A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 357 



terms 



dax{y) ^ _2 T^" ^ sin <p(y, cpo) .^. 



dy IT h " 1 + Cwiv, (po) 



da.Xy) 2 f^" cos<p{y,<po) ..^n 



— 1 — = ~- / d(po , r (.lb; 



dy IT Jo 1 + Cw{y, <po) 



Next we shall calculate the electron motion. We express the acceleration 

 of an electron in the form 



d'z „ /I I /o / ^^ dw{y, <po) 

 ^ = Cuod + Cw{y, M -^^ 



and calculate the circuit field by differentiating F in (8) and B in (12c) 

 with respect to z. One thus obtains from Newton's law 



2[1 + Cw{y, <po)] ^^'^J' ^°^ = (1 + hOMy) sin <p + a,{y) cos <p\ 



dy 



+ ^-^ r^ «in ^ + ^^ cos J - -^ ^. 

 4(1 + 6C) L ^Z/- c?^^ J WomwC^ 



Here Eg is the space charge field, which will be discussed in detail later. 

 Finally a relation between w{y, (po) and <p{y, ^o) is obtained by means of 

 (13) 



difiy, <po) _ ^ ^ ^^(y, <Po) QgN 



dy 1 + Cw{y, <pq) 



Equations (15), (16), (17) and (18) are the four working equations 

 which we have derived for finite C and including space charge. 



Instead of writing the equations in the above form, Rowe, ignoring 

 the backward wave, derives (15) and (16) directly from the circuit 

 equation (1). He obtains an additional term 



C d^tti 



2 dy"" 



for (15) and another term 



C d"ai 



2df 



for (16). It is apparent that the backward wave, though generally a 

 small quantity, does influence the terms involving C. 



