Simple equ circuit for the series-loaded resonant cc with voltage boosting capacitor

Abstract: A simplle, equivalent circuit is resonant converter capacitor. The and straightforward state resonant current ratio; previous analyses techniques requiring numerical solution. an idealised time-domain quantify the errors fundamental frequency results from a 1 MHc accuracy of the technique

A.J. Forsyth S.V. Mollov

fundamental frequency (derived for the series-loaded

with voltage boosting equivalent circuit permits direct

calculation of the steady- and voltage conversion have used time-domain

complex calculation and Comparison with results from

simulation is used to introduced by the

approximation. Practical prototype demonstrate the

for converter design.

Indexing terms: Resonant converlters, Equivalent circuits, Converter design

Principal symbols

Cp = parallel voltage C, = resonant capacitor Io = load current Ixy = resonant current fxy = amplitude of Ixy k = capacitor ratio L = resonant inductor

Q R = load resistance Vcp = voltage across Cp Vi, = input voltage V, = output voltage Vxy = half-bridge outpJt 2 = fundamental

and Cp 2, = characteristic 4 = conduction angl: U) = angular switching w, = normalised wo = natural frequency

= Q factor wo LIR

boosting capacitor

(:,IC,

voltage frequency impedance of rectifier

impedance d(L/C,) of C, frequency

l/d(LCs) switching frequency wiw,

0 IEE, 1998

ary 1998

1 Introduction

By processing electrical power as high frequency sine waves, resonant DC-DC converters permit efficient operation at elevated frequencies thereby allowing min- iaturisation of the transformer and filter components; the device switching losses can be virtually eliminated by simple capacitor snubbers providing operation remains above the natural frequency of the resonant circuit. However, the exploitation of resonant technol- ogy is sometimes impeded by the difficulty of undertak- ing circuit analysis, design and optimisation. Determination of the steady-state operating conditions such as voltage conversion ratio and component ratings frequently requires time domain solution of the circuit equations using numerical methods.

As an alternative, a simple but powerful equivalent circuit analysis technique was proposed in [ 11 for the three most common resonant converter topologies: the series-loaded (Fig. I), parallel-loaded (Fig. 2) and series-parallel-loaded (Fig. 3 ) with the circuits shown in half-bridge configurations. The technique is based on a consideration of the fundamental components of the circuit waveforms, the rectifier, filter and load being replaced by an equivalent resistance. However, the technique is not readily extendible to variants of the three basic resonant converter topologies since the waveform shapes become more complex, as for exam- ple in the modified series-loaded resonant converter, Fig. 4, [2, 31.

1 1771

Fig. 1 Busic half- bridge resonant series-loaded Converter topology

Fig. 2 Busk hapbridge resonant parallel-loaded converter topology

Unlike the series-loaded resonant converter which can only operate in a voltage step-down mode, that is, a voltage conversion ratio less than 0.5 for a half- bridge configuration, the modified circuit of Fig. 4 is

30 1 IEE Proc -Electr Power Appl , VcIl 145, No 4, July 1998

capable of both step-up and step-down operation. Fur- thermore, the output voltage may be regulated at light load [2, 31. The circuit is particularly suited to higher output voltage applications since the rectifier diode reverse voltage is clamped to the output voltage level. The converter has recently been proposed for high power factor rectifier applications [4], but a convenient method of analysis has not been reported. The charac- teristics presented in [2, 31 required numerical solution of the converter equations for each of three different operating modes or topological sequences.

l 4 $ i I T I -

Fig. 3 W Y

Basic haybridge resonant seuies-parallel-loaded converter topol-

I I

of Ixv. The unshaded portion of I,, flows into the recti-

Fig. 4 Sesies-loaded resonant converter with voltage boosting cupucitor Fig. 5 Sketched converter waveJorms

The purpose of this paper is to extend the fundamen- tal frequency analysis in [ 11 to the modified series- loaded resonant converter, Fig. 4, to derive a simple equivalent circuit for steady-state analysis and design. The errors introduced by the fundamental frequency approximation are quantified by comparison with an accurate time-domain solution. The model is verified by practical results from a lMHz, 140W prototype converter.

2 Circuit operation and analysis

Fig. 4 shows the converter in a half-bridge configura- tion. The transistors operate in antiphase with equal duty ratios, the frequency being variable over a narrow band to regulate the power throughput. Compared with the basic series-loaded resonant converter a capac- itor, CI!, is added in parallel with the rectifier input, preventing the instantaneous reversal of the rectifier voltage when the resonant current passes through zero. Inputioutput isolation could be provided by placing a transformer in the resonant circuit, typically between the resonant elements and the rectifier.

The principal circuit waveforms are sketched in Fig. 5, the half-bridge output voltage Vxy, the resonant current -Ixy, and the voltage Vcp across the parallel capacitor Cp. The half-bridge output voltage is a square wave of tV,,/2 at the switching frequency. The rcso- nant current -Ixy is assumed to be a switching frequency sine wave lagging behind Vxy.

While -Ixy is positive and flowing through the recti- fier, Vcp is clamped to +Vu, the output voltage level. As -Ixy falls to zero the current transfers to Cp reversing the capacitor voltage to - V,. The negative half-cycle pair of diodes are then forward biased allowing the res- onant current to flow to the load, Vc, remaining at -Vu. The output voltage ripple is assumed to be negligi- bly small. Angle q5 in Fig. 5 denotes the conduction time of Cp, the current in Cp being the shaded portion

302

Under conditions of low output voltage and high current the reversal of Vcp takes place rapidly, q5 is small and the converter characteristics resemble those of the series loaded converter. With higher values of output voltage and a lower resonant current q5 increases. The effect of C, conducting for a larger part of the cycle is to increase the resonant current above the level expected in the ordinary series-loaded con- verter, thereby producing a higher output voltage. The presence of Cp therefore results in a boosting of the converter output voltage at light load conditions. This is illustrated below in the plots of the converter charac- teristics.

2. I Fundamental frequency analysis A fundamental frequency analysis is undertaken for the converter based on the assumption of a sinusoidal reso- nant current. However, since the rectifier input voltage and current are both nonsinusoidal, the rectifier cannot simply be replaced as in [l] by a fundamental frequency resistance. Instead, this analysis considers the parallel combination ofthe rectifier and capacitor C'. Although the voltage is nonsinusoidal the current is harmonic- free, therefore the active power transfer to C, and the rectifier is only due to the fundamental component of Vcp. The rectifier and Cp may then be represented by a fundamental frequency impedance.

The voltage across Cp may be expressed as a function of angle 8, 8 being zero at the origin in Fig. 5:

0 5 0 5 4

fzy sin 8 d0 - V, v =-1 l e

W C P C P

where w is the angular switching frequency and &, is the amplitude of the resonant current.

IEE PsocElectr. Power Appl.. Vol. 145, No. 4, July 1998

4 I 0 I I T

since Vcp (4) = Vo then

fzy (1 - cos #) V O = I 2 w c p

vcp = v, (2) from eqn. 1

Using eqns. 1 and 2, ar d exploiting the half-wave sym- metry of the waveform the fundamental component of Vcp is

Substituting for Vo using e-1912 and integrating:

[ ( - fq

vcdl) - J 2 T W C P

The fundamental frequency into the parallel may be calculated as 2

f x y sin 8 becomes Therefore,

Vcm - 1 - j& 2 T W C a

Z=---L

To complete the mode. required relating the rent. The average determined as

(4)

eqn. 3, putting cos 8 = (e'' +

1 - cos 24) - j(24 - sin 24)] (5)

impedance 2 seen looking combination of Cp and the rectifier

l,e-J.76/2 = -jIxy in phasor form. = V c P ~ , ~ I x y . ~

[( 1 - cos 24) - j (24 - sin 2 4 ) J

(6) of the converter an equation is

resonant current and output cur- conierter output current I , mdy be

2.2 Equivalent ra ti0 The result of the expressed by the bridge output voltage mental component, The impedance Z is by eqn. 7, and 4 may

IEE Pror.-Electr Power Appl.,

Eqns. 3 and 7 may hie solved for 4, giving

circuit and voltage conversion

fur~damental frequency analysis is equitalent circuit in Fig. 6. The half-

V,, is represented by its funda- ha,ving an amplitude (414 ( Vin/2). giTien by eqn. 6, the load current Io

be calculated from eqn. 8. The

Yol. 145, No. 4, July 1998

1 4 = cos-1 [-I] I T - ~ W ( p R = cos-1 [ IT - 2w,k/Q T + 2w( ?R IT + 2w,k/Q

( 8 ) where the load resista ce R = VolZo. con = colcoo, coo = Ild(LCJ, k = Cp/Cs an Q = cooLIR. k

x Fig. 6 Fundumental frequen

equivalent circuit may be used to determine the con- verter operating conditions:

(9)

substituting for Z using eqn. 6 and rearranging:

4V,,w,k/Z, (1 - cos 24) + j [2nk ( w i - 1) - 24 + sin 241 I,, =

(10) where 2, = d(L/C,J.

The output voltage may be determined using eqn. 7:

eliminating ixy using eqn. 10:

v,- - v,,

4unk(1 + cos dl T r (1 -cos 2#) + j [2nk(w; - 1) - 24 + sin 2411 I

For specific circuit parameters and operating frequency the angle @ may be determined using eqn. 8 and the voltage conversion ratio calculated using eqn. 12.

(12) I [

2.5

2.0

> .- 0 1.5 E

2

c

C ._

z 1.0

c P - s

0.5

C

. . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

0.5 1.0 1.5 2.0 2.5 3.0 3.5

normalised switching frequency w, Voltuge conversion ratio characteristics, k = 0.2, Q = 5.0, 1.0, Fig.7

0.5, 0.1 __ fundamental frequency prediction 0 simulation data

3 circuit

3. I Voltage conversion ratio Figs, 7 and 8 show the voltage conversion ratio plotted against the normalised switching frequency con for a range of Q values, in Fig. 7 the capacitor ratio k = 0.2 while in Fig. 8, k = 1.0. The characteristics plotted in continuous lines were generated from eqns. 8 and 12 using the mathematics environment MATLAB. The diamond symbols denote data points obtained from a

Accuracy and characteristics of the equivalent

303

Saber simulation of the converter. The simulation used lossless components, ideal switches and diodes. The dis- crepancies between the simulation data and the equiva- lent circuit characteristics are therefore solely due to the fundamental frequency assumption in the analysis.

0.5 1.0 1.5 2.0 2.5 2.5 3.0 normalised switching frequency w,

Fig.8 0.5, 0.1 ~ fundamental frequency prediction 0 simulation data

Voltage conversion ratio characteristics, k = 1.0, Q = 5.0, 1.0,

There is close correspondence between the equivalent circuit and simulation data. The agreement is very good with the higher value of k, but with k = 0.2 and lower values of Q there is a noticeable error which is at a maximum of around 12% in the region of the reso- nant peak.

The characteristics are of the expected form [2, 31 and are similar to those of the series-parallel loaded converter [I]. With a high value of Q, heavy load, the circuit operates as a series-loaded converter. There is a sharp resonant peak at CO, = 1, and only step-down operation is possible, the conversion ratio here is less than 0.5 due to the half-bridge configuration. For smaller values of Q, lighter load conditions, the voltage boosting capacitor Cp becomes increasingly dominant permitting step-up operation. The resonant peak moves upwards in frequency towards a value given by the series combination of C,T + Cp and L, that is w, = v'(1 + lik). The sharp resonant peak with small values of (2 permits regulation of the converter operation by small changes in frequency at light and zero load. This is in contrast to the behaviour of the series-loaded con- verter where the voltage conversion ratio against fre- quency characteristics become flat at light load preventing output regulation.

For low loss zero-voltage switching operation the operating point of the converter must be to the right of the resonant peak. With small values of the capacitor ratio k this implies a wider operating frequency range to regulate the output voltage as the load varies.

The characteristics suggest that the converter would be particularly well suited to applications in which a high voltage conversion ratio is required at light load

304

and a lower conversion ratio at heavy load, for exam- ple high power factor rectification systems [4].

3.2 Resonant current fxy, the amplitude of the resonant current, is plotted against switching frequency in Figs. 9 and 10 with k = 0.2 and 1.0, respectively. The resonant current is nor- malised to a base value of VinlZo while the switching frequency is normalised as before to a base value of wo. The continuous lines were generated from the equiva- lent circuit using eqn. 10 while the '+' symbols denote data points from the idealised simulation.

4

3 . . . . . . . . . . . . .:. . . . . . . : . . . . . . . . . . . . . . . . . . .

2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 . . . . . . . . . . . : . . . . . . . . . . . . .

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Q=5.0 I

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .: . . . . .

. . . . . . . . : . . . . . . . . : . . . .

0.5 1.0 1.5 2.0 2.5 3.0 3.5

4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 . . . . . . . . : . . . . . . . . . :.. . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

normalised switching frequency w,

: Q=0.1

Resonant current characteristics, k = 0.2 fundamental frequency prediction simulation data

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ; . . . . . . . . . . . . . . . . . .

I 0.5 1.0 1.5 2.0 2.5 3.0 3.5

: Q=1.0 : 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

0.5 1.0 1.5 2.0 2.5 3.0 3.5

I B ! I

. . . . . . . . . . . .

0.5 1.0 1.5 2.0 2.5 3.0 3.5 normalised switching frequency w,

Fig. 10 ~ fundamental frequency prediction + simulation data

Resonant curreni characteristccs, k = 1.0

IEE Proc.-Electr. Power Appl., Vol. 145. No. 4, July 1998

The correspondence larger value of k , but d k = 0.2, both for small = 0.1 and Q = 1.0. Thc being around 15"/0. Th conversion ratio plots v seen with k = 0.2 and (1

Although not plottec and simulation data fo rent shoys a similar seen for Ixy. Generally a maximum error of 10

The discrepancies ari: intermediate values of having poor selectivity, istic does not exhibit a fore the harmonic distc higher degrading the accuracy is likely to de values of k.

4 Converter design

The design of the convc to that in the series-p: [I], that is, minimising t rent ratio at full load switching frequency ra light load.

In this converter min load current ratio requj duction angle of the c q this implies a high valul small value for k. Increasing Q beyond excessive stored energy in the series-resc of L particularly diffici full load value. The ch between minimising q5 1

maximum switching fre tions.

A 1MHz prototype c on design calculations j converter was designed 1OOV to 200V DC supr the range 0-140W. The and the capacitor ratio to minimum operating onant component value and L = 13.2pH. Cp pa of the rectifier diodes, a itances formed the devil

Table 1 lists the pred tions for the three valuc output voltage being 4 column of data in Tab operating frequency, tl input voltage. The rig1

n the resonant circuit ant current character- 11-defined peak, there-

was constructed based equivalent circuit. The e a 48V output from a ad power varying over Q was chosen to be 5 implying a maximum

ncy range of 2:l. The res-

I I I IY<l

tions at the maximum operating frequency, maximum input voltage and virtually zero load. The centre col- umn of the Table is an intermediate condition. The TableAlists the predicted peak value of the resonant cur- rent I,,, the angle of the resonant current with respect to the half-bridge voltage, and the conduction angle q5 of capacitor Cp. The data was generated from the equivalent circuit with a small resistance of 0.7Q added in series with the resonant circuit to represent the MOSFET and resonant inductor resistances.

I I IY

Fig. 11 Meusured waveforms V,, = 100V, V, = 48V, output power = 140W, Q = 5.0 1 Voltage across lower MOSFET, 100V/div, O.Zp/div. 2 Resonant current I 3 Voltage Vcp 50V/df$l, 0.2pidiv.

2Aidiv, 0.2psidiv.

IEE Prac-Electr. Power AppL. VcL

v, = IOOV vjn = 2oov v, = 2oov output power = 140W output power = 14W output power = 3 W f = 1.03MHz f = 1.77MHz f = 1.99MHz

Q 5.0 0.5 0.1

'xr A 5.3 1.9 1.4

&,,,deg 7 79 88

$,deg 38 113 148

145, No. 4, July 1998

Fig. 12 Measured waveforms V,, = 200V, V;, = 48V, output power = 14W, Q = 0.5 1 Voltage across lower MOSFET. 2OOVidiv. 0.1 usidiv. 2 Resonant current IXy, 1 Aidiv, 0.1 Widiv 3 Voltage Vcp SOVidiv., 0.1 psidiv.

The converter operated as expected. The maximum full-load efficiency of 85% occurred with minimum input voltage, the losses being mainly due to conduc- tion drops in the devices, diodes and resonant compo- nents. Figs. 11, 12 and 13 show, respectively, the measured circuit waveforms for identical load and

.Fable 1: Predicted operating conditions for the prototype converter

305

input voltage conditions to those listed in Table 1. The waveforms shown are the voltage across the lower MOSFET, the resonant current Ixy and the voltage Vcp across capacitor Cp.

Fig. 13 Measuved waveforms VzT$ = 200V, V, = 48V, output power = 3W, Q = 0 1 1 Voltage across lower MOSFET. 200Vidiv. 0.1 usidiv. 2 Resonuant current I,, 1 Aidiv, 0:1 psidiv 3 Voltage Vcp SOVidiv., 0.1 psidiv.

Due to the zero-voltage switching operation the MOSFET switching waveform is free of parasitic ring- ing. However, a small parasitic ringing at around 15MHz is observable on the VcP waveform as the recti- fiers commence conduction, particularly at full load, Fig. 11. This was attributed to the capacitor C, reso- nating with stray inductance between Cp, the rectifiers and the output filter capacitor. Also, a small delay is seen to occur between the zero crossing of the resonant current and the start of the Vcp transition. This was attributed to the reverse recovery effects and the non- linear capacitance of the rectifiers.

The principal features of the measured waveforms are generally seen to correspond closely with the pre- dicted data in Table 1. The peak values of the resonant current agree almost exactly with predicted values, the maximum error being 0.1 A. The predicted angle of the resonant current is accurate to within 5" for the inter- mediate and light load condition, but shows a greater error at full load, the measured angle, Fig. 11, being 20" compared with a prediction of 7". The capacitor conduction angle Q, is predicted with an accuracy of -c5" at all three operating points.

The converter operating conditions listed in Table 1 also illustrate an important property of the converter, the sensitivity of the resonant current to load current. The resonant current varies by almost 4:l over the full load range, implying a good part load efficiency.

Finally, to compare the measured and predicted volt- age conversion ratio characteristics the prototype con- verter was operated with a fixed lOOV input while the frequency was swept over the full operating range. The conversion ratio was measured and plotted with predic- tions from the fundamental frequency analysis, Fig. 14, for three values of load resistance corresponding to Q = 5, 1 and 0.5. The diamonds denote the experimental data points. The conduction loss resistance was again included in the calculations.

The results further confirm the accuracy of the analy- sis, the measured data is within a few percent of the

prediction, the experimental values being slightly below the predictions, mainly due to the circuit losses not included in the analysis.

1.5

c 5 1.0 > 0 ._ 5 c ._ LI 5

0

a, r B E 0.5

0 1.2 1.4 1.6 1 .E 2.0

normalised switching frequency w,

Fig. 14 k = 1.25, V,, = 100 V ~ prediction 0 experimental data

Predicted and measured voltage conversion ratio characteristics,

5 Conclusion

A simple fundamental frequency equivalent circuit has been derived for the series-loaded resonant converter with voltage boosting capacitor. The accuracy of the equivalent circuit has been verified over a wide range of operating conditions by idealised simulations and practical results, confirming the analysis as a basis for design. The prediction of the voltage conversion ratio was very good except with small values of k and small Q. The accuracy of the resonant current predictions was also good, but degraded at low values of k with intermediate values of Q. The reduced accuracy of the analysis with very small values of k is unlikely to be important since operation with a small k implies a wide operating frequency range and this is usually undesira- ble.

6 References

1 STEIGERWALD, R.: 'A comparison of half-bridge resonant converter topologies', IEEE Trans. Power Electron., 1988, 3, (2), pp. 174-182

2 BHAT. A.K.S.. 'Analvsis and desien of a scrics-uarallel resollalit converter with capacitive output Gter', IEEE Tians Ind. Appl , 1991, 27, (3), pp 523-530

3 KIRCHENBERGER, U., and SCHRODER, D.: 'Comparison of multi-resonant half-bridge DC-DC converters for high voltage and high output power'. Proceedings of IEEE Industrial Applica- tions Society meeting, 1992, pp. 902-909

4 BHAT, A.K.S., and ZHANG, Z.: 'Characteristics of a series-par- allel resonant converter with capacitive output filter operating on the utility line'. Proceedings of the international conference on Power electronics, drives and energy systems, 1996, Vol. 1, pp. 403-409

306 IEE Proc.-Elect?. Power Appl,, Vol. 145, No. 4, July 1998