The article presents the calculation of the leakage inductance in power transformers. As a rule, the leakage flux in the transformer window is represented by the short-circuit inductance, which affects the short-circuit voltage, and this is a very important factor for power transformers. This inductance reflects the typical windings of power transformers well, but is insufficient for special transformers or in any case of the internal asymmetry of windings. This paper presents a methodology for calculations of the self- and mutual-leakage inductances for windings arbitrarily located in the air window. It is based on the 2D approach for analyzing the stray field in the air zone only, using discrete partial differential operators. That methodology is verified with the finite element method tested on real transformer data.
Abstract. The Fourier series method is frequently applied to analyze periodical phenomena in electric circuits. Besides its virtues it has many drawbacks. Fourier series usually have slow convergence and fail for fast changing signals, especially for discontinues ones. Therefore they are suitable to describe only quasiharmonic phenomena.For strongly nonsinusoidal signal analysis we propose the L 1 -impulse method. The L 1 -impulse method consists in an equivalent notation of a function belonging to L 1 as a sum of exponential functions. Such exponential functions have rational counterparts with poles in both sides of imaginary axis. With the L 1 -impulse functions we can describe periodical signals, thus we get the homomorfizm between periodical signals and a rational functions sets. This approach is especially adapted to strongly deformed signals (even discontinues ones) in linear power systems, and thanks to that we can easily calculate optimal signals of such systems using the loss operator of the circuit. The loss operator is exactly the rational function with central symmetry of poles [1].In this paper the relation between the L 1 -impulse and the Fourier series method was presented. It was also proved that in the case of strong signal deformation the L 1 -impulse method gains advantage.Key words: periodically time-varying networks, operational calculus, stability, synthesis, optimization.
L 1 -impulses and periodic signalsThe L 1 -impulse is an absolutely summable signaland its periodic extension is a T -period functioñwhere p ∈ Z (integers). Series (1) always converges. It results from the fact that every L 1 -impulse is majorised by an exponential signal:where a, b, c, d -positive numbers, 1(t) -step function. Applying the formula (1) to the (2) we getfor t ∈ [0, T ).The inner product of the L 1 -impulses is defined as followsand the linear operator Hwhere L ∞ -space of bounded signals, R -real numbers. The special case of the (5) operator is the convolution operatorIt results that the sequence of two convolution operators act as L ∞ into L ∞ mapping, which means at the same time thatThus the convolution of the L 1 impulses produce also the L
PurposeDiscrete differential operators (DDOs) of periodic functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary nonlinear differential equations.Design/methodology/approachThe DDOs have been applied to create the finite-difference equations and two approaches have been proposed to reduce the Gibbs effects, which arises in solutions at discontinuities on the boundaries, by adding the buffers at boundaries and applying the method of images.FindingsAn alternative method has been proposed to create finite-difference equations and an effective method to solve the boundary-value problems.Research limitations/implicationsThe proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This can be extended to the 2D or 3D cases with more flexible meshes.Practical implicationsBased on this publication, a unified methodology for directly solving nonlinear partial differential equations can be established.Originality/valueNew finite-difference expressions for the first- and second-order derivatives have been applied.
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