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## Application of higher order forms for the description of electromechanical energy converters

### Abstract

The purpose of that paper is to develop of unified equations of electromechanical energy converters accounting for the magnetic non-linearity of the main magnetic circuit of a converter. The concept of applying higher order forms of winding currents for the description of the co-energy function is introduced in order to derive the structure of converter equations via mathematical analysis. Also, another concept of equivalent magnetizing currents is applied to determine the higher order forms for selected converters designs. The structure of circuital equations for converters with multiple windings has been unified by means of the introduction of matrices of dynamic and nonlinear inductances following the higher order forms of the co-energy function.
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## Influence of commutation conditions on steady states in matrix converter

### Abstract

The paper investigates a significant influence of transients on steady states in a matrix converter with the one-periodic control strategy. Proposed controller can be used as an interconnection device within a power system for a power flow control. However, the presence of inductances in external systems has the significant influence on steady state of a matrix converter operation. The special current injection method has been developed to ensure a proper operation of a matrix converter. Presented analysis of steady states is carried out in a frequency domain using the harmonic balance method. Obtained numerical results, which are confirmed by a time domain analysis, prove the effectiveness of the proposed method.
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## On determination of inductances for circuit models of AC machines

### Abstract

The paper discuss a problem of determination of inductances for AC machine windings when saturation of magnetic circuit is not neglected. For such cases, computation of magnetic field distribution in the machine magnetic circuit is a starting point for post processing procedures leading to various values, among others the co-energy in a given area and linkage fluxes of windings. This paper shows how to determine winding inductances in a nonlinear magnetic circuit from these two values and also how to compute directly nonlinear inductances. Problem is not trivial because such inductances are not uniquely determined as for linear case. In the paper a definition of nonlinear inductances is proposed which makes the choice unique.
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## Algorithm for determining two-periodic steady-states in AC machines directly in time domain

### Abstract

This paper describes an algorithm for finding steady states in AC machines for the cases of their two-periodic nature. The algorithm enables to specify the steady-state solution identified directly in time domain despite of the fact that two-periodic waveforms are not repeated in any finite time interval. The basis for such an algorithm is a discrete differential operator that specifies the temporary values of the derivative of the twoperiodic function in the selected set of points on the basis of the values of that function in the same set of points. It allows to develop algebraic equations defining the steady state solution reached in a chosen point set for the nonlinear differential equations describing the AC machines when electrical and mechanical equations should be solved together. That set of those values allows determining the steady state solution at any time instant up to infinity. The algorithm described in this paper is competitive with respect to the one known in literature an approach based on the harmonic balance method operated in frequency domain.
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## Estimation of coefficients of multivariable power series approximating magnetic nonlinearity of AC machines

### Abstract

Energy based approach was used in the study to formulate a set of functions approximating the magnetic flux linkages versus independent currents. The simplest power series that approximates field co-energy and linked fluxes for a two winding core of an induction machine are described by a set of common unknown coefficients. The authors tested three algorithms for the coefficient estimation using Weighted Least-Squared Method for two different positions of the coils. The comparison of the approximation accuracy was accomplished in the specified area of the currents. All proposed algorithms of the coefficient estimation have been found to be effective. The algorithm based solely on the magnetic field co-energy values is significantly simpler than the method based on the magnetic flux linkages estimation concept. The algorithm based on the field co-energy and linked fluxes seems to be the most suitable for determining simultaneously the coefficients of power series approximating linked fluxes and field co-energy.
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## Modelling concept of N×M matrix converter under periodic control for dynamic states

### Abstract

This paper presents a concept of an NxM Matrix Converter (MC) modeling under periodic control strategy patented in Poland. This strategy allows to change an Nphase input system of voltages and current with the frequency fi to the M-phase output system with the frequency fo, maintaining both systems symmetrical and providing small distortions of voltage and current waveforms at rather high frequencies. In this paper the control strategy is extended for dynamic states when one of the frequencies is changed. Matrix converter equations have been derived using the constrain matrix, which is determined by the switch states. The equations have the hybrid form of a multi-port circuit. To simplify these equations the symmetrical components of input and output voltages and currents have been applied. As a result, rather simple equations have been found. They can be interpreted to an equivalent scheme. All considerations are illustrated using an exemplary 6H3 matrix converter.
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## Description of matrix converter in frequency domain

### Abstract

The paper presents abilities and advantages following from the use of the harmonicbalance method for analysis of steady state of a multiphase system with switching devices on example of a matrix converter. Switching elements are modelled as resistances with step-wise variable parameters, what allows to describe the converter by a linear infinite set of equations. The analysis in frequency domain is presented on example of the one-periodic control strategy. External systems were also added using the Thevenin method approach. The numerical calculation results of a linear equations set were verified by the variable structure method in a time domain and the numerical convergence was confirmed. Furthermore, the exemplary complex system was analysed using the cascade method and current waveforms were obtained.
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## On parameters determination of multi-port equivalent scheme for multi-winding traction transformers

### Abstract

This paper aims to present a new equivalent scheme of multi-windings traction transformers, based on multiport purely inductive circuit. The mathematical background of this equivalent scheme is described. The determination of the different scheme elements is made through a finite-elements calculation of both main and leakage inductances, for the case of a four-winding transformer. A procedure is defined, which allows to estimate the values of these elements from some measurements on the transformer at no-load and short-circuit operations. A specific strategy of short-circuit tests is described, allowing to determine all parameters in a rather simple way.
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## Energy conversion in small water plants with variable speed PM generator

### Abstract

The Small Hydro Power Plants allow to increase the energy amount from renewable sources, especially from small rivers in mountainous areas. This paper presents a new concept of a energy conversion system for application in a Small Hydropower Plant (SHP) which is based on a permanent magnet generator (PM generator) with a propeller turbine integrated with the generator rotor. The PM generator can work at a variable speed and therefore energy produced by the PM generator has to be converted by means of a power electronic unit to fit to the three-phase power grid parameters. For this concept, dimensions and parameters of the PM generator were specially designed on account of integration with water turbine. The paper precisely describes elements of energy conversion system and also presents the results of numerical tests for chosen working conditions. An original algorithm of control strategy for power electronic unit was used to adjust generated energy to the required parameters of the three-phase grid.
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