Analysis and equivalent circuit for accurate wideband calculations of the impedance for a piezoelectric transducer having loss

Others have analyzed the operation of piezoelectric transducers by defining two complex parameters which have the units of frequency to allow for the effects of loss. The present paper presents an analysis in which this procedure is extended to include harmonics as well as the fundamental frequency. In this extension it is seen that both positive and negative extrema in both the resistance and the conductance occur at a series of harmonics. Equivalent circuits are also presented with examples for both high-Q and low-Q materials showing agreement with these simulations.Others have analyzed the operation of piezoelectric transducers by defining two complex parameters which have the units of frequency to allow for the effects of loss. The present paper presents an analysis in which this procedure is extended to include harmonics as well as the fundamental frequency. In this extension it is seen that both positive and negative extrema in both the resistance and the conductance occur at a series of harmonics. Equivalent circuits are also presented with examples for both high-Q and low-Q materials showing agreement with these simulations.


I. INTRODUCTION
Berlincourt et al. 1 and Meeker 2 derived expressions for the electrical impedance of a piezoelectric disk transducer in the thickness modes when the loss is negligible. Sherrit et al. 3 wrote this equation in the following form, and Sherrit and Mukerjee 4 presented other equations having a similar form for modes with other types of piezoelectric transducers.
Here ω is the angular frequency, and the relevant material constants and calculated parameters for Eq. (1) that were defined and evaluated for the first two examples by Sherrit et al. 3 are given in Tables I  and II. For clarity we use the symbol Fp which is defined in Table II instead of fp, where the symbols fp and fs are defined and used later in the present derivation. Berlincourt et al. 1 and Meeker 2 assumed zero loss so they used real values for all of the material parameters which requires that the impedance is purely reactive with singularities at frequencies that are odd integer multiples of Fp. However, Sherrit et al. 3 used measured values for the material parameters and dimensions that are complex to include the effects of loss and are given in Table I which we have used in examples. Table II shows the corresponding calculated parameters.

II. ANALYSIS SHOWING THE RESONANCES
Equation (1) may be written as the difference of two separate terms. The first, which is referred to as the "baseline impedance", varies inversely with the frequency while the second term has a series of resonances that are superimposed on the baseline impedance. The ratio of the second term to the baseline impedance is given by the following expression: Figure 1 shows the magnitude of this ratio for the first 13 resonances (n = 1, 3, 5, . . ., 25) calculated using the material constants and calculated parameters for the high-Q and low-Q examples in Tables I and II. Note that the ratio decreases monotonically with the multiplier and the low-Q example has no peak beyond the 10th harmonic. In narrowband applications, such as with a piezoelectric resonator, it may be sufficient to consider only the first resonance. However, now we generalize and extend the equivalent circuit model ARTICLE scitation.org/journal/adv  to a much greater bandwidth than the single resonance studied by Sherrit et al. 3 The IEEE/ANSI standard on piezoelectricity 5 defines two parameters, fp and fs, as the real values for the frequencies at maximum resistance and maximum conductance, respectively. These two parameters are using Eq. (1) and its reciprocal for the admittance. These two parameters enable defining an equivalent circuit to provide the resonant frequencies for the cases of parallel and series excitation, respectively. We follow the extension of Eq. (1) which was made by Sherrit et al. 6 for the case of lossy materials for which fp and fs are complex numbers.
In searches of the complex frequency plane that we made using Eq. (1) and the corresponding equation for the admittance we have found a sequence of sharply-defined maxima for both the real part of the impedance Re[Z], and the real part of the admittance Re[Y].  Tables I and II. We define the complex parameters γ and α such that fp = γFp and fs = αFp respectively at the maxima for Re [Z] and Re [Y] where Fp was defined in Table II. A. Effect of parameter γ on the real part of the complex impedance For the case of zero loss, at which Z is purely imaginary, there is a singularity when γ is an odd real integer. Thus, to provide continuity to the case of low loss, we examine the behavior of Eq. (1) for the impedance at γ = n + δ when n is an odd integer and δ is small and may be complex.
Equation (6) shows that there is a singularity in the impedance which occurs as the modulus |δ| → 0 regardless of the argument of δ. This is consistent with our numerical simulations and confirms Sherrit's use of Fp as fp when he only considered the first resonance. 3 The singularity in the resistance R at the frequency f = fp requires that the susceptance B must be zero at this frequency. This may be seen by examining the relationship of the impedance to the admittance. For Y = G +jB, Re[Z] = G/(G 2 +B 2 ) is bounded at all values of G when B is non-zero, but Re[Z] may be singular when B = 0.

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Conversely, for Z = R +jX, then Re[Y] = R/(R 2 +X 2 ) is bounded for all values of R when X is non-zero, but Re[Y] may be singular when X = 0. Figure 2 shows the real part of the complex impedance Z in the region of the complex frequency plane near the first resonance for the high-Q example in Tables I and II. The imaginary part of γ has the range of -10 -9 to 10 -9 with zero at the midpoint. The real part of γ has the range of 1-10 -9 to 1+10 -9 with 1 is at the midpoint. Figure 2 shows the neighborhood of the singularity near γ = 1 for the first resonance.
B. Effect of parameter α on the real part of the complex admittance  Figure 3 shows the real part of the complex admittance in the region of the complex frequency plane near the first resonance for the high-Q example in Tables I and II. The center of the complex-Y plane in this figure is at fs = αFp, where Re[α] = 0.8964968342 and Im[α] = 0.0022944067, where each of these parts is specified to 10 decimal places because the values of Y were determined at increments of 10 -10 in both Re[α] and Im [α]. The imaginary part of α has a range of 0.0022944067-10 -9 to 0.0022944067+10 -9 , and the real part of α has a range of 0.8964968342 -10 -9 to 0.8964968342 +10 -9 . Figure 3 verifies that the susceptance Re[Y] has a sharp maximum but is non-singular.
In generating the data for Fig. 3, α was evaluated by maximizing Re[Y] based on Eq. (1), but now a simpler method for evaluating α will be presented. Equation (1) Tables I and II.  Tables I and II. αt, may be used to maximize Re[Y(αt)] to determine α: It is not possible for the denominator in Eq. (7) to be exactly zerofor which Y would be singular. However, α may be approximated by minimizing | πα t 2 − kt 2 tan( πα t 2 )|. Figure 4 shows the imaginary part of α as a function of the harmonic (multiplier) minus the real part of α for the first five values of α. The harmonics are in decreasing order from left to right so the abscissa is 9 -α R , for the high-Q and low-Q examples at the LHS and 1-α R at the RHS. This figure shows that α I ≈ C(nα R ) for all of the harmonics for a particular piezoelectric resonator where C is a constant that depends on the properties of that device. We have shown that accurate and efficient determination of α for multiple resonances with a single device is possible by using the technique described in the previous paragraph with the approximate relationship for the real and imaginary parts of α. Sherrit et al. 7 have made the approximation of determining fp and fs by maximizing Re[fZ] and Re[Y/f] respectively. This simplifies the calculations but this procedure may be questioned because multiplying or dividing by the complex frequency causes a rotation in the complex plane. We have found that this approximation causes a small error which may generally be neglected, but all of the solutions for fp and fs in this paper were determined using Eq. (1) and its equivalent for admittance without further approximations.

III. EXTENSION OF THE SHERRIT EQUIVALENT CIRCUIT
The Butterworth-Van Dyke model for a piezoelectric resonator 8-10 has a resistor, inductor, and capacitor in series (R-L-C), all shunted by a second capacitor C 0 as shown in Fig. 5. Van Dyke was the first to propose this equivalent circuit, and he suggested extending the model by having multiple R-L-C circuits in parallel to include the effects of multiple resonances as shown in Fig. 6. 9 The Sherrit model is shown in Fig. 7 3 and our proposed extension of the Sherrit model for multiple resonances is shown in Fig. 8. Table III gives the parameters for Butterworth-Van Dyke (BVD) and Sherrit equivalent circuits 3 for the high-Q and low-Q examples.   Figures 9 and 10 show the resistance and reactance simulated with these two equivalent circuits and with the analytical solution from Eq. (1). These calculations were only made for the high-Q example in Tables I and II. Figure 9 shows that the resistance for the Sherrit equivalent circuit is consistent with the analytical solution with the exception of missing the higher-order resonances. However, while the BVD model is accurate near the first resonance the resistance is too small above and below the resonance. This may be understood because well below the single resonance the resistance remains approximately constant since the current is divided by C 0 and C 1 , and the only loss is in the resistor R 1 which is in series with C 1 . The BVD model is also inaccurate well above resonance where the inductor L 1 causes a greater fraction of the current to flow through capacitor C 0 instead of through resistor R 1 . Figure 10 shows that the reactance calculated for the BVD and Sherrit equivalent circuits is in good agreement with that from Eq. (1), with the exception of the higher-order resonances that are seen in the analytical solution. This may be understood because well below the single resonance the reactance is approximately that for C 0 and C 1 or C 0 ' and C 1 ' in parallel, and well above the resonance the inductor causes the reactance to approximate that of C 0 , or C 0 ', and the resonance is quite sharp for the high-Q example. Figures 9 and 10, are log-log plots to clarify that the resistive component of the impedance has two components as was previously stated; a baseline impedance varying inversely with the frequency as well as superimposed resonances.
We acknowledge that others have also used single and multibranch equivalent circuits, 11,12 but Sherrit et al. 3 derived complex functions for the circuit elements from Eq. (1), and simulations using their equivalent circuit are consistent with Eq. (1) except for neglecting the higher-order resonances. The introduction of complex expressions for the circuit elements correctly associates the different types of loss with each circuit element. The following discussion considers our proposed extension of the Sherrit model which is shown in Fig. 8.
We begin by requiring that at low-frequencies the impedance of the equivalent circuit must agree with Eq. (1), so that now we must require the following: where N is the number of branches in the equivalent circuit. Thus, once the capacitance in each branch has been determined we evaluate C 0 with the following expression: ARTICLE scitation.org/journal/adv

Model Component
High-Q (Q = 100) Low-Q (Q = 10) BVD C 0 , F 1.95x10 -9 1.94x10 -9 C 1 , F 4.77x10 -10 4.99x10 -10 L 1 , H 5.54x10 -5 5.42x10 -5 R 1 , Ω 3.83 37.7 Sherrit C 0 ', F 1.95x10 -9 /1.20 ○ 1.95x10 -9 /1.42 ○ C 1 ', F 4.76x10 -10 /2.69 ○ 4.71x10 -10 /9.59 ○ L 1 ', H 5.55x10 -5 /1.82 ○ 5.98x10 -5 /21.9 ○ The IEEE/ANSI standard on piezoelectricity 5 defines the parameters fp and fs as real values of the frequencies for maximum resistance and maximum conductance, respectively. Sherrit et al. 6 appear to be the first to treat fp and fs as complex variables to be compatible with lossy piezoelectric resonators and use this procedure to predict multiple resonances; in this case the analysis was for the radial mode instead of the thickness mode which we have studied. Table IV gives the complex values of fp and fs that we calculated for the first 5 resonances using iteration with Eq. (1): To avoid confusion, we have labeled the parameter fp used by Sherrit et al. 3 as Fp in this paper. The parameter Fp in Eq. (1) is a function of the material parameters which is used in Eq. (1) to determine the impedance at any frequency. Notice that f p1 is not exactly equal to Fp, but rather f p1 must also be determined by the process that has just been defined and used to prepare Table V. This difference is more pronounced with the low-Q example.
Now that fpn and fsn have been determined in Table V, the components Cn' and Ln' in the proposed extension of the Sherrit model may be evaluated with Eqs. (10) and (11), and then C 0 ' may be determined with Eq. (9).
The admittance of the equivalent circuit may be calculated as follows: We have used Eq. (13) to determine the impedance of our extension of the Sherrit model for the equivalent circuit because the modular nature of this equation makes it possible to determine the contribution for each branch separately and then these values may be combined to obtain the solution for various values of N. Figure 11 shows the impedance calculated for the equivalent circuit with N = 5 (circles) and using Eq. (1) for the analytical solution (solid lines).  Good agreement of these two methods is seen for both the high-Q and low-Q examples. We have only simulated the thickness mode, and acknowledge that, for this wide of a frequency range, other modes would also be present. Furthermore we have only included the effects of one of the thickness modes where three are possible as described by Balato. 13 Table V gives the values for Cn' and Ln' that were used in these calculations.

IV. CONCLUSIONS
1. The complex frequency for parallel resonance (fp) is equal to an integer multiple of the parameter which we call Fp and need not be found by the general method of iteration. 2. The complex frequency for series resonance (fs) must be determined by iteration since the real part of Y is bounded at this point in the complex frequency plane, which occurs because the imaginary part of Z is non-zero at that same point. However, simpler means have been determined for these iterations. 3. An equivalent circuit has been developed which accurately predicts the impedance from DC through the first five resonances, in agreement with the equation for the impedance, and may be extended for use over a greater frequency range.