654 BELL SYSTEM TECHNICAL JOURNAL 



The nature of this equation indicates that gain may be possible in two 

 ranges of parameters. One is that in which the electron velocity is near to 

 or equal to (as, (15.25)) the circuit phase velocity. In this case there is gain 

 provided that the transverse component of a-c electric field is not zero, and 

 provided that it is related to the longitudinal component as it is for the 

 circuit of Fig. 15.3. It seems likely that this corresponds most nearly to usual 

 magnetron operation. 



The other interesting range of parameters is that near 



;8,//3i = 1 - /3„M (15.31) 



Here /3g refers to the electrons, /8i to the circuit and /3« is the cyclotron fre- 

 quency divided by the electron velocity. When (15.31) holds, there is gain 

 whenever the parameter a, which specifies the ratio of the transverse to the 

 longitudinal fields, is not -f 1. For the circuit of Fig. 15.3, a approaccs +1 

 near the fins if the separation between the fins and the plane is great enough 

 in terms of the wavelength. However, a can be made negative near the fins 



""llllillillll"^^ 



__MM 



lo 



Fig. 15.1 — In a magnetron amplifier a narrow electron stream travels in crossed electric 

 and magnetic fields close to a wave transmission circuit. 



if the potential of the fins is made negative compared with that of the plane, 

 and the electrons are made to move in the opposite direction. 



In either range of parameters, the gain of the increasing wave in db per 

 wavelength is proportional to the square root of the current rather than 

 to the cube root of the current. This means a lower gain than for an ordinary 

 traveling-wave tube with the same circuit and current. 



Increasing and decreasing waves with a negative phase velocity are pos- 

 sible when the magnetic field is great enough. 



15.1 Circuit Equation 

 The circuit equation will be the same as that used in Chapter XIII, that is, 

 T/ - -icoriJC(^p - {h/u,Wy) , 



(r2 _ Y\) ^^-^-^"^ 



It will be assumed that the voltage is given by 



$ = {Ae~'''' + Be''^") (15.1) 



so that 



^'V = -jViAe''^" - Be'^"") (15.2) 



