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MIT 6 001 - PROBLEMS FOR CHAPTER 6

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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.Problems for Chapter 6For Section 6.2:Prob. 6.2.1 Consider the configuration described in Prob. 2.3.3. In the MQS approximation and at lowfrequencies the configuration can be represented by an inductance in series with a resistance. Becausethe current is distributed, and in fact essentially uniform and x-directed, how should the inductancebe computed?(a) One method uses the field in the zero frequency limit to determine the magnetic energy density,and hence by integration the total stored energy. This is then equated to ½Li2 to obtain L. Usethis method to find L and show that it is 1/3 of the value for electrodes without the conductingmaterial but shorted at z = 0.(b) Now, consider an alternative approach which considers the fields as quasistatic with respect tothe magnetic diffusion time o1M. In terms of the driving current, find the zero order fields asif they were static. Then, from Eq. 6.2.7 find the first order fields that result from time varia-tions of the zero order field. Evaluate the voltage at the terminals and show that it has the formtaken for a series inductance and resistance.For Section 6.3:Prob. 6.3.1 Show that Eq. (b)of Table 6.3.1 describes the rotating cylindrical shell shown in thattable.Prob. 6.3.2 Show that Eq. (c)of Table 6.3.1 describes the translating cylindrical shell shown inthat table.Prob. 6.3.3 Show that Eqs. (d) and (e) of Table 6.3.1 describe the rotating spherical shell shown inthat table.Prob. 6.3.4 If a sheet is of extremely high permeability, the normal flux density Bn is not continuous.Consider the sheets of Table 6.3.1 in the limit of zero conductivity but with a very high permeabilityand show that boundary conditions arenx H = 0; A=(V& H) + [[B = 0These boundary conditions are appropriate if wavelengths in the plane of the sheet are long compared tothe sheet thickness. Thus the boundary condition can be used to represent a thin region that wouldotherwise be represented by the flux-potential transfer relations of Sec. 2.16. To see this connection,show that for a planar sheet, the above boundary condition can be written asAk22 + Bix 0Take the long-wave limit of the transfer relations from Table 2.16.1 to obtain this same result.Prob. 6.3.5 In the boundary conditions of Table 6.3.1 representing a thin conducting sheet, Bn iscontinuous while the tangential 1 is not. By contrast, for the condition found in Prob. 6.3.4 for ahighly permeable sheet, Bn is discontinuous and tangential H is continuous. What boundary conditionsshould be used if the sheet is both highly permeable and conducting? To answer this question it isnecessary to give the fields in the sheet some dependence on the normal coordinate. Consider theplanar sheet and assume that the fields within take the formB Bb + (B -) H = Hb + (Ha -Hbx x A x x y y A y yDefine <A> = (Aa+Ab)/2 and show that the boundary conditions areAV <H >+>nIB = 0and Eq. (a) of Table 6.3.1 with B + <B >.x x6.35For Section 6.4:Prob. 6.4.1 A type of tachom-eter employing a permanent magnetis shown in Fig. P6.4.1a. In thedeveloped model, Fig. P6.4.1b, themagnetized material moves to theright with velocity U so that themagnetization is the given func-tion of (y,t). M is a givenconstant. The thickness, a, ofthe conducting sheet is smallcompared to the skin depth.Find the time average force perunit y-z area acting on the con-ducting sheet in the y direction.How would you design the deviceso that the induced force is pro-portional to U?Fig. P6.4.1a Fig. P6.4.1bProb. 6.4.2 Use the electricalterminal relations derived from the model, Eq. 6.4.17, to show that the equivalent circuit of Fig. 6.4.3is valid.Prob. 6.4.3 For the developed induction motor model shown in Fig. 6.4.1b, the time average force inthe direction of motion is calculated. In certain applications, such as the magnetic levitation ofvehicles (see Fig. 6.9.2), the lift force is also of importance. Find the time average lift forceon the stator, <fx>t, with two phase excitation. With single phase excitation, sketch this timeaverage lift force as a function of Sm and explain in physical terms the asymptotic behavior.Prob. 6.4.4 The cross section of a rotating induction machine is shown in Fig. 6.4.1a. The statorinner radius is (a), while the rotor has radius (b) and angular velocity Q. The windings on the statorhave p poles and two phases, as in the planar model developed in the section. For two phase excitation,find the time average torque on the rotor, an expression analogous to Eq. 6.4.11. Define 0 as theclockwise angle from the vertical axis in Fig. 6.4.1a.Prob. 6.4.5 For the rotating machine described in Prob. 6.4.4, find the two phase electrical terminalrelations analogous to Eq. 6.4.17. Determine the parameters in the equivalent circuit, Fig. 6.4.3.Prob. 6.4.6 This problem is intended to illustrate the application of the boundary conditions for athin sheet that is both conducting and highly permeable, as in Prob. 6.3.5. In the plane x=0 there isa surface current density Kf = izRe Ko exp j(wt-ky). The region x < 0 is infinitely permeable. In theplane x=d, a sheet of thickness A, permeability p and conductivity a moves in the y direction withvelocity U. This sheet can shield the magnetic field from the region x > d either by virtue of itsconductivity or its magnetizability. Find the magnetic potential just above the sheet (x=d+). Con-sider p + po and show that for iooA(w-kU)/k large, the field is excluded from the region x>d. Simi-larly, take a + 0 and show that if kA(f/po) >>I, shielding is obtained. Show that the effect of thepermeability is to reduce the effectiveness of conduction shielding. In qualitative physical terms,why is there this conflict between the two types of shielding?Prob. 6.4.7 A linear induction machine has the configuration of Fig. 6.4.1. However, the statorwinding has a finite length k in the y direction. Thus the stator surface current isKs = [u 1(y)-u


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