3.1 INTRODUCTION

This chapter deals with a wide range of electromagnetic devices used for conversion of electrical energy to mechanical energy, or simply a process of transforming electrical energy to a useful form. The electric machine systems for energy processing are:

• Generators, connected to prime movers (steam-, gas-, or hydro-turbine systems), convert mechanical energy to electrical energy for production of electrical power.
• Transformers process (transform) electric energy from the most convenient generation voltage to a voltage level suitable for transmission to a load point and into voltage appropriate for distribution.
• Electric motors are used to convert electrical energy to mechanical energy for drive systems and further energy processing.
• Electronic converters process electric power from AC to DC or DC to AC with controllable voltage, current, or frequency.
• Drive-system controls of motors and converters are capable of controlling the speed or position of mechanical loads.

3.2 MAGNETIC FIELDS

The conversion of power or energy from one machine to another is achieved using magnetic fields that depend on a variety of materials, structures, locations, and intensities. To understand how the magnetic field works or changes as a function of time, we employ Maxwell's equations, and its application to different machine types.

Maxwell's equations may be expressed in differential and integral forms, summarized in the following four laws.

Ampère's Law

(3.1)

The line integral of the magnetic field strength H on a closed geometric-integration loop l (magnetic circulation voltage) is equal to the total electric current flowing through the area A limited by this loop (magneto-motive force and ampere-turns), inclusive of the displacement current, which is neglected. This relates the intensity of the magnetic field to the current producing it; it is the principle behind the generator/motor action, whereas Lenz's law states that the induced emf gives rise to a current whose magnetic field opposes the change in the original magnetic flux.

Faraday's Law

E, the induced voltage = electromotive force (emf). Faraday's law is given as

(3.2)

The line integral of the electric field strength E on a closed geometric integration loop (electric circulation voltage) is equal to the negative, time-dependent variation of the total magnetic flux that penetrates the area A limited by this loop, i.e., induced voltage in a coil of conductor (enclosed flux φ).

For N coils,

(3.3)

(3.4)

(3.5)

Faraday's law leads to transformer action for energy processing.

Gauss's Law of Electricity

(3.6)

The implication of Gauss's law of electricity is that the total electric field penetrating a closed surface of any volume is the electric charges inside this volume.

Gauss's Law of Magnetism

(3.7)

This implies that the total magnetic flux penetrating a closed surface of any volume is zero.

Faraday's law and Ampère's law are very important in electrical machine design. At their simplest, an equation can be employed to determine the voltages induced in the windings of an electrical machine. It is also necessary—for instance, in the determination of losses caused by eddy currents in a magnetic circuit, and when determining the skin effect in copper. With respect to flux linkage in an electric machine's windings, if there are N turns of winding, the flux does not link all these turns ideally, but with a ratio of less than unity. Hence, we may denote the effective turns of winding by k_w N, (k_w < 1). In electrical machines theory, the factor kw is known as the winding factor,

This formulation is essential to electrical machines and is written as

(3.8)

For Ampère's law, consider a toroidal non-ferromagnetic material, shown in Figure 3.1.

Magnetic field is effectively confined to the volume within the toroid (cores). If H_r is the value of flux intensity at the average value path of radius r, it may be expressed as

(3.9)

or

(3.10)

The magnetic field is described by its magnetic flux density, denoted by a vector quantity . For different non-ferrous material, B is related to the magnetic field intensity H by the constant μ₀ given by μ₀ = 4π × 10^{− 7}H/m

(3.11)

(3.12)

(3.13)

3.3 EQUIVALENT MAGNETIC AND ELECTRIC CIRCUITS

To analyze the concept of magnetic force:

(3.14)

(3.15)

where

The constant of proportionality between and φ is related by reluctance ℜ, given by

(3.16)

The relation between the variables may be modeled by the equivalent magnetic circuit seen in Figure 3.2.

The flux leakage λ of the toroidal core

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

So, using equivalent electric circuit

(3.22)

(3.23)

3.4 OVERVIEW OF MAGNETIC MATERIALS

Examples of magnetic materials with high values of relative permeability μ_r are cobalt, aluminum, tungsten, and nickel. These are called ferromagnetic materials because they are relatively easy to magnetize. These ferromagnetic materials are classified as hard or soft materials. Soft materials include: soft steel, iron (Figure 3.3), nickel, cobalt, some rare-earth elements, and many alloys of elements. Hard ferromagnetic materials such as copper–nickel alloys and chromium–steel alloys are alloyed with aluminum, nickel, and cobalt.

3.5 HYSTERESIS LOOPS AND HYSTERESIS LOSSES IN FERROMAGNETIC MATERIALS

For ferromagnetic materials, the magnetizing curve shown in Figure 3.4 is not reversible; that is, if H is increased until the material is saturated, when H is reduced again the value of B does not reduce to zero along the same line. Figure 3.5 illustrates a hysteresis loop.

In Figure 3.5 [11], the point 0, where the axes cross, represents a ferromagnetic material that is unmagnetized. As current in the coil increases, H also increases and between points 0 and 1, flux density, B increases following the magnetization curve. At point 1, the material has reached saturation and B no longer increases. To demagnetize the ferromagnetic material, we gradually reduce the current in the coil. As stated earlier, the graph will not follow the same path it did when the current increased; instead it goes from point 1 through point 2 then down to point 3.

At point 2, H = 0, therefore the current has reached zero, but there is still some remnant flux density so that the material is still partially magnetized. The current is now reversed so that H is in the opposite direction as it was before; during this process, H has a negative value, therefore when the current is increased, the value of H reduces. At point 3, the material is finally demagnetized and the value of H at this point is called the coercive force.

If the reversed current increases further, we reach point 4, where the material saturates so that the magnetic poles of the domains face in the opposite direction to those at point 1. The reversed current is now reduced and reaches zero at point 5; however, once again, some flux remains. If the current is now increased in the original direction, all the flux would have gone at point 6 and saturation is reached once more at point 1.

3.6 DEFINITIONS

• Residual magnetic flux density (B_res): When H is equal to zero, there is a residual value of magnetic flux density B_res in the core. The magnitude of B depends on the material. Thus the residual flux-density effect creates a permanent magnet.
• Coercive force or coercivity: This is the negative value of magnetic field intensity (H_c) required to reduce the flux density B to zero.
• Saturation: Any further increase in H in the negative direction causes the magnetic core to be magnetized with opposite polarity. Increasing the current in a negative direction further results in saturation.
• Hysteresis loop: As the direction of the current in the core reverses back and forth, a loop is formed; this is referred to as a hysteresis loop.

3.7 MAGNETIC CIRCUIT LOSSES

Magnetic materials are subject to time-varying flux, which normally leads to energy loss in the material in the form of magnetic losses. Such magnetic losses are called iron or core losses. There is no core loss if flux does not vary with time.

Core losses are defined as the sum of hysteresis and eddy-current losses, given by

(3.24)

where K_e = proportionate constant depending on the core materials, f = supply frequency, and K_n = property dependency on the magnetic material.

1. Hysteresis Losses

According to Charles Steinmetz, hysteresis loss caused by energy needed to cause the reversal in the hysteresis loop is determined empirically as

(3.25)

where V is the volume of ferromagnetic material,  f is the frequency (Hz), K_n is the property called dependency on the magnetic material. K_n for soft, 1 mm, silicon, sheet steel are 0.025, 0.001, 0.0001, respectively. B_m is maximum flux density and n is the Steinmetz exponent, which it varies from 1.5 to 2.5 or (2.0).

2. Eddy-Current Losses

Because iron is a conductor, time-varying magnetic fluxes induce opposite voltage and currents called eddy currents, which circulate within the core. The undesirable circulating current flow around the flux core is so large that it can demagnetize the magnets. Thus, eddy currents establish flux that opposes the original change imposed by the core.

The equation for the eddy-currents loss calculation is

(3.26)

where K_e is the proportionate constant depending upon the core materials, V is the volume of ferromagnetic material (m³), t_i is the lamination thickness, and B_m is the maximum flux density (T).

3. I²R Losses

I²R losses are found in windings of the machine. By convention, these losses are computed on the basis of the DC resistance of the winding at 75°C. I²R losses depend on the effective resistance of the winding under the operating frequency and flux conditions.

4. Mechanical Losses

Mechanical losses consist of brush and bearing friction and windage losses caused by running a machine at high speed while unloaded and unexcited. Frequently these losses are lumped with core losses and determined at the same time.

5. Stray Losses

Stray losses arise from nonuniform current distribution in the copper winding. There is no formula for its calculation. The usual approach is to take 1 percent of the output of the DC machine. For synchronous and induction machines, we can find it through testing.

In general, the total losses contribute to determining a machine's efficiency, and this is used to compare different electromagnetic-based machines. The contribution of these losses for generators, motors, and transformers is usually determined to calculate their efficiency.

6. Magnetic Fringing

In practice, not all the magnetic flux produced in a magnetic circuit will be concentrated within the core. Apart from the leakage flux, which will appear in the surrounding free space, if a gap exists within the magnetic circuit, the flux tends to spread out, as shown in Figure 3.6 [11]; this effect is known as fringing.

3.8 PRODUCING MAGNETIC FLUX IN AIR GAP

The production of magnetic flux is confined to a curved magnetic path in a toroid, as seen in Figure 3.7 [6]. The creation of flux in an air gap is also necessary for electrifying.

With an iron gap

(3.27)

(3.28)

(3.29)

If φ is the flux in the core of the toroid, are the reluctances of the core and the air gap, respectively then

(3.30)

3.9 RECTANGULAR-SHAPED MAGNETIC CIRCUITS

Let us consider a simple magnetic circuit, as shown in Figure 3.8 [12], with a single core material having uniform cross-sectional area A and mean length of flux path l. Reluctance offered to the flow of flux is ℜ. The corresponding electrical representation is rather simple. The equivalent electrical circuit is also drawn beside the magnetic circuit, as shown in Figure 3.9 for comparison.

Although in the actual magnetic circuit, there is no physical connection of the winding and the core, in the magnetic-circuit representation, magnetomotive force (mmf) and reluctance are shown to be connected.

(3.31)

(3.32)

(3.33)

(3.34)

(3.35)

(3.36)

where and , NI is mmf in ampere-turns, φ is flux in weber, and ℜ is reluctance.

Let permanence of the magnetic circuit

(3.37)

therefore

(3.38)

R can be connected in series or parallel (Figure 3.10) so that the equivalent reluctance is ℜ_eq.

For a system of n series reluctances,

(3.39)

For reluctances in parallel

(3.40)

or permanence

(3.41)

if

(3.42)

and λ = Nφ

If

but

therefore

(3.43)

The inductance L is defined as the flux linkage per ampere of current flow in the coil. It is measured in henrys (H).

Illustrative Problems and Examples

1. What conditions are necessary for a magnetic field to produce a force on a wire?

Solution:

F = BILsin θ

1. 1. There must be a relative angle between the current and the magnetic field.
   2. The electric current must be flowing in the wire path.
   3. The wire must be placed within the magnetic flux of the field.

2. Distinguish between hysteresis and eddy-current loss and describe how to minimize eddy-current loss in a core.

Solution:

• Hysteresis loss is due to the heating of the core as a result of the material molecular-structure reversal, which occurs as the magnetic flux alternates. Example: The power required for the continuous reversal of the elementary magnets of which iron is composed can be minimized using a core with small areas of hysteresis loop.
• Eddy-current loss is due to continuous induced current path in the coil resulting in excessive heating of the coil. It can be minimized by using a core material of high resistivity. It can also be minimized by using some thin sheets of lamination for the core.

3. What is the equivalent expression for an induced voltage?

Solution:

The expression for induced voltage can be written as:

• E = −N(dΦ/dt)
• where E = induced voltage
• N = number of turns
• (dΦ/dt) = rate of change of flux

4. Write the expression for reluctance. What is its unit?

Solution:

The expression for reluctance can be written as:

• reluctance (R) = l/(μ × A)
• where μ = permeability of the material
• A = cross-sectional area
• l = length (mean length).
• The unit is ampere-turn/weber.

5. A magnetic circuit has a continuous core of a ferromagnetic material. The coil is supplied from a battery and draws a certain amount of exciting current, producing flux in the core. If an air gap is then introduced in the core, the exciting current will:

   (a) increase, (b) remain the same, (c) decrease, or (d) become zero.

Solution:

The air gap will not change the exciting current. So, the exciting current remains the same. This is because the air gap is introduced to solve the problem of excessive flux produced by the high level of current.

6. Explain hysteresis in terms of magnetic domain theory.

Solution:

Hysteresis is loss due to the heating of the core as a result of the material molecular structure reverse, which occurs as the magnetic flux alternates.

7. What are eddy-current losses? What can be done to minimize these losses?

Solution:

Eddy-current loss is due to continuous induced current in the core. It results in excessive heating of the core. It can be minimized by using a core material of high resistivity or by using a thin sheet of lamination for the core.

8. What conditions are necessary for a magnetic field to produce a voltage on a wire?

Solution:

The conditions for a magnetic field to produce voltage on a wire is that the conductor must be at an angle with the magnetic field, i.e.,

9. What is Ampère's and Lenz's law?

Solution:

Ampere's law states that the line integral of the magnetic field intensity (H) around a closed path is equal to the total current enclosed by the path.

For a conductor in circular form

Lenz's law states that the direction-induced emf in the coil is such that it tends to oppose the motion of the flux producing it.

3.10 CHAPTER SUMMARY

The magnetic effect of electric current and its applications were the central objectives of this chapter. The centrality of Maxwell's equations to the development of various electromagnetic devices cannot be overemphasized. Working examples and applications of Maxwell's equations to a power energy system were provided. This is important because from generation through transmission down to distribution, critical components of the system are built and operated to provide quality and reliable power based on the principle of magnetic effect of electric current. This fundamental knowledge is essential to students who want to understand next-generation electric power systems as many of the components for power generation, protection, measurement, regulation, and security in a smart/microgrid environment are based on this essential principle. It is therefore essential foundational knowledge the student must embrace. The concepts described in this chapter are used in modeling electromechanical energy processing systems and components.