. DESIGN AND RESULTS

To illustrate the use of analysis presented in designing theresonant converters, two particular cases given in Section IV,are designed in this section. Design procedure is illustratedwith design examples.

A. Example 1, (LC)(LC)-Type SeriesResonant Converter, VS Load

An (LC)(LC)-type resonant converter with VS-type load(Fig. 6) is designed in this section for variable-frequency oper-ation. The specifications of the half-bridge converter designedare:

1) input supply voltage V;2) maximum output power W;3) output load voltage V;4) switching frequency at full load kHz.

Using the analysis presented in Section III, design curvesfor the normalized load current, inverter output rms current(for the 500-W converter), and kilovoltampere rating of tankcircuit per kilowatt of output power, with variation in convertergain for various (ratio of switching frequency to seriesresonant frequency), are plotted in Fig. 9 for and

. Such design curves for other ratios have also beenobtained, but the above ratios were almost optimum (requiringless variation in switching frequency for power control, lowerkilovoltampere rating of tank circuit and inverter rms output

current, and operation in lagging pf mode) for the designexample. From Fig. 9, for the minimum kilovoltampere ratingof tank circuit and minimum inverter rms output current, thefollowing values are chosen in the design: p.u.,

p.u., and .Therefore, V (since

for half bridge) and transformer turns ratio for 48-V outputis . The values of and are calculated using

(31)

The solution of the above equations gives Hand F. Since and ,

H and F.The rms currents through and , rms voltages across

and are A, A,V, and V.

Using the design values and equations for this particularcase, various waveforms obtained at full load are shown inFig. 10(a). The first 33 harmonics (i.e., ) have beenused in the calculations (MATHCAD was used to evaluatethe equations). The SPICE results obtained for the sameconverter are shown in Fig. 10(b). It can be observed thatthese waveforms are in close agreement with those obtainedfrom theory.


(a) (b)

Fig. 11. Fixed-frequency LCL-type converter designed in Section V. Theoretically predicted waveforms at full load with (a)Emin

= 110 V. (b)Emax = 138

V. All the waveforms are referred to the primary side of the HF transformer.

B. Example 2, (LC)(L)-Type Series ResonantConverter, VS Load

This scheme, shown in Fig. 7, is to be designed with thefollowing specifications for fixed-frequency operation (full-bridge configuration):

minimum input supply voltage V;maximum input supply voltage V;maximum output power W;output load voltage V;switching frequency kHz.

Resonant component values are obtained for the worstcase, minimum supply voltage, and maximum load current,with . Similar to the earlier example, various designcurves have been obtained. Appropriate selection of inductorratio and (normalized load current) results in aminimum kilovoltampere rating of tank circuit per kilowattof output power and a minimum peak inverter output currentat full load. In the meantime, a proper choice of(ratio ofswitching frequency to series resonance frequency) also resultsin reduced peak currents at light loads while maintaininglagging pf mode of operation. The following values were foundto be near optimum for the design example:

Load voltage reflected to the primary side is V.Therefore, transformer ratio required to obtain 120-V outputis . The component values calculated for

this design example are

H F

Since

H

The component ratings for V are

A VA A

Fig. 11(a) shows theoretically predicted waveforms formaximum output power ( W) withV and . Operation of the converter at maximumsupply voltage ( V) results in higher stresson the components. Normalized load current for maximumoutput power condition with input supply voltage is

p.u. Therefore, pulse widthwas varied to obtainwhile the load voltage is kept at V

(i.e., p.u. with V). Correspondingtheoretical waveforms obtained are shown in Fig. 11(b).Component ratings for this condition are

A VA A

Therefore, the component ratings are selected based onthe condition. In Fig. 11, the first 53 harmonics (i.e.,

) have been used in the calculations.

VI. EXPERIMENTAL RESULTS

Typical waveforms obtained from an experimental fixed-frequency (LC)(L)-type converter [operating above resonance


(a)

(b)

(c)

Fig. 12. Experimentally obtained waveforms for the fixed-frequency LCL-type converter. With minimum supply voltageEmin

= 110 V. (a) RL = 30

and (b)RL = 60 . (c) With maximum supply voltageEmax = 138 V and RL = 30 . Scale:vAB (90 V/div.) in all the waveforms and inverteroutput currentiLs [4 A/div. in (a) and (b) and 10 A/div. in (c)];vCs (36 V/div.); v

rect:in (90 V/div.); current through parallel inductorLt; iLt [2 A/div.in (a) and 1 A/div. in (b) and (c)]. [Details of the converter: switches used—IXYS Corporation MOSFET’s, IXFH 19N50, feedback diodes—internal toMOSFET’s,Ls = Lr + Ll; Lr = 14:5 �H, Cs = 0:1805 �F, Lt (on secondary side)= 230 �H, rectifier-bridge diodes—MR1386, HF transformer turnsratio, 1:nt = 12:14 having leakage inductancesLlp = L0

ls= Ll=2 = 1:5 �H, and magnetizing inductanceLm = 1000 �H.]


or zero-voltage switching (ZVS) mode] are shown in Fig. 12.Operation of the converter with minimum supply voltage underfull- and half-load conditions are shown in Fig. 12(a) and (b),respectively. Operation of the same converter under full loadand maximum supply voltage conditions is demonstrated inFig. 12(c). These waveforms were stored in an HP54504Adigital oscilloscope and plotted on an HP7440A plotter. Theconverter is delivering a load voltage of approximately 115V. The converter is operating at a fixed frequency of 100kHz in all the waveforms, and the load voltage was heldconstant by varying the phase shift of the gating pulses(which changes the pulse width of ). The details of theconverter used are given in Fig. 12. Internal diodes of theMOSFET’s and capacitive snubbers are used. Leakage andmagnetizing inductances of the HF transformer have beenused as part of series and parallel inductancesand ,respectively. Comparison of the experimental results with theresults calculated from the analysis showed good agreement.It can be easily observed that the peak inverter output currentdecreases with the load current, and this converter maintainsZVS for a wide variation in the load as well as supply voltage.With V, the peak inverter output current decreasedfrom about 6 A at full load to 1.8 A at 10% load. Thecorresponding variation was approximately 8.9–2.8 A with

V.

VII. CONCLUSION

A generalized steady-state analysis of resonant convertersusing a two-port model and Fourier-series approach has beenpresented. Use of the superposition principle and the Fourier-series analysis to the two-port representation of the generalizedtank circuit model for the th harmonic (with VS- or CS-typeload) have been used to derive the steady-state solutions of theresonant converter with a generalized tank circuit. Expressionsfor normalized load current (or the converter gain) have beenderived. Voltage and current stresses for different tank circuitcomponents are obtained easily. The analysis presented isapplicable to variable-frequency as well as fixed-frequencycontrol methods. It has been shown that the equations derivedfor VS-type load can also be used for CS-type load with someminor changes ( and interchanged and parameters re-placed by parameters). The generalized equations have beenused to obtain the equations of particular cases by shorting oropening the impedances of the generalized scheme. Designexamples of (LC)(LC)-type and LCL-type converters havebeen presented in detail with the theoretical results. Theoreticalresults have been verified for the designed converters obtainedfrom SPICE simulation or experimental results.

The number of harmonics required to get accurate resultsdepends on the resonant converter configuration, ratio ofswitching frequency to resonance frequency or pulse width.More harmonics are to be included in the calculations to getsmoother waveforms, especially when the switching frequencyis away from resonance frequency or for reduced pulse widths.It was observed that if computations are done for harmonics upto , reasonably good accuracy and smooth waveformscan be obtained. These computations are very fast and were

implemented in an IBM compatible computer (66-MHz i486DX2 DELL computer) using the MATHCAD program.

REFERENCES

[1] R. L. Steigerwald, “A comparison of half-bridge resonant convertertopologies,” IEEE Trans. Power Electron., vol. 3, no. 2, pp. 174–182,1988.

[2] O. P. Mandhana and R. G. Hoft, “Two port characterization of DCto DC resonant converters,” inIEEE Applied Power Electronics Conf.Rec., 1990, pp. 737–745.

[3] A. K. S. Bhat, “A unified approach for the steady-state analysis ofresonant converters,”IEEE Trans. Ind. Electron., vol. 38, no. 4, pp.251–259, 1991.

[4] M. K. Kazimierczuk and Czarkowski,Resonant Power Converters.New York: Wiley, 1995.

[5] K. K. Sum, Recent Developments in Resonant Power Conversion, In-tertech Communications Inc., CA, 1988.

[6] I. J. Pitel, “Phase-modulated resonant power conversion techniques forhigh-frequency inverters,” inIEEE Industry Applications Conf. Rec.,1985, pp. 1163–1172.

[7] G. Indri, “The calculation of inverters with series and parallel resonantcircuit,” in SPC-PSC Conf. Rec., 1972, pp. 2.3-1–2.3-8.

[8] S. Deb, A. Joshi, and S. R. Doradla, “A novel frequency-domain modelfor a parallel-resonant converter,”IEEE Trans. Power Electron., vol. 3,no. 2, pp. 208–215, 1988.

[9] A. V. Mathew and R. Unnikrishnan, “Simplified steady-state analysisof parallel resonant converters,” inHigh Frequency Power ConversionConf. Proc., May 1988, pp. 319–331.

[10] , “Novel approach to modeling series resonant converters,”Proc.Inst. Elect. Eng.,vol. 136, pt. G, no. 2, pp. 99–104, 1989.

[11] J. Li and Y. Wu, “Closed-form expressions for the frequency-domainmodel of the series resonant converter,”IEEE Trans. Power Electron.,vol. 5, pp. 337–345, July 1990.

[12] P. P. Roy, S. R. Doradla, and S. Deb, “Analysis of the series resonantconverter using a frequency domain model,” inIEEE PESC, 1991, pp.482–489.

[13] P. Jain, D. Bannard, and M. Cardella, “A phase-shift modulated doubletuned resonant DC/DC converter: Analysis and experimental results,”in IEEE APEC, 1992, pp. 90–97.

[14] A. K. S. Bhat, “Frequency domain analysis of a fixed-frequency LCL-type series-resonant converter,” inInt. Power Electronics Motion andControl Conf., Beijing, China, June 27–30, 1994, pp. 354–359.

[15] R. Severns, “Topologies for three element resonant converters,” inIEEEApplied Power Electronics Conf. Rec., Mar. 1990, pp. 712–722.

[16] A. K. S. Bhat and S. B. Dewan, “A generalized approach for the steady-state analysis of resonant inverters,”IEEE Trans. Ind. Applicat., vol. 25,no. 2, pp. 326–338, 1989.

[17] D. E. Johnsonet al., Basic Electric Circuit Analysis. Englewood Cliffs,NJ: Prentice-Hall, 1990.

Ashoka K. S. Bhat (S’82–M’85–SM’87–F’98) re-ceived the B.Sc. degree in physics and math fromMysore University, India, in 1972. He received theB.E. degree in electrical technology and electronicsand the M.E. degree in electrical engineering, bothfrom the Indian Institute of Science, Bangalore, In-dia, in 1975 and 1977, respectively. He also receivedthe M.A.Sc. and Ph.D. degrees in electrical engi-neering from the University of Toronto, Toronto,Ont., Canada, in 1982 and 1985, respectively.

From 1977 to 1981, he worked as a Scientistin the Power Electronics Group of the National Aeronautical Laboratory,Bangalore, and was responsible for the completion of a number of research anddevelopment projects. He was also a Research Scholar at the Indian Instituteof Science from 1980 to 1981. After working as a Post-Doctoral Fellow for ashort time, he joined the Department of Electrical Engineering, University ofVictoria, B.C., Canada, in 1985, where he is currently a Professor of ElectricalEngineering and is engaged in teaching and conducting research in the areaof power electronics. He has been responsible for the development of theElectromechanical Energy Conversion and Power Electronics Laboratories.

Dr. Bhat is a Fellow of the Institution of Electronics and TelecommunicationEngineers (India) and a registered Professional Engineer in the province ofBritish Columbia.

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