ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 557 



Approximate Formula for the Surface Impedance of 

 Tubular Conductors 



The exact formulae (75) for the internal impedances of a tubular 

 conductor are hard to use for numerical computations, but simple 

 approximations can be easily obtained if the modified Bessel functions 

 are replaced by their asymptotic expansions and the necessary division 

 performed as far as the second term. Thus, we have 



Zhh — 



2 Tb 



lira 



coth 0-^ + 7?" ( ~ + T 

 lex \a b 



TT /3 , 1 



coth 0-^ — ;i— T + - 



2a \ b a 



(82) 



Zah = -;= csch at^ 



2Tr^ab 



where t is the thickness of the tube, 

 parts, we have 



Separating the real and imaginary 



Rbb = 



Raa = 7^ 



Rab = |— r 



CjLbb = 



^Laa — 



CoLab = 



1 



2b 



2a 

 1 



|Z„J = 



'/x/ sinh M + sin M o + 36 



TTg cosh u — cos M 

 ixf sinh u + sin u 



leirgab'^ ' 

 b + 3a 



TTg cosh w — cos u 16 TTgba^ ' 



. . u u . , u . u 

 r-? smh - cos 7; + cosh - sm -^ 

 ixj Z 1 2 2 



TTg cosh u — cos u 



jjif sinh u — sin u 



TTg cosh u — cos u ' 



fjif sinh u — sin m 



Tg cosh u — cos u ' 



. , M u . u . u 



. smn - cos -x — cosh - sm -;r 



^ab \ TTg 



cosh u — cos zf 



V7? 



Vrga^ (cosh u — cos i<) 



(83) 



where u = /V2gco;u. 



It is obvious that in the equations for the self-resistances, the second 

 terms represent the first corrections for curvature and vanish altogether 

 if the conductors are plane. Although these formulae were derived by 

 using asymptotic expansions which are valid only when the argument 

 is large, i.e., at high frequencies, the results are good even at low 



