898 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



tance t, these fields are related by the general circuit parameter matrix 

 of a uniform line, namely 



E{0)\ /ch Kt vy sh Kt\ /E{t)\ 



HiO) Y^ chKt \Hit) 



We are now in a position to determine the surface impedance normal 

 to a laminated plane structure composed of layers of which every other 

 one has thickness ti and electrical constants o-i , tji , while the inter- 

 vening layers each have thickness ^2 and electrical constants ao , V2 • 

 Fig. 3 shows the cross section of such a stack in which the total number 

 of double layers is n (2n single layers), while Fig. 4 represents the corre- 

 sponding coaxial stack. Ultimately we shall assume the layers of thickness 

 ti to be good conductors and those of thickness ti to be good insulators, 

 but these assumptions need not be brought in immediately. 



If the fields in the plane stack all vary with z according to e~^^, then 

 when we look in the direction of increasing y each double layer may be 

 regarded as a four-terminal network formed by two sections of uniform 

 transmission line of lengths h and <2, the propagation constants and 

 characteristic impedances of the two sections being given respectively by 



Ki = o-i(l — 7Vo■l)^ Viy = i?i(l — ll<n)^, 



K2 = 0-2(1 — 7 I'^l) , V2y = V2{1 — T /0'2) • 



The matrix of the double layer is the product of the matrices of the two 

 single layers in the proper order. Thus if the tangential field components 

 are Eo , Ho at the lower surface of the first layer and Ei , Hi at the upper 

 surface of the second layer, we have 



EA /a (R\/Ei 

 Ho) \e 3D/Wi 



(57) 



where 



(I -- ch Kl^l ch K2t2 + — Sh K'l^l Sh K2^2 , 

 Viy 



(B = r}2y ch Kit\ sh K^t-l + niy sh Kill ch K2^2 , 



e = — sh K'1^1 ch /V2/2 + — ch Kill sh K2/2 , 

 "niy Viv 



Sj = — sh Ki^i sh K2^2 + ch Ki^i ch ^2^2 • 



Vly 



