GENERAL CIRCUIT CONSIDERATIONS 219 



impedances attained through large values of —dv/d(j: are useful over a nar- 

 row range of frequency only. 



If we consider a broad electron stream of current density /o , the highest 

 effective value of B?/^'^P, and hence the highest value of C, will be attained 

 if there is current everywhere that there is electric field, and if all of the 

 electric field is longitudinal. This leads to a limiting value of C, which is 

 given by (5.23). There Xo is the free-space wavelength. The nearest practical 

 approach to this condition is perhaps a helix of fine wire flooded inside and 

 outside with electrons. 



In many cases, it is desirable to consider circuits for use with a narrow 

 beam of electrons, over which the field may be taken as constant. As the 

 helix is a common as well as a very good circuit, it might seem desirable 

 to use it as a standard for comparison. However, the group velocity of the 

 helix differs a little from the phase velocity, and it seems desirable instead 

 to use a sort of hypothetical circuit or field for which the stored energy is 

 almost the same as in the helix, but for which the group velocity is the same 

 as the phase velocity. This has been referred to in the text as a "forced 

 sinusoidal field." In Fig. 5.3, (E-^/(3~Py'^ for the forced sinusoidal field is 

 compared with {Er/^'Pyi^ for the helix. 



Several other circuits are compared with this: the circular resonators of 

 Fig. 5.4 (the square resonators of Fig. 5.4 give nearly the same impedance) 

 and the resonant quarter-wave and half-wave wires of Figs. 5.6 and 5.7. 

 The comparison is made in Fig. 5.8 for three voltages, which fix three phase 

 velocities. In each case it is assumed that in some way the group velocity 

 has been made equal to the phase velocity. Thus, the comparison is made on 

 the basis of stored energies. The field is taken as the field at radius a (cor- 

 responding to the surface of the helix) in the case of the forced sinusoidal 

 field, and at the point of highest field in the case of the resonators. 



We see from Figs. 5.8 and 5.3 that a helix of small radius is a very fine 

 circuit. 



In circuits made up of a series of resonators, the group velocity can be 

 changed within wide limits by varying the coupling between resonators, as 

 by putting inductive or capacitive irises between them. Thus, even cir- 

 cuits with a large stored energy can be made to have a high impedance by 

 sacrificing bandwidth. 



The circuits of Fig. 5.4 have a large stored energy because of the large 

 opposed surfaces. The wires of Fig. 5.6 have a small stored energy asso- 

 ciated entirely with "fringing fields" about the wires. The narrow strips of 

 Fig. 5.5 have about as much stored energy between the opposed flat sur- 

 faces as that in the fringing field, and are about as good as the half-wave 

 wires of Fig. 5.7. 



An actual circuit made up of resonators such as those of Fig. 5.4 will be 



