Model development for the beveling of quartz crystal blanks

Transcription

1 9t International Congress on Modelling and Simulation, Pert, Australia, 6 December 0 ttp://mssanz.org.au/modsim0 Model development for te beveling of quartz crystal blanks C. Dong a a Department of Mecanical Engineering, Curtin University, GPO Box U987, Pert, WA 6845, Australia c.dong@curtin.edu.au Abstract: A quartz crystal unit is a quartz wafer (also referred to as blank or crystal plate) to wic electrodes ave been applied, and wic is ermetically sealed in a older structure. In order to acieve igly stable and reliable quartz crystal resonators witout spurious response, quartz crystal blanks are finised into sperically contoured sape. Te process of generating a sperical surface on a quartz plate is called beveling or contouring. Crystal manufacturers currently employ a trial-and-error approac because of te insufficient knowledge on te influences of process parameters. Tis limits te development of new generation crystal resonators. Tus, it is desirable to develop a process model, wic can provide te process parameter settings based on te beveled blank design information. A process model for te beveling of quartz crystal blanks is presented in tis paper. For a given crystal blank design, te sape of bevel and beveling time can be predicted from process factors, e.g. powder type, macine rpm and barrel diameter. Te inputs of te beveling model are: blank dimensions: L, W; blank frequency: f; bevel widt: w b ; bevel dept: b ; barrel combination: e.g. φ80, φ30, φ80; average diameter of beveling powder: d p ; macine rpm: ω. First, te model for material removal rate, mrr, was developed. It is sown from our previous experimental study tat te imum material removal rate is dependent on a number of process parameters e.g. barrel diameter, macine rpm, etc. In order to develop a model, te mecanism of material removal was analyzed. Te mrr data of various blanks being beveled using φ30 and φ80 barrels were obtained from experiments. mrr for any macine rpm and barrel diameter is given by ( LH ) ( ω 0) ( D ) mrr = 5 θ () were θ is te contact angle introduced. Second, te initial bevel dept due to corner rounding was modeled. Tis is given by ( LH ) θ ( D 80)( ω 0) t ( δ ) b = () were θ = L D, t = and δ cc is te centerline-corner difference. Tird, te concept of ical gap eigt,, was introduced. Tis is a factor tat determines te sape of bevel. Tis is because tat te gap between a crystal blank and te beveling barrel s internal surface is nonuniform. Material removal increases wit decreasing gap eigt. It is sown from our previous experimental study tat decreases wit average diameter of beveling powder, decreases wit increasing macine rpm, and decreases wit increasing barrel diameter. A model for was developed based on te experimental data, wic is given by were te unit of d p is μm = d H (3) p Tis model as been validated using various blanks and good agreement wit experiments is found. Given te required bevel profile, te optimal process factors can be cosen wit te aid of tis process model. Tus, tis model provides an efficient and cost-effective way to te design of quartz crystal resonators. Keywords: Quartz crystal, bevel, model cc 398

2 Dong, Model development for te beveling of quartz crystal blanks. INTRODUCTION A quartz crystal unit is a quartz wafer (also referred to as blank or crystal plate) to wic electrodes ave been applied, and wic is ermetically sealed in a older structure. It is te main component of a crystal oscillator, wic is an electronic oscillator circuit tat uses te mecanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal wit a very precise frequency. Quartz crystal blanks can be cut from te source crystal in many different ways. AT-cut rectangular quartz crystal resonators are widely used for te ig-precision applications suc as mobile communication and GPS (global positioning system) equipment, because of teir ig temperature-frequency stability, and compatibility for mass production and miniaturization (Koyama et al., 996 and Sekimoto et al., 997). Te most common cut is called AT-cut, wic was developed in 934. In AT-cut, te plate contains te crystal s x axis and is inclined by 35 5 from te z (optic) axis. Te frequency-temperature curve is a sinesaped curve wit inflection point at around 5-35 C. Te frequency constant of AT-cut is.66 MHz mm. In order to acieve igly stable and reliable quartz crystal resonators witout spurious response, quartz crystal blanks are finised into sperically contoured sape. A beveled blank is sown in Figure. Te process of generating a sperical surface on a quartz plate is called beveling or contouring (Virgil, 98). It is called beveling wen one or bot faces of a resonator plate are altered to ave a partially sperical configuration, and contouring wen one or bot faces of a resonator plate are altered to ave a completely sperical configuration. It is easy to see tat tere is no flatness in te center after contouring, and tere is flatness in te center after beveling. Beveling or contouring as a drastic effect on te Figure. A beveled quartz crystal blank decoupling of modes by restricting te vibrating area of te plate to its central region by energy trapping effect. Te frequency separation between te main and overtone modes is inversely proportional to te area of te vibrating region. Terefore, reducing te size of te vibrating area sould cause te troublesome inarmonic modes to move to iger frequencies. Additionally, beveling or contouring offers te advantages of ease of mounting and reduced coupling to unwanted modes at te edge of te blank by restricting te blank s vibrating area to its central region (Salt, 987). Two issues in te beveling of quartz crystal blanks are material removal rate and bevel profile. te material removal of lapping and polising processes as been extensively studied (Evans et al., 003). In lapping, abrasive powders are mixed wit a fluid carrier to form a slurry. Te basic model of polising was investigated by Preston (Preston, 97) in 97. He proposed te polising rate of material to be proportional to a load and relative velocity. A ical size ratio for te caracteristic particle size to film tickness was found by Williams and Hyncica (Williams and Hyncica, 99). In beveling, no fluid is used. Blanks move wit regard to beveling barrels and material is removed by abrasive powders. Te gaps between blanks and a barrel s internal surface are not uniform and play an important role in te sape of bevel. Tere is no reported study on te beveling process design for quartz crystal blanks. Crystal manufacturers currently employ a trial-and-error approac because of te insufficient knowledge on te influences of process parameters. Tis limits te development of new generation crystal resonators. Tus, it is desirable to develop a process model, wic can provide te process parameter settings based on te beveled blank design information. In our previous study (Dong, 00), te beveling process of quartz crystal blanks was studied experimentally and te significant factors affecting te material removal rate and bevel profile were identified. A process model for te beveling of quartz crystal blanks is presented in tis paper. For a given crystal blank design, te sape of bevel and beveling time can be predicted from process factors, e.g. powder type, macine rpm and barrel diameter. Tis model as been validated using various blanks and good agreement wit experiments is found. Given te required bevel profile, te optimal process factors can be cosen wit te aid of tis process model. Tus, tis model provides an efficient and cost-effective way to te design of quartz crystal resonators. 399

3 Dong, Model development for te beveling of quartz crystal blanks As sown in Figure, a centrifugal beveling macine consists of a drum and four cylinders. Eac cylinder can receive a number of beveling barrels. A beveling barrel consists of a number of inter-connected internal speres. Te diameter ranges from 60 to 00 mm. Prior to a beveling process, a number of crystal blanks are prepared and weiged. Te prepared blanks are mixed wit abrasive powder at a certain weigt ratio, and carged into beveling barrels. Te carged beveling barrels are loaded into te four cylinders of te beveling macine. During te beveling process, te main drum rotates at a speed between 0 and 50 rpm, and te direction of rotation reverses intermittently. Because of te drum s rotation, eac cylinder also self-rotates about its own axis. As a result of te drum s rotation, te crystal blanks are pressed against te beveling barrel s internal surface by te centrifugal forces. Te cylinders self-rotation causes te relative movement between te crystal blanks and te beveling barrel s internal surface. Because of te friction, material is removed from te crystal blanks and bevel profiles are formed. Figure. Left: centrifugal beveling macine; rigt: two-spere beveling barrel Te abrasive powders used in tis beveling process are commercially available GC (Green Carborundum) powders of grit sizes from GC800 to GC3000. A tin layer of abrasion powders forms on te beveling barrel internal surface during beveling. A tree-stage beveling was employed in our production. A fixed blank/powder ratio :. by weigt was used. Te powder was canged every twelve ours, wic was called a beveling cycle. During te first stage, te blanks are beveled in φ60 or φ80 barrels for rounding te corners of blanks. Te blanks are beveled in φ30 barrels for several cycles to acieve te target bevel widt in te second stage. During te tird stage, te blanks are beveled in φ80 barrels to deepen te bevel.. MODEL DEVELOPMENT.. Material Removal For te purpose of beveling time estimation, material removal rate needs to be predicted from process parameters. Because a fixed -our powder canging time was used, material removal could only be determined from measuring te bevel profile cange after eac -our cycle. In our experiments, bevel profiles were measured along te centerline of a crystal blank. Tus, material removal was evaluated in -D at te cross-section along te centerline. Te imum material removal occurs at te blank edge, and for cycle i, te imum material removal rate is given by b, i b, i mrr, i = () 400

4 Dong, Model development for te beveling of quartz crystal blanks were mrr,i is te imum material removal rate in cycle i, and b,i and b,i- are te bevel depts after cycles i and i-, respectively. It is sown from te experiments tat imum material removal rate decreases wit time. Tis is because as beveling progresses, te contact area between te blank and te barrel increases, and te pressure and crack dept decreases, wic results in a decreased material removal rate. In order to account for tis decrease of mrr wit beveling cycle, a decay coefficient wit time δ was introduced. It sould also be noted tat if te beveling continues, te gap eigt will approac zero trougout te blank surface area, i.e. te bevel profile matces tat of te internal surface of te beveling barrels. Tus, mrr becomes zero. Te material removal witin a period of time t is given by () t mrr t mr = δ () Te following regression model was fitted to our experimental data. () t = exp( t 5.9) δ (3) In order to develop a model for mrr, te mrr data of various blanks being beveled using φ30 and φ80 barrels were obtained from experiments. For example, te bevel profile development of a 6 MHz mm.768 mm blank is sown in Figure 3. Te diameter of te beveling barrel was 30 mm. mrr was determined to be µm/. Figure 4. mrr Figure 3. Beveling profile development of a 6 MHz for φ30 and φ80 barrels from experiments and model mm.768 mm blank It is sown from our previous experimental study tat te imum material removal rate is dependent on a number of process parameters e.g. barrel diameter, macine rpm, etc. Before a model can be fitted, te mecanism of material removal needs to be analyzed. Wit reference to general lapping processes, material removal rate is governed by Preston s equation, i.e. mrr = Cpv (4) were p is pressure and v is te velocity of movement. For a beveling process, p is due to te centrifugal force and v is te velocity of blanks moving wit respect to te barrel s internal surface. In te -D crosssection along te centerline, p is te combined effect of LH and ω, and v is te combined effect of ω and R. In addition, since blanks contact te barrel s internal surface at an angle, te concept of contact angle θ is introduced. mrr increases wit θ. Since te lengt of a blank is muc less tan te diameter of a beveling barrel, te contact angles are θ = L D for beveling stage and θ 3 = L D3 L D for beveling stage 3. Eqn. 5 is rewritten by taking into account te above mentioned factors as mrr 0 a a a3 ( LH ) ω Dθ = a (5) were a 0, a, a and a 3 are constants to be determined. Tese constants were determined using te experimental data. Te actual regression model is given by 40

5 Dong, Model development for te beveling of quartz crystal blanks ( LH ) ( ω 0) ( D ) mrr = 5 θ (6) Te mrr for φ30 and φ80 barrels from te experiments and model are sown in Figure 4. It is seen good agreement is found... Initial Bevel Dept Wen te material removal is studied in te -D cross-section along te centerline, te initial bevel dept due to corner rounding needs to be predicted. In te beginning of beveling, crystal blanks are beveled for ours to round te corners. After tis -our corner rounding, te centerline of a blank may or may not contact te barrel s internal surface, depending on te centerline-corner difference and material removal rate. Te bevel dept after stage is defined as initial bevel dept, and it is negative wen te centerline does not contact te barrel s internal surface, as sown in Figure 5. Te centerline-corner difference is given by δ cc = 000 = 000 R R ( L ) R ( L ) diag ( L ) R ( L ) ( W ) were 000 is for converting mm to μm. It is also assumed tat te material removal rate causing te initial LH bevel dept is proportional to ( ) ( ) ( ). ω 0 D 30 θ (7). Te total material removal starts at te corners and moves to te centerline. Since te removal of corners is muc faster tan material removal in te centerline, te total material removal is given by mr = b ( c c δ ) cc were c 0 and c are constants to be determined. Te regression model for b is given by ( LH ) θ ( D 80)( ω 0) t ( δ ) b = (8) were θ = L D and t =. Te initial bevel depts from te experiments and model are sown in Figure 6. It is seen good agreement is found. cc Negative initial bevel dept Barrel s internal surface Figure 5. Negative initial bevel dept Figure 6. Initial bevel dept from experiments and model.3. Critical Gap Heigt Critical gap eigt is introduced as a factor tat determines te sape of bevel. Tis is because tat te gap between a crystal blank and te beveling barrel s internal surface is non-uniform. Material removal increases wit decreasing gap eigt. It is sown from our previous experimental study tat decreases wit average diameter of beveling powder, decreases wit increasing macine rpm, and decreases wit increasing 40

6 Dong, Model development for te beveling of quartz crystal blanks barrel diameter. It is assumed tat at te beginning of a beveling process, te beveling barrel is in contact wit te blank edge. At te bevel widt, te initial gap eigt, 0 is given by ( D ) ( L w ) ( D ) ( 0. ) = b L 05 (9) Wit te progress of te beveling process, te blank-barrel gap eigt decreases. Wen te designed bevel widt is acieved, te gap eigt is te ical gap eigt,. can be found by comparing te bevel profile against te beveling barrel surface, i.e. = 0 (0) b Critical gap eigt is dependent on te crack dept distribution on te blank surface. Wen te frequency increases, i.e. te blank is tinner, te difference in te crack dept decreases, and tus te ical gap eigt increases; wen barrel diameter increases, te gap between te blank and te barrel s internal surface decreases, and tus te ical gap eigt decreases. Tus, te process parameters cosen for te ical gap eigt model development are: te tickness of te blank H, te diameter of bevel particles d p, and te imum gap eigt. Te imum gap eigt (μm) between te blank surface and te barrel s internal surface is given by ( D ) ( ) = 000 D L () Te aforementioned analysis postulates tat te regression model for is b b b3 = b d H () 0 p b 0, b, b and b 3 were determined using te experimental data. Te regression model for is given by = d H (3) p were te unit of d p is μm. Te initial bevel depts from te experiments and model are sown in Figure 7. It is seen good agreement is found. Table MHz 4.4 mm.68 mm blank Figure 7. Critical gap eigt from experiments and model Lengt (mm) 4.4 Widt (mm).68 Frequency (MHz) Bevel widt (mm) 0.6 Macine rpm 0 80 Barrel diameter (mm) MODEL VALIDATION Te developed model was validated against te experimental data. Te blank design information is given in Table. Firstly, b was calculated to be μm. Secondly, from te bevel widt and GC00 powder being used, it was calculated tat 0 = 5.9 µm and = 5.3 µm. Tus, te resulting bevel dept was calculated to be b = 8.6 µm. Te material removal rate was determined to be mrr = 0.46 µm/. Tus, te time needed for bevel widt development was calculated to be.4 ours. Te actual time was determined to be 403

7 Dong, Model development for te beveling of quartz crystal blanks (a) Figure 8. Predicted and measured bevel profiles of te MHz 4.4 mm.68 mm blank 4 ours. b was calculated to be 9.3 μm. Te predicted bevel profile curve and te actual curve after φ30 beveling are sown in Figure 8a. Finally, te blanks were beveled for anoter 4 ours using D80 barrels for deeper bevel. mrr 3 and b3 were calculated to be 0.30 µm/ and 4.6 µm, respectively. Te final bevel dept was calculated to be 5.9 μm. Te predicted final bevel profile curve and te actual curve from measurement are sown in Figure 8b. It is seen tat tere is a good agreement between te model and te measurement. 4. CONCLUSIONS A process model for te beveling of quartz crystal blanks is presented in tis paper. For a given crystal blank design, te sape of bevel and beveling time can be predicted from process factors, e.g. powder type, macine rpm and barrel diameter. Tis model as been validated using various blanks and good agreement wit experiments is found. Given te required bevel profile, te optimal process factors can be cosen wit te aid of tis process model. Tus, tis model provides an efficient and cost-effective way to te design of quartz crystal resonators. REFERENCES Becman, R. (958). Elastic and piezoelectric constants of alpa-quartz. Pysical Review 0(5): Dong, C. S. (00). Mecanism analysis and experimental study for te bevelling of quartz crystal blanks. Proceedings of te 6t Australasian Congress on Applied Mecanics, Pert, Australia, Engineers Australia. Evans, C. J., E. Paul, et al. (003). Material removal mecanisms in lapping and polising. Annals of te CIRP 5(): Koyama, M., Y. Watanabe, et al. (996). An experimental study of frequency jumps during te aging of quartz oscillators. IEEE Transactions on Ultrasonic, Ferroelectrics, and Frequency Control 43(5): Mason, W. P. (950). Piezoelectric Crystals and Teir Applications to Ultrasonics. New York, USA, Van Nostrand. Preston, F. W. (97). Te teory and design of plate glass polising macine. J. Soc. Glass Tec. (44): Salt, D. (987). Hy-Q Handbook of Quartz Crystal Devices. Berksire, England, Van Nostrand Reinold (UK) Co. Ltd. Sekimoto, H., S. Goka, et al. (997). Frequency-temperature beavior of spurious vibrations of rectangular AT-cut quartz plates. IEEE International Frequency Control Symposium: Virgil, E. B. (98). Introduction to Quartz Crystal Unit Design. New York, Van Nostrand Reinold Ltd. Williams, J. A. and A. M. Hyncica (99). Mecanisms of abrasive wear in lubricated contacts. Wear 5: (b) 404