object of control. The closest in the formulation is the problem that arises when
controlling the electrophysical properties of a material with a probe in the form of an
open end of a coaxial cable. A description of the analytical model for the interaction of
such a sensor with an object is given, for example, in [25], and an analysis of possible
algorithms for solving the inverse problem (the definition of complex dielectric
permittivity) in [26]. Herewith, there are some differences from the method of
microwave diagnostics considered in this project. First of all, this is the fact that the
proposed method uses the measurement of the shift of the complex value of the
resonance frequency, rather than the complex reflection coefficient. However, this
difference is important in the analysis of the sensitivity of the method and the
measurement error, but it is not important for the mathematical model, since it
calculates parameters interconnected by simple analytical relationships based on the
parameters of the equivalent circuit of a cavity resonator.
A more significant difference is the fact that when using a cavity resonator probe,
the radial component of the electric field in the plane of the aperture is no longer given
by the simple 1/r dependence, but depends on the parameters of the object of control.
In [27], higher modes of oscillations in a coaxial probe are also taken into account, and
a self-consistent problem of field scattering by the object of control is solved. As a
result, it is possible to calculate the solution of the direct problem (finding the complex
admittance of the aperture of the probe) by an analytical-numerical method.
An attempt to solve a similar problem for a cavity resonator probe encounters
significant difficulties in mathematical modeling, and the solutions obtained are even
more cumbersome. In addition, analytical solutions are found suitable only for layered
objects with a finite number of homogeneous layers. With a large number of such layers
or a continuous change in the properties of the material, these methods become
inapplicable in practice.
Practically described models were used primarily to determine the electrophysical
properties of a single layer of material in the environment of several layers with known
characteristics. In fact, the problem was solved of finding two parameters from two
measurements made at different heights of the coaxial sensor or under different
boundary conditions (the opposite side of the object is shielded or not).
The most accurate and simple method for solving the inverse problem is the
method of analytic approximation. To implement it, we must also find solutions to the
direct problem. For this, modeling of objects with parameters that vary within wide
limits is carried out. These parameters include both the electrophysical parameters
measured in the real object and the aperture geometry of the probe, and various
interfering factors. Then, from the obtained data, an analytical approximation is carried
out, as a result of which a family of analytic functions of the signals of the measurement
information is obtained. With its help, the inverse problem is solved.
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