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In a rigorous and exhaustive approach to model multianalyte concentration variations, the authors consider the material under
test to be a homogeneous mixture of the analyte, other constituent chemicals, and air. By varying the proportion of this mixture,
we can vary the relative concentrations of the materials. The effect of variation in proportion of the mixture on its dielectric
property is nonlinear and requires considerable effort to model. The static (frequency-independent) relative dielectric permittivity
for an isotropic ensemble of N equivalent molecules each having a dipole moment, μ, and contained in a spherical volume, V, is given by the following equation:

in which M(0) is the instantaneous dipole moment of the macroscopic sphere at the arbitrary time t=0. The equilibrium value of the scalar product ‹M(0)M(0)› taken over all the complexions of the ensemble will be independent of time for a stationary thermodynamic system. Considering
a system of N constituents in the mixture of volume V

in which P_{
i
}^{
L
} (0) is the instantaneous liquid dipole moment of the ith constituent of the mixture. Detailed analysis of mixtures and their dielectric properties can be found in References 33–37.

Obtaining wideband spectroscopic measurements can be lengthy process because the time available for inline measurements is
limited. Alternatives for such measurements are single-frequency or selected-frequency excitations. In this case, the response
of a sample at a single or selected frequency can be used to monitor changes in physical properties. It should be noted that
the observation of dielectric spectra at just selected frequencies could result in measurement artifacts being misinterpreted
as dielectric phenomena; therefore, such an approach is possible if there is some prior knowledge of the sample response at
the chosen frequency. Off-line studies can establish such characteristics of the sample. To illustrate a single- and selected-frequency
approach, data presented in Figure 3 is analyzed further to establish a dependency between capacitance and coating thickness.
Ideally the mapping between measured capacitance, C, and the coating thickness, T would be of the form

The first term of this equation represents the exponential decrease in the effect of the core layers as the thickness increases.
The second term is a function that accounts for the escaping electronic fields. Because these are mostly owing to the zero-order
field lines and their dispersion depends on the material property of the coating, this loss also can be written as a function
of the thickness, χ(T); α and β are constants. To obtain thickness as a function of capacitance, the previous equation can be rearranged, expanded
as a series in T, and the higher order terms neglected. This gives a polynomial fit. Because there are only four data points and at least
one degree of freedom is needed to validate the fit, a quadratic expansion is chosen. Thus, the thickness of the coating can
be obtained from the measured capacitance using the equation

A. Mathur is a graduate student at the Sensors, Energy, and Automation Laboratory, Department of Electrical Engineering, University of Washington, Seattle.
Articles by A. Mathur

K. Sundara-Rajan is a PhD candidate at the Sensors, Energy, and Automation Laboratory, Department of Electrical Engineering, University of Washington, Seattle, tel. 206.351.8101.
Articles by K. Sundara-Rajan

G. Rowe

G. Rowe is a PhD candidate at the Sensors, Energy, and Automation Laboratory, Department of Electrical Engineering, University of Washington, Seattle
Articles by G. Rowe

A. V. Mamishev

A. V. Mamishev is an associate professor at the Sensors, Energy, and Automation Laboratory, Department of Electrical Engineering, University of Washington, Seattle.
Articles by A. V. Mamishev

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