On Simplified Calculations of Leakage Inductances of Power Transformers
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energies Article On Simplified Calculations of Leakage Inductances of Power Transformers Tadeusz Sobczyk * and Marcin Jaraczewski * Faculty of Electrical and Computer Engineering, Cracow University of Technology, Warszawska 24 Street, 31-155 Cracow, Poland * Correspondence: [email protected] (T.S.); [email protected] (M.J.); Tel.: +48-12-628-2658 (M.J.) Received: 1 August 2020; Accepted: 17 September 2020; Published: 21 September 2020 Abstract: This paper deals with the problem of the leakage inductance calculations in power transformers. Commonly, the leakage flux in the air zone is represented by short-circuit inductance, which determines the short-circuit voltage, which is a very important factor for power transformers. That inductance is a good representation of the typical power transformer windings, but it is insufficient for multi-winding ones. This paper presents simple formulae for self- and mutual leakage inductance calculations for an arbitrary pair of windings. It follows from a simple 1D approach to analyzing the stray field using a discrete differential operator, and it was verified by the finite element method (FEM) calculation results. Keywords: multi-winding transformers; 1D stray field analysis; discrete differential operator; energy-based approach; self- and mutual leakage inductances 1. Introduction Power transformers are rather well-recognized electromagnetic devices. During the almost 150 year history of their development, countless books and papers have been devoted to power transformers, examples of which are [1–4]. Theoretical, design and operation problems have been solved. Besides the classical single-phase and three-phase, new types of power transformers have been designed for special applications: power system control, power electronics units, or electrical traction. Many of them have a lot of windings located in a common magnetic circuit. Three examples of transformers for power system control are shown in Figure1a–c. The connection scheme of phase-shifting transformers to control the level and the phase of voltages is presented in Figure1a. Figure1b presents the connection scheme of multilevel autotransformers, and Figure1c presents the connection of transformer windings for the HVDC transmission lines. Rectifier transformers generating multi-phase voltages or currents can also have many windings. Figure2a,b show the provisional location of windings in the air window for Yyd 3 /6-phase transformers and Figure2b for Y zyd 3/12-phase transformers [5]. The location of the windings loses symmetry that is characteristic for classical three-phase power transformers. Single-phase traction transformers, to ensure various voltage levels on a traction vehicle at different traction supply systems, consist of many (up to 25) separate windings [6,7]. Energies 2020, 13, 4952; doi:10.3390/en13184952 www.mdpi.com/journal/energies Energies 2020, 13, x FOR PEER REVIEW 2 of 14 EnergiesEnergies 20202020,, 1313,, x 4952 FOR PEER REVIEW 22 of of 14 13 (a) (b) (c) Figure 1. Transformers for power system control (a) phase‐shifting transformers (b) multilevel autotransformer (c) HVDC transformer. Rectifier transformers generating multi‐phase voltages or currents can also have many windings. Figure 2a,b show the provisional location of windings in the air window for Yyd 3/6‐phase transformers and(a) Figure 2b for Yzyd 3/12‐phase(b) transformers [5]. The location of(c )the windings loses symmetryFigureFigure that1. 1. Transformers Transformersis characteristic for for power powerfor classical system system threecontrol control‐phase ( (aa)) phase phase-shifting power‐shifting transformers. transformers transformers Single ( (bb)) ‐phasemultilevel multilevel traction transformers,autotransformerautotransformer to ensure ( (cc)) HVDC HVDC various transformer. transformer. voltage levels on a traction vehicle at different traction supply systems, consist of many (up to 25) separate windings [6,7]. Rectifier transformers generating multi‐phase voltages or currents can also have many windings. Figure 2a,b show the provisional location of windings in the air window for Yyd 3/6‐phase transformers and Figure 2b for Yzyd 3/12‐phase transformers [5]. The location of the windings loses symmetry that is characteristic for classical three‐phase power transformers. Single‐phase traction wp1 wp1 i ws1i s1 ws1i s1 i transformers, to wensures1i s1 variousws1i s1voltage levels on a traction2 p vehicle at different traction2 psupply systems, consist of many (up to 25) separate windings [6,7]. wpip wpip wp2pi wp2pi wp1 wp1 ws2i s2 ws2i s2 ws1i s1 w i i w i w i w2 ip s1 s1 w2 p s1 s1 s1 s1 p1 ws2i s2 ws2i s2 p1 2 ip 2 ip wpip wpip wp2pi wp2pi Figure 2. Winding’s location in rectifier transformers (a) 3/6 phase transformers (b) 3/12 (a) (b) phase transformers.ws2i s2 ws2i s2 w w p1 ws2i s2 ws2i s2 p1 Figure 2. Winding’s location in rectifier transformers (a)2 3/6ip phase transformers (b) 3/12 2phaseip transformers.Two approaches are used to recognize the properties of transformers. The first is based on the Maxwell equations for the magnetic vector potential. The second is based on the Kirchhoff equations for magneticallyTwo approaches coupled are used coils. to The recognize first one the is appliedproperties mainly of transformers. for design problems The first ofis transformers,based on the Maxwellwhereas theequations second for is usedthe magnetic for operation vector problems, potential. whenThe second the transformer is based on is the a part Kirchhoff of a multi-object equations forsystem. magnetically These two coupled approaches(a )coils. 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For such cases, the 1D field analysis in the air zone provides rather simple formulae for short-circuit reactances, which can be found in many books and textbooks on transformers, and are still recommended to designers. That approach has been successively applied to calculations of short-circuit impedances of multi-winding traction transformers [6,7]. Unfortunately, it is not valid at asymmetrical locations of windings in the air window. At least a 2D approach is necessary for that, which is presented in [21]. In this paper, an analytical solution of magnetic vector potential has been formulated for the transformer’s air window with arbitrarily located windings at zero Newmann boundary conditions on the ferromagnetic borders. To fulfil such boundary conditions, the ampere-turns of windings in the air window have to be balanced, i.e., their sum must equal zero. This allows us to calculate the short-circuit inductances for an arbitrary pair of windings. However, the leakage inductances of individual windings and the mutual inductances due to the stray field cannot be determined. To find those inductances, 2D or 3D field calculations across the entire magnetic circuit, including both the ferromagnetic core and the air zone, were applied [22]. To overcome the difficulties of 2D or 3D magnetic field computation, the relations between winding linked fluxes n and currents in, necessary for the Kirchhoff’s equations of transformers, can be presented in the form: 82 µ µ µ 3 9 2 3 > L (i) L (i) L (i) 2 σ σ σ 3>2 3 6 1 7 >6 1,1 1,2 1,N 7 6 L1,1 L01,2 L01,N 7>6 i1 7 6 7 >6 µ ··· µ 7 6 σ ··· σ 7>6 7 6 7 >6 L (i) L (i) 7 6 L L 7>6 i 7 6 2 7 <>6 2,2 ··· 2,N 7 6 02,2 02,N 7=>6 2 7 6 . 7 = 6 . 7 + 6 ··· . 7 6 .