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 article, an extended phase shift (EPS) control in full-bridge CLLC resonant converters is proposed to improve the light-load efficiency of the converter.
Abstract: The capacitor–inductor–inductor–capacitor (CLLC) resonant converters are drawing more and more attention for their superiority in soft switching, wide output range, and symmetrically bidirectional operation. However, CLLC converter still suffers the problems of unsatisfactory voltage regulation and low efficiency under light-load conditions. Therefore, this article proposes an optimization for extended phase-shift (EPS) control in full-bridge CLLC resonant converter to improve the light-load efficiency. In order to study the relations between phase-shifts of EPS control and converter efficiency, a detailed circuit model is first established to solve the voltage gain, time-domain expressions of main circuit variables, and root mean square (rms) values of resonant currents. Then, a comprehensive loss evaluation is conducted by calculating and analyzing the main power losses, including conduction loss, switching loss, and core losses of magnetic components. Besides, zero voltage switching conditions of primary and rectifier switches are derived to define the range of soft switching. Finally, based on these analyses, the optimal combination of phase-shifts for EPS control is determined to achieve the maximum conversion efficiency of full-bridge CLLC converter at light-load conditions. The validity of the proposed optimized EPS control is verified on a 18∼25 V/400 V, 200 W GaN-based CLLC resonant converter prototype.
53 citations
TL;DR: In this paper, the authors presented a method to obtain an electric model for transformers and inductors, including both frequency and geometry effects in the windings, which can be linked with existing core models.
Abstract: This paper presents a method to obtain an electric model for transformers and inductors, including both frequency and geometry effects in the windings, which can be linked with existing core models. One-dimensional distributions for magnetic and electric fields are assumed, and from Maxwell's equations an equivalent electric circuit is easily obtained. This equivalent circuit has been included in analog simulators (Spice, AnalogWorkBench, Saber ...), and comparisons between measured and simulated results are shown, both in time domain and in AC sweep, which verify the model accuracy. The model described in this paper allows designers to deal with key issues in the design of high-frequency magnetic components (copper losses, leakage inductance, skin and proximity effects) by using analog simulators, which are usually more familiar to them than finite-element analysis tools.
53 citations
TL;DR: In this article, a lumped circuit model for a winding in a multi-winding transformer is derived for circuit simulation using electrical-network simulators, which is intended to be used in transformer models for circuit simulations.
Abstract: A lumped circuit model is derived for a winding in a multiwinding transformer. The model is intended to be used in transformer models for circuit simulation using electrical-network simulators. A hybrid (partly electrical, partly magnetic) modeling approach is adopted in which magnetic components are described using the capacitance-permeance analogy instead of the widespread resistance-reluctance analogy. The network correctly models energy storage and power dissipation due to DC series wire resistance and to eddy current losses, independent of the way of excitation of the winding (electrical and/or magnetic). All component values are frequency independent and are parameterized by geometrical parameters, winding data and material parameters. The mathematical continued-fraction approximation technique is applied to derive approximating circuits to model eddy current losses. A fourth-order circuit shows acceptably small errors up to a frequency of about a factor of 1500 above the frequency at which eddy-current losses become apparent. The model is applied in a six-layer two-winding transformer model. Calculations both in the frequency domain and in the time domain show good agreement with measurements.
53 citations
TL;DR: In this article, an improved modeling of transformer windings based on bacterial swarming algorithm (BSA) and frequency response analysis (FRA) has been discussed, with the purpose to accurately identify transform windings parameters using a model-based identification approach.
53 citations
TL;DR: This paper builds an analytical model of the transmitting coil, and then, optimizes the parameters of the coil by enlarging the quality factor, and derives an accurate analytical expression of ac resistance.
Abstract: A wireless power transfer system based on the weakly inductive coupling makes it possible to provide the endoscope microrobot (EMR) with infinite power. To facilitate the patients’ inspection with the EMR system, the diameter of the transmitting coil is enlarged to 69 cm. Due to the large transmitting range, a high quality factor of the Litz-wire transmitting coil is a necessity to ensure the intensity of magnetic field generated efficiently. Thus, this paper builds an analytical model of the transmitting coil, and then, optimizes the parameters of the coil by enlarging the quality factor. The lumped model of the transmitting coil includes three parameters: ac resistance, self-inductance, and stray capacitance. Based on the exact two-dimension solution, the accurate analytical expression of ac resistance is derived. Several transmitting coils of different specifications are utilized to verify this analytical expression, being in good agreements with the measured results except the coils with a large number of strands. Then, the quality factor of transmitting coils can be well predicted with the available analytical expressions of self- inductance and stray capacitance. Owing to the exact estimation of quality factor, the appropriate coil turns of the transmitting coil is set to 18–40 within the restrictions of transmitting circuit and human tissue issues. To supply enough energy for the next generation of the EMR equipped with a O9.5×10.1 mm receiving coil, the coil turns of the transmitting coil is optimally set to 28, which can transfer a maximum power of 750 mW with the remarkable delivering efficiency of 3.55%.
52 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