402 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



In these equations (r, 6, z) is the position of an electron at time t ; dots 

 indicate differentiation with respect to /, following the electrons; rj = e/m., 

 where — e is the electronic charge and m its mass; and Vo is the potential 

 describing the steady, axially symmetric electric field. Relativistic ef- 

 fects and the magnetic field resulting from electron motion have been 

 neglected, as our interest is confined to beam velocities which are small 

 compared to that of light. 

 It is readily verified that a solution of the above eciuation is: 



;. = 0, d = do = riBo/2, k = 1(0 (12) 



rj dVo/dr = reo\ dVo/dz = (13) 



Thus all the particles in the beam have the same angular velocity, equal 

 to the Larmor angular frequency, and the same axial velocity Uq . From 

 Poisson's equation, we find the charge density: 



po = - 2ek'/v (14) 



It is convenient to introduce the angular plasma frequency cop , de- 

 fined by: 



cOp = —r]po/e = 2^o" (15) 



In steady-state flow, an electron with initial position (ro, ^o, ^^o) ha^ 

 the position (ro , ^o + di^t, Zo + uJ) at time t. When the beam is mod- 

 ulated by a small ac signal, the electrons suffer small ac displacements 

 from their steady-state trajectories. If we assume that the signal propa- 

 gates along the axis of the beam as exp j(o^t — yz), we can write the 

 perturbed electron coordinates in terms of the Lagrangian coordinates 

 (ro , 9o , Za) as follows: 



r = ro + f(ro)-expj"[co/ - 7(^0 + ud)] (16) 



e = Oo -h dot + e(ro)-expjWt - 7(^0 + WoO] (17) 



z = Zo -{- not + 2(ru) • exp j[a)/ - y{zo + ^'oO] (18) 



where the tildes indicate ac amplitudes, and the dots indicate, as before, 

 time differentiation at fixed ro , do , ^o- Thus the dots are eciuivalent to 

 multipHcation by j(w — 7?/o), when applied to ac quantities. 



The equations of motion for the ac modulated beam differ from the 

 steady-state equations (9) — (11), in that the particle coordinates are 

 now given by (16) — (18), and there are ac fields present in addition to 

 the dc fields —dVo/dr and Bo . As is usual in small-signal theory, only 

 first-order ac quantities are retained in any equation. To this approxi- 



