Qn, Qn+u <2n+2 = constant charge across the condenser during the nonconduction intervals of n, n+lt and n+2 half-cycle of the applied voltage

q — instantaneous value of condenser charge <]n> 9n+i, gn+ 2 = instantaneous value of con-

denser charge during n, n+1, and n+2 half-cycle of the applied volt-age m

Rc = effective control resistance of the series-connected saturable reactor, Rc = 2RWC +RLC

RQ = effective gate resistance of the series-connected saturable reactor, Rg — 2Rwg +RLQ

Rwc = resistance of the control winding of the series-connected saturable reactor

Rwg— resistance of the gate winding of the series-connected saturable reactor

V=constant condenser voltage during non-conduction intervals

Vn> Vn+1, Vn+ 2 — constant voltage across the condenser during the nonconduction intervals of n, n+1, and n+2 half-cycle of the applied voltage

v = instantaneous condenser voltage

AL T H O U G H T H E R E A R E m a n y methods and procedures for ana-

lyzing complex circuits, probably the best-known general approach is to apply an organized, systematic procedure to the combination of the analyt ic expressions for t he voltage-current characteristics of the individual circuit elements. T o be able to apply this me thod of a t tack , i t is essential t h a t the characteristics of t he individual circuit elements, if such ele-ments can be defined, can be represented b y analyt ic expressions, and t h a t an or-ganized me thod exists where the ana-lytic representat ions of the individual circuit elements can be combined to yield the desired result.

In this paper t he foundation for such an analysis will be established for mag-netic amplifiers wi thout diodes. T h e element to be considered as t h e basic mag-netic-amplifier element consists of a 2-winding saturable inductor, one winding being in series with an al ternat ing voltage source. Bo th the saturable inductor and t h e al ternat ing voltage source are funda-menta l to the magnet ic amplifier, because either element wi thout the other cannot provide power amplification.1 By proper choice of variables and the use of certain

ß — instantaneous value of flux density y = steady-state operation, or mean conduc-

tion angle, during half a cycle of the a-c rectangular voltage applied to the gate winding of the series-connected saturable reactor with finite control resistance, during which current flows, and about which transient variation, or oscillation is assumed to take place

7c = critical value of the steady-state opera-tion conduction angle below which the series-connected saturable reactor with capacitive loading and finite control impedance becomes unstable and oscillatory

Ö = total flux ö — effective phase angle of the series-con-

nected saturable reactor of Fig. 1, ( J ^ t a n - 1 l/vC(Rg+M*Re)t degrees or radians

ù = angular frequency of the gate (carrier) voltage applied to the series-con-nected saturable reactor

ojm = angular frequency of the envelope which modulates the average value of

assumptions regarding component char-acteristics and voltage waveforms, i t will be shown t h a t the analyt ic representat ion of the basic magnetic-amplifier element can be made linear.

Analysis of Basic Magnetic-Amplifier Element

T h e basic magnetic-amplifier element, shown in Fig. 1 (A), consists of a saturable inductor and an alternating-voltage source. There are two windings on the inductor : t he gate winding in series with the source and por t 2 ; and the control winding in series wi th por t 1. I n the analysis of the basic element, t he inductor characteristics are considered to be ideal and the two inductor windings are as-sumed to be perfectly coupled. T o sim-plify t he analyt ic expressions obtained, a one-to-one tu rns rat io of the two windings is also assumed. A different tu rns rat io can easily be taken into account b y the use of an ideal transformer. Because of the assumptions of ideal coupling and one-to-one tu rns rat io, t he inductor can be considered as a 1-winding inductor t h a t is ei ther a short circuit or a open circuit, depending upon the flux level in the in-

the gate current flowing in the series-connected saturable reactor, Fig. 6.

References

1. CAPACITIVB LOADNG OP NONLINEAR MAG-NETIC CIRCUITS, J. T. Salihi. Ph.D. Thesis, Uni-versity of California, Berkeley, Calif., 1958. 2. MAGNETIC AMPLIFIERS (book), H. F . Storm. John Wiley & Sons, Inc., New York, N. Y., 1955. 3. ANALYSIS OP 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. 4. TRANSIENTS IN LINEAR SYSTEMS (book), Murray F. Gardner, John L. Barnes. John Wiley & Sons, Inc., 1948. 5. SERIES-CONNECTED SATURABLE REACTOR WITH CONTROL SOURCE OF COMPARATIVELY HIGH IMPEDANCE, H. F . Storm. AIEE Transaction, vol. 69, pt. I I , 1950, pp. 1299-1309. 6. ANALYSIS OF INSTABILITY AND RESPONSE OF REACTORS WITH RECTANGULAR HYSTERESIS LOOP CORE MATERIAL IN SERIES WITH CAPACITORS, Jalal T. Salihi. Ibid., pt. I (Communication and Electronics), vol. 75, July 1956, pp. 296-307.

ductor core. In fact, the two windings are introduced merely to make possible the interconnections of basic elements to be considered la ter ; a t present, the in-ductor can be thought of to advantage as a 1-winding inductor.

T h e analysis assumes, initially, ar-bi t rary waveforms for the supply and por t voltages. I n the course of the anal-ysis the restrictions upon the waveforms t h a t lead to a linear analysis will be estab-lished. A t this stage the only restriction imposed upon the supply voltage is t ha t i ts magni tude be limited so tha t the volt- t ime integrals of the positive and negative par t s are less than the volt-time integral required to drive the core from positive to negative saturation. This requirement can be stated analytically as follows:

J esdt<2n2ABs (1)

ta where

es = source voltage /a = time of the beginning of a positive or

negative half period of es

Paper 59-171, recommended by the AIEE Mag-netic Amplifiers Committee and approved by the AIEE Technical Operations Department for presentation 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. JOHANNBSSEN is with the Applied Re-search Laboratory of the Sylvania Electronics Sys-tem, Waltham, Mass.

The material presented in this paper was sub-mitted to the Massachusetts Institute of Tech-nology, Cambridge, Mass, 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 33(616) -5489, MIT Project No. DSR 7848.

Analysis of Masnetic Amplifiers

Without Diodes

PAUL R. JOHANNESSEN ASSOCIATE MEMBER AIEE

N O V E M B E R 1959 Johannessen—Analysis of Magnetic Amplifiers Without Diodes 471

Control Winding

Gate Winding

ec

Ö (A)

eR, = aei+b(es+e2)

where

(2)

a = voltage transfer ratio between the â÷ and the en terminals, measured with e2 — es=0

b = voltage transfer ratio between the eSl or e2, and the eR terminals, measured with e2 (or es) = ei = 0

In terms of the circuit parameters, ex-pressions for a and b are as follows:

RmRz/JRm -\-Rt) R\ -\-RmRi/(Rm -\-R%)

RmRi/{Rm-\-R\) Ri

R2-\-RmRl/(Rm-l·Rl) R2

(3)

(4)

During the unsaturated period eR

is the voltage across the inductor. By Thevenin's theorem it is also seen that if R is the resistance faced by the inductor, eR'/R is the current in the inductor during the saturated period. The com-ponent ae\ of eRi together with the nega-tive of the component b{es+e2), is plotted in Fig. 3. For reference polarities as

Fig. 1. A—Basic mag-netic-amplifier element. B—The modified ele-

ment

tb = time of the ending of the same half period n2 — number of turns on the gate winding A — cross section of the core Bs = saturation flux density of the core

To avoid difficulties that are usually associated with the analysis when zero and infinite impedances or admittances occur, the resistors Ri, R2, and Rm are added to the basic circuit as shown in Fig. 1(B). The resistors Pi and R2 will later be considered to represent the winding resistances, and the resistor Rm to repre-sent the magnetizing impedance of the inductor. At present, however, these resistors are introduced only for con-venience. The characteristics of the in-ductor in Fig. 1(B) are shown in Fig. 2. The scales of the plot are chosen in such a way that the usual B versus H curve is converted into an / eRdt versus iR curve eR being the ideal saturable-inductor volt-age and iR the inductor current.

To simplify the analysis a fictitious voltage eR' is introduced. The voltage eR' is defined as the voltage that would ap-pear across the inductor if the inductor were open-circuited, and it is given by the following expression:

Ideal Saturable Inductor

in Fig. 1(B), the inductor current eR/R must be positive during saturation on the upper branch of the inductor character-istic. Thus eR must be positive,

aei>—b(es+e2) (5)

that is, saturation on the upper branch of the inductor characteristic can occur only when the voltage component ae\ is above the voltage component — b(es+e2) in Fig. 3. Particular attention should be given to the intersections of the two voltage-component curves, because at these instants, labeled tu t2, and t\ in Fig. 3, both the inductor voltage and the inductor current are zero. If the in-ductor saturates on the upper branch of the characteristic, saturation must end at the intersections t\ and t\ . Similarly, if the inductor saturates on the lower branch of its characteristics, saturation must end at the intersection t2.

Suppose that at time t0 the inductor is saturated and that the operating point P on the inductor characteristic is at point Po in Fig. 2. Saturation continues and the inductor voltage eR is zero until the in-tersection at time h of the component voltages. During the time interval to to tu the operating point P moves from P0

to Pi. After tu the inductor is un-saturated; the inductor voltage eR is

CB)

equal to the fictitious voltage eR which is negative; and the operating point P moves from P\ down to P2 during the interval h to t2. After t2, the inductor voltage is positive and the operating point P moves back up to Pi in the inter-val t2 to t%. Because the flux change from Pi to P 2 is equal to the negative of the flux change from P 2 to Pi, the integral of the inductor voltage during the intervals t\ to t2 and t2 to h must satisfy the equa-tion

I eRdt= — I eRdt Jh Jh

(6)

or the two areas shaded in Fig. 3 must be equal. At h the inductor saturates; the voltage eR drops to zero; and saturation continues until / / , a point in the next period of operation which corresponds to the point h in the first period considered.

The important conclusions from the foregoing considerations are that during the saturated period the inductor voltage is zero and that during the unsaturated period the integral of the inductor voltage is zero; or

r f=0 (7)

Therefore, during the entire period from

Fig. 2. Ideal saturable induc-

tor

-*- ip = ii + i p - i ► i R - I, T i 2 " " ' m

472 Johannessen—Analysis of Magnetic Amplifiers Without Diodes N O V E M B E R 1959

Fig. 3. Voltage waveforms

t\ to t\ called the operating period, the integral of the inductor voltage eR is zero.

The only conditions of the voltage waveforms thus far are:

1. Saturation must occur periodically so that a succession of periods like t\ to / / occur.

2. Saturation must occur at only one end of the inductor characteristic.

For convenience in the remainder of this analysis the following conditions are added:

3. The voltage waveforms must be such that exactly one saturated and one un-saturated period occur in each operating period.

4. The average value of the supply voltage over any complete supply-voltage period must be zero.

Consider now any instant of time t2

within the period t\ to / / . Time t2 divides the operating period into two parts, h to tx and tz tot\\ the first of these periods is called the control period and the second the gating period.

The following relation can now be written

eBdt = 0 eBdt + J h Jtx

Therefore

tx—k Ctx h'—tx Ctv

I «fi<#+7TT I eRdt = 0

tx-hjti h -tXJtx

{tx-k)ERe+{hf-tx)EB

0^0

where

(8)

0»

(10)

EBC — average value of eB in the control

period ER° — average value of eR in the gating

period

If time tx divides the interval from t\ to t\ into two equal parts so that

tx—t\—t\ —tx (11)

the basic relation of equation 10 simplifies to

Å€+Å° = 0 (12)

It should be realized that the choice of tx is entirely arbitrary, and the center position is preferred merely because it simplifies the result of the analysis. I t will be shown later that there are no compensating disadvantages to this choice for practical circuits. It is also of im-portance to realize that time tx does not correspond to time h, the instant of time at which the inductor saturates. As a consequence, the control and gating per-iods do not correspond to the unsaturated and saturated periods. Furthermore, there is no special relation between tx and /2; that is, the interval of time during which eR is negative can be more or less than half the h to t\ period.

If port 2 is short-circuited and zero signal is applied to port 1, equation 2 re-duces to

eB' = bes (13)

Since the voltage es by definition does not contain a direct voltage component, the inductor does not saturate; that is, in Fig. 3, h coincides with / / . If a positive voltage is applied to port 1, /3 moves back from t\ toward t\. Thus, with small sig-nals applied, h may be expected to occur after tXy that is in the gating period; only with very large applied signals at port 1 can /3 occur in the control period. The analysis of interest is concerned primarily with relatively small signals. Also, in practical magnetic amplifiers it is unusual and undesirable for the saturation

period, t* to t\, to extend into the con-trol period. It is, therefore, here assumed that h remains in the gating period. This assumption renders the basic circuit linear throughout the control period so that volt-age-current relations can be calculated easily.

At any instant at all, the following equa-tions hold

ei = Riii+eR

02+£«= #2^2+0«

(14)

(15)

In addition, at any instant in the control period,

eB = aei+b(es+e2) (16)

1 - a -b -b\

Ri Ri Ri

-a 1-b 1 - 6

x?2 -^2 R2 \£s\

1-a -b -b\

Ri Ri Ri

-a l-b l-b\

R% XV2 iv2 |

k*| H

These equations completely determine the relations between port quantities during the control period so that any of the 2-port matrices may be calculated. For example, the equations may be solved to yield the admittance-matrix equa-tion:

(17)

Since these equations are true at each in-stant, they are also true for average values during the control period.

(18)

Here, and in subsequent equations, the notation introduced in equation 10 is used; capital letters with superscript c are used to denote control-period average values of the quantity denoted by the correspond-ing lower-case letters. Superscript g is used to denote the gating-period aver-age value.

Instantaneous-value equations for the gating period are more difficult to obtain, and a different approach, therefore, is used to derive average-value equations for this period. Equations 14 and 15 are valid at any instant, and in terms of average values in the gating period be-come

EJ-RJX'+EB9 (19)

Ef+E9' = RJi'+EB' (20)

B y the use of the basic relation

EBe+EB'«0 (12)

together with equations 16, 18, 19, and 20, equations analogous to equation 18 can be obtained; and these with equation 18 may be written as the admittance-matrix equation of the network:

N O V E M B E R 1959 Johannessen—Analysis of Magnetic Amplifiers Without Diodes 473

Il'\

Il\

hc\

v|

f?» \a 1

—a

* ° *2 °

-b

k ' £°

& 1

R2 ^2

^ °

s · 1-b 1

& 1 R2 -̂ 2

H H ú

H k'll (21)

More generally, any desired network matrix can be obtained from these same equations.

Equation 21 appears to indicate that in terms of half-period average values, the basic element can be represented as a linear network containing a source, Es. Thus, if short circuits are placed at input and output, the following quiescent cur-rents flow:

In the control period, Tc — 77 c

TC _ l Z ^ pC ' 2 g— £Ls

In the gating period,

*2 q — ÔÃ Es ~\~~Z~ Es Ri R2

(22)

(23)

(24)

(25)

As is shown by equations 22 through 25 only alternating currents flow during quiescent condition, inasmuch as Es

c is equal to —Es° in steady-state operation. This result is obvious because no direct-voltage sources are present in the circuit when both network ports are short-circuited.

If now a signal is applied at port 1 with 2 left short-circuited, additional compo-nents of currents flow as follows:

Ri

Att R2

Ef

**'-i *'+£*' R-2

(26)

(27)

(28)

(29)

If port 2 is opened and a load applied, the characteristics of the load determine the voltage and current relations at port 2. If the load characteristics can be formulated as a linear relationship among E2

e, E2ff, I2

C, and I2°f then the equations stating this relationship can be used with the matrix equation, equation 21 (the last two columns of the Y-matrix and

the last two rows of the E-matrix being omitted), to determine expressions for the Ä quantities at the output in terms of the input. Thus, apparently, a linear anal-ysis of the basic element is obtained based on quiescent currents and linear relations among incremental deviations from the quiescent condition, parallel, for exam-ple, to the ordinary vacuum-tube pro-cedure.

There is, however, a hidden difficulty in the foregoing analysis. The half-period average values of the supply volt-age, Es

c and Esff, are not constant, but are

altered by the shift in time of the oper-ating period relative to the power supply-voltage wave and by the change in length of the operating period during transient conditions. This property is seen from Fig. 3, because both t\ and t\ are deter-mined by the intersections of the two waves, — b(es+e2) and aei, representing components of the fictitious voltage eR'. If both ei and e2 are zero, t\ and / / are located at the zero crossings of the es

wave. Existence of the voltages e\ and e2 tends to shift t\ and t\ from these zero crossings. Thus, a nonlinearity in the basic element is hidden in the terms involving Es

c and Es° and their depend-ency on the signal.

During steady-state conditions the period of operation h to t\ is equal to the period T of the power-supply voltage wave so that the nonlinearity in the basic element during steady-state conditions is caused by the shift in the period of operation relative to the power-supply voltage wave only. For practical mag-netic amplifiers consisting of interconnec-tions of basic elements, this nonlinearity is usually canceled. For this cancellation to occur, the terms involving Es

c and Es° in the equations representing the opera-tion of a practical magnetic amplifier must cancel one another. Fortunately, condi-tions under which the terms involving Es

c and Es° cancel can be determined as follows. Equation 21 is a set of linear relations among the half-period average values of the supply voltage and port volt-ages and currents. It should be realized that in the derivation of these relations the restriction that the supply voltage should remain constant was not imposed. Superposition can therefore be used to determine the effect of variations in the quantities Es

c and Es° upon the magnetic-amplifier operation. I t is immaterial what causes the quantities Es

c and Es9 to

vary. Thus, if it can be shown that the changes in Es

c and E8g, caused by the shift

in the period of operation relative to the period of the supply voltage wave, do not affect the quiescent operating conditions,

then the steady-state characteristic of the magnetic amplifier is linear and in-dependent variations in Es

c and Es°. Suppose two basic magnetic-amplifier

elements are interconnected so that the quiescent current cancel. Then two possibilities exist; either the coefficients of Es

c and Esg in the amplifier equations are

zero or they are of the form

hcQ = Kl(Es

c+Es°)

h%=K2{Esc+Es°)

I2cq=Kz{Es

c+Es°)

h% = Ki{Esc+Es

g)

(30)

(31)

(32)

(33)

where Kh K2, K3, and K* are constants. In the quiescent condition

Esc=-Es9 (34)

so that the desired quiescent-current cancellation is ensured. During steady-state operation the changes in Es

c and Es° must satisfy the relation

Ä £ / + Ä Å / = 0 (35)

because the supply voltage contains no direct voltage component and is periodic. Changes in Es

c and Es° during steady-state operation, therefore, do not cause any changes in the short-circuit port currents. Thus, it follows that magnetic amplifiers whose quiescent currents are zero have linear steady-state character-istics independent of variations in Es

c and £ / .

For a single basic element the error introduced by ignoring the variation of the Es

c and Es° terms is not so great as might appear, as is demonstrated in reference 2.

Linear Analytic Representation of the Basic Element

A linear analytic representation of the basic element in terms of incremental half-period average quantities is obtained by eliminating the last two columns in the admittance matrix and the last two rows in the voltage matrix of equation 21, and adding a Ä to the current and voltage symbols. For convenience the added symbol Ä is omitted in writing the fol-lowing equation.

1 - a Ri

a

Ri

—a

~R2

a

R*

0

1 Ri

0

0

-b

Ri

b

Ri

l-b

R2

b

R2

0

0

0

1

Rt

k c

\Åéâ

\E2C

Ä # I

(36)

In the analysis of magnetic amplifiers

474 Johannessen—Analysis of Magnetic Amplifiers Without Diodes N O V E M B E R 1959

not only the static but also the dynamic properties are of interest. The dynamic properties of the basic element are appar-ent from the superscript notation used, which indicates that the short-circuit currents in the gating period depend on the port voltages in both the control period and the gating period. Thus, a time-delay of one-half period exists, for example, between E\ and I2

0. If the nth half cycle of the supply voltage is taken to correspond to a control period, and the (n+l)th to the subsequent gating period, equation 36 can be written as a set of linear difference equations.

Iic(n)=^Elc(n)-~E2

c(n)

7/(^+1) = ̂ - Elc(n)+~ M » + D +

Ri Ri

-E2\n) (37)

Ki K2

I2ff(n+l) = ̂ -E1

c(n)+^E2c(n) +

K2 K2

-£ / («+1) K2

For the analytic representation of the basic element to serve as a means for analysis of magnetic amplifiers, a more convenient representation than offered by equation 37 is desirable. In linear-circuit analysis such a representation is usually obtained by a transformation from the time domain to the frequency domain, accomplished by either the Fourier or Laplace transform, because such a trans-formation reduces the mathematical ma-nipulation to simple algebra. To apply this method, however, it is necessary that the variable quantities be transformable. Clearly, the Laplace transform of an average-value quantity has no meaning. Also, there exists no fixed relation between instantaneous voltages and currents of a magnetic amplifier, because the wave-forms of the dependent variables depend not only upon the waveform of the in-dependent variables but also upon the waveform of the supply voltage. On an instantaneous basis, therefore, the La-place transform does not exist. However, for the purpose of analysis, any waveform can be assigned to the variable quantities in equation 37; and, to permit the use of the Laplace-transform method, it is assumed that the variable quantities consist of impulses whose areas are equal to the magnitude of the average-value quantities. This choice of wave form is by no means the only possible one, but is made for convenience. The Laplace

transform of the difference equations can now be obtained and yields

b?w

k'w

kc(s)\

/.'W

l-a

Ri 0

Ri 0

iL ß JL A I Ri z R\ R\ z

-a l--b 0 R2 R2

a_ 1 b_\_ \_ R2 z 0 R2 z R2

where

Eic(s)\

E2c(s)\

E20(s)\

(38)

(39)

and where a bar over a symbol is used to denote the Laplace transform of the time function represented by the symbol it-self.

The elements of the admittance matrix in equation 38 are the short-circuit driving point and transfer admittances of the basic magnetic-amplifier element. The ad-mittance matrix and its elements may be written

yn

yn

yvi

y-2?

(40)

Because impulses are chosen as the wave-form of the variable quantities, these admittances can be interpreted simply as the Laplace transform of the unit impulse response. For example, the inverse trans-form of the y2i°

c transfer admittance is the short-circuit current at port 2 during the gating periods caused by a voltage applied at port 1 during the first control period (the control period from time 0 to Ã/2).

The admittance matrix of equation 40 can be partitioned into four 2X2 sub-matrices by writing

·*rs — yrs yrs1

Jrs yrs (41)

where r and 5 may each be 1 or 2. Thus, the matrix of equation 40 can be written

Y= 3ºé 3º2 ^21 y22|

(42)

The reader should note that the matrix elements in equation 42 represent sub-matrices of the form shown by equation 41.

Other relations between port voltages and currents useful in magnetic-amplifier analysis are the following:

(43)

Ifiil \EA \E,C< ,l&'

\=z\ \ÉÄ \iA \iA \iA

\ÅË \EA \u\ w\

\=H\

É/.º / l "

\EA

\M

and

= G

EA EA u A'

(44)

(45)

w\ \EA /iel IV

\A

\c -B\ -D\

\EA\ \EA w\ I//I

(46)

The analytic expressions for the matrix components of equations 43 through 46 can readily be obtained from equation 38 and are given in the Appendix.

The fundamental relations of the basic element have been derived in matrix form, this form being particularly useful in the analysis of magnetic amplifiers resulting from interconnections of basic elements. Two ordinary 2-port networks can be interconnected in five fundamental ways:

1. cascade 2. parallel 3. series 4. series-parallel 5. parallel-series

The analysis of these interconnections reduces to simple matrix addition or multiplication if the proper analytic representation for the individual net-works is used.3 The matrices 7, Z, H, G, and ABCD were all derived in order to have available the appropriate analytic representation for the various fundamental interconnections. It is pointed out that there is an important restriction to the matrix method, when interconnections by c, d, and e are of interest. If a potential difference exists between the terminals to be joined in the interconnection of two networks, a circulating current will result and the combined behavior will not be given by superposition of the previous individual behaviors. For the basic am-plifier under consideration, however, this restriction does not apply, since complete isolation exists between the two ports; and superposition of the behavior of the individual elements can always be used to determine the combined behavior. In the following section, the rules of inter-connection of basic elements are de-veloped.

Interconnection of Two Basic Elements

In the previous section it was stated that two 2-port networks can be inter-connected in five fundamental ways.

N O V E M B E R 1959 Johannessen—Analysis of Magnetic Amplifiers Without Diodes 475

' é " (ß÷)é

Éâ,

'ß (en)

Λ *

•À

(in!,

Λ *

■ß

I

I I

♦~ô—

( â ÷ ) 2 |

Ülfo 1 j

(eiifef

f . ;

(á) (c)

(b) (d)

Fig. 4. The four possible subconnections for the series parallel interconnection of 2-port net-works

Each of the fundamental interconnections, in turn, can be made in four possible ways, shown in Fig. 4 for the fundamental series-parallel interconnection. As dem-onstrated by Guillemin,3 the analytic representation of the interconnected net-work can easily be obtained by choosing the proper matrix representations of the individual networks. The various sub-connections of Fig. 4 can be taken into account by a simple transformation of the variables of one of the networks as will be shown next.

The proper analytic representation for the series-parallel interconnection is given by the if-matrix. Let HI be the matrix of network I, and Hn be the matrix of network II when the port voltage and cur-rent polarities are as shown in Fig. 4. The following equations are then valid

elements are the Laplace transforms of half-period average voltages and currents as defined in the previous section.

The constraints imposed upon the port voltages and currents of the individual networks of the various subconnections

(£7)2=±(£7 /)2

( J / ) l = ± ( / / i ) l

(49)

(50)

where the four possible combinations of sign represent the four possible sub-connections.

The port voltages and current of the combination network for the various sub-connections can be written in terms of the port voltages and currents of the in-dividual networks as follows:

For network / ,

|(£/)i| -Ht ( / / ) i

(£/)5

For network II,

]|(£//)il I (///)! I

=H,

(47)

(48)

E / = ( £ / ) i ± ( £ / / ) i

/ / = ( / / ) i

&" = (£/)*

/ / = (/,) ±(J / 7)2

(51)

(52)

(53)

(54)

where the subscript I refers to network I and the subscript II refers to network II. If the two networks under consideration represent ordinary linear networks, the variable quantities in equations 47 and 48 are the Laplace transform of the instantaneous port voltages and currents. If, however, the two networks represent basic magnetic-amplifier elements, then the varible quantities in equations 47 and 48 are 1X2 submatrices in which the

where the double-primed variable quanti-ties are the port voltages and currents of the combination network. Note that for a particular subconnection the last terms in equations 49 and 54 have the same sign and also that the last two terms in equations 50 and 51 have the same sign. The sign of the last terms in equa-tions 49 and 54, however, might differ from the sign of the last terms in equa-tions 50 and 51 depending upon the par-ticular subconnection under considera-tion.

A new set of variables now will be introduced. Let

and

7 i '«±(J„ ) i

(55)

(56)

(57)

(58)

where again, for a particular subconnec-tion, the terms in equations 55 and 57 have the same sign, and the terms in equations 56 and 58 have the same sign. Because of this property equations 55 through 58 can be written as follows:

( £ J I ) I (///)2

and

(7//)i (£/ /)2

= 71

= T\

ES-It'

where for ordinary networks,

± 1 0

o| ± 1

(59)

(60)

(61)

This matrix is called the transformation matrix, and the four possible combinations of sign in this matrix represent the four possible subconnections.

If equations 59 and 60 are substituted in equation 48, there results

Ei' -HnT (62)

The transformation matrix T is equal to its inverse so that premultiplying equation 62 by the matrix T yields

= THnT ; / i , !

!&'!

From equations 49, 50, 56, and 57

/ i ' = ( / / ) i

E2' = (Ej)i

(63)

(64)

(65)

Substitution of equations 55, 58, 64, and 65 in equation 63 yields

±(£ / / ) i | = TH„T (66)

The matrix equation for the subconnec-tions is obtained by adding equations 47 and 66 and substituting equations 51 through 54 into the resulting equation. The following result is obtained

~(H,+TH„T) (67)

Thus, the matrix of the combination net-work is given by the expression

H=HI+TH„T (68)

I t can be shown that a similar matrix can be obtained for any one of the other

476 Johannessen—Analysis of Magnetic Amplifiers Without Diodes NOVEMBER 1959

are

basic interconnections if the proper matrix relation is used. For example, for the series-series interconnection the impedance matrix becomes

Æ = Æé+ÔÆðà (69)

It has already been pointed out that if the networks / and II represent basic magnetic-amplifier elements, the variable quantities in the foregoing equations are 1X2 submatrices. For magnetic ampli-fier elements, therefore, the elements in the transformation matrix T (see equa-tion 61) are 2X2 submatrices. For this reason the transformation matrix is written in the form

T = ±X 0

0 ±X\ (70)

where the zeros are 2X2 zero sub-matrices. From the foregoing analysis it may be expected that the matrix X is a 2X2 unit matrix. However, it will be shown that the matrix X also can take the form

X = 0 1 1 01

(71)

Practical magnetic amplifiers invariably consist of interconnections of two or more identical basic elements. For two or-dinary, linear, identical networks the four possible subconnections reduce to two—subconnections a and b> for example, in Fig. 4—the other subconnections being identical save for reference directions of the final-network port voltages and cur-rents. For basic magnetic-amplifier ele-ments, however, the four possible sub-connections, in general, result in four different magnetic-amplifier circuits even though the two elements are identical, for the following reason. If a d-c signal is applied to the input terminals of basic element I in Fig. 5, the saturable in-ductor of this element saturates during the positive half periods of the supply voltage wave. If the polarity of the input d-c signal is reversed for basic element II in Fig. 5, then the saturable inductor of this element saturates during the negative half periods of the supply voltage wave. Thus, if the supply voltage is common to both elements, the inductors of elements I and II saturate during alternate half periods. For interconnections c and d in Fig. 4 such a polarity reversal of the input d-c signal is obtained because of the reversal of the input terminals of element II. For interconnections a and bt how-ever, no such reversal exists. Thus, if the supply voltage is common to both elements of the interconnection, the satur-able inductors of interconnections a and b saturate during the same half periods

of the supply voltage wave, while the saturable inductors of interconnections c and d saturate during alternate half periods of the supply voltage wave. This is the property which results in four differ-ent magnetic-amplifier circuits for each of the basic interconnections.

The constraints imposed upon the port voltage and currents of the basic ele-ments of interconnections a and b are (see equations 49 and 50)

(72) :l(£/)/

1 =± HEnh°\

and

||(//)ici = ± \a,,),e\\ (73)

For subconnections c and d> however, the control period of element / corresponds to the gating period of element II and vice versa, so that the constraint equations become

(74)

(75)

ll(£/)/ = ± |(£„)2<º

and

= ± a„y\\ (///ë'ÉÉ

Equations 74 and 75 can be rewritten

||(£/)2C|

\\(Ety\ = ± 1°

|l and

I|(//)i1| ll(/i)i'll

= ± | 0 1

If a matrix X is

X=U= 111 1lo

°l 1|

(£//)2

(£//)*

(///V

(76)

(77)

(78)

when the two basic elements inductors saturate during the same half periods, and

X= |o i 1 0

(79)

when the two basic elements inductors saturate during alternate half periods, then the constraints imposed upon the port voltages and currents of the basic elements of the subconnections can be expressed in general as follows:

(£/)* = ± * ( £ / , ) 2 (80)

(81)

where the port voltages and currents are 1X2 column matrices representing the division of the operating period into the control and gating periods. The same procedure as used in the derivation of equation 68 can now be followed. The result is

Fig. 5. Saturation during alternate half periods of supply voltage wave

where H is the matrix of the interconnec-tion, Hj is the matrix of the basic ele-ment, and T is the transformation matrix

à = ±X 0 0 ±X\

(83)

H=Ht+THjT (82)

The interchange of control and gating periods considered in the foregoing anal-ysis is caused by the polarity reversal of the applied d-c signal. This polarity reversal is by no means the only factor that causes interchange of control and gating periods. Direct signals at the output port or a 180-degree change in phase of the supply voltage might also cause inter-change of control and gating periods. When a particular interconnection is to be studied, all these factors must be taken into account to determine if interchange of control and gating periods occurs.

Interconnections of basic elements that result in interchange of control and gating periods impose an additional restriction upon the voltage waveforms. The matrix addition of equation 82 is valid only if the time periods of elements / and II in which the average-value quantities are defined coincide; that is, if the control period of element / corresponds to the gating period of element II, and vice versa. To satisfy this condition it is necessary first to divide the operating period into two equal parts, and second to ensure that the inductor of one element ceases to saturate at the end of the control period of the other element. Clearly, if the waveforms are symmetric, these conditions can be satisfied.

To illustrate the application of the above analysis procedure, the series-connected saturable inductor is analyzed in the next section.

N O V E M B E R 1959 Johannessen—Analysis of Magnetic Amplifiers Without Diodes 477

Analysis of the Series-Connected Saturable Inductor

The circuit diagram of the series-connected saturable inductor is shown in Fig. 6(A). I t is evident from Fig. 6(A) that this circuit can be constructed by interconnecting two basic elements in series-series, as is demonstrated in Fig. 6(B). For this type of interconnection, the proper analytic representation for analysis is given by the impedance matrix. To analyze the series-connected saturable inductor, the analysis procedure outlined in the previous section is used.

The equations relating half-period average voltages and currents of the two basic elements are:

For element I,

Å,-Æ,É,

For element 77,

En = Zidii

(84)

(85)

where the matrices EIy II} Eu> and In are 1X4 column matrices of the form

(86)

The two elements are identical so that

Zt=Zn (87)

The impedance matrix of the basic ele-ment is f

1

£ / = \(Å,)¢ (£/)ii \{E,)A \(Åé)Ë

Zt =

Ri+Rm 0 Rm 0 -Rm/z Ri -Rm/z 0 Rm 0 Ri+Rm 0

-Rm/z 0 -Rm/Z R,

(88)

The next step in the analysis is to determine the transformation matrix T. As demonstrated in the previous section, this transformation matrix is uniquely determined by the constraints imposed upon the port voltages and currents of the two elements by the interconnection. Because the input-voltage polarity of ele-ment 77 is reversed, the inductors of the two elements saturate during alternate half periods (the subconnection of Fig. 6(B) corresponds to subconnection c in Fig. 4 for the series-parallel interconnec-tion). Thus, the control period of ele-ment 7 coincides with the gating period of element 77, and vice versa. The con-straint equation becomes

7// = -X 0

0 X\ Ii = Th

where X is the submatrix

X = \

(89)

(90)

The following transformation of vari-ables is now made:

EU = TE' (91)

Iii = TP (92)

Combining equations 85, 87, 91, and 92 yields

TE'=ZjTl' (93)

The matrix T is equal to its inverse so that premultiplying equation 93 by T yields

E = TZITP (94)

From equations 89 and 92

7' =7/ (95)

Substitution of equation 95 into equation 94 gives

E' = TZlTll (96)

Addition of equations 84 and 96 gives

£ , + £ ' = (Æ,+ÃÆ,Ã)/ , (97)

But

Ej+E' = E" (98)

7,=7" (99)

where the doubled primed quantities are the port voltages and current of the com-bination network. Thus, substitution of equations 98 and 99 into equation 97 yields the following equation;

E'^iZi+TZ^P'^Zl" (100)

Substitution of equation 88 in equation 100 and performing the matrix operation indicated by equation 100 gives

Addition of the first two and last two rows in equation 105 yields

E, = [2RX+Rm{ 1 - l/z)]Ii+Rm( - 1 + 1/*)/* (106)

Et=-Rm(l^l/z)I1+l2Ri+Rm(l + lM]I9

(107)

where

£1 = ( £ ' " ) i c + ( £ " V & = ( £ " ) ? C + ( £ " V /i = ( / " ) i C +( / "V 72 = (/,,,)2

c+(/,',)2<7 (108)

The new variables introduced by equation 108 can be interpreted as follows. The independent current variables with super-script c are by definition zero during the gating periods. Similarly, the independ-ent current variables with superscript g are by definition zero during the control periods. It is evident from equation 105 that the dependent voltage variables also possess the same property because the terms in the expression for a dependent-voltage variable involving currents of superscripts different from that of the voltage variable are shifted in time one half period. Thus, during the control period

E2 = (E"')2C

/i = ( / " V / 2 = (7'")2C (109)

Z=»

2R\-\-Rm —Rm/z — Rm/z 2R}-\-Rn

Rm Rm/z — Rm/z —Rm

Rm Rm/z — Rm/z —Rm 2i?2+^m —Rm/z j -Rm/z 2R2+R m

(101)

A very interesting property is revealed by equation 101 if the matrix is parti-tioned into four 2X2 submatrices. The elements along the two principal diagonals of each of these submatrices are seen to be equal in magnitude. If the transforma-is made,

(102)

(103)

(104)

the matrix equation, equation 105, is obtained.

E» = Á¢ V"

I"=Al'n

where

A = \

1 0

P o

0 1 0 0

0 0

- 1 0

0 0 0 1

and T the desired transformation matrix.

(£ , r ) i c | {E'»)A (£"VI \(E"')i°

\2Ri+Rm

— Rm/z \— Rm \-Rm/z

-Rm/Z 2Ri -{-Rm

-Rm/z — Rm

— Rm Rm/z 2R2~\-Rm Rm/z

Rm/z — Rm Rm/Z 2R2-\-Rm 1

\(I")l

(J"V (/")/ k/"V

and during the gating period

&-GE")/ Á = ( £ " í / i - ( / ' V

/2=(/,,,)2ff (no)

The new variables introduced by equa-tion 108, therefore, are half-period aver-age voltages and currents. These vari-ables are related to the half-period average voltages and currents of the series-con-nected saturable inductor by a trans-formation of the type expressed by equa-tions 102 and 103.

The transformation expressed by equa-

(105)

478 Johannessen—Analysis of Magnetic Amplifiers Without Diodes N O V E M B E R 1959

tions 102 and 103 is fundamental to the analysis of magnetic amplifiers, partic-ularly magnetic amplifiers with diodes. Physically, the transformation can be interpreted as representing the action of a synchronous switch, the switch being operated so that if a d-c signal is applied to it, an a-c output signal is obtained. Or, if an a-c signal is applied, a d-c out-put signal is obtained. Equation 105, therefore, represents the series-connected saturable inductor in cascade with such a synchronous switch.

If the series-connected saturable in-ductor is terminated in a linear, passive load, the steady-state output consists of an a-c signal when a d-c signal is applied to the input. This property can be seen from the circuit diagram of Fig. 6. In steady-state operation the voltage across the saturable inductor terminals cannot contain a direct-voltage component. Thus, the voltage across the series com-bination of the load and the resistance R2 must be an alternating voltage. The transformation of equations 102 and 103, therefore, can be represented physically as the action of a full-wave rectifier con-nected to the output terminals. The full wave rectifier renders the series-connected saturable inductor symmetric so that no distinction can be made between the differ-ent half periods. The four amplifier equations, therefore, can be reduced to two.

To derive the transfer characteristic of the series-connected saturable inductor with resistive load, the following addi-tional equation is used:

£2= —RiJi (HI)

After equation 111 is substituted into equations 106 and 107 and the resulting equations are solved for E2 in terms of Ei, equation 112 is obtained by letting Rm

tend to infinity and R2 tend to zero:

The inverse Laplace transform of equation 112 produces the same difference equation for the series-connected saturable-in-ductor circuit as was previously derived by the author.4

Etin+l)-RL

RL+2RI [Ei(n+1)+Ei(n)] +

RL-2RI

RL+2Rt E2(n) (113)

Since for this type of circuit the magnet-izing resistance, Rm, is very large com-pared with resistance, Rif and the resist-ance, R2, is small compared with load resistance, RL, the limit process performed in the derivation of equation 112 is applicable to practical circuits.

Interconnection of n Basic Elements

The analysis developed previously for a single-phase magnetic amplifier can readily be extended to include the poly-phase case. Characteristic of the single-phase amplifier analysis are the division of the operating period into the control and gating periods, and the selection of half-period average voltages and currents as variables. The characteristic equations written then relate variables in the differ-ent half periods of operation. Equations written for individual basic elements may be combined in a simple fashion to obtain equations for an interconnection of basic elements, provided that the control and gating periods of the individual elements have the same position in time, or occur so that the control period of one element corresponds to the gating period of the other.

An w-phase magnetic amplifier consists of n single-phase magnetic amplifiers each connected to one phase of a symmetrical w-phase power source. The single-phase amplifiers are usually alike and may con-sist of one or more cores. The analysis

to be presented, however, is not restricted to w-phase magnetic amplifiers consisting of n identical single-phase amplifiers but is valid in general. If the single-phase amplifiers are all alike, then the analysis can be greatly simplified. For the case of identical single-phase amplifiers in a power system with even numbers of phases, there exists the possibility that the outputs of the individual amplifiers cancel. The analysis still applies and shows that this cancellation occurs. In regard to the steady-state characteristics, it is of interest to note that, if the quiescent currents cancel in the individual single-phase amplifiers, then the w-phase amplifier has linear steady-state character-istics.

In a symmetrical n-phase system, the operating periods of two single-phase amplifiers in subsequent phases are dis-placed in time from each other by T/n. The control and gating periods of the individual single-phase amplifiers, there-fore, do not have the same position in time nor are they shifted an integral number of half-periods with respect to each other. To overcome this difficulty, it is necessary to divide the control and gating periods into subintervals. In the analysis that follows, it is assumed for an w-phase am-plifier that each of these periods is divided into n subintervals. This subdivision is illustrated in Fig. 7. I t may be seen from the figure that the operating periods of the individual single-phase amplifiers are shifted an integral number of subintervals with respect to each other; thus, time coincidence of the subintervals of the various single-phase amplifiers is obtained. The shift between successive phases in Fig. 7 is two subintervals. Therefore, if n is even, a possible alternative is to divide the control and gating periods into only n/2 subintervals. The larger num-ber, n, of subintervals is used so that the

(A) (B) Fig. 6. Series-connected saturable inductor. B—As an interconnection of two basic elements

N O V E M B E R 1959 Johannessen—Analysis of Magnetic Amplifiers Without Diodes 479

Phase

Phcse 2

y

Phase 3

/

C|

g n -3

Con

y

c 2

y

%

9n-2

rol Period of Phase 1 Amp.

y

c3

s

c l

gn-i

^

c4

c2

>

g n

C5

C3

c l

V

úú

tt

\ V

cn-l

Cn-3

Cn-5

^ ^

cn

cn-2

cn-4

c n- l

Cn-3

Gating Period of Phase 1 Amp.

g 2

c n

cn-2

^3

9|

c n - |

a 4

^2

Cn

1 4" 1 1 - Ί

>

gn -3

gn-5

y 9η-öη-Ι

gn -4 gn-3

c2

y*

9n

t

^ ! 1

c l c2

1 1 >Π

y 9n-2 9n- l 9n

t

y

t

Fig. 7. Division into n subinter-vals of the con-trot and gating periods of the amplifiers in their successive phases of an n-phase magnetic ampli-

fier

analysis may be applicable whether « is even or odd.

To analyze the «-phase magnetic ampli-fier, each of the single-phase magnetic amplifiers is described by a set of equa-tions relating average voltages and cur-rents in the various subintervals, the number of equations being four times (two for control and gating subintervals two for input and output ports) the num-ber of subintervals «. For example, in a symmetrical 3-phase system the number of subintervals is three; thus, twelve equations are needed to describe the operation of the individual single-phase amplifiers. The 4« equations may be written in matrix form. The matrix of the equations may take any of the forms previously introduced (F , Z, G, H, or AB CD), depending upon which currents and voltages are expressed as functions of the other variables. If, for example, the currents are expressed as functions of the voltages, the admittance matrix

F = yn y22

(114)

is obtained, where each yrs ( r , 5 = l , 2 ) i s itself a 2nX2n matrix relating average values of currents and voltages in the sub-intervals chosen.

yrss

yrs

yrs yrs°

r C l C l . CnC\

,.yr*]

• -yrs yrs

yrsc"

yrs' OlQi

-yrs

• yrs1 OiQn]

yrs · · .yrs

(115)

Suppose « single-phase amplifiers each connected to one phase of an «-phase

power source are interconnected parallel-parallel. If YIt YU) . . . YN are the ad-mittance matrices for the individual amplifiers, then the equations for these amplifiers become

Iii-YuEn

where the voltage matrices are 1 X 2 « column matrices of the form

(fi/)/1

■TY— YNEN (116) (£/).-

where the current and voltage matrices are 1 X 4 « column matrices.

As seen in Fig. 7, the following sub-intervals of operation correspond in time:

( £ / ) /

(120)

Ampl. I: Ampl. II: Ampl. Ill:

Ampl. N:

C\ Ci C% ...Cn gl ...gn

gn-l gn Ci . . . C n _ 2 Cn-l · · · g n - 2

gn-l gn-2 gn~l-·-Cn-4 Cn-*. .-gn-i

CA Ch >..g% g \ , .c% (117)

Thus, the constraints imposed upon the port voltages by the interconnection are:

( Å Ë ' 1 ^ / / ) / " - 1 = ( £ / / / ) / " - > = · · ·

- ( A D ) / 4

(Áé)Ë-(£/ / )Ë-*- (£/ / / )Ë-*- . . . - ( 5 * V a (118)

for s—\ for input-port voltages, or 2 for output-port voltages. The constraint equations, equation 118, can be written in matrix form as follows:

{El)^X{Ell)^X\Ell{)^.., =Xn~KEN)s (119)

For equation 119 to be identical to equa-tion 118, the matrix multiplications shown in equation 119 must move all elements of the {En)s matrix two rows down, all elements of the (Ein)s matrix four rows down, and so on for the other matrices. If premultiplying a matrix by a matrix X moves all rows two rows down, then it follows that premultiplying a matrix by the square of this matrix moves all rows four rows down. Thus, the form of equa-tion 119 is correct; it only remains to determine a matrix X that performs the above-mentioned transformation. Such a matrix is readily obtained and is as follows:

480 Johannessen—Analysis of Magnetic Amplifiers Without Diodes N O V E M B E R 1959

Fig. 8. Addition of matrix compo-nents. The elements in the matrix at the left are 2 X 2 sub-matrices of the form:

Yre = l y r e

|yreoJ

y™c5

y r 8c i

X =

|0 0 . . . 0 0 . . . 1 0 . . . 0 1 0 . . 0 0 1 0 .

0 0 . . . ||o o . . .

0

1 Oil 0 1 0 0 0 0 0 0

1 0 0 0 0 1 0 0|

(121)

In order to derive the matrix relation for the interconnection the following new variables are introduced.

Eii' — TEu Ein —I^Em

and

/ / / ' - 2 7 / I

(122)

(123)

where the current and voltage matrices are the 1X4« matrices in equation 116, of the form

i / i - (£//)i (EJJI

and where

(124)

(125)

Substitution of equations 122 and 123 into equation 116 yields

Iti-TY„TTEn'

In the derivation of equation 126 the identity

T-i = TT (127)

is used. The subscript T denotes the transpose of a matrix. The validity of equation 127 can be seen from equations 121 and 125, because

TT = \ \X 0 0 X\ 1 T 1°

From equation 121

XT=X-1

so that

TTA Ix-1

o = r

0 Xi

(128)

(129)

(130)

I t is evident from equations 119, 122, and 125 that the voltage matrices on the right-hand side of equation 126 are equal. Thus, addition of the equations in equa-tion 126 yields

/ = ( r / + 7 T / / 7 Y + . . . +T»^YNT^^)E (131)

where

E> = Ej = EJJ — . . . — £y

(132)

(133)

IN'-T«-iYNTf-*EN' (126)

From equations 123, 132, and 133 it is seen that the variables of equation 131 are the port voltages and currents of the final interconnection. Thus, the ad-mittance matrix of the «-phase amplifier is given by the expression

7 = Õ^ÔÕçÔô+,.. +r»-*Ftf7V»-i (134)

The expressions for the other basic inter-connections of n single-phase amplifiers can be obtained through a similar pro-cedure.

Most practical polyphase magnetic amplifiers consist of interconnections of identical single-phase amplifiers. These single-phase amplifiers are usually inter-connected so that there is no polarity re-versal of port voltages of currents. For such polyphase amplifiers, therefore, the matrices of the individual single-phase amplifiers are identical. Under these con-ditions, equation 134 for the admittance matrix of the parallel-parallel inter-connection becomes

Õ=À1+ÔÕ1Ãô+...+Ôç-éÕÉÔôç-1 (135)

The matrix summation on the right-hand side of equation 135 can easily be per-formed, as will be shown next.

Since the transformation matrix Ã, when written as a 2X2 matrix, is a diagonal one, the submatrices Yrs (see equation 114) of the admittance matrix become

Yrs = (Yl)TS+X(Yl)rsXT+.· · + ÷ç-ÊÕé)ô*×ôç~é (136)

where r and 5 may each be 1 or 2. To per-form the summation of equation 136, the matrices YTS and (77) r s and the matrix X are partitioned into 2X2 submatrices. The partitioning of the admittance

NOVEMBER 1959 Johannessen—Analysis of Magnetic Amplifiers Without Diodes 481

matrices is illustrated in Fig. 8. The partitioned X matrix becomes

X--

0 0 . U 0 . 0 U .

0 0 . 0 0 .

0 U 0 0 0 0

. U 0 0

. 0 U 0

(137)

where the zeros are 2X2 matrices and U is

C7= 1 01 |0 1

(138)

The matrix summation of equation 136 can be performed by determining the effect of pre- and postmultiplying a matrix by X and XTi respectively. It is evident from the matrix of equation 137 that premultiplication by the matrix X moves all rows one row down. Similar-ly, postmultiplication by the trans-pose of the matrix X moves all columns one column to the right. The final ele-ments in the matrix Yrs of equation 136, therefore, are obtained by summing all the diagonal elements of the (F7)rs-matrix, as shown by Fig. 8. The subscript / is omitted in Fig. 8 to simplify the figure. Elements in the same diagonal of the final matrix are identical, and the desired matrix is

Yrs =

Ëé A2

An Á÷

Az AA

A2 At

• An

■ An_{.

■ A, • Ay 1

(139)

where each At (i=l, 2 . . . n) is a 2X2 matrix obtained in the manner indicated by Fig. 8.

The procedure described will be demon-strated in the next section by applying it to a 3-phase amplifier.

Analysis of 3-Phase Amplifier

The amplifier to be analyzed, shown in Fig. 9, consists of three series-connected saturable inductors interconnected in parallel-parallel. For the parallel-parallel interconnection the proper analytic repre-sentation for analysis is the admittance matrix. The first step in the analysis is to determine the short-circuit driving-point and transfer admittances for the single-phase series-connected saturable inductor. When these admittances have been determined, the 3-phase matrix representation of the individual amplifiers will be derived.

The admittance matrix for the series-connected saturable inductor can be found by inverting the impedance matrix of equation 101. To simplify the ex-pressions, it will be assumed that Rm»

Phase I Phase 2 Phase 3

Fig. 9. Three-phase series-connected saturable inductor

Ri and R2. If Rm-matrix is obtained

-oo) the following Y

Y= a b/z c d/z b/z a —d/z —c\ —a —b/z —c— d/z\ b/z a —d/z —c\

(140)

where

d -

b =

c-

J -

(Ri+R*)-

( i ? i + # 2 ) 2 -

2R

(i?!+i?2)2-

-(Ri-

<Ri-

l

<Ri-

{R1+R2)+(R (Ri+Rt) 2 - ( i ? i

2Ri

- # 2 ) / 2 2

/«Vs2

R%)%/**

1~R2)/Z*

-Rtfl*

(Ri+R*)*-(Ri-R*)>/z*

(141)

(142)

(143)

(144)

To determine the 3-phase matrix repre-sentation of the series-connected saturable inductor, it is necessary to establish the subintervals in which saturation of the individual inductors occur. In Fig. 10 these subintervals are shown; in the analysis to be presented the principle mode of operation in which saturation occurs only within the g3 subinterval of each core in each amplifier is considered. If it is desired to analyze the amplifier when saturation occurs in additional sub-intervals, e.g., g2 and g3 in Fig. 10, a new set of equations must be written.

For the individual amplifiers in the different phases, equations can be written as follows:

where m = 1, 2, and 3, and the current and voltage symbols denote Laplace trans-forms of impulses representing average values of currents and voltages in the subinterval denoted by the superscript and at the port denoted by the subscript. During the part of the operating period in which neither core of a particular single-phase amplifier is saturated, that is dur-ing the intervals C\> c% gh and g2, the driving-point and transfer admittances of that amplifier are zero. Hence, the part currents during these intervals must also be zero.

/ ^ l = / ^ 2 = 7 ^ 1 = ^ 2 = 7 / 1 = 7 / 2

= 7 2 ^ 1 = 7 / 2 ^ 0 (149)

To permit the use of the Laplace trans-form in the analysis of magnetic ampli-fiers, it was assumed that the variable quantities consist of impulse. I t was pointed out that for the purpose of anal-ysis any waveform could be assigned to the variable quantities; the impulse function being chosen for convenience only. For the same reason it is assumed in equations 145 through 149, that the port voltages and currents for the 3-phase amplifier consist of impulses occur-ring during the subinterval c1} c2> cs

and gi, g2i gz.

The equations for the single-phase

If» = ^ H y u ^ » ^ * +yuem0nEien +yi2CmenE2

Cn+yi2Cm9nEign\\

« = i

3

w = l

3

If™ = ] T | | y 2 i C m C ^ E i C w + y 2 i C w ' w ^ tt = l

3 If* _ ^ I I D * ! * ^ " ^ +ya

emenE1gn+y,fmfinEten +y22dm0nE2

gn\\

(145)

(146)

(147)

(148)

482 Johannessen—Analysis of Magnetic Amplifiers Without Diodes N O V E M B E R 1959

(n + l)th Half Period

Phase 3

Core I of Phase

Core I of Phase 2

Core Γ of Phase 3

Fig. 10. Inter-vals of operation of the 3-phase magnetic ampli-

fier

k-T/6-H

series-connected saturable inductor now can be used to determine the remainder of the port currents for the individual series-connected saturable inductor in the 3-phase system. Suppose that the half-period quantities are represented by im-pulses that occur during the subinterval Cz (or g3), then

3 (150)

where If1, IS2, and Iic* are the Laplace transform of impulses occurring during the subintervals ch c2, and c3, respec-

tively. The factor 1/3 takes into account the fact that

Iic

is the average current over the complete half period, while

/ Ë / Ë and 7iC3

are the average currents over the sub-intervals cu c2) and cs. Similar relations can also be written for the remainder of the half-period average voltages and currents.

If equation 150 and similar relations for

the voltage variables are substituted into the equations for the single-phase ampli-fier for which equation 140 gives the matrix, the first of the four equations ob-tained is

= > ' l l C C (£ 1C l S~ 2 / 3 +£l C 2 2- 1 / 3 +£l C 3 ) +

ynC0{E^z--^-\-Eiuh-^+El^) +

y12CC{E2Ciz-m+E2Ch-i>*+E*c*) +

yi2Ce{EigK-™+E2d*z-u*+E2

0*) (151)

where the values of the admittances in-dicated may be determined by comparing equation 140 with equation 40. But from equation 149 both

ú^ and hc*

are zero, thus

Iicz = yncz -^Å^ +ynccz -*'»&<* +

ynccElc*+yncez-™Er+

yncoz-u*Elg*+ync°El

(l*+ ynccz-™E2Ci+yi2CCz-u*E2C* + ynccE2C*+ynC0z-u*E201+

y12C°z-u*E20*+y12

CeE2°* (152)

The remaining three equations obtained from equation 140 lead to similar expres-sions for

7 Ë hc\ and Ú203

In Fig. 11 the complete Y matrix represent-ing the series-connected saturable inductor in a 3-phase system is shown.

The matrix of the interconnected 3-phase amplifier can now be derived by following the procedure outlined in the previous section; the result is shown in Fig. 12. For the present, ignore the signs in parentheses.

-2/3

ba -5/3

-2/3

bz -5/3

- 1 / 3

bz - 4 / 3

- 1 / 3

bz -4/3

N O V E M B E R 1959

bz

bz -1

bz -5/3

-2/3

- b z -5/3

-2/3

bz -4/3

- 1 / 3

-bz 4/3

-1/3

bz

-bz

-2/3

-dz -5/3

-2/3

-dz 5/3

-1/3

-ds -4/3

. -1 /3

-ds 4 /3

-dz -1

- d z - 1

dz -5/3

-2/3

- d z ■5/3

-2/3

Fig. 11. Admittance matrix of series-connected saturable inductor in a 3-phase system

Johannessen—Analysis of Magnetic Amplifiers Without Diodes

dz - 4 / 3

-1/3

-dz 4/3

-1/3

dz

-dz

483

- 5 / 3 - 4 / 3 bz

bz

bz

bz

- 1 / 3

- 2 / 3

- 1

- 4 / 3

a

a z - / 3

a J - 2 / 3

„ , - '

b z " 5 / 3

a

- 1 / 3 az

- 2 / 3

bz

bz

-5 /3 - 1 / 3

- 5 / 3

| ( - ) a z " , / 3 ( - ) a

z ~ 2 / 3 - a z - l / 3

. - 4 / 3

- 5 / 3

- b z

( - ) b j

- b z

| ( - ) b z - 1 ( - ) a z " 2 / 3 ( - ) a z - l / 3 ( - ) a

- 2 / 3

1/· ^ ê , - 5 / 3 / a - 4 / 3 , xu -1 |< - ) b z ( - ) b z ( - ) b z ( - ) a z

■4/3

- 5 / 3

- 1 / 3

- 2 / 3

-1

- 4 / 3

- 5 / 3

, - 1 / 3

-2 /3

In the analysis of the single-phase series-connected saturable inductor, a transformation of variables, physically interpreted as the action of a full-wave rectifier connected to the output, reduced the number of equations describing the operation of the amplifier from four to two. It was pointed out that this reduc-tion was possible, because the action of the full-wave rectifier rendered the circuit symmetric so that no distinction could be made between the two intervals of opera-tion. The same procedure can be applied to the 3-phase amplifier, and the addition of a full-wave rectifier at the output re-sults in the transformation

I=DP

and

Å = ¼Å'

(153)

(154)

where D is a 12X12 diagonal matrix in which the 8th, 10th, and 12th diagonal elements are — 1, and the remaining ele-ments are + 1 . (The first six elements of the 7 and E matrices are input quantities and are not changed; the second six are output quantities for the periods clf c% Cs> gu g2, gz and are reversed in alter-nate periods.) This transformation changes the signs of the matrix coefficients as is shown by the signs in parentheses in Fig. 12. Addition of the six first rows and the six last rows after the trans-formation is made and substitution of equations 141 through 144 for the con-stants a, b, c, and d yield the admittance matrix equation:

, z - 2 / 3

b z ' 1

b z - 4 / 3

b z " 5 / 3

a

. z - 1 / 3

- , z " 2 / 3

) b z " '

- b z - 4 / 3

) b z - 5 / 3

- a

. z - 1 / 3

, z - 2 / 3

b z " '

b z " 4 / 3

b z " 5 / 3

a

- 1 / 3 - a z

( - > a z - 2 / 3

- b z " '

• ( - ) b z - ' l / 3

- b z " 5 / 3

c

- a " " 3

c z " 2 / 3

- d z ' 1

d z " 1 / 3

- d z " 5 / 3

- c

< ± > c z - , / 3

- c z " 2 / 3

< ± > d z - '

- d z " 4 / 3

< - ) d z - 5 / 3

<^>c

( - ) c z - 1 / 3

< ± > c z - 2 / 3

< - ) d z ~ '

< i > d z - 4 / 3

< i > d z " 5 / 3

- c

( ♦ > „ - ! / *

- e z - 2 / 3

(±>dz-'

d z - 4 / 3

- d z " 5 / 3

c

- c z - 1 / 3

e z - 2 / 3

- d z " 1

- d z " " 3

( + ) . - 5 / 3 — dz

- c

< i > e i - l / 3

- c z - 2 / 3

< - > d z " '

< ± > d z - , / 3

( - ) d z - 5 / 3

( ^ c

. , - 1 / 3 ( - ) c z

( l ) C 2 - 2 / 3

( i > d z - '

- d z - 4 / 3

<±>dz-5/3

- c

< D C I - l / 3

c r 2 f t

- d z " '

d z " 1 "

- d z - r ' / 3

c

- c z " l / 3

- c z " 2 / 3

( i ) d z - >

- d z - 4 / T

( + ) . - 5 / 3 — dz

- c

< - )< 7.

<*) - 2 / 3

( - ) d z " 1

( ♦) . - 4 / 3 - dz

- r . / 3 ( - ) d z * , / · 1

( + ) — c

<!>.„-'/3

- c z - 2 / 3

^ ' d z - '

- d z " , / 3

« i > d z - 5 / 3

1 / 3 ( + ) . - 5 / 3 . - 4 / 3 ( + ) . - 1 - 2 / 3 i+) - 1 / 3 — dz - d z — dz - r z — cz

FiS- 12 (above). 24 Matrix of 3-phase

amplifier

Fig. 13. Unit step response of 3-phase amplifier

20 30 Number of Subintervals

1 + 2 - l / 3 + 0 - 2 / 3 + 3 - l + z - 4 / 3 + s - 5 / 3 ÷ _ 2 - l / 3 + 0 - 2 / 3 _ 2 - l + £ - 4 / J _ 2 - 6 / 3

(Rl+R<t)+(Ri-R*)/z 1 + 2 - 1 / 3 + 2 - 2 / 3 + 2 - 1 + 2 - 4 / 3 + & - 6 / 3

(Ri+Ra)+(Rl-Ri)/z

(Δi+2W+(Δ-*i)/* 1 _ 2 - l / 3 + 2 - 2 / 3 _ 2 - l + 2 - 4 / 3 _ 2 - 6 / 3

(Rl+R2)+(Ri-R2)/z

(156)

(155)

where Y is the matrix

and

/ i -(/ /) ie i+(/ /) ie iH-(/ ,) ie*+(J r /V1+ {I')in+(I')i0z (157)

and similarly for I2, E\, and E* If satura-tion of the inductors occurs within the specified subintervals, equation 155 com-pletely describes the behavior of the 3-phase amplifier.

To determine the transfer character-istics when resistively loaded, the current /2 is eliminated from equation 155 by using the relation 72= — {\IRL)E% Then, by taking the inverse Laplace transform of equation 155, the following difference equation is obtained.

484 Johannessen—Analysis of Magnetic Amplifiers Without Diodes N O V E M B E R 1959

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