When designing an oscillator in which the required stability is ± 50 ppm or tighter, you need the stability and accuracy of quartz frequency control. Quartz plates show a mechanical movement or strain when subjected to an electrical charge. Conversely, they show a potential difference between the two faces when subjected to a mechanical stress. This relationship is known as the piezoelectric effect. Because of its electro-mechanical properties, a crystal placed in an oscillator circuit can be made to oscillate both mechanically and electrically, with its resonant frequency determined primarily by its mechanical dimensions. Quartz crystal accuracy and stability far surpass the performance obtained by circuits utilizing conventional capacitors, inductors, and resistors. There are many types of crystal cuts used in electrical circuits. The AT-cut is the preferred cut for most applications above 1.0 MHz, and is the only cut discussed in this catalog.

A typical un-canned crystal is shown in Figure 1. This consists of a quartz blank with a metal electrode on each of the two major surfaces. The crystal is mounted in a leaded header and at a later stage is encapsulated by welding a cover over the assembly. This component behaves electrically as the circuit in Figure 2. This circuit is called the equivalent circuit.

The mechanical losses of the crystal appear as an equivalent series resistance, R₁, while the mechanical elasticity of the crystal is equivalent to a series capacitor, C₁. C₀ is the parallel capacitance associated with the holder and the electrode capacitance.

The frequency of a crystal operating at series resonance is given by:

At series resonance the reactance of C₁ and L₁ are equal and opposite thus the net reactance of the series circuit is zero. At this point the crystal appears as a resistance R₁. Because R, is very small relative to the reactance of C₀, in reality, when measured, the apparent series resonant frequency is affected very minimally by the presence of C₀. It is usually ignored. The crystal impedance is a pure resistance having a relatively low value. (20 to 300 Ohms in the frequency range 1 to 200 MHz.)

At frequencies higher than f_s the inductive reactance increases and the capacitive reactance equals that of C₀, the crystal vibrates at the anti-resonant frequency f_a. f_a is given by:

Any external load capacitance C_L added to the crystal circuit becomes a portion of the frequency determining network and the actual frequency is decreased from that of the theoretical frequency f_a. The calculation for the crystal frequency operating with an external load capacitance is given by:

The frequency change or "pullability" created by C_L is given by:

The pullability is useful in TCXO and VCXO applications and in compensating for oscillator circuit variations.

Inherent to the thickness shear mode of vibration there exist frequencies other than the fundamental frequency. These are referred to as harmonic (overtone) and inharmonic modes (spurs). The harmonic modes are related to the fundamental frequency and occur at odd harmonic intervals (1,3,5,7,etc.). The inharmonic modes are not harmonically related to the fundamental mode. The inharmonic can be controlled in amplitude and in frequency separation from the main mode by varying the size and shape of the quartz resonator and of the electrodes.