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[CHAP. 4

Similarly,  R 1  11 R I2 ¼ R  21 R31

V1 V2 V3

 R13  R23  R33 

 R 1  11 I3 ¼ R R  21 R31

R12 R22 R32

 V1  V2  V3 

An expansion of the numerator determinants by cofactors of the voltage terms results in a set of equations which can be helpful in understanding the network, particularly in terms of its driving-point and transfer resistances:       11 21 31 I1 ¼ V1 þ V2 þ V3 ð7Þ     R  R  R 12 22 32 þ V2 þ V3 ð8Þ I2 ¼ V1 R R R       13 23 33 þ V2 þ V3 ð9Þ I3 ¼ V1 R R R Here, ij stands for the cofactor of Rij (the element in row i, column j) in R . signs of the cofactors—see Appendix B.


Care must be taken with the


The network shown in Fig. 4-4(a) contains five nodes, where 4 and 5 are simple nodes and 1, 2, and 3 are principal nodes. In the node voltage method, one of the principal nodes is selected as the reference and equations based on KCL are written at the other principal nodes. At each of these other principal nodes, a voltage is assigned, where it is understood that this is a voltage with respect to the reference node. These voltages are the unknowns and, when determined by a suitable method, result in the network solution.

Fig. 4-4

The network is redrawn in Fig. 4-4(b) and node 3 selected as the reference for voltages V1 and V2 . KCL requires that the total current out of node 1 be zero: V1  Va V1 V1  V2 þ þ ¼0 RA RB RC Similarly, the total current out of node 2 must be zero: V2  V1 V2 V2  Vb þ þ ¼0 RC RD RE (Applying KCL in this form does not imply that the actual branch currents all are directed out of either node. Indeed, the current in branch 1–2 is necessarily directed out of one node and into the other.) Putting the two equations for V1 and V2 in matrix form,