FIELD SOLUTIONS 



649 



earlier theory, from (14.86) with K adjusted according to (14.88), is plotted 

 as a dashed Hne, for 



A = 0.01 



We see that (14.88), which involves the approximations made in our earlier 

 calculations concerning traveling-wave tubes, is a remarkably good fit to the 

 broad-beam expression derived from field theory up very close to the points 

 {de — d) = A, which are the boundaries between real and imaginary argu- 

 ments of the hyperbolic tangent and correspond to the points where the 

 ordinate is zero in Fig. 14.5. 



Fig. 14.9 — These curves compare an exact electronic susceptance for the broad beam 

 case (solid curve) with the approximate expression used earlier (dashed curve). In the 

 approximate expression, the "effective current" was evaluated, not fitted; the space- 

 charge parameter was chosen to give a fit. 



Over the range in which the argument of the hyperbolic tangent in the 

 correct expression is imaginary, the approximate expression of course ex- 

 hibits none of the complex behavior characteristics of the correct expression 

 and illustrated by Fig. 14.6. From (14.88) we see that the multiple excursions 

 of the true curve from — oo to + «2 are replaced in the approximate curve by 

 a single dip down toward and back up again. R. C. Fletcher has used a 

 method similar to that explained above in computing the effective helix 

 impedance and the effective space-charge parameter Q for a solid beam inside 

 of a helically conducting sheet. His work, which is valuable in calculating 

 the gain of traveling-wave tubes, is reproduced in Appendix VI. 



14.5c The Complex Roots 



The propagation constants represent intersections of a circuit curve such 

 as that shown in Fig. 14.8 and an electronic curve such as that shown in Fig. 



