A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 353 



tively the voltages of the forward and the backward waves, we have 

 F{z) = Fi„put e-'"^ + e-^»^ ^« r e'^' p^z) dz (4) 



Z Jo 



Biz) = e^- ^° £ e-^-p„(e) dz (5) 



It can be shown by direct substitution that F and B satisfy respectively 

 the differential equations 



We put (4) and (5) in the form of (6) and (7) simply because the differ- 

 ential equations are easier to manipulate than the integral equations. 

 In fact, we should start the analysis from (6) and (7) if it were not for a 

 physical picture useful to the understanding of the problem. Equations 

 (6) and (7) have the advantage of not being restricted by the boundary 

 conditions at 2; = and D, which we have just imposed to derive (4) 

 and (5). Actually, we can derive (6) and (7) directly from the Brillouin 

 model in the following manner. Suppose Y, I and Zo are respectively 

 the voltage, current and the characteristic impedance of a transmission 

 line system in the usual sense. (V + /Zo) must then be the forward wave 

 and {V — IZo) must be the backward wave. If we substituted F and B 

 in these forms into (1) of the Brillouin' s paper,^^ we should obtain exactly 

 (6) and (7). 



It is obvious that the first and third terms of (2) are respectively the 

 complementary function and the particular solution of (6), and similarly 

 the second and the fourth terms of (2) are respectively the comple- 

 mentary function and the particular solution of (7). From now on, we 

 shall overlook the complementary functions which are far from syn- 

 chronism with the beam and are only useful in matching the boundary 

 conditions. It is the particular solutions which act directly on the elec- 

 tron motion. With these in mind, it is convenient to put F and B in the 

 form 



Fiz, t) = -j~ [aiiij) cos <p - aiiy) sin ^] (8) 



B{z, t) = -^ [hiiy) cos ip - h^iy) sin 9?] (9) 



