614 



BELL SYSTEM TECHNICAL JOURNAL 



It is convenient to write the resulting equations in terms of d-c. 

 and first order a-c. values where the initial values of d-c. velocity 

 and acceleration are given, but initial values of a-c. velocity and 

 conduction current are employed. The first order a-c. relation derived 

 by Llewellyn may be written in the form 



hm time nme 



(57) 



where qa and fXa are the initial values of fluctuation conduction current 

 and velocity, respectively, while A, B and C are defined by: 



A =W - io^^x 4- -^ (2 ■- 2e-'" 



id 



11 



B 



C = 



- -r^ [_aa{iee-^^ + e-'" - 1) + Uaio^ie-^' - 1)] 



hmtisP' 



(58) 



in which ?? is the transit angle, wr, the transit time being r, and /o is 

 the d-c. current. 



In the application of these relations to noise analysis, the initial 

 values of velocity, acceleration, and conduction current must be taken 

 at a point in the ^S-region beyond the potential minimum, but just as 

 close to it as possible without encountering conditions where electrons 

 may be moving toward the cathode, for the equations apply only to 

 cases where the electrons are moving in one direction only. The 

 initial point is, however, located so near to the potential minimum 

 that the d-c. acceleration in (58) may be taken as zero. When this 

 is the case, it may be shown that the initial conduction current is 

 equal to the total current. In other words, the initial value of dis- 

 placement current is zero. Under such conditions (57) and (58) 

 reduce to the following expression for the a-c. anode potential in 

 terms of the a-c. component of current and initial velocity : 



V = 



/i 



nme \ 6 



+ (^""Uaiie + e~^^ - 1) 



-f '=^ [ide-'' + e-'o - 1]. (59) 

 co'e 



The term multiplying the a-c. current h in the above equation is 

 the internal high-frequency impedance z of the planar diode. The 

 last^term may therefore be identified with an internal emf. When 

 the initial velocity /x, is expressed in terms of the fluctuation of electron 



