18 BELL SYSTEM TECHNICAL JOURNAL 



In considering circuits for traveling-wave tubes, and in reformulating 

 the theory in more general terms later on, it is valuable to express C in terms 

 of parameters other than the characteristic impedance. Two physically sig- 

 nificant parameters are the power flow in the circuit and the electric field 

 associated with it which acts on the electron stream. The ratio of the square 

 of the electric field to the power can be evaluated by physical measurement 

 even when it cannot be calculated. For instance, Cutler did this by allowing 

 the power from a wave guide to flow into a terminated helix, so that the 

 power in the helix was the same as the power in the wave guide. He then 

 compared the field in the helix with the field in the wave guide by probe 

 measurements. The field strength in the wave guide could be calculated in 

 terms of the power flow, and hence Cutler's measurements enabled him to 

 evaluate the field in the helix for a given power flow. 



The magnitude of the field is given in terms of the magnitude of the 

 voltage by 



E= \VV\ (2.40) 



Here E is taken as the magnitude of the field. The power flow in the circuit 

 is given in terms of the circuit voltage by 



P = \V \y2K (2.41) 



A quantity which we will use as a circuit parameter is 



£V/32P = 2K (2.42) 



Here it has been assumed that we are concerned with low-loss circuits, so 

 that T\ can be replaced by the phase constant 0^. Usually, /3 can be taken 

 as equal to /3e, the electron phase constant, with small error, and in the 

 preceding work this has been assumed to be exactly true in (2.23). 

 In terms of this new quantity, C is given by 



C = i2K)iIo/SVo) = (E'/0-P){Io/SVo) (2.43) 



If we call Vo/h the beam impedance, C^ is j the circuit impedance divided 

 by the beam impedance. It would have been more sensible to use E-/20-P 

 instead of Er/0P. Unfortunately the writer feels stuck with his benighted 

 first choice because of the number of curves and pubUshed equations which 

 make use of it. 



Besides the circuit impedance, another important circuit parameter is 

 the phase velocity. As the electron velocity is made to deviate from the 

 phase velocity of the circuit, the gain falls off. An analysis to be given later 



^ C. C. Cutler, "Experimental Determination of Helical-Wave Properties," Proc. IRE, 

 Vol. 36, pp. 230-233, February 1948. 





