MIT OpenCourseWare http://ocw.mit.edu Continuum Electromechanics For any use or distribution of this textbook, please cite as follows: Melcher, James R. Continuum Electromechanics. Cambridge, MA: MIT Press, 1981. Copyright Massachusetts Institute of Technology. ISBN: 9780262131650. Also available online from MIT OpenCourseWare at http://ocw.mit.edu (accessed MM DD, YYYY) under Creative Commons license Attribution-NonCommercial-Share Alike. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.4Electromechanical Kinematics:Energy-Conversion Models andProcesses4.1 ObjectivesBeginning with this chapter, progressively more electromechanical "degrees of freedom" are consid-ered. The subject of electromechanical kinematics is first because then the relative mechanical motionsas well as the paths and trajectories of charges and currents are known from the outset. The mechanicsinvolves rigid-body translations or rotations, while charges and currents might be constrained by elec-trodes and wires. Processes in this category can be represented by lumped-parameter models. The fieldapproach of this chapter provides the basis for conceptualizing and interrelating such interactions,for appreciating energy conversion limitations, and for deriving the parameters used in lumped-param-eter models.The representation of total forces and torques in terms of Maxwell stresses is developed in Sec. 4.2,followed in Sec. 4.3 by a classification of common types of energy converters, based on the fundamentalfield interactions. An extension of the transfer relations found in Secs. 2.16 and 2.19 to describeregions occupied by specified distributions of charge and current is made in Secs. 4.5 and 4.8.. Althoughthis chapter is concerned with modeling specific interactions, it is the technique for representingthese systems that is the message. Section 4.4 exemplifies the notation and strategy underlying themethodical formulation of complex systems in not only this chapter, but those to follow. Of the remain-ing sections, only one does not pertain to a specific class of devices. Section 4.12 lends some for-mality to the philosophy underlying quasi-one-dimensional models. Such approximations retain nonlinearinteractions and are illustrated in Secs. 4.13 and 4.14. By contrast, Secs. 4.4, 4.6 -4.9 and 4.11are concerned with field models that are naturally linear, or are linearized. Formally, the linearizedmodel, in which products of amplitudes are ignored compared to terms that are linear in the amplitudes,is the zero-order approximation in an amplitude-parameter expansion for the exact solution. Similarly,the quasi-one-dimensional model is a zero-order approximation to an expansion in a space-rate parameter.The analogies that exist between electric and magnetic field interactions is a theme throughoutthe chapter. This is clear in Sec. 4.3. But a thoughtful comparison of the characteristics of thed-c magnetic machine, considered in more detail in Sec. 4.10, with those of the Van de Graaff machine inSec. 4.14 is worth while.An overview of the chapter is given in Sec. 4.15.4.2 Stress, Force and Torque in Periodic SystemsThe configurations shown in Fig. 4.2.1 typify devices exploiting force or torque producing inter-actions between spatially periodic excitations on a "stator" structure and spatially periodic con-strained or induced sources on a "rotor." In each of these, the interaction is across an air gap, aregion having the electromagnetic characteristics of free space. The planar configuration ofFig. 4.2.1a might represent a linear motor or generator with the relevant force between "stator" (above)and "rotor" (below) z-directed, or it might be a developed model for the cylindrical geometry ofFig. 4.2.1c9(appropriate in the limit where the air-gap spacing is small compared to the radius of therotor). Figure 4.2.1b shows the cross section of either a planar "slab" with the interaction acrosstwo air gaps, or a cylindrical structure having an annular air gap. In either case the relevant netforce is z-directed.. .. .. . . . . . . . . . . . . . .. . . . . .. .. .. .Xor".-•z r T or 'Trrot --------.1Z5^L - "... r SI- °7 .... 7T4T.... .S 's,(a) (b) (c)Fig. 4.2.1. Typical "air-gap" configurations in which a force or torque on a rigid "rotor" resultsfrom spatially periodic sources interacting with spatially periodic excitations on a rigid"stator." Because of the periodicity, the force or torque can be represented in terms of theelectric or magnetic stress acting at the air-gap surfaces S1: (a) planar geometry or devel-oped model; (b) planar or cylindrical beam; (c) cylindrical rotor.Secs. 4.1 & 4.2The total force acting in the z-direction on the "rotor" of Fig. 4.2.1a is conveniently deteby integrating the Maxwell stress, in accordance with Eq. 3.9.4, over the surface S enclosing a poof the rotor having one fundamental length of periodicity. The portion Sl of this surface is at aarbitrary plane x = constant in the air gap. Because the fields and hence the stress components Tzare periodic in z, thq contributions to the integration of the stress over surfaces S2 and S4 cancregardless of where S1 is located in the air gap. The contribution to the integration over S3 canvanish for several reasons. The rolor mny be perfectly permeable, of infinite permittivity or in-finitely conducting, in which case H or E is zero on S3. In Cartesian coordinates, the fields assated with excitations that are periodic in the z-direction decay in the x direction and if S3 is wremoved from the air gap, the contribution on S3 asymptotically vanishes. Yet another possibilitythat the planar model really is a.developed model for the cylindrical configuration of Fig. 4.2.1cin which case the surface S is "pie" shaped and the section S3 does not exist. In any of these cathe z-directed force acting on the rotor of Fig. 4.2.1a is simplyf = A z S (1)where A is the y-z area of the air gap and Tzx is the magnetic or electric stress tensor, as the cmay be. The brackets indicate a spatial average is taken, as discussed in Sec. 2.15.There is no question as to which of the stress tensors in Table 3.10.1 should be used. As dcussed in Sec. 3.10, in the free-space region of the air gap, all of the magnetic and all of the etric stress tensors agree.If Fig. 4.2.1b represents a planar layer, then there are stress contributions from surfaces and S3, and the net force acting on a section of the layer
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