KR100639473B1 - A method for calculating frequency characteristics of piezoelectric material non-linearly according to temperature rising - Google Patents

Abstract

A method for non-linearly founding frequency characteristics of a piezoelectric device according to temperature change is provided to quickly and correctly calculate a change of a material constant and a resonant frequency according to temperature increase when the piezoelectric device is operated. The resonant frequency of the piezoelectric device is found at the specific temperature(S10). The temperature of the piezoelectric device is found under the resonant frequency(S20). The material constant of the piezoelectric device is found in a specific temperature condition(S30). The temperature of the piezoelectric device is found by using the found material constant of the piezoelectric device(S40). Previous two steps are repeated until the found temperature is converged(S50). The changed resonant frequency of the piezoelectric device is found at the converged temperature(S60). The converged temperature is found by performing previous 'S20'-'S50' steps under the found resonant frequency(S70). The last two steps are repeated until the found converged temperature is converged(S80).

Description

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

1A and 1B are schematic diagrams of a planar and vertical section of a contour vibration type piezoelectric transformer used in a simulation for verifying a method according to the present invention, respectively.

2A and 2B show impedance magnitude and phase at no load of the piezoelectric transformer model shown in FIGS. 1A and 1B, respectively, using a three-dimensional finite element method of a piezoelectric body.

3 is an impedance waveform at an input terminal of a piezoelectric transformer when an electrical load of 100 kV is connected to an output terminal of the piezoelectric transformer model shown in FIGS. 1A and 1B.

4a and 4b show the change in the step-down ratio and the resonance frequency of the piezoelectric transformer according to the load, respectively.

5A and 5B are plan and cross-sectional views of a second test model of a 20 W piezoelectric transformer, respectively.

6A and 6B show an impedance waveform at an input stage when an unloaded impedance waveform and an electrical load of 100 Hz for the second test model are connected, respectively.

7A and 7B show changes in resonance frequency and voltage ratio according to the load of the second test model, respectively.

8 illustrates a temperature measurement system for the first test model.

9a and 9b compare the temperature change according to the change of the electrical load, measured at the temperature measuring point of FIG.

10A to 10D illustrate samples for measuring material constant values shown in Equations 1C to 1E.

11A and 11B are graphs showing temperature characteristics of material constant values of piezoelectric bodies.

12 is a flowchart of a nonlinear analysis method of a piezoelectric system in consideration of the temperature rise according to the present invention.

13a and 13b show the results of the temperature distribution analysis for the second test model.

FIG. 14 illustrates nonlinearity due to temperature rise in a second test model according to the flowchart shown in FIG. 12.

The present invention relates to a method for nonlinearly obtaining the frequency characteristic according to the temperature change of the piezoelectric body. According to the method according to the present invention, it is possible to quickly and accurately calculate the nonlinear characteristics (eg, temperature change, material constant change, and resonant frequency change) of the piezoelectric body in response to temperature rise during high field driving.

Currently, piezoelectric materials are widely used in actuators, sensors, and electronic components for precise position control in many industries. When the piezoelectric body is driven in a high electric field, the temperature of the piezoelectric body increases due to mechanical vibration loss and dielectric loss inside the piezoelectric body. This increase in temperature changes the material constant value of the piezoelectric body, resulting in a change in operating frequency of the piezoelectric body and a decrease in the mechanical quality factor.

Therefore, in the case where the piezoelectric body is driven in a high electric field, in order to effectively drive and control the piezoelectric body, the temperature rise of the piezoelectric body is analyzed, and the nonlinear characteristic of the piezoelectric body due to the temperature rise (for example, change of material constant and change of resonance frequency) Should be interpreted.

The present invention is devised to analyze the nonlinear characteristics of the piezoelectric body when driving the piezoelectric body. According to the method of the present invention, the temperature of the piezoelectric body can be obtained by reflecting a material constant value changed according to the temperature change of the piezoelectric body in a state where the resonant frequency is fixed. In addition, by tracking the resonant frequency changed according to the temperature change of the piezoelectric body, it is possible to analyze the temperature change of the piezoelectric body under the tracked resonant frequency conditions. Therefore, it is possible to quickly and accurately calculate the nonlinear characteristics, for example, the change in the material constant and the change in the resonant frequency, due to the temperature rise during the piezoelectric driving.

Accordingly, it is an object of the present invention to provide a method for analyzing the nonlinear characteristics of a piezoelectric body when driving a piezoelectric body.

The present invention relates to a method for nonlinearly obtaining the frequency characteristic according to the temperature change of the piezoelectric body.

More specifically, the method according to the invention,

Obtaining a resonance frequency of the piezoelectric body at a predetermined temperature (a);

Obtaining a temperature of the piezoelectric body under a predetermined resonance frequency (b);

(C) obtaining a material constant value of the piezoelectric body under a predetermined temperature condition;

(D) obtaining a temperature of the piezoelectric body using the material constant value of the piezoelectric body obtained in step (c);

(E) repeating steps (c) and (d) until the temperature obtained in step (d) converges;

Obtaining (f) a changed resonance frequency of the piezoelectric body under the convergence temperature obtained in step (e);

(G) obtaining a convergence temperature by performing steps (b) to (e) under the resonance frequency obtained in step (f); And

And repeating steps (f) and (g) until the convergence temperature obtained in step (g) converges.

Through step (e), the temperature of the piezoelectric body may be obtained by reflecting a material constant value changed according to the temperature change of the piezoelectric body in a fixed resonance frequency. In addition, the resonant frequency changed according to the temperature change of the piezoelectric body is tracked through the step (h), and the temperature change of the piezoelectric body under the tracked resonant frequency condition is analyzed. Therefore, the nonlinear characteristic according to the temperature rise at the time of piezoelectric driving can be calculated quickly and accurately.

In the present invention, the step of obtaining the resonant frequency in the step (a) or step (f),

Obtaining a potential φ based on the material constant value at the predetermined temperature and the characteristic equation of Equation 13 below;

Obtaining an impedance using the potential? Obtained in the above step, the charge amount Q, and the following Equation 16; And

And repeating the above steps while varying the frequency, determining the frequency representing the smallest impedance as the resonance frequency:

[Equation 13]

[Equation 16]

In the above formula,

K _uu is the mechanical stiffness matrix,

D _uu is the mechanical damping matrix,

K _uφ is the piezoelectric coupling coefficient matrix,

K _φφ is the genetic hardness matrix,

M is the mass matrix,

u is mechanical displacement,

φ is potential

F is a mechanical force,

Q is the charge,

Z is impedance.

In the present invention, the predetermined temperature in the step (a) is preferably room temperature.

In the present invention, the step (b) is preferably the step of obtaining the temperature of the piezoelectric body under the resonance frequency obtained in the step (a) or the resonance frequency obtained in the step (f).

At this time, the process of obtaining the temperature of the piezoelectric material in the step (b) or the step (d),

Obtaining mechanical vibration loss P _m and dielectric loss P _d of the piezoelectric body per unit volume using Equations 8a and 8b;

Assuming that an internal heat generation amount Q per unit volume in the heat transfer equation of Equation 9 is equal to the sum of the mechanical vibration loss P _m and the dielectric loss P _d ; And

It is preferable to include obtaining the temperature distribution of the piezoelectric body from Equation 11 below, which is a differential equation applying convective boundary condition and finite element process to Equation 9:

Equation 8a

[Equation 8b]

[Equation 9]

[Equation 11]

In the above formula,

Q _m is the machine quality factor,

ω _r is the resonant frequency

tanδ is the dielectric loss factor,

X ₀ is the absolute value of the stress,

S ₀ is the absolute value of the strain,

D ₀ is the absolute value of the flux density,

E ₀ is the absolute value of the electric field,

k is the heat transfer coefficient,

T is temperature

Q is the internal heat generation amount per unit volume,

Each element of the A _k , A _S , F _Q and F _S matrices is represented by Equation 12:

[Equation 12]

In the above formula, N _{i, j} is a polynomial interpolation function.

In the present invention, the material constant value of the piezoelectric material in step (c) is data obtained by experimentally obtaining the material constant value according to the temperature in advance.

In the present invention, the step (c) is preferably a step of obtaining a material constant value of the piezoelectric material under the temperature obtained in the step (b) or the temperature conditions obtained in the step (d).

In the present invention, in the step (e), the steps (c) and (d) until the temperature change amount of the piezoelectric body according to the change of the material constant value of the piezoelectric body in the state in which the resonance frequency is fixed converges within a predetermined threshold ε ₁ . It is preferable to repeat.

In the present invention, in step (h), (i) the convergence temperature obtained in step (e) under the currently set resonance frequency condition and (ii) the convergence obtained in step (e) under the resonance frequency condition set before. It is preferable to repeat the above steps (f) and (g) until the difference in temperature converges within the predetermined threshold ε ₂ .

Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings. However, the present invention is not limited by the following examples.

First, the basic characteristic equation of a piezoelectric body is demonstrated.

The determinants of Equations 1a to 1e are the most basic governing equations connecting mechanical and electrical phenomena in piezoelectric systems. However, the nonlinearity and loss components of the piezoelectric body are not considered:

Equation 1a

T = c ^E Se ^t E

[Equation 1b]

D = eS + ε ^S E

Equation 1c

Equation 1d

[Equation 1e]

In the above formula,

T stands for mechanical stress,

S represents a mechanical strain,

E represents the electric field,

D stands for electric displacement,

Matrixes representing material properties c ^E , e and ε ^S represent elastic modulus, piezoelectric constant and dielectric constant, respectively.

Applying Hamilton's variation to the piezoelectric body is represented by the following equations (2a) and (2b):

Equation 2a

δ∫Ldt = 0

[Equation 2b]

L = E _kin -E _st + E _d + W

In the above formula,

δ represents the primary variation,

L means the Lagrange term expressed in energies,

E _kin is the kinetic energy, E _st is the elastic energy, E _d is the dielectric energy, W represents the energy given from the outside, can be represented by the following equations 3a to 3d:

Equation 3a

[Equation 3b]

Equation 3c

Equation 3d

In the above formula,

u is the velocity vector [m / s],

V is the volume [m ³ ],

f _B is the bulk force density vector [N / m ³ ],

f _S is the surface force density vector [N / m ² ]

f _P is the force vector [N],

A _F is the area [m ² ]

Q _S is surface charge [A _S ],

Q _P is the point charge [A _S ],

A _C is the area [m ² ] of the region where the charge is applied.

Substituting the polynomial approximation functions of Equations 4a and 4b into Equation 2a, a linear partial differential equation for finite element analysis of the piezoelectric body can be obtained as shown in Equation 5 below:

Equation 4a

[Equation 4b]

[Equation 5]

In the above formula,

Φ is the potential,

는 변위량이며,

Is the displacement,

B is an operator defined as in Equation 6 below,

K _uu is the mechanical stiffness matrix,

D _uu is the mechanical damping matrix,

K _uφ is the piezoelectric coupling coefficient matrix,

K _φφ is the genetic hardness matrix,

M is the mass matrix,

F _B is the force vector acting on the volume,

F _S is the force vector acting on the face,

F _P is the force vector acting on the point,

Q _S is the surface charge,

Q _P is a point charge,

Each can be represented by the following equation:

[Equation 6]

[Equation 7]

In the above formula,

α and β are damping coefficients, respectively

f _B is the bulk force density vector at the element,

f _S is the surface force density vector at the element plane,

f _P is the force vector at the element point.

Damping behavior is determined by the damping matrix D _uu . The damping matrix can be obtained from the damping characteristics of the structure. Even if the equation for D _uu of Equation 7 is used, the damping characteristics of the structure cannot be fully expressed.

In order to accurately represent the damping characteristics for any structure, two or more damping coefficients are required. However, it is difficult to accurately find the damping coefficient. This is because the damping coefficient is measured differently according to the position and the frequency. Therefore, it is almost impossible to obtain the damping coefficient accurately and apply it to the analysis.

Equation 7 for the expression D _uu is a method of practically and realistically considering damping characteristics, which is approximated by two damping coefficients. Using this, one of four damping types is determined by the range of damping coefficients:

(1) without damping: α = 0, β = 0;

(2) viscous damping: α = 0, β> 0;

(3) damping proportional to mass: α> 0, β = 0; And

(4) Rayleigh damping: α> 0, β> 0

The two characteristic equations of Equation 5 obtained through the finite element formulation of the piezoelectric body can be basically expressed as one determinant of frequency as shown in Equation 13 below.

[Equation 13]

In the above formula, K _uu , D _uu , K _uφ , K _φφ and M are as defined above.

When considering the piezoelectric material used in the piezoelectric transformer for the converter, if the load (Z _L ) is connected to the output terminal of the piezoelectric transformer, the voltage and current relationship at the output terminal is expressed by the following equation (14):

[Equation 14]

From equations (13) and (14), an equation of the piezoelectric system for analyzing the load condition at the output electrode of the piezoelectric transformer can be obtained as in equation (15) below:

[Equation 15]

In the following, the results of simulations in which temperature and material constant values are calculated according to the method of the present invention and actual measurement results are compared.

Figures 1a and 1b schematically show the shape of the planar and vertical cross-section of a contour vibration type piezoelectric transformer, respectively, used in the simulation to verify the method according to the present invention. The output stage of the piezoelectric transformer is composed of three layers.

2A and 2B show waveforms of impedance magnitude and phase at no load of the piezoelectric transformer model shown in FIGS. 1A and 1B, respectively, using a three-dimensional finite element method of a piezoelectric body. In FIG. 2A and FIG. 2B, the dotted line represents the actual measured experimental result, and the solid line represents the result obtained by the three-dimensional finite element method. From the figure, it can be seen that the simulation results and the actual measurement results are almost identical.

3 illustrates an impedance waveform at an input terminal of a piezoelectric transformer when an electrical load of 100 kW is connected to an output terminal of the piezoelectric transformer model illustrated in FIGS. 1A and 1B. As can be seen from FIG. 3, it can be seen that the analysis result by the finite element method is almost identical to the actual measurement result.

4A and 4B show changes in the step-down ratio and the resonance frequency of the piezoelectric transformer according to the load, respectively. As shown in Figs. 4A and 4B, it can be seen that the analysis results by the finite element method are almost identical to the experimental results. As the load increased, the gain and resonant frequency of the piezoelectric transformer also increased. Therefore, it is possible to predict the characteristic change of the piezoelectric transformer according to the load variation of the piezoelectric transformer. In addition, when designing a piezoelectric transformer, it can be used to determine the optimum load of the piezoelectric transformer. In addition, instead of considering the electrical load by the conventional one-dimensional equivalent circuit method, by analyzing the electrical load through the combination with the three-dimensional finite element method, it is possible to calculate the characteristics change of the piezoelectric transformer according to the load more accurately.

5A and 5B show a second test model of a 20W piezoelectric transformer. 5A is a plan view of the test model, and FIG. 5B is a cross-sectional view of the test model.

6A and 6B show the finite element analysis results for the second test model. FIG. 6A illustrates an impedance waveform at no load, and FIG. 6B illustrates an impedance waveform at an input terminal of a piezoelectric transformer when an electrical load of 100 kV is connected. As shown in FIG. 6A, it can be seen that the two results in the band about 70 kHz are almost identical.

7A and 7B show changes in resonance frequency and voltage ratio according to the load of the second test model.

Hereinafter, the temperature rise analysis by the three-dimensional heat transfer equation will be described.

Piezoelectric systems have mechanical hysteresis losses and electrical hysteresis losses. These are called mechanical vibration loss and dielectric loss, respectively. In particular, when the piezoelectric system is driven with a high electric field, it results in a temperature rise. Loss analysis by the conventional analytical method is a one-dimensional analysis and is calculated on the assumption that the loss distribution of the system is uniform. However, the piezoelectric system has a limited distribution of losses in the piezoelectric system depending on the operation mode and shape, so that the one-dimensional loss calculation method is limited.

By using the finite element method, it is possible to calculate the mechanical displacement of the piezoelectric body and the voltage generated inside the piezoelectric body when a specific voltage is applied. In addition, mechanical strains S and electric fields E can be obtained by using Equations 4a and 4b, which are solutions of the finite element analysis of the piezoelectric system. Using the calculated mechanical strain (S), electric field (E), stress (X) and flux density (D), mechanical vibration loss and dielectric loss of the piezoelectric body per unit time and volume are calculated using Equations 8a and 8b You can get it by:

Equation 8a

[Equation 8b]

In the above formula,

Q _m is the machine quality factor,

ω _r is the resonant frequency

tanδ is the dielectric loss factor,

P _m is the mechanical vibration loss,

P _d is the dielectric loss,

X ₀ is the absolute value of the stress,

S ₀ is the absolute value of the strain,

D ₀ is the absolute value of the flux density,

E ₀ is the absolute value of the electric field.

Using the law of thermal energy conservation and Fourier's law, a differential equation for the analysis of temperature distribution at steady state can be obtained as shown in Equation 9:

[Equation 9]

In the above formula,

k is the heat transfer coefficient,

T is temperature

Q is the amount of internal heat generation per unit volume.

Considering the boundary conditions due to convection on the piezoelectric surface, the differential equation for the heat transfer analysis is given by:

[Equation 10]

In the above formula,

S represents the surface of the piezoelectric body in contact with air,

h represents the convection coefficient,

T _a represents the ambient air temperature.

When the finite element method is applied to Equation 10, a differential equation for heat transfer analysis is given by Equation 11 below:

[Equation 11]

Where each element of the A _k , A _S , F _Q and F _S matrices is

[Equation 12]

In the above formula, N _{i, j} is a polynomial interpolation function.

The mechanical vibration loss P _m and dielectric loss P _d per unit volume in the piezoelectric body can be calculated using the displacements and electrical potentials at the nodes obtained by using the three-dimensional finite element method of the piezoelectric body. have. In this case, it is assumed that the sum of the two losses P _m and P _d calculated is equal to the calorie Q per unit volume defined in Equations 9 and 10. By assuming such a problem, by solving the heat transfer equation of Equation 11, it is possible to obtain the temperature rise of the piezoelectric transformer caused by the high field driving.

8 illustrates a system for experimentally measuring temperature with respect to the first test model. P1 to P4 represent temperature measuring points.

9A and 9B compare the temperature change according to the change of the electrical load measured at the temperature measuring point of FIG. 8 with the temperature distribution analysis result of the present invention.

Table 1 shows the output power according to the electrical load.

[Table 1] Output power according to electrical load

RL [Ω] 50 100 200 300 500 P0 [W] 5 8.5 9 11 15.9

As shown in Figs. 9A and 9B, the temperature at the temperature measuring point was the lowest when the load was 100 kPa. This means that the 20W piezoelectric transformer has the lowest loss at about 100kW. Therefore, 100 kW is the optimum load of the first test model.

In addition, the results of the temperature calculation by the three-dimensional finite element method and the heat transfer equation at the temperature measurement point P1 (position of the input electrode) where the increase in temperature is most prominent are shown in Table 1 above. As a result, an error from the experimental value appears. This is because the values measured at room temperature are used as material constant values (ie, mechanical stiffness, piezoelectric constant, and dielectric constant) required for finite element analysis.

When driving in the high electric field of the piezoelectric body, the temperature of the piezoelectric transformer increases as the input and output powers increase, thereby changing the material constant value of the piezoelectric body. Therefore, in order to analyze the characteristic change of the piezoelectric transformer caused by the high temperature driving or the increase of the power of the piezoelectric transformer due to the increase in the power of the piezoelectric transformer, the change in the material constant value according to the temperature should be analyzed. .

10A to 10D show piezoelectric samples for experimentally measuring the material constant values shown in Equations 1C to 1E. The material constant value according to the temperature of the piezoelectric body is placed in the temperature chamber of the measurement samples of FIGS. 10A to 10D, and then the resonance and anti-resonance frequencies of each sample are measured by an impedance analyzer while increasing the temperature.

11A and 11B are graphs showing temperature characteristics of material constant values of piezoelectric bodies. As shown in the figure, the mechanical elastic modulus representing the mechanical properties of the piezoelectric body has a relatively small change with temperature. However, it can be seen that the piezoelectric constant, dielectric constant, mechanical quality factor (Q _m ) and dielectric loss factor (tanδ) that require loss analysis vary greatly with temperature. The change in temperature of these material constants results in a change in operating characteristics of the piezoelectric system, such as a change in the resonant frequency of the piezoelectric system and an increase in the voltage transformer of the piezoelectric transformer.

According to the present invention, a method of analyzing the nonlinear characteristics of a piezoelectric system in consideration of a temperature rise includes a finite element analysis of a piezoelectric body, a temperature distribution analysis through a loss analysis and a heat transfer equation, and a change in the piezoelectric material coefficient according to the measured temperature. It is implemented using

12 shows a flowchart of a method for analyzing the nonlinear characteristics of a piezoelectric system in consideration of temperature rise in accordance with the present invention. In the method according to the present invention, first, a resonance frequency of a piezoelectric system is obtained using a material constant value at a predetermined temperature (for example, room temperature) (S10), and a temperature rise of the piezoelectric material at the resonance frequency is calculated ( S20).

The step of obtaining the resonance frequency in the step (S10) or the following step (S60), the step of obtaining the potential φ by the material constant value and the characteristic equation of Equation 13 at a predetermined temperature; Obtaining an impedance using the potential? Obtained in the above step, the charge amount Q, and the following Equation 16; And determining the frequency representing the smallest impedance among the impedances obtained by repeating the above steps while changing the frequency as the resonance frequency:

[Equation 13]

[Equation 16]

In the above formula,

K _uu is the mechanical stiffness matrix,

D _uu is the mechanical damping matrix,

K _uφ is the piezoelectric coupling coefficient matrix,

K _φφ is the genetic hardness matrix,

M is the mass matrix,

u is mechanical displacement,

φ is potential

F is a mechanical force,

Q is the charge,

Z is impedance.

After that, the piezoelectric material value considering the temperature rise is obtained (S30), and the input temperature is obtained to obtain the piezoelectric material (S40). At this time, the resonance frequency is fixed at the resonance frequency obtained in step (a).

The step of obtaining the temperature of the piezoelectric material in the step S20 or the following step S40 may include: calculating mechanical vibration loss P _m and dielectric loss P _d of the piezoelectric body per unit volume using Equations 8a and 8b; Assuming that an internal heat generation amount Q per unit volume in the heat transfer equation of Equation 9 is equal to the sum of the mechanical vibration loss P _m and the dielectric loss P _d ; And a temperature distribution of the piezoelectric body from Equation 11, which is a differential equation applying convective boundary conditions and a finite element process to Equation 9, below.

Equation 8a

[Equation 8b]

[Equation 9]

[Equation 11]

In the above formula,

Q _m is the machine quality factor,

ω _r is the resonant frequency

tanδ is the dielectric loss factor,

X ₀ is the absolute value of the stress,

S ₀ is the absolute value of the strain,

D ₀ is the absolute value of the flux density,

E ₀ is the absolute value of the electric field,

k is the heat transfer coefficient,

T is temperature

Q is the internal heat generation amount per unit volume,

Each element of the A _k , A _S , F _Q and F _S matrices is represented by Equation 12:

[Equation 12]

In the above formula, N _{i, j} is a polynomial interpolation function.

Until the temperature of the piezoelectric body converges within a predetermined error (ε ₁ ), the process of obtaining the material constant value according to the temperature change (S30) and the process of obtaining the temperature change according to the material constant value change are repeated (S50). ).

When the temperature rise converges within a constant error ε ₁ , the change of the resonance frequency with respect to the temperature increase is calculated through the frequency search again for the new temperature (S60).

Then, the temperature rise value is calculated for the new resonance frequency (S20 to S50). At this time, the process of obtaining the change of the resonance frequency according to the temperature change and the temperature change under the resonance frequency until the difference converges within a predetermined error ε ₂ compared with the temperature under the previous resonance frequency. The process of obtaining a temperature rise in consideration of the change in the material constant value is repeated (S70).

If the resonant frequency changes, but the change in the resonant frequency does not have a significant effect on the temperature rise (that is, the temperature rise due to the change in the resonant frequency converges within a certain error ε ₂ ), the loop ( loop). The temperature and the resonance frequency at this time can be interpreted as a change in the resonance frequency due to the temperature rise and the temperature rise in the steady state.

13a and 13b show the results of the temperature distribution analysis for the second test model. FIG. 13A is a result of temperature rise analyzed by the flowchart of FIG. 12. Figure 13b shows the experimental results for the second test model.

As shown in Figs. 13A and 13B, the results of the temperature rise analysis have similar distributions and values as the experimental results. The temperature difference in the piezoelectric transformer with a diameter of 30 mm is about 10 ° C.

Therefore, in order to analyze the nonlinearity of the piezoelectric system having a large characteristic change with the temperature rise, the analysis by the three-dimensional heat transfer equation is appropriate, not the method by the thermal equivalent circuit. In addition, the temperature analysis and nonlinear analysis of the piezoelectric system enable an optimal design for preventing local destruction of the piezoelectric system and suppressing the temperature rise due to the loss. 63.6 ° C in FIG. 13B is the ambient air temperature of the piezoelectric transformer by convection.

FIG. 14 illustrates nonlinearity due to temperature rise of the second test model according to the flowchart shown in FIG. 12. The dotted line represents the voltage ratio to the frequency using the material value obtained by experimenting at room temperature, and the solid line represents the voltage ratio to the frequency obtained using the material value at the increased temperature in consideration of the temperature rise.

As shown in Fig. 14, the temperature rise of the piezoelectric transformer caused by the high field driving resulted in an increase in the resonance frequency and an increase in the voltage ratio. Table 2 shows a comparison between the experimental results and the analysis results for the second test model.

TABLE 2

Resonant frequency Voltage ratio At room temperature When the temperature increases At room temperature When the temperature increases Analysis result 208.8 [kHz] 210.4 [kHz] 0.372 0.42 Experiment result 203.8 [kHz] 205.8 [kHz] 0.378 0.433

As shown in Table 2 above, the temperature rise of about 90 ° C caused by high field driving increases the resonance frequency of the piezoelectric transformer by about 2 kHz and increases the voltage ratio of 15%. It can be seen that the nonlinear analysis considering the temperature rise is almost in agreement with the experimental results.

As described above, in the case of using the method according to the present invention, it is possible to quickly and accurately calculate the nonlinear characteristics of the piezoelectric body due to the temperature rise during high field driving.

Claims (9)

As a method of analyzing the temperature change of the piezoelectric body when driving the piezoelectric body Obtaining a resonance frequency of the piezoelectric body at a predetermined temperature (a); Obtaining a temperature of the piezoelectric body under a predetermined resonance frequency (b); (C) obtaining a material constant value of the piezoelectric body under a predetermined temperature condition; (D) obtaining a temperature of the piezoelectric body using the material constant value of the piezoelectric body obtained in step (c); (E) repeating steps (c) and (d) until the temperature obtained in step (d) converges; Obtaining (f) a changed resonance frequency of the piezoelectric body under the convergence temperature obtained in step (e); (G) obtaining a convergence temperature by performing steps (b) to (e) under the resonance frequency obtained in step (f); And And repeating the steps (f) and (g) until the convergence temperature obtained in the step (g) converges. According to claim 1, wherein the step of obtaining the resonant frequency in the step (a) or the step (f), Obtaining a potential φ based on the material constant value at the predetermined temperature and the characteristic equation of Equation 13 below; Obtaining an impedance using the potential? Obtained in the above step, the charge amount Q, and the following Equation 16; And Determining a frequency representing the smallest impedance among the impedances obtained by repeating the above steps while changing the frequency as a resonance frequency; [Equation 13]

[Equation 16]

In the above formula, K _uu is the mechanical stiffness matrix, D _uu is the mechanical damping matrix, K _uφ is the piezoelectric coupling coefficient matrix, K _φφ is the genetic hardness matrix, M is the mass matrix, u is mechanical displacement, φ is potential F is a mechanical force, Q is the charge, Z is impedance. The method of claim 1 wherein the predetermined temperature in step (a) is room temperature. The method of claim 1, wherein the step (b) is a step of obtaining a temperature of the piezoelectric body under the resonance frequency obtained in the step (a) or the resonance frequency obtained in the step (f). The process of claim 4, wherein the step of obtaining the temperature of the piezoelectric material in the step (b) or the step (d), Obtaining mechanical vibration loss P _m and dielectric loss P _d of the piezoelectric body per unit volume using Equations 8a and 8b; Assuming that an internal heat generation amount Q per unit volume in the heat transfer equation of Equation 9 is equal to the sum of the mechanical vibration loss P _m and the dielectric loss P _d ; And Obtaining a temperature distribution of the piezoelectric body from Equation 11, which is a differential equation applying convective boundary condition and finite element process to Equation 9: Equation 8a

[Equation 8b]

[Equation 9]

[Equation 11]

In the above formula, Q _m is the machine quality factor, ω _r is the resonant frequency tanδ is the dielectric loss factor, X ₀ is the absolute value of the stress, S ₀ is the absolute value of the strain, D ₀ is the absolute value of the flux density, E ₀ is the absolute value of the electric field, k is the heat transfer coefficient, T is temperature Q is the internal heat generation amount per unit volume, Each element of the A _k , A _S , F _Q and F _S matrices is represented by Equation 12: [Equation 12]

In the above formula, N _{i, j} is a polynomial interpolation function. The method according to claim 1, wherein the material constant value of the piezoelectric material in step (c) is data obtained by experimentally obtaining a material constant value according to temperature. 7. The method of claim 6, wherein step (c) is a step of obtaining a material constant value of the piezoelectric material under the temperature obtained in step (b) or the temperature condition obtained in step (d). The method (c) and (d) of claim 1, wherein the step (e) is performed until the temperature change amount of the piezoelectric body according to the change of the material constant value of the piezoelectric body is fixed within a predetermined threshold ε ₁ while the resonance frequency is fixed. ) Is repeated. The method of claim 1, wherein the step (h) comprises: (i) the convergence temperature obtained in the step (e) under the currently set resonance frequency condition and (ii) the convergence temperature obtained in the step (e) under the resonance frequency condition set before. And repeating steps (f) and (g) until the difference is within a predetermined threshold ε ₂ .

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