**Oscilent Corporation
- Technical References**

Introduction to Quartz Frequency Standards

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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.