The System Function
Learning Objectives
Combination of transfer functions and Thevenin equivalent
pulser source term into a single factor - system transfer
function
Relationship between the system transfer function and system
"efficiency" factor
Experimental determination of the system transfer function
by deconvolution
Examples of synthesizing an entire ultrasonic measurement system
The System Function
Vi (ω )
VR (ω )
Ft (ω )
FB (ω )
tA
acoustic/elastic
processes
VR
Vi
tG
s
tA
tA
tR
VR
VR (ω ) = tG (ω ) t R (ω ) Vi (ω ) t A (ω )
= s (ω ) t A (ω )
The System Function
s (ω ) = tG (ω ) t R (ω ) Vi (ω )
… system function
There is a relationship between s(ω) and the system “efficiency”
factor, β(ω), [ Schmerr, Fundamentals of Ultrasonic Nondestructive
Evaluation, 1998] :
pave (ω )
p
t A (ω ) =
VR (ω ) = β (ω ) t Ap (ω )
ρ cvt (ω )
piston velocity
vt (ω )
average incident pressure
pave (ω )
The System Function
VR (ω ) = β (ω ) t Ap (ω )
where
VR (ω ) = s(ω )t A (ω )
pave (ω )
FB (ω )
t (ω ) =
, t A (ω ) =
Ft (ω )
ρ cvt (ω )
p
A
For piston transducers,
high frequency
FB (ω ) = 2 pave (ω )SR
Ft (ω ) = ρcvt (ω )ST
SR , ST … areas of receiving and transmitting transducers
ST
s(ω ) =
β (ω )
2SR
The System Function
Two methods to find s(ω):
A. Measure all the components needed to obtain tR(ω), tG(ω),
Vi(ω) by the methods indicated previously. Then combine
these measurements to give
s(ω ) = t R (ω )tG (ω )Vi (ω )
K Z oe SvIB
Z rA;a SvIA
s ( ω ) = B ;e
V (ω )
B ;e
e
A;e
A; e
e i
( Zin R11 + R12 ) + ( Zin R21 + R22 ) Zo ( Zin T11 + T12 ) + ( Zin T21 + T22 ) Zi
The System Function
VR (ω )
B. Measure VR(ω), model tA(ω), compute s(ω ) =
t A (ω )
(
t A = 2exp[−α (ω ) D]⎧⎨ exp ik f D
⎩
)
[
]
4 π /2 2
⎫
− ∫ sin u exp ik f D2 + 4a 2 cos2 u du ⎬
π 0
⎭
D
2a
2a
VR (ω )
s(ω ) =
t A (ω )
Wiener filter
deconvolution
s (ω ) =
deconvolution
VR (ω ) t *A (ω )
{
t A (ω ) + ε max t A (ω )
2
2
2
}
The System Function
Comparison of system factor obtained with both methods.
method A : synthesized
method B: measured
8
s( f )
volts/MHz
x 10
-2
6
synthesized
4
2
0
0
measured
2
4
6
8
10
12
14
Frequency (MHz)
16
18
20
The System Function
Example setups where tA is known
(a)
(c)
(e)
(b)
(d)
(f)
The System Function
If all the electrical and electromechanical components of an
ultrasonic measurement system are measured to obtain s(ω),
then this s(ω) in conjunction with a knowledge of tA can
used to determine the output voltage of a measurement
system.
The following examples are shown for this setup:
vR ( t )
pulser
receiver
The System Function
0.6
0.4
voltage, volts
0.2
0
-0.2
-0.4
-0.6
2.2
2.4
2.6
2.8
3
3.2
3.4
time, μsec
Fig. 7.3 Directly measured output voltage signal of an ultrasonic pitch-catch
measurement system (solid line) and the voltage synthesized by measurement and
modeling of all the ultrasonic components(dashed-dotted line) for a pair of 5 MHz, 6.35
mm diameter planar transducers.
The System Function
0.4
0.3
voltage, volts
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
2
2.5
3
3.5
4
4.5
time, μsec
Fig. 7.4 Directly measured output voltage signal of an ultrasonic pitch-catch
measurement system (solid line) and the voltage synthesized by measurement and
modeling of all the ultrasonic components(dashed-dotted line) for a pair of 2.25 MHz,
12.7 mm diameter planar transducers.
The System Function
0.6
0.5
0.4
voltage, volts
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
time, μsec
Fig. 7.5 Directly measured output voltage signal of an ultrasonic pitch-catch
measurement system (solid line) and the voltage synthesized by measurement and
modeling of all the ultrasonic components(dashed-dotted line) for a pair of 10 MHz, 6.35
mm diameter planar transducers.