Bhat1993.pdf

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 8. NO.

1,

JANUARY 1993

I

Analysis and Design of a Series-Parallel Resonant Converter Ashoka K. S . Bhat, Senior Member, IEEE

Abstract-A high-frequency link series-parallel resonant converter is analyzed using the state-space approach. Analysis is presented for both the continuous capacitor voltage mode and the discontinuous capacitor voltage mode. Steady-state solutions are derived. Design curves for the converter gain and other component stresses are obtained. A method of optimizing the converter under certain constraints is presented and a simple design procedure is illustrated by a design example. Experimental results are presented to verify the theory.

I. INTRODUCTION

R

ECENTLY, there is an increased interest in the area of resonant converters [ 11-[ 141 due to their advantages, i.e., higher frequency of operation, higher efficiency, small size, light weight, reduced EMI, low-component stresses, etc. Recent research has shown that series-parallel resonant converters (SPRC’s) (also called “using LCC-type commutation” or LCC-type parallel resonant converter) [2]-[6], [SI-[ 141 have a number of desirable features compared to series resonant or parallel resonant converters. The analysis and design of series-parallel resonant converters operating below resonance (leading pf mode) using the state-space approach has been presented in [4]. A similar approach is used in [11]-[13] to obtain the state-plane diagrams and numerical techniques have been used to compute certain parameters. Operation of such converters above resonance (lagging pf mode) results in a number of advantages [6]: elimination of d i / d t inductors and lossy snubbers, use of slow recovery diodes internal to MOSFET’s, reduced size of magnetic components, etc. An approximate analysis of series-parallel resonant converter operating above resonance using complex circuit analysis has been presented in [6] and [lo]. However, this analysis cannot predict properly the different waveforms (especially the resonating inductor voltage and the voltage across Ct ). A conventional parallel resonant converter (PRC) enters the discontinuous capacitor voltage mode (DCVM) under certain conditions [7]. It has been observed that same type of operating mode exists in the case of SPRC. Approximate analysis presented in [ 6 ] , [ 101 cannot predict the DCVM of operation. Analysis in such a mode of operation is also not available in the literature. It is convenient to present the converter gain and other peak component stresses with variation in the load Manuscript received January 14, 1991; revised August 7, 1992. This is a revised version of a paper presented at the IEEE Applied Power Electronics Conference and Exposition, Los Angeles, CA, March 1990. The author is with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, V8W 3P6 Canada. IEEE Log Number 9204148.

current [7]. Such results are not available for SPRC. Also, the optimum point of operation of series-parallel resonant converters for maximum efficiency, minimum inverter output current, or minimum kilovoltampere rating of tank circuit components, is not known (for both below as well as above resonance). Hence, the objectives of this paper are to 1) analyze the series-parallel resonant converter (for both above and below resonance) using the state-spaceapproach for both the continuous capacitor voltage mode (CCVM) and the discontinuous capacitor voltage mode (DCVM); 2) obtain closed-form expressions for steady-state operation; 3) derive the expressions for the converter gain, peak component stresses, and their ratings and to plot the converter gain and peak component stresses with variation in the load current; 4) optimize the converter under certain constraints and to present a simple design procedure; and 5) present experimental results using a laboratory model of the converter These objectives are achieved throughout the paper. Section I1 explains the operation and modeling of the converter. General solutions for the continuous capacitor voltage mode (CCVM) are presented in Section I11 using the statespace approach. Closed-form steady-state solutions are derived from the general solutions and the expressions for the peak component stresses and component ratings are derived. The expression for the critical load current above which the converter enters the discontinuous capacitor voltage mode (DCVM) is derived in Section IV. General solutions and steady-state solutions for the DCVM operation are also presented in this section. Section V presents the optimization of the converter under certain constraints. Design procedure is illustrated with a design example in this section. Experimental results are presented in Section VI, Although the analysis is presented for both below and above resonance modes, results and design presented concentrate on operation above resonance.

11. OPERATION AND MODELING

The original series-parallel resonant converier [2]-[4]. is modified in this paper to include the effect of high-frequency (hf) transformer by placing the capacitor C, on the secondary side (Fig. 1). In this circuit, the leakage inductances of the hf transformer are used profitably as part of resonating inductance. Since the inductor Ld is large enough to maintain

0885-8993/93$03.00 0 1993 IEEE

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 8, NO. I , JANUARY 1993

2

Hioh

freCCCnCv

T

i n v e t -t e r

T:“.t”l

Fig. 1. A dc/dc high-frequency link modified series-parallel resonant converter suitable for operation above resonance. (For operation below resonance, snubber capacitors C , are replaced by RC snubbers, and d i l d t inductors are included in each branch.)

(C) Fig. 3. Equivalent circuits of the converter shown in Fig. 1 during different intervals of operation. (a) Interval B. (b) Interval A. (c) Interval C.

A. Assumptions lnterv

Fig. 2 . Typical waveforms illustrate the operation of series-parallel resonant converter for operation above resonance in the discontinuous capacitor voltage mode. (Interval tc is absent for the continuous capacitor voltage mode.)

a constant current in the load R L , the rectifier input current is a square wave whose polarity changes depending on the instant of change in polarity of capacitor voltage wet. Therefore, a constant current model can be used for modeling the converter [4]. Fig. 2 shows some of the typical operating waveforms for operation above resonance (i.e., for lagging PF operation) mode. In CCVM of operation, there are two basic operating intervals (tc = 0) in each half period denoted by “interval B(t5)”and “interval A(tA).” In the case of DCVM, there is an additional interval (tc) in between intervals B and A. Constant current models used in the analysis for the intervals B and A are shown in Fig. 3. An additional model required for the DCVM (explained in Section IV) is shown in Fig. 3(c).

The following assumptions are used in the analysis. 1) The switches, diodes, inductors, and capacitors used are ideal. 2) The high-frequency (hf) transformer is represented by its leakage inductances (magnetizing inductance is large enough to be neglected) and is used as part of the resonating inductance. 3) The load current can be considered as constant, and hence, the constant current model can be used. 4) The effect of snubber capacitors is neglected. 5) Analysis is presented only for the continuous current mode.

B . Normalization and Notations Used All equations derived in this paper are normalized using the following base quantities:

111. ANALYSIS IN CONTINUOUS CAPACITOR VOLTAGE MODE This section reviews the analysis of the SPRC in the CCVM of operation.Assumptions used in the analysis are given in Section 111-A. Section 111-B presents the base quantities used for the normalization and notations used in the analysis. General and steady-state solutions are presented in Sections 111-C and 111-D, respectively. Gain of the converter is given in Section 111% and the ratings of components are derived in Section 111-F.

where

BHAT ANALYSIS AND DESIGN OF A SERIES-PARALLELRESONANT CONVERTER

L, is the externally added resonating inductance, L1 is the total leakage inductance of the hf transformer, and WO

= l/[LCe]l”

Y = Wt/Wo

In the above equations,

(6) (7)

triggering (or switching) frequency wt = 2 r f t radts. For convenience, all the parameters are referred to the primary side. Thus,the load current is represented by

I = Id/n.

3

(8)

Normalized voltages are represented by m with their subscripts to indicate the component, and the normalized inductor . example, current is represented by j ~For

For a given converter, knowing y (ratio of triggering frequency to resonance frequency, defined in (7)) and the load current J , the performance of the converter can be completely obtained once the above initial conditions are evaluated. However, to evaluate the above initial conditions, the duration of interval B must be known. This is obtained by equating the capacitor voltage equation mctB, at the end of interval B to zero. The duration of interval B thus obtained is

=Kr

-

sin-’

[c/dm]

-

tan-l(b/u)

(19)

where

(20) (21)

Esin(w,T,) b = E(1+ cos(w0T,)) c = b ZI(Ce/Ct) sin(w,T,)

U =

Additional subscripts are used to indicate the interval or the final value of each interval. For example, mcsA and mc,l indicate the voltage across the capacitor C, during interval A and its value at the end of interval A (see Appendices), respectively.

+ + zI(Ce / C s ) ( b / E )(W o T s / 2)

K=O

for y < l

and K = l

(22) for y > l .

It can be shown that the value of normalized inductor current at the end of interval B is given by

C. General Solutions General solutions have been derived in [4], and [14] for CCVM of operation using the constant current model and the state-space approach. The state space representation used in [4] can be represented by three basic differential equations as given in Appendix I. All the normalized (using base quantities of Section III-B) general solutions obtained have been represented by constants A, B , and C; these are summarized in Appendix 11.

j ~ = 1 - sin

+/cos (0,/2).

(23)

The normalized capacitor voltages at the end of interval B are

D.Steady-State Solutions Using the general solutions of Sections III-C, and matching the boundary conditions, closed-form solutions under steadystate operating conditions have been derived in [4], [ 141. These are valid for both above as well as below resonance with some change in the equation for the interval B duration as presented below. These expressions are simplified and are presented below in normalized forms. The inductor current and the capacitor voltages at the beginning of interval B are given by

LO = [J2(Ce/Ct)’- 11tan(8,/2)

Also, it can be shown that

For a converter to operate in the resonance mode (i.e., o be negative. Using lagging PF mode), initial current j ~ must (13), the boundary value of load current above which the converter operates in the lagging PF mode can be obtained as follows:

+ J~ (Ce / Ct )(Ce /cs) (0s/2

(13) m c s ~ = J(C,/C,)(Ce/Ct)[sin+/cos(Bs/2) - $1 (14) mctO = J(Ce/Ct)’sin +/cos(0,/2)

+ J ( c e / cs) (C,/ Ct )4. The combined capacitor initial voltage is

(15)

This is valid for y < 1 and in the CCVM. For y > 1, the converter always operates in the lagging PF mode. If the converter is operating in DCVM and y < 1, then the boundary value must be obtained using the expression for j ~ derived o in Section IV.

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 8, NO. 1, JANUARY 1993

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E . Converter Gain

A , B, and C constants of Appendix B. These expressions can

The average rectified output voltage or the converter gain can be derived as follows:

be

M = (Ce/Ct)(2/OS)[- sin 4/ COS(^',/^)

+ 41

+ 41.

= (Ce/Ct)(2/Os)[jLl

(28)

F . Component Ratings The expressions for the peak component stresses are derived below. I ) Peak Inductor Current: The current through the inductor attains its peak value either during interval B or during interval A, depending on the slope of the inductor current [7]. It can be shown that the value of peak inductor current is given by

2 ) Peak Voltage Across C,: Voltage across the capacitor C, attains its maximum value when the current through the inductor reaches zero. The angle at which the inductor current reaches zero is given by

+ sin-'

[CIA/(AfA

+ B,2A)1/2]

jLo > - -sin-' [ C I B / ( A f B B,2B)1/2] - tan-'(BIB/AIB), j L o < 0. - tan-l(B)lA/AIA),

+

[ c ~ ~ /+(~ 3A2 ~~) '~/

- tan-'(B3~/A3~),

(33)

= 7r +sin-' -

As explained earlier, for values of load current greater than a voltage across the capacitor Ct becomes particular value, Jcrit, zero and all the rectifier diodes start conducting. An equation for Jcritis obtained using the (A3) for interval A,

C,dv,,/dl = i L ( t ) - I .

J . - ' crit - JL1.

(34)

+

Jcrit=

[c~A/(A:~

t a n - l ( B ~ ~ / A 3 ~ ) , j ~
(35)

The value of Octp is substituted in the equation for mCtof interval B ~ ~ ~ e n Bd to i xobtain the peak value W t p . equation is used if j ~ > o - J and the interval A equation is used if j ~ < o -J. Expressions for other component ratings (RMS current rating of switch, etc.) have been derived using equations with

(36)

This equation shows that the slope of u,t is zero at t = t~ and results in

Substituting the values of ~ be shown that

~]

LO > - J

The reason for the occurrence of the discontinuous capacitor voltage mode (DCVM) is the same as that given for the parallel resonant converter [7]. For larger load currents this mode occurs and its analysis is presented in this section. The reason [7] for this is briefly explained as follows: For values of normalized load current J > Jcrit,the value of inductor current i~ at the end of interval B (instant at which the voltage uCt reaches zero) is less than the reflected load current I . This means that the parallel capacitor voltage vCtmust be negative to supply the difference in the current. However, the polarity of voltage across the capacitor cannot change instantaneously. Therefore, the voltage across the parallel capacitor remains at zero value for a certain interval of time ( t c in Fig. 2) during which all the rectifier diodes conduct with the capacitor Ct shorted (Fig. 3(c)). This interval ends when the inductor current Z L equals the load current I and the interval A begins (Fig. 3(b)) with vCthaving the same polarity as V A B (Fig. 2). Section IV-A derives the critical load current above which the DCVM occurs.The general solutions in DCVM are derived in Section IV-B. The steady-state solutions for DCVM are obtained in Section IV-C. Converter gain in the DCVM is derived in Section IV-D and plot of gain versus J is also presented. Plots of peak component stresses are ploted in Section IV-E.

A . Critical Load Current (32)

The value of Ocsp is substituted in the equation for mcs of Appendix B to obtain the peak value Mcsp.The interval A o 0 and the interval B equation is equation is used if j ~ > used if LO < 0. 3 ) Peak Voltage Across Ct: The interval at which the voltage across the capacitor Ct attains its maximum value is dependent on the slope of mCtat t = 0. It can be shown that the angle at which the maximum value of mCtoccurs is given by

Octp = -sin-'

by simp1e

I v . ANALYSIS IN DISCONTINUOUS CAPACITOR VOLTAGEMODE

(29)

The same expression has also been derived in [13]. The converter gain can be evaluated for a given J and y in a closed form using the above equations. A plot of M versus J is presented after the analysis of discontinuous capacitor voltage mode is given in Section IV.

Ocsp = 7r

Obtained

L from I

(37)

(23) and using (26), it can

[ - R + {R2- 4ST}1/2]/(2T)

(38)

where

T=

[(ce/ct) sin(0,/2) + (c=/c,) cos(0,/2)(0,/2)]~

+ c0s2(0,/2)

(39)

R = (ce/ct) sin(Os) + ~ ( C ~ / C S ) ( OCOS2(d,/2) S/~) (40) S = - sin2(0,/2). (404

BHAT: ANALYSIS AND DESIGN OF A SERIES-PARALLELRESONANT CONVERTER

B . General Solutions For DCVM, an additional interval called “interval C” occurs in between intervals B and A (Figs. 3(a) and (b)) presented in Section 111. During this interval the capacitor Ct is shorted, and the equivalent circuit used for the analysis is shown in Fig. 3(c). Using this equivalent circuit, equations for the inductor current and the capacitor voltages for the interval 0 < t’ < tc,t’ = t - t g , can be derived as iLc =

[ ( E- w,.,~)/z] (c,/c,)~/~ sin(w,t’)

+ i L 1 COS(W,t’) W,,C

(41)

= E - ( E - uc,1) COS(W,~’)

+ Z(Ce/Cs)1/2iL1sin(w,t’) VCtC

(59)

(42) (43)

=0

where w, = 1/(LCs)1/2.

(44)

i ~ and 1 wcsl are the values of inductor current and the voltage

across C, at the end of interval B. It must be noted that the resonance frequency of the circuit has changed to w, during this interval. The final values of inductor current, voltages 2 , uct2, across C, and Ct, are represented by i ~ 2 , 1 1 ~ , and respectively. All the normalized general solutions for the three intervals are summarized in Appendix C.

C. Steady-State Analysis Using the general solutions presented in Section IV-B and by matching the boundary conditions at the end of each interval, the values of initial inductor current and capacitor voltages have been derived. The procedure is the same as that used for the CCVM, but here an additional interval is present. Summary of the results obtained are given below. j L 0 = (ClBjl - C 2 b j 2 + C3bj3)/d mcso= (-cibmsi c2bms2 cdms3)/d

+

mcto =

( ~ ~ b m-t lcabmta

+

+ c3bmt3)/d

(45) (46) (47)

where

bjl = 1

+ cos(81c)

-

(Ce/C,)1/2sin(8B) sin(O1c) cos(OA) (48)

bj2 = ( C s / C e ) ’ /sin(8B) 2 sin(81c) sin(8.4)

(49)

(2)

1/2

bst3

=

COS(OB) sin(Olc) sin(Oa)]

-(ct/cs)coS(8A).

(65) (66)

To find the values of initial conditions, it is necessary to . is done evaluate the values of angles O B , Blc, and 0 ~ This as explained below. Due to the presence of a third interval, closed from solutions for OB and Oc can not be obtained. By equating the capacitor voltage, mctBrat the end of interval B to zero and by equating the inductor current at the end of interval C to the load current, J , the following two equations are obtained.

I 6

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 8, NO. 1, JANUARY 1993

Since both the foregoing equations contain terms in

OB and

Ole, they have to be solved simultaneously. But these two equations are nonlinear and are to be solved numerically. Also, initial conditions are to be evaluated at each step using the equations (45) to (66) presented above. The Newton-Raphson method was used to solve them numerically. Using the initial guess values of OBO and Oleo for OB and O ~ Cfor given J and y, improved values of OB and Ole were calculated using

-

4

7

a v

t:

3

C .A

2 2 U 111 4J

:’ C 0

where

U

0

[AOO] = -[JMol-l[Fol

NORMALIZED LOAD CURRENT, J ( P . U . )

Fig. 4. Curves of converter gain versus load current for various values of y for C,/Ct = 1. The DCVM operation occurs to the right of points marked with crosses. The converter operates above resonance for all values of y > 1, whereas such an operation for y < 1 is possible to the right of points marked with D.

Jacobian matrix

In the matrix elements shown above, subscript “0’ represents the evaluation of the functions or partial derivatives at BBO and 0 1 ~ 0 . The foregoing procedure is repeated until the functions FI and Fz approach zero with a specified error.

D.Converter Gain The average rectified output voltage or the converter gain for DCVM can be derived as

The plot of converter gain ( M ) versus normalized load current ( J ) for a capacitor ratio of C,/Ct = 1 is given in Fig. 4 for various values of y (ratio of triggering frequency to resonance frequency). These curves include both the CCVM and the DCVM operations. The value of Jbo has been calculated using (27) for all y except for y = 0.8. It must be noted that for y = 0.8, above-resonance operation occurs in the DCVM. Points on the J axis correspond to the short-circuit condition, whereas points on the M axis represent the opencircuit condition. For a given y and the load current, the operating point of the converter can be obtained from Fig. 4. When the switching frequency is very close to the total resonance frequency, i.e., wt r W O = l/(LCe)1/2,the gain of the converter tends to be very large (ideally infinite at open circuit for y = 1) at light loads since the damping becomes very small (ideally zero damping for open circuit). Therefore,

the gain curves for values of y close to unity are almost parallel to the M axis. This is evident by referring to the gain curve for y = 1.01. When the switching frequency is near the series resonance frequency (i.e.,wt N w, = l/(LC,)l/’), the converter gain is almost independent of the load current. This is verified by referring to the gain curve for y = 0.8, i.e., y, = (Cs/C,)1/2y = 1.13, for C,/Ct = 1. The gain curve for this value of y is almost parallel to the J axis as expected. The gain curves for values of y < 0.8 are not shown since the converter operates in the leading power factor mode up to very large values of load current.

E . Peak Component Stresses The equations for peak component stresses are same as those given by CCVM, but proper care must be taken to use the appropriate interval. Variations of peak inductor current, peak capacitor voltages across C, and Ct are plotted in Fig. 5 for variation in load current for various values of y. These plots include both the CCVM and the DCVM operation. Thus, for a given operating point, peak stresses can be directly read from these curves. The converter gain curves (Fig. 4) have been plotted in the output plane since this type of presentation gives more insight into the operation of the converter. A load line (may be nonlinear) [7] can be easily superimposed in the output plane to show how the frequency varies over the range of possible load condition. Similarly, if the load voltage is regulated at a particular value of hl corresponding to the maximum load current (selected as explained in the design section), then the variation in the frequency required for variations in the load current for the complete load range can be predicted by drawing a straight line parallel to the J axis at the selected value of M . For each operating point ( J and y) that can be easily read from the gain curves, the peak component stresses corresponding to these operating points can be read directly from Fig. 5.

BHAT: ANALYSIS AND DESIGN OF A SERIES-PARALLEL RESONANT CONVERTER

7

E. z a w

2

a

E 0 Zl

2 H

2 [L

NORMALIZED LOAD CURRENT, J ( P . ‘ J . ) (a)

-_

3

a

-

6

a U Ln

“

5 U1

U

w

4

0 U

“

3 V = 0.8 .- - - - - - -

-

2

z 2

i?: W

r:

:a

5 1 NORMALIZEDLOAD CURRENT, J ( P . U . )

NORMALIZED LOAD CURRENT, J ( P . U - 1 (C)

(b)

) as switch peak current) versus Fig. 5. Peak component stresses verses normalized load current J(p.u.). (a) Normalized peak inductor current ( J L ~(same versus load current .J. load current J . (b) Normalized peak voltage acrossC,( Adcsp)versus load current .J. (c) Normalized peak voltage across Ct(-%lctp)

v.

OPTIMIZATION AND

DESIGNOF THE

CONVERTER

The series-parallel resonant converter is optimized for minimum inverter output current, maximum efficiency and minimum kilovoltampere rating of tank circuit components per kilowatt of output power. An optimization function is introduced in order to obtain an optimized design for all the three constraints. Optimization procedure is illustrated by the following example.

A . Design Example The specifications of the converter are as follows: Minimum input supply voltage: Vmin = 230V. Output voltage of the converter: V, = 48 V. Output power of the converter = 1kW. Switching frequency ft = 100 kHz. Base quantities are chosen as per Section 111-B.Motorola MTM15N50 MOSFET’s are used as the switching devices. Using these specifications, the peak inductor current, total kilovoltampere rating of tank circuit per kilowatt of output power, and efficiency of the converter were calculated using a

computer program at the maximum output power for different values of y. The expressions used for calculating the rms and average values (currents and voltages) across different elements were obtained using basic definitions of average and rms values (similar to the steps given in [14], [15]). The expressions for voltages and currents during different intervals given in the Appendices are used for this purpose. The capacitor ratio C,/Ct = 1 was selected, since the output voltage regulation from full load to very small load required very large variation in switching frequency with larger capacitor ratios. In calculating the efficiency of the converter, the following data (in addition to the specifications given earlier) are assumed: On resistance of MOSFET’s = 0.4 0. Q of the resonating inductor used = 90. Voltage drop across the diodes = 1 V. HF transformer losses = 1% of output power. Using the loss calculations procedure given in [15], total losses in the converter was calculated for different values of y and for different load conditions. The variation in efficiency

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 8, NO. 1, JANUARY 1993

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18

17 lC 15 14 13 12 11 10 9

a 7

G 5 4 5 6 6.5 NORMAL1 ZED LOAD CURRENT, J ( P. U. )

(a)

(b)

Fig. 6 . (a) Variation in efficiency with percentage of load current for two values of y. The converter is operating in the DCVM for maximum power output (i.e., 100% load) with .I = 4.3 for y = 0,s. and .I = 3 . 2 5 for y = 0.S5. The load voltage is regulated at the values corresponding to maximum output power. (b) Variation of peak inductor current(lL1,), kilovoltampere rating of tank circuit per kilowatt output, and optimum function, with .I for y = 0.8, for the converter designed at maximum power output.

for power control from full load to 20% load for two values of y are shown in Fig. 6(a). The load voltage is regulated at the full-load value by increasing the switching frequency from the full-load value as the load current decreased. The converter is operating in the DCVM for the full load J chosen. The converter works in the lagging pf mode for the entire load range. The frequency variation required for the voltage regulation from full load to 20% load is 100-142.2 kHz for y = 0.8, whereas for IJ = 0.85, the corresponding frequency change required is 100-134.2 kHz. It can be observed that the decrease in efficiency at light loads is higher for values of y greater than 0.8. Hence y = 0.8 was chosen. From Fig. 4, it can be seen that the converter is operating above resonance for J > 4.15. However, for :y = 0.8. Jcrit= 2.57. Therefore, the converter is operating in the DCVM for operation above resonance. Plots of peak inductor current, efficiency and total kilovoltampere rating of tank circuit per kilowatt of output power with variations in J (for above resonance operation) are shown in Fig. 6(b). In order to optimize all the above parameters, an optimum function [ 1.51 is given as follows: Optimum function = Efficiency/[ILp . (kVA/kW)]. This function is also plotted in Fig. 6(b). For the same power output, an increase in the value of J results in an increase in the inverter output peak current which decreases the efficiency of the converter. The kilovoltampere rating of the tank circuit per kilowatt of output power increases with J whereas the optimum function decreases. Since the converter is required to operate in the above resonance mode, the optimum point of operation at full load is close to the boundary where the converter changes its operation from below resonance to above resonance mode. However, enough tum-off time for the switches ensures safe operation. Therefore, in the present example, J = 4.3 is chosen as the optimum point. Use J = 4.3 = I / ( E / Z ) . Here, I d = 1000/48 = 20.83 A.

For J = 4.3 and y = 0.8, A4 transformer tums ratio is given by

N

1.036 p.u. Hence, the

n = 1.036 x 115/48 = 2.48

and I = 8.4 A.

.'.Z

pv

58.86 R.

Values of L and G, obtained are L = 75 pH and C, = 0.02163 pF. Since C,/Ct = 1,C, = C, = 0.04326 pF. Other component values obtained from the design curves are

IL,= 13.853 A,

Vcsp= 503.3 V,

V,,, = 235.33 V (on primary side).

vr.

EXPERIMENTAL RESULTS

Based on the foregoing design method, an experimental converter was built in the laboratory. Fig. 7 shows the component values used in the experimental converter. This converter was operated above 100 kHz. Different waveforms obtained are shown in Fig. 7. Figs. 7(a)(i) and 7(a)(ii) show the waveforms at an output power of about 720 W (efficiency was about 87%). It can be seen that the converter is operating in the DCVM. However, due to the rectifier drops, their recovery times and wiring inductances, the capacitor voltage during interval C is not exactly zero. The output voltage was about 44 V. This was due to the fact that the component values used were not exactly the same as those obtained from the design. The tums ratio used for the hf transformer was also different from that obtained in the design. Also, the drops in the semiconductors and losses in the circuit were not taken into account in the design. Table I compares the experimentally measured results for the DCVM operation with those predicted by theory. It can be seen that there is a reasonably close agreement between the two. It must be noted that the load voltage obtained is less than the predicted value due to the reasons explained above. Figs. 7(b)(i) and (ii) show the waveforms corresponding to 340

BHAT: ANALYSIS AND DESIGN OF A SERIES-PARALLEL RESONANT CONVERTER

9

Fig. 7. Experimental results obtained with an experimental unit. (a) Discontinuous capacitor voltage mode, RL LV 2.7 12: (i) waveforms 1 ' 1 ~(50 V/div) and inverter output current 11. (1 V/div., 1 V = 5 A). (ii) voltage across capacitor Ct( I',.,, 50 V/div.) and I L (1 V/div., 1 V = 5 A). (b) Continuous capacitor voltage mode: Rr. LV 5 . G S 0 : (i) and (ii) same waveforms as (a)(i) and (ii), voltage scale:(. 1~ (50 V/div), r,.f (20 V/div), current scale = 2.5 A/div. Time scale = 1 ps/div, in all the waveforms. (Details of converter: Switches used-MTM 15N50 MOSFET's, feedback diodes, intemal to the MOSFET's, input dc 230 I - , L , voltage TO pH, L1 = S jrH, C , = 0.012G jtF, Ci LV 0.226 jtF. rectifier diodes BYW31/200, transformer tums ratio LV 16/7.) TABLE I COMPARISON OF EXPERIMENTALLY OBTAINED RESULTS WITH THE THEORETICALLY PREDICTED VALUES FOR OPERATION IN DCVM (FIG. 7(a)), A N D CCVM (FIG. 7(b)) (DETAILS OF THE CONVERTER ARE %'EN IN FIG. 7) DCVM y

Parameter Load voltage, 1 Peak current through L,. i ~ p Peak voltage across C,. I , , Peak voltage across C, . t'<.t,)

*,,

CCVM

= 0.85. .I = 3.7.5 p.u.

y

= 1.029. .I = 1.77.5p.u.

Experiment

Theory

Experiment

Theory

44 v 11 A 455 v 85 V

52 v 10.3 A 441 V 98.4 V

44 v 7.5 A 275 V 70 V

48.1 V 7.13 A 237 V 81 v

W output power (efficiency was about 84%). The switching frequency was increased to keep the output voltage constant. It can be observed that the inductor peak current has decreased to about 7.5 A. Also, the converter is now operating in the CCVM. A comparison between the theoretically predicted values with the measured values for CCVM operation are also presented in Table I.

VII. CONCLUSIONS Analysis of a series-parallel resonant converter for operation in both theCCVM and DCVM has been presented using the state-space approach. Closed-form solutions under steadystate conditions have been derived. In spite of the complexity in the equations, solution of two nonlinear equations result in complete closed-form solutions for the DCVM of operation. Converter gain and peak component stresses have been plotted with variation in the load current aiding simple prediction of performance for a given converter. An optimization pro-

cedure with a design example has been presented. It was observed that the optimum point lies in the DCVM. The converter entered the CCVM of operation as the load current decreased. Peak voltages across Ct and L, are much higher than those predicted by simple complex circuit analysis [6], [lo]. Experimental results have been presented to verify the analysis. APPENDIXA STATE - SPACEREPRESENTATION The models shown in Fig. 3 can be used to obtain the following differential equations [4], [13], [ 141:

D = 1 for interval A and D = -1 for interval B.

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 8, NO. 1, JANUARY 1993

APPENDIXB SOLUTIONS I N SUMMARY OF GENERAL CONTINUOUS CAPACITOR VOLTAGE MODE

A . Interval B Equations During 0

APPENDIXC SOLUTIONS FOR SUMMARY OF GENERAL DISCONTINUOUSCAPACITOR VOLTAGEMODE

< t < tB

A . Interval B , During 0

B. Interval A Equations During 0 < t‘ < tA where t’ = t - tB

LA

+ B ~ cos(w,t’) A + ClA = A 2 ~ (1 cos(w,t’)) + BPAsin(w,t’) = A ~ sin(w,t’) A

+ C2A(wot’-) + mcsl A 3 ~ ( 1 cos(w,t’)) + B34 sin(w,t’) f C3A(wot’) + met1

B. Interval C Equations During 0

< t‘ < t c ,

where t’ = t - t g :

(B4)

+

+

(B5)

j ~ =c A l c sin(w,t’) B l c cos(w,t’) C l c mcsc = Azc(1 - cos(w,t’)) B ~ sin(w,t’) c

036)

mctc = 0.

mcsa4

mctA =

< t < tB

+

+ CZC(WSt’) + mcs1

-

(C4) (C5) (C6)

C. Interval A Equations

where

A1B = [I - (‘%SO + m~to] B1B = j L 0 (Ce/ct)J C1B = - ( c e /ct) J

During 0

+

LA 037)

A2B = (Ce/Cs)AlB B2B = (Ce/C.s)BlB c 2 B = (Ce/Cs)ClB

< t < t A , where t = A ~ sin(w,t) A

mcsA=

mctA =

(B8)

A3B = (Ce/Ct)AlB

The variables j ~ o mcSo, , and mctO are the normalized inductor current, the normalized voltages across the capacitors C, and Ct, at the beginning of interval B. Corresponding quantities at the beginning of interval A are represented by j ~ 1mcSl, . and mctl, respectively.

= t - tB - t c :

+

+

B I Acos(w,t) C I A (C7) A2,4(1- cos(w,t)) B ~ sin(w,t) A

+

+ C 2 A ( W o t ) + mcs2 A 3 ~ ( 1 cos(w,t)) + B3A sin(w,t) + C3A(w0t) f mct2

(C8) (C9)

where interval B constants are defined in the same way as those given in Appendix B. Other constants are defined below.

BHAT: ANALYSIS AND DESIGN OF A SERIES-PARALLEL RESONANT CONVERTER

mcSo,and mctO are the normalized inductor current and the normalized voltages across the capacitors C, and C, at the beginning of interval B. Corresponding quantities at the beginning of intervals C and A are represented by j,, , mc,l, mctl and j ~ 2 mcs2 , mct2,respectively. J’LO,

~

REFERENCES [ l ] K. Kit Sum, “Recent developments in resonant power conversion,” Intertech Communications Inc., 1988. 121 A. K. S. Bhat, “High frequency link photovoltaic power conditioning system,” M.A.Sc. thesis, Dept. of Electrical Engineering, University of Toronto, Toronto, Ontario, Canada, Sept. 1982. [3] J. Chen and R. Bonert, “Load independent ac/dc power supply for high frequencies with sine-wave output,” IEEE Trans. Industry Applications, vol. IA-19, no. 2, pp. 223-227, Mar./Apr. 1983. [4] A. K. S. Bhat and S. B. Dewan, “Analysis and design of a highfrequency converter using LCC-type commutation,” IEEE Trans. Power Electron.. vol. PE-2, no. 4, pp. 291-301, Oct. 1987. “Steady-state analysis of a high-frequency inverter using LCC[5] -, type commutation,” Canadian J . Elec. Computer Eng., vol. 13, no. 2, pp. 153-159, Feb. 1988. 161 R. L. Steigenvald, “A comparison of half-bridge resonant converter topologies,” IEEE Trans. Power Electron., vol. PE-3, no. 2, Apr. 1988, pp. 176182. [7] J. D. Johnson and R. W. Erickson, “Steady-state analysis and design of the parallel resonant converter,” IEEE Trans. Power Electron., vol. 3, no. I , pp. 93-104, Jan. 1989. [SI A. K. S. Bhat and S. B. Dewan, “A generalized approach for the steady state analysis of resonant inverters,” IEEE Trans. Industry Applicarions, vol. 25, no. 2, pp. 326-338, Mar./Apr. 1989. 191 R. Bonert and P. Blanchard, “Design of a resonant inverter with variable voltage and constant frequency,” in IEEE-IAS Con$ Rec., 1988, pp. 1003-1 008. [ 101 A. K. S. Bhat, “A resonant converter suitable for 650V dc bus operation,” in IEEE Applied Power Electron. Cot$ Rec., 1989, pp. 231-239. [ l I ] I. Batarseh, R. Liu. C. Q. Lee, and A. K. Upadhyay, “150 Wand 140 kHz multi-output LCC-type parallel resonant converter,” in IEEE Applied Power Electron. Conf Rec., 1989, pp. 221-230.

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[12] I. Batarseh, R. Liu, and C. Q. Lee, “Design of parallel resonant converter with LCC-type commutation,” Electron. Lett., vol. 24, no. 3, Feb. 1988, pp. 177-179. [13] I. Batarseh and C. Q. Lee, “High-frequency high-order parallel resonant converter,” IEEE Trans. Ind. Electron., vol. 36, no. 4, pp. 485498, Nov. 1989. [14] A. K. S. Bhat, “High frequency resonant converters for dc to utility interface,” Ph.D. dissertation, Dept. of Electrical Engineering, University of Toronto, Toronto, Aug. 1985. [15] A. K. S. Bhat and M. M. Swamy, “Loss calculations in transistorized parallel resonant converters operating above resonance,” IEEE Trans. Power Electron., vol. 4, no. 4, pp. 391401, Oct. 1989.

Ashoka K. S. Bhat obtained the B.Sc. degree in physics and math from Mysore University, India, in 1972. He received the B.E. degree in electrical technology and electronics and the M.E. degree in electrical engineering from the Indian Institute of Science, Bangalore, in 1975 and 1977, respectively. He also obtained the M.A.Sc. and Ph.D. degrees in electrical engineering from the University of Toronto, Ontario, Canada, in 1982 and 1985, respectively From 1977 to 1981. he worked as a scientist in the Power Electronics group of the National Aeronautical Laboratory, Bangalore, India, and was responsible for the completion of a number of research and development projects. He was also a research scholar at the Indian Institute of Science during from 1980 to 1981. After working as a postdoctoral fellow for a short time, he joined the Department of Electrical Engineering, University of Victoria, BC, Canada, where he is currently a Professor of Electrical Engineering and is engaged in teaching and conducting research in the area of power electronics. He has been responsible for the development of the Electromechanical Energy Conversion and Power Electronics Laboratories. Dr. Bhat is a Fellow of the Institution of Electronics and Telecommunication Engineers (India), and a member of the Association of Professional Engineers of British Columbia.