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BELL SYSTEM TECHNICAL JOURNAL 



14.3 Waves in the Presence of Electrons 

 In this section we deal with the equations 



„ -(eiA)"'-'tanh [{e,/ey'\d' - 61)'^'] 



and 



6iA = 1 - 



{6- - eiyi'- 



A 



{e. - ey 



(14.25) 



(14.15) 



We consider cases in which the electron velocity is much less than the 

 velocity of light; hence 



e, » eo (14.33) 



-20 -15 -10 -5 5 10 15 20 



'P 



Fig. 14.4 — If a quantity proportional to Hx/Et at the edge of the central region is 

 plotted vs 4> = —jd, this curve is obtained. There are an infinite number of intersections 

 with a horizontal hne representing the susceptance of the finned structure. These corre- 

 spond to passive modes, for which the field decays exponentially with distance away from 

 the point of excitation. 



In Fig. 14.5, the right-hand side of (14.25) has been plotted vs. 6 for 

 de = 10 00 , corresponding to an electron velocity 1/10 the speed of light. 

 Values oi 6 = 1/10 and A = 1/100 have been chosen merely for conven- 

 ience.* The curve has not been shown in the region from d = .9 \o 6 = 1.1, 

 where ei/e is negative, and this region will be discussed later. 



For a larger value of 7*(| P \ small). Pi in Fig. 14.5, there are 4 intersec- 

 tions corresponding to 4 unaltenuated waves. The two outer intersections 

 obviously correspond to the "circuit" waves we would have in the absence 

 of electrons. The other two intersections near 6 = .9de and B = \Ade we 

 call electronic or space-charge waves. 



* At a beam voltage Vo = 1,000 and for d = 0.1 cm, A = 1/100 means a current density 

 of about .S.SO ma/cm^, which is a current density in the range encountered in practice. 



