902 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



a 



1 + 



1 + 



e = 



1 + 



2-p J 



u + w 



25 



U + t-i 

 2p . 



Ch Kill Ch K2k + — Sh Kltl Sh K2^2 



L2«i 

 ?72p ch Ki^i sh K'd-i + i?ip sh /v'l^i ch ^2^2 



sh Ki^i ch K'lt-y -j- —7 ch K'l/i sh K2I2 



K\p Zk2P 



]. 



+ 



_2k2p 2kip_ 



sh Kid sh /C2/2 , 



■ 1 1 



— sh Kltl ch Kd-^ -f- — ch Kill sh /co/o 



.^p 7j2p 



(73) 



+ 



|_2t72pKiP 277lpK2/C 



sh Kltl sh K2^2 , 



3D = 



1 + 



tl + t. 

 2p 



^72^ 



— sh Ki^i sh /C2/2 + ch Kltl ch k2^2 



+ 



■ 1 1 ^ ■ 



r— sh Ki/a ch /C2^2 + ^^~Z ch m^i sh Kitz 



ZKip ZK2P 



As in the analogous equations (58) for a plane double layer, the sub- 

 scripts 1 and 2 refer to the first and second layers respectively. 



If we have a stack of double layers in which all the layers of the same 

 kind have the same thickness and same electrical constants, then the 

 only term in (73) which varies from one double layer to the next is the 

 mean radius p. Depending on the circumstances, we may wish to use a 

 single value of p for the whole stack, or a few different values, or even, 

 if high-speed computing machinery is available to carry out the matrix 

 multiplications, a different value of p for each double layer. The matrix 

 of the whole stack then becomes a product of powers of as many different 

 matrices as we have chosen values of p. Obviously this method is better 

 adapted to the numerical analysis of special cases than to the general 

 theoretical treatment of a stack whose ratio of outer radius to inner 

 radius is unspecified. 



In principle we are now able to compute the normal surface impedance 

 of any laminated plane or coaxial stack at a given frequency provided 

 that we know the electrical constants and the thickness of each layer, 

 the number of layers, the propagation constant 7 in the z-direction, and 

 the normal impedance Z„ of the material behind the last layer. Since the 

 general formulas even for plane stacks are quite complicated, however, 

 we shall introduce at this point some very good approximations which 

 will be valid for all of the following work. 



