916 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



Given a coaxial Clogston 1 with infiuitesimally thin laminae, having a 

 high-impedance core and a high-impedance sheath of fixed radius b, 

 and in which the total thickness Si -+- S2 of both stacks is a fixed constant 

 2s. Assuming that 2s is small compared to b, what should be the radius 

 a of the core, and how should the total stack thickness be divided between 

 the outer and inner stacks so as to minimize the attenuation constant of 

 the line? Finalh^, what should be the fraction 6 of conducting material 

 in the stacks to minimize the attenuation constant, if the electrical 

 constants of the conducting and insulating layers are fixed, but the 

 properties of the main dielectric are at our disposal? 



If the two inequalities 



si « a, S2 « 6, (132) 



are satisfied (these restrictions will be removed in Section IX, when we 

 discuss Clogston cables having an arbitrary fraction of their total volume 

 filled with laminations), then equation (118) for the attenuation constant 

 of a Clogston 1 with infiuitesimally thin laminae and high-impedance 

 boundaries becomes, approximately. 



1 



asi 0S2J 



(133) 



2r]og log (b/a) 

 If we write 



S2 = 2s - Si , (134) 



and vary Si and S2 in accordance with this relation while holding a and b 

 constant, it is easy to show that the expression on the right side of 

 (133) is a minimum when 



2s\/b 2s-\/a r^o-\ 



Si = —^ ^, S2 = —^ ^. (13o) 



V a + V f> Va + V& 



These equations tell us the most efficient way to divide the stacks in a 

 Clogston 1 when the radii of the core and the outer sheath are a and b re- 

 spectively, still assuming of course that the thickness of each stack is 

 small compared to its mean radius. 



If we introduce the optimum values of Si and So into (133), we get 



1 



1 



■\/a Vb_ 



(136) 



2770^ (si + S2) log (b/a) 



If b is fixed, the last expression is a minimum, considered as a function 

 of a, when 



log (b/a) = 1 -\- \^ctjh. (137) 



