TRAVELING-WAVE TUBES 43 



If we transform this to new coordinates Zi, Xi about an origin at 2 = 0, 

 .V = 7r/4 we obtain 



r + i^ = ,„ ( ;+tanfa+ix.) \ _ / A \ 



\1 — tan (zi -h JXi)/ \sm 2 (21 -\-jxi)/ 



We can now adjust A to give a zero equipotential of diameter 2zi about x — 



xi = 0, Zi = (z = x/4) by letting 



/ I + tan zA 

 \1 — tan Si/ 



A = (sin 221) In ^ ^^" "^ (3.53) 



\1 — tan Si/ 



If /I is so chosen, there will be roughly circular equipotentials of 2-diameter 

 Jzi about 2 = ± 7r/4, etc. There will also be roughly circular equipotentials 



I of the same 2-diameter about z = 0, ±7r/2, etc., of potential ±F. That 



! about 2=0 has a potential 



V = In (i±i^A _A^ (3.54) 



\1 — tan 21/ cos 2zi 



I where A is taken from (3.53). 



j The distance between centers of equipotentials is p — it/A, so that the 

 ratio of 2-diameter of the equipotentials to pitch is 



d/p = 22i/(7r/4) = 2i/(7r/8) (3.55) 



The :v-diameter of the equipotential about 2 = (and of those about 2 = 



± etc.) can be obtained as 2.v by letting V have the value given by (3.54) 



i 2 



i and setting ;: = in (3.51), giving 



i 1' = In tanh x + f , (3.56) 



; cosh 2x 



\ 



The ratio of this .v-diameter to the pitch, di/p, is 



! d,/p = y/{T/S), (3.57) 



j .T is obtained from (3.56). 



[ To obtain the ;r-diameter of the potential electrodes we take the deriva- 

 tive (3.52) with respect to 21, giving the gradient in the 2 direction 



dV .dxp sec^ (21 -\- jxi) sec" (zi + jxi) 



+ ^ ^^ = ^ I .-- /- I •■■^ + 



dzi dzi 1 + tan (si +y.vi) 1 — tan (si + 7.V1) 



(3.58) 

 _ 2A cos 2(zi -\r jxi) 



sin 2(21 -\- jxi) 



