WO2006132451A1

WO2006132451A1 - A method for analyzing frequency characteristics of piezoelectric material

Info

Publication number: WO2006132451A1
Application number: PCT/KR2005/001870
Authority: WO
Inventors: Chang-Hwan Lee; Hyun-Woo Joo
Original assignee: Samhwa Yang Heng Co., Ltd.
Priority date: 2005-06-09
Filing date: 2005-06-17
Publication date: 2006-12-14
Also published as: KR100639472B1

Abstract

Disclosed is a method of calculating an impedance depending on frequencies of a piezoelectric material, comprising steps of: (a) Taylor series-expanding each matrix of a characteristic equation related to a frequency of a piezoelectric material with regard to a start frequency; (b) comparing coefficients of a same order of both terms in the Taylor series- expanded equation to obtain a recursive equation relating to a moment of a matrix exhibiting a mechanical displacement and an electric potential of the characteristic equations; (c) calculating each moment of the matrix from the recursive equation; (d) approximating the matrix to a Pade' approximate function; (e) calculating a coefficient of the Pade' approximate function in the step (d) from the equation obtained in the step (c); (f) calculating a solution of the matrix from the Pade' approximate function obtained in the step (e); and (g) calculating an impedance from the solution.

Description

A METHOD FOR ANALYZING FREQUENCY CHARACTERISTICS OF PIEZOELECTRIC MATERIAL

Technical Field

[1] The invention relates to a method of calculating an impedance depending on frequencies of a piezoelectric material. The method of the invention reduces a calculation time by dozens to hundreds times, as compared to a conventional direct calculation method. Based on such numerical analysis technique, it is possible to obtain an impedance, a material constant, a resonant frequency, an antiresonant frequency and the like of a piezoelectric material. Background Art

[2] In order to calculate a resonant frequency of a piezoelectric material, it is required to calculate an impedance while increasing or decreasing a frequency at an interval for a predetermined frequency band. However, this method is inefficient because a calculation time is significantly increased due to a frequency sweep(varying the frequency) in analyzing and designing the piezoelectric material. Disclosure of Invention Technical Problem

[3] The invention has been made to solve the above problem. The invention uses an approximate function of a matrix to significantly reduce a calculation time required for an impedance analysis of a piezoelectric material.

[4] Accordingly, an object of the invention is to provide a method of calculating an impedance depending on frequencies of a piezoelectric material. Technical Solution

[5] The invention relates to a method of calculating an impedance depending on frequencies of a piezoelectric material.

[6] More specifically, the invention relates to a method of calculating an impedance depending on frequencies of a piezoelectric material, the method comprising steps of:

[7] (a) Taylor series-expanding each matrix of a characteristic equation related to a frequency of a piezoelectric material with regard to a start frequency f ;

[8] (b) comparing coefficients of a same order of both terms in the Taylor series- expanded equation to obtain a recursive equation relating to a moment of a matrix exhibiting a mechanical displacement and an electric potential of the characteristic equations;

[9] (c) calculating each moment of the matrix exhibiting the mechanical displacement and electric potential from the recursive equation; [10] (d) approximating the matrix exhibiting the mechanical displacement and electric potential to a Pade¹ approximate function; [11] (e) calculating a coefficient of the Pade' approximate function in the step (d) from t he equation obtained in the step (c); [12] (f) calculating a solution of the matrix exhibiting the mechanical displacement and electric potential from the Pade¹ approximate function obtained in the step (e), and [13] (g) calculating an impedance depending on frequencies, from the solution of the electric potential obtained.

Brief Description of the Drawings [14] FIG. 1 is a flow chart showing a method of calculating an impedance depending on frequencies of a piezoelectric material, according to an embodiment of the invention; [15] FIG. 2 is a flow chart showing a procedure of selecting a start frequency in a Taylor expansion, according to an embodiment of the invention; [16] FIG. 3 is a graph comparing a calculation result of impedances of a piezoelectric material according to a method shown in FIG. 2 and impedance values experimentally measured; [17] FIG. 4 is a flow chart showing a procedure of searching plural start frequencies when performing a Taylor expansion according to an embodiment of the invention; [18] FIGS. 5 to 13 illustrate examples of setting a search interval in a method shown in

FIG. 4; and [19] FIG. 14 is a graph comparing a calculation result of impedances of a piezoelectric material according to a method shown in FIG. 4 and a calculation result according to the prior art.

Mode for the Invention [20] 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.

[21] First, a basic characteristic equation of a piezoelectric material is described.

[22] 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. [23] [equation Ia]

[24] T=c^ES-eE

[25] [equation Ib]

[26] D=eS+ε^sE

[27] [equation Ic] [28]

' 12 ^13 0 0 0

L L

'12 '1 1 C₁₃ 0 0 0 I Z 0 0 c^E = '13 '13 ^C33 0

I

0 0 0 '44 0 0

I 0 0 0 0 '44 0 0 0 0 0 0 (c_u ^E - '12 )/2

[29] [equation Id] [30]

0 0 0 0 ^15 0 e = 0 0 0 '15 0 0

'31 '31 '33 0 0 0

[31] [equation Ie]

[32]

ε_n 0 0 ε^s = 0 ε_u 0

S

0 0

[33] where, T: mechanical stress,

[34] S: mechanical strain,

[35] E: electric field

[36] D: electric displacement, and

[37] c , e and ε are matrix exhibitir ie mat piezoelectric constant and dielectric constant, respectively. [38] When a variation method of Hamilton is applied to the piezoelectric material, following mathematical equations 2a and 2b are provided. [39] [equation 2a]

[41] [equation 2b]

[42] L = E kin - E st + E d + W

[43] where, δ: linear variation

[44] L: Lagrange term expressed by energies,

[45] E : kinetic energy,

[46] E : elastic energy, st

[47] E : dielectric energy, and

[48] W: energy provided from the exterior.

[49] The above energies can be expressed as follows.

[50] [equation 3a]

[51]

K = \\\P vPdV

[52] [equation 3b]

[53]

[54] [equation 3c]

[55]

E₄

JJjDΕ dV

[56] [equation 3d]

[57]

[58] where, u: velocity vector[m/s],

[59] V: volume[m³],

[60] f : volume force density vector [N/m ],

^B ₂

[61] f : surface force density vector [N/m ],

[62] f : force vector[N], p ₂

[63] A : area to which force is applied[m ],

[64] Q : surface charge[As], [65] Q : point charge[As], and

[66] A : area to which charge is applied[m²].

[67] 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 definite element analysis of the piezoelectric material can be obtained as shown in a following mathematical equation 5.

[68] [equation 4a]

[69]

E = -gradΦ= -grad(N_φΦ) = -B_φΦ

[70] [equation 4b]

[71]

[72] [equation 5]

[73]

- ω Mu + j J ωO uu u + K mi u + K uΦ Φ= F B +F S +F p

[74] where, Φ: electric potential,

[75]

U

: displacement,

[76] B: operator defined as a following mathematical equation 6,

[77] K : mechanical stiffness matrix,

[78] D : mechanical damping matrix,

[79] K : piezoelectric coupling coefficient matrix,

[80] K : dielectric hardness matrix,

[81] M: mass matrix,

[82] F : force vector applied to volume,

B

[83] F : force vector applied to surface,

[84] F : force vector applied to point,

[85] Q : surface charge, and

[86] Q : point charge, wherein they can be expressed as a following mathematical equation 7. [87] [equation 6]

[88]

[89] [equation 7]

[90]

K nil - J fJfJfBV u B u dV

D₁₁₁₁ =

N;_,N_U dV + PW]Ky¹R dV

M = \\\p Kn dV F_B = IJjN:N_FBf_BJF F_s = ||NlN_FSf_sJF F_P = NIf

[91] where, α and β are damping coefficients, [92] f : volume force density vector in an element, [93] f : surface force density vector in an element surface and [94] f : force vector in an element point. [95] The two characteristic equations of the mathematical equation 5 obtained from the definite 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 5a.

[96] [equation 5a] [97] Km₁ - (D² M -^T j⁽D D,,,, K,,_Φ U F

K _Uφ K M Φ Q

[98] In order to simplify the matrix of the mathematical equation 5a, it can be simply expressed as a following mathematical equation 8.

[99] [equation 8] [100]

[101] where, K: sparse matrix which is partially symmetrical, [102] X: matrix exhibiting both the mechanical displacement u and the electric potential

Φ,

[103] F: matrix exhibiting both the mechanical force F and the charge Q, and [104] f: frequency. [105] In order to know a characteristic in an objected frequency band, it should be calculated using reverse matrix which should be calculated in the respective frequencies, one by one. Accordingly, it is required much time for the calculation. However, when the method of the invention is applied, it is possible to calculate the characteristic of the desired frequency band with a single reverse matrix calculation.

[106] In the followings, it will be described a method for obtaining an impedance depending on frequencies of a piezoelectric material with reference to Fig. 1. [107] First, X(f) of the mathematical equation 8 is Taylor series-expanded as a mathematical equation 9.

[108] [equation 9] [109]

n=0

[110] where, f : start frequency of the series expansion, and [111] Q: the number of terms in the Taylor series expansion. [112] When the mathematical equation 9 is substituted to the mathematical equation 8, [X(f)] and [F(T)] are also Taylor series-expanded (SlO) and then it is arranged for a same order of f-f in the both terms, a recursive relation equation capable of calculating moment vectors as mathematical equations 10a and 10b is obtained(S20).

[113] [equation 10a] [114]

[115] [equation 10b] [116]

[117] where, K^ω and F^Cl): i-th differential value. [118] Accordingly, it is possible to calculate the moments of X(f) with the mathematical equations 10a and 10b (S30). According to the mathematical equations 10a and 10b, a reverse matrix for the K matrix is calculated only one time. Accordingly, it is possible to significantly reduce the time required for the calculation.

[119] In the mean time, there is a limitation in the frequency band which can be analyzed with Taylor series only. Accordingly, a Pade' approximate function as a following mathematical equation 11 is used(S40).

[120] [equation 11] [121]

L

[122] where, L: integer equal or larger than 0, [123] M: natural number and [124] L+M=Q. [125] In the above equation, a and b can be obtained by substituting the mathematical equation 9 into the mathematical equation 11 , multiplying both terms by the denominator of the mathematical equation 11 and arranging it with regard to the same order of f-f . The matrix and equation obtained are mathematical equations 12a and 12b.

[126] [equation 12a] [127]

[128] [equation 12b] [129]

[130] It is possible to calculate the coefficients a and b of the Pade' approximate function from the equations 12a and 12b and the moment values obtained in the step S30 (S50). [131] Through the above process, it is possible to obtain the solution X(f) of the matrix of the mathematical equation 8 expressed as a function of frequency (S60). Accordingly, it is possible to analytically express an impulse response to the transfer function K(f) and to obtain a response to any type of input frequency.

[132] In addition, since the solution X(f) exhibits the mechanical displacement and the electric potential, it is possible to obtain the impedance by using the electric potential Φ(f) of the solution X(f)(S70). In other words, when Q is applied to an electrode surface, the impedance can be calculated with a solution at the electrode surface, i.e., electric potential Φ(f) and a mathematical equation 13.

[133] [equation 13] [134]

[135] where, Φ: difference between electric potentials, and [136] Q : charge applied from the exterior. [137] Like this, according to the invention, it is possible to obtain a solution for any input impedances. Accordingly, it is possible to calculate an impedance which is a function of frequency, by using the solution obtained for each frequency.

[138] [139] In the followings, it will be described a binary search algorithm for searching a start frequency f of the Taylor series expansion, with reference to Fig. 2. In general, with regard to any start frequency, it is preferred to select a start frequency f which minimizes a difference between impedance values depending on impedance functions obtained through the steps SlO to S70. [140] In case that search frequency areas of [f , f ] and a tolerance error value ε are given(Sl 10), the binary search algorithm passes through following procedures. [141] First, frequency boundaries are set as f =f and f =f (S120). mm 1 max 2

[142] Then, the f is assumed to be the start frequency f (S132). The steps SlO to S60 mm 0 are performed (S 134) to obtain an impedance function Z (f) (S136). Likewise, an nun impedance function Z (f) is obtained with regard to the f (S133, S135, S137). max max

[143] After that, a middle frequency f = (f +f )/2 between f and f is mid mm max mm max calculated(S140). [144] Then, impedance values are calculated (S 150). At

this time, when a difference between two values is ε(which is a tolerance error) or less, f is selected as the expansion start frequency f (S 160) and then the search is ended. mid 0

[145] However, when the difference of the impedance values is larger than ε in the step

S 150, the search process is repeated for [f , and f ] (S 170).

[146]

[147] Fig. 3 is a graph comparing a calculation result of impedances of a piezoelectric material according to the invention and impedance values experimentally measured. In the experiment, the L and M values of the Pade' approximate function of the mathematical equation 11 were set to be 3 and the Q value was set to be 6. As shown in Fig. 3, it can be seen that the calculation result of the invention and the conventional experiment result are nearly matched. According to the invention, it took about 2 minutes to calculate the impedance of the piezoelectric material. However, according to the conventional method, it took about 2 hours to calculate the impedance of the piezoelectric material.

[148]

[149] When there is substantially one resonant frequency as shown in Fig. 3, it is possible to find out the resonant frequency by performing the Taylor series expansion only for the single start frequency, as described above. However, if the Taylor series expansion is performed only for the single start frequency when the one piezoelectric system has two or more resonant frequencies, it may take much time to find out the resonant frequency.

[150] Accordingly, according to the invention, plural start frequencies are set and subject to the expansion, so that plural resonant frequencies can be simultaneously searched. Fig. 4 is a flow chart showing a method of searching plural start frequencies.

[151] Since steps S210 to S250 are same as the steps SI lO to S150 shown in Fig. 2, the descriptions thereof are omitted. [152] When a difference between impedance values in f , i.e., Z (f ) and Z (f ) is mid mm mid max mid larger than the tolerance error ε (i.e., when it is not converged) in the step S250, it is repeated the search steps S220 to S250 for the interval of [f , (S270).

[153] When the difference between the impedance values is the tolerance error ε or less

(i.e., when it is converged) in the step S250, is selected as the expansion start

frequency f (S260). In addition, a next interval same as the search interval is set as a search interval (S280). Accordingly, the search interval [f , f ] is set as [f , f + | f

1 2 max max max

-f mm I ].

[154] After that, the steps S220 to S240 are performed (S290) to determine whether or not the convergence of the impedance values in the middle frequency f (S 300). mid

[155] When the difference between impedance values in the f is larger than the

tolerance error ε (i.e., when it is not converged) in the step S300, it is repeated the search steps S220 to S250 for the interval of [f , (S270).

[156] When the difference between the impedance values is the tolerance error ε or less

(i.e., when it is converged) in the step S300, is selected as the expansion start

frequency f (S260), likewise. At this time, a next interval twice as large as the search interval is set as a search interval (S310). Accordingly, the search interval [f , f ] is defined as [f , f +2 | f -f I ]. max max max mm

[157] Then, likewise, the steps S220 to S240 are performed to determine whether or not the convergence of the impedance values in the middle frequency (S250).

[158] The above procedures are repeated for the entire interval of [f , f ], so that the plural start frequencies f can be searched. [159] [160] Figs. 5 to 13 illustrate examples of setting a search interval in the method of searching the plural start frequencies as shown in Fig. 4. In Figs., an interval which is indicated by an arrow is a search interval. In the followings, it is described examples of setting a search interval with reference to Figs. 4 and 5 to 13. [161] First, Fig. 5 shows an interval of the search frequency [f , f ] (S210). It is determined whether ] interval is converged (S250). When it is determined

that is diver ed it is a ain determined whether is converged for the search

interval (S270) as shown in Fig. 6. As a result, when it is again determined that f is mid diverged, it is again determined whether is converged for the search interval (S270)

as shown in Fig. 7. As a result, when it is again determined that is diverged, it is

again determined whether is converged for the search interval (S270) as shown in

Fig. 8. [162] As a result, when it is determined that is converged, as shown in Fig. 9, a next interval same as the previous search interval is set as a search interval (S280) to determine whether or not it is converged. As a result, when it is shown that f is ^σ mid converged, as shown in Fig. 10, a next interval twice as large as the previous search interval is set as a search interval (S310) to determined whether or not it is converged(S250). [163] As a result, when it is shown that is diverged, it is determined whether or not it

is converged for the search interval (S270) as shown in Fig. 11 (S250). [164] As a result, when it is shown that f is converged, as shown in Fig. 12, a next mid interval same as the previous search interval is set as a search interval (S280) to determine whether or not it is converged(S300). As a result, when it is shown that

is converged, as shown in Fig. 13, a next interval twice as large as the previous search interval is set as a search interval (S310) to determine whether or not it is converged (S250).

[165] The above procedures are repeated for the entire interval of the original search interval [f , f ]. Accordingly, it is possible to search the plural start frequencies f through the search process.

[166] Fig. 14 is a graph comparing a calculation result of impedances of a piezoelectric material using plural start frequencies as described above, and a calculation result according to the prior art. A blue line shows a calculation result of impedances according to the invention and a red dotted line shows a calculation result of impedances according to the conventional method. As shown in Fig. 14, it can be seen that the two results are nearly matched. When the plural start frequencies are used according to the invention, the calculation time was reduced by about 10 to 20 times. Industrial Applicability

[167] According to the invention, the impedance waveform depending on the frequencies is calculated using the approximate function of a matrix. The method of the invention reduces the calculation time by dozens to hundreds times, as compared to a conventional direct calculation method. Based on such numerical analysis technique, it is possible to obtain an impedance, a material constant, a resonant frequency, an an- tiresonant frequency and the like of a piezoelectric material in a short time. In addition, the method of the invention can be applied to a case of calculating a mechanical displacement depending on frequencies.

Claims

[1] A method of calculating an impedance depending on frequencies of a piezoelectric material, the method comprising steps of:

(a) Taylor series-expanding each matrix of a characteristic equation related to a frequency of a piezoelectric material with regard to a start frequency f ;

(b) comparing coefficients of a same order of both terms in the Taylor series- expanded equation to obtain a recursive equation relating to a moment of a matrix exhibiting a mechanical displacement and an electric potential of the characteristic equations;

(c) calculating each moment of the matrix exhibiting the mechanical displacement and electric potential from the recursive equation;

(d) approximating the matrix exhibiting the mechanical displacement and electric potential to a Pade' approximate function;

(e) calculating a coefficient of the Pade' approximate function in the step (d) from the equation obtained in the step (c);

(f) calculating a solution of the matrix exhibiting the mechanical displacement and electric potential from the Pade' approximate function obtained in the step (e), and

(g) calculating an impedance depending on frequencies, from the solution of the electric potential obtained.

[2] The method according to claim 1 , wherein the characteristic equation in the step

(a) is a following mathematical equation 8: [equation 8]

where, K: sparse matrix which is partially symmetrical,

X: matrix exhibiting both the mechanical displacement and the electric potential, F: matrix exhibiting both the mechanical force and the charge, and f: frequency.

[3] The method according to claim 2, wherein the matrix X in the step (a) is Taylor series expanded with a following mathematical equation 9: [equation 9]

n=0

[4] The method according to claim 3, wherein the recursive equation in the step (b) is following mathematical equations 10a and 10b: [equation 10a]

[equation 10b]

[5] The method according to claim 4, wherein the Pade' approximate function in the step (d) is expressed by a following mathematical equation 11 : [equation 11]

where, L: integer equal or larger than 0, and

M: natural number.

[6] The method according to claim 5, wherein in the step (e), coefficients a and b of the Pade' approximate function are obtained by substituting the mathematical equation 9 and the moment obtained in the step (c) into the mathematical equation 11.

[7] The method according to claim 6, wherein the coefficient b of the Pade¹ ap- proximate function is obtained from a following mathematical equation 12a, and the coefficient a is obtained from the obtained coefficient b and a mathematical equation 12b: [equation 12a]

[equation 12b]

[8] The method according to claim 1 or 7, wherein the start frequency f in the step (a) is selected to be a frequency allowing a difference between impedance values depending on an impedance function to be equal or less than a predetermined tolerance error value, the impedance function being obtained through the steps (a) to (g) from any start frequency.

[9] The method according to claim 8, wherein the start frequency f is selected by a binary search algorithm.

[10] The method according to claim 9, wherein the step of selecting the start frequency f comprises steps of, under conditions of search frequency range [f , f ] and the tolerance error value ε, (h) setting f and f of f range to be f =f and f =f ; nun max 0 mm 1 max 2

(i) applying the steps (a) to (g) for the f and f to obtain impedance functions mm max

Z mm (f) and Z max (f);

(] ) calculating Z (f ) and Z (f ) values for a middle frequency f =(f +f nun mid max mid mid mm max

)/2 of the f and f ; mm max

(k) selecting the f when a difference between the

is equal or smaller than the tolerance error value ε;

(1) setting a frequency range of [f , f ] and [f , f ] to be a new search mm mid mid max frequency range [f , f ], when a difference between the is

larger than the tolerance error value ε; and

(m) repeating the steps (h) to (1) until the start frequency f0 is selected in the step

(k).

[H] The method according to claim 9, wherein plural start frequency f are selected, and the selecting process comprises steps of, under conditions of search frequency range [f , f ] and the tolerance error value ε,

(hh) setting f and f of f range to be f =f and f =f ; mm max 0 mm 1 max 2 (ii) applying the steps (a) to (g) for the f and f to obtain impedance functions mm max

max )/2 of the f mm and f max ;

(kk) setting a frequency range of [f , to be a new search frequency range [f

, f ], when a difference between the Z (f ) and Z (f ) obtained in the step

2 mm mid max mid

(jj) is larger than the tolerance error value ε;

(11) selecting the f and setting a next interval

same as the previous frequency search interval as a search interval, when a difference between the Z (f ) and Z (f ) obtained in the step (jj) is equal or mm mid max mid smaller than the tolerance error value ε;

(mm) repeating the steps (hh) to (jj) for the interval set in the step (11);

(nn) setting the frequency range of [f , f ] to be a new frequency range of [f , f mm max 1

], when a difference between the Z (f ) and Z (f ) obtained in the step

2 mm mid max mid

(mm) is larger than the tolerance error value ε;

(oo) selecting the f and setting a next interval

twice as large as the previous frequency search interval as a search interval, when a difference between the Z (f ) and Z (f ) obtained in the step (mm) mm mid max mid is equal or smaller than the tolerance error value ε; and (pp) repeating the steps (hh) to (11) for the interval set in the step (oo), and wherein the steps (hh) to (pp) are performed for an entire interval of the original search frequency range [f , f ].