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

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Fig. 2.42 Magnetic circuits. (a) closed, (b) open.

respect to materials testing, but also in the design of electric motors and generators and devices containing permanent magnets.

Suppose an iron ring of permeability mr (SI), circumferential length l, and cross section A is uniformly wound with n turns of wire carrying a current of i amperes (Fig. 2.42a). Then the field and the flux are given by

Combining these equations gives

This should be compared with the equation for the current i in a wire of length l, cross secton A, resistance R, resistivity r, and conductivity s ¼ 1=r, when an electromotive force e is acting:

The similarity in form between Equations 2.48 and 2.49 suggests the various analogies between electric and magnetic quantities listed in Table 2.3. The most important of these are the magnetomotive force (mmf ) ¼ (4p/10)ni (cgs), unit gilbert or oersted-centimeter; mmf ¼ ni (SI), unit ampere; and the reluctance ¼ 1/mA (cgs), ¼ 1/mrA (SI).

TABLE 2.3 Circuit Analogies

A magnetic circuit may consist of various substances, including air, in series. We then add reluctances to find the total reluctance of the circuit, just as we add resistances in series:

where l1, l2, . . . are the lengths of the various portions of the circuit, m1, m2, m3, . . . their permeabilities, and A1, A2, . . . their areas. [Linear dimensions are in centimeters for cgs, m for SI; m (cgs) is replaced by mr (SI); and the right-hand side must be multiplied by m0 for SI.) Similarly, if the circuit elements are in parallel, the reciprocal of the total reluctance is equal to the sum of the reciprocals of the individual reluctances.

The open magnetic circuit of Fig. 2.42b, consisting of an iron ring with an air gap, may be regarded as a series circuit of iron and air. Because the permeability of air is so small compared to that of iron, the presence of a gap of length lg greatly increases the reluctance

mA

where m is replaced by mr for SI units.

A value for m or mr of 5000 is typical of iron near the knee of its magnetization curve. If the ring has a mean diameter of 10 cm (circumference 31.4 cm) and a gap length of 1 cm, then the reluctance of the gapped ring is 160 times that of the complete ring, although the gap amounts to only 3% of the circumference. Even if the gap is only 0.05 mm (2 10 3 in), so that it is more in the nature of an imperfect joint than a gap, the reluctance is 1.8 times that of a complete ring. Since the magnetomotive force is proportional to the reluctance (for constant flux), the current in the winding would have to be 160 times as large to produce the same flux in the ring with a 1 cm gap as in the complete ring, and 1.8 times as large for the 0.05 mm gap.

These results may be recast in different language. To say that the current in the winding must be increased to overcome the reluctance of the gap is equivalent to saying that a current increase is required to overcome the demagnetizing field Hd created by the poles formed on either side of the gap. We can then regard the field due to the winding as the applied field Ha. If Ha (and f) are clockwise, as in Fig. 2.42b, Hd will be counterclockwise. This demagnetizing field can be expressed in terms of a demagnetizing coefficient Nd, which we will find to be directly proportional to the gap width. If Htr is the true field,

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But m 1 ¼ 4px ¼ 4pM=Htr, where x is the susceptibility. Substituting and rearranging, we find

The demagnetizing coefficient is thus given by 4p (lg=l). The field in the gap is Hg ¼ Bg, which, because of the continuity of lines of B, is equal to the flux density B in the iron, provided that fringing (widening) of the flux in the gap can be neglected.

Repeating in SI units,

but mr 21 ¼ x ¼ M/H, where x is the volume susceptibility. Substituting and rearranging, we find

The demagnetizing coefficient is given by lg/l. The field in the gap is Hg ¼ Bg/m0, which, because of the continuity of lines of B, is equal to the flux density B in the iron, provided that fringing (widening) of the flux in the gap can be neglected.

Returning to the ungapped ring, we may write Equation 2.46 as

Since l is the mean circumference of the ring, Hl is the line integral of H around the circuit, which we may take as another definition of magnetomotive force:

It is understood here that H is parallel to l, as is usually true. If not, we must write the

Þ

integral as H cosu dl, where u is the angle between H and dl. Although we have extracted Equation 2.55 from a particular case, it is quite generally true and is known as Ampere’s law: the line integral of H around any closed curve equals 4p=10 (cgs) or 1 (SI) times the total current through the surface enclosed by the curve. Thus, if l is the mean circumference of the ring in Fig. 2.42a, the total current through this surface is ni, so that

Hl ¼ (4p=10) ni (cgs) or Hl ¼ ni (SI). Ampere’s law often provides a simple means of

in agreement with Equation 1.10.

Magnetomotive force may also be defined as the work required to take a unit magnetic

pole around the circuit. Since the force exerted by a field H on a unit pole is simply H, the

Þ

work done in moving it a distance dl is H dl, and we again arrive at H dl as the magnetomotive force in the circuit. Pursuing the analogy with electricity still further, we may define the difference in magnetic potential V between two points as the work done in bringing a unit magnetic pole from one point to the other against the field, or

In a circuit, closed or open, composed of permanently magnetized material, flux exists even though the magnetomotive force is zero. Discussion of such circuits will be deferred to Chapter 14.

Although the analogy between magnetic and electric circuits is useful, it cannot be pushed too far. The following differences exist:

1.There is no flow of anything in a magnetic circuit corresponding to the flow of charge in an electric circuit.

2.No such thing as a magnetic insulator exists (except for some superconductors). Thus flux tends to “leak out” of magnetized bodies instead of confining itself to welldefined paths. This fact alone is responsible for the greater difficulty and lower accuracy of many magnetic measurements, compared to electrical.

3.Electrical resistance is independent of the current strength. But magnetic reluctance depends on the flux density, because m or mr varies with flux density B.

2.12.1Permeameter

In a permeameter a closed magnetic circuit is formed by attaching a yoke, or yokes, of soft magnetic material to the specimen in order to provide a closed flux path. A magnetizing winding is applied to the specimen or yoke, or both. A permeameter can be regarded as an electromagnet in which the gap is closed by the specimen.

Many types of permeameters have been made and used, distinguished by the size and shape of the sample, the relative arrangement of specimen, yoke, and magnetizing winding, and by the means of sensing the field. As an example, we may consider the Fahy Simplex permeameter shown in Fig. 2.43. The specimen, usually in the form of a flat bar, is clamped between two soft iron blocks, A and A0, and a heavy U-shaped yoke. The yoke is made of silicon steel (iron containing 2 or 3% Si), a common high-permeability material, laminated to reduce eddy currents. Its cross section is made large relative to

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Fig. 2.43 Fahy Simplex permeameter (sketch).

that of the specimen, so that its reluctance will be low. The lower the reluctance, the greater will be the effect of a given number of ampere-turns on the yoke in producing a field through the specimen.

A B-coil is wound on a hollow form into which the specimen can be easily slipped. If the current through the magnetizing winding is changed, the field H acting on the specimen will change. The resulting change in B in the specimen is measured by means of a fluxmeter connected to the B-coil. The only problem then is the measurement of H, and the same problem exists in any permeameter because none of them has a perfectly closed magnetic circuit. In the Fahy permeameter the field in the specimen is not the same as the field through the magnetizing coil because of leakage; therefore H cannot be calculated from the magnetizing current. Instead, H is measured by means of the H-coil, which consists of a large number of turns on a nonmagnetic cylinder placed near the specimen and between the blocks A and A0. This H-coil is an example of the Chattock potentiometer described on page 42. When the magnetizing current is changed, H is measured by another fluxmeter connected to the H-coil. In effect, the assumptions are made that the magnetic potential difference between the ends of the specimen in contact with the blocks is the same as that between the ends of the H-coil, and that the H-coil accurately measures this difference. This is found to be not quite correct, and rather low accuracy results unless the H-coil is calibrated by means of a specimen with known magnetic properties. All permeameters suffer from uncertainty in measuring the value of the field applied to the sample, which results in part from the fact that the field is not uniform over the length of the sample.

Other permeameters are designed for various shapes and sizes of samples, and may use other methods, such as a Hall probe, to measure the magnetic field. A related class of devices are single-sheet testers, used to measure power loss in samples of transformer and motor steels (see Chapter 13). A closed circuit can be made of interleaved strips of magnetic sheet; this is the basis of the Epstein frame test, which is also described in Chapter 13.

The properties of magnetically soft materials may be sensitive to strain, and this must be kept in mind when clamping a specimen of such material in a permeameter. Too much clamping pressure alters the properties of the specimen, and too little results in a magnetically poor joint and greater leakage.

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and the magnetic flux through the outer coil is

If the coils are connected so that the difference between the two coil signals is measured, the signal due to the sample magnetization disappears, and we are left with

BAo BAi ¼ HAo HAi ¼ H(Ao Ai) (cgs)

(2:59)

BAo BAi ¼ m0(HAo HAi) ¼ m0H(Ao Ai) (SI).

If Ao and Ai are known, integration of the difference signal gives a signal proportional to H. Alternatively, the quantity (Ao 2 Ai) may be determined by calibration in a known field. Equations 2.58 and 2.59 assume that each of the coils has the same number of turns. If this is not true, the number of turns in each coil must be included explicitly to give the area-turns product nxAx for each coil.

This method relies on that fact that the tangential component of field is the same just inside and just outside the sample, and on the assumption that the field is independent of radial position from the center of the sample out to the radius of the outer coil. The comments about drift control, zero offsets, etc., noted above in connection with measurements on ring samples apply here as well, with the added complication that both B and H signals must be integrated.

In samples with length-to-diameter ratios less than about 2, a drop in signal like that caused by the image effect (Fig. 2.41) appears. This is presumably due to non-uniform saturation of the pole tips, with saturation occurring first in the region where the flux from the sample enters the pole tips. This explanation is consistent with the observation that the apparent drop in magnetization occurs at lower fields when the sample saturation magnetization is high. The error fortunately does not influence measurements in the second quadrant, which are of greatest importance for permanent magnets, and can be eliminated by making the length-to-diameter ratio large enough.

Instead of using an analog electronic integrator and recording the integrated signal, it is possible to record the unintegrated voltage signal, either amplified or not, and then carry out the integration digitally in software. This is regularly done in the case of high-speed pulse magnetization, where the short time intervals give large values of dB/dt, but not in the case of slow measurements on bulk permanent magnet samples.

2.13SUSCEPTIBILITY MEASUREMENTS

The chief property of interest in the case of weakly magnetic substances (dia-, para-, and antiferromagnetic) is their susceptibility. The M vs H curve is linear except at very low temperatures and very high fields, so that measurements at one or two values of H are enough to fix the slope of the curve, which equals the susceptibility. Fields of the order of several kOe or several hundred kA/m are usually necessary in order to produce an easily measurable magnetization, and these fields are usually provided by an electromagnet. Because M is small, the demagnetizing field Hd is small, even for short specimens, and usually negligible relative to the large applied fields Ha involved. If a demagnetizing

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Fig. 2.46 Orientations of paraand diamagnetic rods in the field of an electromagnet.

If the field H has components Hx, Hy, Hz, then H2 ¼ H2x þ H2y þ H2z, and the force on the body in the x direction is

It is often necessary to correct for the effect of the medium, usually air, in which the body exists, because the susceptibility x of the body may not be greatly different from the susceptibility x0 of the medium. The force on the body then becomes

because motion of the body in the þx direction must be accompanied by motion of an equal volume of the medium in the 2x direction.

The two most important ways of measuring susceptibility are the Curie method (sometimes called the Faraday method) and the Gouy method.

In the Curie method the specimen is small enough so that it can be located in a region where the field gradient is constant throughout the specimen volume. The pole pieces of an electromagnet may be shaped or arranged in various ways to produce a small region of constant field gradient. Figure 2.47 shows one example. Alternatively, a set of small currentcarrying coils can be placed in the gap of an ordinary electromagnet to produce a local and controllable field gradient. The field is predominantly in the y direction and, in the region occupied by the specimen, Hx and Hz and their gradients with x are small. The variation of H2y with x is shown by the curve superimposed on the diagram, and it is seen that

dHy2 ¼ 2Hy dHy dx dx

is approximately constant over the specimen length. Equation (2.62) therefore reduces to