analysis of magnetic amplifiers with diodes

1 / 3 ) + £ É ( Ë - 2 / 3 ) + £ É ( Ë -

1 ) + £ é ( ç - 4 / 3 ) + Å é ( ç - 5 / 3 ) +

Ä ( * - l / 3 ) - £ r t » - 2 / 3 ) +

4 / 3 ) + £ 2 ( « - 5 / 3 ) ] (158)

To illustrate the dynamic behavior of the amplifier, the response to a unit step input for the case when Ri = R2= 1/8 RL

is shown in Fig. 13.

Conclusion

A general and organized method of magnetic-amplifier analysis has been developed. Although this method of analysis may not be the shortest one for a particular circuit, it provides a general means for analysis and gives a clearer insight to the over-all properties of mag-netic amplifiers. Magnetic amplifiers with diodes can also be analyzed using the same method, as is demonstrated in refer-ence 2. In particular, this method of analysis is useful in determining general properties of magnetic amplifiers.

Appendix. Matrix Relations for the Basic Magnetic-Amplifier

Element

The matrices of equations 43 through 46 are as follows:

Z =

H=

Ri+Rm — Rm/z Rm -Rm/z

0 Ri 0 0

Rm -Rm/z R*-\-Rm -Rm/z

0 0 0 R*

Ri+Rt 0 Rm/(R2+Rm) 0 -Rz/z R} -lRm/(R2+Rm)]/z 0 -R*/R* 0 1/(R2+Rm) 0 [Rt/R2]/z 0 [Rm/(R%+Rm)]/R* 1/R*\

(159)

(160)

where

Rt — RmR%/{Rm"\-R%) (161)

G -

l/(Ri+Rm) 0 RJR, 0 Rm/Ri(Ri+Rm)z 1/Ri -R4/R1S 0 R*/(Ri+R*) 0 R2+R* 0 -RtftRi+RJz 0 -RJz Rt\

where

Ri ~ RmRi/(Rm -\~Ri) (163)

The elements of the A, B, C, and D matrix are given by the following expressions:

£ =

iKRw+RuRm \a/z

0 0

(RiRm -^-RiRm+R\R%)/Rm 0 \aR2/z 0

\VRm 0| \(l+a)/Riz 0

\(Rm-\-Ri)/Rm \Rt(l+a)/R*

0 0

(164)

(165)

(166)

(167) D =

where a is an arbitrary constant. The above expressions for A, B, C, and D were deter-mined by solving the original equations. It was found that any value of the constant a will satisfy these equations. The arbi-

(162)

trary nature of a is a result of the property that port voltages and currents during the gating period do not affect port voltages or currents during the control period.

Ref erences

1. IMPOSSIBLE BEHAVIOR OF NONLINEAR N E T -WORKS, R. J. Duffin. Journal of Applied Physics» New York, N. Y., vol. 26, 1955, pp. 603-05.

2. ANALYSIS OF MAGNETIC AMPLIFIERS, P . R. Johannessen. Report 7848-R-2, Servomechanisms Laboratory, Massachusetts Institute of Tech-nology, Cambridge, Mass., Oct. 1958. 3. COMMUNICATION NETWORKS, VOL. II (book), E. A. Guillemin. John Wiley & Sons, Inc., New York, N. Y.. 1935, pp. 145-50. 4. ANALYSIS OF MAGNETIC AMPLIFIERS BY THE U S E OF DIFFERENCE EQUATIONS, P . R. Johannes-sen. AIEE Transactions, pt. I (Communication and Electronics), vol. 73, 1954 (Jan. 1955 section), pp. 700-11.

Analysis of Magnetic Amplifiers With

Diodes

PAUL R. JOHANNESSEN ASSOCIATE MEMBER AIEE

IT HAS B E E N demonstrated1 that the behavior of magnetic amplifiers without

diodes can be analytically described by a set of linear difference equations relating the average values of voltages and cur-rents at the ports. For most circuits this representation gives values which are accurate to within 1 or 2 % of actual even for a large variety of waveforms of port and supply voltages. The reason for this rather remarkable accuracy of represen-tation can be attributed to three fac-tors:

1. The control circuit resistance Ru the output circuit resistance R2, and the load

resistance RL are small compared to the magnetizing resistance Rm.

2. Almost all practical amplifiers of this type employ an interconnection of basic elements such that quiescent currents tend to cancel at the ports.

3. The state of the saturable inductor, saturated or unsaturated, depends on the integral of the inductor voltage rather than the inductor voltage itself.

The first factor insures that the satura-ble-inductor model is a good approxima-tion to the actual saturable inductor. For all diodeless amplifiers the magnetizing resistance RM can be considered to be infinite; thus, the ideal saturable-in-

ductor model is sufficient to represent an actual inductor. The second factor in-sures that changes in the half-period average values of the supply voltage, caused by a shift in the operating period relative to the supply-voltage period, do not cause any steady-state errors in the linear analytic representation. How-ever, the dynamic properties, as predicted by the linear representation, are affected by changes in the half-period average values of the supply voltage. The third

Paper 59-172, recommended by the AIEE Mag-netic Amplifiers Committee and approved by the AIEE Technical Operations Department for pres-entation at the AIEE Winter General Meeting, New York, N. Y., February 1-6, 1959. Manu-script submitted August 15, 1958; made available for printing January 12, 1959.

PAUL R. JOHANNESSEN is with the Applied Re-search Laboratory of the Sylvania Electronic Sys-tem, Waltham, Mass.

The material presented in this paper represents a portion of the thesis submitted by the author to the Massachusetts Institute of Technology in partial fulfillment of the requirements for the Doctor of Science degree. I t was supported by the U. S. Air Force under Contract No. AF(616)~ 5489, M I T Project No. DSR 7848.

N O V E M B E R 1959 Johannessen—Analysis of Magnetic Amplifiers with Diodes 4 8 5

Saturable es Inductor

(A)

1 ~l íí'

(B)

I R, " f ' " b

Saturable es Inductor

(C)

Fig. 1 . Basic magnetic amplifiers

factor permits a large variety of wave-forms of supply and port voltages to be used without effecting the accuracy of representation.

Magnetic amplifiers with diodes do not possess the aforementioned properties. The back impedance of the diode and the control circuit impedance are usually of the same order of magnitude as the mag-netizing impedance. The accuracy of the saturable-inductor model is thus of paramount importance in designing a good analytical model. Cancellation of quies-cent currents at the ports caused by inter-connection of basic elements is also a property of magnetic amplifiers with diodes; this, however, does not necessar-ily insure linear steady-state character-istics. Probably the most important factor which distinguishes the behavior of these two types of magnetic amplifiers from each other is the manner in which their operation depends on voltage wave-forms. The state of a diode depends upon the instantaneous voltage or cur-rent magnitudes while the state of a saturable inductor depends upon the integral of the inductor voltage, which, in turn, may depend upon the state of one •diode or the state of several diodes, as well as the supply and port voltages. This interrelation greatly complicates the analysis and necessitates a complete re-

evaluation of the behavior of the basic elements.

Magnetic amplifiers with diodes in-variably consist of symmetrical inter-connections of basic elements. Basic ele-ments typical for these types of magnetic amplifiers are shown in Fig. 1. In this paper only element a, of Fig. 1, is analyzed in detail since it is the most common one. Element (B), of Fig. 1, differs from ele-ment (A) only in regard to the physical arrangement of components. Thus, the analysis to be performed of element (A) applies equally well to element (B).

Since the objective of this paper is to demonstrate a new method of magnetic-amplifier analysis, the analysis is restricted to magnetic amplifiers terminated in a resistive load, and to voltage waveforms typical of such amplifiers. In the course of the analysis the restrictions upon the voltage waveforms that lead to a linear analysis are established. The voltage waveforms that satisfy these restrictions are termed ideal.

Analysis of Basic Magnetic Amplifier

The basic magnetic amplifier is shown in Fig. 1(A). The diode characteristic is approximated by a piecewise linear curve as shown in Fig. 2, where Ri' is the impedance of the diode when it is con-ducting and i V is the impedance of the diode when it is nonconducting. Two different models of the saturable inductor will be used: one to determine quiescent operating conditions and another to repre-sent the saturable inductor in an incre-mental linear equivalent circuit. The inductor model which is used to deter-mine the quiescent operating conditions is shown in Fig. 3. The finite loop-width property of the saturable inductor is taken into account by the diode bridge circuit and the constant current source. The diodes in the bridge circuit are assumed to be ideal; that is, having infinite con-ductance in the forward direction and zero conductance in the reverse direction. In Fig. 4, the complete equivalent circuit of the basic magnetic amplifier to be used in determining the quiescent operating

\ Slope i/Rb'

Fig. 2. Diode characteristic

* Slope i/Rf'

- > e

1 f T

1 —°—\ \ i V +

t O

O

Ideal Saturable Inductor

Fig. 3. Saturable inductor model

conditions is shown. Use is made of the inequality Rb'»R/ in Fig. 4 so that

Rb^Rt,'+R2

Rj££R/+Rt (1)

STATES OF THE BASIC MAGNETIC AMPLIFIER

The possible states of the equivalent circuit of Fig. 4 are shown in Table I. Note that each of the three nonlinear ele-ments in Fig. 4 has two possible states. Combination of these states, therefore, should give rise to eight possible states of the basic magnetic amplifier. How-ever, when the inductor is saturated, the state of the diode bridge is immaterial so that only six different combinations of states exist.

TYPICAL VOLTAGE WAVEFORM

The voltage waveforms typical for magnetic amplifiers terminated in a resis-tive load are shown in Fig. 5. The volt-age es passes through zero at time zero with positive slope, is symmetric about the e = 0 axis, and is periodic of period T. The following approximations^can then be made.

Fig. 4. Equiva-lent circuit of basic magnetic

amplifier Ideal Saturable

Inductor

486 Johannessen—Analysis of Magnetic Amplifiers with Diodes N O V E M B E R 1959

Table I

Diode Bridge Bridge

Inductor Non- Non-

Un- Con- con- Con- con- In-satu- Satu- due- due- due- due- ductor

State rated rated tor tor tor tor Voltage

1 1 1 0 2 1 1 0 3 1 1 1 4 1 1 1 5 1 1 1 0 6 1 1 1 0

^ = - U for - Ä Ê / < Ä ß (2)

E.( T\ T T iox--At<t<-+At (3)

2 2

es = — {t-T), for T-At<t<T+At (4)

where

E s = t h e half-period average value of the supply voltage

Ts = constant governed by the slope of es at a zero crossing

Note that the ratio Es/Ts is the slope of the es wave at a zero crossing. For the same time intervals, the input and output waveforms can be approximated as follows:

Ei Si=£i'-f-—,*. for -At<t<At (5)

* - Ä ' - ^ - - J . far--*<*<-+* (6)

el=E1'"+-±-(t-T), for T-At<t<-+At

(7)

and

e2=-—t, for - At<t < 0 T2

et = Q, for 0<*<Äß

"ÜH> for--At<t<-2 2

Ô Ô e2 = 0, f o r - < / < - + A /

2é 2é

â2 = ~—(t-T), for Ã - Ä * < * < 0

e2 = 0, for Ã < * < Ã + Ä *

where

(8)

(9)

(10)

Åé', Ei", and E i ' " = t h e voltage magnitudes of ei at time 0, Ã/2, and Ã

£é /Ãé, Ei/TV, and Å ÷ / Ã é ' - t h e magnitudes of the slope of t\ at time 0, T/2, and Ã

E 2 / r 2 = the magnitude of the slope of e2 just before a zero crossing of the es wave

|T+At

Fig. 5. Voltage waveforms

For reasons of analytical simplicity, all transitions from one state to another are assumed to occur within the time inter-vals in which the waveform approxima-tions are made. Suppose that at some time t-i in the interval — At to 0 the * inductor is saturated, the diode is con-ducting, and the operating point P on the inductor characteristic is at point P-\ in Fig. 6; saturation thus continues and the inductor voltage, eR, is zero until the inductor current is equal to Ic. For t= t-i, the basic magnetic amplifier is in state 1. In Fig. 6, the state of the ampli-fier is indicated by the superscripts on the P's. For t>t-i, a transition occurs to either state 2 or 3. Transition to state 2 occurs if at some time t0>t-i

If, instead, transition from state 1 to state 3 occurs at some time *(/>*-1, then the following conditions must be satisfied at time / = /o'.

es+e2= —-^(ei-RiIc)

es+e2<0

(14)

(15)

Substitution of equation 15 in equation 14 yields

e i - ^ i / c > 0 , for* = V (16)

or in terms of the approximate voltage waveforms

(17)

ei-RJc<0

es+e2 = 0

( Ð )

(12)

For the waveforms chosen it is evident that /o = 0. Substitution of the wave-form approximation for ei in equation 11 yields

As seen from Fig. 5, e2 is assumed to be less than es. This assumption, together with equation 15, leads to a negative value for the time A/,

*o'<0

From equations 17 and 18

ESKRJc (13) E,'>RJC

(18)

(19)

Note that in state 2 the inductor voltage is zero.

An expression for /</ in terms of the approximate waveforms is

Fig. 6. Inductor characteristic

N O V E M B E R 1959 Johannessen—Analysis of Magnetic Amplifiers with Diodes 4 8 7

/ o ' = -Rf/RiW-RxIc)

Es/Ts-E2/r2+(Rf/£iXEi/T>) (20)

W h e n t h e a m p l i f i e r is i n s t a t e 3 , i t c a n b e r e p r e s e n t e d b y t h e e q u i v a l e n t c i r c u i t of F i g . 7. B y u s e of t h i s e q u i v a l e n t c i r -c u i t , t h e i n d u c t o r v o l t a g e c a n b e f o u n d i n t h e fo l l owing m a n n e r . I f t h e s e c t i o n of t h e e q u i v a l e n t c i r c u i t o n t h e le f t s i d e of c u t A i s t r a n s f o r m e d i n t o i t s T h e v e n i n e q u i v a l e n t , t h e c i r c u i t i n F i g . 8 r e s u l t s . B y i n s p e c t i o n of F i g . 8, e q u a t i o n 2 1 for t h e i n d u c t o r v o l t a g e is o b t a i n e d .

eR = a(ei—R1Ic)+b(es+e2)

w h e r e

(21 )

a = t h e v o l t a g e t r ans fe r r a t i o b e t w e e n t h e e\ (or —Rilc) a n d eR (or e{) t e r m i n a l s ( m e a s u r e d w i t h e2 = es = R\IC=0) w h e n t h e d i o d e is c o n d u c t i n g

6 = t h e v o l t a g e t r ans f e r r a t i o b e t w e e n t h e e8 (or e2) a n d eR t e r m i n a l s ( m e a s u r e d with e2 (or es) — e\ —R\IC (or â÷ —R\IC) = 0 ) w h e n t h e d i o d e is c o n d u c t i n g

I n t e r m s of t h e c i r c u i t p a r a m e t e r s , t h e e x p r e s s i o n s fo r a a n d b a r e

RmR mUf

R\Rm+R\Rf-\-RmRf

b = RmRi R>

RiRm-\-R\Rf-\-RmRf Rf

(22)

(23)

I n t e r m s of t h e a p p r o x i m a t e w a v e f o r m s t h e e x p r e s s i o n fo r t h e i n d u c t o r v o l t a g e t h e n b e c o m e s

€ Δ - e ( a ' + ^ < - Δ J e ) + f t ( ^ - ^ ) / (24)

F o r t = h (o r / , ' ) w h e r e h (o r ti)>h ( o r /o')> a t r a n s i t i o n o c c u r s e i t h e r f r o m s t a t e 2 t o s t a t e 4 o r f r o m s t a t e 3 t o s t a t e 4 . T r a n s i t i o n f r o m s t a t e 2 t o s t a t e 4 o c c u r s w h e n

es+e2-- t-^-\ei-RM (25 )

T h i s t r a n s i t i o n m u s t o c c u r fo r t i m e t>0

b e c a u s e t h e a m p l i f i e r e n t e r e d s t a t e 2 a t t i m e z e r o . S u b s t i t u t i o n of t h e e x p r e s -s i o n s fo r t h e a p p r o x i m a t e w a v e f o r m s , e q u a t i o n s 2 - 1 0 , i n e q u a t i o n 2 5 a n d s o l v i n g fo r t\ y i e l d s

*1 = R> RJc-Ei

ΔΔ/r.+ilfc/ΔXJSi/ZY) (26)

Fig. 7 (left). Equiva-lent circuit of basic magnetic amplifier in

state 3

Fig. 8 (right). Equiva-lent circuit obtained by

source transformation

I f t h e a m p l i f i e r i s i n s t a t e 3 , t r a n s i t i o n t o s t a t e 4 o c c u r s a t t i m e t\ w h e n

es-\-e2 — Rm

Ri+R, fr-RJc) (27)

I n s t a t e 3 t h e i n d u c t o r v o l t a g e is p o s i t i v e . T h u s , t h e i n s t a n t of t i m e t\ m u s t o c c u r a f t e r t = 0 . S u b s t i t u t i n g t h e a p p r o x i m a t e e x p r e s s i o n s fo r t h e v o l t a g e w a v e f o r m s i n e q u a t i o n 2 7 a n d s o l v i n g for t\ g i v e s

Rm Ei'—Rilc

Ri+Rm „ /rr R\ (28)

Ri+Rm EJTi

T h e e q u i v a l e n t c i r c u i t of t h e a m p l i f i e r i n s t a t e 4 i s t h e s a m e a s t h a t of F i g . 8 , s t a t e 3 , e x c e p t t h a t t h e d i o d e r e s i s t a n c e i s R* i n s t e a d of Rf. T h e i n d u c t o r v o l t a g e i n s t a t e 4 t h u s b e c o m e s

er = c(ei — RJc) +d(es +e2) (29 )

w h e r e t h e c o n s t a n t s c a n d d a r e v o l t a g e t r a n s f e r r a t i o s s i m i l a r t o a a n d b i n e q u a -t i o n 2 1 e x c e p t t h a t t h e d i o d e i n n o n -c o n d u c t i n g i n s t e a d of c o n d u c t i n g . I n t e r m s of t h e c i r c u i t p a r a m e t e r s e x p r e s s i o n s fo r c a n d d a r e

RmRb

R\Rm -\-RiRb Ë-RmRb

d = -Rm,R\ Ri

(30)

(31) RiRm-hRiRb-hRmRb Rb

T h e t r a n s i t i o n s d i s c u s s e d i n t h e f o r e -g o i n g a r e a s s u m e d t o t a k e p l a c e w i t h i n t h e i n t e r v a l — At t o At. A t t i m e Ä/ t h e a m p l i f i e r is i n s t a t e 4 . I f t h e c o n d i t i o n s

es+e2> — ^ (ej-RJc), for At<t<-- At Ri 2

(32)

7? T es +e2>

w (ei - RJd for At<t<- - Ä/

(33)

a r e sa t i s f i ed , t h e n t h e a m p l i f i e r r e m a i n s i n s t a t e 4 .

I n t h e i n t e r v a l (T/2-At) t o (T/2+

At), t r a n s i t i o n s s i m i l a r t o t h o s e a l r e a d y d i s c u s s e d for t h e i n t e r v a l — Ä/ t o At t a k e p l a c e . I f t h e i n e q u a l i t y of e q u a t i o n 3 2 i s v i o l a t e d i n t h e i n t e r v a l (T/2 —At) t o T/2, t h e n t r a n s i t i o n t o s t a t e 5 o c c u r s a t t i m e

^T^Rj, Rilc-¢' 2 2 R1Es/Ts^E2/T2+(Rb/Rl)(Ei/Tl

f)

(34)

N o t e t h a t s t a t e 5 a n d s t a t e 1 a r e e q u i v -a l e n t i n a s f a r a s c u r r e n t a n d v o l t a g e re la -t i o n s a r e c o n c e r n e d . I f t h e i n e q u a l i t y of e q u a t i o n 3 3 i s v i o l a t e d , t h e n t r a n s i t i o n t o s t a t e 3 o c c u r s a t t i m e

T Rm t 2 ' = - - „ n X

2 Rx+Rn

ES-RJc

Es/ Ts—E2IT2— Rn

(35)

R\-\-Rm EJTX>

N o t e t h a t t2' m u s t o c c u r b e f o r e T/2. I n s t a t e 3 t h e i n d u c t o r v o l t a g e is g i v e n b y e q u a t i o n 2 1 .

N e x t , t r a n s i t i o n f r o m s t a t e 5 o r 3 t o s t a t e 6 o c c u r s . I f t h e a m p l i f i e r i s in s t a t e 5 , t h e n t r a n s i t i o n t o s t a t e 6 o c c u r s a t t i m e

/ • - Ã / 2 (36)

I f t h e w a v e f o r m of e\ h a d b e e n c h o s e n so t h a t E\ i s n e g a t i v e , t h e n t h e t r a n s i -t i o n f r o m s t a t e 5 t o s t a t e 6 m a y o c c u r a f t e r t i m e T/2. T h i s e v e n t t a k e s p l a c e if -E&RJc. T h u s , if -RJjUESK

Rilc, t h e n t r a n s i t i o n t o s t a t e 6 o c c u r s a t t i m e h=T/2.

I f t h e a m p l i f i e r i s i n s t a t e 3 , t h e n t r a n s i t i o n t o s t a t e 6 o c c u r s w h e n

es+e2 = - — (ei -Rilc) (37)

I n s t a t e 3 t h e d i o d e i s c o n d u c t i n g ; t h e i n d u c t o r v o l t a g e m u s t t h e n b e g r e a t e r t h a n es+et. T h u s t h e t i m e of t r a n s i t i o n f r o m s t a t e 3 t o s t a t e 6 o c c u r s a f t e r t i m e T/2, a n d i s g i v e n b y t h e e x p r e s -s i on

h' = T/2+ Rf Ei'—Rilc

RlEs/Ts+(Rf/Ri)(Ei/T1') (38)

T h e n e x t t r a n s i t i o n is f r o m s t a t e 6 t o s t a t e 3 a n d o c c u r s w h e n

* + * - - ~ ( * i + £ i J e )

a n d a t t i m e

(39)

t^T/2+-f ZL±^II ( 4 0 ) 1 R1Es/Ts+(Rf/R1XEi/T1>) K

T h e i n d u c t o r v o l t a g e for s t a t e 3 i s s u p -

488 Johannessen—Analysis of Magnetic Amplifiers with Diodes N O V E M B E R 1 9 5 9

posedly given by equation 21. This equa-tion was derived for the operating point Poz in Fig. 6. For the present case the operating point is on the other side of the inductor characteristic. Thus, for the present case, the current source Ic in the equivalent circuit of Fig. 7 is of opposite polarity so that the term ei—RJc in the equation for the inductor voltage becomes ei+RJc. For the amplifier to remain in state 3 until the inductor saturates at h, es+e2 must satisfy both of the follow-ing inequalities

,+^2<-- i-(ei+RJc)

S+^2< Rm

R>+Rr ■(ei+RJc)

(41)

(42)

Equation 41 insures that the net inductor current is greater than ICi while equation 42 insures that the diode conducts. For the waveforms of Fig. 5, equation 41 is sat-isfied if equation 42 is satisfied. When the inductor saturates at time tb, transition to state 1 occurs and saturation continues until a new period of operation starts. I t is evident that the sequence of transi-tions taking place in the next period of operation may not be the same as in the period just considered. However, if all voltage waveforms are periodic, then the sequence of transitions are the same for all periods of operation.

A diagram illustrating the possible sequences of transitions that may occur during one period of operation is shown in Fig. 9. These sequences of transitions are the only ones that can occur for volt-age waveforms of the type shown in Fig. 5 that satisfy equations 32, 33, 41, and 42. Clearly, for voltage waveforms different from those shown in Fig. 5 that may or may not satisfy equations 32, 33, 41, and 42, there exists a large number of possible sequences of transitions. It is beyond the scope of this work to consider all of these possibilities.

The waveforms of Fig. 5 were chosen in order to demonstrate a method of analysis and to establish restrictions upon the volt-age waveforms that render the equations for the basic amplifier linear. To estab-lish these waveform restrictions, the amplifier equations for the four sequences of transitions in Fig. 9 must be studied. In Appendix I the amplifier equations for sequence of transitions c are derived, and it is shown that these equations are linear if either of the two equations, 163 and 164, together with equations 32, 33, and 42, are satisfied for all periods of opera-tions.

The conditions of equation 164 may appear to be trivial, since they imply

Fig. 9. Possible sequences of transitions during one period of

operation

(á) É - 3 - 4 - 5 - 6 - 3 - É (b) 1 - 3 - 4 - 3 - 6 - 3 - 1 (c) 1 - 2 - 4 - 5 - 6 - 3 - 1 (d) 1 - 3 - 4 - 3 - 6 - 3 - 1

that all the voltage waves are constant near the zero crossings of the es wave. The author, however, has not been able to derive the linearization conditions of equations 163 and 164 without going through the detailed study of possible sequences of transitions as given in this section and the derivation of the amplifier equations as presented in Appendix I. I t is further shown in Appendix I that if these waveform restrictions are satisfied, then the amplifier equations are linear for all four sequences of transitions.

Equations 163 and 164 are not the only linearization conditions that exist. A general study of all possible relations be-tween the various voltage waveforms that render the amplifier equations linear ap-pears to be of little practical value. The conditions of equations 163 and 164, however, are often met in practical ampli-fiers. The linearization condition of equation 163 is of special interest since considerable freedom is allowed for the various voltage waveforms. This condi-tion, together with the conditions of equations 32, 33, and 42, may be for-mulated in terms of four alternate condi-tions as follows:

1. The operation of the amplifier must be periodic and of the same period as the sup-ply voltage.

2. The diode must be nonconducting during the control periods and conducting during the gating periods.

3. The inductor voltage must be positive during the control periods. 4. The input voltage must satisfy the rela-tion.

ei = RxIc, for /= 772 (43)

These conditions need not be satisfied in order to perform a linear, small-signal analysis around the quiescent operating point. This type of analysis is most easily performed by first determining the quiescent operating conditions, as out-lined in Appendix I for the combination of transitions c in Fig. 9, and then linear-izing around the quiescent operating point.

CIRCUIT EQUATIONS FOR BASIC MAGNETIC AMPLIFIERS THAT SATISFY THE FIRST THREE LINEARIZATION CONDITIONS

In basic magnetic amplifiers that satisfy conditions 1,2, and 3, there exist two pos-sible sequences of transitions:

( 1 ) 1 - 4 - 5 - 6 - 3 - 1

(2) 1 - 4 - 6 - 3 - 1

Only sequence 1 need be considered be-cause sequence 2 is actually a special case of 1 for which the time spent in state 5 zero.

During the control period, the follow-ing equations hold,

EXC = RJX

C+ERC (44)

Esc+E2C = Rbh

e+ERc (45)

and during the gating period

E^RJf+Ej? (46)

Eso+E2

0^Rfl2O+EB

Q (47)

For the foregoing, the notation is the same as used in reference 1. To obtain four equations involving the eight port voltages and currents only, two more equations are needed. One of these equa-tions is the well-known relation

EBe+Ea'-0 (48)

the other is obtained by calculating the inductor voltage during the control pe-riod.

The inductor voltage is given by the expressions

â Á = ø é - ^ ) + Ö ß + þ . for 0<f</2 (49) eR = 0, iorh<t<T/2 (50)

Thus,

2 Ch

ERC = ~ I [c(e1-RlIc)+d(es+e2)]dt

2 r /2

e ~ [c(e1-RJc)+d(es+e2)]dt-1 Jo

I 2_ T.

*T/2

[c^-RJJ+d^+e^dt

(51)

which, when integrated, gives

N O V E M B E R 1959 Johannessen—Analysis of Magnetic Amplifiers with Diodes 489

Fig. 10. Linear incremental model of basic magnetic amplifier

es Ideal Inductor

ERc = c(ES

1 Rb

-RJc)+c-^{Esc+E2

c) + Rb

TRL As-A2+(Rb/R1)A1 -W-RJcy

(52)

where

As—Es/Ts = the magnitude of the slope of the es wave at time T/2

A2=E2/T2 = the magnitude of the slope of the es wave immediately before time T/2

A\ —Ei/Ti= the magnitude of the slope of the ei wave at time T/2

If the slopes As, A2, and Á÷', and the volt-age Ei" are independent of the half-period average values of the voltages and cur-rents a t ports 1 and 2, a n d if the supply voltage is constant , then equation 52 can be rewrit ten

Ea^cEf+diEs'+Et^-B (53)

where B is a constant given by the ex-pression

1 Rb B**cR1Ie--zrc

{ES-RJc)* -, (54)

TRi As^A2+(Rb/Rl)A1

If Ei" = Rile, then ERC is independent of

the slopes As, A2, and Ai\ and the ex-pression for B becomes simply

B = cRJc (55)

This is the special case for which the se-quence of t ransi t ion 2 applies.

Subst i tut ion of equations 48 and 53 in equations 44, 45, 46, and 47, with an appropriate rearrangement of terms, yields the admit tance-matr ix equat ion of the amplifier.

Il/i'i

\\Ii\

Ð

H

1

\l-c

Ã ° c 1

Ri Ri

\-k°

IF,· whe r e i

d d

~Ry ° ~Ri

d d

Ri Ri

1-d 1-d

Rb Rb

d 1 d

Rf Rf Rf

? is the column matrix

0

0

0

1

Rf\

M 2"

£ /

+ß (56)

(3 =

1 *1 hj UJ \ Rf 1

(57)

More generally, any desired network mat r ix can be obtained from these same equations.

I N C R E M E N T A L L I N E A R ANALYSIS

Suppose t h a t the half-period average voltages and currents of equat ion 56 are the quiescent currents and voltages, then the admit tance-matr ix equat ion which relates t he Ä quanti t ies a t por ts 1 and 2

Ulic

Uli*

\AI2C\

IATA

=

* - £ n d

Ë

R, Ri

Ri Ri Ri

--T- 0 · — - 0 Rb Rb

c d 1 — 0 — — Rf Rf Rf\

AEi(

AEi

AE2

(58)

I n general, this equat ion is valid only if t he Ä quanti t ies result from incremental changes in the waveforms of the quiescent currents and voltages. Thus , equat ion 58 is a linear relation among incremental deviations from the quiescent condition. I t is of importance to point out t h a t the restriction of small changes in the average values of the por t quanti t ies is not always necessary. If the changes in the wave-forms are such t h a t the to ta l voltage waves a t por ts 1 and 2 satisfy the condi-t ions for ideal waveforms, then the ampli-fier is linear and is represented by equa-tion 58.

A linear-incremental circuit model of t h e basic magnet ic amplifier is shown in Fig. 10. T h e switch S is open during the

positive half periods and closed during the negative half periods of es.

In t he preceding analysis, i t was found tha t , in general, the amplifier equations are not linear. T h e nonlineari ty in the equat ions is introduced by the last term of equat ion 52 and is caused by changes in the slopes of the voltage waves near the zero crossings of the es wave and the quan t i t y Å÷ with applied signals. For practical amplifiers te rminated in a resistive load, the slope of the output voltage wave near the zero crossings of the es wave is governed b y the es wave only. This is t rue in general for single-phase amplifiers te rminated in a resistive load; for 3-phase amplifiers, however, this is t rue only if t he amplifier operates in mode 1 and is resistively loaded. If the es wave is assumed to be invarient, then the following analysis can be made. Le t

Aq=ASQ-A2q+(Rb/Ri)Ai,q (59)

where the subscript q indicates the quies-cent condition. Changes in ER

C caused b y changes in E\ and E2 can then be wri t ten

AER =EB q + AER —ERq

= c(£1c

?-fA£1c-7?1J c) +

d(Escg+E2

cq+AE2

c) +

IRt ( E / g - f - A E / - ^ ! , ) ^ TRX

C Aq+AA

c{Excq - RJC) - d(Es

€q +E2

cq) -

1 Rb {E,%-RJCY TRi

(60)

For small Ä changes the following approxi-mat ion can be made

1 . 1 1 1 / AA\ , x

Aq + AA Aq AA Aq\ Aq

Subst i tut ion of equat ion 61 in equation 60 and cancellation of all Ä2 te rms yield

AERc = cAE1

c+dAE2c +

1-^ï~{Å,%-^É0)ÁÅ^-1 Kl Aq

(62)

T h e quan t i ty AEi" is the change in the magni tude of the ex wave a t t ime T/2. Usually, ÁÅ÷' is linearly related t o AEL

C, Ei" is equal to Å÷ for direct current , and is zero for a full-wave rectified sine wave. Let

AE/^KiAES (63)

where Ki is a constant . T h e quan t i ty Ä^4 is the change in the

4 9 0 Johannessen—Analysis of Magnetic Amplifiers with Diodes N O V E M B E R 1959

slope of the ei wave at time T/2. Clearly, if the waveform of e\ is fixed and only the magnitude is varied, then

7V (64)

where T/ is a constant. Substitution of equations 63 and 64 in equation 62 yields

AERc = c'AEl

c+dAE2c (65)

where

£ '= c{1 + 1rfi ( £ / i-* , 7 c ) x

fc-F'f ô.(Á'·-*Ë)]} (66)

Equation 65, together with equations 44, 45, 46, 47, and 48, leads again to a linear analysis of the basic magnetic amplifier in terms of incremental deviation from the quiescent condition.

It is evident from equation 66 that, for the special cases in which either Aq is infinite, that is, if either of the slopes Asq

and AiQ, or both, are infinite, or E / is equal to Rilc, the quantity c' reduces to c.

The preceding considerations hold not only for incremental deviations from the quiescent condition but also for incre-mental deviations from any predeter-mined operating condition.

BIAS CONSIDERATIONS

In most practical amplifiers a bias, or reset, circuit is provided to adjust the quiescent operating condition as desired. This is accomplished either by an addi-tional bias winding on the inductor con-nected in series with a resistor and a voltage source or by a voltage source in series with the input. These two methods of obtaining the proper bias are shown in Fig. 11 and are referred to as parallel and series bias, respectively. The voltage gain of the amplifier is proportional to the component of the inductor voltage cE{, where the voltage transfer ratio c is given by equation 30. In the case of parallel bias, the resistance Rb effectively shunts the inductor and thus decreases the value of the voltage transfer ratio c, which, in turn, decreases the voltage gain. It is thus desirable to make the resistance Rb

as large as possible. In the limiting case when Rb approaches infinity, the re-sistance Rb and the voltage source eb can be replaced by a constant current source ib. This current source, by Thevinin's theorem, is equivalent to a voltage source of magnitude Rxib in series with the input voltage. Thus the two bias methods be-come identical and no loss in gain results. If

Fig. 11. Bias methods

A—Parallel B—Series

• H — o

Or>

eb = RJc

(A)

(67)

then the bias voltage cancels the equiva-lent voltage source RJC during the con-trol periods. Effectively, this bias volt-age shifts the hysteresis loop in Fig. 6 to the left by an amount Ic so that the apparent loop width is zero during the control periods and equal to 2IC during the gating periods.

With a bias voltage, or current, in the amplifier, the ideal waveform restrictions are altered slightly. When the bias volt-age is equal to R\IC, these restrictions become

£ / = £," = £ / " = ( ) (68)

7? T es+e2>O

n eu for 0 < / < - (69) K) -f-Km Ä

/? IT es+e2<O * (gi+2ftJc), for - <t<tb (70)

Ki-\-Km 2

Kb e9+e2>-- eu for 0<t<T/2 (71)

R\ The expression for the term B in the equa-tion for the inductor voltage, equation 53, becomes

* — 1 * , (E/)*

TRr As-A2+{Rb/R,)Ai

From the relation

E2g = Rfh

a-EBc-Es°

(72)

(73)

obtained by substituting equation 48 in equation 47, it is evident that the rela-tion between the port quantities E2° and h° does not depend upon the inductor characteristics during the gating periods. In fact, any finite resistance can be placed across the inductor during the gating periods without affecting the performance of the amplifier. In addition, if the bias voltage is equal to RJCt the apparent loop width is equal to zero during the con-trol periods. Thus, for ideal waveforms and the bias voltage of equation 67, the amplifier is linear and independent upon the constant loop-width property of the saturable inductor. For this reason, the bias voltage of equation 67 is termed the ideal.

(B)

A SET OF IDEAL VOLTAGE-WAVEFORM RESTRICTIONS

To study voltage waveforms in practical amplifiers, a set of sufficient, although not necessary, restrictions upon the voltage waveforms for the case when

s+22>0, for 0</<^-Δ

s+e2<0, f o r ^ < / < r (74)

are derived from the waveform restric-tions of equations 68 through 71 as follows.

From equation 74,

es-\-e2 = \es+e2\, for 0<t<-

Thus, if

(75)

7? T"

|g«+«*l>p " elt for 0 < / < - (76)

then equation 69 is satisfied. Equation 70 can be alternatively

written

-(es+e2)>^—~- (ei+2RJe), ■tti-T-Km

for ~<t<h (77) Δ

From equation 74,

- ( * . + * ) = k + « i | , for ~<t<T (78)

Thus, if

h + e 2 | > ^ x b - ( - e j ) , f o r J < K i 5 (79)

then equation 70 is satisfied. Through a similar derivation,

É« â +â 2 | > -^âé , for 0<*<Ã/2 (80) R\

then equation 71 is satisfied. Thus, a set of sufficient, but not necessary, restric-tons upon the voltage waveforms is

E 1 '=Jg/ -JE, / , / = 0 (81)

Rn ^4>~T^^u for 0<t<T/2 (76)


h + ' a | > — V ( - « 0 . for T/2<t<h (79)

|e«+^|> - -Ö âé, for 0</<Ã/2 (80)

These waveform restrictions insure that the term B is equal to zero. Thus, if equations 81, 76, 79, and 80 are satisfied, and if ideal bias is used, then the magnetic amplifier is linear.

From equations 76, 79, and 80, a con-siderable amount of information can be obtained regarding the behavior of single-phase magnetic amplifiers terminated in a resistive load. Equation 76 essentially states that the maximum inductor voltage that can exist during the control periods without causing the diode to conduct is equal to the sum of the supply voltage and the output voltage. Because the in-equality

Ri»Rf

usually is satisfied, the sum of the supply and output voltages establishes an upper limit on the inductor voltage during a control period. This limit, in turn, estab-lishes an upper limit on the maximum flux reset that can occur during a control period. If equation 79 is satisfied, the diode is conducting and the component of the inductor voltage caused by the in-put voltage is small. Thus, the inductor is driven toward saturation by the supply and output voltages. However, if the input voltage is large and negative, such that equation 79 is not satisfied, then the diode is nonconducting and the input voltage is effective in driving the induc-tor toward saturation. In conclusion, it apparently is relatively easy to prevent the inductor from being reset during a control period and to drive the inductor toward saturation during a gating period. Difficulties may arise in attempting to reset the inductor during a control period and in preventing it from being driven toward saturation during a gating period. I t may be said that the inductor ' l ikes" to be saturated.

This property is basically responsible for several undesirable magnetic-ampli-fier limitations: the low-efficiency of all fast-response magnetic amplifiers; the impossibility of constructing a full-wave polarity-reversible d-c amplifier of greater than 50% efficiency; and the impossi-bility of constructing a fast-response, highly efficient magnetic amplifier with full-wave a-c output.

T H E VOLTAGE e s+e2 IN PRACTICAL MAGNETIC AMPLIFIERS

A single-phase magnetic amplifier may

consist of 2 or 4 basic magnetic amplifiers interconnected in various ways. For these interconnections the magnitude of the sum e s+e2 must satisfy one of the following cases.

A.

B.

C.

D.

\es-\-e2\£=\es\ -f-]e2! control period |ââ + %!=!âß|+|22| gating period

0*1 + M control period e«' —N gating period

G.

H.

\es+e2 \es+e2\

|2s-r-e2!==|2s! ~~ M control period |ee+g2l=k!+|e2l gating period \e$ +«2l==|e«| — \e2] control period \es+e2]^es\ — \e2\ gating period es-+-e2\=)es\ control period es-\-eJ=\es\+\eJ gating period |es+^2'=ks] control period |es-f-e2'=W— 1021 gating period ks+ß2|=ksl control period |e5-f-e2|==|es|±|22| gating period \es +e2 =Ue| ± | e2\ control period les-r-e2li=W±|e2! gating period

A plus sign or a minus sign on the right-hand side indicates that addition or subtraction always occurs regardless of the polarity of the input voltage. A plus and a minus sign indicates that either addition or subtraction may occur, de-pending on the input voltage polarity. These cases represent the conditions that may exist for a single basic amplifier in a magnetic amplifier consisting of an inter-connection of two or four basic ampli-fiers.

For cases A, B, C, and D, there is out-put during both half periods of operation. Amplifiers with this property are obtained by parallel connecting the output ports of two basic amplifiers that gate during alternate half periods. Inasmuch as output is obtained from the basic ampli-fier under consideration during its gating periods, the output voltage is always subtracted from the supply voltage during the gating periods. Cases A, C, and E, therefore, do not represent practical amplifiers. When the output adds to the supply voltage during the control periods and subtracts from it during the gating periods, the output voltage is always positive because the supply volt-age is an alternating voltage. The out-put voltage is an alternating voltage when it subtracts from the supply volt-age in both control and gating periods. Thus, the output for case B is a direct voltage; and for case D it is an alter-nating voltage. Case F represents the well-known incomplete bridge circuit shown in Fig. 12. This amplifier circuit may be constructed by connecting two basic amplifiers of the type shown in Fig. 1(C) in series-series. Case G repre-sents the so-called fast response, or half-cycle response, magnetic amplifier. These amplifiers are constructed by paral-

leling the output ports of two basic ampli-fiers that gate during the same half periods. Case H represents the various full-wave phase-reversible, or polarity reversible, magnetic amplifiers. These amplifiers result from connecting the output ports of four basic amplifiers in parallel.

The ideal waveform restrictions are easily satisfied for the amplifiers belonging to cases B and F. These amplifiers can be completely reset during one-half period of the supply voltage because of the large diode blocking voltage. They are in-herently efficient, and may be fast or slow response amplifiers; characteristically, they give a unipolar, direct-voltage out-put.

The amplifiers of case D are inherently slow-response high-gain amplifiers. Speed of response may be traded for lower efficiency and lower gain, for example, by connecting resistors in series with the diodes. These resistors decrease the out-put voltage and thus increase the diode blocking voltage. Amplifiers of this type are characterized by a full-wave alternating output voltage.

The fast response, bipolar (or phase reversible if the output is an alternating voltage) amplifiers of case g are character-ized by a low efficiency, which is evident from the magnitude of the sum of the supply and output voltage. This voltage sum is smaller during the control periods than during the gating periods. I t is, therefore, difficult to obtain sufficient reset during the control periods to prevent the inductor from saturating during the gating periods. B ecause the amplifier consists of two basic amplifiers that gate during the same half periods, a time interval exists in which both inductors are saturated; a large circulating current results giving rise to a low efficiency.

The amplifiers belonging to case H have properties common to all the aforemen-tioned amplifiers. The various combina-tions of properties that these amplifiers may possess are left to the imagination of the reader.

Violation of the ideal voltage-waveform restrictions not only prevent a linear anal-ysis from being made, but also result in undesirable properties such as low effi-

Fig. 12. Incomplete bridge circuit


G.

H.

i," (ßé)é

<·é

••■'I (âð

■I

<w.,

•T

I

I I

( i i )2

* _ Ïç

y*ii>z 1 J

(en)«t

T«2

o — o

ß I o o

T

I

11

áé

T è"4"

o-l

t

~] * n

It

(a) (c)

(b) (d)

Fig. 13. The Four possible subconnections for the series parallel interconnection of 2-port networks

ciency, low gain-bandwidth product, and nonlinear transfer characteristics.

Typical voltage waveforms that satisfy the ideal waveform restrictions for prac-tical magnetic amplifiers are

1. Direct input voltage and rectangular supply voltage. 2. Alternating input voltage of the same waveform as the supply voltage. 3. Input voltage of the same waveform as half-wave or full-wave rectified supply volt-age. 4. Input voltage of the same waveform as the waveform of the sum of the supply and output voltages.

N O N I D E A L W A V E F O R M S IN M A G N E T I C AMPLIFIERS

The ideal voltage-waveform restrictions are formulated in terms of certain relations that must be satisfied among the various voltage waves. I t is not logical to speak of an ideal supply, input, or output volt-age, as such, because there are no restric-tions imposed upon each of these voltages separately. If the supply voltage is a rectangular wave, then almost any reason-able input voltage wave will satisfy the ideal waveform restrictions. The com-bination of a sinusoidal supply voltage and a direct input voltage, however, does not satisfy the ideal waveform restrictions, because this combination of waveforms violates the restrictions set forth by equa-tion 81. The amplifier nonlinearities re-sulting from these voltage waveforms are usually not very serious, as will be demon-strated in a later section.

The most frequently encountered input

voltages which violate the ideal waveform restrictions and thus deteriorate the ampli-fier performance are high-frequency noise and, for amplifiers controlled by an alter-nating signal, quadrature signals. Both of these types of input voltages are of particular importance for the fast-ie-sponse phase-reversible magnetic ampli-fiers. Such amplifiers are used exten-sively in feedback control systems in which subtractions of two large, almost equal signals are a common operation, the signal-to-noise ratio of the resultant signal being often as low as 0.01. Thus, proper-ties such as high-frequency noise and quadrature rejection are highly desirable. Ideally, the magnetic amplifier is an average proportional amplifier; because the average value of high-frequency noise, or of random noise if the time interval in which the averaging process takes place is sufficiently long, and quadrature volt-ages are zero, under ideal operating condi-tions the magnetic amplifier should be insensitive to these input signals.

I t has been shown previously that wave-forms that violate the ideal-waveform restrictions tend to cause the inductor to saturate earlier in the gating period than do ideal voltage waveforms of the same half-period average values. This phe-nomenon causes the quiescent currents to increase in each of the basic ampli-fiers of which the magnetic amplifier is constructed. The quiescent currents may still cancel at the output, but the quies-cent power dissipation increases, which, in turn, decreases the amplifier efficiency. This change in the quiescent operating

condition with input noise is quite often a serious problem; not only does the quies-cent power dissipation increase, but the amplifier may be driven to cutoff.

Interconnection of Basic Magnetic Amplifiers

In the previous section it was shown that, if the port and supply-voltage wave-forms satisfy certain restrictions, the basic magnetic amplifier can be repre-sented by a set of linear equations among average values of port voltages and cur-rents. I t was further shown that, even if these waveform restrictions are not satis-fied a linear analysis can be performed in terms of incremental deviations from the quiescent condition.

This section is concerned with the anal-ysis of magnetic amplifiers resulting from various interconnections of basic magnetic amplifiers whose port voltages and cur-rents are linearly related.

INTERCONNECTION OF TWO BASIC AMPLIFIERS

As demonstrated in reference 1, there exist 4 different magnetic-amplifier cir-cuits without diodes for each of the 5 basic interconnections. For magnetic ampli-fiers with diodes there exist 8 different circuits for each of the 5 basic interconnec-tions. Consider, for example, subconnec-tion b of the basic series-parallel inter-connection of Fig. 13; the amplifier resulting is shown in Fig. 14(A). This amplifier belongs to case G of the pre-ceding section since both inductors saturate during the same half period and the output is a half-wave polarity-reversi-ble voltage. Suppose, now, that the diode of basic amplifier II, as seen in Fig. 13, is reversed. The resultant circuit is shown in Fig. 14(B), and it belongs to case B; the inductors saturate during alternate half periods and the output is a positive direct voltage. Thus, for each of the sub-connections of Fig. 13, there exist 2 differ-ent circuits: one is obtained by connecting the ports of 2 ordinary basic amplifiers according to the particular subconnection and the other by reversing either one of the diodes in this circuit. I t is beyond the scope of this work to consider all of the possible interconnections of 2 basic amplifiers; it must suffice to say that there exists a total of 40 different circuits inas-much as there are 5 basic interconnections comprising 8 subconnections each. Not all of these interconnections result in prac-tical amplifiers; for example, the circuits resulting from series connecting the output ports are impractical. However, by connecting a diode across the output port


Ri · I ] ·

-ΛÁΛτ-

Fig. 14 (left). Diode reversal in magnetic

amplifiers

(a)

(b)

of each basic amplifier, some of the im-practical interconnections may become practical amplifiers. An example of such an amplifier is shown in Fig. 12.

Another group of magnetic amplifiers not considered in the foregoing are the bridge circuits, of which Fig. 15 is an ex-ample. Apparently, the bridge circuits cannot be constructed by an interconnec-tion of basic amplifiers. This apparent difficulty can be overcome by splitting the gate winding into two equal parts and by adding another diode in series with the one already in the circuit. The resulting basic magnetic amplifier is shown in Fig. 16. Evidently, the circuit equations of this new basic amplifier are identical to those of the original basic amplifier, since series addition of gate windings and diodes and an interchange in the order of elements connected in series do not effect the opera-tion of the basic amplifier. In Fig. 16 two of these basic amplifiers are connected in series-parallel. The voltage drops be-tween x and x\ and y and y' are given by the equations

&xx —?s — &x— &y

eyyf = es — ex — ev'

(82)

(83)

The voltages ex and ey, and ex and ey' are equal. Writing Kirchoff 's loop equation around the main loop yields

2es-2tx-2exf = 0 (84)

ex+ez' = es (85)

Thus, equations 82 and 83 become

exx' = 0

evy' = 0 (86)

Fig. 15 (risht). Bridge circuit

The points x and xf, and y and y', there-fore, can be joined without disturbing the operation of the amplifier; the magnetic-amplifier circuit of Fig. 15 results.

Quite often a resistance is connected in series with the supply source to limit the supply current when both inductors are saturated simultaneously and to im-prove the amplifier performance. Thus, the voltage across the terminals x and y, and xr and yf depend upon the voltages and currents at the ports. To take into account this effect, the basic amplifier is viewed as a 3-port network, the supply-voltage terminals being the third port. The matrix method of analysis is still applicable since no potential difference exists between the terminals to be joined, as shown by equation 86. In the following subsection the matrix method of determining the behavior of intercon-nected 2-port networks will be extended to w-port networks.

INTERCONNECTION OF NETWORKS WITH n PORTS

The analysis of interconnected basic amplifiers reduces to simple matrix addi-tion or multiplication if the proper rela-tionships among the port voltages and currents are used. These relations are not so evident for an interconnection of n-port networks as for 2-port networks, be-cause only the Y and Z matrices* exist for the w-port while the Y,Z,H,G, and

* The scattering matrix and its inverse, the gather-ing matrix, also exist for an «-port network. To limit the scope of the analysis these matrix relations have been omitted. The usefulness of the scatter-ing matrix in the analysis of magnetic amplifiers have been demonstrated in unpublished notes by G. T. Coate of the Servomechanisms Laboratory at Massachusetts Institute of Technology.

A B CD matrices exist for the 2-port. The matrix method of analysis, however, can easily be extended to interconnections of networks with an arbitrary number of ports. Any two w-port networks where all the ports are connected pair-wise in series or parallel can be analyzed easily by using the impedance or the ad-mittance matrix relations. I t is also evident that a series, or parallel, inter-connection of an arbitrary number, m, of n-port networks can be analyzed in the same manner. A series, or parallel, inter-connection is obtained when one port from each of the m networks is connected in series, or parallel, with all other m— 1 like ports and so forth for the remainder of the n ports.

Consider now the case in which ports 1 of all the m networks are connected in series, ports 2 of all the m networks con-nected in parallel, and so on in a random fashion. The ports can be renumbered so that the port numbers for the series-connected ports run from 1 to p, and from p+l to n for the parallel-connected ports. If the matrix of the interconnected net-work is to be the sum of the individual network matrices, then all the port quanti-ties that are summed, voltages for series connections and currents for parallel connections, must appear on the left-hand side of the matrix equation or must be chosen as the dependent variables. The port quantities that remain invariant during the interconnections, voltages for parallel connections and currents for series connections, must appear on the right-hand side of the equation or must be chosen as the independent variables. Thus, the proper relationship among the port variables of each network for the interconnection in question is shown by equation 87 where M is an m-by-m ma-trix.


or

Fig. 16. Series parallel interconnection of two basic amplifiers of the bridge type

Similarly, for the other m — 1 networks

áð = Ìðâð

ta Ã2

\£p

\ip+i

Hp+2

\in

\=M

\i 1

Ã2

lip \eP+i\ kp+2

\&n \

(87) OLM — MM&M (89B)

Because ports 1 through p are connected in series and ports p+1 through n are con-nected in parallel, the port voltages and

Similar relations can be written for all the currents of the interconnected network other m - 1 networks. If for network / a r e r e l a t e d t o t h e P o r t voltages and cur-the following notation is used,

|(*/)i

\(ei)p («/)»+1

\(il)n

\;ßi=\

|(*/)i I1

\(ii)p

(«/)»+4

|(â/)ç I

rents of the individual networks by the equations:

M \

(88) M

for fc-1, 2, . . ., p (*»

4 = ] £ («*)*

then equation 87 can be written for j=I,IIt ..., M and k=p + l, £+2, . . ., n (91)

I t is tacitly assumed that the polarities of the port voltages and currents of equa-tions 8(A) and (B) are chosen so that no polarity reversal occurs during the inter-connection. Thus, if the matrix equa-tions for the individual networks are added, the matrix equation for the final inter-connection is obtained as follows:

a = (MI+MII+...+MM)ß . (92) Practical magnetic amplifiers consist

of interconnections of identical basic amplifiers. The matrix for these ampli-fiers, therefore, can be written in the form

M=Ml+TnMIT„+... +TMMjTM (93)

where the T"s are the transformation matrices. These matrices account for polarity changes in the port quantities and interchanges of control and gating periods that may occur through inter-connections.

For networks constructed from more than two basic elements, or basic ampli-fiers, there exist other types of intercon-nections than those considered in the foregoing. A typical example of such an interconnection is the 3-phase amplifier analyzed in reference 1. This amplifier was obtained by connecting two basic elements in series-series and then con-necting three of these series-series inter-connections in parallel-parallel. A large number of such interconnections exists. Usually, these interconnections can be divided into groups of interconnections, the first group being the interconnection of basic elements to form a number of identical amplifiers. These new ampli-fiers may then be considered elements for the next group of interconnections, and so forth. The analysis is performed by first determining the relations among port volt-ages and currents of the first group. Then these relations are used to obtain rela-

Fig. 17. Validity test for matrix method of determining composite behavior for 3-port networks Fig. 18. Interconnection of two 3-port basic magnetic amplifiers


tions among port voltages and currents of the second group, and so forth. Be-cause these groups of amplifiers are formed by different types of interconnections, the relations among port quantities obtained for one group must be rewritten to the form desired for the next group of inter-connections in order to apply the matrix method of analysis. This step in the analysis may be involved because matrix inversion of 4-by-4 matrices or larger may be required.

VALIDITY TESTS FOR MATRIX METHOD OF ANALYSIS

The tests that must be performed on a network to insure the validity of the matrix method of determining the com-posite behavior of interconnected sub-networks are described by Guillemin for ordinary 2-port networks.2 These same tests can be generalized for the inter-connection of «-port networks. For the purpose of illustration, the test procedure for 3-port networks is developed here. Consider the series-parallel-parallel inter-connection of Fig. 17, and suppose for the network N that

012 = CL\H -\-atf2 +03^3

e^z = b]^é +^2^2 + &3 3

and for the network N' that

12' = d\ %' -\-a^e%-\-aze%

« l l ' = V * l ' + k V + V«l '

(94)

(95)

(96)

(97)

For no circulating currents to flow the conditions

subject to the restrictions that

(98)

(99)

must be satisfied for all values of «º, e2} and e3. These conditions are satisfied if

hi=-b\ \ b% — b2\ bz — bz (100)

In most practical cases, a check can be made by inspection to determine whether or not these conditions are satisfied. Connect ports 1 and 1' in series so that *'i = *Vi and short-circuit ports 2, 2', 3, and 3 ' so that e2 = e2 = ez=e* = 0. Then v2 = i>3=0, where v2 and vs are the voltages between the ports to be joined, corre-sponds to #é = áé' and äé = äé/. Repeat this procedure by connecting ports 2 and 2 ' in parallel, leaving 1 and 1' open, and short-circuiting ports 3 and 3'. Then ^i = 3 = 0 corresponds to a2 = a2 and b2 = b2. Finally, the same procedure is re-peated with ports 3 and 3 ' connected in parallel, ports 2 and 2 ' short-circuited,

and ports 1 and 1' left open. Then v\ = i>2=0 corresponds to 03=03' and bz=W.

This method can be demonstrated by applying to the bridge circuit of Fig. 16. For convenience the circuit is redrawn, as is shown in Fig. 18. First, ports 1 and 1' are connected in series, and ports 2, 2', and 3, 3 ' are short-circuited. Because the gate circuits of the two basic amplifiers are isolated from each other by trans-former action, either ports 2 and 2 ' or 3 and 3 ' maybe joined without causing any circulating currents. Thus, the voltage v2, or v3, can arbitrarily be assigned a value of zero. If a voltage is applied across the series-connected ports, then cir-culating currents will flow in each gate circuit. Because the voltages induced by transformer action and the impedances in each branch of the gate circuits are respectively equal, the voltage vs must be zero. In the next step, ports 2 and 2 ' are connected in parallel, ports 1 and l ' a r e left open, and ports 3 and 3 ' are short-circuited. By inspection, v\ is zero; and from symmetry consideration, v* must also be zero. The third test is identical to the one just mentioned since there exists complete symmetry between ports 2 and 2', and 3 and 3' . Thus, it has been shown that the matrix method of determining the composite behavior is valid for this particular interconnection. I t is further evident that any series-parallel-parallel interconnection of basic 3-port bridge elements satisfies the valid-ity tests.

For magnetic amplifiers the validity conditions need only be satisfied for half-period average voltages and currents. If the validity conditions are satisfied at any instant, as in the case of the bridge circuit discussed in the foregoing, then it follows that they also are satisfied for half-period average voltages and currents. I t should be pointed out that the instantane-ous-value validity conditions need not be satisfied for magnetic amplifiers. To test for validity in terms of half-period average voltages and currents, however, is a rather formidable task since the num-ber of ports is effectively doubled.

MATRIX EQUATIONS FOR THE BRIDGE CIRCUIT

The proper relations among the half-period average port quantities to be used in the analysis of the interconnected net-work of Fig. 18 are shown in equation 101. The variable quantities of equation 101 are the Laplace transform of the incre-mental changes in the port voltages and currents. For convenience the symbol Ä is omitted in writing this and the following equations.

I£'*l Ei"1

w ø \u ]/,'

=»Af/

i 7 > c II 7i* f \EA \EA \EA \EA

(101)

The matrix Mj may be written in parti-tioned form

Af7 = H B

(102)

where i2"is the ordinary Hmatrix for ports 1 and 2, A is a 2-by-4 matrix, and C is a 2-by-2 matrix. Since ports 2 and 3 are in series, their voltages have the same effect upon the network, and their currents are identical. Thus, the A matrix is identical to the matrix formed by the last two col-umns of the H matrix, the B matrix is identical to the matrix formed by the last two rows, and the C matrix is identical to the h22 submatrix.

A =

B = \

hncc

A l l "

# 2 2 "

h,cg\

h c°\ "22 #22 |

U cc U °0 h cc h c0\ #21 #21 #22 #22

ÄM" hi0" #22" #22 1

Kf = #2!

(103)

|#2i #22|| (104)

(105)

The transformation matrix is obtained by inspection and is

r=

1 0 0 0 0 0

0 - 1

0 0 0 0

0 0

- 1 0 0 0

0 0 0 0 0 0 0 0 0

- 1 0 0 0 1 0 0 0 1|

I r o l 0 u\

(106)

where the matrix T is the ordinary 4-by-4 transformation matrix. In general, the matrix of a series-parallel-parallel inter-connection of two basic 3-port amplifiers becomes

M= Mj+T'MiT'

\H A\ \B C\

\H+T1 \B+B7

+

ºÔ

\T 0

|o u\ Á + º 2C

M

H A\

B C\

\T 0

\c u\

(ic

For the particular interconnection under consideration equation 107 becomes

M = 2 0

(108)

Equation 108 shows that the circuit equa-tions for the fast-response bridge amplifier of Fig. 15 split into two independent sets: one involving the voltages and currents of the input and output ports only; and the other involving the supply voltage and current only. Consequently, the half-period average currents and voltage's at the input and output ports are in-


dependent of changes in the supply volt-age, whatever the cause of the changes. I t also follows t h a t the changes in the half-period average values of the supply voltage and current are independent of the input and ou tpu t por t quanti t ies . If the ideal waveform restrictions are satis-fied and if ideal bias is used, equat ion 108 is valid for the actual values of the half period average voltages and currents a t the ports , as well as for small changes in port quanti t ies.

INTERCONNECTION O F F O U R B A S I C

M A G N E T I C A M P L I F I E R S

For the purpose of analysis, a 4-core magnetic amplifier m a y be viewed as an interconnection of four basic amplifiers. All practical circuits of this type possess complete symmet ry so t h a t the number of useful amplifiers is small compared to the number of possible interconnections. No part icular new features of interest are introduced by these circuits, and they can be analyzed in the manner already de-scribed in the foregoing.

CONCLUSION

T h e mat r ix me thod of analysis discussed in this section is, to the au thor ' s knowl-edge, applicable to all known single-phase magnetic amplifiers. Although this method of analysis m a y no t be the shortest one for a par t icular circuit, i t provides a general means of analysis and gives a clearer insight to the over-all propert ies of magnet ic amplifiers.

Quite often addit ional control windings are provided for summing purposes. T h e effect of such addit ional windings on the amplifier performance can readily be determined by the use of superposition. Feedback is often used to improve the amplifier l ineari ty; i ts effect upon the amplifier can be analyzed b y conven-tional feedback techniques.

T o demons t ra te t h e me thod of analysis developed, t he well-known doubler cir-cuit shown in Fig. 19 will be analyzed in the following section.

A n a l y s i s of D o u b l e r Circuit

T h e analysis of the doubler circuit in this section is divided into two p a r t s : analysis wi th ideal vol tage waveforms, and analysis with nonideal voltage wave-forms. Since this analysis is m a d e for t he explicit purpose of demonst ra t ing t he im-por t an t ideas developed in this work, t he la t te r p a r t of t he analysis is l imited to in-p u t signal variat ions for which sequence of t ransi t ions c occurs. As a result, only pa r t of the amplifier transfer characterist ic when resistively loaded is derived.

T h e doubler circuit may be constructed by connecting two basic amplifiers in series-parallel as shown in Fig. 20. T o obtain the proper circuit configuration, i t is necessary to reverse the polari ty of the supply voltage and the voltage and cur-ren t a t por t 2 of basic amplifier II. As shown in reference 1 reversal of the supply-voltage polari ty m a y cause the inductors to sa tu ra te during a l te rna te half periods. Inspection of the circuit in Fig. 19 indeed shows t h a t the in-ductors sa tura te during a l ternate half periods.

T o apply the mat r ix method of deter-mining the composite behavior, the basic amplifiers m u s t be represented ana-lytically b y the H matr ix . T h e H mat r ix relating incremental signal quant i t ies for the interconnection is of the form

H=>Hi + THnT (109)

where H7 and Hv are the matrices of the basic amplifiers and T the transforma-tion matr ix . T h e matr ix T is obtained b y inspection and is

Ã = 0 1 0 0

1 0 0 0

0 0 0

- 1

0 0

- 1 o|

(110)

I D E A L VOLTAGE W A V E F O R M S

T h e H matr ices of the two basic amplifiers are identical and are given b y equat ion 111

H,=

Ui'+i?»

z

R* Rb

\RZ1

R/z

where

Rl'JRl

* - - RmRb ■> I D .

0

0

0

Rz

Rb

Rz 1

Rb z

Rz

RmRb

22, 1 R/Rb z

0

0

0

1

Rf\

(111)

(112)

( Ð 3 )

Subst i tut ion of equat ions 110, 111, and 112 in equat ion 109 yields

H=

Ri+R*

z

( + ) -S

Rb

Rzl

R/z

- A - 1

Ri+Rz

Rzl (±)-Rz

Rb

Rbz

Rb

22,1 Rbz

Rz 1

(+)

Rzl

Rbz

__Rz

Rb

Because the o u t p u t of the doubler circuit is an al ternat ing vol tage when the inpu t is a direct voltage, the mat r ix of equation 114 exhibits the same symmet ry proper ty as discussed in reference 1 for the series-connected saturable inductor. Reversal of the polari ty of the ou tpu t voltage and current during every other half period corresponds to pre- and post-multiplying the matr ix H by the matr ix

A**

1 0 0 0

0 1 0 0

0 0

- 1 0

ol 0 0 i |

(115)

This operation changes the sign of the matr ix elements as indicated in paren-theses in equation 114. Using these signs, addit ion of the first two rows and the last two rows yields

S+,«,(.-i)[|7,-|(>-i)a

4-(H;>+ (116)

1 , Rz Rz 1

RfRb 2 (H7)

I \Rf RmRb

T o obtain the transfer characteristic with resistive load, the following addi-tional equat ion is needed.

£ 2 = -2?L72 (118)

Eliminating h and J2 from equations 116, 117, and 118 and solving for the ratio £2 / -E\ yield t he expression

RmR. (*>+* À) (Rr\-RL)(RmR1+RbRi+RmRb) +

(Rl +Rm)RfRL - (Rf+RL)RmRb +

1 (Rf+Ri)RmRL -

(119)

If Rm is large compared with other resist-ances in the circuit, and if the approxima-tion

z^l+'s (120)

RmRb Rf

R, 1 (- :

( - ) R, l

R/Rb z

RfRb Z RmRb Rf

(114)


is made, then the following approximate transfer function is obtained.

EC RL(Rf+Rb)

R/Ri X

Rf

Rf+Rb ù -s+ß

1K;)KK:[ (121)

s + 1

T h e transfer function of equat ion 121 can be further simplified b y making use of the inequalit ies: Rb»Rf and Ri»RL

(the la t ter is not always satisfied).

E1 =

RLRI>

"*KX»D; (122)

s + 1

T h e transfer function of equat ion 122 is identical to the one previously derived by the author . 3 A word of caution is necessary in regard to t he derivation of equat ion 122. Equa t ion 122 was ob-tained, among others, b y let t ing Rm-* °° ; this limiting process is no t valid if t he saturable-inductor loop width is small. In Appendix I I i t is shown t h a t for the limiting case when Rm»Rb and the loop width is zero, the diodes conduct during pa r t of the control and block during pa r t of the gating periods of their respective inductors. Bourne a n d Ni tzen have shown t h a t this phenomenon very rarely occurs in practical magnet ic amplifiers because of the finite loop-width proper ty of the saturable inductors .4 I n Appendix I I it is further shown t h a t the magnetiz-ing resistance alone usually prevents this phenomenon from occurring.

Since the advent of silicon diodes, diode back impedances have become consider-ably greater t han the magnetizing im-pedance; the inequalities Rb»Rf and Rb»Rm then apply so t h a t t he transfer function of equat ion 119 can be wri t ten as follows:

ÅÃ RMRL

(Rf-\-RL)Ri R\+Rm IT (123)

Ri s+1

Another interesting relation can be ob-ta ined from equat ion 121. Suppose t h e two diodes in the doubler circuit are short-circuited, then the familiar series-parallel interconnection of two basic elements, basic amplifiers wi thout diodes, results .

Fis- 19 (left). Doubler circuit º

O- -o v w •2

-o—

4 es

-*——( v—o-O -O W V - -

Fig. 20 (right). Doubler circuit as an interconnec-tion of two basic

amplifiers

T h e approximation t h a t Rm is large compared to the other resistances in t he circuit is now valid, and the transfer func-tion of this new amplifier is obtained b y subst i tut ing Rf=Rb = R2 in equat ion 121.

E*

A" 2RL

s+1

" 1(^)KH;I s+1 Ã

(124)

T h e transfer function derived relates changes in the inpu t and o u t p u t voltages from their quiescent values. If ideal bias is used, then the s teady-s ta te transfer characterist ic can be obtained b y making the subst i tut ion

z-+l

E1-+E1, E1-+I1,1\-*lr2

E^-^-E^+Es'

in equat ions 116 and 117. By making use of the relation E2 = —R1J2 and eliminating I 1 and I2 from the resulting equations, t he following expression is obta ined:

R ?/ RL Rf)\Rm Rf/W

=-f;(i+i)£i-\\Rf Rb G w R/)\Es (i25)

T h e vol tage Es' in equat ion 125 is related to t h e supply voltage through a t rans -formation similar to the one used in reducing the four b y four mat r ix of equa-tion 114 to a 2-by-2 matr ix . This t ransformation reversed the polari ty of t h e o u t p u t voltage during the control periods of basic amplifier 1. F r o m this t ransformat ion a n d because the supply vol tage is positive during the control periods and negative during t h e gat ing periods, i t is seen t h a t the voltage Es

!

in equat ion 125 is a lways negative. Mak ing use of this proper ty and solving equat ion 125 for E2 yields

:ιπ |Rm t e

I 6 s

^ ' 2

- - O

E2

( + ) ( - )

RL(Rb+Rf)

Rf(Rm+RL)+Rb(Rf+RL)

Rm

Ri E1 + \ES (126)

T o compare the results of this and the following subsections, a special case will be considered in which the resistance values and the loop width are related by the equat ions:

RL=Rf=0.001R

Rm=Rb = R

Ri=0.1R

Ic = 2R

(127)

Subst i tut ion of equat ion 127 in equation 126 gives

£2 = 3.33£é+0.333\ES (128)

Since RL = Rf} the max imum ou tpu t is 0.5[ßs | . T h e s teady-state transfer char-acteristic is plot ted in Fig. 21 . T h e solid line represents the condition of ideal bias, and the dashed line represents the condi-tion of no bias. T h e two curves are shifted horizontally an amoun t R\IC = 0 . 0 5 | E J from each other.

SINUSOIDAL S U P P L Y VOLTAGE AND

D I R E C T I N P U T VOLTAGE

In order to apply the method of analysis developed in the foregoing sections, the vol tage waveforms t h a t exist a t the inpu t and ou tpu t por ts of each basic amplifier mus t be known. Because of symmetry these voltage waveforms are t he same for bo th basic amplifiers of which the doubler circuit is constructed. T h e inpu t por ts of the two basic amplifiers are connected in series as shown in Fig. 20. Thus , the voltage t h a t exists across the inpu t ports of each basic amplifier is t h e sum of the inpu t voltage applied to t he doubler cir-cuit and the voltage t h a t exists across the inductor of the other basic amplifier. In Appendix I I i t is shown t h a t t he wave-


form approximations apply for this case, and t h a t the constant t e rms in the ex-pressions for t he waveform approxima-tions a re :

For t near zero,

Es 7Ã* E2 Er ÔÃ*

Ts T Tt T1 Ã

E1f=E1

For / near T/2,

(129)

Es TT2 E2 # z

Ã, Ã Ã2 RL+RfT Tx

Ei'=Ex (130)

These waveform approximat ions are valid for sequence of transi t ions c only. If the direct inpu t voltage is greater than RJc, sequence of transi t ions b oc-curs; if i t is less t h a n Rilc, sequence of transit ions c occurs. T o limit the analysis to a reasonable length, only sequence of transi t ions c is considered; and as a result only pa r t of the steady-state transfer characterist ic is derived.

The circuit equat ions for sequence of transitions c a re derived in Appendix I I . Substi tut ing equat ions 129 and 130 in equation 162 yields

ERC = c(E1

c - RJC) +d(Esc +ES) +

the s teady-s ta te characteristic of the amplifier. Since these equations are nonlinear, t he mat r ix method of deter-mining t he composite behavior is not applicable. However, since these equa-tions are valid for any operating point as long as sequence of transit ions c occurs, the slope of the s teady-s ta te transfer characterist ic can be determined by small signal analysis. T h e matr ix method of determining the composite behavior is then applicable since equat ion 131 can be linearized a round any operat ing point . In te rms of incremental signal quant i t ies equat ion 131 becomes

bERc = c8Eic +dbE2

c + cqSEi' (132)

where

\\Ri,RL-\-Rf 9.=

Ri Rf + RbRi

Ri(Ri+Rb)

2

X

7T2ES

(Ei-RJc) (133)

Rb RL+R/ . RbRi

+ R, Rf R1(R1+Rb) X

r2E, (Å,-RJcY (131)

where E\ is the input voltage to the doubler circuit. Equa t ion 131, together with equations 44 through 48, determine

and b is the differential operator. T h e t e rm bEi in equat ion 132 is the

change in the average inpu t voltage to the doubler circuit t h a t occurs during the control periods of basic amplifier / . Thus , hE\ is zero during t he gating periods of basic amplifier / . hEi should not be confused with 6E\C which is the change in the average voltage a t the input por t of amplifier / during the control periods. This la t te r quan t i ty is the sum of bE\ and the change in the average voltage t h a t appears across the control winding of basic amplifier II during its gating periods. Since the voltages Eic, and Ex\ and E\ can be adjusted independently of each

LINEAR CASE WITH NOBIAS-r

NONLINEAR CASE WITH NO BIAS

other, these voltages can be considered independent variables.

I n te rms of increment signal quanti t ies equations 44 through 48 become:

bEf^Ratf+tEz0

äÅ2 =Rt)dl2 -\-8ER

6E1e=R1bIl

e-]-8ERe

bE2g = RfbI2

e+bER0

bERc+bER

g = 0

(134)

(135)

(136)

(137)

(138)

From these equations and equation 132 the H matr ix of basic amplifier / can be obtained. Since these equations are identical to the equations obtained for the linear case, except for the te rm cqbEi in equat ion 132, the matr ix equation of basic amplifier / becomes:

aI=HIßI+PbEl'

where P is the matr ix

(139)

P = q\

c

c

c

jl ú-c c

II 1 - c

1

Rb\\ 1

Rrl

(140)

and Hi is obtained by substi tuting 3 = 1 in equat ion 111. Similarly, for basic amplifier / /

au^Hrfn+PöES (141)

where bEi" is the change in the half-period average value of the input voltage to the doubler circuit during the control periods of basic amplifier / / . Perform-ing the transformation of variables in-dicated by the equation

OLu — TvLll'

ßii = Tßu'

yields

áð', = ÔÇ,ÔâÉÉ

é' + ÔÑäÅ!»

(142)

(143)

To obtain the mat r ix of the series-parallel interconnected amplifier, equa-tions 139 and 143 are added.

aI + au'=HIßi + THiTß„' + ÑäÅé' + ÔÑäÅé" (144)

But , since

ßl=ßll'=ß (145)

where a and ß are the variables of the interconnected amplifier, equation 144 can be rewri t ten

Fig. 2 1 . Steady-state transfer characteristics ll^ + TH^We+PbEi' + TPbES (146)


T h e last two terms in equat ion 146 can be wri t ten al ternat ively:

where

ÑäÅé' + ÔÑäÅ/ =

c

l^c

c

c

1-c

c

( + ) ^ - 1

-l-cRb

c 1_

1-c Rf

= qRS

( + ) - 5 - l | 1-c Rf

c 1 1-c Rb

äÅ÷'

äÅ÷"

(147)

when R and 6" are the matr ices indicated. Ignore, for the t ime being, the signs in parentheses in equat ion 147. Again t he number of variables can be reduced by making use of the t ransformation of var i -ables expressed b y the mat r ix of equat ion 115. Thus , let

a=Aat'

â = Áâ'

(148)

Subst i tut ion of equat ions 147 and 148 in equation 146 yields

a' ^AWHj + THjTWAß'+qARS (149)

Premult iplying the mat r ix R by t he ma-trix A changes the signs in R as indi-cated in parentheses in equat ion 147. Ad-dition of the first two rows a n d the las t two rows of the mat r ix equat ion, equat ion 149, yield

öE^RJh (150)

\\ë+×>(ÁË\\äÅé \[Rf Rb\Rm Rf/\\ +

2R*

Ri áßäÅ^+äÅ/) (151)

From the definitions of äÅ÷ and äÅé", i t is seen t h a t

äÅé'+äÅ÷'^äÅé (152)

T o determine t he s teady-s ta te transfer characteristic wi th resistive load, equa-tion 152 and the relation äÅ2= —RLbI2 are subst i tuted in equat ion 151. Then b y using equation 150 to eliminate ä/é, the following differential equat ion is obtained.

L+J_Ml__l\\\SE R, RLRMM RfJ\\ 2

- ( F . + i ) t « + ^ (153)

T h e quant i ty g, as seen from equation 133, is a function of E i and m a y be wri t ten in t he form

RbRL+Rf

+ RbRi

Rx R/ RL(Ri-{-Rb)

2

r2Es (155)

A relation between E\ and E2 can now be obtained by subst i tut ing equat ion 154 in equat ion 153 and integrat ing the resulting equation.

E2= -K\\(l-2q'R1Ic)E1+q'E1*\\ + constant (156)

where the constant K is the s teady-s ta te gain of the amplifier tor the linear case and is given b y the expression

K=-RL(R*+Rf) Rm

Rf(Rm+RL)+Rb(Rf+RL) Ri (157)

T h e constant of integrat ion m a y be eval-ua ted in the following manner . When the inpu t signal is equal to RJC, t he volt-age waveforms are ideal. Consequently, t he linear and nonlinear s teady-s ta te transfer characteristics should intersect when E\=RiICi provided no bias is used in either case. For the linear case and the chosen parameter values, the ou tpu t voltage with no bias is given by the dashed line in Fig. 21 . As seen from this figure, t he ou tpu t voltage is equal to 0.33 Es when the inpu t voltage is equal to Rilc. For the same parameter values the s teady-s ta te transfer characteristic for the non-linear case becomes

E2 = - 3 . 3 3 0.577£i +4.235 — EA Es

+

constant

q^q'Ex-q'RJc (154)

= - 1 . 9 2 £ ÷ - 1 4 . 1 — E ^ + c o n s t a n t (158) Es

Subst i tu t ing Ei = RiIc = 0.05 Es in equa-t ions 158 and equat ing t he result t o 0.33 Es yields

constant = 0.333E*+1.92 X 0.05E, +

14.1 ^ ( 0 . 0 5 £ , ) 2 (159) Es

=0.464E,

Finally, subst i tut ion of equat ion 159 in equat ion 158 gives

E 2 = - 1 . 9 2 £ ! - 1 4 . 1 — £ , 2 +0 .464£ s (160) Es

T o compare the linear and nonlinear t rans-fer characteristics, equat ion 160 is p lot ted in Fig. 21 . Only a port ion of t he steady-s ta te transfer characterist ic can be pre-dicted by equat ion 160, because i t is valid only for sequence of t ransi t ions c. Sequence of t ransi t ions c occurs for inpu t signals less t h a n Rilc, and sequence b occurs for inpu t signals greater t h a n RJC. Thus , RJc is the upper limit of t he inpu t signal for which equat ion 160 holds. T h e

waveform approximations are used to calculate the average inductor voltage during the intervals 0 to fa and fa to T/2, These t ime intervals, therefore, mus t be small compared to T. When the input vol tage is equal t o zero, Ei is approxi-mate ly 0.057" and fa is approximately T/2 —0.1Ã. Thus , for inpu t signals less t han zero, the val idi ty of equat ion 160 is questionable.

T h e method of determining the steady-s ta te transfer characteristic demonstrated in the foregoing is no t the only one avail-able. For example, the equat ions de-scribing the s teady-s ta te operation of the two basic amplifiers, together with the constraint equations resulting from the interconnection, m a y be solved directly. Since these equations are nonlinear, con-siderable algebraic manipulat ions m a y be involved in obtaining a solution. On the other hand, the method demonstra ted no t only determines the s teady-s ta te charac-teristic, b u t also the small-signal dynamic characteristic.

S u m m a r y

T h e analysis developed in this work is based on five characteristic properties of magnet ic amplifiers:

1. Circuits are symmetrical.

2. Circuits consist of interconnections of basic elements.

3. In steady-state operation all voltages and currents are periodic and possess a com-mon period.

4. Each inductor in an amplifier saturates only once per period.

5. For each basic element the part of the period in which the inductor is saturated is less than or equal to the part in which the inductor is unsaturated.

Propert ies 1 and 2 suggest an analysis in which analytic expressions for the basic elements m a y be combined to yield ana-lytic expressions for the interconnected amplifier. Thus , the analysis of magnet ic amplifiers m a y b e divided into two p a r t s : analysis of basic elements, and analysis of interconnections of basic elements. Three basic elements have been con-sidered, as shown in Fig. 1. F rom an analysis point of view the first two ele-ments are identical; they differ only in regard to the physical arrangements of components connected in series.

Propert ies 3, 4, and 5 allow a great simplification of the analyt ic expressions which determine the s teady-s ta te be-havior of the basic elements. T h e deriva-t ion of the analyt ic expressions describing the bas ic magnetic-amplifier element of reference 1 are, in particular, greatly


simplified because the circuit contains only one nonlinear element.

Because the operational cycle of mag-netic amplifiers is periodic, a period of operation may be defined. For reasons of analytical simplicity, the period of operation is defined as the time interval between two successive instants in time at which the inductor becomes unsat-urated. This period of operation is then divided into two equal parts: the control period and the gating period. Because the output voltage waveform is unrelated to the input voltage waveform, instan-taneous values of voltage and current are almost meaningless; and half-period average voltages and currents are chosen as the circuit variables. This adoption of average-quantity variables, along with the defined period of operation, automati-cally introduces the concept of control-and gating-period average voltages and currents. A control-period average quantity is the average value of the quan-tity during the control periods; thus, it is a train of pulses where each pulse occurs during a control period and whose mag-nitude represents the average value that exists during the same period. Similarly, the gating-period average quantities are trains of pulses defined in a like manner for the gating periods. The sum of a control and the corresponding gating priod average voltage or current is thus equal to the actual half-period average voltage or current. The half period average voltages and currents are thus split into components represented by two pulse trains, these pulse trains being dis-placed from each other in time by T/2 sec. This expansion of a signal into a number of pulse trains may be extended by dividing the period of operation into n subintervals, the average values of the current or voltage during the subinter-vals being the variable quantities. Be-cause the n subintervals repeat periodi-cally with period Ã, these variable quanti-ties are in the form of n pulse trains, the rth pulse train being displaced in time by T/n sec. from the (r+l)th. These n components are orthogonal because each component occupies its own time interval within the period of operation. Friedland employs the same type of expansion in his analysis of periodically time-varying systems, and calls these components "skip-sampled components."5*

During the control periods the inductor is unsaturated and its magnetizing im-pedance can be approximated by a linear

* The author had already developed the method of analysis presented in this work using the skip-sampled components as variable quantities when he became aware of Friedland's work on periodically time-varying systems.

resistance if the high permeability, square-loop-type core materials are used. For amplifiers without diodes this approxima-tion is exceedingly good; for amplifiers with diodes the constant d-c loop width must be taken into account. However, if the voltage waveforms satisfy certain conditions, a linear resistance may be used to represent the magnetizing im-pedance; and, in addition, the diode may be represented as an element which is nonconducting during the control periods and conducting during the gating periods. Thus, during the control periods the basic elements are linear and resistive, and linear relations between control-period average quantities may be written. During the gating periods all components of which the basic elements are composed are linear except for the saturable induc-tor. Because the operation is periodic the average value of the inductor voltage over a complete period must be zero so that the relation

can be used to eliminate the term ER° from the amplifier equations. Thus, the inductor characteristics during the gating periods are immaterial as long as the amplifier possesses properties 3, 4, and 5. The resulting equations are linear relations among control and gating period average voltages and currents at the ports. From these equations, which are based on steady-state properties, the steady-state behavior of magnetic amplifiers can be obtained.

The basic element equations, when recognized as difference equations, can be used to describe the dynamic be-havior of the basic elements. The manip-ulations that are necessary on these equations in order to determine the dy-namic behavior are not as simple as for the steady-state case. If the difference equations are transformed through La-place methods, then all manipulations are reduced to simple algebraic operations, thus simplifying the dynamic analysis. Because the difference equations describ-ing the dynamic properties were derived from steady-state properties, these equa-tions, in general, hold only for small changes in port voltages and currents. However, if certain waveform restrictions are satisfied, these small-change restric-tions need not be adhered to; andt he transformed difference equations may be used to characterize the behavior of a magnetic amplifier for large as well as small variations in the port quantities.

To determine the analytic expressions for magnetic amplifiers resulting from interconnections of basic elements, the

matrix method of determining the com-posite behavior, as described by Guillemin for ordinary 2-port networks, may be used.2 This method is ideally suited for magnetic-amplifier analysis because the input and output ports of basic elements usually are isolated by transformer action. Thus, no circulating current will result from the interconnections of basic ele-ments; this is the validity condition for the matrix method of determining the composite behavior of ordinary linear networks. For magnetic amplifiers re-sulting from interconnections of basic elements, an additional condition must be satisfied in order to apply the matrix method of determining the composite be-havior. The control and gating periods of the individual elements must have the same position in time, or must occur so that the control period of one ele-ment corresponds in time to the gating period of the other. The symmetry property of voltage waveforms in mag-netic amplifiers usually insures that this condition is satisfied. The extension of the analysis to the polyphase case clearly places in evidence the usefulness of the skip-sampled component expansion, together with the matrix method of determining the composite behavior.

For magnetic amplifiers with diodes the condition that the diodes, or diode, conduct during the gating periods and block during the control periods is not always satisfied. By approximating the voltage waveforms within the regions where this condition is violated, a set of nonlinear relations among the half-period average voltages and currents of the ports can be derived. From these nonlinear relations a set of linear differential equa-tions can be obtained relating differential changes in the average port voltages and currents around a predetermined operat-ing point. Since these differential equa-tions are linear, the matrix method of de-termining the composite behavior is valid. The resulting differential equations may then be integrated to yield the steady-state characteristics. The small-signal dynamic characteristics are given by the differential equations. This method holds only if the regions in which the voltage-waveform approximations made are small compared to the period of opera-tion.

Appendix I. Circuit Equations for Sequence of Transitions C

In this appendix the basic magnetic-amplifier circuit equations for sequence of transitions c are derived. The objectives


e R = 0 j

ST. 2

eR=o

ST. 4

eR=eR | I

ST. 5

eR=o I ST. 6

eR=o

CONTROL PERIOD DIODE NONCONDUCTING

T/2 I I I

PERIOD OF OPERATION

ST. 3

e R = e R 3

ST. I

eR=o

ST. 2

GATING PERIOD DIODE CONDUCTING

Fig. 22. Sequence of transitions c

of this derivation are twofold: to demon-strate a method by which the equations determining the steady-state or quiescent operating conditions can be obtained, and to determine the voltage waveform restric-tions that make these equations linear. Even though this analysis is limited to se-quence of transitions c, the method used is equally valid for any one of the other sequences of transitions; sequence of tran-sitions c being chosen because it applies to the doubler circuit of Fig. 19.

For sequence of transitions c, see Fig. 22. Because the diode is nonconducting during the control periods and conducting during the gating periods, equations 44 through 48 are valid. The average induc-tor voltage during the control period, then, is the only remaining unknown; it is cal-culated as follows:

2 r/2

2 CT/%

= rj \\(ei-RiIc)+d(e8+e*)\\dt-

2 Ch

- J c\\(ex-RxIc)+d(es+e2)\\dt-

■ c\\(ex-RxIc)+d(es+e2)\\dt J h

2 7 \

(161)

where t\ and t2 are given by equations 26 and 34, respectively. Substitution of the wave-form approximations for ex, es, and e2 in equation 161 and performing the integra-tion yields

ERC - c(Ex

e - RJC) +d(Esc +E2

C) +

\Rb T,Ti

-2T(Ei'-RlIe)*+

W-RJc)* (162)

Equation 162 shows that the half-period average inductor voltage during the control period is a nonlinear function of the slope of the output voltage (E2/T2) and the slopes of the input voltage (Å÷/Ôë and Å÷/Ô÷') near the zero crossings of the ex wave, and of the magnitude of the input voltage at times 0 and T/2 {Å÷' and E / ) .

The equations for the basic magnetic amplifier are linear if the sum of the last two terms in equation 162 is a constant. For practical amplifiers there are only two cases of interest for which the amplifier equations are linear. Th ese cases are:

E\ =Ei" =RiIc

Å÷' = constant

Ei" = constant

Å÷ -— = constant Ô÷

(163)

Å÷

TV = constant

(164)

E2 —- = constant T2

If a d-c bias voltage is connected in series with port 1, then equation 163 becomes

Å÷' = Å÷''=^É€-Åý (165)

For Et, = RxIc (termed ideal bias) the right-hand side of equation 165 is equal to zero. Thus, if the input voltage is zero when the supply voltage is zero and ideal bias is used, then the amplifier equations for se-quence of transition c are linear. This re-sult can be generalized for all sequences of transitions. Substitution of equation 163 in equations 20, 26, 28, 34, 35, and 36 yields

t2 = t2'=h' = T/2

(166)

(167)

Thus, the four possible sequences of transi-tions in Fig. 9 reduce to the following se-quence

1 - 4 - 6 - 3 - 1

The linearity condition of equation 163, therefore, applies for all four sequences of transitions.

The linearization conditions of equation 164 state tha t both the amplitude and the slope of the input voltage at the zero cross-ings of the supply voltage wave and the slope of the output voltage wave are constant or independent of the half-period average values of the input and output voltages. The output voltage waveform of resistively loaded magnetic amplifiers invariably pos-sesses these properties. Thus, if the input voltage is obtained from another magnetic amplifier, the linearization conditions of equation 164 are usually satisfied.

Similar results may be obtained by con-sidering any one of the other sequences of transitions. For these sequences a slight complication arises since the diode is both conducting and nonconducting during the control and gating periods. Thus, equa-tions 45 and 47 are not valid. This diffi-culty can be overcome, for example, by writing equation 45 as follows:

Esc+E2

e=Rf(hc)'+Rb(I2

cy+ERc (168)

where (72c)' is the half-period average cur-

rent at port 2 which flows during the part

of the control period in which the diode is conducting, and (72

c)" is the half-period average current at port 2 which flows during the par t of the control period in which the diode is nonconducting. The total half-period average current at port 2 is

72c = (7 2

c ) '+(7 2c )" (169)

Solving equation 169 for (72c)" and substitut-

ing the result in equation 168 yields

ES+Ei^iRf-RMW+RJf+Ea' (170)

Because diode unblocking occurs in the inter-vals for which the voltage-waveform approximations are valid, the current (hcY can be expressed in terms of the voltage-waveform approximations. The resulting nonlinear terms can then be combined with the nonlinear terms appearing in the ex-pression for ER.

In steady-state or quiescent operation the port voltages are periodic and of the same period as the supply voltage; equation 162 is thus valid for all periods of operation. From equations 44 through 48 and equa-tion 162 the steady-state or quiescent rela-tions among port voltages and currents can be derived.

If equation 163 is satisfied for all periods of operation, the following operating conditions are met :

1. The operation of the amplifier is periodic and of the same period as the supply volt-age. 2. The diode is nonconducting during the control periods and conducting during the gating periods. 3. The inductor voltage is positive during the control periods. 4. The input voltage satisfies the relation

ex^Rxh, for t= T/2

If these operating conditions are satisfied, then the operation of the amplifier can be described by a set of linear equations among half-period average voltages and currents.

Appendix II. Doubler Circuit

The doubler circuit of Fig. 19 is con-structed of basic amplifiers 7 and 77. In this appendix the sequences of transitions for these basic amplifiers are determined. The conditions chosen are: input of doubler is a direct voltage, supply is a sinusoidal voltage, and a resistive load. The voltage waveform approximations that can be made for sequence of transitions c a r e then deter-mined, and in the end a discussion of the circuit behavior in the limiting case when Rm—*-00 is given.

The state of the basic amplifier depends upon the supply and port voltages. Be-cause the doubler circuit is constructed by connecting two basic amplifiers in series-parallel, the input port voltage of basic amplifier 7 is the sum of the direct input voltage to the doubler circuit and the volt-age that exists across the control winding of basic amplifier 77; the output port voltage of each basic amplifier, however, is the same as the output voltage of the doubler cir-cuit.

The currents tha t flow in the doubler cir-

502 Johannessen—Analysis of Magnetic Amplifiers with Diodes NOVEMBER 1959

Ri.

ii

Fig. 23. Equivalent circuit at time zero

Fig. 24. Approximate equivalent circuit in interval 0 to ti

cuit, when the supply voltage is zero, are caused by the input voltage only. If the input voltage magnitude is less than R\ICt

the inductor voltages are zero. A small in-crease or decrease in the supply voltage, therefore, causes one diode to conduct and the other to block. Thus, the states of the diodes depend entirely upon the supply voltage. A further increase or decrease in the supply voltage causes currents of sufficient magnitude to overcome the loop-width current Ic to flow through the in-ductors. A voltage now appears across each inductor, driving one towards and the other away from saturation. The events just described correspond to sequence of transi-tions c. By a similar argument, it can be shown tha t sequence of transitions b occurs if the input voltage is greater than Rilc-

The voltage-waveform approximations for sequence of transitions c will now be determined. The intervals of interest are from time zero to t\, and from fa to T/2. An equivalent circuit at time 0-f- is shown in Fig. 23. Because Rb is large compared to RL and Rf, the three loop currents are given by the following equations.

H~ Ri

t2=* Rb

es

RL+Rf

(171)

(172)

(173)

At time t\ the net current in inductor / is equal to Ic. The equivalent circuit of Fig. 23, however, is valid only if

k-ii<ii-\-it<Ic (174)

Because RL and Rf are small compared to the other resistances in the circuit, the current %—i\ builds up rapidly and, if ii = Ic, will reach the magnitude Ic at time

4 2IC(RL+RL) „ (175)

The time given by equation 175 is the maxi-

mum time it takes the current H—i\ to reach Ic. For most practical amplifiers IC(RL+R/)<0.01 ES SO tha t

*×<0.002Ã (176)

The time interval from zero to tx, therefore, is negligible. The time fa, then, can be determined from the equivalent circuit shown in Fig. 24. The current * can be expressed as follows

Ri+Rb es+-

(Rl+Rf+RL)Rb R1+RL+Rf

(177)

Substituting es=(v*/T)Etfi, ei=Eu and * = Ic in equation 177 and solving for the time fa yields

Rb

ir*EsRi+Rb

X

(Ri+Rf+Rrilc—-El Ë 1

(178)

If Ri is of the same order of magnitude as R/ and RL, the time fa is small and may be neglected. Thus, only when R\ is large compared to Rf and RL is the time interval from 0 to fa of importance; and the approxi-mation that R\ is large compared to Rf and RL can be made. Equations 177 and 178 can then be rewritten as follows:

i = — *8+ôô (e8+ei) Kb K\

/i = -R>

T*EsRi+Rb

(RJc-EO

(179)

(180)

From equations 179 and 180 the voltage-waveform approximations for the time interval from 0 to t\ become:

¥S'TES

— = — Es 7\ T s

El'^E1

(181)

(182)

(183)

An equivalent circuit for the time inter-val from fa to T/2 is shown in Fig. 25(A). The waveform approximations then be-come:

Ts T °

Es RL

Ts Ri+RfT

2V

El"=El

(184)

(185)

(186)

(187)

These waveform approximations can be used to determine the average inductor voltage during the control period.

An interesting case arises when Rm is large compared to the other resistances in the circuit. In the analysis tha t follows ideal waveforms and zero loop width are assumed. These assumptions normally im-ply tha t the diodes are nonconducting and conducting respectively during the control and gating periods of their associated in-ductors. The half-period average input

+ < ■ËËËô-

i—(~y^ WV f I W

RL

(A) -vw-

Ri ■ËËËô - V W -

êó Rm

-vw-RL

vv\ Rf

(C) Fig. 25. Equivalent circuit

A — I n interval t2 to T /2 B—When the inductors are unsaturated C—When one inductor is saturated

voltage for minimum and maximum output voltage can then be obtained as follows.

Minimum output voltage occurs when neither of the two inductors saturate. This condition is insured if

J S P / = — ER (188)

The average value of the input voltage for which equation 188 is satisfied can read-ily be obtained from the equivalent circuit of Fig. 25(B) and is

( £ i ) m

R^Rb-Rf)

(Rf+Rb)(2RL+Rm+Rb)-Rb(Ri>-Rf)

(189)

where (Ei)min is the value of Å÷ for which the output voltage is a minimum. Making use of the inequality Rb»Rf in equation 189 yields

&)* Ri

~2RL+Rm Es (190)

The average value of ex for which maximum output voltage is obtained can be deter-mined from the equivalent circuit of Fig. 25(C) and the equation

ER - J E P / 7 - 0 (191)

NOVEMBER 1959 Johannessen—Analysis of Magnetic Amplifiers with Diodes 503

The expression for (Ei)m&x is

(Ä) lfi?I+feÄ (192)

where (£i)max is the value of Ex for which the output voltage is a maximum. In the derivation of equation 192 the inequalities Rm»RL and R/»RL and R/ have been used. Furthermore, it has been assumed that the polarities of the currents i2 and H are as shown in Figs. 25(A) and (B). How-ever, if ei is large and positive, the current H may reverse polarity; and if ex is large and negative, the current H may reverse polarity. The range of values of Ex for which the current polarities assumed are correct is as follows

2Rf)Rm -\-RbRi -{-RiRm Es<Ei<

2RLRm+RbRm+Rm2

2RfRm-{-Ri)RiJrRLRm

2RLRm-{-RfRmJrRm't Es (193)

If Ru RL, and R/ are small compared with Rm and Rt,t then equation 193 can be written

Rb+Rm II Rm Rm2\

(194)

The useful range of Ei (range for which the output can be controlled by the input) is obtained from equations 190 and 192, and is

Rf Ri

RÖRL+R/ ES<EX<

Ri

2RL-\-Rm

Es (195)

Because RL«Rm, the right-hand side of equation 194 is always greater than the right-hand side of equation 195. Thus, t2 is al-ways positive within the useful range of e\. The left-hand side of equation 193, however, may, if Rm is sufficiently large, be greater than the left-hand side of equation 195. Thus, i\ may be negative. In the limit when Rm—*■ °° both sides of equation 193 are zero, while only the right-hand side of equation 195 is zero. Consequently, if the saturable inductor is ideal, the diodes conduct during the control periods and block during the gat-

ing periods, regardless of voltage waveforms. In practical circuits this situation very rarely occurs because of the relative values of the resistances in the circuit.

References

1. ANALYSIS OF MAGNETIC AMPLIFIERS WITHOUT DIODES, Paul R. Johannessen. AIEE Trans-actions, see pages 471-85 of this issue.

2. COMMUNICATION NETWORKS, VOL. II (book), E. A. Guillemin. John Wiley & Sons, Inc., New York, N. Y., 1935, pp. 145-50.

3. ANALYSIS OF MAGNETIC AMPLIFIERS BY THE U S E OF DIFFERENCE EQUATIONS, P. R. Johanaes-sen. AIEE Transactions, pt. I {Communicttion and Electronics), vol. 73, 1954 (Jan. 1955 secton), pp. 700-11.

4. DYNAMIC OPERATION OF MAGNETIC AMPLI-FIERS FOR FEEDBACK CONTROL SYSTEMS, H. C. Bourne, Jr., D. Nitzan. Scientific Report No. I, Electronics Research Laboratory, University of California, Berkeley, Calif., May 1956.

5. THEORY OF TIME-VARYING SAMPLED-DATA SYSTEMS, B. Friedland. Technical Report T-191B, Electronics Research Laboratories, New York, N. Y., Apr. 15, 1957.

Standards for Measurement of Brightness

Intensification in Fluoroscopic

Image Intensifiers

WALTER S. LUSBY ASSOCIATE MEMBER AIEE

IT IS convenient to characterize devices for production of intensified X- ray

fluoroscopic images in terms of their brightness as compared with conventional fluoroscopic screens. Thus , an X-ray-image intensifier is described having an intensification factor of 200 or 500 or 1,000 meaning t h a t the image presented to the viewer is 200 or 500 or 1,000 times brighter than the image would be if he used a conventional X- ray screen. These fig-ures, al though widely used today, are meaningless unless all variables are de-fined and s tandardized in their measure-ment . Fur thermore , the same te rm has been used to express the brightness rela-tionship between a light image from a phosphor a t the inpu t end of an image device and the light image ou tpu t . In the case of a fluorescent screen contiguous to a photoelectric surface in a vacuum, the inpu t l ight image used as the base point can be quite different from, and is usually less br ight than, the conventional fluoroscopic screen image under equal st imulus. However, to the best of the author ' s knowledge, no previous a t t e m p t

has been made to propose a set of s tand-ard conditions on which to base a figure of meri t for X- ray image intensifiers.

There are m a n y factors which can lead to var iat ions in the figure of meri t for image intensification. First , based on a comparison with a conventional screen, there is the question of how br ight the conventional X- ray screen is. T h e early l i terature on medical X- r ay intensiffers refers to the intensification factor required for success. Obviously the comparison was wi th a much less br ight screen than the conventional screen of today. T h e indus t ry has progressed from wha t has been posthumously named the Pat terson A fluoroscopic screen to the B, the B2, and the CB2y as well as to the screens of other manufacturers , and a s teady in-crease in the brightness of conventional fluoroscopic screens has been witnessed. For example, since the introduct ion of the Pa t te r son B screen in 1934 the average brightness of Pa t te rson fluoroscopic screens has increased b y approximately 300%.

Second, the effective wavelength or

hardness of the X- ray beam affects the comparisons. Some types of image in-tensifiers employ a construction such as a heavy glass window in the X- ray beam which acts as a substant ia l filter. I t follows t h a t a comparison of speed of the conventional and the new devices can easily va ry b y a factor of 2 depending upon the kilovoltage a t which the meas-urements are made.

Third, one can compare brightness by using an open X- ray beam or by using a simulated or actual pa t ient or industrial object. Again a substant ia l variat ion in results is obtained.

T h e need for s tandards for intensifica-tion factors rests no t only on technical grounds of scientific correctness, b u t also on practical grounds. For example, one m a y a t t e m p t to extrapolate cinefluoro-graphic techniques from one type of image intensifier to another using the intensifica-tion factors published b y the manufac-turers and compensating for all other variables. T h e predicted exposure factors m a y appear to disagree b y more t han 2 to 1. In the opinion of the author , this is a direct result of the lack of an accepted common s t andard and thus there should be unaminous agreement t h a t some form of s tandard is necessary.

Paper 57-316, recommended by the AIEE Elec-trical Techniques in Medicine and Biology Com-mittee and approved by the AIEE Technical Operations Department for presentation at the AIEE Winter General Meeting, New York, N. Y., February 1-6, 1959. Manuscript submitted April 26, 1957; made available for printing December 30, 1958.

WALTER S. LUSBY is with the Westinghouse Electric Corporation, Baltimore, Md.

504 Lusby—Measuring Brightness in Fluoroscopic Image Intensifiers N O V E M B E R 1959

analysis of magnetic amplifiers with diodes

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