3.7 Further Results for Permittivity
75
3.6.1 Return to an Analysis of Fig. 3.10
The blending functions of this figure exhibit a feature that supports the circuit model
of Fig. 3.7 and the analysis surrounding it, namely Eqs. (3.3)–(3.5).
The reciprocal of the conductivities shown in Fig. 3.10 corresponds to the
resistor, R, that shunts the capacitor, C, in Fig. 3.1. These two ‘circuit elements’ are
repeated in the load impedance, Z L , of Fig. 3.7. Hence, the smaller the conductance
value in Fig. 3.10, the larger is R, relative to C, which means that C will dominate
in Z L at a lower frequency. According to our analysis in (3.4) and (3.5) the drivingpoint reactance, δX in , may go positive earlier than for larger conductivities, which
we clearly see in Fig. 3.10. In fact, we see what appears to be a resonance, following
the discussion of (3.5) at 80 MHz.
There is a caveat, however, and that is that δR in in (3.4) is always positive, as
we would expect for a passive system, whereas the resistance in Fig. 3.10 is, for the
most of the frequency range, negative. This is due to the fact that R and X in the
figure correspond to the ‘FAWT Test Region’ in Fig. 3.9, and VIC-3D® treats this
region as an ‘anomaly’ relative to the Graphite-Epoxy Host. Hence, the ‘true’ δZ in
of (3.4) is the sum of the host response plus the responses shown in Fig. 3.10. This
response can be inferred in Fig. 3.13, where we plot the host response together with
the anomalous responses (the blending functions) of Fig. 3.10.
It is clear that when we add the host response to each of the blending functions,
we will get a positive R for all frequencies, but that X will not go positive in this
frequency range, though it is tending that way. When using this model in the lab, we
would subtract the host-only response from the total response to get the anomalous
response which contains the important information for determining the parameter
values in an inverse problem.
3.7 Further Results for Permittivity
To further test our ability to sense capacitive effects in the composite, we model
the problem shown in Fig. 3.9, except that we excite the coil from 0.1–10 GHz, and
we fix σ y = σ z = 25 S/m. We assume the dielectric constant of the composite to
be isotropic, since we have no justification for choosing an anisotropic model. We
choose principal values of the dielectric constant tensor to be [1, 3, 5], and compute
the response, which is shown in Fig. 3.14.
We clearly see a distinct difference in both R and X at the higher frequencies, as
we would expect for the small value of σ y and σ z . Had we chosen larger transverse
values, it is unlikely that we would see dielectric effects at these frequencies. The
oscillations in the responses over this large frequency range are due to the complex
frequency response of (3.6).
75
3.6.1 Return to an Analysis of Fig. 3.10
The blending functions of this figure exhibit a feature that supports the circuit model
of Fig. 3.7 and the analysis surrounding it, namely Eqs. (3.3)–(3.5).
The reciprocal of the conductivities shown in Fig. 3.10 corresponds to the
resistor, R, that shunts the capacitor, C, in Fig. 3.1. These two ‘circuit elements’ are
repeated in the load impedance, Z L , of Fig. 3.7. Hence, the smaller the conductance
value in Fig. 3.10, the larger is R, relative to C, which means that C will dominate
in Z L at a lower frequency. According to our analysis in (3.4) and (3.5) the drivingpoint reactance, δX in , may go positive earlier than for larger conductivities, which
we clearly see in Fig. 3.10. In fact, we see what appears to be a resonance, following
the discussion of (3.5) at 80 MHz.
There is a caveat, however, and that is that δR in in (3.4) is always positive, as
we would expect for a passive system, whereas the resistance in Fig. 3.10 is, for the
most of the frequency range, negative. This is due to the fact that R and X in the
figure correspond to the ‘FAWT Test Region’ in Fig. 3.9, and VIC-3D® treats this
region as an ‘anomaly’ relative to the Graphite-Epoxy Host. Hence, the ‘true’ δZ in
of (3.4) is the sum of the host response plus the responses shown in Fig. 3.10. This
response can be inferred in Fig. 3.13, where we plot the host response together with
the anomalous responses (the blending functions) of Fig. 3.10.
It is clear that when we add the host response to each of the blending functions,
we will get a positive R for all frequencies, but that X will not go positive in this
frequency range, though it is tending that way. When using this model in the lab, we
would subtract the host-only response from the total response to get the anomalous
response which contains the important information for determining the parameter
values in an inverse problem.
3.7 Further Results for Permittivity
To further test our ability to sense capacitive effects in the composite, we model
the problem shown in Fig. 3.9, except that we excite the coil from 0.1–10 GHz, and
we fix σ y = σ z = 25 S/m. We assume the dielectric constant of the composite to
be isotropic, since we have no justification for choosing an anisotropic model. We
choose principal values of the dielectric constant tensor to be [1, 3, 5], and compute
the response, which is shown in Fig. 3.14.
We clearly see a distinct difference in both R and X at the higher frequencies, as
we would expect for the small value of σ y and σ z . Had we chosen larger transverse
values, it is unlikely that we would see dielectric effects at these frequencies. The
oscillations in the responses over this large frequency range are due to the complex
frequency response of (3.6).