Effects of eddy currents in transformer windings
01 Aug 1966-Vol. 113, Iss: 8, pp 1387-1394
TL;DR: In this article, the effect of eddy currents on transformer windings is considered and a method is derived for calculating the variation of winding resistance and leakage inductance with frequency for transformers with single-layer, multilayer and sectionalised windings.
Abstract: The effects of eddy currents in transformer windings are considered, and a method is derived for calculating the variation of winding resistance and leakage inductance with frequency for transformers with single-layer, multilayer and sectionalised windings. The method consists in dividing the winding into portions, calculating the d.c. resistances and d.c. leakage inductances of each of these portions, and then multiplying the d.c. values by appropriate factors to obtain the corresponding a.c. values. These a.c. values are then referred to, say, the primary winding and summed to give the total winding resistance and leakage inductance of the transformer. Formulas are derived and quoted for calculating the d.c. resistances and leakage inductances of the winding portions. Theoretical expressions are derived for the variation with frequency etc. of the factors by which the d.c. values must be multiplied to obtain the corresponding a.c. values. These expressions are presented in the form of graphs, permitting the factors to be read as required.
Citations
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TL;DR: In this paper, the authors compared different levels of detail of two-dimensional finite element (FE) models and investigated the impact of the modeling depth on the resulting copper loss to find a computation time optimized modeling setup.
Abstract: Analysis of ac effects, such as skin and proximity effect as well as the circulating currents and corresponding losses are usually done by finite-element analysis, while mostly only a single slot is considered, neglecting the influence of neighboring phases on the flux density of the stator iron and thus the resulting slot leakage flux. As the simulation of single stranded finite element (FE) models is computationally very demanding and time consuming, this paper compares different levels of detail of two-dimensional FE models and investigates the impact of the modeling depth on the resulting copper loss to find a computation time optimized modeling setup. It is shown that the use of single slot models is sufficient in case of distributed windings. Moreover, the potential increase of ac losses is assessed for two machines that are identical except for the stator winding, so that one and the same machine is investigated at two different voltage levels, namely 400 and 800 V. The reference machine is a 160 kW nominal power and 9000 r/min maximum speed permanent magnet synchronous machine. Since the magnetic circuit has to remain constant, both designs share the same overall winding scheme and total number of strands within a slot but differ in the number of parallel and serial connected strands. Due to the increased bundle cross section of the lower voltage winding, it is more susceptible to ac effects and thus tends to increased winding losses.
26 citations
TL;DR: In this article, a planar coil was fabricated and measured to verify the analytical calculation of the proximity-effect resistance, using litz wire composed of 100 strands with 0.1mm diameter.
Abstract: Planar coils are popularly used in inductive power transfer applications. The coils’ efficiency is improved with litz wire by eliminating the winding's skin-effect loss, but the proximity-effect loss still needs to be considered. To calculate the proximity-effect resistance, the magnetic fields across the windings are required. During the coils’ optimization process, numerical methods of field calculation are too time-consuming due to the parametric sweep of physical parameters. The analytical calculation of the magnetic field is preferred in this scenario, but it becomes complex with the existence of the ferrite plate, which is used to increase the coils’ coupling. In this paper, the method of image is applied to simplify the analytical calculation, in which the magnetic fields across the windings are derived from Ampere's law and Biot–Savart law. The proximity-effect resistance is then calculated with these fields. The study also proves that the size of the ferrite plate has negligible influence on the proximity-effect resistance as long as it is larger than the size of the winding. A planar coil was fabricated and measured to verify the analytical calculation of the proximity-effect resistance, using litz wire composed of 100 strands with 0.1 mm diameter. A square plate of 3C96 ferrite with the dimensions of 100 mm × 100 mm × 5 mm was placed below the winding. The difference between calculation and measurement is less than 15%.
25 citations
23 Dec 2010
TL;DR: In this paper, a hybrid of particle swarm optimization (PSO) and differential evolution (DE) is proposed to solve this multi-objective problem with constraints, which provides diversity to p best of each particle, which is missing in standard PSO.
Abstract: A high frequency transformer is a critical component in a dual active bridge converter (DAB) used in a power electronics-based solid state transformer. Operation of a DAB converter requires its transformer to have a specific amount of winding leakage inductance. The demand for high efficiency requires minimization of transformer copper loss and core loss. Furthermore, available window area limits the winding arrangement of transformer. A hybrid of particle swarm optimization (PSO) and differential evolution (DE) is proposed to solve this multi-objective problem with constraints. DE provides diversity to pbest of each particle, which is missing in standard PSO. The differential evolution particle swarm optimization (DEPSO) algorithm is applied to find optimal transformer designs for DAB converters. Results show the DEPSO method is a generic effective way to find optimal high frequency transformer design.
25 citations
TL;DR: In this paper, the authors proposed a 3D planar-Litz windings planar transformer to achieve low ac resistance and low parasitic capacitance in high-frequency planar transformers, which requires minimal interleaving between primary and secondary windings.
Abstract: This article proposed a new 3-D parallel winding method (also named planar-Litz), which just requires minimal interleaving between primary and secondary windings. Low ac resistance and low parasitic capacitance can both be achieved. A general 1-D analytical model is developed to predict parallel winding circulating/eddy current in planar transformers. The analytical model shows that there is high circulating current in direct-parallel windings in high-frequency planar transformers. Direct-parallel windings and planar-Litz windings have been compared both in finite-element analysis (FEA) and experimental prototypes, which proves that the $d\Phi / dt$ -induced voltage cancellation in planar-Litz winding can minimize circulating current. A 3.0-kW and a 3.2-kW 500-kHz $LLC$ resonant converter are built to compare the direct-parallel winding and planar-Litz winding structures. More than 98.5% peak efficiencies are achieved in both converters. The comparisons of thermal images and efficiencies further verify the effectiveness of planar-Litz winding. The 98.6% peak efficiency and the 97.8% full-load efficiency are measured. The planar-Litz windings planar transformer demonstrated a 26% winding loss reduction and 0.25% converter efficiency improvement over the direct-parallel winding planar transformer on 3 kW. The hot spot inside the transformer can be relieved and hence allow more power delivery.
25 citations
10 May 2015
TL;DR: In this paper, a permanent magnet machine used for electric and hybrid cars showing 6 pole pairs, a maximum power of 80kW and a maximum speed of 12000 rpm is used for this analysis.
Abstract: Due to high range and continious power requirements in electric and hybrid cars the efficiency of electrical machines is one key parameter. In this paper ac copper loss effects, mainly circulating currents, are investigated. A permanent magnet machine used for electric and hybrid cars showing 6 pole pairs, a maximum power of 80kW and a maximum speed of 12000 rpm is used for this analysis. In general parallel strands of small diameter are used to facilitate handling tasks and to minimize skin and proximity effects. But in the design process feasible circulating currents due to unsuitable strand distribution are often not considered. Furthermore the wire distribution is very often unkown due to automatic winding technology. So the main topic of this study is to analyze the impact of the parallel wires' placement based on the described machine and to propose a measurement technique which could be used later to judge automatically wound machines containing unknown strand placement. For this study a good and bad case machine is built up by hand winding to clarify the influence. To realize high quality measurements both machines use exactly the same rotor, about 60 thermocouples and a specific current measurement system in each parallel strand of one phase. It is shown that the measured losses differ for more than 3kW, meaning an additional loss amount of 65% at 11000 rpm. Looking at the measured currents the frequency dependency can be clearly seen. About three times higher current load in some strands is identified compared to ideal wire distribution. So overheating and possible cases of failure are possible. The measured data is used to validate a common analytical as well as a transient FEA model. While the FEA matches the measurement almost perfectly the analytical approach delivers some deviations, especially in bad case conditions. At the end an outlook regarding the temperature dependent loss scaling is given. Addicted to the ac loss amount different scaling behaviors occur.
25 citations
References
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TL;DR: In this article, a multilayer winding carrying an alternating current, such as the windings illustrated in figures 1, 2, and 3, each layer of copper lies in the alternating magnetic field set up by the current in all the other layers.
Abstract: IN any multilayer winding carrying an alternating current, such as the windings illustrated in figures 1, 2, and 3, each layer of copper lies in the alternating magnetic field set up by the current in all the other layers. Eddy currents are set up in each layer in a direction to partly neutralize the magnetic intensities in the interior of the copper wire in each layer. As a result of the eddy-current losses in the copper, the effective resistance of the winding to the alternating current it carries may be many times its resistance to continuous currents.
103 citations
TL;DR: In this article, the authors discuss the more important causes of eddy currents in heavy conductors carrying alternating currents and surrounded on three sides by iron, and propose a method to identify the most important causes.
Abstract: The object of the present paper is the discussion of the more important causes of eddy currents in heavy conductors carrying alternating currents and surrounded on three sides by iron.
93 citations
64 citations
TL;DR: In this article, it is shown that a considerable proportion of the effective resistance of inductive coils when used at radio frequencies is caused by the eddy-currents set up in the wires of the coils by the alternating magnetic field in which they are situated, and that in extreme cases the alternating current resistance may amount to more than one hundred times the direct current resistance.
Abstract: It is well-known that a considerable proportion of the effective resistance of inductive coils when used at radio frequencies is caused by the eddy-currents set up in the wires of the coils by the alternating magnetic field in which they are situated, and that in extreme cases the alternating current resistance may amount to more than one hundred times the direct current resistance. It is therefore important to have reliable formulae for the eddy-current resistance of such coils in order to determine the conditions which will reduce the eddy-current losses to a minimum. The simplest case, that of a long straight cylindrical wire under the action of its own current, has been treated by Kelvin, Rayleigh, Heaviside, and others. The general effect is known as the “skin effect,” because the current tends to concentrate more and more upon the skin of the conductor as the frequency increases.
49 citations
TL;DR: In this article, the authors show how hyperbolic functions of complex angles may be applied to the solution of the problem of heat losses in rectangular conductors that are embedded in open slots.
Abstract: The principal object of this paper is to show how hyperbolic functions of complex angles may be applied to the solution of the problem of heat losses in rectangular conductors that are embedded in open slots. A certain knowledge of the functions themselves is presupposed. Inasmuch, however, as they are handled like trigometric functions of real angles?except in regard to the plus and minus signs?it is a simple matter to acquire the requisite technical skill to use them. The hyperbolic function of a complex angle, consisting as it does of a real and an imaginary part, may represent a vector?the real part being the component of the vector along the horizontal, and the imaginary part, component along the vertical. Thus, for example, A sinh (x + j x) represents a vector just as A e j ? A/?, A (cos ? + j sin ?) represent vectors. Considerable experience has shown that the vector method for handling a-c. problems is much superior to the original method in which simple trigonometric functions were used. With this lesson before us, it should require but little contact with the problem at hand to demonstrate the superiority of the vector method, even though it employs the possibly unfamiliar hyperbolic quantities. These hyperbolic vectors have been used for a number of years in the analysis of problems involving a-c. circuits, which have distributed inductance and capacitance, and have proved their usefulness.
27 citations