624 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



The no space charge condition in the defined region is therefore 



^ ^ Arexp(,yoAT)\ ^, ^ ,kT\ exp(-gFoAr)-l 

 \ q^A / \q/ exp {- qVo/kT) 



To simplify notation define 



expi-qVo/kT] = 70 (D8) 



Next expand both No and u in Fourier series i 



00 

 Nd = ^ As sin sx + Bs cos sx (D9) 



s=0 



00 



u 



= 2Z «3 sin s-x + jSa cos sa: (DIO) 



»=o 



Substitution of (D9) and (DIO) into (D6) and equating coefficients of 

 like terms leads to the set of relations 



/3o = 4^ [^^(-^0 - 1) + ^J (1^11) 



K \1 + (s2/V47r-7o)/ 

 Now the wavelength of the sth component in (D9) is 



X. = 27r/s (D14)' 



If N'd contains no important components of wavelength shorter than 



Vto 



(D15) 



the Bk for such components may be set equal to zero. But then the only- 

 terms which appear in (D12) and (D13) are terms where the denomina- 

 tors which (with the aid of (D14)) may be written as 



may be set equal to k. Thus we have in place of (D12) and (D13) 



a, = ^As= 4^ As (D17) 



