LAMINATED TRANSMISSION LINES. II 1147 



which by equation (365) is proportional to the attenuation constant. 

 We note that the Clogston 2 Hne with fiUing ratio unity lias the low- 

 est attenuation constant of any cable of the same size without mag- 

 netic loading, but that the attenuation constant increases only slowly 

 as the filling ratio decreases, so long as the ratio is greater than about 

 one-half. It also appears that the minimum in xi^, considered as a func- 

 tion of ajh and Si/(si + S2) for a fixed filling ratio, is quite broad, which 

 means that in practice one can attain very nearly optimum perform- 

 ance even while deviating somewhat from the optimum proportions. 



If the filling ratio is at our disposal, then the solution of the optimum 

 problem is as follows: When there is no magnetic loading of the main 

 dielectric relative to the stacks, that is, when mo ^ m, then minimum 

 attenuation is obtained with a complete Clogston 2. If on the other hand 

 there is magnetic loading of the main dielectric, so that juo > m, then 

 minimum attenuation is obtained with a filling ratio less than unity, 

 whose value is a function of the ratio mo/ju. 



According to equation (350), the attenuation constant of a plane 

 Clogston line is 



2 



Xi 



(374) 



2Vm/ c g 



where xi is given by equation (348), 



2m M 

 XI tan xis = — = —77- -. . (375) 



To find the minimum value of xi when a and mo/m are given, we differ- 

 entiate (375) with respect to s and set dx\/ds equal to zero. This gives 



Xi sec^ xi-s = ~7T~ T2 , (375.5) 



which when solved simultaneously with equation (375) leads to 



sin xis = Vm/mo , XI = - sin~^ Vm/mo • (376) 



Substituting this value of xi hito (375) and solving for s in terms of a, 

 we get 



^ _ 1^ MO sin~^ V^o /o77>v 



Mo sin 'Vti/iJ.Q + V m(mo — m) 

 and from (374) the corresponding attenuation constant is 

 2 



« = /^T- - 2 [sii"!"^ Vm/mo + \/m(mo - m)/mo]"- (378) 

 VM/e ga 



