A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 351 



3. The space charge field is computed from a model in which the 

 helix is replaced by a conducting cylinder, and electrons are uniformly 

 charged discs. The discs are infinitely thin, concentric with the helix and 

 have a radius equal to the beam radius. 



4. The circuit is lossfree. 



These are just the assumptions of the Tien-Walker-Wolontis model. 

 In addition, we shall assume a small signal applied at the input end of a 

 long tube, where the beam entered unmodulated. What we are looking 

 for are therefore the characteristics of the tube beyond the point at which 

 the device begins to act non-linearly. Let us imagine a flow of electron 

 discs. The motions of the discs are computed from the circuit and the 

 space charge fields by the familiar Newton's force equation. The elec- 

 trons, in turn, excite waves on the circuit according to the circuit equa- 

 tion derived either from Brillouin's model^ or from Pierce's equivalent 

 circuit. The force equation, the circuit equation, and the equation of 

 conservation of charge in kinematics, are the three basic equations 

 from which the theory is derived. 



3. FORWARD AND BACKWARD WAVES 



In the traveling-wave amplifier, the beam excites forward and back- 

 ward waves on the circuit. (We mean by "forward" wave, the wave 

 which propagates in the direction of the electron flow, and by "back- 

 ward" wave, the wave which propagates in the opposite direction.) 

 Because of phase cancellation, the energy associated with the backward 

 wave is small, but increases with the beam to circuit coupling. It is there- 

 fore important to compute it accurately. In the first place, the waves on 

 the circuit must satisfy the circuit equation 



dH^(z,t) 2d'V{z,t) „ d'p^iz, t) ,v 



Here, V is the total voltage of the waves. Vo and Zo are respectively the 

 phase velocity and the impedance of the cold circuit, z is the distance 

 along the tube and t, the time, p^ is the fundamental component of the 

 linear charge density. V and p„ are functions of z and /. The complete 

 solution of (1) is in the form 



Viz) = Cre'^'' + (726 "^"^ 



+ e —-y— J e " po,{^) dz ^2) 



+ e " —^ j e p^{z) dz 



