This equivalent circuit exhibits both series and parallel resonance. Series resonance occurs at the frequency where the reactances of L1 and C1 are equal. At a slightly higher frequency, parallel resonance occurs when the combination of L1 and C1 exhibit an inductive susceptance which resonates with C2. Assume all frequencies (f) are in radians/sec:
Series resonance: |XL1| = |XC1|
fsL1 = 1/(fsC1)
fs= (L1*C1)-1/2
Parallel Resonance: |XC2| = |XL1 + XC1| = |XL1| - |XC1|
1/(fpC2) = (fpL1)-1/(fpC1)
1/(fpC2) + 1/(fpC1) = fpL1
(1/C1)(1+C1/C2) = (fp)2L1
(1+C1/C2)/(C1*L1) = (1+C1/C2)(fs)2=(fp)2
or,
fp= fs(1+C1/C2)1/2
since C1<<C2, we can approximate (1+C1/C2)1/2= 1+(1/2)(C1/C2), thus:
fp= fs + df
where df = (1/2)(C1/C2)fs
If we assume that the Pierce oscillator circuit operates approximately midway between series and parallel resonances, it is possible to determine values which can make the model of figure 1 useful. Also assume the series resistance of the crystal is about 600 ohms, the packaging capacitance is 8 pF,and the Q is approximately 10,000.
If we accurately measure the frequency of operation of the Pierce oscillator as, say, 75398223 r/s, then
XL= f*L1 = Q*R1
thus
L1 = Q*R1/f = 10,000*600/75398223 = .07957747 Henry
If we say L1 = .08 Henry, then Q will not exactly equal 104 but that's OK. We must find C1 in two steps. First, find an approximate value by recognizing that f is approximately equal to fs. Then C1 can be computed approximately by
C1 = 1/(L1*f2) = 1/((0.08)(75398223)2) = 2.2 fF (approximately)
We can now approximate df,
df = (1/2)(C1/C2)f = (0.5)(2.2E-15/8E-12/)(75.4E6) = 10368 r/s
If the operating frequency of the circuit is midway between series and parallel resonance, then the series resonance frequency must be equal to the operating frequency minus df/2, or 75393039 r/s. We can now use this value of fs to compute a precise value to use for C1:
C1 = 1/(L1*(fs)2) = 1/((0.08)(75393039)2) = 2.199113 fF
Our model thus contains:
L1 = 0.08 H
C1 = 2.199113 fF
R1 = 600 ohms
C2 = 8 pF
fs = 75393039 r/s
df = 10362 r/s
fp = fs + df = 75393039 r/s + 10362 r/s = 75403401 r/s