42

ANALYSIS METHODS

[CHAP. 4

Fig. 4-6 Fig. 4-7

The´venin too). The output resistance is found by dividing the open-circuited voltage to the shortcircuited current at the desired node. The short-circuited current is found in Section 4.6.

4.6

TRANSFER RESISTANCE

A driving voltage in one part of a network results in currents in all the network branches. For example, a voltage source applied to a passive network results in an output current in that part of the network where a load resistance has been connected. In such a case the network has an overall transfer resistance. Consider the passive network suggested in Fig. 4-7, where the voltage source has been designated as Vr and the output current as Is . The mesh current equation for Is contains only one term, the one resulting from Vr in the numerator determinant:      rs Is ¼ ð0Þ 1s þ    þ 0 þ Vr þ 0 þ  R R The network transfer resistance is the ratio of Vr to Is : Rtransfer;rs ¼

R rs

Because the resistance matrix is symmetric, rs ¼ sr , and so Rtransfer;rs ¼ Rtransfer;sr This expresses an important property of linear networks: If a certain voltage in mesh r gives rise to a certain current in mesh s, then the same voltage in mesh s produces the same current in mesh r. Consider now the more general situation of an n-mesh network containing a number of voltage sources. The solution for the current in mesh k can be rewritten in terms of input and transfer resistances [refer to (7), (8), and (9) of Example 4.4]: Ik ¼

V1 Vk1 Vk Vkþ1 Vn þ  þ þ þ þ  þ Rtransfer;1k Rtransfer;ðk1Þk Rinput;k Rtransfer;ðkþ1Þk Rtransfer;nk

There is nothing new here mathematically, but in this form the current equation does illustrate the superposition principle very clearly, showing how the resistances control the eﬀects which the voltage sources have on a particular mesh current. A source far removed from mesh k will have a high transfer resistance into that mesh and will therefore contribute very little to Ik . Source Vk , and others in meshes adjacent to mesh k, will provide the greater part of Ik .

4.7

NETWORK REDUCTION

The mesh current and node voltage methods are the principal techniques of circuit analysis. However, the equivalent resistance of series and parallel branches (Sections 3.4 and 3.5), combined with the voltage and current division rules, provide another method of analyzing a network. This method is tedious and usually requires the drawing of several additional circuits. Even so, the process of reducing

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An