Crystal oscillator parallel and series resonance

A crystal oscillator uses a piezoelectric quartz crystal as the frequency-determining element in place of an LC tank. When an ac voltage is applied, the crystal vibrates mechanically at its natural frequency, and the vibration generates an ac voltage (piezoelectric effect). Because the crystal has a very high Q (10⁴ to 10⁶) and its frequency hardly changes with temperature, the oscillator is very stable (a few ppm).

Equivalent circuit and two resonances

        +-----[ C0 ]-----+     C0 = holder (shunt) capacitance
        |                |
 o------+--[L]--[C]--[R]-+------o

L, C and R represent the mass, stiffness and losses of the crystal (L large, C very small, R small).

• Series resonance (L with C): fs = 1 / (2π √(LC)). The crystal impedance is at its minimum and purely resistive (R).
• Parallel resonance (L with the series combination of C and C0): fp = 1 / (2π √(L Ceq)), with Ceq = C C0/(C + C0). It is slightly higher than fs, and the impedance is at its maximum. Between fs and fp the crystal is inductive.

(a) Series resonance mode

              +VCC
                |
              [RC]
                |
 +---------B-C  NPN ----+----o Output
 |         E            |
[R1]       |            |
 |       [RE]||[CE]     |
 |         |            |
[R2]       GND          |
 |                      |
 +-------[Crystal]------+

The crystal is in the feedback path from the collector to the base. At fs its impedance is minimum, so the feedback is maximum; the circuit oscillates at fs. Away from fs the impedance rises and the feedback drops, so oscillation is sustained only at fs.

(b) Parallel resonance mode (Pierce / Colpitts type)

 Collector o----+--------+
                |        |
               [C1]   [Crystal]
                |        |
 Ground      o--+        |
                |        |
               [C2]      |
                |        |
 Base        o--+--------+

Between fs and fp the crystal behaves as a high-Q inductor, and it replaces the inductor L of the Colpitts oscillator. Together with C1 and C2 it forms the tank, and oscillation occurs at a frequency between fs and fp:

f = 1/(2π √(L Ceq)), where Ceq = C·CL/(C + CL) and CL = C0 + C1C2/(C1 + C2) is the load capacitance seen by the crystal

Parallel mode needs a specified load capacitance (given by the manufacturer) for the correct frequency. Applications: clocks in microprocessors, watches, communication transmitters and frequency counters.