LAMINATIOD TRAXSMLSSIOX LINES. I 915 



//. 



E. 



Z 2(70) I 



'gib' - p') 



+ 



27rp2 L 2p pZb{'Yo)_ 



M Ziiyo)! 



gib' - pO 



+ 



e 27rp2 L ^P pZbi'y(i)_ 



(128) 



^ _ Z2(7o)/ -1 



-C/z ' — ' -^ e 



27rp2 



The average current densitj^ in either stack is uniform and is given by 



7. = gE, , (129) 



tliough in general the current density will not be the same in the two 

 stacks because of the difference in cross-sectional areas. The potential 

 difference between the surface of the inner core and any other point in 

 the same transverse plane is 



Vip) - Via) = -fE, dp. (130) 



If the stacks are thin compared to the thickness of the main dielectric, 

 as we are assuming throughout Part I, then the potential difference 

 across the stacks ^\\\\ be small compared to the potential difference 

 across the main dielectric, and the characteristic impedance Zk of the 

 Clogston 1 cable will be approximately the same as the characteristic 

 impedance of an ideal coaxial cable with perfect conductors of radii 

 Pi and po and the same main dielectric, namely 



Z, = 6O4/S log ^' ohms. (131) 



r ^Or Pi 



We shall defer making any field plots for Clogston-type transmission 

 lines until Section IX of Part II, when we shall discuss the transition 

 from Clogston 1 to Clogston 2 as the space originally occupied by the 

 main dielectric is gradually filled with laminations. Our present results 

 will then appear as the limiting case in which the thickness of the stacks 

 is small compared to the thickness of the main dielectric. 



In conclusion we shall mention briefly the question of how to dispose 

 a given amount of laminated material in a Clogston 1 coaxial cable so as 

 to achieve the minimum attenuation constant. The whole problem of 

 optimum proportions for Clogston cables is a complicated one of which 

 an adequate treatment would require a separate paper in itself, with 

 the results depending to a great extent on engineering considerations 

 which limit the ranges of the parameters that we can vary in any practical 

 case. Here we shall discuss only the following rather highly idealized 

 problem : 



