LAMINATED TRANSMISSION LINES. II 117") 



with no inner core (a = 0), c(iuation (485) takes the lorni 



« = /— . -,2 [1 + O.S9mt\T\n\g\f-]. (480) 



V M/ e go 



It should be emphasized that whereas equation (4G8) was obtained 

 fiom a I'igorous solution of the boundary-value problem for the plane 

 line, eciuation (48.")) for the coaxial cahle has been derixcnl on the l)asis 

 of certain physical assumptions and appi'oximations whose effect on 

 the accuracy of the final result is not very eas.y to estimate. Presumably 

 one might check the acciu-acy of (485) for a particular Clogstou cable 

 l)y setting up the matrix relation between the known surface impedances 

 of the core and the outer sheath and solving numerically for the propa- 

 gation constant. It should not be too difficult to solve the matrix equa- 

 tion by cut-and-try methods for a cable having, say, two hundred double 

 layers, if the matrix of each double layer were assumed to be given by 

 ecjuations (88) of Section III, and high-speed computing machinery were 

 used to perfoi-m the matrix multiplications. In the absence of any such 

 lunnerical results, however, we shall merely assume that equation (485) 

 furnishes a reasonable approximation to the attenuation constant of a 

 coaxial Clogston 2 in the frequency range /i ^ / ^ fs , where /i and /s 

 are the critical frequencies defined by (467). 



The first conclusion which we can draw from (485) is that the maxi- 

 mum permissible thickness of the conducting layers in a coaxial Clog- 

 ston 2 with high-impedance boundaries, if the attenuation constant of 

 the pth mode is not to exceed its "flat" value ao b}' more than a speci- 

 fied small fraction ^a/ao at a top frequency fm , is 



, VS pfp{a/b) /a^ . ..^^. 



h = —^r^ J— A/ — , l-ioO 



or, putting in numerical values for copper, 



_ 36.84p/,(a/6) /K^ 



K'^-l lJmi\s\JmjMc Y Q!o 



For the principal mode in a Clogston cable with no inner core, this 

 becomes 



44.93 /a^ 



(^i)mii3 = /oT ^ /'/ ^ 4/ — • (489) 



{^ilJmiliKjmjMc y ao 



As a second application of equation (485), we shall determine the 

 upper crossover frequency at which the attenuation constant of a 

 Clogston 2 is equal to the attenuation constant of a conventional coaxial 



