408 



BELL SYSTEM TECHNICAL JOURNAL 



These waves correspond to the space-charge waves of Hahn and Ramo, and 

 are quite independent of the circuit impedance, which appears in (8.28) 

 merely as an arbitrary parameter defining the units in which 5 is measured. 

 Equation (8.28) also describes the disturbance we would get if we shorted 

 out the circuit by some means, as by adding excessive loss. 



Practically, we need an estimate of the value of Q for some typical cir- 

 cuit. In Appendix IV an estimate is made on the following basis: The helix 



60 



40 



1.0 

 0.8 



0.04 0.06 



0.1 0.2 



QC 



0.4 0.6 0.8 1.0 



Fig. 8.11— The variation of a quantity proportional to the cube of the gain of the in- 

 creasing wave (ordinate) with a quantity proportional to current (abscissa). For very 

 small currents, the gain of the increasing wave is proportional to the \ power of current, 

 for large currents to the \ power of current. 



of radius c is replaced by a conducting cylinder of the same radius, a thin 

 cylinder of convection current of radius ax and current of i exp{—jl3z) is 

 assumed, and the field is calculated and identified with the second term on 

 the right of (7.1). R. C. Fletcher has obtained a more accurate value of Q 

 by a rigorous method. His work is reproduced in Appendix \T, and in Fig. 

 1 of that appendix, Pletcher's value of () is compared with the approximate 

 value of Appendix IV. 



In I'ig. 8.12, the value (J(j3, y)'' of Appendix I\' is plotted vs. ya for ai/a 

 = .9, .8, .7. For fli/a = 1, ^ = (>■ In a typical 4,()()() mc travcHng-wave 



