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

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Fitzgerald, A. E. (Arthur Eugene), 1909- Electric machinery / A. E. Fitzgerald, Charles Kingsley, Jr., Stephen D. Umans. - -6th ed. 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). resistors, capacitors, and/or inductors and give the values of the components (we will use these in the next

From Eq. 1.51, the rms value of a sine wave can be shown to be 1 / ~/2 times its peak value. Thus the rms value of the induced voltage is

where Volmag is the volume of the magnet, Volair gap is the air-gap volume, and the minus sign arises because, at the operating point of the magnetic circuit, H in the magnet (Hm) is negative. 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. The late Charles Kingsley, Jr. was Professor in the Department of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology, from which he received the S.B. and S.M. degrees. During his career, he spent time at General Electric, Boeing, and Dartmouth College. In addition to Electric Ma- chinery, Professor Kingsley was co-author of the textbook Magnetic Circuits and Transformers. After his retirement, he continued to participate in research activities at M.I.T. He was an active member and Fellow of the IEEE, as well as its predecessor society, the American Institute of Electrical Engineers.

It should be emphasized that, in addition to MATLAB, a number of other numerical-analysis packages, including various spread-sheet packages, are available which can be used to perform calculations and to plot in a fashion similar to that done with MATLAB. If MATLAB is not available or is not the package of preference at your institution, instructors and students are encouraged to select any package with which they are comfortable. Any package that simplifies complex calculations and which enables the student to focus on the concepts as opposed to the mathematics will do just fine. Notice that as H is decreased from its maximum value to zero, the flux density decreases but not to zero. This is the result of the relaxation of the orientation of the magnetic moments of the domains as described above. The result is that there remains a remanant magnetization when H is zero. coercivity, Hc, (approximately - 4 9 kA/m). The remanent magnetization, Br, corresponds to the flux density which would now includes interesting examples which would have otherwise been too mathemat- ically tedious. Similarly, there are now end-of-chapter problems which are relatively straightforward when done with MATLAB but which would be quite impractical if done by hand. Note that each MATLAB example and practice problem has been no- tated with the symbol ~ , found in the margin of the book. End-of-chapter problems which suggest or require MATLAB are similarly notatated. For a magnetic circuit composed of magnetic material of constant magnetic permeability or which includes a dominating air gap, the relationship between q~ and i will be linear and we can define the inductance L as

PROBLEM SOLUTIONS: Chapter 1

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 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. When a magnetic field varies with time, an electric field is produced in space as determined by Faraday's law: 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. I N T R O D U C T I O N TO M A G N E T I C C I R C U I T S The complete, detailed solution for magnetic fields in most situations of practical engineering interest involves the solution of Maxwell 's equations along with various constitutive relationships which describe material properties. Although in practice exact solutions are often unattainable, various simplifying assumptions permit the attainment of useful engineering solutions. 1

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. In general the flux linkage of a coil is equal to the surface integral of the normal component of the magnetic flux density integrated over any surface spanned by that coil. Note that the direction of the induced voltage e is defined by Eq. 1.26 so that if pts) Draw a schematic of this scenario. For now, consider the load as a single impedance, Z, at some In SI units, the magnetic stored energy W is measured in j o u l e s (J). For a single-winding system of constant inductance, the change in magnetic in transformers and rotating machines. The characteristics of ferromagnetic materials are described in Sections 1.3 and 1.4. For the present we assume that/Zr is a known constant, although it actually varies appreciably with the magnitude of the magnetic flux density.For practical magnetic materials (as is discussed in Sections 1.3 and 1.4), Bc and Hc are not simply related by a known constant permeability/z as described by Eq. 1.7. In fact, Bc is often a nonlinear, multivalued function of Hc. Thus, although Eq. 1.10 continues to hold, it does not lead directly to a simple expression relating the mmf and the flux densities, such as that of Eq. 1.11. Instead the specifics of the nonlinear Bc-He relation must be used, either graphically or analytically. However, in many cases, the concept of constant material permeability gives results of acceptable engineering accuracy and is frequently used. 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: Ferromagnetic materials, typically composed of iron and alloys of iron with cobalt, tungsten, nickel, aluminum, and other metals, are by far the most common mag- netic materials. Although these materials are characterized by a wide range of prop- erties, the basic phenomena responsible for their properties are common to them all.

I I So lu t ion a. Since the core permeability is assumed infinite, H in the core is negligible. Recognizing Figure 1.19 shows the magnetization characteristics for a few common permanent magnet materials. Alnico 5 is a widely used alloy of iron, nickel, aluminum, and cobalt, originally discovered in 1931. It has a relatively large residual flux density. Alnico 8 has a lower residual flux density and a higher coercivity than Alnico 5. Hence, it is less subject to demagnetization than Alnico 5. Disadvantages of the Alnico materials are their relatively low coercivity and their mechanical brittleness.

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- and from Eq. 1.20, with the reluctance of the core neglected and assuming that Ac = Ag, the core flux 4~ is Notice that since " at time t" the corresponding values are ~o" and t~. the current is t~, the hysteresis loop is multivalued, it is necessary to be careful to pick the rising-flux values (tp' in the figure) from the rising-flux portion of the hysteresis loop; similarly the falling-flux portion of the hysteresis loop must be selected for the falling-flux values (~o" in the figure). 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. larger than the value of 2900 corresponding to a flux level of 1.8 T. (,.0m) 1 c = ( 6 + 6 + 8 + 8 ) in \ 3 9 " 4 i n = 0 . 7 1 m