A Simple Scheme for Unity Power-Factor

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 5, NO. I , JANUARY 1990

A Simple Scheme for Unity Power-FactorRectification for High FrequencyAC Buses

V A T C H E V O R P E R I A N , M E M B E R , I E E E , A N D R A Y M O N D B. RIDLEY

Abstract-A simple scheme is proposed for off-line unity power fac-tor rectification for high frequency ac buses (20 kHz). In this scheme,a bandpass filter of the series resonant type centered at the line fre-quency is inserted between the line and the full-wave rectified load.The Q = Z, , /R, formed by the load and the characteristic impedanceof the tank circuit determines the power factor, the boundary betweencontinuous and discontinuous conduction modes, the peak stresses andthe transient response of the rectifier. It is shown that for Q > 2 / ~the rectifier operates in continuous conduction mode and the outputvoltage is independent of the load. Also, it is shown that for Q >2 th eline current is nearly sinusoidal with less than five percent third har-monic distortion while the power factor is essentially unity. An in-crease in the value of Q causes an increase in the peak voltages of thetank circuit and a slower transient response of the rectifier circuit. Thedc, small-signal and transient analyses of the rectifier circuit are de-termined and the results are in good agreement with simulation andexperimental results.

I N T R O D U C T I O NHE high frequency ac bus operating at 20 kHz is aT andidate for the power distr ibution system on theU . S . space station and the space platform. Among themany different types of loads co nnected to the bus are dcloads such as computers and other electronic instrumen-tation. T he conven tional full wave rectifier is the schem e

currently proposed for these dc lo ads. Such a rectif icationschem e suffers from po or power facto r and gen erates har-monic currents which are particularly problematic for ahigh frequency ac bus. Although active shaping of the linecurrent can be implemented to improve the power factor[ l ] a n d [2], a much simpler scheme feasible at high fre-quencies is proposed in this paper. The circuit proposedin this paper and its dc analysis have a lso been discussedin [5] and [6]. The proposed circuit is shown in Fig. l(a)and the line voltage and current waveforms are shown inFig. l(c) for a particular design case. I t can be seen thatin this circuit a bandpass filter of the series resonant type,whose cen ter frequency is the same as the line frequency,has been inserted between the line and a conventional full-wave rectifier. The Q =Zo/R, of the filter determines itsbandwidth, BW = uo /Q , which in turn determines theharmonic content of the line current. Hence, a narrower

Manuscript received No vember 28, 1988; revised Septemb er 26, 1989.The authors are with the Bradley Department of Electrical Engineering,Virginia Polytechnic Institute and State University. B lacksbur g, VA 24061.IEEE Log Number 8932855.

4

2

io ?--2 1

4-0 20 40

Time (/,sec)(C)

Fig. 1. (a) Proposed unity power-factor rectifier circuit implemented withdc-to-dc regulated converter. (b) Circuit used for analysis given in text.(c) Input voltage and current waveforms for Q = 2 (also see Fig. 3).

bandwidth results in less harmonic currents which in tumresult in operation closer to unity power factor. A reduc-tion in bandwid th, howev er, is accompa nied by increasedvoltage stresses on the resonant elements and a slowertransient response to load and input voltage variations.The increased voltage stress on the resonant elements isdue to the increase in their imped ance as Q is made larger.An explanation for the slower transient response to lineand load variations as the bandwidth is reduced can begiven by resorting to comm unication circuit theory [3] asfollows. Variations in the input line voltage represent anamplitude modulated (AM) signal that is applied to abandpass filter . Since the response of the bandpass filterto the information content of the AM signal, which re-sides in its envelope, is proportional to its bandwidth, anarrower bandwidth would result in a slower response in0885-8993/90/0100-0077$01OO O 1990 IEEE

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I E E E TRANSACTIONS ON POWER ELECTRONICS. VOL. 5 . NO . I. JANUARY 199078the envelope of the line current and consequently in theoutput voltage. Likewise, load variations generate an AMcurrent signal in the tank circuit to which the response isgoverned by the bandwidth as explained above.In the following sections the dc , small-signal and tran-sient analyses are given. The dc analysis has also beengiven in [5] an d [6], but o ur approach is slightly different.Simple and design-oriented results are obtained in all thecases. Almost all the necessary derivations are given inthe Appendix.

DC A N A L Y S I SThe various steady-state voltage and current waveformsof the circuit in Fig. 1are shoam in F ig . 2. Th e line volt-age is given by

V,, = VP sin wof. (1 )The normalized load Q and the characteristic impedanceZo are given by

- rQ = A ; z ,= J:RI.

Th e resonant frequency is set equal to the line frequencyWO =

Th e rectif ier operating in continuous conduction mo de ( Q1 2 / a ) is analyzed for the following dc characteristics.In continuous conduction mode each pair of diodes con-ducts for the entire half-cycle. If Q is made less than 2 / a ,then the diodes conduct only during the mid portion of thehalf-cycle.

( 3 )1JLOC,'

Rectijication RatioThe rectification ratio is defined as the ratio of the aver-age output voltage to the peak line voltage which for op-eration near unity power factor is given by

a 2VD

diode voltage dropparasitic resistance of the tank circuitesr of the output filter capacitor.The above expression can be approximated as

MR =!! ( 5 )4because the effect of the parasitic elements is usually verysmall at high voltages and low currents.

Fig . 2 . Voltage an d current waveforms of rectifier.

Efic iencyThe rectif ier voltage drops, the parasitic resistances ofthe tank and the output f ilter capacitor contribute to a re-duction in the efficiency which is given by

1n = . ( 6 ),' 1+2-+--+(;-J :a2 o

v o 8 RLLine CurrentFo r Q 2 2 the line current essentially consists of th efundamental which is approximately in phase with the in-put voltage and is given bywhere

zi,=z(') sin wot1" ' =I -" 2

( 7 )

( 8 )T

where Z, is the output current. For the range of 2/a II the effect of the third harmonic becomes noticeableas shown in Fig. 3 and is discussed in the Appendix. Amore accurate expression of the peak inductor currentvalid in the range Q >2 /a s given by

(9)which for Q I 2 approaches I,( a/2)



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VORPERIAN A N D R I D L E Y : U N I TY POWER-FACTOR RECTIFICATION FOR H I G H FR EQ U EN C Y A C 79

402n [p'' )Fig. 3. Line current as function of load parameter Q

Fig. 4. Power factor in continuous conduction mode.Capacitor Voltagevalue is given byThe capacitor voltage is shown in Fig. 2(c) and its peak

Inductor Voltagevalue is given byThe inductor voltage is shown in Fig. 4(d) and its peak

Output Voltage Ripplegiven byThe percent output ripple voltage is approximately

where Fo = wo/2a is the resonant frequency andT L = RLCf. ( 1 3 )

Power Factor Analysisfactor is given byFor continuous conduction mode ( Q 2 2 / a ) t he po w er

( 1 4 )11 + -a4Q2

A plot of the power factor as a function of Q for contin-uous conduction mode is shown in Fig. 4 . Boundary Between Discontinuous and ContinuousConduction Mo des

In order for the rectifier to operate in continuous con-duction mode for a given load RL , the characteristicimpedance should be chosen such that

Comparison of Simulation and P redicted ResultsA 1-kW rectifier operating from a 440-V, 20-kHz acbus was simulated using the simulation program Cosmir

4112fl ( I l V C )( b )Fig. 5 . (a ) Simulated resonant capacitor voltage. (b) Simulated outputripple voltage.

TABLE 1

PREDICTION SIMULATION Q3.2 A 3.18 A 103.36 A 3.27 A 23 .66 A 3 .1 A 21.

vc#.,.i 1535 V 1538 V 2c:, 489 V 490 V 2vr,,,, 5.31 V 5.42 V 2

[4] using the following circuit parameters:RL = 2 3 9 3Cf = 2p F

2Q -, 2, 10.T

The results of the simulation are shown in Figs. 3 and 5.The predictions and simulations are in good agreement asshown in Table I.



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80 I E E E T RANS ACT I ONS ON POWER ELECTRONICS, VOL. 5 . NO . I. J A N U A R Y 1990

SMA LL- SI G N A LN A LY SI SThe only relevant small-signal response is the outputimpedance from which the transient load and line re-sponses can be calculated. Also, the output impedance isimportant in the design of a regulated d c load which maybe connected t o the output of the rectif ier.

Output ImpedanceThe impedance 2; looking into the full-wave bridgeshown in Fig. l(b) is the equivalent lowpass version ofthe narrow-band impedance Z N B looking into the tank cir-cuit scaled by an appro priate constant. An explanation isgiven here by resorting to communication circuit theory

[3 ] while the derivation is given in the Appendix. A mod-ulation in the average rectified current i, ( t ) auses a mod-ulation in the envelope of the line current which in turnbehaves as an AM signal passing through the narrow-bandtank circuit. Since the envelop e of the line current is givenapproximately by the peaks in the line current, the infor-mation in the envelop e is related to the informa tion in theaverage rectif ied current by a simple scaling constantgiven by (9). Now, the information in this AM current isonly affected by the beha vior of the tank im pedance in thevicinity of wo . Hence, the impedance 2; seen by i, ( t )must behave in the vicinity of w 2: 0, or in the base-bandregion, similar to the way ZN B ehaves around w =wo asshown in Fig. 6 . This imp edance is nothing more than thedown-shifted version of ZN B scaled by a constant and isgiven by

(16)7T 22; = (r,, +s 2 L 0 )- r , + sL ,l r 2 7T 2

8where

( 1 7 ), =- o , L, =- o.8 4The equivalent average circuit model of the rectifier cir-cuit can now be obtained as shown in Fig. 7from whichall the small-signal and transient responses can be ob-tained. In this model, the line voltage and any perturba-tions in its peak value are accounted for by their corre-sponding ave rage rectif ied value Vp~ / 4 .he response ofthe output voltage to perturbations in the load and line can

Fig. 6. (a) The narrowband impedance Z N B .Output lowpass impedanceZ:, as shown in Fig. l(b.

Fig.7 . Average equivalent circuit of proposed rectifier circuit.Ic:r -92log 0,

Fig. 8. Output impedance of rectifier circuit as shown in Fig. l(b.

where

be easily determined from this circuit. Also, the respon seof the envelope of the line current can be determ ined fromfied current i, ( t ) and scaling it by a constant factor ofapproximately 7r/2 as given by (9). Hence, all the nec-the circuit in Fig. 7 as will be discussed in the next sec-t ion.

( 2 1his circuit by studying the response in the average recti-

essary responses of the rectifier can be determined from

wf =

WfLO 7T 24 = re +RL(2 2The output impedance 2, is given by A bode plot of the output impedance is shown in Fig. 8 .

20 = (re +SL,) II(rc, + l/SCf)l( RL

20 = r e p L

( 1 8 )which gives TRAN SIENTNALYSISIn this section the rectif ier in Fig. l(b) is analyzed for

transient load and lin e responses using the average equiv-alent circuit model of F ig . 7. The only purpose of thissection is to prove the validity of the average circuit

( +s/szi ) ( +s / s z 2 ) (19)s2.$ s 21 + - + ,Wf Wf



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82 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 5, NO. I , JANUARY 1990m

.m -am c -(ccr.)

"

Fig. 9. Simulated response of output voltage to step line change for threedifferent values of Q. Fig. 10. Predicted response of output voltage to step line change (25) asobtained from the average equivalent circuit model of Fig. 7.

Comparison of Predicted and Simulated ResultsA 1 kW rectifier operating from a 440 V , 20 kHz acbus was simulated on the simulation program Cosmir[4] for transient analysis. The simulated response of theoutput voltage fo r three different values Q for a 2 0 percentstep increase in the line voltage ar e show n in Fig. 9 whilethe predicted response using (25) are shown in Fig. 10.The predicted respon se of the input line current using (31)and the simulated response are shown in Fig. 11. The re-sults in both cases are in very go od agreem ent. Th e circuitparameters used in this simulation were

RL =23 9 QC f = 2 p F

2Q =-, 2, 10a622 sin wot; t 0.uin( t ) =

constant, T ~ ,lso dictates the response time through theterm e --f'27L. The sim ulated response of the output voltageto a 20 percent increase in the load for Q =2 is shownin Fig. 12(a) while the predicted respon se using (33) isshown in Fig. 12(b). The simulated response of the l inecurrent is shown in Fig. 13(a) while the predicted re-sponse using (35) is shown in Fig . 13(b). The predictedand simulated responses are in very good agreement. Thecircuit parameters in this case wereu i n ( t )= 622 sin wot

Q = 2C = 2 p F

23 9 Q ; t 0RL =[AR, = -4 7 Q .

Comparison of Predicted and Experimental ResultAs explained earlier, the response for higher Q can beseen to be slower. It should be clear that the output time An experimental circuit was built using an RF poweramplifier (with an internal impedance of 25 Q ) for the ac



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V O R P E R I A N A N D R I D L E Y : U N I T Y P O W E R - F AC T O R R E C T I F I C A T I O N O R H I G H F REQUENCY AC 83

,- Q = 2 I

(src.) O m0 -(b)

Fig. 11. (a) Simulated response of line current to step line change and (b)predicted response using (31 ). Envelope of current waveform i s obtainedfrom average equivalent circuit model of Fig. 7 .

0 ca2 (SCC.) Dc a .(b)

Fig. 13 . (a ) Simulated response of line current to step load change. (b )Predicted response of line current to step load change using (35). En-velope of current waveform is obtained from equivalent circuit ofFig. 7.

4 1

0 - (SCC.) 0 - Fig. 14 . Experimental circuit. Power source is RF amplifier with internalimpedance of 25 0 operating at 17 kH z and amplitude of 36 V .(a)W !

O = Z i

I m p e d a n c e ( d B ) P h a s e ( D e g r ee )10power source as shown in Fig. 14. The experimental 5waveforms are shown in Fig. 15.The output impedance of the average equivalent circuitverified and the results are shown in Fig. 16. As men-tioned earlier this circuit is one of the fundamental ana-lytical results presented in this paper and its validity hasbeen experimentally verified. The low-frequency asym p-tote isof the RF pow er amplifier.

0-5model shown in Figs. 7 an d 8 has been experimentally - 140

- 1 0

-15

-20 00 1000 10000 100000high because Of the large impedance (25 '1 Fig 16 Experimental and predicted results of output impedance of aver-ag e equivalent circuit (Fig 7 ) of rectifier



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84 IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 5. NO I . JANUARY 1990

CONCLUSIONA simple rectifier circuit which operates at unity powerfactor is proposed for use with high frequency ac buses.Because this rectifier uses only passive elements it pro-vides the cleanest possible almost-sinusoidal input cur-

rents. Active power factor correction circuits, which areused at 60 Hz or 50 Hz, are more complex and introducesome switching noise on the line. Hence, the more suit-able candidate circuit for unity power rectification for highfrequency ac buses is the rectifier proposed in this paper.APPENDIX

In this Appendix, the derivation of many of the equa-tions appearing in the text is given.D C Analysis

Assume that the rectifier is operating in continuous con-duction mo de. The input circuit of the rectifier can be rep-resented by the circuit in Fig. 17where the sq uare voltagesource, V ,( r ) , epresents the voltage across the full-wavebridge on the input side. The time relation between theinput current, the input voltage and the bridge voltage isshown in Fig. 18. An important feature of the waveformsin Fig. 18is that all three waveforms bec ome positive andnegative together in time phas e. Th is can be easily shownby noting that the fundamental component of the bridgevoltage, V04/7r sin war, must be equal to as well as inphase with the input voltage, V,, sin w o t , because theimpedance of the resonant branch at the frequency of thefundamental is zero. (Recall that the line frequency, theresonant frequency and the fundamental frequency are allthe same.) Since V e ( t ) and its fundamental are in timephase , it follows that V , ( t ) nd consequently the line cur-rent must be in time phase with the input voltage. Theideal rectification ratio is obtained by equating the inputvoltage to the fundamental of V , ( r )

This result is independent of the load R L . The rectificationratio in the presence of parasitic elemen ts as given by (4)will be derived after the input current has been deter-mined.The input current i j , l ( r )and its harmonic contents forcontinuous conduction mode (ccm) are determined next.From Fig. 3it can be seen that in ccm the worst shape ofthe input current occurs for values of Q approaching 2 / ~ .Furthermore, it can be seen that the most dominant har-monic is the third harmonic which we proceed to deter-mine as follows (note that the waveform has half-wavesymmetry.) The circuit which corresponds to the nth har-monic can be obtained from Fig. 17 and is shown in Fig.19 where the bridge voltage V s ( t ) has been replaced byits nth harmonic. W e have from Fig. 19:

Fig. 17. Resonant rectifier circuit shown excited by two voltage sources.

Fig. 18 . Input voltage, line current, and bridge voltage have same zero-crossings.

where

where I , is the average output current. The input currentcan now be written taking into account all the harmonicsasi i n ( t )=I " ) sin (war - $1 +C I ( ' ) co s nuor ;

n = 3 , 5 , 7 . . *where I ( ' ) nd II/ of the fundamental are determined asfollows. Since at the beginning of the positive excursionof the input voltage the input current must alwa ys be zero,it follows from the above

where we have made use of the following

When the input is equated to the output power anotherequation in I ( ' ) nd II/ is obtained

p . =In 2

V P Pco s $ = Po,, =V J ,which gives

p, = ;cos $. (39)



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85ORP E RI AN A N D RIDLEY: U N I TY P OWE R- F ACT OR RECTIFICATION FO R H I G H FR EQ U EN C Y AC

Fig. 19. Equivalent circuit from which harmonics in the current are deter-mined.

Solving (38) and (39) we getLtan rl,= ~a 2 Q

It is clear then that the fundamental in continuous con-duction mode ( Q 1 / a ) is essentially given by

The ratio of the third harmonic to the fundamental is givenby

P3 ' 1 1m'= ? s i n $ =From this last equation it can be seen that in continuousconduction mode the third harmonic is suppressed veryquickly as Q is increased. Hence, for Q =2 the thirdharmonic is about f ive percent of the fundamen tal.The power factor can now be calculated using the fol-lowing:

which after som e algebra yields (14) in the text:

The peak inductor current given in (9) in the text forthe range of 2 / a 5 Q 5 2 is obtained empirically bysimply adding the fundamental and the third harmonicpeaks. A rigorous derivation of the peak ind uctor currentin the presence of all the harmonics is rather tedious.With the assumption that the input current consistsmostly of the fundam ental,

a .i i n ( t )= I,, sin coot =I, - in wo t ,2

the effect of the parasitic elements and the diode voltageon the rectification ratio and the efficiency are derived asfollows:

where I : ( ( a 2 / 8 ) - 1) is the rms current in the outputfilter capacitor. The efficiency follows:1

VD a 2 ov, 8 RL1 1 = 1 +2 - +-- +( - I ) ?

The output power can be written as

La+ M R = V - 4

Substitution of (40) in the above yields the exp ression ofthe rectification ratio given in (4) f the text

The capacitive part of the output ripple voltage is deter-mined by referring to Fig. 19:dVCCf- =IP sin W 0 t - , -+ AVc,dt

whereWO a

Carrying o ut the integration and adding the comp onent ofthe ripple due to the equiv alent serie s resistance (ESR) weget (12) of the text.The peak capacitor voltage is determined by realizingthat the average current which takes the capacitor from itsnegative peak to its positive peak is the same as the outputcurrent so that we have1 a= I -o2vco~d,I" 2 * * O

from which (10) of the text follows:

The inductor voltage an d its peak follow imm ediately fromthe waveforms in Fig. 2 by subtracting Vc,,(t) from vi"( )- V,( t ) Note that the jump of 2V, induced in the pri-mary circuit by the switching of the full-wave bridge ap-pears entirely across the inductor.



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86 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL . 5 , NO . I . JANUARY 1990

To determine the boundary between the continuous anddiscontinuous conduction modes, consider the instant, t= O f, when the input voltage is about to turn positive. Ifthe converter has entered dcm, then it is clear that theinductor voltage is zero and conduction does not beginuntil the su m of input voltage an d the peak capac itor volt-age is equal to the output vo ltage (in order to forward biasthe bridge). At the boundary between dcm and ccm thepeak capacitor voltage is equal to the output voltage sothat the critical value of Q is given by

where we have used (41) for the peak capacitor voltage.Derivation of Zh and the Average Circuit Model

The impedance ZNB shown in Figs. l(b) and 19 is sim-ply given by1 + j w c O r O- w ' / w $Z d j w ) =

which in the vicinity of w = w0 behaves asZ N B ( j w ) = r0 + j 2 L , ( o - W O ) ; w = w0

This is the impedance seen by the information in the am-plitude-modulated current or simply the perturbation inthe line current. We know that the information in an AMsignal is up-shifted in frequency by w0 and that it is car-ried by the envelope of the AM signal which has exactlythe same shape as the base-band or the information signal.Hence, the impedance seen by the information, or the en-velope of the line current, [ i i n ] e n v e l o p e ,ust be nothingmore than the down-shifted version of ZNB by wo which issimply given

Z L p ( j w ) =ZNB(j(w +W O ) ) =r0 +jw2L0. (42)This is the equivale nt lowpass version of ZN B.Let us writethe modulated line current as

i in( t>= ( I , + [ 4 n I e n v , ( t ) )in mot (43)that can be written in terms of the modulation in the aver-age rectified cu rrent as

(44)ni i n ( t )=- I , + L r ( t ) ) sin wOtso that the modulation in the envelope and the averagerectified current are related by

2

(45 1T A[ 4 n I e n v ( t )= - i r ( t ) .Now an amplitude modulation of the line current is ac-companied by an amp litude modulation in the bridge volt-ag e V B ) . Since VB ) , nlike the line current, h as many

Fig. 20. Spectral representation of interaction of narrowband and low-pa ssfilters with modulated signals it,, and i ,.

significant harmo nics, the in formatio n is up-shifted in fre-quency by wo, 3w0, etc. Since we are only interested inthe information in the vicinity of wo we con sider the mod-ulation in the fundamental of V , ( t ) given byu L ' ) ( t ) =- (V , + o , ( t ) ) sin woz

n

whereo p ( t ) =- o , ( t ) .

n (47)A comparison of (46) an d (43) reveals that Z L p( s ) s sim-PlY

Substitution of (47) an d (45) in (48) gives

which is (16) in the text. A spectral representation of thederivation given above is shown in Fig. 20.The averageoutput circuit of Fig. 8 follows where the hats in the per-turbation quantit ies have been dropped.

REFERENCES[ I ] M . J . Kocher and R . L. Steigerwald, "An ac to dc converter with highquality input waveforms," in Proc. IEEE 1982 P E S C , pp . 63-75.



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VORPERIAN AND RIDLEY: UNITY POWER-FACTOR RECTIFICATION FOR HIGH FREQUENCY AC 87[2] M. F. Schlecht and B. A. Miwa, Active power factor correction forswitching power supplies, in IEEE Trans. P o w e r E l e c t r o n ., vol. PE-2 no. 4 , pp. 273-281, Oct. 1987.[3] K . K . Clark and D. T. Hess, Communicarion Circuirs: Analysis andD e s i g n . Chapter 3 , New York: Addison-Wesley. This book of coursedoes not discuss the rectifier circuit shown here but the analysis giventhere for the response of narrowhead filters to AM signals can be ex -tended to the rectifier circuit.[4] Chung-jen Hsiao, Circuit-oriented switchmode integration routine forswitching converters, Masters thesis, Virginia Polytechnic Instituteand State University, Blacksburg, Sept. 1987.[SI S . Freeland, I . A unified analysis of resonant converters with reso-nant switches. 11. Input current shaping for single-phase ac-dc powerconverters, Ph.D. dissertation, California Institute of Technology,Pasadena, Oct. 20, 1987.[6] S . Freeland, Input current shaped ac-to-dc converters, NASA Re-port, NASA-CR-176787, May 1986.

Vatchk Vorpkrian (S77-M77-S80-S80-M83) received the Ph.D. degree from the Cali-fornia Institute of Technolo gy, Pasadena, in 1984.He worked for Digital Equipment Corporationfrom 1977 to 1979. Presently, he is an AssistantProfessor at Virginia Polytechnic Institute andState University, Blacksburg. He has taught powerelectronics courses in the industry and has pub-lished 15 papers in the area of modeling and anal-ysis of resonant and PWM converters.

Raymond B. Ridley. Fo r a photograph and biography please see page 39of this issue.

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A Simple Scheme for Unity Power-Factor