偏心电极石英晶体谐振器振动特性研究
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摘要
石英晶体谐振器的电极材料、电极尺寸以及电极位置对其性能的影响非常重要。但在加工制作过程中,经常会出现电极镀层偏离板心现象。而目前仅有少数人员研究具有偏心电极石英晶体谐振器的振动特性。文章运用Stevens和Tiersten提出的二维标量微分方程(S-T方程)的变分形式,利用里兹法研究偏心电极石英晶体谐振器的振动特性。在研究中分析不同电极偏心距、不同质量比对石英晶体谐振器振动模态的影响,以及不同电极偏心距谐振器谐振频率所存在的误差,为石英晶体谐振器的生产和设计提供理论依据。
The electrode material,electrode size and electrode position of quartz crystal resonator are very important to its performance. Due to the process of fabrication,it often occurs that the electrode coating deviates from the plate core. At present,only a few researchers have studied the vibration characteristics of quartz crystal resonators with eccentric electrod. In this paper, the variation form of the twodimensional scalar differential equations( S-T) proposed by Stevens and Tiersten are employed to analyze the thickness vibrations of eccentric electrode quartz crystal resonator. The influence of different electrode eccentricity and different quality on the vibration mode of quartz crystal resonator is analyzed,and the error of resonant frequency of the resonator under different eccentricity is analyzed. This provides theoretical basis for the production and processing of quartz crystal resonator.
The electrode material,electrode size and electrode position of quartz crystal resonator are very important to its performance. Due to the process of fabrication,it often occurs that the electrode coating deviates from the plate core. At present,only a few researchers have studied the vibration characteristics of quartz crystal resonators with eccentric electrod. In this paper, the variation form of the twodimensional scalar differential equations( S-T) proposed by Stevens and Tiersten are employed to analyze the thickness vibrations of eccentric electrode quartz crystal resonator. The influence of different electrode eccentricity and different quality on the vibration mode of quartz crystal resonator is analyzed,and the error of resonant frequency of the resonator under different eccentricity is analyzed. This provides theoretical basis for the production and processing of quartz crystal resonator.
引文
[1]王春雷,李吉超,赵明磊.压电铁电物理[M].北京:科学出版社,2009.73-91.
[2]李辉,陈向东,冯瑛.石英晶体谐振器预应力条件下的模态分析[J].传感器与微系统,2011,1:22-24.
[3]李辉,陈向东,赵静.基于有限元方法研究AT切石英晶体谐振器的动态特性[J].人工晶体学报,2017,46(5):782-785.
[4]何移,高杨,周斌,等.L波段薄膜体声波谐振器纵波谐振频率计算[J].压电与声光,2014,36(3):331-334.
[5]Tiersten H F.Thickness Vibrations of Piezoelectric Plates[J].J.Acoust.Soc.Am.,1963,35:53-58.
[6]Tiersten H F.A Corrected Modal Representation of Thickness Vibrations in Quartz Plates and its Influence on the Transversely Varying Case[J].IEEE Trans.Ultrason.,Ferroelect.,Freq.Contr.,2003,50(11):1436-1443.
[7]Tiersten H F,Mindlin R D.Forced Vibrations of Piezoelectric Crystal Plates[J].Quart.Appl.Math,1962,20:107-119.
[8]Mindlin R D.High Frequency Vibrations of Piezoelectric Crystal Plates[J].Int.J.Solids Struct.,1972,8(7):895-906.
[9]Tiersten H F.Analysis of Intermodulation in Thickness-shear and Trapped Energy Resonators[J].J.Acoust.Soc.Am.,1975,57(3):667-681.
[10]Tiersten H F.Analysis of Trapped-energy Resonators Operating in Overtones of Coupled Thickness-shear and Thickness-twist[J].J.Acoust.Soc.Am.,1976,59(4):879-888.
[11]Sinha B K,Stevens D S.Thickness-shear Vibrations of a Beveled AT-cut Quartz Plate[J].J.Acoust.Soc.Am.,1979,66(1):192-196.
[12]Tiersten H F,Smythe R C.Coupled Thickness-shear and Thickness-twist Vibrations of Unelectroded AT-Cut Quartz Plates[J].J.Acoust.Soc.Am.,1985,78(5):1684-1689.
[13]Stevens D S,Tiersten H F.An Analysis of Doubly Rotated Quartz Resonators Utilizing Essentially Thickness Modes with Transverse Variation[J].J.Acoust.Soc.Am.,1986,79(6):1811-1826.
[14]Wang J.Thickness-shear and Flexural Vibrations of Linearly Contoured Crystal Strips with Multiprecision Computation[J].Comput.Struct.,1999,70(4):437-445.
[15]Lee P C Y,Wang J.Piezoelectrically Forced Thickness-shear and Flexural Vibrations of Contoured Quartz Resonators[J].J.Appl.Phys.,1996,79(7):3411-3422.
[16]Yang J S,Zhou H G.Vibrations of an Isotropic Elastic Wedge[J].IEEE Trans.Ultrason.,Ferroelect.,Freq.Contr.,2005,52(1):139-144.
[17]Yang J S.Shear Horizontal Vibrations of a Piezoelectric/Ferroelectric Wedge[J].J.Acta Mech.,2004,173(1-4):13-17.
[18]He H J,Liu J X,Yang J S.Thickness-shear and Thickness-twist Vibrations of an AT-cut Quartz Mesa Resonator[J].IEEE Trans.Ultrason.,Ferroelect.,Freq.Contr.,2011,58(10):2050-2055.
[19]Cao-Paz A,Rodriguez-Pardo L.Temperature Compensation of QCM Sensors in Liquid Media[J].Sensor.Actuat.B-Chem.,2014,193:78-81.
[20]夏智深,陈玉梅,王骥,等.具有不对称电极的石英晶体板振动效应研究[C].2011年中国西部声学学术交流会论文集,银川:声学技术,2011,401-405.
[21]He H J,Liu J X,Yang J S.Effects of Mismatched Electrodes on an AT-Cut Quartz Resonator[J].IEEE Trans.Ultrason.,Ferroelect.,Freq.Contr.,2012,59(2):281-286.
[22]杜波,马晋毅,蒋欣,等.薄膜体声波谐振器有限元仿真与设计[J].压电与声光,2016,38(4)::531-534.
[23]He H J,Yang J S,Jiang Q.Thickness-shear and Thickness-twist Vibrations of Circular AT-Cut Quartz Resonators[J].Acta Mech.Solida Sin.,2013,26(3):245-254.
[24]He H J,Yang J S,Kosinski J A,et al.Thickness-shear Vibration of a Rectangular Quartz Plate with Partial Electrodes[J].Acta Mech.Solida Sin.,2013,26(2):121-128.
[25]Chen H,Wang J,Du J K,et al.Thickness-shear Modes and Energy Trapping in a Rectangular Piezoelectric Quartz Resonators with Partial Electrodes[J].Ferroelectr.Lett.,2015,42(1-3):10-17.
[26]Zhou Y S,Tiersten H F.On the Normal Acceleration Sensitivity of Contoured Quartz Resonators with the Mode Shape Displaced with Respect to Rectangular Supports[J].J.Appl.Phys.,2008,69(5):2862-2870.
[27]Kosinski J A,Pastore R A.Theory and Design of Piezoelectric Resonators Immune to Acceleration:Present State of the Art[J].IEEE Trans.Ultrason.,Ferroelect.,Freq.Contr.,2001,48(5):1426-1437.
[28]时俊杰.石英晶体谐振器与滤波器自由振动研究[D].郑州:郑州大学,2015.
[2]李辉,陈向东,冯瑛.石英晶体谐振器预应力条件下的模态分析[J].传感器与微系统,2011,1:22-24.
[3]李辉,陈向东,赵静.基于有限元方法研究AT切石英晶体谐振器的动态特性[J].人工晶体学报,2017,46(5):782-785.
[4]何移,高杨,周斌,等.L波段薄膜体声波谐振器纵波谐振频率计算[J].压电与声光,2014,36(3):331-334.
[5]Tiersten H F.Thickness Vibrations of Piezoelectric Plates[J].J.Acoust.Soc.Am.,1963,35:53-58.
[6]Tiersten H F.A Corrected Modal Representation of Thickness Vibrations in Quartz Plates and its Influence on the Transversely Varying Case[J].IEEE Trans.Ultrason.,Ferroelect.,Freq.Contr.,2003,50(11):1436-1443.
[7]Tiersten H F,Mindlin R D.Forced Vibrations of Piezoelectric Crystal Plates[J].Quart.Appl.Math,1962,20:107-119.
[8]Mindlin R D.High Frequency Vibrations of Piezoelectric Crystal Plates[J].Int.J.Solids Struct.,1972,8(7):895-906.
[9]Tiersten H F.Analysis of Intermodulation in Thickness-shear and Trapped Energy Resonators[J].J.Acoust.Soc.Am.,1975,57(3):667-681.
[10]Tiersten H F.Analysis of Trapped-energy Resonators Operating in Overtones of Coupled Thickness-shear and Thickness-twist[J].J.Acoust.Soc.Am.,1976,59(4):879-888.
[11]Sinha B K,Stevens D S.Thickness-shear Vibrations of a Beveled AT-cut Quartz Plate[J].J.Acoust.Soc.Am.,1979,66(1):192-196.
[12]Tiersten H F,Smythe R C.Coupled Thickness-shear and Thickness-twist Vibrations of Unelectroded AT-Cut Quartz Plates[J].J.Acoust.Soc.Am.,1985,78(5):1684-1689.
[13]Stevens D S,Tiersten H F.An Analysis of Doubly Rotated Quartz Resonators Utilizing Essentially Thickness Modes with Transverse Variation[J].J.Acoust.Soc.Am.,1986,79(6):1811-1826.
[14]Wang J.Thickness-shear and Flexural Vibrations of Linearly Contoured Crystal Strips with Multiprecision Computation[J].Comput.Struct.,1999,70(4):437-445.
[15]Lee P C Y,Wang J.Piezoelectrically Forced Thickness-shear and Flexural Vibrations of Contoured Quartz Resonators[J].J.Appl.Phys.,1996,79(7):3411-3422.
[16]Yang J S,Zhou H G.Vibrations of an Isotropic Elastic Wedge[J].IEEE Trans.Ultrason.,Ferroelect.,Freq.Contr.,2005,52(1):139-144.
[17]Yang J S.Shear Horizontal Vibrations of a Piezoelectric/Ferroelectric Wedge[J].J.Acta Mech.,2004,173(1-4):13-17.
[18]He H J,Liu J X,Yang J S.Thickness-shear and Thickness-twist Vibrations of an AT-cut Quartz Mesa Resonator[J].IEEE Trans.Ultrason.,Ferroelect.,Freq.Contr.,2011,58(10):2050-2055.
[19]Cao-Paz A,Rodriguez-Pardo L.Temperature Compensation of QCM Sensors in Liquid Media[J].Sensor.Actuat.B-Chem.,2014,193:78-81.
[20]夏智深,陈玉梅,王骥,等.具有不对称电极的石英晶体板振动效应研究[C].2011年中国西部声学学术交流会论文集,银川:声学技术,2011,401-405.
[21]He H J,Liu J X,Yang J S.Effects of Mismatched Electrodes on an AT-Cut Quartz Resonator[J].IEEE Trans.Ultrason.,Ferroelect.,Freq.Contr.,2012,59(2):281-286.
[22]杜波,马晋毅,蒋欣,等.薄膜体声波谐振器有限元仿真与设计[J].压电与声光,2016,38(4)::531-534.
[23]He H J,Yang J S,Jiang Q.Thickness-shear and Thickness-twist Vibrations of Circular AT-Cut Quartz Resonators[J].Acta Mech.Solida Sin.,2013,26(3):245-254.
[24]He H J,Yang J S,Kosinski J A,et al.Thickness-shear Vibration of a Rectangular Quartz Plate with Partial Electrodes[J].Acta Mech.Solida Sin.,2013,26(2):121-128.
[25]Chen H,Wang J,Du J K,et al.Thickness-shear Modes and Energy Trapping in a Rectangular Piezoelectric Quartz Resonators with Partial Electrodes[J].Ferroelectr.Lett.,2015,42(1-3):10-17.
[26]Zhou Y S,Tiersten H F.On the Normal Acceleration Sensitivity of Contoured Quartz Resonators with the Mode Shape Displaced with Respect to Rectangular Supports[J].J.Appl.Phys.,2008,69(5):2862-2870.
[27]Kosinski J A,Pastore R A.Theory and Design of Piezoelectric Resonators Immune to Acceleration:Present State of the Art[J].IEEE Trans.Ultrason.,Ferroelect.,Freq.Contr.,2001,48(5):1426-1437.
[28]时俊杰.石英晶体谐振器与滤波器自由振动研究[D].郑州:郑州大学,2015.