Representations of switching functions and their application to computers

REPRESENTATIONS OF SWITCHING

FUNCTIONS AND THEIR APPLICATION

TO COMPUTERS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGESCHOOL TE DELFT, OP GEZAG VAN DE RECTOR MAG-NIFICUS, Dr_ R. KRO NlG, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 4 MEI 1960 DES NAMIDDAGS

TE 3 UUR

DOOR

HERMAN

]ACOB HEI]N

ELEKTROTECHNISCH INGENIEUR GEBOREN TE ZAANDAM

-

_{BIBllOl dEEK}

DE

R

iECHNISCHE HOGESCHOOL

DE

LFT

STELLINGEN

I

Is f(t) gedefinieerd voor 0 < t

< 1 en voldoet f(t) aldaar aan de voorwaarden

: 1 (1) f(t»o,

(2)f'(t)

~o

,

(3)f'(t) monotoon niet dalend, terwijl (4)

f

f(t)dt bestaat dan heeft

1

( f(t) cos xt.dt J

o

o

tenminste één nulpunt in elk der intervallen van x,

7T 37T 57T

(- , 7T], (- , 27T], (- , 37T], ....

2 2 2

II

De uitwerking van het voorbeeld, waarmee Creighton Buck aantoont dat het begrip "beste polynoombenadering volgens Tchebycheff" van een meer-di-mensionale functie niet eenduidig behoeft te zijn, is niet volledig.

R. Creighton Buck, On numerical approximation, The Univer-sity of Wisconsin Press, Madison, 1959, p. 12-23.

III

Het is mogelijk het berekenen van Besselfuncties volgens de methode van Stegun en Abramowitz door een eenvoudige overweging aanzienlijk te ver-snellen.

I. A. Stegun, M. Abramowitz, MTAC, 9,255-257, 1957.

IV

Het ruisgedrag van een Esaki-diode (tunnel-diode), die als negatieve weerstand in een schakeling wordt gebruikt, is vooral daarom zo gunstig t.O.V. andere elektronische negatieve weerstanden omdat het negatieve weerstandsdeel van de karakteristiek zich over een klein spanningsgebied uitstrekt.

V

De gebruikelijke methode voor het meten van de effectieve spleetlengte van magnetische weergeefkoppen door bepaling van het eerste nulpunt van de weergeef-karakteristiek bij registratie van een vaste frequentie bij variabele bandsnelheid levert, voor spleetlengten kleiner dan circa 2·5 fJ., onvoldoende gegevens over de dimensionering van de koppen.

VI

De gemiddelde tijd, benodigd voor het doorgeven van de overdrachten in een parallelle opteller, kan door eenvoudige technische hulpmiddelen bekort wor-den.

VII

Het hinderlijke effect van hole-storage bij een transistor kan op sommige plaatsen in de techniek met vrucht gebruikt worden.

VIII

Onder bep~.alde omstandigheden kunnen de in hoofdstuk 2 gevonden repre-sentaties worden toegepast bij de realisatie van schakelfuncties m.b.v. weer-standsnetwerken.

IX

De (Ecc\es-Jordan) flip-flop, een essentieel onderdeel van vele rekenmachines, wordt vaak met tot verwarring aanleiding gevende benamingen aangeduid.

X

Door bij de opteller met magnetische ringen volgens Chen het somcijfer direkt uit de digits x, y en c te vormen kan een actief element per binaire positie bespaard worden.

Mao-Chao Chen, Trans. Inst. Radio Engrs EC, 4, 262-264, 1958. XI

Het verdient aanbeveling ter vergelijking van de interne snelheid van reken-m8.chines, deze aan een aantal internationaal vastgestelde standaardproblemen te onderwerpen.

XII

Het meerling-druksysteem volgens van den Bergh geeft grote bezwaren bij het gebruik van letters met indices en van formules in de tekst en is dus niet bruik-baar om de drukkosten van de meeste boeken voor de Delftse student te ver-lagen.

XIII

Het heeft voordelen bij rechtshoudend verkeer om in het algemeen aan van linkskomend verkeer voorrang te verlenen.

XIV

Alhoewel de door Knapp voorspelde minimum temperaturen op het Zuidpool-plateau redelijk met de gemeten waarden overeenstemmen is de gevolgde be-rekeningsmethode aanvechtbaar.

REPRESENTATIONS OF SWITCHING

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOREN

PROF. Dr. Ir. L. KOSTEN EN

cAan mijn Ouders voor Betsy

CONTENTS

O. INTRODUCTION

0.1. Magnetic-core storage . 0.2. Selection systems . . 0.3. Purpose of the thesis 1. DEFINITIONS

2.

1.1. Logical sum and logical product . . . . . 1.2. Products of numbers and binary quantities . 1.3. Primitive functions . . . .

LINEAR REPRESENTATION OF SWITCHING FUNCTIONS 2.1. Introduction .

2.2. Derivation of the 2n_{- n - l relations}

2.3. Cube representation . 2.4. Representations of B2_k. 2.5. Representations of B3k . 2.6. Representations of B4k. 2.7. Representations of rank m 1 3 4 5 6 7 8 8 10 10 12 14 14 3. REPRESENTATION OF Bn WITH THE AID OF LOGICAL SUMS

3.1. Introduction . 19

3.2. Solution of the problem of 3.1. 19

3.3. Representations of B2_{k . 21}

3.4. Representations of Gn2n -1 22

3.5. Representation of functions of switching functions . 23 4. BINARY QUANTlTlES APPLlED TO THE DESCRIPTION OF

THE BEHA VIOUR OF MAGNETIC CORES

4.1. Introduction . 26

4.2. The equation describing the switching conditions of a corc . 28

4.3. Realization of switching functions . 30

4.4. Current methods 30

4.5. The voltage method . 32

4.6. Realization of logical sums . 32

5. APPLICATION OF THE THEORY

5.1. Concepts and definitions relating to computers 34 5.2. Transferring the contents of a register 35

5.4. Complementing the contents of a register. 39 5.5. Summary and discussion of additional possibilitics 40

5.6. Parallel addition 41

5.7. Multiplication 45

5.8. Counters. 46

5.9. Decoders. 50

6. AVERAGE LENGTH OF THE LONGEST CARRY SEQUENCE

IN AN ADDER HANDLING NUMBERS IN THE 2k_-NUMBER SYSTEM

6.1. Introduction . 53

6.2. Some concepts and definitions 53

6.3. Some values of Pn(v) and Qn(v) . 56

6.4. Generating function for Qn(v) 57

6.5. Explicit expression for Qn(v) and Pn(v). 59

6.6. A single sum for an . 61

6.7. Calculation of an . 62

6.8. Lower and upper limit for an . 63

7. A FAST BINARY ADDER WITH MAGNETIC CORES

7.1. Introduction 71

7.2. The preliminary phase . 71

7.3. Microprogramme of the preliminary phase . 73

7.4. The carry cycles. 74

7.5. Microprogramme of the cycle part of the addition . 75

7.6. Some results 77

SAMENVATTING . 78

o.

INTRODUCTION

0.1. Magnetic-core storage

In recent years storage systems have been developed which consist of nume-rous small rings of a ferromagnetic material having a rectangular hysteresis loop. Impetus was given to this development by the advent of ceramic ferFO-magnetic materials (Ferrites 1)), from which very small cores could be made. Ferrites being insulators, there are no more difficulties with eddycurrents wl).ich. when metallic magnetic materials are used, restrict the speed of reading

and writing. "

The induction in a ferromagnetic toroid, which is fiTSt driven to saturatioQ; and from which the magnetizing field is then removed, can only assume th!i<t~o values +Bo and -Bo (see fig. 1, which shows an idealized loop). If.we call these two remanent states of magnetization "1" and "0"-, respectively. we have an element capable of representing one bit, that .is, cO)ltain~ni one bit of information. Rectangularity of the hysteresis .loop js . not

+80

jX

-80

1858

Fig. 1. Hysteresis loop (induction B as a function of current i) of a magnetic core, idealized as a rectangie.

necessary in principle, for other shapes of hysteresis loops also gi~e_ ris.e to two stabIe states. However, for reasons which will be explained later on, the rectangularity is necessary for a practical storage system.

2

-In order to be able to read the information in the core and if necessary to change it (by reversing the magnetization), thè core is threaded with one or more wires tbrough which currents of the requisite magnitude are fed. The manner in which the wires are threaded through the cores depends on the soIution adopted for the selection problem.

Since the numbers are merely a code for information of different

kinds, they are usually referred to not as numbers but as "words". A word, before being stored, is first recorded in a "buffer register". This is a single series of elements (e.g. flip-flops) which can be in one of two possible states. The store is divided into word registers each of which consists of as many cores as the buffer register contains elements. The word in the buffer register must now be transferred to the particular word register of the store that has been selected for that word. This is one aspect of the selection problem. The other aspect is

how to transfer a word from a given word register of the memory into the buffer register.

The selection problem has been solved by Rajchman 2) and Forrester 3)*) and in a different way by Renwick 4). In order to understand the way in which

the selection cau be performed, the following example will be considered. Suppose that each word consists of one bit only (see fig. 2), so the buffer register too has only one flip-flop. A common wire s passes through n cores 1'1,

1457

Fig. 2. Magnetic cores with selection wires s and Lt. This arrangement constitutes a very simple magnetic core storage with one bit per word.

r2, .•. , rt, ... , r", and wires L1, L2, ... , ~, ... , Ln, respectively, pass through each core separately. Let iX _{be the current needed to change the maguetization} of a core from the one remanent state to the other. We assume that the core material has a rectangular hysteresis loop. To read the contents of core rt we pass through each of the two wires s and ~ a current puise of magnitude tix

in the negative direction, as understood in fig. 1. Only core rt is then subjected

to a field strong enough to reverse its magnetization (tix

+

-tix); the other cores receive merely tix • If core rt had been set to the "I" state previously, its magne-tization reverses and this induces an e.mf in a third wire (not shown in fig. 2) which threads all cores and is connected to the flip-flop of the buffer register.

3

-This wire is called reading wire. Owing to the rectangularity of the hysteresis loop of the core material, the magnetization of the cores su bjected to -tix remains constant. These cores therefore produce no output in the reading wire. The flip-flop having previously been set to the "0" state (i.e. the buffer register having been c\eared), the e.m.! induced in the reading wire brings the flip-flop into the "1" state, which thus takes over the information flOm core rio If ri was ori-ginally in the "0" state its magnetization is not reversed and therefore no e.m.! is induced in the reading wire. The flip-flop remains in state zero and hence again reads the state of rio

Reading the contents of a core evidently amounts to writing an "0" in the core. If the core was in the" 1" state it is therefore necessary, if the original in-formation is not to be lost, to restore the "1" after the reading operation. In order to restore the core to its original state, a positive pulse of -tix is arranged to pass through the wire s after the reading out. If nothing further happens, the core rt remains in state zero. This is correct, if this co re was in the "0" state before the reading out. However if the core was in state" 1", the magnetization must be changed. To ensure that this will be the case, a positive current pulse . of magnitude

li

x _{is sent through wire Lt at the same time as the positive pulse}

passes through the wire S. Whether or not a pulse is sent through Li is controlled by the flip-flop in thebuffer register which took over the information fromcore rio

0.2. Selection systems

Before explaining the difference between Rajchman's selection system and that of Renwick some remarks are necessary. Generally the cores of a store are arranged like the elements of a matrix. Selection wires are woven row by row and column by column through the matrix.

Now in Rajchman's system one matrix holds corresponding bits of every word. There is a number of matrices equal to the number of bits in the word and one word is found in corresponding cores of all matrices. In Renwick's system developed at Cambridge University, however, there is only one common matrix. Here every word makes out one row. For this reason this system is sometimes called word-organized system. In Ra jchman's system every bit of a word must . be read out by coincidence of two current puls es (one row and one column pul se) each of magnitude -tix. All other cores of the selected row or column receive a current pulse of

-tix.

Since the hysteresis loop is never perfectly rect-angular, small flux changes due to the current pulse l

i

x _{will produce a disturbing} pulse in the common reading wire. Although the reading wire can be woven in such a way that the disturbing pulses partly neutralize each other, aresuiting disturbing pulse cannot be wholly avoided. In Renwick's system, ho wever, the disturbing pulses are avoided altogether since a current pulse passes only through the cores under consideration. It is for this reason that this system has been chosen for the design a of computer entirely built up from magnetic cores..

4

-0.3. Purpose of the thesis

It is the purpose of this thesis to derive principles for the design of

a computer using the core system described above as a basic method. In

order to do this, the first part of the thesis is devoted to two representations

of switching functions. The representations are specially adapted to the technique of core circuits as used in storage systems according to Renwick.

In the second part, the theory given in the first part is applied to the design of the main parts of a computer built from magnetic cores.

The intention has been to obtain a very simple computer in which all opera-tions are done sequentially. Therefore, the store as weIl as the arithmetic and the control unit should make use, as far as possible, of the same buffer register and of the same current gates. This leads' to the consideration of the arithmetic and the control unit of the computer as special sections of the store, having only

a different wiring of the registers. In this way it proves possible to design a

com-puter that is reasonably fast whereas the n umber of transistors is kept to a minimum.

In chapter 6 attention is paid to the average length of carry sequences in a

binary adder. It is made clear that dividing the adder into segments of 2 bits gives a very striking advantage over a normal adder.

REFERENCES

1) J. J. Went, E. W. Gorter, Pbilips tech. Rev.13, 181-193,1951/52. 2) J. A. Rajchman, R. C. A. Rev. 13, 183-201, 1952.

3) J. W. Forrester, J. appl. Phys. 22, 44-48, 1951.

1.

DEFINITIONS

1.1. Logical sum and logical product

A binary quantity can assume only two va lues ([J and Q. The set of binary quantities is denoted by X; ([J and Q are constants in

x.

Two binary variables Xi and Xj are equal when Xi = ([J if Xj = ([J, Xi

=

Q if Xj

=

Q. The negation Xi of a binary variabie Xi is defined by

Xi

=

([J if Xi

=

Q,

Xi

=

Q if Xi

=

([J.

The union or logical sum Xi V Xj of two binary variables Xi and Xj is defined

in table 1.

Table 1. Values of Xi V Xj

" " " Xi Xj " " "

Q

It cao be verified from this table that Q Xi V Xi

=

Xi Xi V Xj

=

Xj V Xi XiV Xi

=

Q ([J V Xi = Xi QV Xi

=

Q. Q Q Q (1)

, The intersection or logical product Xi /\ Xj of two binary variables is given

in table IJ.

Table II. Values of Xi /\ Xj

.'. " Xi ([J Q

Xj " "

-([J ([J ([J

'

..

Here we have the relations - 6 -Xi /\ Xi = Xi, Xi /\ Xj

=

Xj /\ Xi, Xi /\ Xi

=

tI>, ti> /\ Xi = tI>, Q /\ Xi

=

Xi· (2)

The intersection will be abbreviated as XiXj. From tables land II it is clear

that Xi V Xj and Xi /\ Xj both belong to X.

Combinations of logical products and logical sums, such as

also belong to X, since Xo /\ Xl and Xz /\ X3 belong to X.

1.2. Products of numbers and binary quantities

Ordinary numbers will be denoted by Greek letters Ài, /Lt, .... The value of a product Ài.Xi I) of a number and a binary quantity Xi is defined as follows:

~ Ài if Xi

=

Q ~

?

°

if Xi

=

ti> ~ (3)

The following relations for the product of a number and a binary variabIe hold:

LXi

=

1 - LXi,

!

Ài.(XiXj)

=

(Ài.Xi).Xj,

Ài.Xt

=

Àt.l.xt,

Ài.Xt = Àt.l.xi = Àt .(1-1.Xi) = Àt - Àt.Xi.

(4)

These relations are easily proved by substitution of the values Xi=tI> or Xt=Q. For examplè, in the fust eq. (4),

Xi = tI> : 1.Q = 1 = 1 - 1. ti> = 1 -

°

= I,

Xi = Q : 1. ti> =

°

= 1 - 1.Q = 1 - 1 = 0.

If a set of 2n numbers (/Lt, Àt) (i = 0,1, ... , n-1) is given, it is possible to define an algebraic sum:

7>--1

pm

=

~ (Àt.Xi

+

/Lt.Xt), i=O

(5)

corresponding to each non-negative integer m

<

2n by giving to the binary

variables Xi the values they have in the binary representation of m: 7>--1

m

=

~ 2i .Xi.

i=O

The sets (/Lt, Ài) (i

=

0,1, ... , n - 1) will be called vectors.

7

-1.3. Primitive functions

A sub-set of X, containing the elements xo, Xl, ... , Xn-l that are independent, is called a sub-set of order n. A product of a number of these variables or their negations wil\ be called a primitive function of the variables. A complete primi-tive function of order n is a primiprimi-tive function in which each vari?ble or its

negation appears once. Tt apparently has the form:

(7) in which i and k together take all values

°

(1) n-l. In all there are 2n _complete primitive functions of order n.

To each number

°

~ m <2n _{there corresponds, from (6), an unique set ofvalues} for Xt (i = 0,1, ... , n-l). Also, (6) can be used to define uniquelyproducts of the type (7): if Xi in (6) has the vaIue Q, Xi occurs in (7); if Xlc in (6) is CP,

:ik occurs in (7). The complete primitive functions wiII be denoted by Gnm .

A binary function Bn(Xi) (switching function) of the n variables Xi(i

=

0,1, ... ,

n - 1) is a binary quantity the value of which depends on the values of the variables Xi.

Tt is known 2) that every binary function Bn(xt) can be written as a logicaI sum *) of a number of different complete primitive functions:

2"-1

Bn(xo,Xl, .. . , Xn-t) = S fm Gnm, (8)

m=O

where the constants fm are binary quantities. In other words, the complete prirnitive functions form a base for the set of switching functions. Since the coefficients fm can be chosen independently, the number of switching functions is 22_".

All switching functions that change into each other by permutation or negation of the variables will be called equivalent. By these equivalence relations the set

of switching functions can be grouped into classes of order n. A representative of such an equivalence c1ass will be called a standard function 3). The number

of these classes, and therefore the number of standard functions, cannot be determined in a simple way and is even unknown for n

>

6.

Also we can map the complete primitive functions Gnm on the vertices of an n-dimensional unit cube. This can be done, for exampIe, by assigning one coor-dinate axis to every variabIe and coorcoor-dinate values 1 and

°

to the values Q and

CP, respectively.

*) A logical sum wil! be denoted by S.

REFERENCES 1) F. M. Pelz, Fernmeldetech. Z., 8,328-334, 1955.

2)( lf 'Serrell, Proc. Inst. Rad\o Engrs, 41, 1366-1380, 1953.

2. LINEAR REPRESENTATION OF SWITCHING

FUNCTIONS

2.1. Introduction

The following problem 1) will now be considered:

Is it possible, with respect to a given switching function Bn, to find according to (5) a vector (fLt, À,)n such that

Pm ~

°

if Bn

=

cp and pm

>

°

if Bn

=

Q. (9)

If so, such a vector (fLt, Àt)n will then be caJled a representation of the switching function Bn.

According to (6), to each value of m there corresponds a certain combination of values of the variables (XO,Xl,X2, .• • ,xn-I) and vice versa. For example for m

=

0, all Xi are equal to CP, and all Xt are equal to Q. With the aid of (3) and (4) we can write:

fLO

+

fL1

+

fL2

+

.

. .

+

fLn-1

=

po.

For m

=

1, Xo has the value Q whereas all other Xt are equal to CP. Therefore,

1.0

+

fLl

+

fL2

+

.

. .

+

fLn-1

=

pl·

Proceeding in this way we cau write 2n _{equations with pt on the right-hand side}

and 2n unknown quantit~es Àt and fLt on the left-hand side. In order to be able

to solvè' tÎ'lesè equations 2~ - n - 1 relations between Pi must be satisfied. These wiU be-derive~ in '~eè.2.2.

A sè-èön.d-sèt of relations -between Pi is given by (9). As already mentioned, a switching function Bn. ca-ti: be written as a logica! sum of complete primitive funcfions of'n variables. -For every set of values of Xi only one term at most of this logical sum can ~ave the'value Q. Then L't(ÀtXi

+

fLtXt) should be positive. 'thereföre, there are as many Pi positive as there are terms in the logical sumo The other pt have to be equal to, or smaller than, zero. If none of these conditions for Pi are contradictory, our problem can be solved.

2.2. Derivation of the 2D

-o-1 relations

First of all we shall derive c<?nditions under which the 2n _{equations can be} solved. In order to do tnis we write the equations in the foJlowing form:

9 -1 -1 -1 -1 -1 1

o

0 0 0 0 .0 !1-0 po

o

1 1 1 1 1 1 0 0 0 0 .0 !1-1 PI 10111 1

o

1 000 .0 !1-n-l Ào À1 1 0 0 0 0 0

o

1 111 1

o

0 0 0 0 .0 1 1 1 1 1 Àn-l _{P2 n_l} or in short: Mn.(!1-t, Àt)n

=

pn (10)

The matrix Mn, which has to be multiplied by the vector (!1-t, Àt)n in order

to get the vector pn, is of a special form. First, the elements of the 2n _{rows only}

have the values one or zero. It is convenient as has been do ne in (10) to split

eacb row into two halves corresponding to !1-i and Ài, respectively. Amongst the

rows, n of them in tbe right-hand half have a unit vector. The right-hand balf

of the fust row contains only zeros. Now we fi.rst subtract the upper row from

the remaining rows:

1 1 1 1 -1 0 0 0 0 0-1 0 0 0 0-1-1-1-1 -1-1-1-1-1 1

o

o

.-1 .-1

o

0 0 0 0 10000

o

1 000

o

1 I 1 I I I 1 .0 .0

o

Àn-l po P l -po

or by cbanging tbe sign of the tirst n columns and of tbe fust row:

1 1 1 1 1 1 0 0 0 0

o

1 000

o

1 1 1 1 1 1 1 1 1 .1 00000 .0 10000 . 0 01000 1 01111 1 11111 . 0

o

o

-po

l

~1 - po (11)

We observe that for eacb row except for tbe upper one the left-hand half and

right-hand half ae equal. Wben we consider the contents of a half-row as a

-

10-started from zero. So the left·hand half and right-hand half <;>f row m contain

the binary number m. Now the conditions for pt under which the equations are solvable can easily be found. From

m

=

~ 2k • Xk

k and by inspection of (11) it follows that

pm - PO = ~ (p2k- PO)' Xk

k

(12)

Of the 2n relations given by (12), n+ 1 are identically fulfilled (m=O and m=2k). The remaining 2n_{-n-l give the required relations for pt.}

2.3. Cube representation

As already mentioned in sec. 1.3, one can identify the complete primitive func-tions Gnm with the coordinates of the vertices of an n-dimensionaJ unit cube. In this way a function Bn determines a number of vertices of the cube. On the other hand the equations (5) with their left-hand si de replaced by zero, determine an (n - 1)-dimensional plane, if Xi are considered as continuous varia bI es and with /kt.Xi

=

fLi - fLt.Xt.

The question whether or not Bn can be represented by a vector (fL, À)n in the way mentioned in the preceding sections then proves to be identical with the question whether or not it is possible to separate the points corresponding to

Bn

=

Q from the remaining points of the cube where Bn

=

CP, by means of an

(n - 1)-dimensional plane. The quantities pm representing the distances from the vertices to this plane should have their signs according to (9).

2.4. Representations of B2k

Now we shall consider B2 and its representations (fLi, Ài)2 in more detail. All switching functions of two variables can be divided into 6 equivalence classes with their representatives chosen as in table 111. We shall denote these standard functions and their possible representations as B2k and (fLi, Àt)2k'

respectively, in which kis the row number in the tabIe.

The fust switching function B20 equals cP for all values of the variables. So, in the representation, pm ,;:;;; 0 must hold for every value of m.

Further, B2! yields:

Hence, P! and P3 must be positive. In the case of B2 2 we can write:

1 1

-Here Pl, P2 and pa have to be positive. Proceeding in this way we find that the following pt have to be positive:

k

=

0: none, k

=

1 : Pl, pa, k

=

2 : Pl, P2, pa, k = 3 : pa, k

=

4: po ,pa, (13) k

=

5 : po, Pb P2, p4·

In order that eq. (10) should be solvable, according to (12) the following relation must hold:

(pl - po)

+

(p2 - po)

=

(pa - po) or

Pl

+

P2 - po

=

pa. (14)

From this it is clear that only B24 has no representation. In this case the left-hand side of (14) is always negative, whereas the right-left-hand side, pa, has to be positive.

Generally one finds a whole set of solutions (!Lt, >.,)2. Those solutions having either >'t or f-tt equal to zero will be caIIed the simplest. and only these will be taken.

As an example we shall work out the case k

=

3.

Condition (13) and eq. (14) are satisfied by, for example,

po

=

-1, Pl = 0, P2

=

0, pa = I.

In order to determine >'i and !Li we write (11) as

!Lo

+

!Ll

=

po

=

-1,

!LO - >'0

=

po - Pl = -1 ,

!Ll- >'1

=

po - P2

=

-1 .

These equations are satisfied by

!L~

=

0, !L1

=

-I, >'0

=

1, >'1

=

0.

Hence we can write

Another solution can be found by putting !LO

=

-1 and !Ll

=

0. It foIIows that

(!Ll, >'i)2a

=

1 . Xo - 1 . Xl is also a representation of B2_a.

In table 111 the switching functions B2k and possible representations (!Li, >'i)2k are given.

-12

-Table 111. B2k and their pos si bie representations (/l-i, Àt)2k

k B2k (p.t, Àt)2k

0 0 0

Xo 1.xo

2 Xo V Xl 1.xo

+ 1.XI

3 XOXI 1.xo - 1.XI; 1.XI - l.xo

4 XOXI V XOXI none

5 1 1

In the geometrical interpretation of the problem for example B23 deter-mines the vertex 11 of the square 00, 10, 11, 01 (fig. 3). The straight line which connects 01 and 10 separates point 11 from 00. If we caU the distance from 11 to this line

+

1, then the distances from 00, 01 and 10 to this line are -1,

o

and 0, respectively. These values agree exactly with the values of po, PI

and p2. X, "'" 10 00 11 Xo Ol"" 1458

Fig. 3. Geometrical interpretation of the function B23 = XOXI. It is possible to separate point 11 from points 00, Ol and 10 by a straight line.

In the case of B2_{4 it is not possible to separate the vertices 00 and 11 from}

the vertices 01 and 10 by a straight line. So there does not exist a representation of B24, as we had already seen.

• t>~ Jî- ",.

2.5. Representations of B3 k

The representation (p.t, Àt)3 k of the switching function B3k can be found in

the same way as that of (p.t, Ài)2k'

Amongst the 402 possible switching functions of 3 varia bles there are only 22 standard functions, which are given in table IV.

1 3

-As an example we shall eonsider B31O: B310

=

XOXl V X1X2 V X2XO

= XOX1X2 V XOX1X2 V XOX1X2 V XOXIX2

=

G33 V G3S V G36 V G37.

It follows that the quantities P3, ps, P6 and P7 have to be positive and the re-maining Pt to be zero or negative.

In the case of 3 variables there are 4 relations between Pj:

Pl

+

P2 - po

=

P3,

l

Pl

+

P4 - po

=

ps,

P2

+

P4 - po

=

P6, (15) . Pl

+

p2

+

P4 -2pO = p7.

These relations and the requirements about the sign of pt ean be satisfied by ehoosing

po

=

-1, Pl

=

0, P2

=

0, P4

=

O.

Bij substitution in (15) it is found that

P3

=

1, P5

=

1, ps

=

1, P7

=

2.

The quantities Ai and /li are found from the following equations:

/Lo

+

/Ll

+

/L2

=

po

=

-1,

/LO - Ao

=

po - Pl

=

-1,

/Ll- Al

=

po- p2

=

-1,

/L2 - A2

=

po - P4

=

-1.

Th~ foJlowing values satisfy these equations:

Hence:

/LO

=

0, /Ll

=

0, /L2

=

-1,

Ao

=

1, Al

=

1, A2

=

O.

(/li, Ai)3 l0

=

1.xl

+

1.X2 - 1.xo. From the symmetry of B3l0 it follows direetly that

1.xo

+

1.X2 - 1.Xl and 1.xo

+

1.Xl - 1.X2 also are amongst the simplest representations of B3l0.

As a seeond example we shall eonsider B3n whieh appears to have no

repre-sentation.

B3n

=

XOXl V X1X2 V XI.\:2

=

G3₂ V G 33 V G34 V G35 V G37.

SO P2, P3, P4, ps and P7 have to be positive, whereas the remaining quantities Pi have to be negative or zero. That these requirements are contradictory eau be proved as follows.

1 4 -In the expression for P6:

P6

=

P2

+

P4 - PO,

p2 an,d P4 are positive and po is zero or negative. Thus it foIIows that P6 should

he positive, which is contrary to the requirement that P6 must be zero or

nega-tive. X2 100 101

110/

_/

000 001 Xo 010

V

vi

X, 1859

Fig. 4. Geometrical interpretation of the function B3u = XOXI V XIX2 V xlxa. It is not possible to separate the vertices marked with a dot from the remaining vertices by aplane.

Geometrically speaking: it is not possible in fig. 4 by means of a plane to separate the vertices marked with a dot from the remaining vertices.

All switching functions B3k , and corresponding representations ("'1, À,)3k if

they exist, are written in table IV. The meaning of the remaining columns wiJl I be explained in sec. 2.7.

2.6. Representations of B4k

For 4 and more variables the method developed for 2 and 3 variables can be

appJied as weIl. However the number of standard functions increases rapidly. For 4 variables there are already 402 and for 5 variables there are even 1 228 158 standard functions. Of the latter group, only the number of standard

functions is known, not the standard functions themselves. In table V the switching functions B4k which have a representation are collected, together with their representation ("'t, Àt)4k.

2.7. Representations of rank m

We saw that only a certain number of switching functions have a represen-tation. If there is a solution ("'t, Àt)n k we shall caU it a direct representation or a

representation ofthe fust rank. If a direct representation cannot be found we can try to find a representation of rank m, which we shall define as follows. The

Table IV. All standard functions B3k and their representation (/Lt, Àt)3k• In column 2 the possible direct representations are given. The remaining f~nctions are all of rank: 2

-B3k direct rank k representation 0 0 0 1 1 Xo 1. x o 1 2 Xo V Xl 1.xo+ 1.X1 1 3 XOX1 1.xo-l.x1 1

4 XOX1 V XOX1 XOX1 V (XOX1) B 33 8 37 2

5 Xo V Xl V X2 l.xo+ 1.X1 + l.X2 1

6 XOX1X2 l.XO-l.x1-l.x2 1

7 XOX1 V X2 l.xo-l.x1+2.x2 1

8 XOX1 V XOX2 l.X1 + l.x2-2.xo 1

9 XOX1 V XOX2 XOX1 V (XOX2) B 33 B3₇ 2

10 XOX1 V XOX2 V X1X2 l.xo+ l.X1-l.X2 1

11 XOX1 V X1X2 V X1X2 (X1XO V X1X2) V X1X2 B3_s B3₇ 2

12 XOX1 V XOX2 V X1X2 (xo V Xl V X2) (xo V Xl V X2) B3S B414 2

13 XOX1 V XOX1 V X2 XOX1 V (XOX1 V X2) B3₇ B3₇ 2

14 XOX1X2 V X1X2 (XOX1X2) V X1X2 B 36 B37 2

15 XOX1X2 V XOX2 V X1X2 XOX1X2 V (XOX2 V X1X2) B3s B41S 2

16 XOX1X~ V XOX1 V XOX2 V X1X2 XOX1X2 V (XoX1 V XOX2 V X1X2) B310 B41S 2

17 XOX1X2 V XOX1X2 XOX1X2 V (XOX1X2) B 36 B41S 2

18 XOXIX2 V XOX1X2 XOXIX2 V (XOXIX2) B36 B41S 2

19 XOXIX2 V XOXIX2 V XOXIX2 (xo V Xl V X2) (XOXI V X1X2 V X2XO) B310 B412 2 20 XOXIX2 V XOXIX2 V XOXIX2 V XOXIX2 X2 (XOXI V XOX1) V X2 (XOX1 V XOXl) B34 1/34 2

21 1 1 1

-Vl

1 6

-representation will be of rank m if the switching function B of n variables x,

can be written in the following form: B(Xt) . .=: BI(BI, Bz) Bz

=

Bz(Bz, B3) ~ ~ Bm-2

=

Bm-z (Bm-2' Bm-I)

-

~ ~ Bm-l

=

Bm-l (Bm-l, Bm)

and if, af ter adjunction of new binary variables Xn, Xn+l, .. . , Xn+m-Z to B2' B3, ... ,Bm, respectively, Bm and Bk (Bk, XnH-I) for k

=

1 (1) m- l have a representation.

We first form the representation of Bm and we adjoin the variabie Xn+m-2 to this representation in the following way. Only if the quantity p that corresponds to the representation of

Îim

is positive, will the quantity Xn+m-Z have the value Q. In the other case Xn+m-Z

=

([J. As a second step we form the representation ·of

Bm-l

=

Bm-l (Bm-l, Xn+m-2)

and we adjoin Xn+m-3 to the representation of Bm-l in the same way as was done with Xn+m-2. By repeating this process we find after m steps:

B = BI (BI, Xn).

In practice we shall try to write B as a logical sum as follows: B

=

BI V Bz V B3 V ... V Bm,

with

Bm-l

=

Bm-l V Xn+m-2, Bm-z

=

Bm-2 V Xn+m-3, ... , BI

=

BI V Xn,

or as a logical product as follows:

B

=

BI Bz B3 ... Bn, with

Bm-l

=

Bm-l xn+m-Z, Bm-z

=

Bm-z Xn+m-3, ... , BI

=

BI Xl1·

Some examples will iIIustrate these procedures. J. Let us consider the function

We first write this in the form:

1 7

-Table V. Standard functions B4k having a representation (fLi, Ài)4k of the fust rank k 0 0 0 1 Xo l.xo 2 Xo V Xl l.XO+ l.XI 3 XOXI l.XO-l.XI 4 Xo V Xl V X2 l.xo+ l.XI

+

l.X2 5 XOXIX2 l.XO-l.XI-l.X2 6 XOXI V X2 l.XO-l.XI + 2.X2

7 XOXI V XOX2 l.XI + l.x2-2.xo

8 XOXI V XOX2 V XIX2 l.xo+ l.XI-l.X2

9 Xo V Xl V X2 V X3 l.xo+ l.XI

+

l.X2+ l.X3

10 XOXIX2X3 . l.XO-l.XI-l.X2-l.x3

11 XOXI V X2 V X3 -l.xo+ 1. Xl +2.X2+2.X3

12 XOX2 V XIX2 V X3 -l.XO-l.XI +2.X2+ 3.X3

13 XOX3 V XIX3 V X2X3 l.xo+ l.XI + l.x2-3.X3

14 XOXI V XOX2 V XIX2 V X3 -l.xo+ l.XI + l.X2 + 2.X3 15 XOX3 V XIX2 V XIX3 V X2X3 -1.xo-2.XI + 2.X2 + 3.X3 16 XOX2 V XOX3 V XIX2 V XIX3 V X2X3 -1.xo-l.XI + 2.X2 + 2.X3 17 XOXI V XOX2 V XOX3 V XIX2 V XIX3 V X2X3 -l.xo+ l.XI + l.X2+ l.X3

18 XOXIX2 V X3 -l.XO-l.XI + l.X2+ 3.X3

19 XOXIX2 V X2X3 -1.xo+l.xl-3.X2+2.X3

20 XOXIX2 V XIX3 V X2X3 -l.XO+ 2.XI + 2.x2-3.X3 21 XOXIX2 V XOX3 V XIX3 V X2X3 l.xo+ 1.XI + l.x2-2.X3

22 XOXIX2 V XIX2X3 l.xo-2.XI-2.X2 + l.x3

23 XOXIX2 V XOXIX3 V X2X3 l.xo-I.XI

+

2.X2-2.X3 24 XOXIX3 V XOX2X3 V XIX2X3 1.xo+ 1.XI-1.X2-2.X3 25 XOXIX2 V XOXIX3 V XOX2X3 V XIX2X3 l.xo+ I.XI-l.X2-1.X3

26 1 1

From table IV:

to which we adjoin X3, thus

B320

=

X2X3 V X2X3

=

B34 (X2,X3).

Hence B320 is of rank 2.

2. We can wTÏte B319 in the form of a product,

- 18-Now,

So adjoining B310 to X3, we get

B3 19

=

(xo V Xl V X2)X3

=

XOX3 V XIX3 V X2X3

=

B413(XO, Xl, X2, X3).

3. As a third example we shall consider

B3lS

=

XOXIX2 V XOXIX2. Here

and we find

Thus we find that these switching functions have representations of rank 2. In the third column of table IV is indicated how the function B3k could be split up. The fourth column contains the switching functions that have to be applied in succession. From the table it is seen that the rank of the represent-ations for 3 variables is 2 at most.

REFERENCE

3. REPRESENTATION OF Bil WITH THE AID OF

LOGICAL SUMS

3.1. Introduction

Consider the elements Xi and the quantities Xi V Xj, Xi V Xj V Xk, Xi V Xj V Xk V Xl. .•• , defined in chap. 1 and all belonging to set X. Moreover

we introduce integers ,\n, ,\ni, ,\nij , ,\nijk , ,\nijkl , ..• all of which with two equal

suflixes are zero.

The algebraic sum ynm will be defined as

in which the suffix m is determined by (6):

n-l

m

=

~ 2i .Xi.

I~O

To every set of values of Xi there corresponds a ynm and vice versa. In particular, if all Xi assume the value CP, we find ynO

=

,\n. In this chapter we shall answer the following question: is it possible to find, for a given switching function Bn, a set (,\n, ,\ni, ,\nij, ,\nijk , ••• ) such that

ynm

=

0 if Bn

=

cP and ynm

=

1 if Bn

=

Q .

If so then Bn is represented by this set (,\n, ,\ni , ,\nij , ,\nijk , ..• ). The

represen-tation will be called An(,\n, ,\ni , ,\nij, ,\nijk , ••• ), or An for short. Often, instead of An, the corresponding function ynm will be given.

3.2. Solution of the problem of 3.1.

The 2n _{relations originating from (16) by giving to Xi all possible values form}

a set of equations, by which the quantities ,\n belonging to a certain Bn must be determined.

In order to find a solution we shall first split up the equations into groups. In the equations of the kth group kvariables Xi have the value Q. Thus there

is only one equation of the oth group: yno

=

,\n. For the kth group the number of equations is

n!

enk

-- (n-k)!k!'

so that the tota! number amounts to

n n!

~ - - = (1+ 1)n = 2n.

k~O (n - k!)k!

2 0

-We can also divide the logical sums Xt V Xj, Xt V Xj V Xk, ••• into groups.

The sums of the kth group contain kvariables Xt. In this way the Xt belong to the

first group, Xt V Xj to the second group, etc. The number of sums of the kth group is also given by (17). Thus here we also have 2n _{different sums, one of} which is equal to Q.

We now arrange the equations (16) according to increasing group number. rn a group the equations are arranged according to increasing values of m. For example with n

=

3 we have the following sequence of equations:

y30

=

,\3 group 0 y3 1

=

,\3

+

,\3 0

+

,\301

+

,\302

+

,\3012

!

y32

=

,\3

+

,\31

+

,\301

+

,\312

+

,\3012 group 1 y3₄ = ,\3

+

,\3 2

+

,\302

+

,\312

+

,\3012 y33

=

,\3

+

,\30

+

,\31

+

,\301

+

,\302

+

,\312

+

,\3012

!

y3S

=

,\3

+ ,\30

+

,\3 2

+

,\301

+

,\302

+

,\312

+

,\3012 group 2 y36

=

,\3

+

_,\31

+

,\32

+

,\301

+

,\302

+

,\312

+

,\3012 y37 = ,\3

+

_,\30

+

_,\31

+

_,\32

+

,\301

+

,\3 02

+

,\312

+

,\3012 group 3 Equivalently, 1 0 0 0 0 0 0 0 ,\3 y3₀ 1 1 0 0 1 1 0 1 ,\3₀ y3₁ 1 0 1 0 1 0 1 1 ,\31 Y23 1 0 0 1 0 1 1 1 ,\32 y3₄ 1 1 0 1 1 1 1 ).3₀₁

-

_{y 33} 1 1 0 1 1 1 1 1 ,\3_{02 y3S} 1 0 1 1 1 1 1 1 ,\312 y3₆ 1 1 1 1 1 1 1 ,\3012 y3₇

or, written in short,

M3. A3 = (r)3,

and generally

Mn. An = (r)n. (18)

..

By application of elementary operations it can be shown that the determil1:'

ant Dn of Mn is equal to 1. This means that the equations (18) are solvable and that every switching function Bn has a representation An.

In order to determine the value (r)n that belongs to a certain function Bn

we write Bn as a logical sum of complete primitive f~nctions Gnm . Every

func-tion Gnm corresponds to a ynm in such a way that if Gn m is present, ynm has the value 1; otherwise ynm = O. In the special case where Bn itself is a complete

2 1

-3.3. Representations of B2k

For n = 2 the equations (18) can he written as:

y20 = ).2,

y21

=

).2

+

).20

+

).201, y22

=

).2

+

).21

+

).201, y 2a = ).2

+

).20

+

).21

+

).201•

With the aid of Cramer's rule ).2, ).20, ).21 and ).201 can he expressed in the

quan-tities y2_{m. However, inspection immediately yields:} ).2 = y20,

).20

=

y 2a - y22, ).21

=

y 2a - y21,

).201 = y21

+

y22 - y 2a - y 20. (19)

According to what was said at the end of sec. 3.2 we now write the switching function Bn as a logical sum of complete primitive functions. To each complete

primitive function of B2 there corresponds one y2k that equals 1. The remaining y2k are zero. By suhstitution of the values of y2k in (19) we find for B2 the values of .,12.

Consider, for example, tahle VI:

whence:

From (19) it follows that

So (20)

As a second example, consider:

Now,

So we find here:

).20

=

-1, ).21

=

-1, ).201

=

2, ).2

=

0,

and

y24

=

-1.xo - LXI

+

2.(xo V xI).

In tahle VI the standard functions B2k and the corresponding functions y2k

2 2

-Table VI. Standard functions of two variables and their representations y2k k

0 0 0

1 Xo l.xo

2 Xo V Xl l.(xo V xr)

3 XOXI l.xo + l.Xl - l.(xo V Xl)

4 XOXI V XOXI -l.xo - 1.Xl + 2.(xo V xl)

5 1 1

3.4. Representations of Gnzn_l

For fin ding the functions ynk, the representations of Gn2n _ l appear to be of special interest. Gn2n_l is the complete primitive function with the property that it has no negations of the variables. First we shall derive the corresponding function yn2"_1.

For 2 variables we already found:

y23

=

1.xo + 1.Xl - 1.(xo V Xl).

For reasons of symmetry, in any sum of (16) the .\'s now are equal. For

COD-venience we shall write .\n as ,_1.0, .\ni as !1-l, .\nij as !1-2, .\niJk as !1-3, etc. Hence we can write yn2"_1, according to (16), in the form:

n-l n-l n-l

!1-0 + !1-l ~ l.Xi + !1-2 ~ 1.(Xi V Xj) + !1-3 ~ l.(Xi V Xj V Xk)

+.

...

(21)

1=0 I,j=o I,}, k=O

1* / i* /,I* k

/* k

If Xi

=

([J, yn2n-l has to be zero, so that !1-0

=

O. Suppose we know the quan-tities !1-1, !1-2, ... !1-n-1, then we can find !1-n by putting in (21) Xn-l

=

Q. Then

yn2n_l changes into yn-l n-l • 2 -1 Now

From (21): y23

=

!1-l(l.XO + 1.xr) + !1-2(XO V Xl). We take

Xl

=

Q,

yll

=

1.xo

=

!1-l.XO + !1-l + !1-2. It follows that

!1-l

=

1, and !1-l + !1-2

=

O. Thus !1-2

=

-1,

2 3

-From (21)

y3?

= 1-'1(1. XO

+

1.X1

+

1.X2)

+

1-'2 {1.(xo V Xl)

+

1.( Xo V X2)

+

1.( Xl V X2)}

+

+

1-'3.(XO V Xl V X2). Now taking X2

=

Q, the result must be equal to y 23:

1.xo

+

1.X1 - l.(xo V Xl) = 1-'1(1.XO

+

1. X1)

+

1-'1

+

1-'2.(XO V xI)

+

2 1-'2

+

1-'3. Hence: 1-'1

=

1, 1-'2

=

-1, 1-'1

+

2 1-'2

+

1-'3

=

0, 1-'3

=

1.

By induction it will be shown that fki(i ;;;: 1) is equal to

+

1 and -1 alter-nately. Putting Xn

=

Q we find fr om (16):

Hence

must be zero. Or:

I-'n

=

-1-'1 - 1-'2 eïl) - ... - I-'n-1 (~=D. We al ready found

1-'1

=

-1-'2

=

1-'3

=

-1-'4

=

..

. =

1. Thus I-'n is determined by:

(n - I)! (n - I)! (n - I)!

I-'n

=

-1

+

(n _ 2)! 1! - (n - 3)! 2!

+

(n - 4)! 3!

+ .

..

+

(- )m(n - I)!

+

_

+ ..

.

=

-(1-1)n_(-1)n

=

(_I)n+1. (n-m-l)!m!

So the quantities 1-'1 (i

>

1) in (21) are alternately

+

1 and -1; 1-'1 being plus one and 1-'0 being zero.

3.5. Representation of functions of switching functions

Now it will be shown that the representation of any switching function can be written as an arithmetic sum of representations of primitive functions of the variables without negations.

First of all some ruies about the representations will be derived. 1. If the switching function B is written in the form

B

=

B1 V B2 (22)

then for the representation of B, in the following denoted by R(E), the reIation

(23)

2 4 -can easily be checked. If B can be written as

B

=

BI V B2 V B3 V ... V Bn (24) we find by repeated application of (23) the following formula for the represen-tation R(B): R(B) = ~ R(Bt) - ~ R(Bt.Bj)

+

~ R(Bt.Bj.Bk)

+ . .

.

(25) 1=1=1 1=1=1 l=I=k Example 1. B24

=

XOXI V XOXI,

R(B'lo4)

=

R(xoxI)

+

R(xox!) - R(XOXI.XOXI)

=

R(xoxI)

+

R(XOXl).

2. In order to eliminate negations from the representation the following rule ean be used:

(26) This ruIe, too, can be proved by substitution. By repeated use of (26) a more generalized formula can be found which, however, is not of much use.

Continuation of example 1. Application of (26) yields:

R(B24)

=

R(xo) - R(XOXI)

+

R(xI) - R(xoxI)

=

R(xo)

+

R(x!) - 2.R(xOXl)

=

1.xo

+

LXI - 2.xo - 2.Xl

+

2(xo V xI).

Thus R(B24)

=

y24

=

-1.xo - LXI

+

2.(xo V xI).

In order to determine the representation of Bnk we must write _Bnk in the form of (24) in which Bi are primitive functions. Next we express the represen-tation of B by means of (25). Then by application of (26) the negations in the primitive functions are eliminated.

Thus, it is proved that the representation of B can be written as an arithmetic sum of representations of primitive functions without negations. And, as from sec. 3.4 we know the representations of primitive functions of this special kind, we have a method of finding the representation of arbitrary switehing functions. Another example will illustrate the method.

Example 2.

Thus

B3n

=

XOXI V XIX2 V XIX2,

R(B3n )

=

R(xoxI)

+

R(XlX2)

+

R(XlX2) - R(XOXlX2)

= R(Xl)

+

R(X2) - 2R(XIX2)

+

R(XOXlX2).

R(B3n)

=

y3n

=

1.xo- 1.(xo V xI)-l.(xo V X2)

+

1.(XI V X2)

+

1.(xo V Xl V X2).

In this way the representations of all standard functions ean be found. Those of 3 varia bles are given in table VII.

Table VII. Standard functions of 3 variables and their representations

rk

k

I

B3

k - - -

I

y3k

o

10

10

1 Xo l.xo

2 XOVXI 1.(xo VxI)

3 XOXI 1.xo

+

1.XI - l.(xo V xI)

4 XOXI V XOXI l.xo

+

l.XI - 2.(xo V Xl)

+

1 5 Xo V Xl V X2 l.( Xo V Xl V X2)

6 XOXIX2 l.xo

+

l.XI

+

l.X2 - 1.(xo V Xl) - l.(xo V X2) - l.(XI V X2)

+

l.(xo V Xl V X2)

7 XOXI V X2 l.(xo V X2)

+

1.(XI V X2) - I.(xo V Xl V X2) 8 XOXI V XOX2 \.xo

+

l.(XI V X2) - 1.(xo V Xl V X2)

9 XOXI V XOX2 1.XI - l.(xo V Xl)

+

l.(xo V X2)

10 XOXI V XOX2 V XIX2 1.(xo V xI)

+

l.(xo V X2)

+

l.(XI V X2) - 2.(xo

V

Xl V X2) 11 XOXI V XIX2 V XIX2 I.xo -l.(xo V xI) -I.(xo V X2)

+

1.(XI V X2)

+

1.(xo V Xl V X2) 12 XOXI V XOX2 V XIX2 -1.xo - 1.xI - 1.X2

+

1.(xo V xI)

+

l.(xo V X2)

+

l.(XI V X2) 13 XOXI V XOXI V X2 1.X2 - 1.(xo V X2) - l.(XI V X2)

+

2.(xo V Xl V X2)

14 XOXIX2 V XIX2 l.xo

+

l.XI

+

1.X2 - l.(xo V Xl) - l.(xo V X2) - 2.(XI V X2)

+

l.(xo V Xl V X2)

+

1

15 XOXIX2 V XOX2 V XIX2 1.xo

+

1.xI

+

1.X2 - l.(xo V xI) - 2.(xo V X2) - 2.(XI V X2)

+

2.(xo V Xl V X2)

+

1 16 XOXIX2 V XOXI V XOX2 V XIX2 1.xo

+

1.xI

+

l.X2 - 2.(xo V xI) - 2.(xo V X2) - 2.(XI V X2)

+

3.(xo V Xl V X2)

+

1 17 XOXIX2 V XOXIX2 l.xo

+

l.XI

+

l.X2 - 2.(xo V Xl) - l.(xo V X2) - 1.(XI V X2)

+

2.(xo V Xl V X2) 18 XOXIX2 V XOXIX2 1.xo

+

1.XI

+

l.X2 - l.(xo V xI) - l.(xo V X2) - l.(XI V X2)

+

1

19 XOXIX2 V XOXIX2 V XOXIX2 - 3.xo - 3.XI - 3.X2

+

4.(xo V Xl)

+

4.(xo V X2)

+

4.(XI V X2) - 5.(xo V Xl V X2) 20 XOXIX2 V XOXIX2 V XOXIX2

Y

XOXIX l.xo

+

1.XI

+

l.X2 - 2.(xo V xI) - 2.(xo V X2) - 2.(XI V X2)

+

4.(xo V Xl V X2)

21 1 1

tv v.

4. BINARY

QUAN1:ITIES

APPLIED TO THE

DESCRIPT-ION

OF

THE BEHA VIOUR OF MAGNETIC CORES

4.1. Introduction

Toroid-shaped cores made of a ferromagnetic material, often caUed cores for short, have been used as binary storage elements for some years. The magne-tization curve is practicaJly rectangular, especially for cores made of certain metallic materials. Ceramic materials however have the advantage that their electric resistance is high. Because of tbis the direction of magnetization in a

ceramic core can be changed faster than in a metal core.

For the sake of simplicity we shall consider the magnetization curve of the cores to be absolutely rectangular as shown in fig. l.

Cores have two properties which make them very suitable for use as storage elements. Firstly when the magnetizing current is zero, the magnetization can be in one of two distinctly different sta bIe states (+ Bo or -Bo). The second property is the existence of a threshold value for the magnetizing current.

It is our intention in this chapter to describe the behaviour of the core with the aid of binary quantities. A binary quantity c may be defiued as one having the value Q if the magnetization in the stabie state is positive (+ Bo), and

having the value f/J in the other stabie state, in which the magnetization is negative (-Bo).

In order to change the magnetization to the opposite stable state (and thus to change the value of c from f/J to Q or vice versa) the core must enclose a temporary current i of sufficient magnitude and in the proper direction. The absolute value of this temporary current (or current pulse) must be at least as large as the threshold value iX _{(see fig. 1).}

If as a result of a current pulse the magnetization has been changed perma-nently we shall say that the magnetization in the core has been reversed or

briefly that the core has been reversed. If the absolute value of current i is

smaller than iX _{or if the current has the wrong direction (i.e. positive direction}

in state c

=

Q and negative direction in state c

=

f/J), then the magnetization returns to the original value after the current has been switched off.

First of aU some definitions and nota ti ons will be given. In the cores a posi-tive direction ofmagnetization has already been chosen. Accordingly, a normal to the plane of the core has a positive direction if it is in right cyclic order with the magnetization. The number of times a wire q passes the interior part of the plane through a core k will be called Wk q; the current through the wire q will be

calied ik q•

2 7

-possible, draw the wires through it horizontally or vertically. Every wire will be marked with a positive or negative number i!ldicating the number of times the positive direction of current passes the plane of the core in the positive sense less the number of times th,e positive direction of current passes the plane in

-2 +1

Fig. 5a Fig. 5b

Example of the symbolism used for threaded eores. Fig. 5b explains the symbol of fig. 5a.

the negative sense. The symbol +1, which is the most common one, will be omitted. Fig. 5b :~ives the manner in which the wires are threaded through the core in the case of the symbolic notation of fig. 5a.

Go

Ge

14bl

Fig. 6. Symbol used for eurrent gates. The gate ean be set by means of a voltage pulse on the eontrol terminal Ge. If the gate was set beforehand, a pulse on the input terminal Gl will eause a eurrent pulse of rr..agnitude a.i'X to arise on the output terminal Go; if not, no output pulse will appear.

The total number of ampere-turns through a core k, Ik, amounts to:

Jk= :Ewkq.ikq. (27)

q

In practice the currents are usually of short duration, and are accordingly called current pulses. These pulses enter the wires through the cores by means of gates, which must be set beforehand. Such a gate (see fig. 6) has a control terminal Ge for setting it (shown as a short bar), an input terminal Gi, which is connected to a pulse generator and an output terminal Go.

Only if a voltage pulse Vq of a certain polarity (e.g. positive) has been fed to the control terminal will a pulse on the input terminal give rise to a current pulse iq at the output terminal. The magnitude (maximum value) of this cur-rent pulse is written within the circle and often expressed in terms of the thres-hold value iX •

2 8

-In order to describe the behaviour of a gate the voltage pulse Vq will be

represented by a binary variabie Xq in accordance with the following rules.

If Vq

>

0 then Xq = Q and if Vq ~ 0 then Xq = 1/>.

The current pulse passed by the gate can now be written: iq • Xq.

Sometimes it is useful to have gates which are set by voltages Vq as well as

~

-gates set by voltages Vq

=

- Vq• The binary variabie belonging to Vq will be called

Xq.

As the current pulse caused by the latter type of gate may often differ in magnitude we will write this current pulse in the form iq • Xq.

Hence the total current pulse through the core k is equal to:

(28) 4.2. The equation describing the switching conditions of a core

When the total number of ampere-turns [k through core k exceeds the thres-hold value iX _{and has the proper sign the magnetization wiU reverse. In order}

to describe the condition under which core k switches, we shall take the binary variables Xkt and x k s, defined as follows, to represent Ik.

If:

[k ~ i X then Xkt

=

Q and Xks

=

1/>,

I[kl

<

i X then Xkt

=

I/> and xks

=

1/>,

[k ~ - iX then Xkt

=

I/> and Xks

=

Q.

(29)

Now, the active current lak will be defined by the foUowing equations.

If [k is positive the active current will be:

[ak

=

~

[k _ iX; [k > iX 0 [k ~ iX. If [k is negative lak

=

~

[k

+

iX ; [k

<

- iX 0 [k ~ - ix.

With the aid of Xkt and xks, lak now can be written

lak

=

[k.(xkt V xks) - iX(1.Xkt - 1. xks). (30)

We now introduce the quantity

(31)

It may easily be shown that this quantity has the value

+

1 when the core switches from c

=

Q to c

=

1/>, -1 when the co re switches in the opposite direction, and 0 if the core does not switch. If the magnetization is in state c

=

1/>, then, to reverse the core, curre,nt pulse [k has to be sufficiently large and positive; hence Xkt

=

Q. The result of (31) is then

+

1.

In the other Ci!Se

-

29-the magnetization is in the state C

=

Q and current lak has to be negative (xks

=

Q).

If a wire is threaded once through core k in the positive sense, the switching of core k from c

=

Q to c

=

<P induces an e.m.f. v which is the same for all wires through core k. If the core switches in the opposite direction the e.m.f. will be -v.

Hence, if a wire pis threaded through core k wkp times, the induced voltage

Vkp can be written with the aid of quantity (31) in the form

(32)

If wire p is threaded through a nu mb er of cores, the magnetizations of which reverse at the same instant, the total induced e.m.f. amounts to:

V p

=

~ Vkp

=

~ v(1.xktC - l.xksC)wkp. (33)

k k

It is to be expected that when the active current increases the e.m.f. v also in-creases. From measurements it appears that these two quantities are linearly dependent. The ratio of the voltage to the current will be called R.

Combining (30) and (32), we find, that:

{lk.(xkt C V Xksc) - iX(1.xktC - l.xksc)}R

=

v(1.xktC - l.xksc) wkp. (34)

The total induced e.m.f. in wire p now can be written

Vp

=

~ {lk.(xktC V xksc) - iX(1.xktC - l.xksc)}R. (35)

k

Fig. 7 is a graph corresponding to formula (35).

rx=bg Ig(R/w~)

- . 1860

Fig. 7. Vp , the e.m.f. per turn induecd in wire p, as a function of the total current pulse Ik

which passes though a core k.

Of ten it is known in which state of magnetization the core is. For exampIe, let us suppose that from a preceding action all cores under consideration are known to be in state c

=

<P. Only by a total positive current lak can the mag-netization of a core k be switched into the state c

=

Q. In this case relation (35) can be simplified to:

Vk = ~ {Ik - iX}.1.xkt.R = ~ lak.R,