20g BELL SYSTEM TECHNICAL JOURNAL 



where 



M = the figure of merit of the crystal 



2 2 



0)2 — CO 



^ = 1 2 



C02 — Wl 



coi = 27r times frequency of resonance of the crystal 



aj2 = 27r times frequency of anti-resonance of the crystal 



CO = the independent variable, lir times frequency 



Co = the static capacitance of the crystal 



The resistive component of Ze is 



M Ri 



Re = 



coCo(l +n^M^) _1_ 4- 2 (12.105) 



In order to express Re in terms of Co and Cj , the quantity n must be expressed 

 in terms of these variables which define the oscillating frequency. This is 

 accomplished as follows: 



The equation of ratio of capacitances of a crystal is 



Cx ^ co^^I ^j2.106) 



t-'O COi 



Similarly, when the capacitance Ct is placed across the crystal 



^' ''' ~ '"' (12.107) 



Co + Ct col 



where C03 is 27r times the anti-resonant frequency of the crystal and Ct in 



parallel. The ratio of (12.107) and (12.106) is 



^ 22 



Co C03 — coi 



Co ~r Ct C02 



2 2 (12.108) 



COl 



The oscillating frequency is given by (12.99) in which m is as defined 

 under (12.97). The oscillating frequency co is therefore given by 



or 



i-/j/rr 



4^ 



P2 (12.109) 



l+//l^ 



4^ 



P2 (12.110) 



