Overview
Factors that affect an oscillator's frequency stability include temperature changes, load variations, and DC supply voltage changes. Choosing appropriate resonant feedback components, including amplifiers, can significantly improve output signal frequency stability. However, the stability achievable with ordinary LC or RC resonant networks is limited.
To maintain high accuracy, oscillators often use a quartz crystal as the frequency-determining element, creating a quartz crystal oscillator (XO). When voltage is applied to a small piece of quartz, it deforms and exhibits the piezoelectric effect. Through this effect, charge is generated when the crystal is mechanically deformed, and conversely, applied charge produces mechanical deformation. Piezoelectric devices are transducers that convert electrical energy to mechanical energy or vice versa. The piezoelectric effect produces mechanical vibration that can replace the LC resonant circuit in an oscillator.
Quartz Material and Crystal Blanks
Various crystal materials can be used for oscillators, but most electronic circuits use quartz because of its favorable mechanical strength. The quartz used in oscillators is a small thin cut or wafer with both parallel surfaces metallized for electrical connection. The physical dimensions and thickness of a crystal are tightly controlled because they determine the oscillator's final frequency, often called the crystal's fundamental frequency. Once cut and shaped, a crystal is intended to operate at that specific frequency. In other words, the size and shape of a quartz crystal determine the oscillator's fundamental frequency.
Equivalent Circuit
The crystal's characteristic frequency is inversely proportional to the physical thickness between its metallized surfaces. Mechanical vibration of the crystal can be modeled by an equivalent electrical circuit consisting of a low series resistance R, a large inductance L, and a small capacitance C. This series RLC branch is shown in the equivalent circuit alongside a parallel capacitance Cp that represents the crystal's electrical connections. Quartz crystals tend to operate near their series resonance.
Series and Parallel Resonance
The crystal's impedance exhibits a series resonance where Cs resonates with Ls at the crystal's series resonant frequency f_s, producing a minimum impedance equal to Rs. Below f_s the crystal appears capacitive. Above f_s the crystal behaves inductively until it reaches a parallel resonant frequency f_p, where the interaction of Ls, Cs, and the parallel capacitance Cp produces a parallel resonance and a maximum impedance across the crystal terminals.
Impedance and Reactance Behavior
The reactance vs frequency slope shows that at f_s the series reactance is dominated by Cs, and the crystal is capacitive below f_s and above f_p. Between f_s and f_p the crystal is inductive because the two capacitances partially cancel. Thus the crystal acts as a combination of series and parallel tuned circuits, oscillating at two closely spaced frequencies determined by the crystal cut. A given oscillator circuit will operate at either the series or the parallel resonance, not both simultaneously.
Resonant Frequency Expressions
The series resonant frequency f_s is given by the usual LC resonance relation for the series branch. A parallel resonant frequency f_p occurs when the series branch reactance matches the reactance of the parallel capacitance Cp. These frequency relationships can be expressed analytically and are often illustrated with equations or diagrams.
Example 1
Given a crystal with Rs = 6.4 Ω, Cs = 0.09972 pF, and Ls = 2.546 mH, and a terminal capacitance Cp = 28.68 pF, calculate the crystal's fundamental oscillation frequency and the secondary resonant frequency.
Quality Factor and Stability
The difference between f_p and f_s in the example is about 18 kHz (10.005 MHz ? 9.987 MHz). Within this range the crystal's Q factor is very high because the inductive reactance is much larger than its capacitive or resistive elements. The example crystal has a Q of about 25,000, since the ratio of XL to R is large. Typical crystal Q values are in the 20,000 to 200,000 range, whereas a good LC tuned circuit usually has a Q well below 1,000. The high Q contributes to excellent frequency stability, making quartz crystals ideal for oscillator circuits, especially at high frequencies.
Frequency Range and Harmonics
Typical crystal oscillators range from roughly 40 kHz to over 100 MHz depending on circuit configuration and the active device used. The crystal cut determines behavior because some cuts allow vibrations at multiple frequencies, producing overtones. If the crystal thickness is not perfectly uniform, it may have multiple resonant frequencies producing harmonics such as second or third harmonics. In practice, the fundamental frequency is usually the strongest and is the intended operating frequency. The equivalent circuit with its two resonant frequencies shows the lower series resonance and the higher parallel resonance.
Oscillation Conditions and Modes
An amplifier will oscillate if the loop gain is greater than or equal to one and the feedback is positive. In a crystal oscillator, the circuit typically oscillates at the crystal's chosen resonance because the crystal provides a strong tendency to vibrate at its resonant frequency. A crystal oscillator can also be tuned to even harmonics of the fundamental (2nd, 4th, 8th, etc.), termed harmonic oscillators, while overtone oscillators operate at odd multiples (3rd, 5th, 7th, etc.). Overtone-mode oscillators often use the crystal's series resonance for operation.
Colpitts Crystal Oscillator
Crystal oscillators commonly use bipolar transistors or FETs because operational amplifiers typically lack the bandwidth to operate successfully at frequencies above about 1 MHz where many crystals operate. One design replaces the LC resonant network of a Colpitts oscillator with a quartz crystal to provide feedback.
This Colpitts-style crystal oscillator is arranged around a common-collector (emitter follower) amplifier. Bias resistors R1 and R2 set a DC bias at the transistor base, while the emitter resistor sets the output voltage level. R2 should be as large as possible to avoid loading the crystal in parallel. A transistor such as the 2N4265 in a common-collector configuration can switch at speeds well above 100 MHz, far exceeding crystal fundamental frequencies in the 1 MHz to 5 MHz range. Capacitors C1 and C2 provide feedback division; the transistor gain limits their maximum values. Output amplitude should be kept low to avoid excessive crystal drive and potential crystal damage from overdrive.
Pierce Oscillator
The Pierce oscillator is another common crystal design. It is similar in concept to the Colpitts but typically uses a JFET or CMOS inverter as the active device. The Pierce configuration typically forms a series-resonant circuit and is well suited to low-component-count implementations.
In the simple Pierce circuit, the crystal operates at its series resonant frequency f_s, presenting a low-impedance path between input and output. The series resonance introduces a 180-degree phase shift providing positive feedback. Output amplitude is limited by the supply and device characteristics. Resistor R1 controls feedback and the drive level applied to the crystal, while an RF choke provides isolation at each cycle. Many digital clocks, watches, and timers use Pierce oscillators because they require few components.
CMOS Crystal Oscillators
Besides transistors and FETs, a basic parallel-resonant crystal oscillator can be built using CMOS inverters as the gain element. The operation is similar to a Pierce oscillator. A simple CMOS crystal oscillator often uses a Schmitt-trigger inverter (for example, TTL 74HC19 or CMOS 40106/4049), one crystal, and two capacitors. The two capacitors set the crystal's load capacitance. A series resistor limits crystal drive current and isolates the inverter output from the crystal-capacitor network.
The crystal oscillates at its series resonance. The inverter is biased via a feedback resistor to its linear region so the inverter's operating point falls in a high-gain region. An additional inverter stage is typically used to buffer the oscillator output. Because CMOS inverters are digital, the output waveform will be a square wave rather than a sine wave, and the maximum operating frequency depends on the switching characteristics of the logic device.
Microprocessor Crystal Clocks
Quartz crystal oscillators are used widely as clock sources for microprocessors, microcontrollers, PICs, and CPUs due to their superior accuracy and frequency stability compared with RC or LC oscillators. A CPU clock determines processing speed; a 1 MHz microprocessor or microcontroller executes one million cycles per second. Typically, producing a microcontroller clock requires only a crystal and two small ceramic capacitors (usually 15 to 33 pF).
Most microcontrollers provide two oscillator pins labeled OSC1 and OSC2 for connection to an external crystal circuit, an RC network, or a ceramic resonator. In these applications the crystal controls a train of square-wave pulses whose fundamental frequency sets the processor's instruction timing and system timing.
Example 2
A crystal has Rs = 1 kΩ, Cs = 0.05 pF, Ls = 3 H, and Cp = 10 pF. Calculate the crystal's series and parallel resonant frequencies. The series resonant frequency and the parallel resonant frequency fall close together, yielding oscillator operation around roughly 411 kHz to 412 kHz for the given values.
