Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 915497, 8 pages

http://dx.doi.org/10.1155/2015/915497

## Perturbation Approach to Reconstructing Deformations in a Coaxial Cylindrical Waveguide

Department of Electromagnetic Engineering, School of Electrical Engineering, KTH Royal Institute of Technology, 10044 Stockholm, Sweden

Received 20 January 2015; Accepted 4 June 2015

Academic Editor: Franklin A. Mendivil

Copyright © 2015 M. Dalarsson et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1], but in order to reduce the risks of power outages, it is desirable to develop online methods that are applicable when the transformer is connected to the power grid. The present authors studied an online method, using microwave antennas inside the transformer, to reconstruct the radial positions of individual winding segments in [2, 3]. In these papers, the reconstruction of the locations of the individual conductors from synthetic measurement data was performed by means of an optimization method, using only up to ten winding segments or turns, in order to reduce the computational complexity. In [4], elliptic, and more importantly, wave-shaped mechanical deformations were studied. It was found that the wave-shaped deformations (to the first order of approximation) can be reduced to radial displacements such as those described in [2, 3], showing that essentially the same mathematical tools can be used to cover a broad range of deformations of individual conductors. It should however be noted that, unlike the approach in the present paper, the analysis in [2–4] uses a discrete conductor model with a number of individual conductors being treated as obstacles in the waveguide and where the mode matching technique is used to handle a limited number of such discrete obstacles. A more realistic number of winding segments, however, make a full-scale numerical model in the optimization procedure prohibitively complex. As the primary interest of our study is the detection of small winding deformations, it is of interest to study whether inversion methods, based on weak scattering, can be used for reconstruction of transformer winding deformations. A step towards such a method was taken in [5], where the transformer winding was not considered in detail but instead modeled as equivalent outer boundary surfaces, whose shape was to be reconstructed. In [5] a parallel-plate waveguide model was assumed, with a piecewise linear shape of the continuous winding deformation. This study was extended in [6], which is similar to the approach pursued here. In the present paper we use the same basic mathematical formalism as in [6], but unlike the treatment in [6] where we used the approximate parallel-plate waveguide model, here we employ the more realistic coaxial waveguide model together with improved numerical techniques, both of which contribute to more accurate reconstruction results in the present paper. Although the present paper can be seen as an extended and improved version of the approximate treatment employed in [6], the abovementioned model change and numerical improvements provide an essentially novel approach to the problem of reconstruction of deformations in the lower coaxial waveguide boundary. Thus we pursue the investigations reported in [5, 6] further by considering a continuous axially symmetric deformation in a coaxial waveguide model of the power transformer winding structure. The inverse problem of reconstructing deformations in the lower coaxial waveguide boundary is solved using a simple and computationally efficient first-order perturbation method. We use synthetic measurement data from the commercial FEM program HFSS to test the model.

#### 2. Problem Formulation and Scattering Analysis

We assume an axially symmetric coaxial waveguide scattering configuration, oriented along the -axis, as shown in Figure 1. The inner boundary cylinder is located at while the outer boundary cylinder is located at giving the radial width of the unperturbed cavity equal to . In the context of a power transformer, the outer boundary represents the transformer tank wall while the lower boundary represents the outermost layer of the winding structure. The winding structure is thus described as an equivalent PEC surface. Using anisotropic boundary conditions [7], it is possible to formulate a more realistic approach to model the winding. Although a realistic transformer is filled with oil and uses also paper and pressboard insulation, in the present paper we assume that the medium inside the waveguide is air (or vacuum). At the inner boundary cylinder along the section there is a local deformation described by The inverse problem here is to reconstruct in the* estimation region * using scattering data obtained when the waveguide is excited from both ends. In order to focus the present study on the primary scattering mechanism, due to the local deformation of the lower boundary, we assume that there are no reflections from the ends of the waveguide.