A 2-D VHDL-AMS Model for Disck-Shape Piezoelectric …bmas.designers-guide.org/2008/5-5_Presentation.pdf · A 2-D VHDL-AMS Model for Disck-Shape Piezoelectric Transducers. Jean-Marc

A 2-D VHDL-AMS Model for Disck-Shape Piezoelectric

Transducers

Jean-Marc GallièrePhilippe Papet

Laurent Latorre

BMAS 2008San Jose

Polytech'Montpellier

Motivation

1

Planar vibrating mode Thickness vibrating mode

Planar/Thickness modelMore realistic model


Outline

Introduction

Behavioral Models (Thickness)

New Unified Model

Conclusion and Future Works

2


Introduction

3

E P E P

Planar Thickness

z

y

x

U. Model ConclusionIntroduction B. Model


Introduction

4

F1

d

F2

i3

u1u2

Impedance Matrix

F2

F1

v3

u1

u2

i3

Z/tan(wd/u) Z/sin(wd/u) h/w

Z/sin(wd/u) Z/tan(wd/u) h/w

h/w h/w 1/wC0

=-j. .

C0=εA/d ; Z=ρuA ; h piezoelectric constant

v3



Introduction

5

v3

F1F2

i3

u1u2

Redwood's Model

F1F2

u1u2

i3

v3

-C0

C0h.C0 : 1

u1+u2



Introduction

6

v3

F1F2

i3

u1u2

Redwood's Model vs Measurement

1

10

100

1000

10000

1M

(Ω)

f (Hz)

Redwood's modelMeasurement

Log(

Z)

3M 5M


A new model is needed


Outline

7

Introduction


New Unified Model



Behavioral Models

8

Redwood's Model

i2

v1

-C0

C0

kt

VHDL-AMS Model


rf rb

i1vte

uti=ite/kt v2

pti=kt.vte

architecture bhv of redwood is

begin

i1 == C0 * v1'dot;

i2 == -C0 * v2'dot;

pti == kt * vte;

uti == ite/kt;

ceramic : entity work.acousticlayer

generic map (Z0=>Z0, td=>td)

port map (p1=>t11,m1=>km,p2=>t22,m2=>km);

rf : entity WORK.resistance

generic map (rnom=>0.08)

port map (plus=>t11 , moins=>t1 );

rb : entity WORK.resistance



end architecture bhv;


Behavioral Models

9

Redwood's Model

i2

v1

-C0

C0

kt

VHDL-AMS Model


rf rb

i1

uti=ite/kt v2

pti=kt.vte


begin

i1 == C0 * v1'dot;

i2 == -C0 * v2'dot;

pti == kt * vte;

uti == ite/kt;











vte


Behavioral Models

10

Redwood's Model

i2

v1

-C0

C0

kt

VHDL-AMS Model


pti=kt.vte

rf rb

i1vte

uti=ite/kt v2


begin

i1 == C0 * v1'dot;

i2 == -C0 * v2'dot;

pti == kt * vte;

uti == ite/kt;










end architecture bhv;Ideal transformer


Behavioral Models

11

Redwood's Model

i2

v1

-C0

C0

kt

VHDL-AMS Model


pti=kt.vte

rf rb

i1vte

uti=ite/kt v2

Z0td


begin

i1 == C0 * v1'dot;

i2 == -C0 * v2'dot;

pti == kt * vte;

uti == ite/kt;










end architecture bhv;Transmission line


Behavioral Models

12

Redwood's Model

i2

v1

-C0

C0

kt

VHDL-AMS Model


pti=kt.vte i1

vte

uti=ite/kt v2


begin

i1 == C0 * v1'dot;

i2 == -C0 * v2'dot;

pti == kt * vte;

uti == ite/kt;











architecture bhva of acousticlayer is

begin

ftt == fi'DELAYED(td) - ftz;

fii == ft'DELAYED(td) - fiz;

fiz == (uiz + utz'DELAYED(td))*Z0/2.0;

ftz == (utz + uiz'DELAYED(td))*Z0/2.0;

end architecture bhva;

fiz ftz

fii ftt

ZO

td

ZO

F. Branin"Transient Analysis of Lossless Transmission Lines," in proceeding of IEEE

ftfi

utzuiz


Behavioral Models

13

Redwood's Model

i2

v1

-C0

C0

kt

VHDL-AMS Model


pti=kt.vte i1

vte

uti=ite/kt v2


begin

i1 == C0 * v1'dot;

i2 == -C0 * v2'dot;

pti == kt * vte;

uti == ite/kt;











rf rb

Z0td

Acoustic impedance : ambient air


Behavioral Models

14

VHDL-AMS Model



begin

i1 == C0 * v1'dot;

i2 == -C0 * v2'dot;

pti == kt * vte;

uti == ite/kt;











1

10

100

1000

10000

= 16mm1mm thickΦ

(Ω)

f (MHz)

Behavioral Redwood model

Measurement

0.5 1 2

Z = v1/(i1+i2)

Log(Z)


Outline

15

Introduction


New Unified Model


Polytech'Montpellier U. Model Conclusion

New Unified ModelIntroduction

16

B. Model

jwC0i

)u(ujw

kh)u(u

jwh

v 4331

2133 ++++=

A. Iula et al., IEEE Trans. On Ultrasonics, Ferroelectrics and Frequency Control, 1998

v

i

with

h : piezoelectric constant

u : particle velocity

k = 2t/r

Anisotropic material : not behave in the same way in all directions



17

B. Model

jwC0i

)u(ujw

kh)u(u

jwh

v 4331

2133 ++++=

P3

1

2v

i

h33with



k = 2t/r

thickness

c33

u = (c/ρ)1/2

c : elastic stiffness



18

B. Model

jwC0i

)u(ujw

kh)u(u

jwh

v 4331

2133 ++++=

P3

1

2v

ic11

with



k = 2t/r

u = (c/ρ)1/2

c : elastic stiffness

h31

thickness planar



19

B. Model

jwC0i

)u(ujw

kh)u(u

jwh

v 4331

2133 ++++=

)uC0(ukh)uC0(uhjwC0vi 43312133 +−+−=

C0

i

jwC0vv i1= h33C0(u1+u2)

i2=kh31C0(u 3+u4)


20


)uC0(ukh)uC0(uhjwC0vi 43312133 +−+−=

fthickness = h33i/jw

⎥⎦

⎤⎢⎣

⎡+−+−= )uu(0Ckh

0jwC1)uu(0Ch

0jwC1v0Chf 4331213333thickness

⎥⎦

⎤⎢⎣

⎡+−+−= )uu(0Ch

0jwC1

)uu(0Ckh0jwC

1v0Chf 2133433131planar

fplanar = h31i/jw

New Unified Model


21


⎥⎦

⎤⎢⎣

⎡+−+−= )uu(0Ckh

0jwC1)uu(0Ch


jwC0v

C0

y

-C0

x

-C0

i

v

fplanar

f thickness

u3+u4

u1+u2

e1=h33C0( v + y + x)

e2=h31C0(v+y+x)

h33C0(u1+u2)

kh31C0(u3+u4)

New Unified Model


22


⎥⎦

⎤⎢⎣

⎡+−+−= )uu(0Ckh

0jwC1)uu(0Ch


jwC0v

C0

y

-C0

x

-C0

i

v

fplanar

f thickness

u3+u4

u1+u2

e1=h33C0( v + y + x)

e2=h31C0(v+y+x)

h33C0(u1+u2)

kh31C0(u3+u4)

New Unified Model


23


⎥⎦

⎤⎢⎣

⎡+−+−= )uu(0Ckh

0jwC1)uu(0Ch


jwC0v

C0

y

-C0

x

-C0

i

v

fplanar

f thickness

u3+u4

u1+u2

e1=h33C0( v + y + x)

e2=h31C0(v+y+x)

h33C0(u1+u2)

kh31C0(u3+u4)

New Unified Model


24


⎥⎦

⎤⎢⎣

⎡+−+−= )uu(0Ckh

0jwC1)uu(0Ch


jwC0v

C0

y

-C0

x

-C0

i

v

fplanar

f thickness

u3+u4

u1+u2

e1=h33C0( v + y + x)

e2=h31C0(v+y+x)

h33C0(u1+u2)

kh31C0(u3+u4)

New Unified Model


25


jwC0v

h33C0(u1+u2)

kh31C0(u3+u4)C0

-

-C0

F4F1 F3F2

i

v

u3+u4

New Unified Model

C0 e1=h33C0(v+y+x)

u1+u2

e2=h31C0(v+y+x)


26


New Unified Model

architecture bhvugp of ugp isbegin

i1 == C0 * v1'dot;

i2 == -C0 * v2'dot;

pti == ktT * (v1-v2-v3);

uti == ite/ktT;

i3 == -C0 * v3'dot;

ptiP == ktP * (v1-v2-v3);

utiP == iteP/ktP;

ite

C0

-

-C0

F4F1 F3F2

i2

v1

utiPC0 pti=ktT(v1-v2-v3)

uti

ptiP=ktP(v1-v2-v3)

v2

i1

iteP

v3

IdT1

IdT2

IdT1

IdT2

.

.

.

thickness planar


27


New Unified Model

ceramicT : entity work.acousticlayer generic map (Z0=>Z0, td=>tdT)

port map (p1=>t11,m1=>km,p2=>t22,m2=>km);rfT : entity WORK.resistance generic map (rnom=>0.08)

port map (plus=>t11 , moins=>t1 );rbT : entity WORK.resistance generic map (rnom=>0.08)


ceramicP : entity work.acousticlayer generic map (Z0=>Z0, td=>tdP)

port map (p1=>t11P,m1=>km,p2=>t22P,m2=>km);rfP : entity WORK.resistance generic map (rnom=>0.08)

port map (plus=>t11P , moins=>t2 );rbP : entity WORK.resistance generic map (rnom=>0.08)

port map (plus=>t22P , moins=>t2 );

end architecture bhvugp;

.

.

.thickness

planar

rfT

rfP

rbT

rbP


28


Φ= 16mm2mm thick

Φ= 16mm1mm thick

(Ω)

f (Hz)

Log(

Z)

f (Hz)

(Ω)

Log(

Z)

1M

New Unified ModelMeasurement

3M 5M 7M

1

10

100

1000

10000

1

10

100

1000

10000

.5M 2M1M

New Unified ModelMeasurement

New Unified Model


Outline

29

Introduction


New Unified Model



Conclusion


New Unified Model

Experimental Validation

Future Works

30


Web :http://www.polytech.univ-montp2.fr

Contact : [email protected]

BMAS 2008San Jose

Documents

A 2-D VHDL-AMS Model for Disck-Shape Piezoelectric …bmas.designers-guide.org/2008/5-5_Presentation.pdf · A 2-D VHDL-AMS Model for Disck-Shape Piezoelectric Transducers. Jean-Marc