E-Alseb ras in Switching Theory

WARREN SEMON N O N M E M B E R A I E E

CE R T A I N CLASSES of switching func-tions, no tab ly the symmetr ic func-

tions, are realizable by i terat ive net-works. 1 ~~3 I t is the purpose of this paper to generalize this concept.

A "switching function of η var iables" is defined as a function from the Cartesian product T=XiXX2X . . . X I « t o Ζ where Χι, X2, ..., Xn, Ζ are sets consisting of two elements each. T h e set of all such func-t ions m a y be identified with the Boolean algebra A of order 22n, consisting of t he subsets of Γ (which has 2n e lements called "configurations") and the opera-t ions of set union, intersection, a n d complementat ion. T h e elements of Γ will sometimes be identified with "con-figuration number s , " integers i in t he interval 0 < i < 2 n — 1, the correspondence being the usual one . 4

E-Algebras

Let Ν be the set of indices of the ar-gument sets Xi, X2, ..., Xn- T h u s N= { l , 2 , . . . , n}. Let (R be an equivalence relation defined on N. A relation R on Γ m a y be defined as follows:

Let x, yeTy so t h a t

X = (xU X2, . . . , Xn)

y=(yu y<i> . . y % )

Define xRy if and only if there exists a

permuta t ion Ρ of the index set Ν such

t h a t if

P ( l , 2 , . . . , w ) = ( * i , * 2 , . . . , * » )

then

and

kGiik k = l,2, ...,n

THEOREM 1

"R is an equivalence relat ion."

Proof: (1). R is reflexive, since Ρ m a y be taken as the ident i ty and k(Rk for all k. (2). R is symmetr ic , since the existence of Ρ implies the existence of P " 1 , and k(Rik implies itGik. (3). R is t ransi-t ive since (R is t ransi t ive, and the existence of Pi and P2 implies the existence of their product .

An equivalence relation on a set corre-sponds biuniquely t o a par t i t ioning of the set in to collectively exhaust ive disjoint subsets, ca l led ' 'equivalence classes.' ' Le t

i t be assumed t h a t (R part i t ions the η indices of Ν in to r equivalence classes and t h a t R par t i t ions Γ into p classes Γι, Γ2, . . . , Tp. T h e Boolean algebra consisting of t he subsets of t he set {Γι, Γ 2 , . . . , Γ Ρ } has 2V elements a n d is isomorphic to the subalgebra of A gen-erated by the p "e lementary funct ions" defined b y the subsets Γ<. A n y sub-algebra of A induced by an equivalence relation & on N, extended to R on Γ will be called an "e lementary ^ -a lgebra . " T h u s a n e lementary Ε-algebra is a sub-algebra induced b y a n equivalence rela-t ion, s

THEOREM 2

" E a c h e lementary function Γ< is in-

var ian t to any permuta t ion of arguments

with indices in a single (R-equivalence

class."

Proof: Let

X = (xi, X2, . . . Xn)

be an arb i t ra ry configuration belonging t o Tf, and let Q be any permuta t ion of the indices in a single (R-equivalence class. Then

Q(x) = ( X i 1 , X i 2 , . . . , * ! „ )

and by hypothesis k(Rik for k=l(l)n. T o show t h a t Q(x)e Tf, choose Ρ as Q~~x. Then P(Q(x))=x and this choice of Ρ clearly satisfies the condition of the defini-tion.

I t follows a t once from Theorem 2 t h a t the equivalence classes into which Ν is divided m a y be thought of as defining permuta t ions of the a rgument sets to which the functions of the elementary E-algebra are invariant . Hence the equiv-alence classes defined by (R correspond to a subgroup Η of the group G of all permuta t ions of the arguments . Fur -thermore, the set of all functions in the elementary £-algebra contains those func-tions which belong to H, i.e., are invar ian t to all permuta t ions of H. If t ransforma-tions of argument complementat ion are now included, then a subgroup c~lHc conjugate to H, where c is an argument complementat ion, induces a subalgebra of A which is isomorphic t o t h a t induced byH.

An example of an elementary E-algebra is t he set S of symmetr ic functions. I t is clear t h a t S comprises the elements of a subalgebra since the sums, products , and complements of symmetr ic functions are

again symmetr ic functions. T h e rela-tion öl on AT" is the universal relation, i.e., *(R; for all i, j in N, whereas the relation R on Γ is the "k out of w" relation. Hence r= 1 and p = n+l (the numbers of equiv-alence classes defined by 01 and R respec-tively). T h e subgroup Η is the group of all permutat ions of the arguments.

THEOREM 3

"Eve ry function / , symmetric in the na

variables a, can be expressed uniquely as

f(a,b) = S(a)g(b)

where S(a) is a symmetr ic function of the variables a only and g(b) is a function of the remaining var iables ."

I t follows from the definitions t ha t every symmetric function is a sum of elementary symmetric functions. I t also follows t h a t the product of an elementary function and any other function of an elementary Ε-algebra is either zero or the elementary function itself.

THEOREM 4

"If Ti is an elementary function and Γ<= S(a)g(b) where S(a) is a symmetr ic func-tion, then S(a) is e lementary ."

Proof: Suppose not , then it is a sum

of elementary symmetr ic functions

5(α) = Σ ^ · ( α ) i

whence

Γ ί = «(δ)2 Skj(a) i

But the function

Ka,b) = Skl(a)g(b)

clearly belongs to the elementary E-

algebra and

h(a,b)Ti(a, b)^0 ^Tt(a, b)

This contradicts t he hypothesis t ha t Ti(a,b) is elementary.

THEOREM 5

"Every elementary function can be

expressed as

r

Ti = J[Sik(ak) k=i

where Sijc(ak) is an elementary symmetric

function of the variables a* of the &th (R-

Paper 60-1266, recommended b y the A I E E Com-puting Devices Commit tee and approved b y the A I E E Technical Operations Department for pres-entat ion at the A I E E Fall General Meeting, Chicago, 111., October 9 -14 , 1960. Manuscript sub-mitted June 7, 1960; made available for printing November 2 3 , 1960.

WARREN SEMON is with Sperry Rand Research Center, Sudbury, Mass .

J U L Y 1961 Semon—Ε-Algebras in Switching Theory 265

,Xj

r

,x,

f Ί

Fig. 1. Subnetwork types

Type A

equivalence class. Here r is the number of such (R-equivalence classes."

Proof: Γ $ is symmetr ic in the variables {α ϊ} . Hence by Theorem 3,

Ti = S(ai)g1(a2, ...,an)

and by Theorem 4, 5(#i) is e lementary, and

^ = (^(«1)^1(^2, . . .,an)

Clearly, gi(a2, . . . , an) is symmetr ic in the variables {a 2 }, so t h a t

gi(a2, . . ., an) = S( 02)22(03, . . . , ß n )

Hence

^ = 5(02)15 (̂01)̂ 2(03, · . . , ö n ) l

and applying Theorem 4 again gives t h a t 5 (a 2 ) is elementary. An obvious induc-tive a rgument can now be used to com-plete the proof. I t is a corollary t h a t the number p of e lementary functions is

i = l

where &ζ· is the number of indices in t he ith (R-equivalence class, i=l(l)r.

THEOREM 6

"All the elementary functions of an elementary Ε-algebra can be realized by a network using

Σ 2

transfers, where

^ • = f [ ( ^ + i ) . "

i = l Proof: By induction on r; if r—1 the

elementary Ε-algebra contains the sym-metric functions, p = n+l, k — n and this is a known resul t . 1 ' 2 Now, every product of a symmetric function of the kr variables in the r th equivalence class and an ele-mentary function of the variables in the remaining equivalence classes is an elementary function by Theorem 5 . All

Type Β

symmetr ic functions of the variables in the r th equivalence class can be realized by using (kr(kr+l))/2 transfers. Each function will be needed pr-i t imes. Hence the to ta l number of transfers re-quired will be

r = i (kr+l)(kr) , kjp,

P r ~ l 2 +

j = i

by the inductive hypothesis. This be-comes

^ 2 + 2 ^ ^ = 2-, 7 = 1 j = 1

as asserted.

kjpj

2

Since each pj<pr, then

npr 2

and except in trivial cases the inequali ty is strong.

T h e two limiting cases of e lementary Ε-algebras are those induced by making (R either t he universal relation or the identi ty relation. T h e first case produces the symmetr ic functions and has already been described. In the second case i(Rj if and only if i =j. Then R becomes the ident i ty relation on Γ, the subsets Tt are the con-figurations themselves, and the induced

Fi 9- 2. Iterative network

elementary Ε-algebra contains all func-tions. The subgroup contains the ident i ty alone and the " i terat ive ne twork" realization is the tree network.

An elementary £-algebra m a y be modi-fied b y superimposing a second equiv-alence relation on the set {1%} whose elements are the equivalence classes in-duced by R. The result ing part i t ioning of Γ will be said to define an "E-algebra ." An example of an £-algebra which is not e lementary is produced by superimposing on the "k out of w" relation which defines the symmetr ic functions, the additional equivalence relation | & | 2 = 1 . (Here \a\m

is defined to be the least positive residue of a modulo m.) The induced E-algebra contains the al ternat ing symmetr ic func-tion, i ts inverse, and the functions 0 and 1. Note t h a t there is no change in (R or H.

Modular Functions

Other examples of nonelementary E-algebras are the algebras of />-modular functions. For these algebras the equiv-alence relation (R is defined by i(Rj if

As rK

ΛΛ rW

S Χ Χ Χ Χ X '

>X4 >X4

^ 4v

Χι ο X

0 OUTPUT

9 A 5

266 Semon—Ε-Algebras in Switching Theory J U L Y 1961

a j \ 0 a j , l a j , 2 · · a j , p - ,

a j + l , 0 a j + l,l a j - H , 2 * " a j + l ,p-l

Fig. 3. Typical subnetwork

and only if i, jeN and \2'\p= \2j\p. T h e

induced relation on Γ m a y have super-

imposed on it a relat ion T, best described

in te rms of configuration numbers , as

kTl if and only if ky leT and |&| p=|/ | j>.

T h e e lementary ^-modular functions are

those defined by the sets of configuration

numbers

ί Γ 2 * - α - ΐ Ί I

Τα = γ,α+ρ,α+2ρ, . . . , α + Ι j p >

where α = 0, 1, 2, . . ., p— 1 (and where χ

the symbol [—1 means the integral pa r t of m

x/tri).

As an example of a modular function,

consider the function of ten variables

defined by the equations

1 0 2 3

f ( x 0 , x u . . . , ^ ) = ^ 2 f i P i ( 2 )

i = 0

where

Λ· = 1 for | i | 8 = 0

and

fi = 0 otherwise (3)

Equat ion 2 is, of course, the canonical

form for a general 10-variable function as

a sum of products , while equat ion 3 de-

fines the part icular function of interest

here. This function can be realized b y a

series chain of subnetworks of t he two

types shown in Fig. 1. T y p e A is used

for variable Xj if \2j\s= 1; t ype Β is used

if | 2 · ? |3=2 . T h e complete network, em-

ploying 48 contacts , is shown in Fig. 2.

T h e crossed lines in the figures are the

usual beginning and ending deletions of

i tera ted networks. T h e function defined

b y equat ion 3 is of course the e lementary

^-modular function obtained b y set t ing

a = 0 in equat ion 1, wi th £ = 3, η = 1 0 .

Since the p-modular functions represent a

hi ther to unremarked set of functions

which are realizable by i terat ive net-

works, a more detailed discussion is given

in what follows.

Suppose t h a t Γ α is the subset of Γ de-

fining the elementary ^-modular function

/ r f l , and p = 2k.c wi th c odd. Then i n / Γ α

there are k multiplicative variables; if c = 1 the remaining n-k variables are vacuous .

For example, consider the 4-variable func-

t ions defined by p = 4. Then c = 1, k = 2,

and

/ r 0 = # 3 X 2 ' * i ' * o ' + XsX2Xi'xo + * 3 * 2 ' * i * 0 + * 3 * 2 * i Xo

= ( X3X2 + ^ 3 ^ 2 + * 3 * 2 + * 3 * 2 ) * [x Q

t t = * 1 * 0

while

/ Γ ι = x i ' t f o , / r 2 = * i * o > / r 3 = * i * o

Now suppose t h a t p = 6, so t h a t k=l, c — 3. Then

jfj^Q = Χ $X <%X χΧ ο ~\~ X 3 * 2 * 1 * 0 "T~ * 3 * 2 * l * 0

= ( x'zXzx'i + Χ%Χ2Χ\ + # 3 * 2 * 1 ) * 0

In general, the following theorem holds.

THEOREM 7

"If p = 2k.c wi th c odd, the e lementary

^-modular function of η variables fra can

be expressed as

f=Xk-iXk-2, · · ·, Xi, Xo

{g(xn-u Xn-2, · · ·,**)}

where each xj} j = 0(1)k — 1, is either x/ or

Xj according as the coefficient of 2j is zero

or one in the dyadic expression of the

integer a; and the function g(xic, . . ., Xn-i) is the c-modular function gTb (as a func-

tion of the remaining n-k variables) where

b=[a/2k]."

Proof: T h e integers a and c in dyadic form are :

a =an_12n-1+an_22n-2+ . . . +ak2

k+

α * _ ι 2 * _ 1 + . . . +α ι2 *+αο2°

c = cn_l2n-1+cn_22

n~2+ . . . +ck2k+

^ - ΐ 2 Α _ 1 + . . . + ί : ι 2 ΐ + ί : ο 2 0

Since it is assumed t h a t 2k.c<2n— 1

2k-c = cn_l2n+k~1+cn_22

n+k-2+ . . . + «&*+0.2*~1+ . . . + 0 . 2 1 + 0 . 2 °

All coefficients of powers of 2 above

those shown are zero.

The set of dyadic integers representing

the integers i for which fi=l m a y be

wri t ten, by hypothesis, as

a, a+l'2k'C,a+2'2k'c, . . .,a+l-2k-c

where

a+l · 2k · c <2n - l<a+(/+1) · 2k · c

Each member of this set has the elements

ao, a>i, • • a,k-i as coefficients of 2°, 2 1 ,

. . ., 2k~l, respectively; hence the corre-

sponding variables are primed and un-

primed in the configuration defined by a in every term. Therefore,

f = Xo, Xu . . ., Xk-i{g(Xk, • • ·, Xn-i)}

Note t ha t cQ= 1 so t h a t exactly k variables

• f

Ρ

Ρ

Ρ

Fig. 4. Realization of a p-modular function

are multiplicative. Moreover, dividing

each i by 2k gives the set of i for which gi= 1, hence this set is

b4 b+c, b+2c, . . ., b+lc

where

& = [ ? ] = α η - ι 2 η ~ Α ~ 1 + · ' ' + Α + ι 2 1 + α Α 2 °

as asserted.

Theorem 7 describes the effect of powers

of 2 in the prime factorization of p on

an elementary ^-modular function fTa. Now consider the case where p is odd. If

2n~1<p then an elementary />-modular

function consists of a single term. The

cases of more interest occur when p is small compared with 2n~1. A p-modular function of η variables, fM> Ρ odd can always be constructed by i terating

subnetworks Aj of the form shown in

Fig. 3. The inputs to Aj are p in num-

ber, and their values depend on the value

modulo p of the dyadic integer correspond-

ing to the configuration of the preceding

variables. The subnetwork Aj contains

gates dependent on the variable Xj and

performs addition modulo p of |#j2^|„.

Since it is assumed t h a t p is odd and

greater t h a n 1, the powers of 2 will either

generate a reduced residue system modulo

p exclusive of zero, or pa r t of such a

system. Hence there will be a t most p—l different subnetworks. More precisely,,

if 2 belongs to t modulo p, t ha t is, if t

is the least positive integer such t h a t |2'|p,

= 1, then there will be / different subnet-

works, assuming of course t h a t η is

J U L Y 1 9 6 1 Semon—E-Algebras in Switching Theory 2 6 7

Fig. 5. Two-variable combination

sufficiently large. I n general 2 <t<<f>(p) 1, where φ(ρ) is Euler 's function.

When the components used for circuit

realization are current-steering devices of

which relays are an example, a combina-

tional circuit for a ^-modular function can

be made of η subnetworks, each using no t

more t h a n p transfer contacts (or the

analogs thereof). This proves the follow-

ing theorem.

THEOREM 8

"Any ^-modular function of η variables

can be realized using not more t h a n np t ransfers."

This result can be sharpened somewhat

by using a tree circuit for the first k variables where 2k~1<p<2k. Then, the

numbers of transfers are as i l lustrated

in Fig. 4. Hence the function can be

realized using no t more than

N=(2k-l)+(n-k)p

transfers. For an elementary ^-modular

function the tree m a y be used for the last

k variables as well as the first, so t h a t

N=2(2k-l)+(n-2k)p

For the elementary ^-modular function

shown in Fig. 2, p = 3, n=10. Hence,

k — 2 and

i V = 2 ( 4 - l ) + ( 1 0 - 4 ) 3 = 2 ( 3 ) + 6 . 3 = 2 4

transfers, as shown.

Since it is known 5 t h a t the network

function defined by / i = l , i<m, fi = 0 otherwise, can be realized using a t most η contacts, the quasi-modular function de-

fined b y / < = 1, for ieTa and i<m, a n d / < = 0

otherwise, can clearly be realized by the

series combination of a ^-modular func-

tion and such a network function.

Similarly, s tep functions 6 can be combined

with ^-modular functions, and, of course

the combinations m a y be either series or

parallel.

For the realization of ^-modular func-

tions using voltage-operated devices such

as vacuum tubes i t is useful to note t h a t if

in Fig. 3 all the inputs and ou tpu t s are

complemented, no physical change need

be m a d e in the circuit. This follows

from the fact t h a t the inputs and ou tpu t s

shown in Fig. 3 are related by the equa-

t ion

I dj,b+xj2j \p = a>j+U \b+xj2J\p

Complement ing any inpu t is equivalent

t o t ak ing the sum of the remaining inputs ,

thus

a,j,j,=ajto-\-ajti+ . ..

These two equat ions m a y be combined to

show t h a t

I a'j,b+zj2J I p = a'j+x, \ b+xj.2J\P

Using this fact, a ^-modular function m a y

be realized using no t more t h a n 4«/>

gr ids , 4 or analogs thereof.

T h e bounds given here assume t h a t

the variables are t rea ted one a t a t ime

and in their na tu ra l order. As in other

types of i tera ted networks nei ther of these

restrictions is necessary. I t m a y easily

be seen t h a t 2-variable combinations Xj a n d # Ä are chosen so t h a t | 2 y + 2 A | J , = 0 have

some advantages , since only three out-

pu ts result from such a pair instead of

the expected four (see Fig. 5). For the

function of six variables, with p=5, t he

straightforward method requires 14 t rans -

fers = 2 8 contacts , since p=5, η = 6, k = 3. However, associating the variables xo, x2,

and Xi, x 3 , Xb leads to the circuit shown

in Fig. 6 , using only 21 contacts , when the

dangling branches (shown crossed) are

deleted.

Applications

T h e circuits required for a computer

using the residue-number s y s t e m 8 - 1 4

would be largely circuits to realize p-modular functions. Fur thermore , i t has

been demons t ra ted 1 2 * 1 3 t h a t the carries

arising in a fixed or mixed radix adder with

composite radices m a y be described in

terms of the carries arising from the same

addit ion carried on modulo the prime fac-

tors of the radices. Hender son 1 4 has

shown how an error-correcting code can be

constructed using residues, which is pre-

served under addit ion and multiplication.

He has also discussed the use of />-modular

functions in the design of circuits for

t ranslat ion and sign determinat ion in the

residue-number system.

Recognition and Identification

Clearly every function which is an

element of an Ε-algebra is invar iant to the

transformations of a subgroup of the

Fig. 6. Modular network

group of a rgument transformations.

Hence any of the m e t h o d s 1 6 , 1 6 developed

for the detection of such invariancies may

be used to discover this fact. Henderson

has noted in a pr ivate communication

t h a t decomposition c h a r t s 1 7 m a y be used

for the identification of ^-modular func-

tions. The method will probably be ap-

plicable to the functions of general E-algebras.

Possession of the requisite symmetries

is a necessary bu t not a sufficient condi-

tion for a function to belong to a part ic-

ular nonelementary Ε-algebra. A s tudy

of the types of equivalence relations which

can be superimposed on the elementary E~ algebras might lead to the classification of

functions according to the Ε-algebras of

which they are elements.

References

1 . THE DESIGN OF SWITCHING CIRCUITS (book), W. Keis ter , A. Ritchie , S. H . Washburn. D . Van Nostrand C o m p a n y , Inc . , Priaceton, N . J., 1 9 5 1 .

2 . SWITCHING CIRCUITS AND LOGICAL DESIGN (book) , S . H . Caldwell . John Wiley & Sons, Inc . , N e w York, Ν . Y. , 1 9 5 8 .

3 . LOGICAL DESIGN OF ELECTRIC CIRCUITS (book), R. Higonnet , R. Grea. McGraw-Hil l Book Com-pany, Inc. , N e w York, Ν . Y. , 1 9 5 8 .

4 . SYNTHESIS OF ELECTRONIC COMPUTING AND CONTROL CIRCUITS (book) . "Annals of the Com-putat ion Laboratory of Harvard Univers i ty ," Cambridge, Mass . , vol . 2 7 , 1 9 5 1 , pp. 1 9 - 2 0 .

5 . SYNTHESIS OF SERIES-PARALLEL NETWORK SWITCHING FUNCTIONS, W. L. S e m o n . Bell System Technical Journal, N e w York, Ν . Y. , vol. 3 7 , July 1 9 5 8 , p. 8 8 1 .

2 6 8 Semon—E-Algebras in Switching Theory J U L Y 1 9 6 1

6. IDENTIFICATION AND OPTIMAL REALIZATION OF A CLASS OF SWITCHING FUNCTIONS, M . Cohn. Report no. BL-21, "Theory of Switching," Harvard Computat ion Laboratory, Dec . 1958.

7. VZNIK KODU A CISELNE SOUSTAVY ZBYTKOVYCH T R i D , M . Valach. Stroje na zPracovani informaci, Nakladete ls tv i ÖSAV, Prague, Czechoslovakia, vol . I l l , 1955.

8. OPERATOROVE OBVODY, M . Valach, A. Svoboda. Ibid.

9. RATIONAL NUMERICAL SYSTEM OF RESIDUAL CLASSES (in Czech) , A. Svoboda. Ibid., vol . V , 1957.

10. THE NUMERICAL SYSTEM OF RESIDUAL CLAS-

SES IN MATHEMATICAL MACHINES, A. Svoboda. Proceedings, Congreso Internacional de Automat ica , Madrid , Spain ( to be published) .

11 . THE RESIDUE NUMBER SYSTEM, H . L. Garner. "Proceedings, Western Joint Computer Confer-ence ," Ins t i tute of Rad io Engineers, N e w York, Ν . Y . , 1959.

12. ADVANCED DIGITAL COMPUTER LOGICS, Η . H . Aiken , W . L. S e m o n . WADC Technical Report, Wright Air D e v e l o p m e n t Center, D a y t o n , Ohio, 1956.

13. ADVANCED DIGITAL COMPUTER LOGICS, Η . H . Aiken , W . L. S e m o n . WA DC Technical Report no. 59-472, Wright Air D e v e l o p m e n t Center, 1959

14. LOGICAL DESIGNS FOR ARITHMETIC UNITS,

D . S. H e n d e r s o n . Ph.D. Thesis, Harvard Univer-s i ty , 1960.

15. A METHOD FOR DETECTING FUNCTIONAL IN-VARIANCE, R. L. Ashenhurst . Report no. BL-2, "Theory of Switching," Harvard Computat ion La-boratory, sect. I I , 1953.

16. DETECTION OF GROUP INVARIANCE AS TOTAL SYMMETRY OF A BOOLEAN FUNCTION, E . J . M c -Cluskey, Jr. Bell System Technical Journal, vol . 35, 1956, pp. 1445-53 .

17. THE DECOMPOSITION OF SWITCHING FUNC-TIONS, R. L. Ashenhurst . "Proceedings of an Inter-national S ym p os iu m on the Theory of Switching" (book) , "Annals of the Computat ion Laboratory of Harvard Univers i ty ," vol. 29, 1959, pp. 74 -116 .

Initiation of Flux Reversal in Magnetic-

Amplifier Cores

F. J. FRIEDLAENDER I. P. LELIAKOV ASSOCIATE MEMBER AIEE NONMEMBER AIEE

FL U X R E V E R S A L in grain-oriented nickel-iron t ape cores has been

described b y several models dur ing re-cent years . 1 · 2 Some difference of opinion, however, still exists with regard to the init iat ion of the process. 1 New evidence is here set forth t h a t nucleation and initial growth of regions or domains of reverse magnetizat ion occur near the tape surface for applied fields sufficiently larger t h a n the coercive force Hc of the material . Fields a t least twice the coercive force will be considered. More evidence is still needed to subs tant ia te models for fields only slightly in excess of Hc. Ini t ial growth will be discussed, bo th for cores previously sa tura ted and for those no t sa tura ted when flux reversal was initiated. T h e la t ter case, though of part icular interest in many applications of ' 'square-loop" magnet ic cores, has no t been t rea ted adequate ly so far.

Paper 60-874, recommended by the A I E E Magnet ic Amplifiers Commit tee and approved by the A I E E Technical Operations Depar tment for presentation at the A I E E Summer General Meet ing , At lant ic Ci ty , N . J., June 1 9 - 2 4 , 1960, and re-presented for discussion on ly at the A I E E Winter General Meet ing , N e w York, Ν . Y . , January 29 -February 3 , 1961. Manuscript submit ted April 15, 1960; made available for printing December 7, 1960.

F . J. FRIEDLAENDER is with Purdue Univers i ty , Lafayette , Ind. , and I. P . LELIAKOV is with Aero-space Corporation, El Segundo, Calif.

T h e work described was made possible through a grant of the Nat ional Science Foundat ion and the Allegheny Ludlum Steel Corporation. Port ions of the paper were abstracted from a dissertation sub-mit ted b y I. P . Lel iakov in partial fulfillment of the requirements for the Doctor of Phi losophy degree at Purdue Univers i ty .

The authors would l ike to thank the Arnold E n -gineering C o m p a n y and especially their chief en-gineer, J. Mi tch , for donat ing the cores used in this work. Professor P. Rauta la of Purdue Univers i ty deserves credit for supplying the metallurgical data , including Fig . 9, through a project carried out b y a group of his s tudents . Also, c o m m e n t s b y Dr. C. P. Bean, General Electric C o m p a n y , Schenectady, Ν . Y. , are gratefully acknowledged.

Assumed Model

T h e ra te of change of flux άφ/dt in a tape core, which has its flux reversed by a step function of magnetomotive force (mmf), is shown in Fig. 1. (The ra te of change is determined by not ing the volt-age across a toroidal winding on the core. T h e initial voltage spike, which is usually observed, is ignored in this paper since it is not of importance in the processes con-sidered here.) If a model of surface growth of domain walls is used, 2 the initial domain wall geometry takes the form of Fig. 2.

Obviously, for a given value of applied field H, άφ/dt—and hence, the induced voltage per wrap of a given core size— should be independent of t ape thickness unt i l domains growing from opposite sur-faces of the t ape "collide," annihi lat ing the domain walls. T h e peak voltage Ep

(volts per wrap) , and hence max imum ra te of change of flux, occurs jus t before merging of adjacent domains is completed; i.e., t he peak occurs jus t before max imum domain wall area is reached. A method permit t ing experimental corroboration of this fact was recently supplied by Becker. 3

Details of the dynamics of domain wall mot ion , 4 which have to be considered to explain this result, as well as the experimental evidence, 5 will be presented in later papers.

If such merging occurs before the collision, then bo th the t ime to peak tv

and the peak voltage per wrap of t ape Ep mus t be independent of t ape thickness for a given value of H. Fig. 3 shows bo th Ep and the initial voltage per wrap Et. Fig. 4 displays l/tp as a function of the field Η which is applied by means of a step

of current in a toroidal winding on a core previously sa tura ted in t he opposite direction. Four cores each of 1/2-mil, 1-mil, 2-mil, and 4-mil thickness were used to obtain the data . Considering the eight-to-one range in t ape thickness and the large number of cores used, t he indicated spread of da t a is tolerably small about the average value given in the figure.

Figs. 3 and 4 supply excellent evidence in support of the surface model for the initiation of flux reversal. Also, Fig. 4 clearly indicates t h a t in 1/2-mil tapes, collision of domain walls growing from opposite t ape surfaces s tar ts to occur just before the peak voltage is reached, resulting in somewhat larger values of l/tP t h a n in the case of the thicker cores, for the same values of H.

--ts Η

Fig. 1. Rate of change of flux in a previously saturated tape core on application of step of

mmf in reverse direction

Fig. 2. Approximate initial domain wall geometry in tape core for H ^ H C

A — 0 < t < t p

B—t^tp C—tp<t<t»

J U L Y 1 9 6 1 Friedlaender, Leliakov—Flux Reversal in Magnetic-Amplifier Cores 269