TRAVELING-WAVE TUBES 11 



the direction of flow, there will be a displacement current / amperes per 

 meter impressed on the transmission circuit. We will assume that the elec- 

 tron beam is very narrow and very close to the circuit, so that the displace- 

 ment current along the stream is negligible compared with that from the 

 stream to the circuit. In this case the displacement current to the circuit will 

 be given by the rate of change of the convection current with distance. 



If the convection current i and the impressed current / are sinusoidal 

 with time, the equations for the network shown in Fig. 2.3 are 



^ = -jBV + J (2.1) 



dz 



i- = -jXI (2.2) 



oz 



Here / and V are the current and the voltage in the line, B and X are the 

 shunt susceptance and series reactance per unit length and / is the im- 

 pressed current per unit length. 



It may be objected that these "network" equations are not valid for a 

 transmission circuit operating at high frequencies. Certainly, the electric 

 field in such a circuit cannot be described by a scalar electric potential. 

 We can, however, choose BX so that the phase velocity of the circuit of 

 Fig. 2.3 is the^ame as that for a particular traveling-wave tube. We can 

 further choose X/B so that, for unit power flow, the longitudinal field acting 

 on the electrons according to Fig. 2.3, that is, —dV/dz, is equal to the true 

 field for a particular circuit. This lends a plausibility to the use of (2.1) and 

 (2.2). The fact that results based on these equations are actually a good ap- 

 proximation for phase velocities small compared with the velocity of light 

 is established in Chapter VI. 



We will be interested in cases in which all quantities vary with distance 

 as exp(— Fs). Under these circumstances, we can replace differentiation 

 with respect to z by multiplication by — F. The impressed current per unit 

 length is given by 



J= -^2 =Ti (2.3) 



dz 



Equations (2.1) and (2.2) become 



-TI = -jBV -\- Ti (2.4) 



-TV = -jXI (2.5) 



If we eliminate /, we obtain 



Vir~ + BX) = -jTXi (2.6) 



