1166 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



wish to take the hnite value of Zn{i) into account, we may calculate 

 K\ and Ki from (446) and then obtain a second approximation to F 

 from (447); and this process may be repeated as often as desired. From 

 the experience gained in treating a particular example we feel that the 

 method of successive approximations is entirely feasible, but it does 

 involve a considerable amount of numerical work. 



In the calculation just described we ha\'e to choose the correct sign 

 of the complex square root occurring in the expressions for K\ and Ki . 

 Without attempting to give a complete discussion of this point here, 

 we observe that it may be shown that 



sh r = sh ©Vig-o' - g0 coth t) + 1. (450) 



Under ordinary circumstances F will be a small complex number with 

 a phase angle of about 90°, and (-) will be a small complex number with 

 a phase angle of 45°. Hence the phase angle of the square root may be 

 expected to be in the neighborhood of 45° rather than 225°. 



In the remainder of this section we shall restrict ourselves to the case 

 of high-impedance sheaths, so that the values of F are given to a suffi- 

 ciently good approximation by equation (449). We shall discuss the 

 principal mode and the higher modes concurrently, but shall assume 

 throughout that the mode number p is small compared to n. From (445) 

 and (449), the value of q for the pth mode is 



_ 2 (oh e - cos I) (^.^^ 



^ sh 



and the propagation constant 7 is obtained by substituting this value 

 of q into equation (443). 



We shall now discuss the variation of the propagation constant of a 

 plane Clogston 2 line with frequency over the full frequency range from 

 zero to very high frequencies." To do this we shall derive approximate 

 expressions for the propagation constant at what may be called, roughly, 

 very low frequencies, low frequencies, high frequencies, and very high 

 frequencies. It will appear presently that the limits of these various 

 frequency ranges depend among other things on the dimensions of the 

 laminated transmission line and the thicknesses of the individual layers, 

 and that the frequency range of greatest engineering importance is 

 what we have called simplj^ "low frequencies". 



From equation (444) we have 



" In this connection see also Reference 1, Appendices A and B, pp. 525-529. 



