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BELL SYSTEM TECHNICAL JOURNAL 



trons are acted upon only when they are very near to the gaps. It is possible 

 to analyze the performance of traveling-wave tubes on this basis'. The chief 

 conclusion of such an analysis is that highly accurate results can be obtained 

 by expressing the field as a sum of travehng waves and taking into account 

 only the wave which has a phase velocity near to the electron velocity. Of 

 course this is satisfactory only if the velocities of the other components are 

 quite different from the electron velocity (that is, different by a fraction 

 several times the gain parameter C). 



As an example, consider a traveling-wave tube in which the electron stream 

 passes through tubular sections of radius a, as shown in Fig. 4.29, and is 

 acted upon by voltages appearing across gaps of length ( spaced L apart. 



->|ih- -AxW JiK JiU 



Vn-1 Vn Vn+i Vn+2 



Fig. 4.29— A series of gaps in a tube of inside radius a. The gaps are ( long and are 

 spaced L apart. Voltages Vn , etc., act across them. 



A wave travels in some sort of structure and produces voltages across the 

 gaps such that that across the «th gap, F, is 



Vn = V,e 



-jnB 



(4.60) 



where n is any integer. 



We analyze this field into traveling-wave components which vary with 

 distance as exp(-j(3mz) where 



(3,n = (^ + 2nnr)/L (4.61) 



where m is any positive or negative integer. Thus, the total field will be 



-C' / J J^m X > •''i ) 



-j?m' 



hiymr) 



(4.62) 



a J - l3o' 



(4.63) 



Here hiymr) is a modified Bessel function, and 7,„ has been chosen so that 

 (4.62) satisfies Maxwell's equations. 



^ J. R. Pierce and Nelson Wax, "A Note on Filter- Type Traveling-Wave Amjjliilers," 

 Froc. I.R.E., Vol. 37, pp. 622-625, June, 1949. 



