LAMINATED TRANSMISSION LINES. II 1149 



about 0.504 times that of the optimum unloaded one. In this example, 

 of course, we have said nothing about the effects of magnetic dissipa- 

 tion. 



In the above Avork we have assumed that the electrical constants 

 M, e, g of the stacks and /zo , fo of the main dielectric were all fixed quanti- 

 ties. We now consider the case in which the electrical constants of the 

 conducting and insulating layers are given, but the fraction 6 of conduct- 

 ing material in the stacks may be varied. We also suppose that the con- 

 stants of the main dielectric are at our disposal, so that Clogston's 

 condition may always be satisfied. When then is the optimum value of 67 



If we express e, m, and g in terms of the constants of the individual 

 layers by equations (268) of Section VIII, we find that the expression 

 for the attenuation constant of the principal mode in a Clogston line 

 becomes 



€2X1 



efW, -f (1 - e)y^,fgx ' 



(382) 



where xi is the lowest root of equation (348) for a plane line or equation 

 (364) for a coaxial cable. We wish to minimize a as a function of 6. 

 If the conducting and insulating la.yers have different permeabilities 

 (mi 7^ M2), then in the general partially filled line xi depends on 6, through 

 the factor ^ in equation (348) or (364), as well as on the geometric pro- 

 portions of the line. In the limiting case of an extreme Clogston 1 line 

 we found in Section IV, equation (145), that the optimum value of d is 



a Ml + (mi + 8MliU2)'' 



3/il + (mi + S/XliUo)' 



(383) 



while in a Clogston 2 with no main dielectric, it turns out from (348) 

 or (364) that xi is independent of 6, and an elementary calculation shows 

 that the value of 6 which minimizes a is 



Q ^ 3 (mi — 2/X2) + (9mi — 4/X1JU2 + 4m2)' /gg.N 



8 (mi — M2) 



For the general partially filled line, however, there seems to be no simple 

 expression for the optimum value of 6. 



If the conducting and insulating layers have equal permeabilities, then 

 the average permeability /I (= mi = M2) is independent of 6, and matters 

 are much simpler. Since xi is also independent of 6, the minimum value 

 of a in equation (382) is achieved when 



6 = 2/3, (385) 



