Frequency-dependent equivalent modelling of broadband air-coupled transducers

The estimation is performed in a constrained piecewise and stepwise manner. The proposed method of frequency-dependent parameters estimation is given step by step in figure 2. The main procedure includes initial values calculation, data reconstruction, estimation and results distribution.

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3.1.1. Initial values calculation.

3.1.2. Data reconstruction.

Before estimation, the original impedance data is reconstructed to facilitate piecewise fitting. Impedance data reconstruction is illustrated in figure 3. The original frequency-impedance data is equally divided into N segments, including the resonant and non-resonant segments. Each segment has a frequency bandwidth of M  ×  Δf  (Δf  is frequency interval). The total number of data points is M  ×  N  +  1. The segments containing resonance frequencies are called resonant segments while the others are non-resonant segments. New impedance data is reconstructed with the combination of the kth (k is integer) segment and the resonant segments chosen from the original impedance data.

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The two resonance frequencies f _s1 and f _s2 are chosen to be fitted on with sufficient accuracy to provide overall constraint to the estimation of the kth segment due to its high importance to the frequency response [23]. Otherwise, improper parameters are likely to be obtained by piecewise fitting without considerations of overall frequency response. Things will get worse as the number of segments increases.

3.1.3. Parameters estimation.

3.1.4. Results distribution.

In [18], a method of calculating initial values of broadband underwater transducers has been proposed. In this section, a more concise method of calculating initial values of air-coupled broadband transducers is presented.

In this study, main focus is given to transducers with double resonances. The input admittance Y of equivalent circuit shown in figure 1(b) with two resonances when n  =  2 is formulated as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle Y=&\,\frac{I}{V}=G+{\rm i}B={\rm i}\omega {{C}_{0}}+\frac{1}{\left( {{R}_{1}}+{\rm i}\omega {{L}_{1}}+\frac{1}{{\rm i}\omega {{C}_{1}}} \right)}\nonumber \\ &+\frac{1}{\left( {{R}_{2}}+{\rm i}\omega {{L}_{2}}+\frac{1}{{\rm i}\omega {{C}_{2}}} \right)},\nonumber \end{align} \tag{ 5 }$$

where G and B are conductance and susceptance, respectively, which can be readily deduced from equation (5):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle G=\frac{{{R}_{1}}}{R_{1}^{2}+{{\left( \omega {{L}_{1}}-\frac{1}{\omega {{C}_{1}}} \right)}^{2}}}+\frac{{{R}_{2}}}{R_{2}^{2}+{{\left( \omega {{L}_{2}}-\frac{1}{\omega {{C}_{2}}} \right)}^{2}}},\nonumber \end{align} \tag{ 6 }$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle B={\rm i}\omega {{C}_{0}}-\frac{\left( \omega {{L}_{1}}-\frac{1}{\omega {{C}_{1}}} \right)}{R_{1}^{2}+{{\left( \omega {{L}_{1}}-\frac{1}{\omega {{C}_{1}}} \right)}^{2}}}-\frac{\left( \omega {{L}_{2}}-\frac{1}{\omega {{C}_{2}}} \right)}{R_{2}^{2}+{{\left( \omega {{L}_{2}}-\frac{1}{\omega {{C}_{2}}} \right)}^{2}}}.\nonumber \end{align} \tag{ 7 }$$

For a single resonance shown in figure 4, the value of equivalent resistor R_n is equal to G_max at which frequency resonance occurs:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{R}_{n}}=\frac{1}{{{G}_{\max }}}.\nonumber \end{align} \tag{ 8 }$$

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In [25], the definition of the quality factor Q relating L_n and R_n is given as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle Q=\frac{{{\omega }_{s}}{{L}_{1}}}{{{R}_{1}}}.\nonumber \end{align} \tag{ 9 }$$

The quality factor Q can be obtained from figure 4:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle Q=\frac{{{\omega }_{s}}}{b}=\frac{{{\omega }_{s}}}{{{\omega }_{2}}-{{\omega }_{1}}},\nonumber \end{align} \tag{ 10 }$$

where ${{\omega }_{1}}$ and ${{\omega }_{2}}$ are frequencies where the susceptance B reaches its maximum and minimum value, respectively, corresponding to frequency points of half G_max.

Combining equations (9) and (10), the inductance L_n is calculated as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{L}_{n}}=\frac{{{R}_{1}}}{{{\omega }_{2}}-{{\omega }_{1}}}.\nonumber \end{align} \tag{ 11 }$$

With the condition on which resonance occurs:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \omega {{L}_{n}}-\frac{1}{\omega {{C}_{n}}}=0,\nonumber \end{align} \tag{ 12 }$$

the equivalent capacitance C_n is obtained as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{C}_{n}}=\frac{1}{\omega _{s}^{2}{{L}_{n}}}.\nonumber \end{align} \tag{ 13 }$$

Parameters of the two branches are calculated independently. The assumption is made that no coupling effect exits between adjacent resonances when calculating R_n, L_n and C_n independently. With the calculated values of R_n, L_n and C_n, C₀ can be calculated by the following expression considering the effects of both resonances:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{C}_{0}}=\frac{{{B}_{{{\omega }_{x}}}}}{{{\omega }_{x}}}+\frac{{{L}_{1}}-\frac{1}{\omega _{x}^{2}{{C}_{1}}}}{R_{1}^{2}+{{\left( {{\omega }_{x}}{{L}_{1}}-\frac{1}{{{\omega }_{x}}{{C}_{1}}} \right)}^{2}}}+\frac{{{L}_{2}}-\frac{1}{\omega _{x}^{2}{{C}_{2}}}}{R_{2}^{2}+{{\left( {{\omega }_{x}}{{L}_{2}}-\frac{1}{{{\omega }_{x}}{{C}_{2}}} \right)}^{2}}},\nonumber \end{align} \tag{ 14 }$$

where ω_x is one of the frequency points with a susceptance value ${{B}_{{{\omega }_{x}}}}$. For simplicity, ω_x is chosen as ω₁ at which frequency susceptance reaches its maximum in the second resonance.

This method as described above takes nine values of G and B while the method in [18] takes 11 values to determine all the seven equivalent parameters. And this method takes the effect of both resonances on static capacitance C₀ into account. Assumption that no coupling effect exits between adjacent resonances when calculating R_n, L_n and C_n independently is more tenable, since a higher Q of air-coupled transducers is expected than that of underwater transducers. This assumption will be examined and verified using sensitivity analysis in section 4.

The objective function is of significant importance to the accuracy of the estimated results. Instead of using admittance (G and B) in optimization process, impedance (modulus |Z| and phase angle T) is adopted due to their relation with all the parameters to be estimated, while G has nothing to do with the static capcitance C₀. Both modulus and phase of impedance should be taken into account in the objective function. As formulated below, the objective function consists of squared errors between measured and estimated results of modulus and phase of impedance:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle F=&\,\frac{1}{J}\sum\nolimits_{j=1}^{J}{\left( {{\left( \frac{{{\left| Z \right|}_{{\rm mea}}}\left(\,j \right)-{{\left| Z \right|}_{{\rm est}}}\left(\,j \right)}{{{Z}_{{\rm mea}}}\left(\,j \right)} \right)}^{2}}\right.}\nonumber \\ &{\left.+{{\left( \frac{{{T}_{{\rm mea}}}\left(\,j \right)-{{T}_{{\rm est}}}\left(\,j \right)}{{{T}_{{\rm mea}}}\left(\,j \right)+\theta } \right)}^{2}} \right)},\nonumber \end{align} \tag{ 15 }$$

where |Z|_mea(j ) and |Z|_est(j ) are the measured and estimated modulus of impedance at j th frequency point, respectively, T_mea(j ) and T_est(j ) are the measured and estimated phase angle of impedance at j th frequency point, respectively, J is the number of frequency points of the measured impedance, θ is an offset angle. The angle θ serves as a offset parameter used to banlancing the enormous difference between the relative errors of phase angle at resonance frequencies and off-resonance frequencies. To this end, the value of θ should be set to avoid near-zero data when calculate the relative errors of phase angle of impedance. And in this case, θ is set to 180°. In practice, without θ, the optimization algorithm could hardly converge to the optimum values and the optimization could not succeed eventually as mentioned in [26]. The objective function is minimized to find optimum values of equivalent parameters based on GA.

The sensitivity analysis provides an effective means of gaining insight into the internal relation between the model-based electrical impedance and equivalent parameters. In this section, sensitivity analysis is performed on both impedance and admittance. To investigate the influence of equivalent circuit parameters on impedance, sensitivity analysis is calculated based on the partial derivative of impedance Z with respect to the mth parameter P_m. Since impedance Z is the inverse of admittance Y, it is more intricate to take the partial derivative of Z with respect to P_m directly. To circumvent this problem, composition function Z(Y(ω, P_m)) can be constructed and the partial derivative of Z(Y(ω, P_m)) with respect to P_m is deduced using the chain rule instead of direct calculation of the partial derivative of Z(ω, P_m) with respect to P_m:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\partial Z\left( \omega ,{{P}_{m}} \right)}{\partial {{P}_{m}}}&=\frac{\partial Z\left( Y\left( \omega ,{{P}_{m}} \right) \right)}{\partial {{P}_{m}}}=\frac{\partial Z\left( Y \right)}{\partial Y}\frac{\partial Y\left( \omega ,{{P}_{m}} \right)}{\partial {{P}_{m}}}\nonumber \\ &=-\frac{1}{{{Y}^{2}}}\frac{\partial Y\left( \omega ,{{P}_{m}} \right)}{\partial {{P}_{m}}}.\nonumber \end{align} \tag{ 16 }$$

A parameter-scaled sensitivity of impedance Z to parameters P_m has been formulated by multiplying the partial derivative of Z(ω, P_m) by the corresponding parameter P_m in [26, 27], resulting in sensitivity coefficients with a unit of Ω. In this study, some large sensitivity coefficients are expected and inconvenience may be caused to compare different parameters effect. For better comparison, normalized sensitivity can be further obtained through dividing the parameter-scaled sensitivity by impedance Z:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{S}_{Z}}\left( \omega \right)=\frac{{{P}_{m}}}{Z}\frac{\partial Z\left( \omega ,{{P}_{m}} \right)}{\partial {{P}_{m}}}=-\frac{{{P}_{m}}}{Z}\frac{1}{{{Y}^{2}}}\frac{\partial Y\left( \omega ,{{P}_{m}} \right)}{\partial {{P}_{m}}}=-\frac{{{P}_{m}}}{Y}\frac{\partial Y\left( \omega ,{{P}_{m}} \right)}{\partial {{P}_{m}}}\nonumber \end{align} \tag{ 17 }$$

where P_m is mth parameter, P_m  ∈  {R_n, L_n, C_n, C₀}, n is the branch number of equivalent circuit (n  =  1,2). The partial derivatives of Y(ω, P_m) with respect to P_m are derived as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\partial Y\left( \omega ,{{P}_{m}} \right)}{\partial {{R}_{n}}}=\frac{{{\left( \omega {{L}_{n}}-\frac{1}{\omega {{C}_{n}}} \right)}^{2}}-R_{n}^{2}}{{{\left( R_{n}^{2}+{{\left( \omega {{L}_{n}}-\frac{1}{\omega {{C}_{n}}} \right)}^{2}} \right)}^{2}}}+j\frac{2{{R}_{n}}\left( \omega {{L}_{n}}-\frac{1}{\omega {{C}_{n}}} \right)}{{{\left( R_{n}^{2}+{{\left( \omega {{L}_{n}}-\frac{1}{\omega {{C}_{n}}} \right)}^{2}} \right)}^{2}}},\nonumber \end{align} \tag{ 18 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\partial Y\left( \omega ,{{P}_{m}} \right)}{\partial {{L}_{n}}}=\frac{2{{R}_{n}}\left( \frac{1}{{{C}_{n}}}-{{\omega }^{2}}{{L}_{n}} \right)}{{{\left( R_{n}^{2}+{{\left( \omega {{L}_{n}}-\frac{1}{\omega {{C}_{n}}} \right)}^{2}} \right)}^{2}}}+j\omega \frac{{{\left( \omega {{L}_{n}}-\frac{1}{\omega {{C}_{n}}} \right)}^{2}}-R_{n}^{2}}{{{\left( R_{n}^{2}+{{\left( \omega {{L}_{n}}-\frac{1}{\omega {{C}_{n}}} \right)}^{2}} \right)}^{2}}},\nonumber \end{align} \tag{ 19 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\partial Y\left( \omega ,{{P}_{m}} \right)}{\partial {{C}_{n}}}=&\,\frac{2{{R}_{n}}\left( \frac{1}{{{\omega }^{2}}{{C}_{n}}}-{{L}_{n}} \right)}{C_{n}^{2}{{\left( R_{n}^{2}+{{\left( \omega {{L}_{n}}-\frac{1}{\omega {{C}_{n}}} \right)}^{2}} \right)}^{2}}}\nonumber \\ &+j\frac{{{\left( \omega {{L}_{n}}-\frac{1}{\omega {{C}_{n}}} \right)}^{2}}-R_{n}^{2}}{\omega C_{n}^{2}{{\left( R_{n}^{2}+{{\left( \omega {{L}_{n}}-\frac{1}{\omega {{C}_{n}}} \right)}^{2}} \right)}^{2}}},\nonumber \end{align} \tag{ 20 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\partial Y\left( \omega ,{{C}_{0}} \right)}{\partial {{C}_{0}}}=j\omega .\nonumber \end{align} \tag{ 21 }$$

The sensitivity of admittance to parameters is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{S}_{Y}}\left( \omega \right)=\frac{{{P}_{m}}}{Y}\frac{\partial Y\left( \omega ,{{P}_{m}} \right)}{\partial {{P}_{m}}},\nonumber \end{align} \tag{ 22 }$$

which is the negative of the sensitivity of impedance to parameters.

A piezoelectric plate and an acoustic matching layer constitute basic component of a typical air-coupled transducer. The use of acoustic matching layer can improve the sensitivity and bandwidth of transducers simultaneously. A homemade air-coupled transducer applied to gas flow measurement shown in figure 5 is used in this study. The impedance of transducer was measured using an impedance analyzer (HP 4294A, Agilent Technologies, Inc., Santa Clara, CA). To eliminate random errors and obtain more precise results, 16 measurements of impedance were averaged and recorded. The vibration velocity was measured by a laser doppler vibrometer (Polytec NLV-2500, Polytech GmbH, Waldbronn, Germany) as shown in figure 6. All measurements were conducted at room temperature.

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Using equations (8), (11), (13) and (14), raw values of R₁, L₁, C₁, R₂, L₂, C₂ and C₀ are calculated and listed in table 1. With GA optimization, a set of parameters values with improved accuracy is further obtained as shown in table 1. Compared with the calculated values, the estimated values witness a visible change. The value of R₁ changes more than the values of L₁ and C₁, while the percentage change in L_n and C_n have similar absolute value but opposite sign. Minor improvement percentage in R₁, L₁ and C₁ was obtained than that in R₂, L₂ and C₂ after GA-based optimization. Similar variation results of equivalent parameters after being refined using regression method can be found in [18].

With initial values refined by GA, frequency-dependent equivalent parameters of the transducer were determined using the proposed method in section 3. The estimation results of frequency-dependent parameters are shown in figure 7. The equivalent parameters show evident frequency-dependency. In some frequency band, the frequency-dependent parameters coincide with constant parameters. This means that BVD with constant parameters can model the electrical impedance accurately over specific frequency range while could not model impedance over broadband without frequency-dependent parameters. In both ends of impedance data, the marginal effect induced by the results distribution procedure is also weakened as the number of segments increases. With the frequency-dependent parameters, improved BVD model could accurately model electrical impedance with more simplified model.

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4.3.1. Measurement uncertainty.

Both magnitude and phase of impedance that are used for estimating the equivalent parameters are measured through the impedance analyzer and consequently are subjected to measurement uncertainties. The sources of uncertainties associated with a measurement include systematic and random errors. The instrument error is considered to be the main source of systematic errors and is given by the manufacturer's specifications, while the random errors can be evaluated through the standard deviation of multiple measurements.

The uncertainties of the impedance measurement are estimated in this subsection accordingly to the methodologies suggested in [28]. Considering that the mean of 16 times measurements is used as final results, the type A evaluation of uncertainty can be calculated by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{u}_{A}}=\frac{{{S}_{x}}}{\sqrt{h}}=\sqrt{\frac{\sum{{{\left( {{x}_{j}}-{{{\bar{x}}}_{j}} \right)}^{2}}}}{h\left( h-1 \right)}},\nonumber \end{align} \tag{ 23 }$$

where ${{x}_{j}}$ is the measurement magnitude and phase at j th frequency point, ${{\bar{x}}_{j}}$ is the mean of the measurement results at j th frequency point, h is the number of measurements.

The type B evaluation of uncertainty is related to the basic impedance accuracy (±0.08%) given in the manufacture's specifications of HP4294A. And on assumption of uniform distribution, the type B evaluation of uncertainty is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{u}_{B}}=\frac{{{{\bar{x}}}_{j}}\times 0.08\%}{\sqrt{3}}.\nonumber \end{align} \tag{ 24 }$$

The combined standard uncertainty is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{u}_{C}}=\sqrt{u_{A}^{2}+u_{B}^{2}}.\nonumber \end{align} \tag{ 25 }$$

The relative combined standard uncertainty is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{u}_{{\rm Cr}}}=\frac{{{u}_{C}}}{{{{\bar{x}}}_{j}}}\times 100\%.\nonumber \end{align} \tag{ 26 }$$

As shown in figure 8, the relative combined standard uncertainties of magnitude and phase are frequency-dependent. The relative combined standard uncertainties of magnitude are within 0.04%–0.1% in the band of interest. It is expected that the relative combined standard uncertainties of phase are unusually large around the resonant frequencies due to the near-zero phase in the denominator when calculated the relative uncertainty. The relative combined standard uncertainties of phase are typically within  ±0.5% at off-resonance frequencies. For a transducer used for gas flow measurement, the working frequency is around 200 kHz. The relative combined standard uncertainties of magnitude and phase at 200 kHz are 0.05709% and 0.05241%, respectively. The small uncertainty values indicate stable and reliable measurements.

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4.3.2. Modelling error.

The effectiveness of the modified BVD model with frequency-dependent parameters is examined by comparison between modelling errors of the two modelling methods. As shown in figure 9, modelling using BVD model with constant parameters achieved poorer fitting results to the measured impedance compared to modelling based on improved BVD with frequency-dependent parameters, especially over the frequency range of 190–200 kHz where spurious resonance locate. For better comparison, relative percentage error between fitted and measured impedance is plotted in figure 10. The more the number of segments, the less the percentage error of the results fitting with frequency-dependent parameters. It is understandable that as the number of segments increases, less data points per segment are used in the estimation. Therefore, with the less strict constraints imposed by local and overall resonant segments as the number of segments increases, more accurate estimation can be obtained. Meanwhile, these constraints can prevent anomalous estimation results. Comparing the impedance fitting errors with constant and frequency-dependent parameters, it can be concluded that the equivalent model with constant parameters are unable to reproduce accurately the impedance data of transducers, indicating that it is necessary to take account of the variation of the equivalent parameters with frequency.

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In addition to visual inspection, quantitative analysis of fitting accuracy is also presented. The relative root-mean-square error (rRMSE) of modulus and phase angle of impedance is calculated by,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm rRMSE}\left( \left| Z \right| \right)=\sqrt{\frac{1}{J}\sum\nolimits_{j=1}^{J}{{{\left( \frac{{{\left| Z \right|}_{{\rm mea}}}\left(\,j \right)-{{\left| Z \right|}_{{\rm est}}}\left(\,j \right)}{{{Z}_{{\rm mea}}}\left(\,j \right)} \right)}^{2}}}},\nonumber \end{align} \tag{ 27 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm rRMSE}\left( T \right)=\sqrt{\frac{1}{J}\sum\nolimits_{j=1}^{J}{{{\left( \frac{{{T}_{{\rm mea}}}\left(\,j \right)-{{T}_{{\rm est}}}\left(\,j \right)}{{{T}_{{\rm mea}}}\left(\,j \right)+\theta } \right)}^{2}}}},\nonumber \end{align} \tag{ 28 }$$

respectively, where j  is index of impedance data and J is the number of impedance data.

The rRMSEs are listed in percentage in table 2 and shown in figure 11. The rRMSEs of modelling with constant parameters is depicted in figure 11 as the first data point. The rRMSEs of both modulus and phase of impedance. The rRMSEs of modulus is higher than that of phase of impedance with  +180° offset when number of segments is within 20. When the number of segments increases to 40, the rRMSE of phase with  +180° offset exceeds that of modulus. When number of segments reaches 280, the rRMSEs of modulus and phase are 0.09% and 0.12%, respectively. Overall, the magnitude of rRMSE of modulus and phase is comparable and decrease exponentially with the increase of number of segments. In this regard, the proposed method can give good fitting on both modulus and phase of impedance. Both visual inspection and the rRMSE in table 2 reveal that the modified BVD model with the obtained frequency-dependent parameters provides an accurate description for the given electrical impedance.

Table 2. rRMSE between measured and fitted impedance in percentage (%).

	Modulus	Phase angle (with offset angle 180°)	Phase angle (without offset angle)
Constant parameters	6.96	2.65	88.46
Segs  =  8	3.11	1.13	20.55
Segs  =  20	0.95	0.91	14.53
Segs  =  40	0.45	0.76	9.75
Segs  =  80	0.29	0.53	3.47
Segs  =  140	0.16	0.27	2.77
Segs  =  280	0.09	0.12	1.74

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In our study, +180° offset is applied to the phase of impedance as a workaround to avoid near-zero data in phase when calculating objective function. The effect of offset angle on the rRMSEs of phase is shown in figure 11. In spite of large numerical difference, the rRMSEs of phase with and without  +180° offset show similar descending trend. The numerical difference mainly due to near-zero data in phase around resonance when calculating relative errors. In view of large difference in rRMSEs of modulus and phase without offset, fitting without phase offset may have difficulty in giving good fitting on both modulus and phase [26]. It can be safely concluded that the use of offset angle is beneficial to the estimation.

The transmitting frequency response (T_R1 and T_R2) of each motional branch modelled based on BVD with constant and frequency-dependent parameters is calculated using equation (4). For a transducer with double resonances, the overall TR is calculated by adding the amplitude of the two T_R1 and T_R2 together and compared with the measurement vibration velocity in figure 12. As the number of segments increases to 20, spurious resonance around 195 kHz can be modelled using the modified BVD with frequency-dependent parameters while could not be modelled based on BVD with constant parameters.

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The computational time cost is also examined using 'tic-toc' tool provided by MATLAB and shown in figure 13. As the number of segments increases from 8 to 280, time cost decreases dramatically from 6594.2 s to 469.1 s. It makes sense that, with the number of segments increases, although more times of optimization are required to give an overall fitting to impedance over the frequency range of interest, much less data points per run lead to less time cost. The time cost is reasonably acceptable since the parameters estimation is an off-line work hence time cost is not a major concern [16].

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With the optimal constant parameters estimated by GA, normalized sensitivity coefficients are calculated by equation (17) and illustrated in figure 14. In figure 14(a), it can be seen that the modulus of impedance is more sensitive to the variation of L_n and C_n than that of R_n (n  =  1, 2), while the variation of L_n and C_n have nearly identical influence on the modulus. In spite of large sensitivity values difference, the response of modulus to the variation of R_n, L_n and C_n show similar trend. It also worth note that R_n, L_n and C_n possess more significant effect on the modulus located in its resonance band than that outside the resonance band. Compared with the effect of the variation of R_n, L_n and C_n, the effect of the variation of C₀ on the modulus shows a unique trend and a semblable tendency with the modulus. In figure 14(b), with overlapped sensitivity curves observed, the variation of L_n and C_n have the same effect on the phase of impedance. It can be safely concluded that the variation of L_n and C_n exerts influence on impedance including both modulus and phase in an analogous manner. It is interesting that the variation of L₂ and C₂ has similar impact on the phase of impedance with the variation of C₀ in the frequency range of 160–190 kHz, while the effect of the variation of L₁ and C₁ on the phase is similar with that of the variation of C₀ in the rest frequency range.

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The sensitivity coefficients of admittance Y to parameters P_m are calculated by equation (22) and plotted in figure 15. As shown in figure 15, the admittance around series resonance corresponding to one branch in the equivalent circuit has little to do with the parameters in the other branch. This is mainly due to the high Q of transducers in air. And with sufficient frequency spacing between the two resonance frequencies, the assumption made in section 3 to obtain initial values is valid reasonably.

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