Method for measuring equivalent circuit parameters and resonant frequency of piezoelectric resonator

Abstract

A method for measuring equivalent circuit parameters and resonant frequency of a piezoelectric resonator, by which the phase-frequency curve of the piezoelectric resonator is measured, and the resonant frequency and the anti-resonant frequency are obtained. Then, the slopes of the phase-frequency curve at the resonant frequency and the anti-resonant frequency are respectively measured. The resonant angular frequency and the anti-resonant angular frequency are also calculated. Finally, the equivalent circuit parameters of the piezoelectric resonator are obtained by solving a system of nonlinear equations.

Claims

1. A method for measuring equivalent circuit parameters and a resonant frequency of a piezoelectric resonator comprising: measuring a phase-frequency curve of the piezoelectric resonator and finding a zero phase point on the phase-frequency curve, namely, obtaining the resonant frequency ƒ.sub.r and an anti-resonant frequency ƒ.sub.a; measuring a slope of the phase-frequency curve at the resonant frequency f.sub.r, recording the slope as Δ.sub.1; and measuring a slope of the phase-frequency curve at the anti-resonant frequency ƒ.sub.a, recording the slope as Δ.sub.2; calculating a resonant angular frequency with a formula ω.sub.r=2πƒ.sub.r, and calculating an anti-resonant angular frequency with a formula ω.sub.a=2πƒ.sub.a; and substituting ω.sub.r, ω.sub.a, Δ.sub.1, and Δ.sub.2 into the following system of nonlinear equations and solving the nonlinear equations to obtain the equivalent circuit parameters of the piezoelectric resonator including a static capacitance C.sub.0, a motional capacitance C.sub.1, a motional inductance L.sub.1, and a motional resistance R.sub.1; $\{\begin{array}{c}\frac{{\omega }_a^2+{\omega }_r^2}{2}=\frac{1}{L_1{C}_1}+\frac{1}{2L_1{C}_0}-\frac{R_1^2}{2L_1^2}\\ {\omega }_r^2{\omega }_a^2=\frac{1}{L_1^2{C}_1^2}\left(1+\frac{C_1}{C_0}\right)\\ \frac{\partial \left(\mathrm{Pashe}\left(f\right)+{\mathrm{Phase}}_{\mathrm{offset}}\right)}{\partial f}{|}_{f=f_r}={\Delta }_1\\ \frac{\partial \left(\mathrm{Pashe}\left(f\right)+{\mathrm{Phase}}_{\mathrm{offset}}\right)}{\partial f}{|}_{f=f_a}={\Delta }_2\end{array}\hspace{1em}$ wherein Phase.sub.offset, is a phase offset value, Pashe(ƒ) is a phase-frequency function, and ƒ is a frequency point on the phase-frequency curve.

2. The method for measuring the equivalent circuit parameters and the resonant frequency of the piezoelectric resonator according to claim 1, wherein the Pashe(ƒ) meets the following formula: $\mathrm{Pashe}\left(f\right)=\frac{180}{\pi }.Math.\arctan \left(\frac{\mathrm{imag}\left(G\left(f\right)\right)}{\mathrm{real}\left(G\left(f\right)\right)}\right)$ $\mathrm{where},G\left(f\right)=\frac{1}{Z\left(f\right)+R_0},$ Z(ƒ) is an impedance of a piezoelectric crystal, and R.sub.0 is an internal resistance of a vector network analyzer.

3. The method for measuring the equivalent circuit parameters and the resonant frequency of the piezoelectric resonator according to claim 1, wherein the piezoelectric resonator comprises a high-Q value crystal piezoelectric resonator, a quartz crystal microbalance piezoelectric resonator, and a microelectromechanical system piezoelectric resonator.

4. The method for measuring the equivalent circuit parameters and the resonant frequency of the piezoelectric resonator according to claim 1, wherein the piezoelectric resonator is made of quartz, lithium tantalate, lanthanum gallium silicate, piezoelectric ceramic lead zirconate titanate (PZT), or aluminum nitride (AlN).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a diagram showing a relation between a measurement error of a resonant frequency and variations of wire length and load according to an IEC-based method;

(2) FIG. 2 is a diagram showing a BVD equivalent model of a piezoelectric crystal;

(3) FIG. 3 is a comparison block diagram of three measurement solutions;

(4) FIG. 4 is a schematic diagram of an ADS software (Advanced Design System) simulation;

(5) FIG. 5 is a diagram showing a simulation result of the ADS software; and

(6) FIG. 6 is a diagram showing an actually measured phase-frequency curve of a QCM (Quartz Crystal Microbalance) subjected to a water load and the phase-frequency curve derived from the calculated parameters.

DETAILED DESCRIPTION OF THE EMBODIMENTS

(7) The specific embodiments of the present invention will be described hereinafter with reference to the drawings, so that those skilled in the art can better understand the present invention. It should be particularly noted that in the following description, when a detailed description of the well-known functions and designs may downplay the main content of the present invention, these descriptions will be omitted herein.

EMBODIMENT

(8) FIG. 2 is a diagram showing a BVD equivalent model of a piezoelectric crystal.

(9) In the present embodiment, the piezoelectric resonator may be a high-Q value crystal piezoelectric resonator, a quartz crystal microbalance piezoelectric resonator, or a microelectromechanical system piezoelectric resonator. As shown in FIG. 2, the piezoelectric crystal is connected to a vector network analyzer to measure S21 characteristics, and the measurement mode is switched to a phase measurement mode. Besides, the piezoelectric resonator may be made of quartz, lithium tantalate, lanthanum gallium silicate, piezoelectric ceramic lead zirconate titanate (PZT), and aluminum nitride (AlN).

(10) The method for measuring equivalent circuit parameters and resonant frequency of a piezoelectric resonator of the present invention includes the following steps:

(11) S1: the phase-frequency curve of the piezoelectric resonator is actually measured through the vector network analyzer and the zero phase point on the phase-frequency curve is found out, i.e., the resonant frequency ƒ.sub.r and the anti-resonant frequency ƒ.sub.a are obtained;

(12) S2: a slope of the phase-frequency curve at the resonant frequency ƒ.sub.r is measured and recorded as Δ.sub.1; and a slope of the phase-frequency curve at the anti-resonant frequency ƒ.sub.a is measured and recorded as Δ.sub.2;

(13) S3: a resonant angular frequency is calculated with a formula ω.sub.r=2πƒ.sub.r, and an anti-resonant angular frequency is calculated with a formula ω.sub.a=2πƒ.sub.a; and

(14) S4: ω.sub.r, ω.sub.a, Δ.sub.1, and Δ.sub.2 are substituted into the following system of nonlinear equations and the nonlinear equations are solved to obtain the equivalent circuit parameters of the piezoelectric resonator: static capacitance C.sub.0, motional capacitance C.sub.1, motional inductance L.sub.1, and motional resistance R.sub.1;

(15) $\{\begin{array}{c}\frac{{\omega }_a^2+{\omega }_r^2}{2}=\frac{1}{L_1{C}_1}+\frac{1}{2L_1{C}_0}-\frac{R_1^2}{2L_1^2}\\ {\omega }_r^2{\omega }_a^2=\frac{1}{L_1^2{C}_1^2}\left(1+\frac{C_1}{C_0}\right)\\ \frac{\partial \left(\mathrm{Pashe}\left(f\right)+{\mathrm{Phase}}_{\mathrm{offset}}\right)}{\partial f}{|}_{f=f_r}={\Delta }_1\\ \frac{\partial \left(\mathrm{Pashe}\left(f\right)+{\mathrm{Phase}}_{\mathrm{offset}}\right)}{\partial f}{|}_{f=f_a}={\Delta }_2\end{array}\hspace{1em}$
where Phase.sub.offset is a phase offset value, Pashe(ƒ) is a phase-frequency function value, and ƒ is a frequency point on the phase-frequency curve.

(16) In the present embodiment, assuming there are N frequency points on the phase-frequency curve, then ƒ may be represented as ƒ.sub.1, ƒ.sub.2, . . . , ƒf.sub.N;

(17) then the phase-frequency function value Pashe(ƒ.sub.i) on the i.sub.th frequency point meets the following equation:

(18) $\mathrm{Pashe}\left(f_i\right)=\frac{180}{\pi }.Math.\mathrm{atan}\left(\frac{\mathrm{imag}\left(G\left(f_i\right)\right)}{\mathrm{real}\left(G\left(f_i\right)\right)}\right)$$\mathrm{where},G\left(f_i\right)=\frac{1}{Z\left(f_i\right)+R_0},$
Z(ƒ.sub.i) is an impedance of the piezoelectric crystal, imag() represents an imaginary part, real( ) represents a real part, and R.sub.0 is an internal resistance of the vector network analyzer;

(19) then a phase difference of each frequency point is:
ΔPashe.sub.i=Pashe(ƒ.sub.i)+Phase.sub.offset−Pashe.sub.measure(ƒ.sub.i)
where Pashe.sub.measure(ƒ.sub.i) represents the actual phase value of the i.sub.th frequency point ƒ.sub.i;

(20) then a root mean square error of the phase differences is calculated as follows:

(21) $\mathrm{RMSE}=\sqrt{\frac{1}{N}\underset{1}{\overset{N}{.Math.}}\Delta {\mathrm{Pashe}}_i}$

(22) The RMSE is minimized by continuously changing the Phase.sub.offset, at that time, the calculated equivalent circuit parameters, resonant frequency and anti-resonant frequency are final results.

(23) FIG. 3 is a comparison block diagram of three measurement solutions.

(24) In the present embodiment, the IEC-based measurement method is shown in FIG. 3(a). The measurement method proposed by Dong Liu in 2017 is shown in FIG. 3(b), and the measurement method of the present invention is shown in FIG. 3(c).

(25) The IEC-based method requires a pi-type network to facilitate impedance matching and has a relatively greater error.

(26) The measurement method proposed by Dong Liu in 2017 requires load capacitance and series resistance, and the measurement error is related to the resistance. This method also needs to switch the state between with and without load-capacitance, which is difficult been integrated in the current scientific instruments as software modules.

(27) The technical solution of the present invention requires no external circuit and no impedance matching, and is directly connected in series to the vector network analyzer. This solution is easy to operate, and has no theoretical error and small measurement error. Simply by measuring four points of the phase-frequency curve, so the resonant frequency, the anti-resonant frequency, and four equivalent circuit parameters can be calculated. The phase-frequency curve inversed from the parameters fits with the actually measured curve at more than 800 points, and the root mean square error of the 800 points is below 0.1107.

Embodiment 1

(28) The simulation experiment was carried out with the ADS (Advanced Design System) software. The schematic diagram is shown in FIG. 4. The phase-frequency curve of the simulation is shown in FIG. 5. The results obtained are shown in Table 1.

(29) TABLE-US-00001 TABLE 1 R.sub.1(Ω) L.sub.1(mH) C.sub.1(fF) C.sub.0(pF) Set value 14.0000 75.0000 3.3600 3.0000 Calculated value 14.0050 75.0000 3.3600 3.0000

Embodiment 2

(30) The solution of the embodiment 2 is similar to embodiment 1. With varying set values of the motional resistance R.sub.1, the calculated motional resistance R.sub.1, which is obtain from the method proposed by the present invention, is shown in Table 2 respectively. The resonant resistance Rr, is also shown in Table 2. The resonant resistance Rr can be obtain from substitution methods similar to IEC.

(31) TABLE-US-00002 TABLE 2 R.sub.r calculated R.sub.1 calculated by the by the Set value IEC-based proposed of R.sub.1 method method 14 14.0094 14.0050 50 55.4998 50.0019 70 90.4585 70.0037

(32) As shown in Table 2, the motional resistance calculated according to the method proposed by the present invention is closer to the set value. The resonant resistance is close to the set value, when the set R1 is small. However, the resonant resistance Rr is different from the motional resistance R1, when the set R1 is bigger.

Embodiment 3

(33) The parameters of a QCM wafer loaded by a drop of water are measured with the vector network analyzer N9913A. The phase-frequency curve measured with the N9913A is shown as a solid line in FIG. 6. According to our nonlinear equations and a MATLAB numerical solution thereof, the obtained equivalent parameters are shown in Table 3.

(34) TABLE-US-00003 TABLE 3 parameter R.sub.r (Ω) R.sub.1(Ω) L.sub.1(mH) C.sub.1(fF) C.sub.0(pF) QCM water 380 321.8300 12.8864 20.0005 4.2462 load

(35) The phase-frequency curve inverted from the parameters measured by the method of the present invention is shown as a dotted line in FIG. 6, and it can be seen that the inversion curve and the measured curve are almost coincident.

(36) The resonant resistance Rr measured by the substitution method, which is similar to the IEC-based method, is also given in Table 3. It can be concluded that the R.sub.1 measured by the method of the present invention is smaller than the Rr which is consistent with the simulation results. Also, it can be concluded from Table 3 that when QCM is loaded with liquid, a large gap exists between Rr and R.sub.1.

Embodiment 4

(37) The phase offsets caused by different loads are measured. The phase offsets of the same QCM wafer with no load, water load, or oil load are measured by the method of the present invention and shown in Table 4.

(38) TABLE-US-00004 TABLE 4 QCM load situation No-load Water load Oil load phase offset −15.0441 −15.6539 −17.0441 value (deg.)

(39) It can be concluded that the present invention can directly measure the zero phase point drifts caused by the variations of load. Therefore, the present invention does not have the measurement errors caused by the variations of load as shown in FIG. 1.

Embodiment 5

(40) The phase offsets caused by different wire lengths are measured. Two quartz crystals with brackets of different lengths are numbered as 1# and 2#, respectively. The phase offsets measured by the method of the present invention are shown in Table 5.

(41) TABLE-US-00005 TABLE 5 Crystal number Crystal 1# Crystal 2# phase offset value −45.0005 −35.1400 (deg.)

(42) It can be concluded that the method of the present invention can directly measure the phase offset caused by the variations of load or the wire length without introducing the measurement errors caused by the variations of load and the wire length as shown in FIG. 1.

(43) The specific embodiments of the present invention are described above to facilitate the understanding of those skilled in the art. It should be noted, however, that the present invention is not limited to the scope of the specific embodiments. As for those of ordinary skill in the art, as long as the various variations fall within the spirit and scope of the present invention as defined and determined by the appended claims, these variations are obvious. All inventions derived from the idea of the present invention are covered by the present invention.