ULTRA-HIGH-FREQUENCY VACUUM TUBES 649 



relative potentials of grid and plate, the size of the grid wires, and the 

 spacing between them. 



In applying the general analysis for parallel planes to this type of 

 structure it is possible to find all of the desired initial conditions at 

 the 5-plane in terms of conditions at the ^ -plane as well as to find the 

 actual a.-c. potential of the grid wires themselves and the relation 

 between plate current and cathode current provided that a very simple 

 condition is fulfilled. This condition is merely that the size of the 

 potential pockets surrounding the grid wires is so small that the planes 

 A and B may be taken close enough together to cause the electron 

 transit time between the two to be negligibly small compared with the 

 transit time from cathode to A or from B to the anode. Without this 

 condition, the problem appears almost hopeless. With it, the pro- 

 cedure is straightforward and simple. 



For the factors afifecting the fulfillment of this condition. Maxwell's 

 analysis fortunately provides a guide that is at least safe, because if the 

 planes can be located near together compared with the distances Xc 

 and Xp on the basis of his analysis, then the smoothing effect of the 

 electrons between grid wires will make the actual operating conditions 

 still better. The adaptation of his equations gives the following 

 expression for the potential distribution and for small diameter grid 



V 



Vo ^ Vp 



Xc ~r Xp 





, - ) — — 2 logf ( 2 sin — ) 

 Xc -{- Xp / a \ a J 



— loge (1 — le^""^'" cos^'^^/" -f e'^'"^'") 



AirXp ( Xc -{- d 



Xc "I Xp 



+ ^-.(|^,)- (53) 



Here d is the distance from the grid to the place where the potential 

 is to be computed, and is considered positive when directed from the 

 grid toward the plate, z is distance measured parallel to the grid, a is 

 the distance between centers of grid wires, c is the wire radius, and the 

 cathode potential, Vc is taken to be zero. The only term here con- 

 tributing to the potential pockets surrounding grid wires is the loga- 

 rithmic one which alone involves z, the distance along plane A or B. 

 The condition for the variations to be smoothed out with either 

 positive or negative values of d is 



p2TrDla 



»2, 



