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Rev. 8.5.3.6Quartz Crystal Resonators and Oscillators For Frequency Control and Timing Applications - A Tutorial          ...
NOTICES                       DisclaimerThe citation of trade names and names of manufacturersin this report is not to be ...
Table of ContentsPreface………………………………..………………………..                           v1.    Applications and Requirements………………………....
Preface                          Why This Tutorial?  “Everything should be made as simple as                   I was frequ...
Notes and References   Notes and references can be found in the “Notes” ofmost of the pages. To view the notes, use the “N...
In all pointed sentences [and tutorials],some degree of accuracy must besacrificed to conciseness.          Samuel Johnson
CHAPTER 1Applications and Requirements              1
Electronics Applications of Quartz CrystalsMilitary & Aerospace            Industrial                 ConsumerCommunicatio...
Frequency Control Device Market                       (estimates, as of ~2006)       Technology                Units      ...
Navigation      Precise time is essential to precise navigation. Historically, navigation has beena principal motivator in...
Commercial Two-way Radio   Historically, as the number of users of commercial two-way radioshave grown, channel spacings h...
Digital Processing of Analog Signals                            The Effect of Timing Jitter(A) Analog*              A/D   ...
Digital Network Synchronization•   Synchronization plays a critical role in digital telecommunication systems.It ensures t...
Phase Noise in PLL and PSK Systems        The phase noise of oscillators can lead to erroneous detection ofphase transitio...
Utility Fault Location                                             Zap!      ta                                           ...
Space ExplorationSchematic of VLBI                             wtTechnique                                                ...
Military Requirements   Military needs are a prime driver of frequency controltechnology. Modern military systems requireo...
Impacts of Oscillator Technology Improvements  •   Higher jamming resistance & improved ability to hide signals  •   Impro...
Spread Spectrum Systems•   In a spread spectrum system, the transmitted signal is spread over a bandwidth that is    much ...
Clock for Very Fast Frequency Hopping Radio                 Jammer                           Example                    J ...
Clocks and Frequency Hopping C3 SystemsSlow hopping ‹-------------------------------›Good clockFast hopping ‹-------------...
Identification-Friend-Or-Foe (IFF)                  Air Defense IFF ApplicationsAWACS                                     ...
Effect of Noise in Doppler Radar System                                                     A                             ...
Bistatic Radar    Conventional (i.e., "monostatic") radar, in which the                               Illuminatorilluminat...
Radar Frequency (GHz)                                                                   0                                 ...
CHAPTER 2Quartz Crystal Oscillators            2
Crystal Oscillator             Tuning             VoltageCrystalresonator                             Output              ...
Oscillation•   At the frequency of oscillation, the closed loop phase shift    = 2nh .•   When initially energized, the on...
Oscillation and Stability•    If a phase perturbation      occurs, the frequency must shift f to maintain the    2n phase ...
Tunability and Stability      Making an oscillator tunable over a wide frequency range degrades itsstability because makin...
Oscillator Acronyms Most Commonly Used:•   XO…………..Crystal Oscillator•   VCXO………Voltage Controlled Crystal Oscillator•   O...
Crystal Oscillator CategoriesThe three categories, based on the method of dealing with the crystal unitsfrequency vs. temp...
Crystal Oscillator Categories                                                      ∆f         Voltage                     ...
Hierarchy of Oscillators        Oscillator Type*             Accuracy**        Typical Applications • Crystal oscillator (...
Oscillator Circuit TypesOf the numerous oscillator circuit types, three of the more common ones, the Pierce, the Colpitts ...
OCXO Block Diagram                                                                  Output             OvenEach of the thr...
Oscillator Instabilities - General Expression                                                                    −1/2     ...
Instabilities due to Sustaining Circuit• Load reactance change - adding a load capacitance to a crystalchanges the frequen...
Oscillator Instabilities - Tuned CircuitsMany oscillators contain tuned circuits - to suppress unwantedmodes, as matching ...
Oscillator Instabilities - Circuit Noise  Flicker PM noise in the sustaining circuit causes flicker FMcontribution to the ...
Oscillator Instabilities - External LoadIf the external load changes, there is a change in the amplitudeor phase of the si...
Oscillator Outputs        Most users require a sine wave, a TTL-compatible, a CMOS-compatible, or an ECL-compatible output...
Silicon Resonator & Oscillator          www.SiTime.comResonator (Si): 0.2 x 0.2 x 0.01 mm3   5 MHz; f vs. T: -30 ppm/oCOsc...
Resonator Self-Temperature Sensing     ff (Hz)                                             ff f 3f1 - f3    172300        ...
Thermometric Beat Frequency GenerationDUAL MODEOSCILLATOR   f1                     X3   M=1            MULTIPLIER         ...
Microcomputer Compensated Crystal Oscillator                 (MCXO)       f1    Dual-    mode              x3     XO      ...
MCXO Frequency Summing Method                                  Block Diagram                                              ...
MCXO - Pulse Deletion Method                                Digital                                circuitry (ASIC)       ...
MCXO - TCXO Resonator Comparison        Parameter                         MCXO            TCXO Cut, overtone              ...
Opto-Electronic Oscillator (OEO)                Bias"Pump Laser"                                      Optical out         ...
CHAPTER 3Quartz Crystal Resonators            3
Why Quartz?  Quartz is the only material known that possesses the followingcombination of properties:• Piezoelectric ("pre...
The Piezoelectric Effect                                      Y                                                           ...
The Piezoelectric Effect in Quartz                                                                 Z                      ...
Modes of Motion          (Click on the mode names to see animation.) Flexure Mode      Extensional Mode Face Shear ModeThi...
Motion Of A Thickness Shear Crystal                     CLICK ON FIGURE                     TO START MOTION
Resonator Vibration Amplitude Distribution                       Metallic                      electrodes                 ...
Resonant Vibrations of a Quartz Plate                  0 db.                                                              ...
Overtone Response of a Quartz Crystal               jX                                Spurious   Reactance                ...
Unwanted Modes vs. Temperature     (3 MHz rectangular AT-cut resonator, 22 X 27 X 0.552 mm)Activity dips occur where the f...
Mathematical Description of a Quartz Resonator•   In piezoelectric materials, electrical current and voltage are coupled t...
Mathematical Description - Continued   •   Number of independent non-zero constants depend on crystal symmetry. For quartz...
Infinite Plate Thickness Shear Resonator                          n c ij                     fn =        , n = 1, 3, 5... ...
Quartz is Highly Anisotropic The properties of quartz vary greatly with crystallographic direction.  For example, when a ...
Zero Temperature Coefficient Quartz Cuts                                           90o                                    ...
Comparison of SC and AT-cuts• Advantages of the SC-cut    • Thermal transient compensated (allows faster warmup OCXO)    •...
Mode Spectrograph of an SC-cut                               1.10               0                         1.0             ...
SC- cut f vs. T for b-mode and c-mode                                 400                                 200   FREQUENCY ...
B and C Modes Of A Thickness Shear Crystal     C MODE                        B MODE                CLICK ON FIGURES       ...
Singly Rotated and Doubly Rotated Cuts’       Vibrational Displacements                        Z                          ...
Resistance vs. Electrode Thickness            AT-cut; f1=12 MHz; polished surfaces; evaporated 1.2 cm (0.490”) diameter si...
Resonator Packaging    Two-point Mount Package            Three- and Four-point Mount Package  Quartz              Electro...
Equivalent CircuitsSpring                 C Mass            LDashpot          R          3-21
Equivalent Circuit of a ResonatorSymbol for crystal unit                   CL                          C0                 ...
Crystal Oscillator f vs. T Compensation                        Uncompensated  Frequency / Voltage     frequency           ...
Resonator Reactance vs. Frequency                          Area of usual                   +      operation in an         ...
Equivalent Circuit Parameter Relationships                                                                 n:  Overtone nu...
What is Q and Why is it Important?                    Energy stored during a cycle             Q≡2π                    Ene...
Decay Time, Linewidth, and Q                                                            Decaying oscillation              ...
Factors that Determine Resonator QThe maximum Q of a resonator can be expressed as:                                     1 ...
Resonator Fabrication Steps  DESIGN       GROW                          SWEEP               CUT       LAP     ROUNDRESONAT...
X-ray Orientation of Crystal Plates                     Shielding                                    Monochromator        ...
Contamination Control    Contamination control is essential during the fabrication ofresonators because contamination can ...
Crystal Enclosure Contamination  The enclosure and sealing process can have importantinfluences on resonator stability.• A...
What is an “f-squared”?   It is standard practice to express the thickness removed by lapping, etching and polishing,and t...
Milestones in Quartz Technology1880    Piezoelectric effect discovered by Jacques and Pierre Curie1905    First hydrotherm...
Quartz Resonators for WristwatchesRequirements: •   Small size •   Low power dissipation (including the oscillator) •   Lo...
Why 32,768 Hz?                32,768 = 215                      32,768                                                  16...
Quartz Tuning Fork                              Z                                      Y                    Xa) natural fa...
Watch Crystal     3-38
Lateral Field Resonator                   Lateral Field               Thickness Field     In lateral field resonators (LFR...
Electrodeless (BVA) Resonator                                                         C                    D2             ...
CHAPTER 4Oscillator Stability         4
The Units of Stability in Perspective• What is one part in 10 10                              ? (As in 1 x 10-10/day aging...
Accuracy, Precision, and Stability    Precise but           Not accurate and           Accurate but         Accurate and  ...
Influences on Oscillator Frequency Time   • Short term (noise)   • Intermediate term (e.g., due to oven fluctuations)   •...
Idealized Frequency-Time-Influence Behavior ∆f    X 108                                                   Oscillator f    ...
Aging and Short-Term Stability                                       Short-term instability                               ...
Aging Mechanisms Mass transfer due to contamination  Since f n 1/t, u f/f = -n t/t; e.g., f Fund M 106 molecular layers, ...
Typical Aging Behaviors                   A(t) = 5 ln(0.5t+1)                                         Time4 f/f           ...
Stresses on a Quartz Resonator PlateCauses:    • Thermal expansion coefficient differences    • Bonding materials changing...
Thermal Expansion Coefficients of Quartz                                 14/K                                             ...
Force-Frequency Coefficient      30              * 10-15 m 0 s / N                                        AT-cut quartz   ...
Strains Due To Mounting ClipsX-ray topograph of an AT-cut, two-point mounted resonator. The topographshows the lattice def...
Strains Due To Bonding Cements                (a)                                            (b)X-ray topographs showing l...
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Vig tutorial jan-2007

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Vig tutorial jan-2007

  1. 1. Rev. 8.5.3.6Quartz Crystal Resonators and Oscillators For Frequency Control and Timing Applications - A Tutorial January 2007 John R. Vig Consultant. Most of this Tutorial was prepared while the author was employed by the US Army Communications-Electronics Research, Development & Engineering Center Fort Monmouth, NJ, USA J.Vig@IEEE.org Approved for public release. Distribution is unlimited
  2. 2. NOTICES DisclaimerThe citation of trade names and names of manufacturersin this report is not to be construed as official Governmentendorsement or consent or approval of commercialproducts or services referenced herein.
  3. 3. Table of ContentsPreface………………………………..……………………….. v1. Applications and Requirements………………………. 12. Quartz Crystal Oscillators………………………………. 23. Quartz Crystal Resonators……………………………… 34. Oscillator Stability………………………………………… 45. Quartz Material Properties……………………………... 56. Atomic Frequency Standards…………………………… 67. Oscillator Comparison and Specification…………….. 78. Time and Timekeeping…………………………………. 89. Related Devices and Applications……………………… 910. FCS Proceedings Ordering, Website, and Index………….. 10 iii
  4. 4. Preface Why This Tutorial? “Everything should be made as simple as I was frequently asked for “hard-copies”possible - but not simpler,” said Einstein. The of the slides, so I started organizing, addingmain goal of this “tutorial” is to assist with some text, and filling the gaps in the slidepresenting the most frequently encountered collection. As the collection grew, I beganconcepts in frequency control and timing, as receiving favorable comments and requestssimply as possible. for additional copies. Apparently, others, too, found this collection to be useful. Eventually, I I have often been called upon to brief assembled this document, the “Tutorial”.visitors, management, and potential users ofprecision oscillators, and have also been This is a work in progress. I plan toinvited to present seminars, tutorials, and include additional material, includingreview papers before university, IEEE, and additional notes. Comments, corrections, andother professional groups. In the beginning, I suggestions for future revisions will bespent a great deal of time preparing these welcome.presentations. Much of the time was spent onpreparing the slides. As I accumulated more John R. Vigand more slides, it became easier and easierto prepare successive presentations. iv
  5. 5. Notes and References Notes and references can be found in the “Notes” ofmost of the pages. To view the notes, use the “NotesPage View” icon (near the lower left corner of thescreen), or select “Notes Page” in the View menu. InPowerPoint 2000 (and, presumably, later versions), thenotes also appear in the “Normal view”. To print a page so that it includes the notes, selectPrint in the File menu, and, near the bottom, at “Printwhat:,” select “Notes Pages”. Many of the references are to IEEE publicationswhich are available online in the IEEE UFFC-S digitalarchive, www.ieee-uffc.org/archive, or in IEEE Xplore,www.ieee.org/ieeexplore . v
  6. 6. In all pointed sentences [and tutorials],some degree of accuracy must besacrificed to conciseness. Samuel Johnson
  7. 7. CHAPTER 1Applications and Requirements 1
  8. 8. Electronics Applications of Quartz CrystalsMilitary & Aerospace Industrial ConsumerCommunications Communications Watches & clocksNavigation Telecommunications Cellular & cordlessIFF Mobile/cellular/portable phones, pagersRadar radio, telephone & pager Radio & hi-fi equipmentSensors Aviation TV & cable TVGuidance systems Marine Personal computersFuzes Navigation Digital camerasElectronic warfare Instrumentation Video camera/recorderSonobouys Computers CB & amateur radio Digital systems Toys & gamesResearch & Metrology CRT displays PacemakersAtomic clocks Disk drives Other medical devicesInstruments Modems Other digital devicesAstronomy & geodesy Tagging/identification AutomotiveSpace tracking Utilities Engine control, stereo,Celestial navigation Sensors clock, yaw stability control, trip computer, GPS 1-
  9. 9. Frequency Control Device Market (estimates, as of ~2006) Technology Units Unit price, Worldwide per year typical market, $/yearQuartz Crystal Resonators & ~ 3 x 109 ~$1 ~$4B Oscillators ($0.1 to 3,000)Atomic Frequency Standards (see chapter 6) Hydrogen maser ~ 20 $100,000 $2M Cesium beam ~ 500 $50,000 $25M frequency standard Rubidium cell ~ 50,000 $2,000 $100M frequency standard 1-2
  10. 10. Navigation Precise time is essential to precise navigation. Historically, navigation has beena principal motivator in mans search for better clocks. Even in ancient times, onecould measure latitude by observing the stars positions. However, to determinelongitude, the problem became one of timing. Since the earth makes one revolutionin 24 hours, one can determine longitude form the time difference between local time(which was determined from the suns position) and the time at the Greenwichmeridian (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 firstperson to produce a clock that allowed the determination of a ships longitude to 30nautical miles at the end of a six week voyage (i.e., a clock accuracy of threeseconds per day). The Englishman John Harrison won the competition in 1735 forhis chronometer invention. Todays electronic navigation systems still require ever greater accuracies. Aselectromagnetic waves travel 300 meters per microsecond, e.g., if a vessels timingwas 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 quartzoscillators in the receivers provide nanosecond-level accuracies. The resulting(worldwide) navigational accuracies are about ten meters (see chapter 8 for furtherdetails about GPS). 1-3
  11. 11. Commercial Two-way Radio Historically, as the number of users of commercial two-way radioshave grown, channel spacings have been narrowed, and higher-frequency spectra have had to be allocated to accommodate thedemand. Narrower channel spacings and higher operating frequenciesnecessitate tighter frequency tolerances for both the transmitters and thereceivers. In 1940, when only a few thousand commercial broadcasttransmitters were in use, a 500 ppm tolerance was adequate. Today,the oscillators in the many millions of cellular telephones (which operateat frequency bands above 800 MHz) must maintain a frequencytolerance of 2.5 ppm and better. The 896-901 MHz and 935-940 MHzmobile radio bands require frequency tolerances of 0.1 ppm at the basestation and 1.5 ppm at the mobile station. The need to accommodate more users will continue to require higherand higher frequency accuracies. For example, a NASA concept for apersonal satellite communication system would use walkie-talkie-likehand-held terminals, a 30 GHz uplink, a 20 GHz downlink, and a 10 kHzchannel spacing. The terminals frequency accuracy requirement is afew parts in 108. 1-4
  12. 12. Digital Processing of Analog Signals The Effect of Timing Jitter(A) Analog* A/D Digital D/A Analog input converter processor converter output Digital* e.g., from an antenna output(B) (C) V(t) V(t) Time DV Analog signal Digitized signal Vt 1-5
  13. 13. Digital Network Synchronization• Synchronization plays a critical role in digital telecommunication systems.It ensures that information transfer is performed with minimal buffer overflow orunderflow events, i.e., with an acceptable level of "slips." Slips causeproblems, e.g., missing lines in FAX transmission, clicks in voice transmission,loss of encryption key in secure voice transmission, and data retransmission.• In AT&Ts network, for example, timing is distributed down a hierarchy of nodes. A timing source-receiver relationship is established between pairs ofnodes containing clocks. The clocks are of four types, in four "stratum levels." Accuracy (Free Running) Stratum 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 1000’s 4 3.2 x 10-5 N.A. XO ~1 million 1-6
  14. 14. Phase Noise in PLL and PSK Systems The phase noise of oscillators can lead to erroneous detection ofphase transitions, i.e., to bit errors, when phase shift keyed (PSK) digitalmodulation 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 thestatistical nature of phase deviations, if the RMS phase deviation is 1.5o,for example, the probability of exceeding the ±7.5o phase deviation is6 X 10-7, which can result in a bit error rate that is significant in someapplications. Shock and vibration can produce large phase deviations even in"low noise" oscillators. Moreover, when the frequency of an oscillator ismultiplied by N, the phase deviations are also multiplied by N. Forexample, a phase deviation of 10-3 radian at 10 MHz becomes 1 radianat 10 GHz. Such large phase excursions can be catastrophic to theperformance of systems, e.g., of those which rely on phase locked loops(PLL) or phase shift keying (PSK). Low noise, acceleration insensitiveoscillators are essential in such applications. 1-7
  15. 15. Utility Fault Location Zap! ta tbSubstation Substation A B Insulator Sportsman X LWhen a fault occurs, e.g., when a "sportsman" shoots out an insulator, a disturbancepropagates down the line. The location of the fault can be determined from the differencesin 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 = xterror=1/2(cl error); therefore, if o error e 1 microsecond, thenxerror r 150 meters d 1/2 of high voltage tower spacings, so, the utility companycan send a repair crew directly to the tower that is nearest to the fault. 1-8
  16. 16. Space ExplorationSchematic of VLBI wtTechnique Mean wavelength w ( t wa Wavefront Local Time & 1 (t) Frequency Standard Microwave Microwave mixer Local mixer Time & Frequency Standard Recorder Data tape Data tape Recorder Correlation cΔ t andΔθ = Integration Lsinθ Amplitude Interference Fringes λ/L sin θ θ( τ) Angle 1-9
  17. 17. Military Requirements Military needs are a prime driver of frequency controltechnology. Modern military systems requireoscillators/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
  18. 18. 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
  19. 19. 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
  20. 20. Clock for Very Fast Frequency Hopping Radio Jammer Example J Let R1 to R2 = 1 km, R1 to t1 t2 J =5 km, and J to R2 = 5 km. Then, since propagation Radio Radio delay =3.3 s/km, R1 tR R2 t1 = t2 = 16.5 . s, tR = 3.3 s, and tm < 30 s.To defeat a “perfect” follower Allowed clock error i mjammer, one needs a hop-rate o 6 s.given by: tm < (t1 + t2) - tR For a 4 hour resynch interval,where tm h message duration/hop clock accuracy requirement is: a 1/hop-rate 4 X 10-10 1-13
  21. 21. Clocks and Frequency Hopping C3 SystemsSlow hopping ‹-------------------------------›Good clockFast hopping ‹------------------------------› Better clockExtended radio silence ‹-----------------› Better clockExtended calibration interval ‹----------› Better clockOthogonality ‹-------------------------------› Better clockInteroperability ‹----------------------------› Better clock 1-14
  22. 22. Identification-Friend-Or-Foe (IFF) Air Defense IFF ApplicationsAWACS FRIEND OR FOE? F-16FAAD PATRIOT STINGER 1-15
  23. 23. Effect of Noise in Doppler Radar System A Moving Decorrelated Transmitter Object Clutter Noise fD Stationary Doppler Signal Receiver Object fD f• 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 radars ability to separate the target signal from the clutter signal. 1-16
  24. 24. Bistatic Radar Conventional (i.e., "monostatic") radar, in which the Illuminatorilluminator and receiver are on the same platform, is vulnerableto a variety of countermeasures. Bistatic radar, in which theilluminator and receiver are widely separated, can greatlyreduce the vulnerability to countermeasures such as jammingand antiradiation weapons, and can increase slow movingtarget detection and identification capability via "clutter tuning” Receiver(receiver maneuvers so that its motion compensates for themotion of the illuminator; creates zero Doppler shift for thearea being searched). The transmitter can remain far from thebattle area, in a "sanctuary." The receiver can remain "quiet.” The timing and phase coherence problems can be ordersof magnitude more severe in bistatic than in monostaticradar, especially when the platforms are moving. The Targetreference oscillators must remain synchronized and syntonizedduring a mission so that the receiver knows when the transmitter emits each pulse, and the phasevariations will be small enough to allow a satisfactory image to be formed. Low noise crystaloscillators are required for short term stability; atomic frequency standards are often required forlong term stability. 1-17
  25. 25. Radar Frequency (GHz) 0 5 10 15 20 25 40 30 10 4km/h - Man or S 100 low Mo ving Ve c hile 1K 100km/ h - V eh1-18 icle, Gro und or A ir 700km/h 10K - S ubs o nic Aircr af t 2, 400 k m/h - M ac h 2 Airc raft Doppler Shifts 100K 1M X-Band RADAR Doppler Shift for Target Moving Toward Fixed Radar (Hz)
  26. 26. CHAPTER 2Quartz Crystal Oscillators 2
  27. 27. Crystal Oscillator Tuning VoltageCrystalresonator Output Frequency Amplifier 2-1
  28. 28. Oscillation• At the frequency of oscillation, the closed loop phase shift = 2nh .• 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
  29. 29. Oscillation and Stability• If a phase perturbation occurs, the frequency must shift f to maintain the 2n phase condition, where c f/f=- t /2QL for a series-resonance oscillator, and QL is loaded Q of the crystal in the network. The "phase slope," do 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 units high Q and high stiffness makes it the primary frequency (and frequency stability) determining element in oscillators. 2-3
  30. 30. Tunability and Stability Making an oscillator tunable over a wide frequency range degrades itsstability because making an oscillator susceptible to intentional tuning alsomakes it susceptible to factors that result in unintentional tuning. The widerthe tuning range, the more difficult it is to maintain a high stability. Forexample, if an OCXO is designed to have a short term stability of 1x10-12 for some averaging time and a tunability of 1 x 10-7, then the crystalsload reactance must be stable to 1 x 10-5 for that averaging time.Achieving such stability is difficult because the load reactance is affectedby stray capacitances and inductances, by the stability of the varactorscapacitance vs. voltage characteristic, and by the stability of the voltage onthe varactor. Moreover, the 1 x 10-5 load reactance stability must bemaintained not only under benign conditions, but also under changingenvironmental conditions (temperature, vibration, radiation, etc.). Whereas a high stability, ovenized 10 MHz voltage controlledoscillator may have a frequency adjustment range of 5 x 10-7 and an agingrate of 2 x 10-8 per year, a wide tuning range 10 MHz VCXO may have atuning range of 50 ppm and an aging rate of 2 ppm per year. 2-4
  31. 31. Oscillator Acronyms Most Commonly Used:• XO…………..Crystal Oscillator• VCXO………Voltage Controlled Crystal Oscillator• OCXO………Oven Controlled Crystal Oscillator• TCXO………Temperature Compensated Crystal OscillatorOthers:• 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
  32. 32. Crystal Oscillator CategoriesThe three categories, based on the method of dealing with the crystal unitsfrequency vs. temperature (f vs. T) characteristic, are:• XO, crystal oscillator, does not contain means for reducing the crystals 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 crystals f vs. T characteristic. Analog TCXOs can provide about a 20X improvement over the crystals 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 crystals f vs. T has zero slope. OCXOs can provide a >1000X improvement over the crystals f vs. T variation. 2-6
  33. 33. Crystal Oscillator Categories ∆f Voltage f +10 ppm Tune 250C -450C +1000C Output TW Crystal Oscillator (XO) -10 ppm ∆f Temperature Compensation Sensor Network or f +1 ppm Computer -450C +1000C T XO -1 ppmX Temperature Compensated (TCXO) Oven XO ∆f Oven f +1 x 10-8 control Temperature -45 C 0 +1000C Sensor T -1 x 10-8O Oven Controlled (OCXO) 2-7
  34. 34. Hierarchy of Oscillators Oscillator Type* Accuracy** Typical Applications • Crystal oscillator (XO) 10-5 to 10-4 Computer timing • Temperature compensated 10-6 Frequency control in tactical crystal oscillator (TCXO) radios • Microcomputer compensated 10-8 to 10-7 Spread spectrum system clock crystal oscillator (MCXO) • Oven controlled crystal 10-8 (with 10-10 Navigation system clock & frequency standard, MTI radar oscillator (OCXO) per g option) • Small atomic frequency 10-9 C3 satellite terminals, bistatic, & multistatic radar standard (Rb, RbXO) • High performance atomic 10-12 to 10-11 Strategic C3, EW standard (Cs)* Sizes range from <5cm3 for clock oscillators to > 30 liters for Cs standards Costs range from <$5 for clock oscillators to > $50,000 for Cs standards.** Including environmental effects (e.g., -40oC to +75oC) and one year of aging. 2-8
  35. 35. Oscillator Circuit TypesOf the numerous oscillator circuit types, three of the more common ones, the Pierce, the Colpitts andthe Clapp, consist of the same circuit except that the rf ground points are at different locations. TheButler and modified Butler are also similar to each other; in each, the emitter current is the crystalcurrent. The gate oscillator is a Pierce-type that uses a logic gate plus a resistor in place of thetransistor in the Pierce oscillator. (Some gate oscillators use more than one gate). b b c c c c c c b Pierce Colpitts Clapp c b c c b c Butler Modified Butler Gate 2-9
  36. 36. OCXO Block Diagram Output OvenEach of the three main parts of an OCXO, i.e., the crystal, the sustainingcircuit, and the oven, contribute to instabilities. The various instabilitiesare discussed in the rest of chapter 3 and in chapter 4. 2-10
  37. 37. Oscillator Instabilities - General Expression −1/2 Δf Δf 1   2ff QL  2  ≈ + 1 +    dφ( ff ) foscillator fresonator 2QL   f     where QL = loaded Q of the resonator, and d= (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
  38. 38. Instabilities due to Sustaining Circuit• Load reactance change - adding a load capacitance to a crystalchanges the frequency by Δf ≅ C1 δf≡ f 2( C0 + CL ) Δ (δ f ) C1 then, ≅− 2( C0 + CL ) 2 Δ CL• Example: If C0 = 5 pF, C1 = 14fF and CL = 20pF, then a • C L = 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
  39. 39. Oscillator Instabilities - Tuned CircuitsMany oscillators contain tuned circuits - to suppress unwantedmodes, as matching circuits, and as filters. The effects of smallchanges in the tuned circuits inductance and capacitance isgiven by:   Δf dφ ( ff )  1  Q  dC dL  ≈ ≈  c  c + c  f oscillator 2QL  1 + 2ff   Q  Cc   L  c   BW where BW is the bandwidth of the filter, ff is the frequencyoffset of the center frequency of the filter from the carrierfrequency, QL is the loaded Q of the resonator, and Qc, Lc andCc are the tuned circuits Q, inductance and capacitance,respectively. 2-13
  40. 40. Oscillator Instabilities - Circuit Noise Flicker PM noise in the sustaining circuit causes flicker FMcontribution to the oscillator output frequency given by: f 2 Losc ( ff ) = Lckt (1Hz ) 3 2 4ff QL and 1 σ y ( τ) = ln2 Lckt (1Hz ) QLwhere ff is the frequency offset from the carrier frequency f, QLis theloaded Q of the resonator in the circuit, Lckt (1Hz) is the flicker PMnoise at ff = 1Hz, and e is any measurement time in the flicker floorrange. For QL = 106 and Lckt (1Hz) = -140dBc/Hz, my(1 ) = 8.3 x 10-14. ( Lckt (1Hz) = -155dBc/Hz has been achieved.) 2-14
  41. 41. Oscillator Instabilities - External LoadIf the external load changes, there is a change in the amplitudeor phase of the signal reflected back into the oscillator. Theportion of that signal which reaches the oscillating loop changesthe oscillation phase, and hence the frequency by Δf dφ( ff )  1  Γ − 1  ≈ ≈  ( sinθ ) isolation f oscillator 2Q  2Q  Γ + 1where c is the VSWR of the load, and f is the phase angle ofthe 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 onlya factor of 3. The problem of load pulling becomes worse athigher frequencies, because both the Q and the isolation arelower. 2-15
  42. 42. Oscillator Outputs Most users require a sine wave, a TTL-compatible, a CMOS-compatible, or an ECL-compatible output. The latter three can besimply generated from a sine wave. The four output types areillustrated below, with the dashed lines representing the supplyvoltage inputs, and the bold solid lines, the outputs. (There is no“standard” input voltage for sine wave oscillators. The inputvoltages for CMOS typically range from 1V to 10V.) +15V +10V +5V 0V -5V Sine TTL CMOS ECL 2-16
  43. 43. Silicon Resonator & Oscillator www.SiTime.comResonator (Si): 0.2 x 0.2 x 0.01 mm3 5 MHz; f vs. T: -30 ppm/oCOscillator (CMOS): 2.0 x 2.5 x 0.85 mm3• ±50 ppm, ±100 ppm; -45 to +85 oC (±5 ppm demoed, w. careful calibration)• 1 to 125 MHz• <2 ppm/y aging; <2 ppm hysteresis• ±200 ps peak-to-peak jitter, 20-125 MHz 2-17
  44. 44. Resonator Self-Temperature Sensing ff (Hz) ff f 3f1 - f3 172300 dfβ = − 14 Hz/ oC = ~ 80 ppm/ oC dT 171300 -35 -15 5 25 45 65 85 Temperature (oC) 170300 2-18
  45. 45. Thermometric Beat Frequency GenerationDUAL MODEOSCILLATOR f1 X3 M=1 MULTIPLIER LOW PASS ff = 3f1 - f3 Mixer FILTER M=3 f3 2-19
  46. 46. Microcomputer Compensated Crystal Oscillator (MCXO) f1 Dual- mode x3 XO Wcom- Reciprocal Correction ff puter f0 f3 Counter Circuit N1 N2 Mixer 2-20
  47. 47. MCXO Frequency Summing Method Block Diagram VCXO 10 MHz PHASE- output 3rd OVERTONE f3 = 10 MHz - fd LOCKEDCRYSTAL LOOP DUAL-MODE Divide by OSCILLATOR 3 fd F Divide f1 by FUNDAMENTAL MODE T Clock 2500 DIRECT Mixer DIGITAL SYNTHESIZER T F fb Clock N2 Divide 1 PPS Clock MICRO- by output COUNTER COMPUTER N1 out 4000 NON-VOLATILE MEMORY T = Timing Mode F = Frequency Mode 2-21
  48. 48. MCXO - Pulse Deletion Method Digital circuitry (ASIC) fc output Dual mode ff oscillator Pulse Counter eliminator f0 corrected SC-cut crystal output for timing Frequency evaluator & correction Microprocessor determination circuitryMicrocomputer compensated crystal oscillator (MCXO) block diagram - pulse deletion method. 2-22
  49. 49. MCXO - TCXO Resonator Comparison Parameter MCXO TCXO Cut, overtone SC-cut, 3rd AT-cut, fund. Angle-of-cut tolerance Loose Tight Blank f and plating tolerance Loose Tight Activity dip incidence 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 2-23
  50. 50. Opto-Electronic Oscillator (OEO) Bias"Pump Laser" Optical out Piezoelectric RF driving port fiber stretcher Filter Electrical output RF coupler Optical Fiber Electrical injection RF Amplifier Photodetector Optical fiber Optical Optical Electrical Injection coupler transmission line 2-24
  51. 51. CHAPTER 3Quartz Crystal Resonators 3
  52. 52. Why Quartz? Quartz is the only material known that possesses the followingcombination of properties:• Piezoelectric ("pressure-electric"; piezein = to press, in Greek)• Zero temperature coefficient cuts exist• Stress compensated cut exists• Low loss (i.e., high Q)• Easy to process; low solubility in everything, under "normal" conditions, except the fluoride and hot alkali etchants; hard but not brittle• Abundant in nature; easy to grow in large quantities, at low cost, and with relatively high purity and perfection. Of the man-grown single crystals, quartz, at ~3,000 tons per year, is second only to silicon in quantity grown (3 to 4 times as much Si is grown annually, as of 1997). 3-1
  53. 53. The Piezoelectric Effect Y Y _ _ _ _ + + + + _ _ _ _ + + + + _ _ _ _ + + + + _ _ _ _ _ _ + + + + + + _ _ _ - + _ _ + + + X _ + X X + + X _ _ _ _ + _ _ + + + + _ _ + + _ _ + + + _ _ + + _ _ + + _ _ + + _ _ + + Undeformed lattice Strained latticeThe piezoelectric effect provides a coupling between the mechanicalproperties of a piezoelectric crystal and an electrical circuit. 3-2
  54. 54. The Piezoelectric Effect in Quartz Z FIELD along: STRAIN X Y Z X X EXTENSIONAL along: Y X X Z X X SHEAR Y X Y about: Z XIn 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 shapedelectrodes. The shear strain about the Z-axis produced by the Y-component of thefield is used in the rotated Y-cut family, including the AT, BT, and ST-cuts. 3-3
  55. 55. Modes of Motion (Click on the mode names to see animation.) Flexure Mode Extensional Mode Face Shear ModeThickness Shear Fundamental Mode Third Overtone Mode Thickness Shear Thickness Shear 3-4
  56. 56. Motion Of A Thickness Shear Crystal CLICK ON FIGURE TO START MOTION
  57. 57. Resonator Vibration Amplitude Distribution Metallic electrodes Resonator plate substrate (the “blank”) u Conventional resonator geometry and amplitude distribution, u 3-5
  58. 58. Resonant Vibrations of a Quartz Plate 0 db. 3555 3200 MHZ Response -10 db. 3507 -20 3742 3383 -30 db. 3642 3652 3802 3852 3707 3256 -40 db. 3200 3400 3600 3800 Frequency, in kHzX-ray topographs (21•0 plane) of various modes excited during a frequencyscan of a fundamental mode, circular, AT-cut resonator. The first peak, at3.2 MHz, is the main mode; all others are unwanted modes. Dark areascorrespond to high amplitudes of displacement. 3-6
  59. 59. Overtone Response of a Quartz Crystal jX Spurious Reactance responses Spurious Spurious responses responses 0 Frequency 5th overtone 3rd overtone -jX Fundamental mode 3-7
  60. 60. 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 intersectthe f vs. T curve of the wanted mode. Such activity dips are highlysensitive to drive level and load reactance. 3-8
  61. 61. 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 [n ] = is the dielectric matrix• For a linear piezoelectric material16 e11 1e21 2e31 T1 c11 c12 c13 c14 c15 c c21 c22 c23 c24 c25 c26 2e12 1e22 2e32 S1 • Elasto-electric matrixSfor -E -E S S S S S quartz -E3 T2 S2 1 2 3 4 5 6 1 2 c31 c32 c33 c34 c35 c36 e13 1e23 2e33 S3 T1 T3 c41 c42 c43 c44 c45 c46 e14 1e24 2e34 et T4 S4 T2 T5 = c51 c52 c53 c54 c55 c56 5e15 1e25 2e35 S5 T3 T6 c61 c62 c63 c64 c65 c66 e16 1e26 2e36 S6 T4 D1 e11 e12 e13 e14 e15 e16 11 1 12 1 13 E1 T5 D2 e21 e22 e23 e24 e25 e26 21 2 22 2 23 E2 T6 CE X D3 e31 e32 e33 e34 e35 e36 31 3 32 3 33 E3 D1 where D2 T1 = T11 S1 = S11 6 e S T2 = T22 S2 = S22 D3 W 2 2 T3 = T33 S3 = S33 LINES JOIN NUMERICAL EQUALITIES 10 EXCEPT FOR COMPLETE RECIPROCITY T4 = T23 S4 = 2S23 ACROSS PRINCIPAL DIAGONAL T5 = T13 S5 = 2S13 INDICATES NEGATIVE OF INDICATES TWICE THE NUMERICAL T6 = T12 S6 = 2S12 EQUALITIES X INDICATES 1/2 (c11 - c12) 3-9
  62. 62. 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 Newtons law of motion for a continuum, and for Maxwells equation* must be solved, with the proper electrical and mechanical ∂Tij ∂Di boundary conditions at the plate surfaces. (F = ma ⇒ = ρ ui ;  ∇ ⋅D = 0 ⇒ = 0, ∂xj ∂xi ∂φ 1 ∂ui ∂u j Ei =− ; S = 2 ( ∂x j + ∂xi ) ; etc.) ∂x i ij • Equations are very "messy" - they have never been solved in closed form for physically realizable three- dimensional 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 canaffect the mounting structure and electrodes. 3-10
  63. 63. 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 c = 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 dTwhere Tf is the linear temperature coefficient of frequency. The temperaturecoefficient 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
  64. 64. 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 10-6/6 C along the Z- direction, and 14.3 x 10-6/6 C perpendicular to the Z-direction; the temperature coefficient of density is, therefore, -36.4 x 10-6/6 C. The temperature coefficients of the elastic constants range from -3300 x 10-6/6 C (for C12) to +164 x 10-6/6 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
  65. 65. Zero Temperature Coefficient Quartz Cuts 90o 60o AT FC IT 30o LC SC z B 0 -30o SBTC BT -60 o -90o 0o 10o 20o 30o z B Do The AT, FC, IT, SC, BT, and SBTC-cuts are ly ing ted S ta ub Ro ly z some of the cuts on the locus of zero temperature Ro t tat coefficient cuts. The LC is a “linear coefficient” Cu Cu ed y t cut that has been used in a quartz thermometer. y Y-cut: t +90 ppm/0C (thickness-shear mode) x xl X-cut: e -20 ppm/0C (extensional mode) 3-13
  66. 66. 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 J f due to edge forces and bending • Lower sensitivity to radiation • Higher capacitance ratio (less r 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 LFRs • The SC-cut is sensitive to electric fields (which can be used for compensation) 3-14
  67. 67. Mode Spectrograph of an SC-cut 1.10 0 1.0 a-mode: quasi-longitudinal mode b-mode: fast quasi-shear mode 1.88 c-mode: slow quasi-shear mode -10Attenuation 3.30 -20 3.0 5.65 5.50 -30 5.0 c(1) b(1) a(1) c(3) b(3) c(5) b(5) a(3) -40 0 1 2 3 4 5 6 Normalized Frequency (referenced to the fundamental c-mode) 3-15
  68. 68. SC- cut f vs. T for b-mode and c-mode 400 200 FREQUENCY DEVIATION (PPM) 0 10 20 Temperature (OC) 0 30 40 50 60 70 -200 c-Mode (Slow Shear) -400 b-Mode (Fast Shear) -600 -25.5 ppm/oC -800 -1000 -1200 3-16
  69. 69. B and C Modes Of A Thickness Shear Crystal C MODE B MODE CLICK ON FIGURES TO START MOTION 3-17
  70. 70. Singly Rotated and Doubly Rotated Cuts’ Vibrational Displacements Z Singly rotated resonator θ gly Si n a t e d Do Ro t t ub Ro ly θ Cu ta Cu ted t Y Doubly rotated resonator ϕ X X’ 3-18
  71. 71. Resistance vs. Electrode Thickness AT-cut; f1=12 MHz; polished surfaces; evaporated 1.2 cm (0.490”) diameter silver electrodes 60 5th 40RS (Ohms) 20 3rd Fundamental 0 10 100 1000 −∆f (kHz) [fundamental mode] 3-19
  72. 72. Resonator Packaging Two-point Mount Package Three- and Four-point Mount Package Quartz Electrodes Quartz blank blank BondingCover area Bonding area Mounting Cover clips Mounting Seal Base clips Pins Seal Pins Base Top view of cover 3-20
  73. 73. Equivalent CircuitsSpring C Mass LDashpot R 3-21
  74. 74. Equivalent Circuit of a ResonatorSymbol for crystal unit CL C0 CL C1 L1 R1 Δf C1 { 1. Voltage control (VCXO) ≈ → fS 2( C0 + CL ) 2. Temperature compensation (TCXO) 3-22
  75. 75. Crystal Oscillator f vs. T Compensation Uncompensated Frequency / Voltage frequency T Compensating Compensated voltage frequency on varactor CL of TCXO 3-23
  76. 76. Resonator Reactance vs. Frequency Area of usual + operation in an oscillator Reactance Resonance, fr Antiresonance, fa 0 Frequency - 1 2πfC0 3-24
  77. 77. Equivalent Circuit Parameter Relationships n: Overtone number A C0 C0: Static capacitance C0 ≅ ε r≡ C1: Motional capacitance t C1 C1n:C1 of n-th overtone L1: Motional inductance L1n:L1 of n-th overtone 1 1 fs fs = 2π L1C1 fa − fs ≅ R1: Motional resistance 2r R1n:R1 of n-th overtone : Dielectric permittivity of quartz e 40 pF/m (average) 1 A: Electrode area Q= 1 ω L1 − 2π fSR1C1 ϕ= ω C1 t: r: Plate thickness Capacitance ratio R1 r’: fn/f1 fs: Series resonance frequency R fa: Antiresonance frequency dϕ 360 Q Q; Quality factor τ 1 = R1C1 ≅ 10 s −14 ≅ df π fs Q1: Motional time constant : : Angular frequency = 2t f : : Phase angle of the impedance k; Piezoelectric coupling factor r C n3L11 3 n R =8.8% for AT-cut, n  2 for SC  π 4.99% C1n ≈ 311 L1n ≈ 3 R1n ≈ 11 2r =   n r r  2k  3-25
  78. 78. What is Q and Why is it Important? Energy stored during a cycle Q≡2π Energy dissipated per cycle Q is proportional to the decay-time, and is inversely proportional to the linewidth of resonance (see next page). • The higher the Q, the higher the frequency stability and accuracy capability of a resonator (i.e., high Q is a necessary but not a sufficient condition). If, e.g., Q = 106, then 10-10 accuracy requires ability to determine center of resonance curve to 0.01% of the linewidth, and stability (for some averaging time) of 10-12 requires ability to stay near peak of resonance curve to 10-6 of linewidth. • Phase noise close to the carrier has an especially strong dependence on Q (L(f) 1/Q4 for quartz oscillators). 3-26
  79. 79. Decay Time, Linewidth, and Q Decaying oscillation of a resonator Oscillation 1 1 = of maximum intensity e 2.7 TIME Excitingpulse ends Max. intensity 1 BW ≅ td πt d Maximum intensity Resonance νo behavior of Q= ≅ νo π t d a resonator ½ Maximum intensity BW BW ν0 FREQUENCY 3-27
  80. 80. Factors that Determine Resonator QThe maximum Q of a resonator can be expressed as: 1 Q max = , 2π fτwhere f is the frequency in Hz, and e is an empirically determined “motionaltime constant” in seconds, which varies with the angles of cut and the mode ofvibration. For example, , = 1 x 10-14s for the AT-cuts c-mode (Qmax = 3.2million at 5 MHz), c = 9.9 x 10-15s for the SC-cuts c-mode, and x = 4.9 x10-15s for the BT-cuts b-mode.Other factors which affect the Q of a resonator include: Overtone  Blank geometry (contour, Surface finish dimensional ratios) Material impurities and defects  Drive level Mounting stresses  Gases inside the enclosure Bonding stresses (pressure, type of gas) Temperature  Interfering modes Electrode geometry and type  Ionizing radiation 3-28
  81. 81. Resonator Fabrication Steps DESIGN GROW SWEEP CUT LAP ROUNDRESONATORS QUARTZ ETCH ORIENT ANGLE X-RAY CLEAN (CHEMICAL CONTOUR IN MASK CORRECT ORIENT POLISH) DEPOSIT PREPARE MOUNT BOND INSPECT CLEANCONTACTS ENCLOSURE TEST FREQUENCY FINAL SEAL PLATE BAKE ADJUST CLEANOSCILLATOR 3-29
  82. 82. X-ray Orientation of Crystal Plates Shielding Monochromator crystal DetectorS X-ray beam Crystal under test Copper target X-ray source Goniometer Double-crystal x-ray diffraction system 3-30
  83. 83. Contamination Control Contamination control is essential during the fabrication ofresonators 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 3-31
  84. 84. Crystal Enclosure Contamination The enclosure and sealing process can have importantinfluences 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/cm3Therefore: In a 1 cm3 enclosure that has a monolayer of contaminationon its inside surfaces, there are ~106 times more adsorbedmolecules than gaseous molecules when the enclosure is sealedat 10-7 torr. The desorption and adsorption of such adsorbedmolecules leads to aging, hysteresis, retrace, noise, etc. 3-32
  85. 85. 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 s f is in kHz and f is in MHz. For example, etching a 10MHz AT-cut plate 1f2 meansthat a thickness is removed that produces o f= 100 kHz; and etching a 30 MHz plate 1f 2means that the s 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.) N f= twhere f is the fundamental mode frequency, t is the thickness of the resonator plate and N isthe frequency constant (1661 MHz•rmode). Therefore, Δf Δt =− f tand, Δf Δt = −N f2So, for example, , f = 1f2 corresponds to the same thickness removal for all frequencies.For an AT-cut, t=1.661 m of quartz (=0.83 o m per side) per f2. An important advantage ofusing units of f2 is that frequency changes can be measured much more accurately thanthickness changes. The reason for expressing h f in kHz and f in MHz is that by doing so, thenumbers of f2 are typically in the range of 0.1 to 10, rather than some very small numbers. 3-33
  86. 86. Milestones in Quartz Technology1880 Piezoelectric effect discovered by Jacques and Pierre Curie1905 First hydrothermal growth of quartz in a laboratory - by G. Spezia1917 First application of piezoelectric effect, in sonar1918 First use of piezoelectric crystal in an oscillator1926 First quartz crystal controlled broadcast station1927 First temperature compensated quartz cut discovered1927 First quartz crystal clock built1934 First practical temp. compensated cut, the AT-cut, developed1949 Contoured, high-Q, high stability AT-cuts developed1956 First commercially grown cultured quartz available1956 First TCXO described1972 Miniature quartz tuning fork developed; quartz watches available1974 The SC-cut (and TS/TTC-cut) predicted; verified in 19761982 First MCXO with dual c-mode self-temperature sensing 3-34
  87. 87. Quartz Resonators for WristwatchesRequirements: • 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 quartztuning forks 3-35
  88. 88. Why 32,768 Hz? 32,768 = 215 32,768 16,3841 In an analog watch, a stepping motor receives 8,192 4,096 one impulse per second which advances the 2,048 second hand by 6o, i.e., 1/60th of a circle, 1,024 every second. 512 256e Dividing 32,768 Hz by two 15 times results 128 in 1 Hz. 64 32i The 32,768 Hz is a compromise among size, 16 power requirement (i.e., battery life) and 8 stability. 4 2 1 3-36
  89. 89. Quartz Tuning Fork Z Y Xa) natural faces and crystallographic axes of quartz Z Y’ 0~50 Y arm baseX b) crystallographic orientation of tuning fork c) vibration mode of tuning fork 3-37
  90. 90. Watch Crystal 3-38
  91. 91. Lateral Field Resonator Lateral Field Thickness Field In lateral field resonators (LFR): 1. the electrodes are absent from theregions of greatest motion, and 2. varying the orientation of the gap betweenthe electrodes varies certain important resonator properties. LFRs can also bemade with electrodes on only one major face. 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) 3-39
  92. 92. Electrodeless (BVA) Resonator C D2 C D1 Quartz bridge Side view of BVA2 Side and top views ofresonator construction center plate C 3-40
  93. 93. CHAPTER 4Oscillator Stability 4
  94. 94. The Units of Stability in Perspective• What is one part in 10 10 ? (As in 1 x 10-10/day aging.) • ~1/2 cm out of the circumference of the earth. • ~1/4 second per human lifetime (of ~80 years).• Power received on earth from a GPS satellite, -160 dBW, is as “bright” as a flashlight in Los Angeles would look in New York City, ~5000 km away (neglecting earth’s curvature).• What is -170 dB? (As in -170 dBc/Hz phase noise.) • -170 dB = 1 part in 1017 7 thickness of a sheet of paper out of the total distance traveled by all the cars in the world in a day. 4-1
  95. 95. Accuracy, Precision, and Stability Precise but Not accurate and Accurate but Accurate and not accurate not precise not precise precisef f f f0 Time Time Time Time Accurate Stable but Not stable and Stable and (on the average) not accurate not accurate accurate but not stable 4-2
  96. 96. Influences on Oscillator Frequency Time • Short term (noise) • Intermediate term (e.g., due to oven fluctuations) • Long term (aging) Temperature • Static frequency vs. temperature • Dynamic frequency vs. temperature (warmup, thermal shock) • Thermal history ("hysteresis," "retrace") Acceleration • Gravity (2g tipover) • Acoustic noise • Vibration • Shock Ionizing radiation • Steady state • Photons (X-rays, n -rays) • Pulsed • Particles (neutrons, protons, electrons) Other • Power supply voltage • Humidity • Magnetic field • Atmospheric pressure (altitude) • Load impedance 4-3
  97. 97. Idealized Frequency-Time-Influence Behavior ∆f X 108 Oscillator f Turn Off Temperature Vibration Shock 2-g Radiation 3 & Step Tipover Turn On Off 2 1 Aging 0 -1 On -2 Short-Term Instability -3 t0 t1 t2 t3 t4 t5 t6 t7 t8 Time 4-4
  98. 98. Aging and Short-Term Stability Short-term instability (Noise) 30 25( f/f (ppm) 20 15 10 5 10 15 20 25 Time (days) 4-5
  99. 99. Aging Mechanisms Mass transfer due to contamination Since f n 1/t, u f/f = -n t/t; e.g., f Fund M 106 molecular layers, 5MHz therefore, 1 quartz-equivalent monolayer f/f 1 ppm Stress relief in the resonators: 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
  100. 100. Typical Aging Behaviors A(t) = 5 ln(0.5t+1) Time4 f/f A(t) +B(t) B(t) = -35 ln(0.006t+1) 4-7
  101. 101. Stresses on a Quartz Resonator PlateCauses: • 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) polishingEffects: • 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
  102. 102. Thermal Expansion Coefficients of Quartz 14/K ZZl -6 0 13.71 W XXl 13 Radial 12 11.63Thermal Expansion Coefficient, 11 T (Thickness) = 11.64 10 Tangential 9.56 9 00 100 200 300 400 500 600 700 800 900 Orientation, , With Respect To XXl 4-9
  103. 103. Force-Frequency Coefficient 30 * 10-15 m 0 s / N AT-cut quartz 25 20 15 Z’ F 10 Δ X’Kf (5 ) 5 F 0 -5 -10 Δf = KF ( Force ) ( Frequency constant ) f ( Diameter ) ( Thickness ) -15 0 0 100 200 300 400 500 600 700 800 900 0 4-10
  104. 104. Strains Due To Mounting ClipsX-ray topograph of an AT-cut, two-point mounted resonator. The topographshows the lattice deformation due to the stresses caused by the mounting clips. 4-11
  105. 105. Strains Due To Bonding Cements (a) (b)X-ray topographs showing lattice distortions caused by bonding cements; (a) Bakelitecement - expanded upon curing, (b) DuPont 5504 cement - shrank upon curing 4-12

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