transducer models 10 - iowa state universitye_m.350/transducer models 10.pdf · two port and three...
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Learning Objectives
Two port and three port transducer modelsSittig modelMason modelKLM Model
Acoustic radiation impedance
Transducer sensitivity, impedance
The sound generation process
Transducer model - fields
Reciprocity :( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )2 1 1 2 2 1 1 2
S
V I V I p p dS− = − ⋅∫ v v n
Consider a P-waveimmersiontransducer:
pressure,velocity fields
p, v
VI
n
S
voltage
current
compressive force
velocity, v(x,ω)
( ) ( )( ) ( )
,
,S
v
F p dS
ω ω
ω ω
=
= ∫v x n
x
Transducer model- ‘lumped’ parameters
so reciprocity becomes:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 1 2 2 1 1 2V I V I F v F v− = −
Forpiston behavior
( ) ( ) ( ) ( ), ,p dS F vω ω ω ω⋅ =∫ x v x n
VI
Fv
pressure, p(x,ω)
[TA]V
I
F
v
Transducer model - transfer matrix
VI
⎧ ⎨ ⎩
⎫ ⎬ ⎭
=T11
A T12A
T21A T22
A⎡
⎣ ⎢
⎤
⎦ ⎥
Fv
⎧ ⎨ ⎩
⎫ ⎬ ⎭
, det T A[ ]= 1
2-Port Transducer Model
From reciprocity
force
VI
F
v velocityA
I
[TA]V F
v
Transducer model - transfer matrix
Sittig model: T A[ ]= TeA[ ]Ta
A[ ]
TeA[ ]=
1 / n n / iω Co
−iω Co 0⎡
⎣ ⎢ ⎤
⎦ ⎥
TaA[ ]= 1
Zba − iZ0
a tan kd / 2( )Zb
a + iZ0a cot kd( ) Z0
a( )2+ iZ0
aZba cot kd( )
1 Zba − 2iZ0
a tan kd / 2( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Transducer model - transfer matrix
0
0 33
33
33 0
33
0
33
0 0
/
/
/,
Dp
D
p
Sss
D
ap
ab
k v
v c
c
n h Ch
C S dS d
Z v S
Z
ω
ρ
ρ
β
β
ρ
=
=
=
=
=
wave number of piezoelectric plate
wave speed of the plate, defined in terms of:
plate elastic constant at constant flux density
plate density
constant, defined in terms of:
plate stiffness
clamped capacitance, defined in terms of:
plate area, thickness
plate dielectric impermeability at constant strain
plate acoustic impedance
backing acoustic impedance
Transducer - three port model
IV
F2 , v2F1 , v1
transducer crystalPlating(thickness neglected)
F1
F2
V
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪ = i
Z0a cot kl( ) Z0
a / sin kl( ) h33 / ωZ0
a / sin kl( ) Z0a cot kl( ) h33 / ω
h33 / ω h33 / ω 1/ ω C0
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
v1
v2
I
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
V I
F2
v2
F1
v1
3x3 impedance matrix
Transducer - Mason equivalent circuit
v1 v2
V
IF1F2
1: n
C0
- C0
iZ0 /sin(kl)a
- iZ0 tan(kl/2)a - iZ0 tan(kl/2)a
Transducer - KLM equivalent circuit model
1 : φ
- i X
C0V
I
F1
v1
F2
v2l/2 l/2
Z0Z0a a
φ =2M sin(kl/2)
1
X = Z0 M2 sin(kl)a
M = h33 / (ωZ0 )a
Sittig model with crystal facing layers
Backing (Zb )a
crystal facing layers(epoxy bonding, wear plate, etc.)
Acoustic layer: F1
v1
F2
v2
[Tl]
Commercial transducer:[TA] = [TA] [TA] [Tl] ...
F1
v1
⎧ ⎨ ⎩
⎫ ⎬ ⎭
=cos kala( ) −iZ0
a sin kala( )−isin kala( ) / Z0
a cos kala( )⎡
⎣ ⎢
⎤
⎦ ⎥
F2
v2
⎧ ⎨ ⎩
⎫ ⎬ ⎭
e a
[TA]V
I
Ft
v
Transducer – radiation into a fluid
At the acoustic port the force and velocity parametersare not independent. We can write ( ) ( ) ( );A a
t rF Z vω ω ω=;A a
rZ … acoustic radiation impedance (a "lumped"parameter that depends on the velocity and pressure distributionat the acoustic port, the port geometry, and the fluid properties)
;A arZ
A
Acoustic radiation impedance
Rayleigh-Sommerfeld integral model of radiation ofwaves into a fluid by a piston transducer
( ) ( ) ( ) ( )exp,
2 S
i v ikrp dS
r
r
ω ρ ωω
π−
=
= −
∫x y
x y
( )( )
( )( ) ( ) ( )
,exp
2a sr
S S
p dSikriZ dS dS
v r
ωωρω
ω π⎧ ⎫−
= = ⎨ ⎬⎩ ⎭
∫∫ ∫
xy x
( )v ω
xyρ… fluid density
c … fluid wave speedk = ω/c
pressure
Greenspan, 1979: showed that for a circular piston transducer of radius a the acoustic radiation impedance obtained from the Rayleigh-Sommerfeld model could be found explicitly in the form
.
J1 … Bessel function
S1 … Struve function
SA = πa2
Acoustic radiation impedance
( ) ( );1 1/ 1 /A a
r AZ cS J ka iS ka kaρ = − −⎡ ⎤⎣ ⎦
Acoustic radiation impedance
>> ka=linspace(0, 25, 100);
>> ka = ka + eps*( ka ==0);
>> Z = 1 -(besselj(1,ka)-i*struve(ka))./ka;
>> plot(ka, abs(Z))
>> xlabel(' ka ')
>> ylabel( ' V/\rhocS')
>> ylabel( ' Z/\rhocS')
0 5 10 15 20 250.5
0.6
0.7
0.8
0.9
1
1.1
1.2
;A ar
A
ZcSρ
ka
Greenspan model of a circular piston transducer
velocity
Fv
Acoustic radiation impedance
Most NDE transducers operate at high frequencies (ka >> 1). At such high frequencies if we canassume piston behavior, for any shaped transducer it can be shown that
;A ar AZ cSρ≅
density, wave speed, area
A
function y = struve(z)num = length(z);y=zeros(1,num);for k = 1:numy(k) = quadl(@struve_arg, 0, 1, [ ],[ ], z(k));end
function y = struve_arg(x, z)
y = (4./pi).*z.*x.^2.*sin(z.*(1-x.^2)).*sqrt(2-x.^2);
Acoustic radiation impedance
( ) ( )
( )
12 2
10
12 2 2
0
2 1 sin 1
4 sin 1 2
zH z t zt dt t x
z x z x x dx
π
π
= − = −
⎡ ⎤= − −⎣ ⎦
∫
∫
this uses
Sensitivity, Impedance
Vin
Iin
;A arZAT⎡ ⎤⎣ ⎦ tF
tv
;A at r tF Z v=
;; 11 12
;21 22
A a A AA e in rin A a A A
in r
V Z T TZI Z T T
+= =
+
inVinI
the electrical characteristics of the transducercan be completely described by its inputimpedance:
Iin
Vin ( );A einZ ω
A
Sensitivity, Impedance
inVinI
The particular sensitivity we will use is:
;21 22
1A tvI A a A A
in r
vSI Z T T
≡ =+
to describe the conversion of electrical signals into acoustic signals, we could use the transducer's sensitivity, SOI, where
tv
tF
OIOSI
=
O … an output (force or velocity)I … an input (voltage or current)
A
Sensitivity, Impedance
;A at r tF Z v=
All the other sensitivities can be found from this sensitivity if the transducer electrical impedance and acoustic radiation impedance are known:
;
;
; ;
/
/
A tvI
in
A A a AtFI r vI
in
A A A etvV vI in
in
A A a A A etFV r vI in
in
vSIFS Z SIvS S ZVFS Z S ZV
=
= =
= =
= =
inVinI
tF
tvA
;A einZ
inI
inV
At vI inv S I=
inVinI
tv
tFA
Sensitivity, Impedance
Thus, we can replace the transfer matrix model of the transducerby a model consisting of an electrical impedance and an ideal "converter" that is defined by the transducer sensitivity:
;A a At r vI inF Z S I=
Entire Sound Generation Process
( )tF ω
transmittingtransducer
cablepulser
( )iV ωoutputforce
( )iV ω ( )tF ω( )Gt ω
Thevenininput voltage
A
( )iV ω
( )eiZ ω
[ ]T ;A einZ
;
At vI in
A at r t
v S I
F Z v
=
=
pulser
cabletransducer
sound generationtransfer function
==
Entire Sound Generation Process
( )tF ω
transmittingtransducer
cablepulser
( )iV ωoutputforce
Thevenininput voltage
A
( )iV ω ( )tF ω( )Gt ω
( ) ( )( ) ( ) ( )
;
; ;11 12 21 22
A a At r vI
G A e A e ei in in i
F Z StV Z T T Z T T Z
ωω
ω= =
+ + +
sound generationtransfer function
References
Ristic, V.M., Principles of Acoustic Devices, John Wiley, 1983
Kino. G.S., Acoustic Waves - Devices, Imaging and Analog Signal Processing, Prentice-Hall, 1987.
Auld, B.A., Acoustic Fields and Waves in Solids, 2nd Ed., Vols. I and II, Krieger Publishing Co. , 1990.
Sacshe, W., and N.N. Hsu,” Ultrasonic transducers for materials testing and their characterization,” in Physical Acoustics, Vol. XIV,Eds. W.P. Mason and R.N. Thurston, 277-406, 1979.
Greeenspan, M., “Piston radiator: some extension of the theory,”J. Acoust. Soc. Am., 65, 608-621, 1979.