Fitzgerald & Kingsley's Electric Machinery (IRWIN ELEC&COMPUTER ENGINERING)

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field problem with simple geometry to a magnetic circuit model. Our limited pur- pose in this section is to introduce some of the concepts and terminology used by engineers in solving practical design problems. We must emphasize that this type of thinking depends quite heavily on engineering judgment and intuition. For example, we have tacitly assumed that the permeabil i ty of the "iron" parts of the magnetic circuit is a constant known quantity, although this is not true in general (see Sec- tion 1.3), and that the magnetic field is confined soley to the core and its air gaps. Although this is a good assumption in many situations, it is also true that the wind- ing currents produce magnetic fields outside the core. As we shall see, when two or more windings are placed on a magnetic circuit, as happens in the case of both transformers and rotating machines, these fields outside the core, which are referred The power at the terminals of a winding on a magnetic circuit is a measure of the rate of energy flow into the circuit through that particular winding. The power, p, is determined from the product of the voltage and the current

However, in electromechanical energy conversion devices, inductances are often time- varying, and Eq. 1.41 must be written as The fraction of the mmfrequired to drive flux through each portion of the magnetic circuit, commonly referred to as the m m f drop across that portion of the magnetic circuit, varies in proportion to its reluctance (directly analogous to the voltage drop across a resistive element in an electric circuit). From Eq. 1.13 we see that high material permeability can result in low core reluctance, which can often be made much smaller than that of the air gap; i.e., for ( I z A c / l c ) >> ( l z o A g / g ) , "R.c << ~'P~g and thus "~-tot ~ ']'~g. In this case, the reluctance of the core can be neglected and the flux and hence B can be found from Eq. 1.16 in terms of f and the air-gap properties alone:

The problem is quite simple: this general treatment is mathematically complex, requiring the solution of a number of simultaneous, complex algebraic equations. This, however, is just the sort of problem at which programs such as MATLAB excel. Thus, this new edition of Electric Machinery includes this general treatment of single-phase induction machines, complete with a worked out quantitative example and end-of-chapter problems. AC E X C I T A T I O N In ac power systems, the waveforms of voltage and flux closely approximate sinusoidal functions of time. This section describes the excitation characteristics and losses associated with steady-state ac operation of magnetic materials under such operating conditions. We use as our model a closed-core magnetic circuit, i.e., with no air gap, such as that shown in Fig. 1.1 or the transformer of Fig. 2.4. The magnetic path length is lc, and the cross-sectional area is Ac throughout the length of the core. We further assume a sinusoidal variation of the core flux ~o(t); thus in a magnetic circuit in the absence of external excitation (such as winding currents). This is a familiar phenomenon to anyone who has afixed notes to a refrigerator with small magnets and is widely used in devices such as loudspeakers and permanent-

material flux density to zero. The significance of remanent magnetization is that it can produce magnetic flux We begin with the assumption that, for the systems treated in this book, the fre- quencies and sizes involved are such that the displacement-current term in Maxwell 's equations can be neglected. This term accounts for magnetic fields being produced in space by time-varying electric fields and is associated with electromagnetic ra- diation. Neglecting this term results in the magneto-quasistatic form of the relevant Maxwell 's equations which relate magnetic fields to the currents which produce them. Although not an electromechanical-energy-conversion device, the transformer is an important component of the overall energy-conversion process and is discussed in Chapter 2. The techniques developed for transformer analysis form the basis for the ensuing discussion of electric machinery. Example 1.9 shows that there is an immense difference between permanent- magnet materials (often referred to as hard magnetic materials) such as Alnico 5 and soft magnetic materials such as M-5 electrical steel. This difference is characterized in large part by the immense difference in their coercivities He. The coercivity can be thought of as a measure of the magnitude of the mmf required to demagnetize the material. As seen from Example 1.9, it is also a measure of the capability of the material to produce flux in a magnetic circuit which includes an air gap. Thus we see that materials which make good permanent magnets are characterized by large values of coercivity He (considerably in excess of 1 kA/m). This seventh edition of Fitzgerald and Kingsley’s Electric Machinery by Stephen Umans was developed recognizing the strength of this classic text since its first edition has been the emphasis on building an understanding of the fundamental physical principles underlying the performance of electric machines.Finally, instructors may wish to select topics from the control material of Chapter 11 rather than include it all. The material on speed control is essentially a relatively straightforward extension of the material found in earlier chapters on the individ- ual machine types. The material on field-oriented control requires a somewhat more sophisticated understanding and builds upon the dq0 transformation found in Ap- pendix C. It would certainly be reasonable to omit this material in an introductory course and to delay it for a more advanced course where sufficient time is available to devote to it. INTERNATIONAL EDITION ISBN 0-07-112193-5 Copyright ~ 2003. Exclusive rights by The McGraw-Hill Companies, Inc., for manufacture and export. This book cannot be re-exported from the country to which it is sold by McGraw-Hill. The International Edition is not available in North America. Note that Eq. 1.59 appears to indicate that one can achieve an arbitrarily large air-gap flux density simply by reducing the air-gap volume. This is not true in practice because as the flux density in the magnetic circuit increases, a point will be reached at which the magnetic core material will begin to saturate and the assumption of infinite permeability will no longer be valid, thus invalidating the derivation leading to Eq. 1.59. A magnetic circuit consists of a structure composed for the most part of high- permeability magnetic material. The presence of high-permeability material tends to cause magnetic flux to be confined to the paths defined by the structure, much as currents are confined to the conductors of an electric circuit. Use of this concept of In Eq. 1.54, the product Aclc can be seen to be equal to the volume of the core and hence the rms exciting voltamperes required to excite the core with sinusoidal can be seen to be proportional to the frequency of excitation, the core volume and the product of the peak flux density and the rms magnetic field intensity. For a magnetic material of mass density Pc, the mass of the core is AclcPc and the exciting rms voltamperes per unit mass, Pa, can be expressed as

Equation 1.2 states that the net magnetic flux entering or leaving a closed surface (equal to the surface integral of B over that closed surface) is zero. This is equivalent to saying that all the flux which enters the surface enclosing a volume must leave that volume over some other portion of that surface because magnetic flux lines form closed loops. At any given time, the value of i~ corresponding to the given value of flux can be found directly from the hysteresis loop. For example, at time t t the flux is ~o t and A simple example of a magnetic circuit is shown in Fig. 1.1. The core is assumed to be composed of magnetic material whose permeability is much greater than that of the surrounding air (/z >>/z0). The core is of uniform cross section and is excited by a winding of N turns carrying a current of i amperes. This winding produces a magnetic field in the core, as shown in the figure. chapter introduces the basic concept of electromechanical energy conversion. The fourth chapter then provides an overview of and on introduction to the various machine types. Some instructors choose to omit all or most of the material in Chapter 3 from an introductory course. This can be done without a significant impact to the understanding of much of the material in the remainder of the book.

PROBLEM SOLUTIONS: Chapter 1

Some ancillaries, including electronic and print components, may not be available to customers outside the United States. Figure 1.9 B-H loops for M-5 grain-oriented electrical steel 0.012 in thick. Only the top halves of the loops are shown here. (Armco Inc.)