An understanding of how a quartz-crystal resonator works can provide design engineers with a better understanding of crystal oscillators. Quartz crystals have very desirable characteristics for use in tuned oscillator circuits, since their natural oscillation frequencies are very stable. In addition, the resonance has a very-high quality factor (Q), ranging from 10,000 to several hundred thousand. In some cases, Q values of two million are possible. Ultimately, however, Q values and stability are the principle limitations of crystal-based oscillators.
The practical frequency range for fundamental-mode AT-cut quartz crystals is 600 kHz to 30 MHz. Crystals for fundamental frequencies higher than 30 to 40 MHz are very thin and fragile. Crystals are used at higher frequencies by operation at overtones of the fundamental frequency. Ninth-overtone crystals are used up to approximately 200 MHz, the practical upper limit of crystal oscillators. This article will limit its discussion to the use of fundamental-mode crystals.
Quartz is a piezoelectric material and when an electric field is placed upon it, a physical displacement occurs. Interestingly, an equivalent electrical circuit can be written to represent the mechanical properties of the crystal. The schematic symbol for a quartz crystal is shown in Fig. 5a and the equivalent circuit for a quartz crystal near fundamental resonance is shown in Fig. 5b.
The equivalent circuit is an electrical representation of the quartz crystal's mechanical and electrical behavior and it does not represent actual circuit components. The crystal is, after all, a vibrating piece of quartz. The circuit elements C1, L1, and R1 are known as the motional arm and represent the mechanical behavior of the crystal element, while capacitor C0 represents the electrical behavior of the crystal element and holder.
The equivalent circuit in Fig. 5b represents one oscillation mode. For the types of crystal oscillators related to this article, the focus will remain on fundamental-mode crystals. A more complex model can represent a crystal through as many overtones as desired. For the sake of simplicity, this simple model is usually employed and different values are used to model fundamental or overtone modes. Spurious resonances occur at frequencies near the desired resonance. In a high-quality crystal, the motional resistance of spurious modes is at least two or three times the primary resonance resistance and the spurious modes may be ignored. In this model, C1 represents motional arm capacitance (in Farads). It represents the elasticity of the quartz, the area of the electrodes on the face, and the thickness and shape of the quartz wafer. Values of C1 are in the range of femtofarads (10−15 F or 10−3 pF).
L1 represents motional arm inductance (in Henrys). It represents the vibrating mechanical mass of the quartz in motion. Low-frequency crystals have thicker and larger quartz wafers and range in a few Henrys. High-frequency crystals have thinner and smaller quartz wafers and range in a few megaHenrys. R1 is the resistance (in Ω). It represents the real resistive losses within the crystal. Values range from 10 Ω for 20-MHz crystals to 200 kΩfor 1-kHz crystals. C0 is the shunt capacitance (in Farads). It is the sum of capacitance due to the electrodes on the crystal plate plus stray capacitances due to the crystal holder and enclosure. Values of range from 3 to 7 pF.