Transformers and Inductors for Power Electronics

Theory, Design and Applications

W.G. Hurley, W.H. Wölfle

• Publisher: Wiley
• Year: 2013
• Print ISBN: 9781119950578
• eBook ISBN: 9781118544679
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Power Resources
• Index: Technology & Engineering

Based on the fundamentals of electromagnetics, this clear and concise text explains basic and applied principles of transformer and inductor design for power electronic applications. It details both the theory and practice of inductors and transformers employed to filter currents, store electromagnetic energy, provide physical isolation between circuits, and perform stepping up and down of DC and AC voltages.

The authors present a broad range of applications from modern power conversion systems. They provide rigorous design guidelines based on a robust methodology for inductor and transformer design.  They offer real design examples, informed by proven and working field examples.

Key features include: 

• emphasis on high frequency design, including optimisation of the winding layout and treatment of non-sinusoidal waveforms
• a chapter on planar magnetic with analytical models and descriptions of the processing technologies
• analysis of the role of variable inductors, and their applications for power factor correction and solar power
• unique coverage on the measurements of inductance and transformer capacitance, as well as tests for core losses at high frequency
• worked examples in MATLAB, end-of-chapter problems, and an accompanying website containing solutions, a full set of instructors' presentations, and copies of all the figures.

Covering the basics of the magnetic components of power electronic converters, this book is a comprehensive reference for students and professional engineers dealing with specialised inductor and transformer design. It is especially useful for senior undergraduate and graduate students in electrical engineering and electrical energy systems, and engineers working with power supplies and energy conversion systems who want to update their knowledge on a field that has progressed considerably in recent years.

1

Introduction

In this chapter, we describe the historical developments that led to the evolution of inductance as a concept in electrical engineering. We introduce the laws of electromagnetism which are used throughout the book. Magnetic materials that are in common use today for inductors and transformers are also discussed.

1.1 Historical Context

In 1820, Oersted discovered that electric current flowing in a conductor produces a magnetic field. Six years later, Ampere quantified the relationship between the current and the magnetic field. In 1831, Faraday discovered that a changing magnetic field causes current to flow in any closed electric circuit linked by the magnetic field, and Lenz showed that there is a relationship between the changing magnetic field and the induced current. Gauss established that magnetic poles cannot exist in isolation. These phenomena established the relationship between electricity and magnetism and became the basis for the science of electromagnetism.

In 1865, Maxwell unified these laws in the celebrated form of Maxwell's equations, which established the basis for modern electrical engineering. He also established the link between phenomena in electromagnetics and electrostatics. Father Nicholas Joseph Callan, who was Professor of Natural Philosophy at the National University of Ireland, Maynooth, in the 1830 s, invented the induction coil. Alexander Anderson was Professor of Natural Philosophy at the National University of Ireland, Galway in the early 1900 s and gave his name to the Anderson Bridge for measuring inductance.

These individuals provide the inspiration for a textbook on magnetic design that focuses on the issues that arise in power electronics. Power electronics is an enabling technology for modern energy conversion systems and inductors and transformers are at the heart of these systems.

Figure 1.1 shows a straight conductor carrying a current, i. The presence of the magnetic field is detected by placing a freely-suspended magnet in the vicinity of the conductor. The direction of the magnetic field (a vector) is given by the direction in which the north pole of the search magnet points. It turns out that the magnitude of the magnetic field is constant on any circle concentric with the conductor, and its direction is tangential to that circle, given by the right hand rule – that is, a conventional (right-handed) cork screw will rotate in the direction of the magnetic field if it is driven in the direction of the current flow. It also turns out that the magnitude of the magnetic field is proportional to the current in the conductor and is inversely proportional to the radial distance from the conductor axis.

Figure 1.1 Magnetic field created by a current.

The magnetic field around a straight conductor is illustrated in Figure 1.2. The direction of the magnetic field as shown complies with the right hand screw rule. An alternative to the right hand screw rule for establishing the direction of the magnetic field created by the current is to point the thumb of your right hand along the conductor in the direction of the current flow, and your fingers will wrap themselves around the conductor in the direction of the magnetic field. The higher density of the lines near the conductor indicates a stronger magnetic field in this area.

Figure 1.2 Magnetic field around a current-carrying conductor.

The magnetic field around the current carrying conductor is described by two vector quantities: the magnetic flux density B and the magnetic field intensity H.

The magnetic field intensity H is best explained by Ampere's law, which expresses these observations about the current-carrying conductor in their most general form:

(1.1)

The closed contour C, the surface S and the normal vector are defined by convention: S is the surface enclosed by C and n is the unit vector normal to S. H is the magnetic field intensity in A/m and J_f is the current density in A/m². The quantity on the right hand side of Equation 1.1 is the current enclosed by the contour.

Figure 1.3 shows a coil with N turns in a magnetic field. The magnetic flux that links each turn of the coil is ϕ and the electromotive force (emf) induced in the coil is given by:

Figure 1.3 Conductor in a changing magnetic field.

(1.2)

This states that the induced electromotive force (emf) in a coil of N turns is proportional to the rate of change of the magnetic flux that links the coil. The negative sign indicates that current flow in the external circuit will create an opposing magnetic field.

In a more general form, Equation 1.2 may be stated as:

(1.3)

The integral in Equation 1.3 represents the flux linking the coil. The surface S and the normal vector are defined as before. The flux density B in Wb/m² or tesla is the flux per unit area inside the coil.

The magnetic field intensity H gives rise to a magnetic flux density B in a medium of permeability μ, so that:

(1.4)

The units for permeability are H/m and for free space μ₀ = 4π × 10^–7 H/m. For magnetic media, μ could be up to 10 000 times greater than μ₀. The permeability is usually presented as the product of μ₀ and the relative permeability μ_r.

(1.5)

Typically, relative permeability ranges from about 400 for ferrites used for power electronics applications to 10 000 for silicon steel that is used in power transformers at 50 Hz or 60 Hz. μ_r is taken as 1 for air. Permeability is treated in Section 1.5.

1.2 The Laws of Electromagnetism

In Maxwell's equations, t...

Table of contents

• Cover
• Title Page
• Copyright
• Dedication
• About the Authors
• Acknowledgements
• Foreword
• Preface
• Nomenclature
• Chapter 1: Introduction
• Section I: Inductors
• Section II: Transformers
• Section III: Advanced Topics
• Appendix A
• Index