**Introduction to Quartz Frequency Standards - Crystal Unit Equivalent Circuit**

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A quartz crystal unit is a quartz wafer to which electrodes have been applied, and which is hermetically sealed in a holder structure. (The wafer is often referred to as the "blank," or the "crystal plate".) Although the design and fabrication of crystal units comprise a complex subject, the oscillator designer can treat the crystal unit as a circuit component and just deal with the crystal unit's equivalent circuit.

The mechanically vibrating system and the circuit shown in Figure 2 are "equivalent," because each can be described by the same differential equation [6]. The mass, spring, and damping element (i.e., the dashpot) correspond to the inductor, capacitor and resistor. The driving force corresponds to the voltage, the displacement of the mass to the charge on the capacitor, and the velocity to the current.

**Figure 2.** Equivalent circuit of a mechanically vibrating system.

A crystal resonator is a mechanically vibrating system that is
linked, via the piezoelectric effect, to the electrical world.
Figure 3 shows a (simplified) equivalent circuit (of one mode
of vibration) of a resonator, together with the circuit symbol
for a crystal unit. A load capacitor C_{L} is shown in
series with the crystal. C_{0}, called the "shunt" capacitance,
is the capacitance due to the electrodes on the crystal plate
plus the stray capacitances due to the crystal enclosure. The
R_{1}, L_{1}, C_{1} portion of the circuit is the "motional arm,"
which arises from the mechanical vibrations of the crystal.

**Figure 3.** Equivalent circuit of crystal unit with load capacitor.

The C_{0} to C_{1} ratio is a measure of the interconversion between
electrical and mechanical energy stored in the crystal, i.e.,
of the piezoelectric coupling factor, k. C_{0}/C_{1}
increases with the square of the overtone number; the relationship
of C_{0}/C_{1} to k and N is 2C_{0}/C_{1}
= [pN^{2}/2k], where N is the
overtone number. When a dc voltage is applied to the electrodes
of a resonator, the capacitance ratio C_{0}/C_{1}
is also a measure of the ratio of electrical energy stored in
the capacitor formed by the electrodes to the energy stored elastically
in the crystal due to the lattice strains produced by the piezoelectric
effect. Figure 4 shows the reactance versus frequency characteristic
of the crystal unit. The C_{0}/C_{1} is also inversely
proportional to the antiresonanceresonance frequency separation
(i.e., the polezero spacing) which is an especially important
parameter in filter applications. The slope of the reactance vs.
frequency curve near f_{S} is inversely proportional to
C_{1}, i.e., DX/(Df/f)
» 1/pfC_{1}
near f_{S}, where X is the reactance. C_{1} is,
therefore, a measure of the crystal's "stiffness," i.e.,
its tunability.

**Figure 4.** Reactance versus frequency of a crystal unit.

When the load capacitor is connected in series with the crystal, the frequency of operation of the oscillator is increased by a Df' where Df' is given by

For the simple RLC circuit of Figure 2, the width of the resonance
curve is inversely proportional to the quality factor Q, but in
a crystal oscillator, the situation is complicated by the presence
of C_{0} and by the fact that the operating Q is lower than the resonator
Q. For a quartz resonator, Q = (2pf_{S}C_{1}R_{1})^{1}.
References 3, 5 and 6 contain further details on the equivalent
circuit.

Some of the numerous advantages of a quartz crystal resonator
over a tank circuit built from discrete R's, C's and L's are that
the crystal is far stiffer and has a far higher Q than what could
be built from normal discrete components. For example, a 5 MHz
fundamental mode ATcut crystal may have C1 = 0.01 pF, L1
= 0.1 H, R1 = 5W, and Q = 10^{6}.
A 0.01 pF capacitor is not available, since the leads attached
to such a capacitor would alone probably contribute more than
0.01 pF. Similarly a 0.1 H inductor would be physically large,
it would need to include a large number of turns, and would need
to be superconducting in order to have a £5
W resistance.

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