FIELD SOLUTIONS 



651 



from those represented by its intersection with the real plane may be very 

 sensitive to the shape of the intersection with the real plane. Thus, we would 

 scarcely be justified by the good fit of the approximations represented in 

 Figs. 14.8 and 14.9 in assuming that the complex roots obtained using the 

 approximations will be good except when they correspond to a near approach 

 of the electronic and circuit curves, as in Fig. 14.10. 



In fact, using the approximate curves, we find that the increasing wave 

 vanishes for electron velocities less than a certain lower limiting velocity. 

 This corresponds to cutting by the circuit curve of the dip down from -\- oo 

 of the approximate electronic curve (the dip is not shown in Fig. 14.9). 

 This is not characteristic of the true solution. An analysis shows, however. 



0.5 



-1.0 -0.5 0.5 1.0 



P 



Fig. 14.11— Complex roots are obtained when curves such as those of Fig. 14.10 do not 

 have the number of intersections required (by the degree of the equation) for real values 

 of the abscissa and ordinate. In this figure, two parabolas narrowly miss intersect ng. 

 Suppose these represent circuit and electronic susceptance curves. We find that the gain 

 of the increasing wave will increase with the square root of the separation at the abscissa 

 of equal slopes, and inversely as the square root of the difference in second derivatives. 



that there will be a limiting electron velocity below which there is no in- 

 creasing wave if there is a charge-free region between the electron flow and 

 the circuit. 



14.6 Some Remarks About Complex Roots 



If we examine our generalized circuit expression (14.60) we see that the 

 circuit impedance parameter {E^/fi-P) is inversely proportional to the slope 

 of the circuit curve at the point where it crosses the horizontal axis. Thus, 

 low-impedance circuits cut the axis steeply and high-impedance circuits cut 

 the axis at a small slope. 



We cannot go directly from this information to an evaluation of gain in 

 terms of impedance; the best course in this respect is to use the methods of 



