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The 1,2-element of the Z-matrix is de ned as: X Z12 (impedance common to I1 and I2 where a summand takes the plus sign if the two currents pass through the impedance in the same direction, and takes the minus sign in the opposite case. It follows that, invariably, Z12 Z21 . In Fig. 9-12, I1 and I2 thread ZB in opposite directions, whence Z12 Z21 ZB Similarly, Z13 Z31 Z23 Z23 X X (impedance common to I1 and I3 0 (impedance common to I2 and I3 ZD

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If two sides of a triangle are congruent, the angles opposite these sides are congruent. (Base angles of an isosceles triangle are congruent.)

The Z-matrix is symmetric. In the V-column on the right-hand side of the equation, the entries Vk (k 1; 2; 3) are de ned exactly as in Section 4.3: X (driving voltage in mesh k Vk where a summand takes the plus sign if the voltage drives in the direction of Ik , and takes the minus sign in the opposite case. For the network of Fig. 9-12, V1 Va V2 0 V3 Vb

Fig. 3-12

(b) Similarly, the integral in (1.21) is no more than the area under the f 2 t curve. Hence, 1 2T T 1=2 4 12 4:25 A T 2 2

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Instead of using the meshes, or windows of the (planar) network, it is sometimes expedient to choose an appropriate set of loops, each containing one or more meshes in its interior. It is easy to see that two loop currents might have the same direction in one impedance and opposite directions in another. Nevertheless, the preceding rules for writing the Z-matrix and the V-column have been formulated in such a way as to apply either to meshes or to loops. These rules are, of course, identical to those used in Section 4.3 to write the R-matrix and V-column.

PRINCIPLE 2:

This result was to be expected, since the average value of a sinusoid over one cycle is zero. Equation (1.21) and the identity sin2 x 1 1 cos 2x provide the rms value of i t : 2 I2 1 2 2

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EXAMPLE 9.6 Suppose that the phasor voltage across ZB , with polarity as indicated in Fig. 9-13 is sought. Choosing meshes as in Fig. 9-12 would entail solving for both I1 and I2 , then obtaining the voltage as VB I2 I1 ZB . In Fig. 9-13 three loops (two of which are meshes) are chosen so as to make I1 the only current in ZB . Furthermore, the direction of I1 is chosen such that VB I1 ZB . Setting up the matrix equation:

Fig. 3-13

Fig. 9-13

2 1 50 66!t 80 cos !t sin 2!t 66 2 2 !t 0 p so that I 66 8:125 A:

PRINCIPLE 3:

Input and Transfer Impedances The notions of input resistance (Section 4.5) and transfer resistance (Section 4.6) have their exact counterparts in the frequency domain. Thus, for the single-source network of Fig. 9-14, the input impedance is Zinput;r Vr z Ir rr

1 2

Principle 3 is a corollary of Principle 1. A corollary of a theorem is another theorem whose statement and proof follow readily from the theorem.

where rr is the cofactor of Zrr in z ; and the transfer impedance between mesh (or loop) r and mesh (loop) s is Ztransfer;rs where rs is the cofactor of Zrs in z . V r z Is rs

Fig. 3-14

Fig. 9-14

PRINCIPLE 4:

As before, the superposition principle for an arbitrary n-mesh or n-loop network may be expressed as V1 Ztransfer;1k Vk 1 Ztransfer; k 1 k Vk Zinput;k Vk 1 Ztransfer; k 1 k Vn Ztransfer;nk

Find the rms (or e ective) value of a current consisting of the sum of two sinusoidally varying functions with frequencies whose ratio is an integer.

Fig. 3-15

CHAP. 9]

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