632 BELL SYSTEM TECHNICAL JOURNAL 



used in earlier chapters, fitted to the true curve in slope and magnitude at 

 -y = 0. Over the range of B of interest in conneclion with increasing waves, 

 the fit is good. 



The difference between HJEz for the central space without electrons 

 (Fig. 14.3) and Hx/Ez for the central space with electrons (Fig. 14.5) can 

 be taken as representing the driving effect of the electrons. The solid curve 

 of Fig. 14.9 is proportional to this difference, and hence represents the true 

 effect of the electrons. The dashed curve is from the ballistical equation 

 used in previous chapters. This has been fitted by adjusting the space- 

 charge parameter Q only; the leading term is evaluated directly in terms of 

 current density, beam width, /5, and variation of field over the beam, which 

 is assumed to be the same as in the absence of electrons. 



Figure 14.10 shows a circuit curve (as, of Fig. 14.8) and an electronic 

 curve (as, of Fig. 14.10). These curves contain the same information as the 

 curves (including one of the dashed horizontal lines) of Fig. 14.5, but dif- 

 ferently distributed. The intersections represent the modes of propagation. 



If such curves were the approximate (dashed) curves of Figs. 14.8 and 

 14,9, the values of 6 for the modes would be quite accurate for real inter- 

 sections. It is not clear that "intersections" for complex values of 6 would be 

 accurately given unless they were for near misses of the curves. In addition, 

 the compHcated behavior near 6 = \ (Fig. 14.6) is quite absent from the 

 approximate electronic curve. Thus, the approximate electronic curve does 

 not predict the multitude of unattenuated space-charge waves near 0=1. 

 Further, the approximate expressions predict a lower limiting electron 

 velocity below which there is no gain. This is not true for the e.xact equations 

 when the electron flow fills the space between the finned structures com- 

 pletely. 



It is of some interest to consider complex intersections in the case of 

 near misses by using curves of simple form (parabolas), as in Fig. 14.11. 

 Such an analysis shows that high gain is to be expected in the case of curves 

 such as those of Fig. 14.10, for instance, when the circuit curve is not steep 

 and when the curvature of the electronic curve is small. In terms of physical 

 parameters, this means a high impedance circuit and a large current density. 



14,1 The System and the Equations 



The system examined is a two-dimensional one closely analogous to that 

 of Fig. 4.4. It is shown in Fig. 14.1. It consists of a central space extending 

 from y = —d\.oy= -\-d, and arrays of thin fins separated by slots ex- 

 tending for a distance // beyond the central opening and short-circuited at 

 the outer ends. An electron flow of current density Jq amperes/w^ fills the 

 open space. It is assumed that the electrons are constrained by a strong 

 magnetic field so that they can move in the z direction only. 



