374 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



Here ^ = Mi + JM2 or Pierce's rri + jiji . From (A-7) the value of Up is 

 determined for a given QC. The ac velocities of the electrons are derived 

 from (A-4), such as, 



= -26 



M 



M + jb 

 1 + jcn 



e"^" cos ( arg 



M 



M + jb 



rvi+iCM/j 



+ M22/ — <Po 



(A-8) 



(A-1), (A-2), (A-7) and (A-8) are the expressions used to calculate the 

 initial conditions at y = 0, Avhen fn and jU2 are solved from Pierce's equa- 

 tion (A-6). 



From (12c), the particular solution of the backward wave at small- 

 signal is found to be 



j^, . ., -2iC(l+iC/x)(M+ib) 



^Ml!/ 



2j — CfjL -\- icb 



r [-2jC{\ - 



cos 



+ iCM)(M+i6)' 



Cn + jcb 



+ M2y — ^0 



which agrees with Pierce's analysis 



17 



3. 



4. 



REFERENCES 



1. J. R. Pierce, Traveling-Wave Tubes, D. Van Nostrand Co., N.Y., 1950, p. 160. 



2. R. L. Hess, Some Results in the Large-Signal Analysis of Traveling-Wave 



Tubes, Technical Report Series No. 60, Issue No. 131, Electronic Research 

 Laboratory, University of California, Berkeley, California. 



C. K. Birdsall, unpublished work. 



J. J. Caldwell, unpublished work. 



5. P. Parzen, Nonlinear Effects in Traveling-Wave Amplifiers, TR/AF-4, Radia- 



tion Laboratory, The Johns Hopkins University, April 27, 1954. 



6. A. Kiel and P. Parzen, Non-linear Wave Propagation in Traveling-Wave 



Amplifiers, TR/AF-13, Radiation Laboratory, The Johns Hopkins Univer- 

 sity, March, 1955. 



7. A. Nordsieck, Theory of the Large-Signal Behavior of Traveling-Wave Ampli- 

 fiers, Proc. I.R.E., 41, pp. 630-637, May, 1953. 



H. C. Poulter, Large Signal Theory of the Traveling-Wave Tube, Tech. Re- 

 port No. 73, Electronics Research Laboratory, Stanford University, Cali- 

 fornia, Jan., 1954. 



P. K. Tien, L. R. Walker and V. M. Wolontis, A Large Signal Theory of Trav- 

 eling-Wave Amplifiers, Proc. LR.E., 43, pp. 260-277 March, 1955. 



J. E. Rowe, A Large Signal Analysis of the Traveling-Wave Amplifier, Tech. 

 Report No. 19, Electron Tube Laboratory, University of Michigan, Ann 

 Arbor, April, 1955. 

 11. P. K. Tien and L. R. Walker, Correspondence Section, Proc. I.R.E., 43, 

 p. 1007, Aug., 1955. 



Nordsieck, op. cit., equation (1). 



L. Brillouin, The Traveling-Wave Tube (Discussion of Waves for Large 

 Amplitudes), J. Appl. Phys., 20, p. 1197, Dec, 1949. 



Pierce, op. cit., p. 9. 



Nordsieck, op. cit., equation (4). 



Pierce, op. cit., equation (7.13). 

 17. J. R. Pierce, Theory of Traveling-Wave Tube, Appendix A, Proc. I.R.E. 

 35, p. 121, Feb., 1947. 



8. 



10 



12 

 13 



14 

 15 

 16 



