Question E6D02:
What is the equivalent circuit of a quartz crystal?  

Choices:  
A: Series RLC in parallel with a shunt C representing electrode and stray capacitance  
B: Parallel RLC, where C is the parallel combination of resonance capacitance of the crystal and electrode and stray capacitance  
C: Series RLC, where C is the parallel combination of resonance capacitance of the crystal and electrode and stray capacitance  
D: Parallel RLC, where C is the series combination of resonance capacitance of the crystal and electrode and stray capacitance    

Correct Answer is  A: Series RLC in parallel with a shunt C representing electrode and stray capacitance  

The quartz crystal is modeled by the Butterworth–Van Dyke (BVD) equivalent: a motional series RLC branch in parallel with a static shunt capacitance from the electrodes and stray capacitance.

1) Why the correct answer is correct (A)
- The crystal’s mechanical resonance is represented by a series branch: $R_m$ (loss), $L_m$ (mass), and $C_m$ (elastic compliance).
- The electrode and holder add a static capacitance $C_0$ that is in parallel with the entire motional branch.
- So the equivalent is a series RLC in parallel with a shunt $C$: exactly choice A.

2) Why the other choices are incorrect
- B: “Parallel RLC ... and C is the parallel combination ...” Incorrect because the resonant mechanical path is a series $RLC$, not parallel. The electrode/stray capacitance is not merged into the motional C; it is a separate shunt $C_0$ in parallel with the whole series branch.
- C: “Series RLC, where C is the parallel combination ...” Incorrect because $C_m$ (motional capacitance) is not a parallel combination with electrode/stray capacitance. $C_0$ is outside the series branch, not combined into $C_m$.
- D: “Parallel RLC, where C is the series combination ...” Incorrect for two reasons: the motional branch is not a parallel RLC, and the electrode/stray capacitance is not in series with the motional elements.

3) Useful formulas and rules of thumb
- Series resonance (minimum impedance):
  $$f_s = \frac{1}{2\pi\sqrt{L_m C_m}}$$
  At $f_s$, the reactances of $L_m$ and $C_m$ cancel and the crystal impedance ≈ $R_m$ (low).
- Parallel/antiresonance (maximum impedance), due to interaction with $C_0$:
  $$f_p \approx f_s \sqrt{1 + \frac{C_m}{C_0}}$$
  Since typically $C_m \ll C_0$, $f_p$ is slightly above $f_s$.
- With added load capacitance $C_L$ (seen in oscillator circuits), the effective shunt capacitance is $C_0 + C_L$, so:
  $$f_{p,\text{load}} \approx f_s \sqrt{1 + \frac{C_m}{C_0 + C_L}}$$
- Typical magnitudes: $C_m$ in femtofarads to tens of fF; $C_0$ in a few pF. Hence small separation between $f_s$ and $f_p$.