ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 563 



Internal Impedances of Laminated Conductors 



So far we have supposed that all conductors were homogeneous. 

 We shall now consider a somewhat more general conductor composed 

 of n coaxial layers of different substances. As before, we are interested 

 in finding expressions for the internal impedances; besides, we may 

 wish to know how the total current is distributed between the different 

 layers of the conductor. 



To begin with, let us suppose that a coaxial return path is provided 

 outside the given conductor. We number our layers consecutively and 

 call the inner layer the first. Let Z«a^"'^ and Zbi'^'^'^ be the surface 

 impedances of the mth layer, the first when the return is internal, the 

 other when it is external ; and let Zgh^'^^ be the transfer impedance 

 from one surface to the other. Formulae for these impedances have 

 already been obtained in the section under "The Surface Impedances 

 of Hollow Cylindrical Shells," page 552. Also, let z^-^^'^^ be the surface 

 impedance of the first m layers with external return ; that is, the ratio 

 of the longitudinal electromotive intensity at the outer surface of the 

 mth. layer to the total current Im in all m layers. ^^ 



By hypothesis, there is no return path inside the laminated con- 

 ductor as a whole. Hence, when we fix our attention on any one 

 layer alone, say the mth, we may say that the current in this layer 

 returns partly through the m — \ layers within it, and partly outside. 

 In the w — 1 inner layers, however, the current is assumed to be 

 Im-i in the outward direction — or what amounts to the same thing 

 — Im-i in the return direction. Hence we conclude that, of the 

 current Im — Im-\ in the layer under discussion, Im returns outside 

 and — Im-i inside. Substituting these values in Theorem 2 on 

 page 555, we find that the electromotive intensity along the inner 

 surface of the layer is Zab^'^^Im — ZaJ-'^^Im~\- 



But the inner surface of the wth layer is the outer surface of the 

 composite conductor comprising the m — \ inner layers, and by 

 Theorem 1, the electromotive intensity on this outer surface is 

 Z66^"'~^^/m-i. As the two must be equal, we obtain an equation from 

 which we can determine the ratio of the current flowing in the first 

 m — \ layers to that flowing in m layers. This is 



h^ ^ ^I^lI . . (92) 



In this formula for the effect of an extra layer on the current dis- 



28 In this notation, the current flowing in the mth layer is /,„ — /,„_i. It should 

 also be noted that Zh6<i> = zw'^^ 



