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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/termsChapter 7MAGNETIC DIFFUSIONAND CHARGE RELAXATION7.0 INTRODUCTIONThere are two important reasons for introducing a distributed or contin-uum description of electromechanical interactions. The most obvious is thatthe mechanical system may be an elastic or fluid medium, with the electric ormagnetic force distributed throughout. In this case the electrical and mechan-ical equations of motion are likely to have both time and space coordinates asindependent variables. As we shall see in later chapters, there is a need for thismodel when the dynamical times of interest (say the period of a sinusoidalexcitation) are on the same order as the time required for a disturbance topropagate from one extreme to another in the mechanical system.*The second reason for introducing a continuum model concerns character-istic times that arise because of competing energy storage and dissipationmechanisms in the electrical system; for example, a lumped inductance L,shunted by a resistance R, constitutes a simple magnetic field system. In theabsence of motion (which could change the inductance L) this circuit has atime constant L/R. The response of the circuit to an excitation dependsgreatly on the time constant (or period) of the excitation relative to LIR.Similarly, the behavior of magnetic field systems involving conductingmaterials depends considerably on the relative values of times associatedwith the motion or with electrical excitation and times that characterize thecompeting energy storage and dissipation phenomena. This point can bestbe made in terms of a simple example.* Note that we neglected similar electromagnetic propagational effects starting in Chapter 1,when the quasi-static electric and magnetic field systems were introduced. This is anextremely good approximation in most systems because the velocity of light is usuallymuch greater than any velocity of propagation for a mechanical or an electromechanicaldisturbance.IntroductionFigure 7.0.1 shows what can be thought of as a one-turn inductor (theperfectly conducting plates short-circuited at the end by a perfectly conductingplate) shunted by a resistance (the block with a conductivity a). Supposethat the plates are excited at the left end by a sinusoidally varying currentsource with the frequency cw.At very low frequencies (essentially directcurrent) we know that this current will flow through the perfectly conductingend plate. As the frequency is raised, however, the rate of change of thetime-varying magnetic field will induce a voltage across the block and acurrent will flow through it between the top and bottom plates. In terms oflumped-parameter models, we could determine this current by attributing aresistance to the block. The block itself, however, has finite dimensions, andwe could think of breaking it into two sections and modeling each section byi(t)e crConstantWel current(b)Fig. 7.0.1 (a)A pair of perfectly conducting plates short-circuited at one end by a perfectlyconducting plate and driven at the other end by a current source: a block of conductivity amakes electrical contact between the plates as shown; (b) equivalent circuit that shows theeffect of magnetic diffusion on the currents induced in the block when the source is time-varying; (c)block moving with a velocity V induces currents that can alter the imposedmagnetic field significantly.s--ei-tMagnetic Diffusion and Charge Relaxationa resistance. We now have the equivalent circuit shown in Fig. 7.0.1b,in which it is easy to see that at very low frequencies i1 = i2 = i,. As thefrequency is raised, the reactances become significant and i1 > i2 > i3.In fact, at very high frequencies we expect that current i3 will be very small.This simple problem points up why a continuum model is required. If thecurrents i1 and i2 are not equal (i.e., if the current through the block is notuniform) to a degree that depends on the dynamical nature of the excitation,what value of resistance do we use to characterize the block? A similarproblem exists in attributing one or more equivalent inductances to thesystem. Our dilemma is brought about by not knowing where the currentsflow or, to put it another way, of not knowing how the magnetic field isdistributed in space. In the following sections a knowledge of the distributionof magnetic field is our objective, as we discuss the physical phenomenon ofmagnetic diffusion.In the situation just discussed the time that determined the appropriatesystem model was 27/rw, or the period of excitation. In electromechanicalproblems the motion is responsible for introducing characteristic timeswhich also play an important role in determining the distribution of magneticfield. This can be illustrated by considering again the system of Fig. 7.0.1,this time with the block moving with the velocity V, as shown in Fig. 7.0.1c,and with constant excitation current.We might model this system by assuming that the constant magnetic fluxdensity Bo, induced by the current source, is the only magnetic field every-where between the plates. Then, because of the perfectly conducting endplate, the electric field E between the plates would be zero. It follows [from(6.3.5)] (Ohm's law for moving media is J' = aE',where field transformations(Table 6.1, Appendix E) require that E' = E + v x B and JX = Jf.) that themotion induces a current density J ofJ = VBor. (7.0.1)(Here for purposes of illustration we assume that the plates have sufficientextent to justify plane parallel geometry.) Now, we could use this current,together with the imposed magnetic field Bo, to compute the magnetic forceon the block. In doing so, however, we assume that the magnetic field inducedby the current J is negligible compared with the imposed field Bo. We canestablish when this assumption is valid by computing the addition Bi to themagnetic flux density to the right of the moving block. Constraints on theproblem require that J flow through the end plate to the right, where there isan addition


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MIT 6 003 - MAGNETIC DIFFUSION AND CHARGE RELAXATION

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