MIT OpenCourseWare http ocw mit edu Electromechanical Dynamics For any use or distribution of this textbook please cite as follows Woodson Herbert H and James R Melcher Electromechanical Dynamics 3 vols Massachusetts Institute of Technology MIT OpenCourseWare http ocw mit edu accessed MM DD YYYY License Creative Commons Attribution NonCommercial Share Alike For more information about citing these materials or our Terms of Use visit http ocw mit edu terms ELECTROMECHANICAL DYNAMICS Part I Discrete Systems ELECTROMECHANICAL DYNAMICS Part I Discrete Systems HERBERT H WOODSON Philip Sporn Professor of Energy Processing Departments of Electrical and Mechanical Engineering JAMES R MELCHER Associate Professor of Electrical Engineering Department of Electrical Engineering both of Massachusetts Institute of Technology JOHN WILEY SONS INC NEW YORK LONDON SYDNEY To our parents PREFACE Part I Discrete Systems In the early 1950 s the option structure was abandoned and a common core curriculum was instituted for all electrical engineering students at M I T The objective of the core curriculum was then and is now to provide a foundation in mathematics and science on which a student can build in his professional growth regardless of the many opportunities in electrical engineering from which he may choose In meeting this objective core curriculum subjects cannot serve the needs of any professional area with respect to nomenclature techniques and problems unique to that area Specialization comes in elective subjects graduate study and professional activities To be effective a core curriculum subject must be broad enough to be germane to the many directions an electrical engineer may go professionally yet it must have adequate depth to be of lasting value At the same time the subject must be related to the real world by examples of application This is true because students learn by seeing material in a familiar context and engineering students are motivated largely by the relevance of the material to the realities of the world around them In the organization of the core curriculum in electrical engineering at M I T electromechanics is one major component As our core curriculum has evolved there have been changes in emphasis and a broadening of the topic The basic text in electromechanics until 1954 when a new departure was made was Electric Machinery by Fitzgerald and Kingsley This change produced ElectromechanicalEnergy Conversion by White and Woodson which was used until 1961 At that time we started the revision that resulted in the present book During this period we went through many versions of notes while teaching the material three semesters a year Our objective has always been to teach a subject that combines classical mechanics with the fundamentals of electricity and magnetism Thus the subject offers the opportunity to teach both mechanics and electromagnetic theory in a context vital to much of the electrical engineering community Our choice of material was to some extent determined by a desire to give the student a breadth of background sufficient for further study of almost any type of electromechanical interaction whether in rotating machinery Preface plasma dynamics the electromechanics of biological systems or magnetoelasticity It was also chosen to achieve adequate depth while maintaining suitable unity but most important examples were chosen that could be enlivened for the engineering student interested in the interplay of physical reality and the analytical model There were many examples from which to choose but only a few satisfied the requirement of being both mathematically lucid and physically demonstrable so that the student could push it or see it and directly associate his observations with symbolic models Among the areas of electrical engineering electromechanics excels in offering the opportunity to establish that all important feel for a physical phenomenon Properly selected electromechanical examples can be the basis for discerning phenomena that are remote from human abilities to observe Before discussing how the material can be used to achieve these ends a review of the contents is in order The student who uses this book is assumed to have a background in electrostatics and magnetostatics Consequently Chapter 1 and Appendix B are essentially a review to define our starting point Chapter 2 is a generalization of the concepts of inductance and capacitance that are necessary to the treatment of electromechanical systems it also provides a brief introduction to rigid body mechanics This treatment is included because many curricula no longer cover mechanics other than particle mechanics in freshman physics The basic ideas of Chapter 2 are repeated in Chapter 3 to establish some properties of electromechanical coupling in lumped parameter systems and to obtain differential equations that describe the dynamics of lumped parameter systems Next the techniques of Chapters 2 and 3 are used to study rotating machines in Chapter 4 Physical models are defined differential equations are written machine types are classified and steady state characteristics are obtained and discussed A separate chapter on rotating machines has been included not only because of the technological importance of machines but also because rotating machines are rich in examples of the kinds of phenomena that can be found in lumped parameter electromechanical systems Chapter 5 is devoted to the study with examples of the dynamic behavior of lumped parameter systems Virtually all electromechanical systems are mathematically nonlinear nonetheless linear incremental models are useful for studying the stability of equilibria and the nature of the dynamical behavior in the vicinity of an equilibrium The second half of this chapter develops the classic potential well motions and loss dominated dynamics in the context of electromechanics These studies of nonlinear dynamics afford an opportunity to place linear models in perspective while forming further insights on the physical significance of for example flux conservation and state functions Preface Chapter 6 represents our first departure from lumped parameter systems into continuum systems with a discussion of how observers in relative motion will define and measure field quantities and the related effects of material motion on electromagnetic fields It is our belief that de rotating machines are most easily understood in this context Certainly they are a
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