Quartz resonator pressure transducers and methods of operation

Abstract

A cylindrical quartz crystal transducer that exhibits a low probability of twinning, and uses a combination of resonator signal inputs at the B-mode and C-mode frequencies to calculate resonator temperature. Crystallographic orientations are disclosed where combinations of B-mode and C-mode resonant frequencies exist that are sufficiently independent of pressure to enable accurate calculation of temperature under transient conditions. Such a transducer is usable at higher pressures and temperatures than conventional quartz pressure transducers. Furthermore, because the structure allows a choice of crystallographic orientation, other characteristics of the transducer, such as increased pressure sensitivity and activity dip-free operation, may be optimized by varying crystallographic orientation.

Claims

1. A method of using a quartz crystal structure of a pressure transducer, the method comprising: stimulating, under transient temperature conditions, a resonator of the quartz crystal structure under external pressure applied to the quartz crystal structure to provide signal inputs to an electronics assembly associated with the resonator, the signal inputs comprising a first signal input from a non-fundamental B-mode resonant frequency and a second signal input from a non-fundamental C-mode resonant frequency; and using a combination of the first signal input and the second signal input to compensate a pressure determined from the second signal input of the non-fundamental C-mode resonant frequency signal input, comprising: receiving with the electronics assembly from the resonator the first signal input of the non-fundamental B-mode resonant frequency that is primarily dependent on temperature; receiving with the electronics assembly from the resonator the second signal input of the non-fundamental C-mode resonant frequency that is primarily dependent on pressure; calculating with the electronics assembly substantially pressure-independent temperature under the transient temperature conditions using a combination of the first signal input and the second signal input and compensating with the electronics assembly the pressure determined from the signal input of the non-fundamental C-mode resonant frequency with data from the calculation of the substantially pressure-independent temperature.

2. The method of claim 1, wherein using a combination of the first signal input and the second signal input comprises using a sum of the first signal input and the second signal input.

3. The method of claim 1, wherein the non-fundamental B-mode resonant frequency and the non-fundamental C-mode resonant frequency is the 3 .sup.rd harmonic of each mode.

4. The method of claim 1, further comprising determining a change in temperature with the following equation: $\begin{array}{cc}T=\frac{\left(f_C+{\mathrm{Kf}}_B-f_{C1}-{\mathrm{Kf}}_{B1}\right)}{\left(f_{C1}{C}_T+{\mathrm{Kf}}_{B1}{B}_T\right)}.& \end{array}$

5. The method of claim 1, further comprising determining a change in pressure with the following equation: $P=\frac{C_T}{C_P}\frac{\left(f_{C1}{C}_P+{\mathrm{Kf}}_{B1}{B}_P\right)}{\left(f_{C1}{C}_T+{\mathrm{Kf}}_{B1}{B}_T\right)}H.$

6. The method of claim 5, further comprising selecting a value of the constant K in the equation defining the change in pressure to minimize temperature dependence of the equation defining the change in pressure.

7. The method of claim 1, wherein stimulating the resonator comprises driving the resonator at least two distinct frequencies with at least one oscillator.

8. The method of claim 1, wherein stimulating the resonator comprises driving the resonator comprising a single crystal with at least one oscillator at a first frequency and a second frequency that is greater than the first frequency to provide both pressure and temperature measurements from the single crystal.

9. The method of claim 1, further comprising determining both pressure and temperature measurements from the resonator comprising a single crystal.

10. A method of using a quartz crystal structure of a pressure transducer, the method comprising: stimulating a resonator of the quartz crystal structure of the pressure transducer with an electronics assembly associated with the pressure transducer under external pressure applied to the quartz crystal structure; receiving with the electronics assembly from the resonator a non-fundamental B-mode resonant frequency signal input primarily dependent on temperature and a non-fundamental C-mode resonant frequency signal input from the resonator primarily dependent on pressure; and determining both a pressure measurement and a temperature measurement from the resonator using both of the non-fundamental B-mode resonant frequency signal input and the non-fundamental C-mode resonant frequency signal input.

11. The method of claim 10, wherein determining both the pressure measurement and the temperature measurement from the resonator comprises determining the temperature measurement using both of the non-fundamental B-mode resonant frequency signal input and the non-fundamental C-mode resonant frequency signal input.

12. The method of claim 11, wherein determining both the pressure measurement and the temperature measurement from the resonator further comprises compensating the pressure measurement using data from the determining of the temperature measurement using both of the non-fundamental B-mode resonant frequency signal input and the non-fundamental C-mode resonant frequency signal input.

13. The method of claim 10, wherein determining both the pressure measurement and the temperature measurement from the resonator comprises determining the temperature measurement under transient temperature conditions using both of the non-fundamental B-mode resonant frequency signal input and the non-fundamental C-mode resonant frequency signal input.

14. The method of claim 10, further comprising utilizing a 3.sup.rd harmonic of each of the non-fundamental B-mode resonant frequency signal input and the non-fundamental C-mode resonant frequency signal input.

15. The method of claim 10, further comprising utilizing a 3.sup.rd harmonic of one of the non-fundamental B-mode resonant frequency signal input and the non-fundamental C-mode resonant frequency signal input and a 5.sup.th harmonic of the other one of the non-fundamental B-mode resonant frequency signal input and the non-fundamental C-mode resonant frequency signal input.

16. A method of using a quartz crystal structure of a pressure transducer, the method comprising: stimulating, under transient conditions, a resonator of the quartz crystal structure of the pressure transducer with an electronics assembly associated with the pressure transducer under external pressure applied to the quartz crystal structure; receiving with the electronics assembly a non-fundamental B-mode resonant frequency signal input and a non-fundamental C-mode resonant frequency signal input from the resonator; and determining temperature-compensated pressure from the non-fundamental B-mode resonant frequency signal input and the non-fundamental C-mode resonant frequency signal input.

17. The method of claim 15, wherein determining the temperature-compensated pressure comprises receiving with the electronics assembly the non-fundamental B-mode resonant frequency signal input that is primarily dependent on temperature.

18. The method of claim 16, wherein determining the temperature-compensated pressure further comprises receiving with the electronics assembly the non-fundamental C-mode resonant frequency signal input that is primarily dependent on pressure.

19. The method of claim 16, wherein determining the temperature-compensated pressure comprises calculating with the electronics assembly substantially pressure-independent temperature under the transient conditions using a combination of the non-fundamental B-mode resonant frequency signal input and the non-fundamental C-mode resonant frequency signal input.

20. The method of claim 19, wherein determining the temperature-compensated pressure further comprises compensating for transient temperature conditions in the pressure determined from the non-fundamental C-mode resonant frequency signal input using the substantially pressure-independent temperature.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1A is a perspective cutaway view of a prior art quartz pressure transducer configuration;

(2) FIG. 1B is a perspective cutaway view of another prior art quartz pressure transducer configuration;

(3) FIG. 1C is a perspective cutaway view of a further prior art quartz pressure transducer configuration;

(4) FIG. 2 is a perspective cutaway view of yet another prior art quartz pressure transducer configuration;

(5) FIG. 3 is a graph of frequency shift versus temperature for phi=26 at various pressures;

(6) FIG. 4 is a graph of slope of frequency shift versus temperature for phi=26 at various pressures;

(7) FIG. 5 is a graph of calculated pressure sensitivity of the B- and C-modes of a round quartz pressure transducer versus phi angle;

(8) FIG. 6 is a graph of the error factor in calculated static pressure for f.sub.C+f.sub.B temperature errors due to the H stress;

(9) FIG. 7 is a graph of slope in Hz/psi an indicated combination of f.sub.C and f.sub.B versus T, calculated using experimental data from a phi=26 quartz pressure transducer; and

(10) FIG. 8 is a schematic diagram of a circuit suitable for use with embodiments of a quartz crystal pressure transducer according to embodiments of the present disclosure.

DETAILED DESCRIPTION

(11) To facilitate a more complete understanding of embodiments of the present disclosure and their operation, it is prudent here to develop a basis for evaluating what errors will occur in a dual-mode transducer given a known level of pressure sensitivity of the B-mode, or in the present case, of the sum of f.sub.C and f.sub.B. This will be done using some conventional methods for describing the relationship of f.sub.C and f.sub.B with pressure and temperature.

(12) Equations for using two modes for computing pressure, P, and temperature, T, are described, for instance, in R. J. Besson et. al., A DUAL-MODE THICKNESS-SHEAR QUARTZ PRESSURE SENSOR, IEEE Trans. Ultrasonics, ferroelectrics, and Frequency Control, Vol. 40, p. 584, 1993, the disclosure of which is hereby incorporated herein in its entirety by this reference. These equations express the pressure and temperature in two-dimensional power series expansions in the two variables, f.sub.C and f.sub.B, which are the measured frequencies of the C-mode and B-mode, respectively. This approach works well because of the smooth behavior of f.sub.C and f.sub.B with pressure and temperature.

(13) FIG. 3 shows the behavior of f.sub.C for a device comprising a transducer structured according to an embodiment of the present disclosure when subjected to a static P and T. The f.sub.C value at atmospheric pressure and a temperature of 25 C. is typically 7.26 MHz. For the behavior illustrated in FIG. 3, the outside diameter of an embodiment of the device configured as shown in FIG. 1C is 0.575 inch, the bore diameter of the end caps is 0.300 inch and bore depth is 0.120 inch. The resonator is a 3.sup.rd overtone blank with a diopter of 2.5 on both sides. The crystallographic orientation is phi=26 and theta is near 34. As known to those of ordinary skill in the art, the angle phi is the angle between the X-axis and the line of intersection of the blank or atomic plane with the XY-plane of a conventionally employed rectangular coordinate system, while theta is the angle between the Z-axis and the plane of the blank or atomic plane. The appropriate theta angle may be chosen such that the first order temperature coefficient of the C-mode is zero. This may be calculated according to the following equation, known to those of ordinary skill in the art:
=35.25(11/180)
The viable pressure range extends to about 30,000 psi and the temperature range is 25 C. to 200 C.

(14) Embodiments of the present disclosure may be physically implemented utilizing the quartz crystal structures illustrated herein in FIGS. 1A and 1C. As noted previously, FIG. 1A depicts a unitary resonator and body with end caps at each end of the body, whereas FIG. 1C depicts a resonator sandwiched between two end caps comprising the body. For example, quartz crystal structures in accordance with the present disclosure may include a convex-convex resonator and two end caps. In other embodiments, the quartz crystal structures in accordance with the present disclosure may include other resonators configurations such as plano-plano and plano-convex.

(15) Referring to FIG. 3, changes in f.sub.C exhibit smooth behavior with shifts in pressure over a wide range of temperatures. The following approach is taken for a theoretical development of the possible errors in using a dual-mode device in a transient situation. At a given static pressure P.sub.1 and static temperature T.sub.1, the f.sub.C and f.sub.B behavior can be described for small excursions in P and T around P.sub.1 and T.sub.1 with a Taylor series expansion. The expansion is limited to terms linear in P and T and any cross-products of P and T are ignored.

(16) $\begin{array}{cc}{f}_C=f_{C1}+\frac{f_C}{T}*\left(T-T_1\right)+\frac{f_C}{P}*\left(P-P_1\right)& \left(1\right)\\ f_B=f_{B1}+\frac{f_B}{T}*\left(T-T_1\right)+\frac{f_B}{P}*\left(P-P_1\right)& \left(2\right)\end{array}$

(17) If the constants C.sub.T, C.sub.P, B.sub.T, and B.sub.P are defined as follows,

(18) $\begin{array}{cc}{C}_T=\frac{1}{f_{C1}}\frac{f_C}{T}{C}_P=\frac{1}{f_{C1}}\frac{f_C}{P}{B}_T=\frac{1}{f_{C1}}\frac{f_B}{T}{B}_P=\frac{1}{f_{C1}}\frac{f_B}{P},& \left(3\right)\end{array}$
then equations 1 and 2 can be written as
f.sub.C=f.sub.C1+f.sub.C1C.sub.T*(TT.sub.1)+f.sub.C1C.sub.P*(PP.sub.1)(4)
f.sub.B=f.sub.B1+f.sub.B1B.sub.T*(TT.sub.1)+f.sub.B1B.sub.P*(PP.sub.1)(5)

(19) Equation 4 can be used in development of an error budget by answering the question: How accurate does one need to know T to calculate P to a given accuracy level? Using Equation 4, we can solve for an error in f.sub.C, f.sub.C , caused by an error in T, T, given that P=P.sub.1.
f.sub.c=f.sub.C1C.sub.TT.(6)
Now, assuming that T=T.sub.1, Equation 4 can be solved for P in terms of f.sub.C.

(20) $\begin{array}{cc}P-P_1=\frac{\left(f_C-f_{C1}\right)}{f_{C1}{C}_P}.& \left(7\right)\end{array}$
If Equation 6 is substituted into Eq. 7, the error in P, P, can be estimated as

(21) $\begin{array}{cc}P=\frac{C_T}{C_P}T.& \left(8\right)\end{array}$
As shown by the equation, a combination of a low temperature sensitivity (small C.sub.T) and a large pressure sensitivity (large C.sub.P) minimizes the P error, P, due to an error in T.

(22) When there is a transient situation involving a temperature shift, there are stresses created in the resonator due to a non-uniform temperature distribution in the resonator. This stress value at the center of the resonator causes a frequency shift that is an error in indicated pressure, which will be termed H. It is conventional to use f.sub.B for the calculation of T. Using Equation 5, the error in T, T, caused by H is

(23) $\begin{array}{cc}T=\frac{B_P}{B_T}H.& \left(9\right)\end{array}$

(24) Thus, an error in the calculated pressure from T caused by H is represented by

(25) $\begin{array}{cc}T=\frac{C_T}{C_P}\frac{B_P}{B_T}H.& \left(10\right)\end{array}$
This is the error that arises from the pressure sensitivity of f.sub.B. It is known in the art to have B.sub.P small and B.sub.T large, as well as small C.sub.T and large C.sub.P. The numbers provided in U.S. Pat. No. 6,111,340 may be used to calculate the coefficient in Equation 10 that the inventors therein considered practical, i.e., substantially insensitive to pressure, (|B.sub.P||C.sub.P|, and |C.sub.T|3 ppm/C. and |B.sub.T|28 ppm/C.). The coefficient is 0.107. This indicates that the error in calculating P due to the non-uniform temperature distribution is approximately ten times (10) less than H, the indicated error in P arising from the non-uniform temperature distribution during a transient event. One may proceed from here assuming that 10 is an approximate threshold for practical dual-mode performance.

(26) FIG. 4 shows the slope of FIG. 3 in ppm/C. for an embodiment of a quartz pressure transducer according to the present disclosure at phi=26. The crystallographic orientation has been adjusted to minimize the magnitude of the slope over the entire pressure and temperature ranges to be 3 ppm/C. This number may be used as one design parameter for evaluating an error budget and this number is approximately the same over the range of phi from 22 to 30. Also, the B.sub.T coefficient is found to be approximately 28 ppm/C. over this phi range. One may proceed with these two numbers assumed to be relatively constant over the phi range under consideration herein for implementation of one embodiment of the present disclosure.

(27) One form of the present disclosure uses f.sub.C+f.sub.B for the temperature calculation. An equation may be derived for this case that is equivalent to Equation 10 for the error in P due to H. Assume that P=P.sub.1. Then, the change in temperature T is calculated from Equations 4 and 5 by

(28) $\begin{array}{cc}T=\frac{\left(f_C+f_B-f_{C1}-f_{B1}\right)}{\left(f_{C1}{C}_T+f_{B1}{B}_T\right)}.& \left(11\right)\end{array}$
Assume that T=T.sub.1. Then if the non-uniform stress is present and there is an error in the indicated pressure of H, the error in frequency for f.sub.C+f.sub.B is given by
(f.sub.C+f.sub.Bf.sub.C1f.sub.B1)=(f.sub.c1C.sub.P+f.sub.B1B.sub.P)H.(12)
When Equations 11 and 12 are combined, the error in T is given by

(29) $\begin{array}{cc}T=\frac{\left(f_{C1}{C}_P+f_{B1}{B}_P\right)}{\left(f_{C1}{C}_T+f_{B1}{B}_T\right)}H.& \left(13\right)\end{array}$
Equation 13 for the error in T, combined with Equation 8, provides us with the equivalent of Equation 10:

(30) $\begin{array}{cc}P=\frac{C_T}{C_P}\frac{\left(f_{C1}{C}_P+f_{B1}{B}_P\right)}{\left(f_{C1}{C}_T+f_{B1}{B}_T\right)}H.& \left(14\right)\end{array}$
Equation 14 represents a significant aspect of the disclosure. Instead of B.sub.P and B.sub.T in Equation 10, which are in ppm/psi and ppm/ C., in the present disclosure the coefficients in the parenthesis are calculated using Hz/psi and Hz/ C. The power of this approach becomes evident when looking at FIG. 5. There, the calculated values are shown for f.sub.C1*C.sub.P and f.sub.B1*B.sub.p, as well as the sum (f.sub.C1*C.sub.P+f.sub.B1*B.sub.p) for a round sensor design according to an embodiment of the disclosure over the range of phi angles from 22 to 30. Since the values are of opposite sign, the sum trends toward zero for phi near 30.

(31) The impact of choice of phi angle is also influenced by the fact that C.sub.P is zero near phi=22 for a round sensor and trends approximately linearly toward 1.5 ppm/psi at phi =30 . Since C.sub.P is in the denominator of Equation 14, the effect of the phi angle on the coefficient in Equation 14 is dramatic. FIG. 6 shows the coefficient in Equation 14 vs. phi angle. It is apparent from FIG. 6 that a round sensor can be used for a dual-mode pressure transducer for phi angles greater than about 25, where the 10 criteria is approximately satisfied. If one is more aggressive and chooses 5 for the criteria, the lower end of the range of usable phi angles falls to 24 . Although the curve in FIG. 6 is theoretical, the experimental point obtained in this work and shown in FIG. 6 supports the theoretical results.

(32) The use of the sum for f.sub.C and f.sub.B is a result of concentrating attention on the phi angle range in FIG. 5. Since C.sub.P passes through zero around phi of 22, at lower phi angles one may use the difference f.sub.Bf.sub.C to reduce the error arising in transient conditions. Here, using the difference, the appropriate equation relating the pressure error to H is given by

(33) 0$\begin{array}{cc}P=\frac{C_T}{C_P}\frac{\left(f_{B1}{B}_P+f_{C1}{C}_P\right)}{\left(f_{B1}{B}_T+f_{C1}{C}_T\right)}H.& \left(15\right)\end{array}$

(34) The form of Equation 14 may be maintained by dividing both numerator and denominator of Equation 15 by 1:

(35) $\begin{array}{cc}P=\frac{C_T}{C_P}\frac{\left(f_{C1}{C}_P+f_{B1}{B}_P\right)}{\left(f_{C1}{C}_T+f_{B1}{B}_T\right)}H.& \left(16\right)\end{array}$

(36) The most general form of the present disclosure is based on the fact that once one has the values for f.sub.C and f.sub.B, one is free to perform almost any desired calculation. Thus, we may use f.sub.C+K*f.sub.B to compute T, where K is a scalar number. The equations for this case may be easily derived. Substituting K*f.sub.B and K*f.sub.B1 for f.sub.B and f.sub.B1, respectively, in Equation 11, the following equation for computing a change in T can be written as

(37) $\begin{array}{cc}T=\frac{\left(f_C+{\mathrm{Kf}}_B-f_{C1}-{\mathrm{Kf}}_{B1}\right)}{\left(f_{C1}{C}_T+{\mathrm{Kf}}_{B1}{B}_T\right)}.& \left(17\right)\end{array}$

(38) If the same substitutions are made into Equation 13, the error in T due to the presence of H may be found:

(39) $\begin{array}{cc}T=\frac{\left(f_{C1}{C}_P+{\mathrm{Kf}}_{B1}{B}_P\right)}{\left(f_{C1}{C}_T+{\mathrm{Kf}}_{B1}{B}_T\right)}H.& \left(18\right)\end{array}$

(40) Equation 18 may be used in Equation 8 to arrive at the most general case of the present disclosure:

(41) $\begin{array}{cc}P=\frac{C_T}{C_P}\frac{\left(f_{C1}{C}_P+{\mathrm{Kf}}_{B1}{B}_P\right)}{\left(f_{C1}{C}_T+{\mathrm{Kf}}_{B1}{B}_T\right)}H.& \left(19\right)\end{array}$

(42) Note that Equation 19 becomes Equation 14 when K=1, and becomes Equation 16 when K=1. However, K may be adjusted to minimize the term f.sub.C1*C.sub.P+K*f.sub.B1*B.sub.P in Equation 19. This has been done for some calibration data of the sensor used for the experimental point in FIG. 5. The result is shown in FIG. 7, where a K of 0.606 was found to reduce the numerator of Equation 19 over the temperature range employed to practically zero. The fact that there is some small T dependence in FIG. 7 arises because f.sub.C and f.sub.B are not perfectly linear with P and T. Since it has already been shown that the value of 0.33 Hz/psi for f.sub.C+f.sub.B is adequate for actual use, the greatly reduced value in FIG. 7 will provide even superior performance.

(43) Thus, one significant benefit of this disclosure is that by proper choice of the combination f.sub.C+K*f.sub.B, one can now choose the angle phi for the crystallographic orientation of the sensor for other reasons. One option is to choose a phi angle far from 22 to obtain a large pressure sensitivity of f.sub.C. Another important consideration is that both the C-mode and B-mode must be free of significant activity dips. Yet another consideration is that the resistance of the two modes changes greatly with phi, so, depending on the circuits to be used in conjunction with the transducer, it may be desirable to adjust phi appropriately.

(44) It should be noted that the use of the deep flats on the transducer body as disclosed in the '340 patent might, if desired, be used to improve the present disclosure in terms of reducing f.sub.C+f.sub.B over that obtained from a round-bodied unit. We can understand this by looking at FIG. 5. With judicious choice of the orientation of the flats, the stresses in the resonator become non-uniform. This can cause the curve for the B-mode to move downward to lower positive values as the P sensitivity decreases, and lower the curve for the C-mode toward larger magnitude, but negative, values for the C-mode as the P sensitivity increases. This reduces the sum f.sub.C+f.sub.B over the value for a round unit. However, the use of flats of any significant depth to create non-uniform stress distributions in the resonator, and in the end caps, may unfortunately increase the potential for twinning, or cracking. In addition, the use of flats may be unnecessary with the present disclosure because the function to be provided by the flats can be effected with f.sub.C+K*f.sub.B.

(45) It should be emphasized that conventional quartz transducer construction practices utilize small exterior flats for alignment purposes during assembly, but such flats are sufficiently small to not cause any appreciable non-uniform stress in the resonator and, accordingly, the term flat as applied to quartz transducer structures means and includes a flat or flats of sufficient magnitude to induce non-uniform stress in a resonator of such transducer structures under applied exterior pressure. For example, a transducer body in accordance with at least one embodiment of the present disclosure may include two large flats and two smaller flats, each being offset about 90 about the body of the transducer as shown in FIG. 2. Such a configuration may aid in the assembly of the transducer body by helping to ensure that the end caps and resonator are assembled in the correct orientation. Accordingly, the term substantially cylindrical as used herein with regard to quartz transducer structures means and includes structures devoid of a flat or flats of sufficient magnitude to induce non-uniform stress in a resonator of such a quartz transducer structure. For example, a substantially cylindrical transducer body may include one or more flats to aid in the assembly of the transducer body as discussed above.

(46) FIG. 8 is a schematic diagram of a circuit 100 suitable for use with embodiments of a quartz crystal pressure transducer according to embodiments of the present disclosure. As shown in FIG. 8, the circuit 100 includes a first oscillator 102 driven by a first amplifier 104 for driving a reference crystal (e.g., one of resonators 1a, 1b, 1c (FIGS. 1A-1C)) at a selected frequency (e.g., about 7.2 MHz). The circuit 100 includes one more oscillators (e.g., oscillator 106 driven by amplifier 108) for driving another crystal (e.g., one of resonators 1a, 1b, 1c) that acts as a dual-mode sensor. For example, the oscillator 106 may drive the dual-mode sensor crystal at two different frequencies (e.g., a C-mode of about 7.24 MHz and a B-mode of about 7.8 MHz) to provide both pressure and temperature measurements from a single crystal. In other embodiments, two oscillators may be utilized to drive the single crystal to provide both pressure and temperature measurements from the single crystal. A frequency signal from the reference crystal may be sent to a processor 110 (e.g., a microcomputer) for further processing, if desired, and that may be outputted to a reference output F.sub.REF.

(47) One or more frequency signals from the dual-mode sensor crystal (e.g., two frequency signals created by the oscillator 106 driving the crystal at two different frequencies) may be may be sent to the processor 110 for further processing, if desired, and for use in the equations for temperature and pressure as set forth above. The results of those calculations may be outputted to output F.sub.TEMP and output F.sub.REF.

(48) In contrast with the state of the art as exemplified by Schodowsky and U.S. Pat. No. 4,545,638 to EerNisse and Ward, embodiments of the present disclosure do not employ the use of harmonically related vibrational modes that require inclusion of the fundamental mode to obtain the required temperature sensitivity and, consequently, avoid the requirement of a relatively large resonator bore diameter and the associate disadvantages indicated above.

(49) For example, in a practical implementation of an embodiment of the present disclosure, any harmonic higher than the fundamental is about the same mode shape and, therefore, usable. Consequently, embodiments of the present disclosure may employ the 3.sup.rd harmonic of both the B- and C-modes, or a 3.sup.rd of one of the B-mode and the C-mode and a 5.sup.th of the other, for temperature calculation and compensation purposes.

(50) While the disclosure may be susceptible to various modifications and alternative forms, specific embodiments have been shown by way of example in the drawings and have been described in detail herein. However, it should be understood that the invention is not intended to be limited to the particular forms disclosed. Rather, the invention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the following appended claims.