Series and parallel resonance of crystal

The nominal operating frequency of the device is indicated on the outer casing of the quartz crystal, but that is only an approximation. In fact, the crystal has multiple resonant frequencies, even under ideal conditions.

Figure 1 shows an equivalent circuit of an ideal crystal in which there are only three circuit elements, a capacitor C1 in series and an inductor L1, and another capacitor C2 in parallel with the L1 C1 in series.

Figure 1: The equivalent circuit of an ideal quartz crystal

At a particular series resonant frequency, the input impedance Z is zero and L1 and C1 are in series resonance. Capacitor C2 has nothing to do with this.

However, at a particular parallel resonant frequency, the input impedance Z reaches infinity. This is the frequency at which the series combination of C2 and L1 C1 occurs in parallel resonance, exhibiting an inductive impedance. In order for the L1 C1 pair to be inductive in parallel resonance, the parallel resonance must occur at a higher frequency than the series resonance.

Therefore, the parallel resonant frequency can only be higher than the series resonant frequency, even if it is only a little higher. The parallel resonant frequency is never lower than the series resonant frequency. As for which resonance is more important to you, it depends on the application.

With uncomplicated mathematical operations, the impedance Z can be expressed as follows:

Figure 2: Impedance formula

Series resonant frequency: Fseries = 1/(2 * pi * sqrt(L1 C1))

Parallel resonant frequency: Fparallel = 1/(2 * pi * sqrt(L1 C1 C2 /(C1 + C2)))

Figure 3: Relative position of series and parallel resonance