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 Chapter 4 ROTATING MACHINES 4 0 INTRODUCTION The most numerous and the most widely used electromechanical device in existence is the magnetic field type rotating machine Rotating machines occur in many different types depending on the nature of the electrical and mechanical systems to be coupled and on the coupling characteristics desired The primary purpose of most rotating machines is to convert energy between electrical and mechanical systems either for electric power generation or for the production of mechanical power to do useful tasks These machines range in size from motors that consume a fraction of a watt to large generators that produce 109 W In spite of the wide variety of types and sizes and of methods of construction which vary greatly most rotating machines fall into two classes defined by their geometrical structures namely smooth airgap and salient pole The analysis of the electromechanical coupling systems in rotating machines can thus be reduced to the analysis of two configurations regardless of the size or type of machine As is to be expected some machines do not fit our classification they are not numerous however and their analyses can be performed by making simple changes in the models and techniques presented in this chapter After defining the two classes of machine geometry smooth air gap and salient pole we establish the conditions necessary for average power conversion and use them as a basis for defining different types of machine We also derive the equations of motion for the different machine types and solve them in the steady state to describe the machines principal characteristics The behavior of machines under transient conditions is covered in Chapter 5 Before starting the treatment of machines it is important to recognize several significant points First as is evident from the treatment a rotating machine is but one specific embodiment of a more general class of electromechanical devices defined in Chapter 3 and as such is conceptually quite simple In a practical configuration such as a polyphase machine the Rotating Machines number of terminal pairs is great enough to make the mathematical description seem lengthy In no case should mathematical complexity be mistaken for conceptual difficulty The analysis of rotating machines is conceptually simple and mathematically complex As our treatment unfolds it will become clear that there are geometrical and mathematical symmetries that imply simplification techniques These techniques have been developed to a high degree of sophistication and are essential in the analysis of machine systems Because our interest here is in the basic physical processes we forego the special techniques and refer the reader to other texts 4 1 SMOOTH AIR GAP MACHINES All rotating machines that fit in the smooth air gap classification can be represented schematically by a physical structure like that shown in end Stator magnetic axis slot rcoil or tooth Rotor Rotor Rotor to Fig 4 1 1a Geometry of smooth air gap rotating machine showing distributed windings on stator and rotor of a single phase machine See for example D C White and H H Woodson ElectromechanicalEnergyConversion Wiley New York 1959 Chapters 4 and 7 to 10 Smooth Air Gap Machines view in Fig 4 1 1a Pictures of a stator and a rotor that fall into this classification are shown in Figs 4 1 2 and 4 1 3 In the structure of Fig 4 1 1a conductors are laid in axial slots that face the air gap The number of conductors in each slot depends on the type and size of the machine and varies from 1 in large turbogenerators to 10 to 12 in small induction machines The conductors of one circuit on one member are in series or series parallel connection at the ends of the machine Note the end turns in Figs 4 1 2 and 4 1 3 The circuits are arranged so that a current in one winding will produce the antisymmetrical pattern about an axial plane indicated by the dots and crosses in Fig 4 1 1a This axial plane is the plane of symmetry of the magnetic field produced by the currents and is therefore called the magnetic Stator axis axis Fig 4 1 1b Schematic rep resentation of the induc tors constituting the rotor and stator windings shown axis The stator and rotor magnetic axes are shown in a in Fig 4 1 1 The example in Fig 4 1 1 has only one circuit winding on the stator and one circuit on the rotor Most machines have more than one circuit on each member In this case a slot will usually contain conductors from different circuits Nonetheless the description given fits each circuit on the rotor or stator The rotor is free to rotate and its instantaneous angular position 0 is by convention the displacement of the rotor magnetic axis with respect to the stator magnetic axis The structure of Fig 4 1 la is called smooth air gap because it can be modeled mathematically with sufficient accuracy by assuming that the magnetic path seen by each circuit is independent of rotor position Such a model neglects the effects of slots and teeth on magnetic path as the angle is changed In a real machine see Figs 4 1 2 and 4 1 3 the slots and teeth are relatively smaller than those shown in Fig 4 1 1a Moreover special construction techniques such as skewing the slots of one member slightly with respect to a line parallel to the axis minimize these effects In any case the essential properties of a machine can be obtained with good accuracy by using a smooth air gap model but slot effects are always present as second order effects in machine terminal characteristics and as first order problems to machine designers For constructional details of rotating machines see for example A E Knowlton ed Standard Handbook for Electrical Engineeers 9th ed McGraw Hill New York 1957 Sections 7 and 8 This also includes numerous references to more detailed design treatments Y lllll I Rotating Machines Fig 4 1 2 Stator armature of an induction motor This is an example of a smooth airgap stator Courtesy of Westinghouse Electric Corporation 4 1 1 Differential Equations In terms of
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