A capacitor bank charger using ahigh-frequency link resonant circuit

S.A. Mirbod, B.Sc. Ph.D., M e m . I E E E . , and W.G. Dunford, B.Sc.(Eng),Ph.D., A.C.G.I., Mem.I.E.E.E.

Indexing terms: Convenors, Industrial applications of power

Abstract: A high-voltage compact capacitor bank charger is required in sonar applications. A resonant high-frequency link is employed which provides fast response to an external command, high efficiency for the conver-tor and isolation between the input and the output of the convertor. The convertor circuit is analysed and thedesign procedure is given. Experimental results obtained from a 1200 J charger are discussed.

List of principal symbols

Cp = communication coupling capacitorL/i, Lf2 ,Cfl, Cf2 = power filter elementsCX,C2,C = capacitors used in resonant circuitSXi, S22, Snn = diodes in high-voltage switchS l 5 S2, Sn . = thyristors in high-voltage switcha = angle of retardy = angular duration of spanning inter-

val (mode III)fi = angular duration of mode I£ = damping factorCo = capacitor used in high voltage dis-

charge circuitCc = Thevenin equivalent capacitanceEb = convertor input voltageE = Thevenin equivalent input voltageR = equivalent series resistance in reson-

ant loopLc = total loop series inductance in reson-

ant loopLd — di/dt inductorLt = value of series inductance including

transformer leakage inductanceKase > hase > ^base = base values for normalisationcoo = undamped natura l frequency

(assumed equal to damped value)t = t imeic(t) = per unit current through Thevenin

capacjtancevc(t) = per unit voltage across Thevenin

capacitancevcl(t), vc2(t) = voltages on capacitors 1 and 2Vcl, Vc2 = peak capacitor voltagesVt = equivalent per unit voltage rep-

resenting loadVtmax = m a x i m u m per uni t load vol tageco = per unit frequencyt0 — t6 = instants of switching between differ-

ent modes of opera t ionTo = total load capacitor charging inter-

valIid = average convertor input current/, = average convertor input current (per

unit)Ir = half cycle average resonant current

(per unit)

Paper 4200B, (P4, S8), first received 11th March and in revised form 14th August1985

Dr. Mirbod is with the Department of Electrical Engineering, Louisiana State Uni-versity, Baton Rouge, LA 70803, USA. Dr. Dunford is with the Department ofElectrical Engineering, University of British Columbia, Vancouver, British Colum-bia, Canada.

KrL

nNWo

w.

= RMS value of resonant current= load capacitor charging current

(assumed constant)= per unit load capacitor charging

current= allowable shoot through current= output resistance of the resonant

convertor= convertor efficiency= turns ratio of isolation transformer= transferred energy to capacitor bank= energy loss in the convertor

Introduction

Many applications require high power, compact, light-weight and reliable capacitor charging units. This paperdescribes a unit intended for use in an undersea sonarapplication. The block diagram is shown in Fig. 1. It con-sists of a high-voltage capacitor charger, a capacitor bank

to tow fish electronics

iDCtoDC i1 rACSMtnrtt ^

Fig. 1

Cresonant

Cfir Cffr jconvertor

I J

Block diagram of submerged sonar source

storage unit, a power switch and an acoustic transducer.The unit is carried in a tow fish close to the ocean bottom.A coaxial cable connects the tow fish to a survey ship. Thiscable acts as a communication link, and supplies DCpower to the tow fish. At the input to the tow fish the DCpower is transmitted through the LC filter shown (Lf Cf),appearing as a DC source voltage Eb at the convertorinput. The power filter blocks the high-frequency commu-nication link, which is capacitively coupled to the tow fishelectronics via Cp.

The convertor charges the capacitor bank to the desiredvoltage (up to 6 kV) and is turned off prior to the dis-charge of the capacitor bank. For normal sonar applica-tions, the required stored energy in the capacitor bank isbetween 500-2000 J and the charging interval is aroundone second.

The series thyristor string comprises the high-voltageswitch. A high-voltage, multisecondary toroidal pulsetransformer is used to ensure simultaneous gating of thesethyristors. Static and dynamic voltage sharing is aided bythe use of matched thyristors and a parallel resistor andsnubber network.

IEE PROCEEDINGS, Vol. 132, Pt. B, No. 6, NOVEMBER 1985 327

As a result of poor air convection in the tow fish, theenergy loss in the convertor must be kept to a minimum.This is also desirable to minimise the power-transmissionrequirement . Also, for proper operat ion of the communica-tion link, the input current should be D C and free of high-frequency noise. In essence, the conver tor must be of highefficiency, lightweight, fast responding, and with low inputripple current.

Several charging methods of the capaci tor bank, such ascons tan t source voltage charging and constant currentcharging, have been discussed in the References [ 1 , 2 ] .These methods are not considered due to their low effi-ciency, and lack of isolation between the capacitor bankand the input source voltage.

This paper discusses a new approach for capacitorcharging by regulating the resonant current of a high-frequency convertor, which charges the capacitor via adiode bridge. By using this approach , the overall powerloss in the resonant circuit is minimised, and the size of theconver tor is reduced. The block diagrams of the convertorand the control circuit are shown in Fig. 2. The resonant

=k V,

C,=r

amplifier filterana control

reference^resonantcurrent

-j~9| iconvertorgating

desireddischarge"energy

Fig. 2itor C,

charge disablecommand

JD2

"n*asi ] discharge

D6

voltage sensingand control

High-frequency DC-DC convertor for charging discharge capac-

circuit transforms the DC input voltage to an AC, sinewave which passes through a boost up transformer. Theoutput voltage of the transformer is rectified and is used tocharge the capacitor bank.

The control circuit uses voltage and current feedback.The voltage sensing circuit monitors the capacitor bankvoltage to sense the required final value. Current feedbackis used to maintain the resonant current of the convertor.The amplitude of this is kept constant by adjustment ofthe angle of retard (a). Some of the advantages of this con-vertor are:

(a) simplicity of the power circuit, with a low number ofreactive components on the high voltage side of the iso-lation transformer

(b) ability to charge the capacitor bank even when thecapacitor is fully discharged, i.e. a short circuit load

(c) ability to use an asymmetrical thyristor (ASCR) withload commutation as the power switch. This switch canoperate reliably up to 50 kHz, switching with speeds of1000 V/tis and 1000 A/^s. Thus, low snubber power lossand high efficiency for the charger are obtained. At highercurrents the reverse conducting thyristor (RCT) can beused. This eliminates the parasitic inductance in series withthe antiparallel diode. This inductance has a serious effecton the available turn off time when heavy currents areinvolved

(d) isolation of the input and the output of the chargerby using a small size, high-frequency transformer

(e) ability to select the source voltage for the chargerindependent of the maximum capacitor bank voltage

(/) reduced electromagnetic interference (EMI) of theresonant circuit due to the quasisinusoidal nature of theswitching currents.

The application of high-frequency series resonant links inDC to DC convertor schemes has been discussed in theliterature [3-6]. However, in all of these References eitherthe effect of the damping factor of the resonant circuit isneglected, or the result of a computer simulation for a par-ticular operating point is reported. The present paperincludes a more detailed study, accounting for thedamping factor of the circuit, along with several resultsthat have not been discussed in the previous literature.Analysis of the circuit operation with continuous and dis-continuous current at a fixed output voltage is included.Moreover, the voltage and current equations of the circuithaving a damping factor of £ are derived, and the impor-tant results are presented in the form of normalised graphs.Furthermore, the procedure for the design of the capacitorcharger and the verification of the theoretical results by theexperimental verification are also given.

2 Operation of the resonant convertorat a fixed output voltage

To follow the operation of the convertor, the sequence ofthe topological modes which occurs during one cycle ofoperation of the convertor is shown in Fig. 3. The loopshown by an arrow indicates the conducting path in thatparticular mode. For simplicity of presentation, the follow-ing simplifications are made:

(a) the source is ideal. In the Thevenin equivalent circuitfor the resonant convertor, shown in Fig. 3, capacitor Cc

and source E have the following values, respectively:

Cc = 2Cl = 2C2 = 2C (1)

(2)

E^*

E±

E-^:

Fig. 3 Modes of operation for a resonant convertorDotted lines indicate conducting pathsa Mode I conduction of thyristor Q,b Mode II conduction of diode D,c Mode III spanning mode (Q2 fired before D, blocks)d Mode IV Q2 conductse Mode V D 2 conductors/ Mode VI Q, and D, spanning mode

328 IEE PROCEEDINGS, Vol. 132, Pt. B, No. 6, NOVEMBER 1985

(b) the high-frequency transformer/fullwave rectifier/capacitor/load combination is represented by a controlledvoltage. The controlled voltage polarity is always in thesense that energy is absorbed from the resonant circuit

(c) the damping factor of the resonant circuit is small. £is expressed as

(3)

where R is the equivalent resistance of the resonant circuitand

Lc = Ll+ Ld (4)

Lx includes the externally added aircore inductor and theleakage inductance of the boost up transformer.

For a small value of a(a < n), the resonant current and thevoltage across the primary of the isolation transformer areshown in Fig. 4. The boundaries between the topological

ic(t)

t = ty, where coty = /?. Angle /? is obtained from eqn. 8 withir set to zero:

vt(t)•v,

-v

1 1 \

o l\ / 1

a

i

1|11

, iii

h cd

1

p

l* -r

t

modes

Fig. 4 Waveforms of resonant current ic(t) and voltage across the iso-lation transformerKey as for Fig. 3

modes are marked by dashed vertical lines. Various con-duction angles, voltages and currents which are used in theanalysis are also defined in Fig. 4. The following basevalues are adopted in normalisation of the circuit par-ameters :

(5)

(6)

(7)

As shown in Fig. 4, mode I begins when thyristor Qx startsto carry the full resonant current. At this instant, the perunit resonant current and capacitor voltage are ic(t0) andvc(t0), respectively, which are expressed as

ic(t) = (1 - vc(t0) - Vfc-f sin cot + ic(t0)

(cos cot - £ sin cot)e~^wl (8)

vc(t) = 1 - Vt — (1 - vc(t0) - V;)(cos cot + f sin coOe"460'

+ iMe~4a>t sin cat (9)

where ^ is the per unit instantaneous voltage across theprimary of the isolation transformer.

Mode I ends as the resonant current drops to zero at

IEE PROCEEDINGS, Vol. 132, Pt. B, No. 6, NOVEMBER 1985

= n — tan-Vt-

(10)

The capacitor voltage at the end of this mode vc{t{) isobtained by substituting from eqn. 10 in eqn. 9.

Mode II begins at t = tx, when the resonant currentreverses its direction through diode Dv This mode ends att = t2, where co(t2 — t1) = ci. As diode Dx conducts, anegative voltage is applied across thyristor Ql 5 ensuringproper turn off of this thyristor. The equivalent circuit isthe same as for mode I, except that the polarity of Vt isreversed, and the initial mode II current is zero. The fol-lowing expressions for the current and voltage at the endof mode II can be obtained by substitution in eqns. 8 and9, with the voltage and current at tl5 being used as theinitial conditions:

ic(t2) = (1 - ve(tx) + Vt)e~^ sin a (11)

ve(t2) = 1 + Vt - (1 - vc(ti) + KXcos a + £ sin a)e~ia

(12)

The duration of mode II can be controlled, and theangular duration a is known as the angle of retard.

Mode III begins when the thyristor Q2 is turned on att = t2. During this mode, the resonant current though Djdecreases linearly. This interval can be compared to theoverlap interval found in phase-contr.olled applications.When mode III ends, Q2 carries the full resonant current.For applications where Ld is small (Ld < Lx), the effect ofthis mode (spanning interval), on the output waveform,can be neglected.

Mode IV starts as thyristor Q2 carries the full resonantcurrent. At this instant, the per unit resonant current andcapacitor voltage are

ve(t3)= -ve(t0)

(13)

(14)

The operation of the circuit during modes IV, V and VI issimilar to that of the first three modes and is not discussed.

As the value of a increases, the resonant currentdecreases. If the time interval between the instants atwhich thyristors Qx and Q2 are fired is greater than theperiod of oscillation T(T = 27iN/Lc CC), the resonantcurrent becomes discontinuous. The operation of thecircuit for discontinuous current is similar to that of thecircuit with continuous current with the following excep-tions:

(i) the per unit initial current ic(t0) is zero and the span-ning interval (y) is nonexistent

y = 0 (15)

= 0 (16)

(ii) mode II ends at angle n, where the resonant currentdrops to zero.

2.1 Voltage and current equationsThe circuit equations for different modes are described byeqns. 8-14. Neglecting the spanning interval and substitu-ting from eqns. 11-16 in eqn. 9, vc{ti) is simplified as

Vc(tl) = 1 + Vt -

+ cos & + £, sin (1

for a < n (17)

329

V e^n — 1ve(tx) = 1 + Vt - 2e^n - ^ for a > 7i (18)

The maximum voltage of the resonant capacitors Ct andC2, in terms of vc(t\), is expressed as

VeX = Vc2 = (1 + vc(h)) -f (19)

substituting from eqns. 17 and 18 in eqns. 11 and 12, ic(to)and vc(t0) are simplified as

lAto) = — 2c o s s i n a

*c(O = 0

sin a for a < n

(20)

for a ^ n

(21)

»c(O = - ( i + K)

Vt e^cos a + <̂ sin a) + cos a cos P + £ sin (a + /?)

yc(fo) = - ( 1 + Jfl + 21 + e 2 ^

The conduction interval /? can be obtained by substitutingfrom eqns. 17-23 in eqn. 10.

^ e4/? + (cos p + £ sin jS)] sin a+ tan

for a < 7i (24)

jff = n for a ^ n (25)

The per unit input average current of the convertor is

c(t) dt (26)

Substituting for the stored charge in the capacitor Cc, /, issimplified as

(27)a + fi

(7r + a)(28)

Similarly, the per half cycle average current of resonantcircuit is:

f

for a < 7r(29)

For a reliable sustained oscillation, the resonant currentduring the conduction of the diodes should be noticeable.Assuming a minimum ratio of 0.1 for the peak diode res-onant current to the peak antiparallel thyristor current,the maximum obtainable Vt during the discontinuousmode of operation is

V =' t max

10 -

1 +(33)

for

for

a

a

< 71

7T

(22)

(23) i

From the above equations those variables of most interest,from a design standpoint, are Vc, / , , Io, rj and 9̂. Forvarious values of V{ and <̂, the calculated curves for themaximum capacitor voltage, (^(tj), / , , /,., rj and j5 areshown in Figs. 5-9. Note that Fig. 5 shows the per unit

V. =0.70.6 0.40.2 0.053.00r

2.80

2.60

2.40

2.20

2.00

1.80

1 .600 40.00 80.00 120.00 160.00 200.00 240.00 280.00

alpha, degrees

1.00

2.80

2.60

=> 2 .40d.

cT 2.20o

2.00

1.80

1.600 40.00 80.00 120.00160.00 200.00 240.00 280.00

alpha, degrees6

Fig. 5 Normalised peak Thevenin capacitor voltage v^tj), as a functionof a, c, and normalised output voltage V,a £, =0.005 b £ = 0.1

a + nfor a ^ n (30)

Neglecting the power loss of the snubbers, the efficiency ofthe convertor supplying a fixed DC output voltage is

rj =2/,. (31)

Substituting from eqns. 27-30 for Ir and /, in eqn. 31,yields:

Y) =Vtvc(h)

(32)

voltage vc{t\) across the resonant capacitor Cc, used in theThevenin model, at time tv The actual maximum capac-itor voltage Vcl or Vc2 can be obtained by using this valuein eqn. 19. Also, note that Figs. 6 and 7 show the per unitcurrents. Assuming a sinusoidal waveform, the RMS valueof the resonant current /„. is

E, [Ccrr r 2 Ju 4

The per unit output current is:

(34)

(35)

330 1EE PROCEEDINGS, Vol. 132, Pt. B, No. 6, NOVEMBER 1985

0.70

0.60

0.50

0A0

7.0.30»

0.20

0.10

00 40.00 80.00 120.00160.00 20000 2A 0.00 280.00

alpha, degreesa

Fig. 6 Normalised convenor input current for conditions of Fig. 5Key as for Fig. 5

u./u

0.60

0.50

a. 0.40

Q30

0.20

0.10

0

r \1 \\

i i i i

0.60.40.20.05

0 A0.00 80.00 120.00 16000 200.00 240.00 280.00alpha degrees

' b

1.A0r

1.20

1 .00

80

60/i

40

. 20

0

Vt : 0 7 0.6 O.A 0 .2 0.051 / / ^

1.20

1.00

1.80

5Q. 1.60

1.40

1.20

0

Vt =0.6 O.A 0.2 0.05

0 A0.00 80.00 120.00 180.00 200.00 240.00 280.00a Ipha .degrees

Vr 0.7

Fig. 7 Normalised half cycle average resonant current Irfor conditions of Fig. 5Key as for Fig. 5

1.00

0.90

0.80

0.70

060

>«050c4;

.<£ 0.A0

"a;0.30

020

0.101

0

0 A0.00 80.00 120.00 160.00 200.00 240.00/60.00alpha , degrees

b

0.05

0.05

0 A0.00 80.00 120.0016000 200.00 2AOX5O 280.00alpha,degrees

a

Fig. 8 Convenor efficiency for conditions of Fig. 5Key as for Fig. 5

0 A0.00 80.00 120.0016000 200.00240.00280.00alpha, degrees

b

IEE PROCEEDINGS, Vol. 132, Pt. B, No. 6, NOVEMBER 1985 331

The actual input current is

(36)

0 20.00 40.00 60.00 80.00 100.00 12000 14000 160.00180.00alpha,degrees

180.00

170.00

160.00

150.00

£ 140.00

•o 130.00a

•o 120.00

110.00

100.00

90.000 20.00 40.00 60.00 80.00 100.00 120.00140.00160.00180.00

alpha degreesb

Fig. 9 Variation of mode I duration (P) for conditions of Fig. 5Key as for Fig. 5

3 Charging considerations using resonantconvertor

As mentioned, the transfer of energy from the resonantconvertor to the capacitor bank is controlled through theadjustment of angle of retard a. Therefore, the equivalentcircuit of the resonant convertor seen from the output ofthe full wave rectifier can be represented by a variable DCvoltage eo(a) with an equivalent source resistance Ro.Assuming a sinusoidal waveform for the resonant current,Ro is given by

(37)

It has been shown that charging the capacitor bank with aconstant current minimises energy loss in the resistor Ro

[6]. Therefore, the angle a should be varied in a manner to

keep the resonant current constant. From Fig. 6, therequired range of variation of a for obtaining a constantoutput current can be approximated. To obtain a higherefficiency for the charger, the resonant current waveformshould be a continuous sine wave; also a and its range ofvariations should be as small as possible. By reducing therange of variation of angle a, the minimum operating fre-quency is increased and the size of the isolation trans-former is reduced. The efficiency of the system is

(38)W0+Wr

where Wo is the transferred energy to the capacitor bankand Wr is the energy loss. By using constant current charg-ing, eqn. 38 is simplified as

(39)2RnCn

In eqn. 38, To is the charging interval, Co is the capacitanceof the capacitor bank and Ro is given by eqn. 37. Note thatfor improving the charger efficiency the charging intervalshould be maximised and the turns ratio of the trans-former should be minimised.

Assuming a large filter capacitor across the input of theconvertor, and neglecting the discharge interval of thecapacitor bank, Fig. 10 shows the different waveforms at

V. *~o

"cf

Fig. 10 Convertor waveforms using constant current capacitor charging

a Capacitor currentb Capacitor voltagec Convertor input powerd Voltage across input filter capacitor Cfl (Fig. I)

the input of the resonant charger. With these assumptions,it can be shown that the voltage ripple across the input ofthe convertor AEb is

1 Wb S(l/2CfE

2b)r, b

where Cf is the total value of the input filter capacitance(Fig. 1), i.e.

= Cn + C (41)n + Cf2

The value of Cf2 is selected according to the maximumpermissible shoot through current in the event of simulta-neous conduction of Qx and Q2. If Lf2 > 2Ld the value of

332 IEE PROCEEDINGS, Vol. 132, Pt. B, No. 6, NOVEMBER 1985

C / 2 can be chosen to allow the shoot through fault currentto ring and turn off Qx and Q 2 . The maximum value ofthe shoot through current is

1

forL/ 2

2Ld (42)

Thyristors Q1 and Q2 turn off during the second half cycleof fault current ringing, whereas Dx and D2 conduct.

The values of Lfl and L / 2 are determined by themaximum high-frequency charging noise acceptable on thecommunication link (Fig. 1).

4.1 Design procedureFor designing the capacitor bank charger, the followingparameters are required:

(a) maximum output stored energy Wo, capacitance ofthe capacitor bank Co, charging interval To and theapproximate value of the damping factor of the circuit

(b) maximum permissible operating frequency of the res-onant convertor and the turn-off time of the thyristors

(c) input source voltage, the low-frequency voltageripple and maximum voltage of the resonant capacitorsVel

(d) maximum allowable shoot-through fault current / s c .

The following example illustrates the design procedure:(a) Wo = 1400 J, Co = 12000 fiF, To = 3.6 s(b) fmax = 22 kHz, tq = 4ns(c) Eb = 240 V, AEb = 50 V, Vcl = 350 V, f = 0.05(d) Isc = 400 A.

Maximum voltage of the capacitor bank is

vn =2 x 1400

= 483.05 V12 x 10"3

The turns ratio of the isolation transformer is

_2K_J 2 x 483.05 J_ _'-'b rt max ^ ^ u u - '

Vtmax is obtained by using eqns. 33 and the given value of<!;. By using eqn. 19 and Fig. 5a, the minimum angle ofretard am, is approximated as

<xm = 100°

By using Fig. 7a, with <xm = 100°, the per unit half cycleaverage resonant current is

7r = 0.8

The average value of the capacitor bank charging currentis

1 ,1.61 A

'—- is obtained as

EhL 240 0.8= 10.37 Q

2I0N 2 1.61 x 5.75

Having the value of am, from Fig. 9 f}m is

K = 125°

using the given value f o r / ^ , JLCCC is

IEE PROCEEDINGS, Vol. 132, Pt. B, No. 6, NOVEMBER 1985

= 5.79 X 10"6 S2(«m + j8J x 2 * ^ 3 6 0

Lc = 60 nH

Cc = 0.56 nF

The available turn-off time is

^ = |5jj x n x 5.79 x 10-6 = 10.1 /is

which is more than the minimum acceptable value. Select-ing the line inductors Ld as 5 ̂ H, inductor Lx is

Lx = Lc - Ld = 60 - 5 = 55 //H

The rate of rise of thyristor current is

di 240= 2 4

at 2Ld 10

The efficiency of the charger is

rj =2R0C0

3.6

3.6 + 2 x ^ - x 0.1 x ^ - (5.75)2 x 12 x 10"3

8 Y 0.56

= 78%

The filter capacitor values can be obtained from eqns.40-42 as follows:

/ /i + /2

1400

AEb 4 x 2 4 0 x 0 . 7 8 x 5 0

= 37.3 mF

// V /400 VCf2 = 2Lj^fJ = 2 x 5 x 10-^—J = 27.

4 Experimental results

The results shown in Figs. 11 and 12 were obtained withthe following circuit components:

i, Q2 RCA - S7310 N

0 1 2 3 t , s

Fig. 11 Experimental discharge capacitor voltage during charging

Cc=0.5piF Ld = 5nH L,

333

with isolation between the input and output of the charger(b) it can keep the charging current constant to mini-

mise the power loss in the convertor(c) the charging current waveform is close to a sine wave

and the range of frequency variation of the resonantcurrent is small. Thus, the bandwidth of the generatednoise is small.

Fig. 12 Resonant current waveforms for conditions of Fig. 11a near start of chargingb near end of chargingHorizontal scale: 10 /is/division, ic: 1 A/division

In Fig. 11, the variation of the capacitor bank voltage isshown. As has been predicted, the capacitor bank ischarged linearly to 480 V in 3.6 s. Fig. 12 shows the capac-itor bank charging current at the start and the end of thecharging interval. From Fig. 12, it can be verified that thenatural frequency of the resonant circuit is 28 kHz and a isaround 100 degrees.

5 Conclusion

A high-frequency convertor which can be used as a high-voltage capacitor charger has been presented in this paper.The advantages of this convertor over other capacitorchargers include the following:

(a) the power circuit is simple. It is also highly efficient

6 Acknowledgment

The support of this work by Texaco Inc. and the NaturalScience and Engineering Research Council of Canada isappreciated.

7 References

1 BROWN, D.R., HUANG, Y.C., BIRL, W.L., and TOLK, K.: 'Charg-ing large capacitor banks for fusion research using HVDC conversionequipment', IEEE Trans., 1984, PAS-103, (2), pp. 398-402

2 KING, R., and STUART, T.A.: 'A normalised model for the half-bridge series resonant convenor', ibid., 1981, AES-17, (2), pp. 190-198

3 ORUGANTI, R., and LEE, F.C.: 'Resonant power processors. Part I:State plane analysis'. IEEE IAS Conference 1984, pp. 860-867

4 ORUGANTI, R., and LEE, F.C.: 'Resonant power processors. Part II:Methods of control'. IEEE IAS Conference 1984, pp. 868-878

5 SCHWARZ, F.C.: 'An improved method of resonant current pulsemodulation for power convenors', IEEE Trans., 1976, IECI-23, (2), pp.133-141

6 ZIOGAS, P.D, RANGANATHAN, V.T., and STEFANOVIC, V.R.:'A four-quadrant current regulated convertor with a high-frequencylink', ibid., 1982, IA-18, (5), pp. 499-506

334 IEE PROCEEDINGS, Vol. 132, Pt. B, No. 6, NOVEMBER 1985