## Characterizing a Crystal

We often discuss the complex designs that oscillators can achieve, but rarely stop and think about a fundamental component: the quartz crystal.  Understanding how the quartz crystal operates can give a design engineer an understanding of how they behave as a reference to a microchip or when embedded in an oscillator circuit, and what their parameters actually represent.  Over the next few blog entries, we will attempt to explain how to define a crystal, and bridge the gap between comparing datasheets and defining performance.  Topics will include:

Equivalent Circuit Crystal Model

Series Vs Parallel Resonant Crystals

Fundamental Vs Third Overtone

Temperature Stability of Crystals and Tuning Forks

Equivalent Circuit Crystal Model

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 a ‘blank’.)  Although the design and fabrication of crystal units comprise a complex subject, a designer can treat the crystal unit as a circuit component and just deal with the crystal unit’s equivalent circuit.  Quartz crystals can be generally modeled as a series LRC circuit in parallel with a shunt capacitance.  In the diagram below, an external load capacitor, CL, is shown in parallel with the crystal. Motional inductance, L1, represents the vibrating mass of the crystal unit. The motional capacitance, C1, represents the elasticity of the quartz. The resistance, R1, represents bulk losses occurring within the quartz.  This portion of the circuit is known as the “motional arm” which arises from the mechanical vibrations of the crystal.  The parallel capacitor, C0, is known as the shunt capacitance, and represents the capacitance due to the electrodes on the crystal plate plus the stray capacitances due to the crystal enclosure.

The equivalent circuit crystal model is particularly useful for evaluating a crystal’s quality factor, Q.  Q represents the amount of energy stored during resonance versus the amount of energy lost.  For resonators higher Q results in better stability and better phase noise performance (especially looking at phase noise close to the carrier).  In a simple RLC circuit, 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 C0 and by the fact that the operating Q is lower than the resonator Q.  For a quartz resonator, Q = (2πfsC1R1)-1 where fs is the series resonant frequency.

This begs the question, “Why use a quartz crystal resonator when I can just connect a few components together?”  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 (lower C1), and has a far higher Q than what could be built from normal discrete components.  For example, a 5 MHz fundamental mode AT-cut crystal may be modeled as having a C1 = 0.01 pF, L1 = 0.1 H, R1 = 5 Ω, and Q = 106.  From a practical perspective, the leads of a 0.01pF capacitor alone would contribute more than 0.01 pF.  Similarly, a 0.1 H inductor would be physically large, need to include a large number of turns, and need to be superconducting in order to have a 5 Ω resistance.  As most crystals have a Q in the 10,000-100,000 range, the discrete component method quickly becomes impossible.

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