220 BELL SYSTEM TECHNICAL JOURNAL 



worse than Fig. 5.8 implies. Thus, there is a decrease of {Er/^'^Py^ due to 

 wall thickness. Thickening the tlat opposed walls of the resonators decreases 

 the spacing between the opposed surfaces, increases the capacitance and 

 hence increases the stored energy for a given gap voltage. In F"ig. 5.9 the 

 factor/ by which {Er/^Py^ is reduced is i)lotted vs. the ratio of the wall 

 thickness / to the resonator spacing L. 



There is a further reduction of effective lield because of the electrical 

 length, 6 in radians, of the space between opposed resonator surfaces. 

 The lower curve in Fig. 5.10 gives a factor by which (Er/^-Py^ is reduced 

 because of this. If the resonator spacing, di in radians, is greater than 2.33 

 radians, it is best to make the opening, or space between the walls, only 

 2.33 radians long by making the opposed disks forming the walls very 

 thick. 



There is of course a further loss in effective field, both in the helix and in 

 circuits made up of resonators, because of the falling-off of the field toward 

 the center of the aperture through which the electrons pass. This was dis- 

 cussed in Chapter IV. 



Finally, it should be pointed out tliat the fraction of the stored energy 

 dissipated in losses during each cycle is inversely proportional to the Q of 

 the circuit or of the resonators forming it. The distance the energy travels 

 in a cycle is proportional to the group velocity. Thus, for a given Q the sig- 

 nal will decay more rapidly with distance if the group velocity is lowered 

 (to increase Er/l^P). Equations (5.38), (5.42) and (5.44) pertain to attenu- 

 ation expressed in terms of group velocity. The table at the end of the 

 chapter shows that a circuit made up of resonators and having a low enough 

 group velocity to give it an impedance comparable with that of a helix can 

 have a very high attenuation. 



5.1 Group and Phase Velocity 



Suppose we use a broad video pulse F{t), containing radian frequencies 

 p lying in the range to />o , to modulate a radio-frequency signal of radian 

 frequency co which is much larger than po , so as to give a radio-frequency 

 pulse /(/) 



/(/) = e'"'Fil) (5.1) 



the functions P{l) and /(/) are indicated in Fig. 5.1. 



F(l), which is a real function of time, can be expressed by means of its 

 Fourier transform in terms of its frequency components 



FiO - r A{p)e'"' dp {S2) 



V— Tin 



