206 BELL SYSTEM TECHNICAL JOURNAL 



12.83 Exact Expressions for PI and Rc 



The error in PI caused by the assumption that the frequency is independ- 

 ent of the crystal resistance R\ , that is, by use of approximate equation 

 (12.86) for the frequency, may be investigated as follows: 



The impedance of the crystal and Ct in parallel is given by 



^ = .(C. + C,)'(l + m'p-) [^ - ^'C + '"^'(» - 1))' (12.97) 

 where 



„ MCo ^3 — W 



P = m = 



r -i- r 2 2 



y^o ^ ^t cos — wi 



COS = 27r times frequency of anti-resonance of the crystal and Ct combina- 

 tion when i?i = 



coi = 27r times frequency of resonance of the crystal and Ct combination 

 when Ri = 



(ji = 2ir times operating frequency 



(Note that P is the figure of merit of the crystal and Ct in parallel.) The 

 imaginary part of Z is 



_ 1 -f mP\m - 1) 

 ^ - co(Co + Ct){\ + m^P^) ^^^-^^^ 



The condition for stable oscillations requires A^ = 0. For this condition 



which defines the exact frequency of oscillation. The negative sign before 

 the radical is used since the effective resistance is greater at this frequency, 

 thus requiring less negative conductance for oscillation. 



With P large {m — > 0) the frequency of oscillations coincides with cos 

 which is the same as given by the approximate frequency equation (12.86). 

 The real part of (12.97) is 



^ = (r 4-rVi ^^^P^\ (^2.100) 



co(Co + Ct)\X + m P ) 



and when m = 



P M , , 



^ = (r A.r\ = ^ ^^2 (12.101) 



coCol 



(-8)' 



f 



