MAGNETRON AMPLIFIER 657 



If we use (15.15) and (15.18) in connection with (15.5) we obtain 



2 o -j^eVxVme - r) + 2j[a/{\ + a'')\^^\H^ 



1 — 1 1 — 



Now let 



If we assume 



^ « 1 (15.23) 



and neglect p in sums in comparison with unity, we obtain 



/>(/3.//3l - 1 - pWe/^l -\- pf - (/3n.M)'] 



= _l [(,./,. _._„+_^]^. 



(15.24) 



We are particularly interested in conditions which lead to an imaginary 

 value of p which is as large as possible. We will obtain such large values of p 

 when one of the factors multiplying p on the left-hand side of (15.24) is 

 small. There are two possibilities. One is that the first factor is small. We 

 explore this by assuming 



^,//3i -1 = (15.25) 



If p is very small, we can write approximately 



■ ^\ (1 + a-) ^1 (15.27) 



P = ±j[a/{l + a2)]i/2(^^/^Ji/2^ 



We see that p goes to zero if a = and is real if a is negative. If we con- 

 sider what this means circuit-wise, we see that there will be gain with the 

 d-c voltage applied between a circuit and a conducting plane as shown in 

 Fig. 15.3. 



Another possible condition in the neighborhood of which p is relatively 

 large is 



^,M - 1 = ± iSj^i (15.28) 



