J. Pyrhönen, T. Jokinen, V. Hrabovcová 2.1

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1 J. Pyrhönen, T. Jokinen, V. Hrabovcová. WINDINGS OF ELECTRICAL MACHINES The oeration rincile 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 oerating modes and the tye of the electrical machine. According to their different functions in an electrical machine, the windings are groued for instance as follows: armature windings, other rotating-field windings (e.g. stator or rotor windings of induction motors) field (magnetizing) windings, damer windings commutating windings, and comensating windings. Armature windings are rotating-field windings, into which the rotating-field-induced voltage required in energy conversion is induced. According to IEC , the armature winding is a winding in a synchronous, DC, or single-hase commutator machine, which, in service, receives active ower from or delivers active ower to the external electrical system. This definition also alies to a synchronous comensator if the term active ower is relaced by reactive ower. The air ga flux comonent caused by the armature current linkage is called the armature reaction. An armature winding determined under these conditions can transmit ower 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 searate magnetizing winding; for instance in asynchronous machines, the stator winding both magnetizes the machine and acts as a winding, where the oerating 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 roduction are created. However, the rotor itself takes only a rotor s dissiation ower (I R) from the air ga ower of the machine, this ower being roortional to the sli; therefore, the machine can be considered stator fed, and deending on the rotor tye, 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 erform the actual ower transmission, the machine being thus rotor fed. Field windings do not normally articiate in energy conversion, double-salient ole reluctance machines maybe excluded: in rincile, they have nothing but magnetizing windings, but the windings also erform the function of the armature. In DC machines, commutating and comensating windings are windings, the urose of which is to create auxiliary field comonents to comensate the armature reaction of the machine and thus imrove its erformance characteristics. Similarly as the reviously described windings, these windings do not articiate in energy conversion in the machine either. The damer windings of synchronous machines are a secial case among different winding tyes. Their rimary function is to dam undesirable henomena, such as oscillations and fields rotating oosite to the main field. Damer windings are imortant during the transients of controlled synchronous drives, in which the damer windings kee the air ga flux linkage instantaneously constant. In the asynchronous drive of a synchronous machine, the damer windings act like cage windings of asynchronous machines. The most imortant windings are categorized according to their geometrical characteristics and internal connections as follows:

2 hase windings, salient ole windings, and commutator windings J. Pyrhönen, T. Jokinen, V. Hrabovcová. Windings, in which searate coils embedded in slots form a single or oly-hase winding, constitute a large grou of AC armature windings. However, a similar winding is also emloyed in the magnetizing of non-salient ole 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 emloyed only as armature windings of DC and AC commutator machines. Salient ole windings are normally concentrated field windings, but may also be used as armature windings for instance in fractional slot ermanent magnet machines and in double-salient reluctance machines. Concentrated stator windings are used as an armature winding also in small shaded-ole motors. In the following, the windings alied in electrical machines are classified according to the two main winding tyes, viz. slot windings and salient ole windings. Both tyes are alicable both to direct and alternating current cases, Table.. Table.. Different tyes of windings or ermanent magnets used instead of a field winding in the most common machine tyes. Stator winding Rotor winding Comensating Commutating Damer winding winding winding Salient ole synchronous machine oly-hase distributed rotating-field slot salient ole winding - - short-circuited cage winding Non-salient ole synchronous machine Synchronous reluctance machine Permanent magnet synchronous machine, PMSM, q > 0.5 Permanent magnet synchronous machine, PMSM, q 0.5 Double-salient reluctance machine Induction motor, IM Solid rotor IM Sli-ring asynchronous motor winding oly-hase distributed rotating-field slot winding oly-hase distributed rotating-field slot winding oly-hase distributed rotating-field slot winding oly-hase concentrated ole winding oly-hase concentrated ole winding oly-hase distributed rotating-field slot winding oly-hase distributed rotating-field slot winding oly-hase distributed rotating-field slot winding slot winding - - solid rotor core or short-circuited cage winding short-circuited cage winding ossible ermanent magnets ermanent magnets - - solid rotor or shortcircuited cage winding, or e.g. aluminium late in the air ga ossible - - Daming should be harmful because of excessive losses cast or soldered cage winding, squirrel cage winding solid rotor made of steel, may be equied with squirrel cage oly-hase distributed rotating-field slot winding

3 J. Pyrhönen, T. Jokinen, V. Hrabovcová.3 DC machine salient ole winding rotating-field commutator slot winding slot winding salient ole winding -. Basic Princiles.. Salient Pole Windings Fig.. illustrates a synchronous machine with a salient ole rotor. To magnetize the machine, direct current is fed through brushes and sli rings to the windings located on the salient oles. The main flux created by the direct current flows from the ole shoe to the stator and back simultaneously enetrating the oly-hase slot winding of the stator. The dotted lines in the figure deict the aths of the main flux. Such a closed ath 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 art of winding that consists of adjacent series-connected turns between the two terminals of the coil. Figure.a illustrates a synchronous machine with a ole with one coil er ole, whereas in Figure.b, the locations of the direct (d) and quadrature (q) axes are shown. q d τ q d a) b) Figure.. a) Salient ole synchronous machine ( = 4). The black areas around two ole bodies form a salient ole winding. b) Single oles with windings, d $= direct axis, q $= quadrature axis. In salient ole machines, these two magnetically different, rotor-geometry-defined axes have a remarkable effect on the machine behaviour; the issue will be discussed later. A grou of coils is a art of winding that magnetizes the same magnetic circuit. In Fig..a, the coils at the different magnetic oles (N and S alternating) form in airs a grou of coils. The number of field winding turns magnetizing one ole is N f. The salient ole 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 art of this current linkage consumed in the air ga, that is, the magnetic otential difference of the air ga U m,δ, may be, for simlicity, regarded as constant between the quadrature axes, and it changes its sign at the quadrature axis q, Fig... A significant field of alication for salient ole windings is doubly salient reluctance machines. In these machines, a solid salient ole is not utilizable, since the changes of flux are raid when oerating at high seeds. At simlest, DC ulses are fed to the ole windings with ower switches. In the air ga, direct current creates a flux that tries to turn the rotor in a direction where the

4 J. Pyrhönen, T. Jokinen, V. Hrabovcová.4 magnetic circuit of the machine reaches its minimum reluctance. The torque of the machine tends to be ulsating, and to reach an even torque, the current of a salient ole winding should be controllable so that the rotor can rotate without jerking. Salient ole windings are emloyed also in the magnetizing windings of the DC machines. All series, shunt, and comound windings are wound on salient oles. The commutating windings are also of the same tye as salient ole windings. Β δ Θ f U m,δ U m,fe 0 Θ f R m,δ U m,fe R m,fe U m,δ Θ f current linkage is created π/ π/ a) b) q d q Figure.. a) Equivalent magnetic circuit. The current linkages Θ f created by two adjacent salient ole windings. Part U m,δ is consumed in the air ga. b) The behaviour of the air ga flux density B δ. Thanks to the aroriate design of the ole shoe, the air ga flux density varies cosinusoidally even though it is caused by the constant magnetic otential difference in the air ga U m,δ. The air ga magnetic flux density B δ has its eak value on the d-axis and is zero on the q- axis. The current linkage created by the ole is accumulated by the amere turns on the ole. EXAMPLE.: Calculate the field winding current that can ensure a maximum magnetic flux density of B δ = 0.8 T in the air ga of a synchronous machine if there are 95 field winding turns er ole. It is assumed that the air ga magnetic flux density of the machine is sinusoidal along the ole shoes and the magnetic ermeability of iron is infinite ( μ Fe = ) in comarison with the 7 ermeability of air μ0 = 4π 0 H/m. The minimum length of the air ga is 3.5 mm. SOLUTION: If μ Fe =, the magnetic reluctance of iron arts and the iron magnetic otential difference is zero. Now, the whole field current linkage Θ f = Nf If is sent in the air ga to create the required magnetic flux density: Bδ Θf = N f If = U m,δ = H δδ = δ = A 7 μ0 4π 0 If the number of turns is N f = 95, the field current is Θf If = = A = 4 7 N 4π 0 95 f A It should be noticed that calculation of this kind is aroriate for an aroximate calculation of the current linkage needed. In fact, about % of the magnetic otential difference in electrical

5 J. Pyrhönen, T. Jokinen, V. Hrabovcová.5 machines is sent in the air ga, and the rest in the iron arts. Therefore, in a detailed design of electrical machines, it is necessary to take into account all the iron arts with aroriate material roerties. A similar calculation is valid for DC machines with the excetion that in DC machines the air ga is usually constant under the oles... Slot Windings Here we concentrate on symmetrical, three-hase AC distributed slot windings, in other words, rotating-field windings. However, first, we discuss the magnetizing winding of a rotor of a nonsalient ole synchronous machine, and finally turn to commutator windings, comensating windings, and damer windings. Because unlike in the salient ole machine, the length of the air ga is now constant, we may create a cosinusoidally distributed flux density in the air ga by roducing a cosinusoidal distribution of current linkage with an AC magnetizing winding, Fig..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..3, the function of the magnetic flux density aroximately follows the curve function of the current linkage distribution Θ ( α ). In machine design, an equivalent air ga δ e is alied, the target being to create a cosinusoidally alternating flux density into the air ga μ0 B ( α ) = Θ ( α ) (.) δ e The concet of equivalent air ga δ e will be discussed later. d α I f rotor current linkage z I f q 0 π α z I f q d q d Figure.3. Current linkage distribution created by two-ole non-salient ole winding and the fundamental of the current linkage. There are z conductors in each slot, and the excitation current in the winding is I f. The height of a single ste of the current linkage is z I f. The slot itch τ u and the slot angle α u are the core arameters of the slot winding. The slot itch is measured in metres, whereas the slot angle is measured in electrical degrees. The number of slots being and the diameter of the air ga D, we may write

6 J. Pyrhönen, T. Jokinen, V. Hrabovcová.6 πd π τ u = ; αu =. (.) The slot itch being usually constant in non-salient ole windings, the current sum (z I f ) in a slot has 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 ga.). 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..3, the number of turns is 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 imrove the steed waveform of the figure to better aroach the cosinusoidal form. The need for this deends on the induced voltage harmonic content in the stator winding. The voltage may be of almost ure sinusoidal waveform desite the fact that the air ga flux density distribution should not be erfectly sinusoidal. This deends on the stator winding factors for different harmonics. In synchronous machines, the air ga is usually relatively large, and corresondingly, the flux density on the stator surface changes more smoothly (neglecting the influence of slots) than the steed current linkage waveform of Fig..3. Here, we aly the well-known finding that if /3 of the rotor surface are slotted and /3 is left slotless, not only the third harmonic comonent but any of its multile harmonics called trilen harmonics are eliminated in the air ga magnetic flux density, and also the low-order odd harmonics (5 th, 7 th ) are suressed...3 End Windings Figure.4 illustrates how the arrangement of the coil end influences the hysical aearance of the winding. The windings a and b in the figure are of equal value with resect 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..4, we note that the coil ends form two searate lanes at the endfaces of the machine. This kind of a winding is therefore called a two-lane winding. The coil ends of the tye are deicted in Fig..4e. In the winding of Fig..4b, the coil ends are overlaing, and therefore, this kind of winding is called a diamond winding (la winding). Figures.4c and d illustrate three-hase stator windings that are identical with resect to the main flux, but in Fig..4c, the grous of coil are non-divided, and in Fig..4d, the grous of coil are divided. In Fig..4c, an arbitrary radius r is drawn across the coil end. It is shown that at any osition, the radius intersects only coils of two hases, and the winding can thus be constructed as a two-lane winding. A corresonding winding constructed with distributed coils (Fig..4d) has to be a three-lane arrangement, since now the radius r may intersect the coil ends of the windings of all the three hases.

7 J. Pyrhönen, T. Jokinen, V. Hrabovcová a) e) b) coil san coil side endwinding f) r r c) d) Figure.4. a) Concentric winding and b) a diamond winding. In a two-lane winding, the coil sans differ from each other. In the diamond winding, all the coils are of equal width. c) A two-lane three-hase four-ole winding with nondivided grous of coil. d) A three-lane three-hase four-ole winding with divided grous of coils. Figures c and d illustrate also a single main flux ath. e) Profile of an end winding arrangement of a two-lane winding. f) Profile of an end winding of a three-lane winding. The radii r in the figures illustrate that in a winding with non-divided grous, an arbitrary radius may intersect only two hases, and in a winding with divided grous, the radius may intersect all the three hases. The two- or three-lane windings will result corresondingly. The art of a coil located in a single slot is called a coil side, and the art of the coil outside the slot is termed a coil end. The coil ends together constitute the end windings of the winding.. Phase Windings Next, oly-hase slot windings that roduce the rotating field of oly-hase AC machines are investigated. In rincile, the number of hases m can be selected freely, but the use of a three-hase suly network has led to a situation in which also most electrical machines are of the three-hase tye. Another, extremely common tye is two-hase electrical machines that are oerated with a caacitor start and run motor in a single-hase network. A symmetrical two-hase winding is in rincile the simlest AC winding that roduces a rotating field.

8 J. Pyrhönen, T. Jokinen, V. Hrabovcová.8 A configuration of a symmetrical oly-hase winding can be considered as follows: the erihery of the air ga is evenly distributed over the oles so that we can determine a ole arc, which covers 80 electrical degrees and a corresonding ole itch, τ, which is exressed in metres πd τ =. (.3) Figure.5 deicts the division of the erihery of the machine into hase zones of ositive and negative values. In the figure, the number of ole airs =, and the number of hases m = 3. 0 o -V U W -U V = m = 3 -W U -V τ τ v -W W 80 V -U o 0 o Figure.5. Division of the erihery of a three-hase four-ole machine into hase zones of ositive and negative values. Pole itch is τ and hase zone distribution τ v. When the windings are located in the zones, the instantaneous currents in the ositive and negative zones are flowing in oosite directions. Phase zone distribution is written as τ τ v =. (.4) m The number of zones will thus be m. The number of slots er each such zone is exressed by the term q, as a number of slots er ole and hase q =. (.5) m Here 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 hase zones are distributed symmetrically to different hase windings so that the hase zones of the hases U, V, W,... are ositioned on the erihery of the machine at equal distances in electrical degrees. In a three-hase system, the angle between the hases is 0 electrical degrees. This is illustrated by the erihery of Fig..5, where we have 360 electrical degrees because of four

9 J. Pyrhönen, T. Jokinen, V. Hrabovcová.9 oles. Now, it is ossible to label every hase zone. We start for instance with the ositive zone of the hase U. The first ositive zone of the hase V shall be 0 electrical degrees from the first ositive zone of the hase U. Corresondingly, the first ositive zone of the hase W shall be 0 electrical degrees from the ositive zone of the hase V etc. In Fig..5, there are two ole airs, and hence we need two ositive zones for each hase U, V, and W. In the slots of each, now labelled hase zones, there are only the coil sides of the labelled hase coil, in all of which the current flows in the same direction. Now, if their direction of current is selected ositive in the diagram, the unlabelled zones become negative. Negative zones are labelled by starting from the distance of a ole itch from the osition of the ositive zones. Now U and U, V and V, W and W are at the distance of 80 electrical degrees from each other..3 Three-Phase Integral Slot Stator Winding The armature winding of a three-hase electrical machine is usually constructed in the stator, and it is satially distributed in the stator slots so that the current linkage created by the stator currents is distributed as sinusoidally as ossible. The simlest stator winding that roduces a noticeable rotating field comrises three coils, the sides of which are divided into six slots, because if m = 3, =, q =, then = mq = 6; see Figs..6 and.7. EXAMPLE.: Create a three-hase, two-ole stator winding with q =. Distribute the hases in the slots and illustrate the current linkage created based on the instant values of hase sinusoidal currents. Draw a hasor diagram of the slot voltage and sum the voltages of the individual hases. Create a current linkage waveform in the air ga for the time instant t when the hase U voltage is in its ositive maximum and for t, which is 30 shifted. SOLUTION: If m = 3, =, q =, then = mq = 6, which is the simlest case of three-hase windings. The distribution of the hases in the slots will be exlained based on Fig..6. Starting from the slot, we insert there the ositive conductors of the hase U forming the zone U. The ole itch exressed in the number of slots er ole, or in other words, the coil san exressed in the number of slot itches y is 6 y = = = 3. Then, the zone U will be one ole itch shifted from U and will be located in the slot 4, because + y = + 3 = 4. The beginning of the hase V is 0 shifted from U, which means the slot 3, and its end V is in the slot 6 (3 + 3 = 6). The hase W is again shifted form V by 0, which means the slot 5, and its end is in the slot ; see Fig..6a. The olarity of instantaneous currents is shown at the instant, when the current of the hase U is in its ositive maximum value flowing in the slot, deicted as a cross (tail of arrow) in U (current flowing away from the observer). Then, U is deicted by a dot (a oint of arrow) in the slot 4 (current flowing towards the observer). At the same instant in V and W, there are also dots, because the hases V and W are carrying negative current values (see.6d), and therefore V and W are ositive, indicated by crosses. In this way, a sequence of slots with inserted hases is as follows: U, W, V, U, W, V, if q =.

10 J. Pyrhönen, T. Jokinen, V. Hrabovcová.0 U V W 6 α = W 5 m = V U a) U b) U U V W W V i U V W t U t c) d) α u = 60 o U W U U U V 4 e) f) Figure.6. The simlest three-hase winding that roduces 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 develoed view of the winding in a lane, and c) a three-dimensional view of the winding. The figure illustrates how the winding enetrates 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 hases U, V, and W at the terminals are denoted U-U, V-V, and W-W. d) The three-hase currents at the observed time instant t when i W = i V = -/ i U. (i U +i V +i W = 0), e) a voltage hasor diagram for the given three-hase system, f) the total hase voltage for individual hases. The voltage of the hase U is created by summing the voltage of the slot and the negative voltage of slot 4, and therefore the direction of the voltage hasor in the slot 4 is taken oosite with the denotation 4. We can see the sum of voltages in both slots and the hase shift by 0 of the V and W hase voltages. The cross-section of the stator winding in Fig..6a shows fictitious coils with current directions resulting in the magnetic field reresented by the force lines and arrows. The hasor diagram in Fig..6e includes six hasors. To determine their number, the largest common divider of and denoted t has to be found. In this case, for = 6 and =, t =, and therefore, the number of hasors is /t = 6. The angle between the voltage hasors in the adjacent slots is given by exression α u = = = 60, 6

11 J. Pyrhönen, T. Jokinen, V. Hrabovcová. which results in the numbering of the voltage hasors in slots as shown in Fig..6e. Now, the total hase voltage for individual hases has to be summed. The voltage of the hase U is created by the ositive voltage in the slot, and the negative voltage in the slot 4. The direction of the voltage hasor in the slot 4 is taken oosite with the denotation 4. We can see the sum of voltages in both slots of the hase U, and the hase shift of 0 of the V and W hase voltages in Fig..6f. The current linkage waveforms for this winding are illustrated in Fig..7b and c for the time instants t and t, between which the waveforms roceed by 30. The rocedure of drawing the figure can be described as follows: We start observation at α = 0. We assume the same constant number of conductors z in all slots. The current linkage value on the left in Fig..7b is changed stewise at the slot, where the hase W is located and is carrying a current with a cross sign. This can be drawn as a ositive ste of Θ with a certain value (Θ(t ) = i uw (t )z ). Now, the current linkage curve remains constant until we reach the slot, where the ositive currents of the hase U are located. The instantaneous current in the slot is the hase U eak current. The current sum is indicated again with a cross sign. The ste height is now twice the height in the slot, because the eak current is twice the current flowing in the slot. Then, in the slot 6, there is again a ositive half ste caused by the hase V. In the slot 5, there is a current sum indicated by a dot, which means a negative Θ ste. The same is reeated with all slots, and when the whole circle has been closed, Fig..7b. When this rocedure is reeated for one eriod of the current, we obtain a travelling wave for the current linkage waveform. Fig..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 and 5. We can also see that the Θ rofiles in b and c are not similar, but the form is changed deending on the time instant at which it is investigated.

12 a) z i U / J. Pyrhönen, T. Jokinen, V. Hrabovcová. Θˆ U 4 iuz = π z i U starting of study z i U / 0 π slotted stator inner erihery U U U slot slot Θ Θ α α O O O i U V W O O O i U V W t b) c) t Figure.7. Current linkages Θ created by a simle three-hase q = winding, a) only the hase U is fed by current and observed. A rectangular waveform of current linkage with its fundamental comonent is shown to exlicate the staircase rofile of the current linkages below. If all three hases are fed and observed in two different current situations (i U +i V +i W = 0) at two time instants t and t, see b) and c) resectively. The figure illustrates also the fundamental of the staircase current linkage curves. The steed curves are obtained by alying Amère s law in the current-carrying teeth zone of the electrical machine. Note that as time elases from t to t, the three hase currents change and also the osition of the fundamental comonent changes. This indicates clearly the rotating-field nature of the winding. The angle α and the numbers of slots refer to the revious figure, in which we see that the maximum flux density in the air ga 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 resent. Figure.7 shows that the current linkage roduced with such a simle winding deviates considerably from sinusoidal waveform. Therefore, in electrical machines, more coil sides are usually emloyed er ole and hase. EXAMPLE.3: Consider an integral slot winding, where = and q =, m = 3. Distribute the hase winding into the slots, make an illustration of the windings in the slots, draw a hasor diagram and show the hase voltages of the individual hases. Create a waveform of the current linkage for this winding and comare it with that in Fig..7. SOLUTION: The number of the slots needed for this winding is = mq = 3 =. The crosssectional area of such a stator with slots and embedded conductors of individual hases is illustrated in Fig..8a. The distribution of the slots into the hases is made in the same order as in Examle., but now q = slots er ole and hase. Therefore, the sequence of the slots for the

13 J. Pyrhönen, T. Jokinen, V. Hrabovcová.3 hases is as follows: U, U, W, W, V, V, U, U, W, W, V, V. The direction of the current in the slots will be determined in the same way as above in Examle.. The coils wound in individual hases are shown in Fig..8b. The ole itch exressed in number of slot itches is y = = = 6 Figure.8c shows how the hase U is wound to kee the full itch equal to 6 slots. In Fig..8d, the average itch is also 6, but the individual stes are y = 5 and 7, which gives the same average result for the value of induced voltage. The hasor diagram has hasors, because t = again. The angle between two hasors of adjacent slots is α u = = = 30 The hasors are numbered gradually around the circle. Based on this diagram, the hase voltage of all hases can be found. Figures.8f and g show that the voltages are the same indeendent of the way how searate coil sides are connected in series. In comarison with the revious examle, the geometrical sum is now less than the algebraic sum. The hase shifting between coil side voltages is caused by the distribution of the winding in more than one slots, here in two slots er each ole. This reduction of the hase voltage is exressed 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..9. We can see that it is much closer to a sinusoidal waveform than in the revious examle with q =. In undamed ermanent magnet synchronous motors, also such windings can be emloyed, the number of slots er ole and hase 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 roduced by the stator winding deviates so much from the fundamental that, because of excessive harmonic losses in the rotor, no other rotor tye comes into question.

14 J. Pyrhönen, T. Jokinen, V. Hrabovcová.4 U V 0 W = m = 3 7 W 4 α 5 6 V U U W V U a) b) U y = 6 U U y = 5 y = 7 U c) d) W V U 0 3 α u = 30 o e) -8-7 U hu U hu -8-7 U hw U hw f) g) U hv U hv Figure.8. Three-hase two-ole winding with two slots er ole and hase, a) a stator with slots, the number of slots er ole and hase q =. b) Divided coil grous, c) full-itch coils of the hase U, d) average full-itch coils of hase U, e) a hasor diagram with hasors, one for each slot, f) sum hase voltage of individual hases corresonding to Fig. c, g) a sum hase voltage of individual hases corresonding to Fig. d. Θ/A Θˆ s α Figure.9. Current linkage Θ = f ( α ) created by the winding on the surface of the stator bore of Fig..8 at a time i s W = i V = / i U. The fundamental Θ of Θ is given as a sinusoidal curve. The numbering of the slots is also given. s s

15 J. Pyrhönen, T. Jokinen, V. Hrabovcová.5 When comaring Fig..9 (q = ) with Fig..7 (q = ), it is obvious that the higher the term q (slots er ole and hase) is, the more sinusoidal the current linkage of the stator winding is. As we can see in Fig..7a, the current linkage amlitude of the fundamental comonent for one fullitch coil is ˆ ˆ 4 ziu Θ U =. (.6) π If the coil winding is distributed into more slots, and q > and N = qz, the winding factor must be taken into account: ˆ ˆ 4 NkwiU Θ U =. (.7) π In a -ole machine (>), the current linkage for one ole is: Nk iˆ ˆ 4 w U Θ U =. (.8) π This exression can be rearranged with the number of conductors in a slot. In one hase, there are N conductors, and they are embedded in the slots belonging to one hase /m. Therefore, the number of conductors in one slot will be: and N mn N z = = = (.9) / m qm q N = qz (.0) Then N/ resented in Eq. (.8) and in the following can be introduced by qz. : ˆ ˆ ˆ 4 NkwiU 4 kwiu Θ U = = qz. (.) π π It can be also exressed with the effective value of sinusoidal hase current if there is a symmetrical system of hase currents: Θ ˆ = 4 Nkw U I π. (.) For an m-hase rotating-field stator or rotor winding, the amlitude of current linkage is m/ times higher: Θ ˆ = m 4 Nkw I π (.3) and for three-hase stator or rotor winding, the current linkage amlitude of the fundamental comonent for one ole is: ˆ 3 4 Nkw 3 Nkw Θ = I = I (.4) π π

16 J. Pyrhönen, T. Jokinen, V. Hrabovcová.6 For a stator current linkage amlitude Θˆ s ν of the harmonic ν of the current linkage of a oly-hase (m > ) rotating-field stator winding (or rotor winding), when the effective value of the stator current is I s, we may write Θˆ m 4 k N π ν mk N πν ˆ wν s wν s sν = I s = is. (.5) EXAMPLE.4: Calculate the amlitude of the fundamental comonent of stator current linkage, if N s = 00, k w = 0.96, m = 3, = and i ( ) ˆ su t = i = A, the effective value for a sinusoidal current I / A = A. being ( ) s = SOLUTION: For the fundamental, we obtain Θ ˆ s = 83.3 A, because: ˆ 3 4 Nk w 3 Nk w Θ = I s = I s = 0.707A = 83.3 A. π π π.4 Voltage Phasor Diagram and Winding Factor Since the winding is satially distributed in the slots on the stator surface, the flux (which is roortional to the current linkage Θ) enetrating the winding does not intersect all windings simultaneously, but with a certain hase shift. Therefore, the electromotive force (emf) of the winding is not calculated directly with the number of turns N s, but the winding factors k wν corresonding to the harmonics are required. The emf of the fundamental induced in the turn is calculated with the flux linkage Ψ by alying Faraday s induction law e = NkwdΦ / dt = dψ/dt (see Eqs..3,.7 and.8) We can see that the winding factor corresondingly indicates the characteristics of the winding to roduce harmonics, and it has thus to be taken into account when calculating the current linkage of the winding (Eq..5). The common distribution of all the current linkages created by all the windings together roduces a flux density distribution in the air ga of the machine, which, when moving with resect to the winding, induces voltages to the conductors of the winding. The hase shift of the induced electromotive force in different coil sides is investigated with a voltage hasor diagram. The voltage hasor diagram is resented in electrical degrees. If the machine is for instance a four-ole one, =, the voltage vectors have to be distributed along two full circles in the stator bore. Figure.0 a) illustrates the voltage hasor diagram of a two-ole winding of Fig..8. In Fig..0a, the hasors and are ositive and 7 and 8 are negative for the hase under consideration. Hence, the hasors 7 and 8 are turned by 80 degrees to form a bunch of hasors. For harmonic ν (excluding slot harmonics that have the same winding factor as the fundamental) the directions of the hasors of the coil sides vary more than in the figure, because the slot angles α u are relaced with the angles να u. According to Fig..0b, when calculating the geometric sum of the voltage hasors for a hase winding, the symmetry line for the bunch of hasors, where the negative hasors have been turned oosite, must be found. The angles α ρ of the hasors with resect to this symmetry line may be used in the calculation of the geometric sum. Each hasor contributes to the sum with a comonent roortional to cosα ρ.

17 J. Pyrhönen, T. Jokinen, V. Hrabovcová.7 3 -U 8 -U 7 α z α ρ z U ρ cosα ρ U -U 8 -U 7 U -U 8 α symmetry line ositive hasors U U a) -U 7 b) negative hasors c) Figure.0. a) and b) Fundamental voltage hasor diagram for the winding of Fig..8 s =, =, q s =. A maximum voltage is induced in the bars in the slots and 7 at the moment deicted 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 hasor diagram. c) General alication of the voltage hasor diagram in the determination of the winding factor (fractional slot winding since the number of hasors is uneven). The hasors of negative coil sides are turned 80, and then the summing of the resulting bunch of hasors is calculated according to Eq. (.6). A symmetry line is drawn in the middle of the bunch, and each hasor forms an angle α ρ with the symmetry line. The geometric sum of all the hasors lies on the symmetry line. We can now write a general resentation for the winding factor k wν of a harmonic ν, by emloying the voltage hasor diagram k wν νπ sin = Z Z ρ= cosα. (.6) ρ Here Z is the total number of ositive and negative hasors of the hase in question, ρ is the ordinal number of a single hasor, and ν is the ordinal number of the harmonic under observation. The π coefficient sin ν in the equation only influences the sign (of the factor). The angle of a single hasor α ρ can be found from the voltage hasor diagram drawn for the secific harmonic, and it is the angle between an individual hasor and the symmetry line drawn for a secific harmonic (c.f. Fig..0b). This voltage vector diagram solution is universal and may be used in all cases, but the numerical values of Eq. (.6) do not always have to be calculated directly from this equation, or with the voltage hasor diagram at all. In simle cases, we may aly equations introduced later. However, the voltage hasor diagram forms the basis for the calculations, and therefore its utilization is discussed further when analyzing different tyes of windings. If we are in Fig..0a considering a currentless stator of a synchronous machine, a maximum voltage can be induced to the coil sides and 7 at the middle of the ole shoe, when the rotor is rotating at no load inside the stator bore (which corresonds to the eak value of the flux density, but the zero value of the flux enetrating the coil), where the derivative of the flux enetrating the coil reaches its eak value, the voltage induction being at its highest at that moment. If the rotor

18 J. Pyrhönen, T. Jokinen, V. Hrabovcová.8 rotates clockwise, a maximum voltage is induced in the coil sides and 8 in a short while, and so on. The voltage hasor diagram then describes the amlitudes of voltages induced in different slots and their temoral hase shift. The series-connected coils of the hase U travel e.g. from the slot to the slot 8 (coil ) and from the slot to the slot 7 (coil ). Thus a voltage, which is the difference of the hasors U and U 8, is induced in the coil. The total voltage of the hase is thus U U = U U + U U. (.7) 8 7 The figure also indicates the ossibility of connecting the coils in the order 7 and 8, which gives the same voltage but a different end winding. The winding factor k w based on the distribution 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 8 + U U 7 k w = = = (.8) sum of absolute values U + U + U + U 8 EXAMPLE.5: Equation (.6) indicates that the winding factor for the harmonics may also be calculated using the voltage hasor diagram. Derive the winding factor for the seventh harmonic of the winding in Fig.8. SOLUTION: We now draw a new voltage hasor diagram based on Fig..0 for the seventh harmonic, Fig... 7 a) τ τ 7 b) 8 c) -U 7,8 U 7, -U 7,7 αu ν = να u -U 7,7 5π/ U 7, U 7, 7 -U 7,8 Fig... Deriving the harmonic winding factor, a) the fundamental and the seventh harmonic field in the air ga over the slots, b) voltage hasors for the seventh harmonic of a full itch q = winding (slot angle α u7 = 0 ), and c) the symmetry line and the sum of the voltage hasors. The hasor angles α ρ with resect to the symmetry line are α ρ = 5π/ or 5 π /. The slots and belong to the ositive zone of the hase U and the slots 7 and 8 to the negative zone measured by the fundamental. In Fig.., we see that the ole itch of the seventh harmonic

19 J. Pyrhönen, T. Jokinen, V. Hrabovcová.9 is one seventh of the fundamental ole itch. Deriving the hasor sum for the seventh harmonic is started for instance with the voltage hasor of the slot. This hasor remains in its original osition. The slot is hysically and by fundamental located 30 clockwise from the slot, but as we are now studying the seventh harmonic, the slot angle measured in degrees for it is 7 30 =0, which can also be seen in the figure. The hasor for the slot is, hence, located 0 from the hasor clockwise. The slot 7 is located at 7 80 =60 from the slot. Since 60 = the hasor 7 remains oosite to the hasor. The hasor 8 is located 0 clockwise from the hasor 7 and will find its lace 30 clockwise from the hasor. By turning the negative zone hasors by π and alying Eq. (.7) we obtain 7π 7π sin 4 sin w7 = + + k cosα ρ = 5π 5π 5π 5π cos + cos + cos + cos = ρ= 4 It is not necessary to aly the voltage hasor diagram, but also simle equations may be derived to directly calculate the winding factor. In rincile, we have three winding factors: a distribution factor, a itch 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 hasors in the case of distributed winding is called the distribution factor with the subscrit d. This factor is always k d. The value k d = can be reached when q =, in which case the geometric sum equals the sum of absolute values, see Fig..6f. If q, then kd <. In fact, it means that the total hase voltage is reduced by this factor (see Examle.6). If each coil is wound as a full-itch winding, the coil itch is in rincile the same as the ole itch. However, the voltage of the hase with full-itch coils is reduced because of the winding distribution with the factor k d. If the coil itch is shorter than the ole itch and the winding is not a full-itch winding, the winding is called a short-itch winding, or a chorded winding (see Fig..5). Note that the winding in Fig..8 is not a short-itch winding, even though the coil may be realized from the slot to the slot 8 (shorter than ole itch) and not from the slot to the slot 7 (equivalent to ole itch). This is because the full-itch coils together roduce the same current linkage as the shorter coils together. A real short-itching is obviously emloyed in the two-layer windings. Shortitching is another reason why the voltage of the hase winding may be reduced. The factor of such reduction is called the itch factor k. The total winding factor is given as: k w = k k. (.9) d Equations to calculate the distribution factor k d will be derived now; see Fig... The equations are based on the geometric sum of the voltage hasors in a similar way as in Figs..0 and..

20 J. Pyrhönen, T. Jokinen, V. Hrabovcová.0 U coil A U U coil3 coil U coil4 B 3 4 U coil5 D U 5 C qα u / ˆB δ ˆB δ5 α ν=5 = 90 o r α u τ 5 τ α ν= = 8 o O a) b) Figure.. a) Determination of the distribution factor with a olygon with q = 5, b) the ole itch for the fundamental and the fifth harmonic. The same hysical angles for the fifth and the fundamental are shown as an examle. The distribution factor for the fundamental comonent is given as geometric sum k = U l d =. (.0) sum of absolute values qucoil According to Fig.., we may write for the triangle ODC qα u sin U l qα u = U l = r sin, (.) r and for the triangle OAB U v α sin u α u = U v = r sin. (.) r We may now write for the distribution factor qα u qα u r sin sin U l k d = = =. (.3) qu α v u α u qr sin qsin This is the basic exression for the distribution factor for the calculation of the fundamental in a closed form. Since the harmonic comonents of the air ga magnetic flux density are resent, the calculation of the distribution factor for the ν th harmonic will be carried out alying the angle να u ; see Fig..b and.b: qα u sinν k dν =. (.4) α u qsinν EXAMPLE.6: Reeat Examle.5 using Eq. (.4)

21 J. Pyrhönen, T. Jokinen, V. Hrabovcová. π qα u 7π sinν sin 7 6 sin SOLUTION: k 6 dν = = = = The result is the same as above. α u π 7π qsinν sin 7 sin 6 The exression for the fundamental may be rearranged: k π sin π/ sin m = m. (.5) π π/6 qsin qsin mq q d = For three-hase machines m = 3, the exression is as follows: k ( π/6) sin =. (.6) π/6 π/6 qsin qsin q q d = This simle exression of the distribution factor for the fundamental is most often emloyed for ractical calculation. EXAMPLE:.7 Calculate the hase voltage of a three-hase four-ole synchronous machine with the stator bore diameter of 0.30 m, the length of 0.5 m, and the seed of rotation 500 rm. The excitation creates the air ga fundamental flux density B ˆ δ = 0.8T. There are 36 slots, in which a one-layer winding with three conductors in each slot is embedded. SOLUTION: According to the Lorentz law, an instantaneous value of the induced electric field strength in a conductor is E = v B o. In one conductor embedded in a slot of an AC machine, we may get the induced voltage by integrating: e c = Bδ l' v where B δ is the local air- ga flux density value of the rotating magnetic field, l' is the effective length of the of stator iron stack, and v is the seed at which the conductor travels in the magnetic field, see Fig.3. travelling direction ˆB δ ˆB δ α B δav l' τ =ˆ T τ τ a) b) Figure.3 a) Flux density variation in the air ga. One flux density eriod travels two ole itches during one time eriod T. b) The flux distribution over one ole and the conductors in slots.

22 J. Pyrhönen, T. Jokinen, V. Hrabovcová. During one eriod T, the magnetic flux density wave travels two ole itches, as it is shown in the figure above. The seed of the magnetic field moving in the air ga is τ v = = τ f. T The effective value of the induced voltage in one conductor is: Bˆ δ Ec = l'τ f. Contrary to a transformer, where aroximately the same value of magnetic flux density enetrates all the winding turns, Fig. shows that in AC rotating-field machines, the conductors are subject to the sinusoidal waveform of flux density, and each conductor is in a different value of magnetic flux density. Therefore, an average value of the magnetic flux density is calculated to unify the value of the magnetic flux for all conductors. The average value of the flux density roduces the maximum value of the flux enetrating a full-itch winding: Φˆ = B δav l' τ. B δ is sread over the ole itch τ, and we get for the average value π B ˆ ˆ δav = Bδ Bδ = Bδav. π Now, the effective average value of the voltage induced in a conductor written by means of average magnetic flux density is: π π E ' ˆ c = Bδavl τ f = Φf. The frequency f is found by the seed n f 500 = n = = 50 Hz The information about three conductors in each slot can be used for calculating the number of turns N in series. In one hase, there are N conductors, and they are embedded in the slots belonging to one hase /m. Therefore, the number of conductors in one slot z will be: N Nm z = = = / m qm N q. In this case, the number of turns in series in one hase is N = 3 q = 3 3 = 8. The effective value N of the induced voltage in one slot is Ec. The number of such slots is q. The linear sum of the q voltages of all conductor belonging to the same hase must be reduced by the winding factor to get N the hase voltage E h = Ecq kw. The final exression for the effective value of the induced q N π voltage in the AC rotating machine is ˆ N E π ˆ h = Ec q kw = Φ f q kw = fφnkw. In q q

23 J. Pyrhönen, T. Jokinen, V. Hrabovcová.3 this examle, there is a full-itch one-layer winding, and therefore k =, and only k d must be calculated: k w = kd = = = 0.96 o o qsin 3sin q 3 The maximum value of the magnetic flux is: Φ ˆ = Bˆ δ τ l' = = Wb, where the π π π D ole itch τ is: s π 0.3 τ = = = m. An effective value of the hase induced voltage is: 4 E = π f Φ ˆ Nk = π V. h d = EXAMPLE.8: A stator of a four-ole three-hase induction motor has 36 slots, and it is fed by 3 400/30 V, 50 Hz. The diameter of the stator bore is D s = 5 cm and the length l Fe = 0 cm. A two-layer winding is embedded in the slots. Besides this, there is a one-layer, full-itch, search coil embedded in two slots. In the no-load condition, a voltage of.3 V has been measured at its terminals. Calculate the air ga flux density, if the voltage dro on the imedance of the search coil can be neglected. SOLUTION: To be able to investigate the air ga flux density, the data of the search coil can be used. This coil is embedded in two electrically oosite slots, and therefore the distribution factor k d = ; because it is a full-itch coil, also the itch factor k =, and therefore k w =. Now, the maximum value of magnetic flux is: ˆ U c.3 Φ = = Wb = Wb π f N ck wc π The amlitude of the air ga flux density is: ˆ ˆ π π Φ π 0.07 Bδ = Bav = = T = T. πds π0.5 l' Winding Analysis The winding analysis starts with the analysis of a single-layer stator winding, in which the number of coils is s /. In the machine design, the following setu is advisable: The erihery of the air ga of the stator bore (diameter D s ) is distributed evenly in all oles, that is, of equal arts, which yields the ole itch τ. Figure.4 illustrates the configuration of a two-ole slot winding ( = ). The ole itch is evenly distributed in all stator hase windings, that is, in m s equal arts. Now we obtain a zone distribution τ sv. In Fig..4, the number of stator hases is m s = 3. The number of zones becomes thus m s = 6. The number of stator slots in a single zone is q s, which is the number of slots er ole and hase in the stator. By using stator values in the general Eq. (.5), we obtain s q s =. (.7) ms