LAMINATED TRANSMISSION LINES. II 1201 



t{3tuuition of tlie cable. Hence the estimate of the increase in uLlenuation 

 which one gets from the present analysis, considering only the variations 

 transverse to the layers at an average cross section, is certain to be opti- 

 mistic in that it neglects completely the effects of variations in other 

 directions. 



XIII. DIELECTRIC AND MAGNETIC LOSSES IN CLOGSTON 2 LINES 



To discuss dielectric and magnetic losses in Clogston 2 lines we may 

 take the electrical constants of the conducting and insulating layers to be 

 complex; thus 



Ml = Ml — ^Mi = Mi(l — i tan fi), 



M2 = M2 — i/2' = M2(l — i tan ^2), (584) 



€2 = €2 — ie-i = €2(1 — '' tan </)2). 



Almost all of the equations of the preceding sections, e.xcept of course 

 those which in\'olve explicit separation of real and imaginary parts, re- 

 mam valid when we introduce complex \^alues of mi , M2 , and et . In par- 

 ticular the propagation constant of the pth mode in a Clogston 2 with in- 

 finitesimally thin laminae and high-impedance walls is given, as in Sections 

 VIII through X, by 



7^ = —o^'fie + (iu€/g)xp, ('585) 



where 



Xp = Pir/a (586) 



for a parallel-plane line, and Xp is the pth root of 



Ji(xa)N,(xh) - Mxb)Ni(xa) = (587) 



for a coaxial line. Taking the square root of the right side of (585) l)y the 

 binomial tlieorem, we have 



2 

 7 = jco\//2e + — .^^ . (588) 



2Vm/€ g 



In the presence of dielectric and/or magnetic dissipation, we write, as 

 in Section VII, 



e =1' - a" = [,',/{! - 6)] - i[e:/{\ - e)i 



(589) 

 M = m' - in" = [en[ + (1 - ^)m2] - i[dix[' + (1 - d)ix^\. 



