68
3 Modeling Composite Structures
We are going to apply this coupled-circuit model to the analysis of carbon-fiber
refinforced polymers (cfrp), which do not exhibit ferromagnetic effects, so we will
drop L μ from here on. Furthermore, our interest is in the transverse effects of the
electromagnetic interaction with the plate, i.e., effects that are orthogonal to the
direction of the fibers, as in Fig. 3.1. Elementary coupled-circuit theory yields an
expression for the driving-point impedance of the loaded coil:
Z in = R 0 + jωL 0 +
ω 2 M 2
0
Z L + jωL 1
.
(3.3)
From this we get the change in impedance due to the presence of the composite
plate:
δZ in = Z in − R 0 − jωL 0 =
ω 2 M 2
0
Z L + jωL 1
=
ω 2 M 2
0 R L − jω 2 M 2
0 (ωL 1 + X L )
R 2
L + (ωL 1 + X L ) 2
,
(3.4)
where R L and X L are the real and imaginary parts of the load impedance, Z L .
If X L ≥ 0 in (3.4), say due to a resistor or inductor, then δX in < 0, which is
Lenz’ law for inductively coupled circuits. If, on the other hand, X L ≤ 0, say due to
a capacitor, then the sign of δX in depends upon the relative value of ωL 1 and X L .
In the case illustrated in Fig. 3.7, which in turn is suggested by the physical picture
of Fig. 3.1, we have a parallel RC circuit, for which
R L =
R
1 + ω 2 R 2 C 2
X L = −
ωCR 2
1 + ω 2 R 2 C 2 .
(3.5)
If ωL 1 >
ωCR 2
1 + ω 2 R 2 C 2 , then δX in is negative in (3.4), as in Lenz’ law, but if
ωL 1 <
ωCR 2
1 + ω 2 R 2 C 2 , then the change in reactance is positive. Finally, we have
the interesting result that if ωL 1 = X L =
ωCR 2
1 + ω 2 R 2 C 2 , then the reactance change
is zero, which is resonance.
A rigorous coupled-circuit model that includes the interactions between the
driver coil, host and anomaly is shown in Fig. 3.8. The analysis of this circuit leads to
the slightly more complicated expression for the driving-point impedance of (3.6),
compared to (3.4).
3 Modeling Composite Structures
We are going to apply this coupled-circuit model to the analysis of carbon-fiber
refinforced polymers (cfrp), which do not exhibit ferromagnetic effects, so we will
drop L μ from here on. Furthermore, our interest is in the transverse effects of the
electromagnetic interaction with the plate, i.e., effects that are orthogonal to the
direction of the fibers, as in Fig. 3.1. Elementary coupled-circuit theory yields an
expression for the driving-point impedance of the loaded coil:
Z in = R 0 + jωL 0 +
ω 2 M 2
0
Z L + jωL 1
.
(3.3)
From this we get the change in impedance due to the presence of the composite
plate:
δZ in = Z in − R 0 − jωL 0 =
ω 2 M 2
0
Z L + jωL 1
=
ω 2 M 2
0 R L − jω 2 M 2
0 (ωL 1 + X L )
R 2
L + (ωL 1 + X L ) 2
,
(3.4)
where R L and X L are the real and imaginary parts of the load impedance, Z L .
If X L ≥ 0 in (3.4), say due to a resistor or inductor, then δX in < 0, which is
Lenz’ law for inductively coupled circuits. If, on the other hand, X L ≤ 0, say due to
a capacitor, then the sign of δX in depends upon the relative value of ωL 1 and X L .
In the case illustrated in Fig. 3.7, which in turn is suggested by the physical picture
of Fig. 3.1, we have a parallel RC circuit, for which
R L =
R
1 + ω 2 R 2 C 2
X L = −
ωCR 2
1 + ω 2 R 2 C 2 .
(3.5)
If ωL 1 >
ωCR 2
1 + ω 2 R 2 C 2 , then δX in is negative in (3.4), as in Lenz’ law, but if
ωL 1 <
ωCR 2
1 + ω 2 R 2 C 2 , then the change in reactance is positive. Finally, we have
the interesting result that if ωL 1 = X L =
ωCR 2
1 + ω 2 R 2 C 2 , then the reactance change
is zero, which is resonance.
A rigorous coupled-circuit model that includes the interactions between the
driver coil, host and anomaly is shown in Fig. 3.8. The analysis of this circuit leads to
the slightly more complicated expression for the driving-point impedance of (3.6),
compared to (3.4).