LAMINATED TRANSMISSION LINES. II 



1165 



wliere « and /x are given by equations (268) of Section VI 11. The i)roi)a- 

 gation constant 7 is thus related to q by 



7 = ?co\//ioe2 



1 + 



dynq 



= /covM^ 



1 + 



(1 - e)Md 



(-443) 



In terms of q and the electrical thickness parameter (-) used in Part I, 

 namely 



= a,h = (1 + i)ii/8i^ K,k, 

 equations (440) and (441) l)ecome, approximately, 

 ch r = ch e - ^50 sh 0, 



and 



A'l 



giti 

 



[k© + Vlq-Q' - qS coth + 1 ], 



(444) 

 (445) 



(446) 



/V2 = — [-k0 + Vi9-0' - 90 coth + 1 ]. 



In the general case when the sheath impedance Zn{y) is a given 

 function of 7, we substitute the expressions for /vi and K2 into equation 

 (438), namely 



^4„r ^ Zl{y) - (A-i + Ko)Zn(y) + K1K2 

 ZU7) + (Kx + A2)Z„(7) + A1A2 ' 



(447) 



and then determine 7 for each mode by simultaneous numerical solution 

 of equations (443), (445), and (447). At least as a first approximation 

 we may neglect the total current in either sheath compared to the one- 

 way current in the stack; to this approximation Zn(y) is effectivel3' 

 infinite and (447) becomes 



inV 



e - = 1. 

 The non-zero I'oots of this equation are 



r = ipir/2n, p = 1, 2, 3, 



(448) 



(449) 



where p = 1 corresponds to the principal mode and tiie higher \alues of 

 p to the higher modes discussed in Section X. (We would get nothing 

 new by including negati\'e \-alues of p.) The ((uantities q and 7 for each 

 mode aic then given by e<iuations (445) and (443) respectively. If we 



