924 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



where Zniyo) is the impedance of the surface behind the stack. If = 0, 

 (170) reduces to (105) of Section IV, that is, 



-^0(70) = ::: , — . ,ry , — r = —ff^ — ; — TTfTl — \ > (171) 



grs + l/Z„(7o) gxTx + l/Zn(yo) 



where Ti is the total thickness of conducting material in the stack. If 

 Zniyo) is infinite, then for all values of and 71. we have 



Zo(7o) = -^ [|© + a©' - © coth + 1)^ coth nT]; (172) 



and if Re nT is large, corresponding to a stack many effective skin depths 

 thick, then for any Znijo) we have 



^1(70) = K, . (173) 



Once Zo(yo) has been computed for a particular frequency, the at- 

 tenuation and phase constants of the plane Clogston 1 line at that fre- 

 quency are given, as in Section II, by 



a = Re Zo(yo)/r,ob, (174) 



= ojaZ/xoTo -f- Im Zo(yo)/rioh. (175) 



Explicit expressions for the surface impedance of a coaxial stack of finite 

 layers have not been derived. However, if in a coaxial Clogston 1 the 

 thickness of each stack is small compared to its mean radius, or if the 

 depth of penetration given by (167) is small compared to the radius of 

 the surface near which the currents flow, then the parallel-plane formula 

 (170) may be used for the stack impedances Zi(yo) and ^2(70) which are 

 to be substituted into the equations of Section II for the attenuation 

 and phase constants, namely 



^ Zi(7o)/pi + Zi{y^lpi , . 



a = Re ^ — , (176) 



27J0 log (P2/P1) 



o / , T^ Zl(7o)/pi + Z2(70)/P2 /.-„x 



jS = coVmo€o + Im — — J— . (177) 



2770 log (P2/P1) 



If the plane approximations are regarded as insufficiently accurate, one 

 can compute the surface impedance of a cylindrical stack by repeated 

 multiplication of matrices similar to the one given by equations (88) of 

 Section III. This procedure would obviously involve considerable numeri- 

 cal computation, but we can hardly expect that it would reveal anything 

 qualitatively new for Clogston cables of the proportions considered in 

 Part I. 



