934 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



(leplli 5i ill the conducting material at tlie operating frecjuency, although 

 if the mismatch parameter k is small, the proportionality constant mul- 

 tiplying 61 will be large. In the general case, the number of double layers 

 in one effective skhi depth is 



N = -A_ = Vlhlh (220) 



^1 + ^2 [VKh/8,y ^ 9k^ + Wi/^iYV 



and the total thickness of conducting material in these layers is 



Ta = Nh = -___^£ -, . (221) 



[VK^iA)* + 9k' + M^i/5i)? 



It is instructive to plot the effective skin depth of a given stack at a 

 fixed frequency as a function of dielectric mismatch. If 



Ao = V3(<i + t2)8l/tl (222) 



denotes the effective skin thickness when there is no mismatch, then the 

 relative skin thickness when the mismatch parameter is k is just 



A V2 



Ao [Vl + SQk^t/tt + if 



(223) 



This ratio is plotted against A- in Fig. 8, a universal curve being obtained 

 by measuring k in units of (ti/diY. It is worth noting that when k = 

 (ti/8i) , the effective skin thickness is only 53 per cent of the skin 

 thickness with perfect dielectric match. 



The surface impedance Zo(yo) of a laminated plane stack at any fre- 

 quency and with any amount of dielectric mismatch is given by eciua- 

 tion (65), 



^( ^ ^ |Z„(7o)(Kie"^ + A-2e""') + KiK, sh nV . . 



'^'^'^ Zn(yo) sh nV + UKxe-^^ + K,e-^) ' ^ -^ ^ 



For a stack with infinitesimally thin layers and total thickness s, the 

 equation becomes 



y f \ zr ^"(to) ch r^s -f iv sh T(S .^r,r\ 



/o(7o) = K ry r \ u T 1 i^ 1 T. J (225) 



Znijo) sh T(S + K ch TiS 



where Ti and K are given by (211) and (212). At zero frequency, 



-^0(70) = Z 1 -, ,r7 t ^ = —fF, ; -i iry / ^ J (226) 



Q8 -f l/Z„(7o) gxTx + l/-^„(7o) 

 while if Zn(7o) is infinite, in general 



