Tsutomu Imai Masako Tanaka, Seiko Epson Corporation, Suwa-city, Nagano

The Datasheet Archive - 100 Million Datasheets from 7500 Manufacturers.

Tsutomu Imai Masako Tanaka, Seiko Epson Corporation, Suwa-city, Nagano

Datasheet Thumbnail

Top Searches for this datasheet

SURFACE CHARGE MEASUREMENT/CALCULATIONS PREDICTION SPURIOUS MODES FREQUENCY JUMPS AT-CUT QUARTZ RESONATORS
Tsutomu Imai Masako Tanaka, Seiko Epson Corporation, Suwa-city, Nagano, JAPAN Yook-Kong Yong, Rutgers University, Piscataway,
Abstract
2-dimentional analysis using 3rd-order Mindlin's plate equations employed prediction spurious modes frequency jumps frequency temperature behavior AT-cut quartz resonators. surface charge over electrode employed means predicting strength coupling spurious modes with fundamental thickness shear mode (main mode). magnitudes frequency jumps also qualitatively related magnitude surface charges spurious modes when compared surface charge main mode. results were verified comparison with experimental data electroded resonator bi-convex resonators. measurements charge distribution spurious vibration modes were good agreement with calculated results flat resonator. 14MHz bi-convex model analyzed. Spurious modes were shown with their surface charge frequency Spectrum. large surface charge spurious mode shown large frequency jump.
frequency-temperature coefficients. temperature change would cause modal branches these spurious modes drift across thickness shear modal branch cause thickness shear frequency "jump". objective this paper improve prediction these frequency jumps this analytical method practical resonator designs. frequency spectra from plate equations showed many more spurious modes than were present actual plate resonator, and, hence, would predict many more frequency jumps than observed AT-cut resonator. solution supplement frequency spectra with surface charge calculations. magnitudes surface charge spurious modes compared magnitude surface charge thickness shear mode, which shall call main mode this paper. found that larger this magnitude when compared that thickness shear mode (main mode) more likely that such mode will interfere with thickness shear mode. This correlation verified experimental measurements surface charges.
Introduction
AT-cut crystal resonators essential devices telecommunication market their frequency stability wide range temperatures. rapid growth telecommunication market makes product development time these crystal devices shorter. addition, miniaturization these devices strongly demanded even frequency resonators. efficient method increasing energy trapping keeping high quality factor AT-cut resonator convex plate. However, spurious modes convex resonator difficult predict, which makes design such resonators very time consuming. have previously shown [1,2] that higher-order Mindlin's plate equations yield accurate frequency spectra fundamental thickness shear mode AT-cut plate resonators with relatively small length thickness ratios (a/b 20). frequency-temperature behavior same plate equations also shown relatively accurate. there only papers concerning convex resonator. common causes frequency jumps stable quartz resonators coupling thickness shear mode with more spurious modes that have large
analysis
analysis equations analysis this paper employed higher order Mindlin's plate equations. shows coordinates AT-cut resonator. According this coordinates, displacements element each axis were expressed equation (1). [2,5,6]
Fig.1 coordinates resonator
desktop used.
Analytical Experimental results
Here, displacement each point, x2/b
order two-dimensional displacement. this analysis, order Mindlin's plate equations were used. correction factors were used obtain more precise solutions thickness shear mode (main mode). correction factors applied following stress components shown below. [3,7]
Rectangular model with uniform thickness studied rectangular model with uniform thickness compared results with experimental results order verify accuracy eigenvalue calculations spurious mode frequencies surface charge distributions. this purpose, electrode free crystal plate with relatively large dimensions 20mm 0.25mm. length- width-to-thickness ratios were directions, respectively. These dimensions were quite large when compared with those manufactured resonators. Fig.2 shows comparison spurious modes with results left (Fig. experimental results right (Fig. 2b). frequency indicated above each surface charge distribution. observe that mode shapes surface charge distribution (number half-waves) along direction direction exactly same measured distribution both vibration modes. measured frequencies were also good agreement with calculated results both cases.
5.447751MHz 5.47076MHz
Tp(n) stress components half thickness crystal plate ;nth strain direction
correction factors frequency-temperature analysis, Lagrangean equations employed obtain frequency-temperature curves. total charge amount each vibration mode calculated equation (3).[8,9,10]
6.178806MHz
6.200795MHz
Calculation Measurement
(u1,3 (u1,
Here, surface charge, surface charge density electric displacement vector direction), e25,e26 piezoelectric constants, ui,j shear strains, respectively. surface charge calculated under electrode area electrode-quartz interface resonators with electrodes. This value used qualitative estimation strength coupling spurious mode with main mode. Mesh model employed both model model flat resonator convex type resonator. 9-node isoparametric element with Lagrangean polynomial shape functions used. thickness also interpolated isoparametrically, hence thickness over each element described quadratic Lagrangean polynomials that finite element mesh accurately follows shape actual convex type resonator. accuracy mesh shape will shown following section. Compaq Alpha ES40 computer (CPU speed 667MHz, Memory 8GB) used computations. pre- post-processing, Fig.2 Surface charge distributions spurious modes Convex mesh model Four factors that determine shape actual blank resonator were employed generating mesh models convex type resonator. These four factors flat area crystal blank along direction, curved area which follows shape along direction, radius, reduced depth which shaved depth along direction. four factors shown schematically fig. Three-dimensional data were employed adjust mesh real shape actual blank. detail adjustment done making following three index frequencies equal. They frequency difference between main mode anharmonic thickness shear mode (3,1,0) (1,1,0)), normalized overtone frequency main mode (1,3,0), frequency first face shear mode direction (0,0,1). These three indexes were very sensitive shape convex curvature, other words, four factors mesh modeling.
(a)-(i) show dependencies index frequencies three factors mesh modeling, here radius fixed These figures show that sensitive flat area insensitive
both curved area reduced depth (see (a), (g)). (1,3,0) sensitive three factors, curved area, reduced depth flat area (see (b), (h)). (0,0,1) more sensitive reduced depth (see (c), (i)). interesting that generally used controlling convex shape because this factor clearly represents energy trapping level, nothing with first face shear mode, which generally believed connected high order face shear mode frequency, which likely cause frequency jump coupling main mode strongly. This means that controlling sufficient determining both energy trap level spurious mode position. mesh model actual blank shapes along directions were compared this model, 4.1mm 1.1mm blank with frequency 14.6MHz calculated. this figure, index frequencies mesh model shown. comparison shapes between mesh model (solid line) real blank (doted line) seem good agreement. Frequency spectrum Then frequency spectrum electroded resonator convex type calculated using mesh model discussed above. center frequency resonator 14.3MHz, blank size 4.1mm 1.1mm, silver electrode formed with size 0.8mm. three index frequencies crystal blank were follows, kHz, F(1,3,0) 2.959, (0,0,1) 2.526 MHz, respectively. width ratio blank thickness changed from
(1,3,0) f(0,0,1) (0,0,1) (MHz)
Four factors convex modeling
(kHz)
(kHz) Curved area
Flat area (micron)
Flat area (micron)
(0,0,1) (MHz)
Flat area
Flat area (micron)
(kHz)
Curved area (micron)
Curved area (micron)
Curved area (micron)
Reduce depth
Reduced depth(micron)
(0,0,1) (MHz)
Reduced depth(micron)
Reduced depth(micron)
Dependencies three index frequencies three mesh factors
10.1. calculation, mesh model used taking into account symmetric shape blank.
Blank size F(1,3,0) (0,0,1)
4.1mm*1.1mm 2.959 2.526
comparison mesh shape with real crystal blank
mesh size 100x30. Fig. shows calculated frequency spectrum. designing resonator, generally choose open area where main mode doesn't cross spurious mode. however know that spurious mode predicted calculation does always cause frequency jump. Some jumps large, some them small negligible cannot detected measurements. Whenever spurious mode crosses over main mode frequency spectrum, there some couplings modes. following reasoning used justify employing total surface charge spurious modes identifying those modes that will cause large frequency jumps. magnitude electric current resonator equal product frequency surface charge electrode. main mode resonance, resonator admittance maximum. This admittance directly related surface charge magnitude because driving voltage frequency relatively constant over range temperatures. concerned with behavior spurious modes immediate neighborhood frequency main mode, which cross over modes. this immediate neighborhood, spurious mode, which large surface charge, would indicate that large current since frequency almost same main mode, driving voltage constant. large current implies that admittance similarly large would indicate that main mode coupling strongly with Fig.7 shows same frequency spectrum that fig. except surface charge calculations. white circles denote main mode colored circles denote spurious modes. Each vibration mode surface charge magnitude, which
Main mode Point
Point
Frequency spectrum convex model (with surface charge information)
Fig. Frequency spectrum convex model (with surface charge information)
expressed, area circle. Since main mode very large surface charge, area circle represent would large that would block view surface charges spurious modes immediate neighborhood. Hence Fig. surface charge main mode divided could from this frequency spectrum, which includes surface charge information that some spurious modes have little effect main mode, while others show strong coupling main mode. example spurious mode Point seems cause large spurious jump spurious modes coupled together crossing point with main mode. another point like point "B", spurious mode doesn't have large surface charge itself, shows strong coupling only when immediate neighborhood main mode. study Points check reasoning using surface charge calculations. Samples were prepared have same index frequencies shown table Frequency temperature curve frequency temperature curve measurement convenient detecting frequency jump that caused spurious mode. Frequency temperature curves were calculated samples Table shows temperature frequency curve which surface charges were considered point sample. From this figure, large spurious modes located near main mode high temperature regions. Also displacement vectors shown spurious modes. Figs. 9(a) show calculated frequency temperature curve curve error. Figs. 9(b) experimental results. predicted Fig. both temperature dependencies show that there spurious mode temperature region high temperature region around Peak temperatures these spurious modes were widely distributed manufactured samples. Since vibration frequency determined dimensions both directions, blank size tolerance both length width will affect vibration
frequency peak temperature frequency jumps. same comparison done point sample. Figs. show calculated results experimental results respectively; observe that agreement surprising good. surface charge information taken into account point sample shown 11(a). surface charge spurious mode crossing point half sample "A". surface charge distribution displacement spurious mode calculated Fig. 11(b), (c). From these figures, observe that this spurious mode flexure mode along direction. frequency strongly dependent width direction. fact, manufactured samples point have smaller peak temperature distributions compare with samples. displacement edge larger than center blank. This distribution displacements resembles results which presented Sekimoto al.[4] calculated thermal coefficient this spurious mode -59ppm/°C, which quite large. measured value -56ppm/°C. These values
Displacement vectors Displacement vectors
Calculated frequency temperature curve with surface charge
Table
Term Calculation Frequency (MHz) Size ratio direction (kHz) (1,3,0) (0,0,1) (MHz) 14.33 9.38 2.959 2.526 Point Experiment 14.32 9.38 2.961 2.526 Calculation 14.33 9.94 2.951 2.431 Point Experiment 14.32 9.94 2.954 2.432
Calculated results sample
Experimental results sample
Fig.9 Frequency temperature curves point sample.
Calculated results sample
Experimental results sample
Fig.10 Frequency temperature curves point sample. good agreement, considers that most spurious modes coefficients around -40ppm/°C. quartz resonators. Relatively good prediction frequency jumps performed using surface charge spurious modes immediate neighborhood main mode identify strength coupling main mode. Good comparisons with experimental results were found both flat convex plate resonators.
Conclusions
two-dimensional analysis using third-order Mindlin's plate equations were employed prediction spurious modes frequency jumps design AT-cut
Frequency temperature curve with surface charge information.
center blank
Displacement vectors zero order order directions Distributions displacement mesh model calculated vibration modes point sample.
Acknowledgment
authors deeply appreciate help from Prof. Watanabe Tokyo metropolitan Univ. splendid work surface charge distribution measurements. also very grateful Wang helping develop useful program; Karaki Sato their excellent work fabricating various shaped resonators their precise measurements; Zhang valuable advise encouragement.
References
"Accuracy Crystal Plate Theories High Frequency Vibrations Range Fundamental Thickness Shear Mode," Yong, Zhang, Hou, IEEE Transactions Ultrasonics, Ferroelectrics, Frequency Control, Vol. No.5, Sept. 1996, pp888-892. Accuracy Mindlin Plate Predictions Frequency-Temperature Behavior Resonant Modes SC-cut Quartz Plates", Yong, Wang Imai, IEEE Transactions Ultrasonics, Ferroelectrics, Frequency Control, Vol. January 1999, 1-14. "Finite Element Analysis Piezoelectric Vibrations Quartz Plate Resonators with Higher Order Plate Theory",
Wang, Yong Imai, International Journal Solids Structures, Vol. 1999, 2303-2319. "Analysis Coupled Vibrations Bi-Convex AT-Cut Quartz Plates with Clamped-Free Edges", Shingaku-Gihou, US88-11, 1988, 47-52. (Written Japanese) "Frequency-temperature Behavior Thickness Vibrations Doubly Rotated Quartz Plates affected Plate Dimensions Orientations," P.C.Lee Y-K. Yong, Appl. Phys., pp2327-2342. "Theoretical Analysis Quartz Resonators Finite Element Applications," Y-K. Yong, Proc. 27th Symposium, 1998,pp51-65. "Higher Order Plate Theory Based Finite Element Analysis Frequency-temperature Relations Quartz Crystal Resonators," Wang, Y-K.Yong, T.Imai, Proc. Freq. Cont. Symp. 1998, pp956-963. "Forced Vibrations Thickness-flexures, Face-shear Face-flexure rectangular AT-cut Quartz Plates," Proc. IEEE Freq. Cont. Symp., 1992, 532-536. "Three Dimensional Analysis Forced Vibrations Rectangular AT-cut Quartz Plates," A.Ishizaki H.Sekimoto, IEEE Freq. Cont. Symp., 1998, pp942-946. "Linear Piezoelectric Plate Vibrations", H.F.Tersten, Plenum, 1969, pp185-187.

Tsutomu Imai Masako Tanaka, Seiko Epson Corporation, Suwa-city, Nagano

Top Searches for this datasheet

Other recent searches