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The purpose of this paper is to create a reliable nonlinear magnetic equivalent circuit (NMEC) of the hybrid magnetic bearing (HMB). Commonly used magnetic equivalent circuits of HMB omit a saturation effect of the magnetic material as well as the leakage and fringing flux. It results in imprecise modelling of the magnetic field distribution. On the other hand, only 3D finite element analysis (FEA) can be used to precisely simulate the magnetic field in this type of the magnetic bearing. The proposed NMEC incorporates the saturation effect of the magnetic material, as well as the leakage and fringing flux.

The magnetic equivalent circuit of presented HMB is proposed to obtain a reliable model that ensures short calculation time. Developed NMEC incorporates the phenomena as the saturation effect, as well as the leakage and fringing flux. The reluctance of the air gap that includes the fringing flux was calculated using 3D FEA. Kirchhoffs’ laws were used to create a set of nonlinear equations that were iteratively solved by Broyden’s method.

Incorporating into NMEC of the HMB a saturation effect of the magnetic material, as well as the leakage and fringing flux, resulted in the accurate model that was in good agreement with 3 D finite element model and the real object. The developed NMEC offers the calculation time in the range of miliseconds, therefore can be successfully used in the engineering design instead of the FEM.

Presented NMEC can be considered as a fundamental model that can be successfully used for accurate and fast simulation of the HMB. Proposed NMEC includes considerable factors that decide about the model accuracy such as the saturation effect of the ferromagnetic material and the leakage and fringing flux. The developed NMEC can be used in the optimization procedures and for simulations of dynamic responses.

The 3D calculation of the monophase flux, due to n‐triple harmonics of the windings currents is considered. For small, three‐phase transformers without ferromagnetic tanks the calculation consists in unbounded problem solution. The 3D magnetic field of the yoke flux outside the magnetic core was analysed numerically by means of integral equations method. Our computer program subroutines, written in Fortran 77, have been implemented to a personal computer. The calculated components of the magnetic flux density were compared with the measured ones. The distribution of the different flux parts was calculated and tested by measurements for a three‐phase transformer.

Discusses the 27 papers in ISEF 1999 Proceedings on the subject of electromagnetisms. States the groups of papers cover such subjects within the discipline as: induction machines; reluctance motors; PM motors; transformers and reactors; and special problems and applications. Debates all of these in great detail and itemizes each with greater in‐depth discussion of the various technical applications and areas. Concludes that the recommendations made should be adhered to.

The purpose of this paper is to examine the calculation of magnetic field distribution in the modular amorphous transformers under short‐circuit state including the flux by the voltage supplying. The magnetically asymmetrical transformer (amorphous asymmetrical transformer – AAT) has been compared also with the symmetrical one (amorphous symmetrical transformer – AST).

3D field problems were analyzed with total ψ and reduced ϕ potentials within the finite element method (FEM). The calculated fluxes have been verified experimentally.

The field method which includes voltage excitation is helpful for flux density (B) calculation and winding reactances determination, as well. Calculations and tests yield similar flux distributions in both AST and AAT constructions. One should emphasize that AAT is better for manufacturing and repairing.

Owing to very thin (80 μm) amorphous ribbon, the solid core has been assumed for computer simulations.

Employment of a field method for calculation of the innovative three‐phase amorphous modular transformers. New construction of amorphous transformer, i.e. AAT, has been manufactured at Opole University of Technology.

The calculation results of the 3D magnetic field in a reactor are examined using the computer packages TRACAL3 as well as OPERA3D. The first own package is based on the boundary integral equations (BIE) and the second commercial package is deduced from finite elements method (FEM). As an example, the magnetic flux density components were calculated. For the reactors with large air gaps, the numerical results obtained by BIE and FEM are relatively close.

Two approaches to 3‐D magnetic field calculations for low‐power special transformers are presented in the paper. The application of fast calculation procedures based on the finite difference method, to direct solving of the Poisson and Laplace equations is given. The calculated example deals with the three‐phase leakage transformer. On the other hand, the iterative solution of regularized integral equations used for calculation of 3‐D field distribution and integral parameters of leakage field in current transformer is also presented. The software packages developed by the authors are useful for computer‐aided design of low‐power special transformers.

The paper discusses the problem of space distribution of zero‐component magnetic flux generated in three‐column transformer. For 3‐D magnetic field calculation the method of integral equations was used. The numerical calculations were made for physical model of the transformer and compared with experimental results. The accuracy of the calculations of the magnetic field, achieved in the work, proves that the modelling may be used as a computer aided designing tool.

The reactor with rectangular shape of the winding turns and open magnetic core has been considered in the work. The integral equation has been solved numerically and components of magnetic flux density as well as leakage inductance of the winding have been computed for the object. A computer program for the analysis of magnetically open problems and a procedure, for numerical calculation of inductance, have been worked out. The computer codes were implemented for IBM PC/AT. The calculations were verified by measurements for physical model of the reactor.

To make easier and faster the designing of transformers using scale models.

The scale modeling in designing of transformers is included. Both computer and physical models of high leakage reactance (HLR) and 3‐phase (TP3C) transformers have been considered. The 3D field computations have been executed for the scaled models, and the results were recalculated to the full‐scaled ones.

It is possible to calculate the scale coefficients for nonlinear models of transformers using finite element method (FEM) software. Obtained coefficients are useful in the designing process. Measurements confirm correctness of the scaling laws.

The calculations were done only for transformers and the eddy current was not taken into account.

Presented formulae for scale model calculation are very useful for designing of transformers by the engineers. It is possible to design a series of transformers. Only one physical model must be manufactured for experimental verification.

This paper offers an innovative approach to non‐linear scaled modelling of transformers using FEM.

In a modular transformer with a wounded amorphous core, the authors should make some cutting to limit the eddy currents in its magnetic ribbon. The purpose of this paper is to deal with 3D magnetic field analysis, including the eddy currents induced by varying frequency of power. The influence of the core leg cutting on the power losses values, in the three variants of a one-phase modular transformer structure, has been presented.

3D field problems including eddy currents of various frequency were analysed using the electrodynamic potentials and V within the finite element method. The wave method and iterative one of the laminated core homogenization, have been employed. The values of the calculated losses have been verified experimentally.

The reduction of the core losses by axial cutting of the transformer legs is an efficient approach for the loss limitation. The wave method is not acceptable for homogenization of the amorphous core for its operation above 1 kHz. The iterative method is the better way to perform the homogenization.

Due to very thin (less than 50 μm) amorphous ribbon, the unhomogenization of the laminated magnetic core should be performed. Thus, the solid core with equivalent parameters has been assumed for the computer simulations. For the frequencies above 1 kHz, the iterative method should be used to determine the equivalent electrical conductivity of the solid substitute core.

Using the wave method with the electrodynamic similarity laws and assuming the wave penetration depth, the equivalent electrical conductivity of the homogenized core, has been determined. This approach is valid for supply frequencies below 1 kHz. For the higher frequencies the authors had to use the iterative method. It seems to be valid for another cores with amorphous and nanocrystalic ribbons. For the modular amorphous core it is only way to calculate the losses in the solid geometry of the homogenized laminated magnetic circuit.