I I So lu t ion a. Since the core permeability is assumed infinite, H in the core is negligible. Recognizing n The analysis of single-phase induction motors has been expanded to cover the general case in which the motor is running off both its main winding and its auxiliary winding (supplied with a series capacitor). Professor Kingsley first asked this author to participate in the fourth edition of Electric Machinery; the professor was actively involved in that edition. He participated in an advisory capacity for the fifth edition. Unfortunately, Professor Kingsley passed away since the publication of the fifth edition and did not live to see the start of the work on this edition. He was a fine gentleman, a valued teacher and friend, and he is missed.

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). Although very little in the way of sophisticated mathematics is required of the reader of this book, the mathematics can get somewhat messy and tedious. This is especially true in the analyis of ac machines in which there is a significant amount of algebra involving complex numbers. One of the significant positive developments in the last decade or so is the widespread availability of programs such as MATLAB which greatly facilitate the solution of such problems. MATLAB is widely used in many universities and is available in a student version. 2

PROBLEM SOLUTIONS: Chapter 1

The relationship between the magnetic field intensity H and the magnetic flux density B is a property of the material in which the field exists. It is common to assume a linear relationship; thus Power electronics are a significant component of many contemporary electric- machine applications. This topic is included in Chapter 10 of this edition of Electric Machinery in recognition of the fact that many electric-machinery courses now include a discussion of power electronics and drive systems. However, it must be emphasized that the single chapter found here is introductory at best. One chapter cannot begin to do justice to this complex topic any more than a single chapter in a power-electronics text could adequately introduce the topic of electric machinery. Clearly, the website will be a living document which will evolve in response to input from users. I strongly urge each of you to visit it frequently and to send in suggestions, problems, and examples, and comments. I fully expect it to become a valuable resource for users of Electric Machinery around the world. In Example 1.9, we found an expression for the flux density in the air gap of the magnetic circuit of Fig. 1.17: Drive-systems based upon power electronics permit a great deal of flexibility in the control of electric machines. This is especially true in the case of ac machines which used to be found almost exclusively in applications where they were supplied from the fixed-frequency, fixed-voltage power system. Thus, the introduction to power electronics in Chapter 10 is followed by a chapter on the control of electric machines.

Transformers are wound on closed cores like that of Fig. 1.1. However, energy conversion devices which incorporate a moving element must have air gaps in their magnetic circuits. A magnetic circuit with an air gap is shown in Fig. 1.2. When the air-gap length g is much smaller than the dimensions of the adjacent core faces, the magnetic flux ~ will follow the path defined by the core and the air gap and the techniques of magnetic-circuit analysis can be used. If the air-gap length becomes excessively large, the flux will be observed to "leak out" of the sides of the air gap and the techniques of magnetic-circuit analysis will no longer be strictly applicable.

As applied to the magnetic circuit of Fig. 1.1, the source of the magnetic field in the core is the ampere-turn product N i. In magnetic circuit terminology N i is the magnetomotive force (mmf) .T" acting on the magnetic circuit. Although Fig. 1.1 shows only a single coil, transformers and most rotating machines have at least two windings, and N i must be replaced by the algebraic sum of the ampere-turns of all the windings. Figure 1.12 Exciting rms voltamperes per kilogram at 60 Hz for M-5 grain-oriented electrical steel 0.012 in thick. (Armco Inc.)

coercivity, Hc, (approximately - 4 9 kA/m). The remanent magnetization, Br, corresponds to the flux density which would In practical systems, the magnetic field lines "fringe" outward somewhat as they cross the air gap, as illustrated in Fig. 1.4. Provided this fringing effect is not excessive, the magnetic-circuit concept remains applicable. The effect of thesefringingfields is to increase the effective cross-sectional area Ag of the air gap. Various empirical methods have been developed to account for this effect. A correction for such fringing fields in short air gaps can be made by adding the gap length to each of the two dimensions making up its cross-sectional area. In this book the effect of fringing fields is usually ignored. If fringing is neglected, Ag = Ac. on both physical insight and analytical techniques. Mastery of the material covered will provide both the basis for understanding many real-world electric-machinery applications as well as the foundation for proceeding on to more advanced courses in electric machinery design and control. The exciting new sixth edition of “Electric Machinery” has been extensively updated while retaining the emphasis on fundamental principles and physical understanding that has been the outstanding feature of this classic book.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- remain in a closed magnetic structure, such as that of Fig. 1.1, made of this material, if the applied mmf (and hence the magnetic field intensity H) were reduced to zero. However, although the M-5 electrical steel also has a large value of remanent magneti- zation (approximately 1.4 T), it has a much smaller value of coercivity (approximately - 6 A/m, smaller by a factor of over 7500). The coercivity Hc corresponds to the value of magnetic field intensity (which is proportional to the mmf) required to reduce the 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. The term which multiplies the mmfis known as thepermeance 7 -9 and is the inverse of the reluctance; thus, for example, the total permeance of a magnetic circuit is that rotor and stator iron have infinite permeability (# --+ c~), find the air-gap flux ~b and flux