9
Magnetization

In Sec. 8.3, we defined the magnitude of the magnetic moment m of a circulating current loop of current i and area a as m = ia. The moment vector, m, was defined as normal to the surface spanning the contour of the loop and pointing in the direction determined by the right-hand rule. In Sec. 8.3, where the moment was in the z direction in spherical coordinates, the loop was found to produce the magnetic field intensity

This field is analogous to the electric field associated with a dipole having the moment p. With p directed along the z axis, the electric dipole field is given by taking the gradient of (4.4.10).

Thus, the dipole fields are obtained from each other by making the identifications

In Sec. 6.1, a spatial distribution of electric dipoles is represented by the polarization density P = Np, where N is the number density of dipoles. Similarly, here we define a magnetization density as

where again N is the number of dipoles per unit volume. Note that just as the analog of the dipole moment p is _o m, the analog of the polarization density P is _o M.

9.2 Laws and Continuity Conditions with Magnetization

Recall that the effect of a spatial distribution of electric dipoles upon the electric field is described by a generalization of Gauss' law for electric fields, (6.2.1) and (6.2.2),

The effect of the spatial distribution of magnetic dipoles upon the magnetic field intensity is now similarly taken into account by generalizing the magnetic flux continuity law.

In this law, there is no analog to an unpaired electric charge density.

The continuity condition found by integrating (2) over an incremental volume enclosing a section of an interface having a normal n is

Suggested by the analogy to the description of polarization is the definition of the quantities on the right in (2) and (3), respectively, as the magnetic charge density _m and the magnetic surface charge density _sm.

Faraday's Law Including Magnetization

The modification of the magnetic flux continuity law implies that another of Maxwell's equations must be generalized. In introducing the flux continuity law in Sec. 1.7, we observed that it was almost inherent in Faraday's law. Because the divergence of the curl is zero, the divergence of the free space form of Faraday's law reduces to

Thus, in free space, _o H must have a divergence that is at least constant in time. The magnetic flux continuity law adds the information that this constant is zero. In the presence of magnetizable material, (2) shows that the quantity _o(H + M ) is solenoidal. To make Faraday's law consistent with this requirement, the law is now written as

Magnetic Flux Density

The grouping of H and M in Faraday's law and the flux continuity law makes it natural to define a new variable, the magnetic flux density B.

This quantity plays a role that is analogous to that of the electric displacement flux density D defined by (6.2.14). Because there are no macroscopic quantities of monopoles of magnetic charge, its divergence is zero. That is, the flux continuity law, (2), becomes simply

and the corresponding continuity condition, (3), becomes simply

A similar simplification is obtained by writing Faraday's law in terms of the magnetic flux density. Equation (7) becomes

If the magnetization is specified independent of H, it is usually best to have it entered explicitly in the formulation by not introducing B. However, if M is given as a function of H, especially if it is linear in H, it is most convenient to remove M from the formulation by using B as a variable.

Terminal Voltage with Magnetization

In Sec. 8.4, where we discussed the terminal voltage of a perfectly conducting coil, there was no magnetization. The generalization of Faraday's law to include magnetization requires a generalization of the terminal relation.

The starting point in deriving the terminal relation was Faraday's integral law, (8.4.9). This law is generalized to included magnetization effects by replacing _o H with B. Otherwise, the derivation of the terminal relation, (8.4.11), is the same as before. Thus, the terminal voltage is again

but now the flux linkage is

In Sec. 9.4 we will see that Faraday's law of induction, as reflected in these last two relations, is the basis for measuring B.

9.3 Permanent Magnetization

As the modern-day versions of the lodestone, which made the existence of magnetic fields apparent in ancient times, permanent magnets are now so cheaply manufactured that they are used at home to pin notes on the refrigerator and so reliable that they are at the heart of motors, transducers, and information storage systems. To a first approximation, a permanent magnet can be modeled by a material having a specified distribution of magnetization density M. Thus, in this section we consider the magnetic field intensity generated by prescribed distributions of M.

In a region where there is no current density J, Ampère's law requires that H be irrotational. It is then often convenient to represent the magnetic field intensity in terms of the scalar magnetic potential \Psi introduced in Sec. 8.3.

From the flux continuity law, (9.2.2), it then follows that \Psi satisfies Poisson's equation.

A specified magnetization density leads to a prescribed magnetic charge density _m. The situation is analogous to that considered in Sec. 6.3, where the polarization density was prescribed and, as a result, where _p was known.

Of course, the net magnetic charge of a magnetizable body is always zero, because

if the integral is taken over the entire volume containing the body. Techniques for solving Poisson's equation for a prescribed charge distribution developed in Chaps. 4 and 5 are directly applicable here. For example, if the magnetization is given throughout all space and there are no other sources, the magnetic scalar potential is given by a superposition integral. Just as the integral of (4.2.2) is (4.5.3), so the integral of (2) is

If the region of interest is bounded by material on which boundary conditions are specified, (4) provides the particular solution.

Example 9.3.1. Magnetic Field Intensity of a Uniformly Magnetized Cylinder

The cylinder shown in Fig. 9.3.1 is uniformly magnetized in the z direction, M = M_o i_z. The first step toward finding the resulting H within the cylinder and in the surrounding free space is an evaluation of the distribution of magnetic charge density. The uniform M has no divergence, so _m = 0 throughout the volume. Thus, the source of H is on the surfaces where M originates and terminates. In view of (9.2.3), it takes the form of the surface charge density

The upper and lower signs refer to the upper and lower surfaces.

In principle, we could use the superposition integral to find the potential everywhere. To keep the integration simple, we confine ourselves here to finding it on the z axis. The integration of (4) then reduces to integrations over the endfaces of the cylinder.

With absolute magnitudes used to make the expressions valid regardless of position along the z axis, these integrals become

The field intensity follows from (1)

where u \equiv 0 for |z| > d/2 and u \equiv 2 for -d/2 < z < d/2. Here, from top to bottom, respectively, the signs correspond to evaluating the field above the upper surface, within the magnet, and below the bottom surface.

The axial distributions of \Psi and H_z shown in Fig. 9.3.1 are consistent with a three-dimensional picture of a field that originates on the top face of the magnet and terminates on the bottom face. As for the spherical magnet (the analogue of the permanently polarized sphere shown in Fig. 6.3.1), the magnetic field intensity inside the magnet has a direction opposite to that of M.

In practice, M would most likely be determined by making measurements of the external field and then deducing M from this field.

If the magnetic field intensity is generated by a combination of prescribed currents and permanent magnetization, it can be evaluated by superimposing the field due to the current and the magnetization. For example, suppose that the uniformly magnetized circular cylinder of Fig. 9.3.1 were surrounded by the N-turn solenoid of Fig. 8.2.3. Then the axial field intensity would be the sum of that for the current [predicted by the Biot-Savart law, (8.2.7)], and for the magnetization [predicted by the negative gradient of (4)].

Example 9.3.2. Retrieval of Signals Stored on Magnetizable Tape

Permanent magnetization is used for a permanent record in the tape recorder. Currents in an electromagnet are used to induce the permanent magnetization, exploiting the hysteresis in the magnetization of certain materials, as will be discussed in Sec. 9.4. Here we look at a model of perpendicular magnetization, an actively pursued research field. The conventional recording is done by producing magnetization M parallel to the tape.

In a thin tape at rest, the magnetization density shown in Fig. 9.3.2 is assumed to be uniform over the thickness and to be of the simple form

The magnetic field is first determined in a frame of reference attached to the tape, denoted by (x, y, z) as defined in Fig. 9.3.2. The tape moves with a velocity U with respect to a fixed sensing "head," and so our second step will be to represent this field in terms of fixed coordinates. With Fig. 9.3.3 in view, it is clear that these coordinates, denoted by (x^' ,y^' ,z^' ), are related to the moving coordinates by

Thus, from the fixed reference frame, the magnetization takes the form of a traveling wave.

If \bf M is observed at a fixed location x^', it has a sinusoidal temporal variation with the frequency = U. This relationship between the fixed frame frequency and the spatial periodicity suggests how the distribution of magnetization is established by "recording" a signal having the frequency .

The magnetization density has no divergence in the volume of the tape, so the field source is a surface charge density. With upper and lower signs denoting the upper and lower tape surfaces, it follows that

The continuity conditions to be satisfied at the upper and lower surfaces represent the continuity of magnetic flux (9.2.3)

and the continuity of tangential \bf H

In addition, the field should go to zero as y \pm .

Because the field sources are confined to surfaces, the magnetic scalar potential must satisfy Laplace's equation, (2) with _m = 0, in the bulk regions delimited by the interfaces. Motivated by the "odd" symmetry of the source with respect to the y = 0 plane and its periodicity in x, we pick solutions to Laplace's equation for the magnetic potential above (a), inside (o), and below (b) the tape that also satisfy the odd symmetry condition of having \Psi (y) = - \Psi (-y).

Subject to the requirement that > 0, the exterior potentials go to zero at y = \pm . The interior function is made an odd function of y by excluding the cosh ( y) cos ( x) solution to Laplace's equation, while the exterior functions are made odd by making the coefficients equal in magnitude and opposite in sign. Thus, only two coefficients remain to be determined. These follow from substituting the assumed solution into either of (13) and either of (14), and then solving the two equations to obtain

The conditions at one interface are automatically satisfied if those at the other are met. This is a proof that the assumed solutions have indeed been correct. Our foresight in defining the origin of the y axis to be at the symmetry plane and exploiting the resulting odd dependence of \Psi on y has reduced the number of undetermined coefficients from four to two.

This field is now expressed in the fixed frame coordinates. With A defined by (16a) and x and y given in terms of the fixed frame coordinates by (10), the magnetic potential above the tape has been determined to be

Next, we determine the output voltage of a fixed coil, positioned at a height h above the tape, as shown in Fig. 9.3.3. This detecting "head" has N turns, a length l in the x^' direction, and width w in the z direction. With the objective of finding the flux linkage, we use (17) to determine the y-directed flux density in the neighborhood of the coil.

The flux linkage follows by multiplying the number of turns N times B_y integrated over the surface in the plane y = h + 1\over 2 d spanned by the coil.

The dependence on l is clarified by using a trigonometric identity to simplify the last term in this expression.

Finally, the output voltage follows from (9.2.12).

The strong dependence of this expression on the wavelength of the magnetization, 2 /, reflects the nature of fields predicted using Laplace's equation. It follows from (21) that the output voltage has the angular frequency = U. Thus, (21) can also be regarded as giving the frequency response of the sensor. The magnitude of v_o has the dependence on either the normalized or shown in Fig. 9.3.4.

Two phenomena underlie the voltage response. The periodic dependence reflects the relationship between the length l of the coil and the wavelength 2 / of the magnetization. When the coil length is equal to the wavelength, there is as much positive as negative flux linking the coil at a given instant, and the signal falls to zero. This is also the condition when l is any multiple of a wavelength and accounts for the sin ( l) term in (21).

The strong decay of the envelope of the output signal as the frequency is increased, and hence the wavelength decreased, reflects a property of Laplace's equation that frequently comes into play in engineering electromagnetic fields. The shorter the wavelength, the more rapid the decay of the field in the direction perpendicular to the tape. With the sensing coil at a fixed height above the tape, this means that once the wavelength is on the order of 2 h, there is an essentially exponential decrease in signal with increasing frequency. Thus, there is a strong incentive to place the coil as close to the tape as possible.

We should expect that if the tape is very thin compared to the wavelength, the field induced by magnetic surface charges on the top surface would tend to be canceled by those of opposite sign on the surface just below. This effect is accounted for by the term [1 + \coth ( d)] in the denominator of (21).

In a practical recording device, the sensing head of the previous example would incorporate magnetizable materials. To predict how these affect the fields, we need a law relating the field to the magnetization it induces. This is the subject of the next section.

9.4 Magnetization Constitutive Laws

The permanent magnetization model of Sec. 9.3 is a somewhat artificial example of the magnetization density M specified, independent of the magnetic field intensity. Even in the best of permanent magnets, there is actually some dependence of M on H.

Constitutive laws relate the magnetization density M or the magnetic flux density B to the macroscopic H within a material. Before discussing some of the more common relations and their underlying physics, it is well to have in view an experiment giving direct evidence of the constitutive law of magnetization. The objective is to observe the establishment of H by a current in accordance with Ampère's law, and deduce B from the voltage it induces in accordance with Faraday's law.

Example 9.4.1. Toroidal Coil

A coil of toroidal geometry is shown in Fig. 9.4.1. It consists of a donut-shaped core filled with magnetizable material with N₁ turns tightly wound on its periphery. By means of a source driving its terminals, this coil carries a current i. The resulting current distribution can be assumed to be so smooth that the fine structure of the field, caused by the finite size of the wires, can be disregarded. We will ignore the slight pitch of the coil and the associated small current component circulating around the axis of the toroid.

Because of the toroidal geometry, the H field in the magnetizable material is determined by Ampère's law and symmetry considerations. Symmetry about the toroidal axis suggests that H is directed. The integral MQS form of Ampère's law is written for a contour C circulating about the toroidal axis within the core and at a radius r. Because the major radius R of the torus is large compared to the minor radius w, we will ignore the variation of r over the cross-section of the torus and approximate r by an average radius R. The surface S spanned by this contour and shown in Fig. 9.4.2 is pierced N₁ times by the current i, giving a total current of N₁ i. Thus, the azimuthal field inside the core is essentially

Note that the same argument shows that the magnetic field intensity outside the core is zero.

In general, if we are given the current distribution and wish to determine H, recourse must be made not only to Ampère's law but to the flux continuity condition as well. In the idealized toroidal geometry, where the flux lines automatically close on themselves without leaving the magnetized material, the flux continuity condition is automatically satisfied. Thus, in the toroidal configuration, the H imposed on the core is determined by a measurement of the current i and the geometry.

How can we measure the magnetic flux density in the core? Because B appears in Faraday's law of induction, the measurement of the terminal voltage of an additional coil, having N₂ turns also wound on the donut-shaped core, gives information on B. The terminals of this coil are terminated in a high enough impedance so that there is a negligible current in this second winding. Thus, the H field established by the current i remains unaltered.

The flux linked by each turn of the sensing coil is essentially the flux density multiplied by the cross-sectional area w²/4 of the core. Thus, the flux linked by the terminals of the sensing coil is

and flux density in the core material is directly reflected in the terminal flux-linkage.

The following demonstration shows how (1) and (2) can be used to infer the magnetization characteristic of the core material from measurement of the terminal current and voltage of the first and second coils.

Demonstration 9.4.1. Measurement of B-H Characteristic

The experiment shown in Fig. 9.4.3 displays the magnetization characteristic on the oscilloscope. The magnetizable material is in the donut-shaped toroidal configuration of Example 9.4.1 with the N₁-turn coil driven by a current i from a Variac. The voltage across a series resistance then gives a horizontal deflection of the oscilloscope proportional to H, in accordance with (1).

The terminals of the N₂ turn-coil are connected through an integrating network to the vertical deflection terminals of the oscilloscope. Thus, the vertical deflection is proportional to the integral of the terminal voltage, to , and hence through (2), to B.

In the discussions of magnetization characteristics which follow, it is helpful to think of the material as comprising the core of the torus in this experiment. Then the magnetic field intensity H is proportional to the current i, while the magnetic flux density B is reflected in the voltage induced in a coil linking this flux.

Many materials are magnetically linear in the sense that

Here _m is the magnetic susceptibility. More commonly, the constitutive law for a magnetically linear material is written in terms of the magnetic flux density, defined by (9.2.8).

According to this law, the magnetization is taken into account by replacing the permeability of free space _o by the permeability of the material. For purposes of comparing the magnetizability of materials, the relative permeability /_o is often used.

Typical susceptibilities for certain elements, compounds, and common materials are given in Table 9.4.1. Most common materials are only slightly magnetizable. Some substances that are readily polarized, such as water, are not easily magnetized. Note that the magnetic susceptibility can be either positive or negative and that there are some materials, notably iron and its compounds, in which it can be enormous. In establishing an appreciation for the degree of magnetizability that can be expected of a material, it is helpful to have a qualitative picture of its microscopic origins, beginning at the atomic level but including the collective effects of groups of atoms or molecules that result when they become as densely packed as they are in solids. These latter effects are prominent in the most easily magnetized materials.

The magnetic moment of an atom (or molecule) is the sum of the orbital and spin contributions. Especially in a gas, where the atoms are dilute, the magnetic susceptibility results from the (partial) alignment of the individual magnetic moments by a magnetic field. Although the spin contributions to the moment tend to cancel, many atoms have net moments of one or more Bohr magnetons. At room temperature, the orientations of the moments are mostly randomized by thermal agitation, even under the most intense fields. As a result, an applied field can give rise to a significant magnetization only at very low temperatures. A paramagnetic material displays an appreciable susceptibility only at low temperatures.

If, in the absence of an applied field, the spin contributions to the moment of an atom very nearly cancel, the material can be diamagnetic, in the sense that it displays a slightly negative susceptibility. With the application of a field, the orbiting electrons are slightly altered in their circulations, giving rise to changes in moment in a direction opposite to that of the applied field. Again, thermal energy tends to disorient these moments. At room temperature, this effect is even smaller than that for paramagnetic materials.

At very low temperatures, it is possible to raise the applied field to such a level that essentially all the moments are aligned. This is reflected in the saturation of the flux density B, as shown in Fig. 9.4.4. At low field intensity, the slope of the magnetization curve is , while at high field strengths, there are no more moments to be aligned and the slope is _o. As long as the field is raised and lowered at a rate slow enough so that there is time for the thermal energy to reach an equilibrium with the magnetic field, the B-H curve is single valued in the sense that the same curve is followed whether the magnetic field is increasing or decreasing, and regardless of its rate of change.

Until now, we have been considering the magnetization of materials that are sufficiently dilute so that the atomic moments do not interact with each other. In solids, atoms can be so closely spaced that the magnetic field due to the moment of one atom can have a significant effect on the orientation of another. In ferromagnetic materials, this mutual interaction is all important.

To appreciate what makes certain materials ferromagnetic rather than simply paramagnetic, we need to remember that the electrons which surround the nuclei of atoms are assigned by quantum mechanical principles to layers or "shells." Each shell has a particular maximum number of electrons. The electron behaves as if it possessed a net angular momentum, or spin, and hence a magnetic moment. A filled shell always contains an even number of electrons which are distributed spatially in such a manner that the total spin, and likewise the magnetic moment, is zero.

For the majority of atoms, the outermost shell is unfilled, and so it is the outermost electrons that play the major role in determining the net magnetic moment of the atom. This picture of the atom is consistent with paramagnetic and diamagnetic behavior. However, the transition elements form a special class. They have unfilled inner shells, so that the electrons responsible for the net moment of the atom are surrounded by the electrons that interact most intimately with the electrons of a neighboring atom. When such atoms are as closely packed as they are in solids, the combination of the interaction between magnetic moments and of electrostatic coupling results in the spontaneous alignment of dipoles, or ferromagnetism. The underlying interaction between atoms is both magnetic and electrostatic, and can be understood only by invoking quantum mechanical arguments.

In a ferromagnetic material, atoms naturally establish an array of moments that reinforce. Nevertheless, on a macroscopic scale, ferromagnetic materials are not necessarily permanently magnetized. The spontaneous alignment of dipoles is commonly confined to microscopic regions, called domains. The moments of the individual domains are randomly oriented and cancel on a macroscopic scale.

Macroscopic magnetization occurs when a field is applied to a solid, because those domains that have a magnetic dipole moment nearly aligned with the applied field grow at the expense of domains whose magnetic dipole moments are less aligned with the applied field. The shift in domain structure caused by raising the applied field from one level to another is illustrated in Fig. 9.4.5. The domain walls encounter a resistance to propagation that balances the effect of the field.

A typical trajectory traced out in the B-H plane as the field is applied to a typical ferromagnetic material is shown in Fig. 9.4.6. If the magnetization is zero at the outset, the initial trajectory followed as the field is turned up starts at the origin. If the field is then turned down, the domains require a certain degree of coercion to reduce their average magnetization. In fact, with the applied field turned off, there generally remains a flux density, and the field must be reversed to reduce the flux density to zero. The trajectory traced out if the applied field is slowly cycled between positive and negative values many times is the one shown in the figure, with the remanence flux density B_r when H = 0 and a coercive field intensity H_c required to make the flux density zero. Some values of these parameters, for materials used to make permanent magnets, are given in Table 9.4.2.

In the toroidal geometry of Example 9.4.1, H is proportional to the terminal current i. Thus, imposition of a sinusoidally varying current results in a sinusoidally varying H, as illustrated in Fig. 9.4.6b. As the i and hence H increases, the trajectory in the B-H plane is the one of increasing H. With decreasing H, a different trajectory is followed. In general, it is not possible to specify B simply by giving H (or even the time derivatives of H). When the magnetization state reflects the previous states of magnetization, the material is said to be hysteretic. The B-H trajectory representing the response to a sinusoidal H is then called the hysteresis loop.

Hysteresis can be both harmful and useful. Permanent magnetization is one result of hysteresis, and as we illustrated in Example 9.3.2, this can be the basis for the storage of information on tapes. When we develop a picture of energy dissipation in Chap. 11, it will be clear that hysteresis also implies the generation of heat, and this can impose limits on the use of magnetizable materials.

Liquids having significant magnetizabilities have been synthesized by permanently suspending macroscopic particles composed of single ferromagnetic domains. Here also the relatively high magnetizability comes from the ferromagnetic character of the individual domains. However, the very different way in which the domains interact with each other helps in gaining an appreciation for the magnetization of ferromagnetic polycrystalline solids.

In the absence of a field imposed on the synthesized liquid, the thermal molecular energy randomizes the dipole moments and there is no residual magnetization. With the application of a low frequency H field, the suspended particles assume an average alignment with the field and a single-valued B-H curve is traced out, typically as shown in Fig. 9.4.4. However, as the frequency is raised, the reorientation of the domains lags behind the applied field, and the B-H curve shows hysteresis, much as for solids.

Although both the solid and the liquid can show hysteresis, the two differ in an important way. In the solid, the magnetization shows hysteresis even in the limit of zero frequency. In the liquid, hysteresis results only if there is a finite rate of change of the applied field.

Ferromagnetic materials such as iron are metallic solids and hence tend to be relatively good electrical conductors. As we will see in Chap. 10, this means that unless care is taken to interrupt conduction paths in the material, currents will be induced by a time-varying magnetic flux density. Often, these eddy currents are undesired. With the objective of obtaining a highly magnetizable insulating material, iron atoms can be combined into an oxide crystal. Although the spontaneous interaction between molecules that characterizes ferromagnetism is indeed observed, the alignment of neighbors is antiparallel rather than parallel. As a result, such pure oxides do not show strong magnetic properties. However, a mixed-oxide material like Fe₃O₄ (magnetite) is composed of sublattice oxides of differing moments. The spontaneous antiparallel alignment results in a net moment. The class of relatively magnetizable but electrically insulating materials are called ferrimagnetic.

Our discussion of the origins of magnetization began at the atomic level, where electronic orbits and spins are fundamental. However, it ends with a discussion of constitutive laws that can only be explained by bringing in additional effects that occur on scales much greater than atomic or molecular. Thus, the macroscopic B and H used to describe magnetizable materials can represent averages with respect to scales of domains or of macroscopic particles. In Sec. 9.5 we will make an artificial diamagnetic material from a matrix of "perfectly" conducting particles. In a time-varying magnetic field, a magnetic moment is induced in each particle that tends to cancel that being imposed, as was shown in Example 8.4.3. In fact, the currents induced in the particles and responsible for this induced moment are analogous to the induced changes in electronic orbits responsible on the atomic scale for diamagnetism^[1].

9.5 Fields In The Presence Of Magnetically Linear Insulating Materials

9.7 Magnetic Circuits

The availability of relatively inexpensive magnetic materials, with magnetic susceptibilities of the order of 1000 or more, allows the production of high magnetic flux densities with relatively small currents. Devices designed to exploit these materials include compact inductors, transformers, and rotating machines. Many of these are modeled as the magnetic circuits that are the theme of this section.

A magnetic circuit typical of transformer cores is shown in Fig. 9.7.1. A core of high permeability material has a pair of rectangular windows cut through its center. Wires passing through these windows are wrapped around the central column. The flux generated by this coil tends to be guided by the magnetizable material. It passes upward through the center leg of the material, and splits into parts that circulate through the legs to left and right.

Example 9.6.2, with its highly permeable sphere excited by a small coil, offered the opportunity to study the trapping of magnetic flux. Here, as in that case with _b /_a \gg 1, the flux density inside the core tends to be tangential to the surface. Thus, the magnetic flux density is guided by the material and the field distribution within the core tends to be independent of the exterior configuration.

In situations of this type, where the ducting of the magnetic flux makes it possible to approximate the distribution of magnetic field, the MQS integral laws serve much the same purpose as do Kirchhoff's laws for electrical circuits.

The MQS form of Ampère's integral law applies to a contour, such as C₁ in Fig. 9.7.2, following a path of circulating magnetic flux.

The surface enclosed by this contour in Fig. 9.7.2 is pierced N times by the current carried by the wire, so the surface integral of the current density on the right in (1) is, in this case, Ni. The same equation could be written for a contour circulating through the left leg, or for one circulating around through the outer legs. Note that the latter would enclose a surface S through which the net current would be zero.

If Ampère's integral law plays a role analogous to Kirchhoff's voltage law, then the integral law expressing continuity of magnetic flux is analogous to Kirchhoff's current law. It requires that through a closed surface, such as S₂ in Fig. 9.7.2, the net magnetic flux is zero.

As a result, the flux entering the closed surface S₂ in Fig. 9.7.2 through the central leg must be equal to that leaving to left and right through the upper legs of the magnetic circuit. We will return to this particular magnetic circuit when we discuss transformers.

Example 9.7.1. The Air Gap Field of an Electromagnet

The magnetic circuit of Fig. 9.7.3 might be used to produce a high magnetic field intensity in the narrow air gap. An N-turn coil is wrapped around the left leg of the highly permeable core. Provided that the length g of the air gap is not too large, the flux resulting from the current i in this winding is largely guided along the magnetizable material.

By approximating the fields in sections of the circuit as being essentially uniform, it is possible to use the integral laws to determine the field intensity in the gap. In the left leg, the field is approximated by the constant H₁ over the length l₁ and cross-sectional area A₁. Similarly, over the lengths l₂, which have the cross-sectional areas A₂, the field intensity is approximated by H₂. Finally, under the assumption that the gap width g is small compared to the cross-sectional dimensions of the gap, the field in the gap is represented by the constant H_g. The line integral of H in Ampère's integral law, (1), is then applied to the contour C that follows the magnetic field intensity around the circuit to obtain the left-hand side of the expression

The right-hand side of this equation represents the surface integral of J d a for a surface S having this contour as its edge. The total current through the surface is simply the current through one wire multiplied by the number of times it pierces the surface S.

We presume that the magnetizable material is operated under conditions of magnetic linearity. The constitutive law then relates the flux density and field intensity in each of the regions.

Continuity of magnetic flux, (2), requires that the total flux through each section of the circuit be the same. With the flux densities expressed using (4), this requires that

Our objective is to determine H_g. To that end, (5) is used to write

and these relations used to eliminate H₁ and H₂ in favor of H_g in (3). From the resulting expression, it follows that

Note that in the limit of infinite core permeability, the gap field intensity is simply Ni/g.

If the magnetic circuit can be broken into sections in which the field intensity is essentially uniform, then the fields may be determined from the integral laws. The previous example is a case in point. A more general approach is required if the core is of complex geometry or if a more accurate model is required.

We presume throughout this chapter that the magnetizable material is sufficiently insulating so that even if the fields are time varying, there is no current density in the core. As a result, the magnetic field intensity in the core can be represented in terms of the scalar magnetic potential introduced in Sec. 8.3.

According to Ampère's integral law, (1), integration of H ds around a closed contour must be equal to the "Ampère turns" Ni passing through the surface spanning the contour. With H expressed in terms of \Psi, integration from (a) to (b) around a contour such as C in Fig. 9.7.4, which encircles a net current equal to the product of the turns N and the current per turn i, gives \Psi_a - \Psi_b \equiv \Psi = Ni. With (a) and (b) adjacent to each other, it is clear that \Psi is multiple-valued. To specify the principal value of this multiple-valued function we must introduce a discontinuity in \Psi somewhere along the contour. In the circuit of Fig. 9.7.4, this discontinuity is defined to occur across the surface S_d.

To make the line integral of H ds from any point just above the surface S_d around the circuit to a point just below the surface equal to Ni, the potential is required to suffer a discontinuity \Psi = Ni across S_d. Everywhere inside the magnetic material, \Psi satisfies Laplace's equation. If, in addition, the normal flux density on the walls of the magnetizable material is required to vanish, the distribution of \Psi within the core is uniquely determined. Note that only the discontinuity in \Psi is specified on the surface S_d. The magnitude of \Psi on one side or the other is not specified. Also, the normal derivative of \Psi, which is proportional to the normal component of H, must be continuous across S_d.

The following simple example shows how the scalar magnetic potential can be used to determine the field inside a magnetic circuit.

Example 9.7.2. The Magnetic Potential inside a Magnetizable Core

The core of the magnetic circuit shown in Fig. 9.7.5 has outer and inner radii a and b, respectively, and a length d in the z direction that is large compared to a. A current i is carried in the z direction through the center hole and returned on the outer periphery by N turns. Thus, the integral of H ds over a contour circulating around the magnetic circuit must be Ni, and a surface of discontinuity S_d is arbitrarily introduced as shown in Fig. 9.7.5. With the boundary condition of no flux leakage, \Psi /\partial r = 0 at r = a and at r = b, the solution to Laplace's equation within the core is uniquely specified.

In principle, the boundary value problem can be solved even if the geometry is complicated. For the configuration shown in Fig. 9.7.5, the requirement of no radial derivative suggests that \Psi is independent of r. Thus, with A an arbitrary coefficient, a reasonable guess is

The coefficient A has been selected so that there is indeed a discontinuity Ni in \Psi between = 2 and = 0.

The magnetic field intensity given by substituting (9) into (8) is

Note that H is continuous, as it should be.

Now that the inside field has been determined, it is possible, in turn, to find the fields in the surrounding free space regions. The solution for the inside field, together with the given surface current distribution at the boundary between regions, provides the tangential field at the boundaries of the outside regions. Within an arbitrary constant, a boundary condition on \Psi is therefore specified. In the outside regions, there is no closed contour that both stays within the region and encircles current. In these regions, \Psi is continuous. Thus, the problem of finding the "leakage" fields is reduced to finding the boundary value solution to Laplace's equation.

This inside-outside approach gives an approximate field distribution that is justified only if the relative permeability of the core is very large. Once the outside field is approximated in this way, it can be used to predict how much flux has left the magnetic circuit and hence how much error there is in the calculation. Generally, the error will be found to depend not only on the relative permeability but also on the geometry. If the magnetic circuit is composed of legs that are long and thin, then we would expect the leakage of flux to be large and the approximation of the inside-outside approach to become invalid.

Electrical Terminal Relations and Characteristics

Practical inductors (chokes) often take the form of magnetic circuits. With more than one winding on the same magnetic circuit, the magnetic circuit serves as the core of a transformer. Figure 9.7.6 gives the schematic representation of a transformer. Each winding is modeled as perfectly conducting, so its terminal voltage is given by (9.2.12).

However, the flux linked by one winding is due to two currents. If the core is magnetically linear, we have a flux linked by the first coil that is the sum of a flux linkage L₁₁ i₁ due to its own current and a flux linkage L₁₂ due to the current in the second winding. The situation for the second coil is similar. Thus, the flux linkages are related to the terminal currents by an inductance matrix.

The coefficients L_ij are functions of the core and coil geometries and properties of the material, with L₁₁ and L₂₂ the familiar self-inductances and L₁₂ and L₂₁ the mutual inductances.

The word "transformer" is commonly used in two ways, each often represented schematically, as in Fig. 9.7.6. In the first, the implication is only that the terminal relations are as summarized by (12). In the second usage, where the device is said to be an ideal transformer, the terminal relations are given as voltage and current ratios. For an ideal transformer,

Presumably, such a device can serve to step up the voltage while stepping down the current. The relationships between terminal voltages and between terminal currents is linear, so that such a device is "ideal" for processing signals.

The magnetic circuit developed in the next example is that of a typical transformer. We have two objectives. First, we determine the inductances needed to complete (12). Second, we define the conditions under which such a transformer operates as an ideal transformer.

Example 9.7.3. A Transformer

The core shown in Fig. 9.7.7 is familiar from the introduction to this section, Fig. 9.7.1. The "windows" have been filled up by a pair of windings, having the turns N₁ and N₂, respectively. They share the center leg of the magnetic circuit as a common core and generate a flux that circulates through the branches to either side.

The relation between the terminal voltages for an ideal transformer depends only on unity coupling between the two windings. That is, if we call the magnetic flux through the center leg, the flux linking the respective coils is

These statements presume that there is no leakage flux which would link one coil but bypass the other.

In terms of the magnetic flux through the center leg, the terminal voltages follow from (14) as

From these expressions, without further restrictions on the mode of operation, follows the relation between the terminal voltages of (13).

We now use the integral laws to determine the flux linkages in terms of the currents. Because it is desirable to minimize the peak magnetic flux density at each point throughout the core, and because the flux through the center leg divides evenly between the two circuits, the cross-sectional areas of the return legs are made half as large as that of the center leg.

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³ To optimize the usage of core material, the relative dimensions are often taken as in the inset to Fig. 9.7.7. Two cores are cut from rectangular sections measuring 6h x 8h. Once the windows have been removed, the rectangle is cut in two, forming two "E" cores which can then be combined with the "I's" to form two complete cores. To reduce eddy currents, the core is often made from varnished laminations. This will be discussed in Chap. 10. As a result, the magnitude of B, and hence H, can be approximated as constant throughout the core. [Note that we have now used the flux continuity condition of (2).]

With the average length of a circulating magnetic field line taken as l, Ampère's integral law, (1), gives

In view of the presumed magnetic linearity of the core, the flux through the cross-sectional area A of the center leg is

and it follows from these last two expressions that

Multiplication by the turns N₁ and then N₂, respectively, gives the flux linkages ₁ and ₂.

Comparison of this expression with (12) identifies the self- and mutual inductances as

Note that the mutual inductances are equal. In Sec. 11.7, we shall see that this is a consequence of energy conservation. Also, the self-inductances are related to either mutual inductance by

Under what conditions do the terminal currents obey the relations for an "ideal transformer"?

Suppose that the (1) terminals are selected as the "primary" terminals of the transformer and driven by a current source I(t), and that the terminals of the (2) winding, the "secondary," are connected to a resistive load R. To recognize that the winding in fact has an internal resistance, this load includes the winding resistance as well. The electrical circuit is as shown in Fig. 9.7.8.

The secondary circuit equation is

and using (12) with i₁ = I, it follows that the secondary current i₂ is governed by

For purposes of illustration, consider the response to a drive that is in the sinusoidal steady state. With the drive angular frequency equal to , the response has the same time dependence in the steady state.

Substitution into (23) then shows that the complex amplitude of the response is

The ideal transformer-current relation is obtained if

In that case, (25) reduces to

When the ideal transformer condition, (26), holds, the first term on the left in (23) overwhelms the second. What remains if the resistance term is neglected is the statement

We conclude that for ideal transformer operation, the flux linkages are negligible. This is crucial to having a transformer behave as a linear device. Whether represented by the inductance matrix of (12) or by the ideal relations of (13), linear operation hinges on having a linear relation between B and H in the core, (17). By operating in the regime of (26) so that B is small enough to avoid saturation, (17) tends to remain valid.