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 4 2 4 Problems starting it also damps out transients in torque angle Operation as an induction motor brings the speed to near synchronous speed The torque oscillations resulting from the interaction between the rotor field due to dc excitation and the rotating stator field occur at the slip frequency which is quite low This allows the oscillating torque ample time to accelerate the rotor inertia and pull it into step at synchronous speed during one half cycle In a turbogenerator the solid steel rotor provides enough induction motor action for adequate damping and no separate damper winding is used see Fig 4 1 10 4 3 DISCUSSION At this point it is worthwhile to re emphasize several points made in this chapter First although we have treated two geometrical configurations the techniques are applicable to other rotating machines by simple extensions and modifications Thus we should understand the basic concepts that are quite simple physically Second we have considered in some detail the steady state characteristics of some standard machine types for two purposes to illustrate how the transition is actually made from basic concepts to practical descriptions of steady state terminal behavior and to present the characteristics of some of the most important rotating machines Next when the reader thinks back through the material presented in this chapter he will realize that the basic concepts of energy conversion in rotating machines are quite simple though the mathematics sometimes becomes lengthy As we indicated earlier the symmetries that exist in rotating machines have led to orderly mathematical procedures forhandling the manipulation 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 greatest numbers and in the widest variety of sizes and types Thus they form an important part of any study of electromechanics PROBLEMS 4 1 The object of this problem is to analyze a physical configuration that yields the electrical terminal relations of 4 1 6 and 4 1 7 almost exactly The system of Fig 4P 1 consists of two concentric cylinders of ferromagnetic material with infinite permeability and zero conductivity Both cylinders have length I and are separated by the air gap g As indicated in the figure the rotor carries a winding of N turns distributed sinusoidally and Rotating Machines N 2 R g sin 0 into paper Ni sin p 0 Fig 4P 1 having negligible radial thickness The stator carries a winding of N turns distributed sinusoidally and having negligible radial thickness Current through these windings leads to sinusoidally distributed surface currents as indicated In the analysis we neglect the effects of end turns and assume g R so that the radial variation of magnetic field can be neglected 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 representable by surface current densities Ni Nsis R g for O i r for n for O y for 27r r o O r 2r Problems In 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 the stator winding is sinusoidally distributed as in Problem 4 1 the mutual inductance contains only 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 n A Li r M cos nO i n odd where L L r 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 amplitude of the fundamental component of stator voltage c Plot one complete cycle of v t found in a Fig 4P 4 4 5 Calculate the electromagnetic torque TO of 4 1 8 by using the electrical terminal relations 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 rotor winding is superconducting and the rotor has moment of inertiaJ The machine is constructed so that the electrical terminal relations are A L i Mi r 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 the current i is raised to 1 b The rotor r terminals are short circuited to conserve the flux Ar regardless of 0 t and i t c The current i is constrained by the independent current source i t I Rotating Machines i t I Fig 4P 6 Write 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 may be ignored 4 7 A smooth air gap machine with one winding on the rotor and one on the stator see Fig 4 1 1 has the electrical terminal relations of 4 1 1 and 4 1 2 4 1 1 A Li Lsr O ir 4 L8 0 i Li 4 1 2 The mutual inductance L r contains two spatial harmonics the fundamental and the third Thus L r M 1 cos 0 M s cos 30 where M 1 and M s are constants a Find the torque of electric origin as a function of i i 0 M 1 and M s b Constrain the machine with the current sources i I sin co t i Irsin wrt and the position source 0 w t 7 where I I w w and y are constants Find the values of w at which the machine can produce an average torque and find an expression for the average torque for each value of ow found 4 8 The smooth air gap machine of Fig 4
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