REDUCTION OF SKIN EFFECT LOSSES 509 



ductors yet large compared to the effective skin depth 8^. In the following 

 sections, we will deal with several situations in which the stacks are much 

 smaller in depth than 8^. The size of the stack, then, reflected in the imag- 

 inary part of the propagation constant k has more effect on a than anything 

 else and we may consider W/8 and, all the more, Cy to be zero. 



Under these conditions, we shall calculate the attenuation of the parallel 

 plane transmission Hne shown in Fig. 7. In that figure we have two parallel 

 plates or shields of conductivity a separated a distance 2d. Inside each 

 plate there is a thickness 5 of laminated conductor of average conductivity 

 ff and average transverse dielectric constant e. The interior of the line is 

 filled with a dielectric of thickness 2h = 2{d — s) and having a dielectric 

 constant €i . The calculations to be made will be valid down to some low 

 frequency at which the skin depth in the outer shields becomes equal to 

 their thickness. 



With reference to equations (II-8) and (IT9), we can write down the 

 following expressions for the fields in the various parts of the line: 



In shield: H. = Ae-"^'''^ (IV-1) 



E.^-A'^ e-'^'-'^ (IV-2) 



rj = ^/uofjLocr (IV-3) 



In laminae: H, = B cosh ^{y - d) + C sinh ^{y - d) (IV-4) 



E,=^z[B sinh ^ {y - d) + C cosh ^{y - d)] (IV-5) 



= i/^Ak' -coVoe) (IV-6) 



In dielectric : Hs = cosh ^y (IV-7) 



(IV-8) 



(IV-9) 



where A, B and C are constants. 



The fields H, and E^ must match at the boundaries y = h and y = d. 

 Imposing these conditions, we find the characteristic equation for deter- 

 mining k to be 



1 + ( A)-tanh^/? 

 tanh r. = - , ^^Wr; ^^^_^^^ 



\ico€i/ \<t/ r \*7 V 



