1. technology and computing
2. electronic components

SLCET-TR-88-1 (Rev.8.4.2)
AD-A328861 (revised)
QUARTZ CRYSTAL
RESONATORS AND OSCILLATORS
For Frequency Control and Timing Applications
A Tutorial
John R. Vig
U.S. Army Communications-Electronics Command
Attn: AMSEL-RD-C2-PT
Fort Monmouth, NJ 07703, USA
J.Vig@IEEE.org
January 2000
Approved for public release.
Distribution is unlimited.
NOTICES
Disclaimer
The findings in this report are not to be construed as an
official Department of the Army position, unless so
designated by other authorized documents.
The citation of trade names and names of manufacturers
in this report is not to be construed as official Government
endorsement or consent or approval of commercial
products or services referenced herein.
Report Documentation Page
Table of Contents
Preface………………………………..………………………..
v
1.
Applications and Requirements……………………….
1
2.
Quartz Crystal Oscillators……………………………….
2
3.
Quartz Crystal Resonators………………………………
3
4.
Oscillator Stability…………………………………………
4
5.
Quartz Material Properties……………………………...
5
6.
Atomic Frequency Standards……………………………
6
7.
Oscillator Comparison and Specification……………..
7
8.
Time and Timekeeping………………………………….
8
9.
Related Devices and Applications………………………
9
10.
FCS Proceedings Ordering, Website, and Index…………..
10
iii
Preface
Why This Tutorial?
“Everything should be made as simple as
possible - but not simpler,” said Einstein. The
main goal of this “tutorial” is to assist with
presenting the most frequently encountered
concepts in frequency control and timing, as
simply as possible.
I was frequently asked for “hard-copies”
of the slides, so I started organizing, adding
some text, and filling the gaps in the slide
collection. As the collection grew, I began
receiving favorable comments and requests
for additional copies. Apparently, others, too,
found this collection to be useful. Eventually, I
assembled this document, the “Tutorial”.
I have often been called upon to brief
visitors, management, and potential users of
precision oscillators, and have also been
invited to present seminars, tutorials, and
review papers before university, IEEE, and
other professional groups. In the beginning, I
spent a great deal of time preparing these
presentations. Much of the time was spent on
preparing the slides. As I accumulated more
and more slides, it became easier and easier
to prepare successive presentations.
This is a work in progress. I plan to
include new material, including additional
notes. Comments, corrections, and
suggestions for future revisions will be
welcome.
John R. Vig
iv
Don’t Forget the Notes!
In the PowerPoint version of this document, notes and
references can be found in the “Notes” of most of the pages.
To view the notes, use the “Notes Page View” icon (near the
lower left corner of the screen), or select “Notes Page” in the
View menu. In PowerPoint 2000 (and, presumably, later
versions), the notes also appear in the “Normal view”.
To print a page so that it includes the notes, select Print in
the File menu, and, near the bottom, at “Print what:,” select
“Notes Pages”.
The HTML version can be viewed with a web browser (best
viewed at 1024 x 768 screen size). The notes then appear in
the lower pane on the right.
v
CHAPTER 1
Applications and Requirements
1
Electronics Applications of Quartz Crystals
Industrial
Communications
Telecommunications
Mobile/cellular/portable
radio, telephone & pager
Aviation
Marine
Navigation
Instrumentation
Computers
Digital systems
Research & Metrology CRT displays
Disk drives
Atomic clocks
Modems
Instruments
Astronomy & geodesy Tagging/identification
Utilities
Space tracking
Sensors
Celestial navigation
Military & Aerospace
Communications
Navigation
IFF
Radar
Sensors
Guidance systems
Fuzes
Electronic warfare
Sonobouys
1-1
Consumer
Watches & clocks
Cellular & cordless
phones, pagers
Radio & hi-fi equipment
Color TV
Cable TV systems
Home computers
VCR & video camera
CB & amateur radio
Toys & games
Pacemakers
Other medical devices
Automotive
Engine control, stereo,
clock
Trip computer, GPS
Frequency Control Device Market
(as of ~1997)
Technology
Units
per year
Unit price,
typical
Worldwide
market, $/year
Crystal
~ 2 x 109
~$1
($0.1 to 3,000)
~$1.2B
Hydrogen maser
~ 10
$200,000
$2M
Cesium beam
frequency standard
Rubidium cell
frequency standard
~ 300
$50,000
$15M
~ 20,000
$2,000
$40M
Atomic Frequency Standards
(see chapter 6)
1-2
Navigation
Precise time is essential to precise navigation. Historically, navigation has
been a principal motivator in man's search for better clocks. Even in ancient times,
one could measure latitude by observing the stars' positions. However, to determine
longitude, the problem became one of timing. Since the earth makes one revolution
in 24 hours, one can determine longitude form the time difference between local time
(which was determined from the sun's position) and the time at the Greenwich
meridian (which was determined by a clock):
Longitude in degrees = (360 degrees/24 hours) x t in hours.
In 1714, the British government offered a reward of 20,000 pounds to the first
person to produce a clock that allowed the determination of a ship's longitude to 30
nautical miles at the end of a six week voyage (i.e., a clock accuracy of three
seconds per day). The Englishman John Harrison won the competition in 1735 for
his chronometer invention.
Today's electronic navigation systems still require ever greater accuracies. As
electromagnetic waves travel 300 meters per microsecond, e.g., if a vessel's timing
was in error by one millisecond, a navigational error of 300 kilometers would result.
In the Global Positioning System (GPS), atomic clocks in the satellites and quartz
oscillators in the receivers provide nanosecond-level accuracies. The resulting
(worldwide) navigational accuracies are about ten meters (see chapter 8 for further
details about GPS).
1-3
Commercial Two-way Radio
Historically, as the number of users of commercial two-way radios
have grown, channel spacings have been narrowed, and higherfrequency spectra have had to be allocated to accommodate the
demand. Narrower channel spacings and higher operating frequencies
necessitate tighter frequency tolerances for both the transmitters and the
receivers. In 1940, when only a few thousand commercial broadcast
transmitters were in use, a 500 ppm tolerance was adequate. Today,
the oscillators in the many millions of cellular telephones (which operate
at frequency bands above 800 MHz) must maintain a frequency
tolerance of 2.5 ppm and better. The 896-901 MHz and 935-940 MHz
mobile radio bands require frequency tolerances of 0.1 ppm at the base
station and 1.5 ppm at the mobile station.
The need to accommodate more users will continue to require higher
and higher frequency accuracies. For example, a NASA concept for a
personal satellite communication system would use walkie-talkie-like
hand-held terminals, a 30 GHz uplink, a 20 GHz downlink, and a 10 kHz
channel spacing. The terminals' frequency accuracy requirement is a
few parts in 108.
1-4
Digital Processing of Analog Signals
The Effect of Timing Jitter
(A)
Analog*
input
A/D
A/D
converter
converter
Digital
Digital
processor
processor
D/A
D/A
converter
converter
Analog
output
Digital
output
* e.g., from an antenna
(C)
(B)
V(t)
V(t)
Time
Analog signal
Digitized signal
1-5
∆V
∆t
•
Digital Network Synchronization
Synchronization plays a critical role in digital telecommunication systems.
It ensures that information transfer is performed with minimal buffer overflow or
underflow events, i.e., with an acceptable level of "slips." Slips cause
problems, e.g., missing lines in FAX transmission, clicks in voice transmission,
loss of encryption key in secure voice transmission, and data retransmission.
•
In AT&T's network, for example, timing is distributed down a hierarchy of
nodes. A timing source-receiver relationship is established between pairs of
nodes containing clocks. The clocks are of four types, in four "stratum levels."
Stratum
Accuracy (Free Running)
Long Term Per 1st Day
Clock Type
Number Used
1
1 x 10-11
N.A.
GPS W/Two Rb
16
2
1.6 x 10-8
1 x 10-10
Rb Or OCXO
~200
3
4.6 x 10-6
3.7 x 10-7 OCXO Or TCXO
4
3.2 x 10-5
N.A.
1-6
XO
1000’s
~1 million
Phase Noise in PLL and PSK Systems
The phase noise of oscillators can lead to erroneous detection of
phase transitions, i.e., to bit errors, when phase shift keyed (PSK) digital
modulation is used. In digital communications, for example, where
8-phase PSK is used, the maximum phase tolerance is ±22.5o, of which
±7.5o is the typical allowable carrier noise contribution. Due to the
statistical nature of phase deviations, if the RMS phase deviation is 1.5o,
for example, the probability of exceeding the ±7.5o phase deviation is
6 X 10-7, which can result in a bit error rate that is significant in some
applications.
Shock and vibration can produce large phase deviations even in
"low noise" oscillators. Moreover, when the frequency of an oscillator is
multiplied by N, the phase deviations are also multiplied by N. For
example, a phase deviation of 10-3 radian at 10 MHz becomes 1 radian
at 10 GHz. Such large phase excursions can be catastrophic to the
performance of systems, e.g., of those which rely on phase locked loops
(PLL) or phase shift keying (PSK). Low noise, acceleration insensitive
oscillators are essential in such applications.
1-7
Utility Fault Location
Zap!
ta
tb
Substation
Substation
AA
Substation
Substation
BB
Insulator
Sportsman
X
L
When a fault occurs, e.g., when a "sportsman" shoots out an insulator, a disturbance
propagates down the line. The location of the fault can be determined from the differences
in the times of arrival at the nearest substations:
x=1/2[L - c(tb-ta)] = 1/2[L - c∆t]
where x = distance of the fault from substation A, L = A to B line length, c = speed of light,
and ta and tb= time of arrival of disturbance at A and B, respectively.
Fault locator error = xerror=1/2(c∆terror); therefore, if ∆terror ≤ 1 microsecond, then
xerror ≤ 150 meters ≤ 1/2 of high voltage tower spacings, so, the utility company
can send a repair crew directly to the tower that is nearest to the fault.
1-8
Space Exploration
∆t
Schematic of VBLI
Technique
Mean
wavelength λ
∆θ
∆θ
Wavefront
Local
Local
Time &
Time &
Frequency
Frequency
Standard
Standard
θ(t)
Microwave
mixer
Recorder
Recorder
∆θ =
Local
Local
Time &
Time &
Frequency
Frequency
Standard
Standard
Data tape
Data tape
Microwave
mixer
Recorder
Recorder
Correlation
Correlation
and
and
Integration
Integration
c∆ t
Lsin θ
Amplitude
λ/L sin θ
1-9
θ (τ ) Angle
Interference
Fringes
Military Requirements
Military needs are a prime driver of frequency control
technology. Modern military systems require
oscillators/clocks that are:
• Stable over a wide range of parameters (time,
temperature, acceleration, radiation, etc.)
• Low noise
• Low power
• Small size
• Fast warmup
• Low life-cycle cost
1-10
Impacts of Oscillator Technology Improvements
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Higher jamming resistance & improved ability to hide signals
Improved ability to deny use of systems to unauthorized users
Longer autonomy period (radio silence interval)
Fast signal acquisition (net entry)
Lower power for reduced battery consumption
Improved spectrum utilization
Improved surveillance capability (e.g., slow-moving target detection,
bistatic radar)
Improved missile guidance (e.g., on-board radar vs. ground radar)
Improved identification-friend-or-foe (IFF) capability
Improved electronic warfare capability (e.g., emitter location via TOA)
Lower error rates in digital communications
Improved navigation capability
Improved survivability and performance in radiation environment
Improved survivability and performance in high shock applications
Longer life, and smaller size, weight, and cost
Longer recalibration interval (lower logistics costs)
1-11
•
•
•
•
Spread Spectrum Systems
In a spread spectrum system, the transmitted signal is spread over a bandwidth that is
much wider than the bandwidth required to transmit the information being sent (e.g., a
voice channel of a few kHz bandwidth is spread over many MHz). This is
accomplished by modulating a carrier signal with the information being sent, using a
wideband pseudonoise (PN) encoding signal. A spread spectrum receiver with the
appropriate PN code can demodulate and extract the information being sent. Those
without the PN code may completely miss the signal, or if they detect the signal, it
appears to them as noise.
Two of the spread spectrum modulation types are: 1. direct sequence, in which the
carrier is modulated by a digital code sequence, and 2. frequency hopping, in which
the carrier frequency jumps from frequency to frequency, within some predetermined
set, the order of frequencies being determined by a code sequence.
Transmitter and receiver contain clocks which must be synchronized; e.g., in a
frequency hopping system, the transmitter and receiver must hop to the same
frequency at the same time. The faster the hopping rate, the higher the jamming
resistance, and the more accurate the clocks must be (see the next page for an
example).
Advantages of spread spectrum systems include the following capabilities: 1. rejection
of intentional and unintentional jamming, 2. low probability of intercept (LPI), 3.
selective addressing, 4. multiple access, and 5. high accuracy navigation and ranging.
1-12
Clock for Very Fast Frequency Hopping Radio
Example
Jammer
J
t1
Radio
R1
Let R1 to R2 = 1 km, R1 to
J =5 km, and J to R2 = 5 km.
Then, since propagation
delay =3.3 µs/km,
t1 = t2 = 16.5 µs,
tR = 3.3 µs, and tm < 30 µs.
Allowed clock error ≈ 0.2 tm
≈ 6 µs.
t2
tR
Radio
R2
To defeat a “perfect” follower
jammer, one needs a hop-rate
given by:
For a 4 hour resynch interval,
clock accuracy requirement is:
4 X 10-10
tm < (t1 + t2) - tR
where tm ≈ message duration/hop
≈ 1/hop-rate
1-13
Clocks and Frequency Hopping C3 Systems
Slow hopping ‹-------------------------------›Good clock
Fast hopping ‹------------------------------› Better clock
Extended radio silence ‹-----------------› Better clock
Extended calibration interval ‹----------› Better clock
Othogonality ‹-------------------------------› Better clock
Interoperability ‹----------------------------› Better clock
1-14
Identification-Friend-Or-Foe (IFF)
Air Defense IFF Applications
AWACS
FRIEND OR FOE?
F-16
FAAD
STINGER
PATRIOT
1-15
Effect of Noise in Doppler Radar System
A
Transmitter
Transmitter
Moving
Moving
Object
Object
Decorrelated
Clutter Noise
Stationary
Stationary
Object
Object
Doppler Signal
fD
Receiver
fD
•
Echo = Doppler-shifted echo from moving target + large "clutter" signal
•
(Echo signal) - (reference signal) --› Doppler shifted signal from target
•
Phase noise of the local oscillator modulates (decorrelates) the clutter
signal, generates higher frequency clutter components, and thereby
degrades the radar's ability to separate the target signal from the clutter
signal.
1-16
f
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
Radar Frequency (GHz)
20
15
0
25
or Slow
30
Moving
Vechile
40
10
100
1K
1-18
10
10K
2,400 k
m/h
2 Aircra
ft
bsonic A
ircraft
- Mach
700km/h
- Su
100km/
h - Veh
icle, Gro
und or A
ir
4km/h Man
Doppler Shifts
X-Band RADAR
5
100K
1M
Doppler Shift for Target Moving Toward Fixed Radar (Hz)
CHAPTER 2
Quartz Crystal Oscillators
2
Crystal Oscillator
Tuning
Voltage
Crystal
resonator
Output
Frequency
Amplifier
2-1
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
Oscillation and Stability
•
•
•
If a phase perturbation ∆φ occurs, the frequency must shift ∆f to maintain the
2nπ
π phase condition, where ∆f/f=-∆φ
∆φ/2Q
∆φ
L for a series-resonance oscillator,
φ/df
and QL is loaded Q of the crystal in the network. The "phase slope," dφ
is proportional to QL in the vicinity of the series resonance frequency (also
see "Equivalent Circuit" and "Frequency vs. Reactance" in Chapt. 3).
Most oscillators operate at "parallel resonance," where the reactance vs.
frequency slope, dX/df, i.e., the "stiffness," is inversely proportional to C1,
the motional capacitance of the crystal unit.
For maximum frequency stability with respect to phase (or reactance)
perturbations in the oscillator loop, the phase slope (or reactance slope) must
be maximum, i.e., C1 should be minimum and QL should be maximum. A
quartz crystal unit's high Q and high stiffness makes it the primary frequency
(and frequency stability) determining element in oscillators.
2-3
Tunability and Stability
Making an oscillator tunable over a wide frequency range degrades
its stability because making an oscillator susceptible to intentional tuning
also makes it susceptible to factors that result in unintentional tuning. The
wider the tuning range, the more difficult it is to maintain a high stability.
For example, if an OCXO is designed to have a short term stability of
1 x 10-12 for some averaging time and a tunability of 1 x 10-7, then the
crystal's load reactance must be stable to 1 x 10-5 for that averaging time.
Achieving such stability is difficult because the load reactance is affected
by stray capacitances and inductances, by the stability of the varactor's
capacitance vs. voltage characteristic, and by the stability of the voltage
on the varactor. Moreover, the 1 x 10-5 load reactance stability must be
maintained not only under benign conditions, but also under changing
environmental conditions (temperature, vibration, radiation, etc.).
Whereas a high stability, ovenized 10 MHz voltage controlled
oscillator may have a frequency adjustment range of 5 x 10-7 and an
aging rate of 2 x 10-8 per year, a wide tuning range 10 MHz VCXO may
have a tuning range of 50 ppm and an aging rate of 2 ppm per year.
2-4
Oscillator Acronyms
•
XO…………..Crystal Oscillator
•
VCXO………Voltage Controlled Crystal Oscillator
•
OCXO………Oven Controlled Crystal Oscillator
•
TCXO………Temperature Compensated Crystal Oscillator
•
TCVCXO..…Temperature Compensated/Voltage Controlled
Crystal Oscillator
•
OCVCXO.….Oven Controlled/Voltage Controlled Crystal Oscillator
•
MCXO………Microcomputer Compensated Crystal Oscillator
•
RbXO……….Rubidium-Crystal Oscillator
2-5
Crystal Oscillator Categories
The three categories, based on the method of dealing with the crystal unit's
frequency vs. temperature (f vs. T) characteristic, are:
•
•
•
XO, crystal oscillator, does not contain means for reducing the crystal's
f vs. T characteristic (also called PXO-packaged crystal oscillator).
TCXO, temperature compensated crystal oscillator, in which, e.g., the
output signal from a temperature sensor (e.g., a thermistor) is used to
generate a correction voltage that is applied to a variable reactance (e.g., a
varactor) in the crystal network. The reactance variations compensate for
the crystal's f vs. T characteristic. Analog TCXO's can provide about a 20X
improvement over the crystal's f vs. T variation.
OCXO, oven controlled crystal oscillator, in which the crystal and other
temperature sensitive components are in a stable oven which is adjusted to
the temperature where the crystal's f vs. T has zero slope. OCXO's can
provide a >1000X improvement over the crystal's f vs. T variation.
2-6
Crystal Oscillator Categories
∆f
f +10 ppm
250C
Voltage
Tune
+1000C
-450C
Output
T
• Crystal Oscillator (XO)
Temperature
Temperature
Sensor
Sensor
Compensation
Compensation
Network or
Network or
Computer
Computer
-10 ppm
-450C
∆f
f
+1 ppm
+1000C
T
XO
XO
-1 ppm
• Temperature Compensated (TCXO)
Oven
Oven
Oven
control
control
XO
XO
Temperature
Temperature
Sensor
Sensor
-450C
∆f
f
+1 x 10-8
+1000C
T
-1 x 10-8
• Oven Controlled (OCXO)
2-7
Hierarchy of Oscillators
Oscillator Type*
• Crystal oscillator (XO)
• Temperature compensated
Accuracy**
Typical Applications
10-5 to 10-4
Computer timing
10-6
Frequency control in tactical
radios
10-8 to 10-7
Spread spectrum system clock
10-8 (with 10-10
per g option)
Navigation system clock &
frequency standard, MTI radar
• Small atomic frequency
10-9
C3 satellite terminals, bistatic,
& multistatic radar
• High performance atomic
10-12 to 10-11
Strategic C3, EW
crystal oscillator (TCXO)
• Microcomputer compensated
crystal oscillator (MCXO)
• Oven controlled crystal
oscillator (OCXO)
standard (Rb, RbXO)
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
Oscillator Circuit Types
Of the numerous oscillator circuit types, three of the more common ones, the Pierce, the Colpitts and
the Clapp, consist of the same circuit except that the rf ground points are at different locations. The
Butler and modified Butler are also similar to each other; in each, the emitter current is the crystal
current. The gate oscillator is a Pierce-type that uses a logic gate plus a resistor in place of the
transistor in the Pierce oscillator. (Some gate oscillators use more than one gate).
b
c
∈
Pierce
c
∈
∈
b ∈
c
b
Colpitts
b
c
Clapp
c
∈
b
Butler
Modified
Butler
2-9
Gate
OCXO Block Diagram
ϕ
Output
Oven
Each of the three main parts of an OCXO, i.e., the crystal, the sustaining
circuit, and the oven, contribute to instabilities. The various instabilities
are discussed in the rest of chapter 3 and in chapter 4.
2-10
Oscillator Instabilities - General Expression
∆f
foscillator
≈
∆f
fresonator
1
+
2QL
  2ff QL 

1 + 
  f 
2



−1/2
dφ(ff )
where QL = loaded Q of the resonator, and dφ(ff) is a small
change in loop phase at offset frequency ff away from carrier
frequency f. Systematic phase changes and phase noise within
the loop can originate in either the resonator or the sustaining
circuits. Maximizing QL helps to reduce the effects of noise and
environmentally induced changes in the sustaining electronics.
In a properly designed oscillator, the short-term instabilities are
determined by the resonator at offset frequencies smaller than
the resonator’s half-bandwidth, and by the sustaining circuit and
the amount of power delivered from the loop for larger offsets.
2-11
Instabilities due to Sustaining Circuit
• Load reactance change - adding a load capacitance to a crystal
changes the frequency by
C1
∆
f
≅
δf≡
f
2(C0 + CL )
then,
∆ (δ f )
C1
≅−
2
∆ CL
2(C0 + CL )
• Example: If C0 = 5 pF, C1 = 14fF and CL = 20pF, then a ∆CL = 10 fF
(= 5 X 10-4) causes ≈1 X 10-7 frequency change, and a CL aging of
10 ppm per day causes 2 X 10-9 per day of oscillator aging.
• Drive level changes: Typically 10-8 per ma2 for a 10 MHz 3rd SC-cut.
• DC bias on the crystal also contributes to oscillator aging.
2-12
Oscillator Instabilities - Tuned Circuits
Many oscillators contain tuned circuits - to suppress unwanted
modes, as matching circuits, and as filters. The effects of
small changes in the tuned circuit's inductance and
capacitance is given by:
∆f
f oscillator

dφ (ff )  1
≈
≈
2QL
 1 + 2ff

BW

 Q  dC
dL 
 c  c + c 
L 
 Q  Cc
c 

where BW is the bandwidth of the filter, ff is the frequency
offset of the center frequency of the filter from the carrier
frequency, QL is the loaded Q of the resonator, and Qc, Lc and
Cc are the tuned circuit's Q, inductance and capacitance,
respectively.
2-13
Oscillator Instabilities - Circuit Noise
Flicker PM noise in the sustaining circuit causes flicker FM
contribution to the oscillator output frequency given by:
2
f
L osc (ff ) = L ckt (1Hz ) 3 2
4ff QL
and
1
σ y (τ) =
ln2 L ckt (1Hz )
QL
where ff is the frequency offset from the carrier frequency f, QLis the
loaded Q of the resonator in the circuit, Lckt (1Hz) is the flicker PM
noise at ff = 1Hz, and τ is any measurement time in the flicker floor
range. For QL = 106 and Lckt (1Hz) = -140dBc/Hz, σy(τ) = 8.3 x 10-14.
( Lckt (1Hz) = -155dBc/Hz has been achieved.)
2-14
Oscillator Instabilities - External Load
If the external load changes, there is a change in the amplitude
or phase of the signal reflected back into the oscillator. The
portion of that signal which reaches the oscillating loop changes
the oscillation phase, and hence the frequency by
∆f
dφ(ff )  1  Γ − 1 
≈
≈

(sinθ ) isolation
f oscillator
2Q
 2Q  Γ + 1
where Γ is the VSWR of the load, and θ is the phase angle of
the reflected wave; e.g., if Q ~ 106, and isolation ~40 dB
(i.e., ~10-4), then the worst case (100% reflection) pulling is
~5 x 10-9. A VSWR of 2 reduces the maximum pulling by only
a factor of 3. The problem of load pulling becomes worse at
higher frequencies, because both the Q and the isolation are
lower.
2-15
Oscillator Outputs
Most users require a sine wave, a TTL-compatible, a CMOScompatible, or an ECL-compatible output. The latter three can be
simply generated from a sine wave. The four output types are
illustrated below, with the dashed lines representing the supply
voltage inputs, and the bold solid lines, the outputs. (There is no
“standard” input voltage for sine wave oscillators, and the input
voltage for CMOS typically ranges from 3V to 15V.)
+15V
+10V
+5V
0V
-5V
Sine
TTL
CMOS
2-16
ECL
Resonator Self-Temperature Sensing
fβ ≡ 3f1 - f3
fβ (Hz)
172300
dfβ
= − 14 Hz/oC
dT
171300
-35
-15
5
25
170300
2-17
45
65
85
Temperature (oC)
Thermometric Beat Frequency Generation
DUAL MODE
OSCILLATOR
f1
X3
X3
MULTIPLIER
MULTIPLIER
M=1
Mixer
M=3
f3
2-18
fβ = 3f1 - f3
LOW
PASS
LOW PASS
FILTER
FILTER
Microcomputer Compensated Crystal Oscillator
(MCXO)
f1
DualDualmode
mode
XO
XO
x3
x3
fβ
f3
Reciprocal
Reciprocal
Counter
Counter
N1
Mixer
2-19
µµcomcomputer
puter
N2
Correction
Correction
Circuit
Circuit
f0
MCXO Frequency Summing Method
Block Diagram
VCXO
3rd OVERTONE
CRYSTAL
f3 = 10 MHz fd
DUAL-MODE
OSCILLATOR
FUNDAMENTAL
MODE
fd
F
f1
Divide
by
2500
TClock
DIRECT
DIGITAL
SYNTHESIZER
fb
Clock
Clock
N1 out
10 MHz
output
LOCKED
LOOP
Divide by
3
Mixer
COUNTER
PHASE-
MICROCOMPUTER
N2
T
F
Divide
by
4000
NON-VOLATILE
MEMORY
T = Timing Mode
F = Frequency
Mode
2-20
1 PPS
output
MCXO - Pulse Deletion Method
fc
f
Dual
Dualmode
mode β
oscillator
oscillator
Digital
circuitry (ASIC)
output
output
Pulse
Pulse
eliminator
eliminator
Counter
Counter
SC-cut crystal
Frequency
Frequency
evaluator
evaluator
&&correction
correction
determination
determination
f0
corrected
output
for timing
Microprocessor
circuitry
Microcomputer compensated crystal oscillator (MCXO) block diagram - pulse deletion method.
2-21
MCXO - TCXO Resonator Comparison
Parameter
MCXO
Cut, overtone
TCXO
SC-cut, 3rd
AT-cut, fund.
Angle-of-cut tolerance
Loose
Tight
Blank f and plating tolerance
Loose
Tight
Low
Significant
Hysteresis (-550C to +850C)
10-9 to 10-8
10-7 to 10-6
Aging per year
10-8 to 10-7
10-7 to 10-6
Activity dip incidence
2-22
Opto-Electronic Oscillator (OEO)
Bias
"Pump Laser"
Optical out
Piezoelectric
fiber stretcher
RF driving port
Electrical
output
Electrical
injection
Filter
Optical
Fiber
RF coupler
RF Amplifier
Photodetector
Optical
Injection
Optical fiber
Optical
coupler
Electrical
transmission
line
2-23
CHAPTER 3
Quartz Crystal Resonators
3
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 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
The Piezoelectric Effect
Y
Y
_
+
+
_
_
_
_
+
+
+
+
+
_
_
+
• • +
+
_
+
_
+
Undeformed lattice
+
+
+
X
+
_
_
_
+
_
+
_
+
_
_
_
+
+
_
+
+
_ -
_
+
+
_
X
_
_
_
+
_
_
_
+
_
_
_
+
+
+
+
+
_
_
_
_
+
+
+
+
+
+
_
_
_
_
+
+
+
+
_
_
_
_
_
+
Strained lattice
The piezoelectric effect provides a coupling between the mechanical
properties of a piezoelectric crystal and an electrical circuit.
3-2
The Piezoelectric Effect in Quartz
Z
FIELD along:
STRAIN
X
X
EXTENSIONAL
along:
Y
Y
√
√
X
Z
X
SHEAR
about:
Z
√
Y
√
Z
√
Y
In quartz, the five strain components shown may be generated by an electric field.
The modes shown on the next page may be excited by suitably placed and shaped
electrodes. The shear strain about the Z-axis produced by the Y-component of the
field is used in the rotated Y-cut family, including the AT, BT, and ST-cuts.
3-3
Modes of Motion
Flexure Mode
Extensional Mode Face Shear Mode
Thickness Shear Fundamental Mode Third Overtone
Thickness Shear Thickness Shear
Mode
3-4
Resonator Vibration Amplitude Distribution
Metallic
electrodes
Resonator
plate substrate
(the “blank”)
u
Conventional resonator geometry
and amplitude distribution, u
3-5
Resonant Vibrations of a Quartz Plate
3555
3200
MHZ
3507
-10 db.
3383
3742
-20
3852
3802
3707
3652
3642
-30 db.
3256
Response
0 db.
-40 db.
3200
3400
3600
3800
Frequency, in kHz
X-ray topographs (21•0 plane) of various modes excited during a frequency
scan of a fundamental mode, circular, AT-cut resonator. The first peak, at
3.2 MHz, is the main mode; all others are unwanted modes. Dark areas
correspond to high amplitudes of displacement.
3-6
Overtone Response of a Quartz Crystal
Reactance
jX
Spurious
responses
Spurious
responses
Spurious
responses
0
Frequency
5th overtone
-jX
3rd overtone
Fundamental mode
3-7
Unwanted Modes vs. Temperature
(3 MHz rectangular AT-cut resonator, 22 X 27 X 0.552 mm)
Activity dips occur where the f vs. T curves of unwanted modes intersect
the f vs. T curve of the wanted mode. Such activity dips are highly
sensitive to drive level and load reactance.
3-8
Mathematical Description of a Quartz Resonator
•
In piezoelectric materials, electrical current and voltage are coupled to elastic displacement and stress:
{T} = [C] {S} - [e] {E}
{D} = [e] {S} + [∈
∈] {E}
where {T} = stress tensor, [C] = elastic stiffness matrix, {S} = strain tensor, [e] = piezoelectric matrix
{E} = electric field vector, {D} = electric displacement vector, and [∈
∈] = is the dielectric matrix
•
For a linear piezoelectric material
T1
T2
T3
T4
T5
T6
D1
D2
D3
=
C11
C21
C31
C41
C51
C61
C12
C22
C32
C42
C52
C62
C13
C23
C33
C43
C53
C63
C14
C24
C34
C44
C54
C64
C15
C25
C35
C45
C55
C65
C16 −e11
C26 −e12
C36 −e13
C46 −e14
C56 −e15
C66 −e16
−e21 −e31
−e22 −e32
−e23 −e33
−e24 −e34
−e25 −e35
−e26 −e36
e11 e12 e13 e14 e15 e16 ∈11 ∈12 ∈13
e21 e22 e23 e24 e25 e26 ∈21 ∈22 ∈23
e31 e32 e33 e34 e35 e36 ∈31 ∈32 ∈33
S1
S2
S3
S4
S5 ,
S6
E1
E2
E3
• Elasto-electric matrix for quartz
S1 S2 S3 S4 S5 S6 -E1 -E2 -E3
T1
et
T2
T3
T4
T5
T6
CE
X
D1
where
T1 = T11,
T2 = T22,
T3 = T33,
T4 = T23,
T5 = T13,
T6 = T12,
D2
S1 = S11,
S2 = S22,
S3 = S33,
S4 = 2S23,
S5 = 2S13,
S6 = 2S12,
D3
e
∈
S
LINES JOIN NUMERICAL EQUALITIES
EXCEPT FOR COMPLETE RECIPROCITY
ACROSS PRINCIPAL DIAGONAL
INDICATES NEGATIVE OF
INDICATES TWICE THE NUMERICAL
EQUALITIES
X INDICATES 1/2 (c11 - c12)
3-9
6
2
2
10
Mathematical Description - Continued
•
Number of independent non-zero constants depend on crystal symmetry. For quartz (trigonal, class 32),
there are 10 independent linear constants - 6 elastic, 2 piezoelectric and 2 dielectric. "Constants” depend
on temperature, stress, coordinate system, etc.
•
To describe the behavior of a resonator, the differential equations for Newton's law of motion for a
continuum, and for Maxwell's equation* must be solved, with the proper electrical and mechanical
boundary conditions at the plate surfaces.
∂Tij
∂Di
Ei
∂φ
=− ; S =
ij
∂x i
1 ∂ui
2 ( ∂x j
(F = ma ⇒
+
∂u j
∂xi ) ; etc.)
∂xj
&&i ;
=ρu
∇ ⋅D = 0 ⇒
∂xi
= 0,
•
Equations are very "messy" - they have never been solved in closed form for physically realizable threedimensional resonators. Nearly all theoretical work has used approximations.
•
Some of the most important resonator phenomena (e.g., acceleration sensitivity) are due to nonlinear
effects. Quartz has numerous higher order constants, e.g., 14 third-order and 23 fourth-order elastic
constants, as well as 16 third-order piezoelectric coefficients are known; nonlinear equations are extremely
messy.
* Magnetic field effects are generally negligible; quartz is diamagnetic, however, magnetic fields can
affect the mounting structure and electrodes.
3-10
Infinite Plate Thickness Shear Resonator
n c ij
fn =
, n = 1, 3, 5...
2h ρ
Where fn = resonant frequency of n-th harmonic
h = plate thickness
ρ = density
cij = elastic modulus associated with the elastic wave
being propagated
d(log fn ) 1 dfn − 1 dh 1 dρ
1 dc ij
Tf =
=
=
−
+
dT
fn dT
h dT 2ρ dT 2c ij dT
where Tf is the linear temperature coefficient of frequency. The temperature
coefficient of cij is negative for most materials (i.e., “springs” become “softer”
as T increases). The coefficients for quartz can be +, - or zero (see next page).
3-11
Quartz is Highly Anisotropic
The properties of quartz vary greatly with crystallographic direction.
For example, when a quartz sphere is etched deeply in HF, the
sphere takes on a triangular shape when viewed along the Z-axis, and
a lenticular shape when viewed along the Y-axis. The etching rate is
more than 100 times faster along the fastest etching rate direction (the
Z-direction) than along the slowest direction (the slow-X-direction).
The thermal expansion coefficient is 7.8 x 10-6/°C along the Zdirection, and 14.3 x 10-6/°C perpendicular to the Z-direction; the
temperature coefficient of density is, therefore, -36.4 x 10-6/°C.
The temperature coefficients of the elastic constants range from
-3300 x 10-6/°C (for C12) to +164 x 10-6/°C (for C66).
For the proper angles of cut, the sum of the first two terms in Tf on the
previous page is cancelled by the third term, i.e., temperature
compensated cuts exist in quartz. (See next page.)
3-12
Zero Temperature Coefficient Quartz Cuts
90o
60o
AT
30o
θ
z
FC
IT
SC
LC
0
SBTC
-30o
BT
-60o
-90o
θ
g ly d
n
i
S t a te
Ro t
Cu
Do
ub
Ro ly
ta t
Cu e d
t
0o
θ
y
10o
30o
The AT, FC, IT, SC, BT, and SBTC-cuts are some
of the cuts on the locus of zero temperature
coefficient cuts. The LC is a “linear coefficient”
cut that has been used in a quartz thermometer.
φ
x
φ
20o
Y-cut: ≈ +90 ppm/0C
(thickness-shear mode)
xl
X-cut: ≈ -20 ppm/0C
(extensional mode)
3-13
Comparison of SC and AT-cuts
• Advantages of the SC-cut
• Thermal transient compensated (allows faster warmup OCXO)
• Static and dynamic f vs. T allow higher stability OCXO and MCXO
• Better f vs. T repeatability allows higher stability OCXO and MCXO
• Far fewer activity dips
• Lower drive level sensitivity
• Planar stress compensated; lower ∆f due to edge forces and bending
• Lower sensitivity to radiation
• Higher capacitance ratio (less ∆f for oscillator reactance changes)
• Higher Q for fundamental mode resonators of similar geometry
• Less sensitive to plate geometry - can use wide range of contours
• Disadvantage of the SC-cut : More difficult to manufacture for OCXO (but is
easier to manufacture for MCXO than is an AT-cut for precision TCXO)
• Other Significant Differences
• B-mode is excited in the SC-cut, although not necessarily in LFR's
• The SC-cut is sensitive to electric fields (which can be used for
compensation)
3-14
Mode Spectrograph of an SC-cut
1.10
0
1.0
1.88
-10
Attenuation
a-mode: quasi-longitudinal mode
b-mode: fast quasi-shear mode
c-mode: slow quasi-shear mode
3.30
-20
3.0
5.50
-30
5.0
c(1)
-40
5.65
0
b(1)
1
a(1)
2
c(3)
b(3)
3
c(5)
4
b(5)
5
Normalized Frequency (referenced to the fundamental c-mode)
3-15
a(3)
6
SC- cut f vs. T for b-mode and c-mode
400
FREQUENCY DEVIATION (PPM)
200
0
10
Temperature (OC)
20
0
40
30
50
60
70
c-Mode (Slow Shear)
-200
-400
b-Mode (Fast Shear)
-25.5 ppm/oC
-600
-800
-1000
-1200
3-16
Singly Rotated and Doubly Rotated Cuts’
Vibrational Displacements
Z
θ
g ly
Sin ated
t
Ro
t
Cu
Do
u
Ro bly
ta
Cu ted
t
θ
Y
ϕ
X
X’
3-17
Resonator Packaging
Two-point Mount Package
Quartz
blank
Three- and Four-point Mount Package
Electrodes
Quartz
blank
Bonding
area
Cover
Mounting
clips
Seal
Base
Bonding
area
Cover
Mounting
clips
Seal
Pins
Top view of cover
3-18
Pins
Base
Equivalent Circuits
Spring
C
Mass
L
Dashpot
R
3-19
Equivalent Circuit of a Resonator
CL
Symbol for crystal unit
C0
CL
C1
L1
R1
∆f
C1
≈
→
fS 2(C0 + CL )
3-20
1. Voltage control (VCXO)
2. Temperature compensation
(TCXO)
Crystal Oscillator f vs. T Compensation
Frequency / Voltage
Uncompensated
frequency
T
Compensating
voltage
on varactor CL
Compensated
frequency
of TCXO
3-21
Resonator Frequency vs. Reactance
Antiresonance
Reactance
Area of usual
+
“parallel”
resonance”
fS
0
Frequency
Series
resonance
-
fA
1
2πfC0
3-22
Equivalent Circuit Parameter Relationships
C0 ≅ ε
1
fs =
2π
A
t
r≡
fs
fa − fs ≅
2r
1
L1C1
1
Q=
2π fSR1C1
ϕ=
τ 1 = R1C1 ≅ 10 s
−14
r' C
C1n ≈ 311
n
C0
C1
L1n
ω L1 −
1
ω C1
R1
dϕ 360 Q
≅
df
π fs
n3L11
≈ 3
r'
R1n
3-23
n:
C 0:
C 1:
C1n:
L1:
L1n:
R 1:
R1n:
ε:
A:
t:
r:
r’:
fs:
fa:
Q;
τ1:
ω:
ϕ:
k;
n3R11
≈
r'
Overtone number
Static capacitance
Motional capacitance
C1 of n-th overtone
Motional inductance
L1 of n-th overtone
Motional resistance
R1 of n-th overtone
Dielectric permittivity of quartz
≈40 x 10-13 pF/mm (average)
Electrode area
Plate thickness
Capacitance ratio
f1/fn
Series resonance frequency ≈fR
Antiresonance frequency
Quality factor
Motional time constant
Angular frequency = 2πf
Phase angle of the impedance
Piezoelectric coupling factor
=8.8% for AT-cut, 4.99% for SC
π n 
2r = 

2k


2
What is Q and Why is it Important?
Q ≡ 2π
Energy stored during a cycle
Energy lost during the 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).
3-24
Decay Time, Linewidth, and Q
Decaying
oscillation
of a resonator
Oscillation
1
e
=
1
of maximum intensity
2.7
TIME
Exciting
pulse ends
Max.
intensity
td
Maximum intensity
½ Maximum intensity
BW
Q=
ν0
νo
≅ νo π t d
BW
FREQUENCY
3-25
1
BW ≅
πt d
Resonance
behavior of
a resonator
Factors that Determine Resonator Q
The maximum Q of a resonator can be expressed as:
1
,
Q max =
2π fτ
where f is the frequency in Hz, and τ is an empirically determined “motional
time constant” in seconds, which varies with the angles of cut and the mode
of vibration. For example, τ = 1 x 10-14s for the AT-cut's c-mode (Qmax = 3.2
million at 5 MHz), τ = 9.9 x 10-15s for the SC-cut's c-mode, and τ = 4.9 x 10-15s
for the BT-cut's b-mode.
Other factors which affect the Q of a resonator include:
Blank geometry (contour,
dimensional ratios)
Drive level
Gases inside the enclosure
(pressure, type of gas)
Interfering modes
Ionizing radiation
Overtone
Surface finish
Material impurities and defects
Mounting stresses
Bonding stresses
Temperature
Electrode geometry and type
3-26
Resonator Fabrication Steps
DESIGN
RESONATORS
GROW
QUARTZ
SWEEP
CUT
LAP
ROUND
ORIENT
IN MASK
CLEAN
ETCH
(CHEMICAL
POLISH)
CONTOUR
ANGLE
CORRECT
X-RAY
ORIENT
DEPOSIT
CONTACTS
PREPARE
ENCLOSURE
MOUNT
BOND
INSPECT
CLEAN
SEAL
FREQUENCY
ADJUST
PLATE
BAKE
FINAL
CLEAN
TEST
OSCILLATOR
3-27
X-ray Orientation of Crystal Plates
Shielding
Monochromator
crystal
Detector
S
X-ray beam
Crystal under test
Copper target
X-ray source
Goniometer
Double-crystal x-ray diffraction system
3-28
Contamination Control
Contamination control is essential during the fabrication of
resonators because contamination can adversely affect:
•
Stability (see chapter 4)
- aging
- hysteresis
- retrace
- noise
- nonlinearities and resistance anomalies (high starting
resistance, second-level of drive, intermodulation in filters)
- frequency jumps?
•
•
Manufacturing yields
Reliability
3-29
Crystal Enclosure Contamination
The enclosure and sealing process can have important
influences on resonator stability.
• A monolayer of adsorbed contamination contains ~ 1015
molecules/cm2 (on a smooth surface)
• An enclosure at 10-7 torr contains ~109 gaseous
molecules/cm3
Therefore:
In a 1 cm3 enclosure that has a monolayer of contamination
on its inside surfaces, there are ~106 times more adsorbed
molecules than gaseous molecules when the enclosure is sealed
at 10-7 torr. The desorption and adsorption of such adsorbed
molecules leads to aging, hysteresis, noise, etc.
3-30
What is an “f-squared”?
It is standard practice to express the thickness removed by lapping, etching and polishing,
and the mass added by the electrodes, in terms of frequency change, ∆f, in units of “f2”, where
the ∆f is in kHz and f is in MHz. For example, etching a 10MHz AT-cut plate 1f2 means that a
thickness is removed that produces ∆f= 100 kHz; and etching a 30 MHz plate 1f2 means that
the ∆f= 900 kHz. In both cases, ∆f=1f2 produces the same thickness change.
To understand this, consider that for a thickness-shear resonator (AT-cut, SC-cut, etc.)
f=
N
t
where f is the fundamental mode frequency, t is the thickness of the resonator plate and N is
the frequency constant (1661 MHz•µm for an AT-cut, and 1797 MHz•µm for a SC-cut’s cmode). Therefore,
∆f
∆t
=−
f
t
and,
∆t = −N
∆f
f2
So, for example, ∆f = 1f2 corresponds to the same thickness removal for all frequencies.
For an AT-cut, ∆t=1.661 µm of quartz (=0.83 µm per side) per f2. An important advantage of
using units of f2 is that frequency changes can be measured much more accurately than
thickness changes. The reason for expressing ∆f in kHz and f in MHz is that by doing so, the
numbers of f2 are typically in the range of 0.1 to 10, rather than some very small numbers.
3-31
Milestones in Quartz Technology
1880
1905
1917
1918
1926
1927
1927
1934
1949
1956
1956
1972
1974
1982
Piezoelectric effect discovered by Jacques and Pierre Curie
First hydrothermal growth of quartz in a laboratory - by G. Spezia
First application of piezoelectric effect, in sonar
First use of piezoelectric crystal in an oscillator
First quartz crystal controlled broadcast station
First temperature compensated quartz cut discovered
First quartz crystal clock built
First practical temp. compensated cut, the AT-cut, developed
Contoured, high-Q, high stability AT-cuts developed
First commercially grown cultured quartz available
First TCXO described
Miniature quartz tuning fork developed; quartz watches available
The SC-cut (and TS/TTC-cut) predicted; verified in 1976
First MCXO with dual c-mode self-temperature sensing
3-32
Quartz Resonators for Wristwatches
Requirements:
•
Small size
•
Low power dissipation (including the oscillator)
•
Low cost
•
High stability (temperature, aging, shock,
attitude)
These requirements can be met with 32,768 Hz quartz
tuning forks
3-33
Why 32,768 Hz?
32,768 = 215
• In an analog watch, a stepping motor receives
one impulse per second which advances the
second hand by 6o, i.e., 1/60th of a circle,
every second.
• Dividing 32,768 Hz by two 15 times results
in 1 Hz.
• The 32,768 Hz is a compromise among size,
power requirement (i.e., battery life) and
stability.
3-34
32,768
16,384
8,192
4,096
2,048
1,024
512
256
128
64
32
16
8
4
2
1
Quartz Tuning Fork
Z
Y
X
a) natural faces and crystallographic axes of quartz
Z
Y’
0~50
Y
arm
base
X
b) crystallographic orientation of tuning fork
3-35
c) vibration mode of tuning fork
Watch Crystal
3-36
Lateral Field Resonator
Lateral Field
Thickness Field
In lateral field resonators (LFR): 1. the electrodes are absent from the
regions of greatest motion, and 2. varying the orientation of the gap between
the electrodes varies certain important resonator properties. Advantages of
LFR are:
• Ability to eliminate undesired modes, e.g., the b-mode in SC-cuts
• Potentially higher Q (less damping due to electrodes and mode traps)
• Potentially higher stability (less electrode and mode trap effects, smaller C1)
3-37
Electrodeless (BVA) Resonator
C
D2
C
D1
Quartz
bridge
Side view of BVA2
resonator construction
Side and top views of
center plate C
3-38
CHAPTER 4
Oscillator Stability
4
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).
• 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 total distance traveled by all
the cars in the world in a day.
4-1
Accuracy, Precision, and Stability
f
Accurate but
not precise
Not accurate and
not precise
Precise but
not accurate
f
f
Accurate and
precise
f
0
Time
Stable but
not accurate
Time
Time
Accurate but
not stable
Not stable and
not accurate
4-2
Time
Stable and
accurate
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)
• Vibration
• Acoustic noise
• Shock
Ionizing radiation
• Steady state
• Pulsed
• Photons (X-rays, γ-rays)
• Particles (neutrons, protons, electrons)
Other
• Power supply voltage
• Atmospheric pressure (altitude)
• Humidity
• Load impedance
4-3
• Magnetic field
Idealized Frequency-Time-Influence Behavior
∆f
X 108
f
3
Temperature
Step
Vibration
Shock
Oscillator
Turn Off
2-g
&
Tipover
Turn On
Radiation
Off
2
Aging
1
0
-1
On
-2
Short-Term
Instability
-3
t0
t1
t2
t3
t4
4-4
t5 t6
t7
t8 Time
Aging and Short-Term Stability
Short-term instability
(Noise)
30
∆f/f (ppm)
25
20
15
10
5
10
15
20
4-5
25
Time (days)
Aging Mechanisms
Mass transfer due to contamination
Since f ∝ 1/t, ∆f/f = -∆t/t; e.g., f5MHz ≈ 106 molecular layers,
therefore, 1 quartz-equivalent monolayer ⇒ ∆f/f ≈ 1 ppm
Stress relief in the resonator's: mounting and bonding structure,
electrodes, and in the quartz (?)
Other effects
Quartz outgassing
Diffusion effects
Chemical reaction effects
Pressure changes in resonator enclosure (leaks and outgassing)
Oscillator circuit aging (load reactance and drive level changes)
Electric field changes (doubly rotated crystals only)
Oven-control circuitry aging
4-6
Typical Aging Behaviors
A(t) = 5 ln(0.5t+1)
∆f/f
Time
A(t) +B(t)
B(t) = -35 ln(0.006t+1)
4-7
Stresses on a Quartz Resonator Plate
Causes:
• Thermal expansion coefficient differences
• Bonding materials changing dimensions upon solidifying/curing
• Residual stresses due to clip forming and welding operations,
sealing
• Intrinsic stresses in electrodes
• Nonuniform growth, impurities & other defects during quartz
growing
• Surface damage due to cutting, lapping and (mechanical) polishing
Effects:
• In-plane diametric forces
• Tangential (torsional) forces, especially in 3 and 4-point mounts
• Bending (flexural) forces, e.g., due to clip misalignment and
electrode stresses
• Localized stresses in the quartz lattice due to dislocations,
inclusions, other impurities, and surface damage
4-8
Thermal Expansion Coefficient, α, of AT-cut Quartz, 10-6/0K
Thermal Expansion Coefficients of Quartz
14
ZZl
13.71
Ψ
13
XXl
Radial
12
11.63
α (Thickness) = 11.64
11
10
Tangential
9.56
9
00
100
200
300
400
500
600
Orientation, Ψ, With Respect To XXl
4-9
700
800
900
Force-Frequency Coefficient
30
* 10-15 m • s / N
AT-cut quartz
25
20
15
Z’
F
10
Ψ
Kf (Ψ
Ψ)
5
X’
F
0
-5
-10
-15 0
0
(Force ) (Frequency constant )
∆f
= KF
(Diameter ) (Thickness )
f
100
200
300
400
4-10
Ψ
500
600
700
800
900
Strains Due To Mounting Clips
X-ray topograph of an AT-cut, two-point mounted resonator. The topograph
shows the lattice deformation due to the stresses caused by the mounting clips.
4-11
Strains Due To Bonding Cements
(a)
(b)
X-ray topographs showing lattice distortions caused by bonding cements; (a) Bakelite
cement - expanded upon curing, (b) DuPont 5504 cement - shrank upon curing
4-12
Mounting Force Induced Frequency Change
The force-frequency coefficient, KF (ψ), is defined by
Z’
(
Force) (Frequency − constant )
∆f
= KF
(Diameter ) (Thickness)
f
F
Ψ
Maximum KF (AT-cut) = 24.5 x 10-15 m-s/N at ψ = 0o
Maximum KF (SC-cut) = 14.7 x 10-15 m-s/N at ψ = 44o
F
As an example, consider a 5 MHz 3rd overtone, 14 mm diameter resonator.
Assuming the presence of diametrical forces only, (1 gram = 9.81 x 10-3
newtons),
-8
 ∆f 
  = 2.9 x 10 -8 per gram for an AT-cut resonator
1.7 x 10 per gram for an SC-cut resonator
 f Max
 ∆f 
  = 0 at ψ = 61o for an AT-cut resonator, and at ψ = 82o for an
 f Min
{
SC-cut.
4-13
X’
Apparent angle shift (minutes)
Bonding Strains Induced Frequency Changes
6’
5’
• Blank No. 7
•
•
Z’
Blank No. 8
4’
Ψ
3’
X’
•
2’
•
1’
•
0’
•
-1’
•
•
-2’
300
600
Bonding orientation, Ψ
900
When 22 MHz fundamental mode AT-cut resonators were reprocessed so as to vary the
bonding orientations, the frequency vs. temperature characteristics of the resonators changed
as if the angles of cut had been changed. The resonator blanks were 6.4 mm in diameter
plano-plano, and were bonded to low-stress mounting clips by nickel electrobonding.
4-14
Bending Force vs. Frequency Change
AT-cut resonator
SC-cut resonator
fo = 10Mz
fo = 10Mz
5gf
•
•
• •
• •
•
•
•
20
•
+10
•
•
•
•
•
•
• •
•
10
•
5gf
Frequency Change (Hz)
Frequency Change (Hz)
30
•
•
•
•
•
•
••
•
•
• •
•
•
•
• ••
••
•
•
•
•
•
•
•
•
• •
••• • 360
• 60 120• 180 240 • •300
• ••
•
••
•
Azimuth angle ψ (degrees)
•
•
••
-10
0
60
120
180
240
300
Azimuth angle ψ (degrees)
360
Frequency change for symmetrical
bending, AT-cut crystal.
Frequency change for symmetrical
bending, SC-cut crystal.
4-15
Short Term Instability (Noise)
Stable Frequency (Ideal Oscillator)
V
1
-1
T1
1
V -1
T1
Φ(t)
T3
T2
V(t) = V0 sin(2πν0t)
Φ(t) = 2πν0t
Time
Unstable Frequency (Real Oscillator)
Φ(t)
T2
T3
Time
Φ(t) = 2πν0t + φ(t)
V(t) =[V0 + ε(t)] sin[2πν0t + φ(t)]
Instantaneous frequency,
ν(t ) =
1 d Φ(t )
1 dφ(t )
= ν0 +
2π d t
2π d t
V(t) = Oscillator output voltage, V0 = Nominal peak voltage amplitude
ε(t) = Amplitude noise,
ν0 = Nominal (or "carrier") frequency
Φ(t) = Instantaneous phase, and φ(t) = Deviation of phase from nominal (i.e., the ideal)
4-16
Instantaneous Output Voltage of an Oscillator
- Voltage +
0
Amplitude
instability
Phase
instability
Frequency
instability
Time
4-17
Time Domain - Frequency Domain
A
f
(a)
Amplitude - Time
Amplitude - Frequency
t
(b)
(c)
A(f)
A(t)
4-18
Causes of Short Term Instabilities
Johnson noise (thermally induced charge fluctuations, i.e., "thermal emf” in
resistive elements)
Phonon scattering by defects & quantum fluctuations (related to Q)
Noise due to oscillator circuitry (active and passive components)
Temperature fluctuations- thermal transient effects
- activity dips at oven set-point
Random vibration
Fluctuations in the number of adsorbed molecules
Stress relief, fluctuations at interfaces (quartz, electrode, mount, bond)
Shot noise in atomic frequency standards
???
4-19
Short-Term Stability Measures
Measure
Symbol
Two-sample deviation, also called “Allan deviation”
Spectral density of phase deviations
Spectral density of fractional frequency deviations
Phase noise
σy(τ)*
Sφ(f)
Sy(f)
L(f)*
* Most frequently found on oscillator specification sheets
f2Sφ(f) = ν2Sy(f); L(f) ≡ ½ [Sφ(f)]
and
2
σ ( τ) =
(πντ)2
2
y
∫
∞
0
(per IEEE Std. 1139),
S φ (f)sin4 ( πfτ)df
Where τ = averaging time, ν = carrier frequency, and f = offset or
Fourier frequency, or “frequency from the carrier”.
4-20
Allan Deviation
Also called two-sample deviation, or square-root of the "Allan
variance," it is the standard method of describing the short term
stability of oscillators in the time domain. It is denoted by σy(τ),
σ 2y ( τ ) =
where
1
< ( y k +1 - y k )2 > .
2
∆f
The fractional frequencies, y =
are measured over a time
f
interval, τ; (yk+1 - yk) are the differences between pairs of
successive measurements of y, and, ideally, < > denotes a time
average of an infinite number of (yk+1 - yk)2. A good estimate
can be obtained by a limited number, m, of measurements
(m≥100). σy(τ) generally denotes σ2y ( τ, m), i.e.,
1
σ (τ) = σ ( τ, m) =
m
2
y
2
y
4-21
∑ (yk +1 − yk )
j=1
m
1
2
2
j
Why σy(τ)?
Classical variance:
σ 2yi =
(
)
1
yi − y 2,
∑
m -1
diverges for some commonly observed noise
processes, such as random walk, i.e., the variance
increases with increasing number of data points.
Allan variance:
• Converges for all noise processes observed in precision
oscillators.
• Has straightforward relationship to power law spectral
density types.
• Is easy to compute.
• Is faster and more accurate in estimating noise
processes than the Fast Fourier Transform.
4-22
Frequency Noise and σy(ττ)
3 X 10-11
∆f
f
0.1 s averaging time
0
100 s
-3 X 10-11
3 X 10-11
1.0 s averaging time
∆f
f
0
100 s
-3 X 10-11
σy(τ) 10
-10
10-11
10-12
0.01
1
0.1
4-23
10
100
Averaging time, τ, s
Time Domain Stability
σy(ττ)
Aging* and
random walk
of frequency
Frequency noise
1s
Short-term
stability
1m
1h
Sample time τ
Long-term
stability
*For σy(τ) to be a proper measure of random frequency fluctuations,
aging must be properly subtracted from the data at long τ’s.
4-24
Power Law Dependence of σy(ττ)
σy(ττ)
τ-1
τ-1
τ-1
White
Noise type: phase
Flicker
phase
White
freq.
τ0
τ1
Flicker Random
freq.
walk freq.
Below the flicker of frequency noise (i.e., the “flicker floor”) region,
crystal oscillators typically show τ-1 (white phase noise) dependence.
Atomic standards show τ-1/2 (white frequency noise) dependence down to
about the servo-loop time constant, and τ-1 dependence at less than that
time constant. Typical τ’s at the start of flicker floors are: 1 second for a
crystal oscillator, 103s for a Rb standard and 105s for a Cs standard.
4-25
Pictures of Noise
Sz(f) = hαfα Noise name
Plot of z(t) vs. t
α=0
White
α = -1
Flicker
α = -2
Random
walk
α = -3
Plots show fluctuations of a quantity z(t), which can be,e.g., the output of a counter (∆f vs. t)
or of a phase detector (φ[t] vs. t). The plots show simulated time-domain behaviors
corresponding to the most common (power-law) spectral densities; hα is an amplitude
coefficient. Note: since S∆f = f 2Sφ, e.g. white frequency noise and random walk of phase are
equivalent.
4-26
Spectral Densities
V(t) = V0 + ε (t) sin [ 2πν0 t + φ(t)]
In the frequency domain, due to the phase deviation, φ(t), some of
the power is at frequencies other than ν0. The stabilities are
characterized by "spectral densities." The spectral density, SV(f), the
mean-square voltage <V2(t)> in a unit bandwidth centered at f, is not a
good measure of frequency stability because both ε(t) and φ(t) contribute
to it, and because it is not uniquely related to frequency fluctuations
(although ε(t) is often negligible in precision frequency sources.)
The spectral densities of phase and fractional-frequency fluctuations,
Sφ(f) and Sy(f), respectively, are used to measure the stabilities in the
frequency domain. The spectral density Sg(f) of a quantity g(t) is the
mean square value of g(t) in a unit bandwidth centered at f. Moreover,
2
the RMS value of g2 in bandwidth BW is given by g RMS
(t) = ∫ S g(f)d f .
BW
4-27
Mixer Functions
V1 = A 1sin(ω1t + ϕ1 )
V1V2
V2 = A2sin (ω2t + ϕ2 )
Filter
Filter
V0
Trigonometric identities: sin(x)sin(y) = ½cos(x-y) - ½cos(x+y)
cos(x±
±π/2) = sin(x)
Let ω1 = ω 2 ;Φ1 ≡ ω1t + φ 1, and Φ2 ≡ ω 2 t + φ 2 .
•
•
Phase detector:
AM
When Φ1 = Φ2 + π / 2 and A 1 = A 2 = 1, then
1
1
V0 = sin(φ 1 − φ 2 ) = (φ 1 − φ 2 ) for small φ ' s
2
2
detector: When A 2 = 1 and the filter is a low − pass filter, then
V0 =
•
Then the mixer can become :
1
2
A 1cos(φ 1 − φ 2 ); if φ 1 ≈ φ 2 , then V0 ≈
1
A1
2
When V1 = V2 and the filter is bandpass at 2ω1
Frequency multiplier:
1 2
then, V0 = A 1 cos(2ω1t + 2φ 1 ) ⇒ doubles the frequency and phase error.
2
4-28
Phase Detector
DUT
V(t)
~
fO
Sφ(f)
LPF
∆Φ =
~
Reference
900
VO(t)
Vφ(t)
Low-Noise
Amplifier
VR(t)
Quadrature
Monitor*
* Or phase-locked loop
4-29
Spectrum Analyzer
Phase Noise Measurement
RF
RFSource
Source
V (t ) = Vo sin[2π0 t + Φ(t )]
Phase
PhaseDetector
Detector
VVφ(t)
(t)==kΦ(t)
kΦ(t)
φ
Vφ(t)
RF
Voltmeter
Spectrum Analyzer
Oscilloscope
Φ(t)
ΦRMS(t) in BW of meter
4-30
Sφ(f) vs. f
Frequency - Phase - Time Relationships
t
1 dφ(t )
= " instantane ous" frequency; φ(t ) = φ0 + ∫ 2π[ν(t' ) − ν 0 ]dt'
ν(t ) = ν 0 +
2π dt
0
•
y (t ) ≡
ν(t ) − ν 0
φ(t )
2
=
= normalized frequency; φRMS
= ∫ S φ (f )dt
ν0
2πν 0
2
2
φRMS
ν 
S φ (f ) =
=  0  S y (f );
BW  f 
(
σ (τ ) = 1/2 < y k +1 − y k
2
y
)
2
L (f ) ≡ 1/2 S φ (f ), per IEEE Standard 1139 − 1988
2
>=
(πν0 τ)2
∞
4
(
)
(πfτ) df
S
f
sin
φ
∫
0
The five common power-law noise processes in precision oscillators are:
S y (f ) = h2 f 2
(White PM)
+
h1f
+
h0
h −1f −1
+
+
h −2 f −2
(Flicker PM) (White FM) (Flicker FM) (Random-walk FM)
t
Time deviation = x (t ) = ∫ y (t' )dt' =
o
4-31
φ(t )
2πν
Sφ(f) to SSB Power Ratio Relationship
Consider the “simple” case of sinusoidal phase modulation at frequency fm. Then,
φ(t) = φo(t)sin(2πfmt), and V(t) = Vocos[2πfct + φ(t)] = Vocos[2πfct + φ0(t)sin(πfmt)],
where φo(t)= peak phase excursion, and fc=carrier frequency. Cosine of a sine
function suggests a Bessel function expansion of V(t) into its components at
various frequencies via the identities:
cos(X + Y ) = cosX cosY − sinX sinY
cosXcosY = 1/2[cos(X + Y ) + cos(X − Y )]
− sinXsinY = [cos(X + Y ) − cos(X − Y )]
cos(BsinX ) = J0 (B) + 2 ∑ J2n (B )cos(2nX )
∞
sin(BsinX ) = 2 ∑ J2n+1 (B )sin[(2n + 1)X]
n =0
After some messy algebra, SV(f) and Sφ(f) are as shown on the next page. Then,
2 2
V0 J1 [Φ(fm )]
SSB Power Ratio at fm =
∞
2
2
2
V0 J0 [Φ(fm )] + 2∑ Ji [Φ(fm )]
if Φ(fm ) << 1, then J0 = 1, J1 = 1/2Φ(fm ), Jn = 0 for n > 1, and
Φ2 (fm ) S φ (fm )
=
SSB Power Ratio = L (fm ) =
4
2
i=1
4-32
Sφ(f), Sv(f) and L (f)
Sφ (f )
Φ(t ) = Φ(fm )cos(2πfm t )
0
Φ2
(fm )
2
f
fm
V (t ) = V0cos[2πfC t + Φ(fm )]
2
2
V0 J0
[Φ(fm )]
2
2 2
V0 J1
[Φ(fm )]
2
2
2
V0 J2
[Φ(fm )]
2
2
2
V0 J3
[Φ(fm )]
2
SV(f)
fC-3fm
fC-2fm
fC-fm
fC
fC+2fm
V0 J1 [Φ(fm )]
2
SSB Power Ratio =
fC+fm
2
∞
V0 J0 [Φ(fm )] + 2∑ J 2 Φ f 

m 
i=1 i  
2
2
4-33
fC+3fm
≅ L (fm ) ≡
f
Sφ(fm )
2
Types of Phase Noise
L(ff)
40 dB/decade (ff-4)
Random walk of frequency
30 dB/decade (ff-3)
Flicker of frequency
20 dB/decade (ff-2)
White frequency; Random walk of phase
10 dB/decade (ff-1)
Flicker of phase
0 dB/decade (ff0)
White phase
ff
~BW of resonator
Offset frequency
(also, Fourier frequency,
sideband frequency,
or modulation frequency)
4-34
Noise in Crystal Oscillators
The resonator is the primary noise source close to the carrier; the oscillator sustaining
circuitry is the primary source far from the carrier.
Frequency multiplication by N increases the phase noise by N2 (i.e., by 20log N, in dB's).
Vibration-induced "noise" dominates all other sources of noise in many applications
(see acceleration effects section, later).
Close to the carrier (within BW of resonator), Sy(f) varies as 1/f, Sφ(f) as 1/f3, where f =
offset from carrier frequency, ν. Sφ(f) also varies as 1/Q4, where Q = unloaded Q. Since
Qmaxν = const., Sφ(f) ∝ ν4. (Qmaxν)BAW = 1.6 x 1013 Hz; (Qmaxν)SAW = 1.05 x 1013 Hz.
In the time domain, noise floor is σy(τ) ≥ (2.0 x 10-7)Q-1 ≈ 1.2 x 10-20ν, ν in Hz. In the
regions where σy(τ) varies as τ-1 and τ-1/2 (τ-1/2 occurs in atomic frequency standards),
σy(τ) ∝ (QSR)-1, where SR is the signal-to-noise ratio; i.e., the higher the Q and the signalto-noise ratio, the better the short term stability (and the phase noise far from the carrier,
in the frequency domain).
It is the loaded Q of the resonator that affects the noise when the oscillator sustaining
circuitry is a significant noise source.
Noise floor is limited by Johnson noise; noise power, kT = -174 dBm/Hz at 290°K.
Higher signal level improves the noise floor but not the close-in noise. (In fact, high drive
levels generally degrade the close-in noise, for reasons that are not fully understood.)
Low noise SAW vs. low noise BAW multiplied up: BAW is lower noise at f < ~1 kHz,
SAW is lower noise at f > ~1 kHz; can phase lock the two to get the best of both.
4-35
Low-Noise SAW and BAW Multiplied to 10 GHz
(in a nonvibrating environment)
0
BAW = bulk-acoustic wave
oscillator
SAW = surface acoustic
wave oscillator
-20
L(f) in dBc/Hz
-40
-60
-80
BAW
5 MHz x 2000
-100
-120
-140
-160
10-1
200
100
101
BAW is
lower noise
102
5500
103
Offset frequency in Hz
4-36
SAW is
lower noise
104
105
BAW
100 MHz x 100
SAW
500 MHz x 20
106
Low-Noise SAW and BAW Multiplied to 10 GHz
(in a vibrating environment)
0
Vibration induced phase noise dominates the phase
noise of both (whichever has lower acceleration
sensitivity will have lower phase noise; currently,
BAW can provide lower sensitivity than SAW.)
Illustration assumes 1 x 10-9/g acceleration
sensitivity for both BAW and SAW, and 0.01
g2 / Hz random vibration power spectral
density at all vibration frequencies
-20
L(f) in dBc / Hz
-40
-60
-80
5 MHz x 2000
BAW
BA
W
an
dS
AW
-100
-120
-140
100 MHz x 100
-160
500 MHz x 20
10-1
100
101
102
103
Offset frequency in Hz
4-37
104
105
106
Effects of Frequency Multiplication
fi ≡ fin
fo ≡ fout = Mfi
∆ fi
∆ fo = M∆ fi
∆ fi
≡y
fi
∆ fo ∆ fi
=
fo
fi
∆φ i
∆ φ o = M ∆φ i
L (f )i
Sφ (f )i
fi x M = fo
Noiseless
Multiplier
L (f )o = L (f )i + 20 log M
Sφ (f )o = M2S φ (f )i
S y (f )i
S y (f )o = S y (f )i
σ y (τ)i
σ y (τ )o = σ y (τ )i
Note that y =
∆f
, Sy(f), and σy(τ) are unaffected by frequency multiplication.
f
4-38
TCXO Noise
The short term stabilities of TCXOs are temperature (T) dependent, and are generally
worse than those of OCXOs, for the following reasons:
• The slope of the TCXO crystal’s frequency (f) vs. T varies with T. For example,
the f vs. T slope may be near zero at ~20oC, but it will be ~1ppm/oC at the T extremes. T
fluctuations will cause small f fluctuations at laboratory ambient T’s, so the stability can be
good there, but millidegree fluctuations will cause ~10-9 f fluctuations at the T extremes. The
TCXO’s f vs. T slopes also vary with T; the zeros and maxima can be at any T, and the
maximum slopes can be on the order of 1 ppm/oC.
• AT-cut crystals’ thermal transient sensitivity makes the effects of T fluctuations
depend not only on the T but also on the rate of change of T (whereas the SC-cut crystals
typically used in precision OCXOs are insensitive to thermal transients). Under changing T
conditions, the T gradient between the T sensor (thermistor) and the crystal will aggravate the
problems.
• TCXOs typically use fundamental mode AT-cut crystals which have lower Q and
larger C1 than the crystals typically used in OCXOs. The lower Q makes the crystals inherently
noisier, and the larger C1 makes the oscillators more susceptible to circuitry noise.
• AT-cut crystals’ f vs. T often exhibit activity dips (see “Activity Dips” later in this
chapter). At the T’s where the dips occur, the f vs. T slope can be very high, so the noise due
to T fluctuations will also be very high, e.g., 100x degradation of σy(τ) and 30 dB degradation of
phase noise are possible. Activity dips can occur at any T.
4-39
Quartz Wristwatch Accuracy vs. Temperature
Time Error per Day (seconds)
Temperature coefficient of frequency = -0.035 ppm/0C2
0
10
20
-550C
Military
“Cold”
-100C
Winter
+280C +490C
Wrist Desert
Temp.
4-40
+850C
Military
“Hot”
Frequency vs. Temperature Characteristics
Frequency
f (LTP)
Inflection Point
f (UTP)
Lower
Turnover
Point (LTP)
Temperature
4-41
Upper
Turnover
Point (UTP)
Resonator f vs. T Determining Factors
Primary: Angles of cut
Secondary:
• Overtone
• Blank geometry (contour, dimensional ratios)
• Material impurities and strains
• Mounting & bonding stresses (magnitude and
direction)
• Electrodes (size, shape, thickness, density, stress)
• Drive level
• Interfering modes
• Load reactance (value & temperature coefficient)
• Temperature rate of change
• Thermal history
• Ionizing radiation
4-42
Frequency-Temperature vs. Angle-of-Cut, AT-cut
25
∆θ
∆f
f (ppm)
35¼
R
20
8
7’
10
’
BT-cut
R
Y
R -1’
m
r
6
0’
Z
Y-bar quartz
5’
5
49o
o
r
m
R
’
15
Z
AT-cut
1’
4’
0
3’
-5
3’
2’
4’
1’
-10
0’
-15
’
-1
-20
-25
2’
5’
θ = 35o 20’ + ∆θ,
∆θ ϕ = 0
for 5th overtone AT-cut
6’
θ = 35o 12.5’+ ∆θ,
∆θ ϕ = 0 for
fundamental mode plano-plano AT-cut
7’
8’
-45 -40 -35 -30 -25
-20 -15 -10
-5
0
5
10
15
20
25
30
35
Temperature (oC)
4-43
40
45
50
55
60
65
70
75
80
85
90
Desired f vs. T of SC-cut Resonator
for OCXO Applications
Frequency Offset (ppm)
20
15
10
5
0
-5
Frequency remains within ± 1 ppm
over a ± 250C range about Ti
-10
-15
-20
20
40
60
80
100
Temperature (0C)
4-44
120
140
160
Frequency
OCXO Oven’s Effect on Stability
TURNOVER
POINT
OVEN SET POINT
TURNOVER
POINT
OVEN
OFFSET
2∆ To
OVEN CYCLING RANGE
Typical f vs. T characteristic
for AT and SC-cut resonators
Temperature
Oven Parameters vs. Stability for SC-cut Oscillator
Assuming Ti - TLTP = 100C
Ti - TLTP =
100C
Oven Offset
(millidegrees)
10
10-12
Oven Cycling Range
(millidegrees)
1
0.1
0.01
100
4x
10
6 x 10-13
4 x 10-14
4 x 10-15
4 x 10-16
1
2 x 10-13
6 x 10-15
4 x 10-16
4 x 10-17
0.1
2 x 10-13
2 x 10-15
6 x 10-17
4 x 10-18
0
2 x 10-13
2 x 10-15
2 x 10-17
2 x 10-19
4x
10-13
4x
10-14
4 x 10-15
A comparative table for AT and other non-thermal-transient compensated cuts of oscillators
would not be meaningful because the dynamic f vs. T effects would generally dominate the
static f vs. T effects.
4-45
Oven Stability Limits
• Thermal gain of 105 has been achieved with a feed-forward
compensation technique (i.e., measure outside T of case &
adjust setpoint of the thermistor to anticipate and
compensate). For example, with a 105 gain, if outside
∆T = 100oC, inside ∆T = 1 mK.
• Stability of a good amplifier ~1µK/K
• Stability of thermistors ~1mK/year to 100mK/year
• Noise < 1µK (Johnson noise in thermistor + amplifier noise
+ shot noise in the bridge current)
• Quantum limit of temperature fluctuations ~ 1nK
• Optimum oven design can provide very high f vs. T stability
4-46
Fractional Frequency Deviation From Turnover Frequency
Warmup of AT- and SC-cut Resonators
10-3
10-4
10-5
Deviation from static f vs. t =
10-6
where, for example,
~
a
dT
~
a ,
dt
≈-2 x 10-7 s/K2
10-7
for a typical AT-cut resonator
10-8
0
3
6
9
-10-8
Oven Warmup Time
-10-7
-10-6
4-47
12
Time (min)
15
TCXO Thermal Hysteresis
Fractional Frequency Error (ppm)
1.0
0.5
0.0
-25
-5
15
35
55
Temperature (0C)
75
-0.5
-1.0
TCXO = Temperature Compensated Crystal Oscillator
4-48
Normalized frequency change (ppm)
45
40
35
30
25
20
15
10
5
0
-55
-35
-25
-15
-5
Temperature (C)
4-49
5
15
25
35
45
55
65
75
85
Apparent Hysteresis
-45
OCXO Retrace
15
14 days
10
∆f
-9
f X 10
5
OVEN
OFF
OVEN ON
(a)
OSCILLATOR ON
(b)
0
15
14 days
10
5
OSCILLATOR
OFF
0
In (a), the oscillator was kept on continuously while the oven was cycled
off and on. In (b), the oven was kept on continuously while the oscillator
was cycled off and on.
4-50
TCXO Trim Effect
2
-6 ppm aging adjustment
-1
T (0C)
75
55
35
15
-5
0
-25
∆f
(ppm)
f
1
+6 ppm aging adjustment
In TCXO’s, temperature sensitive reactances are used to compensate for f vs. T
variations. A variable reactance is also used to compensate for TCXO aging. The effect
of the adjustment for aging on f vs. T stability is the “trim effect”. Curves show f vs. T
stability of a “0.5 ppm TCXO,” at zero trim and at ±6 ppm trim. (Curves have been
vertically displaced for clarity.)
4-51
Why the Trim Effect?
C1
∆f
≈
fS 2(C0 + CL )
∆f
fS
Compensated f vs. T
CL
Compensating CL vs. T
4-52
Effects of Load Capacitance on f vs. T
12
* 10-6
SC-cut
8
4
∆f
f
0
-4
-8
r = Co/C1 = 746
α = 0.130
-12
-50
200
450
700
950
T
4-53
1200
1450
DEGREES CELSIUS
1700
1950
Effects of Harmonics on f vs. T
50
40
30
20
∞
∆f 10
f 0
3
(ppm)-10
1
-20
M
AT-cut
Reference angle-of-cut (θ
θ) is about
8 minutes higher for the overtone modes.
(for the overtone modes of the SC-cut,
the reference θ-angle-of-cut is about 30
minutes higher)
-30
-40
-50
-100
5
-80
-60
-40
-20
-0
4-54
20
40
60
80
∆T, 0C
Normalized current amplitude
Amplitude - Frequency Effect
10 µ W
400 µ W
100 µ W
4000 µ W
10 -6
Frequency
At high drive levels, resonance curves become asymmetric
due to the nonlinearities of quartz.
4-55
Frequency Change (parts in 109)
Frequency vs. Drive Level
80
5 MHz AT
60
3 diopter
10 MHz SC
40
2 diopter
10 MHz SC
20
1 diopter
10 MHz SC
0
-20
10 MHz BT
100
200
300
400
500
600
Crystal Current (microamperes)
4-56
700
Drive Level vs. Resistance
Drive level
effects
Normal
operating
range
Resistance R1
Anomalous
starting
resistance
10-3
10-2
10-1
1
IX (mA)
4-57
10
100
Second Level of Drive Effect
Activity (current)
C
B
O
A
D
4-58
Drive level (voltage)
Activity Dips
fL2
Resistance
Frequency
∆f
10 X10-6
f
fL1
fR
RL2
RL1
R1
-40
-20
0
20
Temperature
40
(0C)
60
80
100
Activity dips in the f vs. T and R vs. T when operated with and without load
capacitors. Dip temperatures are a function of CL, which indicates that the dip is
caused by a mode (probably flexure) with a large negative temperature coefficient.
4-59
Frequency Jumps
2.0 x 10-11
Frequency deviation (ppb)
30 min.
4.0
3.0
2.0
No. 2
1.0
No. 3
0.0
No. 4
-1.0
0
2
4
6
Elapsed time (hours)
4-60
8
10
Acceleration vs. Frequency Change
∆f
f
Z’
O
X’
Y’
Crystal
plate
G
Supports
Frequency shift is a function of the magnitude and direction of the
acceleration, and is usually linear with magnitude up to at least 50 g’s.
4-61
2-g Tipover Test
(∆
∆f vs. attitude about three axes)
Axis 3
10.000 MHz oscillator’s tipover test
∆f
X 10 −9
f
(f(max) - f(min))/2 = 1.889x10-09 (ccw)
(f(max) - f(min))/2 = 1.863x10-09 (cw)
delta θ = 106.0 deg.
4
g
-4
4-62
360
270
315
360
315
360
315
-2
135
45
0
90
2
270
(f(max) - f(min))/2 = 1.882x10-09 (ccw)
(f(max) - f(min))/2 = 1.859x10-09 (cw)
delta θ = 16.0 deg.
4
Axis 1
225
Axis 2
225
-4
270
-2
135
45
0
90
2
225
(f(max) - f(min))/2 = 6.841x10-10 (ccw)
(f(max) - f(min))/2 = 6.896x10-10 (cw)
delta θ = 150.0 deg.
4
180
Axis 2
Axis 1
180
-4
180
-2
135
45
0
90
2
f0 - ∆f
f0 - ∆f
f0 - ∆f
f0 - ∆f
f0 + ∆f
t=0
t=
π
2fv
t=
π
fv
t=
3π
2fv
f0 + ∆f
f0 + ∆f
f0 + ∆f
f0 + ∆f
t=
Acceleration
f0 - ∆f
Time
Voltage
Time
Sinusoidal Vibration Modulated Frequency
Time
2π
fv
4-63
Acceleration Levels and Effects
Environment
∆f
Acceleration
typical levels*, in g’s
x10-11,
for 1x10-9/g oscillator
Buildings**, quiesent
0.02 rms
2
Tractor-trailer (3-80 Hz)
0.2 peak
20
Armored personnel carrier
0.5 to 3 rms
50 to 300
Ship - calm seas
0.02 to 0.1 peak
2 to 10
Ship - rough seas
0.8 peak
80
Propeller aircraft
0.3 to 5 rms
30 to 500
Helicopter
0.1 to 7 rms
10 to 700
Jet aircraft
0.02 to 2 rms
2 to 200
Missile - boost phase
15 peak
1,500
Railroads
0.1 to 1 peak
10 to 100
* Levels at the oscillator depend on how and where the oscillator is mounted
Platform resonances can greatly amplify the acceleration levels.
** Building vibrations can have significant effects on noise measurements
4-64
Acceleration Sensitivity Vector
Γ = γ1î + γ2 ĵ + γ3k̂
Axis 3
γ3
Γ = γ12 + γ22 + γ32
Γ
γ1
γ2
Axis 2
Axis 1
4-65
Vibration-Induced Allan Deviation Degradation
σy (τ)
10-9
σy
(τ) =
−12
10
τ
+
γA τν
π τ
10-10
σy
10-11
(τ) =
10
−12
τ
10-12
0.001
0.01
τ(sec)
0.1
Example shown: fv = 20, Hz A = 1.0 g along Γ, Γ = 1 x 10-9/g
4-66
1
Vibration-Induced Phase Excursion
The phase of a vibration modulated signal is
∆ f 
φ(t ) = 2πf0 t +  sin(2πfv t )
 fv 
When the oscillator is subjected to a sinusoidal vibration, the peak
phase excursion is
∆ φ peak
(
)
∆ f Γ • A f0
=
=
fv
fv
Example: if a 10 MHz, 1 x 10-9/g oscillator is subjected to a 10 Hz
sinusoidal vibration of amplitude 1g, the peak vibration-induced phase
excursion is 1 x 10-3 radian. If this oscillator is used as the reference
oscillator in a 10 GHz radar system, the peak phase excursion at 10GHz
will be 1 radian. Such a large phase excursion can be catastrophic to
the performance of many systems, such as those which employ phase
locked loops (PLL) or phase shift keying (PSK).
4-67
Vibration-Induced Sidebands
0
NOTE: the “sidebands” are spectral
lines at ±fV from the carrier frequency
-10
(where fV = vibration frequency). The
lines are broadened because of the finite
bandwidth of the spectrum analyzer.
-20
L(f)
-30
10g amplitude @ 100 Hz
Γ = 1.4 x 10-9 per g
-40
-50
-60
-70
-80
-90
4-68
f
250
200
150
100
50
0
-50
-100
-150
-200
-250
-100
Vibration-Induced Sidebands
After Frequency Multiplication
Each frequency multiplication
by 10 increases the sidebands
by 20 dB.
L(f) 0
-10
-20
-30
-40
10X
-50
-60
-70
1X
-80
-90
4-69
f
250
200
150
100
50
0
-50
-100
-150
-200
-250
-100
Sine Vibration-Induced Phase Noise
Sinusoidal vibration produces spectral lines at ±fv from the
carrier, where fv is the vibration frequency.
 Γ • Af0 

L' (fv ) = 20 log 
 2f v 
e.g., if Γ = 1 x 10-9/g and f0 = 10 MHz, then even if the
oscillator is completely noise free at rest, the phase “noise”
i.e., the spectral lines, due solely to a sine vibration level of
1g will be;
Vibr. freq., fv, in Hz
1
10
100
1,000
10,000
4-70
L’(fv), in dBc
-46
-66
-86
-106
-126
Random Vibration-Induced Phase Noise
Random vibration’s contribution to phase noise is given by:
 Γ • Af0 
,
L (f ) = 20 log 
 2f 
where lAl = [(2)(PSD )]
1
2
e.g., if Γ = 1 x 10-9/g and f0 = 10 MHz, then even if the
oscillator is completely noise free at rest, the phase “noise”
i.e., the spectral lines, due solely to a vibration
PSD = 0.1 g2/Hz will be:
Offset freq., f, in Hz
1
10
100
1,000
10,000
4-71
L’(f), in dBc/Hz
-53
-73
-93
-113
-133
Random-Vibration-Induced Phase Noise
L( f
-70
-80
)u
nd
e
L (f) (dBc)
-90
r th
er
an
do
m
-100
vib
ra
-110
-120
L(
f)
w
-130
PSD (g2/Hz)
Phase noise under vibration is for Γ = 1 x 10-9 per g and f = 10 MHz
tio
ns
ho
wn
ibr
ati
on
-140
-150
-160
5
300
1K 2K
4-72
.04
5
45 dB
ith
ou
tv
.07
300
1K 2K
Frequency (Hz)
Typical aircraft
random vibration
envelope
Acceleration Sensitivity vs. Vibration Frequency
Vibration Sensitivity (/g)
10-8
10-9
10-10
100
it
nge lim
a
r
c
i
m
na
yzer dy
l
a
n
a
m
Spectru
200
300
400
500
Vibration Frequency (Hz)
4-73
1000
Acceleration Sensitivity of Quartz Resonators
Resonator acceleration sensitivities range from the low parts in 1010 per g for
the best commercially available SC-cuts, to parts in 107 per g for tuning-fork-type
watch crystals. When a wide range of resonators were examined: AT, BT, FC, IT,
SC, AK, and GT-cuts; 5 MHz 5th overtones to 500 MHz fundamental mode inverted
mesa resonators; resonators made of natural quartz, cultured quartz, and swept
cultured quartz; numerous geometries and mounting configurations (including
rectangular AT-cuts); nearly all of the results were within a factor of three of 1x10-9
per g. On the other hand, the fact that a few resonators have been found to have
sensitivities of less than 1x10-10 per g indicates that the observed acceleration
sensitivities are not due to any inherent natural limitations.
Theoretical and experimental evidence indicates that the major variables yet
to be controlled properly are the mode shape and location (i.e., the amplitude of
vibration distribution), and the strain distribution associated with the mode of
vibration. Theoretically, when the mounting is completely symmetrical with respect
to the mode shape, the acceleration sensitivity can be zero, but tiny changes from
this ideal condition can cause a significant sensitivity. Until the acceleration
sensitivity problem is solved, acceleration compensation and vibration isolation can
provide lower than 1x10-10 per g, for a limited range of vibration frequencies, and at
a cost.
4-74
Phase Noise Degradation Due to Vibration
• Data shown is for a 10 MHz,
2 x 10-9 per g oscillator
Impacts on Radar Performance
• Lower probability of detection
• Lower probability of identification
• Shorter range
• False targets
• Radar spec. shown is for a coherent
radar (e.g., SOTAS)
-50
“Good’ oscillator on
vibrating platform (1g)
Required to
“see” 4km/hr
target
-100
53 dB
dBc/Hz
“Good’
oscillator
at rest
Radar oscillator
specification
-150
100K 10K
1K
100
10
1
1
OFFSET FROM CARRIER (Hz)
4-75
10
100
1K
10K 100K
Coherent Radar Probability of Detection
To “see” 4 km/h targets, low phase noise 70 Hz from the
carrier is required. Shown is the probability of detection of
4 km/h targets vs. the phase noise 70 Hz from the carrier
of a 10 MHz reference oscillator. (After multiplication to 10
GHz the phase noise will be at least 60 dB higher.) The
phase noise due to platform vibration, e.g., on an aircraft,
reduces the probability of detection of slow-moving targets
to zero.
Probability of Detection (%)
100
80
60
40
20
Low Noise
-140
-135
High Noise
-130
-125
-120
-115
Phase Noise (dBc/Hz)
at 70 Hz from carrier, for 4 km/h targets
4-76
-110
Vibration Isolation
Limitations
Transmissibility
1
• Poor at low frequencies
Region of
Amplification
• Adds size, weight and cost
Region of
Isolation
0.2
1
• Ineffective for acoustic
noise
2
Forcing Freq./Resonant Freq.
4-77
Vibration Compensation
Stimulus
Compensated Oscillator
Response
f
-9
7x
OSC.
10
/ Vo
lt
ACC
ACC
OSC.
AMP
V
DC Voltage on Crystal
5 MHz fund. SC
Vibration Compensated
Oscillator
fv
ACC = accelerometer
OSC.
Response to Vibration
AC Voltage on Crystal
fv
fO - fV
fO
fO - fV
fO - fV
fO
fO - fV
OSC.
Crystal Being Vibrated
4-78
fO - fV
fO
fO - fV
Vibration Sensitivity Measurement System
Controller
Controller
Plotter
Plotteror
or
Printer
Printer
Spectrum
Spectrum
Analyzer
Analyzer
Signal
Signal
Generator
Generator
Frequency
Frequency
Multiplier
Multiplier
(x10)
(x10)
Synthesizer
Synthesizer
(Local
(LocalOscillator)
Oscillator)
Accelerometer
Accelerometer
Test
TestOscillator
Oscillator
Shake
ShakeTable
Table
Power
Power
Amplifier
Amplifier
4-79
Vibration
Vibration
Level
Level
Controller
Controller
fVf
V
Shock
The frequency excursion during a shock is due to
the resonator’s stress sensitivity. The magnitude of
the excursion is a function of resonator design, and
of the shock induced stresses on the resonator
(resonances in the mounting structure will
amplify the stresses.) The permanent frequency
offset can be due to: shock induced stress
changes, the removal of (particulate) contamination
from the resonator surfaces, and changes in the
oscillator circuitry. Survival under shock is primarily
a function of resonator surface imperfections.
Chemical-polishing-produced scratch-free resonators
have survived shocks up to 36,000 g in air gun tests,
and have survived the shocks due to being fired from
a 155 mm howitzer (16,000 g, 12 ms duration).
4-80
∆f
X 10 8
f
3
2
Shock
1
0
-1
-2
-3
tO
t1
Radiation-Induced Frequency Shifts
Frequency
fO
fO = original, preirradiation
frequency
∆fSS
fSS
fSS = steady-state frequency
(0.2 to 24 hours after
exposure)
ft
∆fSS = steady-state frequency
offset
fT = frequency at time t
t0
t
Time
∆fSS/rad* =
{
10-11 for natural quartz (and R increase can stop the oscillation)
10-12 for cultured quartz
10-13 for swept cultured quartz
* for a 1 megarad dose (the coefficients are dose dependent)
Idealized frequency vs. time behavior for a quartz resonator following a pulse
of ionizing radiation.
4-81
Effects of Repeated Radiations
Fractional Frequency, ppb
10
2. Reirradiation (after 2.5 x 104 rad)
0
3. Reirradiation (after >106 rad)
-10
-20
Five irradiations; responses during the 4th and 5th
irradiations repeated the results of the 3rd. At
least 2 days elapsed between successive
irradiations.
1. Initial irradiation
-30
-40
10
Initial slopes:
1st:
-1 x 10-9/rad
2nd: +1 x 10-11/rad
3rd: +3 x 10-12/rad
4th: +3 x 10-12/rad
5th: +5 x 10-12/rad
102
103
104
Dose, rad(SiO2)
4-82
105
106
Radiation Induced ∆f vs. Dose and Quartz Type
10 MeV electrons,
5 MHz 5th overtone
AT-cut resonators
Frequency Change (Hz)
50
30
Z-growth
cultured
10
0
-10
Swept Z-growth
cultured
-30
-50
104
Natural
5
105
5
106
5
Reds (Si)
4-83
107
Annealing of Radiation Induced f Changes
Frequency change, Hz
-∆f
∆fS = 41 Hz
40
x
x
x
x
x
x
x
x
20
X T= 4330K (1600C)
X T= 4540K
x
X T= 4680K
x
x
X T= 4880K (2150C)
T = 5130K(2400C)
x
0
100
200
Annealing time, minutes
x
300
• For a 4 MHz AT-cut resonator, X-ray dose of 6 x 106 rads produced ∆f = 41 Hz.
• Activiation energies were calculated from the temperature dependence of the
annealing curves. The experimental results can be reproduced by two
processes, with activation energies E1 = 0.3 ± 0.1 eV and E2 = 1.3 ± 0.3eV.
• Annealing was complete in less than 3 hours at > 2400C.
4-84
Transient ∆f After a Pulse of γ Radiation
X
4
X
X
0
X
X
X
X
X
∆f/f (pp 108)
-4
X
X
X
X
X
-8
XX
X
X
X
X
X
X
X
X
-12
-16
-20
X
X
X
X
X
Experimental data, dose = 1.3 x 104 rads, SC-cut
X
Experimental data, dose = 2.3 x 104 rads, AT-cut
Model Calculation: AT-cut
-24
0.1
1.0
10
100
Time
1000
(seconds after event)
4-85
Effects of Flash X-rays on RS
32 MHz AT-cut resonators
80
13
N-4
N-4 (Natural)
C-7 (Unswept synthetic)
C-22 (Unswept synthetic)
S-25 Swept synthetic)
S-26 Swept synthetic)
60
C-7
4.5 x 104 R
4 x 104 R
3.5 x 104 R
4 x 104 R
4 x 104 R
11
10
9
50
8
40
7
C-22
Value of Q-1 x 106
12
70
RS in Ohms
14
6
30
0.001
(S-25)
(S-26)
Preirradiation value RS
0.01
0.1
5
1.0
10
100
1000
Time following exposure (seconds)
The curves show the series resonance resistance, RS, vs. time following a
4 x 104 rad pulse. Resonators made of swept quartz show no change in RS
from the earliest measurement time (1 ms) after exposure, at room
temperature. Large increases in RS (i.e., large decrease in the Q) will stop the
oscillation.
4-86
Frequency Change due to Neutrons
Frequency Deviation, f
∆f
X106
1000
5 MHz
AT-cut
900
800
700
600
Slope = 0.7 x 10-21/n/cm2
500
400
300
200
100
0
0
1
2
3
4
5
6
7
8
Fast Neutron Exposure (nvt)
4-87
9
10 11 12
x1017
Neutron Damage
(1)
(2)
(3)
(4)
A fast neutron can displace about 50 to 100 atoms before it comes
to rest. Most of the damage is done by the recoiling atoms. Net result
is that each neutron can cause numerous vacancies and interstitials.
4-88
Summary - Steady-State Radiation Results
Dose vs. frequency change is nonlinear; frequency change per rad is larger at low doses.
At doses > 1 kRad, frequency change is quartz-impurity dependent. The ionizing radiation
produces electron-hole pairs; the holes are trapped by the impurity Al sites while the
compensating cation (e.g., Li+ or Na+) is released. The freed cations are loosely trapped along
the optic axis. The lattice near the Al is altered, the elastic constant is changed; therefore, the
frequency shifts. Ge impurity is also troublesome.
At a 1 MRad dose, frequency change ranges from pp 1011 per rad for natural quartz to pp 1014
per rad for high quality swept quartz.
Frequency change is negative for natural quartz; it can be positive or negative for cultured and
swept cultured quartz.
Frequency change saturates at doses >> 106 rads.
Q degrades upon irradiation if the quartz contains a high concentration of alkali impurities; Q of
resonators made of properly swept cultured quartz is unaffected.
High dose radiation can also rotate the f vs. T characteristic.
Frequency change anneals at T > 240°C in less than 3 hours.
Preconditioning (e.g., with doses > 105 rads) reduces the high dose radiation sensitivities upon
subsequent irradiations.
At < 100 rad, frequency change is not well understood. Radiation induced stress relief & surface
effects (adsorption, desorption, dissociation, polymerization and charging) may be factors.
4-89
Summary - Pulse Irradiation Results
For applications requiring circuits hardened to pulse irradiation, quartz resonators are the
least tolerant element in properly designed oscillator circuits.
Resonators made of unswept quartz or natural quartz can experience a large increase in Rs
following a pulse of radiation. The radiation pulse can stop the oscillation.
Natural, cultured, and swept cultured AT-cut quartz resonators experience an initial
negative frequency shift immediately after exposure to a pulse of X-rays (e.g., 104 to
105 Rad of flash X-rays), ∆f/f is as large as -3ppm at 0.02sec after burst of 1012 Rad/sec.
Transient f offset anneals as t-1/2; the nonthermal-transient part of the f offset is probably
due to the diffusion and retrapping of hydrogen at the Al3+ trap.
Resonators made of properly swept quartz experience a negligibly small change in Rs
when subjected to pulsed ionizing radiation (therefore, the oscillator circuit does not require
a large reserve of gain margin).
SC-cut quartz resonators made of properly swept high Q quartz do not exhibit transient
frequency offsets following a pulse of ionizing radiation.
Crystal oscillators will stop oscillating during an intense pulse of ionizing radiation because
of the large prompt photoconductivity in quartz and in the transistors comprising the
oscillator circuit. Oscillation will start up within 15µsec after a burst if swept quartz is used
in the resonator and the oscillator circuit is properly designed for the radiation environment.
4-90
Summary - Neutron Irradiation Results
When a fast neutron (~MeV energy) hurtles into a crystal lattice and collides with an
atom, it is scattered like a billiard ball. The recoiling atom, having an energy (~104
to 106 eV) that is much greater than its binding energy in the lattice, leaves behind
a vacancy and, as it travels through the lattice, it displaces and ionizes other atoms.
A single fast neutron can thereby produce numerous vacancies, interstitials, and
broken interatomic bonds. Neutron damage thus changes both the elastic
constants and the density of quartz. Of the fast neutrons that impinge on a
resonator, most pass through without any collisions, i.e., without any effects on the
resonator. The small fraction of neutrons that collide with atoms in the lattice cause
the damage.
Frequency increases approximately linearly with fluence. For AT- and SC-cut
resonators, the slopes range from +0.7 x 10-21/n/cm2, at very high fluences (1017 to
1018n/cm2) to 5 x 10-21/n/cm2 at 1012 to 1013n/cm2, and 8 x 10-21/n/cm2at 1010 to
1012n/cm2. Sensitivity probably depends somewhat on the quartz defect density
and on the neutron energy distribution. (Thermonuclear neutrons cause more
damage than reactor neutrons.)
Neutron irradiation also rotates the frequency vs. temperature characteristic.
When a heavily neutron irradiated sample was baked at 500°C for six days, 90% of
the neutron-induced frequency shift was removed (but the 10% remaining was still
93 ppm).
4-91
Other Effects on Stability
Electric field - affects doubly-rotated resonators; e.g., a voltage on the electrodes of a 5 MHz
fundamental mode SC-cut resonator results in a ∆f/f = 7 x 10-9 per volt. The voltage can also
cause sweeping, which can affect the frequency (of all cuts), even at normal operating
temperatures.
Magnetic field - quartz is diamagnetic, however, magnetic fields can induce Eddy currents, and
will affect magnetic materials in the resonator package and the oscillator circuitry. Induced ac
voltages can affect varactors, AGC circuits and power supplies. Typical frequency change of a
"good" quartz oscillator is <<10-10 per gauss.
Ambient pressure (altitude) - deformation of resonator and oscillator packages, and change in
heat transfer conditions affect the frequency.
Humidity - can affect the oscillator circuitry, and the oscillator's thermal properties, e.g.,
moisture absorbed by organics can affect dielectric constants.
Power supply voltage, and load impedance - affect the oscillator circuitry, and indirectly, the
resonator's drive level and load reactance. A change in load impedance changes the amplitude
or phase of the signal reflected into the oscillator loop, which changes the phase (and
frequency) of the oscillation. The effects can be minimized by using a (low noise) voltage
regulator and buffer amplifier.
Gas permeation - stability can be affected by excessive levels of atmospheric hydrogen and
helium diffusing into "hermetically sealed" metal and glass enclosures (e.g., hydrogen diffusion
through nickel resonator enclosures, and helium diffusion through glass Rb standard bulbs).
4-92
Interactions Among Influences
In attempting to measure the effect of a single influence, one often encounters
interfering influences, the presence of which may or may not be obvious.
Interfering Influence
Measurement
Resonator aging
∆T due to oven T (i.e., thermistor) aging
∆ drive level due to osc. circuit aging
Short term stability
Vibration
Vibration sensitivity
Induced voltages due to magnetic fields
2-g tipover sensitivity
∆T due to convection inside oven
Resonator f vs. T
(static)
Thermal transient effect, humidity
T-coefficient of load reactances
Radiation sensitivity
∆T, thermal transient effect, aging
4-93
CHAPTER 5
Quartz Material Properties
5
Hydrothermal Growth of Quartz
•
Cover
Growth
zone, T1
•
Closure
area
The baffle localizes the temperature gradient
so that each zone is nearly isothermal.
Autoclave
•
Seeds
Baffle
Nutrient
dissolving
zone, T2
The autoclave is filled to some
predetermined factor with water plus
mineralizer (NaOH or Na2CO3).
The seeds are thin slices of (usually)
Z-cut single crystals.
•
The nutrient consists of small (~2½ to 4 cm)
pieces of single-crystal quartz (“lascas”).
•
Solutenutrient
Nutrient
T2 > T1
5-1
The temperatures and pressures are
typically about 3500C and 800 to 2,000
atmospheres; T2 - T1 is typically 40C to 100C.
•
The nutrient dissolves slowly (30 to 260 days
per run), diffuses to the growth zone, and
deposits onto the seeds.
Deeply Dissolved Quartz Sphere
Anisotropic Etching
Z
Y
+X
+X
Looking along Y-axis
Looking along Z-axis
5-2
Etching & Chemical Polishing
Diffusion Controlled Etching:
Etchant Must:
1. Diffuse to Surface
2. Be Adsorbed
3. React Chemically
Lapped surface
Reaction Products Must:
4. Be Desorbed
5. Diffuse Away
Chemically polished surface
5-3
Left-Handed and Right-Handed Quartz
X
X
Y
Y
r
z
r
z
r
z
S
X
r
S
z
X
Z
Z
Y
m
m
z
m
m
r
z
Left-Handed
Y
r
Right-Handed
5-4
The Quartz Lattice
Si
Si
Z
O
O
144.2o
109o
Y
O
Si
O
Si
Si
5-5
Quartz Properties’ Effects on Device Properties
Quartz Property
Device and Device-Fabrication Property
Q
Oscillator short-term stability, phase noise
close to carrier, long-term stability, filter loss
Purity (Al, Fe, Li, Na, K,
-OH, H2O)
Radiation hardness, susceptibility to twinning,
optical characteristics
Crystalline Perfection,
Strains
Sweepability, etchability for chem. polishing
and photolithographic processing, optical
properties, strength, aging(?), hysteresis(?)
Inclusions
High-temperature processing and applications,
resonator Q, optical characteristics, etchability
5-6
Ions in Quartz - Simplified Model
a
A)
B)
= Oxygen
Axis of channel
H
H
Al
= Si4+
Al
C)
D)
E)
Na
Li
K
Al
Al
Al
0.143 eV
0.089 eV
0.055 eV
5-7
0.2 eV
Aluminum Associated Defects
Al-M+ center
Al-OH center
Ox
Ox
Ox
Ox
Al3+
M+
H+
OH molecule
Al-hole center
Interstitial
Alkali
Ox
Ox
Ox
Ox
Ox
Ox
Al3+
Ox
Hole trapped in nonbonding
oxygen p orbital
5-8
h+
Al3+
Sweeping
Oven
T = 500OC
Z
Thermometer
Ammeter
Cr-Au
Quartz bar
E = 1000 V/cm
High voltage power
supply
I
0.5 µa/cm2
Time
5-9
Quartz Quality Indicators
•
•
•
•
•
•
•
•
•
•
•
•
Infrared absorption coefficient*
Etch-channel density *
Etch-pit density
Inclusion density *
Acoustic attenuation
Impurity analysis
X-ray topography
UV absorption
Birefringence along the optic axis
Thermal shock induced fracture
Electron spin resonance
???
* EIA Standard 477-1 contains standard test method for this quantity
5-10
Infrared Absorption of Quartz
2.5
100
3.0
3.5
E parallel to Z
3500
E parallel to X
80
Transmission (%)
4.0
3410
60
3585
40
3200
3300
20
0
4000
3500
3000
Wave number (cm-1)
5-11
2500
Infrared Absorption Coefficient
One of the factors that determine the maximum achievable resonator Q is the
OH content of the quartz. Infrared absorption measurements are routinely used
to measure the intensities of the temperature-broadened OH defect bands.
The infrared absorption coefficient α is defined by EIA Standard 477-1 as
A (3500 cm-1) - A (3800 cm-1)
α=
Y-cut thickness in cm
where the A’s are the logarithm (base 10) of the fraction of the incident beam
absorbed at the wave numbers in the parentheses.
Grade
A
B
C
D
E
α, in cm-1
0.03
0.045
0.060
0.12
0.25
Approx. max. Q*
3.0
2.2
1.8
1.0
0.5
* In millions, at 5 MHz (α is a quality indicator for unswept quartz only).
5-12
Quartz Twinning
>>
>>>
>>>>
>
>>>> >>>
> >>>
>>>
>> >
> > >>>>>
> >> > >
>>>>>>>>
>>
>
> >>>> > >>>> >> >
>
>
>
>>> >> >>>>> >> >>>>>
> > > > >>>
>
>
>> > >> >>> >
>
>
>> > > >> > >>
>>>>>> >>>>
> > >>>>>
>> > >
∧ ∧ ∧ ∧∧ ∧ ∧ ∧∧ ∧ ∧
∧
∧∧
∧ ∧ ∧ ∧∧ ∧∧
∧∧∧
∧∧ ∧
∧
∧∧
∧∧
∧∧ ∧∧∧∧∧∧
∧∧
∧∧ ∧∧
∧∧∧ ∧∧
∧ ∧∧ ∧∧ ∧∧
∧
∧
∧ ∧ ∧∧ ∧ ∧∧
∧
∧ ∧∧ ∧ ∧ ∧∧∧
∧∧∧∧
∧
∧
∧
∧
∧
∧
∧
∧
∧
∧
∧
∧∧
∧
∧∧ ∧∧ ∧∧ ∧ ∧∧ ∧
∧ ∧∧
∧ ∧∧∧ ∧∧∧ ∧∧
∧ ∧ ∧ ∧ ∧
>>
>>
>> >
>>
>>>> >
>
>
∧ ∧
∧∧ ∧∧∧∧ ∧∧∧∧∧
Electrical twinning
•
•
•
•
Optical Twinning
The X-axes of quartz, the electrical axes, are parallel to the line bisecting
adjacent prism faces; the +X-direction is positive upon extension due to
tension.
Electric twinning (also called Dauphiné twinning) consists of localized reversal
of the X-axes. It usually consists of irregular patches, with irregular
boundaries. It can be produced artificially by inversion from high quartz,
thermal shock, high local pressure (even at room temperature), and by an
intense electric field.
In right-handed quartz, the plane of polarization is rotated clockwise as seen
by looking toward the light source; in left handed, it is counterclockwise.
Optically twinned (also called Brazil twinned) quartz contains both left and
right-handed quartz. Boundaries between optical twins are usually straight.
Etching can reveal both kinds of twinning.
5-13
Twinning - Axial Relationships
Z
+
r
r
-
Z
r
Electrical
(Dauphine)
+
-
Z
r
Z
Z
+
r
r
Z
Z
+
-
+
+
r
r
-
+
r
Z
-
Z
-
-
+
-
r
+
r
r
r
r
-
+
Z
Z
+
-
r
r
-
Optical
(Brazil)
+
-
Z
+ Combined
Z
Z
+
Z
+
Z
+
r
Z
r
Z
+
+
The diagrams illustrate the relationship between the axial system
and hand of twinned crystals. The arrows indicate the hand.
5-14
Quartz Lattice and Twinning
Z-axis projection showing electric (Dauphiné)
twins separated by a twin wall of one unit cell
thickness. The numbers in the atoms are
atom heights expressed in units of percent of
a unit cell height. The atom shifts during
twinning involve motions of <0.03 nm.
Domain
Wall
00
79
21
88
45
Silicon
Oxygen
67
88
33
45
55
67
12
00
79
00
79
21
21
88
33
45
67
5-15
Quartz Inversion
•
•
•
•
Quartz undergoes a high-low inversion (α
α - β transformation) at
5730C. (It is 5730C at 1 atm on rising temperature; it can be 10 to 20C
lower on falling temperature.)
Bond angles between adjoining (SiO4) tetrahedra change at the
inversion. Whereas low-quartz (α
α-quartz) is trigonal, high quartz
(β
β -quartz) is hexagonal. Both forms are piezoelectric.
An abrupt change in nearly all physical properties takes place at the
inversion point; volume increases by 0.86% during inversion from
low to high quartz. The changes are reversible, although Dauphiné
twinning is usually acquired upon cooling through the inversion
point.
Inversion temperature decreases with increasing Al and alkali
content, increases with Ge content, and increases 10C for each 40
atm increase in hydrostatic pressure.
5-16
Phase Diagram of Silica (SiO2)
12
Stishovite
10
P(gpa)
8
Coesite
6
4
Liquid
2
0
Low or
α-quartz
0
500
High or
β-quartz
Tridymite
Cristobalite
1000
2000
1500
T(oC)
5-17
2500
Internal Friction of Quartz
100
60
Value of Q, in millions
40
Most probable internal
friction curve for quartz;
excluding mounting losses
20
10
8
6
4
90 mm
30 mm
2
Diameter of
shaped quartz
plates, in
vacuum
1
0.8
0.6
15 mm
0.4
Flat quartz
plates, in
air
0.2
0.1
0.1
0.2
0.4 0.6
1.0
2
4
6 8 10
20
40 60
100
Frequency in MHz
Empirically determined Q vs. frequency curves indicate that the maximum achievable
Q times the frequency is a constant, e.g., 16 million for AT-cut resonators, when f is in MHz.
5-18
Langasite and Its Isomorphs
•
La3Ga5SiO14
Langasite (LGS)
La3Ga5.5Nb0.5O14
Langanite (LGN)
La3Ga5.5Ta0.5O14
Langatate (LGT)
Lower acoustic attenuation than quartz (higher Qf than
AT- or SC-cut quartz)
•
No phase transition (melts at ~1,400 oC vs. phase transition
at 573 oC for quartz)
•
Higher piezoelectric coupling than quartz
•
Thicker than quartz at the same frequency
•
Temperature-compensated
5-19
CHAPTER 6
Atomic Frequency Standards*
* There are two important reasons for including this chapter: 1. atomic frequency standards are
one of the most important applications of precision quartz oscillators, and 2. those who study or
use crystal oscillators ought to be aware of what is available in case they need an oscillator with
better long-term stability than what crystal oscillators can provide.
6
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)
Trapped ions (f0 > 10 GHz, Q > 1011)
Cesium fountain (f0 = 9.2 GHz, Q ~ 5 x 1011)
6-1
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
Hydrogen-Like Atoms
MF =
3
Electron
spin and
dipole
Closed
electronic
shell
2
Nucleus
Electron
1
N
F=2
S
∆W
N
Electron
3
4
X
-1
MF =
-2
-3
Hydrogen-like (or alkali)
atoms
2
F=1
S
Nuclear
spin and
dipole
2
1
0
-1
-2
-1
0
1
Hyperfine structure of 87Rb, with nuclear spin I=3/2,
ν0=∆W/h=6,834,682,605 Hz and X=[(-µJ/J) +(µI/I)]H0/∆W
calibrated in units of 2.44 x 103 Oe.
6-3
Atomic Frequency Standard
Block Diagram
Atomic
Atomic
Resonator
Resonator
Multiplier
Multiplier
Feedback
Feedback
Quartz
Quartz
Crystal
Crystal
Oscillator
Oscillator
5 MHz
Output
6-4
Generalized Atomic Resonator
Prepare
PrepareAtomic
Atomic
State
State
Apply
Apply
Microwaves
Microwaves
Detect
DetectAtomic
Atomic
State
StateChange
Change
B
hν0
Tune
TuneMicrowave
MicrowaveFrequency
Frequency
For
ForMaximum
MaximumState
StateChange
Change
6-5
A
Atomic Resonator Concepts
• The energy levels used are due to the spin-spin interaction between the atomic nucleus and
the outer electron in the ground state (2S1/2) of the atom; i.e., the ground state hyperfine
transitions.
• Nearly all atomic standards use Rb or Cs atoms; nuclear spins I = 3/2 and 7/2, respectively.
• Energy levels split into 2(I ± 1/2)+1 sublevels in a magnetic field; the "clock transition" is the
transition between the least magnetic-field-sensitive sublevels. A constant magnetic field, the
"C-field," is applied to minimize the probability of the more magnetic-field-sensitive transitions.
• Magnetic shielding is used to reduce external magnetic fields (e.g., the earth's) at least
100-fold.
• The Heisenberg uncertainty principle limits the achievable accuracy: ∆E∆t ≥ h/2π, E = hν,
therefore, ∆ν∆t ≥1, and, long observation time → small frequency uncertainty.
• Resonance linewidth (i.e., 1/Q) is inversely proportional to coherent observation time ∆t; ∆t is
limited by: 1.) when atom enters and leaves the apparatus, and 2.) when the atom stops
oscillating due to collisions with other atoms or with container walls (collisions disturb atom's
electronic structure).
• Since atoms move with respect to the microwave source, resonance frequency is shifted due
to the Doppler effect (k•v); velocity distribution results in "Doppler broadening"; the secondorder Doppler shift (1/2 v2/c2) is due to relativistic time dilation.
6-6
Rubidium Cell Frequency Standard
Energy level diagrams of 85Rb and 87Rb
F=3
F=2
F=2
363 MHz
52P1/2
816 MHz
F=1
795 nm
795 nm
F=3
F=2
3.045 GHz
52S1/2
F=2
6.834,682,608 GHz
F=1
87Rb
85Rb
6-7
Rubidium Cell Frequency Standard
Atomic resonator schematic diagram
Magnetic shield
“C-Field”
87Rb
lamp
85Rb
+ buffer
gas
Light
Absorption
cell
Rb-87
+ buffer
gas
Photo
cell
Filter
Cell
Cavity
Frequency
input
6.834,685 GHz
rf
lamp
exciter
Power supplies
for lamp, filter
and absorption
cell thermostats
C-field
power
supply
6-8
Detector
output
Cs Hyperfine Energy Levels
Energy (Frequency) (GHz)
(F, mF)
(4,4)
(4,3)
(4,2)
(4,1)
(4,0)
(4,-1)
(4,-2)
(4,-3)
(4,-4)
9.2
9.192,631,770 GHz
(3,-3)
(3,-2)
(3,-1)
(3,0)
(3,1)
(3,2)
(3,3)
0
Magnetic Field
HO Energy states
at H = HO
6-9
Cesium-Beam Frequency Standard
Cs atom detection
Atomic state selection
O
CT
TE
E
D
NO SIGNAL
S
Cs VAPOR, CONTAINING AN EQUAL
AMOUNT OF THE TWO
KINDS OF Cs ATOMS
KIND 1 - ATOMS
(LOWER STATE) STATE SELECTED
ATOMIC BEAM
MAGNET
R
TO
C
TE
DE
MICROWAVE SIGNAL
(OF ATOMIC RESONANCE
FREQUENCY)
N
ATOMIC BEAM
SOURCE
S
MAGNET
(STATE SELECTOR)
VACUUM CHAMBER
KIND 2 - ATOMS STATE SELECTED
(UPPER STATE) ATOMIC BEAM
MICROWAVE
CAVITY
6-10
NO SIGNAL
N
MICROWAVE
CAVITY
S
ATOMIC BEAM
R
N
MAGNET
MAXIMUM
SIGNAL
Cesium-Beam Frequency Standard
Cs atomic resonator schematic diagram
DC
MAGNETIC SHIELD
“C-FIELD”
GETTER
C-FIELD
POWER SUPPLY
B-MAGNET
HOT WIRE
IONIZER
Cs-BEAM
A-MAGNET
CAVITY
GETTER
ION
COLLECTOR
VACUUM
ENVELOPE
OVEN
OVEN
HEATER
HEATER
POWER
POWER
SUPPLY
SUPPLY
FREQUENCY
INPUT
9,192,631,770 Hz
6-11
DETECTOR PUMP
SIGNAL
PUMP
POWER
SUPPLY
DETECTOR
POWER
SUPPLY
Atomic Hydrogen Energy Levels
(F, mF)
(1, +1)
(1, 0)
W
(1, -1)
1.42040…GHz
(0, 0)
H’
Ground state energy levels of atomic hydrogen as a function of magnetic field H’.
6-12
Passive H-Maser Schematic Diagram
Teflon coated
storage bulb
Microwave
cavity
Microwave
output
Microwave
input
State
selector
Hydrogen
atoms
6-13
Atomic Resonator Instabilities
• Noise - due to the circuitry, crystal resonator, and atomic resonator. (See next
page.)
• Cavity pulling - microwave cavity is also a resonator; atoms and cavity behave
as two coupled oscillators; effect can be minimized by tuning the cavity to the
atomic resonance frequency, and by maximizing the atomic resonance Q to cavity
Q ratio.
• Collisions - cause frequency shifts and shortening of oscillation duration.
• Doppler effects - 1st order is classical, can be minimized by design; 2nd
order is relativistic; can be minimized by slowing the atoms via laser cooling - see
“Laser Cooling of Atoms” later in this chapter.
• Magnetic field - this is the only influence that directly affects the atomic
resonance frequency.
• Microwave spectrum - asymmetric frequency distribution causes frequency
pulling; can be made negligible through proper design.
• Environmental effects - magnetic field changes, temperature changes,
vibration, shock, radiation, atmospheric pressure changes, and He
permeation into Rb bulbs.
6-14
Noise in Atomic Frequency Standards
If the time constant for the atomic-to-crystal servo-loop is to, then
at τ < to, the crystal oscillator determines σy(τ), i.e., σy (τ) ~ τ-1. From
τ > to to the τ where the "flicker floor" begins, variations in the atomic
beam intensity (shot-noise) determine σy(τ), and σy(τ) ~ (iτ)-1/2, where
i = number of signal events per second. Shot noise within the
feedback loop shows up as white frequency noise (random walk of
phase). Shot noise is generally present in any electronic device
(vacuum tube, transistor, photodetector, etc.) where discrete particles
(electrons, atoms) move across a potential barrier in a random way.
In commercial standards, to ranges from 0.01 s for a small Rb
standard to 60 s for a high-performance Cs standard. In the regions
where σy(τ) varies as τ -1 and τ -1/2, σy(τ) ∝ (QSR)-1, where SR is the
signal-to-noise ratio, i.e., the higher the Q and the signal-to-noise
ratio, the better the short term stability (and the phase noise far from
the carrier, in the frequency domain).
6-15
Short-Term Stability of a Cs Standard
σy(ττ)
10-9
STANDARD TUBE
10-10
τ0 = 1 SECOND
10-11
τ0 = 60 SECONDS*
10-12
OPTION 004 TUBE
10-13
τ0 = 1 SECOND
10-14
10-15
SYSTEM BW = 100 kHz
10-16
10-3 10-2 10-1 100 101 102 103 104 105
AVERAGING TIME τ (SECONDS)
6-16
* The 60 s time constant
provides better shortterm stability, but it is
106 usable only in benign
environments.
Short-Term Stability of a Rb Standard
σy(ττ)
fL (LOOP BW) =
1
2πτ
fL = 100 Hz (STANDARD)
OTHERS OPTIONAL
fL = 100 Hz
10-9
RUBIDIUM - WORST CASE
10-10
fL = 0.01 Hz
fL = 1 Hz
10-11
10-12
.001
VCXO
.01
.1
1
6-17
10
100
τ(seconds)
Acceleration Sensitivity of Atomic Standards
Atomic
Atomic
Resonator
Resonator
Multiplier
Multiplier
Feedback
Feedback
Quartz
Quartz
Crystal
Crystal
Oscillator
Oscillator
5 MHz Output
Let the servo loop time constant = t0, let the atomic standard's Γ = ΓA, and the
crystal oscillator’s (VCXO's) Γ = ΓO. Then,
πt0), ΓA = ΓO
• For fast acceleration changes (fvib >> 1/2π
• For slow acceleration changes, (fvib << 1/2π
πt0), ΓA << ΓO
• For fvib ≈ fmod, 2fmod, servo is confused, ΓA ≈ ΓO, plus a frequency offset
• For small fvib, (at Bessel function null), loss of lock, ΓA ≈ ΓO
6-18
Atomic Standard Acceleration Effects
In Rb cell standards, high acceleration can cause ∆f due to light shift,
power shift, and servo effects:
• Location of molten Rb in the Rb lamp can shift
• Mechanical changes can deflect light beam
• Mechanical changes can cause rf power changes
In Cs beam standards, high acceleration can cause ∆f due to changes
in the atomic trajectory with respect to the tube & microwave cavity
structures:
• Vibration modulates the amplitude of the detected signal.
Worst when fvib = f mod.
• Beam to cavity position change causes cavity phase shift
effects
• Velocity distribution of Cs atoms can change
• Rocking effect can cause ∆f even when fvib < f mod
In H-masers, cavity deformation causes ∆f due to cavity pulling effect
6-19
Magnetic Field Sensitivities of Atomic Clocks
Clock transition frequency ν = νo + CHHo2, where CH is the quadratic
Zeeman effect coefficient (which varies as 1/νo).
Atom
Transition Frequency
C-field*
Shielding Sensitivity
(milligauss)** Factor* per gauss**
Rb v=6.8 GHz + (574 Hz/G2) Bo 2
250
5k
10-11
Cs
v=9.2 GHz + (427 Hz/G2) Bo 2
60
50k
10-13
H
v=1.4 GHz + (2750 Hz/G2) Bo 2
0.5
50k
10-13
* Typical values
** 1 gauss = 10-4 Tesla; Tesla is the SI unit of magnetic flux density.
6-20
Crystal’s Influences on Atomic Standard
• Short term stability - for averaging times less than the atomic-tocrystal servo loop time constant, τL, the crystal oscillator determines
σy(ττ).
• Loss of lock - caused by large phase excursions in t < τL (due to
shock, attitude change, vibration, thermal transient, radiation pulse).
At a Rb standard's 6.8 GHz, for a ∆f = 1 x 10-9 in 1s, as in a 2g
tipover in 1s, ∆φ
φ ~ 7π
π. Control voltage sweeping during
reacquisition attempt can cause the phase and frequency to change
wildly.
• Maintenance or end of life - when crystal oscillator frequency
offset due to aging approaches EFC range (typically ~ 1 to 2 x 10-7).
• Long term stability - noise at second harmonic of modulation f
causes time varying ∆f's; this effect is significant only in the highest
stability (e.g., H and Hg) standards.
6-21
Optically Pumped Cs Standard
Fluorescence Detector
F=5
2
•
6 P 3/2
F = 4 453 MHz
Tuned laser
diode pumps
F=3
Oven
852 nm
2
Pump
laser(s)
Detection
laser
6 S 1/2
Spontaneous
decays
(~350 THz)
F=4
9.192631770 GHz
F=3
State selection
Essential Elements of an Optically Pumped
Cesium Beam Standard
State detection
Atomic Energy Levels
The proper atomic energy levels are populated by optical pumping with
a laser diode. This method provides superior utilization of Cs atoms,
and provides the potential advantages of: higher S/N, longer life, lower
weight, and the possibility of trading off size for accuracy. A miniature
Cs standard of 1 x 10-11 accuracy, and <<1 liter volume, i.e., about
100x higher accuracy than a Rb standard, in a smaller volume (but not
necessarily the same shape factor) seems possible.
6-22
Rubidium - Crystal Oscillator (RbXO)
Rubidium
Frequency
Standard
(≈
≈25W @ -550C)
RbXO
Interface
6-24
Low-power
Crystal
Oscillator
RbXO Principle of Operation
Output
Rb Reference
Power Source
RF
Sample
OCXO
Control
Voltage
Rb
Reference
Control
Signals
Tuning
Memory
OCXO and Tuning
Memory Power Source
6-25
CHAPTER 7
Oscillator Comparisons and
Specifications
7
Oscillator Comparison
Quartz Oscillators
Atomic Oscillators
TCXO
MCXO
OCXO
Rubidium
RbXO
Cesium
Accuracy *
(per year)
2 x 10-6
5 x 10-8
1 x 10-8
5 x 10-10
7 x 10-10
2 x 10-11
Aging/Year
5 x 10-7
2 x 10-8
5 x 10-9
2 x 10-10
2 x 10-10
0
Temp. Stab.
(range, 0C)
5 x 10-7
(-55 to +85)
3 x 10-8
(-55 to +85)
1 x 10-9
(-55 to +85)
3 x 10-10
(-55 to +68)
5 x 10-10
(-55 to +85)
2 x 10-11
(-28 to +65)
Stability,σ
σy(ττ)
(ττ = 1s)
1 x 10-9
3 x 10-10
1 x 10-12
3 x 10-12
5 x 10-12
5 x 10-11
Size
(cm3)
10
30
20-200
200-800
1,000
6,000
Warmup Time
(min)
0.03
(to 1 x 10-6)
0.03
(to 2 x 10-8)
4
(to 1 x 10-8)
3
(to 5 x 10-10)
3
(to 5 x 10-10)
20
(to 2 x 10-11)
Power (W)
0.04
0.04
0.6
20
0.65
30
10 - 100
<1,000
200-2,000
2,000-8,000
<10,000
50,000
(at lowest temp.)
Price (~$)
* Including environmental effects (note that the temperature ranges for Rb
and Cs are narrower than for quartz).
7-1
Clock Accuracy vs. Power Requirement*
(Goal of R&D is to move technologies toward the upper left)
10-12
Cs
Cs
1µ
µs/day
1ms/year
Accuracy
10-10
Rb
Rb
10-8
1ms/day
1s/year
OCXO
OCXO
TCXO
TCXO
10-6
10-4
0.001
1s/day
XO
0.01
0.1
1
Power (W)
10
100
* Accuracy vs, size, and accuracy vs. cost have similar relationships
Clock Accuracy vs. Power Requirement*
(Goal of R&D is to move technologies toward the upper left)
Accuracy
10-12
10-10
Cs
Cs
Mini-Cs
= in production
= developmental
Rb
Rb
RbXO
10-8
1ms/day
1s/year
MCXO TMXO
OCXO
OCXO
10-6
TCXO
TCXO
10-4
0.001
1µ
µs/day
1ms/year
1s/day
XO
XO
0.01
0.1
1
Power (W)
10
100
* Accuracy vs, size, and accuracy vs. cost have similar relationships
7-3
Short Term Stability Ranges of Various
Frequency Standards
-9
rtz
a
Qu
Log (σ
σy(ττ))
-10
-11
Ru
-12
m
u
i
bid
-13
C
-14
-15
Hydrogen Maser
-16
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
Log (ττ), seconds
7-4
4.0
5.0
6.0
7.0
1 day1 month
um
i
s
e
Phase Instabilities of Various Frequency
Standards
Ru
-30
um
si
-60
-70
-90
en
-100
g
ro
yd
-80
H
-110
Quartz
er
as
M
10*Log(Sφ(f))
Ce
-50
um
di
bi
-40
-120
-130
-140
-150
-160
-3
-2
-1
0
Log (f)
1
2
3
4
Typical one-sided spectral density of phase deviation vs. offset
frequency, for various standards, calculated at 5 MHz. L(f) = ½ Sφ
7-5
Weaknesses and Wearout Mechanisms
Quartz
Wearout Mechanisms
Weaknesses
None
Aging
Rad hardness
Rubidium depletion
Buffer gas depletion
Glass contaminants
Rubidium Life
Power
Weight
Cesium
Cesium supply depletion
Spent cesium gettering
Ion pump capacity
Electron multiplier
Life
Power
Weight
Cost
Temp. range
7-6
Why Do Crystal Oscillators Fail?
Crystal oscillators have no inherent failure mechanisms. Some have operated
for decades without failure. Oscillators do fail (go out of spec.) occasionally for reasons
such as:
•
•
•
•
•
•
•
•
Poor workmanship & quality control - e.g., wires come loose at poor quality solder
joints, leaks into the enclosure, and random failure of components
Frequency ages to outside the calibration range due to high aging plus insufficient
tuning range
TCXO frequency vs. temperature characteristic degrades due to aging and the
“trim effect”.
OCXO frequency vs. temperature characteristic degrades due to shift of oven set
point.
Oscillation stops, or frequency shifts out of range or becomes noisy at certain
temperatures, due to activity dips
Oscillation stops or frequency shifts out of range when exposed to ionizing
radiation - due to use of unswept quartz or poor choice of circuit components
Oscillator noise exceeds specifications due to vibration induced noise
Crystal breaks under shock due to insufficient surface finish
7-6
Oscillator Selection Considerations
• Frequency accuracy or reproducibility requirement
• Recalibration interval
• Environmental extremes
• Power availability - must it operate from batteries?
• Allowable warmup time
• Short term stability (phase noise) requirements
• Size and weight constraints
• Cost to be minimized - acquisition or life cycle cost
7-7
Crystal Oscillator Specification: MIL-PRF-55310
MIL-PRF-55310D
15 March 1998
----------------------SUPERSEDING
MIL-0-55310C
15 Mar 1994
PERFORMANCE SPECIFICATION
OSCILLATOR, CRYSTAL CONTROLLED
GENERAL SPECIFICATION FOR
This specification is approved for use by all Departments and Agencies of the Department of Defense.
1. SCOPE
1.1 Statement of scope. This specification covers the general requirements for
quartz crystal oscillators used in electronic equipment.
---------------------------------Full text version is available via a link from <http:\\www.ieee.org/uffc/fc>
7-8
CHAPTER 8
Time and Timekeeping
8
What Is Time?
• "What, then, is time? If no one asks me, I know; if I wish to explain to
him who asks, I know not." --- Saint Augustine, circa 400 A.D.
• The question, both a philosophical and a scientific one, has no entirely
satisfactory answer. "Time is what a clock measures." "It defines the
temporal order of events." "It is an element in the four-dimensional
geometry of space-time.” “It is nature’s way of making sure that
everything doesn't happen at once.”
• Why are there "arrows" of time? The arrows are: entropy,
electromagnetic waves, expansion of the universe, k-meson decay,
and psychological. Does time have a beginning and an end? (Big
bang; no more "events", eventually.)
• The unit of time, the second, is one of the seven base units in the
International System of Units (SI units)*. Since time is the quantity
that can be measured with the highest accuracy, it plays a central role
in metrology.
8-1
Dictionary Definition of “Time”
(From The Random House Dictionary of the English Language 1987)
tim e (tim ), n ., a d j., u , tim e d , tim ~ in g . — n . 1 . th e s y s te m
o f th o s e s e q u e n tia l re la tio n s th a t a n y e v e n t h a s to a n y
o th e r, a s p a s t, p re s e n t, o r fu tu re , in d e fin ite a n d c o n tin u o u s d u ra tio n re g a rd e d a s th a t in w h ic h e v e n ts s u c c e e d o n e a n o th e r. 2 . d u ra tio n re g a rd e d a s b e lo n g in g to
th e p re s e n t life a s d is tin c t fro m th e life to c o m e o r fro m
e te rn ity ; fin ite d u ra tio n . 3 . (s o m e tim e s c a p .) a s y s te m
o r m e th o d o f m e a s u rin g o r re c k o n in g th e p a s s a g e o f
tim e : m e a n tim e ; a p p a re n t tim e ; G re e n w ic h T im e . 4 . a
lim ite d p e rio d o r in te rv a l, a s b e tw e e n tw o s u c c e s s iv e
e v e n ts : a lo n g tim e . 5 . a p a rtic u la r p e rio d c o n s id e re d
a s d is tin c t fro m o th e r p e rio d s : Y o u th is th e b e s t tim e o f
life . 6 . O fte n , tim e s . a . a p e rio d in th e h is to ry o f th e
w o rld , o r c o n te m p o ra ry w ith th e life o r a c tiv itie s o f a
n o ta b le p e rs o n : p re h is to ric tim e s ; in L in c o ln 's tim e . b .
th e p e rio d o r e ra n o w o r p re v io u s ly p re s e n t: a s ig n o f
th e tim e s ; H o w tim e s h a v e c h a n g e d ! c . a p e rio d
c o n s id e re d w ith re fe re n c e to it’s e v e n ts o r p re v a ilin g
c o n d itio n s , te n d e n c ie s Id e a s , e tc .: h a rd tim e s ; a tim e o f
w a r. 7 . a p re s c rib e d o r a llo tte d p e rio d , a s o f o n e 's life ,
fo r p a y m e n t o f a d e b t, e tc . 8 . th e e n d o f a p re s c rib e d o r
a llo tte d p e rio d , a s o f o n e 's life o r a p re g n a n c y : H is tim e
h a d c o m e , b u t th e re w a s n o o n e le ft to m o u rn o v e r
h im . W h e n h e r tim e c a m e , h e r h u s b a n d a c c o m p a n ie d
h e r to th e d e liv e ry ro o m . 9 . a p e rio d w ith re fe re n c e to
p e rs o n a l e x p e rie n c e o f a s p e c ifie d k in d : to h a v e a g o o d
tim e ; a h o t tim e in th e o ld to w n to n ig h t. 1 0 . a p e rio d o f
w o rk o f a n e m p lo y e e , o r th e p a y fo r it; w o rk in g h o u rs o r
d a y s o r a n h o u rly o r d a ily p a y ra te . 1 1 . In fo rm a l. a te rm
o f e n fo rc e d d u ty o r im p ris o n m e n t: to s e rv e tim e in th e
a rm y , d o tim e in p ris o n . 1 2 . th e p e rio d n e c e s s a ry fo r o r
o c c u p ie d b y s o m e th in g : T h e tim e o f th e b a s e b a II g a m e
w a s tw o h o u rs a n d tw o m in u te s . T h e b u s ta k e s to o
m u c h tim e , s o I'll ta k e a p la n e . 1 3 . Ie is u re tim e ;
s u ffic ie n t o r s p a re tim e : to h a v e tim e fo r a v a c a tio n ; I
h a v e n o tim e to s to p n o w . 1 4 . a p a rtic u la r o r d e fin ite
p o in t in tim e , a s in d ic a te d b y a c lo c k : W h a t tim e is it?
1 5 . a p a rtic u la r p a rt o f a y e a r, d a y , e tc .; s e a s o n o r
p e rio d : It's tim e fo r lu n c h . 1 6 . a n a p p o in te d , fit, d u e , o r
p ro p e r in s ta n t o r p e rio d : A tim e fo r s o w in g ; th e tim e
w h e n th e s u n c ro s s e s th e m e rid ia n T h e re is A tim e fo r
e v e ry th in g . 1 7 . th e p a rtic u la r p o in t in tim e w h e n a n
e v e n t Is s c h e d u le d to ta k e p la c e : tra in tim e ; c u rta in
tim e . 1 8 . a n in d e fin ite , fre q u e n tly p ro lo n g e d p e rio d o r
d u ra tio n in th e fu tu re : T im e w ill te ll if w h a t w e h a v e
d o n e h e re to d a y w a s rig h t. 1 9 . th e rig h t o c c a s io n o r
o p p o rtu n ity : to w a tc h o n e 's tim e . 2 0 . e a c h o c c a s io n o f
a re c u rrin g a c tio n o r e v e n t: to d o a th in g fiv e tim e s , It's
th e p itc h e r's tim e a t b a t. 2 1 . tim e s , u s e d a s a
m u ltip lic a tiv e w o rd in p h ra s a l c o m b in a tio n s e x p re s s in g
h o w m a n y in s ta n c e s o f a q u a n tity o r fa c to r a re ta k e n
to g e th e r: T w o g o e s in to s ix th re e tim e s , fiv e tim e s
fa s te r 2 2 . D ra m a . o n e o f th e th re e u n itie s . C f. u n ity
(d e f. 8 ) 2 3 . P ro s . a u n it o r a g ro u p o f u n its in th e
m e a s u re m e n t o f m e te r. 2 4 . M u s ic a . te m p o , re la tiv e
ra p id ity o f m o v e m e n t. b . th e m e tric a l d u ra tio n o f a n o te
o r re s t. c . p ro p e r o r c h a ra c te ris tic te m p o . d . th e g e n e ra l
m o v e m e n t o f a p a rtic u la r k in d o f m u s ic a l c o m p o s itio n
w ith re fe re n c e to its rh y th m , m e tric a l s tru c tu re , a n d
te m p o . e . th e m o v e m e n t o f a d a n c e o r th e lik e to m u s ic
s o a rra n g e d : w a ltz tim e . 2 5 . M il. ra te o f m a rc h in g ,
c a lc u la te d o n th e n u m b e r o f p a c e s ta k e n p e r m in u te :
d o u b le tim e ; q u ic k tim e . 2 6 . M a n e g e . e a c h c o m p le te d
a c tio n o r m o v e m e n t o f th e h o rs e . 2 7 . a g a in s t tim e , in
a n e ffo rt to fin is h s o m e th in g w ith in a lim ite d p e rio d : W e
w o rk e d a g a in s t tim e to g e t o u t th e n e w s p a p e r. 2 8 .
a h e a d o f tim e , b e fo re th e tim e d u e ; e a rly : T h e b u ild in g
w a s c o m p le te d a h e a d o f tim e . 2 9 . a t o n e tim e , a .
o n c e ; in a fo rm e r tim e : A t o n e tim e th e y o w n e d a
re s ta u ra n t. b . a t th e s a m e tim e ; a t o n c e : T h e y a ll trie d
to ta lk a t o n e tim e . 3 0 . a t th e s a m e tim e , n e v e rth e le s s ;
y e t: I'd lik e to try it b u t a t th e s a m e tim e I'm a little
a fra id 3 1 . a t tim e s a t in te rv a ls ; o c c a s io n a lly : A t tim e s
th e c ity b e c o m e s in to le ra b le . 3 2 . b e a t s o m e o n e 's
tim e . S la n g . to c o m p e te fo r o r w in a p e rs o n b e in g
d a te d o r c o u rte d b y a n o th e r; p re v a il o v e r a riv a l: H e
a c c u s e d m e , h is o w n b ro th e r, o f try in g to b e a t h is tim e .
3 3 . b e h in d th e tim e s , o ld -fa s h io n e d ; d a te d : T h e s e
a ttitu d e s a re b e h in d th e tim e s . 3 4 . fo r th e tim e b e in g ,
te m p o ra rily ; fo r th e p re s e n t: L e t's fo rg e t a b o u t it fo r
th e tim e b e in g . 3 5 . fro m tim e to tim e , o n o c c a s io n ;
o c c a s io n a lly ; a t in te rv a ls : S h e c o m e s to s e e u s fro m
tim e to tim e . 3 6 . g a in tim e , to p o s tp o n e in o rd e r to
m a k e p re p a ra tio n s o r g a in a n a d v a n ta g e d e la y th e
o u tc o m e o f: H e h o p e d to g a in tim e b y p u ttin g o ff
s ig n in g th e p a p e rs fo r a fe w d a y s m o re . 3 7 . in g o o d
tim e , a . a t th e rig h t tim e ; o n tim e , p u n c tu a lly . b . in a d v a n c e o f th e rig h t tim e ; e a rly : W e a rriv e d a t th e a p p o in te d s p o t in g o o d tim e . 3 8 . in n o tim e , in a v e ry
b rie f tim e ; a lm o s t a t o n c e : W o rk in g to g e th e r, th e y
c le a n e d th e e n tire h o u s e in n o tim e . 3 9 . in tim e , a .
e a rly e n o u g h : to c o m e in tim e fo r d in n e r. b . in th e fu tu re ; e v e n tu a lly : In tim e h e 'll s e e w h a t is rig h t. c . in th e
c o rre c t rh y th m o r te m p o : T h e re w o u ld a lw a y s b e a t
le a s t o n e c h ild w h o c o u ld n 't p la y in tim e w ith th e m u s ic
4 0 . k e e p tim e , a . to re c o rd tim e , a s a w a tc h o r c lo c k
8-2
d o e s h to m a rk o r o b s e rv e th e te m p o . c .to p e rfo rm
rh y th m ic m o v e m e n ts in u n is o n . 4 1 . k ill tim e , to o c c u p y
o n e s e lf w ith s o m e a c tiv ity to m a k e tim e p a s s q u ic k ly :
W h ile I w a s w a itin g , I k ille d tim e c o u n tin g th e c a rs o n
th e fre ig h t tra in s . 4 2 . m a k e tim e , a . to m o v e q u ic k ly ,
e s p . in a n a tte m p t to re c o v e r lo s t tim e . b . to tra v e l a t a
p a rtic u la r s p e e d . 4 3 . m a k e tim e w ith , S la n g . to p u rs u e
o r ta k e a s a s e x u a l p a rtn e r. 4 4 . m a n y a tim e , a g a in
a n d a g a in ; fre q u e n tly : M a n y a tim e th e y d id n 't h a v e
e n o u g h to e a t a n d w e n t to b e d h u n g ry . 4 5 . m a rk tim e ,
a . to s u s p e n d p ro g re s s te m p o ra rily , a s to a w a it
d e v e lo p m e n ts ; fa il to a d v a n c e . b . M il. to m o v e th e fe e t
a lte rn a te ly a s in m a rc h in g , b u t w ith o u t a d v a n c in g . 4 6 .
o n o n e 's o w n tim e , d u rin g o n e 's fre e tim e ; w ith o u t
p a y m e n t: H e w o rk e d o u t m o re e ffic ie n t p ro d u c tio n
m e th o d s o n h is o w n tim e . 4 7 . o n tim e , a . a t th e
s p e c ifie d tim e ; p u n c tu a B y . b . to b e p a id fo r w ith in a
d e s ig n a te d p e rio d o f tim e , a s in in s ta llm e n ts : M a n y
p e o p le a re n e v e r o u t o f d e b t b e c a u s e th e y b u y
e v e ry th in g o n tim e . 4 8 . o u t o f tim e , n o t in th e p ro p e r
rh y th m : H is s in g in g w a s o u t o f tim e w ith th e m u s ic 4 9 .
P a s s th e tim e o f d a y . to c o n v e rs e b rie fly w ith o r g re e t
s o m e o n e : T h e w o m e n w o u ld s to p in th e m a rk e t to
p a s s th e tim e o f d a y . 5 0 . ta k e o n e 's tim e , to b e s lo w
o r le is u re ly ; d a w d le : S p e e d w a s im p o rta n t h e re , b u t h e
ju s t to o k h is tim e . 5 1 . tim e a fte r tim e . a g a in a n d a g a in ;
re p e a te d ly ; o fte n : I'v e to ld h im tim e a fte r tim e n o t to
s la m th e d o o r. 5 2 . tim e a n d tim e a g a in , re p e a te d ly ;
o fte n : T im e a n d tim e a g a in I w a rn e d h e r to s to p
s m o k in g . A ls o , tim e a n d a g a in . 5 3 . tim e o f llfe ,
(o n e 's ) a g e : A t y o u r tim e o f life y o u m u s t b e c a re fu l n o t
to o v e rd o th in g s . 5 4 . tim e o f o n e 's life , In fo rm a l. a n
e x tre m e ly e n jo y a b le e x p e rie n c e : T h e y h a d th e tim e o f
th e ir liv e s o n th e ir trip to E u ro p e .. 5 5 . o f. p e rta in in g to ,
o r s h o w in g th e p a s s a g e o f tim e 5 6 . (o f a n e x p lo s iv e
d e v ic e ) c o n ta in in g a c lo c k s o th a t it w ill d e to n a te a t th e
d e s ire d m o m e n t a tim e b o m b 5 7 . C o m . p a y a b le a t a
s ta te d p e rio d o f tim e a fte r p re s e n tm e n t: tim e d ra fts o r
n o te s . 5 8 . o f o r p e rta in in g to p u rc h a s e s o n th e
in s ta llm e n t p la n , o r w ith p a y m e n t p o s tp o n e d . v .t. 5 9 . to
m e a s u re o r re c o rd th e s p e e d , d u ra tio n , o r ra te o f: to
tim e a ra c e . 6 0 . to fix th e d u ra tio n o f: T h e p ro c to r tim e d
th e te s t a t 1 6 m in u te s . 6 1 . to fix th e in te rv a l b e tw e e n
(a c tio n s , e v e n ts , e tc .): T h e y tim e d th e ir s tro k e s a t s ix
p e r m in u te . 6 2 . to re g u la te (a tra in , c lo c k e tc .) a s to
tim e . 6 3 . to a p p o in t o r c h o o s e th e m o m e n t o r o c c a s lo n
fo r; s c h e d u le : H e tim e d th e a tta c k p e rfe c tly . — v .i. 6 4 .
to k e e p tim e ; s o u n d o r m o v e in u n is o n . [b e f. 9 0 0 ; (n .)
The Second
• The SI unit of time is the second (symbol s).
• The second was defined, by international agreement, in October,
1967, at the XIII General Conference of Weights and Measures.
• The second is "the duration of 9,192,631,770 periods of the
radiation corresponding to the transition between the two
hyperfine levels of the ground state of the cesium atom 133."
• Prior to 1967, the unit of time was based on astronomical
observations; the second was defined in terms of ephemeris time,
i.e., as "1/31,556,925.9747 of the tropical year..."
• The unit of frequency is defined as the hertz (symbol Hz). One
hertz equals the repetitive occurrence of one "event" per second.
8-3
Frequency and Time
1
f =
τ
where f = frequency (= number of “events” per unit time), and
τ = period (= time between “events”)
Total number of events
Accumulate d clock time =
Number of events per unit of time
3 rotations of the earth
Example :
= 3 days
1 rotation/d ay
Frequency source + counting mechanism → clock
Examples of frequency sources: the rotating earth, pendulum,
quartz crystal oscillator, and atomic frequency standard.
8-4
Typical Clock System
Counting
Mechanism
Frequency
Source
Setting
Mechanism
Display or
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
Evolution of Clock Technologies
• Sundials, and continuous flow of:
• Water (clepsydra)
• Sand (hour glass)
• Falling weights, with frictional control of rate
• Vibrating, but non-resonant motion - escapement
mechanisms: falling weight applies torque through train of
wheels; rate control depends on moments of inertia,
friction and torque; period is the time it takes to move from
one angular position to another.
• Resonant control
• Mechanical: pendulum, hairspring and balance wheel
• Mechanical, electrically driven: tuning fork, quartz
resonator
• Atomic and molecular
8-6
Progress in Timekeeping
Time Period
Clock/Milestone
Accuracy Per Day
4th millennium B.C. Day & night divided into 12 equal hours
Up to 1280 A.D.
Sundials, water clocks (clepsydrae)
~1 h
~1280 A.D.
~30 to 60 min
Mechanical clock invented- assembly time for prayer
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
10-9 to 10-4 s
8-7
Clock Errors
t
t
0
0
T(t) = T0 + ∫ R(t)dt + ε(t) = T0 + (R0t + 1/2At2 + …) + ∫ Ei(t)dt +εε(t)
Where,
T(t) = time difference between two clocks at time t after synchronization
T0 = synchronization error at t = 0
R(t) = the rate (i.e., fractional frequency) difference between the two clocks
under comparison; R(t) = R0 + At + …Ei(t)
ε(t) = error due to random fluctuations = τσy(τ)
R0 = R(t) at t = 0
A = aging term (higher order terms are included if the aging is not linear)
Ei(t) = rate difference due to environmental effects (temperature, etc.)
-------------Example: If a watch is set to within 0.5 seconds of a time tone (T0 = 0.5 s), and the
watch initially gains 2 s/week (R0 = 2 s/week), and the watch rate ages -0.1 s per
week2, (A = -0.1 s/week2), then after 10 weeks (and assuming Ei(t) = 0):
T (10 weeks) = 0.5 (2 x 10) + 1/2(-0.1 x (10)2) = 15.5 seconds.
8-8
t
1
2
3
1
2
3
t
fr
t
1 2 3
Time Error
Frequency
t
fr
Time Error
fr
t
Frequency
t
Time Error
fr
Time Error
Frequency
Frequency
Frequency Error vs. Time Error
fr = reference (i.e., the “correct”) frequency
8-9
t
t
1 2 3
Time Error (ms)
Clock Error vs. Resynchronization Interval
= 5 x 10-10
= 2 x 10-8
= 4 days
Aging/Day
Temp Stability
Resync Interval
25
20
15
10
5
0
10
20
30
40
50
60
70
80
90
100
Days Since Calibration
TCXO
OCXO
MCXO
RbXO
Avg. Temp. Stab.
1 x 10-6
2 x 10-8
2 x 10-8
2 x 10-8
Aging/Day
1 x 10-8
1 x 10-10
5 x 10-11
5 x 10-13
Resynch Interval*
(A/J & security)
Recalibr. Interval *
(Maintenance cost)
10 min
4 hrs
6 hours 4 days 6 hours 4 days
10 yrs 80 days 50 years 1.5 yrs 94 yrs
3 yrs
6 hours 4 days
None
needed 300 yrs
* Calculated for an accuracy requirement of 25 milliseconds. Many modern systems need much better.
8-10
Time Error vs. Elapsed Time
10 s
-4
Aging rates are per day
1s
G
A
Accumulated Time Error
-5
100 ms
T
SE
F
OF
T
SE
F
OF
10 ms
T
SE
F
OF
1 ms
G
A
10 µs
30
1
hour
2
G
IN
-6
-5
1
X
10
G
A
G
IN
1
X
-8
10
G
A
G
IN
-7
1
X
G
A
10
1X
G
A
-7
10
X
1
G
IN
G
IN
G
IN
G
A
-8
4
10
G
A
-6
0
X1
1
T
SE
F
OF
9
01
1X
T
E
FS
F
O
100 µs
1 µs
10
minutes
10
X
1
G
IN
1
X
10
T1
E
FS
OF
8
16
1
day
2
1
ET
S
F
OF
3 4 5 61
week
Elapsed Time
8-11
X
10
-
X
10
10
G
IN
1
-
10
X
-
11
G
IN
1
X
10
12
11
10
01
X
X
-9
1
1
AG
1
10
2
01
X
3 1
2
MONTH
4
1
YEAR
On Using Time for Clock Rate Calibration
It takes time to measure the clock rate (i.e., frequency) difference
between two clocks. The smaller the rate difference between a clock to be
calibrated and a reference clock, the longer it takes to measure the
difference (∆t/t ≈ ∆f/f).
For example, assume that a reference timing source (e.g., Loran or GPS)
with a time uncertainty of 100 ns is used to calibrate the rate of a clock to
1 x 10-11 accuracy. A frequency offset of 1 x 10-11 will produce1 x 10-11 x
3600 s/hour = 36 ns time error per hour. Then, to have a high certainty that
the measured time difference is due to the frequency offset rather than the
reference clock uncertainty, one must accumulate a sufficient amount (≥100
ns) of time error. It will take hours to perform the calibration. (See the next
page for a different example.) If one wishes to know the frequency offset to
a ±1 x 10-12 precision, then the calibration will take more than a day.
Of course, if one has a cesium standard for frequency reference, then, for
example, with a high resolution frequency counter, one can make frequency
comparisons of the same precision much faster.
8-12
Calibration With a 1 pps Reference
Let
A = desired clock rate accuracy after calibration
A' = actual clock rate accuracy
∆τ = jitter in the 1 pps of the reference clock, rms
∆τ' = jitter in the 1 pps of the clock being calibrated, rms
t = calibration duration
∆t = accumulated time error during calibration
Then, what should be the t for a given set of A, ∆t, and ∆t'?
Example: The crystal oscillator in a clock is to be calibrated by
comparing the 1 pps output from the clock with the 1 pps output from a standard. If
A = 1 x 10-9; ∆τ = 0.1 µs, and ∆τ' = 1.2 µs, then, [(∆τ)2 + (∆τ')2]1/2 ≈ 1.2 µs, and
when A = A', ∆t = (1 x 10-9)t ≡ (1.2 µs)N, and t = (1200N) s. The value of N to be
chosen depends on the statistics of the noise processes, on the confidence level
desired for A' to be ≤ A, and on whether one makes measurements every second
or only at the end points. If one measures at the end points only, and the noise is
white phase noise, and the measurement errors are normally distributed, then,
with N = 1, 68% of the calibrations will be within A; with N = 2, and 3, 95% and
99.7%, respectively, will be within A. One can reduce t by about a factor 2/N3/2 by
making measurements every second; e.g., from 1200 s to 2 x (1200)2/3 = 225 s.
8-13
Time Transfer Methods
Method
Accuracy
~ Cost (‘95)
Portable Cs clock
10 - 100 ns
$45K - 70K
GPS time dissemination
GPS common view
20 - 100 ns
5 - 20 ns
$100 - 5K
Two-way via satellite
~1 ns
$60k
Loran-C
100 ns
$1K - 5K
HF (WWV)
2 ms
$100 - 5K
Portable quartz & Rb
clocks
Calibration interval
dependent
$200 - 2K
8-14
Global Positioning System (GPS)
GPS Nominal Constellation:
24 satelites in 6 orbital planes,
4 satelites in each plane,
20,200 km altitude, 55 degree inclinations
8-15
GPS
GPS can provide global, all-weather, 24-hour, real-time, accurate navigation
and time reference to an unlimited number of users.
•
GPS Accuracies (2σ
σ)
Position:
Velocity:
Time:
120 m for Standard Positioning Service, SPS
40 m for Precise Positioning Service, PPS
1 cm + 1ppm for differential, static land survey
0.3 m/s (SPS), 0.1 m/s (PPS).
350 ns to < 10 ns
•
24 satellites in 6 orbital planes; 6 to 10 visible at all times; ~12 h period
20,200 km orbits.
•
Pseudorandom noise (PRN) navigation signals are broadcast at L1 =
1.575 GHz (19 cm) and L2 = 1.228 GHz (24 cm); two codes, C/A and
P are sent; messages provide satellite position, time, and atmospheric
propagation data; receivers select the optimum 4 (or more) satellites to
track. PPS (for DoD users) uses L1 and L2, SPS uses L1 only.
8-16
Oscillator’s Impact on GPS
• Satellite oscillator’s (clock’s) inaccuracy & noise are major sources of
navigational inaccuracy.
• Receiver oscillator affects GPS performance, as follows:
Oscillator Parameter
GPS Performance Parameter
Warmup time
Time to first fix
Power
Mission duration, logistics costs (batteries)
Size and weight
Short term stability
(0.1 s to 100 s)
Manpack size and weight
∆ range measurement accuracy, acceleration
performance, jamming resistance
Short term stability
(~15 minute)
Time to subsequent fix
Phase noise
Jamming margin, data demodulation, tracking
Acceleration sensitivity
See short term stability and phase noise effects
8-17
Time Scales
• A "time scale" is a system of assigning dates, i.e., a "time," to events; e.g., 6
January 1989, 13 h, 32 m, 46.382912 s, UTC, is a date.
• A "time interval" is a "length" of time between two events; e.g., five seconds.
• Universal time scales, UT0, UT1, and UT2, are based on the earth's spin on its
axis, with corrections.
• Celestial navigation: clock (UT1) + sextant
position.
• International Atomic Time (TAI) is maintained by the International Bureau of
Weights and Measures (BIPM; in France), and is derived from an ensemble of
more than 200 atomic clocks, from more than 60 laboratories around the world.
• Coordinated Universal Time (UTC) is the time scale today, by international
agreement. The rate of UTC is determined by TAI, but, in order to not let the
time vs. the earth's position change indefinitely, UTC is adjusted by means of
leap seconds so as to keep UTC within 0.9 s of UT1.
8-18
Clock Ensembles and Time Scales
• An ensemble of clocks is a group of clocks in which the time outputs of
individual clocks are combined, via a “time-scale algorithm,” to form a
time scale.
• Ensembles are often used in mission critical applications where a
clock’s failure (or degraded performance) presents an unacceptable
risk.
• Advantages of an ensemble:
- system time & frequency are maintained even if a clock in the
ensemble fails
- ensemble average can be used to estimate the characteristics of
each clock; outliers can be detected
- performance of ensemble can (sometimes) be better than any of
the contributors
- a proper algorithm can combine clocks of different characteristics,
and different duty cycles, in an optimum way
8-19
Relativistic Time
• Time is not absolute. The "time" at which a distant event takes
place depends on the observer. For example, if two events, A and
B, are so close in time or so widely separated in space that no
signal traveling at the speed of light can get from one to the other
before the latter takes place, then, even after correcting for
propagation delays, it is possible for one observer to find that A
took place before B, for a second to find that B took place before
A, and for a third to find that A and B occurred simultaneously.
Although it seems bizarre, all three can be right.
• Rapidly moving objects exhibit a "time dilation" effect. (“Twin
paradox”: Twin on a spaceship moving at 0.87c will age 6 months
while twin on earth ages 1 year. There is no "paradox" because
spaceship twin must accelerate; i.e., there is no symmetry to the
problem.)
• A clock's rate also depends on its position in a gravitational field. A
high clock runs faster than a low clock.
8-20
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 = +44 µsec/day
• In precise time and frequency comparisons, relativistic effects must be
included in the comparison procedures.
8-21
Relativistic Time Corrections
The following expression accounts for relativistic effects, provides for clock rate
accuracies of better than 1 part in 1014, and allows for global-scale clock
comparisons of nanosecond accuracy, via satellites:
∆t = −
(
)
 1 v 2 − v 2 − (Φ − Φ ) dt + 2 ω A
s
g
S
g 
E
c 2 ∫0  2
c2

1
T
Where ∆t = time difference between spacecraft clock and ground clock, tS-Tg
VS = spacecraft velocity (<<c), Vg = velocity of ground station
ΦS = gravitational potential at the spacecraft
Φg = gravitational potential at the ground station
ω = angular velocity of rotation of the earth
AE = the projected area on the earth’s equatorial plane swept out by the vector
whose tail is at the center of the earth and whose head is at the position
of the portable clock or the electromagnetic signal pulse. The AE is taken
positive if the head of the vector moves in the eastward direction.
Within 24 km of sea level, Φ = gh is accurate to 1 x 10-14 where g = (9.780 + 0.052
sin2Ψ )m/s2, Ψ = the latitude, h = the distance above sea level, and where the sin2Ψ
term accounts for the centrifugal potential due to the earth's rotation. The "Sagnac
effect," (2ω/c2)AE = (1.6227 x 10-21s/m2)AE, accounts for the earth-fixed coordinate
system being a rotating, noninertial reference frame.
8-22
Some Useful Relationships
• Propagation delay = 1 ns/30 cm = 1 ns/ft = 3.3 µs/km ≈ 5 µs/mile
• 1 day = 86,400 seconds; 1 year = 31.5 million seconds
• Clock accuracy: 1 ms/day ≈ 1 x 10-8
• At 10 MHz: period = 100 ns; phase deviation of 1° = 0.3 ns of time
deviation
• Doppler shift* = ∆f/f = 2v/c
--------------------------------------
* Doppler shift example:
if v = 4 km/h and f = 10 GHz (e.g., a slowmoving vehicle approaching an X-band radar), then ∆f = 74 Hz, i.e.,
an oscillator with low phase noise at 74Hz from the carrier is
necessary in order to "see" the vehicle.
8-23
One Pulse-Per-Second Timing Signal
(MIL-STD-188-115)
10 Volts
(±
± 10%)
Rise Time
< 20 Nanoseconds
0 Volts
(±
± 1 Volt
Fall Time
< 1 Microseconds
20 µsec ± 5%
"The leading edge of the BCD code (negative going transitions after extended high
level) shall coincide with the on-time (positive going transition) edge of the one
pulse-per-second signal to within ±1 millisecond." See next page for the MIL-STD
BCD code.
8-24
BCD Time Code
(MIL-STD-188-115)
Example: Selected Time is 12:34:56
20
msec
Level
Held
Hi
Until
Start
of
CodeWord
LLLH
8421
LLHL
8421
1
2
Hours
LLHH LHLL
8421 8421
3
4
Minutes
Rate: 50 Bits per Second
Bit Pulse Width: 20 msec
H = +6V dc ± 1V
L = -6V dc ± 1V
LHLH
LHHL
8421
8421
5
6
Seconds
8 4 2 1 8 4 2 1 8 4 2 1 8 4 2 1 8 4 2 1 8 4 2 1
Level
Held
Hi
Until
Start
of
Next
CodeWord
24 Bit BCD Time Code*
* May be followed by 12 bits for day-of-year and/or 4 bits for figure-of-merit
(FOM). The FOM ranges from better than 1 ns (BCD character 1) to greater
than 10 ms (BCD character 9).
8-25
Time and Frequency Subsystem
Oscillator
Oscillator and
and
Clock
Clock Driver
Driver
Time
Time Code
Code
Generator
Generator
Frequency
Frequency
Distribution
Distribution
f1
f2
Power
Power
Source
Source
f3
TOD
8-26
1 pps
The MIFTTI Subsystem
MIFTTI = Modular Intelligent Frequency, Time and Time
Interval
Battery
Battery
Oscillator/
Oscillator/
Clock-Driver
Clock-Driver
Microcomputer
Microcomputer
Compensation
Compensationand
and
Control
Control
External
External
Reference
Reference
Frequency
Frequency
Distribution
Distribution
f1
f2
f3
Clock
Clockand
and
Time
code
Time code
Generator
Generator
TOD 1pps
V1
V2
DC-DC
DC-DC
Converter
Converter
Vin
* The microcomputer compensates for systematic effects (after filtering
random effects), and performs: automatic synchronization and calibration
when an external reference is available, and built-in-testing.
8-27
"Time" Quotations
3 o'clock is always too late or too early for anything you want to do..................Jean-Paul Sartre
Time ripens all things. No man's born wise............Cervantes.
Time is the rider that breaks youth............George Herbert
Time heals all wounds..................Proverb
Time is on our side..................William E. Gladstone
Time wounds all heels..................Jane Ace
The hardest time to tell: when to stop.....Malcolm Forbes
It takes time to save time.............Joe Taylor
Time, whose tooth gnaws away everything else, is powerless against truth..................Thomas H. Huxley
Time has a wonderful way of weeding out the trivial..................Richard Ben Sapir
Time is a file that wears and makes no noise...........English proverb
The trouble with life is that there are so many beautiful women - and so little time..................John Barrymore
Life is too short, and the time we waste yawning can never be regained..................Stendahl
Time goes by: reputation increases, ability declines..................Dag Hammarskjöld
Remember that time is money...............Benjamin Franklin
The butterfly counts not months but moments, and has time enough..................Rabindranath Tagore
Everywhere is walking distance if you have the time..................Steven Wright
The only true time which a man can properly call his own, is that which he has all to himself; the rest, though in
some sense he may be said to live it, is other people's time, not his..................Charles Lamb
It is familiarity with life that makes time speed quickly. When every day is a step in the unknown, as for children, the
days are long with gathering of experience..................George Gissing
Time is a great teacher, but unfortunately it kills all its pupils..................Hector Berlioz
To everything there is a season, and a time to every purpose under the heaven..................Ecclesiastes 3:1
Time goes, you say? Ah no! Time stays, we go..................Henry Austin Dobson
Time is money - says the vulgarest saw known to any age or people. Turn it around, and you get a precious truth Money is time..................George (Robert) Gissing
8-28
Units of Measurement Having Special Names
in the International System of Units (SI)
SI Base Units
Mass
kilogram
Length
meter
kg
m
Time
second
s
Electric
Current
ampere
Temperature
kelvin
Luminous
Intensity
candela
Amount of
Substance
mole
A
K
cd
mol
K
cd sr
Luminous Flux
lumen
lm
S
Coordinated Time
international atomic time
SI Derived Units
TAI
s-2
kg m
Force
newton
N
m2s-2
kg
Energy
joule
J
Celsius
Temperature
0
Celsius
0
C
s-1
Frequency
hertz
Hz
m-2cd sr
Illuminance
SA
lux
Electric charge
lx
coulomb
C
kg m2s-3A-2
2
-3
-1
kg m s A
sr: the steradian is the supplementary
Resistance
Electric Potential
ohm
SI unit of solid angle (dimensionless)
volt
Ω
-1
rad: the radian is the supplementary
V
s
kg m2s-2A-1
kg-1 m2s4 A2
Activity
SI unit of plane angle (dimensionless)
Magnetic Flux
Capacitance
becquerel
weber
farad
Bq
Wb
F
m2s-1
kg m2s-2A-2
kg-1 m2s3 A2
Absorbed Dose
Inductance
Conductance
gray
henry
siemens
Gy
H
m2s-2
S
Dose Equivalent
kg s-2 A-1
sievert
Conductance
Sv
siemens
Electromagnetic
S
Health related
measurement units
kg m-1s-2
Pressure
pascal
Pa
kg m s-3
Power
Non-SI units
watt
recognized
W
for use with SI
day: 1 d = 86400 s
hour: 1 h = 3600 s
minute: 1 min = 60 s
liter: 1 l = 10-3 m3
ton: 1 t = 103 kg
degree: 10 = (π/180) rad
minute: 1’ = (π/10800)rad
second: 1” = (π/648000)rad
electronvolt: 1 eV ≈ 1.602177 x 10-19 J
unified atomic mass unit: 1 u ≈ 1.660540 x 10-27 kg
measurement units
8-29
Units of Measurement Having Special Names in the SI Units,
NOT Needing Standard Uncertainty in SI Average Frequency
SI Base Units
Temperature
Mass
kilogram
kelvin
kg
K
K
SI Derived Units
Celsius
Temperature
0
Celsius
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
8-30
Amount of
Substance
mole
mol
CHAPTER 9
Related Devices and Application
9
Discrete-Resonator Crystal Filter
A Typical Six-pole Narrow-band Filter
Layout
Circuit
9-1
Monolithic Crystal Filters
Two-pole filter and its response
25.0
Four-pole filter
electrode arrangement
Attenuation (dB)
20.0
15.0
10.0
5.0
0
Frequency
9-2
Surface Acoustic Wave (SAW) Devices
λ 2
λ 2
BAW
SAW, One-port
SAW, Two-port
Simplified Equivalent Circuits
C0
C1
L1
L1
C1
R1
C0
R1
BAW and One-port SAW
C0
Two-port SAW
9-3
Quartz Bulk-Wave Resonator Sensors
In frequency control and timekeeping applications, resonators are
designed to have minimum sensitivity to environmental parameters.
In sensor applications, the resonator is designed to have a high
sensitivity to an environmental parameter, such as temperature,
adsorbed mass, force, pressure and acceleration.
Quartz resonators' advantages over other sensor technologies are:
• High resolution and wide dynamic range (due to excellent shortterm stability); e.g., one part in 107 (10-6 g out of 20 g)
accelerometers are available, and quartz sorption detectors are
capable of sensing 10-12 grams.
• High long-term accuracy and stability, and
• Frequency counting is inherently digital.
9-4
Fo
rc
e
Tuning Fork Resonator Sensors
Beam
Motion
Fo
rc
e
Tine
Motion
Photolithographically produced tuning forks, single- and double-ended (flexural-mode
or torsional-mode), can provide low-cost, high-resolution sensors for measuring
temperature, pressure, force, and acceleration. Shown are flexural-mode tuning forks.
9-5
Dual Mode SC-cut Sensors
•Advantages
- Self temperature sensing by dual mode operation allows
separation/compensation of temp. effects
- Thermal transient compensated
- Isotropic stress compensated
- Fewer activity dips than AT-cut
- Less sensitive to circuit reactance changes
- Less sensitive to drive level changes
•Disadvantage
- Severe attenuation in a liquid
- Fewer SC-cut suppliers than AT-cut suppliers
9-6
Separation of Mass and Temperature
Effects
• Frequency changes
∆f (m, T, x ) ∆f (m)
∆f (T )
=
+
+
f
f
f
o
o
o
total
mass
temperature
∆f ( x )
f
o
other effects
• Mass: adsorption and desorption
∆f ( m )
∆m
≅−
fo
mo
• Temperature/beat frequency
∆f (T )
=
fo
∑ ci ⋅ ∆fβ
i
fβ ≡ 3fc1(T ) − fc 3 (T )
i
fo
9-7
Dual-Mode Pressure Sensor
End Caps
Electrode
Resonator
9-8
CHAPTER 10
Proceedings Ordering Information,
Website, and Index
10
IEEE International Frequency Control Symposium
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Prior to 1992, the Symposium’s name was the “Annual Symposium on Frequency Control,” and in 1992, the name was IEEE Frequency
Control Symposium (i.e., without the “International”).
10-1
Frequency Control Web Site
Frequency control information can be found on the
World Wide Web at
http://www.ieee-uffc.org/fc
Available at this site are the abstracts of all the papers
ever published in the Proceedings of the Frequency
Control Symposium, i.e., since 1956, reference and
tutorial information, historical information, and links to
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sites.
10-2
INDEX (A-D)
Acceleration, 3-37, 4-4, 4-58 to 4-77
Accuracy, 2-23, 4-2, 7-2
Activity Dips, 3-8, 4-59
Aging, 4-3 to 4-14, 4-55, 4-90, 6-24, 7-1
Allan Variance & Allan Deviation, 4-16 to 4-24, 4-30 to 4-36, 4-64, 6-14 to 6-17, 7-1, 7-3
Altitude, 4-3, 4-90
Amplitude, 3-6, 4-57 to 4-60
Amplitude Distribution, 3-6
Angles of Cut, 3-3, 3-13, 4-41
Applications, 1-1 to 1-20, 4-73 to 4-74, 8-15, 8-25, 8-26
ASFC Proceedings, 10-2
AT-Cut, 3-8, 3-13, 3-23, 4-10 to 4-14, 4-41, 4-43, 4-45, 4-46, 4-59, 4-72, 4-81 to 4-85, 4-88, 4-89
Atmospheric Pressure, 4-3, 4-90
Atomic Frequency Standards, 2-8, 2-22 to 2-23, 3-37, 6-23- to 6-24, 6-1 to 7-5
BCD Time Code, 8-23
Bending Strains, 4-15
Bistatic Radar, 1-18
Bonding Strains, 4-12, 4-14
Calibration, 8-12, 8-13
Ceramic Flatpack, 3-5, 3-37, 6-24
Cesium-Beam Frequency Standard, 2-8, 6-1, 6-9 to 6-11, 6-14, 6-15, 6-19, 6-20, 6-22, 7-1 to 7-5
Clocks, 4-38, 8-4 to 8-12, 8-19 to 8-22, 8-25, 8-26
Commercial vs. Military Markets, 1-2, 1-3
Conversions, 8-21
Doppler, 1-10, 1-17o 1-19, 8-21
Doubly Rotated Cuts, 3-13
Drive Level, 4-57 to 4-60, 4-91
10-3
INDEX (E-I)
Electric Field, 4-90
Emerging/Improving Technologies, 3-37, 6-25
Environmental Effects, 4-3, 4-4
Equivalent Circuit, 3-19, 3-20, 3-21
f-squared, 3-31
Fabrication Steps, 3-27
Failures, 7-6
Filters, 9-1, 9-2
Force-Frequency Effect, 4-10 to 4-14
Frequency Hopping, 1-13 to 1-15
Frequency Jumps, 4-60
Frequency Multiplication, 4-27, 4-34 to 4-36, 4-66, 4-73, 4-74
Frequency vs. Temperature, 3-11 to 3-13, 3-21, 4-37 to 4-51, 4-91, 6-24
G-sensitivity, 4-4, 4-56, 4-58 to 4-77, 4-91, 3-34
Global Positioning System (GPS), 1-4, 8-14 to 8-17
Harmonics, 3-4, 3-7, 3-8, 4-45
Historical Information, 3-32, 8-6, 8-7
Humidity, 4-3, 4-90
Hydrogen Maser, 6-1, 6-12, 6-13
Hysteresis, 2-23, 4-47,
Identification Friend or Foe (IFF), 1-12, 1-16,
Impacts of Technology, 1-12 to 1-19, 4-17, 4-71, 9-17
Influences on Frequency, 4-3, 4-4, 4-91
Infrared Absorption, 5-11 to 5-13
Interactions, 4-91
Inversion, quartz, 5-17, 5-18
10-4
INDEX (J-O)
Jitter, 4-64, 8-13
Johnson Noise, 4-19, 4-34
Jumps, 4-60
Laser Cooling of Atoms, 6-23
Lateral Field Resonator, 3-37, 3-38,
Load Capacitance, 3-20, 3-21, 4-40, 4-44, 4-51 to 4-55
Load, external, 2-15
Load Impedance, 3-20, 3-21, 4-3, 4-40, 4-44, 4-51 to 4-55
Loran, 8-12, 8-14
Magnetic Field, 4-3, 4-89, 6-9, 6-14, 6-20
Mathematical Description of a Resonator, 3-10 to 3-12
MCXO, 2-5, 2-8, 2-19 to 2-21, 3-36, 3-38, 6-24, 7-1, 8-10
MIFTTI, 8-26
Milestones in Quartz Technology, 3-32
Milestones in Timekeeping, 8-7
Military Requirements, 1-11 to 1-19
Mixer, 4-27, 4-28
Modes of Motion, 3-3, 3-4, 3-7, 3-8
Monolithic Crystal Filter, 9-2
Motional Parameters, 3-16, 3-23
Mounting Stress Effects, 4-8 to 4-14
Navigation, 1-4, 8-15 to 8-17
Neutrons, Effects of, 4-85, 4-86, 4-89
Noise, 2-14, 4-3, 4-5, 4-16 to 4-19, 4-17 to 4-26, 4-29 to 4-36 6-14 to 6-21
Nonlinearity, 2-2, 3-10
OCXO, 2-5 to 2-8, 2-11, 4-43, 4-44, 4-46, 4-48, 7-1, 7-2, 8-10
One Pulse Per Second, 8-13, 8-23
10-5
INDEX (O-P)
Optically Pumped Cs Standard, 6-22
Opto-Electronic Oscillator, 2-23
Oscillation, 2-2 to 2-4
Oscillator Categories, 2-6
Oscillator Circuit Caused Instabilities, 2-10 to 2-15
Oscillator Circuits, 2-9
Oscillator Comparison, 7-1 to 7-5
Oscillator Failures, 7-6
Oscillator Hierarchy, 2-8
Oscillator Outputs, 2-10
Oscillator Selection, 7-7
Oscillator Specification, 7-7
Oscillator Types, 2-5 to 2-8
Oscillators, 2-1 to 2-8, 2-19 to 2-23, 6-23 to 6-24, 7-1 to 7-7
Oven, 4-43, 4-44, 4-45
Packaging, 3-18, 3-38
Phase, 2-2, 2-3, 4-16, 4-25, 7-1 to 7-5
Phase Detector, 4-28 to 4-29
Phase Diagram of Silica, 5-18
Phase Jitter, 4-65
Phase Noise, 1-8, 1-17, 4-16 to 4-36, 4-62 to 4-69, 4-73, 4-74
Piezoelectricity, 3-1 to 3-3, 3-9 to 3-11
Power Requirements, 6-24, 7-1, 7-2
Power Supply, 2-9, 4-3, 4-89
Precision, 4-2
Proceedings, ASFC, 10-2
10-6
Propagation Delay, 8-22
INDEX (Q-S)
Q, 3-23 to 3-26, 4-19, 4-34, 4-87, 4-88, 5-5, 5-13, 6-1, 6-6, 6-14, 6-15
Quartz (material), 3-1, 3-12, 5-1 to 5-18
Quartz Cuts, 3-13, 4-41
Quartz Inversion, 5-17, 5-18
Quartz Lattice, 5-5, 5-7, 5-16
Quartz Phases, 5-18
Quartz Twinning, 5-14 to 5-16
Quotations, time, 8-1, 8-27
Radar, 1-17 to 1-20, 4-18, 4-73, 4-74, 8-22
Radiation Effects, 4-4, 4-79 to 4-89, 4-91, 5-5, 6-25, 7-5
Radio, two-way, 1-6
Random vibration, 4-19, 4-69, 4-71
Random walk,4-5, 4-26, 4-33
RbXO, 2-5, 2-8, 2-22 to 2-23, 3-37, 7-1, 7-2
Relativistic Time, 8-19 to 8-21
Resistance, 3-19, 3-20, 3-23, 4-52, 4-56, 4-84
Resonator Packaging, 3-4, 3-5, 4-90
Resonator Theory, 3-9 to 3-11
Resonators, 3-2 to 3-36
Restart, 4-48
Retrace, 4-47, 4-48
Rubidium Frequency Standard, 2-8, 2-22, 3-37, 6-23 to 6-24, 6-1, 6-3, 6-5, 6-7, 6-8, 6-15, 6-17, 7-1 to 7-5
SAW, 4-35 to 4-37, 9-3
SC-Cut, 2-19 to 2-20, 3-13, 3-35 to 3-37, 4-13, 4-42 to 4-44, 4-46, 4-55, 4-60, 4-72, 4-76, 4-83, 4-88, 6-25
Sensors, 9-4, 9-5
Shock, 4-3, 4-4, 4-78, 6-24
Short Term Instability Causes, 4-19, 6-15
10-7
INDEX (S-T)
Short Term Stability, 4-3 to 4-5, 4-16 to 4-36, 6-14 to 6-21, 7-1 to 7-4
Short-Term Stability Measures, 4-17 to 4-21, 4-25
Shot Noise, 4-18, 6-15
Signal-to-noise ratio, 4-34, 6-15
Silica, 5-18
Space Exploration, 1-10
Specifications, 7-7, 7-8
Spectral Densities, 4-25, 4-26, 4-29 to 4-37
Spread Spectrum, 1-12 to 1-16
Stability, 2-3, 2-4, 4-2 to 4-90, 7-3, 7-4
Strains, 4-10 to 4-14
Surface Acoustic Wave Devices, 4-35 to 4-37, 9-3
Sweeping, 5-9, 5-10
TCXO, 2-5 to 2-8, 3-20, 3-21, 4-39, 4-47, 4-49, 4-51, 7-1, 7-2, 8-10
Telecommunication, 1-7
Thermal Expansion of Quartz, 4-9
Thermal Hysteresis, 4-47, 6-23
Thermal Transient Effect, 4-43, 4-46, 4-88, 4-91
Time & Frequency Subsystem, 8-25, 8-26
Time, 8-1 to 8-28
Time Code, 8-23, 8-24
Time Constant, 3-23 to 3-27
Time Domain - Frequency Domain, 4-17
Time Errors, 8-8 to 8-11
Time Quotations, 8-27
10-8
INDEX (T-Z)
Time Scales, 8-18
Time Transfer, 8-12 to 8-16, 8-20, 8-21
Timing Signal, 8-23, 8-24
Trim Effect, 4-49, 4-51
Tunability, 2-4, 3-20
Tuned circuit, 2-13
Tuning fork, 3-33 to 3-36, 4-40
Twinning, 5-14 to 5-16
Two-way Radio, 1-6
Universal Time, 8-18
UTC, 8-18
Utility Fault Location, 1-9
VCXO, 2-5, 3-19
Vibration, 4-4, 4-18, 4-54 to 4-77
Vibration-Induced Sidebands, 4-65 to 4-68, 4-77, 4-91
Vibration Compensation, 4-76, 4-77
Vibration Isolation, 4-75
Vibration Resonance, 4-71
Very Long Baseline Interferometry (VLBI), 1-10
Warmup, 4-46, 7-1
Wearout Mechanisms, 7-5
Wristwatch, 3-32 to 3-36, 4-40
Zero Temperature Coefficient Cuts, 3-11 to 3-13
10-9
Advanced Phase-Lock Techniques
James A. Crawford
2008
Artech House
510 pages, 480 figures, 1200 equations
CD-ROM with all MATLAB scripts
ISBN-13: 978-1-59693-140-4
ISBN-10: 1-59693-140-X
Chapter
1
2
3
4
5
6
7
8
9
10
Brief Description
Phase-Locked Systems—A High-Level Perspective
An expansive, multi-disciplined view of the PLL, its history, and its wide application.
Design Notes
A compilation of design notes and formulas that are developed in details separately in the
text. Includes an exhaustive list of closed-form results for the classic type-2 PLL, many of
which have not been published before.
Fundamental Limits
A detailed discussion of the many fundamental limits that PLL designers may have to be
attentive to or else never achieve their lofty performance objectives, e.g., Paley-Wiener
Criterion, Poisson Sum, Time-Bandwidth Product.
Noise in PLL-Based Systems
An extensive look at noise, its sources, and its modeling in PLL systems. Includes special
attention to 1/f noise, and the creation of custom noise sources that exhibit specific power
spectral densities.
System Performance
A detailed look at phase noise and clock-jitter, and their effects on system performance.
Attention given to transmitters, receivers, and specific signaling waveforms like OFDM, MQAM, M-PSK. Relationships between EVM and image suppression are presented for the first
time. The effect of phase noise on channel capacity and channel cutoff rate are also
developed.
Fundamental Concepts for Continuous-Time Systems
th
A thorough examination of the classical continuous-time PLL up through 4 -order. The
powerful Haggai constant phase-margin architecture is presented along with the type-3 PLL.
Pseudo-continuous PLL systems (the most common PLL type in use today) are examined
rigorously. Transient response calculation methods, 9 in total, are discussed in detail.
Fundamental Concepts for Sampled-Data Control Systems
A thorough discussion of sampling effects in continuous-time systems is developed in terms
th
of the z-transform, and closed-form results given through 4 -order.
Fractional-N Frequency Synthesizers
A historic look at the fractional-N frequency synthesis method based on the U.S. patent
record is first presented, followed by a thorough treatment of the concept based on ∆-Σ
methods.
Oscillators
An exhaustive look at oscillator fundamentals, configurations, and their use in PLL systems.
Clock and Data Recovery
Bit synchronization and clock recovery are developed in rigorous terms and compared to the
theoretical performance attainable as dictated by the Cramer-Rao bound.
Pages
26
44
38
66
48
71
32
54
62
52
Phase-Locked Systems—A High-Level Perspective
3
are described further for the ideal type-2 PLL in Table 1-1. The feedback divider is normally present
only in frequency synthesis applications, and is therefore shown as an optional element in this figure.
PLLs are most frequently discussed in the context of continuous-time and Laplace transforms. A
clear distinction is made in this text between continuous-time and discrete-time (i.e., sampled) PLLs
because the analysis methods are, rigorously speaking, related but different. A brief introduction to
continuous-time PLLs is provided in this section with more extensive details provided in Chapter 6.
PLL type and PLL order are two technical terms that are frequently used interchangeably even
though they represent distinctly different quantities. PLL type refers to the number of ideal poles (or
integrators) within the linear system. A voltage-controlled oscillator (VCO) is an ideal integrator of
phase, for example. PLL order refers to the order of the characteristic equation polynomial for the
linear system (e.g., denominator portion of (1.4)). The loop-order must always be greater than or equal
to the loop-type. Type-2 third- and fourth-order PLLs are discussed in Chapter 6, as well as a type-3
PLL, for example.
Phase
Detector
Loop
Filter
θout
θ ref
Kd
VCO
1
N
Feedback
Divider
Figure 1-2 Basic PLL structure exhibiting the basic functional ingredients.
Table 1-1
Block Name
Phase Detector
Loop Filter
Basic Constitutive Elements for a Type-2 Second-Order PLL
Laplace Transfer Function
Description
Kd, V/rad
Phase error metric that outputs a voltage that is proportional
to the phase error existing between its input θref and the
feedback phase θout/N. Charge-pump phase detectors output
a current rather than a voltage, in which case Kd has units of
A/rad.
Also called the lead-lag network, it contains one ideal pole
1 + sτ 2
and one finite zero.
sτ 1
VCO
Kv
s
The voltage-controlled oscillator (VCO) is an ideal
integrator of phase. Kv normally has units of rad/s/V.
Feedback Divider
1/N
A digital divider that is represented by a continuous divider
of phase in the continuous-time description.
The type-2 second-order PLL is arguably the workhorse even for modern PLL designs. This PLL
is characterized by (i) its natural frequency ωn (rad/s) and (ii) its damping factor ζ. These terms are
used extensively throughout the text, including the examples used in this chapter. These terms are
separately discussed later in Sections 6.3.1 and 6.3.2. The role of these parameters in shaping the timeand frequency-domain behavior of this PLL is captured in the extensive list of formula provided in
Section 2.1. In the continuous-time-domain, the type-2 second-order PLL3 open-loop gain function is
given by
3
See Section 6.2.
4
Advanced Phase-Lock Techniques
2
 ω  1 + sτ 2
GOL (s ) =  n 
 s  sτ 1
(1.1)
and the key loop parameters are given by
Kd Kv
Nτ 1
(1.2)
1
ζ = ωnτ 2
2
(1.3)
ωn =
The time constants τ1 and τ2 are associated with the loop filter’s R and C values as developed in
Chapter 6. The closed-loop transfer function associated with this PLL is given by the classical result
 2ζ 
ωn2  1 +
s
ωn 
1 θ out (s )

H1 ( s ) =
=
N θ ref (s ) s 2 + 2ζωn s + ωn2
(1.4)
The transfer function between the synthesizer output phase noise and the VCO self-noise is given by
H2(s) where
H 2 ( s ) = 1 − H1 (s )
(1.5)
A convenient frequency-domain description of the open-loop gain function is provided in Figure
1-3. The frequency break-points called out in this figure and the next two appear frequently in PLL
work and are worth committing to memory. The unity-gain radian frequency is denoted by ωu in this
figure and is given by
ωu = ωn 2ζ 2 + 4ζ 4 + 1
(1.6)
A convenient approximation for the unity-gain frequency (1.6) is given by ωu ≅ 2ζωn. This result is
accurate to within 10% for ζ ≥ 0.704.
The H1(s) transfer function determines how phase noise sources appearing at the PLL input are
conveyed to the PLL output and a number of other important quantities. Normally, the input phase
noise spectrum is assumed to be spectrally flat resulting in the output spectrum due to the reference
noise being shaped entirely by |H1(s)|2. A representative plot of |H1|2 is shown in Figure 1-4. The key
frequencies in the figure are the frequency of maximum gain, the zero dB gain frequency, and the –3
dB gain frequency which are given respectively by
FPk =
1 ωn
2π 2ζ
F0 dB =
F3dB =
1 + 8ζ 2 − 1 Hz
1
2π
2ωn Hz
ωn
1
1 + 2ζ 2 + 2 ζ 4 + ζ 2 + Hz
2π
2
(1.7)
(1.8)
(1.9)
Phase-Locked Systems—A High-Level Perspective
5
10log10 ωn4 + 4ωn2ζ 2 
Gain, dB (6 dB/cm)
-12 dB/octave
-6 dB/octave
40log10 (2.38ζ )
0 dB
1
2
3
ωn
2ζ
5
7
10
ωn
2
3
ωu
5
Frequency, rad/sec
Figure 1-3 Open-loop gain approximations for classic continuous-time type-2 PLL.
6
4
H1 Closed-Loop Gain
FPk
F0dB
2
Gain, dB
0
F3dB
GPk
-2
-4
-6
-8
Asymptotic
-6 dB/octave
-10
-12
0
10
1
2
10
Frequency, Hz
10
Figure 1-4 Closed-loop gain H1( f ) for type-2 second-order PLL4 from (1.4).
The amount of gain-peaking that occurs at frequency Fpk is given by

8ζ 4
GPk = 10 log10 
 8ζ 4 − 4ζ 2 − 1 + 1 + 8ζ 2


 dB


(1.10)
For situations where the close-in phase noise spectrum is dominated by reference-related phase noise,
the amount of gain-peaking can be directly used to infer the loop’s damping factor from (1.10), and the
4
Book CD:\Ch1\u14033_figequs.m, ζ = 0.707, ωn = 2π 10 Hz.
6
Advanced Phase-Lock Techniques
loop’s natural frequency from (1.7). Normally, the close-in (i.e., radian offset frequencies less than
ωn /2ζ) phase noise performance of a frequency synthesizer is entirely dominated by reference-related
phase noise since the VCO phase noise generally increases 6 dB/octave with decreasing offset
frequency5 whereas the open-loop gain function exhibits a 12 dB/octave increase in this same
frequency range.
VCO-related phase noise is attenuated by the H2(s) transfer function (1.5) at the PLL’s output for
offset frequencies less than approximately ωn. At larger offset frequencies, H2(s) is insufficient to
suppress VCO-related phase noise at the PLL’s output. Consequently, the PLL’s output phase noise
spectrum is normally dominated by the VCO self-noise phase noise spectrum for the larger frequency
offsets. The key frequency offsets and relevant H2(s) gains are shown in Figure 1-5 and given in Table
1-2.
Closed-Loop Gain H2
5
GH2max
GFn
0
-3 dB
H2 Gain, dB
-5
FH2max
Fn
FH2-3dB
-10
ζ =0.4
-15
Fn =10 Hz
-20
-25
0
10
1
2
10
Frequency, Hz
10
Figure 1-5 Closed-loop gain6 H2 and key frequencies for the classic continuous-time type-2 PLL.
Table 1-2
Key Frequencies Associated with H2(s) for the Ideal Type-2 PLL
Frequency, Hz
Associated H2 Gain, dB
GH 2 _ 1rad / s = −10 log10 ωn4 + ωn2 (4ζ 2 − 2 ) + 1
1/2π
FH 2 − 3 dB =
2

ωn  2ζ − 1 +


2π  2 − 4ζ 2 + 4ζ 4 
FH 2 _ 0 dB =
1
2π
Fn = ωn / 2π
FH 2 _ max =
5
6
1
2π
ωn
1 − 2ζ 2
—
1/ 2
–3
ωn
2 − 4ζ 2
Constraints
on ]
0
GH 2 _ ωn = −10 log10 (4ζ 2 )
GH 2 _ max = −10 log10 (4ζ 2 − 4ζ 4 )
Leeson’ s model in Section 9.5.1; Haggai oscillator model in Section 9.5.2.
Book CD:\Ch1\u14035_h2.m.
—
ζ <
2
2
—
ζ <
2
2
Phase-Locked Systems—A High-Level Perspective
19
Assuming that the noise samples have equal variances and are uncorrelated, R = σn2I where I is the
K×K identity matrix. In order to maximize (1.43) with respect to θ, a necessary condition is that the
derivative of (1.43) with respect to θ be zero, or equivalently
∂L
∂
=
∂θ ∂θ
∑  r
k
k
− A cos (ωo tk + θ ) = 0
2
= ∑ 2  rk − A cos (ωo tk + θ ) A sin (ωo tk + θ ) = 0
(1.44)
k
Simplifying this result further and discarding the double-frequency terms that appear, the maximumlikelihood estimate for θ is that value that satisfies the constraint
∑ r sin (ω t
)
k
+ θˆ = 0
k
k
o k
(1.45)
The top line indicates that double-frequency terms are to be filtered out and discarded. This result is
equivalent to the minimum-variance estimator just derived in (1.40).
Under the assumed linear Gaussian conditions, the minimum-variance (MV) and maximumlikelihood (ML) estimators take the same form when implemented with a PLL. Both algorithms seek to
reduce any quadrature error between the estimate and the observation data to zero.
1.4.3 PLL as a Maximum A Posteriori (MAP)-Based Estimator
The MAP estimator is used for the estimation of random parameters whereas the maximum-likelihood
(ML) form is generally associated with the estimation of deterministic parameters. From Bayes rule for
an observation z, the a posteriori probability density is given by
p (θ z ) =
p ( z θ ) p (θ )
(1.46)
p (z )
and this can be re-written in the logarithmic form as
log e  p (θ z ) = log e  p ( z θ ) + log e  p (θ ) − log e  p ( z )
(1.47)
This log-probability may be maximized by setting the derivative with respect to θ to zero thereby
creating the necessary condition that27
{
}
d
log e  p ( z θ ) + log e  p (θ )
dθ
θ =θˆMAP
=0
(1.48)
If the density p(θ ) is not known, the second term in (1.48) is normally discarded (set to zero) which
degenerates naturally to the maximum-likelihood form as
{
}
d
log e  p ( z θ )
dθ
27
[15] Section 6.2.1, [17] Section 2.4.1, [18] Section 5.4, and [22].
θ =θˆML
=0
(1.49)
30
Advanced Phase-Lock Techniques
Time of Peak Phase-Error with Frequency-Step Applied
 1− ζ 2 
1

T fstep =
tan −1 
 ζ

ωn 1 − ζ 2


Note.1 See Figure 2-19 and Figure 2-20.
Time of Peak Phase-Error with Phase-Step Applied
Tθ step =
1
(
(2.29)
)
tan −1 2ζ 1 − ζ 2 , 2ζ 2 − 1 =
 1−ζ 2
tan −1 
 ζ
1−ζ 2

2
ωn 1 − ζ
ωn
See Figure 2-19 and Figure 2-20.
Time of Peak Frequency-Error with Phase-Step Applied
1

θu
ζ ≤ 2 :
1
Tpk =

ωn 1 − ζ 2 ζ > 1 : θ + π
u

2
with θu = tan −1 (1 − 4ζ 2 ) 1 − ζ 2 ,3ζ − 4ζ 3 


See Figure 2-21 and Figure 2-22.
Tpk corresponds to the first point in time where dfo/dt = 0.
Maximum Frequency-Error with Phase-Step Applied
Use (2.31) in (2.28).
Time of Peak Frequency-Error with Frequency-Step Applied
 1− ζ 2 
2

tan −1 
Tpk =
 ζ

ωn 1 − ζ 2


% Transient Frequency Overshoot for Frequency-Step Applied

 −ζω T
ζ
OS% = cos 1 − ζ 2 ωnTpk −
sin 1 − ζ 2 ωnTpk  e n pk × 100%
2


1−ζ
 1− ζ 2 
2

tan −1 
Tpk =
 ζ

ωn 1 − ζ 2


2
Note. See Figure 2-23 and Figure 2-24.
Linear Hold-In Range with Frequency-Step Applied (Without Cycle-Slip)
 ζ
 1 − ζ 2 
  Hz
∆Fmax = ωn exp 
tan −1 
 ζ

 1 − ζ 2


See Figure 2-25.
Linear Settling Time with Frequency-Step Applied (Without Cycle-Slip) (Approx.)
 ∆F
1
1 
 sec
TLock ≤
log e 
 δ F 1−ζ 2 
ζωn


for applied frequency-step of ∆F and residual δ F remaining at lock
See Figure 2-26.
2
(
1
)
(
)




(2.30)
(2.31)
(2.32)
(2.33)
(2.34)
(2.35)
(2.36)
(2.37)
(2.38)
The peak occurrence time is precisely one-half that given by (2.34).
See Figure 2-24 for time of occurrence Tpk for peak overshoot/undershoot with ωn = 2π. Amount of overshoot/undershoot in
percent provided in Figure 2-23.
2
44
Advanced Phase-Lock Techniques
2.3.2.2 Second-Order Gear Result for H1(z) for Ideal Type-2 PLL
v
θin
+
θe
Σ
2Ts ω 2
3 n
4ζ
ωnTs
D
D
+
_
+
Σ
Σ
+
+
Σ
+
_
+
1
3
Figure 2-32 Second-order Gear redesign of H1(s) (2.4).
 2ω T 
GOL ( z ) =  n s 
 3 
θ o (k ) =
a0 = 1 +
3ζ
ωnTs
4ζ
ωnTs
ζ
a2 =
ωnTs
a1 = −
1+
1
3
3ζ  4 −1 1 −2 
1− z + z 
ωnTs  3
3

2
 4 −1 1 −2 
1 − z + z 
3
 3

(2.52)
2
4
1 2

a
θ
k
−
n
+
b
v
k
−
n
+
cnθ o (k − n )
(
)
(
)
∑
∑
∑
n
in
n

D  n =0

n=0
n =1
b0 =
(2.54)
2
3
2ωn2Ts
2
ωn2Ts
1
b2 =
2ωn2Ts
b1 = −
c1 =
(2.55)
6
(ωnTs )
2
c2 = −
c3 =
(2.53)
4ζ
ωnTs
+
11
2 (ωnTs )
2
2
−
ζ
ωnTs
(2.56)
(ωnTs )
c4 = −
 3 
3ζ
+
D = 1+

ωnTs  2ωnTs 
D
4
3
D
4
3
1+ 3ζ
ωnTs
θo
D
ζ
ωnTs
Σ
D
-
2Ts
3
_
2
1
(2ωnTs )
2
2
(2.57)
2.3.3 Higher-Order Differentiation Formulas
In cases where a precision first-order time-derivative f (xn+1) must be computed from an equally
spaced sample sequence, higher-order formulas may be helpful.8 Several of these are provided here
in Table 2-2. The uniform time between samples is represented by Ts.
8
Precisions compared in Book CD:\Ch2\u14028_diff_forms.m.
52
Advanced Phase-Lock Techniques
2.5.5 64-QAM Symbol Error Rate
64-QAM Symbol Error Rate
-2
10
o
σφ = 2 rms
o
σφ = 1.5 rms
-3
10
o
σφ = 1 rms
-4
SER
10
-5
10
-6
10
o
σφ = 0.5 rms
Proakis
-7
10
No Phase Noise
-8
10
15
16
17
18
19
20
Eb/No, dB
21
22
23
24
25
Figure 2-37 64-QAM uncoded symbol error rate with noisy local oscillator.13 Circled datapoints are from (2.87).
13
Book CD:\Ch5\u13159_qam_ser.m. See Section 5.5.3 for additional information. Circled datapoints are based on Proakis
[3] page 282, equation (4.2.144), included in this text as (2.87).
Fundamental Limits
83
A more detailed discussion of the Chernoff bound and its applications is available in [9].
Key Point: The Chernoff bound can be used to provide a tight upper-bound for the tail-probability
of a one-sided probability density. It is a much tighter bound than the Chebyschev inequality given
in Section 3.5. The bound given by (3.43) for the complementary error function can be helpful in
bounding other performance measures.
3.7 CRAMER-RAO BOUND
The Cramer-Rao bound16 (CRB) was first introduced in Section 1.4.4, and frequently appears in
phase- and frequency-related estimation work when low SNR conditions prevail. Systems that
asymptotically achieve the CRB are called efficient in estimation theory terminology. In this text,
the CRB is used to quantify system performance limits pertaining to important quantities such as
phase and frequency estimation, signal amplitude estimation, bit error rate, etc.
The CRB is used in Chapter 10 to assess the performance of several synchronization
algorithms with respect to theory. Owing to the much larger signal SNRs involved with frequency
synthesis, however, the CRB is rarely used in PLL-related synthesis work. The CRB is developed in
considerable detail in the sections that follow because of its general importance, and its widespread
applicability to the analysis of many communication system problems.
The CR bound provides a lower limit for the error covariance of any unbiased estimator of a
deterministic parameter θ based on the probability density function of the data observations. The
data observations are represented here by zk for k = 1, . . ., N, and the probability density of the
observations is represented by p(z1, z2, . . ., zN) = p(z). When θ represents a single parameter and θ hat represents the estimate of the parameter based on the observed data z, the CRB is given by three
equivalent forms as
(
)
(
)
2
var  θˆ − θ  = E  θˆ − θ 





2
   ∂
 
≥ E   log e p (z | θ ) 
 
   ∂θ
2
 
  ∂
≥ − E  2 log e p (z | θ ) 
  ∂θ
 
2
+∞
  ∂

 1
≥∫ 
p (z )
dz 
 p (z ) 
 −∞  ∂θ
−1
−1
(3.46)
−1
The first form of the CR bound in (3.46) can be derived as follows. Since θ -hat is an unbiased
(zero-mean) estimator of the deterministic parameter θ, it must be true that
+∞
( ) ∫ θˆ− θ  p (z )dz = 0
E θ =
−∞
in which dz = dz1dz2 . . . dz N . Differentiating (3.47) with respect to θ produces the equality
16
See [10]–[14].
(3.47)
86
Advanced Phase-Lock Techniques
σ
var bˆo ≥
M
{}
2
for all cases
(3.62)

σ2
Phase known, amplitude known or unknown

b02 Q

var {ωˆ oTs } ≥ 
12σ 2

Phase unknown, amplitude known or unknown
 b02 M (M 2 − 1)

(3.63)

σ2
Frequency known, amplitude known or unknown

b02 M

var θˆo ≥ 
12σ 2 Q

Frequency unknown, amplitude known or unknown
 b02 M 2 (M 2 − 1)

(3.64)
{ }
In the formulation presented by (3.55), the signal-to-noise ratio ρ is given by ρ = b02 / (2σ 2).
For the present example, the CR bound is given by the top equation in (3.63) and is as shown
in Figure 3-9 when the initial signal phase θo is known a priori. Usually, the carrier phase θo is not
known a priori when estimating the signal frequency, however, and the additional unknown
parameter causes the estimation error variance to be increased, making the variance asymptotically
4-times larger than when the phase is known a priori. This CR variance bound for this more typical
unknown signal phase situation is shown in Figure 3-10.
Beginning with (3.57), a maximum-likelihood17 frequency estimator can be formulated as
described in Appendix 3A. It is insightful to compare this estimator’s performance with its
respective CR bound. For simplicity, the initial phase θo is assumed to be random but known a
priori. The results for M = 80 are shown in Figure 3-11 where the onset of thresholding is apparent
for ρ ≅ –2 dB. Similar results are shown in Figure 3-12 for M = 160 where the threshold onset has
been improved to about ρ ≅ –5 dB.
CR Bound for Frequency Estimation Error Variance
0
10
ρ = -20 dB
-1
Estimator Variance, var ( ωoT )
10
ρ = -10 dB
-2
10
ρ = 0 dB
-3
10
ρ = 10 dB
-4
10
ρ = 20 dB
-5
10
-6
10
-7
10
0
10
1
2
10
10
3
10
Number of Samples
Figure 3-9 CR bound18 for frequency estimation error with phase θo known a priori (3.63).
17
18
See Section 1.4.2.
Book CD:\Ch3\u13000_crb.m. Amplitude known or unknown, frequency unknown, initial phase known.
Noise in PLL-Based Systems
135
would be measured and displayed on a spectrum analyzer. Having recognized the carrier and
continuous spectrum portions within (4.65), it is possible to equate29
/ ( f ) ≅ Pθ ( f ) rad 2 /Hz
(4.66)
Power
/ ( fx )
Carrier
1
Pθ ( f −ν o )
2
νo
−ν o
Frequency
fx
Figure 4-17 Resultant two-sided power spectral density from (4.65), and the single-sideband-to-carrier ratio
( f ).
Both /( f ) and Pθ ( f ) are two-sided power spectral densities, being defined for positive as well as
negative frequencies.
The use of one-sided versus two-sided power spectral densities is a frequent point of confusion
in the literature. Some PSDs are formally defined only as a one-sided density. Two-sided power
spectral densities are used throughout this text (aside from the formal definitions for some quantities
given in Section 4.6.1) because they naturally occur when the Wiener-Khintchine relationship is
utilized.
4.6.1 Phase Noise Spectrum Terminology
A minimum amount of standardized terminology has been used thus far in this chapter to
characterize phase noise quantities. In this section, several of the more important formal definitions
that apply to phase noise are provided.
A number of papers have been published which discuss phase noise characterization
fundamentals [34]–[40]. The updated recommendations of the IEEE are provided in [41] and those
of the CCIR in [42]. A collection of excellent papers is also available in [43].
In the discussion that follows, the nominal carrier frequency is denoted by νo (Hz) and the
frequency-offset from the carrier is denoted by f (Hz) which is sometimes also referred to as the
Fourier frequency.
One of the most prevalent phase noise spectrum measures used within industry is /( f ) which
was encountered in the previous section. This important quantity is defined as [44]:
/( f ): The normalized frequency-domain representation of phase fluctuations. It is the ratio
of the power spectral density in one phase modulation sideband, referred to the carrier
frequency on a spectral density basis, to the total signal power, at a frequency offset f. The
units30 for this quantity are Hz–1. The frequency range for f ranges from –νo to ∞. /( f ) is
therefore a two-sided spectral density and is also called single-sideband phase noise.
29
30
It implicitly assumed that the units for
Also as rad2/Hz.
( f ), dBc/Hz or rad2/Hz, can be inferred from context.
Noise in PLL-Based Systems
163
α

zi = pi exp  ∆p 
2

(4B.10)
A minimum of one filter section per frequency decade is recommended for reasonable accuracy. A
sample result using this method across four frequency decades using 3 and 5 filter sections is shown
in Figure 4B-3.
f -α Power Spectral Density
45
40
α=1
Relative Spectrum Level, dB
35
30
5 Filter Sections
25
20
15
10
3 Filter Sections
5
0
0
10
1
10
2
3
10
10
Radian Frequency
4
5
10
10
Figure 4B-3 1/f noise creation using recursive 1/f 2 filtering method4 with white Gaussian noise.
1/f Noise Generation Using Fractional-Differencing Methods
Hosking [6] was the first to propose the fractional differencing method for generating 1/f α noise. As
pointed out in [3], this approach resolves many of the problems associated with other generation
methods. In the continuous-time-domain, the generation of 1/f α noise processes involves the
application of a nonrealizable filter to a white Gaussian noise source having s–α/2 for its transfer
function. Since the z-transform equivalent of 1/s is H(z) = (1 – z–1)–1, the fractional digital filter of
interest here is given by
1
Hα ( z ) =
(4B.11)
α /2
(1 − z −1 )
A straightforward power series expansion of the denominator can be used to express the filter as an
infinite IIR filter response that uses only integer-powers of z as

α α

1 − 
 α

2
2  −2
H α ( z ) ≈ 1 − z −1 − 
z − . . .
2!
 2





−1
in which the general recursion formula for the polynomial coefficients is given by
4
Book CD:\Ch4\u13070_recursive_flicker_noise.m.
(4B.12)
System Performance
185
Strong
Interferer
Local Oscillator
Spectrum
∆LdB
Sideband
Noise
Desired
Channel
∆FSep
∆FSep
Figure 5-9 Strong interfering channels are heterodyned on top of the desired receive channel by local oscillator sideband
noise.
Relative PSDs at Baseband
10
5
Desired Signal
0
PSD, dB
-5
-10
-15
+35 dB
-20
+40 dB
-25
-30
-35
+45 dB
-2
0
2
4
6
Baseband Frequency, MHz
8
10
Figure 5-10 Baseband spectra10 caused by reciprocal mixing between a strong interferer that is offset 4B Hz higher in
frequency than the desired signal and stronger than the desired signal by the dB amounts shown.
The first term in (5.28) 2BLFloor is attributable to the ultimate blocking performance of the receiver
as discussed in Section 5.3. The resultant output SNR versus input SNR is given by
SNRout
 1
σ2 
=
+ MFX 
 SNRin 2 BLIQ 
−1
(5.29)
It is worthwhile to note that the interfering spectra in Figure 5-10 are not uniform across the
matched-filter frequency region [–B, B]. Multicarrier modulation like OFDM (see Section 5.6) will
potentially be affected differently than single-carrier modulation such as QAM (see Section 5.5.3)
when the interference spectrum is not uniform with respect to frequency.
The result given by (5.29) is shown for several interfering levels versus receiver input SNR in
Figure 5-11.
10
Book CD:\Ch5\u13157_rx_desense.m. Lorentzian spectrum parameters: Lo = –90 dBc/Hz, fc = 75 kHz, LFloor = –160
dBc/Hz, B = 3.84/2 MHz.
System Performance
221
of 3° rms phase noise is shown in Figure 5B-8. The tail probability is worse than the exact
computations shown in Figure 5-17 but the two results otherwise match very well.
Channel Cutoff Rate Ro
4
0o
3.8
3.6
Bits per Symbol
3.4
3.2
3
2.8
4o
2.6
10o
6o
8o
2.4
2.2
5
7
9
11
13
Eb/No, dB
15
17
19
20
Figure 5B-6 Channel cutoff rate,7 Ro, for 16-QAM with static phase errors as shown, from (5B.16).
Channel Cutoff Rate Ro Including Phase Noise
4
3.75
3.5
Bits per Symbol
3.25
3
2.75
2.5
2.25
2
1.75
1.5
1
3
5
7
9
11
13
Eb/No, dB
15
17
19
21
23
Figure 5B-7 Ro for8 16-QAM versus Eb/No for 5° rms phase noise from (5B.18) (to accentuate loss in Ro even at high SNR
values).
7
8
Book CD:\Ch5\u13176_rolo.m.
Ibid.
Fractional-N Frequency Synthesizers
371
required, however, because the offset current will introduce its own shot-current noise contribution,
and the increased duty-cycle of the charge-pump activity will also introduce additional noise and
potentially higher reference spurs. Single-bit ∆-Σ modulators are attractive in this respect because
they lead to the minimum-width phase-error distribution possible.
Ideal
I CP
θ PD
Figure 8-70 Charge-pump (i) dead-zone and (ii) unequal positive versus negative error gain.
-40
MASH 2-2 Output Spectrum with Nonlinear PD
-50
-60
Phase Noise, dBc/Hz
-70
-80
-90
-100
-110
-120
-130
-140
3
10
4
10
5
10
Frequency, Hz
6
10
7
10
Figure 8-71 Phase error power spectral density48 for the MASH 2-2 ∆-Σ modulator shown in Figure 8-55 with M = 222, P =
M/2 + 3,201, and 2% charge-pump gain imbalance. Increased noise floor and discrete spurs are clearly apparent compared to
Figure 8-56.
Classical random processes theory can be used to provide several useful insights about
nonlinear phase detector operation. In the case of unequal positive-error versus negative-error phase
detector gain, the memoryless nonlinearity can be modeled as
θ pd = φin + α (φin > 0 )φin
(8.39)
where α represents the additional gain that is present for positive phase errors. The instantaneous
phase error due to the modulator’s internal quantization creates a random phase error sequence that
can be represented by
48
Book CD:\Ch8\u12735_MASH2_2_nonlinear.m.
Clock and Data Recovery
457
the sampling-point within each symbol-period after the datalink signal has been fully acquired. In
the example results that follow, the data source is assumed to be operating at 1 bit-per-second,
utilizing square-root raised-cosine pulse-shaping with an excess bandwidth parameter β = 0.50 at
the transmitter. The eye-diagram of the signal at the transmit end is shown in Figure 10-15. The
ideal matched-filter function in the CDR is closely approximated by an N = 3 Butterworth lowpass
filter having a –3 dB corner frequency of 0.50 Hz like the filter used in Section 10.4. The resulting
eye-diagram at the matched-filter output is shown in Figure 10-16 for Eb/No = 25 dB.
Sample Gate
alk
Matched
Filter
y (t )
y f (t )
 2

tanh 
y f (t )
 No

A (t )
H MF ( f )
VCO
v (t )
Loop
Filter
Sample Gate
B (t )
( j 2π f ) H MF ( f )
Figure 10-14 ML-CDR implemented with continuous-time filters based on the timing-error metric given by (10.21).
Square-Root Raised-Cosine Eye-Diagram
1.5
1
Normalized
0.5
0
-0.5
-1
-1.5
-1
-0.5
0
Time, Symbols
0.5
1
Figure 10-15 Eye diagram15 at the data source output assuming square-root raised-cosine pulse shaping with an excess
bandwidth parameter β = 0.50.
A clear understanding of the error metric represented by v(t) in Figure 10-14 is vital for
understanding how the CDR operates. The metric is best described by its S-curve behavior versus
input Eb/No as shown in Figure 10-17. Each curve is created by setting the noise power spectral
density No for a specified Eb/No value with Eb = 1, and computing the average of v( kTsym+ ε ) for k =
[0, K] as the timing-error ε is swept across [0, Tsym]. The slope of each S-curve near the zero-error
steady-state tracking value determines the linear gain of the metric that is needed to compute the
closed-loop bandwidth, loop stability margin, and other important quantities. For a given input SNR,
15
Book CD:\Ch10\u14004_ml_cdr.m.
458
Advanced Phase-Lock Techniques
the corresponding S-curve has only one timing-error value εo for which the error metric value is zero
and the S-curve slope has the correct polarity. As the gain value changes with input Eb/No, the
closed-loop parameters will also vary. For large gain variations, the Haggai loop concept explored
in Section 6.7 may prove advantageous.
Eye Diagram at Matched-Filter Output
1.5
1
Amplitude
0.5
0
-0.5
-1
-1.5
-1
-0.5
0
Time
0.5
1
16
Figure 10-16 Eye diagram at the CDR matched-filter output for Eb/No = 25 dB corresponding to the data source shown in
Figure 10-15 and using an N = 3 Butterworth lowpass filter with BT = 0.50 for the approximate matched-filter.
S-Curves
0.4
0.3
0.2
V
0.1
6 dB
9 dB
0
3 dB
15 dB
0 dB
-0.1
-0.2
-0.3
-0.4
Eb/No= 25 dB
0
0.1
0.2
0.3
0.4
0.5
0.6
Timing, symbols
0.7
0.8
0.9
1
Figure 10-17 S-curves17 versus Eb/No corresponding to Figure 10-16 and ideal ML-CDR shown in Figure 10-14. Eb = 1 is
assumed constant.
A second important characteristic of the timing-error metric is its variance versus input Eb/No
and static timing-error ε. For this present example, this information is shown in Figure 10-18. The
variance understandably decreases as the input SNR is increased, and as the optimum timealignment within each data symbol is approached. The variance of the recovered data clock σclk2 can
be closely estimated in terms of the tracking-point voltage-error variance from Figure 10-18 denoted
by σve2 (V2), the slope (i.e., gain) of the corresponding S-curve (Kte, V/UI) from Figure 10-17, the
symbol rate Fsym (= 1/Tsym), and the one-sided closed-loop PLL bandwidth BL (Hz) as
16
17
Ibid.
Ibid.