910 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



ohmic losses in a stack carrying a fixed total current the current density 

 should be uniform across the stack, and that we can achieve uniform cur- 

 rent density by adjusting the mo«o product of the main dielectric so as to 

 make Tf equal to zero. If in equation (93) we set 



7 = To = iu-\/tioeo , (101) 



then Ti will be zero if 



M0€0 = fii = [dn, + (1 - e)M2][62/(l - d)]. (102) 



Equation (102) will be referred to henceforth as Clogston's condition' 

 If the permeabilities of the various materials are all equal, the condition 

 reduces to 



eo = 6 = 62/(1 - e) , (103) 



which is the form employed by Clogston in Reference 1. 



When Clogston's condition is satisfied, r^ = and the effective skin 

 depth of the stack is infinite ;^^ that is, the current density is uniform in 

 any stack of finite total thickness. The quantities Tt and K vanish 

 simultaneously, but the limiting value of their ratio is finite; and the 

 matrix of the plane stack, as given by (92), takes the form 



(104) 



Accordingly we obtain, for the surface impedance Zo(7o) of the stack, 



which is, as might have been expected, just the impedance between 

 opposite edges of a unit square of material of conductivity g and thick- 

 ness s through which the current density is uniform, in parallel with the 

 sheath impedance Zniyo). It follows from equations (20) and (21) of 

 Section II that the attenuation and phase constants of the principal mode 

 in a plane Clogston 1 line with infinitesimally thin laminae, Clogston's 

 condition being satisfied exactly, are 



" This statement is certainly accurate enough for all practical purposes, al- 

 though an exact calculation which takes into account the small terms that were 

 neglected in the approximate formula (93) for Tf shows that the effective skin 

 depth is \J'2-Kd, where Xo is the length of a free wave in the main dielectric. The 

 exact result is derived by Clogston in Reference 1, equation (11-26). In practice, 

 finite lamina thickness will restrict us to effective skin depths much smaller than 

 this theoretical limit. 



