144 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



means of a length of overlap between them an integral number of half 

 beat-wavelengths long. Ob^'iously, one will design the helices in such a 

 way as to take advantage of this situation. 



Optimiun conditions are easily obtained by dijfferentiating ^c with 

 respect to (3 and setting d^c/d^ equal to zero. This gives for the optimum 

 conditions 



^opt — 



1 



b — a 



(2.7.5) 



or 



Pc opt 



2e 



h — a 



= 2e ')8opt 



(2.7.6) 



Equation (2.7.5), then, determines the ratio of the helix radii if it is re- 

 quired that deviations from a chosen operating frequency shall have 

 least effect. 



2.8 Field Solutions 



In treating the problem of coaxial coupled helices from the transmis- 

 sion line point of view one important fact has not been considered, 

 namely, the dispersive character of the phase constants of the separate 

 helices, /3i and fS-i . By dispersion we mean change of phase velocity with 

 frequency. If the dispersion of the inner and outer helices were the same 

 it would be of little consequence. It is well known, however, that the 

 dispersion of a helical transmission line is a function of the ratio of helix 

 radius to wavelength, and thus becomes a parameter to be considered. 

 When the theory of wave propagation on a helix was solved by means of 

 Maxwell's equations subject to the boundary condition of a helically 

 conducting cylindrical sheath, the phenomenon of dispersion first made 

 its appearance. It is clear, therefore, that a more complete theory of 



/i 





'V^ 'TV 





Fig. 2.3 — ShoMtli liolix arrangement on which the field equations are based. 



