Viscosity Models and Vibration Solutions of Piezoelectric Resonators

Viscosity Models and Vibration Solutions of Piezoelectric Resonators J. Wang 1, H. Chen 2, X. W. Xu 3, J. K. Du 4, J. L. Shen 5 and M. C. Chao 6 1,2,3,4 Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University, 818 Fenghua Road, Ningbo, Zhejiang 315211, China 5,6 TXC (Ningbo) Corporation, 189 West Huangshan Road, Beilun District, Ningbo, Zhejiang 315800, China Viscosity is ubiquitous in structures through material combinations individually or collectively with required or inevitable effects to be considered with the identical approach of energy consideration. The presence and consequently results of viscosity are shown in different forms such as temperature increase from friction and vibration reduction due to energy absorption. In case of device structures operating in an electric field, energy absorption is also shown as the increase of electric impedance and consequently degradation of the so-called quality factor, which is a measure of energy dissipation in an oscillatory element in electronic devices nowadays. Apparently, the consideration of mechanical viscosity in device structures is needed for the estimation of device parameters such as impedance or resistance to be precise, offers key parameters in device applications and design through accurate predictions of performance in addition to known ones. The essence of such approach is the calculation of energy dissipation of resonant piezoelectric devices under alternating driving voltage with the presence of resistance that can be obtained based on the imagery part of elastic constants, which are related to energy dissipation from the detion of strain energy. We have been exploring accurate and simple methods to calculate the energy dissipation in piezoelectric devices through the consideration of material and structural viscosity/damping in high frequency vibrations of structures with recent finding that the introduction of equivalent viscosity in quartz crystal resonators can provide reasonably closer resistance needed in characterization of resonators including tuning fork resonators. These studies are showing that the introduction of structural and material viscosity in a proper manner can enhance the analysis of piezoelectric resonant devices through accurate consideration of energy dissipation mechanism. In turn, such models and analytical approaches can be further generalized in the analysis of similar devices operating in an electrical field with energy dissipation features representing a perfect combination of mechanical models and couplings of mechanical and electrical fields from practical problems. We believe such an approach can be also used in many other electronic and electrical devices such as MEMS and smart structures for the property estimation and prediction. Keywords: viscosity, damping, piezoelectric, wave, resonator, vibration, impedance 1 Introduction Material viscosity, or viscous damping, exists in all structural materials with important effects in structural vibrations through modification of structural performance parameters such as the vibration amplitude and frequency. As a result, the effect of viscosity can be positive or adverse if functions and viscosity are combined in analysis and applications. For instance, structural viscosity or damping is frequently required in the vibration control and reduction through the internal consumption of strain energy in vibrations for eventual reduction of amplitudes in structures as in earthquake 1 Professor, wangji@nbu.edu.cn 4 Professor, dujianke@nbu.edu.cn engineering. The widely used materials such as rubber and soft layers will increase structural damping and enhance the energy dissipation in structural vibrations to delay the structural response and shorten the duration of vibrations upon excitation. For such purposes, the viscosity of structures is modified through additions of layers or mixtures of materials with relatively high damping parameters to enhance the energy dissipation feature. Similar applications can be widely found in many engineering practices such as cushion, safety, protection, packaging, and so on. Another important application in the opposite direction focuses on reducing viscosity through material selection and structural optimization so structural vibrations can be easily excited for measurement and signal detection purpose, which emphasizes high sensitivity without structural and material damping and viscosity. In such

applications, like resonant devices and structures we frequently encounter in today s electronic products and smart devices, we need to reduce the viscosity and damping through material selection and process improvement such as clean-room technology and fine process for polishing and packaging. On the other hand, accurate estimation of intrinsic and internal damping will allow the determination of structural properties for the further calculation of vibration frequency, amplitude, and eventually electrical parameters such as the resistance. With such capability in the design process, electronic devices can be improved with modifications of structural parameters through detailed analysis without going through prototype and fabrication processes which are expensive and time consuming. Of course, the computer-analysis based design and improvement process are expected in both occasions for the consideration of viscosity without too much emphasis on vibration control and sensitivity enhancement because many techniques such as the analytical methods and material combinations of viscous solids are common and applicable without checking the background. As the advances of design methods, products, and solution development process, improvement and refinement of material models and analytical methods will be beneficial to all engineering applications facing common challenges. It is hoped that viscosity models and analytical procedures in this study will also provide practical concepts and paths for the improvement in similar applications. Apparently viscosity and damping in structures and devices are topics with many focuses in disversified technical disciplines we cannot possibly cover with our research and findings. In this study, we use the piezoelectric resonators such as quartz crystal and tuning fork types to start our analysis with the introduction of viscosity models and coupling of mechanical and electrical fields in high frequency vibrations. As stated before, the material viscosity is ubiquitous, and the structural viscosity can be significant for certain materials and devices in the fabrication process. It is also verified from our research that in some applications, the material damping itself is not important, or the effect on devices is negligible. But in certain other applications, the same material can behave differently through analytical models. Nonetheless, the viscosity is considered as the energy dissipation mechanism through the complex elastic constants. The introduction of complex elastic constants is the simplest approach for the analysis of vibrations through both analytical and te element methods, although we need to carry out operations of complex numbers by using doubled storage and arithmetic operations. We found that the results are actually accurate for extracting electrical parameters of resonators we are interested. The complex procedure of vibration analysis, namely the eigenvalue extraction and determination of eigenvectors, can be done without much difficulty with available mathematical software tools such as Matlab and Mathematica. In the case of te element analysis (FEA) with commercial codes such as Ansys and Abaqus, the complex matrix analysis can also be done automatically. It is our finding that the approach presented in this paper is simple, reliable, and useful in the analysis of acoustic wave resonators. 2 Viscosity Models and Considerations For the analysis of acoustic wave devices, we always start with field and constitutive equations for linear piezoelectric materials in the form of 1, 2] (1) where,,,,,,,, and are the stress tensor, density of materials, displacements, electrical displacements, electrical field, strain tensor, elastic constants, piezoelectric constants, and dielectric constants, respectively. In the classical formulation of wave propagation in ideal elastic solids without viscosity, material constants above are all real numbers. Consequently, essential solutions in terms of vibration frequency, electrical field, deformation, and other variables are also in real numbers. In this case, energy dissipation in the structure will not occur and some characteristics of structures such as electrical impedance cannot be obtained and many dependent variables are absent from outcomes of vibration analysis. To obtain such parameters to enable formulation of more resonator properties from vibration analysis, we have no choice but considering the material and structure viscosity so the energy dissipation can be calculated in the vibration analysis. This objective can be achieved with the introduction of viscous damping, or viscosity, in the piezoelectric structure. This is to suggest that stress is not only proportional to strain but strain rate as well through (2) where are the viscosity constants of material. For time-harmonic motions, this implies that the stresses and elastic constants are redefined by (3) where are now the complex elastic constants. Therefore, the constitutive relations of viscosity models updated to (4) These are the general formulation of vibrations of piezoelectric solids and structures with complex elastic constants. For piezoelectric resonators at relatively low

frequency, we do not need to consider complex piezoelectric and dielectric constants. The introduction of complex elastic constants will not change the boundary conditions and they will not be discussed here. (9) with as the fundamental thickness-shear frequency. As a numerical example, now we consider an ATcut quartz plate with a fundamental frequency of. With the standard procedure of vibration analysis, we have plotted dispersion and spectrum relations in Figs. 2 and 3. The viscosity constants of the quartz crystal 4], which supplements the elastic constants 5], will be essential in the and calculation. Figure 1: A typical plate model of crystal resonators. Now we consider an AT-cut quartz plate shown in Fig. 1. Its thickness is. We have the constitutive relations of the first-mindlin plate in abbreviated notations 3] without considering the effect of piezoelectricity as, ( )] ( )] (5) Figure 2: Dispersion relations of AT-cut quartz plate with dissipation. where the modified elastic constant and correction factor are (6) The corresponding stress equations of motion of the first-order Mindlin plate are (7) Figure 3: Frequency vs. length/thickness ratio of ATcut quartz plate with dissipation. These are the first-order Mindlin plate equations of a piezoelectric plate for the thickness-shear (TSh) vibrations of a simple resonator model with the consideration of viscosity. Analytical solutions are usually obtained by the assumption of cylindrical deformations. The displacements are assumed as 3 Equivalent Viscosity of Quartz Crystal Resonators (8) where and are amplitudes and wavenumber, respectively. The normalized variables are utilized as Figure 4: A partially electrode, AT-cut quart resonator.

Now we consider the partially electroded AT-cut quartz plate in Fig. 4. Its thickness is. The thickness and mass density of electrodes are and. The mass ratio between the electrodes and the crystal plate is. The corresponding equations of motion with the consideration of electrodes are 6] (10) One-dimensional constitutive relations for an AT-cut quartz crystal plate take the form ( ( ( ) ( ) ( ) ) ) ] ] ] (11) where are the complex constants and take the following form 7-9] ] (12) where the real elastic constants of quartz crystal is from Bechmann 5], is the equivalent viscosity coefficient which is used for the description of viscosity of electrode regions, and are the elastic constants of electrode material. This modification of elastic constants reflects the stiffness effects of electrodes. The modified elastic, piezoelectric, and dielectric constants and correction factors are The charge on the upper electrode and the current flowing into the top electrode of the resonator are 10] where (15) ( ) (16) In our plate model in Fig. 4, under the driving voltage, we have the electrical impedance as (17) The real part of impedance is the dynamic resistance. The static and motional capacitance is (18) For regions without electrodes, we neglect the weak electrical field and treat the plate as an anisotropic elastic plate. As a result, equations (10), (11), and (14) can still apply with mass ratio and the applied voltage. The boundary conditions include the traction-free and charge-free ends and continuity conditions at ] ] ] ] ] ] ] ] are (19) (20) where the square brackets represent the jump of a field across the intersection. Since (14) is inhomogeneous, the general solution can be written as the sum of the corresponding homogeneous equation and a particular solution, i.e., (13) Then, equations of motion in displacements and the charge equation in electrical potential are obtained by the substitution of (11) into (10) ( ) ( ) ) ( ] ( ) ] (14) (21) where the particular solutions given by

(22) For parts of the quartz crystal plate without electrode, we assume the displacements and electrical potential as these constants and use the command line (APDL) scripts as Table 1. As a result, we can see from Table 2 that the frequency of the tuning fork resonator model based on actual product CX-4 of Statek (Orange, CA, USA) with material viscosity is closer to the actual frequency of the product. ] ] ] ] (23) The substitution of (21) and (23) into the four boundary conditions in (19) and the eight continuity conditions in (20) leads to twelve linear algebraic equations, as in Refs. 11-13], for the twelve constants,,. We consider an AT-cut quartz plate with and the mass ratio. We have resistance versus the length of electrode in Fig. 5 and capacitance ratio versus normalized frequency in Fig. 6. Figure 5: Motional resistance vs. with partially plated Au,,,. 4 Viscosity Model for Tuning Fork Resonators With the successful consideration of viscosity in AT-cut quartz crystal resonators, we are interested in looking further into similar analysis of other types of resonators such as tuning fork resonators working at relatively low frequency and surface acoustic wave (SAW) resonators functioning at higher frequency.the core concept is still the viscosity through materials and structures. However, it is better to use te element method for the analysis of such resonators because analytical approximation has been proven difficult in these cases. Now we utilize the feature of current ANSYS for viscosity consideration through including complex elastic constants to make it possible to fulfill the objective of electrical parameter prediction of a quartz crystal tuning fork. We now consider the FEA model of a tuning fork in Fig. 7, which is created with ANSYS. The material element type is solid226, and two fingers are plated by driving electrode and detection electrode without the presence of wires between electrodes. The electrode is not considered in the modeling due to the thinner thickness, and nodes with electrode have equal electrical potential. Now we consider the material and structural viscosity in the modeling process and use the complex elastic constants as explained in Section 2. Then, complex constants such as and should be added into ANSYS as usual by changing the form of Figure 6: Capacitance ratio vs. with partially plated Au,,,,. Figure 7: FEA model of a quartz tuning fork resonator. 5 Conclusions Viscosity of materials and structures have been known and utilized for practical studies of vibrations of

elastic solids and structures with objectives of obtain optimal parameters of performance. In applications, Table 1: The APDL Script for Ansys Implementation.! ==Modal analysis /SOLU antype,modal! Use modal anlysis modopt,qrdamp,4! Use damped eigensolver solve! ==Harmonic analysis /SOLU antype,harm! Harmonic response analysis dmprat,damprato! Set damping harfrq,,32768! The specified load frequency outres,all,all! Output all results solve /POST1! The General Postprocessor set,1,1 /dscale,1,6 plns,uz! Plot map anharm! Animate complex displacements /POST1 PLNSOL,EF,SUM,0!Electric-field distribution PLNSOL,VOLT,,0!Potential distribution Table 2: The comparison of frequencies between calculated results with and without damping and measurements of STATEK CX-4. Frequency (KHz) Without damping With damping CX-4 32.174 32.698 32.768 constitutive relations with the consideration of material viscosity or damping is needed and the complex elastic constants are utilized to represent the energy dissipation feature. The challenge of such a formulation is shown through the complex material properties and consequently complex solutions of vibrations. The general principle based on the introduction of complex material properties have been suggested before and now we are using the concept for the analysis of quartz crystal resonators. We found that through adding equivalent viscosity in electrodes, complex vibration solutions give reasonable solutions of resistance of a resonator which is in good agreement with actual measurement. Then through a curve fitting, we can determine the value of viscosity and consequently other circuit parameters needed in resonator design and applications. Since the quartz crystal resonator fabrication process is relatively stable and unique, we can use accumulated data to decide basic parameters related to process for eventual determination of resonator properties. Our procedure has been tested and more experimental data are being collected for the completion of model utilization of viscosity for the estimation of energy dissipation to optimize and enhance structural improvement. We have also used this approach with the te element analysis of tuning fork resonators as part of the effort to extend the analytical technique to all types of acoustic wave resonators. References 1] Tiersten, H.F., Linear Piezoelectric Plate Vibrations, Plenum Press, New York, 1969. 2] Yang, J.S., An Introduction to the Theory of Piezoelectricity, Springer, Berlin, 2005. 3] Mindlin, R.D. (edited by Yang, J.S.), An Introduction to the Mathematical Theory of Vibrations of Elastic Plates, World Scientific, New Jersey, 2006. 4] Lamb, J. and Richter, J., Anisotropic acoustic attenuation with new measurements for quartz at room temperatures, Proc. R. Soc. Lond., Vol. A293, p. 479, 1966. 5] Bechmann, R., Elastic and piezoelectric constants of alpha quartz, Phys. Rev., Vol. 110, p. 1060, 1958. 6] Mindlin, R.D., High frequency vibrations of piezoelectric crystal plates, Intl. J. Solids Struct., Vol. 8, p. 895, 1972. 7] Reed, C.E., Knanzawa, K.K., and Kaufman, J.H., Physical description of a viscoelastically loaded AT-cut quartz resonator, J. Appl. Phys. Vol. 68, p. 1993, 1990. 8] Wang, J., Chen, Y.M., Wang, L.H., Wu. R.X., and Du, J.K., The calculation of resistance of quartz crystal resonators with consideration of electrode viscosity, Proc. of 2011 IEEE Intl. Freq. Contr. Symp., p. 387, 2011. 9] Chen, H., Wang, J., and Ma, T.F., The calculation of electrical circuit parameters of quartz crystal resonators with the consideration of equivalent viscous dissipation, Proc. of 2013 IEEE Intl. Freq. Contr. Symp., p. 1, 2013. 10] Zhang, C.L., Chen, W.Q., and Yang, J.S., Electrically forced vibrations of a rectangular piezoelectric plate of monoclinic crystals, Intl. J. Appl. Electrom., Vol. 31, p. 207, 2009. 11] Wang, J., Zhao, W.H., and Du, J.K., The determination of electrical parameters of partially plated quartz crystal plates with the consideration of dissipation, Proc. of 2006 IEEE Intl. Freq. Contr. Symp., p. 32, 2006. 12] Wang, J., Zhao, W.H., and Du, J.K., The determination of electrical paramenters of quartz crystal resonators with the consideration of dissipation, Ultrasonics, Vol. 44, p. 869, 2006. 13] Wang, J., Zhao, W.H., Du, J.K., and Hu, Y.T., The calculation of electrical parameters of AT-cut quartz crystal resonators with the consideration of material viscosity, Ultrasonics, Vol. 51, p. 65, 2011.