1164 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



inV 



(437) is solved for e " , it takes the form 



4„r ^ [Zn(y) — /vi][Zn(7) - Ko] , . 



[Zn(y) + K,][ZM + K,] ' ^^'"''^ 



In order to simplify the general expressions for T, ivi , and K2 , we 

 make the following approximations: 



(i) We neglect coe/gi compared to unity, where e represents the di- 

 electric constant of either the conducting or the insulating layers. As 

 we have said before, this is an exceedingly good approximation at all 

 engineering frequencies. 



(ii) We neglect y la\ {= y /iwniQi) compared to unity. It turns out 

 that not only is this approximation valid in the frequency range of 

 greatest interest, where y is approximately equal to icoy/fie, but also 

 it is valid all the way down to zero frequency, so that in the present 

 section we can easily derive results for the complete frequency range 

 do^vn to dc. So long as tVo-i <$C 1, we have from equations (56) 'of 

 Section III, 



Ki ^ ai , V\y ^ VI ' (439) 



(iii) We suppose that the thickness ^2 of the layers of insulation is so 

 small that we may replace sh k2^2 by k2^2 and ch koI^ by unity. These 

 approximations wdll be amply justified if (2 is not greater than a few 

 times the skin depth in the conducting layers at the highest operating 

 frequency. 



The foregoing approximations lead to results identical with equations 

 (86) and (87) of Section III, namelj^ 



ch r = "^ sh K,h + ch K,h , (440) 



2r]iy 



and 



A'l = — ^r]2yK2t2 + \/ {^V2yK-lto)~ + "niylTjiyK^i COth Ki^i + T^fj, 



Ki = -{-hmyfi'it-i + '\/ {hViyK--J''i)'^ + ViyV-iyK^ti COth Kiti + TJly . 



To simplify the notation, we introduce the symbol q defined bj^ 



_ V2yK2t2 



(441) 



(442) 



