IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, VOL. IECI-15, NO. 1, NOVEMBER 1968

the weight. The power used in this vacuum tube modelwas less than 150 watts. In addition, solid-state opera-tional amplifiers have much less noise or drift problemsso increased accuracy would result. The accuracy (± 1percent) of the input resistors and integrating capacitorsmay be questioned. Increased accuracy of these com-ponents should not have made any difference since thecomputer was calibrated for a given pulse rate. Themain requirement here is that the values chosen remainconstant.The largest error was that due to the nonlinear rela-

tionship between input and output. To improve thisweakness, two possibilities are available. One would be

to use better operational amplifiers since the amplifiersused are considered utility grade. The other improve-ment would be to use an expanded scale for the input.This would reduce the range of voltages at which largeerrors were introduced. One additional improvementwould be to change relays. The use of mercury-wettedrelays would eliminate all bounce problems as experi-enced in this inodel.

BIBLIOGRAPHY[11 P. Lefferts, "Operational trigger for precise control," Electronics,

pp. 50-55, November 2, 1964.[21 C. F. Morrison, Jr., Generalized Instrumentation for Research and

Teaching. Pullman, Wash.: Washington State University,1964, pp. 11-56.

Resonance Commutated Thyristor de ChopperDesign Criteria

IOAN I. PONNER

Abstract-A procedure is developed for determining optimumvalues of the LC commutating circuit components in the resonancecommutated thyristor dc switch or chopper circuit. Based on an ap-proximate analysis of the circuit by Gutzwiller et al. [1], a more accu-rate expression is obtained in parametric form and solved graphically.With this solution, values for the series inductor and commutatingcapacitor can be selected which minimize component size and costfor given values of thyristor turn-off time and peak load current. It isshown that for a given reverse-biasing time, the influence of the valueof commutating capacitor on rate of rise of voltage across the thyris-tor is not pronounced.

INTRODUCTION

N a previous work [1] a resonance commutated dcI switch with series inductor was presented (Fig. 1).

In operation the polarity of the voltage across thecapacitor C corresponds to the SCR, conducting state.When SCR2 is triggered to interrupt load current, ca-pacitor C is connected across SCR,. Thus the capacitorvoltage reverse-biases SCR1, and the current through itis interrupted. Capacitor C resonantly dischargesthrough CR1 and L, reversing its voltage. With loadvoltage now applied to R2, the capacitor charges withopposite polarity from that shown in Fig. 1. WhenSCR, is triggered the action is reversed and SCR2 isturned off. For an inductive load, the free-wheelingrectifier CR1 must be connected across the load to dis-charge the inductive energy in the load when its circuit

Manuscript received August 11, 1967; revised May 3, 1968.The author is with the Polytechnic Institute of Bucharest,

Bucharest, Romania.

is interrupted. For satisfactory turn-off of SCR1, thereverse-biasing time must be greater than the maximumrequired turn-off time of SCRj.

In the cited work it is assumed that 1) the thyristorsand diodes are perfect; that is, infinite resistance in theOFF state and zero resistance in the ON state; 2) reverserecovery of SCRs is instantaneous and "sweep out"current is negligible; 3) the inductor has zero seriesresistance; and 4) during resonant discharge of thecapacitor C, R2 is assumed to be infinite.On the basis of these simplifying hypotheses the volt-

age across capacitor C from the beginning of the turn-off of SCR, is given by

= E(ot ID sL t_\v, = E tcos +, sin _J . (1)

Turn-off time is the interval from t =0 until the volt-age v, reaches zero. For v0=0, from (1), t1 is given by

t = V7LC tan-' K/tC) (2)

The supply voltage E, the maximum load current ID,and the required turn-off time tc of the SCRs are usuallyknown, and the circuit constants L and C are to be de-termined.

In [1] these circuit constants are determined as fol-lows.

28

PONNER: RESONANCE COMMUTATED THYRISTOR DC CHOPPER

Fig. 1. Resonance commutated dc switch withinductor in series with load.

1) It is assumed that tI0jVLC<0.5; then one canconsider with an error less than 10 percent thattan (to/+/LC) ;tc/ V/L C, and from (2) one gets

pc2 7r (&(Percent) = tan-,1-- 1) X ioo7r 2 (6)

= - 36.2 percent.

It is interesting to remark that this error is constantand independent on the circuit parameters. This sug-gests the idea of making adequate corrections in (3) and(4), so that the resulting theoretical reverse-biasing timeof SCR1 may be quite the imposed one.From (6) a modified turn-off time t,* can be computed,

which if considered in the previous design formulas forL and C, the actual reverse-biasing time t,' will be equalto the desired turn-off time t4. For this modified turn-offtime we find

*-= t_ __- 1.57tc.1 + 8/'100 (7)

UcIDE (3)

2) It is assumed that the reverse-biasing time of theSCR1 is one-fourth of the resonance period of the cir-cuit; t, = (r/2) V\LC, which yields

4Et.L= * . (4)

CORRECTION TO REVERSE BIAS TIME

Considering for L and C the values given by (3) and(4), the actual reverse-biasing time t1' of the SCR1 willalways be smaller than calculated. As a result, there is arisk that the triggering of SCR2 will not turn off SCR1,and both SCR1 and SCR2 will conduct simultaneously.The difference between the actual and the desired

reverse-biasing time of the SCR1 is due to the fact thatthe hypotheses on which (3) and (4) are based are con-tradictory. Indeed, if in order to deduce (4) we assumethat the reverse-biasing time of SCR1 is t, = T/4, thent4/VNLC=7r/2 and tan (t,/V\LC) = cc. Obviously theassumption tan (t4/VLC)O t0j/VLC on which (3) isbased is far from being valid. One may note that thecapacitor value resulting from (3), as will be shownlater, represents the minimum value of this capacitor,corresponding to L= cc.

With these two contradictory hypotheses, the errorin the actual reverse-biasing time of the SCR1 is

IC' - to&(percent) =---X 100

_ E !C\(5)VLC tan-1 V- - (t5

-___- - x 100.tC

When introduced into (3) and (4), this yields

IDt,C= 1.57 -

E(8)

and

Et,L = 0.64 -

ID(9)

M\IORE ACCURATE CALCULATION OF L AND C VALUES

For a more accurate calculation of the circuit parame-ters we shall consider also the losses in the resonant cir-cuit, assuming that the equivalent resistance R of thecircuit is located in the inductor L and in the SCR whichis turned on. We shall go on considering that the diodesCR1 and CR2 are ideal.When SCR2 is triggered, current ID is flowing in L and

capacitor C is charged to E, as indicated in Fig. 1. Forthis case the equivalent circuit of Fig. 2 holds. Withthese initial conditions, the capacitor voltage v, is found[2 ] to be given by

VC = Ee-t [cos wt + (- - ) sin wt] (10)

where1 RL=- -

2 L

IO=

LC

W; = -\/Co02 - 0-2.

(1 1)

(12)

(13)

Likewise, the turn-off time t4, when v, =0, is given bythe relation

By replacing the values of L and C given by (3) and(4), we obtain

1

tnct=(IDIwCE) - (lo(14)

29

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION, NOVEMBER 1968

R

f -at t=o I at t=O

Fig. 2. Equivalent circuit of the dischargingof capacitor C.

It will be noted that for the case of R = 0, a = 0, and (14)reduces to (2).From (14) an infinity of L-C pairs can be found which

will result for a desired value of t, with given values ofID, E, and R. Likewise, the values of L and C given by(8) and (9) represent only one pair out of an infinitywhich satisfy the simplified (2). An optimum L-C pair,in terms of minimum component sizes and costs for agiven application, can be determined by a trial and errorprocedure. To facilitate this procedure it is convenientto re-express (14) in parametric form and plot the solu-tion in graphical form.

PARAMETRIC RELATION BETWEEN L AND C

Considering the L-C pairs which satisfy the simplified(2) it is seen that L may vary from 0 to oo, while C isrestricted to values greater than a minimal value Cmin.From (2), for L- >oo, we obtain

IDtcCmin = E * (15)

E

Obviously, for a given application, the optimum valuefor the capacitor must be larger than its minimal value.Thus, it is convenient to define a parameter a, withvalue greater than unity, by

CEC =aC..min or a =

IDtc(16)

Fig. 3. Diagram for the calculation of the L-C pair.

tanVI + 4A

-\/aX

I

13 2a(1 +,) 4A

2(l + ) X

(20)

This equation may be solved by conventional graphicalmeans for X =f(a) with f as a parameter. The results aregiven in Fig. 3 for B = 0, 0.01, and 0.03.

It will be noted that for the special case of R=0,j3=0, and (2) reduces to

1 / atan - -lVc/aX X

PARAMETRIC EXPRESSION FOR LC PRODUCT

(21)

It is likewise convenient to define a parameter X whichrelates the value of inductance L to the approximationof (4):

Etc LIDL = X- or X =- X

ID EtC(17)

Since the value of L depends on the value chosen for C,the parameters a and X are interrelated.

If, in addition, a parameter is defined by

R

and it is assumed that the maximum value of ID is

given by

E E 1ID = (19)

R1+R R11+13

then it is possible to express (14) in parametric form as

As a further aid in determining optimum values forL and C, a parametric expression for the LC product isuseful. From (15), (16), and (17) which define a and X,the relation

LC = aXt02

is obtained. If pt is defined by A =aX, this becomes

LC =-t2.

(22)

(23)

In Fig. 4, , is plotted as It=f(a), for the simplifiedcase of lossless inductor, diodes, thyristors; e.g., R=1=0.

INTERPRETATION OF PARAMETRIC CURVES

From Fig. 3 it is seen that X falls rapidly for increasinga in the range 1 <a < 3. With increased losses (increas-ing values of ,B), the values of X and the required induc-tance are reduced. This reduction in L is of course accom-

panied by an increase in circuit losses and correspondingreduction in efficiency.

30

/a

x

PONNER: RESONANCE COMMUTATED THYRISTOR DC CHOPPER

A5

4

3

2

I

0

Lv

:12

4 6 9

It2d-~~~~~~~~~~~~~~~~~~~. I t. af

2 4 6 d fo a

Fig. 4. Diagram for the calculation of the product LC and the rateof voltage rise across the turn-off SCR.

Referring to the plot of A vs. a in Fig. 4, it is seen thatA falls very rapidly for increasing values of a in therange I <a< 2. For a> 3, A is approximately constantat a value of approximately 0.5.

It is probable for most applications that the optimumvalue of a will be greater than 3. It is, however, neces-sary to resort to trial and error procedures to determinethe component values which will result in minimumtotal size and cost for any given application.A factor not considered in the preceding analysis is

the "sweep-out" or reverse recovery current in the thy-ristor. The loss of commutating capacitor charge dueto this current is usually small. If required, it can becompensated for in the determination of L and C valuesby considering the initial value of capacitor voltage tobe correspondingly reduced.

RATE OF REAPPLICATION OF POSITIVE ANODE VOLTAGE

In addition to the requirement that reverse-biasingtime interval be in excess of the maximum thyristorturn-off time, it is also necessary that the rate of reap-plication of forward voltage across the thyristor bewithin the device ratings. An excessive rate of increaseof forward voltage can cause the thyristor to re-fire,with resultant loss of chopper action [3]- [5].An expression for the rate of reapplication can be

readily obtained for the simplified case of 0=0, as fol-lows. Taking the time derivative of (1), vc' (t) is given by

V7'Q(t) - -+E j + sin( - + 4)where

_ ID /L1 - tan-'-l-

E 'Vc

The maxi'mum value of v,' occurs for

2

(24)

Thus

Vtc maxttenal -E |LC + CE)

Expressing this 'in parametric form

EVIc max - -,

IC

where

v I 1+

Y

= - +-

(26)

(28)

Values of v vary over the range I < v <Xr/2 as a variesfrom 1<a< oo (see Fig. 4 for plot of v =f(a)). Fromthe curve it can be seen that for the range of a> 3, whichis of greatest interest for most applications, the value ofv varies only slightly from 1.35 to 1.5. Thus the value ofV,' is essentially independent of the particular values ofL and C, but is inversely proportional to t,. Dependenton the ratings of the thyristor devices being used for aparticular application, it may be necessary to determinethe required value for t on the basis of v' ratings ratherthan turn-off time.

CONCLUSIONSA resonance commutated dc switch with inductor in

the load circuit is discussed. Approximate design for-mulas given in a previous work are based on contradic-tory hypotheses, and if they are applied directly anerror of -36.2 percent results for the reverse-biasingtime of the turn-off SCR. These design formulas areconveniently corrected in order to obtain an accuratereverse-biasing time. However, it must be noted thatthese corrected design formulas give the values of onlyone L-C pair out of an infinite number that satisfy thereverse-biasing condition.A parametric solution for the L-C pair is developed

and solved graphically (Flig 3), allowing the choice ofan optimal L-C pair with convenient cost or size andminimal energy losses in the series inductor. This is ofinterest in the case of powerful dc switches.The influence of the commutating capacitor value on

the rate of voltage rise across the turn-off SCR is ex-amined and given in a general form (Fig. 4). It may beseen, this influence is not pronounced, but v' is inverselyproportional to the selected reverse-biasing time.

REFERENCES[1] F. E. Gentry, F. W. Gutzwiller, N. Holonyak, Jr., and E. E. von

Zastrow, Semiconductor Controlled Rectifiers: Principles and Ap-plications of p-n-p-n Devices. Englewood Cliffs, N. J.: Prentice-Hall, 1964, pp. 284-287.

(25) [2] W. H. Chen, The Analysis of Linear Systems. New York:McGraw-Hill, 1963, p. 277.

[31 "Silicon controlled rectifier manual," General ElectricCo., Auburn, N. Y., 1967.

[4] 'Silicon controlled rectifier designer's handbook," West-inghouse Electric Corp., Youngwood, Pa., 1964.

[5] A. Hoffmann and K. Stocker, 'Thyristor-Handbuch. Der Thy-ristor als Bauelement der Leistungselektronik," Siernens-Schuck-ertwerke Aktiengesellschaft, Berlin-Erlangen, 1966.

+ 41-\/ LC