138 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



and call this the "difference phase-constant," i.e., the hase constant cor- 

 responding to two uncoupled waves of the same frequency but differing 

 phase velocities. We can thus state the relation between these phase 

 constants : 



^b' = &I + ^c (2.4.6) 



This relation is identical (except for notation) with expression (33) in 

 S. E. Miller's paper. ^ In this paper Miller also gives expressions for the 

 voltage amplitudes in two coupled transmission systems in the case of 

 unequal phase velocities. It turns out that in such a case the power trans- 

 fer from one system to the other is necessarily incomplete. This is of 

 particular interest to us, in connection with a number of practical 

 schemes. In our notation it is relatively simple, and we can state it by 

 saying that the maximum fraction of power transferred is 



(2.4.7) 



or, in more detail, 



iS/ + iSc- (^1 - iS2)- + ^Kh + xY 



This relationship can be shown to be a good approximation from (2.2.6), 

 (2.3.4), (2.4.2), on the assumption that h ^ x and Zx 'PH Z2 , and the 

 further assumption that the system is lossless; that is, 



I 72 I ^ + I Fi I ^ = constant (2.4.8) 



We note that the phase velocity difference gives rise to two phenomena : 

 It reduces the coupling w^avelength and it reduces the amount of power 

 that can be transferred from one helix to the other. 



Something should be said about the case where the two helix imped- 

 ances are not equal, since this, indeed, is usually the case with coupled 

 concentric helices. Equation (2.4.8) becomes: 



I F2 1 _^ \Vx\_ ^ (3Qj^g^^j^^ (2.4.9) 



Z2 Z\ 



Using this relation it is found from (2.3.4) that 



F2 , /Zi 



FiT z, 



(1 ± Vl - /^) (2.4.10) 



When Ihis is combined with (2.2.6) it is found that the impedances droj) 

 out with the voltages, and that "F" is a function of the |S's only. In other 



