An Introduction to Electrical Circuit Theory by G. Williams (auth.)

By G. Williams (auth.)

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8 Graphs of dual circuits G METHODS OF ANALYSIS AND NETWORK THEOREMS 3I The dual situation is that the conductance of 5 S has a voltage of 3 V across it and a current of IS A through it. Thus the power dissipated is G 2 V which is 45 W. Consider the circuit shown in fig. 8(a). Suppose it is necessary to construct its dual. By observation the network has three meshes (A, B, and C); ten branches (1-I 0) and 7 independent nodes (a, b, c, d, e, f and h with reference to the ground node g). The dual network must therefore consists of three independent nodes (A, Band C with reference to a ground node G); ten branches (1-IO) and seven meshes (a, b, c, d, e, f and h with an outer space g).

To replace a delta by an equivalent star the star resistances are product of adjacent ~ resistances . , R 13 and R-y then R CJ. = Rc 'Y which may be summarised as follows. To replace a star by an equivalent delta the delta resistances are sum of the products of the Y resistances taken in pairs . _the opposite Y resistance Similar results may be obtained if the elements are conductances. An important result occurs when the resistances of either the Y or Ll are equal. Substituting RCJ. /3 Thus the equivalent star resistances are !

If now a branch 3 is added between nodes a and b no further nodes are added but an independent loop is created, loop 312. 3 Construction of a topological graph branches 4, 1, 2 and 4, 3. These loops, however, are not independent because loop 4, 1, 2 can be obtained by adding loop 1, 3, 2 to loop 4, 3. Hence the addition of branch 4 adds only one independent loop. It now becomes clear that every time a branch is added either a new node or an independent loop is created. Thus, apart from the ground node, the number of branches must equal the number of nodes plus the number of independent loops and this is the basic theory of topology, that is b=(n-1)+!

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