WO2007013702A1 - A method for calculating frequency characteristics of piezoelectric material non-linearly according to temperature rising - Google Patents

A method for calculating frequency characteristics of piezoelectric material non-linearly according to temperature rising Download PDF

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Publication number
WO2007013702A1
WO2007013702A1 PCT/KR2005/001874 KR2005001874W WO2007013702A1 WO 2007013702 A1 WO2007013702 A1 WO 2007013702A1 KR 2005001874 W KR2005001874 W KR 2005001874W WO 2007013702 A1 WO2007013702 A1 WO 2007013702A1
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Prior art keywords
temperature
equation
piezoelectric material
resonant frequency
converged
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PCT/KR2005/001874
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French (fr)
Inventor
Chang-Hwan Lee
Hyun-Woo Joo
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Samhwa Yang Heng Co., Ltd.
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Publication of WO2007013702A1 publication Critical patent/WO2007013702A1/en

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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01K—MEASURING TEMPERATURE; MEASURING QUANTITY OF HEAT; THERMALLY-SENSITIVE ELEMENTS NOT OTHERWISE PROVIDED FOR
    • G01K7/00—Measuring temperature based on the use of electric or magnetic elements directly sensitive to heat ; Power supply therefor, e.g. using thermoelectric elements
    • G01K7/32—Measuring temperature based on the use of electric or magnetic elements directly sensitive to heat ; Power supply therefor, e.g. using thermoelectric elements using change of resonant frequency of a crystal
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • H—ELECTRICITY
    • H10—SEMICONDUCTOR DEVICES; ELECTRIC SOLID-STATE DEVICES NOT OTHERWISE PROVIDED FOR
    • H10N—ELECTRIC SOLID-STATE DEVICES NOT OTHERWISE PROVIDED FOR
    • H10N30/00—Piezoelectric or electrostrictive devices
    • H10N30/40—Piezoelectric or electrostrictive devices with electrical input and electrical output, e.g. functioning as transformers

Definitions

  • the invention relates to a method of obtaining a frequency characteristic depending on temperature chagne of a piezoelectric material in a non-linear manner. According to the method of the invention, it is possible to quickly and accurately calculate nonlinear characteristics (for example, temperature variation, and variations of a material constant and a resonant frequency) of a piezoelectric material depending on an increase in temperature when it is driven in a high electric field.
  • nonlinear characteristics for example, temperature variation, and variations of a material constant and a resonant frequency
  • a piezoelectric material is widely used for an actuator, a sensor and an electronic part so as to accurately control a position in many industrial fields.
  • a temperature of the piezoelectric material is increased due to mechanical vibration loss and dielectric loss in the piezoelectric material.
  • the temperature increase causes a change of a material constant of the piezoelectric material, thereby changing an operating frequency of the piezoelectric material and deteriorating a mechanical quality factor thereof.
  • the invention has been made to analyze non-linear characteristics of a piezoelectric material. According to a method of the invention, it is possible to calculate a temperature of a piezoelectric material reflecting a value of a material constant which is varied as a temperature of the piezoelectric material is changed under state that a resonant frequency is fixed. In addition, by tracing the resonant frequency varied as the temperature of the piezoelectric material is changed, it is possible to analyze the temperature change of the piezoelectric material under condition of the traced resonant frequency. Accordingly, it is possible to quickly and accurately calculate the non-linear characteristics depending on the temperature increase when driving the piezoelectric material, for example the variations of the material constant and the resonant frequency.
  • an object of the invention is to provide a method of analyzing nonlinear characteristics of a piezoelectric material when driving the piezoelectric material.
  • the invention relates to a method of obtaining a frequency characteristic depending on a temperature change of a piezoelectric material in a non-linear manner.
  • the method of the invention comprises steps of: (a) calculating a resonant frequency of a piezoelectric material at a predetermined temperature; (b) calculating a temperature of the piezoelectric material under predetermined resonant frequency; (c) obtaining a material constant of the piezoelectric material under predetermined temperature condition; (d) calculating a temperature of the piezoelectric material using the material constant of the piezoelectric material obtained in the step (c); (e) repeating the steps (c) and (d) until the temperature obtained in the step (d) is converged; (f) calculating a changed resonant frequency of the piezoelectric material under the converged temperature obtained in the step (e); (g) calculating a converged temperature by performing the steps (b) to (e) under the resonant frequency obtained in the step (f); and (h) repeating the steps (f) and (g) until the converged temperature obtained in the step (g) is converged.
  • step (e) it is possible to calculate a temperature of the piezoelectric material reflecting the material constant changed as the temperature of the piezoelectric material is changed under state that the resonant frequency is fixed.
  • step (h) it is possible to trace the resonant frequency changed as the temperature of the piezoelectric material is changed and to analyze the temperature change of the piezoelectric material under condition of the traced resonant frequency. Accordingly, it is possible to quickly and accurately calculate the non-linear characteristics depending on the temperature increase when driving the piezoelectric material.
  • the process of calculating the resonant frequency in the step (a) or (f) may comprise steps of calculating an electric potential ⁇ by the material constant under predetermined temperature and a characteristic equation of a mathematical equation 13; calculating an impedance using the electric potential ⁇ obtained in the above step, a charge Q, and a following mathematical equation 16; and determining a frequency as a resonant frequency, which frequency exhibits the smallest impedance of the impedances obtained by repeating the above steps while changing the frequency.
  • the step (b) may be a step of calculating the temperature of the piezoelectric material under resonant frequency obtained in the step (a) or (f).
  • the process of calculating the temperature of the piezoelectric material in the step (b) or (d) preferably comprises steps of: calculating a mechanical vibration loss P and a dielectric loss P of the piezoelectric material per unit volume using m d following mathematical equations 8a and 8b; assuming that heat generation Q of an interior per unit volume is equal to a summation of the mechanical vibration loss P and the dielectric loss P in a heat transfer equation of a following mathematical equation 9; and calculating a temperature distribution of the piezoelectric material from a mathematical equation 11 which is a differential equation applying a convection boundary condition and a finite element method to the mathematical equation 9.
  • each element of A , A , F and F matrix are as following mathematical equation k s Q s
  • N polynomial interpolation function
  • the material constant of the piezoelectric material obtained in the step (c) is data that a material constant depending on temperatures is previously obtained in an experiment.
  • the step (c) is preferably a step of obtaining a material constant of the piezoelectric material under temperature condition obtained in the step
  • the steps (c) and (d) are preferably repeated until the temperature variation of the piezoelectric material resulting from a change of the material constant of the piezoelectric material under state that the resonant frequency is fixed is converged within a predetermined critical value ⁇ .
  • the steps (T) and (g) are preferably repeated until a difference between (i) a converged temperature obtained in the step (e) under condition of resonant frequency currently set and (ii) a converged temperature obtained in the step (e) under condition of resonant frequency previously set is converged within a predetermined critical value ⁇ .
  • FIGS. 1 and 2 are schematic views of plane and horizontal cross-sections of an outline vibration type piezoelectric transformer which is used a simulation test for verifying a method of the invention, respectively;
  • FIGS. 3 and 4 show amplitude and phase of no load impedances of the piezoelectric transformer model shown in FIGS. 1 and 2, using a three-dimensional finite element method of the piezoelectric material;
  • FIG. 5 shows a waveform of impedances at an input terminal of the piezoelectric transformer, when electric load of 100 ⁇ is connected to an output terminal of the piezoelectric transformer model shown in FIGS. 1 and 2;
  • FIGS. 6 and 7 show variations of a voltage drop ratio and a resonant frequency of a piezoelectric transformer depending on loads;
  • FIGS. 8 and 9 are plane view and section view of a second test model of a
  • FIGS. 10 and 11 show a waveform of no-load impedances regarding a second test model and an impedance waveform at an input terminal when electric load of 100 ⁇ is connected, respectively;
  • FIGS. 12 and 13 show variations of a resonant frequency and a voltage ratio depending on loads of the second test model, respectively;
  • FIG. 14 shows a temperature measuring system for the first test model
  • FIGS. 15 and 16 are graphs comparing temperature variations depending on a change of electric loads, which are measured at temperature measurement points in
  • FIG. 14 [61] FIGS. 17 to 20 show samples for measuring a material constant shown in mathematical equations Ic to Ie; [62] FIGS. 21 to 24 are graphs showing a temperature characteristic of material constants of a piezoelectric material; [63] FIG. 25 is a flow chart showing a non-linear analysis method of a piezoelectric system considering a temperature increase, according to an embodiment of the invention; [64] FIGS. 26 and 27 show analysis results of a temperature distribution for the second test model; and [65] FIG. 28 shows a non-linearity due to the temperature increase of the second test model according to the flow chart shown in FIG. 25.
  • T mechanical stress
  • S mechanical strain
  • E electric field
  • D electric displacement
  • c E , e and ⁇ s are matrix exhibiting material properties, and are elastic modulus, piezoelectric constant and dielectric constant, respectively.
  • V volume[m 3 ]
  • A area to which charge is applied[m ].
  • K piezoelectric coupling coefficient matrix
  • ⁇ and ⁇ are damping coefficients
  • [138] f volume force density vector in an element
  • [139] f surface force density vector in an element surface
  • [140] f force vector in an element point.
  • D is a damping matrix.
  • the damping matrix can be obtained from the damping behavior of a structure. Even when the equation for D of the mathematical equation 7 is used, it is not possible to uu perfectly express the damping behavior of the structure.
  • K , D , K , K and M are defined as described above.
  • FIGs. 1 and 2 are schematic views of plane and horizontal cross-sections of an outline vibration type piezoelectric transformer which is used a simulation test for verifying a method of the invention, respectively.
  • An output terminal of the piezoelectric transformer consists of three layers.
  • Figs. 3 and 4 show waveforms for amplitude and phase of no load impedances of the piezoelectric transformer model shown in Figs. 1 and 2, using a three-dimensional finite element method of the piezoelectric material.
  • the dotted lines in Figs. 3 and 4 show an actual measurement result and the solid lines show a result obtained by a three-dimensional finite element method. As shown, it can be seen that the simulation test result and the actual measurement result are nearly matched.
  • Fig. 5 shows a waveform of impedances at an input terminal of the piezoelectric transformer, when electric load of 100 ⁇ is connected to an output terminal of the piezoelectric transformer model shown in Figs. 1 and 2.
  • the analysis result by the finite element method and the actual measurement result are nearly matched.
  • Figs. 6 and 7 show variations of a voltage drop ratio and a resonant frequency of the piezoelectric transformer depending on loads, respectively. As shown in Figs. 6 and 7, it can be seen that the analysis result by the finite element method and the actual measurement result are nearly matched. When the load is increased, a gain and a resonant frequency of the piezoelectric transformer are also increased. Accordingly, it is possible to predict a characteristic change of the piezoelectric transformer in accordance with a load change of the piezoelectric transformer. In addition, when designing the piezoelectric transformer, the method can be used to determine an optimal load of the piezoelectric transformer.
  • the electric load is analyzed by combining it with a three-dimensional finite element method, so that it is possible to calculate the characteristic change of the piezoelectric transformer depending on loads, more accurately.
  • Figs. 8 and 9 show a second test model of a 20W-piezoelectric transformer.
  • Fig. 8 is a plane view of the test model and
  • Fig. 9 is a sectional view of the test model.
  • Figs. 10 and 11 show a result of a finite element analysis for the second test model.
  • Fig. 10 shows a waveform of no-load impedances and Fig. 11 shows an impedance waveform at an input terminal of the piezoelectric transformer when electric load of 100 ⁇ is connected. As shown in Fig. 10, it can be seen that two results in a band of about 7OkHz are nearly matched.
  • Figs. 12 and 13 show variations of a resonant frequency and a voltage ratio depending on loads of the second test model, respectively.
  • An piezoelectric system has a mechanical hysteresis loss and a electrical hysteresis loss. They are referred to as mechanical vibration loss and dielectric loss, respectively.
  • the loss analysis using a conventional analytical method is a one-dimensional analysis and assumes that a loss distribution of the system is uniform.
  • the loss distribution in the piezoelectric system is not uniform depending on operating modes and shapes, the one-dimensional loss calculation method has a limitation.
  • E absolute value of an electric field.
  • T temperature of surrounding air.
  • N polynomial interpolation function.
  • Fig. 14 show a temperature measuring system for the first test model in an experimental manner. Pl to P4 show temperature measuring points.
  • Figs. 15 and 16 are graphs comparing temperature variations depending on a change of electric loads, which are measured at temperature measurement points in Fig. 14 with a result of a temperature distribution analysis according to the invention.
  • Table 1 shows output power depending on electric loads. [214] Table 1
  • the temperature at the temperature measuring point is lowest when the load is 100 ⁇ . This means that the 20W-piezoelectric transformer has the smallest loss at about 100 ⁇ . Accordingly, 100 ⁇ is the optimal load of the first test model.
  • Figs. 17 to 20 show piezoelectric samples for experimentally measuring a material constant shown in mathematical equations Ic to Ie.
  • the material constant depending on the temperatures of the piezoelectric material is measured by positioning the sample shown in Figs. 17 to 20 in a temperature chamber and measuring resonant and an- tiresonant frequencies of the respective samples using an impedance analyzer while increasing the temperature of the sample.
  • Figs. 21 to 24 are graphs showing a temperature characteristic of material constants of a piezoelectric material. As shown, an elastic modulus exhibiting the mechanical characteristic of the piezoelectric material is relatively little sensitive to the temperature change. However, the piezoelectric constant, the dielectric constant, the mechanical quality factor (Q ) and the dielectric loss factor (tan ⁇ ) for which the loss analysis is necessary are much changed depending on the temperatures. The change of the material constants depending on the temperatures causes a change of the operating characteristic of the piezoelectric system, such as a change of the resonant frequency of the piezoelectric system and an increase in voltage of the piezoelectric transformer.
  • the method of analyzing the non-linear characteristic of the piezoelectric system in consideration of the temperature increase is embodied by using the finite element analysis of the piezoelectric material, the temperature distribution analysis through the loss analysis and the heat transfer equation analysis, and the change of the coefficient of the piezoelectric material depending on the measured temperatures.
  • Fig. 25 is a flow chart showing a method of analyzing a non-linear characteristic of a piezoelectric system considering a temperature increase, according to an embodiment of the invention.
  • a resonant frequency of the piezoelectric system is calculated using the material constant at a predetermined temperature (for example, room temperature) (SlO) and then a temperature increase is calculated at the resonant frequency (S20).
  • the process of calculating the resonant frequency in the step (S 10) or a later step (S60) may comprise steps of calculating an electric potential ⁇ by the material constant under predetermined temperature and a characteristic equation of a following mathematical equation 13; calculating an impedance using the electric potential ⁇ obtained in the above step, a charge Q and a following mathematical equation 16; and determining a frequency as a resonant frequency, which frequency exhibits the smallest impedance of the impedances obtained by repeating the steps while changing the frequency.
  • K , D , K , K , M, u, ⁇ , F, Q and Z are defined as described above.
  • the process of calculating the temperature of the piezoelectric material in the step (S20) or a later step (S40) may comprises steps of: calculating a mechanical vibration loss P and a dielectric loss P of the piezoelectric material per unit volume using m d following mathematical equations 8a and 8b; assuming that heat generation Q of an interior per unit volume is equal to a summation of the mechanical vibration loss P and the dielectric loss P in a heat transfer equation of a following mathematical d equation 9; and calculating a temperature distribution of the piezoelectric material from a mathematical equation 11 which is a differential equation applying a convection boundary condition and a finite element process to the mathematical equation 9. [231] [equation 8a]
  • N polynomial interpolation function
  • Figs. 26 and 27 show analysis results of a temperature distribution for the second test model.
  • Fig. 26 is a temperature increase result analyzed according to the flow chart shown in Fig. 25.
  • Fig. 27 shows an experiment result for the second test model.
  • Fig. 28 shows a non-linearity due to the temperature increase of the second test model according to the flow chart shown in Fig. 25.
  • the dotted line shows a voltage ratio depending on frequency obtained by using the material constant obtained in the experiment at the room temperature
  • the solid line shows a voltage ratio depending on frequency obtained by using the material constant at a temperature increased in consideration of the temperature increase.

Abstract

The invention relates to a method of obtaining a frequency characteristic depending on temperature change of a piezoelectric material in a non-linear manner, comprising steps of: (a) calculating a resonant frequency of a piezoelectric material at a predetermined temperature; (b) calculating a temperature of the piezoelectric material under predetermined resonant frequency; (c) obtaining a material constant of the piezoelectric material under predetermined temperature condition; (d) calculating a temperature of the piezoelectric material using the material constant; (e) repeating the steps (c) and (d) until the temperature is converged; (f) calculating a changed resonant frequency under converged temperature; (g) calculating a converged temperature by performing the steps (b) to (e) under resonant frequency obtained in the step (f); and (h) repeating the steps (f) and (g) until the converged temperature is converged.

Description

Description
A METHOD FOR CALCULATING FREQUENCY CHARACTERISTICS OF PIEZOELECTRIC MATERIAL NON- LINEARLY ACCORDING TO TEMPERATURE RISING
Technical Field
[1] The invention relates to a method of obtaining a frequency characteristic depending on temperature chagne of a piezoelectric material in a non-linear manner. According to the method of the invention, it is possible to quickly and accurately calculate nonlinear characteristics (for example, temperature variation, and variations of a material constant and a resonant frequency) of a piezoelectric material depending on an increase in temperature when it is driven in a high electric field. Background Art
[2] Currently, a piezoelectric material is widely used for an actuator, a sensor and an electronic part so as to accurately control a position in many industrial fields. When the piezoelectric material is driven in a high electric field, a temperature of the piezoelectric material is increased due to mechanical vibration loss and dielectric loss in the piezoelectric material. The temperature increase causes a change of a material constant of the piezoelectric material, thereby changing an operating frequency of the piezoelectric material and deteriorating a mechanical quality factor thereof.
[3] Accordingly, when the piezoelectric material is driven in the high electric field, it is required to analyze temperature increase of the piezoelectric material and to analyze non-linear characteristics (for example, variations of material constant and resonant frequency) due to the temperature increase, so as to effectively drive and control the piezoelectric material. Disclosure of Invention Technical Problem
[4] The invention has been made to analyze non-linear characteristics of a piezoelectric material. According to a method of the invention, it is possible to calculate a temperature of a piezoelectric material reflecting a value of a material constant which is varied as a temperature of the piezoelectric material is changed under state that a resonant frequency is fixed. In addition, by tracing the resonant frequency varied as the temperature of the piezoelectric material is changed, it is possible to analyze the temperature change of the piezoelectric material under condition of the traced resonant frequency. Accordingly, it is possible to quickly and accurately calculate the non-linear characteristics depending on the temperature increase when driving the piezoelectric material, for example the variations of the material constant and the resonant frequency.
[5] Accordingly, an object of the invention is to provide a method of analyzing nonlinear characteristics of a piezoelectric material when driving the piezoelectric material. Technical Solution
[6] The invention relates to a method of obtaining a frequency characteristic depending on a temperature change of a piezoelectric material in a non-linear manner.
[7] More specifically, the method of the invention comprises steps of: (a) calculating a resonant frequency of a piezoelectric material at a predetermined temperature; (b) calculating a temperature of the piezoelectric material under predetermined resonant frequency; (c) obtaining a material constant of the piezoelectric material under predetermined temperature condition; (d) calculating a temperature of the piezoelectric material using the material constant of the piezoelectric material obtained in the step (c); (e) repeating the steps (c) and (d) until the temperature obtained in the step (d) is converged; (f) calculating a changed resonant frequency of the piezoelectric material under the converged temperature obtained in the step (e); (g) calculating a converged temperature by performing the steps (b) to (e) under the resonant frequency obtained in the step (f); and (h) repeating the steps (f) and (g) until the converged temperature obtained in the step (g) is converged.
[8] Through the step (e), it is possible to calculate a temperature of the piezoelectric material reflecting the material constant changed as the temperature of the piezoelectric material is changed under state that the resonant frequency is fixed. In addition, through the step (h), it is possible to trace the resonant frequency changed as the temperature of the piezoelectric material is changed and to analyze the temperature change of the piezoelectric material under condition of the traced resonant frequency. Accordingly, it is possible to quickly and accurately calculate the non-linear characteristics depending on the temperature increase when driving the piezoelectric material.
[9] According to a preferred embodiment of the invention, the process of calculating the resonant frequency in the step (a) or (f) may comprise steps of calculating an electric potential Φ by the material constant under predetermined temperature and a characteristic equation of a mathematical equation 13; calculating an impedance using the electric potential Φ obtained in the above step, a charge Q, and a following mathematical equation 16; and determining a frequency as a resonant frequency, which frequency exhibits the smallest impedance of the impedances obtained by repeating the above steps while changing the frequency.
[10] [equation 13] [H]
Figure imgf000005_0001
[12] [equation 16]
[13]
Φ(ω)
Z(ω) = jωQ
[14] where, K : mechanical stiffness matrix, [15] D : mechanical damping matrix, [16] K : piezoelectric coupling coefficient matrix, uφ [17] K : dielectric hardness matrix, φφ [18] M: mass matrix, [19] u: mechanical displacement, [20] φ: electric potential, [21] F: mechanical force, [22] Q: charge, and [23] Z: impedance. [24] According to a preferred embodiment of the invention, the step (b) may be a step of calculating the temperature of the piezoelectric material under resonant frequency obtained in the step (a) or (f).
[25] At this time, the process of calculating the temperature of the piezoelectric material in the step (b) or (d) preferably comprises steps of: calculating a mechanical vibration loss P and a dielectric loss P of the piezoelectric material per unit volume using m d following mathematical equations 8a and 8b; assuming that heat generation Q of an interior per unit volume is equal to a summation of the mechanical vibration loss P and the dielectric loss P in a heat transfer equation of a following mathematical equation 9; and calculating a temperature distribution of the piezoelectric material from a mathematical equation 11 which is a differential equation applying a convection boundary condition and a finite element method to the mathematical equation 9.
[26] [equation 8a] [27]
Figure imgf000006_0001
[28] [equation 8b] [29]
pd = - x (D0) x (E0) x fi>r x tan δ [Watt/m3]
[30] [equation 9]
[31]
Figure imgf000006_0002
[32] [equation 11]
[33]
Figure imgf000006_0003
[34] where, Q : mechanical quality factor, m
[35] ω : resonant frequency, r
[36] tanδ: dielectric loss factor,
[37] X : absolute value of stress,
[38] S : absolute value of strain,
[39] D : absolute value of an electric displacement
[40] E : absolute value of an electric field,
[41] k: heat transfer coefficient,
[42] T: temperature,
[43] Q: heat generation of an interior per unit volume, and
[44] each element of A , A , F and F matrix are as following mathematical equation k s Q s
12.
[45] [equation 12] (Ak =
Figure imgf000007_0001
(As ),j = N, (hNj )dS
Figure imgf000007_0002
(Fs )1 = N,hTadS
[47] where, N : polynomial interpolation function.
1O
[48] According to the invention, the material constant of the piezoelectric material obtained in the step (c) is data that a material constant depending on temperatures is previously obtained in an experiment. [49] According to the invention, the step (c) is preferably a step of obtaining a material constant of the piezoelectric material under temperature condition obtained in the step
(b) or (d). [50] According to the invention, in the step (e), the steps (c) and (d) are preferably repeated until the temperature variation of the piezoelectric material resulting from a change of the material constant of the piezoelectric material under state that the resonant frequency is fixed is converged within a predetermined critical value ε . [51] In addition, according to the invention, in the step (h), the steps (T) and (g) are preferably repeated until a difference between (i) a converged temperature obtained in the step (e) under condition of resonant frequency currently set and (ii) a converged temperature obtained in the step (e) under condition of resonant frequency previously set is converged within a predetermined critical value ε .
Brief Description of the Drawings [52] FIGS. 1 and 2 are schematic views of plane and horizontal cross-sections of an outline vibration type piezoelectric transformer which is used a simulation test for verifying a method of the invention, respectively; [53] FIGS. 3 and 4 show amplitude and phase of no load impedances of the piezoelectric transformer model shown in FIGS. 1 and 2, using a three-dimensional finite element method of the piezoelectric material; [54] FIG. 5 shows a waveform of impedances at an input terminal of the piezoelectric transformer, when electric load of 100Ω is connected to an output terminal of the piezoelectric transformer model shown in FIGS. 1 and 2; [55] FIGS. 6 and 7 show variations of a voltage drop ratio and a resonant frequency of a piezoelectric transformer depending on loads; [56] FIGS. 8 and 9 are plane view and section view of a second test model of a
20W-piezoelectric transformer; [57] FIGS. 10 and 11 show a waveform of no-load impedances regarding a second test model and an impedance waveform at an input terminal when electric load of 100Ω is connected, respectively; [58] FIGS. 12 and 13 show variations of a resonant frequency and a voltage ratio depending on loads of the second test model, respectively;
[59] FIG. 14 shows a temperature measuring system for the first test model;
[60] FIGS. 15 and 16 are graphs comparing temperature variations depending on a change of electric loads, which are measured at temperature measurement points in
FIG. 14; [61] FIGS. 17 to 20 show samples for measuring a material constant shown in mathematical equations Ic to Ie; [62] FIGS. 21 to 24 are graphs showing a temperature characteristic of material constants of a piezoelectric material; [63] FIG. 25 is a flow chart showing a non-linear analysis method of a piezoelectric system considering a temperature increase, according to an embodiment of the invention; [64] FIGS. 26 and 27 show analysis results of a temperature distribution for the second test model; and [65] FIG. 28 shows a non-linearity due to the temperature increase of the second test model according to the flow chart shown in FIG. 25.
Mode for the Invention [66] Hereinafter, a preferred embodiment of the invention will be specifically described with reference to the drawings. However, it should be noted that the invention is not limited to the embodiment.
[67] First, a basic characteristic equation of a piezoelectric material is described.
[68] Matrix of mathematical equations Ia to Ie are the most basic dominant equations connecting a mechanical phenomenon and an electrical phenomenon in a piezoelectric system. However, it is not considered a non-linearity and a loss component of the piezoelectric material. [69] [equation Ia]
[70] T=cES-eE
[71] [equation Ib] [72] D=eS+ε E [73] [equation Ic] [74]
Figure imgf000009_0001
[75] [equation Id] [76]
0 0 0 0 el5 0 e = 0 0 0 «15 0 0
β31 *3 , e33 0 0 0
[77] [equation Ie] [78]
"11 0 0 εs = 0 0
0 0 :33
[79] where, T: mechanical stress, [80] S: mechanical strain, [81] E: electric field [82] D: electric displacement, and [83] cE, e and εs are matrix exhibiting material properties, and are elastic modulus, piezoelectric constant and dielectric constant, respectively.
[84] When a variation method of Hamilton is applied to the piezoelectric material, following mathematical equations 2a and 2b are provided,
[85] [equation 2a]
Figure imgf000009_0002
[87] [equation 2b] [88] L = E - E + E + W
Mn st d
[89] where, δ: linear variation
[90] L: Lagrange term expressed by energies,
[91] E : kinetic energy, kin
[92] E : elastic energy,
[93] E : dielectric energy, and
[94] W: energy provided from the exterior.
[95] The above energies can be expressed as follows.
[96] [equation 3a]
[97]
Figure imgf000010_0001
[98] [equation 3b]
[99]
Figure imgf000010_0002
[100] [equation 3c]
[101]
Figure imgf000010_0003
[102] [equation 3d]
[103]
Figure imgf000010_0004
[104] where, u: velocity vector [m/s],
[105] V: volume[m3],
[106] f : volume force density vector [N/m ],
B 2
[107] f : surface force density vector[N/m ],
[108] f P: force vector[N],
[109] A : area to which force is applied[m ],
[110] Q : surface charge [As],
[111] Q : point charge[As] , and 2
[112] A : area to which charge is applied[m ].
[113] When polynomial approximate functions of following mathematical equations 4a and 4b are substituted into the mathematical equations 2a, a linear partial differential equation for a finite element analysis of the piezoelectric material is obtained as shown in a following mathematical equation 5.
[114] [equation 4a]
[115]
E = -gradΦ= -grad(NφΦ) = -BΦΦ
[116] [equation 4b]
[117]
S = Bu = BN u = B U
[118] [equation 5]
[119]
- ω 2Mu + j J ωO uu u + K uu u + K uΦ Φ= F B + FS + Fp
KUi + K Φ= Q + Q
[120] where, Φ: electric potential,
[121]
U
: displacement,
[122] B: operator defined as a following mathematical equation 6,
[123] K : mechanical stiffness matrix,
[124] D : mechanical damping matrix, uu
[125] K : piezoelectric coupling coefficient matrix,
Figure imgf000011_0001
[126] K : dielectric hardness matrix,
ΦΦ [127] M: mass matrix,
[128] F : force vector applied to volume,
B
[129] F : force vector applied to surface,
[130] F : force vector applied to point,
[131] Q : surface charge, and
[132] Q : point charge, wherein they can be expressed as a following mathematical equation 7. [133] [equation 6] [134] d/dx 0 0
0 d/dy 0
τ> 0 0 d/dz r> — d /dy d /dx 0
0 d /dz d/dy d/dz 0 d/dx
[135] [equation 7]
[136]
K UU = J fJfJfB U'cEB U dV
D UU =
Figure imgf000012_0001
N1 UNU dV + β " J JJJJfB >1C1B U dV
K = JJJBVB dF
K <]>Φ = J fJfJfB ψ'ε Ε <ι> dV
Figure imgf000012_0002
FB - JJlN:NFBfBJK Fs = JjNlNFSfs dV FP = NT,
Q = -NO
[137] where, α and β are damping coefficients, [138] f : volume force density vector in an element, [139] f : surface force density vector in an element surface and [140] f : force vector in an element point. [141] [142] The damping behavior is determined by D which is a damping matrix. The damping matrix can be obtained from the damping behavior of a structure. Even when the equation for D of the mathematical equation 7 is used, it is not possible to uu perfectly express the damping behavior of the structure.
[143] In order to accurately express the damping behavior of any structure, it is required two or more damping coefficients. However, it is difficult to accurately calculate the damping coefficient. The reason is because the damping coefficient is measured to be different depending on positions and frequencies. Accordingly, it is nearly impossible to accurately measure the damping coefficient and to apply it to the analysis.
[144] The D -related equation of the mathematical equation 7 is approximated with two uu damping coefficients, which is a method of practically and actually considering the damping behavior. When it is used, one of following four damping types is determined, depending on a range of the damping coefficient.
[145] (1) in case of no damping : α=0, β=0 [146] (2) viscous damping : α=0, β>0 [147] (3) damping proportional to mass : α>0, β=0, and [148] (4) Rayleigh damping : α>0, β>0. [149] [150] The two characteristic equations of the mathematical equation 5 obtained from the finite element method of the piezoelectric material can be basically expressed by a single matrix relating to a frequency, as shown in a following mathematical equation 13.
[151] [equation 13] [152]
Figure imgf000013_0001
[153] where, K , D , K , K and M are defined as described above.
UU UU uφ φφ [154] [155] Considering the piezoelectric material used for a piezoelectric transformer for a converter, when load (Z ) is connected to an output terminal of the piezoelectric transformer, a relation of a voltage and a current at the output terminal is as follows:
[156] [equation 14] [157]
φ = IZι = dQ zL
1 dt L
[158] From the equations 13 and 14, it is possible to obtain an analysis equation for analyzing a load condition at the output terminal electroode of the piezoelectric transformer, as shown in a mathematical equation 15:
[159] [equation 15] [160] K1111 - CO
Figure imgf000014_0001
[161] [162] In the followings, it is compared a result of a simulation test for calculating a temperature and a material constant and an actual measurement result, according to the method of the invention.
[163] Figs. 1 and 2 are schematic views of plane and horizontal cross-sections of an outline vibration type piezoelectric transformer which is used a simulation test for verifying a method of the invention, respectively. An output terminal of the piezoelectric transformer consists of three layers.
[164] Figs. 3 and 4 show waveforms for amplitude and phase of no load impedances of the piezoelectric transformer model shown in Figs. 1 and 2, using a three-dimensional finite element method of the piezoelectric material. The dotted lines in Figs. 3 and 4 show an actual measurement result and the solid lines show a result obtained by a three-dimensional finite element method. As shown, it can be seen that the simulation test result and the actual measurement result are nearly matched.
[165] Fig. 5 shows a waveform of impedances at an input terminal of the piezoelectric transformer, when electric load of 100Ω is connected to an output terminal of the piezoelectric transformer model shown in Figs. 1 and 2. As can be seen from Fig. 5, the analysis result by the finite element method and the actual measurement result are nearly matched.
[166] Figs. 6 and 7 show variations of a voltage drop ratio and a resonant frequency of the piezoelectric transformer depending on loads, respectively. As shown in Figs. 6 and 7, it can be seen that the analysis result by the finite element method and the actual measurement result are nearly matched. When the load is increased, a gain and a resonant frequency of the piezoelectric transformer are also increased. Accordingly, it is possible to predict a characteristic change of the piezoelectric transformer in accordance with a load change of the piezoelectric transformer. In addition, when designing the piezoelectric transformer, the method can be used to determine an optimal load of the piezoelectric transformer. Further, instead of considering an electric load by a conventional one-dimensional equivalent circuit method, the electric load is analyzed by combining it with a three-dimensional finite element method, so that it is possible to calculate the characteristic change of the piezoelectric transformer depending on loads, more accurately.
[167] Figs. 8 and 9 show a second test model of a 20W-piezoelectric transformer. Fig. 8 is a plane view of the test model and Fig. 9 is a sectional view of the test model.
[168] Figs. 10 and 11 show a result of a finite element analysis for the second test model.
Fig. 10 shows a waveform of no-load impedances and Fig. 11 shows an impedance waveform at an input terminal of the piezoelectric transformer when electric load of 100Ω is connected. As shown in Fig. 10, it can be seen that two results in a band of about 7OkHz are nearly matched.
[169] Figs. 12 and 13 show variations of a resonant frequency and a voltage ratio depending on loads of the second test model, respectively.
[170]
[171] In the followings, it is described a temperature increase analysis through a three- dimensional heat transfer equation.
[172] An piezoelectric system has a mechanical hysteresis loss and a electrical hysteresis loss. They are referred to as mechanical vibration loss and dielectric loss, respectively. In particular, when the piezoelectric system is driven in a high electric field, it causes a temperature increase. The loss analysis using a conventional analytical method is a one-dimensional analysis and assumes that a loss distribution of the system is uniform. However, since the loss distribution in the piezoelectric system is not uniform depending on operating modes and shapes, the one-dimensional loss calculation method has a limitation.
[173] It is possible to calculate a mechanical displacement of the piezoelectric material and a voltage occurring in the piezoelectric material when a specified voltage is applied, by using the finite element method. In addition, it is possible to obtain the mechanical strain (S) and the electric field (E) by using the mathematical equations 4a and 4b which are solutions of the finite element analysis of the piezoelectric system. By using the calculated mechanical strain (S), electric field (E), stress (X) and electric displacement (D), it is possible to obtain the mechanical vibration loss and dielectric loss of the piezoelectric material per unit time and unit volume, from following mathematical equations 8 a and 8b.
[174] [equation 8a]
[175]
Pm = -±- x (X0) x (S0) x ωr [WattW]
[176] [equation 8b] [177]
Pd = - X (D0) X (E0) x ωr x t<m δ [Watt/m3] [178] where, Q : mechanical quality factor, m
[179] ω : resonant frequency, r
[180] tanδ: dielectric loss factor,
[181] P : mechanical vibration loss, m
[182] P : dielectric loss, d
[183] X : absolute value of stress,
[184] S : absolute value of strain,
[185] D : absolute value of an electric displacement, and
[186] E : absolute value of an electric field.
[187]
[188] By using a heat energy conservation law and Fourier law, it is possible to obtain a differential equation for analyzing a temperature distribution under steady state, as a following mathematical equation 9: [189] [equation 9]
[190]
*V T = -Q
[191] where, k: heat transfer coefficient,
[192] T: temperature and
[193] Q: heat generation of an interior per a unit volume.
[194]
[195] Considering a boundary condition by convection at a surface of the piezoelectric material, a differential equation for a heat transfer analysis is as follows:
[196] [equation 10] [197]
Figure imgf000016_0001
[198] where, S: surface of the piezoelectric material in contact with air,
[199] h: convection coefficient, and
[200] T : temperature of surrounding air. [201]
[202] When the finite element method is applied to the mathematical equation 10, a differential equation for a heat transfer analysis is as follows:
[203] [equation 11] [204]
Figure imgf000017_0001
[205] where, each element of A , A , F and F matrix is a following mathematical equation 12:
[206] [equation 12] [207]
(FQ\ = [N1QdV
(Fs\ = [NτhTadS
[208] where, N : polynomial interpolation function. [209] [210] By using the displacement and electrical potential at a nodal point which are obtained by the three-dimensional finite element method of the piezoelectric material, it is possible to calculate the mechanical vibration loss P and the dielectric loss P per m d a unit volume in the piezoelectric material, as the mathematical equations 8a and 8b. At this time, it is assumed that the summation of the two losses, i.e., P and P is same m d as the quantity of heat (Q) per a unit volume defined in the mathematical equations 9 and 10. By solving the heat transfer equation of the mathematical equation 11 under the assumption, it is possible to calculate the temperature increase of the piezoelectric transformer which is caused when it is driven in the high electric field.
[211] Fig. 14 show a temperature measuring system for the first test model in an experimental manner. Pl to P4 show temperature measuring points. [212] Figs. 15 and 16 are graphs comparing temperature variations depending on a change of electric loads, which are measured at temperature measurement points in Fig. 14 with a result of a temperature distribution analysis according to the invention.
[213] Table 1 shows output power depending on electric loads. [214] Table 1
Figure imgf000017_0003
[215] As shown in Figs. 15 and 16, the temperature at the temperature measuring point is lowest when the load is 100Ω. This means that the 20W-piezoelectric transformer has the smallest loss at about 100Ω. Accordingly, 100Ω is the optimal load of the first test model.
[216] In addition, according to the temperature calculation result by the three-dimensional finite element method and the heat transfer equation at the temperature measuring point Pl (position of an input electrode) at which the temperature increase is dominated, there is an error with the experiment value as output power is increased in accordance with the load increase, as shown in the table 1. The reason is that the value measured at the room temperature is used as the material constant (i.e., mechanical stiffness, piezoelectric constant and dielectric constant) necessary for the finite element method.
[217] When the piezoelectric material is driven in the high electric field, the temperature of the piezoelectric material is increased as the input and output power are increased, so that the material constant of the piezoelectric material is changed. Accordingly, in order to analyze the characteristic change of the piezoelectric material which is caused when it is driven in the high electric field or when the temperature of the piezoelectric material is increased due to the power increase of the piezoelectric transformer, the change of the material constant depending on the temperatures should be reflected.
[218] Figs. 17 to 20 show piezoelectric samples for experimentally measuring a material constant shown in mathematical equations Ic to Ie. The material constant depending on the temperatures of the piezoelectric material is measured by positioning the sample shown in Figs. 17 to 20 in a temperature chamber and measuring resonant and an- tiresonant frequencies of the respective samples using an impedance analyzer while increasing the temperature of the sample.
[219] Figs. 21 to 24 are graphs showing a temperature characteristic of material constants of a piezoelectric material. As shown, an elastic modulus exhibiting the mechanical characteristic of the piezoelectric material is relatively little sensitive to the temperature change. However, the piezoelectric constant, the dielectric constant, the mechanical quality factor (Q ) and the dielectric loss factor (tanδ) for which the loss analysis is necessary are much changed depending on the temperatures. The change of the material constants depending on the temperatures causes a change of the operating characteristic of the piezoelectric system, such as a change of the resonant frequency of the piezoelectric system and an increase in voltage of the piezoelectric transformer.
[220]
[221] The method of analyzing the non-linear characteristic of the piezoelectric system in consideration of the temperature increase is embodied by using the finite element analysis of the piezoelectric material, the temperature distribution analysis through the loss analysis and the heat transfer equation analysis, and the change of the coefficient of the piezoelectric material depending on the measured temperatures.
[2221 Fig. 25 is a flow chart showing a method of analyzing a non-linear characteristic of a piezoelectric system considering a temperature increase, according to an embodiment of the invention. According to the invention, a resonant frequency of the piezoelectric system is calculated using the material constant at a predetermined temperature (for example, room temperature) (SlO) and then a temperature increase is calculated at the resonant frequency (S20).
[223] The process of calculating the resonant frequency in the step (S 10) or a later step (S60) may comprise steps of calculating an electric potential Φ by the material constant under predetermined temperature and a characteristic equation of a following mathematical equation 13; calculating an impedance using the electric potential Φ obtained in the above step, a charge Q and a following mathematical equation 16; and determining a frequency as a resonant frequency, which frequency exhibits the smallest impedance of the impedances obtained by repeating the steps while changing the frequency.
[224] [equation 13] [225]
Figure imgf000019_0001
[226] [equation 16]
[227]
v j∞Q
[228] where, K , D , K , K , M, u, φ, F, Q and Z are defined as described above.
UU UU uφ φφ [229] After that, a value of the piezoelectric material is calculated considering the temperature increase (S30) and then a temperature of the piezoelectric material is obtained by inserting the calculated value (S40). At this time, the resonant frequency is fixed as the resonant frequency obtained in the step (a).
[230] The process of calculating the temperature of the piezoelectric material in the step (S20) or a later step (S40) may comprises steps of: calculating a mechanical vibration loss P and a dielectric loss P of the piezoelectric material per unit volume using m d following mathematical equations 8a and 8b; assuming that heat generation Q of an interior per unit volume is equal to a summation of the mechanical vibration loss P and the dielectric loss P in a heat transfer equation of a following mathematical d equation 9; and calculating a temperature distribution of the piezoelectric material from a mathematical equation 11 which is a differential equation applying a convection boundary condition and a finite element process to the mathematical equation 9. [231] [equation 8a]
[232]
Figure imgf000020_0001
[233] [equation 8b] [234]
Pd = - x (D0) x (E0) x ωr x tan £ [Watt/m3]
[235] [equation 9]
[236] k = -Q
[237] [equation H]
[238]
{Ak - A8)T = FQ + FS
[239] where, Q , ω tanδ, X , S , D , E , k, T, and Q are defined as described above, and m r, 0 0 0 0 each element of A , A , F and F matrix are as following mathematical equation 12. [240] [equation 12]
[241]
(Ah - >( .d όNlV,, d oNi\ J,, d cWN7, S oNi\ 1 dN, 9N1
+ J + , (M1 ™J )dV dx dx dy dy dz dz
Figure imgf000020_0002
(FQ)1 - N1QdV
(Fs\ = N1HTJS
[242] where, N : polynomial interpolation function.
[243] [244] The process (S30) of calculating the material constant depending on the temperature change and the process (S40) of calculating the temperature change depending on the material constants are repeated until the temperature variation of the piezoelectric material is converged within a predetermined error (ε ) (S50).
[245] When the temperature increase is converged within the error (ε ), a change of the resonant frequency is calculated in accordance with the temperature increase through the frequency search for a new temperature (S 60).
[246] After that, it is calculated a temperature increase for the new resonant frequency
(S20 to S50). At this time, the process (S60) of calculating the change of the resonant frequency in accordance with the temperature increase and the process (S70) of calculating the temperature increase in consideration of the change of the material constant in accordance with the temperature change under the resonant frequency are repeated until a difference compared to the temperature under previous resonant frequency is converged within a predetermined error (ε ) (S80).
[247] Even though the resonant frequency is changed, when the change of the resonant frequency little affects the temperature increase (i.e., the temperature increase in accordance with the change of the resonant frequency is converged within the predetermined error (ε )), the loop is ended. The temperature and the resonant frequency at this time can be analyzed as a temperature increase under steady state and the change of the resonant frequency due to the temperature increase.
[248]
[249] Figs. 26 and 27 show analysis results of a temperature distribution for the second test model. Fig. 26 is a temperature increase result analyzed according to the flow chart shown in Fig. 25. Fig. 27 shows an experiment result for the second test model.
[250] As shown in Figs. 26 and 27, the result of the temperature increase analysis has the distribution and value similar to the experiment result. A temperature differenece in the piezoelectric transformer having a diameter of 30 mm is about 100C.
[251] Accordingly, in order to analyze the non-linearity of the piezoelectric system having a large characteristic change depending on the temperature increase, the analysis using the three-dimensional heat transfer equation is suitable, rather than the heat equivalent circuit method. In addition, the temperature analysis and non-linear analysis of the piezoelectric system allow an optimal design for prevention of a local failure of the piezoelectric system and suppression of the temperature increase due to the loss. 63.60C in Fig. 27 is a surrounding air temperature of the piezoelectric transformer by the convection.
[252] Fig. 28 shows a non-linearity due to the temperature increase of the second test model according to the flow chart shown in Fig. 25. The dotted line shows a voltage ratio depending on frequency obtained by using the material constant obtained in the experiment at the room temperature, and the solid line shows a voltage ratio depending on frequency obtained by using the material constant at a temperature increased in consideration of the temperature increase.
[253] As shown in Fig. 28, the temperature increase of the piezoelectric transformer caused by the driving of the high electric field caused the increases of the resonant frequency and the voltage ratio. A comparison of the experiment result and the analysis result for the second test model is shown in table 2.
[254] Table 2
Figure imgf000022_0001
[255] As shown in the table 2, the temperature increase of about 900C caused by the driving of the high electric field increases the resonant frequency of the piezoelectric transformer by about 2kHz and the voltage ratio by 15%. It can be seen that the nonlinear analysis considering the temperature increase is nearly matched to the experiment result. Industrial Applicability
[256] As described above, according to the invention, it is possible to quickly and accurately calculate the non-linear characteristic of the piezoelectric material depending on the temperature increase when it is driven in the high electric field.

Claims

Claims
[1] A method of analyzing a temperature change of a piezoelectric material when the material is driven, the method comprising steps of:
(a) obtaining a resonant frequency of a piezoelectric material at a predetermined temperature;
(b) obtaining a temperature of the piezoelectric material under predetermined resonant frequency;
(c) obtaining a material constant of the piezoelectric material under predetermined temperature condition;
(d) obtaining a temperature of the piezoelectric material using the material constant of the piezoelectric material obtained in the step (c);
(e) repeating the steps (c) and (d) until the temperature obtained in the step (d) is converged;
(f) obtaining a changed resonant frequency of the piezoelectric material under converged temperature obtained in the step (e);
(g) obtaining a converged temperature by performing the steps (b) to (e) under resonant frequency obtained in the step (f); and
(h) repeating the steps (T) and (g) until the converged temperature obtained in the step (g) is converged.
[2] The method according to claim 1, wherein the process of calculating the resonant frequency in the step (a) or (f) comprises steps of: obtaining an electric potential Φ by the material constant under predetermined temperature and a characteristic equation of a mathematical equation 13; obtaining an impedance using the electric potential Φ obtained in the above step, a charge Q and a following mathematical equation 16; and determining a frequency as a resonant frequency, which frequency exhibits the smallest impedance of the impedances obtained by repeating the above steps while changing the frequency.
[equation 13]
K,m - ω M + jω D,m K, u F
Figure imgf000023_0001
[equation 16]
Φ(ω)
Z(ω) = jωQ where, K : mechanical stiffness matrix,
D : mechanical damping matrix,
K : piezoelectric coupling coefficient matrix, uφ
K : dielectric hardness matrix,
M: mass matrix, u: mechanical displacement, φ: electric potential,
F: mechanical force,
Q: charge, and
Z: impedance. [3] The method according to claim 1, wherein the predetermined temperature in the step (a) is a room temperature. [4] The method according to claim 1, wherein the step (b) is a step of obtaining the temperature of the piezoelectric material under resonant frequency obtained in the step (a) or (f). [5] The method according to claim 4, wherein the step of obtaining the temperature of the piezoelectric material in the step (b) or (d) comprises steps of: calculating a mechanical vibration loss P and a dielectric loss P of the m d piezoelectric material per unit volume using following mathematical equations 8a and 8b; assuming that heat generation Q of an interior per unit volume in a heat transfer equation of a following mathematical equation 9 is equal to a summation of the mechanical vibration loss P and the dielectric loss P ; and m d calculating a temperature distribution of the piezoelectric material from a mathematical equation 11 which is a differential equation applying a convection boundary condition and a finite element process to the mathematical equation 9. [equation 8 a]
P"
Figure imgf000024_0001
o~ (X° {S° °l [Watt/m3]
[equation 8b]
Pd = - X (D0) x (E0) x ωr x tan £ [Watt/m3]
[equation 9]
[equation 11]
Figure imgf000025_0001
where, Q : mechanical quality factor, ω : resonant frequency, r tanδ: dielectric loss factor,
X : absolute value of stress, o
S : absolute value of strain, o
D : absolute value of an electric displacement
E : absolute value of an electric field, o k: heat transfer coefficient,
T: temperature,
Q: heat generation of an interior per a unit volume, and each element of A , A , F and F matrix are as following mathematical equation k s Q s
12. [equation 12] r BN ^N1 dN dN, dN SN1 (Ak\ = k(^- — L + ^lL — L + ^l± — L)dV
•r dx dx dy dy dz dz
Figure imgf000025_0002
(Fs)1 = N1HTJS
where, N : polynomial interpolation function.
[6] The method according to claim 1 , wherein the material constant of the piezoelectric material obtained in the step (c) is data of a material constant depending on temperature previously obtained in an experiment.
[7] The method according to claim 6, wherein the step (c) is a step of obtaining a material constant of the piezoelectric material under temperature condition obtained in the step (b) or (d).
[8] The method according to claim 1, wherein in the step (e), the steps (c) and (d) are repeated until the temperature variation of the piezoelectric material resulting from a change of the material constant of the piezoelectric material under state that the resonant frequency is fixed is converged within a predetermined critical value ε . i [9] The method according to claim 1, wherein in the step (h), the steps (f) and (g) are repeated until a difference between (i) a converged temperature obtained in the step (e) under condition of resonant frequency currently set and (ii) a converged temperature obtained in the step (e) under condition of resonant frequency previously set is converged within a predetermined critical value ε .
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