Cullity B.D. Introduction to Magnetic Materials. Second Edition (2008)

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Fig. 2.19 SQUID (superconducting quantum interference device) flux sensor.

2.6MAGNETIC MEASUREMENTS IN CLOSED CIRCUITS

Lines of magnetic induction B are continuous and form closed loops. The region occupied by these closed loops is called a magnetic circuit. Sometimes the flux follows a welldefined path, sometimes not. When the flux path lies entirely within strongly magnetic material, except possibly for a small amount of leakage flux, the circuit is said to be closed. If the flux passes partially through “nonmagnetic” material, usually air, the circuit is said to be open.

An important property of a closed and homogeneous magnetic circuit is that the material comprising it can be magnetized without the production of magnetic poles, and therefore without the production of any magnetic fields due to the material itself. As we shall see, this circumstance considerably simplifies the determination of the field H, which causes the magnetization.

The simplest example of a closed magnetic circuit is a ring, with a uniform circular or square or rectangular cross-section, magnetized circumferentially. We will now consider how the normal induction curve and hysteresis loop of a ring specimen can be determined. A search coil of N turns is wound directly on the ring, often over a thin layer of electrically insulating tape to protect the electrical insulation on the wire. This is called the secondary winding, and since it will carry no significant current, it can be made using the smallest practical wire size. If the material is known to be homogeneous and nondirectional, this winding need not extend around the entire circumference of the sample. However, not

Fig. 2.20 Arrangement for measuring the magnetic properties of a ring sample.

many samples can be reliably known to meet these conditions, so a complete circumferential winding is usually best. This winding should consist of a single layer if possible (and it usually is). Over this is placed a magnetizing winding of n turns. This is the primary winding. It must carry the magnetizing current, and its wire diameter must be chosen accordingly (see Fig. 2.20). The primary winding must be distributed uniformly around the sample circumference, and may consist of multiple layers. A current i through this winding subjects the material of the ring to a field H, given by

where C1 is defined at Equation 2.1. The quantity L is the circumference of the ring (cm for cgs, m for SI). Note that the field will be larger around the inside circumference and smaller around the outside circumference, as the value of L varies. A common recommendation is that

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where the Ds are the diameters of the sample ring. This ensures that the field is uniform within +5% over the volume of the ring sample. The value of L in Equation 2.15 is then taken as the mean circumference of the ring.

The general procedure is to vary the current through the primary winding and measure its magnitude (usually from the voltage drop across a low-value shunt resistor in series), while simultaneously integrating the output voltage from the secondary winding with a fluxmeter. The primary winding current can be converted directly to magnetic field by means of Equation 2.15, and the integrator output is proportional to changes in the flux density in the sample. The two voltages are plotted as x and y signals to give the hysteresis loop of the sample material. The plot may be produced directly using an x2y recorder, but more commonly the voltages are converted to digital values using an analog-to-digital (A2D) converter and a computer. The hysteresis loop may then be plotted on the computer screen and/or on a printer, using the software that controls the A2D converter or some other program. An ordinary spreadsheet program works very well. A complete setup for measuring and recording hysteresis loops is called a hysteresigraph, or sometimes a hysteresisgraph.

Before beginning the measurement, the fluxmeter controls should be adjusted for minimum drift, and the fluxmeter should be reset to give zero output. If only a complete hysteresis loop is needed, it is best to start the integration at the maximum field (þ or 2). The field is then varied from its maximum value, through zero, to its maximum value in the opposite direction, and then back to its original value. Use of a bipolar power supply allows the magnetizing current to be varied smoothly through zero; otherwise a reversing switch is required, with the direction of current flow reversed at zero current. The field sweep may be manually controlled, or controlled by a computer driving a programmable power supply. It may be desirable to decrease the rate of field change while traversing the steepest parts of the loop. Unless the loop shape is bizarre, it is normally sufficient to acquire about 50 data points for the entire loop, although with most software and hardware setups it is easy to acquire many more points.

If there is drift in the integrator during the measurement, the plotted loop will not close perfectly at the starting/ending tip. The usual practice is to correct for this by assuming the drift rate is constant throughout the measurement, and applying a linearly increasing (positive or negative) correction to each recorded point such that the plotted loop closes.

Since the zero setting of the fluxmeter will not in general coincide with the demagnetized state of the sample, the plotted loop will be displaced from zero in the y direction and a constant value must be added to (or subtracted from) each measured y value to center the loop about the x axis. Finally, it is necessary to convert the recorded voltages to values of field and flux density, using the dimensions of the sample, the value of the series resistor in the primary circuit, the number of turns in the two windings, and the calibration factor of the fluxmeter. Some or all of these corrections and calibrations may be made automatically in the software if routine measurements are being made on similar samples.

Note that this procedure measures the magnetic flux density B, not the magnetization M. Since it is not possible to apply very large circumferential fields to a ring sample, the method is generally limited to measurements on soft magnetic materials, in which B is large relative to H (or m0H ) and the distinction between B and 4pM (or m0M ) is not significant. However, the correction from B to M is easily made if necessary, using 4pM ¼ B H (cgs) or m0M ¼ B m0H (SI).

To measure the normal magnetization curve, we must start with a demagnetized sample at zero field, so that H ¼ 0 and B ¼ 0 simultaneously. As noted in Chapter 1, there are two ways to demagnetize a ring sample: thermal and cyclic. Thermal demagnetization is achieved by heating the sample above its Curie temperature and cooling in zero field. This is tedious at best, and furthermore once any significant field is applied to the sample, the demagnetized state is lost and can only be regained by another thermal cycle. So usually a demagnetized state is achieved by subjecting the sample to a series of decreasing positive and negative fields, as indicated in Fig. 1.16. The sample then traverses a series of smaller and smaller symmetrical hysteresis loops collapsing toward the origin. At H ¼ 0 and B ¼ 0, the fluxmeter is set to zero and the field increased to its maximum value to record the normal curve. Before the introduction of the electronic fluxmeter, when point by point readings were necessary, a series of symmetrical hysteresis loops with increasing maximum field could be measured, and the normal curve taken as the line joining the tips of the loops.

In principle, the thermally demagnetized state is not the same as the cyclically demagnetized state, and the resulting normal magnetization curves might differ. Thermal demagnetization is hardly ever used in practice, mainly because it is time-consuming but also because the thermally demagnetized state will rarely be achieved in the actual operation of a magnetic device, and so is of limited practical interest. Of course, the cyclically demagnetized state is also rarely achieved in working devices.

Sometimes minor hysteresis loops, in which the field limits do not correspond to magnetic saturation, and may not be symmetrical about H ¼ 0, are of interest. They are easy to record using the system described above. See Fig. 1.15.

The time to record a nominally dc hysteresis loop is generally a minute or less. Ac loops can be measured up to some limiting frequency; the limit may be set by the power supply, the A2D converter, or the eddy-currents in the material (see Chapter 12).

Ring specimens, although free from magnetic poles, have some disadvantages. As noted above, it is generally not possible to apply very high magnetic fields to the sample. Primary and secondary windings must be applied to each specimen to be tested, and this can be time-consuming. There are toroidal coil winding machines to speed this procedure. Some specimens cannot be formed into a satisfactory ring. For example, if a wire or rod is bent into a circle, there will be a significant gap at the joint. If the joint is welded, it is not in the same magnetic state as the rest of the ring and this can lead to erroneous results. Sheet material, on the other hand, is often quite satisfactory; rings can be stamped out, and if necessary a number of these can be stacked together to form a composite, laminated ring. Small ring samples of thin sheet material may be placed in a snugly fitting protective plastic cover, called a core box, before the windings are applied. Note that sheet material is usually magnetically anisotropic; it has different properties in directions at different angles to the direction in which the sheet was originally rolled. Therefore, measurements on rings cut or stamped from such sheets reveal only the average properties over the various directions in the sheet. Thin strip material may be coiled like a roll of masking tape to make a laminated ring. Such a sample is known as a tape-wound core.

A “ring” sample need not be circular, but may be cut in the form of a hollow square or other closed geometrical figure. This is appropriate, for example, when the sample is a single crystal, and the properties in a particular crystallographic direction are required. Such a sample is generally called a picture-frame sample.

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2.7DEMAGNETIZING FIELDS

Before considering magnetic measurements in open circuits, we must examine the nature of the fields involved. A magnetic field H can be produced either by electric currents or by magnetic poles. If due to currents, the lines of H are continuous and form closed loops; for example, the H lines around a current-carrying conductor are concentric circles. If due to poles, on the other hand, the H lines begin on north poles and end on south poles.

Suppose a bar sample is magnetized by a field applied from left to right and subsequently removed. Then a north pole is formed at the right end, and a south pole at the left, as shown in Fig. 2.21a. We see that the H lines, radiating out from the north pole and ending at the south pole, constitute a field both outside and inside the magnet which acts from north to south and which therefore tends to demagnetize the magnet. This selfdemagnetizing action of a magnetized body is important, not only because of its bearing

Fig. 2.21 Fields of a bar magnet in zero applied field. (a) H field, and (b) B field. The vectors in the center indicate the values and directions of B, Hd, and 4pM (cgs units) at the center of the magnet.

Fig. 2.22 Variation of the demagnetizing field along the length of a bar magnet.

on magnetic measurements, but also because it strongly influences the behavior of magnetic materials in many practical devices. We will therefore consider it in some detail.

The demagnetizing field Hd acts in the opposite direction to the magnetization M which creates it. In Fig. 2.21a, Hd is the only field acting, and the relation B ¼ H þ 4pM becomes

B ¼ 2Hd þ 4pM (cgs), or B ¼ m0(H þ M) becomes B ¼ m0Hd þ m0M (SI). The flux density B inside the magnet is therefore less than 4pM (m0M ) but in the same direction,

because Hd (m0Hd) can never exceed 4pM (m0M ) in magnitude. These vectors are indicated in Fig. 2.21, along with a sketch of the B field of the magnet. Note that lines of B are continuous and are directed from south to north inside the magnet. Outside the magnet, B ¼ H (cgs) or B ¼ m0H (SI) and the external fields in Fig. 2.21a and b are therefore identical. The magnet of Fig. 2.21b is in an open magnetic circuit, because part of the flux is in the magnet and part is in air.

As Fig. 2.21b shows, the flux density of a bar magnet is not uniform: the lines diverge toward the ends, so that the flux density there is less than in the center. This results from the fact that Hd is stronger near the poles, and Fig. 2.22 shows why: the dashed lines show the H field due to each pole separately, and the resultant curve has a minimum at the center.

The variation in induction along a bar magnet is easily demonstrated experimentally. A closely fitting but moveable search coil, connected to a fluxmeter, is placed around the magnet at a particular point and then removed to a distance where the field is negligible; the resulting deflection is proportional to B at that point. The distribution of B shown in Fig. 2.23a was measured on a steel bar magnet. Newer and better permanent magnets are more resistant to demagnetization, so the same experiment using an alnico magnet of almost the same length-to-area ratio gives a different result (Fig. 2.23b), but still shows the reduction in flux at the ends of the sample due to the demagnetizing field. Ferrite and rare-earth permanent magnets would show even less drop in flux, but these materials are not normally made as rods or bars magnetized lengthwise.

When a soft magnetic body is placed in a field, it alters the shape of that field. Thus, in Fig. 2.24, suppose that Fig. 2.24a is a uniform field, such as the field of a solenoid. It may be regarded as either an H field or a B field. The B field of a magnet in zero applied field is shown in Fig. 2.24b. The B field in Fig. 2.24c is the vector sum of the fields in Fig. 2.24a and b. The flux tends to crowd into the magnet, as though it were more permeable than the surrounding air; this is the origin of the term permeability for the quantity m. At points outside the magnet near its center, the field is actually reduced. The same general result

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Fig. 2.23 (a) Measured flux density vs position in a steel bar magnet. (b) Same for an alnico bar magnet. The two magnets had similar length/area ratios.

is obtained if the body placed in the field is originally unmagnetized, because the field itself will produce magnetization. Figure 2.24 applies to a material like iron, with m 1. The opposite effect occurs for a diamagnetic body: the flux tends to avoid the body, so that the flux density is greater outside than inside. (Because lines of B are continuous, the B lines of Fig. 2.24 must close on themselves outside the drawing. If the field in which the body was placed was generated by a solenoid, then the manner in which the lines close is suggested in Fig. 1.8.)

The extent to which a body, originally unmagnetized, disturbs the field in which it is placed depends on its permeability. For strongly magnetic materials (ferroand ferrimagnetic) the disturbance is considerable; for weakly magnetic materials it is practically negligible. Steel ships produce appreciable disturbance of the Earth’s magnetic field at a considerable distance from the ship, and the magnetic mines used in warfare make use of this fact. As the ship passes, the change in field at the position of the mine is sensed by some kind of magnetometer which then actuates an electrical circuit to activate the mine.

Fig. 2.24 Result of placing a magnetized body in an originally uniform field.

2.8MAGNETIC SHIELDING

If a high-permeability ring or cylinder is placed in a field, it tends to shield the space inside from the field, as suggested by Fig. 2.25a. The field lines tend to follow the magnetic material around the perimeter and emerge from the other side. The difficulty with this explanation of shielding is that it suggests that the part of the cylinder normal to the field plays a primary role in diverting the flux. Actually, a field normal to the center of a flat plate passes right through, undeviated, as shown in Fig. 2.25b. It is the sides of the cylinder parallel to the applied field that have the greatest effect. These become magnetized, with poles as shown in Fig. 2.25c, and they reduce the field inside the ring by exactly the same mechanism as that by which the bar magnet of Fig. 2.24 reduces the field in the region adjacent to its center. (The portions of the cylinder normal to the field acquire little magnetization, because of their very large demagnetizing factor. See below.) Two or more concentric thin cylinders, separated by air gaps, are more effective than one thick cylinder. Components of some electronic circuits and devices need to be shielded from external magnetic fields, and this is done by enclosing them in one or more thin sheets of a high-permeability material, usually a Ni–Fe alloy.