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By G. Williams (auth.)

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Again, it is also possible to reduce the number of voltage variables by expressing some voltages in terms of others. This is done by writing Kirchhoffs voltage law for a loop and using the resulting equation to express one of the loop voltages in terms of the others. It is therefore possible to eliminate as many voltage variables as there are independent loops and the number of voltage variables required to solve the network is reduced from b to b - !. The basic law of topology states that b - l = n - 1 and thus the number of voltage variables required is equal to the number of independent nodes (that is, n - 1) and a similar number of equations is required for a solution.

4. Similarly, the equations relating the branch currents to the mesh currents from one network are identical to the equations relating the branch voltages and node voltages in the other. Thus dual elements in the dual networks obey the same law and the electrical behaviour of one of two duals may be obtained from the other by interchanging voltages and currents. The simplest pair of dual elements are resistance and conductance. Thus if a resistance in a network has a value of 5 n its volt-ampere equation is v = 5i The dual element in the dual network has a conductance of 5 S and its volt-ampere equation is i = 5v If the resistance of 5 n has a current of 3 A flowing through it then the voltage across it is 15 V.

The topological graph of a network is a 'line and point' diagram depicting how the nodes of a network are interconnected by the branches. Hence the circuit shown in fig. 11 has the graph shown in fig. 2. h Graph of circuit shown in fig. 11 reveals the relationship which exists between the branches, nodes and independent loops of the circuit. Consider the construction of such a graph as shown in fig. 3. Beginning with the ground (or reference) node g and adding to it branch 1 a second node a is added.

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