Accurate Clocks and Their Applications

November 2011

John R. Vig
Consultant.
Much of this Tutorial was prepared while the author was employed by the
US Army Communications-Electronics Research, Development & Engineering Center
Fort Monmouth, NJ, USA
J.Vig@IEEE.org

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Rev. 8.5.3.9

Quartz Crystal Resonators and

Oscillators
For Frequency Control and Timing Applications - A Tutorial
November 2008

John R. Vig
Consultant.
Most of this Tutorial was prepared while the author was employed by the
US Army Communications-Electronics Research, Development & Engineering Center
Fort Monmouth, NJ, USA
J.Vig@IEEE.org
Approved for public release.
Distribution is unlimited

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.9, by John R. Vig, November 2008.
 Applications

Quartz for the National Defense Stockpile, Report of the Committee on Cultured Quartz for the
National Defense Stockpile, National Materials Advisory Board Commission on Engineering and
Technical Systems, National Research Council, NMAB-424, National Academy Press,
Washington, D.C., 1985.

J. R. Vig, "Military Applications of High Accuracy Frequency Standards and Clocks," IEEE
Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 40, pp. 522-527, 1993.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.9, by John R. Vig, November 2008.
 Global Positioning System (GPS)
Navigation Timing

GPS Nominal Constellation:

24 satellites in 6 orbital planes,
4 satellites in each plane,
20,200 km altitude, 55 degree inclinations
8-16

The Global Positioning System (GPS) is the most precise worldwide navigation system
available. It is also capable of providing nanosecond-level timing accuracies, so, it is also one
of the most accurate time sources.
GPS is a satellite-based radio navigation and positioning system that is designed to
provide global, all-weather, 24-hour, accurate navigation to an unlimited number of users. Each
of the satellites contain atomic clocks. The satellites transmit a navigation message that
provides satellite position, time, and atmospheric propagation correction data. The GPS
receiver, which contains a quartz crystal clock, measures the transit time of the satellite signal
and multiplies that time by the speed of light to compute range to the satellite. The satellite
clocks are more accurate than the receiver clocks. Therefore, although three satellites can
provide latitude, longitude and altitude, the signal from a fourth satellite is used to correct for the
navigational error caused by the receiver clock's inaccuracy, i.e., the receivers calculate their x,
y, z, and t from receiving each of four satellites’ x, y, z, and t. Velocity is determined from the
Doppler shifts of the the transmitted carrier frequencies.

A. J. Van Dierendonck and M. Birnbaum, "Time Requirements in the NAVSTAR Global

Positioning System (GPS)," Proc. 30th Annual Symposium on Frequency Control, pp. 375-383,
1976, AD046089.

F. E. Butterfield, "Frequency Control and Time Information in the NAVSTAR/Global Positioning

System," Proc. 30th Annual Symposium on Frequency Control, pp. 371-374, 1976, AD046089

Much information is available on the Internet, e.g., see “Navstar GPS Internet Connections” at
http://gauss.gge.unb.ca/GPS.INTERNET.SERVICES.HTML, and
“Global Positioning System Overview” by Peter H. Dana (from which the above illustration was
“borrowed,” with permission from Peter H. Dana, The University of Texas at Austin) at
http://www.utexas.edu/depts/grg/gcraft/notes/gps/gps.html

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Electronics Applications of Clocks
Military & Aerospace Industrial Consumer
Communications Communications Watches & clocks
Navigation Telecommunications Cellular & cordless
IFF Mobile/cellular/portable phones, pagers
Radar radio, telephone & pager Radio & hi-fi equipment
Sensors Aviation TV & cable TV
Guidance systems Marine Personal computers
Fuzes Navigation Digital cameras
Electronic warfare Instrumentation Video camera/recorder
Sonobouys Computers CB & amateur radio
Flash drives Toys & games
Research & Metrology Digital systems Pacemakers
Atomic clocks CRT displays Other medical devices
Instruments Disk drives Other digital devices
Astronomy & geodesy Modems Automotive
Space tracking Tagging/identification Engine control, stereo,
Celestial navigation Utilities clock, yaw stability
Sensors control, trip computer,
GPS
1-1

Quartz for the National Defense Stockpile, Report of the Committee on Cultured Quartz for the
National Defense Stockpile, National Materials Advisory Board Commission on Engineering and
Technical Systems, National Research Council, NMAB-424, National Academy Press,
Washington, D.C., 1985.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Clock for Very Fast Frequency Hopping Radio

Jammer Example
J

Let R1 to R2 = 1 km, R1 to
t1 t2 J =5 km, and J to R2 = 5 km.
Then, since propagation
Radio Radio
delay =3.3 µs/km,
R1
tR R2 t1 = t2 = 16.5 µs,
tR = 3.3 µs, and tm < 30 µs.
To defeat a “perfect” follower Allowed clock error ≈ 0.2 tm
jammer, one needs a hop-rate ≈ 6 µs.
given by:
For a 4 hour resynch interval,
tm < (t1 + t2) - tR
clock accuracy requirement is:
where tm ≈ message duration/hop
≈ 1/hop-rate 4 X 10-10

1-13

With the availability of fast spectrum analyzers and synthesizers, it is possible to jam
frequency hopping systems. If a jammer is fast enough, it can detect the frequency of
transmission and tune the jammer to that frequency well before the radio hops to the next
frequency. However, with a good enough clock, it is possible to defeat such “follower” jamming.
As illustrated above, even a "perfect" follower jammer can be defeated if a good enough clock is
available. (A perfect jammer is defined here as one that can identify the frequency of a received
signal, tune a synthesizer to that frequency, and transmit the jamming signal in zero time.)
Because radio waves travel at the speed of light, the radio-to-jammer-to-radio (R1 to J to
R2) and radio-to-radio (R1 to R2) propagation delays are 3.3 µs per km. Therefore, if the
hopping rate is fast enough for the propagation delay difference to be greater than 1/hop-rate,
i.e., if the radios can hop to the next frequency before the jamming signal reaches the receiver,
then the radios are jamming-proof (for follower jammers). In the example above, the
propagation delays t1, t2, and tR imply that the message duration tm be less than 30 µs. Since
the clock accuracies required by frequency hopping systems are usually 10% to 20% of tm, the
allowed clock error is about 6 µs. In a military environment, such accuracies can be maintained
for periods of hours and longer only with atomic clocks.

A. D. Robertson and F. C. Painter, "Tactical Jamming," Defense Science and Engineering, pp.
20-28, September 1985.

J. R. Vig, "Military Applications of High Accuracy Frequency Standards and Clocks," IEEE
Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 40, pp. 522-527, 1993.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Identification-Friend-Or-Foe (IFF)
Air Defense IFF Applications
AWACS

FRIEND OR FOE?

F-16

FAAD

PATRIOT STINGER
1-15

In a modern battle, when the sky is filled with friendly and enemy aircraft, and a variety of
advanced weapons are ready to fire from both ground and airborne platforms, positive
identification of friend and foe is critically important. For example fratricide due to identification
errors has been a major problem in all 20th century wars.
Current IFF systems use an interrogation/response method which employs
cryptographically encoded spread spectrum signals. The interrogation signal received by a
friend is supposed to result in the "correct" code being automatically sent back via a
transponder on the friendly platform. The "correct" code must change frequently to prevent a
foe from recording and transmitting that code ("repeat jamming"), thereby appearing as a friend.
The code is changed at the end of what is called the code validity interval (CVI).
The better the clock accuracy, the shorter can be the CVI, the more resistant the system
can be to repeat jamming, and the longer can be the autonomy period for users who cannot
resynchronize their clocks during a mission.

J. R. Vig, "Military Applications of High Accuracy Frequency Standards and Clocks," IEEE
Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 40, pp. 522-527, 1993.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Bistatic Radar
Conventional (i.e., "monostatic") radar, in which the Illuminator
illuminator and receiver are on the same platform, is vulnerable
to a variety of countermeasures. Bistatic radar, in which the
illuminator and receiver are widely separated, can greatly
reduce the vulnerability to countermeasures such as jamming
and antiradiation weapons, and can increase slow moving
target detection and identification capability via "clutter tuning” Receiver
(receiver maneuvers so that its motion compensates for the
motion of the illuminator; creates zero Doppler shift for the area
being searched). The transmitter can remain far from the battle
area, in a "sanctuary." The receiver can remain "quiet.”
The timing and phase coherence problems can be orders
of magnitude more severe in bistatic than in monostatic
radar, especially when the platforms are moving. The
Target
reference oscillators must remain synchronized and syntonized
during a mission so that the receiver knows when the transmitter emits each pulse, and the phase
variations will be small enough to allow a satisfactory image to be formed. Low noise crystal
oscillators are required for short term stability; atomic frequency standards are often required for
long term stability.

1-17

Similar requirements exist in electronic warfare applications. The ability to locate radio and
radar emitters is important in modern warfare. One method of locating emitters is to measure
the time difference of arrival of the same signal at widely separated locations. Emitter location
by means of this method depends on the availability of highly accurate clocks, and on highly
accurate methods of synchronizing clocks that are widely separated. Since electromagnetic
waves travel at the speed of light, 30 cm per nanosecond, the clocks of emitter locating systems
must be kept synchronized to within nanoseconds in order to locate emitters with high accuracy.
(Multipath and the geometrical arrangement of emitter locators usually results in a dilution of
precision.) Without resynchronization, even the best available militarized atomic clocks can
maintain such accuracies for periods of only a few hours. With the availability of GPS and using
the "GPS common view" method of time transfer, widely separated clocks can be synchronized
to better than 10 ns (assuming that GPS is not jammed). An even more accurate method of
synchronization is "two-way time transfer via communication satellites," which, by means of
very small aperture terminals (VSATs) and pseudonoise modems, can attain subnanosecond
time transfer accuracies.
Another important application for low-noise frequency sources is the ELINT (ELectronic
INTelligence) receiver. These receivers are used to search a broad range of frequencies for
signals that may be emitted by a potential adversary. The frequency source must be as noise-
free as possible so as not to obscure weak incoming signals. The frequency source must also
be extremely stable and accurate in order to allow accurate measurement of the incoming
signal's characteristics.

N. J. Willis, "Bistatic Radar," in Radar Handbook, M. I. Skolnik, editor, Chapter 25, Mc-Graw-Hill
Publishing Co., 1990.

W. Lewandowski and C. Thomas, "GPS Time Transfer," Proc. IEEE, Vol. 79, pp. 991-1000, July
1991.

G. Lippermeier and R. Vernon, "IFFN: Solving the Identification Riddle," Defense Electronics,
“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS
pp. Timing
For Frequency Control and 83-88,Applications
1988. - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Relativistic Time Effects
• Transporting "perfect" clocks slowly around the surface of the earth along
the equator yields ∆t = -207 ns eastward and ∆t = +207 ns westward
(portable clock is late eastward). The effect is due to the earth's rotation.

• At latitude 40o, for example, the rate of a clock will change by 1.091 x 10-13
per kilometer above sea level. Moving a clock from sea level to 1km
elevation makes it gain 9.4 nsec/day at that latitude.

• In 1971, atomic clocks flown eastward then westward around the world in
airlines demonstrated relativistic time effects; eastward ∆t = -59 ns,
westward ∆t = +273 ns; both values agreed with prediction to within the
experimental uncertainties.

• Spacecraft Examples:
• For a space shuttle in a 325 km orbit, ∆t = tspace - tground = -25 µsec/day
• For GPS satellites (12 hr period circular orbits), ∆t = +38.5 µsec/day

• In precise time and frequency comparisons, relativistic effects must be

included in the comparison procedures.

8-22

N. Ashby & M. Weiss, “Global Positioning System Receivers and Relativity,” NIST Technical
Note 1385, March 1999.

C. Alley, "Relativity and Clocks," Proc. 33rd Annual Symposium on Frequency Control, pp. 4-
39, 1979.

G. M. R. Winkler, “Synchronization and Relativity,” Proc. IEEE, vol. 79, pp. 1029-1039, 1991

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.9, by John R. Vig, November 2008.
 Units of Measurement Having Special Names
in the International System of Units (SI)
SI Base Units
Electric Temperature Luminous Amount of
Mass Length Time
Current kelvin Intensity Substance
kilogram meter second
ampere candela mole

kg m s A K cd mol
S K cd sr
Coordinated Time Celsius Luminous Flux
international atomic time Temperature
SI Derived Units 0Celsius lumen
TAI 0C lm
s-1
kg m s-2 Frequency
Force hertz
newton Hz
N m-2cd sr
kg m2s-2 kg m-1s-2 SA Illuminance
Pressure Electric charge lux
Energy pascal lx
joule coulomb
J Pa C
kg m s-3 kg m 2s-3A-2
Non-SI units Power kg m2s-3 A-1 Resistance sr: the steradian is the supplementary
watt Electric Potential ohm
recognized volt SI unit of solid angle (dimensionless)
W Ω
for use with SI s-1 V rad: the radian is the supplementary
Activity kg-1 m2s4 A2 kg m2s-2A-1 SI unit of plane angle (dimensionless)
day: 1 d = 86400 s becquerel Capacitance Magnetic Flux
hour: 1 h = 3600 s Bq farad weber
minute: 1 min = 60 s m2s-1 F Wb
Absorbed Dose kg-1 m2s3 A2 kg m2s-2A-2
liter: 1 l = 10-3 m3 gray Conductance Inductance
ton: 1 t = 103 kg Gy siemens henry
m2s-2 S H
degree: 10 = (π/180) rad Dose Equivalent kg s-2 A-1
minute: 1’ = (π/10800)rad sievert Conductance
second: 1” = (π/648000)rad Sv siemens Electromagnetic
Health related S measurement units
electronvolt: 1 eV ≈ 1.602177 x 10-19 J measurement units
unified atomic mass unit: 1 u ≈ 1.660540 x 10-27 kg

8-31

Time interval (frequency) is the quantity that can be determined with the highest accuracy.
It can be measured with an accuracy greater than 1 part in 1013. With the help of satellites, it is
possible to compare the time scales kept by the national laboratories, worldwide, to an accuracy
of ~1 ns. Time, therefore, plays a central role in metrology and in the definitions of SI units.
The SI consists of seven base units and a number of derived units, as shown above.
Shown on the next page are the units that do NOT depend on the unit of time.
-------------------------------
R. J. Douglas, et. al, "Frequency Standards, Timekeeping, and Traceable Services at the
National Research Council of Canada," Proc. 28th Ann. Precise Time & Time Interval (PTTI)
Applications & Planning Meeting, pp. 65-80, 1996.

The chart above, and the one on the next page, were provided by R.J. Douglas, National
Research Council Canada, 1997.

E. R. Cohen & B. N. Taylor, “The Fundamental Physical Constants,” Physics Today, pp. BG7-
BG14, August 1997.

B. W. Petley, "Time and Frequency Fundamental Metrology," Proceedings of the

IEEE, vol. 79, pp. 1070-1076, 1991.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.9, by John R. Vig, November 2008.
 Units of Measurement Having Special Names in the SI Units,
NOT Needing Standard Uncertainty in SI Average Frequency

SI Base Units

Amount of
Mass Temperature
Substance
kilogram kelvin
mole
kg K mol
K
SI Derived Units Celsius
Temperature
0Celsius
0C

Non-SI units
recognized
for use with SI

ton: 1 t = 103 kg
degree: 10 = (π/180) rad
minute: 1’ = (π/10800)rad
second: 1” = (π/648000)rad
unified atomic mass unit: 1 u ≈ 1.660540 x 10-27 kg

8-32

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.9, by John R. Vig, November 2008.
 Clocks

Quartz for the National Defense Stockpile, Report of the Committee on Cultured Quartz for the
National Defense Stockpile, National Materials Advisory Board Commission on Engineering and
Technical Systems, National Research Council, NMAB-424, National Academy Press,
Washington, D.C., 1985.

J. R. Vig, "Military Applications of High Accuracy Frequency Standards and Clocks," IEEE
Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 40, pp. 522-527, 1993.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.9, by John R. Vig, November 2008.
 Typical Clock

Counting
Mechanism
Frequency Setting Display or
Source Mechanism Time code
Synchronization
Mechanism

t = t0 + Σ∆τ
Where t is the time output, t0 is the initial setting,
and ∆τ is the time interval being counted.

8-5

A clock may or may not have a display. In many consumer applications, clocks display
the time of day. In many other applications, clocks are used internally only; their output is
typically a one-pulse-per-second (1 pps) or a time code signal which are used for sequencing or
time-tagging events (see “One Pulse-Per-Second Timing Signal” and “BCD Time Code” later in
this chapter.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.9, by John R. Vig, November 2008.
 Progress in Timekeeping
Time Period Clock/Milestone Accuracy Per Day
4th millennium B.C. Sumarians divided day & night into 12 equal hours
Up to 1280 A.D. Sundials, water clocks (clepsydrae) ~1 h
~1280 A.D. Mechanical clock invented - assembly time for prayer ~30 to 60 min
was first regular use
14th century Invention of the escapement; clockmaking becomes ~15 to 30 min
a major industry
~1345 Hour divided into minutes and seconds
15th century Clock time used to regulate people’s lives (work hours) ~2 min
16th century Time’s impact on science becomes significant ~1 min
(Galileo times physical events, e.g., free-fall)
1656 First pendulum clock (Huygens) ~100 s
18th century Temperature compensated pendulum clocks 1 to 10 s
19th century Electrically driven free-pendulum clocks 10-2 to 10-1 s
~1910 to 1920 Wrist watches become widely available
1920 to 1934 Electrically driven tuning forks 10-3 to 10-2 s
1921 to present Quartz crystal clocks (and watches. Since ~1971) 10-5 to 10-1 s
1949 to present Atomic clocks 10-10 to 10-4 s

8-7

Human beings’ use of clocks is a relatively recent phenomenon in terms of human history.
Modern humans (Homo sapiens) are believed to have originated somewhere around 200,000
years ago. http://en.wikipedia.org/wiki/History_of_the_Earth#2_Ma:_Human_evolution
Sumarians divided night & day into 12 equal hours each, whose length varied as the daylight
hours did. Except for astronomical purposes, equal hours were useless because people lived
by the sun until the invention of mechanical clocks.

-------------------------
H. Tait, Clocks and Watches, British Museum Publications, 1983

W. A. Marrison, “The Evolution of the Quartz Crystal Clock,” The Bell System Technical
Journal, Vol. XXVII, pp. 510-588, 1948. Reprinted at <http://www.ieee.org/uffc/fc>

<http://www.horology.com/>

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.9, by John R. Vig, November 2008.
 Frequency Control Device Market
(estimates, as of ~2006)

Technology Units Unit price, Worldwide

per year typical market, $/year
Quartz Crystal Resonators & ~ 3 x 109 ~$1 ~$4B
Oscillators ($0.1 to 3,000)

Atomic Frequency Standards

(see chapter 6)

Hydrogen maser ~ 20 $100,000 $2M

Cesium beam ~ 500 $50,000 $25M

frequency standard
Rubidium cell ~ 50,000 $2,000 $100M
frequency standard

1-2

The estimates are based on occasional informal surveys of industry leaders. The
numbers are probably accurate to a factor of two.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Resonator Vibration Amplitude Distribution
Metallic
electrodes

Resonator
plate substrate
(the “blank”)

Conventional resonator geometry

and amplitude distribution, u

3-5

In an ideal resonator, the amplitude of vibration falls off approximately exponentially outside
the electrodes. In a properly designed resonator, a negligible amount of energy is lost to the
mounting and bonding structure, i.e., the edges must be inactive in order for the resonator to be
able to possess a high Q. The displacement of a point on the resonator surface is proportional to
the drive current. At the typical drive currents used in (e.g., 10 MHz) thickness shear resonators,
the peak displacement is a few atomic spacings.
The peak acceleration of a point on the surface is often more than a million ‘g’s. To show
this, if the displacement u = uo sin ωt, then, the acceleration = d2u/dt2 = - ω2 uo sin ωt, and the
peak acceleration = - ω2 uo. If we assume that uo = two lattice spacings ~1 x 10-9 m, then, at 10
MHz, ω2 uo = (2π x 107)2 (10-9) ~ 106 g.

B. Capelle, J. Detaint, A. Zarka, Y. Zheng and J. Schwartzel, “Mode Shape Analysis Techniques
Using Synchrotron X-ray Topography,” Proc. 44th Ann. Symp. On Frequency Control, pp. 416-
423, 1990.

J. S. Yang and H. F. Tiersten, “An Analysis of Contoured Quartz Resonators with Beveled
Cylindrical Edges,” Proc. 1995 Int’l Frequency Control Symp., pp. 727-739, 1995.

H. F. Tiersten and D. S. Stevens, “The Evaluation of the Coefficients of Nonlinear Resonance for
SC-cut Quartz Resonators,” Proc. 39th Annual Symposium on Frequency Control, pp. 325-332,
1985, IEEE Catalog No. 85CH2186-5.

V. E. Bottom, Introduction to Quartz Crystal Unit Design, Van Nostrand Reinhold Company,
1982.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Modes of Motion
(Click on the mode names to see animation.)

Flexure Mode Extensional Mode Face Shear Mode

Thickness Shear Fundamental Mode Third Overtone

Mode Thickness Shear Thickness Shear

3-4

Shown above are the bulk acoustic wave (BAW) modes of motion. For example, AT-cut
and SC-cut resonators vibrate in the thickness shear mode. Above 100 MHz, overtone units
that operate at a selected harmonic mode of vibration are often used (e.g., third overtone or 5th
overtone). Higher than 100 MHz fundamental mode units can be manufactured by, e.g.,
chemical polishing (diffusion controlled wet etching), plasma etching, and ion milling techniques.
Below 1 MHz, tuning forks, X-Y and NT bars (flexure mode), +5° X-cuts (extensional mode), or
CT-cut and DT-cut units (face shear mode) can be used. Tuning forks have become the
dominant type of low-frequency units due to their small size and low cost (see “Quartz
Resonators for Wristwatches” and the following pages later in this chapter).

The velocities of acoustic waves in solids are typically ~3,000 m/s (~10-5 times the velocity
of light). For the shear waves in AT-cut quartz, for example, the velocity of propagation in the
thickness direction is 3,320 m/s; the fundamental mode frequency ~ v/2h, where v is the
acoustic wave velocity and h is the plate thickness. (The thickness of the plate is one half the
wavelength.)

Animations are courtesy of Raymond L. Filler

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Crystal Oscillator
Tuning
Voltage

Crystal
resonator

Output
Frequency
Amplifier

2-1

Above is a simplified circuit diagram that shows the basic elements of a crystal oscillator
(XO). The amplifier of an XO consists of at least one active device, the necessary biasing
networks, and may include other elements for band limiting, impedance matching, and gain
control. The feedback network consists of the crystal resonator, and may contain other
elements, such as a variable capacitor for tuning.

W. L. Smith, "Precision Oscillators," in E. A. Gerber and A. Ballato, Precision Frequency

Control, Vol. 2, pp. 45-98, Academic Press, 1985.

B. Parzen, Design of Crystal and Other Harmonic Oscillators, John Wiley and Sons, Inc., 1983.

M. E. Frerking, "Temperature Control and Compensation," in E. A. Gerber and A. Ballato,

Precision Frequency Control, Vol. 2, pp. 99-111, Academic Press, 1985.

M. E. Frerking, Crystal Oscillator Design and Temperature Compensation, Van Nostrand

Reinhold Company, 1978.

"Fundamentals of Quartz Oscillators," Hewlett-Packard application note AN 200-2, Hewlett-

Packard Company,
<http://www.tmo.hp.com/@@2ZcNpBcQ240oRhrt/tmo/Notes/English/5965-7662E.html>

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Oscillation
• At the frequency of oscillation, the closed loop phase shift
= 2nπ.
• When initially energized, the only signal in the circuit is
noise. That component of noise, the frequency of which
satisfies the phase condition for oscillation, is propagated
around the loop with increasing amplitude. The rate of
increase depends on the excess; i.e., small-signal, loop
gain and on the BW of the crystal in the network.
• The amplitude continues to increase until the amplifier gain
is reduced either by nonlinearities of the active elements
("self limiting") or by some automatic level control.
• At steady state, the closed-loop gain = 1.

2-2

See “Decay Time, Linewidth, and Q” in chapter 3 for further information on oscillator
startup time.
In addition to noise, switching on the DC power supply is another oscillation trigger.

W. L. Smith, "Precision Oscillators," in E. A. Gerber and A. Ballato, Precision Frequency

Control, Vol. 2, pp. 45-98, Academic Press, 1985.

M. Toki and Y. Tsuzuki, “Analysis of Start-up Characteristics of CMOS Crystal Oscillators,”

Proc. 1992 IEEE Frequency Control Symposium, pp. 448-452, 1992.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Crystal Oscillator Categories
∆f
Voltage f +10 ppm
Tune
250C
-450C +1000C
Output T

• Crystal Oscillator (XO) -10 ppm

∆f
Temperature
Temperature Compensation
Compensation
Sensor Network or f +1 ppm
Sensor Network or
Computer -450C +1000C
Computer
T
XO
XO -1 ppm

• Temperature Compensated (TCXO)

Oven
XO
∆f
Oven XO +1 x 10-8
Oven
-450C f
control
control Temperature +1000C
Temperature
Sensor T
Sensor
-1 x 10-8
• Oven Controlled (OCXO)
2-7

A wide temperature range XO has a typical f vs. T stability of ~10 to 50 ppm. A TCXO
can reduce that to ~1 ppm. An OCXO can reduce that stability to 1 x 10-8 or better (but at the
cost of much higher power consumption). High-end (SC-cut) OCXOs can stay within 1 x 10-10
over a wide temperature range.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Hierarchy of Oscillators
Oscillator Type* Accuracy** Typical Applications
• Crystal oscillator (XO) 10-5 to 10-4 Computer timing
• Temperature compensated 10-6 Frequency control in tactical
crystal oscillator (TCXO) radios
• Microcomputer compensated 10-8 to 10-7 Spread spectrum system clock
crystal oscillator (MCXO)
Navigation system clock &
• Oven controlled crystal 10-8 (with 10-10
frequency standard, MTI radar
oscillator (OCXO) per g option)
• Small atomic frequency 10-9 C3 satellite terminals, bistatic,
standard (Rb, RbXO) & multistatic radar

• High performance atomic 10-12 to 10-11 Strategic C3, EW

standard (Cs)

* Sizes range from <5cm3 for clock oscillators to > 30 liters for Cs standards
Costs range from <$5 for clock oscillators to > $50,000 for Cs standards.
** Including environmental effects (e.g., -40oC to +75oC) and one year of
aging.

2-8

See also chapter 7 for more detailed comparisons of various oscillators.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Why Quartz?
Quartz is the only material known that possesses the following
combination of properties:

• Piezoelectric ("pressure-electric"; piezein = to press, in Greek)

• Zero temperature coefficient cuts exist

• Stress compensated cut exists

• Low loss (i.e., high Q)

• Easy to process; low solubility in everything, under "normal" conditions,

except the fluoride and hot alkali etchants; hard but not brittle

• Abundant in nature; easy to grow in large quantities, at low cost, and

with relatively high purity and perfection. Of the man-grown single
crystals, quartz, at ~3,000 tons per year, is second only to silicon in
quantity grown (3 to 4 times as much Si is grown annually, as of 1997).

3-1

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Hydrothermal Growth of Quartz

• The autoclave is filled to some

Cover predetermined factor with water plus
mineralizer (NaOH or Na2CO3).

Closure • The baffle localizes the temperature gradient

area so that each zone is nearly isothermal.

• The seeds are thin slices of (usually)

Autoclave Z-cut single crystals.
Growth
zone, T1 Seeds
• The nutrient consists of small (~2½ to 4 cm)
pieces of single-crystal quartz (“lascas”).
Baffle
• The temperatures and pressures are
Nutrient typically about 3500C and 800 to 2,000
Solute- atmospheres; T2 - T1 is typically 40C to 100C.
dissolving
nutrient
zone, T2
• The nutrient dissolves slowly (30 to 260 days
Nutrient per run), diffuses to the growth zone, and
deposits onto the seeds.
T2 > T1

5-1

Prior to ~1956, the material used for quartz resonators was natural quartz, i.e., mined
quartz. Today, it is “cultured quartz,” i.e., quartz grown in factories. Although this quartz is
often referred to as “synthetic quartz,” nobody has yet found a way to synthesize single crystal
quartz directly from silicon and oxygen. Large quartz bars (typically ~15 cm long) of uniform
size and shape are grown from small, irregularly shaped pieces of quartz (called “lascas” by the
culturing process described above. So, strictly speaking, the quartz is “cultured quartz”.
Quartz is a common material in the earth’s crust (e.g., sand is mostly quartz), however,
the high purity crystals needed for quartz growing are not so common. Most of the nutrient
materials used by quartz growers are mined in Brazil and the USA (near Jessieville, Arkansas).
The autoclave is a long, thick-walled ~25 to 100 cm inner diameter steel tube that can
withstand the high temperatures and pressures of the growth process.
The anisotropy of quartz is discussed on the next page, and in chapter 3, where it is
pointed out that the highest etching rate direction is the Z-direction. Similarly, during quartz
growing, the Z-direction is the fastest direction of growth.

R. A. Laudise and R. L. Barns, “Perfection of Quartz and Its Connection to Crystal Growth,”
IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 35, No. 3, pp.
277-287, May 1988, IEEE Catalog 88CH2588-2.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Deeply Dissolved Quartz Sphere
Anisotropic Etching

Z
Y
+X
+X

Looking along Y-axis Looking along Z-axis

5-2

Polished quartz spheres, when deeply etched in concentrated HF, dissolve in a highly
anisotropic manner. The partially dissolved spheres become “triangular, lenticular,” as shown
above - the shape is triangular when observed along the Z-axis, and lenticular when observed
along the Y-axis. The etching rate along the fastest etching direction, the Z-direction, is nearly
1000 times faster than the rate along the slowest direction, the -X direction.

A good review of the early etching studies can be found in C. Frondel, The System of
Mineralogy, Vol. III, “Silica Minerals”, John Wiley and Sons, Inc., New York, 1962.

R. W. Ward, “Etching of Quartz Crystal Spheres,” Proc. 1993 IEEE Int’l Frequency Control
Symposium, pp. 390-396, 1993.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.9, by John R. Vig, November 2008.
 Zero
ZeroTemperature
Temperature Coefficient QuartzCuts
Coefficient Quartz Cuts
90oo
60o
AT FC IT
30oo
LC SC
z
θ 0
-30oo SBTC
SBTC
BT
-60oo
-60
-90oo oo
0 10oo 20oo 30oo
θ φ
Dou The AT, FC, IT, SC, BT, and SBTC-cuts are some
ggllyy θ
Sinn taatteedd
bl of the cuts on the locus of zero
zero temperature
temperature
o t Root lyy
R
R o tt taat coefficient cuts. The LC is a “linear coefficient”
coefficient”
CCuu CCu teedd y
utt cut that has been used in a quartz
quartz thermometer.
thermometer.
φ Y-cut: ≈ +90 ppm/0C
(thickness-shear mode)

x xll X-cut: ≈ -20 ppm/0C

(extensional mode)

3-13

------------------------
The locus of zero-temperature-coefficient cuts in quartz is shown above. The cuts usually have two-
A. letter
Ballato, “Doubly rotated thickness mode plate vibrators,” in Physical Acoustics, Vol.
names, where the "T" in the name indicates a temperature-compensated cut; for instance, the AT-cut
XIII, pp. 115-181, Academic Press,cut
was the first temperature-compensated 1977.
discovered. The FC, IT, BT, and SBTC-cuts are other cuts
along the zero-temperature coefficient locus. These cuts were studied in the past (before the discovery of
the SC-cut) for some special properties, but are rarely used today. The highest-stability crystal oscillators
employ SC-cut crystal units. The X, Y, and Z directions have been chosen to make the description of
properties as simple as possible. The Z-axis is an axis of threefold symmetry in quartz; in other words, the
physical properties repeat every 120° as the crystal is rotated about the Z-axis.
------------------------------------
A. Ballato, "Doubly Rotated Thickness Mode Plate Vibrators," in Physical Acoustics, Vol. XIII, W. P. Mason
and R. N. Thurston, Eds., pp. 115-181, Academic Press, New York, 1977.

J. P. Buchanan, Handbook of Piezoelectric Crystals for Radio Equipment Designers, WADC Technical
Report 56-156, October 1956 (692 pages), available from NTIS, AD 110448.

D. L. Hammond, C. A. Adams and P. Schmidt, "A Linear, Quartz Crystal, Temperature Sensing Element,"
ISA Transactions, vol. 4, pp. 349-354, 1965.

M. Valdois, B. K. Sinha, and J. J. Boy, “Experimental Verification of Stress Compensation in the SBTC-
Cut,” IEEE Trans. on Ultrasonics, Ferroelectrics and Frequency Control, vol. 36, pp. 643-651, 1989.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Crystal Oscillator f vs. T Compensation
Uncompensated
frequency
Frequency / Voltage

Compensating Compensated
voltage frequency
on varactor CL of TCXO

3-23

When the load capacitor is connected in series with the crystal, the frequency of
operation of the oscillator is increased by a ∆f', where ∆f' is given by the equation on the
previous page. When an inductor is connected in series with the crystal, the frequency of
operation is decreased. The ability to change the frequency of operation by adding or changing
a reactance allows for compensation of the frequency versus temperature variations of crystal
units in TCXOs, and for tuning the output frequency of voltage controlled crystal oscillators
(VCXO). In both, the frequency can be changed, e.g., by changing the voltage on a varactor.
Other means of temperature compensation include the use of a temperature sensitive
reactance element such that the variations of the reactance with temperature compensate for
the f vs. T variations of the resonator, and the use digital compensation techniques. The
microcomputer compensated crystal oscillator (MCXO), which uses a high-accuracy digital
compensation technique, is discussed in chapter 2.

M. E. Frerking, “Temperature Control and Compensation,” in E. A. Gerber and A. Ballato,

Precision Frequency Control, Vol. 2, pp.1-44, Academic Press, 1985.

M. E. Frerking, Crystal Oscillator Design and Temperature Compensation, Van Nostrand

Reinhold Company, 1978.

S. Okano, T. Mitsuoka & T. Ohshima, “Direct-temperature Compensated Crystal Oscillator for

Advanced VHF/UHF Radio Communication Systems,” Proc. 34th Ann. Frequency Control
Symposium, pp. 488-497, 1980.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Silicon Resonator & Oscillator

www.SiTime.com

Resonator (Si): 0.2 x 0.2 x 0.01 mm3

5 MHz; f vs. T: -30 ppm/oC
Oscillator (CMOS): 2.0 x 2.5 x 0.85 mm3
• ±50 ppm, ±100 ppm; -45 to +85 oC
(±5 ppm demoed, w. careful calibration)
• 1 to 125 MHz
• <2 ppm/y aging; <2 ppm hysteresis
• ±200 ps peak-to-peak jitter, 20-125 MHz
2-17

The resonator, including its hermetically sealed enclosure, is made of single crystal silicon.
The resonator is ~200um on a side by ~10um thick. The trench gap is nominally
0.4um. Resonant frequency is ~5 MHz. Q is ~75k at room temperature. The resonator is
integrated into standard silicon CMOS chips. The oscillator is, therefore, inexpensive to
produce.
All oscillator frequencies are derived from the same 5MHz resonator. The oscillator is
compensated by measuring the temperature with a bandgap thermometer on the CMOS die
and adjusting a delta-sigma fractional PLL. The resolution is ~0.05C, giving a frequency
resolution of ~1.5ppm. The spec is +/- 100ppm and +/-50ppm depending on the
grade. Carefully calibrated over temperature, a f vs. T of about +/- 5ppm can be obtained (with
a cubic residue). With a simpler one-temperature calibration, typically +/-30 ppm f vs. T is
obtained. Because each part is calibrated after packaging, the initial frequency offset is small,
for example under about 5ppm.
The resonator is driven with five lines. The electrodes are doped silicon. At the center there is
a bias contact to the resonator element at the anchor that is used to bias it, presently at 5V, the
eight electrostatic drive and sense electrodes around the quad are driven at DC of zero. The
four drive electrodes are divided into pairs of drive minus (on the inside of the quad) and drive
plus (on the outside of the quad. Two electrodes in each pair are wired in parallel. The sense
electrodes are wired in a similar way (in fact the drive and sense are interchangeable). The
pairs are organized on adjacent sides so that there is minimal net drive and sense for the
wineglass mode. A substrate ground connection is tied to the cover and the substrate.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 What is Q and Why is it Important?
Energy stored during a cycle
Q≡2π
Energy dissipated per cycle

Q is proportional to the decay-time, and is inversely

proportional to the linewidth of resonance (see next page).

• The higher the Q, the higher the frequency stability and

accuracy capability of a resonator (i.e., high Q is a
necessary but not a sufficient condition). If, e.g., Q = 106,
then 10-10 accuracy requires ability to determine center of
resonance curve to 0.01% of the linewidth, and stability (for
some averaging time) of 10-12 requires ability to stay near
peak of resonance curve to 10-6 of linewidth.

• Phase noise close to the carrier has an especially strong

dependence on Q (L(f) ∝ 1/Q4 for quartz oscillators).

3-26

See the next page for other definitions of Q, and see chapter 5 for additional information
about the Q of quartz resonators. When the signal is decaying, as shown on the next page, the
energies in the definition above are averaged over the cycle. Close to the carrier, a factor of
two difference in Q results in a factor of 16 difference in phase noise.

IEEE Std 100-1996, The IEEE Standard Dictionary of Electrical and Electronics Terms,
http://shop.ieee.org/store/

H. Hellwig, "Frequency Standards and Clocks: A Tutorial Introduction," NBS Technical Note
616, 1977, Time and Frequency Division, NIST, Boulder, CO 80303.

V. B. Braginsky, V. P. Mitrofanov & V. I. Panov, Systems with Small Dissipation, The

University of Chicago Press, 1985.

E. I. Green, "The Story of Q," American Scientist, pp. 584-595, 1955.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Precision Frequency Standards

• Quartz crystal resonator-based (f ~ 5 MHz, Q ~ 106)

• Atomic resonator-based
Rubidium cell (f0 = 6.8 GHz, Q ~ 107)
Cesium beam (f0 = 9.2 GHz, Q ~ 108)
Hydrogen maser (f0 = 1.4 GHz, Q ~ 109)
Cesium fountain (f0 = 9.2 GHz, Q ~ 5 x 1011)
Trapped ions (f0 > 10 GHz, Q > 1011)

6-1

A high Q is necessary (but not sufficient) for high frequency stability - see Chapter 3 for a
discussion of Q. The higher the Q, the higher the frequency stability and accuracy capability of
a resonator. If, e.g., Q = 106, then 10-10 accuracy requires the ability to determine the center of
the resonance curve to 0.01% of the linewidth, and stability (for some averaging time) of 10-12
requires the ability to stay near the peak of the resonance curve to 10-6 of linewidth.
A high Q is not sufficient for high stability because a high Q resonator may, for example,
have a poor temperature stability. Sapphire resonators, for example, can have a very high Q,
but their poor temperature stability prevents their use in clocks.
The Q, or line width of an atomic transition is determined by the observation time. The
atomic resonance Qs listed above are typical values. Laser cooling of atoms can significantly
extend the observation time and Q (see “Laser Cooling of Atoms” later in this chapter. Laser
cooling is necessary to achieve a Cs fountain).

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Atomic Frequency Standard Basic Concepts
When an atomic system changes energy from an exited state to a
lower energy state, a photon is emitted. The photon frequency ν is given
by Planck’s law E2 − E1
ν =
h
where E2 and E1 are the energies of the upper and lower states,
respectively, and h is Planck’s constant. An atomic frequency standard
produces an output signal the frequency of which is determined by this
intrinsic frequency rather than by the properties of a solid object and how it
is fabricated (as it is in quartz oscillators).
The properties of isolated atoms at rest, and in free space, would not
change with space and time. Therefore, the frequency of an ideal atomic
standard would not change with time or with changes in the environment.
Unfortunately, in real atomic frequency standards: 1) the atoms are
moving at thermal velocities, 2) the atoms are not isolated but experience
collisions and electric and magnetic fields, and 3) some of the components
needed for producing and observing the atomic transitions contribute to
instabilities.

6-2

Atomic frequency standards must be understood in terms of the concepts of quantum

mechanics. The properties of simple atomic systems cannot assume arbitrary values. For
example, the energies of the bound states of an atomic system are constrained to discrete
values called energy levels. When an atomic system changes energy from an excited state to a
state with lower energy, it emits a quantity of electromagnetic energy called a photon, the
frequency of which is determined by the energy difference between the two states, in
accordance with Planck’s law, shown above.
Atomic systems can be isolated from unwanted perturbations, which result in small
sensitivities to temperature, pressure, and other environmental conditions. The low level of
interaction also results in extremely sharp resonance features, and reduces errors due to
imperfections in the electronics. All atoms of an element are identical, and atomic properties
are time invariant, which makes it possible to build very stable devices.
Atomic frequency standards are categorized in several ways; most often, they are
referred to by the type of atom: hydrogen, rubidium, or cesium. Actually, these three devices
are based on the same type of atomic interaction, but there are great practical differences in
their implementation. Some atomic frequency standards, called oscillators, are active, in which
case the output signal is derived from the radiation emitted by the atom. Others are passive;
the atoms are then employed as a discriminator to measure and control the frequency of an
electronic oscillator, such as a quartz oscillator. The third classification follows the method of
interaction. In atomic beams, the atoms are observed "on the fly"; they pass through the
interaction region and are not used again. In contrast, storage devices contain some type of
cell that holds the atoms to be observed indefinitely (ideally).

S. R. Stein and J. R. Vig, "Frequency Standards for Communications," U. S. Army Laboratory

Command Research and Development Technical Report SLCET-TR-91-2 (Rev. 1), October
1991, AD-A243211. This report is a reprint of a chapter, "Communications Frequency
Standards," in The Froehlich/Kent Encyclopedia of Telecommunications, Vol. 3, pp. 445-500,
Marcel Dekker, Inc., 1992.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Atomic Frequency Standard*
Block Diagram

Multiplier
Multiplier Quartz
Quartz
Atomic
Atomic Crystal
Crystal
Resonator
Resonator Feedback
Feedback Oscillator
Oscillator

5 MHz
Output

* Passive microwave atomic standard (e.g., commercial Rb and Cs standards)

6-4

Atomic resonators are inherently noisy due to the discreet nature of atomic transitions.
The short term stabilities, σy(τ) vs. τ, vary as the square-root of the measurement interval, i.e.,
as τ1/2, for short intervals. This is due to the statistics of counting atomic transitions; σy(τ) varies
as the square-root of the number of transitions. Crystal oscillators are less noisy at small τ.
Therefore, in all commercial atomic standards, the atomic resonator frequency is generated
from the crystal oscillator’s frequency (by frequency multiplication or frequency synthesis), and
the crystal oscillator frequency is locked to the frequency of the atomic resonator with a servo
loop time constant that is selected to provide optimum performance for the intended application.
Of the many atomic transitions available, the ones selected are those which are least sensitive
to environmental effects and which can be conveniently locked to the VCXO.
The atomic standard behaves as the crystal oscillator for measurement times shorter than
the time constant (which, for example, is typically 100 ms to 500 ms for a Rb standard, longer in
Cs standards), and it behaves as an atomic oscillator for measurement times longer than the
time constant.
Since all atomic frequency standards derive their output signal from quartz oscillators, the
performance of the atomic standards is significantly affected by the capabilities of the crystal
oscillators. In particular, the short-term frequency stability, the vibration sensitivity, the radiation
pulse sensitivity, and the sensitivity to thermal transients depend on the performance of the
crystal oscillator. The atomic resonator’s superior long term stability and lower sensitivity to
environmental changes is used to “servo out” the crystal oscillator’s aging and some of the
crystal oscillator’s environmental sensitivities.
----------------------------
L. L. Lewis, “An Introduction to Frequency Standards,” Proc. IEEE, vol. 79, pp. 927-935,
1991.
H. Hellwig, "Frequency Standards and Clocks: A Tutorial Introduction," NBS Technical
Note 616, 1977, Time and Frequency Division, NIST, 325 Broadway, Boulder, Colorado, 80303.
H. Hellwig, "Microwave Frequency and Time Standards," in E. A. Gerber and A. Ballato,
Precision Frequency Control, Vol. 2, pp. 113-176, Academic Press, 1985.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Chip Scale Atomic Clock (CSAC)

120mW power consumption

16cm3 volume (1.6” x 1.39” x 0.45”)
35g weight
±5.0E-11 accuracy at shipment
σy < 5 x 10-12 at τ = 1 hour short-term stability (Allan Deviation)
<3.0E-10/month aging rate
4-1

http://www.symmetricom.com/products/frequency-references/chip-scale-atomic-clock-
csac/

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Laser Cooling of Atoms
1 Direction of motion

Light Light

Atom

2 3 4
Direction
of force

6-14

Laser cooling of atoms can create atoms that move very slowly (equivalent to
temperatures of microkelvins). This allows long observation times. The slow speed virtually
eliminates Doppler shifts, and the long observation times allow high accuracy determinations of
atomic transition frequencies, per the Heisenberg uncertainty principle, i.e., ∆E∆t ~ h and E =
hν, so ∆ ν ~ 1/∆t. Laser cooling promises frequency accuracies of parts in 1016. The
explanation of laser cooling is as follows. The numbers correspond to the numbers in the
illustration above:
1. Consider two rays of light that bombard an atom. One ray travels in the same direction as
the atom,; the other moves in the opposite direction. The frequency of the light is slightly lower
than the frequency that the atom readily absorbs.
2. From the atom’s perspective, the ray moving in the same direction as the atom is shifted
down in frequency; the other ray is shifted up in frequency.
3. The atom is likely to absorb the high-frequency light but not the low. It is therefore pushed in
a direction opposite its motion and slows down.
4. The emission of the absorbed light pushes the atom in some random direction, but if the
process is repeated many times, the emission exerts no net force.

Chu, Steven, ”Laser Trapping of Neutral Particles," Scientific American, February 1992, pp. 71-
76.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Cesium Fountain

Click here for animation

• Accuracy ~1 x 10-15 or
1 second in 30 million years
• 1 x 10-16 is achievable

6-15

A fountain clock, such as NIST-F1 shown above, uses a fountain-like movement of atoms to measure
frequency and time interval. First, a gas of cesium atoms is introduced into the clock's vacuum chamber.
Six infrared laser beams then are directed at right angles to each other at the center of the chamber. The
lasers gently push the cesium atoms together into a ball. In the process of creating this ball, the lasers slow
down the movement of the atoms and cool them to temperatures near absolute zero.
Two vertical lasers are used to gently toss the ball upward (the "fountain" action), and then all of the lasers
are turned off. This little push is just enough to loft the ball about a meter high through a microwave-filled
cavity. Under the influence of gravity, the ball then falls back down through the microwave cavity.
The round trip up and down through the microwave cavity lasts for about 1 second. During the trip, the
atomic states of the atoms might or might not be altered as they interact with the microwave signal. When
their trip is finished, another laser is pointed at the atoms. Those atoms whose atomic state were altered by
the microwave signal emit light (a state known as fluorescence). The photons, or the tiny packets of light
that they emit, are measured by a detector.
This process is repeated many times while the microwave signal in the cavity is tuned to different
frequencies. Eventually, a microwave frequency is found that alters the states of most of the cesium atoms
and maximizes their fluorescence. This frequency is the natural resonance frequency of the cesium atom
(9,192,631,770 Hz), or the frequency used to define the second.
The combination of laser cooling and the fountain design allows NIST-F1 to observe cesium atoms for
longer periods, and thus achieve its unprecedented accuracy. Traditional cesium clocks measure room-
temperature atoms moving at several hundred meters per second. Since the atoms are moving so fast, the
observation time is limited to a few milliseconds. NIST-F1 uses a different approach. Laser cooling drops
the temperature of the atoms to a few millionths of a degree above absolute zero, and reduces their
thermal velocity to a few centimeters per second. The laser cooled atoms are launched vertically and pass
twice through a microwave cavity, once on the way up and once on the way down. The result is an
observation time of about one second, which is limited only by the force of gravity pulling the atoms to the
ground.
As you might guess, the longer observation times make it easier to tune the microwave frequency. The
improved tuning of the microwave frequency leads to a better realization and control of the resonance
frequency of cesium. And of course, the improved frequency control leads to what is one of the world's
most accurate clocks.
Credits
NIST-F1 was developed by Steve Jefferts and Dawn Meekhof of the Time and Frequency Division of
NIST's Physics Laboratory in Boulder, Colorado. It was constructed and tested in less than four years.
-----------------------------------
The above figure, text and animation were copied from the NIST Time & Frequency Division website,
http://www.boulder.nist.gov/timefreq/cesium/fountain.htm, with the permission of Steve Jefferts.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 The Units of Stability in Perspective

• What is one part in 1010 ? (As in 1 x 10-10/day aging.)

• ~1/2 cm out of the circumference of the earth.
• ~1/4 second per human lifetime (of ~80 years).
• Power received on earth from a GPS satellite, -160 dBW, is
as “bright” as a flashlight in Los Angeles would look in New
York City, ~5000 km away (neglecting earth’s curvature).
• What is -170 dB? (As in -170 dBc/Hz phase noise.)
• -170 dB = 1 part in 1017 ≈ thickness of a sheet
of paper out of the total distance traveled by all
the cars in the world in a day.

4-1

The human mind is limited in its ability to understand very small and very large numbers.
Above is an attempt to make the small numbers used in the frequency and time field a bit more
understandable.

GPS analogy is courtesy of Raymond Filler, March 2004.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Accuracy, Precision, and Stability

Precise but Not accurate and Accurate but Accurate and

not accurate not precise not precise precise
f f f f

Time Time Time Time

Accurate
Stable but Not stable and Stable and
(on the average)
not accurate not accurate accurate
but not stable

4-2

The terms accuracy, stability, and precision are often used in describing an oscillator's
quality. Above is an illustration of the meanings of these terms for a marksman and for a
frequency source. (For the marksman, each bullet hole's distance to the center of the target is
the "measurement.") Accuracy is the extent to which a given measurement, or the average of
a set of measurements for one sample, agrees with the definition of the quantity being
measured. It is the degree of "correctness" of a quantity. Reproducibility is the ability of a
single frequency standard to produce the same frequency, without adjustment, each time it is
put into operation. From the user's point of view, once a frequency standard is calibrated,
reproducibility confers the same advantages as accuracy. Stability describes the amount
something changes as a function of parameters such as time, temperature, shock, and the like.
Precision is the extent to which a given set of measurements of one sample agrees with the
mean of the set. (A related meaning of the term is used as a descriptor of the quality of an
instrument, as in a "precision instrument." In that context, the meaning is usually defined as
accurate and precise, although a precision instrument can also be inaccurate and precise, in
which case the instrument needs to be calibrated.)
The military specification for crystal oscillators, MIL-PRF-55310D*, defines “Overall
Frequency Accuracy” as “6.4.33 Overall frequency accuracy. The maximum permissible
frequency deviation of the oscillator frequency from the assigned nominal value due to all
combinations of specified operating and nonoperating parameters within a specified period of
time. In the general case, overall accuracy of an oscillator is the sum of the absolute values
assigned to the following:
a. The initial frequency-temperature accuracy (see 6.4.24).
b. Frequency-tolerances due to supply voltage changes (see 6.4.17) and other environmental
effects
(see 6.4.12).
Total frequency change from an initial value due to frequency aging (see 6.4.11) at a specified
temperature.”
The International System (SI) of units for time and frequency (the second and Hz,
respectively) are obtained in laboratories using very accurate frequency standards called
primary standards. A primary standard operates at a frequency calculable in terms of the SI
definition ofAND
“QUARTZ CRYSTAL RESONATORS the OSCILLATORS
second**: "the duration of 9,192,631,770 periods of the radiation
corresponding
For Frequency Control and to the
Timing Applications - Atransition
TUTORIAL”between the two hyperfine levels of the ground state of the
Rev. 8.5.3.6, by John R.cesium
Vig, January
atom 2007.
133”.
 Influences on Oscillator Frequency

Time
• Short term (noise)
• Intermediate term (e.g., due to oven fluctuations)
• Long term (aging)

Temperature
• Static frequency vs. temperature
• Dynamic frequency vs. temperature (warmup, thermal shock)
• Thermal history ("hysteresis," "retrace")

Acceleration
• Gravity (2g tipover) • Acoustic noise
• Vibration • Shock

Ionizing radiation
• Steady state • Photons (X-rays, γ-rays)
• Pulsed • Particles (neutrons, protons, electrons)

Other
• Power supply voltage • Humidity • Magnetic field
• Atmospheric pressure (altitude) • Load impedance

4-3

Many factors influence the frequency stability of an oscillator. Changes in the

environment can cause especially large instabilities. For example, orders of magnitude (tens of
dBs) changes can be observed when the phase noise of an oscillator is measured in a quiet
laboratory environment, and in a vibrating environment, such as a moving vehicle.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Idealized Frequency-Time-Influence Behavior
∆f
X 10 8 Oscillator
f Turn Off 2-g
Temperature Vibration Shock & Radiation
3 Step Tipover
Turn On
Off
2

1 Aging

-1
On

-2
Short-Term
Instability
-3

t0 t1 t2 t3 t4 t5 t6 t7 t8 Time

4-4

Shown above are the major types of oscillator frequency instabilities. The pages that
follow show each of the changes, and some others, in more detail.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Aging and Short-Term Stability
Short-term instability
(Noise)

30
25
∆ f/f (ppm)

20

15

10

5 10 15 20 25 Time (days)

4-5

The difference between aging and short-term instability, i.e., noise, is illustrated above.
One is a systematic effect that is observed over long periods of time (days to years), whereas
the other is random, observed over periods that are typically measured in fractions of a second
to minutes. Over periods of hours, a combination of systematic and random effects are usually
observed. The frequency vs. time characteristics over such periods often appear to be random
walk (at least some of which is usually environmentally caused).
"Aging" and "drift" have occasionally been used interchangeably in the frequency control
literature. However, in 1990, recognizing the "need for common terminology for the
unambiguous specification and description of frequency and time standard systems," the CCIR
adopted a glossary of terms and definitions. According to this glossary, aging is "the systematic
change in frequency with time due to internal changes in the oscillator." Added to the definition
is: "Note - It is the frequency change with time when factors external to the oscillator
(environment, power supply, etc.) are kept constant." Drift is defined as "the systematic change
in frequency with time of an oscillator." Drift is due to a combination of factors, i.e., it due to
aging plus changes in the environment and other factors external to the oscillator. Aging is
what one specifies and what one measures during oscillator evaluation. Drift is what one
observes in an application. For example, the drift of an oscillator in a spacecraft is due to (the
algebraic sum of) aging and frequency changes due to radiation, temperature changes in the
spacecraft, and power supply changes.

CCIR Recommendation No. 686, “[TF.686-1] Glossary,” CCIR 17th Plenary Assembly, Vol. VII,
"Standard Frequency and Time Signals (Study Group 7)," 1990. Consultative Committee on
International Radio (CCIR); copies available from: International Telecommunications Union,
General Secretariat - Sales Section, Place des Nations, CH-1211 Geneva, Switzerland.
http://www.itu.int/itudoc/itu-r/rec/tf/

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Frequency Noise and σy(τ)
3 X 10-11 0.1 s averaging time

∆f
0
f
-3 X 10-11 100 s

3 X 10-11
1.0 s averaging time
∆f
0
f 100 s

-11
-3 X 10

σy(τ) 10 -10

10-11

10-12
0.01 0.1 1 10 100
Averaging time, τ, s

4-24

“The noise” is a function of the averaging time (also called “measurement time” or “tau”),
as is illustrated above. For the same oscillator, the fluctuations in the frequency vs. time plot
measured with a 0.1 second averaging time are larger than when measured with a 1 second
averaging time. Also shown are the corresponding Allan deviations.
At short averaging times, the longer the averaging time, the lower the noise, up to the
“flicker floor,” i.e., for certain noise processes (see the next four pages), the hills and valleys in
the frequency vs. time data average out. Longer averaging does not help when the dominant
noise process is flicker of frequency. At the flicker floor, the Allan deviation is independent of
averaging time. At longer averaging times, the Allan deviation increases because the dominant
noise process is random walk of frequency, for which the longer the averaging time, the larger
the Allan deviation.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Clock Accuracy vs. Power Requirement*
(Goal of R&D is to move technologies toward the upper left)

10-12
Cs
Cs 1µs/day
1ms/year
10-10
Rb
Rb
Accuracy

10-8 1ms/day
OCXO
OCXO 1s/year

10-6 TCXO
TCXO
1s/day

10-4 XO
0.001 0.01 0.1 1 10 100
Power (W)
* Accuracy vs., size, and accuracy vs. cost have similar relationships
7-2

Commercially available frequency sources cover an accuracy range of several orders of

magnitude - from the simple XO to the cesium-beam frequency standard. As the accuracy
increases, so does the power requirement, size, and cost. Shown above is the relationship
between accuracy and power requirement. (Note that it is a log-log scale.) Accuracy versus
cost would be a similar relationship, ranging from about $1 for a simple XO to about $40,000 for
a cesium standard (1997 prices).

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.6, by John R. Vig, January 2007.
 Your Head is Older Than Your Feet
• Einstein: time passes faster at higher elevations; moving
clock runs slower

• Super-precise, 1 x 10-17, atomic clocks (e.g., trapped Al ion

optical clocks) can measure “time dilation” effects on a
“tabletop:”

o When one of two clocks at NIST was raised 33 cm, it ran

faster than the other, as predicted by Einstein

o When, e.g., a clock is moving at 10 km/h, it runs faster

than a stationary clock, as predicted by Einstein

o Heads age faster than feet by ~0.5 µs per human lifetime

8-23

C.W. Chou, D.B. Hume, T. Rosenband and D.J. Wineland. Optical clocks and relativity.
Science. Sept. 24, 2010.

http://www.nist.gov/pml/div688/clocks_092810.cfm

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.9, by John R. Vig, November 2008.
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“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.9, by John R. Vig, November 2008.
Thank you - Questions

44
 Thank you.

“QUARTZ CRYSTAL RESONATORS AND OSCILLATORS

For Frequency Control and Timing Applications - A TUTORIAL”
Rev. 8.5.3.9, by John R. Vig, November 2008.