LAMINATED TRANSMISSION LINES. I 891) 



The stack of double layers may be regarded as a chain of iterated four- 

 poles; such chains have an extensive literature. The relation between 

 the tangential fields £"„ , //"„ at the upper surface of the nth double layer 

 and Eo , Ho at the lowei- surface of the first double layer is 



where M is the (KBCSD-matrix appearing in eiiuation (57). TToweN'ci- tiuM'e 

 is a simple expression for the /?th j)ower of a scjuare matrix of oi-der 

 two, namely 



^. ^ ^^,,„_, shnr^_ ^^u -^h in- l)r (g^^ 



sh r sh r 



wliere I is the unit matrix of order two, F is the propagation constant 

 per section of the chain of four-poles, defined by 



ch r = (a + 2D)/2iir, (01) 



and M is the determinant of the matrix M, that is, 



M = aSD - (Be. (62) 



The determinant of the matrix whose elements are given by (58) is unity, 

 as may easily be verified ; but this may not be the case for all the matrices 

 which occur in our study of cylindrical structures. M wall therefore be 

 carried explicitly in the following equations. 



We now introduce the iterative impedances Ki and K2 , defined by 



„ (a - D) + Via + 30)- - 4iif 



Ai = 



/V2 = 



26 



-(g - £>) + Via + ^y - ui 



26 



(03) 



/vi is the impedance seen when we look into a semi-infinite stack of 

 double layers if the first layer is of type 1, while K2 is the impedance seen 

 if the first layer is of type 2. In calculations relating to Clogston 1 lines 

 with dissipative walls, the real parts of Ki and K2 will both be positive. 

 By a straightforward procedure we may express the matrix elements 

 a, (B, 6, 3D in terms of Ki , K2 , T, and M, and then transform e( [nation 



'See, for example, E. A. Guillemiii, Comtnunication Networks, 2, Wiley, New 

 York. 1935, pp. 161-166. 



'OF. Abeles. Comptes Rendus, 226, 1872 (1948). This result was called to the 

 author's attention by Mr. J. G. Kreer. 



