656 BELL SYSTEM TECHNICAL JOURNAL 



For large values of transit angle it is convenient to have these im- 

 pedances expressed in closed instead of in series form. By using (41) 

 instead of (41a) throughout the analysis the result may be obtained. 

 The expressions for Zg and Z\ will not differ from those given above, 

 but Zc, Zi and Zz may be written in the following forms: 



Z2 = ^ [^ - hUe-^^^ + 1) - 2(6-"^^ _ 1) 1, (83a) 



Z3=^[^4(/. + ,F) 



+ I2{h - gh - g)g-(W)^c -I- 2(1 - h + gh)e-''^' + Ige'^^ - 2] 

 + |- (1 - g)(e-(^+^'^^ - e-^^^ - e-^^ + 1)1- (84a) 



Pc J 



Of the various impedances Zz is the only one that is really trouble- 

 some. The values of Zc and Z^ may be obtained from data given in 

 published papers ^' ^ when it is noticed that Z2/h* may be calculated 

 from Zc if jSc is replaced by jSp — h^c- In the treatment of Zz there 

 seems to be no easy road, although the series form (84) may be ex- 

 panded with comparative ease. 



Further steps consist in the transposition of these equations to 

 obtain convenient forms and to show how they harmonize with low- 

 frequency theory. A useful expression may be obtained from the 

 fundamental relations (76), (77), (78), (79) by eliminating Va, h and 

 Ig. The result is 



T r [(^^ + ^2 + Zz)Zg 4- (Zi + Z2)Zc'] ^ .^^. 



This begins to look like the familiar low-frequency equation 



(Vc - F,)i + m(^c - FJi = I,Z,„ (86) 



where /x is the amplification factor and Z,, is the internal plate impe- 

 dance. The two are equivalent at all frequencies if Ip is interpreted 

 as the density of the total plate current, and not the conduction com- 



