Femtet Seminar

Understanding

Piezoelectric Analysis&

Acoustic Analysis
202009All Rights Reserved, Copyright ⓒ Murata Software Co., Ltd.

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Table of Contents☆Piezoelectric Analysis
1. Case Studies
2. Functions
3. Points☆Acoustic Analysis
4. Case Studies
5. Functions
6. Points☆Piezoelectric-Acoustic Coupled Analysis
7. Case Studies
8. Points
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Table of Contents☆Piezoelectric Analysis
1. Case Studies
2. Functions
3. Points☆Acoustic Analysis
4. Case Studies
5. Functions
6. Points☆Piezoelectric-Acoustic Coupled Analysis
7. Case Studies
8. Points
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Piezoelectric Analysis

Case Studies
Compressive Force-

Tensile Force
Surface Acoustic Wave+

Electric
SignalElectric
SignalIDT

Compressive Force+

Piezoelectric Substrate
Tensile Force

Force⇒Electricity

Electricity⇒Force

Ultrasonic Sensor
Acceleration Sensor
Piezoelectric Gyro
Shock Sensor
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Electricity⇒Force⇒Electricity

CrystalPiezoelectric Buzzer
Ultrasonic Motor
Actuator

SAW Filter
Crystal Oscillator4Force to Electricity
Pressure Sensor

Acceleration Sensor

PiezoelectricMaterial

Acceleration

Metal
PiezoelectricMaterial

Voltage
Distribution

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Metal

Displacement
Distribution5

Electricity to Force
Buzzer
Piezoelectric Material

Metal

Ultrasonic Motorsin (ωt+4π/3)[V]sin (ωt+5π/3)[V]sin (ωt+π)[V]sin (ωt+2π/3)[V]sin (ωt+π/3) [V]sin(ωt)[V]PiezoelectricMaterial

Metal

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Electricity to Force to Electricity
Crystal Oscillator

Piezoelectric Transformer
Driving Electrode

Polarization
Polarization

DetectingElectrode

Impedance Characteristics

Voltage Distribution
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2.Functions of Piezoelectric Analysis(1) Analysis Condition(2) Boundary Condition(3) Material Property(4) Body Attribute(5) Results Display

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(1) Analysis Condition
Analysis Type・Static
・Harmonic
・Resonant・Transient
Options・Acceleration
・Thermal Load・Angular Velocity
・Initial Stress Taken into AccountAnalysis Plane
Used for 2D analysis
Variables to Constrain
Select [Voltage] to couple withacoustic analysis without taking
piezoelectricity into account
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Harmonic Analysis

Frequency setting
for calculation[Discrete Sweep] is recommended.
[Fast Sweep] is selectable but the convergence
will be deteriorated in the piezoelectric analysis.
If your PC has enough memory,[Parallel Discrete Sweep] can be used.*

If selected, the phase of the electrodecan be changed after calculation.
* Option for Accelerator is required.
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Resonant Analysis
6.862kHz

7.020kHz

8.322kHz

8.378kHz29.753kHz

30.146kHz[The Number of Modes]: Calculations done by this number[Approximated Frequency]: Calculates frequencies near to this value

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11

Transient Analysis
Input Voltage

Displacement

About 1/10 of the reciprocal of the resonant frequency

The modes obtained in the resonant analysis are used.
[Inverse Fourier transform] is applied.

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Angular Velocity
By specifying the angular velocity, Coriolis force can be taken into account.

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Analysis with Initial Stress Taken into AccountTensile Force

Tensile Force

Tensile reverting force istaken into accountAnalysis Model

Resonant Freq [KHz]

Tensile Force and Frequency

At first, static analysis is executed.
Its results are imported by [Result Import].
Harmonic and resonant analysis are executed.
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Pressure [GPa]14

(2) Boundary Condition
MechanicalElectric

Symmetry and Continuity

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Electric Boundary Condition
Magnetic Wall
Electric field is parallel to the wall

Example: Piezoelectric Transformer

Magnetic Wall
Electric Field

Electric Wall (Voltage Specified)
Magnetic Wall

Electric Wall
Electric field is perpendicular to the wall
Electric Wall
Electric Field
Electric Wall
(Floating Electrode)

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V
Voltage Meter

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Mechanical Boundary Condition
u=0 (Constrained)

u=1[mm] Forced Displacement

T=p[Pa]Pressure, Load

Acoustic Impedance (Resistance of Media such as Air and Water)

Vibration

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Air, Water, etc.

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Example: Acoustic Impedance
Acoustic impedance represents the resistance of media such as air or water.
By applying the acoustic impedance as a boundary condition to the face that is incontact with the media, the resistance of air or water can be taken into account.
In the harmonic analysis, acoustic impedance is expressed by Z=ρc where ρ isdensity, c is sound speed of medium.

Log Z[Examples]
Air: Z = 1.205[kg/m3]*340[m/s]=409.7[kg/(m2・s)]=409.7 [N・s/m3]
Water: Z = 997[kg/m3]*1500[m/s]=1.496e6[kg/(m2・s)]=1.4596e6[N・s/m3]

Freq [KHz]
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Symmetric/Periodic Boundary Condition
Symmetric Model

Full Model
Symmetric
Boundary

Periodic Boundary

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(3) Material Property
Piezoelectric Constants
Elasticity
Piezoelectricity
Permittivity
Loss (1/Qm, tanδ)Density

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Piezoelectric Constante-type (often used for single-crystal)d-type (often used for ceramic)

h-typeg-typewhere
T：stress,
S：strain,
D: electric flux density,
E：electric field・Set the value for each vibration mode
・The resonant amplitude is proportional to Qm
・Resonant resistance is inversely proportional to Qm

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21

Piezoelectric Materials in the Database

Piezoelectric Ceramic
made by
Murata Mfg. Co., Ltd.
(not for sale)Single-crystal
Click on “+”.
The list of materials shows up.

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22

(4) Body Attribute
Thickness/Width
The thickness of sheet body
is for the 3D analysis.

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Direction
The axis direction of polarization andcrystallization is specified.

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Crystallization Axis Setting by Vector
Default Setting
Piezoelectric Transformer

Polarization
Polarization

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Crystallization Axis Setting by Euler Angle

Euler angle (θ, φ, ψ) is an angle rotated around Z axis byangle θ, around X’ axis by angle φ, and around Z” axisby angle ψ.

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(5) Results Display - Field
・Displacement [m]・Strain・Mechanical Stress [Pa]Displacement

・Potential [V]・Electric Field [V/m]・Electric Flux Density [C/m2]

Electric Flux Density

Potential

The charge on the electrode is not uniform.
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Results Display
Numerical Summary of Harmonic Analysis“Admittance” Tab
・Charge of electrode with voltage specified [C]・Current of electrode with voltage specified [A]・Voltage of floating electrode [V]*[Add a resistor across to ground] is deselected.

“Charge [C]” Tab“Floating electrode potential [V]” Tab
The sum of charges on the floating
electrode is zero.
The current is zero as well.
“Current [A]” Tab

The data of the numerical table can beexported to csv file and used on Excel
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Results Display
Numerical Summary of Resonant Analysis“Damping Capacitance [F]” Tab

・Damping capacitance: Cd [pF]
・Free capacitance: Cf [pF]
・Resonant frequency: Fr[Hz]・Difference between resonant freq.
and anti-resonant freq.: DF[Hz]・Coupling coefficient: k [%]・Resonant resistance: Rn[ohm]・Equivalent capacitance: Cn[pF]
・Equivalent inductance: Ln[ H]“Resonant Frequency [Hz]” Tab“Coupling Coefficient [%]” Tab“Equivalent Capacitance [F]” Tab

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28

Equivalent Circuit
Equivalent circuit is obtainedin the resonant analysis.

C1
Cd

C2

C3

L1

L2

L3

R1

R2

R3

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Ck : equivalent series capacitance
Lk : equivalent series inductance
Rk : equivalent series resistance
Cd : damping capacitanceC f : free capacitance29

Equivalent Circuit & Resonancef r  1 / 2 L1C1

Equivalent circuit is obtainedin the resonant analysis.

f r  1 / 2 L2 C 2
1.0E+09

Cd

C2

C3

L1

L2

L3

R1

R2

R3

1.0E+08

[Ω]Ω
|Z|
抵抗

ｆ

C1

1.0E+07

1.0E+06

R1
R2
1.0E+050④ ⑥ ⑧② ③ ⑤ ⑦①200400
周波数 [KHz]
KHｚ
Frequency

600

800

The equivalent circuit has multiple resonators. The number of resonators corresponds to the number of modes
which is defined on the [Resonant Analysis] tab of the analysis condition setting. In the right figure aboveshows 8 resonant points, which means the calculation was performed with at least 8 modes.
The harmonic analysis results take into account all modes included in the range of analysis frequencies.
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30

Damping Capacitance & Free Capacitance
Equivalent circuit is obtainedin the resonant analysis.

C d : Damping CapacitanceC f : Free Capacitance

Cannot move freely

Freely moving
C f  Cd   Ckk

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31

Other Parameters
In the resonant analysis, resonant frequency, difference of the resonant
frequency and anti-resonant frequency, and coupling coefficient are output.

Resonant Frequency
The resistance is at its minimum.

1.0E+09

fr

Coupling Coefficient
Conversion efficiency between
electrical and mechanicalk  Ck / C f

[Ω] Ω
Z 抵抗

Anti-resonant Frequency
The resistance is at is maximum.
f a  f r 1  Ck / Cdfa

1.0E+08

1.0E+07

1.0E+06fr

1.0E+05
0

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50

100
Freq
[KHz]
周波数
KHｚ

150

200

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Complex Resonant Frequency & Q

Q is the ratio of the real part and the imaginary part.

f k  f k ' jf k ' '
Q

fk '
1

f k  ' f k  ' R k

Lk
f '
 k
Ck 2 f k ' 'Displacement1/√21

In the resonant analysis, complex resonant frequency is obtained byentering Q of material, tanδ, and acoustic impedance. Frequency
response of displacement is obtained in the harmonic analysis.

Q is large (large loss)

f k ' f k ' f k 'Amplitude

Frequency

Q is small (small loss)
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33

3.Points of Piezoelectric Analysis(1) Mesh Size
(2) Symmetric Model

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(1) Mesh SizeElongation of a Bar

Displacement
変位

Frequency of the vertical vibration

1st
2nd
3rd
4th

3 divisions of a wavelength:1%
6 divisions of a wavelength: 0.1%xFrequency Error of the 2nd Order Element

1.0E+00
1.0E-01

1st
2nd
3rd
4th

1.0E-02
1.0E-03

1.0E-01

1st
2nd
3rd
4th

1.0E-02
1.0E-03
Error
Numerical
誤差

Numerical
誤差ErrorDivision Numbers of a Wavelength and Error

1.0E+00

1.0E-04
1.0E-05
1.0E-06

1.0E-04
1.0E-05
1.0E-06

1.0E-07

1.0E-07

1.0E-08

1.0E-08

1.0E-09

1.0E-09
1

10
Numberof Division
分割数

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100

1

10
Number of 分割数/波長
Division / Wave Length

100

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(2) Symmetric Model
By analyzing a part of the domain, calculation is performedin shorter time with less memory usage.

Symmetric Model

Full Model
DomainMemory

Time

1/2

1/2

1/4

1/4

1/4

1/16 ( 6.3%)

1/8

1/8

1/64 ( 1.6%)

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(25.0%)

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(2) Symmetric Model
Set symmetric boundary condition on the faces of symmetry.
Symmetric Boundary

Calculation with a Full Model

Symmetric Boundary
Z displacement
matchesConvert the symmetric modelto the full modelCalculation with a symmetric model
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37

(2) Symmetric ModelIn the axisymmetric analysis of XZ plane, the analysis time is greatly reduced.

3D ModelAxisymmetric Model
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38

Table of Contents☆Piezoelectric Analysis
1. Case Studies
2. Functions
3. Points☆Acoustic Analysis
4. Case Studies
5. Functions
6. Points☆Piezoelectric-Acoustic Coupled Analysis
7. Case Studies
8. Points
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39

Acoustic Analysis

Case Studies
Reflection and Diffraction

Open Boundary

No Reflection

Reflection

Diffraction

ObstacleSound Source

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40

Speaker and Interference of Sound Waves
Interference of Sound Waves

Sound Pressure of Speaker

1[kHz]

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5. Functions of Acoustic Analysis(1) Analysis Condition(2) Boundary Condition(3) Material Property(4) Body Attribute(5) Results Display

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42

(1) Analysis Condition
Analysis Type
Harmonic
Transient

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43

Harmonic Analysis

Setting of Frequency
for Calculation

If selected, the phase of the wave source can be changed after calculation.

Select [Discrete Sweep] or[Fast Sweep].
If using high-spec computerwith enough memory,[Parallel Discrete Sweep] isavailable**Option for Accelerator is required.

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44

Transient Analysis
In the acoustic transient analysis, if timestep is set[Automatic], timestep is set based on the CFL
number so as to prevent numerical instability.

*CFL Number is the number of meshes that travel during
one timestep. If the number is too large, numericalinstability occurs. The number is determined by the meshsize and the sound speed.

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45

(2) Boundary Condition
Open Boundary
Rigid Wall

Speed/Pressure

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46

(3) Material Property
DensitySound Speed

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47

Damping of Sound
Damping
Material

By setting imaginary part of the sound Speed
Boundary
speed, the damping of the soundwaves can be expressed.
Air
Fig 1 shows the damping material
between air.

Acoustic
ImpedanceAir

Fig 1. Sound Pressure [Pa]As Fig 2 shows, the sound waves aredamped in the damping material.
Fig 2. Sound Pressure [Pa]Fig 3 shows the level on the left sideof the damping material is not
uniform. It indicates the reflection is
occurring.
Fig 3. Sound Pressure Level [dB]

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48

Damping of Sound
Loss of the Medium

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49

(4) Body Attribute
Analysis Domain
Example 8:
View by the Directivity Calculation Method

Calculated domain with thedirectivity calculation method

Calculation Domain
for the Finite
Element Method

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50

Calculation Method of Directivity
Huygens' Principle

Analysis Domain for
Finite Element Method

Radiations from the soundsources in the open boundary
are calculated.
An observation point of thedirectivity is placed outside theanalysis domain.
The outside sound waves arecalculated with the Huygens’
principle.

Open Boundary

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51

(5) Results Display - Field
Sound Pressure [Pa]Sound Pressure Level[dB]
Particle Velocity [m/sec]
Acoustic Intensity [W/m2]

Sound PressureSound Pressure Level

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Particle Velocity

52

Results Display - Numerical Table
Radiation Energy：Power[W]Radiation Impedance：Zr[Ns/m]
ExamplePower [W] = 2.60245624e-005 7.47853134e-005j
Zr [Ns/m] = 4.33208345e+000 1.24488632e+001j

Real Part of Power
Energy lost in radiation

Imaginary part of Power
Energy stored in space

Zr: ratio of power of sound source and velocity
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53

Results Display - Graph of Directivity[XY Plane]
Φ：min 0/max 360/step 100
θ：min 0/max 0/step 0
Horizontal axis： Φ[XZ Plane]
Φ：min 0/max 0/step 0
Θ：min -180/max 180/step 100
Horizontal axis: θ[YZ Plane]
Φ：min 90/max 90/step 0
Θ：min -180/max 180/step 100
Horizontal axis: θBased on the Huygens’ principle, the soundpressure outside the domain is calculated,
where the finite element method is notapplied.

Example of Directivity of [XZ Plane]
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54

Results Display
Graph of Frequency Response

Frequency Response of Sound Pressure

Outside of Analysis Domain

Inside of Analysis
DomainInside of Analysis Domain
Use Graph function of the contour diagram.
Outside of Analysis Domain
Refer to [Acoustic Analysis Example 8].
Or use macro in the Technical Note [Frequency
Response of Sound Pressure Level].
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55

6. Points of Acoustic Analysis(1) Analysis Domain(2) Mesh Size
(3) Multiple Sound Sources(4) Frequency-dependent Sound Speed

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56

(1) Analysis Domain
Create air body around the sound source
Air Body
The radius of the air body is R.
R is 0.2 times the wavelength of thefrequency of your interest.
If the value is small, a warning will show
up as below.
** Warning **Distance between the wave source and theopen boundary is too short. Separate themat least 0.2 wavelength.

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57

(2) Mesh Size
Mesh size is somewhere between
a fourth and sixth of the wavelength.

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Finer mesh must be applied to thecomplicated form near the sound source.

58

(2) Mesh Size
Ultrasonic Analysis
To prepare the mesh of a sixth of the wavelength, the number of meshes becomeslarge if the analysis domain is wide.
In such case, a quarter model or axisymmetric model are helpful.
[Example]
Axisymmetric Model (1MHz)
Ultrasonic waves show strong straightness

Wave Source

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59

(3) Multiple Sound Sources
Wave Source[V1]

Wave Source[V2]
Select [Enable each port’s individual weight
setting for the superposed field display]

Acoustic Analysis Example 3

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60

(3) Multiple Sound Sources
Scale factor and phasecan be shifted for eachboundary condition of
the results.

Phase Difference: 0

Phase Difference: 90

Phase Difference: 180

Sound
Pressure

Sound
Pressure
Level

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61

(4) Frequency-dependent Sound Speed
Project Tree > Right-click on Variables > Select [New Variables]

Prepare variables f and t.
Initial value can be any values.
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Analysis Condition Setting > Harmonic Analysis > Set f to Frequency.
Edit Material Property > Set t to Sound Speed.

62(4) Frequency-dependent Sound Speed
Run Mesher/Solver > Submenu > Parametric Analysis

By setting in the [Sweep Table],analysis can be performed whilechanging frequencies (f) and soundspeed (t).

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63

Table of Contents☆Piezoelectric Analysis
1. Case Studies
2. Functions
3. Points☆Acoustic Analysis
4. Case Studies
5. Functions
6. Points☆Piezoelectric-Acoustic Coupled Analysis
7. Case Studies
8. Points
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64

Piezoelectric-Acoustic Coupled Analysis

Case Studies
Ultrasonic Sensor
Piezoelectric Analysis
Resonating Element

Acoustic Analysis

Metal Plate

Base

Piezoelectric
Ceramic
Terminal

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65

8. Points of Piezoelectric-Acoustic
Coupled Analysis

Analysis Condition

Select [Piezoelectric Analysis]and [Acoustic Analysis]

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Select [Strong-coupling analysis].
Select [Voltage] if [piezoelectricity] is
not taken into account.

66

8. Points of Piezoelectric-Acoustic
Coupled Analysis

Body Attribute

Select [Piezoelectric Analysis] for solid.
Select [Acoustic Analysis] for media.
Both cannot be selected at the same time.

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67

Strong-coupling Analysis
Weak-coupling Analysis

Strong-coupling Analysis(Unidirectional from piezoelectric to acoustic)(Bidirectional between piezoelectric and acoustic)

The vibrating effect of the structure by the soundwaves cannot be calculated. The sound waves outside
the shielding plate cannot be calculated either.

The vibrating effect of the structure by the soundwaves can be calculated and the sound waves outside
the shielding plate can be calculated as well.
However the calculation time is long.

CasingCavityPiezoelectric Material
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68

Thank You

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69