An improved equivalent circuit of piezoelectric transducers including the effect of dielectric loss

Transcription

1 J. Acoust. Soc. Jpn.(E) 6, 3 (1985) An improved equivalent circuit of piezoelectric transducers including the effect of dielectric loss Hiroshi Shimizu* and Shigemi Saito** *Faculty of Engineering, Tohoku University, Aramaki, Sendai, 980Japan **Faculty of Marine Science and Technology, Tokai University, , Orido, Shimizu, 424Japan (Received19July1984) A dielectric loss of a piezoelectric transducer decreases the mechanical Q under the electric terminal condition in short-circuit, and makes it small in comparison to that under the open-circuit condition. Even if the loss angle ƒâ is small, this effect cannot be neglected in transducers with high electromechanical coupling. However, the conventional equivalent circuit of piezoelectric transducers fails to represent this kind of effect due to the dielectric loss by only adding a shunt conductance to the clamped capacitance Cd. This inconvenience comes from the fact that all the elements in the conventional circuit, including Cd, are required to take complex values when ƒâ 0. In this paper, the equivalent circuit applicable to the case of ƒâ 0 is theoretically derived. The improved circuit consists of only real constant elements, independent of the frequency. Validity of the obtained circuit is demonstrated by the experiments on radial mode transducers. PACS number: Fx, Ct, Ar INTRODUCTION is usually very small. Recently many kinds of highcoupling piezoelectric material have been developed. As is well known, in the presence of finite dielectric loss-angle ƒâ (=- ÚƒÃ, s is the dielectric constant), the mechanical Q of a piezoelectric transducer resonating under the electric terminal condition of open-circuit is higher than that under the short-circuit condition. In relation to this effect of maximum electroacoustic conversion-efficiency is obtained at the anti-resonance, i. e., the mechanical resonance under the open-circuit condition. And it becomes lower at the resonance, that is the mechanical resonance under the short-circuit condition.1,2) The similar problem in magnetostrictive transducers is already well known, since the loss angle of effective permeability is especially large in metallic transducers because of eddy current. However the effect of loss angle ƒâ in piezoelectric transducers has seldom been discussed, because ƒâ In piezoelectric transducers using such a high electromechanical coupling material, their strong piezoelectric reaction causes the mechanical Q to be significantly affected by ƒâ, even if ƒâ is fairly small. Since, in addition, the effective dielectric loss-angle of ferroelectric materials becomes larger with increasing the driving-voltage amplitude, problems related to dielectric loss are very important, in piezoelectric transducers working at high power level, such as intense ultrasound radiators and piezoelectric ceramic transformers. An equivalent circuit is useful for analyzing the performance of piezoelectric transducers. Figure1 shows the conventional equivalent circuit, consisting of a series resonance circuit connected in parallel with a clamped capacitance, which has been widely adopted. In the conventional circuit, the dielectric 225

2 J. Acoust. Soc. Jpn.(E) 6, 3 (1985) capacitance Cd in the circuit of Fig.1. One question arises: Isn't it possible to obtain a reasonable circuit similar to Fig. 1, by decomposing all the complex-constant elements into resistive and reactive components? Let us investigate this problem. Based on the analogy to the theory of magneto- Conventional equivalent circuit. strictive constant by Kikuchi, 3) the clamped. capacitance Cd is proportional to the dielectric constantƒã, and the force factor A is proportional to the loss is neglected. If one tries to introduce the effect dielectric susceptibility x ( ߃Ã-ƒÃ0, ƒã0 is the of finite dielectric loss-angle into this circuit by only adding a shunt conductance to the clamped capacitance Cd, it will result in a value of Q under the short-circuit condition that is greater than the Q under the open-circuit condition. This result is opposed to the truth. In the present paper, the main reason that the conventional equivalent circuit fails to express effects of finite dielectric loss, is discussed first. Then a generalized equivalent circuit without such a problem is derived theoretically. The result of an experimental study is also shown, which demonstrates the validity of the new circuit. dielectric constant in vacuum). Then ÚA= ÚX. Expressing as Cd and A are written as The relation between ƒâ and ƒæ is given by When ƒæ áƒî/2,eq.(5) becomes (3) (4) (5) (6) DISCUSSION ON CONVENTIONAL EQUIVALENT CIRCUIT From Eqs.(1), (2), (3), and (4), the free admittance Y, defined by (I/V)F=0can be represented as follows: Neglecting the resistance and inductance in the connection circuit, fundamental equations of piezoelectric transducers are expressed as follows: (1) I is the terminal current, V is the terminal voltage, v is the velocity, F is the exerted force, co is the angular frequency, A is the force factor, and za is the mechanical impedance under the shortcircuit condition. Let us assume that (2) (7) From Eqs.(1) and (2), the conventional equivalent circuit as shown in Fig. 1, which represents the admittance of the electric terminals, in the case of F=0, is easily obtained. In the presence of dielectric loss, each element of the series resonance circuit also must take a complex value as well as Cd, because the force factor A takes a complex value. Therefore, the effect of dielectric loss-angle cannot be represented by only adding a shunt conductance to the (8) From Eqs.(7) and (8), an equivalent circuit shown 226

3 H. SHIMIZU and S. SAITO: EQUIVALENT CIRCUITIOF PIEZOELECTRIC TRANSDUCERS (11) Conventional equivalent circuit consisting of classified resistive and reactive components for case of finite dielectric loss. (12) in Fig.2 is obtained, in which all the elements have real-number values. This circuit looks to suggest that the value of Q under the open-circuit condition is smaller than that under the short-circuit condition owing to addition of the shunt conductance Gd. This is contrary to the true phenomenon. In the circuit of Fig.2, the value of R'remarkably depends on the angular frequency co and severely varies in the vicinity of the resonance angular-frequency (DA. frequency dependence of R' is taken into account, the equivalent circuit of Fig.2 explains the fact that the Q under the open-circuit condition is higher than that under the short-circuit condition. However, it is not easy to foresee intuitively the transducer behavior through the equivalent circuit of Fig.2.Thus, the conventional equivalent circuit is considered to be unsuitable for transducers with finite dielectric loss-angle. We need an equivalent circuit composed of the elements of which values are real and independent of the frequency. and zb is the mechanical impedance when I'=0. The impedance 1/jwCdx, is divided into the resistive and reactive component as follows: Introducing the voltage V'expressed by Eq.(11) is rewritten as (13) (14) (15) IMPROVED EQUIVALENT CIRCUIT Four Terminal Description Let us divide the clamped capacitance Cd into two components, i. e., capacitances due to ƒã0 and x, as follows: (9) Furthermore, Eq.(15) can be rewritten in the following form similar to Eq.(1): (16) An equivalent circuit representing Eqs.(10), (14), and and (16) is shown in Fig.3, all the elements are denoted by real constants. In the absence of dielectric loss, namely when ƒæ=0, ra is equal to rb Introducing the current I' defined by the following equations can be derived from Eq.(1). (10) and the circuit of Fig.3is reduced to the conventional circuit. Since the value of ƒæ in piezoelectric transducers is usually much smaller than ƒî/2, cos ƒæ à 1and sinƒæ àƒæ. For materials with large dielectric constant such as piezoelectric ceramics, it can be 227

4 J. Acoust. Soc. Jpn.(E) 6, 3 (1985) Improved four-terminal equivalent circuit including effect of dielectric loss. Approximate equivalent circuit for cases of 0 áƒî/2 and ƒã àx. Another representation (impedance-type equivalent circuit). assumed that Cdx=Cd and ƒæ=ƒâ, and then Cd0 can be neglected. In this case, the circuit of Fig.3is simplified as shown in Fig.4with good accuracy. It is notable that the new circuit has the same form as the conventional circuit except that a resistance is connected in series to the electric terminal. In the case tan ƒâ á1and1/qb á1, another representation of the equivalent circuit as shown in Fig. 5 can be derived approximately. The derivation will be described in Appendix A. We call it the impedance-type equivalent circuit. The equivalent circuit of piezoelectric transducers with two electrode-pairs, which will be available for the analysis of ceramic transformers, is derived easily as shown in Fig.6. For the case of magnetostrictive transducers, the effect of magnetic loss-angle has formerly been stud- Equivalent circuit of transducer with two electrode-pairs. ied, since the loss angle of the effective permeability is large in metallic materials, due to eddy current. However, the equivalent circuit of which each element has a value regarded as constant over the wide 228

5 H. SHIMIZU and S. SAITO: EQUIVALENT CIRCUIT OF PIEZOELECTRIC TRANSDUCERS frequency range has not been discussed completely, probably because of its low electromechanical coupling. The improved equivalent circuit of magnetostrictive transducers can be obtained in the same manner (see Appendix B). Two Terminal Description From Fig.3, we can derive a two-terminal equivalent circuit shown in Fig.7, which represents the free admittance Yf. This circuit is approximately reduced Fig. Improved two-terminal equivalent circuit including effect of dielectric loss. as shown in. Fig.8 for the case of ƒæ sƒî/2and x àƒã Recently, Toki et al. reported that the experimental results for admittance characteristics of radial-mode ceramic-disk transducers could be well explained by a new circuit having an additional resistance connected in series to the conventional circuit of Fig. 1, 4) but they did not give the theoretical derivation and the physical explanation. Their circuit is supposed to be just the same as the circuit shown in 8. They also found from experimental results Fig. Fig. Approximate equivalent circuit for cases of ƒæ sƒî/2and ƒã àx. in Ref. 4) that the value of the additional series resistance was proportional to the thickness of disk transducer, and inversely proportional to the diameter. These facts can be explained by the relation Rd=ƒÂ/(ƒÖ bcd b) in the present paper, because I bcd b is inversely proportional to the thickness, and (ƒöa bcd b is proportional to the diameter. Setting the condition of F=0by shorting out the secondary terminals of the circuit in Fig.5, we obtain another representation of the two-terminal equivalent circuit as shown in Fig.9. This circuit satisfies our intuition that the conductance Gd due to dielectric loss should be connected in parallel with the clamped capacitance Cd. Furthermore, while the capacitance C and resistance R of the series resonance circuit in Figs. 7and8are determined by the stiffness SA under the short-circuit condition and the mechanical resistance rb under the open-circuit condition respectively, the whole of the parallel resonance circuit in Fig.9are determined only by the mechanical impedance zb observed under the open-circuit condition. In this sense, a correspondence to the physics in transducers is more clear in Fig. Impedance-type two-terminal equivalent circuit. 9than in Fig.8. Fig. EXPERIMENT The big difference of the new circuit from the conventional circuit is that it has one more resistance element in series to the electrical terminal. The resistance value is proportional to the dielectric loss angle as theoretically derived above. Some experiments focused on the test of this point were carried out on three samples of circular disks resonating at the fundamental radial mode. The disks were made 229

6 J. Acoust. Soc. Jpn.(E) 6, 3 (1985) Equivalent circuit constants. element of the conventional equivalent circuit of Fig.1 can be obtained from each of the measured characteristics of Figs. 10 (a) and10 (b). A broken line in Fig.10 (a) shows a resonance curve calculated (a) from the conventional equivalent circuit obtained using the measured anti-resonance characteristic in Fig.10 (b). Another broken line in Fig.10 (b) expresses an anti-resonance curve calculated from the conventional equivalent circuit obtained using the measured resonance characteristics in Fig.10 (a). We find great discrepancy between the calculated and measured. This means that the conventional circuit, which neglects the dielectric loss, cannot express the transducer characteristics over a wide frequency range. On the other hand, the improved equivalent circuit shown in Fig.8 has both the admittance and impedance characteristics that agree with the experimental results, as shown by the solid lines in Figs. 10 (a) and10 (b). Table1shows the values of (b) Comparison of measured and calculated immitance characteristics of sample new circuit,----: calculated from conventional circuit.(a) free admittance, (b) free impedance. from a PZT-type ceramic material N-21 (Tohoku Metal Industrial Co.), which has a high electromechanical coupling factor and a low mechanical Q. The three samples are referred to as #1, #2, and #3, whose thicknesses are2.1, 3.1, and4.0mm, respectively. Each sample has the same radius of19.7 mm, and the resonance frequency is around50khz. Ring plots in Figs.10 (a) and10 (b) show a measured free-admittance characteristic near the resonance frequency and a measured free-impedance characteristic near the anti-resonance frequency, respectively. If we neglect the dielectric loss, each improved-circuit's elements obtained for the three disks. In the course of the present determination of the improved circuit elements, the values of L, C, and Cd were kept equal to the respective values in the conventional circuit obtained from the experimental plots in Fig.10 (a), for simplicity. The values of R and Rd were determined through trial and error so that a calculated impedance characteristic agrees well with the experimental one, keeping a sum of R and Rd equal to the value of R in the conventional circuit derived from the experimental plots in Fig. 10 (a). We have to pay attention to the fact that the value of Cd in Table1is actually different from the true value of clamped capacitance, because the measured Cd contains the capacitive component brought by the higher order mode resonances. Also, the dielectric loss angle observed at very low frequency, which is denoted by80, is different from the intrinsic loss angleĉ, that is the loss angle of clamped capacitance Cd. Taking these facts into considerations, the two 230

7 H. SHIMIZU and S. SAITO: EQUIVALENT CIRCUIT OF PIEZOELECTRIC TRANSDUCERS Table2 Comparison of calculated and measured values of tan ƒâ0and tan ƒâ. Fig.11 Equivalent circuit of radial mode N-21ceramic transducer for low frequency range. Schering bridge employed for measuring ƒâ0. terminal equivalent circuit of radial mode transducers at very low frequencies is derived as shown in Fig.11, using the element values listed in Table 1 (see Appendix C). The loss angle of this circuit, which is equal to ƒâ0, is given by (17) ƒöa is the fundamental resonance angular frequency, and Cd'=Cd-0.27C, that is the true clamped capacitance. Substituting the values listed in Table1into Eq.(17), the value of tan 60 is obtained as shown in Table2. This value is fairly large in comparison to the value of tan a obtained from the relation Rd=ƒÂ/(ƒÖ)Cd') by using the values of Rd, Cd, and C in Table1. Another evaluation of tan ƒâ0was carried out by direct measurement at frequencies in the range10 ` 25kHz, utilizing Schering bridge with Wagner's earthing device shown in Fig.12. A silvered-mica capacitor (tanƒâ<0.1%) and air capacitor were provided for the variable capacitances Cl and C2, respectively. The loss angle is obtained from the bridge-balance condition ƒâ0=ƒöc2r2. Figure13 shows an example of experimental results. The values ofƒâ0at ƒö=0were determined by extrapolating the approximate lines which were obtained by Result of direct measurement of ƒâ0. the least squares method. In order to verify the new equivalent circuit, we compare the directly measured values of loss angle mined through the above-mentioned process involving the experimental derivation of improved equivalent circuit elements. As shown in Table2, these values agree reasonably well. This agreement confirms that the new equivalent circuit correctly expresses the effect of the dielectric loss. CONCLUSION In piezoelectric transducer with high electromechanical coupling or very high mechanical Q, even 231

8 J. Acoust. Soc. Jpn.(E) 6,3 (1985) if the loss angle ĉ of the dielectric constant is fairly small, the excess mechanical resistance due to the piezoelectric reaction cannot be neglected. When the conventional equivalent circuit neglecting the the following equation. (A1) Section2is equivalent to an imaginary gyrator, since the four terminal matrix (F matrix) of this section is described as dielectric loss is applied to such a case, a value of each circuit-element becomes complex or strongly dependent on the frequency. Therefore, the conventional circuit is often inadequate for analyses. In order to improve the representation, an equivalent circuit, whose elements take all the real and frequency-independent values, was theoretically derived in the present paper. Experimental validation of the new circuit on piezoceramic disks was presented. Although the discussion in this paper was focused on vibrations with single degree of freedom, the results may be easily expanded to cases of multiple Therefore, the F matrix for the cascade connection of Sections2, 3, and4is transformed as follows: degrees of freedom. REFERENCES 1) Y. Kikuchi and H. Shimizu,"On the B-type resonance of magnetostriction and electrostriction transducers," Sci. Rep. Res. Inst. Tohoku Univ. B 4-1, (1952). 2) H. Shimizu,"Effective attenuation of piezo-active transducers and its universal chart," in Ultrasonic Transducers, Y. Kikuchi, ed. (Corona, Tokyo, 1969), Chap. 8, p ) Y. Kikuchi,"Theoretical consideration of piezoactive constants for longitudinal vibration," ibid., Chap. 3, p.33. 4) M. Toki, Y. Tsuzuki, and O. Kawano,"Precise measurement of equivalent circuit parameters of ceramic resonators," Trans. IECE J62-A, (A2) (1979)(in Japanese). 5) T. Suzuki,"Fundamental type piezoelectric and electrostrictive transducers," in Ultrasonic Transducers, Y. Kikuchi, ed. (Corona, Tokyo, 1969), Chap. 7, p.236. APPENDIX A: DERIVATION OF IMPEDANCE-TYPE FOUR-TERMINAL EQUIVALENT CIRCUIT By connecting two capacitances bcd b I and bcd b The last matrix in the right side of Eq.(A2) expresses an imaginary gyrator. From the above discussion, the equivalent circuit of Fig.4can be transformed as shown in Fig.A2. In this circuit, each of the elements representing (Ě/Ěb)2zb has a value dependent on the frequency. It is desired that all the elements have values independent of the frequency. Then, (Ěb/Ě)2/zb,, is rewritten as in series to the resistance Rd, the admittance-type equivalent circuit shown in Fig.4can be modified as Fig.A1. This circuit is considered to be composed of four sections connected in cascade as illustrated in the figure. Each section may be modified in the form suitable for impedance-type equivalent circuit by the following considerations. The series connection of Rd and bcd b in Section1 can be transformed to the parallel connection, using (A3) 232

9 H. SHIMIZU and S. SAITO: EQUIVALENT CIRCUIT OF PIEZOELECTRIC TRANSDUCERS Illustration of four sections composing the circuit of Fig.4. Impedance type equivalent circuit with frequency-dependent elements. Thus, another representation of the four-terminal equivalent circuit is obtained as shown in Fig.5. APPENDIX B: ADVANCED EQUIVALENT CIRCUIT OF MAGNETOSTRICTIVE TRANSDUCERS Improved equivalent circuits of magnetostrictive transducers are shown in Figs. B1and B2. These Equivalent circuit of magnetostrictive transducers which has dual relation with circuit of Fig.3. Equivalent circuit of magnetostrictive transducers which has dual relation with circuit of Fig

10 circuits are in the duality relation with the equivalent circuits of piezoelectric transducers shown in Figs.3and6. EQUIVALENT CIRCUIT AT LOW FREQUENCY The motional admittance Ym of radial-mode circular-disk transducer is expressed as follows5): Considering the mechanical loss by denoting as EG=EG1+jEG2, one obtains the ratio of the motional resistance near the fundamental resonance, RA, and that at very low frequency, R0, as follows: (C3) (C1) er is the apparent piezoelectric stress constant, EG is the Young's modulus, ƒðg is the Poisson's ratio, a is the radius, t is the thickness, and k is the wave number. The subscript G in EG and ƒðg. means that they should be the values under the short-circuit condition. From Eq.(C1), the ratio of the motional capacitance near the fundamental resonance, CA, and that at very low frequency, C0, is expressed as follows: (C2) ƒ 0=k0a, which is the smallest root of the following characteristic equation; In the present case of the material N-21in which reduced to Therefore, it can be seen that the clamped capacitance Cd observed around the fundamental radialmode resonance includes the capacitive component due to the higher-order resonances, which is0.27 times as large as the motional capacitance C. (C4) Using Eq.(C4) and an assumption that the dielectric loss tan ƒð is independent of the frequency, the equivalent circuit for radial-mode type N-21 ceramic transducers at sufficiently low frequencies is obtained as shown in Fig.11of the present paper. 234