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.

Effects of eddy currents in transformer windings

The one well-known one-dimensional method for calculating the AC resistance of multilayer transformer windings contains a built-in orthogonality which has not been reported previously. Orthogonality between skin effect and proximity effect makes a more generalized approach for the analytical solution of AC resistance in windings possible. This includes a method to calculate the AC resistance of round conductor windings which is not only convenient to use, but gives more accurate answers than the basic one-dimensional method because the exact analytical equations for round conductors can be used. >

Improved analytical modeling of conductive losses in magnetic components

AC losses due to nonsinusoidal current waveforms have been found by calculating the losses at harmonic frequencies when the Fourier coefficients are known. An optimized foil or layer thickness in a winding may be found by applying the Fourier analysis over a range of thickness values. This paper presents a new formula for the optimum foil or layer thickness, without the need for Fourier coefficients and calculations at harmonic frequencies. The new formula requires the RMS value of the current waveform and the RMS value of its derivative. It is simple, straightforward and applies to any periodic waveshape.

Optimizing the AC resistance of multilayer transformer windings with arbitrary current waveforms

The two best-known methods for calculating high-frequency winding loss in round-wire windings-the Dowell method and the Ferreira method-give significantly different results at high frequency We apply 2-D finite-element method (FEM) simulations to evaluate the accuracy of each method for predicting proximity-effect losses We find that both methods can have substantial errors, exceeding 60% The Ferreira method, which is based on the exact Bessel-function solution for the eddy current in an isolated conducting cylinder subjected to a time-varying magnetic field, is found to be most accurate for loosely packed windings, whereas the Dowell method, which approximates winding layers comprising multiple turns of round wire with a rectangular conducting sheet, is most accurate for closely-packed windings To achieve higher accuracy than is possible with either method alone, we introduce a new formula, based on modifying the Dowell method Parameters in the new formula are chosen based on fitting our FEM simulation data By expressing the results in terms of normalized parameters, we construct a model that can be used to determine proximity-effect loss for any round-wire winding with error under 2%

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An improved calculation of proximity-effect loss in high-frequency windings of round conductors

This study presents an approximate model for multi-strand wire winding, including litz-wire winding. The proposed model is evaluated using Dowell's equation. The model takes into account the existence of proximity effect within the litz-wire bundle between the strands and between the bundles, as well the skin effect. The expressions for optimum strand diameter and number of strands at which minimum winding AC resistance is obtained for the litz-wire windings are derived. The boundary frequency between the low-frequency and the medium-frequency ranges are given for both litz-wire and solid-round wire inductors. Hence, the low-frequency ranges of both wire windings are determined. It is shown that litz-wire is better than the solid wire only in specific frequency range. The model has been verified by the measurements, and the theoretical results were in good agreement with those experimentally measured. Comparison of the theoretical predictions of the proposed approximate litz-wire model with models proposed in other publications and with experimental results is given.

Winding resistance of litz-wire and multi-strand inductors

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.

Effective resistance to alternating currents of multilayer windings

The features or characteristics of the corona around wires brought out by oscillograms of the charging and leakage currents may be thus summarized: 1. The distortion in the current wave form, which results from copious ionization at the time of the voltage peak, is observed simultaneously with the hissing sound and bluish discharge from the wire. 2. At air pressures below 20 cm. of mercury, copious ionization,?as indicated by the hump on the current oscillograms,?sets in at a lower voltage during the half cycle for which the wire is the cathode than during the half cycle for which it is the anode. 3. From the oscillograms thus far obtained at atmospheric pressure it has been impossible to determine whether there is any difference in the voltage at which the distortion appears in the two half cycles, at pressures near atmospheric. 4. The boundary of the ionized region does not gradually expand outward from the wire with the increase in voltage during the half cycle. The loss of insulating properties in the air surrounding the wire must be conceived of as taking place with extreme suddenness. The evidence of this is the oscillation in the current shown on the oscillograms. 5.

An Oscillograph Study of Corona

(1) It is pointed out that by the use of a current transformer having a primary to secondary current ratio of the order of 1 to 100, oscillograms can be obtained of the charging current of a single high-tension insulator or of a few feet of high-tension transmission line; that is, oscillograms of currents of the order of 0.1 to 0.5 milliampere may be obtained. (2) The drawings and specification are given for a transformer for this purpose. (3) The transformer relations are discussed; the methods of determining the transformer constants are outlined, and the performance of transformers constructed in accordance with the specifications is determined. (4) A series of oscillograms is given to illustrate some of the applications of the transformer, such as to the study of corona, high-tension insulators, and leakage currents in evacuated lamps.

http://ieeexplore.ieee.org/iel7/6530810/6661271/06661289.pdf

A Milliampere Current Transformer

The proximity effect, usually considered as undesirable and wasteful, may be used to concentrate heating currents in predetermined strips of conducting bodies. Both the current density and shape of the strip may be so controlled as to make this method valuable in a number of heating processes.

http://ieeexplore.ieee.org/iel5/6413714/6429924/06429933.pdf

Concentration of heating currents

A power amplifier may be so connected to the circuit supplying the power to be amplified that it introduces terms into the equation for the current in the power circuit which subtract from the resistance terms. Upon this fact depends the operation of the device as a generator of sustained oscillations, as a regenerative amplifier, and as a resistance neutralizer. In the treatment of oscillating audion or triode circuits this fact has been pointed out by a number of writers but the more general significance of these subtractive terms has never been discussed. This paper points out that, by making use of the idea of resistance neutralization, circuits and systems having all the properties of low-resistance systems and also a number of other unique properties may be obtained. The first part of the paper derives and discusses the current and power relations which obtain in circuits having a resistance neutralizer associated with them. One of the things brought out is that not only does the neutralizer supply power to the circuit but it also causes the generator or source of driving e. m. f. to furnish more power to the circuit than it would were the neutralizer not present. Thus if the source of driving e. m. f. is the impinging of electromagnetic waves upon a wireless antenna the neutralizer not only amplifies the power received but it actually causes the impinging waves of the correspondent station to give up more power to the antenna circuit while it causes the waves from detuned station to give up less power to the receiving circuit. The power relations cited above while important are not the most important results obtained by resistance neutralization. At resonance, the power delivered to a wireless receiving system by impinging waves is inversely proportional to the net resistance, while the power received from detuned stations is practically independent of the net resistance. The neutralizer, by lowering the net resistance of the receiving system, thus causes the ratio of signal to interfered power to increase. In a simple series antenna circuit the ratio of signal to interferent power, where the interferent source may be either atmospheric strays or interferent station, is a function of (L/R) n where n is a positive number. The neutralizer thus increases the selectivity of the receiving station against all types of disturbances. A general physical argument is given (paragraph 7) to show that a triode may be made to function as a resistance neutralizer. This physical argument is illustrated by a mathematical treatment of the steady and transient states for a particular method of associating the triode with the circuit in which neutralization is desired. This mathematical treatment shows that under all conditions both in the steady and transient state, the neutralizer circuit functions so as to reduce the net resistance of the circuit of interest to some predetermined value. A numerical example is given to illustrate the mathematical theory. In this particular circuit the ratio of the signal to interferent power in the steady state is increased 200 fold by the insertion of the neutralizer. Section (II) of the paper treats of the optimum conditions obtaining in receiving circuits containing a resistance neutralizer, and gives relations for the designing of an antenna and its circuits so as to obtain the maximum selectivity and maximum power abstraction from impinging waves. The last topic discussed (Section 12) deals with a circuit which neutralizes resistance for a narrow range of frequencies near a desired frequency and introduces resistance and reactance into the circuit for frequencies removed fom this band of frequencies. A numerical example is given to illustrate the theory. The performance curves of the neutralizer are given by Figs. 12 and 13. Fig. 14 shows the ratio of the power received at resonance to the power received from a detuned station by a given antenna circuit; first, without a neutralizer, second, with a pure resistance neutralizer, and third, with a selective neutralizer. For example, if the interferent source is detuned by 3 per cent the circuit without the neutralizer gives practically 1 for the ratio of signal to interferent power; with pure resistance neutralization the ratio is about 1300; with the selective neutralizer the ratio is increased to 3000.

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Edward Bennett

Papers

Effective resistance to alternating currents of multilayer windings

An Oscillograph Study of Corona

A Milliampere Current Transformer

Concentration of heating currents

Resistance neutralization: An application of thermionic amplifier circuits