LAMINATED TRANSMISSION LINES. I 1)21 



froquoncy Inuul. In practice, too, the noeo product of the main dioloctric 

 must be held very close to the Clojiston value or the benefit of the lai-jic 

 effective skin depth is lost; and the stacks must be extremely uniform oi- 

 ajiain the depth of penetration is fj;reatly reduced. We shall take up all 

 these matters in later sections, and shall see that while the n^sulls just 

 given represent ultimate limits of performance, the practical impioxe- 

 ments which can be achie\-ed o\'er conv(Mitional cables depend upon the 

 degree to which one can solve the manufacturing ])roblems that tend to 

 make every actual Clogston cable differ morc^ or less from the ideal striic- 

 tuvc considered above. 



V. EFFECT OF FINITE LAMINA THICKNESS. FREQUENCY DEPENDENCE OF 

 ATTENU.\TION IN CLOGSTON 1 LINES 



The principal effect of finite lamina thickness in a Clogston cable is to 

 introduce a frequency dependence into the propagation constant, and 

 in particulai" to cause the attenuation constant to increase, with increas- 

 ing frequency, above the value which we have found for infinitesimally 

 thin laminae (or for finite laminae at low frequencies). The increased 

 losses are associated with the fact that the penetration depth in a lami- 

 nated stack decreases with increasing freciuency, even when Clogston's 

 condition is exactly satisfied, if the laminae are of finite thickness. We 

 shall now obtain expressions for the surface impedance of a plane lami- 

 nated stack of n double layers, such as is shown in Fig. 3, when Clogston's 

 condition is satisfied but the individual layers are of finite thickness. 



We first observe that Clogston's condition (102) implies 



. r. 0M1 + (1 - ^)m2' 



= tCOUo i — ~ — 



L (1 - ^)M2 . 



= —io^md = —vicTih 



(1 - d)h 



6 (15G) 



where in the last step we have used the fact that in the conducting 

 layers rjiy is equal to 771 and ki is equal to ai to a very good approximation. 

 We now introduce the dimensionless parameter 



= a,h = (1 + i)h/8i ^ K,U , (157) 



which may be regarded as a measure of the electrical thickness of the 

 individual conducting layers. From (86) and (156) we have, for the prop- 

 agation constant per double layer, 



