506 BELL SYSTEM TECHNICAL JOURNAL 



The total capacitance Cout includes two contributions: from the active 

 electrode area inside the tube (C22) and from the passive resonating circuit 

 (Cp2). It is convenient to consider these separately, writing the figure of 

 merit as follows: 



I r. fB = iKi- 7 ^-V7 T^x (4) 



47rGii C22 



(' + l')C + §^') 



The first factor is the "intrinsic" electronic figure of merit of the active 

 transducer alone, while the second factor expresses the deterioration 

 caused by input passive circuit loss Gpi and output passive circuit ca- 

 pacitance Cp2, both of which should ideally be held as small as possible. 



Consider the first factor, the intrinsic electronic gainband product 

 which depends only upon the properties of the electron stream and the 

 electrode dimensions in the regions occupied by the electron stream. 



It is the responsibility of the tube design engineer to maximize this 

 product consistent with any limitations which may be imposed by me- 

 chanical, emission, thermal or circuital considerations. 



On the other hand, in maximizing this intrinsic gain-band product, the 

 tube engineer must not proceed in ignorance of the effect of his actions 

 on the possibility of obtaining a favorable value for the second factor. 

 For example, he may attempt to make C22 so small (in order to maximize 

 the first factor) that it becomes physically impossible to obtain an effec- 

 tive circuit capacitance Cpi which is not large compared to €21 ■ In such a 

 case, the actual gain-band product would be much smaller than the in- 

 trinsic product of which the tube would be capable if circuit capacitance 

 were negligible. Such a balancing of effects will become apparent from 

 the subsequent discussion. 



It is desired, therefore, to express the transadmittance, input conduct- 

 ance and output capacitance of the electronic transducer in terms of such 

 parameters as cathode current density, electrode dimensions, frequency 

 and potentials in such a way that it will become clear how a maximizing 

 process may be carried out b}' adjusting these parameters. 



As a first approximation let us use the results of Llewellyn and Peter- 

 son's analysis of plane-parallel flow^, which makes the following assump- 

 tions: 



1. All electrons are emitted with zero velocity. 



2. All electrons in a given plane have the same velocity. 



' F. B. Llewellyn and L. C. Peterson, "Vacuum Tube Networks," Proc. L R. £., i2, ' 

 144-166 (1944). 



