272
10 Characterization of Atherosclerotic Lesions by Inversion of Eddy-. . .
10.9 Determining the Constitutive Parameters of Saline
With the results of Fig. 10.19, we can use VIC-3D ® to determine the constitutive
parameters, σ and , of saline. We first develop a model of the experiment, and
then run VIC-3D ® with various values of these parameters in a simple inversion
process. We model the current loop as a single-turn coil of inner radius 8.8 mm
and an outer radius 8.9 mm, with a height of 0.1 mm. The coil is placed 0.55 mm
above the saline, which is modeled as a half-space. We then applied NLSE to
determine the best values of the electromagnetic parameters to be σ = 1.016 S/m,
= 219.79 0 . Figure 10.20 shows the result of this inversion. The result suggests
that there are two independent mechanisms at work in the saline, one that produces
the two resonances that were described above, and one that apparently produces a
third resonance beyond 500 MHz. We now want to discuss these phenomena from a
circuit-theoretic viewpoint, mixed in with a little field analysis.
10.10 Comments and Discussion
Maxwell’s second equation, which is Ampere’s circuital law in differential vector
form, is
∇ × H = σ E + jωωE
= J c + J d ,
(10.7)
where σ is the conductivity of the medium, the dielectric constant, J the conduction current density, and J d the displacement current density. When we induce
a current into a workpiece for Eddy-current NDE, we can develop an analogous
equivalent circuit, as in Fig. 10.21, to help understand what is happening. This figure
shows a simple coupled circuit whose secondary has a load impedance, Z L , and
whose primary is the exciting coil. When the workpiece is weakly conducting, and
has a significant dielectric constant, then the total current, which is the sum of the
conduction and displacement currents, can be thought of as flowing in the equivalent
load impedance that consists of a shunt capacitor and conductance. These elements
represent, respectively, the displacement current and conduction current of (10.7).
The equations for the circuit of Fig. 10.21 are:
E 0 = I 0 (R 0 + jωL 0 ) − jωMI 1
0 = −jωMI 0 + I 1 (j ωL 1 + Z L ) ,
(10.8)
10 Characterization of Atherosclerotic Lesions by Inversion of Eddy-. . .
10.9 Determining the Constitutive Parameters of Saline
With the results of Fig. 10.19, we can use VIC-3D ® to determine the constitutive
parameters, σ and , of saline. We first develop a model of the experiment, and
then run VIC-3D ® with various values of these parameters in a simple inversion
process. We model the current loop as a single-turn coil of inner radius 8.8 mm
and an outer radius 8.9 mm, with a height of 0.1 mm. The coil is placed 0.55 mm
above the saline, which is modeled as a half-space. We then applied NLSE to
determine the best values of the electromagnetic parameters to be σ = 1.016 S/m,
= 219.79 0 . Figure 10.20 shows the result of this inversion. The result suggests
that there are two independent mechanisms at work in the saline, one that produces
the two resonances that were described above, and one that apparently produces a
third resonance beyond 500 MHz. We now want to discuss these phenomena from a
circuit-theoretic viewpoint, mixed in with a little field analysis.
10.10 Comments and Discussion
Maxwell’s second equation, which is Ampere’s circuital law in differential vector
form, is
∇ × H = σ E + jωωE
= J c + J d ,
(10.7)
where σ is the conductivity of the medium, the dielectric constant, J the conduction current density, and J d the displacement current density. When we induce
a current into a workpiece for Eddy-current NDE, we can develop an analogous
equivalent circuit, as in Fig. 10.21, to help understand what is happening. This figure
shows a simple coupled circuit whose secondary has a load impedance, Z L , and
whose primary is the exciting coil. When the workpiece is weakly conducting, and
has a significant dielectric constant, then the total current, which is the sum of the
conduction and displacement currents, can be thought of as flowing in the equivalent
load impedance that consists of a shunt capacitor and conductance. These elements
represent, respectively, the displacement current and conduction current of (10.7).
The equations for the circuit of Fig. 10.21 are:
E 0 = I 0 (R 0 + jωL 0 ) − jωMI 1
0 = −jωMI 0 + I 1 (j ωL 1 + Z L ) ,
(10.8)