Rotating Machines Lecture Notes

what is rotating electrical machines and what is a rotating machine that transforms electrical energy, rotating machines examples and basic concepts
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CHAPTER Introduction to Rotating Machines he object of this chapter is to introduce and discuss some of the principles underlying the performance of electric machinery. As will be seen, these prin- ciples are common to both ac and dc machines. Various techniques and approx- T imations involved in reducing a physical machine to simple mathematical models, sufficient to illustrate the basic principles, will be developed. 4.1 ELEMENTARY CONCEPTS Equation 1.27, e = d)/dt, can be used to determine the voltages induced by time- varying magnetic fields. Electromagnetic energy conversion occurs when changes in the flux linkage result from mechanical motion. In rotating machines, voltages are generated in windings or groups of coils by rotating these windings mechanically through a magnetic field, by mechanically rotating a magnetic field past the winding, or by designing the magnetic circuit so that the reluctance varies with rotation of the rotor. By any of these methods, the flux linking a specific coil is changed cyclically, and a time-varying voltage is generated. A set of such coils connected together is typically referred to as an armature winding. In general, the term armature winding is used to refer to a winding or a set of windings on a rotating machine which carry ac currents. In ac machines such as synchronous or induction machines, the armature winding is typically on the station- ary portion of the motor referred to as the stator, in which case these windings may also be referred to as stator windings. Figure 4.1 shows the stator winding of a large, multipole, three-phase synchronous motor under construction. In a dc machine, the armature winding is found on the rotating member, referred to as the rotor. Figure 4.2 shows a dc-machine rotor. As we will see, the armature winding of a dc machine consists of many coils connected together to form a closed loop. A rotating mechanical contact is used to supply current to the armature winding as the rotor rotates. 173 . 174 CHAPTER 4 Introduction Machines Rotating Figure 4.1 Stator of a 190-MVA three-phase 12-kV 37-r/min hydroelectric generator. The conductors have hollow passages through which cooling water circulated. Synchronous and dc machines typically include a second winding (or set of windings) which carry dc current and which are used to produce the main operating flux in the machine. Such a winding is typically referred to as field winding. The field winding on a dc machine is found on the stator, while that on a synchronous machine is found on the rotor, in which case current must be supplied to the field winding via a rotating mechanical contact. As we have seen, permanent magnets also produce dc magnetic flux and are used in the place of field windings in some machines. In most rotating machines, the stator and rotor are made of electrical steel, and the windings are installed in slots on these structures. As is discussed in Chapter the use of such high-permeability material maximizes the coupling between the coils and increases the magnetic energy density associated with the electromechanical interaction. It also enables the machine designer to shape and distribute the magnetic fields according to the requirements of each particular machine design. The time- varying flux present in the armature structures of these machines tends to induce currents, known as eddy currents, in the electrical steel. Eddy currents can be a large source of loss in such machines and can significantly reduce machine performance. In order to minimize the effects of eddy currents, the armature structure is typically built from thin laminations of electrical steel which are insulated from each other. This is illustrated in Fig. 4.3, which shows the stator core of an ac motor being constructed as a stack of individual laminations. In some machines, such as variable reluctance machines and stepper motors, there are no windings on the rotor. Operation of these machines depends on the 1, Corporation.) Boveri (Brown is to 4.1 Elementary Concepts 175 Figure 4.2 Armature of a dc motor. Electric (General Figure 4.3 Partially completed stator core for an ac motor. Electric Corporation.) (Westinghouse Company.) 176 CHAPTER 4 Introduction to Rotating Machines nonuniformity of air-gap reluctance associated with variations in rotor position in conjunction with time-varying currents applied to their stator windings. In such ma- chines, both the stator and rotor structures are subjected to time-varying magnetic flux and, as a result, both may require lamination to reduce eddy-current losses. Rotating electric machines take many forms and are known by many names: dc, synchronous, permanent-magnet, induction, variable reluctance, hysteresis, brush- less, and so on. Although these machines appear to be quite dissimilar, the physical principles governing their behavior are quite similar, and it is often helpful to think of them in terms of the same physical picture. For example, analysis of a dc machine shows that associated with both the rotor and the stator are magnetic flux distributions which are fixed in space and that the torque-producing characteristic of the dc machine stems from the tendency of these flux distributions to align. An induction machine, in spite of many fundamental differences, works on exactly the same principle; one can identify flux distributions associated with the rotor and stator. Although they are not stationary but rather rotate in synchronism, just as in a dc motor they are displaced by a constant angular separation, and torque is produced by the tendency of these flux distribution to align. Certainly, analytically based models are essential to the analysis and design of electric machines, and such models will be derived thoughout this book. However, it is also important to recognize that physical insight into the performance of these devices is equally useful. One objective of this and subsequent chapters is to guide the reader in the development of such insight. 4,2 INTRODUCTION TO AC AND DC MACHINES 4.2.1 AC Machines Traditional ac machines fall into one of two categories: synchronous and induction. In synchronous machines, rotor-winding currents are supplied directly from the sta- tionary frame through a rotating contact. In induction machines, rotor currents are induced in the rotor windings by a combination of the time-variation of the stator currents and the motion of the rotor relative to the stator. Synchronous Machines A preliminary picture of synchronous-machine perfor- mance can be gained by discussing the voltage induced in the armature of the very much simplified salient-pole ac synchronous generator shown schematically in Fig. 4.4. The field-winding of this machine produces a single pair of magnetic poles (similar to that of a bar magnet), and hence this machine is referred to as a two-pole machine. With rare exceptions, the armature winding of a synchronous machine is on the stator, and the field winding is on the rotor, as is true for the simplified machine of Fig. 4.4. The field winding is excited by direct current conducted to it by means of stationary carbon brushes which contact rotatating slip rings or collector rings. Practical factors usually dictate this orientation of the two windings: It is advantageous 4.2 Introduction to and Machines 177 Figure 4.4 Schematic view of a simple, two-pole, single-phase synchronous generator. to have the single, low-power field winding on the rotor while having the high-power, typically multiple-phase, armature winding on the stator. The armature winding, consisting here of only a single coil of N turns, is indicated in cross section by the two coil sides a and -a placed in diametrically opposite narrow slots on the inner periphery of the stator of Fig. 4.4. The conductors forming these coil sides are parallel to the shaft of the machine and are connected in series by end connections (not shown in the figure). The rotor is turned at a constant speed by a source of mechanical power connected to its shaft. The armature winding is assumed to be open-circuited and hence the flux in this machine is produced by the field winding alone. Flux paths are shown schematically by dashed lines in Fig. 4.4. A highly idealized analysis of this machine would assume a sinusoidal distribu- tion of magnetic flux in the air gap. The resultant radial distribution of air-gap flux density B is shown in Fig. 4.5a as a function of the spatial angle (measured with respect to the magnetic axis of the armature winding) around the rotor periphery. In e ° I Figure 4.5 (a) Space distribution of flux density and (b) corresponding waveform of the generated voltage for the single-phase generator of Fig. 4.4. (b) (a) o: t Oa paths Flux nding ld DC AC 178 CHAPTER 4 Introduction to Rotating Machines practice, the air-gap flux-density of practical salient-pole machines can be made to approximate a sinusoidal distribution by properly shaping the pole faces. As the rotor rotates, the flux-linkages of the armature winding change with time. Under the assumption of a sinusoidal flux distribution and constant rotor speed, the resulting coil voltage will be sinusoidal in time as shown in Fig. 4.5b. The coil voltage passes through a complete cycle for each revolution of the two-pole machine of Fig. 4.4. Its frequency in cycles per second (Hz) is the same as the speed of the rotor in revolutions per second: the electric frequency of the generated voltage is synchronized with the mechanical speed, and this is the reason for the designation "synchronous" machine. Thus a two-pole synchronous machine must revolve at 3600 revolutions per minute to produce a 60-Hz voltage. A great many synchronous machines have more than two poles. As a specific example, Fig. 4.6 shows in schematic form a four-pole single-phase generator. The field coils are connected so that the poles are of alternate polarity. There are two complete wavelengths, or cycles, in the flux distribution around the periphery, as shown in Fig. 4.7. The armature winding now consists of two coils al, -al and a2, a2 connected in series by their end connections. The span of each coil is one wavelength of flux. The generated voltage now goes through two complete cycles per revolution of the rotor. The frequency in hertz will thus be twice the speed in revolutions per second. When a machine has more than two poles, it is convenient to concentrate on a single pair of poles and to recognize that the electric, magnetic, and mechanical conditions associated with every other pole pair are repetitions of those for the pair under consideration. For this reason it is convenient to express angles in electrical degrees or electrical radians rather than in physical units. One pair of poles in a multipole machine or one cycle of flux distribution equals 360 electrical degrees or 2Jr electrical radians. Since there are poles/2 complete wavelengths, or cycles, in one Figure 4.6 Schematic view of a simple, four-pole, single-phase synchronous generator. 4.2 Introduction to AC and Machines 179 al ' \ \ 0 Figure 4.7 Space distribution of the air-gap flux density a idealized, four-pole synchronous generator. complete revolution, it follows, for example, that (p°les) (4.1) = 2 where is the angle in electrical units and is the spatial angle. This same rela- tionship applies to all angular measurements in a multipole machine; their values in electrical units will be equal to (poles/2) times their actual spatial values. The coil voltage of a multipole machine passes through a complete cycle every time a pair of poles sweeps by, or (poles/2) times each revolution. The electrical frequency fe of the voltage generated in a synchronous machine is therefore fe (ples) n Hz (4.2) where n is the mechanical speed in revolutions per minute, and hence n/60 is the speed in revolutions per second. The electrical frequency of the generated voltage in radians per second is = (poles/2) where is the mechanical speed in radians per second. The rotors shown in Figs. 4.4 and 4.6 have salient, or projecting, poles with con- centrated windings. Figure 4.8 shows diagrammatically a nonsalient-pole, or cylin- drical rotor. The field winding is a two-pole distributed winding; the coil sides are distributed in multiple slots around the rotor periphery and arranged to produce an approximately sinusoidal distribution of radial air-gap flux. The relationship between electrical frequency and rotor speed of Eq. 4.2 can serve as a basis for understanding why some synchronous generators have salient-pole ro- tor structures while others have cylindrical rotors. Most power systems in the world operate at frequencies of either 50 or 60 Hz. A salient-pole construction is character- istic of hydroelectric generators because hydraulic turbines operate at relatively low speeds, and hence a relatively large number of poles is required to produce the desired frequency; the salient-pole construction is better adapted mechanically to this situa- tion. The rotor of a large hydroelectric generator is shown in Fig. 4.9. Steam turbines and gas turbines, however, operate best at relatively high speeds, and turbine-driven alternators or turbine generators are commonly two- or four-pole cylindrical-rotor corn corn We 0a 0ae 0ae Oa in radians electrical Oae, radians 2zr zr// mechanical 0a, a2 DC 180 CHAPTER 4 Introduction to Rotating Machines Figure 4.8 Elementary two-pole cylindrical-rotor field winding. Figure 4.9 Water-cooled rotor of the 190-MVA hydroelectric generator whose stator is shown Fig. 4.1. Boveri Corporation.) (Brown in 4.2 Introduction to AC and Machines t81 Figure 4.10 Rotor of a two-pole 3600 r/min turbine generator. Electric machines. The rotors are made from a single steel forging or from several forgings, as shown in Figs. 4.10 and 4.11. Most of the world's power systems are three-phase systems and, as a result, with very few exceptions, synchronous generators are three-phase machines. For the production of a set of three voltages phase-displaced by 120 electrical degrees in time, a minimum of three coils phase-displaced 120 electrical degrees in space must be used. A simplified schematic view of a three-phase, two-pole machine with one coil per phase is shown in Fig. 4.12a. The three phases are designated by the letters a, b, and c. In an elementary four-pole machine, a minimum of two such sets of coils must be used, as illustrated in Fig. 4.12b; in an elementary multipole machine, the minimum number of coils sets is given by one half the number of poles. The two coils in each phase of Fig. 4.12b are connected in series so that their voltages add, and the three phases may then be either Y- or A-connected. Figure 4.12c shows how the coils may be interconnected to form a Y connection. Note however, since the voltages in the coils of each phase are indentical, a parallel connection is also possible, e.g., coil (a, -a) in parallel with coil (a', -a'), and so on. When a synchronous generator supplies electric power to a load, the armature current creates a magnetic flux wave in the air gap which rotates at synchronous speed, as shown in Section 4.5. This flux reacts with the flux created by the field current, and electromechanical torque results from the tendency of these two magnetic fields to align. In a generator this torque opposes rotation, and mechanical torque must be applied from the prime mover to sustain rotation. This electromechanical torque is the mechanism through which the synchronous generator converts mechanical to electric energy. The counterpart of the synchronous generator is the synchronous motor. A cut- away view of a three-phase, 60-Hz synchronous motor is shown in Fig. 4.13. Alter- nating current is supplied to the armature winding on the stator, and dc excitation is supplied to the field winding on the rotor. The magnetic field produced by the Corporation.) (Westinghouse DC 182 CHAPTER 4 Introduction Machines Rotating Figure 4.11 Parts of multipiece rotor for a 1333-MVA three-phase 1800 r/min turbine generator. The separate forgings will be shrunk on the shaft before final machining and milling slots for the windings. The total weight of the rotor 435,000 (Brown Boveri o a a a l 0 Figure 4.12 Schematic views of three-phase generators: (a) two-pole, (b) four-pole, and (c) Y connection of the windings. armature currents rotates at synchronous speed. To produce a steady electromechan- ical torque, the magnetic fields of the stator and rotor must be constant in amplitude and stationary with respect to each other. In a synchronous motor, the steady-state speed is determined by the number of poles and the frequency of the armature current. Thus a synchronous motor operated from a constant-frequency ac source will operate at a constant steady-state speed. (c) (b) (a) Corporation.) lb. is to 4.2 Introduction to AC and DC Machines 183 Figure 4.13 Cutaway view of a high-speed synchronous motor. The excitor shown on the left end of the rotor is a small ac generator with a rotating semiconductor rectifier assembly. In a motor the electromechanical torque is in the direction of rotation and balances the opposing torque required to drive the mechanical load. The flux produced by currents in the armature of a synchronous motor rotates ahead of that produced by the field, thus pulling on the field (and hence on the rotor) and doing work. This is the opposite of the situation in a synchronous generator, where the field does work as its flux pulls on that of the armature, which is lagging behind. In both generators and motors, an electromechanical torque and a rotational voltage are produced. These are the essential phenomena for electromechanical energy conversion. Induction Machines A second type of ac machine is the induction machine. Like the synchronous machine, the stator winding of an induction machine is excited with alternating currents. In contrast to a synchronous machine in which a field winding on the rotor is excited with dc current, alternating currents flow in the rotor windings of an induction machine. In induction machines, alternating currents are applied directly to the stator windings. Rotor currents are then produced by induction, i.e., transformer action. The induction machine may be regarded as a generalized transformer in which electric power is transformed between rotor and stator together with a change of frequency and a flow of mechanical power. Although the induction motor is the most Company.) Electric (General 184 CHAPTER 4 Introduction to Rotating Machines common of all motors, it is seldom used as a generator; its performance characteristics as a generator are unsatisfactory for most applications, although in recent years it has been found to be well suited for wind-power applications. The induction machine may also be used as a frequency changer. In the induction motor, the stator windings are essentially the same as those of a synchronous machine. However, the rotor windings are electrically short-circuited and frequently have no external connections; currents are induced by transformer action from the stator winding. A cutaway view of a squirrel-cage induction motor is shown in Fig. 4.14. Here the rotor "windings" are actually solid aluminum bars which are cast into the slots in the rotor and which are shorted together by cast aluminum rings at each end of the rotor. This type of rotor construction results in induction motors which are relatively inexpensive and highly reliable, factors contributing to their immense popularity and widespread application. As in a synchronous motor, the armature flux in the induction motor leads that of the rotor and produces an electromechanical torque. In fact, we will see that, just as in a synchronous machine, the rotor and stator fluxes rotate in synchronism with each other and that torque is related to the relative displacement between them. However, unlike a synchronous machine, the rotor of an induction machine does not itself rotate synchronously; it is the "slipping" of the rotor with respect to the synchronous armature flux that gives rise to the induced rotor currents and hence the torque. Induction motors operate at speeds less than the synchronous mechanical speed. A typical speed-torque characteristic for an induction motor is shown in Fig. 4.15. Figure 4.14 Cutaway view of a squirrel-cage induction motor. Electric Corporation.) (Westinghouse 4.2 Introduction to and Machines 185 ,,,._ r of Figure 4.15 Typical induction-motor speed-torque characteristic. Figure 4.16 Cutaway view a typical integral-horsepower dc motor. 4.2.2 DC Machines As has been discussed, the armature winding of a dc generator is on the rotor with current conducted from it by means of carbon brushes. The field winding is on the stator and is excited by direct current. A cutaway view of a dc motor is shown in Fig. 4.16. A very elementary two-pole dc generator is shown in Fig. 4.17. The armature winding, consisting of a single coil of N turns, is indicated by the two coil sides Boveri.) Brown (ASEA of speed synchronous percent in Speed 100 80 6O 40 2O DC AC 186 CHAPTER 4 Introduction to Rotating Machines N Figure 4.17 Elementary dc machine with commutator. a and -a placed at diametrically opposite points on the rotor with the conductors parallel to the shaft. The rotor is normally turned at a constant speed by a source of mechanical power connected to the shaft. The air-gap flux distribution usually approximates a flat-topped wave, rather than the sine wave found in ac machines, and is shown in Fig. 4.18a. Rotation of the coil generates a coil voltage which is a time function having the same waveform as the spatial flux-density distribution. Although the ultimate purpose is the generation of a direct voltage, the voltage induced in an individual armature coil is an alternating voltage, which must there- fore be rectified. The output voltage of an ac machine can be rectified using external semiconductor rectifiers. This is in contrast to the conventional dc machine in which rectification is produced mechanically by means of a commutator, which is a cylinder formed of copper segments insulated from each other by mica or some other highly insulating material and mounted on, but insulated from, the rotor shaft. Stationary carbon brushes held against the commutator surface connect the winding to the exter- nal armature terminals. The commutator and brushes can readily be seen in Fig. 4.16. The need for commutation is the reason why the armature windings of dc machines are placed on the rotor. For the elementary dc generator, the commutator takes the form shown in Fig. 4.17. For the direction of rotation shown, the commutator at all times connects the coil side, which is under the south pole, to the positive brush and that under the north pole to the negative brush. The commutator provides full-wave rectification, transforming the voltage waveform between brushes to that of Fig. 4.18b and making available a unidirectional voltage to the external circuit. The dc machine of Fig. 4.17 is, of course, simplified to the point of being unrealistic in the practical sense, and later it will be essential to examine the action of more realistic commutators. The effect of direct current in the field winding of a dc machine is to create a magnetic flux distribution which is stationary with respect to the stator. Similarly, the segments commutator Copper Rotation brush Carbon turns :oil, rmature A, 4,3 of Distributed Windings t87 _ ...... an Anglearoun d . __ . ........... p 2 ca) / V V t Figure 4.18 (a) Space distribution of air-gap flux density an elementary dc machine; (b) waveform of voltage between brushes effect of the commutator is such that when direct current flows through the brushes, the armature creates a magnetic flux distribution which is also fixed in space and whose axis, determined by the design of the machine and the position of the brushes, is typically perpendicular to the axis of the field flux. Thus, just as in the ac machines discussed previously, it is the interaction of these two flux distributions that creates the torque of the dc machine. If the machine is acting as a generator, this torque opposes rotation. If it is acting as a motor, the electrome- chanical torque acts in the direction of the rotation. Remarks similar to those already made concerning the roles played by the generated voltage and electromechanical torque in the energy conversion process in synchronous machines apply equally well to dc machines. 4.3 MMF OF DISTRIBUTED WINDINGS Most armatures have distributed windings, i.e., windings which are spread over a number of slots around the air-gap periphery, as in Figs. 4.2 and 4.1. The individual coils are interconnected so that the result is a magnetic field having the same number of poles as the field winding. The study of the magnetic fields of distributed windings can be approached by examining the magnetic field produced by a winding consisting of a single N-turn coil which spans 180 electrical degrees, as shown in Fig. 4.19a. A coil which spans in (b) Time (a) periphery zn ;i-a a density flux of distribution Space MMF 188 CHAPTER 4 Introduction to Rotating Machines N-turn coil carrying current ux lines Magnetic axis of stator coil (a) Fundamental 'agl Ni x .. " ' I I Oa Ni / 27r 2 Rotor surface Stator surface .... ii Figure 4.19 (a) Schematic view of flux produced by a concentrated, full-pitch winding a machine with a uniform air gap. (b) The air-gap mmf produced by current this winding. 180 electrical degrees is known as a full-pitch coil. The dots and crosses indicate cur- rent flow towards and away from the reader, respectively. For simplicity, a concentric cylindrical rotor is shown. The general nature of the magnetic field produced by the current in the coil is shown by the dashed lines in Fig. 4.19a. Since the permeability of the armature and field iron is much greater than that of air, it is sufficiently accurate for our present purposes to assume that all the reluctance of the magnetic circuit is in the air gap. From symmetry of the structure it is evident that the magnetic field intensity in the air gap at angle under one pole is the same in magnitude as that at angle + under the opposite pole, but the fields are in the opposite direction. Around any of the closed paths shown by the flux lines in Fig. 4.19a the mmf is N i. The assumption that all the reluctance of this magnetic circuit is in the air gap leads to the result that the line integral of H inside the iron is negligibly small, and thus it is reasonable to neglect the mmf drops associated with portions of the magnetic circuit inside the iron. By symmetry we argued that the air-gap fields on opposite sides of the rotor are equal in magnitude but opposite in direction. It follows that the air-gap mmf should be similarly distributed; since each flux line crosses the air gap twice, the mmf drop across the air gap must be equal to half of the total or Ni/2. Figure 4.19b shows the air gap and winding in developed form, i.e., laid out flat. The air-gap mmf distribution is shown by the steplike distribution of amplitude Hag zr 0a 0a Hag in in (b) ...... Jr %% 4.3 of Distributed Windings 189 Ni/2. On the assumption of narrow slot openings, the mmfjumps abruptly by Ni in crossing from one side to the other of a coil. This mmf distribution is discussed again in Section 4.4, where the resultant magnetic fields are evaluated. 4.3.1 AC Machines Fourier analysis can show that the air-gap mmf produced by a single coil such as the full-pitch coil of Fig. 4.19 consists of a fundamental space-harmonic component as well as a series of higher-order harmonic components. In the design of ac machines, serious efforts are made to distribute the coils making up the windings so as to minimize the higher-order harmonic components and to produce an air-gap mmf wave which consists predominantly of the space-fundamental sinusoidal component. It is thus appropriate here to assume that this has been done and to focus our attention on the fundamental component. The rectangular air-gap mmf wave of the concentrated two-pole, full-pitch coil of Fig. 4.19b can be resolved into a Fourier series comprising a fundamental component and a series of odd harmonics. The fundamental component .Tagl is (4.3) .)E'ag 1 COS Oa T where 0a is measured from the magnetic axis of the stator coil, as shown by the dashed sinusoid in Fig. 4.19b. It is a sinusoidal space wave of amplitude (4.4) (Fagl)peak __ T with its peak aligned with the magnetic axis of the coil. Now consider a distributed winding, consisting of coils distributed in several slots. For example, Fig. 4.20a shows phase a of the armature winding of a somewhat simplified two-pole, three-phase ac machine. Phases b and c occupy the empty slots. The windings of the three phases are identical and are located with their magnetic axes 120 degrees apart. We direct our attention to the air-gap mmf of phase a alone, postponing the discussion of the effects of all three phases until Section 4.5. The winding is arranged in two layers, each full-pitch coil of turns having one side in the top of a slot and the other coil side in the bottom of a slot a pole pitch away. In a practical machine, this two-layer arrangement simplifies the geometric problem of getting the end turns of the individual coils past each other. Figure 4.20b shows one pole of this winding laid out fiat. With the coils connected in series and hence carrying the same current, the mmf wave is a series of steps each of height 2Ncia (equal to the ampere-turns in the slot), where is the winding current. Its space-fundamental component is shown by the sinusoid. It can be seen that the distributed winding produces a closer approximation to a sinusoidal mmf wave than the concentrated coil of Fig. 4.19. The amplitude of the fundamental-space-harmonic-component of the mmf wave of a distributed winding is less than the sum of the fundamental components of the ia Nc MMF 190 CHAPTER 4 Introduction to Rotating Machines Axis of phase a A I p S ac jfu?d e aavmtnta, 2ncia a 0 o ii a a Figure 4.20 The mmf of one phase of a distributed two-pole, three-phase winding with full-pitch coils. individual coils because the magnetic axes of the individual coils are not aligned with the resultant. The modified form of Eq. 4.3 for a distributed multipole winding having series turns per phase is 4 (po,es) • "agl ia COS a 0 (4.5) :r poles 2 in which the factor 4/7r arises from the Fourier-series analysis of the rectangular mmf wave of a concentrated full-pitch coil, as in Eq. 4.3, and the factor takes into account the distribution of the winding. This factor is required because the mmf's produced by the individual coils of any one phase group have different magnetic axes. kw winding Nph (b) ......__...A ,, ff (a) 4.3 Distributed Windings 191 When they are connected in series to form the phase winding, their phasor sum is then less than their numerical sum. (See Appendix B for details.) For most three-phase windings, kw typically falls in the range of 0.85 to 0.95. The factor kw Nph is the effective series turns per phase for the fundamental mmf. The peak amplitude of this mmf wave is (fagl)peak = ia (4.6) Jr poles "XAMPLE 4. The phase-a two-pole armature winding of Fig. 4.20a can be considered to consist of 8 Nc-turn, full-pitch coils connected in series, with each slot containing two coils. There are a total of 24 armature slots, and thus each slot is separated by 360°/24 = °. Assume angle is measured from the magnetic axis of phase a such that the four slots containing the coil sides labeled a are at = 67.5 °, 82.5 °, 97.5 °, and 112.5 °. The opposite sides of each coil are thus found in the slots found at - 112.5 °, -97.5 °, -82.5 ° and -67.5 °, respectively. Assume this winding to be carrying current (a) Write an expression for the space-fundamental mmf produced by the two coils whose sides are in the slots at = 112.5 ° and -67.5 °. (b) Write an expression for the space- fundamental mmf produced by the two coils whose sides are in the slots at 67.5 ° and -112.5 °. (c) Write an expression for the space-fundamental mmf of the complete armature winding. (d) Determine the winding factor for this distributed winding. I Solution a. Noting that the magnetic axis of this pair of coils is at (9 a = (112.5 ° - 67.5°)/2 = 22.5 ° and that the total ampere-turns in the slot is equal to 2Ncia, the mmf produced by this pair of coils can be found from analogy with Eq. 4.3 to be 4 (2Ncia) (,'agl)22.5 o COS 22.5 °) rr 2 b. This pair of coils produces the same space-fundamental mmf as the pair of part (a) with the exception that this mmf is centered at 22.5 °. Thus 4 (2Ncia) (,L"agl)_22.5o = COS ((9 a -Jr- 22.5 °) c. By analogy with parts (a) and (b), the total space-fundamental mmf can be written as (,L-'agl)total = (,E'agl)-22.5o "31-()L-'agl)-7.5o -Jr-(,E'agl)7.5o + (,E'agl)22.5o 4 + 7.5 °) + 22.5°) 4 2 4.88Ncia cos (9 a Jr (7"66Nc)iaCOSOa (0a COS COS(0a (2Nc)iaCOS(Oa+22.S°,+cos(Oa-k-7.5°) 0a (0a kw 0a 0a ia. 0a 0a 15 of MMF 192 CHAPTER 4 Introduction to Rotating Machines d. Recognizing that, for this winding = 8Nc, the total mmf of part (c) can be rewritten as 4 ()E'agl)total = ia COS 0a 2 Comparison with Eq. 4.5 shows that for this winding, the winding factor is = 0.958. Practice Problem 4. Calculate the winding factor of the phase-a winding of Fig. 4.20 if the number of turns in the four coils in the two outer pairs of slots is reduced to six while the number of turns in the four coils in the inner slots remains at eight. Solution = 0.962 Equation 4.5 describes the space-fundamental component of the mmf wave pro- duced by current in phase a of a distributed winding. If the phase-a current is sinusoidal in time, e.g., = cot, the result will be an mmf wave which is stationary in space and varies sinusoidally both with respect to and in time. In Section 4.5 we will study the effect of currents in all three phases and will see that the application of three-phase currents will produce a rotating mmf wave. In a directly analogous fashion, rotor windings are often distributed in slots to reduce the effects of space harmonics. Figure 4.21 a shows the rotor of a typical two- pole round-rotor generator. Although the winding is symmetric with respect to the rotor axis, the number of turns per slot can be varied to control the various harmonics. In Fig. 4.21b it can be seen that there are fewer turns in the slots nearest the pole face. In addition, the designer can vary the spacing of the slots. As for distributed armature windings, the fundamental air-gap mmf wave of a multipole rotor winding can be found from Eq. 4.5 in terms of the total number of series turns Nr, the winding current Ir and a winding factor kr as 4 (krNr)(poles ) f'agl - Ir (4.7) poles 2 where is the spatial angle measured with respect to the rotor magnetic axis, as shown in Fig. 4.21 b. Its peak amplitude is 4 (4.8) (Fagl)peak Ir poles 4.3.2 DC Machines Because of the restrictions imposed on the winding arrangement by the commu- tator, the mmf wave of a dc machine armature approximates a sawtooth waveform Jr Or Jr Or COS 0a COS Im ia kw kw zr Nph

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