Electric motor Windings

winding function for electrical machine analysis design and analysis of windings of electrical machines types of windings in electrical machines
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Published Date:14-07-2017
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J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.1 2 WINDINGS OF ELECTRICAL MACHINES The operation principle of electrical machines is based on the interaction between the magnetic fields and the currents flowing in the windings of the machine. The winding constructions and connections together with the currents and voltages fed into the windings determine the operating modes and the type of the electrical machine. According to their different functions in an electrical machine, the windings are grouped for instance as follows: • armature windings, • other rotating-field windings (e.g. stator or rotor windings of induction motors) • field (magnetizing) windings, • damper windings • commutating windings, and • compensating windings. Armature windings are rotating-field windings, into which the rotating-field-induced voltage required in energy conversion is induced. According to IEC 60050-411, the armature winding is a winding in a synchronous, DC, or single-phase commutator machine, which, in service, receives active power from or delivers active power to the external electrical system. This definition also applies to a synchronous compensator if the term ‘active power’ is replaced by ‘reactive power’. The air gap flux component caused by the armature current linkage is called the armature reaction. An armature winding determined under these conditions can transmit power between an electrical network and a mechanical system. Magnetizing windings create a magnetic field required in the energy conversion. All machines do not include a separate magnetizing winding; for instance in asynchronous machines, the stator winding both magnetizes the machine and acts as a winding, where the operating voltage is induced. The stator winding of an asynchronous machine is similar to the armature of a synchronous machine; however, it is not defined as an armature in the IEC standard. In this material, the asynchronous machine stator is therefore referred to as a rotating-field stator winding, not an armature winding. Voltages are also induced to the rotor of an asynchronous machine, and currents significant in the torque production are created. However, the rotor itself 2 takes only a rotor’s dissipation power (I R) from the air gap power of the machine, this power being proportional to the slip; therefore, the machine can be considered stator fed, and depending on the rotor type, the rotor is called either a squirrel cage rotor or a wound rotor. In DC machines, the function of a rotor armature winding is to perform the actual power transmission, the machine being thus rotor fed. Field windings do not normally participate in energy conversion, double-salient pole reluctance machines maybe excluded: in principle, they have nothing but magnetizing windings, but the windings also perform the function of the armature. In DC machines, commutating and compensating windings are windings, the purpose of which is to create auxiliary field components to compensate the armature reaction of the machine and thus improve its performance characteristics. Similarly as the previously described windings, these windings do not participate in energy conversion in the machine either. The damper windings of synchronous machines are a special case among different winding types. Their primary function is to damp undesirable phenomena, such as oscillations and fields rotating opposite to the main field. Damper windings are important during the transients of controlled synchronous drives, in which the damper windings keep the air gap flux linkage instantaneously constant. In the asynchronous drive of a synchronous machine, the damper windings act like cage windings of asynchronous machines. The most important windings are categorized according to their geometrical characteristics and internal connections as follows: J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.2 • phase windings, • salient pole windings, and • commutator windings Windings, in which separate coils embedded in slots form a single or poly-phase winding, constitute a large group of AC armature windings. However, a similar winding is also employed in the magnetizing of non-salient pole synchronous machines. In commutator windings, individual coils contained in slots form a single or several closed circuits, which are connected together via a commutator. Commutator windings are employed only as armature windings of DC and AC commutator machines. Salient pole windings are normally concentrated field windings, but may also be used as armature windings for instance in fractional slot permanent magnet machines and in double-salient reluctance machines. Concentrated stator windings are used as an armature winding also in small shaded-pole motors. In the following, the windings applied in electrical machines are classified according to the two main winding types, viz. slot windings and salient pole windings. Both types are applicable both to direct and alternating current cases, Table 2.1. Table 2.1. Different types of windings or permanent magnets used instead of a field winding in the most common machine types. Stator winding Rotor winding Compensating Commutating Damper winding winding winding Salient pole poly-phase salient pole - - short-circuited cage synchronous machine distributed winding winding rotating-field slot winding Non-salient pole poly-phase slot winding - - solid rotor core or distributed short-circuited cage synchronous machine rotating-field slot winding winding Synchronous poly-phase - - - short-circuited cage reluctance machine distributed winding possible rotating-field slot winding Permanent magnet poly-phase permanent - - solid rotor or short- synchronous machine, distributed magnets circuited cage PMSM, rotating-field slot winding, or e.g. q 0.5 winding aluminium plate in the air gap possible Permanent magnet poly-phase permanent - - Damping should be synchronous machine, concentrated magnets harmful because of PMSM, pole winding excessive losses q ≤ 0.5 Double-salient poly-phase - - - - reluctance machine concentrated pole winding Induction motor, IM poly-phase cast or soldered - - - distributed cage winding, rotating-field slot squirrel cage winding winding Solid rotor IM poly-phase solid rotor made - - - distributed of steel, may be rotating-field slot equipped with winding squirrel cage Slip-ring poly-phase poly-phase - - - asynchronous motor distributed distributed rotating-field slot rotating-field slot winding winding J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.3 DC machine salient pole rotating-field slot winding salient pole - winding commutator slot winding winding 2.1 Basic Principles 2.1.1 Salient Pole Windings Fig. 2.1 illustrates a synchronous machine with a salient pole rotor. To magnetize the machine, direct current is fed through brushes and slip rings to the windings located on the salient poles. The main flux created by the direct current flows from the pole shoe to the stator and back simultaneously penetrating the poly-phase slot winding of the stator. The dotted lines in the figure depict the paths of the main flux. Such a closed path of a flux forms the magnetic circuit of a machine. One turn of a coil is a conductor that constitutes a single turn around the magnetic circuit. A coil is a part of winding that consists of adjacent series-connected turns between the two terminals of the coil. Figure 2.1a illustrates a synchronous machine with a pole with one coil per pole, whereas in Figure 2.1b, the locations of the direct (d) and quadrature (q) axes are shown. d q q τ p d a) b) Figure 2.1. a) Salient pole synchronous machine (p = 4). The black areas around two pole bodies form a salient pole winding. b) Single poles with windings, d = direct axis, q = quadrature axis. In salient pole machines, these two magnetically different, rotor-geometry-defined axes have a remarkable effect on the machine behaviour; the issue will be discussed later. A group of coils is a part of winding that magnetizes the same magnetic circuit. In Fig. 2.1a, the coils at the different magnetic poles (N and S alternating) form in pairs a group of coils. The number of field winding turns magnetizing one pole is N . f The salient pole windings located on the rotor or on the stator are mostly used for the DC magnetizing of a machine. The windings are then called magnetizing or sometimes excitation windings. With a direct current, they create a time-constant current linkage Θ. The part of this current linkage consumed in the air gap, that is, the magnetic potential difference of the air gap U , m,δ may be, for simplicity, regarded as constant between the quadrature axes, and it changes its sign at the quadrature axis q, Fig. 2.2. A significant field of application for salient pole windings is doubly salient reluctance machines. In these machines, a solid salient pole is not utilizable, since the changes of flux are rapid when operating at high speeds. At simplest, DC pulses are fed to the pole windings with power switches. In the air gap, direct current creates a flux that tries to turn the rotor in a direction where the J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.4 magnetic circuit of the machine reaches its minimum reluctance. The torque of the machine tends to be pulsating, and to reach an even torque, the current of a salient pole winding should be controllable so that the rotor can rotate without jerking. Salient pole windings are employed also in the magnetizing windings of the DC machines. All series, shunt, and compound windings are wound on salient poles. The commutating windings are also of the same type as salient pole windings. Β δ Θ f 0 U U m,δ m,Fe R R m,δ m,Fe Θ f −π/2 π/2 Θ f U U m,Fe m,δ q d q a) b) Figure 2.2. a) Equivalent magnetic circuit. The current linkages Θ created by two adjacent salient pole windings. Part f U is consumed in the air gap. b) The behaviour of the air gap flux density B . Thanks to the appropriate design of the m,δ δ pole shoe, the air gap flux density varies cosinusoidally even though it is caused by the constant magnetic potential difference in the air gap U . The air gap magnetic flux density B has its peak value on the d-axis and is zero on the q- m,δ δ axis. The current linkage created by the pole is accumulated by the ampere turns on the pole. EXAMPLE 2.1: Calculate the field winding current that can ensure a maximum magnetic flux density of B = 0.82 T in the air gap of a synchronous machine if there are 95 field winding turns δ per pole. It is assumed that the air gap magnetic flux density of the machine is sinusoidal along the pole shoes and the magnetic permeability of iron is infinite ( μ = ∞) in comparison with the Fe −7 permeability of air μ = 4π ⋅10 H/m. The minimum length of the air gap is 3.5 mm. 0 SOLUTION: If μ = ∞, the magnetic reluctance of iron parts and the iron magnetic potential Fe difference is zero. Now, the whole field current linkage Θ = N I is spent in the air gap to create f f f the required magnetic flux density: B 0.82 3 δ Θ = N I = U = H δ = δ = 3.5⋅10 A f f f m,δδ −7 μ 4π ⋅10 0 If the number of turns is N = 95, the field current is f Θ 0.82 1 −3 f I = = 3.5⋅10 A = 24 A f −7 N 4π ⋅10 95 f It should be noticed that calculation of this kind is appropriate for an approximate calculation of the current linkage needed. In fact, about 60–90 % of the magnetic potential difference in electrical current linkage is createdJ. Pyrhönen, T. Jokinen, V. Hrabovcová 2.5 machines is spent in the air gap, and the rest in the iron parts. Therefore, in a detailed design of electrical machines, it is necessary to take into account all the iron parts with appropriate material properties. A similar calculation is valid for DC machines with the exception that in DC machines the air gap is usually constant under the poles. 2.1.2 Slot Windings Here we concentrate on symmetrical, three-phase AC distributed slot windings, in other words, rotating-field windings. However, first, we discuss the magnetizing winding of a rotor of a non- salient pole synchronous machine, and finally turn to commutator windings, compensating windings, and damper windings. Because unlike in the salient pole machine, the length of the air gap is now constant, we may create a cosinusoidally distributed flux density in the air gap by producing a cosinusoidal distribution of current linkage with an AC magnetizing winding, Fig. 2.3. The cosinusoidal distribution, instead of sinusoidal, is used because we want the flux density to reach its maximum on the direct-axis, where α = 0. In the case of Fig. 2.3, the function of the magnetic flux density approximately follows the curve function of the current linkage distribution Θ (α ). In machine design, an equivalent air gap δ is e applied, the target being to create a cosinusoidally alternating flux density into the air gap μ 0 B() α = Θ() α (2.1) δ e The concept of equivalent air gap δ will be discussed later. e d α I f rotor current linkage z I Q f q α 0π I f q dq z d Q Figure 2.3. Current linkage distribution created by two-pole non-salient pole winding and the fundamental of the current linkage. There are z conductors in each slot, and the excitation current in the winding is I . The height of a single step f Q of the current linkage is z I . Q f The slot pitch τ and the slot angle α are the core parameters of the slot winding. The slot pitch is u u measured in metres, whereas the slot angle is measured in electrical degrees. The number of slots being Q and the diameter of the air gap D, we may write J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.6 πD 2π τ = ; α = p. (2.2) u u Q Q The slot pitch being usually constant in non-salient pole windings, the current sum (z I ) in a slot has Q f to be of a different magnitude in different slots (in a sinusoidal or cosinusoidal manner to achieve a sinusoidal or cosinusoidal variation of current linkage along the surface of the air gap.). Usually, there is a current of equal magnitude flowing in all turns in the slot, and therefore, the number of conductors z in the slots has to be varied. In the slots of the rotor in Fig. 2.3, the number of turns is Q equal in all slots, and a current of equal magnitude is flowing in the slots. We may see that by selecting z slightly differently in different slots, we can improve the stepped waveform of the figure Q to better approach the cosinusoidal form. The need for this depends on the induced voltage harmonic content in the stator winding. The voltage may be of almost pure sinusoidal waveform despite the fact that the air gap flux density distribution should not be perfectly sinusoidal. This depends on the stator winding factors for different harmonics. In synchronous machines, the air gap is usually relatively large, and correspondingly, the flux density on the stator surface changes more smoothly (neglecting the influence of slots) than the stepped current linkage waveform of Fig. 2.3. Here, we apply the well-known finding that if 2/3 of the rotor surface are slotted and 1/3 is left slotless, not only the third harmonic component but any of its multiple harmonics called triplen harmonics are th th eliminated in the air gap magnetic flux density, and also the low-order odd harmonics (5 , 7 ) are suppressed. 2.1.3 End Windings Figure 2.4 illustrates how the arrangement of the coil end influences the physical appearance of the winding. The windings a and b in the figure are of equal value with respect to the main flux, but their leakage inductances diverge from each other because of the slightly different coil ends. When investigating the winding a of Fig. 2.4, we note that the coil ends form two separate planes at the endfaces of the machine. This kind of a winding is therefore called a two-plane winding. The coil ends of the type are depicted in Fig. 2.4e. In the winding of Fig. 2.4b, the coil ends are overlapping, and therefore, this kind of winding is called a diamond winding (lap winding). Figures 2.4c and d illustrate three-phase stator windings that are identical with respect to the main flux, but in Fig. 2.4c, the groups of coil are non-divided, and in Fig. 2.4d, the groups of coil are divided. In Fig. 2.4c, an arbitrary radius r is drawn across the coil end. It is shown that at any position, the radius intersects only coils of two phases, and the winding can thus be constructed as a two-plane winding. A corresponding winding constructed with distributed coils (Fig. 2.4d) has to be a three-plane arrangement, since now the radius r may intersect the coil ends of the windings of all the three phases. J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.7 1 2 3 4 5 67 8 1 2 3 45 67 8 a) e) 12 3 4 5 6 7 8 12 3 4 5 6 7 8 b) coil span f) coil side r r 1 1 24 24 2 2 23 23 3 3 22 22 4 4 21 21 5 5 20 20 6 6 19 7 19 7 18 18 8 8 17 9 17 9 16 10 16 10 15 11 15 11 14 14 12 12 13 13 c) d) Figure 2.4. a) Concentric winding and b) a diamond winding. In a two-plane winding, the coil spans differ from each other. In the diamond winding, all the coils are of equal width. c) A two-plane three-phase four-pole winding with non- divided groups of coil. d) A three-plane three-phase four-pole winding with divided groups of coils. Figures c and d illustrate also a single main flux path. e) Profile of an end winding arrangement of a two-plane winding. f) Profile of an end winding of a three-plane winding. The radii r in the figures illustrate that in a winding with non-divided groups, an arbitrary radius may intersect only two phases, and in a winding with divided groups, the radius may intersect all the three phases. The two- or three-plane windings will result correspondingly. The part of a coil located in a single slot is called a coil side, and the part of the coil outside the slot is termed a coil end. The coil ends together constitute the end windings of the winding. 2.2 Phase Windings Next, poly-phase slot windings that produce the rotating field of poly-phase AC machines are investigated. In principle, the number of phases m can be selected freely, but the use of a three-phase supply network has led to a situation in which also most electrical machines are of the three-phase type. Another, extremely common type is two-phase electrical machines that are operated with a capacitor start and run motor in a single-phase network. A symmetrical two-phase winding is in principle the simplest AC winding that produces a rotating field. endwindingJ. Pyrhönen, T. Jokinen, V. Hrabovcová 2.8 A configuration of a symmetrical poly-phase winding can be considered as follows: the periphery of the air gap is evenly distributed over the poles so that we can determine a pole arc, which covers 180 electrical degrees and a corresponding pole pitch, τ , which is expressed in metres p πD τ = . (2.3) p 2 p Figure 2.5 depicts the division of the periphery of the machine into phase zones of positive and negative values. In the figure, the number of pole pairs p = 2, and the number of phases m = 3. τ p -U V W -W o 120 τ v U -V p = 2 U -V m = 3 -W W o 180 V -U o 120 Figure 2.5. Division of the periphery of a three-phase four-pole machine into phase zones of positive and negative values. Pole pitch is τ and phase zone distribution τ . When the windings are located in the zones, the instantaneous p v currents in the positive and negative zones are flowing in opposite directions. Phase zone distribution is written as τ p τ =. (2.4) v m The number of zones will thus be 2pm. The number of slots per each such zone is expressed by the term q, as a number of slots per pole and phase Q q = . (2.5) 2 pm Here Q is the number of slots in the stator or in the rotor. In integral slot windings, q is an integer. However, q can also be a fraction. In that case, the winding is called a fractional slot winding. The phase zones are distributed symmetrically to different phase windings so that the phase zones of the phases U, V, W, ... are positioned on the periphery of the machine at equal distances in electrical degrees. In a three-phase system, the angle between the phases is 120 electrical degrees. This is illustrated by the periphery of Fig. 2.5, where we have 2×360 electrical degrees because of four J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.9 poles. Now, it is possible to label every phase zone. We start for instance with the positive zone of the phase U. The first positive zone of the phase V shall be 120 electrical degrees from the first positive zone of the phase U. Correspondingly, the first positive zone of the phase W shall be 120 electrical degrees from the positive zone of the phase V etc. In Fig. 2.5, there are two pole pairs, and hence we need two positive zones for each phase U, V, and W. In the slots of each, now labelled phase zones, there are only the coil sides of the labelled phase coil, in all of which the current flows in the same direction. Now, if their direction of current is selected positive in the diagram, the unlabelled zones become negative. Negative zones are labelled by starting from the distance of a pole pitch from the position of the positive zones. Now U and –U, V and –V, W and –W are at the distance of 180 electrical degrees from each other. 2.3 Three-Phase Integral Slot Stator Winding The armature winding of a three-phase electrical machine is usually constructed in the stator, and it is spatially distributed in the stator slots so that the current linkage created by the stator currents is distributed as sinusoidally as possible. The simplest stator winding that produces a noticeable rotating field comprises three coils, the sides of which are divided into six slots, because if m = 3, p = 1, q = 1, then Q = 2pmq = 6; see Figs. 2.6 and 2.7. EXAMPLE 2.2: Create a three-phase, two-pole stator winding with q = 1. Distribute the phases in the slots and illustrate the current linkage created based on the instant values of phase sinusoidal currents. Draw a phasor diagram of the slot voltage and sum the voltages of the individual phases. Create a current linkage waveform in the air gap for the time instant t when the phase U voltage is 1 in its positive maximum and for t , which is 30° shifted. 2 SOLUTION: If m = 3, p = 1, q = 1, then Q = 2pmq = 6, which is the simplest case of three-phase windings. The distribution of the phases in the slots will be explained based on Fig. 2.6. Starting from the slot 1, we insert there the positive conductors of the phase U forming the zone U1. The pole pitch expressed in the number of slots per pole, or in other words, ‘the coil span expressed in the number of slot pitches y ’ is Q Q 6 y = = = 3 . Q 2 p 2 Then, the zone U2 will be one pole pitch shifted from U1 and will be located in the slot 4, because 1 + y = 1 + 3 = 4. The beginning of the phase V1 is 120°shifted from U1, which means the slot 3, and Q its end V2 is in the slot 6 (3 + 3 = 6). The phase W1 is again shifted form V1 by 120°, which means the slot 5, and its end is in the slot 2; see Fig. 2.6a. The polarity of instantaneous currents is shown at the instant, when the current of the phase U is in its positive maximum value flowing in the slot 1, depicted as a cross (tail of arrow) in U1 (current flowing away from the observer). Then, U2 is depicted by a dot (a point of arrow) in the slot 4 (current flowing towards the observer). At the same instant in V1 and W1, there are also dots, because the phases V and W are carrying negative current values (see 2.6d), and therefore V2 and W2 are positive, indicated by crosses. In this way, a sequence of slots with inserted phases is as follows: U1, W2, V1, U2, W1, V2, if q = 1. J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.10 U1 1 V2 W2 α 2 6 12 34 5 6 p = 1 m = 3 W1 V1 5 3 4 U2 U1 U2 a) b) U1 V2 W2 UV W i W1 V1 t t U2 1 c) d) 1 U U o -4 α = 60 u 6 2 1 5 3 -2 -6 5 U U 3 W V 4 e) f) Figure 2.6. The simplest three-phase winding that produces a rotating field. a) A cross-sectional surface of the machine and a schematic view of the main flux route at the observation instant t , b) a developed view of the winding in a plane, 1 and c) a three-dimensional view of the winding. The figure illustrates how the winding penetrates the machine. The coil end at the rear end of the machine is not illustrated as in reality, but the coil comes directly from a slot to another without travelling along the rear endface of the stator. The ends of the phases U, V, and W at the terminals are denoted U1-U2, V1-V2, and W1-W2. d) The three-phase currents at the observed time instant t when i = i = -1/2 i . (i +i +i = 0), 1 W V U U V W e) a voltage phasor diagram for the given three-phase system, f) the total phase voltage for individual phases. The voltage of the phase U is created by summing the voltage of the slot 1 and the negative voltage of slot 4, and therefore the direction of the voltage phasor in the slot 4 is taken opposite with the denotation −4. We can see the sum of voltages in both slots and the phase shift by 120° of the V and W phase voltages. The cross-section of the stator winding in Fig. 2.6a shows fictitious coils with current directions resulting in the magnetic field represented by the force lines and arrows. The phasor diagram in Fig. 2.6e includes six phasors. To determine their number, the largest common divider of Q and p denoted t has to be found. In this case, for Q = 6 and p = 1, t = 1, and therefore, the number of phasors is Q /t = 6. The angle between the voltage phasors in the adjacent slots is given by expression 360° p 360° ⋅1 α = = = 60° , u Q 6J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.11 which results in the numbering of the voltage phasors in slots as shown in Fig. 2.6e. Now, the total phase voltage for individual phases has to be summed. The voltage of the phase U is created by the positive voltage in the slot 1, and the negative voltage in the slot 4. The direction of the voltage phasor in the slot 4 is taken opposite with the denotation −4. We can see the sum of voltages in both slots of the phase U, and the phase shift of 120° of the V and W phase voltages in Fig. 2.6f. The current linkage waveforms for this winding are illustrated in Fig. 2.7b and c for the time instants t and t , between which the waveforms proceed by 30°. The procedure of drawing the figure can be 1 2 described as follows: We start observation at α = 0. We assume the same constant number of conductors z in all slots. Q The current linkage value on the left in Fig. 2.7b is changed stepwise at the slot 2, where the phase W is located and is carrying a current with a cross sign. This can be drawn as a positive step of Θ with a certain value (Θ(t ) = i (t )z ). Now, the current linkage curve remains constant until we 1 uW 1 Q reach the slot 1, where the positive currents of the phase U are located. The instantaneous current in the slot 1 is the phase U peak current. The current sum is indicated again with a cross sign. The step height is now twice the height in the slot 2, because the peak current is twice the current flowing in the slot 2. Then, in the slot 6, there is again a positive half step caused by the phase V. In the slot 5, there is a current sum indicated by a dot, which means a negative Θ step. The same is repeated with all slots, and when the whole circle has been closed, Fig. 2.7b. When this procedure is repeated for one period of the current, we obtain a travelling wave for the current linkage waveform. Fig. 2.7c shows the current linkage waveform after 30 degrees. Here we can see that if the instantaneous value of a slot current is zero, the current linkage does not change, and the current linkage remains constant; see the slots 2 and 5. We can also see that the Θ profiles in b and c are not similar, but the form is changed depending on the time instant at which it is investigated. J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.12 i z 4 U1 Q ˆ Θ = U1 π 2 a) z i /2 z i Q U Q U 0 π z i /2 Q U starting of study slotted stator inner periphery U1 U2 U1 slot 2 1 6 5 4 3 2 slot 2 1 6 5 4 3 2 ΘΘ αα OO O OO O 0 180 360 0 180 360 U V W U V W i i t t 1 2 b) c) Figure 2.7. Current linkages Θ created by a simple three-phase q = 1 winding, a) only the phase U is fed by current and observed. A rectangular waveform of current linkage with its fundamental component is shown to explicate the staircase profile of the current linkages below. If all three phases are fed and observed in two different current situations (i +i +i = 0) at two time instants t and t , see b) and c) respectively. The figure illustrates also the fundamental of the U V W 1 2 staircase current linkage curves. The stepped curves are obtained by applying Ampère’s law in the current-carrying teeth zone of the electrical machine. Note that as time elapses from t to t , the three phase currents change and also the 1 2 position of the fundamental component changes. This indicates clearly the rotating-field nature of the winding. The angle α and the numbers of slots refer to the previous figure, in which we see that the maximum flux density in the air gap lies between the slots 6 and 5. This coincides with the maximum current linkage shown in this figure. This is valid if no rotor currents are present. Figure 2.7 shows that the current linkage produced with such a simple winding deviates considerably from sinusoidal waveform. Therefore, in electrical machines, more coil sides are usually employed per pole and phase. EXAMPLE 2.3: Consider an integral slot winding, where p = 1 and q = 2, m = 3. Distribute the phase winding into the slots, make an illustration of the windings in the slots, draw a phasor diagram and show the phase voltages of the individual phases. Create a waveform of the current linkage for this winding and compare it with that in Fig. 2.7. SOLUTION: The number of the slots needed for this winding is Q = 2 pmq = 2 ⋅ 3⋅ 2 = 12 . The cross- sectional area of such a stator with 12 slots and embedded conductors of individual phases is illustrated in Fig. 2.8a. The distribution of the slots into the phases is made in the same order as in Example 2.2, but now q = 2 slots per pole and phase. Therefore, the sequence of the slots for the J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.13 phases is as follows: U1, U1, W2, W2, V1, V1, U2, U2, W1, W1, V2, V2. The direction of the current in the slots will be determined in the same way as above in Example 2.2. The coils wound in individual phases are shown in Fig. 2.8b. The pole pitch expressed in number of slot pitches is Q 12 y = = = 6 Q 2 p 2 Figure 2.8c shows how the phase U is wound to keep the full pitch equal to 6 slots. In Fig. 2.8d, the average pitch is also 6, but the individual steps are y = 5 and 7, which gives the same average result Q for the value of induced voltage. The phasor diagram has 12 phasors, because t = 1 again. The angle between two phasors of adjacent slots is 360° p 360° ⋅1 α = = = 30° u Q 12 The phasors are numbered gradually around the circle. Based on this diagram, the phase voltage of all phases can be found. Figures 2.8f and g show that the voltages are the same independent of the way how separate coil sides are connected in series. In comparison with the previous example, the geometrical sum is now less than the algebraic sum. The phase shifting between coil side voltages is caused by the distribution of the winding in more than one slots, here in two slots per each pole. This reduction of the phase voltage is expressed by means of a distribution winding factor; this will be derived later. The waveform of the current linkage for this winding is given in Fig. 2.9. We can see that it is much closer to a sinusoidal waveform than in the previous example with q = 1. In undamped permanent magnet synchronous motors, also such windings can be employed, the number of slots per pole and phase of which is clearly less than one, for instance q = 0.4. In that case, a well-designed machine looks like a rotating-field machine when observed at its terminals, but the current linkage produced by the stator winding deviates so much from the fundamental that, because of excessive harmonic losses in the rotor, no other rotor type comes into question. J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.14 U1 2 1 3 V2 W2 12 4 α 11 5 1234 56 8 9 10 11 12 1 12 7 p = 1 m = 3 10 6 V1 W1 9 7 8 U2 U1 W2 V1 W1 V2 U1 U2 a) b) 1 12 2 11 12 1 2 3 4 5 6 7 8 9 12 1 2 3 4 5 6 7 8 9 3 o 10 α = 30 u y = 5 Q1 4 9 5 U1 y = 7 U2 8 y = 6 Q2 U1 U2 Q 6 7 c) d) e) U phU -8 -7 U phU 2 2 -8 -7 1 1 U phW U phW U phV g) f) U phV Figure 2.8. Three-phase two-pole winding with two slots per pole and phase, a) a stator with 12 slots, the number of slots per pole and phase q = 2. b) Divided coil groups, c) full-pitch coils of the phase U, d) average full-pitch coils of phase U, e) a phasor diagram with 12 phasors, one for each slot, f) sum phase voltage of individual phases corresponding to Fig. c, g) a sum phase voltage of individual phases corresponding to Fig. d. Θ/A 200 ˆ 100 Θ s1 α 5 4 3 2 1 12 11 10 9 8 7 6 5 4 3 2 1 Figure 2.9. Current linkage created by the winding on the surface of the stator bore of Fig. 2.8 at a time i = Θ = f (α ) W s i = −1/2 i . The fundamental is given as a sinusoidal curve. The numbering of the slots is also given. Θ of Θ V U s1 s J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.15 When comparing Fig. 2.9 (q = 2) with Fig. 2.7 (q = 1), it is obvious that the higher the term q (slots per pole and phase) is, the more sinusoidal the current linkage of the stator winding is. As we can see in Fig. 2.7a, the current linkage amplitude of the fundamental component for one full- pitch coil is ˆ z i 4 Q U ˆ Θ = . (2.6) 1U π 2 If the coil winding is distributed into more slots, and q 1 and N = pqz , the winding factor must be Q taken into account: ˆ 4 Nk i w1 U ˆ Θ = . (2.7) 1U π 2 In a 2p-pole machine (2p2), the current linkage for one pole is: ˆ 4 Nk i w1 U ˆ Θ = . (2.8) 1U π 2 p This expression can be rearranged with the number of conductors in a slot. In one phase, there are 2N conductors, and they are embedded in the slots belonging to one phase Q/m. Therefore, the number of conductors in one slot will be: 2N 2mN N z = = = (2.9) Q Q / m 2 pqm pq and N = qz (2.10) Q p Then N/p presented in Eq. (2.8) and in the following can be introduced by qz : Q. ˆ ˆ 4 Nk i 4 k i w1 U w1 U ˆ Θ = = qz . (2.11) 1U Q π 2 pπ 2 It can be also expressed with the effective value of sinusoidal phase current if there is a symmetrical system of phase currents: 4 Nk w1 ˆ Θ = 2I. (2.12) 1U π 2 p For an m-phase rotating-field stator or rotor winding, the amplitude of current linkage is m/2 times higher: m 4 Nk w1 ˆ Θ = 2I (2.13) 1 2π 2 p and for three-phase stator or rotor winding, the current linkage amplitude of the fundamental component for one pole is: 3 4 Nk 3 Nk w1 w1 ˆ Θ = 2I = 2I (2.14) 1 2π 2 pπ pJ. Pyrhönen, T. Jokinen, V. Hrabovcová 2.16 ˆ For a stator current linkage amplitude Θ of the harmonic ν of the current linkage of a poly-phase sν (m 1) rotating-field stator winding (or rotor winding), when the effective value of the stator current is I , we may write s m 4 k N 1 mk N wν s wν s ˆ ˆ Θ = 2I = i. (2.15) sν s s 2π pν 2πpν EXAMPLE 2.4: Calculate the amplitude of the fundamental component of stator current linkage, if ˆ N = 200, k = 0.96, m = 3, p = 1 and i (t) = i = 1 A , the effective value for a sinusoidal current s w1 sU being I = (1/ 2) A = 0.707 A. s ˆ SOLUTION: For the fundamental, we obtain Θ = 183.3 A, because: s1 3 4 Nk 3 Nk 3 200 ⋅ 0.96 w1 w1 ˆ Θ = 2I = 2I = ⋅ 2 ⋅ 0.707A = 183.3 A. 1 s s 2π 2 pπ pπ 1 2.4 Voltage Phasor Diagram and Winding Factor Since the winding is spatially distributed in the slots on the stator surface, the flux (which is proportional to the current linkage Θ) penetrating the winding does not intersect all windings simultaneously, but with a certain phase shift. Therefore, the electromotive force (emf) of the winding is not calculated directly with the number of turns N, but the winding factors k s wν corresponding to the harmonics are required. The emf of the fundamental induced in the turn is calculated with the flux linkage Ψ by applying Faraday’s induction law e = −Nk dΦ / dt = −dΨ/dt w1 (see Eqs. 1.3, 1.7 and 1.8) We can see that the winding factor correspondingly indicates the characteristics of the winding to produce harmonics, and it has thus to be taken into account when calculating the current linkage of the winding (Eq. 2.15). The common distribution of all the current linkages created by all the windings together produces a flux density distribution in the air gap of the machine, which, when moving with respect to the winding, induces voltages to the conductors of the winding. The phase shift of the induced electromotive force in different coil sides is investigated with a voltage phasor diagram. The voltage phasor diagram is presented in electrical degrees. If the machine is for instance a four-pole one, p = 2, the voltage vectors have to be distributed along two full circles in the stator bore. Figure 2.10 a) illustrates the voltage phasor diagram of a two-pole winding of Fig. 2.8. In Fig. 2.10a, the phasors 1 and 2 are positive and 7 and 8 are negative for the phase under consideration. Hence, the phasors 7 and 8 are turned by 180 degrees to form a bunch of phasors. For harmonic ν (excluding slot harmonics that have the same winding factor as the fundamental) the directions of the phasors of the coil sides vary more than in the figure, because the slot angles α are u replaced with the angles να . u According to Fig. 2.10b, when calculating the geometric sum of the voltage phasors for a phase winding, the symmetry line for the bunch of phasors, where the negative phasors have been turned opposite, must be found. The angles α of the phasors with respect to this symmetry line may be ρ used in the calculation of the geometric sum. Each phasor contributes to the sum with a component proportional to cosα . ρ J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.17 Uρ ∑ z α z 2 1 -U 7 -U 3 cosα 8 α ρ ρ 12 4 U 2 U symmetry 2 11 line 5 α 2 -U 8 -U 7 10 positive 6 phasors 9 7 U -U 1 8 8 -U U 7 2 negative phasors a) b) c) Figure 2.10. a) and b) Fundamental voltage phasor diagram for the winding of Fig. 2.8 Q = 12, p = 1, q = 2. A s s maximum voltage is induced in the bars in the slots 1 and 7 at the moment depicted in the figure, when the rotor is rotating clockwise. The figure illustrates also the calculation of the voltage in a single coil with the radii of the voltage phasor diagram. c) General application of the voltage phasor diagram in the determination of the winding factor (fractional slot winding since the number of phasors is uneven). The phasors of negative coil sides are turned 180°, and then the summing of the resulting bunch of phasors is calculated according to Eq. (2.16). A symmetry line is drawn in the middle of the bunch, and each phasor forms an angle α with the symmetry line. The geometric sum of all the ρ phasors lies on the symmetry line. We can now write a general presentation for the winding factor k of a harmonic ν , by employing wν the voltage phasor diagram νπ sin Z 2 k = cosα. (2.16) wν ∑ρ Z ρ=1 Here Z is the total number of positive and negative phasors of the phase in question, ρ is the ordinal number of a single phasor, and ν is the ordinal number of the harmonic under observation. The νπ sin coefficient in the equation only influences the sign (of the factor). The angle of a single 2 phasor α can be found from the voltage phasor diagram drawn for the specific harmonic, and it is ρ the angle between an individual phasor and the symmetry line drawn for a specific harmonic (c.f. Fig. 2.10b). This voltage vector diagram solution is universal and may be used in all cases, but the numerical values of Eq. (2.16) do not always have to be calculated directly from this equation, or with the voltage phasor diagram at all. In simple cases, we may apply equations introduced later. However, the voltage phasor diagram forms the basis for the calculations, and therefore its utilization is discussed further when analyzing different types of windings. If we are in Fig. 2.10a considering a currentless stator of a synchronous machine, a maximum voltage can be induced to the coil sides 1 and 7 at the middle of the pole shoe, when the rotor is rotating at no load inside the stator bore (which corresponds to the peak value of the flux density, but the zero value of the flux penetrating the coil), where the derivative of the flux penetrating the coil reaches its peak value, the voltage induction being at its highest at that moment. If the rotor J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.18 rotates clockwise, a maximum voltage is induced in the coil sides 2 and 8 in a short while, and so on. The voltage phasor diagram then describes the amplitudes of voltages induced in different slots and their temporal phase shift. The series-connected coils of the phase U travel e.g. from the slot 1 to the slot 8 (coil 1) and from the slot 2 to the slot 7 (coil 2). Thus a voltage, which is the difference of the phasors U and U , is 1 8 induced in the coil 1. The total voltage of the phase is thus U = U −U + U −U. (2.17) U 1 8 2 7 The figure also indicates the possibility of connecting the coils in the order 1–7 and 2–8, which gives the same voltage but a different end winding. The winding factor k based on the distribution w1 of the winding for the fundamental is calculated here as a ratio of the geometric sum and the sum of absolute values as follows: geometric sum U −U +U −U 1 8 2 7 k = = = 0.966 ≤ 1. (2.18) w1 sum of absolute values U + U + U + U 1 8 2 7 EXAMPLE 2.5: Equation (2.16) indicates that the winding factor for the harmonics may also be calculated using the voltage phasor diagram. Derive the winding factor for the seventh harmonic of the winding in Fig 2.8. SOLUTION: We now draw a new voltage phasor diagram based on Fig. 2.10 for the seventh harmonic, Fig. 2.11. τ p1 a) τ p7 10 11 12 1 2 3 4 5 6 7 8 9 10 8 1 U -U b) c) -U 7,2 7,8 7,7 U 7,1 -U 7,7 α =να 5π/12 uν u U 7,2 -U 7 7,8 2 Fig. 2.11. Deriving the harmonic winding factor, a) the fundamental and the seventh harmonic field in the air gap over the slots, b) voltage phasors for the seventh harmonic of a full pitch q = 2 winding (slot angle α = 210°), and c) the u7 symmetry line and the sum of the voltage phasors. The phasor angles α with respect to the symmetry line are α = ρ ρ 5π/12 or −5 π /12. The slots 1 and 2 belong to the positive zone of the phase U and the slots 7 and 8 to the negative zone measured by the fundamental. In Fig. 2.11, we see that the pole pitch of the seventh harmonic J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.19 is one seventh of the fundamental pole pitch. Deriving the phasor sum for the seventh harmonic is started for instance with the voltage phasor of the slot 1. This phasor remains in its original position. The slot 2 is physically and by fundamental located 30° clockwise from the slot 1, but as we are now studying the seventh harmonic, the slot angle measured in degrees for it is 7×30°=210°, which can also be seen in the figure. The phasor for the slot 2 is, hence, located 210° from the phasor 1 clockwise. The slot 7 is located at 7×180°=1260° from the slot 1. Since 1260° = 3×360° + 180° the phasor 7 remains opposite to the phasor 1. The phasor 8 is located 210° clockwise from the phasor 7 and will find its place 30° clockwise from the phasor 1. By turning the negative zone phasors by π and applying Eq. (2.17) we obtain 7π 7π sin sin 4 − 5π + 5π − 5π + 5π ⎛ ⎞ 2 2 k = cosα = cos + cos + cos + cos = − 0.2588. ⎜ ⎟ ∑ w7ρ 4 4 12 12 12 12 ⎝ ⎠ ρ=1 It is not necessary to apply the voltage phasor diagram, but also simple equations may be derived to directly calculate the winding factor. In principle, we have three winding factors: a distribution factor, a pitch factor, and a skewing factor. The latter may also be taken into account by a leakage inductance. The winding factor derived based on the shifted voltage phasors in the case of distributed winding is called the distribution factor with the subscript ‘d’. This factor is always k≤ d1 1. The value k = 1 can be reached when q = 1, in which case the geometric sum equals the sum of d1 absolute values, see Fig. 2.6f. If q ≠ 1, then k 1.In fact, it means that the total phase voltage is d1 reduced by this factor (see Example 2.6). If each coil is wound as a full-pitch winding, the coil pitch is in principle the same as the pole pitch. However, the voltage of the phase with full-pitch coils is reduced because of the winding distribution with the factor k . If the coil pitch is shorter than the pole pitch and the winding is not a d full-pitch winding, the winding is called a short-pitch winding, or a chorded winding (see Fig. 2.15). Note that the winding in Fig. 2.8 is not a short-pitch winding, even though the coil may be realized from the slot 1 to the slot 8 (shorter than pole pitch) and not from the slot 1 to the slot 7 (equivalent to pole pitch). This is because the full-pitch coils together produce the same current linkage as the shorter coils together. A real short-pitching is obviously employed in the two-layer windings. Short- pitching is another reason why the voltage of the phase winding may be reduced. The factor of such reduction is called the pitch factor k . The total winding factor is given as: p k = k ⋅k. (2.19) w d p Equations to calculate the distribution factor k will be derived now; see Fig. 2.12. The equations are d based on the geometric sum of the voltage phasors in a similar way as in Figs. 2.10 and 2.11. J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.20 U coil3 U U coil4 coil2 ˆ B B δ1 3 4 2 U coil5 U o coil1 U α = 90 ˆ 5 ν=5 D B 1 δ5 1 A C qα /2 u τ α p5 u r τ p1 o α = 18 ν=1 O a) b) Figure 2.12. a) Determination of the distribution factor with a polygon with q = 5, b) the pole pitch for the fundamental and the fifth harmonic. The same physical angles for the fifth and the fundamental are shown as an example. The distribution factor for the fundamental component is given as geometric sum U l k = = . (2.20) d1 sum of absolute values qU coil1 According to Fig. 2.12, we may write for the triangle ODC U l qα qα u 2 u sin = ⇒ U = 2r sin , (2.21) l 2 r 2 and for the triangle OAB U v1 α α u u 2 sin = ⇒ U = 2r sin . (2.22) v1 2 r 2 We may now write for the distribution factor qα qα u u 2r sin sin U l 2 2 k = = = . (2.23) d1 α α qU u u v1 q2r sin qsin 2 2 This is the basic expression for the distribution factor for the calculation of the fundamental in a closed form. Since the harmonic components of the air gap magnetic flux density are present, the th calculation of the distribution factor for the ν harmonic will be carried out applying the angle να ; u see Fig. 2.11b and 2.12b: qα u sinν 2 k = . (2.24) dν α u qsinν 2 EXAMPLE 2.6: Repeat Example 2.5 using Eq. (2.24)

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