IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 35, NO. 3, MAY 1988                        

© 1996 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

The Acceleration Sensitivity of Quartz Crystal Oscillators: A Review

Invited Paper

RAYMOND L. FILLER, MEMBER, IEEE

Manuscript received March 24, 1987; revised May 4, 1987. (Corrected by the author, 1999)

The author is with the U.S. Army Electronics Technology and Devices Laboratory (LABCOM), Attn: SLCET-EQ, Fort Monmouth, NJ 07703-5000.
IEEE Log Number 8719296.

Abstract–The acceleration sensitivity of quartz crystal oscillators is assuming increasing significance in modern communications, navigation, radar, and identification systems. This paper is a tutorial and a review of the various aspects of acceleration sensitivity. The topics to be discussed are the consequences of acceleration sensitivity in crystal oscillators on the Allan deviation, including the effects of sinusoidal and random vibration, phase noise and integrated phase jitter; the vector nature of quartz resonator sensitivity; the theoretical description of the cause of the acceleration sensitivity of quartz resonators; techniques for he measurement of acceleration sensitivity; and the effect of frequency multiplication on acceleration effect. In addition, various techniques currently being used or developed for reducing the effective acceleration sensitivity will be discussed. The techniques fall into three general categories: reduction of the acceleration sensitivity of the resonator passive techniques that use compensating elements in the oscillator feedback loop, e.g., a second resonator or an acceleration sensitivity capacitor; and active acceleration compensation schemes that sense the acceleration and feedback a compensating signal to a tuning network.

INTRODUCTION

THERE has been an awareness of acceleration effects in frequency sources at least since the advent of missile and satellite applications [1]-[6], Doppler radars [7], [8], and other systems requiring extremely low noise [9], [10]. There has not, however, been a general appreciation of the magnitude of the problem. As a case in point, there is little or no mention of acceleration sensitivity in the general texts on quartz crystal oscillators and resonators or of vibration induced phase noise in radar textbooks.

    Time-dependent acceleration, i.e., vibration, can cause a large increase in the noise level of a quartz crystal oscillator. In fact, in frequency sources operating on mobile platforms, the vibration-induced phase noise is usually greater than all other noise sources combined.

    High-stability frequency sources, including atomic standards, contain quartz crystal resonators. One result of the evolution of electronics, i.e., the transition from tubes to transistors, and from point-to-point wiring to printed circuits, is the establishment of the quartz crystal resonator as the most acceleration-sensitive component in frequency sources. This paper will review the causes and effects of acceleration sensitivity of bulkwave quartz crystal resonators, and the methods that reduce or compensate for that sensitivity. Most of what is discussed is equally relevant to surface-acoustic-wave (SAW) and shallow-bulk-acoustic-wave (SBAW) devices.

THE EFFECT OF ACCELERATION ON A CRYSTAL RESONATOR

    A quartz crystal resonator subject to a steady acceleration has a slightly different series resonant frequency than the same resonator experiencing zero acceleration [1]. Furthermore, it has been observed that the magnitude of the frequency shift is proportional to the magnitude of the acceleration, and is also dependent upon the direction of the acceleration relative to a coordinate system fixed to the resonator [11]. It has been shown, empirically, that the acceleration sensitivity of a quartz crystal resonator is a vector quantity [12]. Therefore, the frequency during acceleration can be written as a function of the scalar product of two vectors

where is the resonant frequency of the resonator experiencing acceleration a, is the frequency with no acceleration (referred to as the carrier frequency), and is the acceleration-sensitivity vector. It can be seen from (1) that the frequency of an accelerating resonator is a maximum when tile acceleration is parallel to the acceleration-sensitivity vector; it is a minimum when the acceleration is antiparallel to the acceleration-sensitivity vector. An important result of (1) is that the frequency shift, is zero for any acceleration in the plane normal to the acceleration-sensitivity vector.

    The frequency shift described in (1) is also induced by the acceleration due to gravity (even without motion). This is commonly referred to as "2 g-tipover." During "2-g-tipover," the magnitude of the acceleration is 1 g in the direction towards the center of the earth. (The magnitude of acceleration given in this paper will be in units of g, i.e., the magnitude of the earth's gravitational acceleration at sea level, 980 cm/s2.) When a resonator is rotated 180^o about a horizontal axis, the scalar product of the acceleration and the unit vector normal to the initial "top" of the resonator goes from -from -1 g to +1 g, which is a difference of 2 g. Fig. 1 shows actual data of the fractional frequency shift of a resonator (operating in an oscillator) when the oscillator is rotated about three mutually perpendicular axes in the earth's gravitational field. For each curve, the axis of rotation is held horizontal. The sinusoidal shape of each curve is a consequence of the scalar product being proportional to the cosine of the angle between the acceleration-sensitivity vector and the acceleration due to gravity.

Fig. 1.  "2-g tipover," frequency changes versus rotation in earth's gravitational field for three mutually perpendicular axes.

THE EFFECT OF VIBRATION ON A CRYSTAL RESONATOR

    In most applications, the magnitude of the acceleration is time-dependent. The magnitude of the acceleration-sensitivity vector, for acceleration amplitudes commonly encountered, is independent of acceleration amplitude [11]. The time-dependent frequency shift due to a complex vibration can, therefore, be determined from the sum of the individual sinusoidal components, i.e., the system is linear and superposition holds. Simple sinusoidal vibration will be discussed first; the extension will then be made to random vibration.

    Simple harmonic motion will be assumed, with an acceleration given by

where is the peak acceleration vector in units of g, , is the frequency of vibration in Hertz, and t is time in seconds. The variation of the frequency with time can be determined by combining (2) with (1), resulting in

The behavior of the device can be described by defining a rectangular coordinate system fixed to the resonator. The acceleration-sensitivity vector and the acceleration vector can then be described in terms of the three unit vectors defined by that coordinate system. Therefore (3) can be transformed into a scalar equation containing the three (i, j, and k) components of and , i.e.

This can be rewritten as

where

is the peak frequency shift due to the acceleration .

    There are three quantities with units of frequency to keep in mind: It can be seen from (5) that the output frequency deviates from the center frequency, by the amount at a rate of This is shown schematically in Fig. 2. Each plot is the instantaneous output frequency of a crystal oscillator while undergoing a vibration at frequency Fig. 3(b) is the voltage versus time at the output of the same crystal oscillator showing, in a much exaggerated way, the variation in the frequency with acceleration amplitude. Fig. 3(a) is the acceleration waveform.

Fig. 2.  Instantaneous "carrier" frequency for several instants during one cycle of vibration.

Fig. 3.  (a) Acceleration level versus time. (b) Resulting oscillation output showing frequency modulation.

FREQUENCY DOMAIN

    It is very useful to transform the effect of vibration into the frequency domain. This will allow the formulation of a convenient measurement scheme and allow comparison of vibration effects to more familiar forms of phase noise.

    The voltage appearing at the output of an oscillator is given by

where the phase is derived from the frequency by

When the oscillator frequency is varying due to simple harmonic acceleration modulating the resonant frequency of the resonator, the phase in (7) becomes, using (5) and (8),

When (9) is inserted into (7), the result is

Equation (10) is the expression for a frequency-modulated signal. It can be expanded in an infinite series of Bessel functions [13] resulting in [14]

where is the modulation index (from standard FM theory).

    The first term in (11) is a sine wave at the carrier frequency with an amplitude, relative to of The other terms are vibration-induced sidebands at frequencies etc. In analogy with common phase-noise notation [16], the ratio of the power in the nth vibration-induced sideband to the power in the carrier, denoted by is given by

or, more commonly expressed in decibels as

where dBc refers to dB relative to the carrier and the subscript v refers to the vibration.

SMALL MODULATION INDEX

    Several approximations can be made if the modulation index is less than 0.1. This is the case for most frequency standards in the HE band experiencing "normal" accelerations of 10 g or less at acceleration frequencies above a few hertz. The approximations are

Therefore, after combining (132) and (14):

For small modulation index, therefore, most of the power is in the carrier and a small amount is in the first sideband pair (upper and lower). The amplitudes of the higher sidebands are negligible. The condition of small modulation index permits to be equated to [16].

    As an example, consider a 5-MHz oscillator with a vibration sensitivity of 2 x 10-9/g. If the oscillator experiences an acceleration of 5 g along the direction of the acceleration-sensitivity vector, the value of is (5 x 106 Hz) * (5 g) * (2 x 10-9/g) = 0.05 Hz. The modulation index and for several vibration frequencies are shown in Table I. It can be seen that falls off at 6 dB per octave (20 dB per decade). A consequence of linearity at low vibration levels is that these sinusoidal accelerations are independent. The amplitude at any frequency is the same if the bright lines are excited separately or in combination. The first three of these vibration induced sidebands and the carrier are shown in a typical spectrum analyzer output in Fig. 4.

TABLE I

First Sideband Level For A 5-MHz Oscillator
With An Acceleration Sensitivity Of 2 x 10^-9/g
Experiencing An Acceleration Of 5 g Peak

Fig. 4.  Vibration-induced sidebands and carrier resulting from sinusoidal acceleration at 5 Hz, 25 Hz, and 50 Hz.

FREQUENCY MULTIPLICATION

    Phase noise during vibration is of great concern in systems such as radar, navigation, and satellite communications, where the frequency of the crystal oscillator is multiplied up to the microwave region. Upon frequency multiplication by a factor of N. the vibration frequency is unaffected since it is an external influence. The peak frequency change due to vibration, , however, becomes

The modulation index is therefore increased by a factor of N. Expressed in decibels, frequency multiplication by a factor N increases the phase noise by 20 log N.

    The relationship between the vibration-induced phase noise of two oscillators with the same vibration sensitivity and different carrier frequencies is

Where is the sideband level, in dBc, of the oscillator at frequency and is the sideband level of the oscillator at frequency This is identical to the "20 log N" term associated with the increase in phase noise due to frequency multiplication. Again, this relationship holds only if

LARGE MODULATION INDEX

    If the modulation index is larger than about 0.1, the approximations made in (14) are not valid. This often occurs in UHF and higher frequency systems [9]. Frequency multiplication to a higher frequency is indistinguishable from direct frequency generation at the higher frequency. For example, when a 2 x 10-9/g sensitivity 5 MHz oscillator's frequency is multiplied by a factor of 308 to generate a frequency of 1575 MHz, its output will have identical vibration-induced sidebands to a 1575 MHz SAW oscillator with a
sensitivity of 2 x 10-9/g. At 5 g acceleration, for example, the vibration-induced sidebands produced by the 1575 MHz oscillator and the 5 MHz oscillator multiplied by 308 are shown in Table II. It can be seen that the values of in Table II are much greater than those in Table I for the same vibration frequency. This is a consequence of the ratio of output frequency to vibration frequency being much larger. It is possible, as can be seen in this example, for the sidebands to be larger than the carrier. Also, there are even conditions where the carrier disappears and the value of goes to infinity, e.g., when, equals 2.4, that is, all of the power is in the sidebands and none is in the carrier.

ALLAN DEVIATION

    The effect of sinusoidal phase modulation on the Allan deviation of a frequency standard has been shown [15] to be given by

where is the peak phase deviation, is the frequency of the phase modulation, and t is the measurement time. It can be seen from (9) that the magnitude of the maximum phase deviation for a single vibration-induced sideband is Therefore, the vibration-induced Allan deviation of a frequency standard with an acceleration sensitivity of subjected to acceleration at a frequency of is

This effect is shown in Fig. 5. The frequency standard is assumed to have an Allan deviation of (1 x 10^-12/t)² when not being accelerated.

    A plausibility argument for the occurrence of the peaks and valleys in Fig. 5 is that the average frequency, as given in (5), is for an averaging time equal to an integer multiple of the period of the vibration. Since this is a constant, the variance is zero. The average frequency departure from is a maximum when the averaging time is an integer multiple of one-half the period of the vibration. Therefore, the peaks in Fig. 5 occur when and the valleys occur when where n = 0, 1, 2, 3,.

TABLE II

First Sideband Level For A 1575-MHz Oscillator
With An Acceleration Sensitivity Of 2 x 10^-9/g
Experiencing An Acceleration Of 5 g Peak

Fig. 5.  Sigma of y of oscillator with acceleration-sensitivity of 1 ppb/g during 1-g vibration at 20 Hz.

RANDOM VIBRATIONS

In most situations the acceleration experienced by a frequency standard is not simple harmonic motion; it is random, i.e., the vibratory power is randomly distributed over a range of frequencies, phases, and amplitudes. The acceleration can be described by its power spectral density G( f ). The power spectral density of frequency fluctuations Sv(f) can be obtained by multiplying the power spectral density of acceleration by the square of the acceleration sensitivity in (Hz/g)2. is related to Sv, if the modulation index is small, by [16]

Therefore, for random vibration is given by

As an example, consider the random acceleration spectrum given in the upper right of Fig. 6. G(f), in units of g2/Hz, is given by

Fig. 6.  Ramdom vibration-induced phase noise and random vibration envelope.

If the vibration sensitivity is 1 x 10-9/g and the oscillator is operating at 10 MHz, the peak frequency deviation is 0.01 Hz. Therefore, is

which is shown in Fig. 6. Note that outside of the vibration frequency range defined by the given acceleration spectrum, the phase noise is identical to that of a nonaccelerated device.

INTEGRATED PHASE NOISE AND PEAK PHASE EXCURSION

Specialists in crystal resonators and oscillators generally characterize phase noise by Many users of crystal oscillators, however, characterize phase noise in terms of "phase jitter." Phase jitter is the phase noise integrated over the system bandwidth. Similarly, in phase-lock loops, it is the magnitude of the phase excursions that determines whether or not the loop will break lock under vibration.

One can use the previous example to investigate the effect of vibration on integrated phase noise. Integrated phase noise is defined, for the band as

where is the spectral density of phase, equal to 2 In the frequency band of 1 Hz to 2 KHz, the phase noise of the nonvibrating oscillator from Fig. 6 is given by

and the integrated phase noise in the same band is

Therefore

While the oscillator is vibrating, the phase noise is given by (25) in the band from 1 Hz to 5 Hz, and by (23) in the band from 5 Hz to 2000 Hz. The integrated phase noise is

Therefore

While the oscillator is vibrating, it can be seen that the integrated phase noise is 4000 times that of the noise when it is not vibrating and the rms phase deviation, is about 60 times larger during vibration.

When the oscillator is subjected to a simple sinusoidal vibration, the peak phase excursion follows from (9), i.e.,

For example, if our 10 MHz, 1 x 10-9/g oscillator is subjected to a 10 Hz sinusoidal vibration of amplitude 1 g, the peak vibration-induced phase excursion is 1 x 10-3 radians. If this oscillator is used as the reference oscillator in a 10 GHz radar system, the peak phase excursion at 10 GHz will be 1 red. Similarly, when an oscillator's frequency is multiplied, the integrated phase noise is in creased by the multiplication factor. At 10 GHz, for example, the integrated phase noise in the above example increases from 9 x 10-4 red to 0.9 rad! Such large phase excursions are detrimental to the performance of many systems, such as those which employ phase lock loops or phase-shift keying.

MEASUREMENT

The sidebands generated by vibration can be used to measure the acceleration sensitivity. Equation (15) can be rearranged to get

where is the component of the acceleration sensitivity vector in the i direction. Three measurements, along mutually perpendicular axes, are required to characterize which becomes

with a magnitude of

One scheme for measuring is shown in Fig. 7. The local oscillator is used to mix the carrier frequency down to the range of the spectrum analyzer. If the local oscillator is not modulated, the relative sideband levels are unchanged by mixing. The frequency multiplier is used to overcome dynamic range limitations of the spectrum analyzer, using the "20 log N" enhancement discussed previously. The measured sideband levels must be adjusted for the multiplication factor prior to insertion into (31). It must be stressed that (17) is valid only if < 0.1. A sample measurement output and calculation is given in Fig. 8.

Fig. 7.  Typical equipment configuration for measurement of acceleration-sensitivity using sinusoidal acceleration.

Fig. 8.  Sample acceleration-sensitivity measurement using sinusoidal acceleration.

    Other measurement schemes include passive excitation [17] of the resonator, the use of random vibration [18], and the 2-g tipover test. Many oven-controlled oscillators are not suitable for characterization by the 2-g tipover test, however, because rotation of the oscillator results in temperature changes inside the oven that can mask the effects due to the acceleration changes. Vibration tests are also subject to pitfalls since resonances in the oscillator or shake table assembly can produce false results. It is. therefore, important to perform the test at more than one frequency.

THEORY

    The theoretical understanding of acceleration sensitivity does not yet enable one to predict the acceleration sensitivity of '`real" resonator designs. Theoretical activity started with the study of in-plane forces on simplified resonator models. The first studies were concerned with the closely related force-frequency effect. This effect, first reported by Bottom [19] in 1947, is the change in frequency induced by a pair of opposed forces in the plane of a resonator plate applied at the rim. Those forces distort the quartz plate and, because of the nonlinear elastic behavior, change the acoustic velocity [11]. Since the frequency of a resonator is a function of the acoustic velocity and the dimensions of the quartz plate, the forces change the frequency.

    The first attempt at an analytical solution to the force frequency problem was made by Mingins, Barcus, and Perry in 1962 [20]. They assumed an AT-cut crystal plate of infinite lateral dimension and used a perturbation technique with linear elastic coefficients. In 1963, they discussed nonlinear theory and the need to include the third order elastic coefficients [21]. Those coefficients were measured in 1966 by Thurston, McSkimin, and Andreatch [22]. The first calculation to use the nonlinear theory was performed by Keyes and Blair in 1967 [23]. In 1973, Lee, Wang, and Markenscoff [24] published the first of a continuing series of papers (by Lee and his students), in which they calculated the force-frequency coefficient (as defined by Ratajski [25] as a function of azimuth, using the general theory of incremental elastic deformations superimposed on finite initial deformations. The next year, Lee et al. investigated the frequency change due to cantilever bending of the plate [26]. In 1977, a variational analysis of the force-frequency effect, including the effect of material anisotropy on initial stress, was given by EerNisse, Ballato, and Lukaszek [27] for doubly rotated cuts. In 1978, Janiaud, Nissim, and Gagnepain [28] obtained analytic solutions for the biasing stress in singly and doubly rotated plates subject to diametric forces. Those results allowed them to calculate Ratajski's force-frequency coefficient, and also the in-plane acceleration sensitivity. Lee and Wu [29] extended the work of [24] to treat plates of any cut. In that paper, the solution for an arbitrary number of ribbon supports was obtained.

    Under acceleration, body forces in the quartz plate are balanced by reaction forces from the mounting structure. As in the force-frequency effect, distortion of the crystal lattice causes the resonant frequency to change. For bulkwave resonators, only the in-plane case has been treated. The first analysis was by Valdois, Besson, and Gagnepain [11] in 1974, who showed that the effect is linear. Several papers by Lee and Wu [29]-[31] considered resonators with three- and four-point mounts. Two-point mounts were considered by Janiaud, Nissim, and Gagnepain [28]. For SAW resonators, recent work by Shick and Tiersten [32] and by Sinha and Locke [33] has treated acceleration both in the plane of the plate and normal to it.

    There seems to be a significant dependence of the acceleration sensitivity on small variations in the mounting. Analyses of bulkwave resonators, to-date, have considered only point supports. Real resonators have supports that are distributed over a finite area. In addition, initial stress conditions are difficult to determine and may not be well-reproduced. There is an ongoing effort by Lee and Tang [34] to use finite element analysis to more accurately model the mounting structure. It is hoped that this will allow computer simulations that will permit the determination of the optimum design parameters for minimizing acceleration sensitivity.

EXPERIMENTAL RESULTS

    Several papers have reported experimental results on the force-frequency effect [20], [25], [35] and the effects of bending moments [36]. All used point mounts in special fixtures, and results agree fairly well with the theoretical analyses. The reported experimental observations of the actual acceleration sensitivity of real resonators, on the other hand, is remarkable in that it defies simple explanation. ,

    The experimental effort on acceleration-sensitivity started in the 1960's with Smith, Spencer, and Warner [1]-[5]. Valdois, Besson, and Gagnepain [11] made measurements on resonators and oscillators to demonstrate that the resonator was the acceleration-sensitive element. Results from 2-g tipover experiments were reported for SC cut resonators by Kusters, Adams, Yoshida, and Leach [37] in 1977. In 1979, Warner, Goldfrank, Meirs, and Rosenfeld [38] reported results on SC- and FC-cut devices. In 1981, the first of a series of papers intended to explore a wide range of resonator fabrication parameters was published by Filler and Vig tam. In that same year, Nakazawa, Lukaszek, and Ballato [40] reported constant acceleration results that suggested an acceleration sensitivity that is not linear with acceleration level. The only significant parameter for the reduction of acceleration sensitivity was reported by Filler, Kosinski, and Vig [41] in 1982. As one makes piano-convex and biconvex AT-cut resonators flatter, the acceleration sensitivity decreases. It must be noted, however, that other effects, as yet undetermined, cause large scatter in the data. A survey of a large number of resonator parameters, such as SC-cut contour, thickness, drive level, temperature, and angle of cut, was published in 1983 by Filler, Kosinski, and Vig [42]. No significant correlations were found.

    The most recent paper, by Weglein [43] in 1984, reported the acceleration sensitivity of VHF AT- and SC-cut resonators. All of the resonators in that study were disassembled after the acceleration sensitivity measurements to examine fabrication details. That investigation, which seems to summarize the efforts to date, showed "little dependence on any recognizable parameter," except that the lowest average acceleration sensitivity was found in the group of AT units. The large spread in the data, typical of all experimental observations of acceleration-sensitivity, seems to he caused by a combination of sometimes offsetting subtle effects, which are difficult to) control during fabrication.

REDUCTION OF THE ACCELERATION SENSITIVITY OF RESONATORS

    The introduction of the stress-compensated cut of quartz, i.e., the SC-cut, was accompanied by the widespread expectation that the SC-cut would have significantly lower acceleration sensitivity than other cuts. Unfortunately, that expectation has not been realized. The lowest acceleration sensitivity achieved with AT-cut resonators [41]-[43] equals that achieved with SC-cut resonators; both cuts can have low parts in 1010/g sensitivity.

    Efforts to reduce the sensitivity of individual resonators to the effects of acceleration have stressed the support structure. Lukaszek and Ballato [44] proposed a plate geometry that would assure the proper support configuration to reduce the force-frequency effect. Besson, Gagnepain, Janiaud and Valdois [45] proposed a support structure that insured symmetry with the median plane of the resonator plate. Debaisieux, Aubry, anal Groslambert [46] and Aubry and Debaisieux [47] have reported results using QAS resonators, a variation of the BVA [48], which insures symmetry of mount as well as accurate mount locations for reducing the force-frequency effect. Their results show a marked reduction in the scatter of the measured acceleration sensitivity.

ACCELERATION COMPENSATION OF OSCILLATORS

    The lack of progress in reducing the acceleration sensitivity of the resonator below the low parts in 1010/g level has spawned several techniques for compensation of the effect. There are two general classes of compensation, passive and active. The first compensation results were published by Gagnepain and Walls in 1977 [49]. They used the passive method of mechanically arranging two resonators such that the components of the acceleration sensitivities normal to the plates were antiparallel. The resonators were electrically connected in series in a single oscillator. Przyjemski [12] and Emmons [50] used an active technique. They sensed the acceleration magnitude with an accelerometer aligned with the direction of the acceleration sensitivity vector of the resonator. The accelerometer signal was fed into a tuning circuit in the oscillator in order to counter the acceleration-induced frequency changes. A limitation of this technique is the requirement on the linearity of the tuning network at all operating points. Emmons also employed the dual-resonator technique and suggested using an acceleration-sensitive capacitor in the tuning network of the oscillator. The latter technique is available as an option on a commercial cesium-beam frequency standard.

    Ballato [51] suggested a method for compensation in all directions using a resonator pair made of enantiomorphs. He argued that opposite handedness is the only way to have all three crystallographic axes line up antiparallel. This was extended by a series of patents by Ballato and Vig [52]-[54]. A simplification was patented by Filler [55], who showed that, since the acceleration-sensitivity has vector properties, all that is required is that the vectors be aligned antiparallel, independent of the handedness of the quartz. Vig and Walls [56] extended this work by suggesting a method to accommodate resonators with different acceleration sensitivity magnitudes.

    Rosati patented [57] an active technique of compensation that was further developed by Rosati and Filler [58]. This method makes use of the polarization effect in doubly rotated resonators, i.e., that the resonant frequency of a doubly rotated resonator is a function of the voltage applied to the electrodes [59]. If one senses the acceleration using an accelerometer and feeds that signal with appropriate amplification and phase reversal directly to the resonator electrodes, compensation can be achieved. The advantage of feeding the correction signal to the resonator electrodes rather than a varactor is that the polarization effect has superior linearity. When the correction signal is applied to a varactor, the nonlinearity of the varactor and of the frequency-capacitance function causes sidebands at harmonics of the vibration frequency. One implementation of this technique was used to compensate a rubidium oscillator [60].

CONCLUSION

    Vibration effects are a significant problem in modern communication, navigation, and radar systems. Progress has been made in understanding the causes of acceleration sensitivity but a full explanation for real devices has been elusive. A great deal of effort has been expended on compensating for acceleration-sensitivity, but more work is needed to improve the level and bandwidth of the compensation.

ACKNOWLEDGMENTS

    For many helpful discussions, the author thanks Mr. Robert Smythe of Piezo Technology, Inc., and Dr. John Vig and Mr. Vincent Rosati of U.S. Army LABCOM. In addition, the author thanks Mr. Martin Bloch for generating the initial interest in this problem.

REFERENCES

Raymond L. Filler (M'75) was born in Brooklyn, NY, in 1948: He received the B.S. degree in physics from Rensselaer Polytechnic Institute, Troy, NY, in 1969 and the Ph.D. degree from Rutgers, New Brunswick, NJ, in 1975.

    He is currently the leader of the Crystal Resonator Team of the Frequency Control and Timing Branch of the Electronics Technology and Devices Laboratory (LABCOM), Fort Monmouth, NJ. His research interests include techniques to improve long and short term stability, and shock and vibration sensitivity of quartz crystal resonators. He has five patents in the frequency control field.

    Dr. Filler is a member of the American Physical Society. He has served as the Publicity Chairman of the IEEE co-sponsored Annual Symposium on Frequency Control since 1986, and as Chairman of a committee to establish traceability of crystal parameter measurement to the National Bureau of Standards.

© 2000 IEEE