Piezoelectric Analysis Acoustic Analysis

All Rights Reserved, Copyright ⓒ Murata Software Co., Ltd. 1

Understanding

Piezoelectric Analysis&

Acoustic Analysis

Femtet Seminar

202009

All Rights Reserved, Copyright ⓒ Murata Software Co., Ltd.

☆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

2

Table of Contents



1. Case Studies

2. Functions

3. Points


4. Case Studies

5. Functions

6. Points


7. Case Studies

8. Points

Table of Contents


Force⇒Electricity

Ultrasonic Sensor

Acceleration Sensor

Piezoelectric Gyro

Shock Sensor

Compressive ForceTensile Force

+

-

-

+Compressive Force Tensile Force

Piezoelectric Buzzer

Ultrasonic Motor

Actuator

Electricity⇒Force

SAW Filter

Crystal Oscillator

Electricity⇒Force⇒Electricity

Crystal

Electric

SignalElectric

Signal

Surface Acoustic Wave

IDT

Piezoelectric Substrate

Case StudiesPiezoelectric Analysis


Acceleration Sensor Pressure Sensor

Metal

Piezoelectric

MaterialMetal

Piezoelectric

MaterialAcceleration

Voltage

Distribution Displacement

Distribution

5

Force to Electricity


Metal

Piezoelectric

Material

sin(ωt)[V]sin (ωt+π/3) [V]

sin (ωt+2π/3)[V]sin (ωt+π)[V]

sin (ωt+4π/3)[V]

sin (ωt+5π/3)[V]Piezoelectric Material

Metal

6

Electricity to Force

Ultrasonic MotorBuzzer

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Crystal Oscillator Piezoelectric Transformer

Driving Electrode

Detecting

Electrode

Polarization

Polarization

Voltage Distribution

7

Electricity to Force to Electricity

Impedance Characteristics


2.Functions of Piezoelectric Analysis

(1)Analysis Condition

(2)Boundary Condition

(3) Material Property

(4) Body Attribute

(5) Results Display


Options

・Acceleration

・Thermal Load

・Angular Velocity

・Initial Stress Taken into Account

Analysis Type

・Static

・Harmonic

・Resonant

・Transient

Analysis Plane

Used for 2D analysis

Variables to Constrain

Select [Voltage] to couple with

acoustic analysis without taking

piezoelectricity into account

9

(1) Analysis Condition


Frequency setting

for calculation

If selected, the phase of the electrode

can be changed after calculation.

10

* Option for Accelerator is required.

Harmonic Analysis

[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.*


[The Number of Modes]: Calculations done by this number

[Approximated Frequency]: Calculates frequencies near to this value

6.862kHz 7.020kHz

8.322kHz8.378kHz

29.753kHz 30.146kHz

11

Resonant Analysis

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Input Voltage

Displacement

About 1/10 of the reciprocal of the resonant frequency

12

The modes obtained in the resonant analysis are used.

[Inverse Fourier transform] is applied.

Transient Analysis


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

Angular Velocity


Pressure [GPa]

Res

on

an

t F

req

[K

Hz]

Tensile reverting force is

taken into account

Analysis Model

Tensile Force and Frequency

14

Tensile Force Tensile Force

At first, static analysis is executed.

Its results are imported by [Result Import].

Harmonic and resonant analysis are executed.

Analysis with Initial Stress Taken into Account


Electric

Symmetry and Continuity

15

(2) Boundary Condition

Mechanical


Magnetic Wall

Electric field is parallel to the wall

Electric Wall

Electric field is perpendicular to the wall

Example: Piezoelectric Transformer

Electric Wall (Voltage Specified)

Electric Wall

(Floating Electrode)

Magnetic Wall

16

V

Voltage Meter

Electric Field

Electric Wall

Electric Field

Magnetic Wall

Electric Boundary Condition


u=0 (Constrained) u=1[mm]

Pressure, Load

Vibration Air, Water, etc.

Forced Displacement

T=p[Pa]

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

17

Mechanical Boundary Condition


Freq [KHz]

Lo

g Z

Example: Acoustic ImpedanceAcoustic 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 in

contact 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 is

density, c is sound speed of medium.

[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]


Full Model Symmetric Model

Periodic Boundary

Symmetric

Boundary

19

Symmetric/Periodic Boundary Condition


Density

Piezoelectric Constants

Elasticity

Piezoelectricity

Permittivity

Loss (1/Qm, tanδ)

20



e-type (often used for single-crystal)

d-type (often used for ceramic)

・Set the value for each vibration mode

・The resonant amplitude is proportional to Qm

・Resonant resistance is inversely proportional to Qm

where

T：stress,

S：strain,

D: electric flux density,

E：electric field

h-type

g-type

21

Piezoelectric Constant


Piezoelectric Ceramic

made by

Murata Mfg. Co., Ltd.

(not for sale)

Single-crystal

Click on “+”.

The list of materials shows up.

22

Piezoelectric Materials in the Database


Thickness/Width Direction

The thickness of sheet body

is for the 3D analysis.

The axis direction of polarization and

crystallization is specified.

23

(4) Body Attribute


Polarization

Polarization

Piezoelectric Transformer

Default Setting

24

Crystallization Axis Setting by Vector


Euler angle (θ, φ, ψ) is an angle rotated around Z axis by

angle θ, around X’ axis by angle φ, and around Z” axis

by angle ψ.

Crystallization Axis Setting by Euler Angle


・Displacement [m] ・Potential [V]

・Strain ・Electric Field [V/m]

・Mechanical Stress [Pa] ・Electric Flux Density [C/m2]

Displacement Electric Flux Density Potential

The charge on the electrode is not uniform.

26

(5) Results Display - Field


The sum of charges on the floating

electrode is zero.

The current is zero as well.

27

“Admittance” Tab

“Charge [C]” Tab

“Floating electrode potential [V]” Tab

“Current [A]” Tab

Results Display

Numerical Summary of Harmonic Analysis

・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.

The data of the numerical table can be

exported to csv file and used on Excel


“Damping Capacitance [F]” Tab

“Resonant Frequency [Hz]” Tab

“Coupling Coefficient [%]” Tab

“Equivalent Capacitance [F]” Tab

Results Display

Numerical Summary of Resonant Analysis

・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]


C1

Cd

C2 C3

L1 L2 L3

R1 R2 R3

Ck : equivalent series capacitance

Lk : equivalent series inductance

Rk : equivalent series resistance

Cd : damping capacitance

C f : free capacitance

29

Equivalent CircuitEquivalent circuit is obtained

in the resonant analysis.


Equivalent circuit is obtained


C1

Cd

C2 C3

L1 L2 L3

R1 R2 R3

ｆ

1.0E+05

1.0E+06

1.0E+07

1.0E+08

1.0E+09

0 200 400 600 800

周波数　KHｚ

抵抗　Ω

R1R2

112/1 CLf r

222/1 CLf r

① ② ③④⑤⑥⑦⑧

|Z| [

Ω]

Frequency [KHz]

Equivalent Circuit & Resonance

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 above

shows 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.


k

kdf CCC

: Damping Capacitance : Free CapacitancefCdC

31

Damping Capacitance & Free Capacitance

Equivalent circuit is obtained


Cannot move freely Freely moving


1.0E+05

1.0E+06

1.0E+07

1.0E+08

1.0E+09

0 50 100 150 200

周波数　KHｚ

抵抗　Ω

Freq [KHz]

Z [

Ω]

Resonant Frequency

The resistance is at its minimum.rf

Anti-resonant Frequency

The resistance is at is maximum.

dkra CCff /1

Coupling Coefficient

Conversion efficiency between

electrical and mechanical

fk CCk / rf

af

32

Other Parameters

In the resonant analysis, resonant frequency, difference of the resonant

frequency and anti-resonant frequency, and coupling coefficient are output.


''2

'1

''

'

'''

k

k

k

k

kkk

k

kkk

f

f

C

L

Rff

fQ

jfff

Am

pli

tude

Q is large (large loss)

Q is small (small loss)

33

'kf

11/√

2

'kf 'kf

Frequency

Dis

pla

cem

ent

Complex Resonant Frequency & Q

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

In the resonant analysis, complex resonant frequency is obtained by

entering Q of material, tanδ, and acoustic impedance. Frequency

response of displacement is obtained in the harmonic analysis.


(1) Mesh Size

(2) Symmetric Model

3.Points of Piezoelectric Analysis


1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1 10 100

分割数

誤差

1st

2nd

3rd

4th

Num

eric

al E

rro

r

Number of Division

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1 10 100

分割数/波長

誤差

1st

2nd

3rd

4th

Num

eric

al E

rro

r

Number of Division / Wave Length

x

変位

1st

2nd

3rd

4thDis

pla

cem

ent

Elongation of a Bar

35

Frequency Error of the 2nd Order Element Division Numbers of a Wavelength and Error

3 divisions of a wavelength:1%

6 divisions of a wavelength: 0.1%

Frequency of the vertical vibration

(1) Mesh Size


Full ModelSymmetric Model

Domain Memory Time

1/2 1/2 1/4 (25.0%)

1/4 1/4 1/16 ( 6.3%)

1/8 1/8 1/64 ( 1.6%)

36

By analyzing a part of the domain, calculation is performed

in shorter time with less memory usage.

(2) Symmetric Model


Symmetric Boundary

Symmetric Boundary

Z displacement

matches

37

Set symmetric boundary condition on the faces of symmetry.

Calculation with a Full Model

Calculation with a symmetric model

(2) Symmetric Model

Convert the symmetric model

to the full model


3D Model

Axisymmetric Model

In the axisymmetric analysis of XZ plane, the analysis time is greatly reduced.

38

(2) Symmetric Model


Table of Contents


1. Case Studies

2. Functions

3. Points


4. Case Studies

5. Functions

6. Points


7. Case Studies

8. Points


Sound Source

Open Boundary

Obstacle

No Reflection DiffractionReflection

Reflection and Diffraction

40

Case StudiesAcoustic Analysis

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Sound Pressure of Speaker

41

1[kHz]

2[kHz]

Interference of Sound Waves

Speaker and Interference of Sound Waves

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5. Functions of Acoustic Analysis

(1)Analysis Condition

(2)Boundary Condition


(4) Body Attribute

(5) Results Display


Analysis Type

Harmonic

Transient

(1) Analysis Condition


Setting of Frequency

for Calculation

Select [Discrete Sweep] or

[Fast Sweep].

If using high-spec computer

with enough memory,

[Parallel Discrete Sweep] is

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

44

*Option for Accelerator is required.

Harmonic Analysis


Transient Analysis

*CFL Number is the number of meshes that travel during

one timestep. If the number is too large, numerical

instability occurs. The number is determined by the mesh

size and the sound speed.

In the acoustic transient analysis, if timestep is set

[Automatic], timestep is set based on the CFL

number so as to prevent numerical instability.


Speed/Pressure

Open Boundary

Rigid Wall

46

(2) Boundary Condition


Density

Sound Speed



By setting imaginary part of the sound

speed, the damping of the sound

waves can be expressed.

Fig 1 shows the damping material

between air.

As Fig 2 shows, the sound waves are

damped in the damping material.

Fig 3 shows the level on the left side

of the damping material is not

uniform. It indicates the reflection is

occurring.

Air Air

Damping

Material Acoustic

Impedance

Speed

Boundary

Fig 2. Sound Pressure [Pa]

Fig 3. Sound Pressure Level [dB]

Damping of Sound

Fig 1. Sound Pressure [Pa]


Loss of the Medium

49

Damping of Sound


Calculation Domain

for the Finite

Element Method

Calculated domain with the

directivity calculation method

Analysis Domain

Example 8:

View by the Directivity Calculation Method

(4) Body Attribute


An observation point of the

directivity is placed outside the

analysis domain.

The outside sound waves are

calculated with the Huygens’

principle.

Huygens' Principle

Radiations from the sound

sources in the open boundary

are calculated.

Open Boundary

Analysis Domain for

Finite Element Method

Calculation Method of Directivity


Sound Pressure [Pa]

Sound Pressure Level[dB]

Particle Velocity [m/sec]

Acoustic Intensity [W/m2]

Sound Pressure Sound Pressure Level Particle Velocity

52

(5) Results Display - Field


Example

Power [W] = 2.60245624e-005 7.47853134e-005j

Zr [Ns/m] = 4.33208345e+000 1.24488632e+001j

Radiation Energy：Power[W]

Radiation Impedance：Zr[Ns/m]

53

Zr: ratio of power of sound source and velocity

Real Part of Power

Energy lost in radiation

Imaginary part of Power

Energy stored in space

Results Display - Numerical Table


Based on the Huygens’ principle, the sound

pressure outside the domain is calculated,

where the finite element method is not

applied.

[XY Plane]

Φ：min 0/max 360/step 100

θ：min 0/max 0/step 0

Horizontal axis： Φ

[XZ Plane]


Θ：min -180/max 180/step 100

Horizontal axis: θ

[YZ Plane]


Θ：min -180/max 180/step 100

Horizontal axis: θ

Results Display - Graph of Directivity

Example of Directivity of [XZ Plane]


Inside 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].

55

Frequency Response of Sound Pressure

Results Display

Graph of Frequency Response

Inside of Analysis

Domain

Outside of Analysis Domain


6. Points of Acoustic Analysis

(1)Analysis Domain

(2)Mesh Size

(3)Multiple Sound Sources

(4)Frequency-dependent Sound Speed


The radius of the air body is R.

R is 0.2 times the wavelength of the

frequency of your interest.

If the value is small, a warning will show

up as below.

** Warning **

Distance between the wave source and the

open boundary is too short. Separate them

at least 0.2 wavelength.

Create air body around the sound source

Air Body

57

(1) Analysis Domain


Mesh size is somewhere between

a fourth and sixth of the wavelength.Finer mesh must be applied to the

complicated form near the sound source.

58

(2) Mesh Size


Ultrasonic Analysis

59

Wave Source

(2) Mesh Size

To prepare the mesh of a sixth of the wavelength, the number of meshes becomes

large 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


Acoustic Analysis Example 3

60

Wave Source

[V1]

Wave Source

[V2]

Select [Enable each port’s individual weight

setting for the superposed field display]

(3) Multiple Sound Sources


Sound

Pressure

Sound

Pressure

Level

Phase Difference: 0 Phase Difference: 90 Phase Difference: 180

Scale factor and phase

can be shifted for each

boundary condition of

the results.

61

(3) Multiple Sound Sources


Project Tree > Right-click on Variables > Select [New Variables]

Prepare variables f and t.

Initial value can be any values.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 while

changing frequencies (f) and sound

speed (t).

63

(4) Frequency-dependent Sound Speed



1. Case Studies

2. Functions

3. Points


4. Case Studies

5. Functions

6. Points


7. Case Studies

8. Points

Table of Contents


Terminal

Piezoelectric

Ceramic

Resonating ElementMetal Plate

Base

Ultrasonic Sensor

Piezoelectric Analysis Acoustic Analysis

65

Case StudiesPiezoelectric-Acoustic Coupled Analysis

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Select [Piezoelectric Analysis]

and [Acoustic Analysis]

66

Analysis Condition

Select [Strong-coupling analysis].

Select [Voltage] if [piezoelectricity] is

not taken into account.

8. Points of Piezoelectric-Acoustic

Coupled Analysis


Select [Piezoelectric Analysis] for solid.

Select [Acoustic Analysis] for media.

Both cannot be selected at the same time.

67

8. Points of Piezoelectric-Acoustic

Coupled Analysis

Body Attribute


Weak-coupling Analysis(Unidirectional from piezoelectric to acoustic)

Strong-coupling Analysis(Bidirectional between piezoelectric and acoustic)

68

Casing

Cavity

Piezoelectric Material

The vibrating effect of the structure by the sound

waves cannot be calculated. The sound waves outside

the shielding plate cannot be calculated either.

The vibrating effect of the structure by the sound

waves can be calculated and the sound waves outside

the shielding plate can be calculated as well.

However the calculation time is long.

Strong-coupling Analysis

アニメ/圧電音波連成.avi

アニメ/圧電音波連成.avi


Thank You

69

Piezoelectric Analysis Acoustic Analysis

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