234 BELL SYSTEM TECHNICAL JOURNAL 



value for thin resonators with infinitely thin walls by a factor given by 

 (5.32), which is plotted vs. di in Fig. 5.10. 



If there are edge effects, the optimum gap spacing and the reduction in 

 {F?/^Pyi^ will be somewhat different. However, Fig. 5.10 should still be a 

 useful guide. 



In case of wide gap separation (large dt), there would be some gain in 

 using reentrant resonators, as shown in Fig. 4.11, in order to reduce the 

 capacitance. How good can such a structure be? Certainly, it will be worse 

 than a helix. Consider merely the sections of metal tube with short gaps, 

 which surround the electron beam. The shorter the gaps, the greater the 

 capacitance. The space outside the beam has been capacitively loaded, 

 which tends to reduce the impedance. This capacitance can be thought of 

 as being associated with many spatial harmonics in the electric field, which 

 do not contribute to interaction with the electrons. 



5.5 Attenuation 



Suppose we have a circuit made up of resonators with specified unloaded 

 Q.\ The energy lost per cycle is 



W^ = IwWs/Q (5.33) 



In one cycle, however, a signal moves forward a distance L, where 



L = vjj (5.34) 



The fractional energy loss per unit distance, which we will call 2q', is 



la = ^1-^ \ (5.35) 



whence 



0^ = 7^ (5.36) 



So defined, a is the attenuation constant, and the amplitude will decay 

 along the circuit as exp( — as). 

 The wavelength, X, is given by 



X = v/f = 27rVco (5.37) 



'J1ie loss per wavelength in db is 



db/wavelength = 20 logio exp(«X) 



db/wavelength = ~ 



t Disregarding coupling losses, the circuit and the resonantors will both have this 

 same Q. 



