416 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



circuit. By means of two approximations, the normal-mode parameters 

 of this circuit have been evakiated. 



The first approximation amounts to neglecting the TE fields coupled 

 to the TM wave, and is valid for most low-power traveling-wave tubes. 

 The second approximation consists of replacing the circuit admittance 

 function by an algebraic expression with the same zero and pole, and the 

 same slope at the zero. Although thin-beam theory predicts small devi- 

 ations of complex roots (of the admittance equation) from the natural 

 propagation wave number, it is difficult to judge whether any such roots 

 might occur outside of the region in which this approximation holds, 

 for the finite beam. 



The space-charge parameter Qb is found to be the same as for a tliin 

 hollow beam with rectilinear flow (Fig. 1 of Reference 4, or Fig. A6.1 of 

 Reference 1). The gain parameter Cb can be computed from Eciuation 

 (85), Fig. 3.4 of Reference 1, and Fig. 3 of this paper. The gain of the 

 cylindrical beam with Brillouin flow is found to be greater than that of 

 a similar cylindrical beam with rectilinear flow, presumably because of 

 transverse electron motion in the former. Its gain, however, is less than 

 that of a thin hollow cylindrical beam with rectilinear floAv, for the same 

 radius, current, and voltage (Fig. 4). 



ACKNOWLEDGEMENTS 



The writers are indebted to J. R. Pierce for suggesting the general 

 approach used in this paper, to H. Suhl and C. F. Quate for their helpful 

 criticisms, and to Mrs. C. Lambert for computing the graph material. 



REFERENCES 



1. J. R. Pierce, Traveling -Wave Tubes, D. Van Nostrand, N. Y., Chapters VII, 



VIII, (1950). 



2. L. J. Chu and J. D. Jackson, Field Theory of Traveling-Wave Tubes, Proc. 



I.R.E., 36, p. 853, 1948. 



3. E. H. Rydbeck, Theorv of tlie Traveling-Wave Tube, Ericsson Tech. 46, p. 3, 



1948. ' 



4. R. C. Fletcher, Helix Parameters Used in Traveling-Wave Tube Theorv, Proc. 



I.R.E., 38, p. 413, 1950. 



5. L. Brillouin, A Theorem of Larmor and Its Importance for Electrons in INIag- 



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