LAMINATED TRANSMISSION LINES. I 



923 



where we have expanded coth by D wight 657.5 and chosen the sign 

 of the square root to make Re Ki and Re Ko both positive. 



Our first observation is that when the lamina thickness is finite the 

 effective skin depth of the stack is also finite. We have, from (157) and 

 (163), 



,4 



r = -L r^ + i^ 



V3 L^i 155 J 



8^1 

 5255? 



(165) 



niid (he average propagation constant pvv unit distance into tlic stack is 



r 1 



i\ = 



^tl 



ih + ^>) Vsdi + /o) 



+ 



ill 



M 1561 



5256"i 



(166) 



If as usual we define the effecti\'e skin depth A to be the distance, meas- 

 ured into an infinitely deep stack, at which the current density has fallen 

 to 1/e of its value at the surface, then keeping only the first term in (166) 

 we have 



. 1 Vsih + t2)8l V3(^i + t2) .._. 



A = :=; = 2 = 72 — J ^^"^^ 



Re r^ ^1 TTiJugifti 



a result also given by Clogston.^^ The number A'' of double layers in one 

 effective skin depth is 



A V35? VS 



N = 



T^Mifk 



ih + ^2) tl 



while the total thickness Ta of conducting material in these layers is 



(168) 



Ta = Nh = 



^1 



TruiQift] 



(169) 



T^ is essentially the thickness of conducting material in each stack which 

 is effectively carrying current; it is evident that for small values of ^1/61 

 this efTective thickness is inversely proportional to the frequency / and 

 to the thickness h of the individual conducting layers, but independent 

 of the thckness /2 of the insulating layers, provided that <2 is very small 

 compared to the length of a free wave in the insulating material. 



In the general case, still assuming of course that Clogston's condition 

 is satisfied, the surface impedance Zoijo) of a plane Clogston stack is 

 given by equation (65) of Section III, namely 



Zoijo) = 



|Zn(7o)(A>"" + AV"') + K,K, sh nr 

 Z„(7o) sh nr + UKif"^ + ^26""^) 



(170) 



'^Reference 1, equation (III-44). 



