Equivalent Electrical Simulation of High -Power Ultrasonic Piezoelectric Transducers by Using Finite Element Analysis

Transcription

1 Equvalent Electrcal Smulaton of Hgh -Power Ultrasonc Pezoelectrc Transducers by Usng Fnte Element Analyss Amr Abdullah 1, Abbas Pak, Alreza Shahd 3 1Department of Mechancal Engneerng, Amrkabr Unversty of Technology, Tehran, Iran P.O. Box , Tel: () , Fax: +98- () , E-mal: amrah@aut.ac.r Manufacturng Engneerng Group, School of Engneerng, Tarbat Modarres Unversty, Tehran, Iran. 3 MSc and Senor Researcher n Manufacturng Engneerng Abstract Fnte element method (FEM) has been employed extensvely for vbraton modal analyss of pezoelectrc transducers and devces. In recent years, there has been a growng nterest n the smulaton and analyss of pezoelectrc transducers by usng equvalent electrcal crcut models. Ths paper has been devoted to such approach for two desgned and fabrcated ultrasonc sandwch transducers havng longtudnal vbratons for hgh power applcatons. By usng analytcal analyss, the dmensons of components of two hgh power ultrasonc transducers, whch are wdely used n ultrasonc weldng and cleanng, were determned for the assumed resonance frequences of KHz and 3 KHz, respectvely. By usng a D fnte element model, and regardng two dfferent modelng technques for the transducers and by applcaton of a current source whch was connected drectly to the models, transent and harmonc analyss was performed by ANSYS for extracton of resonance and ant-resonance frequences as well as voltage, current and equvalent mpedance varatons. Fnally, smulaton and expermental results were compared. Keywords: Ultrasonc Transducer, Equvalent Electrcal rcut, Fnte Element Modelng, Modal Analyss, Harmonc Analyss, Transent Analyss, Statc Analyss 1. Introducton Hgh power ultrasound s nowadays used n a wde varety of applcatons rangng from medcal devces, ultrasonc cleanng, ultrasonc weldng and machnng to sono-chemstry [1]. Snce Prof. Langevn developed the frst sandwch ultrasonc transducer by embeddng pezoelectrc rngs between two metallc peces and employed the assembly for hgh ntensty vbraton, there have been great efforts n modelng and formulatng such transducers [-6]. One of the well known works s the Mason s analyss for the PZT transducers;.e. the Equvalent rcut Method (EM) [7, 8]. In ths method, the mpedance propertes of a pezoelectrc transducer are presented near an solated resonance frequency by a lumped-parameter 1

2 equvalent crcut, the smplest forms of whch are shown n fgure 1. In ths fgure, two of the most wdely used lumped-parameters pezoelectrc converter mpedance models have been shown Smulaton of a pezoelectrc transducer by these crcuts s useful only f the crcut parameters are constant and ndependent of frequency. In general, the parameters are approxmately ndependent of frequency only for narrow range of frequences near the resonance frequency and only f the mode n queston s suffcently solated from other modes. The elements gven n fgure 1 are: s ; statc capactance of pezoelectrc, L 1 and 1 ; equvalent motonal mass and complance elements of the transducer respectvely and R 1 ; relates to dsspatve power loss whch s attrbuted to the jont losses (from planar frcton losses between pezoelectrc and metal parts), and to the materal hysteretc-related losses (nternal mechancal dampng n all transducer parts). Equvalent electrcal crcut models are expressed n terms of easy measurable or quantfable parameters lke resstance, capactance, nductance, voltages and currents. Ths paper presents and compares the results of electrcal soluton of two desgned and fabrcated hgh power pezoelectrc-transducers by four methods; usng analytcal relatonshps, usng ANSYS capablty for solvng the proposed equvalent electrcal model by the authors, employng pezoelectrc transducer modelng capablty on the ANSYS tself and usng expermental results. Fg. 1- Typcal equvalent electrcal crcuts for pezoelectrc transducers In the present work, by usng the analytcal method, the dmensons of components of two hgh power ultrasonc transducers whch are wdely used n ultrasonc weldng and cleanng were determned by assumng resonance frequences of KHz and 3 KHz, respectvely. Then, the fnte element D models of these transducers were created n ANSYS and afterwards an electrcal power supply was connected drectly across the pezoelectrcs of the models. These analyses were used to determne the values of the equvalentcrcut elements lke capactance and nductance. Then by transent and harmonc analyss, voltage and current varatons aganst frequency were determned and also to obtan resonance and ant-resonance frequences were obtaned. Ths s can be a good tool for optmzaton of the power supply behavor and transducer qualty, knowng how to model and calculate the mechancal load, realzng optmal resonant frequency and to have a control on the out put power.

3 . FEM modelng of the ultrasonc transducer The weldng transducer was composed of sx PZT-4 pezoelectrc rngs, a steel cylnder-shaped back mass (St 34) and an Alumnum stepped front mass (Al 775-T6) wth 3KW power. The cleanng transducer was composed of two PZT-SA pezoelectrc rngs, a steel cylnder-shaped back mass (St 34) and an Alumnum front mass (Al 775-T6) wth 5 Watt power. The bolt materal was taken from steel. As the exact value of densty and sound speed of the materals must be utlzed n the desgn process, these two propertes were accurately measured for backng and matchng. Measurement of sound speed was made n the NDT Laboratory by usng ultrasonc equpment ASANWIN, E.58,. The tme of flght (TOF) of the pulse whch was transmtted and receved by a sngle probe of MHz, Φ4 was measured. By knowng the thckness of the specmens, the sound speed could be obtaned by a smple calculaton. The materal propertes of the components are as shown n table 1. Fgures and 3 show the desgned and manufactured transducers. Table 1-Materal propertes of transducer s components AL 775-T6 St 34 Steel bolt Nckel%99 Standard ode alculated alculated (1.9 class) electrodes Determned sound bar speed (m/s) Measured sound dlatatonal speed (m/s) Modulus of elastcty N Brass electrodes m Major Posson s rato Measured Densty Kg m For the transducers desgn dscussed n ths paper, PZT-4 and PZT-SA whch are produced by Morgan Matroc Inc. and TAMURA o., respectvely were chosen as pezoelectrc materals. The materal propertes of these pezoelectrcs are shown n table [1, 11]. Morgan Matroc Inc., a popular manufacturer of pezoelectrc ceramcs, lsts the materal propertes of PZT-4 as [1]: S Delectrc Relatve Permttvty Matrx at onstant Stran, [ ] S [ ] ε r 73 = ε (Polarzaton axs along Y-axs): Pezoelectrc Stress Matrx (Stress developed/electrc feld appled at constant stran), [e] (Polarzaton axs along Y-axs): [] e = 1.7 m omplance Matrx [s] under constant electrc feld, [s E ] (Polarzaton axs along Y-axs): r 3

4 [ s ] E 1 = 1 m N The same matrxes were produced for PZT-SA of TAMURA o. for 5W ultrasonc transducer. Fg.- Modeled 3KW ultrasonc weldng transducer (Nomnal frequency KHz) Fg. 3- Modeled 5W ultrasonc cleanng transducer (Nomnal frequency 3 KHz) No structural constrant was used for the modal analyss. Ths produces a smulaton of an unrestraned transducer assembly. Ths state s smlar to the state of physcal testng where the transducer s hanged from node pont by a thread wth no restrcton. Ths model gnores the presence of the electrcally-nsulatng mechancally-algnng polymer (PTFE) bushes, normally used nsde the ceramc dsk holes around clampng bolt shank, as they are free and not stressed durng the assembly. Furthermore, although transducer performance has been observed to drft slghtly durng operaton as the ceramc peces warm up due to losses, the temperature effects were gnored n ths study. 4

5 Table - Materal propertes of pezoelectrcs, PZT-4 from Morgan Inc. and PZT-SA from TAMURA o. Item Desgnaton Unt PZT-4 PZT-SA Densty ρ Kg 3 m Delectrc Relatve Permttvty Pezoelectrc Stress constant Elastc omplance constant Pezoelectrc Stran constant S ε r S ε r e m m m e e 15 E m 1 s 11 1 N E m 1 s 1 1 N E m 1 s 13 1 N E m 1 s 33 1 N E m 1 s 44 1 N E m 1 s 1 66 N m 1 d31 1 V m 1 d 1 33 V m 1 d15 1 V To observe the vbraton behavor through ts smulaton by modal analyss, the fnte element method provded by commercal ANSYS was employed for D ax-symmetrc modelng and analyss by usng PLANE3 elements used for pezoelectrc and PLANE13 elements used for other components. PLANE3 has a -D Structural-Thermal, Structural Thermoelectrc, Pezoresstve, Pezoelectrc, Thermal- Electrc, Thermal-Pezoelectrc feld capablty and has eght nodes wth up to four degrees of freedom per node (Fgure 4). PLANE13 has a D magnetc, thermal, electrcal, pezoelectrc and structural feld capablty. Ths element s defned by four nodes wth up to four degrees of freedom per node. For ths modelng the element sze was selected to be 1 mm. Boundary condtons VOLT D.O.F Boundary condtons VOLT D.O.F (a) (b) Fg. 4- D ax-symmetrc modelng wth PLANE3 elements used for pezoelectrc and PLANE13 elements used for other components a) 3KW transducer b) 5W transducer 5

6 3. FEM-crcut coupled smulaton A FEM-crcut coupled pezoelectrc smulaton method, and analyss based on electrc charge balance, has been descrbed n reference [1]. In ths analyss the postve faces of the pezoelectrc rngs are electrcally connected together and also the negatve faces are electrcally coupled together (pezoelectrc rngs are electrcally connected n parallel). The nodes on these faces are coupled together as equ-potental ponts (voltage D.O.F 1 - see fgure 4). Ths s a good assumpton as the pezoelectrc peces actually have a thn slver coatng to nsure excellent electrcal contact and conductvty. The modeled transducer was used n statc and dynamc transent smulatons. Also modal and harmonc analyss were performed to understand the transducer s electromechancal behavor and to nspect ts resonance frequency and ts voltage and current varatons under dfferent frequences [13]. 3.1 Statc analyss The statc analyss was mplemented to determne the total equvalent statc capactance of the total pezoelectrc rngs. The pezoelectrc rngs were modeled wth PLANE3 elements. The negatve faces (poles) of the pezoelectrc rngs were connected to zero voltage of common ground and a unt voltage (1V) was appled to other poles (postve faces) connecton. After statc analyss, the total stored electrc charge on postve poles was determned by ANSYS. Then, by usng equaton (1) the equvalent statc capactances; s for the two transducers was calculated as: Q s = (1) V Q V 8 ( ) = = nf s KW = 4 3 Q s W = 31 5 V 8 ( ) = = nf Where Q s the total electrc charge stored and V s the appled voltage (V). Alternatvely, n the statc state, by usng analytcal relatonshp under constant stress (e.g. free state), the T T statc capactance of the pezoelectrc could be calculate as ( ( ε ) 13, ( ) 13 = ε 3KW 5W T r33ε. A = l = = (.5. ) π 4.6 π 4.5 = 3.16nF ( ) = 1.835nF r = 33 3KW ε [1, 11]): r = 33 5KW Then for sx and two pezoelectrc rngs whch were used for two hgh power transducers, total : 9 ( total ) = 6 = = nF 3 KW 9 ( ) = = nf total = 67 5 W : apactance of one pezoelectrc (F) 1 Degree Of Freedom 6

7 T ε r 33 : Delectrc relatve permttvty at constant stress (e.g. free state) along polarzaton axs 1 ε : Vacuum permttvty ( F/m) A: Pezoelectrc surface area (m ) l: Pezoelectrc thckness (m) As t s seen, there s a dfference between the statc analyss and analytcal results. Ths dfference can result from the real condton of the statc analyss by ANSYS where the pezoelectrc rngs used n the transducer are stressed and bounded by two metal masses and an elastc clampng screw. Therefore, they are not T completely free smlar to analytcal condton [e.g. for PZT-4, ( ε ) 13 s [ ( ) 635 ε ]. r = 33 3KW r = 33 3KW ] and not completely clamped 3. Transent analyss of the drect model of pezoelectrc transducer n ANSYS and the electrcal equvalent crcuts Ths s a drect analyss by ANSYS for a crcut n whch the pezoelectrc transducer s connected parallel to a resstor (R) and excted suddenly by an ndependent electrc current source (I) as shown n fgure 5. IRU94 elements were used to model the electrcal components and PLANE3 elements were used to model the pezoelectrc. IRU94 s a crcut element for use n pezoelectrc-crcut analyses. The element has two or three nodes to defne the crcut components and one or two degrees of freedom to model the crcut response. KEYOPT (1) settngs and the correspondng real constants defne the crcut components. Real constant nput s dependent on the element crcut opton used. A summary of the element optons has been gven n table 3. Alternatvely, ANSYS analyzed an equvalent electrcal crcut n whch the pezoelectrc transducer was approxmated and replaced by the equvalent capactor ( s ) the value of whch had been determned from the statc analyss of the pezoelectrc by ANSYS (See fgure 6). In both cases the current source and parallel resstor were dentcal. Transent analyss was performed for determnaton of the varaton of the current passng through the resstor wth tme when the power supply s a D constant-current source. There s another possblty to calculate the current varaton analytcally by equaton (). t / R I = I (1 e ) () Fg 5-Drect model of the pezoelectrc transducer n ANSYS n the transent analyss 7

8 Fg 6- a) Drect model of the pezoelectrc transducer n ANSYS n the transent analyss and b) Equvalent electrcal crcut n the transent analyss by ANSYS Table 3 -IRU94 element optons [13] rcut omponent KEYOPT(1) Real onstants Resstor (R) R1 = Resstance (RES) Inductor (L) 1 R1 = Inductance (IND) R = Intal nductor current (ILO) apactor () Independent urrent Source (I) Independent Voltage Source (V) 3 4 R1 = apactance (AP) R = Intal apactor Voltage (VO) For KEYOPT() = : R1 = Ampltude (AMPL) R = Phase angle (PHAS) For KEYOPT() = : R1 = Ampltude (AMPL) R = Phase angle (PHAS) In ths analyss, real constant nput for the resstance (R) was.1/ s Ohm and analyss tme (t) was selected as R s second. urrent level for the current source (I ) was selected.1 A. The varaton of the current passng through the resstor n the real modeled pezoelectrc crcut and n the resstor of the equvalent electrc crcut (both obtaned by FEM transent analyss) and the varaton obtaned by analytcal soluton (Equaton ()) are shown n fgure 7. (a).8 1 x 1-3 ANSYS Pezoelectrc Model (FEM) Equvalent Electrc rcut (FEM) Analytcal Soluton (b).8 1 x 1-3 ANSYS Pezoelectrc Model (FEM) Equvalent Electrc rcut (FEM) Analytcal Soluton urrent (A).6.4 urrent (A) Tme (Sec) x 1-4 Tme (Sec) x 1-4 Fg. 7- The current passng through the resstor under three methods of soluton a) 3KW transducer b) 5W transducer 8

9 3.3 Modal analyss To obtan the crcut element values of electrcal equvalent crcut of the transducers modal analyss of the pezoelectrc transducers was performed by ANSYS. Ths analyss determnes the resonance frequences, f, and stored average electrc charge, Q, on the postve poles n each resonance frequency (the condton resembles electrc short crcut condton where the dynamc capactve load and the dynamc nductve load cancel each other). In ths attempt the pezoelectrc transducer was approxmated wth capactors and an nductor ( s,, and L ) as shown n fgure 8. To obtan the equvalent dynamc capactance,, and dynamc nductance, L, the followng equatons were used [1]. Q ω = (3) ω = πf (4) 1 = (5) L ω Where: Q = Average electrc charge stored on the postve poles of the pezoelectrc peces n th pezoelectrc transducer resonance frequency ω = Angular frequency n th pezoelectrc transducer resonance condton Fg 8-Equvalent electrcal crcut used n modal analyss near the th pezoelectrc-transducer resonance frequency In ths model R element, whch represents dsspatve power loss, was not consdered, as t was gnored n FEM modelng. Ths loss s composed of jont and frcton losses between pezoelectrc rngs and metal parts, and nternal mechancal dampng n transducer s components. Results from modal analyss have been summarzed n table 4. Table 4-Results obtaned from modal analyss Mode No. () Frequency (Hz) (nf) L (H) 3KW 5W 3KW 5W 3KW 5W

10 To represent the pezoelectrc transducer more accurately, t would be approprate to add branches of capactor-nductor to the reduced order model. For example, use nne capactor-nductor branches as shown n fgure 9. The nne -L ( = 1,... 9) branches correspond to the frst nne resonance modes of the pezoelectrc transducer. The equvalent statc capactance and resstance values can be adjusted to [13]: 9 = = s (6) = 1.9 R = ω Where ω = Angular frequency of the second resonance mode Fg. 9- Equvalent electrcal crcut used n harmonc analyss correspondng to the frst nne transducer resonance frequency 3.4 Harmonc Analyss The harmonc analyss n ANSYS was performed on two models. The frst was on the electrcal equvalent crcut element value of whch was obtaned n statc and modal analyss of the above models. The second was on the drect model created by ANSYS tself. The results of ths analyss are the voltage and current varatons aganst frequency (the voltage s taken across the transducer and the current s passng through the transducer). For the frst models t was assumed that the structure of the transducer s under no constrant. The harmonc analyss was performed over a frequency span nsde whch the longtudnal-mode resonance frequency was expected (.95 to 1.1 tmes the frequency obtaned by modal analyss). Ths frequency span was dvded nto 1 steps for determnng voltage and current varatons n drect model created by ANSYS transducer crcut and n the equvalent electrc crcut. The resonance frequences for 3KW and 5W transducers obtaned from harmonc analyss were 17.3 khz and 9.96 khz respectvely and antresonance frequences obtaned were khz and 3.78 khz respectvely. The voltage and current varatons have been shown n fgures 1 and 11. Fgure 1 shows the electrcal mpedance of the drect pezoelectrc transducer model created by ANSYS and also the electrcal mpedance of the equvalent electrc crcut whch was calculated by dvdng voltage over current (V/I). 1

11 (a) urrent Ampltude (A) x 1-3 Drect ANSYS Modelng (FEM) Equvalent Electrc rcut (FEM) (b) Ampltude x 1-3 Drect ANSYS Modelng (FEM) Equvalent Electrc rcut x Frequency (Hz) x 1 4 Ferequency (Hz) Fg. 1- urrent ampltude varatons n the drect electrcal modelng by ANSYS and n the equvalent electrc crcut modelng aganst frequency, both obtaned from ANSYS harmonc analyss a) 3KW transducer b) 5W transducer (a).8.7 Drect ANSYS Modelng (FEM) Equvalent Electrc rcut(fem) (b) Drect ANSYS Modelng (FEM) Equvalent Electrc rcut (FEM) Voltage Ampltude (V) Voltage Ampltude (V) Ferequency (Hz) x Frequency (Hz) x 1 4 Fg. 11- Voltage ampltude varatons n the drect electrcal modelng by ANSYS and n the equvalent electrc crcut modelng aganst frequency, both obtaned from ANSYS harmonc analyss a) 3KW transducer b) 5W transducer (a) Impedance Ampltude Drect ANSYS Modelng (FEM) Equvalent Electrc rcut (FEM) (b) Impedance Ampltude Drect ANSYS Modelng (FEM) Equvalent Electrc rcut (FEM) Frequency (HZ) x Ferequency (Hz) x 1 4 Fg. 1- Equvalent mpedance ampltude varatons n the drect electrcal modelng by ANSYS and n the equvalent electrc crcut modelng aganst frequency, both obtaned from ANSYS harmonc analyss a) 3KW transducer b) 5W transducer 4. Test of the fabrcated Transducers To measure the actual resonant frequency of the desgned and fabrcated ultrasonc transducer a Network Analyzer of ROHDE & SHWARZ was employed. The sweepng frequency of ths devce was wthn 9kHz-4GHz wth a resoluton of 1Hz. The sweepng frequency was set between 1 khz to 3 khz and the system was calbrated. Then, phase-versus-frequency dagrams were drawn (See fgure 13). Both the 11

12 seres and parallel frequences are observed n the dagrams. The measurement was made wth the transducers under loaded and unloaded condtons. It should be noted, however, that smulaton of the loadng condton s greatly dependent upon the transducers applcaton. The tests showed that, under free state, the resonance and ant-resonance frequences of 3KW transducer were fs=17hz and f p =1897Hz respectvely. The measured resonance frequency of 5W transducer was 9 Hz. (a) (b) Fg 13- The fabrcated transducers and the dagram of phase versus frequency generated by Network Analyzer n unloaded condton for a) 3KW transducer b) 5W transducer 5. onclusons The fnte element approach presented n ths paper was mplemented by the general-purpose fnte element package ANSYS release 1. The study presented concludes: 1. FEM modelng by ANSYS proved that ths software s a good tool for equvalent electrcal crcut smulaton of hgh power ultrasonc pezoelectrc-transducers and also t shows a good conformty wth the drect pezoelectrc mpedance modelng.. As shown n fgures 7, 1, 11 and 1 there s a good conformty between the results obtaned from the drect pezoelectrc transducer electrcal crcut modelng by ANSYS and the equvalent electrc crcut smulatons created by crcut elements values determnaton. These also show a very close resonance and ant-resonance frequences wth the network analyzer test results. 1

13 3. Fgure 7 shows that, under sudden connecton of a D current source to the models, the current passng through the parallel resstor under analytcal and the two FEM modelng methods have a very close conformty. 4. The FEM model s a computer approxmaton of an actual structure. The error of ths approxmaton depends on the consderaton of all the system components, ther accurate propertes, approprate selecton and connecton of the elements and refnement of the model. 5. The nfluence of an external acoustc load on the transducer can be modeled by addng a resstance, R, to the equvalent electrc crcut. However, t s very dffcult to model the effect of an external load on ultrasonc transducer. Nomenclature A Pezoelectrc surface area m Adjusted dynamc capactance F Dynamc capactance F s Equvalent statc capactance F e Pezoelectrc Stress Matrx /m f Resonance frequency Hz f p Ant-resonance frequency Hz I urrent A I Maxmum current A l Pezoelectrc thckness m L Dynamc nductance H R Resstance Ω s E omplance Matrx under constant electrc feld m /N t Analyss tme Sec V Voltage V Q Average electrc charge of the pezoelectrc peces n th pezoelectrc transducer resonance frequency ω Angular frequency n th pezoelectrc transducer resonance condton Rad/Sec T ε r 33 Delectrc Relatve Permttvty Matrx at onstant Stress - s ε r 33 Delectrc Relatve Permttvty Matrx at onstant Stran - ε Vacuum permttvty F/m Acknowledgment Ths work was funded by Persan Keyan Technologes o. Thanks should also be gven to Electrcal Engneerng Faculty of Amrkabr Unversty of Technology. Authors express ther sncere apprecaton to Dr. Prokc and hs colleagues at MPInterconsultng for provdng nvaluable gudes and helpful comments. 13

14 References [1] R. Frederck (1965) Ultrasonc Engneerng, John Wely & Sons Inc., New York. [] P. Langevn, French Patent No (9.519); 5573 (5.8.19); ( ). [3] W.P. Mason (194) Electromechancal Transducers and Wave Flters, D. Van Nostrand, New York. [4] R. Krmholtz, D.A. Leedom, G.L. Mattae (197) New equvalent crcuts for elementary pezoelectrc transducer, Electron., [5] M. Redwood (1964) Experments wth the electrcal analog of a pezoelectrc transducer, J. Acoust. Soc. Am. 36 (1) [6] Kagawa, Y., Yamabuch, T., Mar (1979) Fnte Element Smulaton of a omposte Pezoelectrc Ultrasonc Transducer IEEE Transactons on Soncs and Ultrasoncs, Volume 6, Issue, Page(s): [7] W. P. Mason (1948) Electromechancal Transducers and Wave Flters, Prnceton,, Van Nostrand. [8] W. P. Mason (195) Pezoelectrc rystals and ther Applcatons to Ultrasonc, Prnceton, Van Nostrand-Renhold. [9] matweb.com [1] Morgan Matroc Inc. (6) Pezoelectrc Technology Data for Desgners, Morgan Matroc Inc., Electro eramcs Dvson. [11] TAMURA O. (7) Pezoelectrc eramcs, Pezoelectrc eramcs for Hgh Power Applcatons data sheet. [1] Jan S. Wang and Dale F. Ostergaard (1999) A Fnte Element-Electrc rcut oupled Smulaton Method for Pezoelectrc Transducer, IEEE Ultrasonc Symposum, [13] ANSYS Instructons Release 1 (July 5), ANSYS Inc. 14