274
10 Characterization of Atherosclerotic Lesions by Inversion of Eddy-. . .
+
−
Z L
M
Z in
E
0
I 0
R 0
L 0
L 1
I 1
Z L
G
L
j C L
ω
Ζ L
G L
C L
R
a
L
a
C a
R b
L b
C b
To Account For Resonances
Fig. 10.21 Equivalent circuit for Eddy-current NDE. The resistor accounts for the conduction
current, and the capacitor for the displacement current of (). The second load circuit includes
elements that account for resonances (if they exist)
which can be easily solved to yield
Z in =
E 0
I 0
= R 0 + jωL 0 +
ω 2 M 2
jωL 1 + Z L
.
(10.9)
The first term in (10.9) is the freespace impedance of the coil, whereas the second
term is δZ W , the change in impedance due to the presence of the workpiece. This is
the term that we’re interested in:
δZ W =
ω 2 M 2
jωL 1 + Z L
=
ω 2 M 2
Z ∗
L − jωL 1
|Z L | 2 + ω 2 L 2
1 + 2(j ωL 1 Z ∗
L )
,
(10.10)
where * denotes the complex conjugate, and denotes the ’real part’ of a complex
number.
If, as in the upper-right of Fig. 10.21, Z L = 1/ (G L + jωC L ), then
δZ W =
ω 2 M 2
G L + jωC L
1 − Z 2
0 G 2
L − (ω/ω 0 ) 2
1 − (ω/ω 0 ) 2
2 − Z 2
0 G 2
L
+ (ω 2 /ω 2
0 ) 2
,
(10.11)