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 7 MAGNETIC DIFFUSION AND CHARGE RELAXATION 7 0 INTRODUCTION There are two important reasons for introducing a distributed or continuum description of electromechanical interactions The most obvious is that the mechanical system may be an elastic or fluid medium with the electric or magnetic force distributed throughout In this case the electrical and mechanical equations of motion are likely to have both time and space coordinates as independent variables As we shall see in later chapters there is a need for this model when the dynamical times of interest say the period of a sinusoidal excitation are on the same order as the time required for a disturbance to propagate from one extreme to another in the mechanical system The second reason for introducing a continuum model concerns characteristic times that arise because of competing energy storage and dissipation mechanisms in the electrical system for example a lumped inductance L shunted by a resistance R constitutes a simple magnetic field system In the absence of motion which could change the inductance L this circuit has a time constant L R The response of the circuit to an excitation depends greatly on the time constant or period of the excitation relative to LIR Similarly the behavior of magnetic field systems involving conducting materials depends considerably on the relative values of times associated with the motion or with electrical excitation and times that characterize the competing energy storage and dissipation phenomena This point can best be 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 an extremely good approximation in most systems because the velocity of light is usually much greater than any velocity of propagation for a mechanical or an electromechanical disturbance Introduction Figure 7 0 1 shows what can be thought of as a one turn inductor the perfectly conducting plates short circuited at the end by a perfectly conducting plate shunted by a resistance the block with a conductivity a Suppose that the plates are excited at the left end by a sinusoidally varying current source with the frequency cw At very low frequencies essentially direct current we know that this current will flow through the perfectly conducting end plate As the frequency is raised however the rate of change of the time varying magnetic field will induce a voltage across the block and a current will flow through it between the top and bottom plates In terms of lumped parameter models we could determine this current by attributing a resistance to the block The block itself however has finite dimensions and we could think of breaking it into two sections and modeling each section by i t e Constant cr current Wel s ei t b Fig 7 0 1 a A pair of perfectly conducting plates short circuited at one end by a perfectly conducting plate and driven at the other end by a current source a block of conductivity a makes electrical contact between the plates as shown b equivalent circuit that shows the effect of magnetic diffusion on the currents induced in the block when the source is timevarying c block moving with a velocity V induces currents that can alter the imposed magnetic field significantly Magnetic Diffusion and Charge Relaxation a 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 the frequency 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 the currents i1 and i2 are not equal i e if the current through the block is not uniform to a degree that depends on the dynamical nature of the excitation what value of resistance do we use to characterize the block A similar problem exists in attributing one or more equivalent inductances to the system Our dilemma is brought about by not knowing where the currents flow or to put it another way of not knowing how the magnetic field is distributed in space In the following sections a knowledge of the distribution of magnetic field is our objective as we discuss the physical phenomenon of magnetic diffusion In the situation just discussed the time that determined the appropriate system model was 27 rw or the period of excitation In electromechanical problems the motion is responsible for introducing characteristic times which also play an important role in determining the distribution of magnetic field 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 flux density Bo induced by the current source is the only magnetic field everywhere between the plates Then because of the perfectly conducting end plate 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 the motion induces a current density J of J VBor 7 0 1 Here for purposes of illustration we assume that the plates have sufficient extent to justify plane parallel geometry Now we could use this current together with the imposed magnetic field Bo to compute the magnetic force on the block In doing so however we assume that the magnetic field induced by the current J is negligible compared with the imposed field Bo We can establish when this assumption is valid by computing the addition Bi to the magnetic flux density to the right of the moving block Constraints on the problem require that J flow through the end plate to the right where there is an addition to the surface current K of Ki J1 7 0 2 Hence the induced magnetic field in the region to the right is Bi PoJl 7 0
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