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

over a very wide range. Transformers are a necessary part of electric power grids around the world.

The alternating flux induces an emf, not only in the secondary coil, but also in the core itself and, if the core is a conductor, this emf will cause eddy currents. At a time when the flux is increasing from left to right in the top leg, the eddy currents in the section AA will have the direction shown in Fig. 13.1b, i.e., the direction which will produce an eddycurrent field from right to left, opposing the main field. Eddy currents are objectionable, not only because they decrease the flux, but also because they produce heat, a direct power loss, proportional to i2R, where i is the eddy current and R the resistance of its path. The primary way to decrease this loss is to make the core out of thin sheets, or laminations, as in Fig. 13.1c, rather than from a solid piece. If these sheets are electrically insulated from one another, the eddy currents are forced to circulate within each lamination. The path length in each lamination is now shorter, thus decreasing R, but the cross-sectional area A of the path is also reduced. The induced emf e is therefore reduced, according to Equation 12.1, and the net effect is a decrease in the current i and in the eddy-current power loss (¼ei ¼ i2R). The effect is a strong one (Problem 13.3), and laminated construction is standard for the cores of transformers, motors, and generators made from metallic (conducting) materials.

How thin should the laminations be made? We saw in Section 12.2 that, when a body is exposed to an alternating field, eddy currents are generated which partially shield the interior of the body from the field. Thus the value of H and hence B inside the body can be much less than the value at the surface. This phenomenon is referred to as the skin effect, which is the confinement of the flux to the surface layers of a body at a “high” frequency. This effect can be very strong in a material like transformer steel at ordinary power frequencies (50 or 60 Hz), which are normally considered “low.” Figure 13.2 shows that the flux density B at the center of a sheet of transformer steel containing 3.25% silicon, 1.6 mm or 161 inch thick, is only about half the value at the surface at 60 Hz. Extrapolating the data of Fig. 13.2 to a sheet 6.4 mm or 14 inch thick, one can conclude that the flux at its center is virtually zero. The material at the center of such a sheet is contributing nothing to

Fig. 13.2 Calculated amplitude of alternating flux in transformer steel sheets of various thicknesses as a fraction of the value at the surface. Frequency ¼ 60 Hz, skin depth d ¼ 0.5 mm, sheet thicknesses as indicated.

442 SOFT MAGNETIC MATERIALS

the desired multiplication of the applied field and might as well not be there; it is simply wasted material.

Eddy-current shielding can be convincingly demonstrated by a simple experiment. A cylindrical tube is made of annealed steel. The tube is 5 inches long, 0.707 inches outside diameter, and 0.5 inch inside diameter. Into this tube can be inserted a solid rod, 0.5 inch outside diameter, made of the same steel. The tube and rod dimensions are such that the cross-sectional area of the central rod is the same as that of the tube. If the parts are made of medium-carbon steel (0.35 wt% C), the permeability will be about 2000 at a field of 10 Oe (800 A/m); any steel with similar or higher permeability can be used. The tube is wound with primary and secondary coils, and an ac voltage is applied to the primary. At frequencies of 25, 60, and 100 Hz, the output voltage from the secondary coil will be exactly the same, whether or not the inner rod is present in the tube. So the value of B falls to zero within the thickness of the tube wall [(0.707 2 0.50) ¼ 0.207 inch ¼ 5.26 mm]. The central rod is completely shielded even at 25 Hz.

The way in which the flux decreases with depth below the surface of a sheet sample can be calculated exactly only by making certain simplifying assumptions: that the permeability m is constant, independent of H, and that it has the same value in all parts of the sheet. The field is applied parallel to the surface of the sheet and is assumed to vary sinusoidally with time t, so that H ¼ H0 cos 2p ft, where f is the frequency in Hz. Then the field amplitude Hx and the flux density amplitude Bx (¼mHx) at a distance x from the center of the sheet are

where r is the resistivity in ohm-cm. With the assumptions given, Hx and Bx vary sinusoidally with time, but they lag in time behind H0 and B0 at the surface. Equation 13.1 does not give the ratio of the interior value of H or B to the surface value at a given time, but rather the ratio of the amplitude of the interior value to the amplitude of the surface value. These amplitudes (maximum values) are reached at different times.

Flux penetration increases as m and f decrease and as r increases, for a constant plate thickness. (Note that m and f enter as the product mf. Thus the flux penetration for m ¼ 1000 and f ¼ 60 Hz is the same as for m ¼ 100 and f ¼ 600 Hz.) The quantity d is called the skin depth and is the depth under the surface at which H or B falls to 1/e (¼37%) of its value at the surface. A sheet is “thick” if d is small compared to the sheet thickness d. (Note that d is measured from the sheet surface, whereas x is measured from the center of the sheet.)

The curves of Fig. 13.2 were calculated from Equation 13.1 with m ¼ 6750 and r ¼ 40 1026 ohm-cm, which are appropriate for Fe with 3.25% Si, and they apply for a frequency of 60 Hz. This gives a skin depth d of 0.5 mm. Alternatively, they apply to any material, thickness, or frequency for which the dimensionless constant d/d has the values 0.8, 1.6, 3.2, and 6.4, respectively. The flux penetration in each sheet of a stack

Fig. 13.3 Calculated relative flux amplitude in a stack of insulated sheets, each 0.75 mm thick, for the same conditions as in Fig. 13.2.

of sheets, as in the laminated core of a transformer, is the same as in a single sheet, provided the sheets are electrically insulated from one another. This point is illustrated in Fig. 13.3.

Flux penetration in cylindrical rods is given by the following relation, equivalent to Equation 13.1:

where x is the distance from the rod axis, r is the rod radius, and d is given by Equation 13.2 as before. The functions ber and bei are particular kinds of Bessel functions defined in mathematical handbooks. Rods are not used in transformer cores, but Equation 13.3 is given here because magnetic measurements may be made on rod specimens, and it is sometimes necessary to estimate flux penetration in such specimens in alternating fields.

The eddy-current power loss P per unit volume of sheet can be calculated by integrating I2r, where I is the current density, over the volume of the sheet and the length of a cycle. The result is

This is the “classical” eddy-current loss equation; it assumes complete flux penetration, or d/d less than 1, and constant permeability. The thickness d is in cm, the flux amplitude at the surface B0 in gauss, the frequency f in Hz, and the resistivity r in ohm-cm. To obtain the power loss in watts/cm3, multiply Equation 13.4 by 1027. The strong dependence of the eddy-current loss on d, f, and B0 should be noted. However, the loss is independent of m, because complete flux penetration is assumed.

Equations 13.1, 13.3, and 13.4 are all “classical,” in that the permeability m is assumed to be constant both in time and in space. The permeability does not depend on the magnitude of H as H changes with time, and is the same everywhere in the sample—which is to say that the domain structure is ignored. These assumptions are strictly valid only for para-, dia-, and antiferromagnetic materials. They are certainly not true for strongly magnetic

induction Bs. The dashed line shows the higher losses that occur when the domain walls move so far that the sheet is saturated once in each half cycle. The walls are assumed to remain flat, so that the calculation applies only to low frequencies where the walls do not bow out. (The lower the frequency, the lower is the average wall velocity, for constant B0.) We see from this calculation that Ped becomes equal to the classical value Pec as the domain size approaches zero; this is reasonable in that m then becomes homogeneous on a microscopic scale. But when the domain size becomes equal to the sheet thickness, as it does in some electrical steels, the eddy-current loss can become almost double the value calculated on a classical model.

The calculation of Pry and Bean was made for a very special arrangement of domain walls, but their qualitative conclusions are generally true, namely, that the spatial inhomogeneity of m, which is the inevitable result of a domain structure, must lead to higher eddy-current losses than those calculated classically. In a material containing domains, the eddy currents are localized at the moving domain walls, as indicated in Fig. 12.6, and the eddy-current density I can reach very large values at the walls. The average value of I2r over the specimen is then larger than if the eddy currents were evenly distributed throughout. A larger domain size means more widely spaced domain walls, so each wall must move faster to produce a given flux change at a given frequency. Since the losses depend on the square of the domain wall velocity, the total losses increase as the number of domain walls decreases.

13.3LOSSES IN ELECTRICAL MACHINES

The machines to be considered here are transformers, motors, and generators.

13.3.1Transformers

These are among the most efficient machines ever made. Efficiency generally increases with size, and large transformers have efficiencies exceeding 98%. Efficiency is defined as

output ¼ input losses : input input

Transformer losses are mainly composed of:

1.Core losses, sometimes called iron losses, occurring in the magnetic core, as discussed above.

2.Copper losses, occurring in the windings and given simply by i2R.

The ratio of core loss to copper loss in a transformer can be varied over a fairly wide range, by such things as choice of magnetic material, peak flux density, diameter of copper wire, etc. Usually the two losses are more or less equal. The core losses occur continuously as long as the transformer is connected to a power supply, whether or not any power is drawn from the secondary winding. The copper losses, on the other hand, vary as the square of the power output. So a transformer that is lightly or intermittently loaded should be designed to minimize core losses, while a transformer for continuous operation at full power should be designed for equal core and copper losses.

difference between the two is larger the larger the domain size, as shown by Fig. 13.4. The difficulty is that the details of the domain structure and of the domain wall motion are not known in an actual sample, so that calculations can only be made for assumed and simplified models.

A physical approach to core losses not only rejects the “anomalous loss” as a fiction, but it also rejects the conventional separation of the total loss into hysteresis and eddy-current components as artificial. This separation is based on the idea that the static or dc hysteresis arises from a different source than the ac eddy-current losses. This is incorrect; the magnetization process under dc conditions consists of domain wall motion in low fields and domain rotation at high fields, with an intermediate region where both mechanisms occur. Nothing in this basic picture changes as the frequency of magnetization reversal is increased. The detailed configuration of domain walls may change, as domain walls are annihilated and new domains nucleated, and at sufficiently high frequency the skin effect may shield the interior of the sample and confine magnetization changes to the surface. But there is no physical distinction between the processes that occur when a magnetic material is magnetized under dc conditions and under ac conditions: the distinction between hysteresis loss and ac or eddy-current loss is imaginary. Despite the general acceptance of this picture of losses based on domain wall motion, the “separation of losses” dogma continues to appear in the engineering literature.

In a transformer steel lamination made of oriented 3% Si–Fe, the domain structure may be reasonably represented as an array of equally spaced parallel 1808 walls. If all the domain walls move, magnetic saturation is reached when each wall has moved a distance equal to half the domain wall spacing distance d. When the wall has moved this distance, it meets another wall moving the other way, and the two walls annihilate. This happens in the time of one-quarter cycle Dt ¼ (1/4 . 1/f ), where f is the frequency, as the field goes from zero to maximum. The average velocity of a domain wall is then

v ¼ d=2 ¼ 2df :

1=4f

The voltage induced by the moving domain wall is proportional to dB/dt, and hence to v¯. The power dissipated by the moving wall, which is the source of core loss, is proportional to the voltage squared [P ¼ E2/R] and therefore is proportional to v¯2 or to (df )2. The loss per cycle P/f should then be proportional to f, or more specifically to d2f.

What is usually observed is that the loss per cycle rises less rapidly than linearly with increasing frequency f, giving a concave-downward curve, as in Fig. 13.7. The simplest explanation is that the domain spacing d is not constant, but decreases with increasing frequency. That is, more domain walls are nucleated and move; the distance moved by each wall in a quarter-cycle is less, the average velocity v¯ is reduced, and the loss increases less rapidly than it would for a fixed number of domain walls. There is clear experimental evidence for this, at least in the case of simple parallel-wall domain structures [G. L. Houze, J. Appl. Phys., 38 (1967) p. 1089; T. R. Haller and J. J. Kramer, J. Appl. Phys. 41 (1971) p. 1034].

It is also observed that domain wall motion tends to be jerky, not smooth, as the walls are pinned by various internal obstacles. This means the wall velocity during a Barkhausen jump is higher than the average velocity. And since the loss depends on the square of the velocity, jerky wall motion produces higher losses than smooth wall motion.

Some of the data reported by Houze is shown in Table 13.1. Note that the magnetization was not driven to saturation, but to Bmax/Bs 0.45, or less than half of saturation. When the