Evaluation of transducer for cryogenic actuators by equivalent circuit model

The mechanism of preload application is shown in Fig. 1. External metal shrinks thermally as the temperature decreases. The internal metals shrink in the same way. In this case, if the thermal expansion coefficient of the internal metals is smaller than that of the external metal, the amount of shrinkage of the internal metal will be smaller. This difference in thermal contraction between the external and internal metals applies a preload to the piezoelectric element.

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Structure of the transducer is shown in Fig. 2. This transducer is a longitudinal vibrator that uses vibration in the thickness direction of the piezoelectric element. The transducer consists of a body made of SUS304, inner parts made of 42 invar, nickel alloy, electrode plates made of cupper, and piezoelectric elements made of lead zirconate titanate, PZT (Fuji Ceramics Company, Ltd., C-213). The 42 invar has a lower thermal expansion coefficient than SUS304. Thus, preload can be applied to the PZT due to this difference in thermal contraction. ^36–38)

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By using a FEM, static structural analysis was conducted to estimate the von-Mises stress applied to the piezoelectric element due to thermal contraction of the parts caused by the temperature decrease. The linear expansion coefficients of the materials used in the analysis ^36–38) and the static structural analysis result is shown in Figs. 3, 4. In Fig. 4, the relationship between stress and temperature converged to a specific value when several analytical mesh densities of the piezoelectric element were considered. Therefore, the analytical results are shown when the number of divisions in the thickness direction is 4, which is the smallest analytical mesh density among those considered. Figure 4(b) shows the averaged von-Mises stresses on the top surface of piezoelectric element. It also shows that the preload increased with decreasing temperature, reaching a maximum at 45 K.

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Next, modal analysis was conducted to obtain the resonance frequencies in the longitudinal vibration mode of the transducer. The modal analysis result is shown in Fig. 5. The result shows that the resonance frequency at RT, 295 K and 4.5 K are 49.0 kHz and is 48.9 kHz.

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The measured admittance values at 295 K and 4.5 K are shown in Fig. 10. The admittance gain at low temperature is larger than that at RT, and the phase difference is also larger. Compared to the FEM simulation results, the resonance frequency at RT is smaller. The above admittance $Y$ is obtained by the following equation as

$$\begin{eqnarray}&&Y=G+{jB},\end{eqnarray} \tag{ 1 }$$

where $G$ is conductance and $B$ is susceptance, and the admittance loops are defined with ${G}$ on the real axis and $B$ on the imaginary axis. ³⁹⁾ The admittance loops obtained from the measured admittance are shown in Fig. 11. The Q factor can be obtained from the admittance loop by the following equation as

$$\begin{eqnarray}&&{Q}_{m}=\frac{{\omega }_{s}}{{\omega }_{2}-{\omega }_{1}},\end{eqnarray} \tag{ 2 }$$

where ${\omega }_{s}$ is the series resonance angular frequency, ${\omega }_{1}$ is the quadrant angular frequency that gives the maximum susceptance, and ${\omega }_{2}$ is the quadrant angular frequency that gives the minimum susceptance. ³⁹⁾ Figure 12 shows the estimated Q factors when the chamber temperature was changed. The admittance loops at low temperatures are larger than those at RT. The increase in Q factors with decreasing temperature is due to the increase in elasticity of the material that constitutes the transducer and the decrease in damping. The physical meaning of the Q factor is how much more reactive energy is stored than the energy consumed by the mechanical resistance, ⁴⁴⁾ and high conversion efficiency can be expected as an electromechanical conversion device in low-temperature environments. The constants of simplified electrical equivalent circuit can be obtained from Figs. 11 and 12. $R,$ $L,$ $C,$ and ${C}_{d}$ of the simplified electrical equivalent circuit can be obtained by the following equations as

$$\begin{eqnarray}&&R=\frac{1}{{G}_{\max }},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&L=\frac{{Q}_{m}R}{{\omega }_{s}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&C=\frac{1}{{\omega }_{s}^{2}L},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{C}_{d}=\frac{{\omega }_{s}^{2}C}{{\omega }_{s}^{2}-{\omega }_{p}^{2}},\end{eqnarray} \tag{ 6 }$$

where G_max is the maximum value of conductance and ${\omega }_{p}$ is the parallel resonance angular frequency. ³⁹⁾ The calculated constants are shown in Table I. Values except reactance are smaller at low temperatures than at RT.

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Table I. Equivalent circuit constants when the chamber temperature was changed.

Temperature [K]	$R[{\rm{\Omega }}]$	$C[{\rm{F}}]$	$L[{\rm{H}}]$	${C}_{d}[{\rm{F}}]$
295	$1.69\times {10}^{3}$	$6.45\times {10}^{-12}$	$2.02$	$6.42\times {10}^{-9}$
77	$1.55\times {10}^{2}$	$6.37\times {10}^{-12}$	$1.73$	$3.88\times {10}^{-9}$
20	$3.25\times {10}^{2}$	$2.83\times {10}^{-12}$	$3.92$	$2.54\times {10}^{-9}$
4.5	$2.55\times {10}^{2}$	$1.53\times {10}^{-12}$	$7.25$	$2.15\times {10}^{-9}$

Next, the measured transducer end surface vibration velocities at ambient temperatures of 295 K and 4.5 K and applied voltages of 20 V and 120 V as peak to peak values are shown in Fig. 13. Figure 13 shows that the nonlinearity of the vibration velocity tends to become stronger as the applied voltage increases. Additionally, the relationship between the applied voltage and the vibration velocity of the transducer end surface when the chamber temperature was changed is shown in Fig. 14. At all temperatures, the vibration velocity increases as the applied voltage increases, indicating a linear relationship between applied voltage and the vibration velocity. Then, Fig. 15 shows the relationship between the vibration velocity of the transducer's end surface and temperature when the applied voltage was changed. For all applied voltages, the vibration velocity increases with decreasing temperature compared to RT. These results show that sufficient preload is applied to the piezoelectric element as the temperature decreases. The finite element analysis results show that the preload applied to the piezoelectric element increased as the temperature decreased, and the admittance loops were larger at the lower temperature. This suggests that the vibration velocity is larger at low temperatures than at RT. However, the temperatures with maximum properties differ between the analytical and experimental results. There are two possible reasons for this problem. The physical properties of 42 invar at low temperatures have not been evaluated. In the FEM simulation, those of 36 invar, which has a similar composition to that of 42 invar, was used, and the data was obtained from the literature. ³⁸⁾ Because the physical properties used in the analysis differ from those of the obtained material, there may be a difference between the analytical and the experimental results. In addition, the FEM results are about the temperature characteristics of the von-Mises stress in the piezoelectric element. In those results, the stress applied to the piezoelectric element is largest at the temperature of 45 K. On the other hand, in experiments, measured admittance and vibration velocity values of the transducer are maximum at 77 K. It is necessary to take into account parameters other than the preload on the piezoelectric element for the vibration characteristics.

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The force factor was calculated from the values in Table I and the measured vibration velocity. The force factor $A$ can be obtained by the following equation as

$$\begin{eqnarray}&&A=\frac{{I}_{m}}{v},\end{eqnarray} \tag{ 7 }$$

where ${I}_{m}$ is the dynamic current in Fig. 9, and $v$ is the vibration velocity. ³⁹⁾ As ${I}_{m}$ cannot be measured experimentally, it can be obtained from the values in Table I. The vibration velocity at 20 V as peak to peak values is used to calculate the force factor. The calculated force factors when the chamber temperature was changed are shown in Table II. Table II shows that the change in force factors with decreasing temperature is small. This tendency indicates that the efficiency of converting electrical energy into mechanical energy is maintained at the same level as at RT even in a low-temperature environment by applying a preload to the piezoelectric element.

Table II. Force factors estimated by using equivalent circuit.

Temperature [K]	$A[{\rm{A}}\bullet {\rm{s}}/{\rm{m}}]$
295	$5.36\times {10}^{-1}$
77	$9.21\times {10}^{-1}$
20	$5.01\times {10}^{-1}$
4.5	$6.87\times {10}^{-1}$

A figure of merit can be implemented as another evaluation indicator. The figure of merit indicates how well the transducer performs, the higher the value, the better the performance of the transducer. The figure of merit $M$ can be calculated as

$$\begin{eqnarray}&&M=\frac{{Q}_{m}}{\gamma },\end{eqnarray} \tag{ 8 }$$

where $\gamma$ is the capacitance ratio, expressed as $\gamma ={C}_{d}/C.$ ⁴⁵⁾ From the equation, the larger ${Q}_{m}$ and the smaller $\gamma$ lead to a larger $M.$ The calculated $\gamma$ and $M$ values are shown in Table III. From Table III, the figure of merit $M$ is larger at low temperatures than at RT. Those results indicate that the transducers fabricated in this study have high conversion efficiency as electromechanical conversion elements in a low-temperature environment.

Table III. Figure of merit and capacitance ratio estimated by using equivalent circuit.

Temperature [K]	$\gamma$	$M$
295	995	0.33
77	609	5.51
20	898	4.03
4.5	1405	6.07