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Chapter 12

Magnetic Materials for Inductors and Transformers

Abstract

This chapter discusses the materials and physical characteristics of magnetic materials, such as ferrites and iron dust cores. Although not essential reading, it does contain useful information, even if the reader has no intention of making his own inductors or transformers. Topics, such as magnetic saturation and efficiency are discussed.

Keywords

ferrite

iron-dust core

torroid

E-core

bobbin

magnetic saturation

induction

remnance

coercivity

air gap

AL factor

Standard off-the-shelf transformers and inductors were described in Chapter 11. This chapter will describe magnetic materials and techniques for constructing custom transformers and inductors. If you have no interest in using anything other than standard off-the-shelf parts, this chapter can be skipped. The primary design requirement is to minimize losses, but to do this we have to consider copper losses, core losses, magnetic saturation, size, and construction. As this book is about designing light-emitting diode (LED) drivers, only the basics of magnetic materials will be given here. For more detail, the reader should consult specialist books on the subject.

An inductor can be made from a coil of wire, wound on a bobbin, and surrounded by a soft-magnetic core material. By soft, the meaning is that magnetization is easy and demagnetization occurs when the magnetizing force is removed. A hard-magnetic core is like a permanent magnet; it has high “remnance” (magnetic field remaining once the magnetizing force has been removed). Most magnetic materials have some remnance and it takes some reverse magnetic field to return the magnetic flux to zero; the field strength required to overcome any remnance, is called the “coercivity.” On a graph showing magnetic flux versus field strength, the curve follows an italic “S” shape. But when the magnetic field direction is reversed, the flux does not follow the same curve; it needs more field strength (more energy) to return to the same point and thus forms a “fat S” shape. The fatter the S: the higher the magnetizing losses.

The core can be rectangular or cylindrical in cross-section with two halves that separate to allow the bobbin to be inserted. When the inductor is assembled, two spring steel clips (or adhesive) hold the two halves of the core together. This form of inductor is suitable for values of a few microhenries up to about 1 H. Some cores are toroidal (doughnut shaped) and are formed as a single piece, so that special coil winding machines are needed to wind a coil around the core.

The advantage of making a custom inductor is that they can be made to any value. Remembering that inductance is proportional to the number of turns squared, the number of turns required is given by the simple formula:

N = \sqrt{\frac{L (nH)}{A L}}

$N = \sqrt{\frac{L (nH)}{A L}}$

. Here L is the required inductance in nanohenries and AL is the core’s inductance factor (nanohenries per turn). Each core type has an AL value determined by the core manufacturer, which will be given in the manufacturer’s datasheet or catalog. The AL factor is the inductance in nanohenries that will be produced for a single turn of wire.

The core’s AL value is related to the permeability of the magnetic material used. Different magnetic materials are used, depending on the frequency at which the inductor is operating. If a particular AL value is required, it can be obtained by removing some of the magnetic material from the center of the core, thus creating an air gap. Note, an air gap in the center of the core, rather than in the outer material, reduces the emission of magnetic fields because the outer material behaves like a shield. The air gap has a lower permeability than the ferrite material, so increasing the gap reduces the overall AL value. A typical core gap is 0.1–0.5 mm, although it may be larger or smaller depending on the magnetic material permeability and the required AL value. The larger the air gap, the higher the magnetizing force that be achieved without saturating the core.

The presence of an air gap in inductor and transformer cores makes them suitable for high-magnetic saturation levels. An example application for this is inductors in power factor correction (PFC) circuits, which have a discontinuous magnetizing force. In PFC circuits, the current is switched on and off at high frequency with zero current flow between each pulse. The amplitude of the current pulse is made to rise and fall in proportion to the instantaneous AC voltage, so the average current is sinusoidal. Thus the power factor is close to unity (true sine wave).

Transformer cores that have no air gap are prone to saturate easily; their AL is normally far higher than an inductor core made from a similar magnetic material, but with an air gap. Gap-less inductor cores are often used in forward converters, in which the secondary current flows at the same time as the primary current. There is no stored energy in a transformer used in a forward converter.

If the coupling between windings must be very close, bifilar winding is often used. A bifilar winding has two insulated strands of wire twisted together before winding. Trifilar and higher order windings use multiple strands. However, if high-voltage insulation is required between the windings, bifilar techniques cannot be used unless special winding wire with high-voltage insulation is available, for example, Rubadue wire.

Sometimes multiple winding strands are used to reduce the equivalent series resistance because at high-switching frequencies the skin effect must be considered. Remember that the skin effect forces current to flow through the outside surface of a conductor, so if insulated strands are used the effective surface area is very large. A type of winding wire with multiple twisted strands is called Litz wire; each strand has a thin polymer film surrounding the conductor, for insulation.

12.1. Ferrite Cores

Ferrite cores are available in many shapes and material types. These cores are quite brittle and can break if dropped or struck with a hard object. Ferrite is usually a compound made from magnesium and zinc, or from nickel and zinc. Most ferrites have very poor-electrical conductivity, which limits any eddy currents in the core.

Nickel–zinc ferrites are used in inductors intended for EMI filters because they have high losses at high frequency; the core absorbs most of the energy above 20 MHz, up to about 1 GHz.

Manganese–zinc cores have losses that rise above 10 MHz, but have little effect on signals above 80 MHz. This characteristic makes them almost useless for EMI filtering.

Manufacturers data should be studied for details of the switching losses and optimum switching frequency. Ferrite is less effective at very low or very high frequencies. Generally, frequencies in the range 10 kHz–1 MHz are suitable for ferrite cores.

12.2. Iron Dust Cores

Iron dust cores (also called iron powder cores) are sometimes made toroidal shaped. The iron dust is ferrous oxide and is mixed in with clay-like slurry, which sets when baked. The result is ceramic material with soft-magnetic properties and with high-magnetic saturation levels.

These cores are good for switching frequencies up to about 400 kHz. From about 10 MHz, up to 20 MHz, the core is very lossy. Above 20 MHz the core has little effect and so cannot be used in EMI-filtering applications.

12.3. Special Cores

Proprietary compounds are used to make special cores. An example is molypermalloy powder (MPP). This has the ability to operate with high-flux density of typically 800 mT, rather than 200 mT of conventional ferrite cores.

MPP cores are distributed air gap toroidal cores made from a 79% nickel, 17% iron, and 4% molybdenum alloy powder for the lowest core losses of any powder core material.

MPP cores possess many outstanding magnetic characteristics, such as high-electrical resistance (thus, low-eddy current losses), low hysteresis (magnetizing) losses, excellent inductance stability after high-DC magnetization or under high-DC bias conditions, and minimal inductance shift when subject to flux densities up to 2000 G (200 mT) under AC conditions.

12.4. Core Shapes and Sizes

For custom inductors and transformers, E-cores are popular. An E-core has two halves that look like a capital E. The center segment is designed to pass through the middle of a bobbin on which the windings are wound. This center segment can be machined to create an air gap, as shown in Fig. 12.1, to allow high-magnetic flux without saturation of the core.

Variations on E-cores are EF and EFD cores. The EFD core is shaped so that the center segment is thinner than the main body of the core, so that the bobbin has a rectangular cross section, rather than square.

Pot-cores have a round body with a central spigot, so that a round bobbin drops inside the cavity. However, the area on the circuit board is essentially square. This means that the ferrite core has less material and does not provide the maximum use of the space. These cores are rarely used except in a tuned filter, when an adjuster is provided in the central spigot.

Toroidal cores are good from an EMI point of view because the magnetic field is fairly well kept in the ferrite core; there are no “corners” in the core where magnetic flux is prone to leak out. However, toroid shapes are difficult to wind, as the wire must loop many times through the central hole. Special coil winders are available for toroidal cores. Magnetic saturation can be a problem, so MPP and iron powder tend to be used because they have the ability to carry a high-flux density. A toroidal core is shown in Fig. 12.2.

12.5. Magnetic Saturation

Magnetizing (core) losses are also present and are due to the energy required to make the magnetic fields in the core to align with each other. In a switching circuit these losses are continuous and can cause core heating. These losses increase rapidly if the magnetization is forced to operate outside its linear region. Generally, the magnetic flux density should be limited to about 200 mT (200 Wb/m²).

If an inductor or transformer has a large-discontinuous current flow, as in certain fly-back transformers and input inductors, the magnetic flux density may need to be lower than 200 mT. Ferrite core manufacturers recommend that flux variation due to ripple current or discontinuous mode operation should be limited to 50 mT. Inductors requiring the ability to handle high levels of flux variation sometimes use special cores with low losses at high-flux density, in which flux levels much greater than 50 mT are used. This allows a much smaller-inductor size.

The flux density is given by the equation:

B = \frac{L I}{N A_{e}}

$B = \frac{L I}{N A_{e}}$

. Here, L is the inductance, I is the peak current, N is the number of turns, and A_e is the effective core area. The inductance and peak current are calculated in the design of the LED driver circuit. We do not know the core area or the number of turns at this stage, but through iteration we can find something suitable.

The approach for choosing a suitable core is to select a core with a known effective area (A_e value), find the number of turns, and then calculate the maximum AL value that can be used with that size core. The number of turns can be found by transposing the previous equation:

N = \frac{L I}{B A_{e}}

$N = \frac{L I}{B A_{e}}$

. The equation for the maximum AL value is:

A L_{\max} = \frac{L \cdot 10^{9}}{N^{2}}

$A L_{\max} = \frac{L \cdot 10^{9}}{N^{2}}$

Cores are usually available with standard AL sizes. If a core is available with a slightly lower-AL value than the maximum previously calculated, it should be selected and then a new value for the number of turns should be used,

N_{1} = \frac{L I}{B A_{e}}

$N_{1} = \frac{L I}{B A_{e}}$

. However, if a lower-AL value is not available, a larger-core size with a higher-A_e value should be selected and the aforementioned process repeated. A simple spreadsheet can be created to make this process quick and simple.

12.6. Copper Losses

Copper loss is the term used to describe the energy dissipated by resistance in the wire used to wind a coil. In 99.9% of cases this wire will be made of copper, whose resistivity at 20°C is about 1.73 × 10^–8 Ω m. However, coils often have to operate above room temperature and will be heated by the operating losses in any case. The wire resistance at any temperature can be estimated from Table 12.1, developed by Mullard (now Philips).

Table 12.1

Wire resistance versus temperature

Temperature (°C)	Multiplying Factor
20	1.000
40	1.079
60	1.157
80	1.236
100	1.314

Unfortunately, the resistance of wire also increases as the frequency of signals passing through it increases. The phenomenon of the “skin effect” is when the magnetic field caused by the current flow tends to force the electrons to flow down the outside of the wire. An alternating magnetic field produced by the current in the wire induces an electric field, strongest at the center of the wire, which repels the electrons and forces them to the outside surface of the wire. Thus changes in current produces a force that opposes those changes, which is inductance on a small scale.

The skin depth is given in Table 12.2.

Table 12.2

Skin depth versus frequency

Frequency	Skin Depth (mm)
50 Hz	9.36
1 kHz	2.09
100 kHz	0.209
1 MHz	0.0662
10 MHz	0.0209

Fortunately, Terman has created a formula for a wire gauge (in millimeters) where the skin effect increases resistance by 10%, which is a nominal limit that allows reasonable losses:

$D = \frac{200}{\sqrt{f}} mm.$ $D = \frac{200}{\sqrt{f}} mm.$

For example, suppose we are operating at 100 kHz, then D = 0.63 mm. Using a larger-diameter wire than this does not give much benefit because the current will not be carried in the center of the wire. In fact, in an LED driver (or any PWM power supply) harmonics are present at many times the switching frequency. In the earlier case, a significant proportion of the signal will have a frequency of 300 kHz.

In some cases, it is necessary to suffer higher-copper losses that are desirable, to have a transformer of a reasonable size. The use of Litz wire may be justified (although it is expensive) if low-copper loss is essential at high-switching frequency.

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Table of Contents for Chapter 12: Magnetic Materials for Inductors and Transformers

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