308 BELL SYSTEM TECHNICAL JOURNAL 



The quantities in which we are specifically interested are electric field, 

 amplification factor, and current density. The electric field is equal to the 

 negative gradient of the potential function. The amplification factor is 

 found by taking the ratio of partial derivatives of the electric field at the 

 cathode with respect to grid and anode voltages. The current density may 

 then be studied for any assumed operating values of grid and plate voltages. 



To calculate the gradient we note that since V{x, y) is the real part of an 

 analytic function W{z) = V -\- iU, it follows from the Cauchy-Riemann 

 equations, 



W'(z) = ^ - i ^ = -£, + ,£, (16) 



ox oy 



where Ex and Ey are the x- and y- components of the electric intensity. 

 From (1), 



AW{z) = [{di + d^) F{z) - lird^iiz + d<,)/a]V\ 



+ [B{di - iz) + liri d.z/a - diF{z)]\'\ 



(17) 



where 



/^(.)-fa^^^"^%~^;f^"^ (18) 



Calculating the derivative and making use of the relation, 

 d[{z - ttt) d[{^S) 



i?i(2 - xr) ??i(2) 



we find at the cathode surface 



+ 2i, (19) 



where 



F'{x-id{)= ^'[1+C(.v)] (20) 



a 



, . dxU{x + idi)a] , . 



C{x) = Im — - — ^-^ -j- (21) 



§1 7r(.T + I di) a] 



It follows that when y ^ —di,\ve must have Ex = and 

 a^£^/27r = [di + {di + d2)C{x)]V, 



+ [^2 - di - aB/lw - diC{x)]Vp 

 The ampUfication factor is then given by 



dEy/dV, _ di-\- {di + d2)C{x) 



(22) 



(23) 



'^ dEy/dVp di- di- aB/2T - diC{x) 



Numerical calculation from these formulas can be made by means of (5) 

 and (6). When d^ and d^ are both large compared with unity, Eq. (23) 



