Piezoelectric Transducer Thesis

8/9/2019 Piezoelectric Transducer Thesis

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Master of Science in ElectronicsJanuary 2011

Trond Ytterdal, IET

Tonni Franke Johansen, ISB

Submission date:

Supervisor:

Co-supervisor:

Norwegian University of Science and Technology

Department of Electronics and Telecommunications

Finite Element Modeling ofPiezoelectric Ultrasonic Transducers

Morten Wollert Nygren


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Problem DescriptionA series of piezoelectric ultrasonic transducer with increasing complexity is to be modeled in the

finite element analysis program COMSOL Multiphysics, and compared to a one-dimentional model

as long as such a comparison is reasonable. The final model will be a thick film piezoelectric layer

deposited on a spherical cap.

Assignment given: 26. August 2010

Supervisor: Trond Ytterdal, IET


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TC

e

e

x

w=

tPZT =

tmatch =

tload = tPML =

w=

tPZT =

tmatch =

tload = tback = tPML,l = tPML,b =


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r

z

x1 x3


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=

w =

tload =tPML =

=

=

=

=


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i

j

i = j

i =j


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Tij T ji i = j

T11=T1 T23 = T32 =T4T22=T2 T13 = T31 =T5T33=T3 T12 = T21 =T6

i

j

Sij =1

2

ixj

+j

xi

T1T2T3T4T5T6

=

c11 c12 c13 c14 c15 c16c21 c22 c23 c24 c25 c26c31 c32 c33 c34 c35 c36c41 c42 c43 c44 c45 c46c51 c52 c53 c54 c55 c56c61 c62 c63 c64 c65 c66

S1S2S3S4S5S6


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c

T1T2T3T4T5T6

=

+ 2 0 0 0 + 2 0 0 0 + 2 0 0 00 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2

S1S2S3S4S5S6

T3

x3 +

T4

x2 +

T5

x1 =

23

t2

t

3

S5

T5

T5x1

= 23t2

23x21

=23t2

3(x3, t) = 3,0ej(tkx3)


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3

+3

= 2f k = c

cs=

= 0

T3x3

=23t2

T3 S3

T3= ( + 2)S3 = ( + 2)3x3


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23x23

=

+ 2

23t2

3(x3, t) = 3,0ej(tkx3)

cl =

+ 2

=

M

M

= 0

cl =

=

B

B

3

u3

u3 =3

t =j3

p= ( + 2)3x3

3

p = jk( + 2)3 = jcl3 = clu

3


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Zl = p

u3= cl

((C2H2F2)n)

(P b[ZrxT i1x]O3, 0 < x < 1)


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TC

D1D2

D3

=0

E1E2

E3

P1P2

P3

, D= 0

D 0

E P


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T1T2T3T4T5T6

=

cE11 cE12 c

E13 c

E14 c

E15 c

E16

cE21 cE22 c

E23 c

E24 c

E25 c

E26

cE31 cE32 c

E33 c

E34 c

E35 c

E36

cE41 cE42 c

E43 c

E44 c

E45 c

E46

cE51 cE52 c

E53 c

E54 c

E55 c

E56

cE61 cE62 c

E63 c

E64 c

E65 c

E66

S1S2S3S4S5S6

e11 e21 e31e12 e22 e32e13 e23 e33e14 e24 e34e15 e25 e35e16 e26 e36

E1E2

E3

D1D2D3

=e11 e12 e13 e14 e15 e16

e21 e22 e23 e24 e25 e26e31 e32 e33 e34 e35 e36

S1S2S3S4S5S6

S11 S12 S13

S21 S22 S23S31

S32

S33

E1E2E3

E

e

S T

E D

D= eS+ SE

T = cES etE

D= dT+ TE

S= sET+ dtE

E= gT+ (T

)1

D

S= sDT+ gtD

E= hS+ (S)1D

T = cDS htD

t

s


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x3

T1T2T3T4T5T6

=

cE11 cE12 c

E13 0 0 0

cE12 cE11 c

E13 0 0 0

cE13 cE13 c

E33 0 0 0

0 0 0 cE44 0 00 0 0 0 cE44 00 0 0 0 0 cE66

S1S2S3S4S5S6

0 0 e310 0 e310 0 e330 e15 0

e15 0 00 0 0

E1E2

E3

cE66=

12

(cE11 cE12)

D1D2

D3

=

0 0 0 0 e15 00 0 0 e15 0 0

e31 e31 e33 0 0 0

S1S2S3S4S5S6

S11 0 00 S11 0

0 0 S33

E1E2

E3


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c = c jc = c(1 j tanm) =c(1

j

Qm)

c

c

tanm

Qm


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Ni

i

ua

u

u u=na=1

Naua

n

u

ui


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u

A(u)

B(u)

A(u)=

A1(u)

A2(u)

=0 B(u)=

B1(u)

B2(u)

=0

u

n

ua

Gb gb

Gb(u)d +

gb(u)d = 0, b= 1, 2, 3,...,n

e

e

me=1

eGb(u)d +

egb(u)d

= 0, e= 1, 2, 3,...,m


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e

e

m

u

A(u)

vTA(u)d = 0

v n v

u

NTb A(u)d +

NbB(u)d = 0, b= 0, 1, 2, 3,...,n


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h33

S33/0

22n

n

n

22

Y Htt

Htt YM

Y

Htt

YM


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p

1

0(p q)

2p

0c2 =Q

p(x, t) = p(x)eit

0 c q

Q


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R= 0

V

x

y

z

E=

u(x, t) =u(x)ejt (x, t) = (x)ejt

u


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R= 0


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cE e

S/0


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eikx

0 1 2 3 4 5 6 7 8 9 101

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

real part of x

imaginarypartofx

x

0 1 2 3 4 5 6 7 8 9 101

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

real part of x

imaginarypartofx

x

0 1 2 3 4 5 6 7 8 9 101

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

real part of x

am

plitudeofwave

eikx

0 1 2 3 4 5 6 7 8 9 101

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

real part of x

am

plitudeofwave

eikx

x

x


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w = tPZT =

tmatch = tload =

tPML=


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w tload

=

w =

tload =tPML=

=

=

=

=


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MAEv = 1

N

Ni=1

| cv,i xv,i |

MAEv v cv xv

N

i

cY real,N

MAEY real M AEY imag MAEHttabs MAEHttangle

2

2

2


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7.4 7.5 7.6 7.7 7.8 7.95000

5020

5040

5060

5080

5100

5120

Frequency [MHz]

[S/m2]


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w =

tPZT =

tmatch = tload = tback =

tPML,l = tPML,b =

0 2 4 6 8 10 1 2 14 16 1 8 200

1

2

3

4

5

6

7

8

9

10x 10

4

Frequency [MHz]

[S/m2]

0 2 4 6 8 10 1 2 14 16 1 8 204

3

2

1

0

1

2

3

4

5

6x 10

4

Frequency [MHz]

[S/m2]

0 2 4 6 8 10 1 2 14 16 1 8 200

0.05

0.1

0.15

0.2

0.25

Frequency [MHz]

[m/(s*

V)]

0 2 4 6 8 10 1 2 14 16 1 8 20300

250

200

150

100

50

0

50

100

150

200

Frequency [MHz]

[degre

es]


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0 2 4 6 8 10 1 2 14 16 1 8 200

0.5

1

1.5

2

2.5

3

3.5x 10

4

Frequency [MHz]

[S/m2]

0 2 4 6 8 10 1 2 14 16 1 8 201

0.5

0

0.5

1

1.5

2

2.5

3x 10

4

Frequency [MHz]

[S/m2]

0 2 4 6 8 10 1 2 14 16 1 8 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency [MHz]

[m/(s*V)]

0 2 4 6 8 10 1 2 14 16 1 8 20300

250

200

150

100

50

0

50

100

Frequency [MHz]

[degrees]


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cE


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0 2 4 6 8 10 12 14 16 18 2 00

2000

4000

6000

8000

10000

12000

14000

16000

18000

Frequency [MHz]

/m

]

0 2 4 6 8 10 12 14 16 18 2 00

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Frequency [MHz]

[S/m2]

0 2 4 6 8 10 12 14 16 18 2 00

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Frequency [MHz]

[m/s*V]

0 2 4 6 8 10 12 14 16 18 2 0300

250

200

150

100

50

0

50

100

Frequency [MHz]

[degrees]

0 2 4 6 8 10 12 14 16 18 2 00

0.5

1

1.5

2

2.5

3

3.5x 10

4

Frequency [MHz]

[S/m2]

0 2 4 6 8 10 12 14 16 18 2 01

0.5

0

0.5

1

1.5

2

2.5

3x 10

4

Frequency [MHz]

[S/m2]

0 2 4 6 8 10 12 14 16 18 2 00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Frequency [MHz]

[m/s*V]

0 2 4 6 8 10 12 14 16 18 2 0300

250

200

150

100

50

0

50

100

Frequency [MHz]

[degrees]


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cD

cE

cE =cD eth

cE

0 2 4 6 8 10 1 2 14 16 1 8 200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Frequency [MHz]

[m/s*V]

0 2 4 6 8 10 1 2 14 16 1 8 200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Frequency [MHz]

[m/s*V]


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0 2 4 6 8 10 1 2 14 16 1 8 200.02

0

0.02

0.04

0.06

0.08

0.1

0.12

Frequency [MHz]

[S]

0 2 4 6 8 10 1 2 14 16 1 8 200.04

0.02

0

0.02

0.04

0.06

0.08

0.1

Frequency [MHz]

[S]

0 2 4 6 8 10 1 2 14 16 1 8 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency [MHz]

[m/(s*V)]

0 2 4 6 8 10 1 2 14 16 1 8 20300

250

200

150

100

50

0

50

100

Frequency [MHz]

[degrees]


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r0 = 0 z0 = 1000m


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0 2 4 6 8 10 1 2 14 16 1 8 200.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Frequency [MHz]

[S]

0 2 4 6 8 10 1 2 14 16 1 8 200.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Frequency [MHz]

[S]

0 2 4 6 8 10 1 2 14 16 1 8 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Frequency [MHz]

[m/(s*V)]

0 2 4 6 8 10 1 2 14 16 1 8 20100

80

60

40

20

0

20

40

60

80

100

Frequency [MHz]

[degrees]


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0 2 4 6 8 10 1 2 14 16 1 8 202

0

2

4

6

8

10

12

14

16

18x 10

4

Frequency [MHz]

[S]

0 2 4 6 8 10 1 2 14 16 1 8 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency [MHz]

[S]

0 2 4 6 8 10 1 2 14 16 1 8 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Frequency [MHz]

[m/(s*V)]

0 2 4 6 8 10 1 2 14 16 1 8 20220

200

180

160

140

120

100

80

60

40

20

Frequency [MHz]

[degrees]


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0 2 4 6 8 10 1 2 14 16 1 8 202

0

2

4

6

8

10

12

14

16

18x 10

4

Frequency [MHz]

[S]

0 2 4 6 8 10 1 2 14 16 1 8 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency [MHz]

[S]

0 2 4 6 8 10 1 2 14 16 1 8 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Frequency [MHz]

[m/(s*V)]

0 2 4 6 8 10 1 2 14 16 1 8 20220

200

180

160

140

120

100

80

60

40

20

Frequency [MHz]

[degrees]


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= 1210 kg

m3

cE11 = 8.7e9P acE12 = 5.5e9P acE44 = 1.6e9P a

Sr = 10

vl = 2681.4329ms

Z= 3.244533MRayl

= 1050 kg

m3

cE11 = 6e9P acE12 = 5.3e9P acE44 = 2.735e9P a

Sr = 10

vl = 2390.4572

ms

Z= 2.509980MRayl


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= 7500 kg

m3

cE11 = 1.27205e11P acE12 = 8.02122e10P acE13 = 8.46702e10P acE33 = 1.17436e11P acE44 = 1.6e9P a

e31 = 6.62281

Cm2

e33

= 23.2403 Cm

2

e15 = 17.0345 Cm2

Sr11 = 1704.4Sr33 = 1433.6

vl = 4618.9008

ms

Z= 34.641756MRayl

h33 = 18.3176e8

Vm

= 5425 kg

m3

cE11 = 5.42787e11P a cE12 = 1.72871e11P a cE13 = 1.44000e11P acE33 = 4.51816e10P acE44 = 1.59683e10P a

e31 = 0.213342 Cm2e33 = 6.74731

Cm2

e15 = 2.05839 Cm2

Sr11 = 438 Sr33 = 370

vl = 3300

ms

Z= 17.9MRayl

h33 = 20.5959e8

Vm


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= 6660 kgm3

cE11 = 4.9636e10P acE12 = 2.1272e10P acE44 = 2.8363e10P a

Sr = 10

vl = 2730

ms

Z= 18.2MRayl

= 999.6 kg

m3

vl = 1481

ms

Z= 1.480871MRayl

= 1.204 kg

m3

vl = 343.2

ms

Z= 0.004133MRayl


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ejc(1j)z

c

c

c =

c (1 j)

j

c c(1 +j)

M = c2 c2(1 + 2j)

M

G= (1 +jS)G

S

S= 2

tan


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c

1 jln(10)

40

=

c(1 j)

= ln(10)

40

Q= 1

2

10

20f z104 =e

2fc z

= ln(10) c

4105


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x1 x2 x3 x2

x1

R

x3

r0, z0

R=

(r r0)2 + (z z0)2

x1 x3

x1 x3

0 R 0 0 1 1 06

3R2

R2

12

32

32

12

4

R2

R2

12 1

212

12

3

R2

3R2

32 1

212

32

2

0 R 1 0 0 1


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r

z

x1 x3

x1 x3

x1

r= zz0

R

z= rr0R

x3

r= rr0

R

z= zz0R

Piezoelectric Transducer Thesis

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