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Transformers and Inductors for Electronics Applications


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Thayer School of Engineering

Dartmouth College

Selected Publications, by Topic

With notes. Topics:
Magnetics design.
Microfabricated magnetics.
Microprocessor power delivery.
Other topics.

Related links: Chronological list. Download experimental BibTeX file: dmcper.bib (read the notes at the top of the file.).

Magnetics Design

2D models for simulations and gap reluctance formulas

Using a 2D model for numerical modeling (i.e., finite-element analysis) of a 3D structure (such as an E-core) is not as straightforward as it sounds, as shown in “An Improved Two-Dimensional Numerical Modeling Method for E-Core Transformers .” The straightforward approach can lead to substantial errors. In the course of developing a better way to do this, we found that we needed better accuracy from an analytical approximation to the fringing reluctance around a gap. Although the literature contains exact formulas for the 2D situation, we needed to develop an accurate approximation for the actual 3D situation, and this can be found in the same paper. For more detail on any of this, you can download Anderson Hoke's complete thesis on the same topics.

Litz wire

"Optimal Choice for Number of Strands in a Litz-Wire Transformer Winding" gives general background and explains how to choose a strand number and diameter for the absolute minimum loss. Unfortunately, this minimum loss design is usually very expensive. Thus, "Cost-Constrained Selection of Strand Wire and Number in a Litz-Wire Transformer Winding." becomes very important. This second paper shows how to generate a curve showing the possible trade-offs between loss and cost. Both of these first two papers address only simple geometries such as standard transformer windings. With geometries in which 2- and 3-D fields are important, the “SFD” method described in "Computationally Efficient Winding Loss Calculation with Multiple Windings, Arbitrary Waveforms, and Two- or Three-Dimensional Field Geometry" can calculate loss accurately. Optimization, considering cost, for such geometries is addressed in "Optimization of a flyback transformer winding considering two-dimensional field effects, cost and loss." A general purpose version of this optimization technique has been implemented in a free CAD tool that is detailed in "Easy-To-Use CAD Tools for Litz-Wire Winding Optimization," and is available from our software page. The optimization software is a great way to reduce the cost of litz wire. Another approach is to skip the insulation on individual strands, as analyzed in "Stranded Wire With Uninsulated Strands as a Low-Cost Alternative to Litz Wire.”

See also inductor design for low AC resistance.

Inductor design for low AC resistance

The effect of an air gap used in an inductor on the field distribution can sometimes result in disastrous winding AC resistance. One mitigation strategy is to use a low-permeability "distributed gap" to avoid the high flux one gets near a single lumped gap. This requires special materials, but we discuss a popular alternative using multiple small gaps in conventional material, in "The Quasi-Distributed Gap Technique for Planar Inductors--Design Guidelines". We developed a set of reasonably simple equations that can be used to predict ac resistance in a wide range of planar inductors, including ones with single gaps, and set of even simpler design guidelines.

For wire-wound, as opposed to planar, inductors, the quasi-distributed gap technique can also be used. However, in "Optimization of Shapes for Round Wire, High Frequency Gapped Inductor Windings" we show that a less expensive alternative has better performance. The position of the turns in the winding window, as well as the number of strands (in litz wire) or diameter of wire, can be optimized to result in lower loss than that in an ideal distributed-gap inductor. In "Analytical Method for Generalization of Numerically Optimized Inductor Winding Shapes" we show how to generalize the results to from one set of numerical optimizations to make them useful in any design with the same geometry. These first two papers do not explicitly consider cost; in "Analysis of Minimum Cost in Shape-Optimized Litz-Wire Inductor Windings," we analyze the effect of cost on the shape optimization. The use of shape optimized windings is further examined in "Optimal Core Dimensional Ratios for Minimizing Winding Loss in High-Frequency Gapped-Inductor Windings."

High-Frequency Winding Loss Analysis

For a given winding and current, losses increase with frequency. The best known cause of this is skin effect, but proximity effect can cause losses to increase dramatically even when the wire is small compared to skin depth. In most cases, optimum designs fall in this intermediate frequency range. This allows simplifications that allow relatively easy analysis of complex situations such as 2- or 3-D field geometries and non-sinusoidal waveforms.

Analysis of winding loss with non-sinusoidal currents in simple situations can be accomplished using an "effective frequency," as discussed in the appendix of "Optimal Choice for Number of Strands in a Litz-Wire Transformer Winding ." This has the advantage of allowing the direct use of relatively simple optimization techniques that were developed for sinusoidal waveforms. For situations involving different waveforms in different windings or 2- or 3-D geometries, the "SFD" method described in "Computationally Efficient Winding Loss Calculation with Multiple Windings, Arbitrary Waveforms, and Two- or Three-Dimensional Field Geometry" is required. Optimization, considering cost, for such geometries is addressed in "Optimization of a flyback transformer winding considering two-dimensional field effects, cost and loss ."

When the skin depth become small compared to the wire diameter, loss analysis becomes more complex. The Dowell method is the most common method used that covers this higher frequency range. It is an approximate method. Attempts have been made to use an exact Bessel function solution for higher accuracy. However, the approximation involved in the use of Bessel functions leads to even greater error. But since the Dowell gives 5% error at low frequencies and up to 60% error at high frequencies, an improvement is needed. We developed a more accurate equation that gives predictions of proximity-effect loss that are accurate to within a few percent over a wide range of conditions; the version of this model presented in "Simplified High-Accuracy Calculation of Eddy-Current Loss in Round-Wire Windings" is easier to use than the version presented in the original paper, "An Improved Calculation of Proximity-Effect Loss in High-Frequency Windings of Round Conductors", although the original is still useful as it supplies additional background and evaluation of other methods.

Our method calculates the loss based on the magnetic field. For simple 1-D geometries, the field is easy to calculate, as described in the first paper listed above. For 2-D geometries, a 2-D field calculation is also needed. This field calculation is complicated by the effect of eddy currents in the wire on the field. From our loss model, we have also developed "A Two-Dimensional Equivalent Complex Permeability Model of Round-Wire Windings" which simplifies the required field calculations. This model is applied to litz wire in "An Equivalent Complex Permeability Model for Litz-Wire Windings".

Microfabricated Magnetics

Core Loss in Ferrites with Arbitrary Flux Waveforms

Core loss in ferrites for power electronics applications is conventionally predicted using the Steinmetz equation or manufacturers' datasheet plots. Unfortunately, the Steinmetz equation and manufacturers' data are based only on tests with sinusoidal waveforms. But many power electronics circuits use other waveforms. In "Improved Calculation of Core Loss With Nonsinusoidal Waveforms" we review existing methods, discuss their limitations, and introduce a new method that overcomes some of the limitations. The method is substantially improved in "Accurate Prediction of Ferrite Core Loss with Nonsinusoidal Waveforms Using Only Steinmetz Parameters." The new method is highly practical, requiring no new characterization of a material beyond the Steinmetz equation parameters that are often provided by the core manufacturer.

Microprocessor Power Delivery

Microfabricated Magnetics for Microprocessor Power Delivery

A brief two-page overview of this work is available in "Thin-Film Inductor Designs and Materials for High-Current Low-Voltage Power."

Microprocessor power requirements are rapidly getting very difficult to meet. Not only are power levels increasing, but voltages are dropping, driving currents up even faster than power. Particularly difficult is maintaining a tightly regulated voltage despite rapid changes in load current. High switching frequencies and miniaturized components that can be placed very close to the load can overcome these challenges. We are developing high-frequency (8 MHz) power inductors fabricated by thin-film deposition and photolithography. They are described in "Design of Microfabricated Inductors for Microprocessor Power Delivery" and "Converter and Inductor Design for Fast-Response Microprocessor Power Delivery", and in the first and second papers titled "Fabrication of Thin-Film V-Groove Inductors Using Composite Magnetic Materials." Our first measured results are described in "Measured Electrical Performance of V-Groove Inductors for Microprocessor Power Delivery." One of the new high-frequency magnetic materials for this application is described in "Evaporatively Deposited Co-MgF2 Granular Materials for Thin-Film Inductors."

Circuit Design for Microprocessor Power Delivery

In "Coupled-Inductor Design Optimization for Fast-Response Low-Voltage DC-DC Converters," we show how magnetic coupling can be used to circumvent the tradeoff between efficiency and fast response that otherwise constrains circuit designs for the challenging new requirements of microprocessor power delivery (described in the section above). Circuits for this application are also discussed in "Converter and Inductor Design for Fast-Response Microprocessor Power Delivery".

Other Topics

Capacitor Modeling

Conventionally, non-idealities in capacitors are models by adding ESR (effective series resistance) and ESL (effective series inductance). In "Physically-Based Distributed Models for Multi-Layer Ceramic Capacitors," we show that ESL is a poor approximation to the behavior of an actual capacitor and we develop more accurate models. The poster from this presentation is also available. Earlier papers on this work address film capacitors as well as ceramic capacitors: "Capacitors with Fast Current Switching Require Distributed Models" and "Improved Distributed Model for Capacitors in High-Performance Packages."

Power Electronics Education

The tools and movies discussed in “Three-Dimensional Animations of Power-Electronics Circuits Visualize Voltage and Current.” are available on the animation Web site.

For comments or questions e-mail: Charles.R.Sullivan@dartmouth.edu.
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