A Symbolic Analysis of Relay and Switching Circuits, Shannon C.E., 1936

8/12/2019 A Symbolic Analysis of Relay and Switching Circuits, Shannon C.E., 1936

1/72

A S Y l ~ B O L I C A l ~ A L Y S I S

OF

by

Claude

Elwood

Shannon

B . S .

U n iv ersity o f E ic hig an

Submitted

in Par t i a l Ful f i l l m e nt of

th e

Requirements f o r th e

Degree

of

lLASTER OF SCIE JCE

from

tile

: :assachusetts Inst i tute o f Technolcgy

1940

Sigl18.ture 1 Author _

Department of

Electr ical Engineering August 10

9 7

i g n t ~ r e of

P ro fesso r

in Char38 o f R es e a r c h

S i ~ Y J c l t u r e o f

Cnclirman

of Deuartment

COfficittee on Graduate Students ~ L ~ ~ ~ ~ _ . ~ ~ ~ ~

y


2/72

; \

~

T BLE OF O ~ T T E N T S

page

I n t r o d u c t i o n ; Types o f Problems

1

ser ies Para l le l

Two Terminal

ircui ts

Fundamental Def in i t ions a n d P o s t u l a t e s

Theorems 6

A nal og ue W ith th e c al c u lu s o f

P r o p o s i t i o n s

8

I I I Mul t l

I

rarminal and

N o n S a r i e s P a r a l l e l

Networks

18

Equivalence o f n Term1nel

Networks

18

star Ivlash an d Delts Nye T r a n s f o r m a t i o n s 19

Hinderance

Function

o f a

N o n S a r l e s P a r a l l e l

Network 21

S1roul t a n a o u s

Equa

t ions

~

Matrix Methods

25

Spec ia l

Types o f R e l a y s

an d

S ~ i t o h e s

8

\

IV

~ ~ t h a s s

o f

Networks

31

G eneral Theore ms on Networks and

Functions

31

Due 1 N e tw or ks

36

s y n t h e s i s o f th e G apersl Symmetric Function

39

Equations

from Given

O perating harac t e r i s t i c s

47

v

I l lus t r a t ive Examples

A s e l e c t i v e

ircu i t

An Elec t r ic Comb1nat1 on Lock

A Vote O U ~ t ng C

1 r c u 1 t

An

Adder

to th e

Ba sa

Two

5 1

52

55

58

59

A

F a c t o r

References

69


3/72

KNOvVLEDG ffiNT

The

au

thor i s

in e te

PrOfeSSOI F.

L.

Hitchcock

who supervised the

thesis for helpful

cri t icism

and

advice.


4/72

In t roduc t ion : Types o f Prob lams

I n

th e

c o n t r o l

an d

p r o t e c t i v e

c i rcu i t s

o f

com

p l e x

e lec t r i ca l

systems t

i s f r e q u e u t l y

n e c e s s a r y

to

make in t r i ca te i n t e r c o n n e c t i o n s o f relay o n ~ t s and

swi tches Examples o f

these c i r cu i t s

o c c u r

in

au to-

ma.tic t e l e p h o n e e x c h a n g e s , indust r5 a l m ot or con t ro l

equipment

an d in almost any c i r c u i t s designed to

g e r

f or m c or np le x

o p e r a t i o n s

a . u t o m a t i c a l l y .

wu

problems

tha t

oc c ur in

c on ne ct io n w it h such n e ~ w o r k s o f switches

w i l l

be t r e a t ed

he r e rb e

f i r s t

w h i c h w i l l be

c9.11ed

ana lys i s

i s to cietermine

th e

o p e r ~ i n g charac te r j s-

t i c s of a g i v e n c i r cu i t t i s

of

course alwa.ys p o s

s ib le

to analyze any given c i r cu i t y set t ing up

a l l

p o s s i b l e

s e t s

o f

i n i t i a l

c o n d i t i o n s

p o s i t i o n s o f

switches and r e l a y s )

a n d following thr ough th e

c h a i n

o f

event

so

ins t iga ted T h i s

method

i s

h o w e v e r ,

very tedious a n d open GO frequent e r r o r .

The

s ec on d p ro ble m i s tha t

o f syn thes i s

Given

c e r t a i n c h a r a c t e r i s t i c s t

i s r e q u i r e d

t o f ind

c i r c u i t

ir Lco r p o r atin g

t h e s e

c h a r a c t e r i s t i c s

s o l u t i o n

o f th i s

type of problem i s n o t unique and

t

i s

-cI-J.erefore

addi t io l la l ly

d e s i r a . b l e

t h a t the c i r c u i t

r equ i r ing the

l e a s t number o f

sVii

t

h

blades an d r e l ay


5/72

C O L ~ a c s be

~ o u n d Although

a solut ion

can

usual ly

be

obta i ned. by a

lieu

t and

t ry

m tho d f i rs t sa s fy i

ng

one

requirement

and

then

making

addi t ions

un t i l

a l l

are sa t i s f i ed the

c i rcu i t so

obtained

wi l l seldom

be

the

s i m p l e s ~ .mis

method 9.1so ha.s

the disadvan-

tages of

being long,

and

the

resul t ing design

often

conta ins hidden sneak c i r cu i t s .

The

method o f so lu tio n

o f

these problems which

wi l l be developed here

may

be described br ie f ly

as

fo l lows: An; c i r eu i t

i s

represented a

se t o f

equa-

t i ons the terms

of

th e e qu atio ns

represent ing

the

v a r i o t ~ s re lays an d sw itch es o f the

c i r c u i t .

A ca l -

culus is developed for

manipulating these equations

simple

mathematical

processes ,

most of

which

are

s imi l a r to

ordinary

a lgebra ic

a lgor i sms .

This Ca.l-

culus i s shown

to

be

exact ly

analogous

to

the Qalcu-

lUs

o f

P ropos i t ions \ l sed i n

the s ymbolic

s tudy

o f

log ic .

For

the synthesis problem

the

desired charac-

t e r i s t i c s

a re f ~ r s t wri t ten as a s ~ r s t e m o f e au atio ns

and the equat ions are then ffianipulated into tha form

represent ing the

simplest c i rcu i t .

The

c i r c u i t

may

then

be inwediately

drawn from the equat ions.

th i s nethod

i s

always

possible to

find

the

simplest

c i r c ~ i t

containing

only series and 9ara l l e l connec t ions


6/72

and for c e r t a in

types of funct ions t

i s 9 3s ib le

to

find the simplest c i r c u i t conta in ing

any

type of con-

nec t ion In

the

analys is

problem

the

equations

r e ~ r e -

sen t ing th e given

c i r c u i t a re wri t t en

and

may

then be

in terpre ted in terms

of

the

ogerating c h a r a c t e r i s t i c s

o f th e c i r c u i t t i s alGo

possible

with the ca lcu lus

to ob ta in any

number

o f c i r c u i t s

equ iva len t

to

given

c i r c u i t

phraseology

wil l

be

borrowed

frJm

ordinary

network theory

fo r con 6pts in swi tch ing c i r cu i t s

tha t are r o u g h ~ y ~ l o g o u s to those of

iffipedencJ

networks


7/72

4

I I ~ e r i e s p a r a l l e l

wo

Terminal J i rcu i t s

u n d a ~ e n t a l Definit ions and

Postula tes .

Tve sha

11

l imi t our tre atrn8n t

to c i rcu i t s

conta ining only

re

lay contacts

and

switches,

and

therefore a t any given

t ime

the

r u t between any

tVifo

termi

nals

must be

e i the r open

inf in i te

impedance) or Closed (zero

impedance). Let us associa te a

symbol

X

ao

or more

simply

X,

with the terminals a ana

b.

This var iable ,

funct ion

o f

t ime, lNill e

ca l l ed the hinderance

of the

two

terminal ~ i r c u i t

a-b. The

symbol

0

(zero)

wil l

be

used to represent th e h inde ranc e

of

a

closed

c i r cu i t , and th e

s-ymbol

1 un i ty to represent the

hinderance o f an open c i r c u i t . trhus when the c i r -

\ cu l t a-b

i s open

X

ab

1

and

when closed X

h

=

O.

wo hinderances X

ab

and

Xed

will oe said to

be equal

\ Vhenever the

c i r c u i t

a -b

i s o'pen, tl1.8 c i r cu i t c -d

i s open,

qnd v Jhenevc:r

a-b i s

c losed ,

c -d

i s

c losed .

oW

l e t the

symbol (plUS)

be

defined

to mean the

ser ies connect ion o f the tvY terr l inal c i r c u i t s

whose

hinderanc as a re ad1ied

toge thel

Thus

X

a

b

i:3

the Ilderanc e

0 f the

e i r e u i

t a.cd

when

b a n d c a r e

c o ~ n e c t e d together .

Similarly the

product of

two

hinderances

X

ab

Xed)

wil l

be defined to mean

the


8/72

5

hinderance of

the

i r ~ u i t

formed

by connecting the

c i r cu i t s

a -b

and c -d

in

pa r a l l e l .

A re lay

con tac t

o r

swi tch wi l l

be

rep resen ted in a c i r c u i t

by

the

symbol in Fig . 1 the l e t t e r being the corresponding

hinderance

func t ion .

Fig . 2

shows

the

i n t e r p r e t a -

t ion of the plus

s ign

and Fig . the r:lul t i p l ic q t io n

s i gn .

Xab

a ..._IIIlIftIO Do---..b

x y _ X+Y

.......

0 0

e o

x

-Ci}.

_ Xy

0

Fi

g.

1

Fig .

2

Fig.

This cho ice o f

symbols makes

th e manipu la t ion o f

hinderancea very

s imilar

to ordinary

numerical l g e ~

bra .

It

i s evident tha \vi th

the

a b o ~ l e de f i n i

t ions

th e fo l luwing postula tes wil l ho ld :

Po s tula. te s

1 .

a .

0 -0

= 0

b.

1

at

1

=

2.

a .

1 0

=

0 1

=

1

o 0-1 ~ ~ =

A

closed Ci r c u i t

in

para l l e l

vii th a c los e

d

c

i

rcu

i

tis

c lo s ed e i re ui t

An

aoen

c ir c u i t in se r ies

vv1th an open

c i r c u i t

i s an

open c ir c u i

t.

An

o?en ~ i r u i t

in

ser ies

with

a

c lo sed

c i r c u i t

in

e i ~ h e r

order

i s

an

ogen

e i re u i t .

A c losed c ir c u i t in para l le l

with an open ~ i r u i t

i n

e i t h e r o rde r is a

c losed

~ i r u i t .


9/72

3 . ~

0

0

=

0

A closed ~ i r c u i t in 8eries

with a closed c i r c u i t i s a

c l as ad

r C \ J . i

t

6

b.

1-1

=

1 An ODen c i r c u i t in p a r a l l e l

with

an open c i r c u i t i s an

open c i r e u i

t .

At

g i

van tj.me

e l

t h e r

X

=

0

or

X :

These are

s u f f i c i e n t

to develop a l l the theo-

rems ;J hich w i l l be

\ l s e d

i n

connect ion \vith c i r

4

c u i t 3

c o n t a i n i n g

only

s e r i e s and

p a r a l l e l c o n n e c t i o n s . The

p o s ~ L 1 1 a t e s a r e

a r r a n g e d

i n p a i r s to emphasize a d u a l i t y

r e l a t i o n s r J . i p betweerl the

o p e r a t i o n s

o f a d d i t i o I l

and

ml11t ipl icat ion ar Ld th e q .u an ti t ie s

zero

ind o n ~ Thus

y

i i n any

of the B p O s t u l a t e s the

z e r o s

a r e replaced

t;y o n e s and the m u l t i p l i c a t i o n s by a d d i t i o n s

and

vice

v9rs

a

, tb_e corresponding

b

i

p o s t u l a t e

wil l . reS' tllt.

This

f g c t

i s

of

g r e a t

importance.

I t

gives

e ach theorem

e

t l l ~

it being n e c e s s a r y

t o prove only

o ne

to

e s t a -

b l i s h both. T ~ e

only

one of

these p o s t u l a t e s which

d i f f e r s from

ordinary algebra

i s l b . However,

t h i s

ens b l

e s

g J ~ e f t

s i

mpl

i f i

os

ti

on

sin

th_e

me lipula t i on

of

Theorems.

Irl.

t h i s

s e c t i o n a

numbar

of theorems gov-

erning the combination of hinderances w i l l be ~ i v e n

Ina smucl:1 a s an;T of the theorems ma - he ruved y a

very

s imple p r o c e s s tl e proofs

w i l l not be

g:iver


10/72

7

except fo r

an

i l l u s t r ~ t i v e example.

Tbe

method of

Proof i s

t h a t

of per fec t induc t ion i e the ve r i -

f ica t ion o f

the theorem

fo r a l l possi

ble

cases .

Since

o s ~ l l p t e 4 each

var iab le

i s l imi ted to the values

o Bn

d

1

th i s i s

B

simple r t t e r . Some of the

the orems

may

be

Droved

more

e legant ly by recourse

to

p ~ e v i o u s

theorems, Cut

the method

o f pe rfec t induct ion i s

so un i -

versal

tha

i s pro ba b ly to be pre fe r red .

1 .

8 .

X

Y

y

x

b . xy

=

yx

2 . a . x y = x y g

b.

X(YIi)

xy)

3 .

a

x y

ii )

xy

X5

-

b.

x +

yfll

-

x

y

x

a-

4.

a

l-x

x

-

b.

0

x

-

x

-

5.

a

1

x

1

b.

Ox

=

0

For

example,

to

Drove

theo

rem

4A,

note

the t

X i s e i t he r o r

1.

If it

i s 0, the theorem f ollows

from 1)ostll1ate

2b; if 1 it folLOWS from

rOs tu la te 3b.

Je

sha l l now def ine a

new

ope at ion to be

oal l ed negs t ion . rhe negat ive

o f

a

~ i n d e r n c e

X Wil l

be

Writ ten X

t

and i s defined as

a var iable Which

i s

equa l to

1 When

X equals

0 and

equal to

0 When X


11/72

8

equals

1

X

is the

}1 ind =Jr 3TIce

of the make contac t s

o f a

rela:T:i

then

XI

i s

trte hinderance o f the break con-

t ac t s

of

the same re lay.

The def in i t ion

of

the

nega-

t ive

of

8

hinder8nce

gives th e fo llowing theorems:

6.

8

X

XI

1

b.

V

0

n ~ \

7 .

a .

0

-

1

-

b.

1

1

-

0

-

8 .

X t

I

-

X

-

Analogue ~ f l i t h the ca l cu lus o f P r o p o s t i o t L ~ s Te Rre

now

in

A posi t ion

to

demonstrate

the

equ iv alence o f

th i s calculus vvith

cer ta in elementary p3r t s of th e

calCtl1u_s of propos i t i ons . Yne a lgebra o f o ~ (1)

2 ) , 3 ) or ig ina t ed

e o r ~ e

Ecole , i s

a

symoolic

method

of

i nve st ig a ti ng l og ic a l r e la ti on sh i p s.

The

symbols o f

Boolean algebra admit of

two log ica l i n t e r -

pl e ta t i6ns .

i n t e rp r e t ed in terms

of

c lasses ,

th e

varta

b1

33

are no t l im i t ed to

tn.G

p os sib le v alu es

o and 1 . This in te rp re ta t ion is k n o ~ v n as the algebra

of c la s se s . I f , hoVJ8Ver, the terms a re taken

to

r e p r ~ e

sen t

propos i t ions , \ve

have

th e calClll1 . .1S o f p,roposi

t i ons

in W ~ i c h

var iables

are

l imi ted

to

the

values 0 and

1*,

*I his

r e f 3 r s

on ly to the

c18

s s i c8 l

theory o f

the oa1

cul J.s

o f Propos i t ions Recent ly some ~ v o r k ha

s been

done

vvi

trt l og i ca l systems in vVhich

pro

posi

t ion s ma T

ha

ve

more than

tvvo

t t ru th vallIe s .


12/72

9

as

a re

th e h i n d e r a n c e functions above. U sually

~

two

sUbjects a r e developed simultaneously from ta e Same

se t

01

postula tes except

f o r

the

a dd iti o n in

~ case o f

th e

Cctlculus of

Pr o p o s i t i o n s of

a

p o s t u l a t e

~ u i v l n t

to p o s t u l a t e

4

aoove. S.V. Huntington

4) gives tne

followin5

se t

of

p o s t u l a t e s f o r symbolic l o g i c :

1 . Tne c l a s s K contains a t lea s t two dis t inc t

elements.

2. I f a and b

are

in tne c l a s s K tnen

a+

b

i s

in tl e c l a s s

K

a b z b a

4.

a .

b)

C = a + b

c)

5.

a a a

6. ab

ab

::

a

where

ab

is

defined a s

a + b

)

I f we l e t ~ class K be ~

c la s s

c o n s i s t i n g of the

two

l m ~ n t s

0

an d

1 , taen tnese

p o s t u l a t e s follow

from

those

given

on

pages

5

and

6.

Also

p o s t u l a t e s

1

2,

and 3 given t n e r e can be deduced from

Huntington s

p o s t u l a t e s . Aduing 4 an d res t r ic t ing

our

d i s c u s s i o n

to tile CEi.lculus of p ro p o s i t ions i t is evident t ha t a

p e r f e c t tine.logy exis ts between tn e

calculus

fo r swi t c n -

ing

c i r e u i

ts B.Jlli

t I l i s

br2J1Ch of s y mb o l i c

loSlc

The

tw o

in terpre tc t ions

o f

t ~

symbols

a re

sh:wn

in

Table

1 .

* T h i s 8.nalogy lllay also be s een from a s l i sh t l y d i f fe ren t

view point. I n s t e a d o f

a s s o c i a t i n g Xab di rec t ly wltfi

th e

c i rcu i t a-b

l e t

Xab r e p r e s e n t

t ~ g r o o o s i t i o n t n a t

the


13/72

10

O le to t h i s anal ogy any theorem 0

f

tre c a l c u l u s

o f


i s a l s o a

t r u e

theorem

if

i nt e rp re te d in

terms o f ~ a l a y c i r c u i t s . Th e

remaining

theorems

in t h i s

s e c t i o n a re

taken d i r e c t l y

from

t h i s f i e l d .

I

De M o r ~ n s theorem:

9 .

X Y

) t = X t . y . Z

b X Y Z

) =

X y Z

This

theorem

gives the

n e g a t i v e

o f

a

sum o r product in

terms

o f the n e g a t i v e s o f th e summands or f a c t o r s . I t

may

be

e a s il y v e r if ie d

f o r

two

terms by ~ ~ b s t l t u t n g

a l l

p o s s i b l e v a l u e s and then

extended

to any n u m b e ~ n

o f v a r i a b l e s

by mathematicsl

i n d u c t i o n .

A f u n c t i o n o f c e r t a i n v a r i a b l e S

X l ,

~ - - - Xx . 1 s

any e x p r e s s i o n

formed fl om t h e

v a r i a b l e s

w i t h th e o p a r a -

t10n

S 0

f

a

ddi

t 1 o n ,

mul

t i p l

i

ca

t 1 o n ,

and ne

ga t 1 o n .

The

n o t a t i o n

t X

l

, X

2

,

Xu

w i l l b e

u s e d to r e p r e s e n t a

f l J : ~ t o n Thus

we

m @:ht h a v e f{X, Y , Z ) ) = XY +

X

y Z ) .

I n i n f i n i t e s i m a l

c a l c u l u s

it i s shown t h a t

any

r u n ~ t o n

p r o v i d i n g i t

is

continuous an d

a l l

d e r i v a t i v e s e r a eon-

,

t i n u o u s ) may be ex p an d ed

i n

8 T8:rlor S e r i e

s A

somewha t

s i m i l a r expansion i s p o s s i b l e i n the calculus o f propos1-

t i o n s .

To

develop the

s e r i e s

expansion of

f u n c t i o n s

F oo tn ote c on tin ue d from p r e c e d i n g page)

c i r c u i t a - b i s open. Then

a l l

the

symbols

a r e

d i r e c t l y

i nt er p r et ed a s P:-- oposit1ons and

th e

o p e r a t i o n s o f a d d i t i o n

an d

~ u l t i p l l c a t o n w i l l seen to

r e p r e s e n t

s e r i e s

and

p a r a l l e l c o n ~ e c t i o n s ~


14/72

11

TABLE I

n l o ~ u e

Between

the

Calculus

o f


and

the Symbolic

Relay

A n s l y s i s

s ymbol

x

o

X Y

=

n terpre ta t ion

in

relay

clrcu1cs

The

c 1 r c u l

X.

The c1rcl A i

1 s c losed

The c l r ru 1 i s open.

fhe

ser ies

connection of

c1reu1

s X and Y

The para l le l

connection

of c 1 r e u 1 t s X and Y

rhe

c i r e u 1

whic

h

1

S

pen

when

X

i s c l o s e d , and

c l o s e d when X i s open.

The

a t

rcu1 t s open and

c l o s e

s i m u l t a n e o u s l y .

n terpre ta t ion

in

th e

c a l c u l u s o f P ro po sitio ns

Th e p r o P o s i t i o n X.

The p ro p o sitio n 1s

fa l se

The p r o p o s i t i o n 1 s

t rue

Th e p r o p o s i t i o n which

1

s t r u e a1 the r

X o:r

Y

1s

tru e . ,

The p r o p o s i t i o n

Which

1 s

true b o t h

X an d

y a r e t rue

The

c o n t r a d i c t o r y

o f

p r o p o s i t i o n

X.

Each p ro p o sitio n

i m p l i e s th e

other


15/72

10.

12

f i r s t

note

th e fo llowing

equations:

a .

f X l

X

2

Y ::

{ l X

2

y + X r 0 , X2 X

n

-0 ,

f ,

11

I ,

b. f(Xl X

n

> [f(O,X2

Xn

+

x1l.(f(l,X2

Xn +Xi

These reduce to iden t i t i e s i f . we le t Xl aqual

ei ther

o or

1 .

In

these

equations the fUnction f is sa id to

be expanded

arout

Xl. The coeff ic ients of

X

end

Xi

1 1

in

~ r func tio ns of

the n- l

variati les X

2

and may

thus

be expanded

smu t any

o.f these var iables

in

the

same

manner. The addi t ive terms in \a:ke1so may

be

exnanded in th i s

manner.

Thus

we

get :

11. a . f(XI X

n

=X

I

X

2

f l ~ l , X 3

Xn

+

X I X ~

f(I,O,X3 Xn

+

X

1

X

2

f(O,1,X

3

X

n

+ X X ~ f O,O,X

5

X

n

b. f(Xl

X

n

) e

[Xl

+ + f O.O,X

3

X

n

]

[Xl +

X ~ + f(O, l l .X

n

] - [Xi + X

2

+ f(l.O X

n

]

lX

+

XI

+

f l , l ,X

x

)]

1 2 3

n

Continuing

th is

process n t imes we wil l arive a t the

complete

ser ies expansion haVing

the fo

I m:

1

1) X

t

X

2

X + + f O,O,O O

1 n

XIX

x

1 2

n

b.

f(XI

X

n

: [Xl

+

X

2

+

+ f O,O,O

O ]

- [X i + X

2

+ X

n

+ f(l,O O O

[Xl

+ Xl

+

x

t

+ f l , l ,

l ]

2

n


16/72

13

By 1 2 9 ,

f

1s e q u a l

to

th e

sum

o f t he produ

c ts formed

by p er mu tin g pri m es on th e

t e r m s of

X

1

X

2

in a l l

P o s s i b le ways a n d g i Vi n g e a c h p r o d u c t a coe f f i c ien t

equal to th e va lu e o f the fU.nction when

tha t

product

i s S im ilarly fo r 1 2 b .

As

an

appl icat ion

of

th e

se rie s

expansion

t

should be

n o t e d

tha t we wish

to f i nd

a c i rcu i t

represent ing

any g i

van f u n c t i o n

we ca n a l w a y s expand

th e

f u n c t i o n by

a t

t he r

lO a o r

lO b

in s u c h

a

way t ha t

an y

g iv en variable

appear s a t m ost

t w i c e ,

onc e as

a

make

c o n t a c t

and

onc e

8 S 8 b r e a k contac t T h i s

i s

shown in F i fl 4:

x

1

x

={

Fig

Simi la r ly

by

11

any

o the r

var iab le need

a p J E s r no

more

than

times two make

an d

tw o br eak c ont a c t s )

e t c .

A

general izat ion of

De Morgans theorem

i s

rep resen ted symbolically

in th e following equation:

13. [ r X l , x 2 1

. ]1.

=

f X i X ~ X r i , + )

By th is we mean

t ha t

th e n eg ativ e o f any function ma y


17/72

l ~

be obtained

b y r e p l a c i n ~

each

v a r i a b l e by i t s negat ive

and l n t a r c h

a

nging

th e and s YUlools. E X P l i c i t

and

i m p l i c i t parentheses w i l l , o f course, remain

i n the

same

plaees . For example, th e n eg ativ e of

X +

y .

(Z

fIX

w i l l be XI

y

Z X).

Soma other theorems usefUl

i n

s i m p l i y i n g

express ions e re

g1

van below:

14.

8

X

X

+ X

=

X

+ X

X

-

e t c .

-

b.

X

-

X

X

X

X

X

e t c .

-

-

15a

S .

X

XY

=

X

b.

X X y

=

X

16.

XY+

X ~

-

Y

x t ~ ye

b.

X

Y ) X f

=

X

Y)

(XI

~ } y

1 7 .

a

Xf X)

=

Xf(l)

b. X

f(X)

=X

f{O)

18.

X f(X).

=

X1f(O)

b

X

f (X)

=X

.

f l )

...

A n ~ e x p r e s s i o n

formed

w i t h the o p e r a t i o n s

o f

a d d i t i o n , m u l t i p l i c a t i o n , and n eg atio n

represents

eXPlic i t ly a

c 1 r ~ ~ 1 t containing

only ser ies

and

pa

r a l l s l

connec

tior . s

Su

ch a

c i rou1

t w i l l be

c a l l a

d

a

s e r i e s - p a r a l l e l

c i r c u i t .

Each l e t t e r

in

an axpres-

sian

of

t h i s s o r t represents a

make or

break r e l a y

conts

c t ,

o r e swi toh

blade and

conts c t

o f i n d th e

c i r c u i t r e q u i r i n g

the

l e a s t

number o f c o n t a c t s , it i s


18/72

15

t h e r e f o r e n ec es sa ry to

man ip u late th e

e x p r e s s i o n i n t o

th e form i n w h i c h th e l e a s t n u m b ~ r

o f l e t t e r s a p p e a r .

The theorems ~ i v e n above

a re always

s u f f i c i e n t to do

t h i s . A ] 1 t t l e

p r a c t i c e

i n t he m a ni pu la tio n

o f t h e s e

s;rm :o s i s

811 t h a t i s r e q u i r e d .

F o r t u n a t e l y

m o s t o f

th e

t h eo rems a r e e x ac t l y th e

same a s t h o s e o f

n u m e r i -

cal

a l ~ e o r a - - t h e

a s s o c i a t i v e

commutative, an d d i s t r i b -

u t 1 v e

laws

o f a l g e b r a

h o l d h e r e . Th e

w r i t e r

h a s found

theorems

3 , 6 , 9 1 4 1 5 , 1 6 a ,

1 7 ,

and 1 8 to be

e s -

p e c l a l l y

u s e f u l

i n

t h e

s i m p l i f i c a t i o n

o f

complex

ax-

p r e s s i an s .

AS a n examp l e

o f

th e

81

mp

f i c a t i o n

o f e x -

p r e s s i o n s

c o n s i d e r th e

c i r c u i t shoWn

i n F i ~ .

5 .

5

~

v y

. . . . .

x

F i Q; .

5

o

z

. . . . . .

0

Z

The h in d 3 r a n c e f u n c t i o n

X

ab

f o r t h i s c i r c l l i t w i l l

b e :

X

ab

=W+\\II X+Y) X + ~ H S + W + e ) ~ + Y + S V )

i\ [

=

~ + X + Y + X + ~ ) S + l + g ) g l + Y + s t V )

= W + X + y + g ~ + S V )

lthesa r e d u c t i o n s

walee

made

\ V1 tth

1 7 b

u s i n g f i r s t then X and


19/72

16

y as the

XU

o f

17b . }IO\iV mul t ip ly ing ou t :

X

ab

=

W X Y

gel ~ ~ V

: W

x y

~ S V

The c i rcu i t corresponding to

th is expression

i s

shown

in F ig . 6 . Note the la rge reduct ion

i n

the

number

o f elemen ts.

z

w Y

a _ lI O Vi

l

_ .n .o a a

Fig .

6

It i s convenient in drawing c i r c u i t ~ to l abel

a re lay

With the

same l e t t e r

as

the

hinderance

of

make contacts of the

re lay .

Thus i f a relay i s

con

neoted to

e

source

of

voltBQ:6 through

a network whose

hlnder8nce

funct ion i s

X

the

relay

and any make

con

t ec ta on i t would be labeled

X.

Break aontects would

be

labeled

XI.

This assumes t r ~ t the relay

operates

instarl t lY and tha t the make oontacts

close

end the

break

contacts

open

s1multaneousl

y . Cases in which

there

1s time delay wi l l be

t rea ted

l a t e r .

It i s also poss ib le to use

th e

analogy between

Booleian

algebra

and

re lay c i r c u ~ s

in

th9 opposite

di rec t ion i . e .

to

represent

logical re la t ions by


20/72

7

m ns of e le c tr ic c ir cu its

om

interest ing resul tz

have been

obtained

o n ~

th i s

l i ne h l t are of

no im-

portance here


21/72

18

I I I Multi terminal

end

Non aeries paral lel Circui ts

Equivalence o f

n Tetlminal

Networks

control i r r n ~ t

wil l take the

form

of

Fig

7 where

x X X are re lays or other devices controlled

1 2

by the e i r eu i t and N i s a network o f relay

contacts

and

w t ches

Fig 7

I t

i s desirable

to f ind t ransformations tha t ~ be

applied

to

N which

wil l

keep the operation of a l l

the rela:v

s

Xl X

n

the same So f r he ve only

cons ide red t rans fo rma ti on

s Which m y

e

apPl ied to

a

two ;terminel

ne twork keeping

the

opera t ion 0 one

re lay in s 3 1 ~ i e s With th is

network

the

sama To

t h i s en d we s h al l de fine

equivalence

o f two n term ..

ina 1

networks

s fol lows:

Def in i t i on :

TvvQ

n termina 1

networks

h

an

d N

wi

11

be sa id

to

be

equi

valen t

wi

th

respec t

to th e se


22/72

19

t e rmina l s

if

and only if X jk =Y jk .j, k =

1

2 :3 -. n

r 7 r ~ e r e X i s the

hinderance

on

network T e t ~ v e e n

t e rmi

jk

ne ls

and

k

and

Y

i s

tha t

fo r

between

the

co r

jk

r e s p o n i n ~ terminals .

Thus

under t h i s de f in i t i on th e

equ_ivelenc3s

o f the preceding sec t ions

were

\\ 1 th respec t to two

Star-Mesh end Delta-vVye Transformat ions .

-

As in ordi

nary

network

theory

there ex ie t

l;l:1r to me

fJh

2nd

de l t a

to vvy-e

t ransforms

t i on s .

The

de l t a

to

y t l ~ n

sforms-

t ion

i s

shown

in Fig .

8 .

These

two

n e t w o ~ k s are

equivalent

with respect to the three t e r m i n l ~ a

b,

and c ,

since by

the

d i s t r l l n t i v e

law X

ab

= R S

T

=RS

RT and

s imi lar ly for

the

o the r pai rs of

termi-

nels

a-c

end

b-c.

b-

-

b

1

R

S

RS

-

ReT

ST

c

T

Fig . 8


23/72

20

The y

to

de l

ta

t ran sformation

i s shown in

Fig

9 This

follovJS

from the f ac t

tha t X

ab

= R S :

R

S R

T

T

S .

R ~ S

a

Tot-a

c

An n po in

t

s ta r a1

so

he s a me s h e qu

1

va 1 en

t

w ith the cent ra l node el iminated

This

i s formed

axe

c t ly

8 s

in

the simple

th ree pain

t s

ta

r by

Con-

nect1ng each

pair

of

terminals

of the mesh through

8 h1nderan ce which

i s

the

sum

f

the

co

z ~ e s p o n d i n g

For n

::

5 t h i s i s s h..Q\vn i n ~ F i g 10

b

arms of

the

s t a r

b

R

e

c

F ig 10

a

c

- - - - . ~ ~

t I I ~ - - - . .

e


24/72

21

Hinderance Function of

a Non-Ser ies-paral le l

Network

The

methods of par t I I were

not

s u f f i c i e n t

to

handle

c i r c u i t s which

contained

connections o t h e r

than those

of

a

s e r i e s - p a r a l l e l t y p e .

The

bridge

o f Fig.

11

fo r

examPle

i s a

n o n - a e r i e s - p a r a l l e l

network.

These

n e t

works v i l l l e handled

y reducing

t o

en

e q u i v a l e n t


c i r e u i t .

rhree methods

have

baen

developed fo r

f i n d i n g the

equivalent

of a network

such e s the br idge .

v

s

F i g .

The

f i r s t

i s

the

obVious method

of

aPPlying

the t ransformst ions

u n t i l

the network i s o f

the


typ

an d

then

wr1 t1ng the h1nderan ce

function v inspect ion. This process i s

exact ly

th e

same a s

i s used in s i m p l i f y i n g

complex impedal1

c

e

networks.

apply

t h i s

to

the c i r c u i t of F i g . 11

f i r s t

el iminate

the

node

c

y

applying

the

s t a r

to

mesh

t ransformation t o the

s t a r

a - c

b-c d-c.

This ves the network o f

F i g . 1 2 .


25/72

22

Fig .

12

The hinderance function may be wri t ten dovvn

from

inspect ion

for t h i s network.

X

ab

= R

S)[U R T V T S }

S l m p f y i n ~

by the

theorems gives :

x = RU S

TV

SID

ab

The second

method

of anal:rs1 s

i s

to

draw

sll Pas

8

ble

paths between the

points

under oonsid-

ere t ion throu.gh

the network. These paths ere

drawn

l o n ~

the l i n e s represent ing th component hinder-. .

Bnce

eleJllents

o f

the c i r cu i t .

I f anyone of these

pa ths h8 s zero hinderen as the

requ

i red :f\ln et len

must be

zero.

Hence

the

~ e s u l t is writ ten 8S

a

product

the

hirlderanes

o f

each path

vl i l l

a

f ac to r

of t h i s

product. The required

resul t may

therefore

be

wr1 t t an

as

the product

of the hlnder

ances

o f

a l l

pass i

b le

pe

th s b9tween the

two po in t s .

P ath s whioh

touCh the

sarna po in t

more than

once need


26/72

23

no t be

con sid-3red. In F i ~ 13 t h i s

method

i s

apPl ied

to

the b r idge .

The

paths are marked in

red.

~

Fig . 13

The

:f\.lnction

1s therefore

g

van by:

X

ab

=

R

sHu

V) R

T

V U

T

S

v

:; U

SY

RTV

UTS

The

same

re su l t

is th us obta in ed as

with the f i r s t

method.

The tl1.ird

method, the

dual o f the second,

is

to draw a l l poss ib le l ines Which VJould br-e8k t

he

c i r

cui

t

between the

point s

under

cons dara

t i on

making

the

l i ne s go through th e hinderances of

the

c i rcu i t .

The ras111t i s writ ten as sum

each

term corres

pending to

a

Qdrtain

l i n e . The

sa t arms

a re

the J:Jrod-

ucts

of

a l l

the

hinderances on

the

l i ne . This

method

i s ap nlied to

the

bridge in F ~ 14, the l ines b e n ~

drawn

in

red .

b

F ig .

14


27/72


28/72

25

Sometimes

the re la t ion

ab

t

=

0 ob t a ins between

two

re lays 8

end

b. This 1s t ru e , fo r example,

in a

sequ

ent1Bl

sy stem

whe

re

ee

ch

re lay of

the

sequen

ce

1.oCks i t s e l f

in

and

a

precedes

b in

t he sequence .

Nhenever

b

i s o perated

8

i s

operated.

In such

a

case

the following s impl i f i ca t i ons may be

made:

a

b

l

= 0

Then

a

b

t

=

s , b

l

a b -

b

l

-

ab

-

eb

a b

l

=

8

-

8 I

b = 1

(a

b

l

(a

l

b ){e

-

a

l

-

a

= (e

8

=

b

Matrix

Methods.

I t i s also poss ib le to t r sa t mult1-

terminal networks by means of matr ices .

Although

use -

fu l

fo r

t h eo re t i c a l

work

the

method

i s

cumbersome

ox

prac t i ca l

problems and wi l l

th 3r

l

3fore on ly

e

br i e f ly

sketched . e sha l l

as

mma

th e

same ru19s

of m n i p u l a t ~ o n

o f m atr ic es

as

usuell-:T def ined in v/orks on

higher

a lge -

bra,

the only

difference

b a n ~

tha t the elements

of

our matr ices wi l l be

hinderance

funct ions r a t he r

than

ordina ry

a lgebra ic

numbers

o r

va r i ab le s . The

XI

matr ix

o f

8 ne two rkw i t h n nodes wi l l be de f in ed as th e fo l -


29/72

6

I

I

I

1

X

12

X

13

X

n

I

I

X

21

1

X

23 X

n

X

I

1

n

where

X j

is the negative

o f the hinder ance

common to

nodes

j

and k

Theorem:

The X matrix

o f

a network

formed

by

con

n ectin g two n

node

ne t works

p a ~ e l oorrespond-

ing n o d e s

t o g e t h s I )

i s the sum

of

tr18 1nd1 V i d u a l XI

m a t r i c e s .

This

theorem 1s

more g e n e r a l

thaD might

appear a t f i r s t

since any i n t e r c o n n e c t i o n

of

two n e t -

works

may be

oonsidered S

a

paral le l

connection o f

two ne t works wi th

th e same

numb3r o f node s

by

a ddi ng

nodes S ~

whose mutual

hinderances

to

th e o t h e r

n o d e s

i s

o n e .

ow

d efin e

s

matrix

to be

c e l l e d th e U m

t r ix

o f a

network

S

f o l l o w s :

~

U

12

Ul

n

U

21

1 U

2n


30/72

7

Wh A-rA

TIl

1 g th e n e ~ a t i v e

of

trle hinderance fu n ctio n

.- . - - jk

from node j

to

k , th e

network

considered 8 S

a

two t ~

1 n a l

c i re l l i

t .

Thus fo r the t h r e e

node

ne twork of

F i g . 16 th e

X and TIl

m a t r i c e s

a re as shown

a t th e

r igh t .

2

xl\y

1

X l

z

1 X y Z I

z x1y

x

1

y

x y z

1

y X 7 .

l ~

Z l

y

1

z x y

: f l X Z

1

z

Fig . 1 6

X

Matrix

U

Matrix

T h e o ~ e m Any p ~ e r o f th e XI matrix o f 8 network

g i v e s

a netvlork which i s e q u i v a l e n t With r e s p e c t

to 811

nodes. The matrix is r a i s e d

to a

powsr

by

th e

u s u a l

r u le f o r

m u l t i p l i c a t i o n

o f m a tr ic es .

Theorem:

I

t

I

1

U12

.

.

.

.

U

1n

1

X

12

..

.

X

1n

s

I

t

Xl

U

21

1

...

.

U

n

X

21

1

.

.

.

-

2rl

-

.

.. . . . . . . . ..

.

..

.

~

...... .

X

n

1

s ~

n-l

Theorem: An y node., s a y th e kth may be alirnina ted

l es t l i n g th e ne twork e q u i v a l e n t with respec t t o

remaining no des by

a d d i n g

to

each eleraent

X ~ s of

th e


31/72

28

Xi

m a t r i x

th e o r o d u c t o f th e

elemoo.ts

X k and X

k

a n d

r _8

s c r i k i n g

a u t

t h e

k t h r o w a n d

column.

Thu s e l i m i n a t i n t h e 3 r d node o f F i g . 16 we p;et:

L z

z

t

x z y

l y y

I:

1

x J1 z

X y 1 z l

1

The p ro o fs o f these theorems a re

o f

a simple

n a t u r e ,

ut

q u i t e

l o n ~ e n d

w i l l no t

be

g i v e n .

S p e c i a l

TyP3

s o f

R e l a y s Band SVrl

t c h e s .

I n c e r t a i n type s

o f

c i r c u i t s it i s n e c e s s a r y to p r e s e r v e d e f i n i t e

s e q u e n t i a l

r e l a t i o n

in t he o p e r a t i o n o f

th e

c o n t a c t s

o f a

r e l a y . T h i s

is

d o n e w i t h

make-barare-break

o r

con t i n u i ty an d brae

k-make

o t

t r a n s f e r )

con t a c t s .

I n

hand11np;

t h i s t y p e

o f

01 r o u t t t h e simple

s t

t h a d

seems

to

be

to assume i n

s e t t i n g u p t h e

e q u a t i o n s

t h a t

th e make

and

br eak contaots

o p e r a t e s1 mu ltan e-

o u s l y , a n d aft:3I a l l s i m p l i f i c a t i o n s

o f t h e

e q u a t i o n s

have

been

made

en d

th e

r e s u l t i n g c i I c u l t drawn

th e

r e q u i r e d ty p e o f c o n t a c t sequence i s found

from

i n -

s p e c t i o n .

R e la ys h a Vi n g a

t i m e

d e l a y i n o n s r a t 1 n g

o r

d e o n e r 8 t i n g may

be t r e a t e d s i m i l a r l y

o r

by s i t i n ~


32/72

29

tha t iu t ix is .

'rnus

'elay co i l i s

Con_naeted

to

battery through

a

hinderance

X

and the re lay has

a

de lay of

seconds

in

opere t ing and r e l ea s ing , then

th a h in deran ce fUnction

of

the

eontacts

o f the

re lay

wi l l also be

X,

but a t

a

time seconds

l a t e r . This

may

be ind ica ted

wri t ing

X t for the

hinderance in

se r i e s With

the

r e l ay ,

and X t-p) fo r

t h a t

o f the r e

l a t oontacts .

There Bre

many

spec ia l types of re lays end

sWitches fo r

pa r t i cu l a r

Plr-poses, such

as the

s t e p p n ~

switches

and

selector switches

of

various

s o ~ t s

multi-winding re lays ,

cross-bar switches, e t c . The

opera t ion of

a l l

these

types

may be descr ibed with

the words

or , and,n

i f ,

l1oparated,

and not

opera ted.

This i s a su f f i c i en t eondi t lon t ha t

may

be

desc r i bed

in

terms o f hinderance fUnctions with

the

operat ions

of

addi t ion ,

mul t iP l ica t ion , nega

t i on , end equa l i ty . Thus two w i n d i n ~ re lay might

be

const ructed tha t

t

i s operated

the f i r s t

or the second winding

i s

opera ted ac t iva ted)

and

the

f i r s t

the

seeond windings are

not operated.

Usual ly ,

however,

these

specia l

relays

occur

only

a t

the end of

a

complex a i rcu i t and

may

be omitted en

t i re ly

from

the

o alc ule tio ns to

be added a f t e r the

re s t of the c i r cu i t i s designed.


33/72

m ~ ~ ~ m e s a r e l a y X

1s

to opera te when

B c i r -

cu i t r c l o s e s and to

remain closed

i nde pe nde nt o f r

un t i l

a c i r cu i t S

opens

Suoh

c i r cu i t

i s known as

e l ock i n

c i r cu i t

I t s e q u a t i o n

i s :

X

=

rX

S

R ep lacin g

X b y X.

v e s :

= rX S

o r

X :

l

X S

In

t h i s c a sa X i s

opened when

r

closes and rem ains

open

unt i l

S

opens.


34/72

31

IV S y n t h e s i s

o f

Networks

Some

Gen er al

Theorems on }letworks and

F U n c t i o n s .

he

3 bee n shown

_that

an y furl

e t t o n may be

e xpa nde d i n a

s e r 1 e s con 81 s t 1 n

g 0

f a sum o f produ e t a , e a c h prodU a t

beinp; o f

t h e

form XlX

2

X

n

wi

t h

some p e r m u t a t i o n o f

primes

on

th e

l e t t e r s ,

and

each

p ro du ct h av in g

th e

co

e f f i c i e n t

0 o r

1 .

ow

since

each

o f th e

n

v a ria b le s

m a y o r may

n o t

have a

pr ime,

t h e r e 1 s 8 t o t a l o f 2

n

d i f f e r e n t

products o f t h i s form .

S i m i l a r l y each prod

u c t ma y have

th e

c o e f f i ~ a n t

0 o r

th e c o e f f i c i e n t

1

2

2n

t h

o t h e r e

a r e

p o s s i b l e

sums

o f

1 s S O I t .

E ach o f

t h e s e sums w i l l r e p r e s e n t a u n i q u e f u n c t i o n , b ut

some

o f t h e f u n c t i o n s may

a c t u a l l y involve

l e s s t h a n n v a r i -

a

b l e s i . e . ,

t h e y e r e of

su ch a form thQ

t fo r

one o r

more

o f

t h e

n

v a r i a b l e s ,

say

X ~

we

have

i d e n t i o a l l y

f Xl , k ~ l 0 ,

Xk+l X

n

=f X1.Xk-1J 1 , X

k

1

X

n

so

th e

t

u n d e r no

oo.ndi t n s

do

as th e va

lue o f

th e fu nc tio n

depend

on the value o f

X

k

Hence we have th e

theorem:

Theorem: The number o f fu n ctio n s

o f

n v a r i a b l e s o r

2

n

l e s s

i s

2

To

f i n d

th e number o f f u n c t i o n s

W h i c h

8 0 t u

a

l l y

i n v o l v e

n

v a r i a b l e s

we

p r o ce e d a s f o l l o w s . L e t r/ n

be

th e

r e q u i r e d

number.

Then Q th e theorem

j u s t

given:


35/72

32

where

~ :

nl /k

Hn-k) i s

the

number of comb1.nation s

of

n t h ings taken k

a t

a t ime .

S o l v 1 n ~

fo r ~ n

gives:

2n

; n = 2 ~ R);{k)

k=O

y

5Ubst1 tu t ing

fo r

, n-l)

on th e r i g h t the

s imi la r

expression found by replacing n by n -l

in

th is

equation,

x

then s imi lar ly

sUbsti tut ing fo r

~ n - 2 in the expres-

s ian thus obta ined ,

e tc ,

an

equat ion

m y be obtained

involv ing only

~ n .

This equat ion may

than

be

slm

p11

f i

ad

to t he form:

~

2

k

n

; n) :

[ k 2 -1)

]

k :

As n increases th i s

x ~ s s o n

approaches i t s

leading

term 2

2

asymptot ical ly .

The e r ro r in uSing o nly

t h i s

term

fo r n :

5

i s

l e ss

than

.01 .

e

sha l l now

determine

those fUnctions of

n

vert.s ble s which requi re tb.e mo

s t

re lay

con tac ts to re -

e l iza ,

and find th e

number

of

contacts r equ i r ed .

In

o rde r

to do t h i s , 1s

necessary

to define a

func t ion

of two var iab les krtown

s

th e sum modulo

two

o r d i s -

junct

of

the ~ l a r i e b l e s .

This

funct ion

1s wr i t t en

l

ex

2

end i s def ined

by

the equa t ion:

X l ~ X 2 =

X

X

2

X X

2


36/72

33

i s easy to sPw tha t the sum modulo two obeys the

commutat ive , asso }ia t ive ,

and the d i s t r i h l t i v e

law

with

resoect

to

mult ip l ica t ion

i . e .

x

1

2

: ~ x

X

1

eX

2

eX

3

=X

1

8 X

2

e

3

Also:

x el :

1 1

Since the sum modulo two

obeys

the

assoc ia t ive law,

we may omit

parentheses

in

a

sum

of

several terms

Without ambiguity_ The sum modulo

two

of

th e

n var1-

ables

1

n

wi l l

fo r

convenience

be

wri t t en :

n

X l e x e x e ~ = ~ X k

Theorem:

The two

funct ions

of n variables which re -

quire the most elements

re lay

contacts in a ser ies -

n n

pa r a l l e l

r ea l i za t ion Bre

~ X a n d ~ X ~ ) I , each o f wlUch

2 1

requi res

32

n

-

1

_2 elements .


37/72

This

wil l

be proved by mathematical induct ion.

F i r s t

note

tha t

t

i s

t rue fo r

n

= 2 .

There

Bre 10

fUnctions

of

2 variaQles,

namely,

r

X Y

Xty, XI+Y,

XY',

X+Y X'Y' J

XI

+Y', XY

X'Y,

XY+X'Y'. All of

these

but

the l a s t two require two elements; the l e s t

two r ~ r four elements

and ara

XfY and X8Y)

respec t ive ly . Thus the

theorem

i s t r u ~

fo r

n = 2 .

oW

8 SBuming 1 t true fo r n - l , we sha l l

prove

1

t t rue

fo ' n and thus complete the induet lon Any

function

o f

n

var iab les

may be V /rl t t an by

lOa :

l ~ o w the terms f (1 ,X

2

X

n

) and f O,Xe:> X ) are f\1nc

n .

t iona of n - l va r iab le s ,

and

t hey

ind iv idua l ly

re -

quire

the most elements

for n - l

varia

b le s , ' then f wi l l

require

the

most

elements

fo r

n

var iables ,

providing

there

i s no other method

of writ ing

f so tha t le ss

elements ere required. t ~ J e

have

assumed tha t the

most

elements for these n - l var iab les are required by

~ X k

and ~ X k f - I f we therefore su bs t i tu te for

n

f{1,X

2

-- .X

n

) th e

funct ion

and

for f{O,X

2

- _X

n

)

n

k

the

~ u n t o n

t eXk ) f

we

get:

2

n n n

f =Xl. Xk

I

X if

2Xk

t =

~ 2 X k

I


38/72

35

From

the

symetry

of th i s funct lon

there

is

no

other Vv y

o f

e x p a n d i n ~

vhich

v ~ i l l

reduce

the number of elements .

the r11nctions ere s t lbs t i tu ted in the o th er o rd er ,

w ~ t

This

oomvletes

the proof tha t

these functions require

the

most

elements- To show that

each

requires

3_2

n

_2)

elements, l e t

the

number

of

elements required

be de

noted

by s n . Then from 19) w

~ a t

the

differenoe

equat ion:

s n : 2s n-l) 2

With s 2 = 4 . This

i s

l i n ea r , ~ v t h cons tan t coe f f i

c i en t s ,

end

may

be solved by

the usua l

thods 5 .

The

solution

i s :

n - l

s n

=

3 .2 -2

a s may be ~ a s Uy verlf1 ad by

su

bst1

tu

t in g in the d l

te rence

equation

and toundary condi t ion .

Note

t hat the above only apPl ies

to 8

s s r i e s -

para l le l r e a l i z a t i on . In a l a t e r sect ion it Wil l be

n

shown

t h a t the

f l l n c t i o n ~ X k

and i t s negative may be

r es l i zed

with

4: n-l)

elements

u s i n ~

8

more

p:eneral

type

of c i r c u i t . The

fUnction

requir ing the most

elements u s i n ~

any

type

of c i r c u i t

has

no t as yet

been

determined.


39/72

Dual

l{et\vorks.

36

The n e ~ e t v e of any network m87 be

found y De ~ l o r g n l s theorem, bu ne t v ~ o r k must

f i r s t

be transformed into an eQUivalent ser ies-para l le l

c i r c u i t un less i s a l ready o f t h i s t ype . A theorem

Will

be

developed With which th e

nega t ive o f

any

planar

two-terminal c i rou i t may

be round

di rec t ly . As B

coro

l l a ry a method o f f ind ing a constsn t current 1

rcu i

t

equivalent to e

~ i v e n

constant voltage c i rcu i t and

vice

versa Wil l

be g i van.

Let

N represent a planer network of hinder

snoas , With the function

X

ab

between th e

terminals

a and b Which are on the outer

edge

of

the

network.

For def in i teness

eon sld er th e

netwo k of Fig . 17

(here

the

hinderances

are shown merely as

l ine s .

NoW

l e t

M rep rese nt th e

dual of N, as

found

the

follow np pro cess;

fo

r

as ch

c on

tour or me

sh

N

assign a n9de o r junction point of M For eaoh

element of

N

say

X

k l

s e p i r 8 t i n ~

the

contours

r a nd

s

there

corresponds

an e l ~ m e n t

X

k

connecting the

nodes

r a n d

s o f M

The

area

e x t 3 r o ~

to N i s to

be considered as tVlQ meshes, c and

d,

corresponding

to nodes

c

end

d

of

M

Thus th e dual o f F ~

17

i s

the

network of F i ~ 18.


40/72

a

m esh c

s

mesh d

Fig .

17

b

Fig .

c

Theorem: I f M

and

N

bear th i s

dua l i ty

re la t ionship

then

X

a

b =

~

To

pro

va th i S J

l t

t

he networks M end N

be

superimposed, the nodes o f

M

within

the

corresponding

meshes

of M and

corresponding elements

cross ing. For

the

network of Fig . 17 ,

th i s

1s shoWn

in

Fig. 19,

With

N in

black and

M

in

red .

Inc identa l ly the

sa s i e

s t me

thad 0 f f inding the

dual

of a ne two rk .

Whether of t h i s type

or

an 1mpedlnce nstwork

1s

to

draw the required

ne two

rk

superlmpo sed

on

t

h.e g van

networtk.

Now

i f

X

ab

: 0 ,

then

there

must

be

some

-path

fI om

to

b l o n ~ the l ines

of

N

such

th t

every

element on th i s

path

equals z ero . But th i s path

repre-

sents

a pa th across

M d1

v i

ding the c i r cu i t

from c to d

along w n i ~ every element of M 1s ona. Hence Xed =

1 .

Similar ly ,

i f Xed

=

0, then X

ab

=

1 and follows tha t

X

V

8b

-

ad-

a

Fig .

b


41/72

38

In a eonstan t-vo lt age re lay system

a l l

the

re lays

are in paral le l

across

the

l ine

To open a

relay a

ser ies conneotion 1s

o p e n ~ d The

general con-

s tant-vol tage system

1s shown in Fig.

20. In

a constant-

currant

system the re lay s

a re

a l l

in

se r ies

in the

l i ne

To

d e ~ o p e r a t e

a

re lay

t

i s

shor t c l rou i tad

The

gen-

e ra l constant-current

c i rcu i t

corresponding to

Fig. 20

i s

shown

in

F ~ 21. I f

the relay

Y

k

of

F i ~

21 is

to

be

operated

whenever

the relay

X

k

of F i ~ 20

i s

opera ted

and not

otherwi

se than eVidentl y the

hin-

der8tlCe in pa ra l l e l

wi

th Y

k

whi_ch Shorts 1.t out mus

t

be

the na ga t va

f

the hinderan

ce .

in s a r i

as vii

th

X

k

Which

connects

t

across

the

vol tage sou rc e. I f

t h i s i s

t rue

fo r a l l th e re lay s we sha l l say t ha t the

oonstant-currant and constant-voltage systems

are

equiv-

a l en t The ove theorem y be used to f ind equivalent

C ircuits of

th is sor t

For

we make the networks

N end M of Figs.

20

and 21 duels in the

sense

described

than

the

condit ion

wi l l

be

sa t i s f i ed

E

constant voltage

source.

Fig 20

Constant

I current

t

source.

~ Y

n

l

Fig 21


42/72

A simple example o f t h i s i s shown i n F i g s . 22 and

2 3 .

R

F i g . 22

R

F ~

2 3

y

, S y n t h e s i s

o f th e G e n e r a l S x m e t ~ c FUn ctio n .

As ha s

been shown any f u n c t i o n r e p r e s e n t s e x p l i c i t l y a

s e r i e s p a r a l l e l c i r c u i t . The s e r i e s p a r a l l e l ~ e 8 z 8

t i o n may r e q u i r e re e l e m e n t s J howev61 , th e n some

o t h e r

c i r c u i t

re p re se n tin g . th e same

f u n c t i o n .

I n

t h i s sectio n a m e t h o d w i l l be

g i v e n

f o r f i n d i n g a c i r -

o u i t r e pr e s e nt i ng B c e rta in type o f f Un ct io n which

in

~ e n e r a l i s much more economical o f elements

t h a n

th e

b e s t

s e r i e s p e r a l l e l

c i r c u i t .

This type

o f

fUnc-

t 1 0 n

f r e q u e n t l y a p p e a r s in r e l a y c i r c u i C s

and

i s o f

much i m p o r t a n c e .

A f u n c t ion

0

f

th e

n v a r i a b l

e s X l

X

2

.X

n

i s s a i d t o be symmetric in

t h e s e v a r i a b l e s i

any

t n t e r c h a n ~ e

o f

th e

sa

v ar ia b le s l e a ve s th e f u n c t i o n


43/72

ident ica l ly th e same. Thus

XY XZ y z i s symmetr1a

i n

th e

var iab les

X,

Y , and

Z .

Si nc e a ny p e r m u t a t i o n

o f var iab les may be o b t a i n e d y s u c ae s s iv e 1 n te rc h en g as

o f

tw o

var iab les , a n e c e s s a r y

an d

su f f i c i en t c o n d i t i o n

tha t f u n c t i o n

be

symmetric

i s tha t

any n t e r c h a n ~ e

of

two v a r i a b l e s l e a v e s th e fUnction u n a l t e r e d .

now give a theorem Which

forms

th e b e s i s

of

th e method o f s y n t h e s i s

to

be d ~ a c r b e d

Theorem:

Th e

n e c e s s a r y a nd

suf f ic ien t c o n d i t i o n

tha t 8 fUnction

be

sym metric 1 s

t ha t

t

may

be

spec1-

t ied f s ta t ing a

se t o f numbers

8 1

8

2

,

8

k

such

thB

t

i f axa

c

t y

a

j ( j

=

1 , 2 ,

:3,

k

0 f t he va r a b e s

a r e z ero ,th en th e fUnction

i s zero and n o t

o t h e r w i s e .

T h i s f o l l o w s eas i ly f ~ o m the de f in i t ion . F o r th e ex-

ample

g1 van

t h e

sa num b a r s a re 2

an d

3 .

Theorem:

There

a re

2n

+

1 symma tric

functions

o f

n

v a t - 1 a b l e s . F o r eV8 I y se lec tio n o f 8 s e t

of

numbal s

from

the

numbers 0 , 1 ,

2 ,

n

cor r esponds to on e

a n d

only

ona

s ~ ~ e t r i c

f u n c t i o n .

Since t he r e a re n+ l numbers

e a c h

o f Which ma y be a1 t he r t a k e n o r no t i n our se l ec

n + l

t1on,

th e

to ta l number o f

f u n c t i o n s

i s

2

wo o f

these

fUnctions

a re

t r iv ia l , however,

namely

the

se -

la ctia ns in

Which

none and a l l o f th e numbers a re

t a k e n .

These g i v e

th e func t ions l

and 0

respec t ive ly .


44/72

41

B

proper s e l e c t i o n o f the

v a r i a l b e s

many

a p p a r e n t l y u n s ~ r m m e t r i c

f u n c t i o n s

may t

made symmetric.

For e xamp le ,

XY Z

X YZ

1

X Y Z ,

al though

n o t

symmetric

i n

Y, end

Z, i s

symnetl io

in

X, Y, and Z .

s e t

of

~ m e r s a

l

, a

2

, sk

wil l fo r con

venience

be

c a l l e d

the 8-n'umbers

of t funct ion.

The theorems concerning comtlnations

of

symmetric

functions

ere

most

e a s i l y

s t a t e d in terms

of the

0 1 8

s

sa

S 0

f

8

-num

bar

s

For

t h i

s

rea

son

we

dena

t a

the

c l e s s of a-numbers

by

a s n ~ l e l e t t e r

A. I f two

d i f f e r -

ent

s e t s

of a-numbers are under

consideration

they

will

be denoted

by A

1

and

A

The symmetric function

of n

v a r i a ble s he

ving th e

a -num bel 'S 8

1

, 82

s k w i l l be

written Sn{a

l

, 8

2

a

k

) or

an(A).

Theorem: 3

n

(A

l

)

Sn(A

2

)

=

Sn(A

l

+ A

2

)

where

A

1

A

2

means the l06 c

a

l sum

or

the classes Al

and A

2

i.e.,

the c la ss of tho

sa numbers

which B

re members

of e i t h e r A

l

o r A

2

or

both. Thus 36(1,

2,

3 ) . 8

6

(2 , 3 , 5)

i s equa 1 t S6 1 , 2 , 3 , 5).

Theorem: 3

n

(A

l

) + 5n(A

a

)

Sn(A

1

,A2)

where

AlwA

2

i s

the

l o g i c a l

product

of

the

mlassas

i . e . ,

the

a la s s 0 f

numbers Which

are

common

t o

A

1

and

A2.

Thus

5

6

1 ,

2 , 3) + S6 2 , 3 , 5) C 3 6 2 , 3 ) .

These

theorems follow from the f a c t t h a t 8 product i s


45/72

42

zero

i

e i ther factor i s zero, while 8 sum i s zero

only

i both

terTI1S

are zex o. The negs t i

va

of

8

se t o f

a

-numbers

wi l l be \ I1 1tten

AI

and

meBns

the

c la ss o f

a l l

the numbers

from

to n 1nclus i

va which

a

re

not members

o f

A.

Thus

i A

i s

the se t of

numbers 2 ,

3 , and 5, and n 6 then

AI i s the set of numbers 0 , 1 , ~ and 6.

Theorem:

These thaorams

are useful

i

several

symmetric

functions

are

to

be

obtained

simultaneously

Before w

study the synthesis

of

8 network for

the

general

symmetric fUnction consider

the

c i rcu i t 8-b

of

Fig .

2 ~

Th1 s c i rcu i t

represents 33 2 .

L o X ~

2

L

:n

3

The

l ine o m ~ g

in s t a f i r s t

encounters

a

pa i r o f

h1nderancas

Xl

and xl I f

Xl = 0, the

l ine i s switched


46/72

up to th e l evel

marked

1 meaning tha t 1

o f

th e var1eQles

1 s

z e r o .

I f Xl

=

1 th e l ine stays

on th e leve l marked

0 ;

n e x t

h i n d e r a n a a s

X

2

and

~

Bra

e n c o u n t e r e d .

I f

X

2

i s ~ e r o th e l ine is

switched

up

a leve l ;

i not it

s tays a t

the

same

l e ve l .

Fir ls l ly r e a c h i n g th e r igh t

hand

s e t o f

te rmina l s

the

l i ne h a s

been

s w i t c h e d u p

to

l eve l

represent ing

th e

number o f var iab les w h i ch

;

a re

dqua]. to

ze ro .

T e r min a l

b 1 s

co n n ected to leva l

2

an d

therefore

th e c i r cu i t a b

wi l l be comPleted i

and only i 2 o f the v a r i a b l e s

are

z e r o .

Thus

th e

funct ion 2 i s r e p ~ e s e n t e 3 3 0 , 3 )

h ad been

des i red t e rminal b would be c o n n e c t e d to

b o t h

l eve l s

o end

3 .

I n

f igure

24 cer tain o f th e

elements

Bre

e v l d e n

t ly

Sll p e r f l u o u s . The c1 r o u t t may

be

s impl i f ied

to th e

form o f F ig . 2 5 .

Fig .

25

For th e general function e x a c t l y the same

m e t h o d

i s follov Jed. U s i n g th e genera l CirC1 1it fo r n

b


47/72

var ia

bles

o f

Fig. 26, the

te rminal

b

i s cOllnected

t

the

l eve l s

c o r r e s ~ o n d i n g

to the a-numbers of the

desired

svmmetric

funct ion.

In

Fig . 26

th e h in de ra nc es

Bra

represented

by

simple l i ne s ,

and

the

t t ~ s

are omitted

from the

c i r cu i t ,

u

the

hinderance of

each l ine may

eas i l y

be seen y gene r a l i z i ng

Fig .

24:.

NOTE:

All

s loping

l i n e s

have hinderance of the

var i a ble \vri t ten below;

hor izon ta l l i nes

have

negative

of

th is

hinder-

ance.

~ b

to

a

numbers

o

n-:-l)

n

a

Fig . 26

Aftar terminal b i s

connected,

a l l

superfluous ele-

ments

may be

deleted.

In

cer ta in

c ses

i s

poss i b le

to grea t ly

s impl i fy

th e

c i r cu i t

y sh i f t i ng the

l eve l s

down.

Supnose the rbnction 3

6

0 ,3 ,5 1s

des i red .

Ins tead

of continuing

the c i rcu i t up to the 6th

l eve l ,

we


48/72

oonnect the

2nd l eve l

ts down to the zero l eve l

as

shown in F ig

27. The

zero l eve l

then

a lso becomes

the 3rd

leve l and

the 6th

l eve l

2 5

1 4

0 3 6

a . j . . . . . ~ I 1 1 ~ ~ b

th t e rmina l

b

connected to th is le ve l

we have r e a -

l1zed the

function

with a

great

saving

of

elements.

El iminat ing unnecessary elements tIlo c i r c u i t o f Fig 28

1s

obtained.

This deV ice 1 s e 3p ec ia l ly usefu l

i

the

8-numbers form an r i t ~ m e t c progression,

although

it

can

sometimes

be applied

in

other cases

The fUnctions

n n

l :2

X

k end

2

X

k

1 Which were shown to

require

the most

io

elements for a s e r e s ~ p r l l e l real izat ion

have

very

simple c i rcu i t s when developed in t h i s

mann,er.

t

n

Jan

be

eas i ly shown

t h a t

i n i s

even,

then :2Xk i s

the

symmetric function

with a l l

the even numbers

fo r

a-numbers ,

i

n

i s

odd

it

has

a l l

the

odd numbers

n

fo r a-numbers . The funct ion ~ k i s o f course ,

jus t the opcos1te.

Using the sh i f t i ng down process


49/72

46

the

c i rcui t s

are- as shown

in Fig.

29.

~ ~ _ w : ~ _

n

~ X

k

for n odd;

1

~

b

Xl X

3

xn X

n

n n

l:2

X

k

for n even; 1;2

Xk 1 fo r

n odd

1 1

Fig

These

circui ts

each

require

4{n-l

elements. They

wi l l be

re co gn iz ed a s

the f ami l i a r

c i r cu i t

fo r con

t ro llinJ2; a 11 ght from n pain

t s ; I f

a t

an:,

one of

the

points the posi t ion

of the switch

1s changed,

the

to ta l number

of

vAriables which

e q u l ~

~ e r o i s changed

by one, so t h a t i

the

l igh t 1s

on,

it will- be turned

o ff end i a l r ~ a d ~ r o f f it wi l l be

turned

on


50/72

47

The g e n e r a l

n etwo r k o f Fig .

2 6 c o n t a i n s

n n

1

elemen ts. t can be shown t ha t fo r any ~ i v e n s ~ l e c t i o n

o f

a-numbers a t l ea s t

n o f

the e l e m e n t s wil l be super

f l u o u s . I t f o l l o w s t ha t any symmetric

f u n c t i o n

o f n

var iab les

can be raa l ized With a t most n

e l e m e n t s .

E q u a t i o n s from Given o p e r a t i n ~ Charac te r i s t i c s .

I n

gen-

e ra l

t h e r e i s a c e r t a i n

s e t

o f i n d e p e n d e n t

variables

A,

B

a Which

may

be s \n tches e x t e ~ l e l l o p e r a t e d

or protect ive

re lays .

Thera

i s

also

a

se t

o f

d e p e n d ~ n t

v a r i a b l e s

x

y , z

Which

r e p r e s e n t re lays

motors or

o the r d e v i ~ e s

to be cont ro l l ed by

th e c 1 rc u i

t .

t 1 s

r e q u i r e d

to

f ind a n et wo rk

which

~ v a s fo r e a c h poss ible

c omta ne tion of v a lu e s o f

th e

independent var iab les th e

cor rec t v al u e

s

fo r a l l th e d e p e n d e n t va r i ab le s . Th e

f o l l o w i n ~

pr inc ip l e s

g i

va

th e

genera l

m e t h o d

o f

s o l u -

t ion .

1 .

A d d i t i o n a l de pe nde nt v a r i a b l e s

must be

i n t r o d u c e d fo r e a c h

a d d e d

p h a s e

o f opera t ion of 8

sequen t i a l s y s t e m .

Thus i

it

i s des i red to

c o n s t r u c t

8

syste m

wh i ch op era tes in th ree s teps tw o addi t iona l

v 8 r i a b l e s must be in tro d u ced to r e p r e s e n t t h ~ b e g i n n i r ~

of the l a s t t\ Vo st 3ps T h e se add i t i ona l var iab les

may r e p ~ e s e n t c o n t a c t s on

a

s t e p p i n g s w i t c h

o r

r e l a y s

Which l o c k in

sequentia l ly . Simi la r ly

e a c h

r e q u i r e d

t ime

d e ~ T

wl11

reqt1.ire

a nevv va r i ab l e represent ing


51/72

8

a t i m e delay relB;T o f s o n a so r t

O t h e r

fo rm s o f r e l ays

which may

be u s c e s s a r y wi l l usua l ly be obvious from

the

nature

o f the p r o b l e m .

2 . Th e h i n d e r a n c e

equat ions

fo r each

of the

dependent

var iab les s h o u l d

o

be writ ten down. These

t ~ n c t i o n s may i n v o l v e

any

o f

th e

var iab les , de pe nde nt

or i n d e p e n d e n t

i n c l u d i n g th e

var iab le whose

fUnction

i s

b ein g

de t e rmi ne d

a s ,

for

ex amp le in a l o ~ k in

c i rCUi t ) . Th e c o n d i t i o n s ma y be ei ther c o n d i t i o n s

fc\r

o p e r a t i o n or

fo r

n o n - o p e r a t i o n . E q u a t i o n s are

w r i t t e n from o p e r a t i n g

charac te r i s t i c s

c c o r d i n ~ to

T a b l e I I . T i l l u s t r a t e th e

use

o f t h i s

t ab le s t lP -

pose a r e l a y A i s to

o p e r a t e

i x 1s o p ~ r t e d a n d y

or

z

i s operated

end x o r w

o r

z

1s

n ot o pe ra te d.

The

e x p r e s s i o n fo r

A wi l l 1 8:

A

=

x yz

X W Z I

Lock

in

r e l ay e q u a t i o n s h av e a lr -g ad v been d i s o u s s e d .

t does no t , o f c o u r s e

matter

i the

seme c o n d i t i o n s

are

t

in t e e x p r e s s i o n more

t ha n once - - s l l

s u p e r

f luous

mater ia l

wi l l d i s a p p e a r in th e f in a l s im p lif i

ca t ion .

3 .

The

e x p r e

s

s i a n

s

fo r

th e

va

r ious

d e p e n d e n t

var iab les sh o u ld n e x t

be

s impl i f i ed

a s

much

s

poss ib le

by means o f

th e

t he or e m s on

m a n i p U l a t i o n o f

these q u a n

ti t i e s . Jus t ho\v

mtlch t h i s 0811

be

done

d ep en d s somewhat


52/72

T LE

I I

R E L T I O ~ OF OPERATlllG F ~ R T E R I S T I S

tTD

EQUArIOrIS

Symool

X

XI

+

,

In Terms o f o p e r a t i o n

The switch or r e l a y

X is 0 pe ra ted .

I f .

The switoh

o r

r e l a y

X

i s

no t

o p e r a t e d .

Or

And.

The

c i rcu i t

- -

i s n o t

c l o s e d o r

a p p l y De

Morgan s Theorem.

In Terms o f Non- oper ation

The switch

or re la y

X

s

n o t

o p e r a t e d .

I f .

The swi tch o r relay X

s

0

ra

t ad .

And.

O r.

Th e c i rc u it - - i s

c l o s e d o r apply De

Morgans Theorem.


53/72

5

on the in g n u ~

the de s1

er

4

The resul t ing c i rcu i t should now be

drawn Any

necessary

addit ions

dic ta ted

pract ica l

oonsidera t ions such as

current

oarrying ab i l i t y se -

quence of contact operat ion e t c shculd be made


54/72

51

v

I l lus t r a t ive

Examples

In

th i s sect ion several problems

wil l

be

so lv ed with

the methods

whiah

have

been

developed.

he examples are

intended

more to show the versa t i l -

i t y of

relay

and sWitching

c i rcui t s and to

i l lus t ra te

the

us e

o f

the ca lcu lus in

aotual

pro blems

than

to

de

s -

cr ibe

prac t ica l

devices.

i s

p oss ib le to ~ r o r

complex mathematical

or;>eretions

y

means o f re lay c i r cu i t s . lTumbers

may e

represented

the

pos i t i ons

of

relays

or s t epp ing

sWitches ,

and i n t e rconnec t ions

between

se t s

o f

re lays

can

be

m de

to

represent various mathematioal opera

Documents

A Symbolic Analysis of Relay and Switching Circuits, Shannon C.E., 1936