1172 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



tlie vai-ious possible modes. If the coaxial laj^ers are of finite thickness, 

 howe\"er, the relation between the surface impedances of the stack in- 

 \'olves the product of as many different matrices as there are layers in 

 the whole stack, and this matrix product is not suited to analytic treat- 

 ment. We shall therefore approach the problem from another point of 

 view. 



We have seen that if the conducting layers in a laminated transmis- 

 sion line are sufficiently thin compared to the skin depth, the attenua- 

 tion constant is essentially independent of frequency. In practice it is 

 important to know how rapidly the attenuation constant of a Clogston 

 cable with finite laminae begins to deviate from its low-frequency value 

 as the frequency is increased. In accordance with the results foi- the 

 parallel-plane line, we expect the initial increase to be proportional to 

 the square of the frequency. We shall derive the term proportional to 

 /" in the attenuation constant of the coaxial line on the basis of the 

 following assumptions : 



We assume that the macroscopic current distribution in a coaxial 

 Clogston 2 is independent of frequency, and hence is given by the ex- 

 pressions which have already been derived for the case of infinitesimally 

 thin laminae. (It is eas}^ to show that this assumption is valid for a 

 ylane Clogston 2.) If the conducting layers are of finite thickness, then 

 each carries a definite finite fraction of the total current in the line. At 

 low frequencies the current density in any given layer is approximately' 

 uniform, but as the frequency is increased it becomes nonuniform be- 

 cause of the development of skin effect, and the power dissipated in the 

 layer is increased. We shall calculate the total power dissipated in the 

 stack, and the corresponding attenuation constant, up to terms in /". 



Let the jth conducting layer in the stack be a hollow cylinder of 

 conductivity gi , inner radius py_i , and thickness ^i . Thus if there are 

 2n double layers we have po = a and p2« = b, where as usual a and h 

 denote the iimer and outer radii of the whole stack. Let the total cur- 

 rent flowing in the positive ^-direction inside p = pj-i be /y_i , and let 

 the current flowing in the jth. conducting layer be AIj . It is shown in 

 Appendix III that the average power dissipated per unit length in the 

 jth conductor is approximately 



AP,. = 1 



4:irgitipj^i 



,4 



|A/y|-^ + 4 l^i-ip] . (^72) 



381 



up to terms in (ti/8i) , Avhere curvature corrections of the order of 

 fi/pj-i have been neglected in comparison with unity. Presumably the 

 only layers for which it may not be justifiable to neglect curvatuj'e cor- 



