FUNDAMENTALS OF MAGNETIC THEORY Basic Laws of Magnetic Theory Magnetic Materials Magnetic Circuits References FAST DESIGN APPROACH INCLUDING EDDY CURRENT LOSSES Fast Design Approach Examples Conclusions Appendix 2.A.1: Core Size Scale Law for Ferrites in Non-Saturated Thermal Limited Design Appendix 2.A.2: Eddy Current Losses for Wide Frequency Appendix 2.A.3: MathCAD Example Files References SOFT MAGNETIC MATERIALS Magnetic Core Materials Comparison and Applications of the Core Materials in Power Electronics Losses in Soft Magnetic Materials Ferrite Core Losses with Non-Sinusoidal Voltage Waveforms Wide Frequency Model of Magnetic Sheets Including Hysteresis Effects Appendix 3.A: Power and Impedance of Magnetic Sheets References COIL WINDING AND ELECTRICAL INSULATION Filling Factor Wire Length Physical Aspects of Breakdown Insulation Requirements and Standards Thermal Requirements and Standards Magnetic Component Manufacturing Sheet References EDDY CURRENTS IN CONDUCTORS Introduction Basic Approximations Losses in Rectangular Conductors Quadrature of the Circle Method for Round Conductors Losses of a Current Carrying Round Conductor in 2-D Approach Losses of a Round Conductor in a Uniform Transverse AC Field Low Frequency 2-D Approximation Method for Round Conductors Wide Frequency Method for Calculating Eddy Current Losses in Windings Losses in Foil Windings Losses in Planar Windings Appendix 5.A.1: Eddy Current 1-D Model for Rectangular Conductors Appendix 5.A.2: Low Frequency 2-D Models for Eddy Current Losses in Round Wires Appendix 5.A.3: Field Factor For Inductors References THERMAL ASPECTS Fast Thermal Design Approach (Level 0 Thermal Design) Single Thermal Resistance Design Approach (Level 1 Thermal Design) Classic Heat Transfer Mechanisms Thermal Design Utilizing a Resistance Network Contribution to Heat Transfer Theory of Magnetic Components Transient Heat Transfer Summary Appendix 6.A: Accurate Natural Convection Modeling for Magnetic Components References PARASITIC CAPACITANCES IN MAGNETIC COMPONENTS Capacitance Between Windings: Inter Capacitance Self-Capacitance of a Winding: Intra Capacitance Capacitance Between the Windings and the Magnetic Material Practical Approaches for Decreasing the Effects of Parasitic Capacitances References INDUCTOR DESIGN Air Coils and Related Shapes Inductor Shapes Typical Ferrite Inductor Shapes Fringing in Wire-Wound Inductors with Magnetic Cores Eddy Currents in Inductor Windings Foil Wound Inductors Inductor Types Depending on Application Design Examples of Different Types of Inductors Fringing Coefficients For Gapped-Wire-Wound Inductors Analitical Modeling of Combined Litz Wire-Full Wire Inductors References TRANSFORMER DESIGN Transformer Design in Power Electronics Magnetizing Inductance Leakage Inductance Using Parallel Wires and Litz Wires Interleaved Windings Superimposing Frequency Components Superimposing Modes References OPTIMAL COPPER/CORE LOSS RATIO IN MAGNETIC COMPONENTS Simplified Approach Loss Minimization in the General Case Loss Minimization Without Eddy Current Losses Loss Minimization Including Low-Frequency Eddy Current Losses Summary Examples References MEASUREMENTS Introduction Temperature Measurements Power Losses Measurements Measurement of Inductances Core Loss Measurements Measurement of Parasitic Capacitances Combined Measuring Instruments References APPENDIX A: RMS VALUES OF WAVEFORMS Definitions RMS Values of Some Basic Waveforms RMS Values of Common Waveforms APPENDIX B: MAGNETIC CORE DATA ETD Core Data (Economic Transformer Design Core) EE Core Data Planar EE Core Data ER Core Data UU Core Data Ring Core Data (Toroid Core) P Core Data (Pot Core) PQ Core Data RM Core Data APPENDIX C: COPPER WIRES DATA Round Wire Data American Wire Gauge Data Litz Wire Data APPENDIX D: MATHEMATICAL FUNCTIONS References INDEX

Inductors and Transformers for Power Electronics

The squared-field-derivative method for calculating eddy-current (proximity-effect) losses in round-wire or litz-wire transformer and inductor windings is derived. The method is capable of analyzing losses due to two-dimensional and three-dimensional field effects in multiple windings with arbitrary waveforms in each winding. It uses a simple set of numerical magnetostatic field calculations, which require orders of magnitude less computation time than numerical eddy-current solutions, to derive a frequency-independent matrix describing the transformer or inductor. This is combined with a second, independently calculated matrix, based on derivatives of winding currents, to compute total AC loss. Experiments confirm the accuracy of the method.

Computationally efficient winding loss calculation with multiple windings, arbitrary waveforms, and two-dimensional or three-dimensional field geometry

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

The momentum toward high efficiency, high frequency, and high power density in power supplies limits wide use of conventional wire-wound magnetic components This paper gives an overview of planar magnetic technologies with respect to the development of modern power electronics The major advantages and disadvantages in the use of planar magnetics for high-frequency power converters are covered, and publications on planar magnetics are reviewed A detailed survey of winding conduction loss, leakage inductance, and winding capacitance for planar magnetics is presented so power electronics engineers and researchers can have a clear understanding of the intrinsic properties of planar magnetics

Overview of Planar Magnetic Technology—Fundamental Properties

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

The paper compares two methods for ohmic losses calculation in switch-mode power supply transformers: Dowell's 1D approach and 2D finite element method simulations (validated by measurements). Dowell's method underestimates losses at the fundamental frequency for foil windings, whereas harmonic losses are generally overestimated (depending on conductor type). More significant differences are found with high leakage inductance transformers.

Ohmic losses calculation in SMPS transformers: numerical study of Dowell's approach accuracy

A new formula aimed at calculating ohmic losses in switch-mode power supply (SMPS) transformers is presented. It is based on intensive two-dimensional (2-D) finite element method (FEM) simulations, the results of which have been summarized in a closed-form formula following a "semi-empirical" approach. The main benefit of this new formula, specifically intended for industrial designers, is to combine the precision of 2-D models with the ease-of-use and speed of calculation of one-dimensional (1-D) models, on the whole frequency range. It accurately covers cases where the classical Dowell's formula significantly underestimated the losses, specifically those with significant edge effect in foil windings. Experimental validation and discussion of accuracy is provided. At the moment, the formula is only valid for one layer of foil located between a zero and a maximum of the magnetomotive force but a similar approach could be applied with success to other types of windings. Furthermore, the analytical expression proposed in the article, based on Maxwell equations, can be used as a stand-alone tool to model the real behavior of any type of winding. More accurate understanding of the 2-D field is also possible thanks to the direct link established between the losses and the geometrical data of the winding.

A closed-form formula for 2-D ohmic losses calculation in SMPS transformer foils

In design of high-frequency magnetic devices, various approaches have been used to provide calculation tools that are able to predict winding losses, including skin effect and proximity effect, in their two-dimensional (2-D) aspects. Up to now, these approaches have been mainly focused on giving quantitative results for specific cases. Because the final purpose of designers is not to perform single calculations, but to obtain optimal designs, we have looked for a complementary approach providing more general 2-D guidelines and understanding of the phenomena. In this paper, we examine the case of a foil winding undergoing edge effect, a typical situation where classical one-dimensional theories underestimate the real losses. Using a 2-D closed-form formula we developed in a previous work and the associated database of 400 2-D numerical models, we systematically analyze the dependence of the 2-D field and the corresponding extra losses with the geometrical design variables describing the dimension and position of the winding.

Two-dimensional analysis of the edge effect field and losses in high-frequency transformer foils

A new closed-form formula aimed at calculating ohmic losses in switch-mode power supply transformers is presented. It is based on intensive 2D finite element method simulation. Only valid for one-layer-foil windings at the moment, it accurately covers cases where the classical Dowell's formula significantly underestimated the losses.

J.-P. Schauwers

Papers

Ohmic losses calculation in SMPS transformers: numerical study of Dowell's approach accuracy

A closed-form formula for 2-D ohmic losses calculation in SMPS transformer foils

Two-dimensional analysis of the edge effect field and losses in high-frequency transformer foils

A closed-form formula for 2D ohmic losses calculation in SMPS transformer foils

Eddy current losses in SMPS transformers: a full-frequency-range review of 2D effects inside the windings