922 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 





[+!© + a®' - ecoth0 + I)*], 

 [- ^0 + (\e' - coth + 1)*], 



(158) 



(159) 



ch r = ch - i0 sh 0, 

 and from (87), for the iterative impedances, 

 _©_ 



@_ 



since rjiy = ki/qi = S/gik . 



If the thickness ^i of each conducting layer is moderately small com- 

 pared to the skin depth 5i at the highest frequency of interest, the quanti- 

 ties r, Ki , and K2 may conveniently be expanded in powers of 0. The 

 identity 



2 



ch a:; — 1 = 2 sh la; 

 enables us to transform (158) into 



sh' ir = i(ch - 1) - i0 sh 



0^ 



48 







0^ 



1 + 15+560 + 



(160) 



(161) 



after we expand sh and ch by Dwight 657.1 and 657.2 and collect 

 terms. Taking the square root by the binomial theorem gives 



sh hV = - 



4V3 



0* 



170' 



30 50400 



+ 



(162) 



the negative sign being introduced because from (157) is a positive 

 imaginary number and we want Re F > 0. Then 



2sh" 



L WS \ 30 ^ 50400 



+ 



■©_' 0* 0^ 

 V3 L 2 60 + 525 



(163) 



provided that we expand the sh ^ function by Dwight 706. From (159) 

 we get 



Ki = 



Ko = 



1 

 J_ 



■(3 - iVs) 



&' + 



i\/S r\i ^'V3 



45 



0* - 



1575 



0'-f 



(3 -f V3) 



0' + 



iV3 

 45 



0^ 



^VS .2.6 



(164) 



1575 



^ © + 



