A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 373 



ever, the ac velocities of the electrons are no longer small as compared 

 I with their average speed. The velocity spread of the electrons becomes 

 , an important factor in determining the efficiency. Its effect is to loosen 

 the bunching, and consequently it lowers the saturation level and re- 

 duces the limiting efficiency. It is seen from Figs. 5 and 6 that the 

 . velocity spread increases sharply with C and also steadily with b and QC. 

 \ This explains the fact that in the present calculation the saturation 

 Eff./C decreases with C and is almost constant with h whereas in the 

 1 1 small-C theory it is constant with C and increases steadily with b. 



12. ACKNOWLEDGEMENTS 



The writer wishes to thank J. R. Pierce for his guidance during the 

 course of this research, and L. R. Walker for many interesting discus- 

 sions concerning the working equations and the method of calculating 



I the backward wave. The writer is particularly grateful to Miss D. C. 

 Leagus who, under the guidance of V. M. Wolontis, has carried out the 



^ numerical work presented with endless effort and enthusiasm. 



APPENDIX 



The initial conditions at i/ = are computed from Pierce's linearized 

 theory. For small-signal, we have 



ai(?/) = 4A(y) cos (6 -f ^2)2/ (A-1) 



«2(2/) = -4A(y) sin (6 + ju2)y (A-2) 



A(y) = ee"'' (A-3) 



Here e is taken equal to 0.03, a value which has been used in Tien-Walker- 

 ' Wolontis' paper. Define 



; ^ = wiy, <po) (A-4) 'X = pe-^'" + p*e^'^'> (A-5) 



dy 



where p* is the conjugate of p. After substituting (A-1) to (A-5) into the 

 working equations (15) to (18) and carrying out considerable algebraic 

 work, we obtain exactly Pierce's equation. 



2 (1 + jC/i)(l + bC) innn \ ah \^ r\ r\ 



(j - >iCfi -h j}/ibC)(ti + jb) 



provided that 



+ CO 



—k\((>(.y ,<po+<t>)—<p(.y ,Vo)l['^+Cw(.y ,ipo+(t>)] 

 



(A-7) 

 • di^ sgn (^(?/, .i?o + «/)) - 9?(^, <Po)) = 8eQC 



(1 -f 3Cy){ii ^ jb) I e''" cos (arg [(1 -f jCm)(m + jb)] + my - ^0) 



