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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
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; \
~
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
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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.
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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
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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
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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
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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
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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 .
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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
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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
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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 .
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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
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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
P r o p o s i t i o n s
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 ~
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11
TABLE I
n l o ~ u e
Between
the
Calculus
o f
P r o p o s i t i o n s
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
s e r i e s - p a r a l l e l
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
s e r i e s - p a r a l l e l
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 .
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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
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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
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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 -
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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
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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
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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 ~
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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.
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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.
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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:
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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
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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 .
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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
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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.
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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.
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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
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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
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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
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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 .
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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