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

To produce magnetic flux in the core requires current in the exciting winding known as the exciting current, i~o. 6 The nonlinear magnetic properties of the core re- quire that the waveform of the exciting current differs from the sinusoidal waveform of the flux. A curve of the exciting current as a function of time can be found graphically from the magnetic characteristics of the core material, as illustrated in Fig. 1.1 la. Since Bc and Hc are related to ~o and i~ by known geometric constants, the ac hys- teresis loop of Fig. 1.1 lb has been drawn in terms of ~o = BcAc and is i~0 = Hclc/N. Sine waves of induced voltage, e, and flux, ~o, in accordance with Eqs. 1.48 and 1.49, are shown in Fig. 1.11 a. In steady-state ac operation, we are usually more interested in the root-mean- square or rms values of voltages and currents than in instantaneous or maximum values. In general, the rms value of a periodic function of time, f ( t ) , of period T is defined as properties of the core material will have little effect on the terminal properties of the inductor. Although much of the material from the previous editions has been retained in this edition, there have been some significant changes. These include: 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.Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright (~) 2003, 1990, 1983, 1971, 1961, 1952 by The McGraw-Hill Companies, Inc. All rights reserved. Copyright renewed 1980 by Rosemary Fitzgerald and Charles Kingsley, Jr. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. 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. 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. the magnetic circuit is illustrated in this section and will be seen to apply quite well to many situations in this book. 2

One additional benefit is derived from the introduction of MATLAB into this edition of Electric Machinery. As readers of previous editions will be aware, the treatment of single-phase induction motors was never complete in that an analytical treatment of the general case of a single-phase motor running with both its main and auxiliary windings excited (with a capacitor in series with the auxiliary winding) was never considered. In fact, such a treatment of single-phase induction motors is not found in any other introductory electric-machinery textbook of which the author is aware. The total magnetic stored energy at any given value of ~ can be found from setting X~ equal to zero: 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).

PROBLEM SOLUTIONS: Chapter 1

where Hc is average magnitude of H in the core. The direction of Hc in the core can be found from the right-hand rule, which can is negligible and Eq. 1.20 (with g replaced by the total gap length 2g) can be used to find the flux Note also that a curve of constant B-H product is a hyperbola. A set of such hyperbolas for different values of the B-H product is plotted in Fig. 1.16a. From these curves, we see that the maximum energy product for Alnico 5 is 40 kJ/m 3 and that this occurs at the point B = 1.0 T and H = - 4 0 kA/m. The magnetic circuit of Fig. 1.6a consists of an N-turn winding on a magnetic core of infinite permeability with two parallel air gaps of lengths g~ and g2 and areas A~ and A2, respectively. where Hg and Hm are the magnetic field intensities in the air gap and the magnetic material, respectively.

In Example 1.9, we found an expression for the flux density in the air gap of the magnetic circuit of Fig. 1.17: Figure 1.3 Analogy between electric and magnetic circuits. (a) Electric circuit, (b) magnetic circuit. 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. 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 The core dimensions are such that the path length of any flux line is close to the mean core length lc. As a result, the line integral of Eq. 1.5 becomes simply the scalar product Hclc of the magnitude of H and the mean flux path length Ic. Thus, the relationship between the mmf and the magnetic field intensity can be written in magnetic circuit terminology ascurrents, commonly referred to as eddy currents, which circulate in the core material and oppose changes in flux density in the material. To counteract the correspond- ing demagnetizing effect, the current in the exciting winding must increase. Thus the resultant "dynamic" B-H loop under ac operation is somewhat "fatter" than the hysteresis loop for slowly varying conditions, and this effect increases as the excita- tion frequency is increased. It is for this reason that the characteristics of electrical steels vary with frequency and hence manufacturers typically supply characteristics over the expected operating frequency range of a particular electrical steel. Note for example that the exciting rms voltamperes of Fig. 1.12 are specified at a frequency of 60 Hz. From which we see that the inductance of a winding in a magnetic circuit is propor- tional to the square of the turns and inversely proportional to the reluctance of the magnetic circuit associated with that winding.

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 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.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. Samarium-cobalt represents a significant advance in permanent magnet technol- ogy which began in the 1960s with the discovery of rare earth permanent magnet materials. From Fig. 1.19 it can be seen to have a high residual flux density such as is found with the Alnico materials, while at the same time having a much higher coer- civity and maximum energy product. The newest of the rare earth magnetic materials is the neodymium-iron-boron material. It features even larger residual flux density, coercivity, and maximum energy product than does samarium-cobalt. In general, these losses depend on the metallurgy of the material as well as the flux density and frequency. Information on core loss is typically presented in graphical form. It is plotted in terms of watts per unit weight as a function of flux density; often a family of curves for different frequencies are given. Figure 1.14 shows the core loss Pc for M-5 grain-oriented electrical steel at 60 Hz.