1 1 1 þ þ 6 RA RB RC 6 4 1 RC
32 3 2 3 1 Va =RA V1 76 7 6 7 RC 76 7 ¼ 6 7 5 1 1 1 54 5 4 þ þ V2 Vb =RE RC RD RE
Note the symmetry of the coeﬃcient matrix. The 1,1-element contains the sum of the reciprocals of all resistances connected to note 1; the 2,2-element contains the sum of the reciprocals of all resistances connected to node 2. The 1,2- and 2,1-elements are each equal to the negative of the sum of the reciprocals of the resistances of all branches joining nodes 1 and 2. (There is just one such branch in the present circuit.) On the right-hand side, the current matrix contains Va =RA and Vb =RE , the driving currents. Both these terms are taken positive because they both drive a current into a node. Further discussion of the elements in the matrix representation of the node voltage equations is given in Chapter 9, where the networks are treated in the sinusoidal steady state. EXAMPLE 4.5 Solve the circuit of Example 4.2 using the node voltage method. The circuit is redrawn in Fig. 4-5. With two principal nodes, only one equation is required. currents are all directed out of the upper node and the bottom node is the reference,
V1 20 V1 V1 8 þ ¼0 þ 5 2 10 from which V1 ¼ 10 V. Then, I1 ¼ ð10 20Þ=5 ¼ 2 A (the negative sign indicates that current I1 ﬂows into node 1); I2 ¼ ð10 8Þ=2 ¼ 1 A; I3 ¼ 10=10 ¼ 1 A. Current I3 in Example 4.2 is shown dotted.
INPUT AND OUTPUT RESISTANCES
In single-source networks, the input or driving-point resistance is often of interest. Such a network is suggested in Fig. 4-6, where the driving voltage has been designated as V1 and the corresponding current as I1 . Since the only source is V1 , the equation for I1 is [see (7) of Example 4.4]: 11 I1 ¼ V1 R The input resistance is the ratio of V1 to I1 : Rinput;1 ¼
The reader should verify that R =11 actually carries the units . A voltage source applied to a passive network results in voltages between all nodes of the network. An external resistor connected between two nodes will draw current from the network and in general will reduce the voltage between those nodes. This is due to the voltage across the output resistance (see