LAMINATED TRANSMISSION LINES. I 



895 



The magnetic field lines of the principal mode will of course be circles 

 and the electric held will he largely radial, l)ut with a small longitudinal 

 compoiuMil unless the wall impedances are rigorously zero. The general 

 exj)r(>ssi()ns (33) for the fields may be reduced to simple approximate 

 fornuilas if we use the fact that kq is given \)y (41) and kop is small com- 

 pared to unity. The ratio A H may be obtained from either of equations 

 (3()). Introducing the approximations (39) for the Bessel functions and 

 carrying out a little algebra, we get the following approximate expres- 

 sions for the fields: 



H, 



E. 



E. 



(46) 



'?i(2^1og^+?!^hog^ 



.pi P P2 P J 



27r log (po/pi) 



where the amplitude factor / is equal to the total current flowing in the 

 inner cylinder. Incidentally we note that the above results might have 

 been tlerived from more elementary arguments if we had started with the 

 fields in a coaxial line with perfectly conducting walls and treated the 

 effect of finite wall impedance as a small perturbation. 



If we consider an ordinary coaxial cable with solid metal walls at a 

 frequency high enough so that there is a well-developed skin effect on 

 both conductors, then to a good approximation 



Zi(7o) = ^2(70) = (1 + i)/giBr , 



(47) 



where r/i and 5i are the conductivity and the skin thickness of the metal; 

 and the attenuation and phase constants are given by the well-known 

 expressions 



1/pi + 1/P2 



(3 = u 



2vogi8i log (P2/P1) ' 



1/pi + 1/P2 



(48) 

 (49) 



2r]ogi8i log (P2/P1) 



If necessary we may take account of dissipation in the main dielectric 

 of either a plane or a coaxial transmission line by assigning complex 

 values to eo and /xo , say 



' See, for example, C. G. Montgomery, Principles of Microwave. Circuits, M. I. T. 

 Rad. Lab. Series, 8, McGraw-Hill, New York, 1948, pp. 365-369 and 382-385. 



