We study a detection method for continuous mechanical deformations of coaxial cylindrical waveguide boundaries, using perturbation theory. The inner boundary of the waveguide is described as a continuous PEC structure with deformations modeled by suitable continuous functions. In the present approach, the computation complexity is significantly reduced compared to discrete conductor models studied in our previous work. If the mechanically deformed metallic structure is irradiated by the microwave fields of appropriate frequencies, then, by means of measurements of the scattered fields at both ends, we can reconstruct the continuous deformation function. We apply the first-order perturbation method to the inverse problem of reconstruction of boundary deformations, using the dominant TEM-mode of the microwave radiation. Different orders of Tikhonov regularization, using the L-curve criterion, are investigated. Using reflection data, we obtain reconstruction results that indicate an agreement between the reconstructed and true continuous deformations of waveguide boundaries.

Power transformers are fundamental components of an electric power grid that require careful monitoring and fault assessment. Mechanical deformations of power transformer windings, mainly due to the heavy mechanical forces from short-circuit currents, increase the risks of serious electrical power outages in the grid. In order to reduce the risks, it is of interest to investigate suitable online early detection methods for local mechanical winding deformations. One available method to diagnose various degradation phenomena in power transformers is the frequency response analysis (FRA) method. It is, however, only applicable when a transformer is disconnected from the power grid. FRA has been proposed for detection of winding deformations [

We assume an axially symmetric coaxial waveguide scattering configuration, oriented along the

Geometry of the coaxial waveguide model.

We restrict our analysis to the TM-modes only and to mechanical deformations that possess cylindrical symmetry (i.e., are independent on the azimuthal angle

The inversion scheme (Section

In the present study, a full-wave FEM model, implemented in the commercial program HFSS, is used to generate synthetic measurement data. For simplicity, the unperturbed waveguide cavity radial size was chosen as

Let us now consider the perturbation function

In practice however, the measurements of

The coefficients

In this section we present the reconstruction results for five different perturbation shape functions: (a) continuous intrusion, (b) continuous extrusion, (c) continuous intrusion/extrusion, (d) discontinuous intrusion, and (e) discontinuous extrusion. In all the reconstruction examples, we use

Reconstruction results for

In Figure

Reconstruction results for

From Figures

Reconstruction results for different heights of sinusoidal intrusions (30%, 40%, and 50%). The true shapes are represented by the black lines and the reconstructed shapes by the blue (

Reconstruction results for different heights of square extrusions (10%, 20%, and 30%). The true shapes are represented by the black lines and the reconstructed shapes by the green (

Finally, in Figures

Reconstruction results for deformation length 1.5 m and (a) 20% intrusion, (b) 20% extrusion, and (c) 20% intrusion and extrusion. In all three graphs, the true shape is represented by the black line and the reconstructed shapes are represented by the blue (

Reconstruction results for deformation length 1.5 m and (a) 20% intrusion and (b) 20% extrusion. In both graphs, the true shape is represented by the black line and the reconstructed shapes are represented by the blue (

It should be noted here that the high accuracy of the reconstructions in cases of continuous intrusions (see e.g., Figure

A simple and computationally efficient first-order perturbation approach to the inverse problem of reconstructing deformations in a lower coaxial waveguide boundary, based on the contributions from the dominant mode only, has been investigated. Using a full-wave FEM model implemented in the commercial program HFSS, as the generator of synthetic measured reflection data, we obtained reconstruction results indicating an agreement between the reconstructed and true continuous deformations of waveguide boundaries. The cases presented in the paper, as well as other results omitted for sake of brevity, show that the method works well for continuous and discontinuous deformations and is able to distinguish between these two types of shapes. The method is also stable under different deformation heights and lengths. As a proposal for continued efforts, other regularization techniques based on Tikhonov regularization could be considered. Furthermore, improved methods to include the contributions from higher-order modes would also be of interest.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was funded by the Swedish Energy Agency, Project no. 34146-1.