1148 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



As mo/m increases from unity to very large values, the optimum value of 

 s decreases from ^a toward \a, so that the filling ratio decreases from 

 unity toward one-half. If mo/m < 1, equation (376) does not yield a real 

 solution, but the complete Clogston 2 is still the physical structure hav- 

 ing the lowest attenuation. 



For a coaxial Clogston line without magnetic loading the optimum 

 filling ratio is unity, as we have seen above, while in the presence of 

 magnetic loading a smaller filling ratio is optimum. This filling ratio 

 and the optimum distribution of the laminated material in the cable 

 can be determined by numerical analysis for any given value of juo/m- 

 It is reasonably evident on physical grounds, and can be proved mathe- 

 matically by a variational argument applied to the lowest eigenvalue 

 of equations (369), that whatever may be the radii pi and p2 of the main 

 dielectric, the lowest attenuation constant is achieved when a = 0, 

 that is, when there is no core inside the inner stack. (This is only a 

 mathematical limit; from a practical standpoint, the use of a small core 

 in the manufacturing process is not likely to make any significant in- 

 crease in the attenuation of the cable.) For each value of the loading 

 ratio no/fi, therefore, we have merely to minimize the value of (xi&)' as a 

 function of the two ratios pi/6 and pz/b, which can be done by the double 

 interpolation procedure mentioned earlier. We find that as mo/m in- 

 creases from unity to very large values, the optimum value of pi de- 

 creases from 0.62766 toward 0.39306, while p2 increases from 0.62766 to- 

 ward 0.82266, so that the filling ratio decreases from unity toward 0.5704. 

 The limiting values of pi and p2 when mo/m ^ 1 are derived from equa- 

 tion (364) by the method shown in Appendix II. 



As a numerical example we have considered a Clogston cable with 

 Ho = 3/i. The optimum proportions of this cable and the corresponding 

 value of xi are approximately 



Pi = 0.4266, p2 = 0.7966, xi = 2.720/6; (379) 



and the minimum attenuation constant is 



3.699 , , 



(380) 



Vm/c gb~ ' 



The attenuation constant of a complete Clogston 2 with the same stack 

 parameters jl and e is, from (318), 



7.341 , , 



(381) 



V m/€ gb" ' 

 so that the attenuation constant of the optimum loaded cable is only 



