142 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



.if they lie in the same surface, it is clear that both coupling factors h and x 

 will go to unity. 



As pointed out earlier in Section 2.3, the choice of sign for h is arbi- 

 trary. However, once a sign for h has been chosen, the sign of x is neces- 

 sarily the opposite when the helices are wound in the same direction, and 

 vice versa. We shall choose, therefore, 



the sign of the latter depending on whether the helices are wound in the 

 same direction or not. 



In the case of unequal velocities, (5, the propagation constant, would 

 be given by 



1^ = VM~2 (2.6.2) 



2.7 Strength of Coupling versus Frequency 



The exponential variation of coupling factors with respect to frequency 

 (since /3 = co/y) has an important consequence. Consider the expression 

 for the coupling phase constant 



/3. = I3{b + x) (2.3.8) 



or 



l/3e| = 2/3^"^^'""^ (2.7.1) 



The coupling wavelength, which is defined as 



Ac 

 is, therefore, 



27r 



(2.7.2) 



or 



Xc- -e 



X, = ;^ g(2./x)u.-«) (2.7.3) 



where X is the (slowed-down) RF wavelength on either helix. It is con- 

 venient to multiply both sides of (2.7.1) with a, the inner helix radius, 

 in order to obtain a dimensionless relation between /3c and /3: 



^,a = 2/3ac~^''"''°^"" (2.7.4) 



This relalion is j)l()Ued on Fig. 2.2 for several values of b/a. 



