Crystal Oscillators: History, Parameters, and Types
By John Esterline, Greenray Industries
Precision quartz oscillators have been a mainstay of embeddable timing solutions for over half a century. Crystal oscillators are flexible and rugged, making them ideal for a wide range of applications. Because of the versatility, there are many types of crystal oscillators with unique attributes. This paper covers oscillator types and defines and explains many of the parameters associated with crystal oscillators. Attributes and parameters such as phase noise, jitter, short term stability, long term stability, and temperature performance are all covered.
Time bases are a staple of modern electronics. They come in a variety of forms and offer an even wider variety of performance characteristics. Everything from discrete LC oscillators to atomic clocks are available for engineers to choose from as their application’s time base. Because of their inherent stability, small size, and relatively low power consumption, quartz crystal oscillators have been the industry’s main solution for embeddable time bases for over half a century.
When dealing with crystal oscillators there are a variety of parameters that need to be considered and specified to ensure that the user is receiving an oscillator that meets their needs. Temperature stability, long and short term stability, frequency, and pullability are but a few of the many parameters that need to be defined when considering which crystal oscillator is appropriate. Crystal oscillators come in a variety of types and forms, with each archetype possessing its own very unique attributes. It is important to understand the differences among these archetypes and what that means in terms of performance.
The following paper will discuss several topics, including:
1. Brief background on quartz as time base
2. Parameter definitions
3. Crystal oscillator archetypes and their properties
Within these sections it will be shown why crystal oscillators are so widely used. The multiple forms and varying levels of stability and accuracy make quartz a very robust and flexible choice for designing a host of solutions for a wide range of applications. For these reasons, quartz will be with us for a long time into the foreseeable future, and with constant advances, the quality of the timing provided by quartz will continue to improve and footprint and power consumption will continue to decrease.
The following sections are also intended to provide information in a manner such that someone with little or limited understanding of crystal oscillators will gain enough understanding of the subject to intelligently understand basic oscillator specifications.
Quartz crystals have been in wide scale use in electronics since 1939. “During the summer of 1939 the army performed comparative testing on radios with and without crystal control. As a result of those experiments it was decided to convert all military radio equipment to crystal control. At that time there were serious disadvantages to using quartz; its lack of flexibility, cost, and unavailability.” The benefits of quartz crystals over traditional discrete LC circuitry far outweighed its disadvantages. This can be seen in a statement by Major General Roger B. Colton in July of 1944 only 5 years after changing the military standard for radio equipment, where he said, “Our decision to go into crystal controlled radios for widespread tactical use has been more than justified by the results obtained. The Army had radio before they had crystals. Now the Army has communications. That’s the difference. Crystals gave us communications.”
The implications of that statement show just how beneficial quartz crystals are as frequency generating devices. With their extremely high Q and superb temperature performance, crystals transformed the radio industry into a communications industry capable of supporting mass communications on a global scale.
Countless advances since 1944 have made quartz crystal oscillators even more robust and accurate than ever before. Despite having a high Q, the quartz crystal oscillator is not perfect. Ideally, a sinusoidal oscillator would produce a voltage as in:
Where Vo is the amplitude, fo is the frequency, and t is time.
However, real world oscillators have some amplitude fluctuations and phase fluctuations present within them and behave as in Equation 2. 
Where Vo is the amplitude, fo is the frequency, t is time, λ(t) is the amplitude fluctuation and Φ(t) is the phase fluctuation
Equations 1 and 2 deal with the voltage of the oscillation at any instant in time, with Equation 1 being the ideal output and Equation 2 accounting for amplitude and phase fluctuations within the oscillator.
Quartz provides a very predictable, stable, and low noise time source via the piezoelectric effect. Piezoelectric effect is, in essence, a transformation of energy types. By applying a stress or force to the crystal, we induce electrical charge into the crystal. The electrical equivalent of a quartz crystal can be seen in Figure 1.
As can be seen in Figure 1, if R1 is kept small, then C1 and L1 will form a resonant circuit, which will produce very stable oscillations. This signal is very small and must be amplified and sustained to be of a useful nature. Using the crystal as the focal point, circuits can be designed that produce all forms of output wave shapes and power levels to meet any application’s needs.
With Qs ranging from 30,000 to more than 1,000,000, quartz crystals are by far the most stable and accurate time bases for embeddable applications. However, despite their high Q, the quartz crystal is not perfect. The oscillator’s performance is affected by many different factors. Temperature and aging are the two biggest contributors to oscillator frequency deviation, but other factors such as phase noise and short term stability are also critical in evaluating overall oscillator performance.
There are a host of electrical parameters associated with crystal oscillators that define the performance of a particular oscillator. In the following sections various oscillator parameters will be discussed. Each section should provide the reader with an elementary understanding of the parameter and how it affects the performance of the oscillator.
Frequency and Pullability
The oscillator will have a nominal frequency at which it operates, and that is determined primarily by the crystal, but reactive elements must be put in the feedback loop to ensure that output frequency is “fine tuned” to the desired nominal frequency. Depending on oscillator type and crystal type, the frequency may be the fundamental mode of the crystal or possibly the 3rd or 5th overtone. Beyond that, the frequency could be further multiplied or divided to provide the desired output frequency. In the oscillator industry (and for the remainder of this text), the frequency is referred to in parts per million (ppm) or parts per billion (ppb), relative to the nominal frequency. Equation 3 shows the fractional frequency difference formula.
where Δf is the change in frequency and f is the nominal frequency
For example, if the nominal frequency of a given oscillator is 10 MHz and the measured frequency is 9.999990 MHz, then Δf is 10 and f is 10 MHz. This results in an S of 1 X 10-6 , or 1ppm. This nomenclature makes it easy to discuss relative accuracies and stabilities in general and not at a given frequency.
For some oscillators there needs to be the ability to move the frequency of the oscillator. This is generally achieved in one of two ways. Electrcical Frequency Control (EFC) provides the user with the ability to input a control voltage that will change the frequency either proportionally or inversely proportional to the applied voltage. Mechanical adjust provides the user with access to an adjust trimmer inside the unit. By physically changing the value of the trimmer the frequency is moved.
Both of these methods have their place and are discussed in greater detail below, but the EFC is generally used if the user will be actively steering the frequency in their application, such as a PLL or GPS receiver. The mechanical adjust is mainly to recalibrate the oscillator for its natural aging drift, discussed below.
Short Term Stability
Short term stability or Allan Variance (AVAR) is a measurement of the short term frequency variations from the oscillator. Generally AVAR is specified relative to a particular gate time. For example, a 20ms gate time may be selected and 100 samples taken and applied to the following formula:
Where f(i)-f(i-1) is the difference between successive frequency measurements and N is the number of samples (as per MIL-PRF-55310D) 
The result gives us insight for how stable the oscillator is reading to reading at the given gate time. One might ask why not use classical variance, which is defined as:
Where f(i)-f is the difference between the ith frequency measurement and the mean of the frequency data set and N is the number of samples (as per MIL-PRF-55310D) 
The reason Allan Variance is used for this measurement instead of classical variance is because classical variance “diverges for commonly observed noise processes, such as random walk. Allan Variance is known to converge for all noise types observed in precision oscillators, and also has a straightforward relationship to power law spectral density.”
Looking at short term stability, it can be seen that the instantaneous frequency of the oscillator is not constant, but rather it varies slightly about the nominal frequency, which creates an uncertainty in the frequency at any given point in time. This frequency change can be viewed as a change in time of the waveform edge from the ideal nominal frequency edge. The change in timing of the edge is called jitter. Figure 2  illustrates the jitter effect on a square wave.
Jitter can be measured in the time domain and is expressed in a peak to peak time variation of the edge. This method, however, may not be very useful in some applications because the variations in the edges are coming from the entire frequency band, thus exaggerating the magnitude of the jitter. Most real world applications will operate within a well defined band and therefore, the jitter only needs to be considered within that band. To effectively see and measure jitter over a particular band, conversion to the frequency domain must take place in order to isolate the band of interest .
This conversion to the frequency domain results in a measurement called phase noise. It is commonly expressed as a graph of dBc/Hz vs. Hz. To see how jitter is converted phase noise Equations 5 and 6 are used in conjunction. Jitter can be defined several ways and expressed as several different quantities. One way to measure jitter is to measure the variance of each period from the average period, as per formula 6 . This quantity is called RMS cycle jitter.
Where τn is the period of the nth sample and τavg is the average period of all N samples
This RMS cycle jitter can then be used to calculate the phase noise at a given frequency, as can be seen in Equation 7 .
where fosc is the frequency of the oscillator and f is the frequency away from the carrier
This computation can be repeated at many discrete frequencies and compiled into a graph of dBc/Hz vs. Hz. The above computations (Equations 5 and 6) are simplified by assuming that there is no 1/f noise or burst noise . However, in real world oscillators these noise sources are present and need to be considered.
If we take into account real world components and the noise that is generated in practical circuits, the phase noise calculation becomes more complex. The Leeson Equation 8 shows us how circuit noise and circuit elements factor into the phase noise measurement .
Where Ql is the loaded Q of the circuit, fm is the frequency from the carrier, fc is the flicker noise corner frequency, fo is the carrier (oscillator) frequency, T is the temperature in Kelvin, Pavs is the power through the resonator, F is the noise factor of the active device, and k is the Boltzmann constant
Figure 3 shows each portion of the phase noise plot is affected by the various elements of the Leeson equation. It can be seen that close to the carrier, flicker noise dominates the curve and has a cutoff frequency equal to the corner frequency of the active device. The middle portion of the phase noise plot follows Leeson’s equation and is a combination of loaded Q, noise factor, power and temperature . For frequencies above fo/(2Ql) the floor is determined by noise factor, temperature and power.
From this plot, it is fairly easy to come up with guidelines for minimizing phase noise in oscillator designs. One guideline is to use devices with low flicker noise. Since the 9dB/octave section is dominated by this quantity, reduction of circuit flicker noise is of great concern. BJTs have a much lower flicker noise than FETs, making them more suitable for low phase noise applications. The 6dB/octave section implies that the Q of the circuit being higher is of great interest, as is noise factor and power. Higher drive power is also desirable because that is the driving factor for the phase noise floor (frequencies above fo/(2Ql)). This comes with a trade-off because higher drive levels usually equal some phase noise degradation close in to the carrier .
When manufacturing military grade oscillators (and some high end commercial GPS applications) another factor comes into play that must be analyzed to understand how oscillator performance is affected.
As stated above, crystal oscillators operate via the piezoelectric effect. This pressure transduction means that we can apply a voltage or electrical pressure to the crystal and receive a stable signal in return. However, any secondary pressures applied to the crystal will in turn affect its operating point. For most commercial electronics this is not an issue because when the oscillator is mounted within the device it is then kept relatively stable, as in sitting on a table top or maybe traveling slowly in a benign environment. In the military world however, the oscillators are used for communications, clocking, and radar in aircraft and missiles. This means the oscillators are subjected to acceleration forces or “G-forces,” as they are more commonly referred to. This in effect is adding additional pressure on the crystal, which results in a shift in frequency from its static operating point. If the acceleration is constant, an offset in frequency will be seen. If the acceleration is a vibration spectrum, this will result in a worsening of the noise band close to the carrier, which is commonly viewed as phase noise. This effect is undesirable, and at the same time inherent to crystal oscillators because of their fundamental mode of operation. This long-standing problem has been characterized and examined for many decades. Crystal manufacturers continually come up with ways to reduce the G-sensitivity of their crystals, however, due to being a piezoelectric device, it can never truly be eliminated on the crystal level.
Long Term Drift/Aging
Long term drift, also called aging, is a natural phenomenon of the quartz and has been well characterized and understood for some time. There are several key factors that induce different modes of aging. “Stress relief in the resonator’s electrodes, mounting and bonding structure, and the quartz itself, are causes of positive aging (frequency increasing). Mass loading onto the crystal blank is a cause of negative aging (frequency decrease).” Mass loading can come from raw contamination inside the crystal package or from outgassing of materials inside the crystal package. Generally speaking, positive aging is considered “good” aging and negative aging is considered “bad” aging. This is only a general statement because you can have a negative aging crystal that will flatten and slow. However, if mass loading is occurring, it all too often leads to continued downward slope that does not slow in a reasonable amount of time. Positive aging, on the other hand, seems to generally slow and produce good aging fits, as shown below. The aging characteristic is defined as:
Where t is time in days and A, B, and fo are constants determined from least squares fit (as per MIL-PRF-55310D) 
As can be seen in Figure 4, the aging characteristic of quartz is a natural log function that slows over time. This means that the frequency drift of the oscillator will diminish as time passes. This is a desirable effect in terms of long term performance. The oscillator will drift, but that rate of change will slow and the oscillator in effect will become more stable.
Temperature Stability and Oscillator Archetypes
In the end application, the oscillator must perform over a temperature range and still be within specification. For a cell phone, we want to be able to receive calls and have a quality conversation whether we are skiing in the Alps or exploring the pyramids of Egypt. For military applications it can be even more extreme. The oscillator must remain within tolerance when affixed to a hard point on wing at 30,000 ft., and also at the upper end of its temperature range as it gets ramped from cold to hot in only a few minutes. The temperature stability required will dictate what type of oscillator can be used. Because of the wide variety of applications and the need for different levels of temperature stability, several different classes of oscillators exist to accommodate this need.
Temperature stability is a classic catchall phrase, however in oscillator specification it should be treated with care. There are several ways of defining and specifying temperature performance, and misinterpretation of the terminology can lead to problems for both the customer and the oscillator manufacturer.
Temperature stability, which is the specification most often used, is defined as in Equation 10 .
Where fmax is the maximum frequency measured during the temperature run and fmin is the minimum frequency measured during the temperature run
Temperature stability is the peak to peak deviation of frequency over temperature, however it is not referenced to any frequency at a given temperature. There are times when the deviation referenced to the frequency at a given temperature (most commonly 25˚ C) is desired. Therefore another method of specifying temperature performance is needed. In this case, frequency versus temperature accuracy is used. This is specified in Equation 11 as follows: 
Where MAX is the maximum of the set [δfmax, δfmin]and δfmax = (fmax-fnom)/fnom δfmin = (fmin-fnom)/fnom
To show how this can produce a very different specification, a sample frequency versus temperature curve is shown in Figure 5. Figure 5 shows a non- symmetrical frequency temperature curve. The sample graph is for a 10 MHz oscillator whose frequency versus temperature curve passes through 9.999995 MHz at 25 ˚C. If we apply the above formulas to this curve we can see that frequency versus temperature stability is +/-0.5ppm. However, if we apply the frequency versus temperature accuracy criteria with a reference temperature of 25 ˚C, +/- 1ppm is the result. This illustrates how adding a reference temperature to the specification can make the interpretation of the data completely different. Therefore, it is important to understand the language of the specification so that misunderstandings and misinterpretations can be minimized.
XO (Crystal Oscillator)
XOs, which stands for crystal oscillators, are simply a quartz crystal and driving circuitry. These bare bones clocks are cheap and small but offer limited accuracy as the crystal will wander in frequency approximately +/-30ppm over temperature. However, due to the fact that they are not compensated and offer no external electrical frequency control (EFC) to the customer, they can be made to have excellent noise performance. Therefore, if temperature stability is not a big concern, a low noise XO could be a viable solution.
VCXO (Voltage Controlled Crystal Oscillator)
VCXOs, (Voltage Controlled Crystal Oscillators) also include a means for electrically adjusting the reactance of the circuit to shift the frequency of oscillation. This control voltage is generally referred to as EFC (Electrical Frequency Control) and is specified by how many ppm the frequency can be pulled over the input voltage range, or alternatively as ppm/Volt. They offer similar temperature performance to the XO since they have no compensation of any kind. By adding the ability for the user to pull the frequency, the noise performance of the oscillator is degraded. Therefore, a VCXO would be useable in an application that was not concerned about tight temperature stability but would like voltage adjustment for recalibration due to aging, or for narrow bandwidth PLL applications. VCXOs with some temperature compensation are available (TCVCXOs) however, it is a trade-off between the amount of pullability needed and the level of compensation needed.
TCXO (Temperature Compensated Crystal Oscillator)
TCXOs, which stand for Temperature Compensated Crystal Oscillators, use a compensation voltage to correct for the crystal’s natural temperature drift. This is accomplished with a classic thermistor resistor network or a polynomial generator. TCXOs provide tighter stability over temperature (+/-0.25ppm can be obtained). TCXOs can be found in many package types as small as 2.0mm X 2.5mm and with very low power consumption (≤2.0mA). TCXOs can be designed to have excellent noise performance, although not rivaling an OCXO. TCXOs have become very popular for precision oscillator applications and can be found in everything from cell phones to missiles.
MCXO (Microprocessor Compensated Crystal Oscillator)
MCXOs (Microprocessor Compensated Crystal Oscillator) use a microprocessor to correct for the crystal’s natural temperature drift by sensing the temperature of operation and using that data to correct the frequency of the oscillator. These oscillators can achieve stabilities of +/-0.1ppm over temperature, but have a slightly larger footprint, consume more power, and have degraded noise characteristics due to the microprocessor running in the oscillator. MCXOs are found in a variety of applications where temperature stability better than that of a TCXO is needed yet noise performance is not a serious issue.
OCXO (Oven Controlled Crystal Oscillator)
OCXOs (Oven Controlled Crystal Oscillators) and DOCXOs (Double Oven Controlled Crystal Oscillators) provide the greatest stability that quartz crystal oscillators have to offer. By heating with a proportional controller, the circuitry and the crystal stay at an almost constant temperature, nearly eliminating the natural temperature drift of the crystal. Stabilities of parts in 10-10 over temperature are achievable, but at the trade-off of footprint (at least 1 inch square), and power consumption (possibly over 1A). OCXOs are built for stability and as a result generally have very excellent noise characteristics. OCXOs are found in applications where a very stable embeddable frequency base is needed, such as frequency counters and spectrum analyzers. They are also found in a host of military applications ranging from radar, radios, missiles, and GPS applications.
Crystal oscillators are a versatile and hardy solution to the embeddable timing needs of many industries and applications. Because of this versatility, many different types of crystal oscillators are available and each of these has an almost endless amount of configurations it can exist in. Contained above is an attempt to highlight some of the more important aspects of crystal oscillators and provide some background information so that someone with little or limited knowledge of the subject could intelligently look at and understand crystal oscillator specification sheets. This by no means encompasses all of the parameters or possibilities for crystal oscillators. In fact, many solutions involving crystal oscillators are a custom design made in conjunction with the end user to fit their specific application.
It should also be clear that because of their miniature size, durable qualities, and extremely high Q, quartz crystal oscillators will continue to be the premier embeddable timing solution for the foreseeable future. Advances in crystal manufacture and oscillator design are constantly pushing the envelope of performance for nearly every parameter. The industries’ need for ever smaller, tighter stability, and lower noise timing sources will fuel even more advancements in oscillator technology in the years to come.
About the Author
John Esterline is an oscillator design engineer at Greenray Industries in Mechanicsburg, PA. John holds a BSEE, and is currently pursuing a Masters of Engineering in Electrical Engineering from the Pennsylvania State University. John has been involved in the testing and design of precision crystal oscillators for 6 years. He may be reached at firstname.lastname@example.org.
 Virgil E. Bottom, “A History of the Quartz Crystal Industry in the USA,” 35th Annual Frequency Control Symposium, 1981.
 Barry Kleinle, “Phase Noise in Crystal Oscillators,” Greenray Industries, 2004.
 Oscillator, Crystal Controlled General Specification For, MIL-PRF-55310D, 1998.
 John R. Vig, “Quartz Crystal Resonators and Oscillators,” U.S. Army Electronics Technology and Devices, Fort Monmouth, NJ, 1990.
 Steve Fry, “The Design and Performance of Precision Miniature TCXOs, RF Design,” September 2006.
 Hugo Fruehauf, “‘G’-Compensated, Miniature, High Performance Quartz Crystal Oscillators,” April 2007.
 Neil Roberts, “Phase Noise and Jitter – A Primer for Digital Designers,” EE Design, July 14, 2003.
 John Esterline, “Phase Noise: Theory versus Practicality,” Microwave Journal, April 2008.
 Rick Poore, “Phase Noise and Jitter,” Agilent, 2001.
 “Phase Noise,” [Online]. Available http://www.zen118213.zen.co.uk/Systems_And_Devices_Files/PhaseNoise.pdf
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