652 BELL SYSTEM TECHNICAL JOURNAL 



Chapter VIII. We can, however, show a relation between gain and the 

 properties of the circuit and electronic curves for cases in which the curves 

 almost touch (an electron velocity just a little lower than that for which gain 

 appears). Suppose the curves nearly touch at = 0i , as indicated in Fig. 

 14.11. Let 



e = e^-\- p (14.91) 



Let us represent the curves for small values of p by the first three terms of a 

 Taylor's series. Let the ordinate y of the circuit curve be given by 



y=a, + hp^ cip^ (14.92) 



and let the ordinate of the electronic curve be given by 



y = (h+b2p-\- C2f (14.93) 



Then, at the intersection 



(ci - C2)/»2 + (bi - b2)p + (a, - oa) = 



If we choose dx as the point at which the slopes are the same 



bi- b2 = (14.95) 



and we see that the imaginary part of p increases with the square root of the 

 separation, and at a rate inversely proportional to the difference in second 

 derivatives. This is exemplified by the behavior of Xi and X2 for b a little small 

 than (3/2)(2)i/» in Fig. 8.1. 



Now, referring to Fig. 14.10, we see that a circuit curve which cuts the 

 axis at a shallow angle (a high-impedance circuit curve) will approach or be 

 tangent to the electronic curve at a point where the second derivative is 

 small, while a steep (low impedance) circuit curve will approach the elec- 

 tronic curve at a point where the second derivative is high. This fits in with 

 the idea that a high impedance should give a high gain and a low impedance 

 should give a low gain. 



