LAMINATED TRANSMISSION LINES. I U17 



The root of tliis transcendental equation is 



b/a = 4.3827, a = 0.228176. (138) 



Substitutiiiii' this xahio of h a into (135), we find 



si = 1.3535s, 



S2 = 0.r)465s, (139) 



si/so = 2.0935; 



wiiile from (130) and (138) the minimum \-aluo of the attenuation 

 constant is 



- 3.238 ^^^^^ 



Vog{si + S2)h 



To find the ()i)tinium value of 9, we observe that e(iuat ion (118) forthe 

 attenuation constant of a Clogston 1 cable with infinitesimaily thin 

 laminae and high-impedance boundaries may be written in the form 



(eo/Mo)' t-r 1 \ r-i ^^\ 



« = —^ .f(a, 6, Si , .So), (141) 



where the first factor depends on the electrical constants of the com- 

 ponents of the cable, while /(a, 6, Si , S2) is a function only of the geometry. 

 By (110) the attenuation constant of a plane Clogston 1 has the same 

 form, only with a different dependence on the geometrical factors. Now 

 assume that the geometrical proportions of the line are fixed, and that 

 the electrical constants ijli , gi , ^2 , and €2 of the conducting and insula- 

 ting layers are given, but that the constants /xo , to of the main di- 

 electric and the fraction of space 9 occupied by conducting layers in 

 the stacks are at our disposal. The M(tfn product of the main dielectric 

 is to be codetermined with 9 so that Clogston's condition (102) is always 

 satisfied. Solving (102) for 9 gives 



- M060 - M2e2 (^^2) 



jUoeo + I'm! — ^2)^2 

 Hence the first factor in the expression (141) for a may be written 



(co/jLto)' _ coUoeo + (mi ~ M2)g2] 

 ^S'l gifjL~n\n(>€a — M2€2] 



(143) 



If we minimize the right side of (143) with respect to to , all other quanti- 

 ties being held constant, by equating to zero the derivative with respect 



