Stray flux in a transformer enters the outer packages of a core perpendicularly and induces significant eddy currents which cause ohmic losses. Local thermal conditions near the core are affected by these losses which dissipate as heat and can cause local overheating in the oil near the outer packages. Stray losses in the outer packages should be considered during the power transformer design process, and it is therefore important to model them properly. This type of losses can be solved using 3D FEM; however, that approach is time-consuming which makes it impractical for the loss calculation in real-time. Another approach is 2D modelling, which is faster, but it is limited to axisymmetric and planar geometries only. Due to its complex geometry, a transformer cannot be modelled using a single 2D model. Instead, each quantity of interest, such as a magnetic field or eddy currents, can be obtained from a suitable 2D model. These models include certain simplifications, which should not affect the accuracy of the results. In this paper, a hybrid approach that utilizes various 2D models for the calculation of stray losses in the outer packages is proposed. The advantage of the proposed methodology is in obtaining the same level of accuracy while considerably reducing the calculation time when comparing to the 3D FEM approach. The estimation of the temperature rise of the outer packages, based on the calculated losses, will be a part of further research.

This paper presents a quasi-3D method for calculating stray losses in the outer lamination stacks of a power transformer core. The method uses multiple 2D FEM models in several steps to capture the 3D nature of the problem. This paper builds on a previous study that presented a quasi-3D calculation of the magnetic field on the surface of the outer stack. Here, the focus is on the magnetic field and eddy currents within the stack. The magnetic field penetration is analyzed in two distinct regions. To analyze the first region, located at the stack edges where magnetic field penetration is significant, a magnetostatic 2D FEM model with non-linear and anisotropic material properties is used. This model incorporates the concept of equivalent permeability to capture the impact of eddy currents on the field distribution. The equivalent permeability is derived from a 1D analytical model of a metallic plate subjected to a time-varying magnetic field. For the second region, near the surface of the stack where eddy currents strongly affect the field, a basic 1D analytical model is used. This model describes the field behavior using an exponentially decaying function characterized by the skin depth parameter, and its linearization is based on the results of the previous model. The magnetic field distributions serve as inputs to two additional 2D FEM models used to calculate the distribution of eddy currents. By combining the results from these models, the total stray losses in the stack are obtained. The calculation results are then validated experimentally. The experimental setup consists of an analyzed lamination stack, excitation windings, a U-shaped core, and thermal insulation. Typically, magnetically induced losses are evaluated by comparing losses in a system with and without the analyzed metallic component. However, this conventional approach cannot isolate losses in a single metallic component when other metallic components are present. The presented experimental evaluation, based on temperature measurements, overcomes this limitation. Thermal insulation is used to minimize the losses needed to reach a target temperature range and to improve temperature distribution uniformity, facilitating more accurate average temperature measurements. In all test cases, the deviation between the calculated and experimentally evaluated losses remains within the expanded uncertainty of approximately 7%, corresponding to a 95% confidence level.

The paper describes a high-frequency model for transient calculation in transformer windings with segmentation on a turn level. The windings are represented by lumped RLC parameters given by the analytical approach, whereas the model is solved in the time domain. The accuracy of the analytical calculation is improved with correction factors obtained with a set of numerical calculations. The presented model is verified by measurements on the example of power transformer disc winding.

A multi-step approach for the fast calculation of the magnetic field inside transformer tank shields, based on the 2D FEM, is presented in the paper. Due to the limitations of the 2D FEM, the proposed approach utilizes several 2D FEM models and calculates the magnetic field in multiple steps to account for the 3D geometry of the problem. In the first step, a distribution of the magnetic flux density that enters the tank shields is calculated using the quasi-3D model of the transformer. This quasi-3D model is obtained by superimposing the solution of multiple axisymmetric 2D FEM models, and assumes a considerably simplified transformer geometry. To account for the tank shield geometry that is neglected in the quasi-3D FEM model, an additional 2D FEM model with tank shields is introduced. After the distribution of the magnetic flux density that enters the tank shields is calculated, it is imposed in the final 2D FEM model with a non-linear tank shield which is used to calculate the magnetic flux density distribution inside the tank shields. The proposed approach enables a fast calculation of magnetic field distributions, both in the vertical and horizontal directions. The results of the proposed approach are compared against the 3D FEM. The relative error of the maximum magnetic flux density is under 2%, while the NRMSE of the magnetic flux density distribution within the tank shields is under 10%. The key contribution of the proposed approach is a low computation time. In the presented case study, the total computation time of the proposed approach is ~30 s, while the computation time of the 3D FEM is ~1 h. As the computation time is significantly reduced, while the accuracy is acceptable, the proposed approach can be a good alternative to the 3D FEM for design purposes. Therefore, it has industrial value.