A Symbolic Analysis of Relay and Switching Circuits

m.esh c
s
c
37
a
b
mesh d
Fig. 17
Fig.
Theorem:
If M and N bear this duality relationship,
then Xa b = X~d·
To pro va thi S J let t he networks M end N be
superimposed, the nodes of M within the corresponding
meshes of M and corresponding elements crossing.
the network of Fig. 17, this 1s shoWn in Fig. 19,
With N in black and M in red. Incidentally, the
For
sa siest me thad 0 f finding the dual of a ne two rk .
(Whether of this type or an 1mpedlnce nstwork) 1s to
draw the required ne two rk superlmposed on t h.e
g1. van
networtk. Now, if' X ab
: 0, then there must be some
-path fI'om 8 to b alon~ the lines of N such that every
element on this path equals zero.
But this path represents
a
pa th across M d1 viding the circuit from c to d
along wni~h every element of M 1s ona. Hence Xed =1.
Similarly, if Xed =0, then X =1, and it follows that
ab
X8b -
-VI
"''"''ada
b
Fig.