WAVE PUOl'AGATIOX ALONG AX KLlXTliON UKAM 411 



Then, dividing l)oth sides of (()2) by 2Trb, we may i-e\vrite it as a wave- 

 admittance equation : 



■2jQ 



-ti.y!i.R = -c.vM-A' 



- rl "^ (3e 



(64) 



With the aid of (59), we can suh-e the a(hnittance ecjuation (58) for 

 R, and re-write it as follows: 



Ye = Y, (65) 



with 



r. = -tiy±R (66) 



7 2 



^jaseyh hjyb) / 6o \ 



The solid-c\'lindrical Brilloiiin beam in a helix is thus equivalent to a 

 tliin beam whose circuit admittance is Yb . By equating Yb to the right 

 side of (64), we can evaluate the normal-mode parameters for this ad- 

 mittance, and thereby use all the results of previous thin-beam calcula- 

 tions. ' ' The equivalence of the two circuit expressions, however, requires 

 that we replace the transcendental expression (67) by an algebraic one, 

 with no more than three arbitrary constants. This can be done very 

 effectively, in the region of interest, by means of the approximation:^ 



Ys^-(y,-y,)(?L') -1^^' (68) 



\ dy /7=7(, 7 — Tp 



in which 70 and 7^ are the zero and pole, respecti\'el3^, of Yb : 



60(70) = 



'«^^^^ ^ (- K;S) + 76-A(76VXo(76)l=.. ^^^^ 



If we were to neglect the term containing F in (70), the error in the 

 magnitude of 60(7^) would be measured by: 



(«, .ot i) ('^) WP^, (71) 



yh-Ii(yb)-In{yh) \ c / Io{yh) ■ Ko{ya) 



In most low-power traveling-wa\e tubes, the first factor in parentheses is 

 usually less than 3; the second factor less than O.Ol; and the last factor 

 always less than unity. The error in evaluating yp , moreover, is less than 



