LAMINATED TRANSMISSION LINES. I 



905 



ch r = 





sli Kiti + ch Kid 



(86) 



while from (03) the iteiatue impedances are 



A'l = — hrjiyK-J-i + \/ ihmvKiii)' + VlymyKiti COth Kid + vlv ) 



A'o = +I7?2J/^•2/•2 + \/{hV2vK2t2)- + T?]yr72j,K2^2 COth Kj^i + r}ly . 



If we make the same simplifications in the approximate expressions 

 (73) for the matrix elements of a coaxial double layer, we obtain 



(87) 



a = 



(B = 



1 + 1 



-pj 



ch Kifi — ~ ; sh Kiti , 

 Zkip 



1 + 



tl + /2 ' 



2-p . 



V-loK-'lti C'h K\ti 



+ [l + .^ + (2 - 'i=^) ^" 



e = 



SD = 



1 + 



2pJ 



'yip 



sh Klti , 



77ip sh Kid , 

 (88) 



^1 + 1-2 _ Vl 



^P 



2772p/ClK2^2pJ Vlp 



"n-ipf^it 



sh m/i 



+ 



1 + 



d + 2^; 



2p 



ch Kid ■ 



In the preceding equations no restrictions have been laid on the 

 thicknesses d and h except the trivial requirement that d shall be small 

 compared to a wavelength. We shall now consider the limiting case in 

 which both ^1 and ti are infinitesimally small. When we make this last 

 and most drastic approximation we do not expect that the idealized 

 structure thus obtained will show all of the features which are of interest 

 in a physical transmission line with finite layers; but the results of the 

 simplified analj'sis will be useful in some cases nevertheless. It need 

 scarcely be pointed out that we are dealing here only with a mathematical 

 limiting process, in which we assume that each layer, no matter how 

 thin, always exhibits the same electrical properties as the bulk material. 

 If this assumption be regarded as mu'ealistic, it may be observed that 

 the (juantity which we actually allow to tend to zero is the ratio of layer 

 thickness to skin depth. The skin depth may be made as large as desired 

 by lowering the frequency, so that the formulas which we derive by 



