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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/terms_ _ ·starting, it also damps out transients in torque angle. Operation as an induc-tion motor brings the speed to near synchronous speed. The torque oscillationsresulting from the interaction between the rotor field due to dc excitationand the rotating stator field occur at the slip frequency, which is quite low.This allows the oscillating torque ample time to accelerate the rotor inertiaand pull it into step at synchronous speed during one half cycle. In a turbo-generator the solid steel rotor provides enough induction-motor actionfor adequate damping and no separate damper winding is used (see Fig.4.1.10).4.3 DISCUSSIONAt this point it is worthwhile to re-emphasize several points made in thischapter.First, although we have treated two geometrical configurations, thetechniques are applicable to other rotating machines by simple extensionsand modifications. Thus we should understand the basic concepts that arequite simple physically.Second, we have considered in some detail the steady-state characteristicsof some standard machine types for two purposes: to illustrate how thetransition is actually made from basic concepts to practical descriptions ofsteady-state terminal behavior and to present the characteristics of some ofthe most important rotating machines.Next, when the reader thinks back through the material presented in thischapter he will realize that the basic concepts of energy conversion in rotatingmachines are quite simple, though the mathematics sometimes becomeslengthy. As we indicated earlier, the symmetries that exist in rotating ma-chines have led to orderly mathematical procedures forhandling the manipula-tion. Thus rotating machine theory may appear formidable at first glance,but we, you and the authors, know that this is not so.Finally, we want to state again that among all electromechanical devices,past, present, and forseeable future, rotating machines occur in the greatestnumbers and in the widest variety of sizes and types. Thus they form animportant part of any study of electromechanics.PROBLEMS4.1. The object of this problem is to analyze a physical configuration that yields theelectrical terminal relations of (4.1.6) and (4.1.7) almost exactly. The system of Fig. 4P.1consists of two concentric cylinders of ferromagnetic material with infinite permeabilityand zero conductivity. Both cylinders have length I and are separated by the air gap g. Asindicated in the figure, the rotor carries a winding of N, turns distributed sinusoidally and4.2.4ProblemsRotating MachinesN sin 02(R +g)into paper--Ni, sin (;p -0)Fig. 4P.1having negligible radial thickness. The stator carries a winding of N, turns distributedsinusoidally and having negligible radial thickness. Current through these windings leadsto sinusoidally distributed surface currents as indicated. In the analysis we neglect theeffects of end turns and assume g << R so that the radial variation of magnetic field can beneglected.(a) Find the radial component of air-gap flux density due to stator current alone.(b) Find the radial component of air-gap flux density due to rotor current alone.(c) Use the flux densities found in parts (a) and (b) to find A, and Ar in the form of(4.1.6) and (4.1.7). In particular, evaluate L,, L,, and M in terms of given data.4.2. Rework Problem 4.1 with the more practical uniform winding distribution representableby surface current densities{. Nsis(Ni (R + g)for O< i < r,for n < < 27r,for O<(•y-O) <r,for r < (o -- ) < 2r.ProblemsIn part (c) you will find the mutual inductance to be expressed as an infinite series like(4.1.4).43. With reference to Problems 4.1 and 4.2, show that if either the rotor winding or thestator winding is sinusoidally distributed as in Problem 4.1, the mutual inductance containsonly a space fundamental term, regardless of the winding distribution on the other member.4.4. The machine represented schematically in Fig. 4P.4 has uniform winding distributions.As indicated by Problem 4.2, the electrical terminal relations are ideally, = LiO + i -cosnO,nodd nA = Lir + i, M cos nO,n oddwhere L,, Lr, and Mo are constants. We now constrain the machine as follows: if= I =constant; 0 = ct, w = constant, stator winding open-circuited i, = 0.(a) Find the instantaneous stator voltage v,(t).(b) Find the ratio of the amplitude of the nth harmonic stator voltage to the amplitudeof the fundamental component of stator voltage.(c) Plot one complete cycle of v,(t) found in (a).Fig. 4P.44.5. Calculate the electromagnetic torque TO of (4.1.8) by using the electrical terminalrelations (4.1.6) and (4.1.7) and the assumption that the coupling system is conservative.4.6. A schematic representation of a rotating machine is shown in Fig. 4P.6. The rotorwinding is superconducting and the rotor has moment of inertiaJ. The machine is constructedso that the electrical terminal relations are A, = L,i, + Mir cos 0, A, = Mi, cos 0 + Li,.The machine is placed in operation as follows:(a) With the rotor (r) terminals open-circuited and the rotor position at 0 = 0, thecurrent i, is raised to 1,.(b) The rotor (r) terminals are short circuited to conserve the flux Ar, regardless of0(t) and i,(t).(c) The current i, is constrained by the independent current source i(t).__··_I·___·_·Rotating Machinesi(t)IFig. 4P.6Write the equation of motion for the shaft with no external mechanical torque applied.Your answer should be one equation involving 0(t) as the only unknown. Damping maybe ignored.4.7. A smooth-air-gap machine with one winding on the rotor and one on the stator (seeFig. 4.1.1) has the electrical terminal relations of (4.1.1) and (4.1.2).A, = Li, + Lsr(O)ir, (4.1.1)4,= L8,(0)i, + Li,. (4.1.2)The mutual inductance L,,r() contains two spatial harmonics, the fundamental and thethird. Thus L,,r() = M1 cos 0 + Ms cos 30, where M1 and Ms are constants.(a) Find the torque of electric origin as a function of i, i,, 0, M1, and Ms.(b) Constrain the machine with the current sources i, = I, sin


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