Jan Kocbach Dr. Scient. Thesis Department of Physics University of Bergen November 2000 Finite Element modeling of Ultrasonic Piezoelectric Transducers Influence of geometry and material parameters on vibration, response functions and radiated field

Outline  Motivation  Piezoelectric materials  Piezoelectric transducers  FE modeling of transducers and surrounding fluid  Results: –Systematic analysis for piezoelectric disks –Systematic analysis for piezoelectric disks with a front layer  Conclusions - noe om objectives med arbeidet må sies tidlig! NB! Fokuser mere på hva som er nytt i dette arbeidet, istedenfor å vise masse forskjellige enkeltting. Se på konklusjoner, objectives osv. for å se hva det er som er nytt!! - Egenmoder: Nytt at ny mode klassifiserings system. Resp.fu: Nytt at systematisk vist. - Frontlag: Mye nytt og spennende - noe om objectives med arbeidet må sies tidlig! NB! Fokuser mere på hva som er nytt i dette arbeidet, istedenfor å vise masse forskjellige enkeltting. Se på konklusjoner, objectives osv. for å se hva det er som er nytt!! - Egenmoder: Nytt at ny mode klassifiserings system. Resp.fu: Nytt at systematisk vist. - Frontlag: Mye nytt og spennende

Motivation Accurate simulation tool few repetitions required Repeat until prototype has desired properties  Choose a basic design based on needs and experience  Vary design parameters in simulation tool until desired properties  Build a prototype (expensive!)  Increased understanding and control in a transducer design process A typical example of a combined numerical/experi mental trnasducer design process. Without appropriate simulation tools, all of the variation of design parameters must be made using measurements only. This is a time- consuming and expensive process. Candidate

–AC field alternate in size sound radiated Piezoelectric materials  Piezoelectric effect: –Mechanical stress applied voltage produced –Voltage applied mechanical distortion surface electrode piezoelectric bar New figure here?

Piezoelectric materials  Equations governing a piezoelectric material: Sjekk disse ligningene p[ tensor form!!!

Piezoelectric transducers  Found in a wide range of different applications –e.g. medical ultrasound, ultrasonic measurement systems, sonars, ultrasonic distance measurement, non- destructive testing, process instrumentation..... Need som more here. Fra sluttrapport! Need som more here  Many different types of transducer structures –e.g. disk transducers, bar transducers, ring transducers, transducer arrays, unimorph/bimorph transducers.....  Present work restricted to axisymmetric piezoelectric disk transducers

+ Transducer housing Piezoelectric transducers  Axisymmetric piezoelectric disk transducers Backing layer Frontlayer Piezoelectric disk + Fluid medium The central part of a piezoelectric disk transducer is the piezoelectric disk. There is a large impedance mismatch between typical piezoelectric materials applied in a piezoelectric disk transducer and air or water, and therefore a front layer is applied for better matching between the piezoelectric disk and the fluid medium. A backing layer is applied at the back of the disk to damp down vibrations. A transducer housing is also often applied, to protect the transducer from the operating environment.

+ Transducer housing Piezoelectric transducers  Axisymmetric piezoelectric disk transducers Piezoelectric disk Frontlayer Backing layer Present work: Focus on the basic parts of the transducer: disk + frontlayer + fluid medium + Fluid medium The central part of a piezoelectric disk transducer is the piezoelectric disk. There is a large impedance mismatch between typical piezoelectric materials applied in a piezoelectric disk transducer and air or water, and therefore a front layer is applied for better matching between the piezoelectric disk and the fluid medium. A backing layer is applied at the back of the disk to damp down vibrations. A transducer housing is also often applied, to protect the transducer from the operating environment.

Piezoelectric transducers  Transducer properties studied: –Transducer vibration –Transducer response functions - Impedance/Admittance - Source sensitivity response –Radiated sound field - Directivity pattern - Near and farfield pressure field Nevne andre ”transducer properties”, som for eksempel mottakerfølsomhet o.l.

Transducer modeling  Possible approaches:  Use commercial simulation tool  Develop new simulation tool  New simulation tool developed: FEMP  Why develop new? ¬ Research tool: •Implement/evaluate different methods and identify best approach ­ Systematic analysis •Commercial tools difficult to tailor for efficient systematic analyses  3D approach required for accurate modeling

FEMP: Modeling approach –Desired properties of simulation tool •Model complete axisymmetric piezoelectric transducer + fluid medium •Efficient calculation of vibration, response functions and sound field •Efficient calculation in a large frequency band –Modeling approach for piezoelectric medium: •Finite element method / Finite difference method / Boundary element method –Modeling approach for infinite fluid medium : •Fluid finite elements + infinite elements / Fluid boundary elements / Fluid finite elements + dampers / Fluid finite elements + analytical solution / etc. •conjugated Astley-Leis inf. el. / unconjugated Astley-Leis inf. el. / conjugated or unconjugated Burnett inf. el. / Olson-Bathe inf. el. / etc. –Harmonic analysis method : •Mode superposition method / Direct harmonic solution / FFT from time domain –Loss model for piezoelectric medium : •Complex material constants / Structural friction force / Rayleigh damping \posding No need to mesh fluid region & \negding Part of fluid region must be meshed \posding Small BE matrices & \negding Large FE matrices \negding BE matrices are full & \posding FE matrices are sparse \negding Problems at characteristic frequencies & \posding No problems at any frequencies \negding Evaluation of singular integrals& \posding All integrals may be evaluated easily \negding Computationally intensive to calculate field & \posding Field is calculated automatically \negding BE matrices must be set up for every frequeny & \posding FE matrices set up only once Astley-Leis inf. el: Applied in near field. Field easily calculated in far field. Say something about which properties are important for the method implemented here! \posding No need to mesh fluid region & \negding Part of fluid region must be meshed \posding Small BE matrices & \negding Large FE matrices \negding BE matrices are full & \posding FE matrices are sparse \negding Problems at characteristic frequencies & \posding No problems at any frequencies \negding Evaluation of singular integrals& \posding All integrals may be evaluated easily \negding Computationally intensive to calculate field & \posding Field is calculated automatically \negding BE matrices must be set up for every frequeny & \posding FE matrices set up only once Astley-Leis inf. el: Applied in near field. Field easily calculated in far field. Say something about which properties are important for the method implemented here! Si en del om de forskjellige metodene, ulemper/fordeler osv. her!  Different modeling approaches evaluated to find best approach

 FE formulation set up using Galerkin method  Set up weak formulation  variational formulation  Divide region of analysis into elements and nodes  Set up FE equations from variational formulation FEMP: Theory

 Set up weak formulation FEMP: Theory –apply boundary conditions –results in variational formulation (fluid) (piezoelectric) ( )

 Divide region of analysis into elements and nodes FEMP: Theory Fluid infinite elements Fluid finite elements Piezoelectric finite elements

 Divide region of analysis into elements and nodes FEMP: Theory –Unknowns approximated by nodal values in each element using interpolation functions û4û4 û3û3 û2û2 û1û1 Interpolation functions (quadratic + variable order) û7û7 û8û8 û5û5 û6û6

 Set up FE equations : set of matrix equations FEMP: Theory  FE matrices calculated for each local element  Assembled to global FE matrices  Solved for unknown quantities for each frequency Calculation of one matrix as example?? Size of system? Boundary conditions? Calculation of one matrix as example?? Size of system? Boundary conditions? When the region of analysis is divided into nodes and elements, a set of matrix equations, also called the FE equations, may be set up.

f [kHz] Input conductance [S] FEMP: Verification  Comparison with other FE codes •ABAQUS, ANSYS, CAPA: < 5 ppm difference (resonance freq.)  Comparison with measurements •good qualitative agreement (resonance freq., response funct.) Input conductance of PZT-5A disk with D/T=12 in water.

normalized distance S normalized pressure p FEMP: Verification  Comparison with other FE codes •ABAQUS, ANSYS, CAPA: < 5 ppm difference (resonance freq.)  Comparison with measurements •good qualitative agreement (resonance freq., response funct.)  Comparison with analytical solution •good quantitative agreement (plane piston pressure field) On-axis pressure field from plane-piston radiator

FEMP: Verification  Comparison with other FE codes •ABAQUS, ANSYS, CAPA: < 5 ppm difference (resonance freq.)  Comparison with measurements •good qualitative agreement (resonance freq., response funct.)  Comparison with analytical solution •good quantitative agreement (plane piston pressure field)  Comparison with literature results •good qualitative agreement (resonance freq., response funct.)

Piezoelectric disk Frontlayer D T T front Fluid medium Analysis results  Focus of investigations: –How does systematic variation of geometry and material parameters influence on transducer properties? •D/T ratio of disk •Thickness of front layer •Disk material •Frontlayer material Dette kan evt. gjøres annerledes ved at man viser hvordan design parametrene endres grafisk, kutter ut menypunkter på at man kun ser på disk+frontlag (sagt før). •Fluid medium (air/water)  Present analysis made for piezoelectric disks and piezoelectric disks with a front layer

Challenges related to analysis  All of the analysis: •Quantify the accuracy in the results  Piezoelectric disks: •refine previous mode classification scheme •relation vibrational modes  peaks in response functions •relation disk vibration  radiated sound field  Piezoelectric disks with a front layer: •analysis for thick disks •analysis for radial mode transducers •differences in optimal front layer thickness and materials •how to change geometry and material parameters to avoid unwanted peaks and dips in response functions Komme med konklusjonene av analysen her, og så trekke ut eksempler etterpå! NB! Ta med konkl. av FE program o.l. tidligere et sted, eventuelt skrive det inn her! Komme med konklusjonene av analysen her, og så trekke ut eksempler etterpå! NB! Ta med konkl. av FE program o.l. tidligere et sted, eventuelt skrive det inn her! Forklare bakgrunnen for valg av kvartbølgelag, og hva som forventes av et kvartbølgelag, hva de 1D modellene sier, og at det stort sett har blitt brukt 1D modeller i bestemmelse av optimal frontlags tykkelse og materialvalg.

Accuracy considerations  Literature: no clear answer for piezoelectric media –2 -10 elements per  needed for “sufficient accuracy”  What is “sufficient accuracy”? –Application dependent •5-10% error tolerated in some applications •Resonance freq. (e.g. for material constant evaluation):ppm level  Challenge: –How many elements to get a specified accuracy? •Answer through convergence tests •Accuracy found as function of elements per wavelength

Accuracy considerations  Example: Convergence test for resonance freq. (quadratic elements)  Corresponding convergence tests made for response functions and radiated field 5 elements per wavelength max 0.25% error 10 elements per wavelength max 100 ppm error

Analysis results: Piezoelectric disks  Mode classification scheme  Relation between eigenmodes and radiated field  Relation between eigenmodes and response functions Komme med konklusjonene av analysen her, og så trekke ut eksempler etterpå! NB! Ta med konkl. av FE program o.l. tidligere et sted, eventuelt skrive det inn her! Komme med konklusjonene av analysen her, og så trekke ut eksempler etterpå! NB! Ta med konkl. av FE program o.l. tidligere et sted, eventuelt skrive det inn her! Forklare bakgrunnen for valg av kvartbølgelag, og hva som forventes av et kvartbølgelag, hva de 1D modellene sier, og at det stort sett har blitt brukt 1D modeller i bestemmelse av optimal frontlags tykkelse og materialvalg.

Piezoelectric disks: mode classification The refined mode classification scheme developed in the present work is a basis for the remainder of the work.  Mode classification scheme developed based on: –resonance frequency spectra –vibration of disks for varying geometry and materials  Vibrational modes classified into –R modes –E modes –A modes –L modes  Thickness extensional modes and thickness shear modes special types of A and L modes

Diameter/Thickness ratio frequency-thickness product [kHz mm] Piezoelectric disks: mode classification The refined mode classification scheme developed in the present work is a basis for the remainder of the work. Example: PZT-5A disks –Resonance frequency spectrum of PZT-5A disks –Resonance frequencies of disks shown as a function of D/T ratio Thin disks: D/T=20 Thick disks: D/T=1

Piezoelectric disks: R modes Diameter/Thickness ratio frequency-thickness product [kHz mm] Mode classification difficult due to strong mode coupling to E, A and L modes R modes, associated with 1st order symmetric Lamb wave in infinite plate

Piezoelectric disks: R modes Diameter/Thickness ratio R1: D/T=5 R1: D/T=10 R1: D/T=20 frequency-thickness product [kHz mm] Fundamental radial mode: Disk expands in thickness direction when it contracts in radial direction.

Piezoelectric disks: R modes Diameter/Thickness ratio R2: D/T=10 R3: D/T=10 R4: D/T=10 frequency-thickness product [kHz mm] Higher order radial modes: number of nodal circles with zero radial displacement increases with order of radial modes.

Piezoelectric disks: R modes Diameter/Thickness ratio R1: D/T=5 (water) frequency-thickness product [kHz mm]

Diameter/Thickness ratio Piezoelectric disks: E mode E mode: D/T=5 E mode: D/T=10 frequency-thickness product [kHz mm] E mode Large axial displacement at circular edge of disk.

Piezoelectric disks: E mode Diameter/Thickness ratio frequency-thickness product [kHz mm] E-mode: D/T=5 (water)

Piezoelectric disks: A modes Diameter/Thickness ratio A1 (TS1) mode: D/T=8 A1 (TS1) mode: D/T=5 A2 mode: D/T=10 frequency-thickness product [kHz mm] A modes, associated with 2nd order symmetric Lamb mode in infinite plate PZT-5A: A1 = TS1 New numbering scheme for A modes suggested A1 A2 A3 A4 A5A6 A7

Piezoelectric disks: L modes Diameter/Thickness ratio L1(TE1) mode: D/T=14 L2 mode: D/T=10 L1(TE1) mode: D/T=10 frequency-thickness product [kHz mm] L modes, associated with 3rd order symmetric Lamb mode in infinite plate PZT-5A: L1 = TE1 ideal: no R mode coupling

Piezoelectric disks: L modes Diameter/Thickness ratio frequency-thickness product [kHz mm] L1 (TE1) mode D/T=5 (water)

Piezoelectric disks: response functions  Relation between peaks in response functions and eigenmodes: –Found by comparing resonance frequency spectrum and response functions –Example for electrical input conductance of PZT-5A disk with D/T=10

–Relation: Resonance frequency spectrum  response functions Conductance [mS] frequency-thickness product [kHz mm] Diameter/Thickness ratio Piezoelectric disks: response functions R1 R2 R3 R4 R5 E R6 R7 R8 A3 A4 L1 (TE1) R12 A1 A2 R9 L2

Analysis results: Piezoelectric disk with a frontlayer  Why to use a frontlayer - general theory  1D model results valid for thin disks  3D model results for thick disks  3D model results for radial mode transducers Komme med konklusjonene av analysen her, og så trekke ut eksempler etterpå! NB! Ta med konkl. av FE program o.l. tidligere et sted, eventuelt skrive det inn her! Komme med konklusjonene av analysen her, og så trekke ut eksempler etterpå! NB! Ta med konkl. av FE program o.l. tidligere et sted, eventuelt skrive det inn her! Forklare bakgrunnen for valg av kvartbølgelag, og hva som forventes av et kvartbølgelag, hva de 1D modellene sier, og at det stort sett har blitt brukt 1D modeller i bestemmelse av optimal frontlags tykkelse og materialvalg.

Fluid medium: Z  400 rayl -1.5 Mrayl Piezoelectric disk: Z  35 Mrayl D T Piezoelectric disks with a front layer: General theory  Piezoelectric disk in water/air/gas: –large acoustic impedance mismatch: •low bandwidth •low acoustic transmission coefficient Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance

Fluid medium: Z  400 rayl -1.5 Mrayl Piezoelectric disk: Z  35 Mrayl D T Piezoelectric disks with a front layer: General theory  Piezoelectric disk in water/air/gas: –large acoustic impedance mismatch: •low bandwidth •low acoustic transmission coefficient Frontlayer: Z = ??? T front Solution: matching layer for better transmission Important parameters for bandwidth and transmission Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance

Piezoelectric disks with a front layer: Example  Most previous work: 1D, thin disks/plates, TE1 mode –Quarterwave thick frontlayer for optimal transmission –Optimal acoustic impedance of front layer given by: –Z front = (Z piezo Z fluid ) 1/2 –Z front =2 1/3 (Z piezo ) 1/3 (Z fluid ) 2/3 –Z front = (Z piezo ) 1/3 ( Z fluid ) 2/3  Other transducer configurations and modes: –Other optimal values for frontlayer thickness and acoustic impedance Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance  Example: –PZT-5A (D/T=5) with epoxy frontlayer (Z front =4.17 Mrayl ) in water –Comparison 1D model and 3D model for TE1 mode region

normalized frontlayer thickness T front /T  /4,TE1 normalized frequency f/f TE1 conductance [dB re 1 mS] Piezoelectric disks with a front layer: Thick disk, TE1 mode region Without frontlayer Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance –Without frontlayer: One narrow peak –Two peaks in response functions with positions varying with T front. 1D model result

normalized frontlayer thickness T front /T  /4,TE1 normalized frequency f/f TE1 conductance [dB re 1 mS] Piezoelectric disks with a front layer: Thick disk, TE1 mode region T front = 1.0 T  /4,TE1 Z front = 4.17 Mrayl Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance –Equal height at quarterwave thickness –Low Z front  one wide peak at quarterwave thickness

normalized frontlayer thickness T front /T  /4,TE1 normalized frequency f/f TE1 conductance [dB re 1 mS] Piezoelectric disks with a front layer: Thick disk, TE1 mode region TE1 A2 A1 Without front layer Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance

normalized frontlayer thickness T front /T  /4,TE1 normalized frequency f/f TE1 conductance [dB re 1 mS] Piezoelectric disks with a front layer: Thick disk, TE1 mode region TE1 A2 A1 T front = 0.6 T  /4,TE1 Z front = 4.17 Mrayl Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance

normalized frontlayer thickness T front /T  /4,TE1 normalized frequency f/f TE1 conductance [dB re 1 mS] Piezoelectric disks with a front layer: Thick disk, TE1 mode region TE1 A2 A1 T front = 0.8 T  /4,TE1 Z front = 4.17 Mrayl Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance

normalized frontlayer thickness T front /T  /4,TE1 normalized frequency f/f TE1 conductance [dB re 1 mS] Piezoelectric disks with a front layer: Thick disk, TE1 mode region A2 T front = 1.0 T  /4,TE1 Z front = 4.17 Mrayl Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance problem

Piezoelectric disks with a front layer: Thick disk, TE1 mode region Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance  Corresponding systematic analysis made for the source sensitivity response (acoustic response): –Several different front layer materials –Avoid dips in otherwise flat response by minor change of D/T ratio –Highest bandwidth found for frontlayer which is 10-15% thinner than a quarterwave thick

Piezoelectric disks with a front layer: R1 mode region  No previous systematic analysis for R1 mode region  Questions: –Similar behaviour as for TE1 mode region? –Influence from flexural modes in the disk? –Influence of changing shear velocity of front layer? Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance  Example: R1 mode region (in-air analysis) –PZT-5A (D/T=3 / D/T=10) with epoxy frontlayer (Z front =2.64 Mrayl ) –Source sensitivity response (Rayleigh integral)

normalized frequency f/f R1 sensitivity [dB re 1 Pa/V] Piezoelectric disks with a front layer: R1 mode region D/T=3, without frontlayer normalized frontlayer thickness T front /T  /4,R1 R1 F Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance

T front = 1.0 T  /4,R1 Z front = 2.64 Mrayl, D/T=3 normalized frequency f/f R1 sensitivity [dB re 1 Pa/V] Piezoelectric disks with a front layer: R1 mode region normalized frontlayer thickness T front /T  /4,R1 R1 F Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance Similar to TE1 mode region: Splitting into two symmetric peaks with equal height.

T front = 1.0 T  /4,R1 Z front = 2.64 Mrayl, D/T=10 normalized frequency f/f R1 sensitivity [dB re 1 Pa/V] Piezoelectric disks with a front layer: R1 mode region normalized frontlayer thickness T front /T  /4,R1 R1 F D/T ratio of piezoelectric disk changed from 3 to 10 Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance

T front = 1.0 T  /4,R1 Z front = 2.64 Mrayl, D/T=10 normalized frequency f/f R1 sensitivity [dB re 1 Pa/V] Piezoelectric disks with a front layer: R1 mode region normalized frontlayer thickness T front /T  /4,R1 R1 F Shear velocity of front layer lowered 30% Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt?). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! quarterwave transformator? -> of intermediate acoustic impedance quarterwave transformator? -> of intermediate acoustic impedance

Conclusions  FE code developed : –based on evaluation of different modeling approaches –accuracy in FE results quantified  Contribute to increased control and understanding in a transducer design process –based on varying design parameters for simple piezoelectric disk transducers  Suggestions for further work: –Corresponding analysis for more complex transducers –Include fluid flow and receiving transducer in analysis