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

Transcription

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

24 J. Pyrhönen, T. Jokinen, V. Hrabovcová.4 If q s is an integer, the winding is termed an integral slot winding, and if q s is a fraction, the winding is called a fractional slot winding. The hase zones are labelled symmetrically to the hase windings, and the directions of currents are determined so that we obtain a number of m s magnetic axes at the distance of 360o/ m s from each other. The hase zones are labelled as stated in.. The ositive zone of the hase U, that is, a zone where the current of the hase U is flowing away from the observer, is set as an examle (Figure.4). Now the negative zone of the hase U is at the distance of 80 electrical degrees, in other words, electrically on the oosite side. The conductors of resective zones are connected so that the current flows as desired. This can be carried out for instance as illustrated in the figure below. In the figure, it is assumed that there are three slots in each zone, q s = 3. The figure shows that the magnetic axis of the hase winding U is to the direction of the arrow drawn in the middle of the illustration. Because this is a three-hase machine, the directions of the currents of the hases V and W have to be such that the magnetic axes of the hases V and W are at the distance of 0 (electrical degrees) from the magnetic axis of the hase U. This can be realized by setting the zones of the V and W hases and the current directions according to Fig..4. U Θ τ τ sv +U -V -W U V W +W +V α 0 0 U -U α -U τ U a) b) Figure.4. a) Three-hase stator diamond winding =, q s = 3, s = 8. Only the coil end on the side of observation is visible of the U hase winding. The figure also illustrates the ositive magnetic axes of the hase windings U, V, and W. The current linkage creates a flux in the direction of the magnetic axis when the current is enetrating the winding at its ositive terminal, e.g. U. The current directions of the figure deict a situation in which the current of the windings V and W is a negative half of the current in the winding U. b) The created current linkage distribution Θ s is shown at the moment, when its maximum is in the direction of the magnetic axis of the hase U. The small arrows in Fig..4a at the end winding indicate the current directions and the transitions from coil to coil. The same winding will be observed in Fig..3. The way how the conductors of different zones are connected roduces different mechanical winding constructions, but the air ga remains similar irresective of the mechanical construction. However, the arrangement of connections has a significant influence on the sace requirements for the end windings, the amount of coer and the roduction costs of the winding. The connections also have an effect on certain electric roerties, such as the leakage flux of the end windings. The oly-hase winding in the stator of a rotating-field machine creates a flux wave when a symmetric oly-hase current flows in the winding. A flux wave is created for instance when the current linkage of Fig..4 begins to roagate in the direction of the ositive α-axis, and the currents of the oly-hase winding are alternating sinusoidally as a function of time. We have to note, however, that the roagation seeds of the harmonics created by the winding are different from the seed of the fundamental (n sν = ± n s /ν), and therefore the shae of the current linkage curve changes as a function of time. However, the fundamental roagates in the air ga at a seed defined by the fundamental of the current and by the number of ole airs. Furthermore, the fundamental is usually dominating (when q ), and thus the oeration of the machine can be analyzed basically

25 J. Pyrhönen, T. Jokinen, V. Hrabovcová.5 with the fundamental. For instance in a three-hase winding, time-varying sinusoidal currents with a 0 hase shift in time create a temorally and ositionally alternating flux in the windings that are distributed at the distances of 0 electrical degrees. The flux distribution roagates as a wave on the stator surface. (See e.g. Fig. 7.7 illustrating the fundamental ν = of a six-ole and a two-ole machine.).6 Short-Pitching In a double-layer diamond winding, the slot is divided into an uer and a lower art, and there is one coil side in each half slot. The coil side at the bottom of the slot belongs to the bottom layer of the slot, and the coil side adjacent to the air ga belongs to the uer layer. The number of coils is now the same as the number of slots s of the winding; see Fig..5b. A double-layer diamond winding is constructed like the single-layer winding. As illustrated in Fig..5, there are two zone rings, the outer illustrating the bottom layer and the inner the uer layer, Fig..5a. The distribution of zones does not have to be identical in the uer and bottom layer. The zone distribution can be shifted by a multile of the slot itch. In Fig..5a, a single zone shift equals to a single slot itch. Figure.5b illustrates one of the coils of the hase U. By comaring the width of the coil to the coil san of the winding in Fig..4, we can see that the coil is now one ole itch narrower; the coil is said to be short itched. Because of short-itching, the coil end has become shorter, and the coer consumtion is thus reduced. On the other hand, the flux linking the coil decreases somewhat because of short-itching, and therefore the number of coil turns at the same voltage has to be higher than for a full-itch winding. The short-itching of the coil end is of more significance than the increased number of coil turns, and as a result, the consumtion of coil material decreases. Short-itching also influences the harmonics content of the flux density of the air ga. A correctly short-itched winding roduces a more sinusoidal current linkage distribution than a full-itch winding. In a salient ole synchronous generator, where the flux density distribution is basically governed by the shae of ole shoes, a short-itch winding roduces a more sinusoidal ole voltage than a full-itch winding. to +W -V +U +U -V V -W U W +W +V -U -W W bottom +V -U a) b) Fig..5. a) Three-hase double-layer diamond winding s = 8, q s = 3, =. One end winding is shown to illustrate the coil san. The winding is created from the revious winding by dividing the slots into uer and bottom layers and by shifting the bottom layers clockwise by a single ole itch. The magnetic axes of the new short-itched winding are

26 J. Pyrhönen, T. Jokinen, V. Hrabovcová.6 also shown. b) A coil end of a double-layer winding roduced of reformed coer, coil san W or exressed in numbers of slot itches y. The coil ends start from the left at the bottom of the slot and continue to the right to the to of the slot. Figure.6 illustrates the basic difference of a short-itch winding and a full-itch winding U U 7 7 U -7 W = τ W < U 7 U -6 U 6 y π y 6 π y y τ a) b) Figure.6. Cross-sectional areas of two machines with slots. The basic differences of a) a two-ole full-itch winding and b) a two-ole short-itch winding. In the short-itch winding, the width of a single coil W is less than the ole itch τ or exressed in the number of slots the short itch y is less than a full itch y. The voltage U -6 is lower than U -7. The short-itched coil is located on the chord of the erihery, and therefore the winding tye is also called a chorded winding. The coil without short-itching is located on the diameter of the machine. The short-itch construction is now investigated in more detail. Short-itching is commonly created by winding ste shortening (Fig..7b), coil side shift in a slot (Fig..7c), and coil side transfer to another zone (Fig..7d). In Fig..7, the zone grahs illustrate the configurations of a full-itch winding and of the short-itch windings constructed with the above-mentioned methods. Of these methods, the ste shortening can be considered to be created from a full-itch winding by shifting the uer layer left for a certain number of slot itches. Coil side shift in a slot is generated by changing the coil sides of the uer and bottom layers in certain slots of a short itch winding. For instance if in Fig..7 b, the coil sides of the bottom layer in the slots 8 and 0 are removed to the uer layer, and in the slots and 4, the uer coil sides are shifted to the bottom layer, we get the winding of Fig.7c. Now, the width of a coil is again W = τ, but because the magnetic voltage of the air ga does not deend on the osition of a single coil side, the magnetizing characteristics of the winding remains unchanged. The windings with a coil side shift in a slot and winding ste shortening are equal in this resect. The average number of slots er ole and hase for windings with a coil side shift in a slot is s q + q qs = =, q = q + j; q = q j (.8) m where j is the difference of q (the numbers of slots er ole and hase) in different layers. In Fig..7, j =.

27 J. Pyrhönen, T. Jokinen, V. Hrabovcová.7 W=6τ u =τ W Full-itch winding W=τ a) τ v=τ u W=5τ u x W Winding ste shortening W=(5/6)τ b) τ v=τ u W=6τ u W x Coil side shift in a slot W=τ c) τ v =3τ u W=6τ u τ v =τ u Coil s. transfer W=τ to another zone d) Figure.7. Different methods of short-itching for a double-layer winding; a) full-itch winding, b) winding ste shortening, c) coil side shift in a slot, d) coil side transfer to another zone. τ v $= zone, τ u $= slot itch, W $= coil san. x $= coil san decrease. In the figure, a cross sign indicates one coil end of the hase U, and the dot indicates the other coil end of the hase U. In the grah for a coil side transfer to another zone, the grid indicates the arts of slots filled with the windings of the hase W. Coil side transfer to another zone (Fig.7d) can be considered to be created from the full-itch winding of Fig..7a by transferring the side of the uer layers of and 4 to a foreign zone W. There is no general rule for the transfer, but racticality and the urose of use decide the solution to be selected. The method may be adoted in order to cancel a certain harmonic from the current linkage of the winding. The above-described methods can also be emloyed simultaneously. For instance, if we shift the uer layer of the coil with a coil side transfer of Fig..7d left for the distance of one slot itch, we receive a combination of a winding ste shortening and a coil side transfer. This kind of winding is double short itched, and it can eliminate two harmonics. This kind of short-itching is often emloyed in machines where the windings may be rearranged during drive to form another ole number. When different methods are comared, we have to bear in mind that when considering the connection, a common winding ste shortening is always the simlest to realize, and the consumtion of coer is often the lowest. A winding ste shortening is an advisable method down

28 J. Pyrhönen, T. Jokinen, V. Hrabovcová.8 to W = 0.8τ without an increase in coer consumtion. In short two-ole machines, the ends are relatively long, and therefore it is advisable to use even shorter itches to make the end winding area shorter. Short-itchings down to W = 0.7τ are used in two-ole machines with refabricated coils. The most crucial issue concerning short itching is, however, how comletely we wish to eliminate the harmonics. This is best investigated with winding factors. The winding factor k wν was already determined with Fig..0 and Eq. (.6). When the winding is short itched, and the coil ends are not at a distance of 80 electrical degrees from each other, we can easily understand that short-itching reduces the winding factor of the fundamental. This is described with a itch factor k ν. Further, if the number of slots er ole and hase is higher than, we can see that in addition to the itch factor k ν, the distribution factor k dν is required as it was discussed above. Thus, we can consider the winding factor to consist of the itch factor k ν and the distribution factor k dν and is some cases, of a skewing factor (c.f. Eq..35). The full itch can be exressed in radians as π, as a ole itch τ, or in the number of slot itches y covering the ole itch. The itch exressed in the number of slots is y, and now the relative shortening is y / y.therefore, the angle of the short itch is (y / y ) π. A comlement to π, which is the sum of angles in a triangle, is: y y π π = π.this value will be divided equally to the y y π other two angles as in Fig.6b: y. Also the itch factor is defined as a ratio of the y geometric sum of hasors and the sum of the absolute values of the voltage hasors, see Fig..6b: The itch factor is U total k =. (.9) U slot When cos π y y is exressed in the triangle analyzed, it will be found that it equals with the itch factor defined: π y total / total cos U U = = y U slot U slot = k. (.30) After rearranging the final exression for the itch factor will be obtained: y π = W π k = sin sin. y τ (.3) In a full-itch winding, the itch is equal to the ole itch, y = y, W = τ and the itch factor is k =. If the itch is less than y, k <. In a general resentation, the distribution factor k dν and the itch factor k ν have to be valid also for harmonic frequencies. We may write for the ν th harmonic, the itch factor k ν, and the distribution factor k dν k ν W π = y π = sin ν sin ν, (.3) τ y

29 J. Pyrhönen, T. Jokinen, V. Hrabovcová.9 kd ν sin = qsin ( νqαu / ) ( να / ) u = π sin ν m. (.33) π sin ν m Here α u is the slot angle, α u = π /. The skewing factor will be develoed in Chater 4, but it is shown here k sqν s π sin ν π sinν τ mq = =. (.34) s π π ν ν τ mq Here the skew is measured as s/τ (c.f. Fig..6). The winding factor is thus a roduct of these factors k wν ν π π sin sinν W π m mq = kνkdνkskν = sin ν. (.35) τ π sin νπ ν m mq EXAMPLE.9: Calculate a winding factor for the two-layer winding = 4, = 4, m = 3, y = 5, (see Fig.7 b). There is no skewing, i.e. k skν =. 4 SOLUTION: The number of slots er hase er ole is q = = =.The distribution factor m 4 3 for the three-hase winding is k d = = = The number of slots er ole, or qsin sin q 4 in the other words, the ole itch exressed in the number of slots: y = = = 6. If the itch is 4 5 slots, it means that it is a short itch, and it is necessary to calculate the itch factor: π 5 π sin y k = = sin = y 6 k k k k = w = d skν = The winding factor is: EXAMPLE.0: A two-ole alternator has on the stator a three-hase two-layer winding embedded in 7 slots, two conductors in each slot, with a short itch of 9/36. The diameter of the stator bore is D s = 0.85 m, the effective length of the stack is l' =.75 m. Calculate the fundamental comonent of the induced voltage in one hase of the stator winding, if the amlitude of the fundamental comonent of the air ga flux density is ˆB = 0.9 T and the seed of rotation is 3000 min -. δ

30 J. Pyrhönen, T. Jokinen, V. Hrabovcová.30 SOLUTION: The effective value of the induced hase voltage will be calculated from the exression ˆ 3000 Eh = π f Φ Nskw at the frequency of f = n = = 50 Hz. The ole itch is π D s π 0.85m τ = = =.335 m. The maximum value of the magnetic flux is Φ ˆ ˆ = Bδτ l' = 0.9T.335m.75m =.368 Wb. The number of slots er ole and hase is π π s 7 q s = = =. The number of turns in series N s will be determined from the number of m 3 N conductors in the slot: z = = N = q = = 4. There is a two-layer distributed q short-itch winding, and therefore both the distribution and itch factors must be calculated (c.f. Eq..6): k d = = = π / 6 π / 6, π 9 π sin y k = = sin =, because the full qs sin sin y 36 qs 7 itch is y = = = 36. The winding factor yields kw = kdk = = 0. 9and the induced hase voltage of the stator is: E ˆ h = π f Φ Nskw = π Vs = 6637 V. s On the other hand, as it was shown reviously, the winding constructed with the coil side shift in a slot in Fig..7c roved to have an identical current linkage as the winding ste shortening.7b, and therefore also its winding factor has to be same. The distribution factor k dν is calculated with an average number of slots er ole and hase q =, and thus the itch factor k ν has to be the same as above, although no actual winding ste shortening has been erformed. For coil side shift in a slot, Eq. (.3) is not valid as such for the calculation of itch factor (because sin(ν π/) = ). When comaring magnetically equivalent windings of Fig..7 that aly winding ste shortening and coil side shift in a slot, it is shown that an equivalent reduction x of the coil san for a winding with coil side shift is x = τ W = ( q q) τ u. (.36) The substitution of slot itch τ u = τ / in the equation yields W τ = ( q q ). (.37) In other words, if the number of slots er ole and hase of the different layers of coil side shift are q and q, the winding corresonds to the winding ste shortening in the ratio of W/τ. By substituting (.37) in Eq. (.3) we obtain a itch factor k wν of the coil side shift in a slot

31 k w J. Pyrhönen, T. Jokinen, V. Hrabovcová.3 π ν = sin ν ( q q). (.38) In the case of the coil side shift of Fig..7c, q = 3 and q =. We may assume the winding to be a four-ole construction as a whole ( =, s = 4). In the figure, a basic winding is constructed of the conductors of the first twelve slots (the comlete winding may comrise an undefined number of sets of twelve-slot windings in series), and thus we obtain for the fundamental winding factor π 5 π kw = sin (3 ) = sin = Which is the same result of Eq. (.3) in the case of a winding ste shortening of Fig..7b. Because we may often aly both the winding ste shortening and coil side transfer in a different zone in the same winding, the winding factor has to be rewritten k = k k k. (.39) wν dν wν ν With this kind of a doubly short-itched winding, we may eliminate two harmonics, as stated earlier. The elimination of harmonics imlies that we select a double-short itched winding for which for instance k w5 = 0 and k 7 = 0. Now we can eliminate the undesirable fifth and seventh harmonics. However, the fundamental winding factor will get smaller. The distribution factor k dν is now calculated with the average number of slots er ole and hase q = (q + q )/. When analyzing comlicated short-itch arrangements, it is often difficult to find a universal equation for the winding factor. In that case, a voltage hasor diagram can be emloyed, as mentioned earlier in the discussion of Fig..0. Next, the coil side transfer to another zone of Fig..7 is discussed. Figure.8a deicts the fundamental voltage hasor diagram of the winding with coil side transfer to another zone in Fig..7, assuming that there are = 4 slots and = 4 oles. The slot angle is now α u = 30 electrical degrees, and thus the hase shift of the emf is 30. Figure.8b deicts the hasors of the hase U in a olygon according to the Fig..7d. The resultant and its ratio to the sum of the absolute values of the hasors is the fundamental winding factor. By drawing also the resultants of the windings V and W from oint 0, we obtain an illustration of the symmetry of a three-hase machine. The other harmonics (ordinal ν) are treated equally, but the hase shift angle of the hasors is now να u.

32 Figure.8. Voltage hasor diagram of a four-ole winding with a coil side transfer (Fig..7d) and b) the sum of slot voltage hasors of a single hase. Note that the voltage hasor diagram for a four-ole machine ( = ) is doubled, because it is drawn in electrical degrees. The arts of the figure are in different scales. In the voltage hasor diagram, there are in rincile two layers (one for the bottom layer and another for the to layer), but only one of them is illustrated here. Now two consequent hasors, e.g. and 4, when laced one after another, form a single radius of the voltage hasor diagram. The winding factor is thus defined with the voltage hasor diagram by comaring the geometrical sum to the sum of absolute values. J. Pyrhönen, T. Jokinen, V. Hrabovcová a) b) U hu.7 Current Linkage of a Slot Winding Current linkage of a slot winding refers to a function Θ = f(α), created by the winding and its currents to the equivalent air ga of the machine. The winding of Fig..8 and its current linkage in the air ga are investigated at a time when i W = i V = ½i U, Fig..9. The curve function in the figure is drawn at time t = 0. The hasors of the hase currents rotate at an angular seed ω, and thus after a time π/(3ω) = /(3f) the current of the hase V has reached its ositive eak, and the function of current linkage has shifted three slot itches to the right. After a time /(3f), the shift is six slot itches and so on. The curve function shifts constantly to the direction +α. To be exact, the curve waveform is of the form resented in the figure only at times t = c/(3f), when the factor c is of the values c = 0,,, 3,.... As the time elases, the waveform roceeds constantly. The Fourier analysis of the waveform, however, roduces harmonics that remain constant. Figure.9 illustrates the current linkage Θ (α) roduced by a single coil at a single ole itch. A flux that asses through the air ga at an angle γ returns at an angle β = π γ. In the case of a nonfull-itch winding, the current linkage is distributed in the ratio of the ermeances of the aths. This gives us a air of equations Θγ β z i = Θ γ + Θβ; =, (.40) Θ γ β from which we obtain two constant values for the current linkage waveformθ (α) β γ Θ γ = zi; Θβ = zi. (.4) π π

33 J. Pyrhönen, T. Jokinen, V. Hrabovcová.33 The Fourier series of the function Θ (α) of a single coil is ( α ) = Θˆ cosα + Θˆ cos3α + Θˆ cos5α Θˆ cosνα... Θ ν (.4) The magnitude of the ν th term of the series is obtained from the equation by substituting the function of the current linkage waveform for a single coil π ˆ = νγ Θν Θ ( α ) cos( να ) dα = zisin π νπ. (.43) As γ / = π/τ 0 W, and hence / = ( / τ ) ( π/) γ W the last factor of Eq. (.43) νγ W π sin = sin ν = k τ ν (.44) is the itch factor of the harmonic ν for the coil observed. For the fundamental ν =, we obtain k. We can thus see that the fundamental is just a secial case of the general harmonicν. While the electrical angle of the fundamental is α, the corresonding angle for the harmonic ν is always να. If now νγ/ is a multile of the angle π, the itch factor becomes k ν = 0. Thus, the winding does not roduce such harmonics, neither are voltages induced to the winding by the influence of ossible flux comonents at this distribution. However, voltages are induced to the coil sides, but in the whole coil these voltages comensate each other. Thus, with a suitable short-itching, it is ossible to eliminate harmful harmonics. γ Θ γ iz π/ 0 +π/ α Θ β z β a) b) β π γ a) b) Figure.9. a) Currents and schematic flux lines of a short-itch coil in a two-ole system. b) Two wavelengths of the current linkage created by a single, narrow coil. In Fig..0, there are several coils... q in a single ole of a slot winding. The current linkage of each coil is z i. The coil angle (in electrical degrees) for the harmonic ν of the narrowest coil is νγ, the next being ν(γ + α u ) and the broadest ν(γ + (q - )α u ). For an arbitrary coil g, the current linkage is according to Eq. (.43) Θ γ = z isin ν + ( q ) α u π. (.45) ν νg

34 J. Pyrhönen, T. Jokinen, V. Hrabovcová.34 When all the harmonics of the same ordinal generated by all coils of one hase are summed, we obtain er ole q q Θν tot = Θνg = qzi kw ν = g = νπ g = νγ γ γ = zi sin + sinν + αu sinν + νπ ( q ) αu. (.46) The sum k wν in brackets can be calculated for instance with the geometrical figure of Figs..0 and.. The line segment AC is written as νqα u AC = r sin. (.47) The arithmetical sum of unit segments is να u q AB = qr sin (.48) and we may find for the harmonic ν νqα u sin AC k dν = =. (.49) qab να u qsin να u r iz q iz iz νγ νγ / α = 0 C' A C q ν (q-)α u / να u νγ B / α = 0 Figure.0. Concentric coils of a single ole; the calculation of winding factor for a harmonic ν. This equals to the distribution factor of Eq. (.33). Now, we see that the line segment AC = qk dν. We use Fig. a again: The angle BAC is obtained from Fig.. as a difference of the angles OAB and OAC. It is ν(q )α u /. The rojection AC' is thus ( γ + ( q ) α ) ν u AC ' = AC sin = ACkν. (.50)

35 J. Pyrhönen, T. Jokinen, V. Hrabovcová.35 Here we have a itch factor influencing the harmonic ν, because in Fig..0, ν[γ + (q )α u ] is the average coil width angle. Eq. (.46) is now reduced where kw ν Θν tot = qzi, (.5) π ν k = k k. (.5) wν ν dν This is the winding factor of the harmonic ν. This is an imortant observation. The winding factor was originally found for the calculation of the induced voltages. Now we understand that the same distribution and itch factors also affect the current linkage harmonic roduction. By substituting the harmonic current linkage Θ ν in Eq. (.4) with the current linkage Θ νtot, we obtain the harmonic ν generated by the current linkage of q coils Θ = Θ cosνα. (.53) ν νtot This is valid for a single hase coil. The harmonic ν created by a oly-hase winding is calculated by summing all the harmonics created by different hases. By its nature, the itch factor is zero, if να u = ±cπ (since sin( ν qα u / ) = sin( ± cπq) = 0 ) (i.e., the coil sides are in the same magnetic otential), when the factor c = 0,,, 3, 4... It allows thereby only the harmonics π ν c. (.54) α u The slot angle and the hase number are interrelated π qα u =. (.55) m The distribution factor is zero, if ν = ±cm. The winding thus roduces harmonics ν = +± cm. (.56) EXAMPLE.: Calculate which ordinals of the harmonics can be created by a three-hase winding. SOLUTION: A symmetrical three-hase winding may create harmonics calculated based on Eq. (.56), by inserting m = 3. These are listed in Table.. Table.. Ordinals of the harmonics created by a three-hase winding (m = 3). c ν ositive sequence negative sequence We see that ν =, and all even harmonics and harmonics divisible by three are missing. In other words, a symmetrical oly-hase winding does not roduce for instance a harmonic roagating to an oosite direction at the fundamental frequency. Instead, a single-hase winding m = creates

36 J. Pyrhönen, T. Jokinen, V. Hrabovcová.36 also a harmonic, the ordinal of which is ν =. This is a articularly harmful harmonic, and imedes the oeration of single-hase machines. For instance a single-hase induction motor, because of the field rotating to negative direction, does not start without assistance because the ositive and negative sequence fields are equally strong. EXAMPLE.: Calculate the itch and distribution factors for ν =, 5, 7 if a chorded stator of an AC machine has 8 slots er ole and the first coil is embedded in the slots and 6. Calculate also the relative harmonic current linkages. SOLUTION: The full itch would be y = 8 and a full-itch coil should be embedded in the slots and 9. If the coil is located in the slots and 6, the coil itch is shorted to y = 5. Therefore, the itch factor for the fundamental will be: W π y π 5 π k = sin = sin = sin = τ y 8 harmonics: 0.966, and corresondingly for the fifth and seventh k π 5 π 5 = sin = sin 5 = 0.59 ν y y 8, π 5 π sin y k sin = ν = = y The number of slots er ole and hase is q = 8/3 = 6 and the slot angle is α u = π/8. Now, the distribution factor is νqα u 6π /8 5 6π /8 sin sin sin k dν =, k = d = 0.956, k = d 5 = 0.97, να u π /8 5π /8 qsin 6sin 6sin 7 6π /8 sin k = d7 = π /8 6sin k w = kdk = = 0.93, k w 5 = kd 5k 5 = 0.97 ( 0.59) = k w7 = kd7k7 = = kw ν k The winding creates current linkages Θν tot = qzi. Calculating wν for the harmonics, 5, 7 π ν ν k w kw kw = 0.93, = = 0. 0, = = Here we can see that because of the chorded winding, the current linkages of the fifth and seventh harmonics will be reduced to.% and 0.58 % of the fundamental, as the fundamental is also reduced to 9.3 % of the full sum of the absolute values of slot voltages. EXAMPLE.3: A rotating magnetic flux created by a three-hase 50 Hz, 600 min - alternator has a satial distribution of magnetic flux density given by the exression: B = Bˆ ϑ ˆ ϑ ˆ sin + B3 sin 3 + B5 sin 5ϑ = 0.9sinϑ + 0.5sin 3ϑ + 0.8sin 5ϑ [T]. The alternator has 80 slots, the winding is wound with two layers, and each coil has three turns with the san of 5 slots. The armature diameter is 35 cm and the effective length of the iron core 0.50 m. Write the exression for the instantaneous value of the induced voltage in one hase of the winding. Calculate the effective value of hase voltage and also the line-to-line voltage of the machine.

37 J. Pyrhönen, T. Jokinen, V. Hrabovcová.37 SOLUTION: The number of ole airs is given by the seed and the frequency: n 60 f f = = = = 5, and the number of oles is 0. The area of one ole is: 60 n 600 π D π.35 τ l' = l' = 0.50 = 0. m. 0 From the exression for the instantaneous value of the magnetic flux density, we may derive B ˆ = 0. 9 T, B ˆ3 = 0. 5 T and B ˆ5 = 0. 8 T. The fundamental of the magnetic flux on the τ is ˆ ˆ Φ = Bτ l' = 0.9T 0.m = 0.4 Vs. To be able to calculate the induced voltage, it is π π necessary to make a reliminary calculation of some arameters: 80 The number of slots er ole and hase is q = = = 6, m 0 3 π π The angle between the voltages of adjacent slots α u = =, 8 The distribution and itch factors for each harmonics: α u π π sin q sin 6 sin 3 6 π sin 5 6 k d = = = 0.956, kd 3 = = α u π π, k d 5 = = qsin 6sin 6sin 3 π sin The number of slots er ole is = = 8, which would be a full itch. The coil san is 5 slots, 0 which means the chorded itch y = 5, and the itch factors are: π 5 π sin y π 5 π k = = sin = 0.966, sin y k sin = 3 = = y 8 y 8 π 5 π sin y k 5 = 5 = sin 5 = y , which results in the following winding factors: k = k k = , k = ( 0.707) = , k = w d = w3 w5 = Now, it is ossible to calculate the effective values of the induced voltages of the harmonics. The hase number of turns is determined as follows: the total number of coils in a 80-slot machine in a two-layer winding is 80. It means that the number of coils er hase is 80 / 3 = 60, each coil has three turns, and therefore N = 60 3 = 80. E ˆ = π fφnkw = π Vs = 448 V. s The induced voltage of harmonics can be written similarly as follows: E ν ˆ π ˆ = π f νφ ν N kwν = πf ν Bντ ν N kwν = πνf Bν N kwν = π f Bντ N kwν π ˆ τ ν π ˆ

38 J. Pyrhönen, T. Jokinen, V. Hrabovcová.38 The ratio of the ν th harmonic and fundamental is E E ν = π f Bˆ ντ Nk π π f ˆ Bτ Nk π wν w = Bˆ νk Bˆ k wν w results for E ν : Bˆ 3k w3 0.5 ( ) E 3 = E = 448V = 6.9 V, Bˆ k w Bˆ 5k w E 5 = E = 448V = 49.5V. Bˆ k w Finally, the exression for the instantaneous value of the induced voltage is: e t = Eˆ sinωt + Eˆ sin 3ωt + Eˆ sin 5ωt = E sinωt + E sin 3ωt + E sin 5ωt e () () t = 448sinωt 6.9sin 3ωt sin 5ωt = sinωt sin 3ωt + 70sin 5ωt The total value of the effective hase voltage is: Eh = E + E3 + E5 = V = 454 V, and its line-to-line voltage is E = 3 E + E5 = V = V. The third harmonic comonent does not aear in the line-to-line voltage, which will be demonstrated later on. EXAMPLE.4: Calculate the winding factors and er unit magnitudes of the current linkage for ν = y, 3, 5 if = 4, m = 3, q =, W/τ = = 5/6. y SOLUTION: The winding factor is used to derive the er unit magnitude of the current linkage. In Fig.., we have a current linkage distribution of the hase U of a short-itch winding ( = 4, m y = 3, q =, = 4, W/τ = = 5/6), as well as its fundamental and the third harmonic at time t = 0, y when i iˆ U =. The total maximum height of the current linkage of a ole air is at that moment qz iˆ. A half of the magnetic circuit (involving a single air ga) is influenced by a half of this current linkage. The winding factors for the fundamental and lowest harmonics and the amlitudes of the current linkages according to Eq. (.5), (.5) and Examle.3 are: ν = k w = k k d = = 0,93 Θ ˆ =.85 Θ max ν = 3 k w3 = k 3 k d3 = = 0,5 $ Θ 3 = 0. Θ max ν = 5 k w5 = k 5 k d 5 = = $ Θ 5 = 0.07 Θ max The negative signs for the third and fifth harmonic amlitudes mean that, if starting at the same hase, the third and fifth harmonics will have a negative eak value as the fundamental is in its ositive eak, see Fig.. and.. For instance, the fundamental is calculated with (.5); see also (.5) ˆ kw 0.93 = qzi = zi. 85z π ν π = Θ ν Only the fundamental and the third harmonic are illustrated in Fig... i

39 J. Pyrhönen, T. Jokinen, V. Hrabovcová.39 The amlitude of harmonics is often exressed as a ercentage of the fundamental. In this case, the amlitude of the third harmonic is 7.8 % (0../.85) of the amlitude of the fundamental. However, this is not necessarily harmful in a three-hase machine, because in the harmonic current linkage created by the windings together, the third harmonic is comensated. The situation is illustrated in Fig.., where the currents i U = and i V = i W = ½ flow in the winding of Fig... In salient ole machines, the third harmonic may, however, cause circulating currents in delta connection, and therefore the star connection in the armature is referred. +U -W +V -U +W -V +U Θ max Θ max Θ ( α ) 0 π ^ Θ α Figure.. Short-itch winding ( s = 4, =, m = 3, q s = ) and the analysis of its current linkage distribution of the hase U. The distribution includes a notable amount of the third harmonic. In the figure, the fundamental and third harmonic are illustrated with dashed lines. In single- and two-hase machines, the number of slots is referably selected higher than in threehase machines, because in these coils, at certain instants only a single hase coil alone creates the whole current linkage of the winding. In such a case, the winding alone should roduce as sinusoidal current linkage as ossible. In single- and double-hase windings, it is sometimes necessary to fit a different number of conductors in the slots to make the steed line Θ(α) aroach sinusoidal form.

40 J. Pyrhönen, T. Jokinen, V. Hrabovcová.40 +U -W +V -U +W -V +U Θ... U α V α W 0 α Figure.. Comensation of the third harmonic in a three-hase winding. There are currents i U = i V = i W flowing in the winding. We see that when we sum the third harmonics of the hases V and W with the harmonic of the hase U, the harmonics comensate each other. A oly-hase winding thus roduces harmonics, the ordinals of which are calculated with Eq. (.56). When the stator is fed at an angular frequency ω s, the angular seed of the harmonicν with resect to stator is ω ν ωs s =. (.57) ν The situation is illustrated in Fig..3, which shows that the shae of the harmonic current linkage changes as the harmonic roagates in the air ga. The deformation of the harmonic indicates the fact that harmonic amlitudes roagate at different seeds and in different directions. A harmonic according to Eq. (.57) induces the voltage of the fundamental frequency to the stator winding. The ordinal of the harmonic indicates how many wavelengths of a harmonic are fitted to a distance τ of a single ole air of the fundamental. This yields the number of ole airs and the ole itch of a harmonic ν = ν, (.58) τ =. ν (.59) τ ν The amlitude of the ν th harmonic is determined with the ordinal from the amlitude of the current linkage of the fundamental, and it is calculated in relation of the winding factors

41 J. Pyrhönen, T. Jokinen, V. Hrabovcová.4 Θˆ ˆ k Θ wν ν =. (.60) νkw Θ τ Θ U β =ωt τ -U -V U +W +U -W V W +V -U τ v Θ β =ωt τ τ t t 3 t 5 Θ β =ωt 3 τ i U i W i V t t 4 Θ β =ωt 4 τ β =ωt 5 Figure.3. Proagation of a harmonic current linkage and the deformations caused by harmonics. If there is a current flowing only in the stator winding, we are able to set the eak of the air ga flux density at β. The flux roagates but the magnetic axis of the winding U remains stable. The winding factor of the harmonic ν can be determined with Eqs. (.3) and (.33) by multilying the itch factor k ν and the distribution factor k dν. k = k wν k ν dν W π ν π sin ν sin τ m =. (.6) sin νπ m Comared with the angular velocity ω s of the fundamental comonent, a harmonic current linkage wave roagates in the air ga at a fractional angular velocity ω s /ν. The synchronous seed of the harmonic ν is at the same very angular seed ω s /ν. If a motor is running at about synchronous seed, the rotor is travelling much faster than the harmonic wave. If we have an asynchronously

42 J. Pyrhönen, T. Jokinen, V. Hrabovcová.4 running motor with a er-unit sli s ( ωs Ωr )/ωs =, (ω s is the stator angular frequency and Ω r is the rotor mechanical angular rotating frequency), the sli of the rotor with resect to the ν th stator harmonic ( ) sν = ν s. (.6) The angular frequency of theν th harmonic in the rotor is thus ( ν ( s) ) ω ν = ω. (.63) r s If we have a synchronous machine running with the sli s = 0, we immediately observe from Eq. (.6) and (.63) that the angular frequency created by the fundamental comonent of the flux density of the stator winding is zero in the rotor co-ordinate. However, harmonic current linkage waves ass the rotor at different seeds. If the shae of the ole shoe is such that the rotor roduces flux density harmonics, they roagate at the seed of the rotor and thereby generate ulsating torques with the stator harmonics travelling at different seeds. This is a roblem articularly in low-seed ermanent magnet synchronous motors, in which the rotor magnetizing often roduces a quadratic flux density and the number of slots in the stator is small, for instance q = or even lower, the amlitudes of the stator harmonics being thus notably high..8 Poly-Phase Fractional Slot Windings If the number of slots er ole and hase q of a winding is a fraction, the winding is called a fractional slot winding. The windings of this tye are either concentric or diamond windings with one or two layers. Some advantages of fractional slot windings when comared with integer slot windings are: great freedom of choice with resect to the number of slots oortunity to reach a suitable magnetic flux density with the given dimensions multile alternatives for short-itching if the number of slots is redetermined, the fractional slot winding can be alied to a wider range of numbers of oles than the integral slot winding segment structures of large machines are better controlled by using fractional slot windings oortunity to imrove the voltage waveform of a generator by removing certain harmonics The greatest disadvantage of fractional slot windings is subharmonics, when the denominator of q (slots er ole and hase) is n z q = =. (.64) m n Now, q is reduced so that the numerator and the denominator are smallest ossible integers: the numerator being z and the denominator n. If the denominator n is an odd number, the winding is said to be a first-grade winding, and when n is an even number, the winding is of the second grade. The most reliable fractional slot winding is constructed by selecting n =. An esecially interesting winding of this tye can be designed for fractional slot ermanent magnet machines by selecting q = ½.

43 J. Pyrhönen, T. Jokinen, V. Hrabovcová.43 In integral slot windings, the base winding is of the length of two ole itches (the distance of the fundamental wavelength), whereas in the case of fractional slot windings, a distance of several fundamental wavelengths has to be travelled before the hasor of a voltage hasor diagram meets the same exact oint of the waveform again. The difference of an integral slot and fractional slot winding is illustrated in Fig..4. B B 0 π π α 0 π π α ' π slot ositions slot ositions 5... s s,... = s, s '+... B base winding base winding a) b) = 5 0 slot ositions and coil sides of hase U α s = base winding c) Figure.4. Basic differences of a) an integral slot stator winding and b) a fractional slot winding. The number of stator slots is s. In an integral slot winding, the length of the base winding is s / slots (a: slots, q s = ), but in a fractional slot winding, the division is not equal (b: q s < ). In the observed integer slot winding, the base winding length is s = and after that, the magnetic conditions for the slots reeat themselves equally; observe the slots and 3. In the fractional slot winding, the base winding is notably longer and contains s ' slots. Figure c illustrates an examle of a fractional slot winding with s = and = 5. Such a winding may be used in concentrated wound ermanent magnet fractional slot machines, where q = 0.4. In a two-layer system, each of the stator hases carries four coils. The coil sides are located in the slots,, 6 7, and 7 8. In a fractional slot winding, we have to roceed the distance of ' ole airs before a coil side of the same hase meets exactly the eak value of the flux density again. Then, we need a number of s ' hasors of the voltage hasor diagram, ointing at different directions. Now, we can write s s '= ', s ' < s, > '. (.65) Here the voltage hasors s ' +, s ' +, 3 s ' + and (t-) s ' + are in the same osition in the voltage hasor diagram as the voltage hasor of the slot. In this osition, the cycle of the voltage hasor diagram is always started again. Either a new erihery is drawn, or more slot numbers are added to the hasors of the initial diagram. In the numbering of a voltage hasor diagram, each layer of the diagram has to be circled ' times. Thus, t layers are created to the voltage hasor diagram. In other words, in each electrical machine, there are t electrically equal slot sequences, the slot number of which is s ' = s /t and the number of ole airs ' = /t. To determine t, we have to find the

44 J. Pyrhönen, T. Jokinen, V. Hrabovcová.44 smallest integers s ' and '. t is thus the largest common divider of s and. If s /(m) N (N is a set of integers, N even is a set of even integers, and N odd is a set of odd integers), we have an integral slot winding, and t =, s ' = s / and ' = / =. Table.3 shows some arameters of a voltage hasor diagram. To generalize the reresentation, the subscrit s is left out in the following. Table.3. Parameters of voltage hasor diagrams. t the largest common divider of and, the number of hasors of a single radius, the number of layers of a voltage hasor diagram ' = /t ' =/t (/t) the number of radii, or the number of hasors of a single turn in a voltage hasor diagram. (the number of slots in a base winding) the number of revolutions around a single layer when numbering a voltage hasor diagram the hasors skied in the numbering of the voltage hasor diagram If the number of radii in the voltage hasor diagram is ' = /t, the angle of adjacent radii, that is, the hasor angle α z is written π α z = t. (.66) The slot angle α u is corresondingly a multile of the hasor angle α z α u = α z = 'α z. (.67) t When = t, we obtain α u = α z, and the numbering of the voltage hasor diagram roceeds continuously. If > t, α u > α z, a number of (/t) hasors have to be skied in the numbering of slots. In that case, a single layer of a voltage hasor diagram has to be circled (/t) times when numbering the slots. When considering the voltage hasor diagrams of harmonics ν, we see that the slot angle of the ν th harmonic is να u. Also the hasor angle is να z. The voltage hasor diagram of theν th harmonic differs from the voltage hasor diagram of the fundamental with resect to the angles, which are ν-fold. EXAMPLE.5: Create voltage hasor diagrams for two different fractional slot windings: a) = 7 and = 3, b) = 30, = 4. SOLUTION: a) = 7, = 3, / = 9 N, q s =.5, t = = 3, ' = 9, ' =, α u = α z = 40. There are, hence, nine radii in the voltage hasor diagram, each having three hasors. Because α u = α z, no hasors are skied in the numbering, Fig..5a. b) = 30, = 4, / = 7.5 N, q s =.5, t =, ' = 5, ' =, Z = /t = 30/ =5, α z = 360 /5 = 4, α u = α z = 4 = 48, (/t) - =. In this case, there are 5 radii in the voltage hasor diagram, each having two hasors. Because α u = α z, the number of hasors skied will be [(/t) - = ]. Both of the layers of the voltage hasor diagram have to be circled twice in order to number all the hasors, Fig..5b.

45 J. Pyrhönen, T. Jokinen, V. Hrabovcová α z = α u α u 6 8 α z a) b) Figure.5. Voltage hasor diagrams for two different fractional slot windings. Left, the numbering is continuous, whereas on the right, certain hasors are skied. a) = 7, = 3, t = 3, ' = 9, ' = α u = α z = 40 ; b) = 30, = 4, t =, ' = 5, ' = α u = α z = 48 ; α u is the angle between voltages in the slots in electrical degrees and the angle α z is the angle between two adjacent hasors in electrical degrees..9 Phase Systems and Zones of Windings Phase systems Generally seaking, windings may involve single or multile hases, the most common case being a three-hase winding, which has been discussed here also. However, various other winding constructions are ossible, as illustrated in Table.4. On a single magnetic axis of an electrical machine, there may be located only one axis of a single hase winding. If another hase winding is laced on the same axis, no genuine oly-hase system is created, because both windings roduce arallel fluxes. Therefore, each hase system that involves an even number of hases is reduced to involve only a half of the original number of hases m' as illustrated in Table.4. If the reduction roduces a system with an odd number of hases, we obtain a radially symmetric oly-hase system, also known as a normal system. If the reduction roduces a system with an even number of hases, the result is called a reduced system. In this sense, an ordinary two-hase system is a reduced four-hase system, as illustrated in Table.4. For an m-hase normal system, the hase angle is α h = π/m. (.68) Corresondingly for a reduced system, the hase angle is α h = π/m. (.69) For examle in a three-hase system, α h3 = π/3 and for a two-hase system, α h = π/.

46 J. Pyrhönen, T. Jokinen, V. Hrabovcová.46 If there is even a single odd number as a multilicand of the hase number in the reduced system, a radial-symmetric winding can be constructed again by turning the direction of the suitable hasors by 80 electrical degrees in the system, as shown in Table.4 for a six-hase system (6 = 3). With this kind of a system, a non-loaded star oint is created exactly like in a normal system. In a normal reduced system, the star oint is loaded, and thus for instance a star oint of a reduced twohase machine requires a conductor of its own, which is not required in a normal system. Without a neutral conductor, a reduced two-hase system becomes a single-hase system, because the windings cannot oerate indeendently, but the same current that roduces the current linkage is always flowing in them, and together they form only a single magnetic axis. An ordinary threehase system also becomes a single-hase system for instance if the voltage suly of one hase ceases for some reason. Table.4. Phase systems of the windings of electrical machines. The fourth column introduces radially symmetric winding alternatives. Number of hases m Non-reduced winding systems have searate windings for ositive and negative magnetic axes Reduced system: loaded star oint needs a neutral line unless radial-symmetric (e.g. m = 6) - Normal system: non-loaded star oint and no neutral line, unless m = m' = - 3 m' = 4-4 m' = 6-5 m' = 8 - m' = m' = - m' = 4

47 Reading instructions for Table.4 J. Pyrhönen, T. Jokinen, V. Hrabovcová.47 L L loaded star oint L N L L3 non-loaded star oint Of the winding systems in Table.4, the three-hase normal system is dominant in industrial alications. Five- and seven-hase windings have been suggested for frequency converter use to increase the system outut ower at a low voltage. Six-hase motors are used in large synchronous motor drives. In some larger high-seed alications, six-hase windings are also useful. In ractice, all hase systems divisible by three are ractical in inverter sulies. Each of the three-hase artial systems is sulied by its own three-hase frequency converter having a temoral hase shift π/m', in a -hase system e.g. π/. For examle a -hase system is sulied with four three-hase converters having a π/ temoral hase shift. Single-hase windings may be used in single-hase synchronous generators and also in small induction motors. In the case of a single-hase-sulied induction motor, the motor, however, needs starting assistance, which is often realized as an auxiliary winding with a hase shift of π/. In such a case, the winding arrangement starts to resemble the two-hase reduced winding system, but since the windings are usually not similar, the machine is not urely a two-hase machine Zones of Windings In double-layer windings, both layers have searate zones; an uer-layer zone and a bottom-layer zone, Fig..6. This double number of zones means also a double number of coil grous when comared with single-layer windings. In double-layer windings, one coil side is always located in the uer layer, and the other in the bottom layer. In short-itched double-layer windings, the uer layer is shifted with resect to the bottom layer, as it is shown in Figs..5 and.7. The san of the zones can be varied between the uer and bottom layer, as shown in Fig..7 (zone variation). With double-layer windings, we can easily aly systems with a double zone san, which usually occur only in machines, where the windings may be rearranged during the drive to roduce another number of oles. In fractional slot windings, zones of varying sans are ossible. This kind of zone variation is called natural zone variation. In a single-layer winding, each coil requires two slots. For each slot, there is now one half of a coil. In double-layer windings, there are two coil sides in each slot, and thus, in rincile, there is one coil er slot. A total number of coils z c is thereby for single-layer windings: z c =. (.70) for double-layer windings: z c =. (.7)

48 J. Pyrhönen, T. Jokinen, V. Hrabovcová.48 -V +U +U -W -X +W -V U +W W -U +U V -W +V -W +W -V U V -W W -U -V +Z U V +Y Z W +V Y X +X +W -Y -U +V +V -U -Z a) b) c) Figure.6. Zone formation of double-layer windings, m = 3, =. a) a normal zone san b) a double zone san, c) the zone distribution of a six-hase radial-symmetric winding with a double zone san. The tails and heads of the arrows corresond to a situation in which there are currents I U = -I V = - I W flowing in the windings. The winding in Fig..6a corresonds to a single-layer winding, which is obtained by unifying the winding layers by removing the insulation layer between the layers. The single-layer windings and double-layer windings with double-width zones form m coil grous er ole air. Double-layer windings with a normal zone san form m coil grous er ole air. The total numbers of coil grous are thus m and m resectively. Table.5 lists some of the core arameters of hase windings. Table.5. Phase winding arameters Winding Number of coils z c Number of grous of coil Average zone san Average zone angle α zav Single-layer / m τ /m π/m Double-layer, m τ /m π/m normal zone san Double-layer, double zone san m τ /m π/m.0 Symmetry Conditions A winding is said to be symmetrical, if it, when fed from a symmetrical suly, creates a rotating magnetic field. Both of the following symmetry conditions must be fulfilled. a) The first condition of symmetry: Normally, the number of coils er hase winding has to be an integer: For single-layer windings: = q N. (.7) m For double-layer windings: = q N. (.73) m The first condition is met easier by double-layer windings than by single-layer windings, thanks to a wider range of alternative constructions. b) The second condition of symmetry: In oly-hase machines, the angle α h between the hase windings has to be an integral multile of the angle α z. Therefore for normal systems, we can write

49 J. Pyrhönen, T. Jokinen, V. Hrabovcová.49 α α h z π = mπt = mt N, (.74) and for reduced systems α α h z π = mπt = N mt. (.75) Let us now consider how the symmetry conditions are met with integral slot windings. The first condition is always met, since and q are integers. The number of slots in integral slot windings is = qm. Now the largest common divider t of and is always. When we substitute = t to the second symmetry condition, we can see that it is always met, since mt = = q N m. (.76) Integral slot windings are thus symmetrical. Because t =, also α u = α z, and hence the numbering of the voltage hasor diagram of the integral slot winding is always consecutive, as can be seen for instance in Figures.0 and.8. Symmetrical Fractional Slot Windings Fractional slot windings are not necessarily symmetrical. A successful fulfilment of symmetry requirements starts with the correct selection of the initial arameters of the winding. First, we have to select q (slots er ole and hase) so that the fraction resenting the number of slots z q = (.77) n is indivisible. Here the denominator n is a quantity tyical of fractional slot windings. a) The first condition of symmetry: For single-layer windings (Eq..7), it is required that in the equation m = q = z n, N. (.78) n Here z and n constitute an indivisible fraction, and thus and n have to be evenly divisible. We see that when designing a winding, with the ole air number usually as an initial condition, we can select only certain integer values for n. Corresondingly, for double-layer windings (Eq..73), the first condition of symmetry requires that in the equation m z = q =, N. (.79) n n When comaring Eq. (.78) with Eq. (.79), we can see that we achieve a wider range of alternative solutions for fractional slot windings by alying a double-layer winding than a single-layer winding. For instance for a two-ole machine =, a single-layer winding can be constructed only when n =, which leads to an integral slot winding. On the other hand, a fractional slot winding, for which n = and =, can be constructed as a double-layer winding.

50 J. Pyrhönen, T. Jokinen, V. Hrabovcová.50 b) The second condition of symmetry: To meet the second condition of symmetry (Eq..74), the largest common divider t of and n has to be defined. This divider can be determined from the following equation: = qm = mz and = n. n n According to Eq. (.78), /n N, and thus this ratio is a divider of both and. Because z is indivisible by n, the other dividers of and can be included only in the figures m and n. These dividers are denoted generally with c, and thus t = c. (.80) n Now the second condition of symmetry can be rewritten for normal oly-hase windings in a form that is in harmony with Eq. (.74) mt mz = n mc n z = N c. (.8) The divider c of n cannot be a divider of z. The only ossible values for c are c = or c =. For normal oly-hase systems, m is an odd integer. For reduced oly-hase systems, according to Eq. (.75), it is written mt mz = n z = N mc c n. (.8) For c, this allows only the value c =. As shown in Table.4, for normal oly-hase windings, the hase number m has to be an odd integer. The divider c = of m and n cannot be a divider of m. For reduced oly-hase systems, m is an even integer, and thus the only ossibility is c =. The second condition of symmetry can now be written simly in the form: n and m cannot have a common divider n / m N. If m = 3, n cannot be divisible by three, and the second condition of symmetry reads: n N. (.83) 3 The conditions (.78) and (.83) automatically determine that if includes only the figure three as its factor ( = 3, 9, 7,...), a single-layer fractional slot winding cannot be constructed at all. Table.6 lists the symmetry conditions of fractional slot windings. Table.6. Conditions of symmetry for fractional slot windings.

51 J. Pyrhönen, T. Jokinen, V. Hrabovcová.5 Number of slots er ole and hase q = z/n, where z and n cannot be mutually divisible Tye of winding, number of hases Single-layer windings Double-layer windings Two-hase m = Three-hase m = 3 Condition of symmetry /n N /n N n/m N n/ N n/m N n/3 N As shown, it is not always ossible to construct a symmetrical fractional slot winding for certain numbers of ole airs. However, if some of the slots are left without a winding, a fractional slot winding can be carried out. In ractice, only three hase windings are realized with emty slots. Free slots o have to be distributed on the erihery of the machine so that the hase windings become symmetrical. The number of free slots has thus to be divisible by three, and the angle between the corresonding free slots has to be 0. The first condition of symmetry is now written as: 6 o N. (.84) The second condition of symmetry is N. (.85) 3t Furthermore, it is also required that o odd 3 N. (.86) Usually, the number of free slots is selected to be three, because this enables the construction of a winding, but does not leave a considerable amount of the volume of the machine without utilization. For normal zone-width windings with free slots, the average number of slots of a coil grou is obtained from the equation av o o o = = = q. (.87) m m m m. Base Windings It was shown reviously that in fractional slot windings, a certain coil side of a hase winding occurs at the same osition with the air ga flux always after ' = /t ole airs, if the largest common divider t of and is larger than one. In that case, there are t electrically equal slot sequences containing ' slots in the armature, each of which includes a single layer of the voltage hasor diagram. Now it is worth considering whether it is ossible to connect a system of t equal sequences of slots containing a winding as t equal indeendent winding sections. This is ossible when all the slots ' of the slot sequence of all t electrically equal slot grous meet the first condition of symmetry. The second condition of symmetry does not have to be shown.

52 J. Pyrhönen, T. Jokinen, V. Hrabovcová.5 If '/m is an even number, both a single-layer and a double-layer winding can be constructed in ' slots. If '/m is an odd number, only a double-layer winding is ossible in ' slots. When constructing a single-layer winding, q has to be an integer. Thus, two slot itches of t with ' slots altogether are required for a smallest indeendent symmetrical single-layer winding. The smallest indeendent symmetrical section of a winding is called a base winding. When a winding consists of several base windings, the current and voltage of which are due to geometrical reasons always of the same hase and magnitude, it is ossible to connect these basic windings in series and in arallel to form a comlete winding. Deending on the number of '/m, that is, whether it is an even or odd number, the windings are defined either as first- or second-grade windings. Table.7 lists some of the arameters of base windings. Table.7. Some arameters of fractional slot base windings. Base winding of first grade Parameter q Denominator n Parameter Parameter '/ m / tm Divider t, the largest common divider of and Slot angle α u exressed with voltage hasor angle α z Tye of winding Base winding of second grade q = z/n n N n N odd even ' N m N tm even even ' N m N tm t = t = n n π n π α u = nαz = n t α = α = n t u z single-layer windings double-layer windings single-layer windings odd odd double-layer windings Number of slots * of a base * = * = * = winding t t t Number of ole airs * of a n base winding * = = n t * = = n t = t Number of layers t* in a voltage hasor diagram for a base winding t* = t* = t* = The asterix * indicates the values of the base winding... First-Grade Fractional Slot Base Windings In first-grade base windings, ' m = N mt even. (.88) There are * slots in a first-grade base winding, and the following is valid for the arameters of the winding: * =, * = = n, t* =. (.89) t t

53 J. Pyrhönen, T. Jokinen, V. Hrabovcová.53 The both conditions of symmetry (.7) (.75) are met under these conditions... Second-Grade Fractional Slot Base Windings A recondition of the second-grade base windings is that ' m = N mt odd. (.90) According to Eqs. (.8) and (.8), Eq. (.90) is valid for normal oly-hase windings, when c = only for the even values of n. Thus we obtain t = /n and α u = nα z /. The first condition of symmetry is met with the base windings of the second grade only when * = '. Only now we obtain * ' = = m m mt N. (.9) The second-grade single-layer base winding comrises thus two consequent t th arts of a total winding. Their arameters are written as * =, * = = n, t* =. (.9) t t With these values, also the second condition of symmetry is met, since * ' = = mt * m mt N. (.93) The second-grade double-layer base winding meets the first condition of symmetry immediately when the number of slots is * = '. Hence * ' = = m m mt The arameters are now N. (.94) n * =, * = =, t* =. (.95) t t The second condition of symmetry is now also met...3 Integral Slot Base Windings For integral slot windings, t =. Hence, we obtain for normal oly-hase systems mt = = q N even. (.96) m

54 J. Pyrhönen, T. Jokinen, V. Hrabovcová.54 For a base winding of the first grade, we may write * =, * = =, t* =. (.97) Since also the integral slot windings of reduced oly-hase systems form base windings of the first grade, we can see that all integral slot windings are of the first grade, and that integral slot base windings comrise only a single ole air. The design of integral slot windings is therefore fairly easy. As we can see in Fig..7, the winding construction is reeated without changes always after one ole air. Thus, to create a comlete integral slot winding, we connect a sufficient number of base windings with a single ole air either in series or in arallel. EXAMPLE.6: Create a voltage hasor diagram of a single-layer integral slot winding, for which = 36, =, m = 3. SOLUTION: The number of slots er ole and hase is q = = 3. m A zone distribution, Fig..7, and a voltage hasor diagram, Fig..8 are constructed for the winding. U V W U -U +W -V +U -W +V -U +W -V +U V -W +V W τ Figure.7. Zone distribution for a single-layer winding. = 36, =, m = 3, q = 3.

55 J. Pyrhönen, T. Jokinen, V. Hrabovcová.55 -U α z = α u +V -W U α h +W V Figure.8. Comlete voltage hasor diagram for a single-layer winding. =, m = 3, = 36, q = 3, t =, ' = 8, α z = α u = 0. The second layer of the voltage hasor diagram reeats the first layer and it may, hence, be omitted. The base winding length is 8 slots. A double-layer integral slot winding is now easily constructed by selecting different hasors of the voltage hasor diagram of Fig..8 for instance for the uer layer. This way, we can immediately calculate the influence of different short-itchings. The voltage hasor diagram of Fig..8 is alicable to the definition of the winding factors of the short-itched coils of Fig..5. Only the zones labelled in the figure will change lace. Figure.8 is directly alicable to a full-itch winding of Fig..4.. Fractional Slot Windings.. Single-Layer Fractional Slot Windings Fractional slot windings with extremely small fractions are oular in brushless DC machines and ermanent magnet synchronous machines (PMSM). Machines oerating with sinusoidal voltages and currents are regarded as synchronous machines even though their air ga flux density might be rectangular. Figure.9 deicts the differences between single-layer and double-layer windings in a case where the ermanent magnet rotor has four oles and q = ½.

56 J. Pyrhönen, T. Jokinen, V. Hrabovcová.56 + U -W +U - W - U +W -V -U +V + W + V +V -U -V +W - V +U -W Figure.9. Comarison of a single-layer and a double-layer fractional slot winding with concentrated coils. s = 6, m = 3, =, q = ½. When the number of slots er ole and hase q of a fractional slot machine is larger than one, the coil grous of the winding have to be of the desired slot number q in average. In rincile, the zone distribution of the single-layer fractional slot windings is carried out either based on the voltage hasor diagram or the zone diagram. The use of a voltage hasor diagram has often roved to lead to an uneconomical distribution of coil grous, and therefore it is usually advisable to aly a zone diagram in the zone distribution. The average slot number er ole and er hase of a fractional slot winding is hence q, which is a fraction that gives the average number of slots er ole and hase q av. This kind of an average number of slots can naturally be realized only by varying the number of slots in different zones. The number of slots in a single zone is denoted by q k. Now qk qav = q N. (.98) q av has thus to be an average of the different values of q k. Now we write z' q = g +, (.99) n where g is an integer, and the quotient is indivisible so that z' < n. Now we have an average number of slots er ole and hase q = q av, when the width of z zones in n coil grous is set g + and the width of n z zones is g. ( g + ) + ( n z' ) n z' g z' qav = qk = = g + = q. (.00) n n n k= The divergences from a totally symmetrical winding are at smallest when the same number of slots er ole and hase occurs in consequent coil grous as seldom as ossible. The best fractional slot winding is found with n =, when the number of slots er ole and hase varies constantly when travelling from one zone to another. To fulfil Eq. (.00), at least n grous of coil are required. Now we obtain the required number of coils * q avnm = q * m =. (.0) This number corresonds to the size of a single-layer base winding. Now we have shown again that a base winding is the smallest indeendent winding for single-layer windings. When the second

57 J. Pyrhönen, T. Jokinen, V. Hrabovcová.57 condition of symmetry for fractional slot windings is considered, it makes no difference how the windings are distributed in the slots (n zones, q k coil sides in each), if only the desired average q av is reached (e.g gives an average of / 5 ). The nm coil grous of a base winding have to be distributed to m hase windings so that each hase gets n single values of q k (a local number of slots), in the same order in each hase. The coil numbers of consequent coil grous run through the single values of q k n times in m equal cycles. This way, a cycle of coil grous is generated. The first column of Table.8 shows m consequently numbered cycles of coil grous. The second column consists of nm coil grous in running order. The third column lists n single values of q k m times in running order. Because the consequent coil grous belong to consequent hases, we get a corresonding running hase cycle in the fourth column. During a single cycle, the adjacent, fifth column goes through all the hases U, V, W, m of the machine. The sixth column reeats the numbers of coil grous. Table.8. Order of coil grous for symmetrical single-layer fractional slot windings. Cycle of coil Phases grous from to m m Number of coil grou. This column runs m times from to n (the divider of the fraction q = z/n) Local number of slots er ole and hase (equals local number of slots er ole and hase q k ) Phase cycle, All the hases are introduced once (for a threehase system we have U, V, W) Number of coil grou U V 3 3 W 3 K q k m m U m+ V m+ W m+3 N q n n+ n+ n+k q k Dn q n d+ dn+ c c C m Cm dn+k q k c+ U cm+ V W m (m-)n+k q k m m m m m mn q n N m Nm EXAMPLE.7: Comare two single-layer windings, an integral slot winding and a fractional slot winding, having the same number of oles. The arameters for the integral slot winding are: = 36, =, m = 3, q = 3 and for the single-layer fractional slot winding = 30, =, m = 3, q = ½.

58 J. Pyrhönen, T. Jokinen, V. Hrabovcová.58 ( g + ) + ( n z' ) n z' g z' SOLUTION: For the fractional slot winding, qav = qk = = g + = q. We n k= n n see that in this case g =, z' =, n =. Now, a grou of coils with z' = is obtained, in which there are q = g + = 3 coils, and another n z' = grou of coils with q = g = coils. As = * = n =, we have here a base winding with three (m = 3) cycles of coil grous of both the coil numbers q and q. They occur in n = cycles of three hases. In Table.9, examle of Table.8 is alied to Fig..30. For the fractional slot winding q = z/n = 5/. Table.9. Examle of Table.8 alied to Fig..30. For the fractional slot winding q = z/n = 5/. Cycle of coil grous Number of coil grou Number of coils (q k ) Phase cycle Phase Number of coil grou 3 = q U (= n) = q V + = 3 3 W 3 + = 4 U 4 3 (= m) (+)+ = 5 3 V 5 3 (= m) (+)+ = 6 = nm = 3 W 6 = nm The above table can be resented simly as: q k hase U V W U V W Each hase is comrised of a single coil grou with two coils, and one coil grou with three coils. Figure.30 comares the above integral slot winding and a fractional slot winding. Fractional slot windings create more harmonics than integral slot windings. By dividing the ordinal number ν of the harmonics of a fractional slot winding by the number of ole airs *, we obtain ν ν ' =. (.0) * In integral slot windings, such relative ordinal numbers of the harmonics are the following odd integers: ν ' =, 3, 5, 7, 9,... For fractional slot windings, when ν =,, 3, 4, 5,... the relative ordinal number gets the values ν ' = /*, ν ' = /*, ν ' = 3/*,... in other words, values for which ν ' < ; ν ' N and ν ' N even. The lowest harmonic created by an integral slot winding is the fundamental ( ν ' = ), but a fractional slot winding can roduce also subharmonics ( ν ' < ). There occur also harmonics, the ordinal number of which is a fraction or an even integer. These harmonics cause additional forces, unintended torques and losses. These additional harmonics are the stronger, the greater is the zone variation, in other words, the divergence of the current linkage distribution from the resective distribution of an integral slot winding. In oly-hase windings, not all the integer harmonics are resent. For instance, of the sectrum of three-hase windings, those harmonics are absent, the ordinal number of which is divisible by three, because α str,ν = α str, = ν 360 /m = ν 0, and thus, because of the dislacement angle of the hase windings α str = 0, they do not create a voltage between different hases.

59 J. Pyrhönen, T. Jokinen, V. Hrabovcová.59 Θ subharmonic 0 τ τ 3τ 4τ -U +W -V +U -W +V -U +W -V +U -W +V Figure.30. Zone diagrams and current linkage distributions of two different windings (q = 3, q = ½). The integral slot winding is fully symmetrical, but the current linkage distribution of the fractional slot winding (dotted line) differs somewhat from the distribution of the integral slot winding (continuous line). The current linkage clearly contains a subharmonic, which has a double ole itch comared with the fundamental. EXAMPLE.8: Design a single-layer fractional slot winding of the first grade, for which = 68, = 0, m = 3. What is the winding factor of the fundamental? SOLUTION: The number of slots er ole and hase is 68 q = =, We have a fractional slot winding with n = 5 as a divider. The conditions of symmetry (Table.6) /n = 0/5 = 4 N and n/3 = 5/3 N are met. According to Table.7, when n is an odd number n = 5 N odd, a first-grade fractional slot winding is created. When t = /n = 4, its arameters are * = /t = 68/4 = 4, * = n = 5; t* =. * The diagram of coil grous, according to Eq. (.0), q avnm = q * m = consists of *m = nm = 5 3 = 5 grous of coils. The coil grous and the hase order are selected according to Table.8 q k hase U V W U V W U V W U V W U V W m = 3 cycles of coil grous with n = 5 consequent numbers of coils q k, yield a symmetrical distribution of coil grous for single hase coils q k q q q 3 q 4 q 5 q q q 3 q 4 q 5 q q q 3 q 4 q 5 hase U hase V hase W In each hase, there is one grou of coils q n. The average number of slots er ole and hase q av of the coil grou is written according to Eq. (.00) using the local q k value order of hase U

60 5 qk = ( ) = = q. 5 k = 5 5 J. Pyrhönen, T. Jokinen, V. Hrabovcová.60 We now obtain a coil grou diagram according to Fig..3 and a winding hasor diagram according to Fig..3. q k hase U V W U V W U V W U V W U V W Figure.3. Coil grou diagram of single-layer fractional slot winding. * = /t = 68/4 = 4, * = n = 5; t* =. When calculating the winding factor for the winding, the following arameters are obtained for the voltage hasor diagram: Number of layers in the voltage hasor diagram t* = Number of radii ' = */t* = 4 Slot angle α u = 360 */* = 360 5/4 = 4 6 / 7 Phasor angle α z = 360 t*/* = 360 /4 = 8 4 / 7 Number of hasors skied in the numbering (*/t*) = 5 = 4. Number of hasors for one hase Z = /m = 4/3 = 4 The voltage hasor diagram is illustrated in Fig..3. -W +V U U V +W a) b)

61 J. Pyrhönen, T. Jokinen, V. Hrabovcová.6 Figure.3 a) Voltage hasor diagram of a first-grade single-layer base winding. * = 5, m = 3 ' = * = 4, q = /5, t* =, α u = 5α z = 4 6 / 7, α z = 8 4 / 7 The hasors of the hase U are illustrated with a continuous line, b) the hasors of hase U turned to form a bunch of hasors for the winding factor calculation and the symmetry line. The winding factor may now be calculated using Eq. (.6) k wν νπ sin = Z Z ρ= cosα The number of hasors Z = 4 for one hase and the angle between the hasors in the bunch is α z = 8 4 / 7. The fundamental winding factor is found after having determined the angles α ρ between the hasors and the symmetry line, hence k ( cos( ) + cos( ) + cos( 8 4 ) + cos( 8 4 ) + ) 7 7 w = = As a result of the winding design based on the zone distribution given above, we have a winding in which, according to the voltage hasor diagram, certain coil sides are transferred to the zone of the neighbour hase. By exchanging the hasors 9 36, 5, and 8 33 we would also receive a functioning winding but there would be less similar coils than in the winding construction resented above. This kind of a winding would lead to a technically inferior solution, in which undivided and divided coil grous would occur side by side. Such winding solutions are favourable, in which the variation of coil arrangements is ket to a minimum. This way, the best shae of the end winding is achieved. EXAMPLE.9: Is it ossible to design a winding with a) = 7, = 5, m = 3, b) = 36, = 7, m = 3, c) = 4, = 3, m = 3? SOLUTION: a) Using Table.6, we check the conditions of symmetry for fractional slot windings. The number of slots er ole and hase is q = z/n = 7/( 5 3) = /5. z = and n = 5, which are not mutually divisible. As /n = 5/5 = N a single-layer winding should be made and as n/m = 5/3 N the symmetry conditions are OK. And as n N odd we will consider a first-grade base winding as follows: * = 7, * = 5, m = 3, q = /5. q k hase U V W U V W U V W U V W U V W ( ) = q q 3 =. This is a feasible winding. = av 5 5 SOLUTION: b) The number of slots er ole and hase is q = z/n = 36/( 7 3) = 6 /7. z = 6 and n = 7, which are not mutually divisible. As /n = 7/7 = N a single layer winding can be made, and as n/m = 7/3 N the symmetry conditions are OK. And as n N odd we will consider a first-grade base winding as follows: * = 36, * = 7, m = 3, q = 6 /7. q k hase U V W U V W U V W U V W U V W U V W U V W 6 ( ) = 7 q. av = 7 ρ

62 J. Pyrhönen, T. Jokinen, V. Hrabovcová.6 The number of slots er ole and hase can thus also be smaller than one q <. In such a case, there occur coil grous with no coils. These non-existent coil grous are naturally evenly distributed in all hases. SOLUTION: c) The number of slots er ole and hase is q = z/n = 4/( 3 3) = /3. z = 7 and n = 3, which are mutually divisible, the condition n/3 N is not met, and the winding is not symmetric. If we, desite the non symmetrical nature, considered a first-grade base winding, we should get a result as follows: * = 4, * = 3, m = 3, q = /3. q k hase U V W U V W U V W We can see that all coil grous with two coils now belong to the hase W. Such a winding is not functional. EXAMPLE.0: Create a winding with = 60, = 8, m = 3. SOLUTION: The number of slots er ole and hase is q = 60/( 8 3) = /4. z = 5, n = 4. The largest common divider of and is t = /n = 6/4 = 4. As t also indicates the number of layers in the hasor diagram we get ' = /t = 60/4 = 5 which is the number of radii in the hasor diagram in one layer. '/m = 5/3 = 5 N odd. The conditions of symmetry /n = 8/4 = N and n/3 = 4/3 N are met. Because n = 4 N even, we have according to the arameters in Table.7 a second-grade single-layer fractional slot winding. We get the base winding arameters * = /t = 60/4 = 30, * = n = 4, t* = The second-grade single-layer fractional slot windings are designed like the first-grade windings. However, the voltage hasor diagram is now doubled. The coil grou diagram of the base winding comrises * m = n m = 4 3 = coil grous. The coil grou hase diagram is selected as follows: q k hase U V W U V W U V W U V W A coil grou diagram for a base winding corresonding to this case is illustrated in Fig..33. q k hase U V W U V W U V W U V W U slots base winding base winding

63 J. Pyrhönen, T. Jokinen, V. Hrabovcová.63 Figure.33. Coil grou diagram of a base winding for a single-layer fractional slot winding = 8, m = 3, = 60, q = / 4. A voltage hasor diagram for the base winding is illustrated in Fig..34. Number of layers in the voltage vector diagram t* = (second-grade winding) Number of radii ' = */t* = 30/ = 5 Slot angle α u = 360 */* = 360 4/30 = 48 Phasor angle α z = 360 t*/* = 360 /30 = 4 Number of hasors skied in the numbering (*/t*) = 4/ =. +V -U W -W W U V a) b) Figure.34 a). Voltage hasor diagram of the base winding * = 4, * = 30, t* =, ' = 5, α u = α z = 48 of a singlelayer fractional slot winding =, m = 3, = 60, q = /4. The hasors belonging to the hase U are illustrated with a continuous line. b) The hasors of the hase U turned for calculating the winding factor and for illustrating a symmetrical bunch of hasors. The number of hasors Z = 0 for one hase and the angle between the hasors in the bunch is α z = 4. After having found the angles α ρ with resect to the symmetry line the fundamental winding factor may be calculated using Eq. (.6) ( cos( 6 ) + cos( 6 + ) + cos( )) k w = = Double-Layer Fractional Slot Windings In double-layer windings, one of the coil sides of each coil is in the uer layer of the slot, and the other coil side is in the bottom layer. The coils are all of equal san. Consequently, when the ositions of the left coil sides are defined, also the right sides will be defined. Here double-layer fractional slot windings differ from single-layer windings. Let us now assume that the left coil sides are ositioned in the uer layer. For these coil sides of the uer layer, a voltage hasor diagram of a double-layer winding is valid. Contrary to the voltage hasor diagram illustrated in Fig..34, there is only one layer in the voltage hasor diagram of the double-layer fractional slot winding. Therefore, the design of a symmetrical double-layer fractional slot winding is fairly straightforward

64 J. Pyrhönen, T. Jokinen, V. Hrabovcová.64 with a voltage hasor diagram of a base winding. Now, symmetrically distributed closed bunches of hasors are comosed of the hasors of single hases. This hasor order roduces the minimum divergence when comared with the current linkage distribution of the integral slot winding. First, we investigate first-grade double-layer fractional slot windings. It is ossible to divide the hasors of such a winding into bunches of equal size, in other words, into zones of equal width. EXAMPLE.: Design the winding reviously constructed as a single-layer winding = 68, = 0, m = 3, q = /5 now as a double layer winding. SOLUTION: We have a fractional slot winding for which the divider n = 5. The conditions of symmetry (Table.6) /n = 0/5 = 4 N and n/3 = 5/3 N are met. According to Table.7, if n is an odd number n = 5 N odd, a first-grade fractional slot winding is created. The arameters of the voltage hasor diagram of such a winding are: Number of layers in the voltage hasor diagram t* = Number of ole airs in the base winding * = 5 Number of radii ' = */t* = 4 Slot angle α u = 360 */* = 360 5/4 = 4 6 / 7 Phasor angle α z = 360 t*/* = 360 /4 = 8 4 / 7 Number of hasors skied in the numbering (*/t*) = 5 = 4. Since t* =, the number of radii ' is the same as the number of hasors *, and we obtain */m = 4/3 = 4 hasors for each hase, which then are divided into negative Z- and ositive Z + hasors. The number of hasors er hase in the first-grade base winding is */m = /mt N even. In normal cases, there is no zone variation, and the hasors are evenly divided into ositive and negative hasors. In the examle case, the number of hasors of both tyes is seven, Z- = Z + = 7. By emloying a normal zone order -U, +W, -V, +U, -W, +V we are able to divide the voltage hasor diagram into zones with seven hasors in each, Fig..35. When the voltage hasor diagram is ready, the uer layer of the winding is set. The ositions of the coil sides in the bottom layer are defined when an aroriate coil san is selected. For fractional slot windings, it is not ossible to construct a full-itch winding, because q N. For a winding in question, the full-itch coil san y of a full-itch winding would be y in slot itches y = y = mq = 3 = 4 N, 5 5 which is not ossible in ractice because the ste has, of course, to be an integer number of slot itches. Now the coil san may be decreased by y v = /5. The coil san thus becomes an integer, which enables the construction of the winding. y = mq y v = 3 = 4 N. 5 5 Double-layer fractional slot windings are thus short-itched windings.

65 J. Pyrhönen, T. Jokinen, V. Hrabovcová.65 -W α u α z +V U U V +W U U 5 U -U 5 5 U U 6 U -U 6 a) b) Figure.35 a) Voltage hasor diagram of a first-grade double-layer base winding * = 5, m = 3, ' = * = 4, q* = /5, t* =, α u = 5α z = 4 6 /7, α z = 8 4 / 7, b) A coule of examles of coil voltages in the hase U. When constructing a two-layer fractional slot winding, there are two coil sides in each slot. Hence, we have as many coils as slots in the winding. In this examle, we first locate the U-hase bottom coil side in slot. The other coil side is laced according to the coil san of y = 4 at a distance of four slots in the uer art of slot 5. Similarly, coils run from to 6. The coils to be formed are 5, 8, 35 39, 0 4, 7 3, 6, and 9 3. Starting from the +U zone, we have coils 6, 39, 4 8, 3 35, 6 0, 3 7, and 40. Now, six coil grous with one coil in each and four coil grous with two coils in each are created in each hase. The average is 4 q = ( ) = = A section of the base winding of the constructed winding is illustrated in Fig U...U Figure.36. Base winding of a fractional slot winding. = 0, m = 3, = 68, q = /5. The U end of the base winding is laced in the slot 40. Next, a configuration of a second-grade double-layer fractional slot winding is investigated. Because now '/m = */m = /mt N odd, a division Z- = Z + is not ossible. In other words, all the zones of the voltage hasor diagram are not equal. The voltage hasor diagram can nevertheless be constructed so that hasors of neighbour zones are not located inside the zones of each other. EXAMPLE.: Create a second-grade double-layer fractional slot winding with = 30, = 4, m = 3. SOLUTION: The number of slots er ole er hase is written

66 J. Pyrhönen, T. Jokinen, V. Hrabovcová q = = ½. 3 As n = N even, we have a second-grade double-layer fractional slot winding. Because t = /n =, its arameters are: * = /t = 30/ = 5, * = n/ = / =, t* =. This winding shows that a base winding of a second-grade double-layer fractional slot winding can only be of the length of one ole air. The arameters of the voltage hasor diagram are: Number of layers in the voltage hasor diagram t* = = * Number of radii ' = */t* = 5 Slot angle α u = 360 */* = 360 /5 = 4 Phasor angle α z = 360 t*/* = 360 / = 4 Number of hasors skied in the numbering (*/t*) = 0. For each hase, we obtain '/m = */m = 5/3 = 5 hasors. This does not allow an equal number of negative and ositive hasors. If a natural zone variation is emloyed, we have to set either Z + = Z - + or Z + = Z-. In the latter case, we obtain Z- = 3 and Z + =. With the known zone variation, electrical zones are created in the voltage hasor diagram, for which the number of hasors varies: Z- = 3 hasors of zone -U, Z+ = hasors of zone +W, Z- = 3 hasors of zone V, and so on, Fig Figure.37. Voltage hasor diagram of a second-grade double-layer fractional slot winding. * =, m = 3, q = ½, ' = */t* = 5, α u = 360 */* = 360 /5 = 4, α z = 360 t*/* = 360 /5 = 4, (*/t*) = W 4 +V -U 3 +W U -V When the coil san is decreased by y v = ½, the coil san becomes an integer y = mq y = 0½ ½ 0. v = The winding diagram of Fig..38 shows that all the ositive coil grous consist of three coils, and all the negative coil grous comrise two coils, which yields an average of q = ½. Since all negative and all ositive coil grous are comrised of an equal number of coils, resectively, the

67 J. Pyrhönen, T. Jokinen, V. Hrabovcová.67 winding can be constructed as a wave winding. A wave is created that asses through the winding three times in one direction and two times in the oosite direction. The waves are connected in series to create a comlete hase winding U a) U b) U U Figure.38. Winding diagram of a double-layer fractional slot winding. =, m = 3, = 30 q = ½. a) la winding, b) wave winding. To simlify the illustration, only a single hase is shown. EXAMPLE.3: Create a fractional slot winding for the three hase machine, where the number of slots is, and the number of rotor oles is 0, Fig..39. SOLUTION: The number of slots er ole and hase is q = /(3 0) = /5 = z/n = 0.4. Hence, n = 5. We should thereby find a base winding of the first kind. According to Table.6, /n N. In this case 5/5 N. In a three-hase machine n/m N n/3 N. Now 5/3 N and the symmetry conditions are met. Let us next consider Table.7 arameters. The largest common divider of and is t = /n = 5/5 =, /tm = /( 3) = 4 which is an even number. The slot angle in the voltage hasor diagram is π π 5π α u = nαz = n t, α u = 5α z = 5 = 6 The number slots in the base windings is /t = and the number of ole airs in the base winding is /t = n = 5. The winding may be realized either as a single- or double-layer winding, and in this case, a double-layer winding is found. In drawing the voltage hasor diagram, the number of hasors skied in the numbering is (*/t*) = ((/t)/t*) = ((5/)/) = 4.

68 J. Pyrhönen, T. Jokinen, V. Hrabovcová.68 -U -U +V 8 6 +W +V 3 -V -V -W +V +V -U +U +U -U +W 3 to layer bottom layer o α = 30 U 8 U 8 -U 7 +W 0 5 -V -V 7 +U +U 4 -W 9 -W 0 +W +W -W +U 9 -U -U 8 +U -V 7 -W -W 4 -V +W 5 +V +V 6 k o cos30 + = 4 w = -U 9 -U 3 -U U U -U a) b) c) Figure.39. a) Phasors of a -slot 0-ole machine, b) the double-layer winding of a -slot 0-ole machine, c) the hasors of the hase U for the calculation of the winding factor. First, hasors are drawn (a number of ', when ' = */t*). The hasor number is ositioned to oint straight uwards, and the next hasor, number, is located at an electrical angle of 360 / from the first hasor, in this case 360 5/ = 50 degrees. The hasor number 3 is, again, located at an angle of 50 degrees from the hasor etc. The first coil ( U, +U) will be located on the to layer of the slot and on the bottom layer of the slot. The other coil (+U, U) 3 will be located on the to layer of the slot and on the bottom layer of the slot 3. The hase coils are set in the order U, V, W, U, V, W. In the examle, a single hase zone comrises four slots, and thus a single winding zone includes two ositive and two negative slots. Based on the voltage hasor diagram of.39a and the winding construction of.39b, the fundamental winding factor of the machine can be solved, Fig..39c. First, the olarity of the coils of hase U in Fig..39b are checked and the resective hasors are drawn. In the slots,, and 3, there are four coil sides of the hase U in total, and the number of hasors will thus be four. Now, the angles between the hasors and their cosines are calculated. This yields a winding factor of EXAMPLE.4: Create a fractional slot winding for the three-hase machine, in which the number of slots is, and the number of rotor oles is, Fig..40. SOLUTION: The number of slots er ole and hase is thus only q = /66 = z/n = 7/ = As n N even, we have a fractional slot winding of the second grade. Although a winding of this kind meets the symmetry conditions, it is not an ideal construction, because in the winding, all the coils of a single hase are located in the same side of the machine. Such a coil system may roduce harmful unbalanced magnetic forces in the machine. Figure.40a. hasors are drawn (a number of, when = */t*). The hasor number is laced at the to and the next hasor at the distance of 360 / from it. In this examle, the distance is thus 360 / = 88.6 degrees. The hasor number is thus set at an angle of 88.6 degrees from the hasor. The rocedure is reeated with the hasors 3, 4,... etc. The hase coils are set in the order W, U, -V, W, -U, V. Here a single hase consists of seven slots, and therefore we cannot lace an equal number of ositive and negative coils in one hase. In one hase, there are four ositive and three negative slots. Note that we are now generating just the to winding layer, and when also the bottom winding is inserted, we have an equal number of ositive and negative coils.

69 J. Pyrhönen, T. Jokinen, V. Hrabovcová.69 +U -U +U +V -V -U +U -U +V 0 -U -V +U -V 3 +U +V 9 4 +V -U 8 -U 5 -V -V 7 6 +U +U 6 +V +V 7 -U +W 5 8 -V 4 9 -W -V -W -W 3 0 +W +V +W -W +W -W +W +W -W -W +W +V -U -U +V 0 3 -U 8 +V 5 -U 6 7 -W 4 9 +W -W +W -W 0 3 +W 8 5 -W 6 7 -V +U 4 9 -V +U +U -V -V k w = a) b) c) Figure.40. a) The winding of a -slot -ole machine, b) Phasors of a -slot -ole machine, and c) The hasors of the hase U for the calculation of the winding factor. Figure.40 b. The coils are inserted in the bottom layer of the slots in Fig..40 b) according to the hasors of Fig..40 a). The hasor number of Fig..40 a) is U, and it is located in the to layer of the slot. Corresondingly, the hasor, +U, is mounted in the to layer of the slot. The bottom winding of the machine reeats the order of the to winding. When the to coil sides are transferred by a distance of one slot forward and the ± sign of each is changed, a suitable bottom layer is obtained. The first coil of the hase U will be located in the bottom of the slot and on the surface of the slot, and so on. Table.0 introduces some arameters of double-layer fractional slot windings, when the number of slots q 0.5 (Salminen 004). Table.0 Winding factors k w of the fundamental and numbers of slots er ole and hase q for double-layer threehase fractional slot concentrated windings (q 0.5). The boldface figures are the highest values in each column. Reroduced by ermission of Pia Salminen. s Number of oles k 6 w ** ** q k 9 w * * q k w ** q k 5 w ** * * q k 8 w q k w q k 4 w q 0.5 * not recommended as single base winding because of unbalanced magnetic ull * * ** not recommended, because the denominator n (q = z/n) is an integral multile of the number of hases m. ** ** ** ** **

70 .3 Single- and Two-Phase Windings J. Pyrhönen, T. Jokinen, V. Hrabovcová.70 The above three-hase windings are the most common rotating-field windings emloyed in olyhase machines. Double- and single-hase windings, windings ermitting a varying number of oles, and naturally also commutator windings are common in machine construction. Of commutator AC machines, nowadays only single-hase sulied series-connected commutator machines occur for instance as motors of electric tools. Poly-hase commutator AC machines will eventually disaear as the ower electronics enables an easy rotation seed control of different motor tyes. Since there is no two-hase suly network, two-hase windings occur mainly as auxiliary and main windings of machines sulied from a single-hase network. In some secial cases, for instance small auxiliary automotive drives such as fan drives, two-hase motors are also used in ower electronic suly. A two-hase winding can also be constructed on the rotor of low-ower sli-ring asynchronous motors. As known, a two-hase system is the simlest ossible winding that roduces a rotating field, and it is therefore most alicable to rotating-field machines. In a two-hase suly, however, there exist time instants when the current of either of the windings is zero. This means that each of the windings should alone be caable of creating as sinusoidal a suly as ossible to achieve low harmonic content in the air ga and low losses in the rotor. This makes the design of high-efficiency two-hase winding machines more demanding than three-hase machines. The design of a two-hase winding is based on the rinciles already discussed in the design of three-hase windings. However, we must always bear in mind that in the case of a reduced olyhase system, when constructing the zone distribution, the signs of the zones do not vary in the way they do in a three-hase system, but the zone distribution will be -U, -V, +U, +V. In a single-hase asynchronous machine, the number of coils of the main winding is usually higher than the number of coils of the auxiliary winding. EXAMPLE.5: Create a 5/6 short-itched double-layer two-hase winding of a small electrical machine, =, =, m =, q = 3. SOLUTION: The required winding is illustrated in Fig..4, where the rules mentioned above are alied U V U V Figure.4. Symmetrical 5/6 short-itched double-layer two-hase winding, =, =, m =, q = 3.

71 J. Pyrhönen, T. Jokinen, V. Hrabovcová.7 When considering a single-hase winding, we must bear in mind that it does not, as a stationary winding, roduce a rotating field, but a ulsating field. A ulsating field can be resented as a sum of two fields rotating in oosite directions. The armature reaction of a single-hase machine has thus a field comonent rotating against the rotor. In synchronous machines, this comonent can be damed with the damer windings of the rotor. However, the damer winding coer losses are significant. In single-hase squirrel cage induction motors, the rotor also creates extra losses when daming the negative-sequence field. Also the magnetizing windings of the rotors of non-salient ole machines belong to the grou of single-hase windings, as exemlified at the beginning of the chater. If a single-hase winding is installed on the rotating art of the machine it, of course, creates a rotating field in the air ga of the machine contrary to the ulsating field of a single-hase stator winding. Large single-hase machines are rare, but for instance in Germany, single-hase synchronous machines are used to feed the suly network of 6 /3 Hz electric locomotives. Since there is only one hase in such a machine, there are only two zones er ole air, and the construction of an integral slot winding is usually relatively simle. In these machines, damer windings have to cancel the negative sequence field. This, however, obviously is roblematic because lots of losses are generated in the damer. The core rincile also in designing single-hase windings is to aim at as sinusoidal distribution of the current linkage as ossible. This is even more imortant in the single-hase windings than in the three-hase windings, the current linkage distribution of which is by nature closer to ideal. The current linkage distribution of a single-hase winding can be made to resemble the current linkage distribution of a three-hase winding instantaneously in a osition where a current of one hase of a three-hase winding is zero. At that instant, a third of the slots of the machine are in rincile currentless. The current linkage distribution of a single-hase machine can best be made to aroach a sinusoidal distribution when a third of the slots are left without conductors, and a different number of turns of coil are inserted in each slot. The magnetizing winding of a non-salient ole machine of Fig..3 is illustrated as an examle of such a winding. EXAMPLE.6: Create various kinds of zone distributions to aroach a sinusoidal current linkage distribution for a single-hase winding with m =, =, = 4, q =. SOLUTION: Figure.4 deicts various methods to roduce a current linkage waveform with a single-hase winding. current linkage c) a) b) a) -U +U -U b) c) -U +W -V +U -W +V Figure.4. Zone diagram of a single-hase winding =, = 4, q =, and current linkage distributions roduced by different zone distributions. a) A single-hase winding covering all slots. b) A /3 winding, with the zones of a corresonding three-hase winding. The three hase zones +W and -W are left without conductors. The distribution of

72 J. Pyrhönen, T. Jokinen, V. Hrabovcová.7 the current linkage is better than in the case a). c) A /3 short-itched winding, roducing a current linkage distribution closer to an ideal (dark steed line)..4 Windings Permitting a Varying Number of Poles Here, windings ermitting a variable ole number refer to such single- or oly-hase windings that can be connected via terminals to a varying number of oles. Windings of this tye occur tyically in asynchronous machines, when the rotation seed of the machine has to be varied in a certain ratio. The most common examle of such machines is a two-seed motor. The most common connection for this kind of an arrangement is a Lindström-Dahlander connection that enables the alteration of the ole number of a three-hase machine in the roortion of to. Figure.43 illustrates the winding diagram of a single hase of a 4-slot machine. It can be arranged as a double-layer diamond winding, which is a tyical Dahlander winding. Now the smaller number of ole airs is denoted ' and the higher '', where '' = '. The winding is divided into two sections U U and U 3 U 4, both of which consist of two coil grous with the higher number of oles. The Dahlander winding is normally realized for the higher ole air number as a double-layer double zone width winding. The number of coil grous er hase is equal to ", which is always an even number. Deriving a double-layer integral slot full-itch (short-itch or fractional slot Dahlander windings are not ossible at all) Dahlander winding starts with creating m" negative zones with a double width. All the negative zones U, -V, -W are located in the to layer and all the ositive zones in the bottom layers of slots. U4 V V a) b) U U U0 W4 W0 V0 V4 W W U V W U3 U0 U U4 U4 V c) d) W U0 V0 V U W0 V4 W4 U W U V U3 W U0 U U4 flux going flux coming +W +U +V +W +U +V +W +U +V +W +U +V +W +U e) zone lan for " = 4 -V -W -U -V -W -U -V -W -U -V -W -U -V -W +U -W +V -U +W -V +U -W +V -U +W -V f) zone lan for ' = +V -U +W -V +U -W +V -U +W -V +U -W

73 J. Pyrhönen, T. Jokinen, V. Hrabovcová.73 Figure.43. Princile of a Dahlander winding. The uer connection a) roduces eight oles and the lower c) four oles. The number of oles is shown by the flux arrowheads and tails b) and d) equivalent connections. When also the number of the coil turns of the hase varies inversely roortional to the seed, the winding can be sulied with the same voltage at both seeds. Network connections are U, V, and W. The figure illustrates only a winding of one hase. In Fig..43d between U and U4 there is the connection W, between W4 and W there is U, and between V and V4 there is V to kee the same direction of rotation, e) and f) zone lans for " = 4 and ' =. The hase U is followed by the winding V at a distance of 0. For a number of ole airs '' = 4, the winding V has to be laced at a distance 4 τ u = τ u = τ u 3' ' 3 4 from the winding U. The winding V starts thus from the slot 3 in the same way as the winding U starts from the slot. The winding W starts then from the slot 5. When considering the ole air number ' =, we can see that the winding V is laced at a distance 4 τ u = τ u = 4τ u 3' 3 from the winding U and starts thus from the slot 5, and the winding W from the slot 9. External connections have to be arranged to meet these requirements. At simlest, the shift of the abovementioned ole air from one winding to another is carried out according to the right-hand circuit diagrams. To kee the machine rotating to the same direction, the hases U, V, and W have to be connected according to the illustration. There is also another method to create windings with two different ole numbers. Pole-Amlitude- Modulation (PAM) is a method with which even other ratios than : may be found. PAM is based on the following trigonometric equation sin bα sin mα = ( cos( b m ) α cos( b + m ) α ). (.03) The current linkage is roduced as a function of the angle α running in the erimeter of the air ga. A hase winding might be realized with a base ole air number b and a modulating ole air number m. In ractice, this means that if for instance b = 4 and m =, the PAM method roduces ole airs 4 or 4 +. The winding must be created so that one of the harmonics is damed and the other dominates..5 Commutator Windings A characteristic of oly-hase windings is that the hase windings are, in rincile, galvanically searated. The hase windings are connected via terminals to each other, in star or in a olygon. The armature winding of commutator machines does not start nor ends at terminals. The winding is comrised of turns of conductor soldered as a continuum and wound in the slots of the rotor so that the sum of induced voltages is always zero in the continuum. This is ossible if the sum of slot voltages is zero. All the coil sides of such a winding can be connected in series to form a continuum without causing a current flow in the closed ring as a result of the voltages in the coil sides. An external electric circuit is created by couling the connection oints of the coils to the commutator segments. A current is fed to the winding via brushes dragging along the commutator. The commutator switches the coils in turns to the brushes acting thus as a mechanical inverter or

74 J. Pyrhönen, T. Jokinen, V. Hrabovcová.74 rectifier deending on the oerating mode of the machine. This is called commutating. In the design of a winding, the construction of a reliable commutating arrangement is a demanding task. Commutator windings are always double-layer windings. One coil side of each coil is always in the uer layer and the other in the bottom layer aroximately at a distance of a ole air from each other. Because of roblems in commutating, the voltage difference between the commutator segments must not be too high, and thus the number of segments and coils has always to be high enough. On the other hand, the number of slots is restricted by the minimum width of the teeth. Therefore, usually more than two coil sides are laced in each slot. In the slot of the uer diagram of Fig..44, there are two coil sides, and in the lower diagram, the number of coil sides is four. The coil sides are often numbered so that the sides of the bottom layer are even numbers, and the slots of the uer layer are odd numbers. If the number of coils is z c, z c coil sides have to be mounted in slots, and thus there are u = z c / sides in a slot. The symbol u gives the number of coil sides in one layer. In each side, there are N v conductors. The total number of conductors z in the armature is Here z u z c z = z = un = z N. (.04) v is the number of slots, is the number of conductors in a slot, is the number of coil sides in a layer, is the number of coils N v, number of conductors in a coil side, un v = z, because c v z un v z = = = un v, see (.04) a) 4 Figure.44. Two examles of commutator winding coil sides mounted in the slots. a) Two coil sides in a slot, one side in a layer, u = b) four coil sides in a slot, two coil sides in a layer, u =. Even-numbered coil sides are located at the bottom of the slots. There has to be a large enough number of coils and commutator segments to kee the voltage between commutator segments small enough. b) Commutator windings may be used both in AC and DC machines. Multi-hase commutator AC machines are, however, becoming rare. DC machines, instead, are built and used also in the resentday industry even though DC drives are gradually relaced by ower electronic AC drives. Nevertheless, it is advisable to briefly look also at the DC windings. The AC and DC commutator windings are in rincile equal. For simlicity, the configuration of the winding is investigated with a voltage hasor diagram of a DC machine. Here, it suffices to investigate a two-ole machine, since the winding of machines with multile oles is reeated unchanged with each ole air. The rotor of Figure.45, with = 6, u =, is assumed to rotate clockwise at an angular seed Ω in a constant magnetic field between the oles N and S. The magnetic field rotates to the ositive direction with resect to the conductors in the slots, that is, counterclockwise. Now, a coil voltage hasor diagram is constructed for a winding, in which we

75 J. Pyrhönen, T. Jokinen, V. Hrabovcová.75 have already calculated the difference of the coil side voltages given by the coil voltage hasor diagram. By alying the numbering system of Fig..44, we have in the slot the coil sides and 3, and in the slot 9 the coil sides 6 and 7. With this system, the coil voltage hasor diagram can be illustrated as in Fig..45b. N q Ω S d a) b) Figure.45. a) Princile of a two-ole double-layer commutator armature. The armature rotates at an angular seed Ω clockwise generating an emf to the conductors in the slots. The emf tends to create the current directions illustrated in the figure. b) A coil voltage hasor diagram of the armature. It is a full-itch winding, which does not normally occur as a commutator winding. Nevertheless, a full-itch winding is here given as a clarifying examle. = 6, u = (one coil side er layer) ω = Ω Figure.45 shows that if the induced emf decides the direction of the armature current, the roduced torque is oosite to the direction of rotation (counterclockwise in Fig..45), and mechanical ower has to be sulied to the machine, which is acting as a generator. Now, if the armature current is forced to flow against the emf with the assistance of an external voltage or current source, the torque is to the direction of rotation, and the machine acts as a motor. There are z c = u = 6 = 6 coils in the winding, the ends of which should next be connected to the commutator. Deending on the way they are connected, different kinds of windings are roduced. Each connection oint of the coil ends is connected to the commutator. There are two main tyes of commutator winding s, viz. la windings and wave windings. A la winding has coils, creating loo-like atterns. The ends of coils are connected to adjacent commutator segments. A wave winding has a wave-like drawing-attern when resented in a lane. The number of commutator segments is given by K = u, (.05) because each coil side begins and ends at the commutator segment. The number of commutator segments, therefore, deends on the conductor arrangement in the slot, eventually on the number of coil sides in one layer. Further imortant arameters of commutator windings are: y y coil san exressed in the number of slots er ole back end connector itch which is a coil san exressed in the number of coil sides. For the winding the coil sides of which are numbered with odd figures in the to layer and with even figures in the bottom layer is: where y = uy, (.06) m

76 J. Pyrhönen, T. Jokinen, V. Hrabovcová.76 sign stands for a coil side numbering as seen in Fig..44, and the + sign for a numbering, where in the slot there are coil sides,, in the slot, coil sides 3, 4, etc, if u =, or in the to layer of the slot coil sides, 3, and in the bottom layer,4, etc., if u =. y y y c front end connector itch; it is a itch exressed in the number of coil sides between the right coil side of one coil and the left coil side of the next coil. total winding itch exressed in the number of coil sides between two left coil sides of the two adjacent coils commutator itch between the beginning and end of one coil exressed in the number of commutator segments. The equation for the commutator itch is a basic equation for the winding design because this itch must be an integer nk ± a yc =, (.07) where a is the number of arallel aths er half armature in a commutator winding, which means a arallel aths for the whole armature. The most often emloyed windings are characterized on the basis of n: a ) If n = 0, it results in a la winding. The commutator itch will be: y c = ±, which means that a is an integer multile of to get an integer for the commutating itch. If for a la winding a =, this means a =, y c = ±. Such a winding is called a arallel one. The ositive sign is for a rogressive winding, moving from left to right, and the negative sign for a retrogressive winding, moving from right to left. If a is a k-multile of the ole air number, a = k, then it is a k-multilex arallel winding. For examle for a =, the commutator itch is y c = ±, and this winding is called a dulex arallel winding. ) If n =, it results in a wave winding and a commutator itch K ± a u ± a yc = = (.08) must be an integer. The ositive sign is for rogressive and the negative sign for retrogressive winding. In the wave winding the number of arallel aths is always, there is only one air of arallel aths, irresective of the number of oles: a =, a =. Not all the combinations of K, a, result in an integer. It is a designer s task to choose a roer number of slots, coil sides, number of oles, and tye of winding to ensure an integer commutator itch. If the number of coils equals the number of commutator segments, then, if the coil sides are numbered with odd figures in the to layer and even figures in the bottom layer, we may write: y = = y + y yc. (.09) Therefore, if the commutator itch is determined, the total itch exressed as a number of coil sides is given

77 J. Pyrhönen, T. Jokinen, V. Hrabovcová.77 y = y c (.0) and after y is determined based on the numbers of slots er ole y and number of coil sides in a layer u: y, (.) y = u y. (.) m The front end connector itch can be determined as: y = y. (.3) y.5. La Winding Princiles The rinciles of the la winding can best be exlained by an examle: EXAMPLE.7: Make a layout of a la winding for a two-ole DC machine with 6 slots and one coil side in a layer. SOLUTION: Given: = 6, =, u =, the number of commutator segments K is a K = u = 6 = 6, and for a la winding a = =. The commutator itch is y c = ± = ±. We choose a rogressive winding, which means that y c = + (The winding roceeds from left to right), and the total itch is y = y c =. The coil san y in the number of slots is given by the number of slots er ole: y 6 = = = 8. The same itch exressed in the number of coil sides is: y = uy = 8 = 5. The front end connector itch is: y = y y = 5 = 3, which is illustrated in Fig..46. The coils can be connected in series in the same order they are inserted in the slots of the rotor. The neighbouring coils are connected together, the coil 6 to the coil 3 8, this again to 5 0 and so on. This yields a winding diagram of Fig..46.

78 J. Pyrhönen, T. Jokinen, V. Hrabovcová.78 d q d y = 5 y = y = 3 S N H I R H Figure.46. Diagram of a full-itch la winding. The winding is connected via brushes to an external resistance R. The ole shoes are also illustrated above the winding. In reality, they are laced above the coil sides. The las illustrated with a thick line have been short-circuited via the brushes during commutation. The direction of current changes during the commutation. The numbers of slots ( 6) are given. The numbers of coil sides ( 3) in the slots are also given, the coil sides and 3 are located in the slot and e.g. the coil sides 8, 9 are located in the slot 5. The commutator segments are numbered ( 6) according to the slots. It is said that the brushes are on the quadrature axis; this is nevertheless valid only magnetically. In this figure, the brushes are hysically laced close to the direct axes. When we follow the winding by starting from the coil side, we can see that it roceeds one ste of san y = 5 coil sides. In the slots and 9, a coil with a large enough number of winding turns is inserted. Finally, after the last coil turn the winding returns left a distance of one ste of connection y = 5= 3 coil sides, to the uer coil side 3. We continue this way until the comlete winding has been gone through, and the coil side 4 is connected to the uer coil side through the commutator segment. The winding has now been closed as a continuum. The winding roceeds in las from left to right; hence the name la winding. The itches of a commutator winding are thus calculated by the number of coil sides, not by the number of slots, because there can be more than two coil sides in one layer, for instance four coil sides in a slot layer (u =,, 3, 4 ). In the la winding of Fig..46, all the coil voltages are connected in series. The connection in series can be illustrated by constructing a olygon of the coil voltages, Fig..47. The figure illustrates the hasors at time t = 0 in the coil voltage hasor diagram of Fig..45. When the rotor rotates at an angular seed Ω, also the coil voltage hasor diagram rotates in a two-ole machine at an angular seed ω = Ω. Also the olygon rotates around its centre at the same angular seed. The real instantaneous value of each coil voltage may be found as a rojection of the hasor on the real axis (see figure). According to the figure, the sum of all coil voltages is zero. Therefore, no circulating currents occur in the continuum.

79 J. Pyrhönen, T. Jokinen, V. Hrabovcová Figure.47. Polygon of coil voltages of the winding in Fig..46. The sum of all the voltages is zero and hence the coils may be connected in series. The instantaneous value of a coil voltage u(t) will be the rojection of the hasor on the real axis, e.g. u(t) u(t) H 9- A H 9-4 Re The highest value of the sum of the instantaneous values of coil voltages is equal to the diameter H H arallel to the real axis. This value remains almost constant as the olygon rotates, and thus the hasor H H reresents a DC voltage without significant rile. The voltage aroaches a constant value, when the number of coils aroaches infinite. A DC voltage can be connected to an external electric circuit via the brushes that are in contact with the commutator segments. At the moment t = 0, as illustrated, the brushes have to be in touch with the commutator segment airs 5 6 and 3 4, that are connected to coils 9 4 and 5 8. According to Fig..46, the magnetic south ole (S) is at the slot and magnetic north ole (N) at the slot 9. According to Fig..46, the direction of the magnetic flux is towards the observer at the south ole, and away from the observer at the north ole. As the winding moves left, a ositive emf is induced to the conductors under the south ole, and a negative emf under the north ole, to the direction ointed by the arrows. By following the las from the segment 4 to the coil 7 0, we end u at the commutator segment 5, then gradually to the segments 6,,, 3, 4, and at last the coil 7, is brought to the segment 5, touched by the brush H. We have just described one arallel ath created by coils connected in series via commutator segments. An induced emf creates a current in the external art of the electric circuit from the brush H to the brush H, and thus in generator drive, the H is a ositive brush with the given direction of rotation. A half of the current I in the external art of the circuit flows the above-described ath, and the other half via the coils , via commutator segments,, 7 to the brush H and further to the external art of the circuit. In other words, there are two arallel aths in the winding. In the windings of large machines, there can be several airs of aths in order to revent the cross-sectional area of the conductors from increasing imractically. Because the ends of different airs of aths touch the neighbouring commutator segments and they have no other galvanic contact, the brushes have to be made wider to kee each air of aths always in contact to the external circuit. If for instance in the coil voltage hasor diagram of Fig..45 every other coil 6, 5 0, is connected in series with the first air of aths, the la is closed after the last turn of coil side by connecting the coil to the first coil 6 (, from to ). The coils that remain free are connected in the order 3 8, , and the la is closed at the osition 4 3. This way, a doubly-closed winding with two aths a = is roduced. In the voltage olygon, there are two revolutions, and its diameter, that is, the brush voltage is reduced to a half of the original olygon of one revolution illustrated in Fig..47. The outut ower of the system remains the same, because the current can be doubled when the voltage is cut into half. In general, the number of airs of aths a always requires that a hasors are left between the hasors of series-connected coils in a coil voltage hasor diagram. The hasors may be similar. Because u is the number of coil sides

80 J. Pyrhönen, T. Jokinen, V. Hrabovcová.80 er layer, each hasor of the coil voltage hasor diagram reresents u coil voltages. This makes it ossible to ski comletely similar voltage hasors. This takes lace for instance when u =. The winding of Fig..46 is wound clockwise, because the voltages of the coil voltage hasor diagram are connected in series clockwise starting from the hasor 6. Were the coil 6 connected via the commutator segment 6 to the coil 3 4, the winding would have been wound counterclockwise. The number of brushes in a la winding is always the same as the number of oles. The brushes of the same sign are connected together. According to Fig..46, the brushes always short-circuit those coils, the coil sides of which are located at the quadrature axis (in the middle, between two stator oles) of the stator, where the magnetic flux density created by the ole magnetizing is zero. This situation is also described by stating that the brushes are located at the quadrature axis of the stator indeendent of the real hysical osition of the brushes..5. Wave Winding Princiles The winding of Fig..46 can be turned into a wave winding by bending the coil ends of the commutator side according to the illustration of Fig..48, as a solution of the Examle.8 (see also Fig..49). EXAMPLE.8: Make a layout of a wave winding for a two-ole DC machine with 6 slots and one coil side in a layer. SOLUTION: It is given that = 6, =, u =. The number of commutator segments is K = u = 6 = 6, and for a wave winding a =. The commutator itch is K ± a 6 ± y c = = = 7, or 5. We choose y c = + 5 (winding roceeds from right to left), and the total winding itch is y = y c = 30. The coil san y in the number of slots is given by the number 6 of slots er ole: y = = = 8. The same itch exressed in the number of coil sides is: y = uy = 8 = 5. The front end connector itch is: y = y y = 30 5 = 5, which is shown in Fig..48.

81 J. Pyrhönen, T. Jokinen, V. Hrabovcová.8 y = 5 y = 30 y = 5 y = 5 S N H H I R Figure.48. Full-itch la winding of Fig..46 turned into a wave winding. The currents of waveforms indicated with a thick line are commutating at the moment illustrated in the figure. The commutator itch is y c = 5, which means almost two ole itches. For instance the wave coil that starts at the commutator segment 4 ends at the segment 3, because 4 + y c = = 9, but there are only 6 commutator segments, and therefore 9 6 = 3. In the above wave winding, the uer coil side is connected to the commutator segment 0, and not to the segment as in the la winding. From the segment 0, the winding roceeds to the bottom side 8. The winding receives thus a waveform. In the figure, the winding roceeds from the right to the left, and counterclockwise in the coil voltage hasor diagram. The winding is thus rotated to the left. If the winding were turned to the right, the commutator itch would be y c = 7, y = 34, y = 5, y = 9. The coil from the bottom side 6 would have to be bent to the right to the segment 0, and further to the uer side 3, because 6 + y = = 35. But there are only 3 coil sides, and therefore the coil will roceed to 35 3 = 3 rd coil side. The commutator ends would in that case be even longer, which would be of no use. The itch of the winding for a wave winding follows the illustration y = y + y. (.4) The osition of brushes in a wave winding is solved in the same way as in a la winding. When comaring the la and wave windings, we can see that the brushes short-circuit the same coils in both cases. The differences between the windings are merely structural, and the winding tye is selected basically on these structural grounds. As it was written above, the itch of winding for regular commutator windings, which equals commutator itch is obtained from nk ± a nu ± a yc = =, (.5) + in the equation is used for rogressive la winding and retrogressive wave winding, and is used for retrogressive la and rogressive wave winding, n is zero or a ositive integer. If n is zero, it results in a la winding, if n =, it results in a wave winding. The commutator itch y c must be an

82 J. Pyrhönen, T. Jokinen, V. Hrabovcová.8 integer, otherwise the winding cannot be constructed. Not all combinations of K,, and a result in y c as an integer, and therefore a designer must solve this roblem in its comlexity..5.3 Commutator Winding Examles, Balancing Connectors Nowadays, the conductors are tyically inserted in slots that are on the surfaces of the armature. These windings are called drum armature windings. Drum-wound armature windings are in ractice always double-layer windings, in which there are two coil sides in the slot on to of each other. Drum armature windings are constructed either as la or wave windings. As discussed reviously, the term la winding describes a winding that is wound in las along the erihery of the armature, the ends of one coil being connected to adjacent segments, Fig..49a. EXAMPLE.9: Make a layout of the la winding for a four-ole DC machine with 3 slots and one coil side in a layer. SOLUTION: It is given that = 3, = 4, u =, number of commutator segment is a K = u = 3 = 3, and for la winding a = = 4. The commutator itch is y c = ± = = ±. We choose a rogressive winding, which means that y c = + (winding roceeds from left to right), and the total winding itch is y = yc =. The coil san y in the number of slots is given by the 3 number of slots er ole: y = = = slots. The same itch exressed in the number 4 of coil sides is: y = uy m = 6 + = 3. The negative sign is used because of the coil side arrangement in the slots according to Fig..44a. The front end connector itch is: y = y y = 3 =, which is shown in Fig..49a. In the winding of Fig..49a, there are 3 armature coils (46 coil sides, two in each slot), with one turn in each, four brushes, and a commutator with 3 segments. There are four current aths in the winding (a = 4), which thereby requires four brushes. By following the winding starting from the first brush, we have to travel a fourth of the total winding to reach the next brush of oosite sign. From the segment, the left coil side is ut to the uer layer number in the slot (see Fig..44a). Then, the right side is ut to the bottom layer +y = + = in the slot 7, from where it is led to the segment number + y c = + =. From the segment to the uer layer 3 in the slot, because 9 = 3. Then it roceeds to the bottom layer 4 in the slot 8, because 3 + = 4, and. then to the segment 3, and continuing to the coil side 5 in the slot 3 and via 6 in the slot 9 to the segment 4, and so on. In the figure, the brushes are broader than the segments of the commutator, the las illustrated with thick lines being short-circuited via the brushes. In DC machines, a roer commutating requires that the brushes cover several segments. The coil sides of short-circuited coils are aroximately in the middle between the oles, where the flux density is small. In these coils, the induced voltage is low, and the created short-circuit current is thus insignificant. EXAMPLE.30: Make a layout of a wave winding for a four-ole DC machine with 3 slots and one coil side in a layer.

83 J. Pyrhönen, T. Jokinen, V. Hrabovcová.83 SOLUTION: It is given that = 3, = 4, u =. The number of commutator segments is K = u = 3 = 3, and for a wave winding a =. The commutator itch is K ± a 3 ± y c = = =, or. We choose y c = + and the total winding itch is y = y c = 4. The coil san y in the number of slots is given by the number of slots er ole: 3 y = = = The same itch exressed in the number of coil sides is: 4 y = uy = 6 + = 3. The negative sign is used because of the coil side arrangement in the slots according to Fig..44a. The front end connector itch is: y = y y = 4 3 =, which is shown in Fig..49b. Fig.49b illustrates the same winding as in Fig..49a develoed to a wave winding. In wave windings, there are only two current aths, a = regardless of the number of oles. A wave winding and a la winding can also be combined as a frog-leg winding. We can see in Fig..49b that from the commutator segment the coil left side is ut to the uer layer number 3 in the slot 7. Then the right side is in the lower layer 3 + y = = 6 in the slot 3, from where it is led to the segment number + y c = + = 3. From the segment 3 to the uer layer 37 (slot 9), because 4 + y = = 37. Then we roceed to the lower layer in the slot, because 37 + y = 37 + = 48, which is over the number coil sides 46 in 3 slots; therefore, it is necessary to make a correction 48 ( 3) = =. From here we continue to the segment, because 3 + y c = 3 + = 5, after the correction 5 3 =, and so on.

84 J. Pyrhönen, T. Jokinen, V. Hrabovcová.84 y = y = y = 9 N S N S H H H3 H a) + - y = 3 + = 4 y =3 y = N S N S b) 3 H H H3 H y c = Figure.49 a) Four-ole double-layer la winding resented in a lane. The winding moves from left to right and acts as a generator. The coils belonging to the commutator circuit are illustrated with a thick line. This winding is not a fullitch winding unlike the revious ones. The illustrated winding commutates better than a full-itch winding. b) The same winding develoed to a wave winding. The wave under commutation is drawn with a thicker line than the others. When assing through a wave winding from one brush to another brush of the oosite sign, a half of the winding and a half of the segments of the commutator are gone through. The current has thus only two aths irresective of the number of oles. As a matter of fact, in a wave winding, only one air of brushes is required, which is actually enough for small machines. Nevertheless, usually as many brushes are required as there are oles in the machine. This number is selected in order to reach a maximum brush area with the shortest commutator ossible. One coil of a wave winding is always connected to the commutator at about a distance of two ole itches. A wave winding is a more common solution than a la winding for small (< 50 kw) machines, since it is usually more cost-efficient than a la winding. In a machine designed for a certain seed, a number of ole airs and a flux, the wave winding requires less turns than a la winding, a two-ole machine excluded. Corresondingly, the cross-sectional area of conductors in a wave winding has to

85 J. Pyrhönen, T. Jokinen, V. Hrabovcová.85 be larger than the area of a la winding. Therefore, in a machine of a certain outut, the coer consumtion is the same irresective of the tye of winding. The revious windings are simle examles of the various alternative constructions for commutator windings. In articular, when numerous arallel aths are emloyed, we must ensure that the voltages in the aths are equal, or else there will occur comensating currents flowing through the brushes. These currents create sarks and wear out the commutator and the brushes. The commutator windings have to be symmetric to avoid extra losses. If the number of arallel airs of aths is a, there are also a revolutions in its voltage olygon. If the revolutions are comletely overlaing in the voltage olygon, the winding is symmetrical. In addition to this condition, the diameter H H has to slit the olygon into two equal halves at all times. These conditions are usually met when both the number of slots and the number of oles are evenly divisible with the number of arallel aths a. Figure.50 illustrates a winding diagram of a four-ole machine. The number of slots is = 6, and the number of arallel aths is a = 4. Hence, the winding meets the above conditions of symmetry. The coil voltage hasor diagram and the voltage olygon are deicted in Fig..5. Since a =, there has to be one hasor a = of the coil voltage hasor diagram between the consequent hasors of the olygon. When starting with the hasor 8, the next hasor in the voltage olygon is 3 0. In between, there is a hasor 7 4, which is of the same hase as the revious one, and so on. The first circle around the voltage olygon ends u at the oint of the hasor 5, in the winding diagram at the commutator segment 9. However, the winding is not yet closed at this oint, but continues for a second, similar revolution formed by the hasors The winding is fully symmetrical, and the coils short-circuited by the brushes are laced on the quadrature axes. S N S A B C D I a I a Figure.50. Balancing connectors or equalizer bars (Bars A, B, C, and D) of a la winding. For instance the coil sides 9 and 3 are located in the similar magnetic ositions if the machine is symmetric. Hence, the commutator segments 5 and 7 may be connected together with a balancing connector.

86 J. Pyrhönen, T. Jokinen, V. Hrabovcová A A: comm segments and 9 B: comm segments 3 and B H D D: comm segments 7 and 5 C: comm segments 5 and C H a) b) Figure.5. a) Coil voltage hasor diagram of the winding of Fig..50, b) the coil voltage olygon of the winding of Fig..50 and the connection oints of the balancing connectors A, B, C, and D. There are two overlaing olygons in the illustrated voltage olygon. The hasor 8 is the first hasor and 3 0 the next hasor of the olygon. The hasor 7 4 is equal to the hasor 8 because both have their ositions in the middle of oles, but it is skied when constructing the first olygon. The first olygon is closed at the ti of the hasor 5. The winding continues to form another similar olygon using hasors The winding is comletely symmetrical, and its brush-shortcircuited coils are on the quadrature axes. The hasors 3 0 and 9 6 have a common ti oint B in the olygons created by the commutator segment 3 and, as shown in Fig..50 and in Fig..5b, and the oints can thus be connected by balancing connectors. The three other balancing connector oints are A, C, and D. The otential at different ositions of the winding is now investigated with resect to an arbitrary osition, for instance a commutator segment. In the voltage olygon, this zero otential is indicated by the oint A of the olygon. At t = 0, the instant deicted by the voltage olygon, the otential of the segment amounts to the amlitude of the hasor 8, otherwise it is a rojection on the straight line H H. Resectively, the otential at all other oints in the olygon with resect to the segment is at every instant the hasor drawn from the oint A to this oint, rojected on the straight line H H. Since for instance the hasors 3 0 and 9 6 have a common oint in the voltage olygon, the otential of the resective segments 3 and of the commutator is always the same, and the otential difference between them is zero at every instant. Thus, these commutator segments can be connected with conductors. All those oints that corresond to the common oints of the voltage olygon, can be interconnected. Figure.50 also deicts three other balancing connectors. The urose of these comensating combinations is to conduct currents that are created by the structural asymmetries of the machine, such as the eccentricity of the rotor. Without balancing connectors, the comensating currents, created for various reasons, would flow through the brushes, thus imeding the commutation. There is an alternating current flowing in the comensating combinations, the resulting flux of which tends to comensate the asymmetry of the magnetic flux caused by the eccentricity of the rotor. From this we may conclude that comensating combinations are not required in machines with two brushes. The maximum number of comensating combinations is obtained from the number of equiotential oints. In the winding of Figs..50 and.5, we could thus assemble eight combinations; however, usually only a art of the ossible combinations are needed to imrove the oeration of the

87 J. Pyrhönen, T. Jokinen, V. Hrabovcová.87 machine. According to the illustrations, there are four ossible combinations: A, B, C, and D. In machines that do not commutate easily, it may rove necessary to emloy all the ossible comensating combinations. In small and medium machines, the comensating combinations are laced behind the commutator. In large machines, ring rails are laced to one end of the rotor, while the commutator is at the other end..5.4 AC Commutator Windings The equiotential oints A, B, C, and D of the winding in Fig..5 are connected with rails in A, B, C, and D in Fig..50. The voltages between the rails at time t = 0 are illustrated by the resective voltages in the voltage olygon. When the machine is running, the voltage olygon is rotating, and therefore the voltages between the rails form a symmetrical four-hase system. The frequency of the voltages deends on the rotation seed of the machine. The hase windings of this system, with two arallel aths in each, are connected in a square. With the same rincile, with taings, we may create other oly-hase systems connected in a olygon, Fig..5. A Figure.5. Equiotential oints A, B, and C of the voltage olygon, which reresents coils of the commutator winding, are connected as a triangle to form a oly-hase system. The resective voltages are connected via sli rings and brushes to the terminals of the machine. C B If z c coils are connected as a closed commutator winding with a arallel ath airs, it is transformed with taings into a m-hase AC system connected into a olygon by couling the taings at a distance of a ste z y = c m ma (.6) from each other. In a symmetrical oly-hase system, both y m and z c /a are integers. Windings of this tye have been emloyed for instance in rotary converters, the windings of which have been connected both to the commutator and to the sli rings. They convert direct current into alternating current and vice versa. Closed commutator windings cannot be turned into star-connected windings, but only olygons are allowed..5.5 Current Linkage of the Commutator Winding and Armature Reaction The curve function of the current linkage created by the commutator winding is comuted in the way illustrated in Fig..9 for a three-hase winding. When defining the slot current I u, the current of the short-circuited coils can be set zero. In short-itched coils, there may be currents flowing in oosite directions in the different coil sides of a single slot. If z b is the number of brushes, the armature current I a is divided into brush currents I = I a /(z b /). Each brush current in turn is divided into two aths as conductor currents I s = I/ = I a /a, when a is the total number of ath airs of the winding. In a slot, there are z conductors, and thus the sum current of a slot is

88 I J. Pyrhönen, T. Jokinen, V. Hrabovcová.88 z I a = zi s, (.7) a u = where z is the number of conductors in the comlete winding. All the ole airs of the armature are alike, and therefore it suffices to investigate only one of them, that is, a two-ole winding, Fig..46. The curve function of the magnetic voltage follows thus the illustration of Fig..53. Θ S N Θ ma Θ Θ δa = ma slots Figure.53. Current linkage curve of the winding of Fig..46, when the commutating takes lace in the coils in the slots 5 and 3. The number of brushes of a commutator machine normally equals to the number of oles. The number of slots between the brushes is q =. (.8) This corresonds to the number of slots er ole and hase of AC windings. The effective number of slots er ole and hase is always somewhat lower, because a art of the coils are always short circuited. The distribution factor for an armature winding is obtained from Eq. (.33). For a fundamental, and m = it is rewritten in the form kda =. (.9) π sin Armature coils are often short itched, and the itch factor is thus obtained from Eq. (.3). The fundamental winding factor of a commutator winding is thus k wa = kdaka. (.0) π When the number of slots er ole increases, k da aroaches the limit /π. This is the ratio of the voltage circle (olygon) diameter to the circle erimeter. In ordinary machines, the ratio of shortitching is W/τ > 0,8, and therefore k a > 0,95. As a result, the aroximate value k wa = / π is an adequate starting oint in the initial manual comutation. More thorough investigations have to be based on the analysis of the curve function of the current linkage. In that case, the winding has to be observed in different ositions of the brushes. Fig..45 shows that at the right side of the quadrature axis q, the direction of each slot current is towards the observer, and on the left, away from the observer. In other words, the rotor becomes an electromagnet with the north ole at the bottom and the south ole at the to. The ole air current linkage of the rotor is

89 J. Pyrhönen, T. Jokinen, V. Hrabovcová.89 z I a z Θ ma = qi u = = I a = N a I a. (.) a 4a The member z N = a 4a (.) of the equation is the number of coil turns er ole air in a commutator armature in one arallel ath, that is, turns connected in series, because z/ is the number of all armature turns; z/(a) is the number of turns in one arallel ath, in other words, connected in series, and finally, z/(a) is the number of turns er ole air. The current linkage calculated according to Eq. (.) is slightly higher than in reality, because the number of slots er ole and hase includes also the slots with short-circuited coil sides. In calculation, we may emloy the linear current density I N I A = u a a a = = NaIa. (.3) πd πd τ The current linkage of the linear current density is divided into magnetic voltages of the air gas, the eak value of which is Θˆ δa τ z Ia NaIa Θma = A adx = Aaτ = = =. (.4) a 0 Θ δa = Θ ma / = z I a Θ ma A aτ = = Na I a = = Θδa. (.5) a In the diagram, the eak value Θˆ δa is located at the brushes (in the middle of the oles) the value varying linearly between the brushes, as illustrated with the dashed line in Fig..53. Θˆ δa is the armature reaction acting in the quadrature axis under one ti of a ole shoe, and it is the current linkage to be comensated. The armature current linkage also creates commutating roblems because of which the brushes have to be shifted from the q-axis by and angle ε to a new osition as shown in Fig.54. The figure gives also the ositive directions of the current I and the resective current linkage. The current linkage can be divided into two comonents Θ Θ Θma = sin ε Θδa sin ε, (.6) Θma = cosε Θδa cosε. (.7) md = mq = The former is called a direct comonent and the latter a quadrature comonent. The direct comonent magnetizes the machine either into the arallel or oosite direction with the actual field winding of the main oles of the machine. There is demagnetizing effect, if brushes are shifted in the direction of rotation in generator mode, or in the oosite direction of rotation in motoring oeration, the magnetizing effect in contrary, in generating oeration oosite direction of rotation, in motoring mode in the direction of rotation. The quadrature comonent distorts the magnetic field

90 J. Pyrhönen, T. Jokinen, V. Hrabovcová.90 of the main oles, but neither magnetizes nor demagnetizes it. This is not a henomenon restricted to commutator machines, but the reaction is in fact resent in all rotating machines. d Figure.54. Current linkage of a commutator armature and its comonents. To ensure better commutation, the brushes are not laced on the q-axis. Θ md Θ ma I ε q I.6 Comensating Windings and Commutating Poles As it was stated above, the armature current linkage (called also armature reaction) has some negative influence on the DC machine oeration. The armature reaction may create commutating roblems and must, therefore, be comensated. There are different methods to mitigate the armature reaction caused roblems: ) to shift brushes from their geometrical neutral axis to the new magnetically neutral axis, ) to increase the field current to comensate the main flux decrease caused by the armature reaction, 3) to build commutating oles, and 4) to build comensating winding. The urose of comensating windings in DC machines is to comensate harmful flux comonents created by armature windings. Flux comonents are harmful, because they create an unfavourable air ga flux distribution in DC machines. The dimensioning of comensating windings is based on the current linkage that has to be comensated by the comensating winding. The conductors of a comensating winding have therefore to be laced close to the surface of the armature, and the current flowing in them has to be oosite to the armature current. In DC machines, the comensating winding is inserted in the slots of the ole shoes. The comensating effect has to be created in the section αiτ of the ole itch, as illustrated in Fig..55. If z is the total number of conductors in the armature winding, and the current flowing in them is I s, we obtain an armature linear current density A a z I s =. (.8) Dπ The total current linkage Θ Σ of the armature reaction and the comensating winding has to be zero in the integration ath. It is ossible to calculate the required comensating current linkage Θ k by evaluating the corresonding current linkage of the armature to be comensated. The current linkage of the armature Θ a occurring under the comensating winding at the distance α DC τ /, as it is shown in Fig..55 is

91 J. Pyrhönen, T. Jokinen, V. Hrabovcová.9 αiτ αdcτ αdcτ Aa Θ a = zis =. (.9) Dπ Since there is an armature current I a flowing in the comensating winding, we obtain the current linkage of the comensating winding accordingly Θ = N I, (.30) k k a where N k is the number of turns of the comensating winding. Since the current linkage of the armature winding has to be comensated in the integration ath, the common current linkage is written αiτ Aa Θ = Θk + Θa = NkIa + = 0, (.3) z I a where Θ a = α i A aτ = α i = Na I a. a (.3) Now, we obtain the number of turns of the comensating winding to be inserted in the ole shoes roducing demagnetizing magnetic flux in the q-axis comensating the armature reaction flux: N α τ A i a k =. (.33) Ia As N k has to be an integer, Eq. (.33) is only aroximately feasible. To avoid large ulsating flux comonents and noise, the slot itch of the comensating winding is set to diverge 0 5 % from the slot itch of the armature. Since a comensating winding cannot comletely cover the surface of the armature, also commutating oles are utilized to comensate the armature reaction although their function is just to imrove commutation. These commutating oles are located between the main magnetizing oles of the machine. There is an armature current flowing in the commutating oles. The number of turns on the oles is selected such that the effect of the comensating winding is strengthened aroriately. In small machines, commutating oles alone are used to comensate the armature reaction. If commutation roblems still occur desite a comensating winding and commutating oles, the osition of the brush rocker of the DC machine can be adjusted so that the brushes are laced on the real magnetic quadrature axis of the machine.

92 J. Pyrhönen, T. Jokinen, V. Hrabovcová.9 N commutating ole N S i a i a comensating winding armature S N N S S i a i a T i f i f field winding i a N S i a commutating ole N k - turns comensating winding N a - turns er ole air ατ i armature winding τ a) b) Figure.55. a) Location of the comensating windings and the commutating oles, b) definition of the current linkage of a comensating winding. In rincile, the dimensioning of a commutating ole winding is straightforward. Since the comensating winding covers the sectionαiτ of the ole itch and includes N k turns that carry the armature current I a, the commutating ole winding should comensate the remaining current linkage of the armature ( α i ) τ. The number of turns in the commutating ole N c should be αi Nc = Nk. (.33) αi When the same armature current I a flows both in the comensating winding and in the commutating ole, the armature reaction will be fully comensated. If there is no comensation winding, the commutating ole winding must be dimensioned and the brushes ositioned so that the flux in a commutating armature coil is at its maximum, and no voltage is induced in the coil..7 Rotor Windings of Asynchronous Machines The simlest rotor of an induction machine is a solid iron body, turned and milled to a correct shae. In general, a solid rotor is alicable to high-seed machines and in certain cases also to normalseed drive. However, the comutation of the electromagnetic characteristics of a steel rotor is a demanding task, and it is not discussed here. A solid rotor is characterized by a high resistance and a high leakage inductance of the rotor. The hase angle of the aarent ower created by a waveform enetrating a linear material is 45, but the saturation of the steel rotor reduces the hase angle. A tyical value for the hase angle of a solid rotor varies between 30 and 45, deending on the saturation. The characteristics of a solid-rotor machine are discussed for instance in Pyrhönen (99), Huunen (004), and Aho (007). The erformance characteristics of a solid rotor can be imroved by slotting the surface of the rotor, Fig..56. Axial slots are used to control the flow of eddy currents in a direction favourable to the torque roduction. Radial slots increase the length of the aths of the eddy currents created by certain high-frequency henomena. This way, eddy currents are damed and the efficiency of the machine is imroved. The structure of the rotor is of great significance in torque roduction, Fig..57. An advantage of common cage winding rotors is that they roduce the highest torque with small values of sli, whereas solid rotors yield a good starting torque.

93 J. Pyrhönen, T. Jokinen, V. Hrabovcová.93 axial slots radial slots a) b) c) Figure.56. Different solid rotors. a) A solid rotor with axial and radial slots (in this model, short-circuit rings are required. They can be constructed either by leaving the art of the rotor that extends from the stator without slots, or by equiing it with aluminium or coer rings, b) a rotor equied with short-circuit rings in addition to slots, c) a slotted and cage-wound rotor. A comletely smooth rotor can also be emloyed. Figure.57. Torque curves of different induction rotors as a function of mechanical angular seed Ω. a) A normal double-cage winding rotor, b) a smooth solid rotor without short-circuit rings, c) a smooth solid rotor equied with coer short-circuit rings, d) axially and radially slitted solid rotor equied with coer short-circuit rings. T 0 0 d ) c ) b ) a ) Ω In small machines, a Ferraris rotor can be emloyed. It is constructed of a laminated steel core covered with a thin layer of coer. The coer covering rovides a suitable ath for eddy currents induced to it. The coer covering takes u a certain sace in the air ga, the electric value of which increases notably because of the covering, since the relative ermeability of coer is μ r = As a diamagnetic material, coer is thus even a somewhat weaker ath for the magnetic flux than the air. The rotor of an induction machine can be roduced as a normal slot winding by following the rinciles discussed in the revious sections. A wound rotor has to be equied with the same number of ole airs as the stator, and therefore it is not in ractice suitable for machines ermitting a varying number of oles. The hase number of the rotor may differ from the hase number of the stator. For instance, a two-hase rotor can be emloyed in sli-ring machines with a three-hase stator. The rotor winding is connected to an external circuit via sli rings. The most common short-circuit winding is the cage winding, Fig..58. The rotor is roduced of electric steel sheets and it is rovided with slots containing non-insulated bars, the ends of which are connected either by welding or brazing to the end rings, that is, to the short-circuit rings. The short-circuit rings are often equied with fins that together act as a cooling fan as the rotor rotates. The cage winding of small machines is roduced of ure aluminium by simultaneously ressure casting the short-circuit rings, the cooling ribs and the bars of the rotor.

94 J. Pyrhönen, T. Jokinen, V. Hrabovcová.94 bars short circuit rings Figure.58. Simle cage winding. Cooling fans are not illustrated. r = 4. Fig..59 illustrates a full-itch winding of a two-ole machine observed from the rotor end. Each coil of the rotor constitutes also a comlete hase coil, since the number of slots in the rotor is r = 6. The star oint 0 forms, based on symmetry, a neutral oint. If there is only one turn in each coil, the coils can be connected at this oint. The magnetic voltage created by the rotor deends only on the current flowing in the slot, and therefore the connection of the windings at the star oint is of no influence. However, the connection of the star oint at one end of the rotor turns the winding into a six-hase star connection with one bar, that is, half a turn, in each hase. The six-hase winding is then short circuited also at the other end. Since also the shaft of the machine takes some room, the star oint has to be created with a short-circuit ring as illustrated in Fig..58. We can now see that Fig..59 deicts a star-connected, short-circuited oly-hase winding, for which the number of hase coils is in a two-ole case equal to the number of bars in the rotor: m r = r. Figure.59. Three-hase winding of a two-ole rotor. The number of turns in the hase coil is N r =. If the winding is connected in star at oint 0 and it is shortcircuited at the other end, a six-hase, short-circuited winding is created, for which the number of turns is N r = ½, k wr =. 3 0 N r = In machine design, it is often assumed that the analysis of the fundamental ν = alone gives an adequate descrition of the characteristics of the machine. However, this is valid for cage windings only if we consider also the conditions related to its number of bars. A cage winding acts differently with resect to different harmonics ν. Therefore, a cage winding has to be analyzed with resect to the general harmonic ν. This is discussed in more detail in Chater 7, in which different tyes of machines are investigated searately.

95 J. Pyrhönen, T. Jokinen, V. Hrabovcová.95.8 Damer Windings The damer windings of synchronous machines are usually short-circuit windings, which in nonsalient ole machines are contained in the same slots with magnetizing windings, and in salient ole machines in articular, in the slots at the surfaces of ole shoes. There are no bars in the damer windings on the quadrature axes of salient ole machines, but only the short-circuit rings encircle the machine. The resistances and inductances of the damer winding of the rotor are thus quite different in the d-direction and q-direction. In a salient ole machine constructed of solid steel, the material of the rotor core itself may suffice as a damer winding. In that case, an asynchronous oeration resembles the oeration of a solid rotor induction machine. Figure.60 illustrates a damer winding of a salient ole synchronous machine. Damer windings imrove the erformance characteristics of synchronous machines esecially during the transients. Like in asynchronous machines, thanks to damer windings, synchronous machines can in rincile be started direct-on-line. Also a stationary asynchronous drive is in some cases a ossible choice. Esecially in single-hase synchronous machines and in the unbalanced load situations of three-hase machines, the function of damer windings is to dam the counterrotating fields of the air ga which otherwise cause great losses. In articular, the function of damer windings is to dam the fluctuation of the rotation seed of a synchronous machine when rotating loads with ulsating torques, such as iston comressors. damer bar coer late q-axis connector coer late d-axis Figure.60. Structure of the damer winding of a six-ole salient ole synchronous machine. The coer end lates are connected with a suitable coer connector to form a ring for the damer currents. Sometimes also real rings connect the damer bars. The effective mechanisms of damer windings are relatively comlicated and diverse, and therefore their mathematically accurate design is difficult. That is why damer windings are usually constructed by drawing uon emirical knowledge. However, the inductances and resistances of the selected winding can usually be evaluated with normal methods to define the time constants of the winding. When the damer windings of salient ole machines are laced in the slots, the slot itch has to be selected to diverge 0 5 % from the slot itch of the stator to avoid the ulsation of the flux and

96 J. Pyrhönen, T. Jokinen, V. Hrabovcová.96 noise. If the slots are skewed (usually for an amount of a single stator slot itch), the same slot itch can be selected both for the stator and the rotor. Damer winding comes into effect only when the bars of the winding are connected with short-circuit rings. If the ole shoes are solid, they may, similarly as the solid rotor of a non-salient ole machine, act as a damer winding as long as the ends of the ole shoes are connected with durable short-circuit rings. In non-salient ole machines, an individual damer winding is seldom used. However, in non-salient ole machines, conductors may be mounted under slot wedges, or the slot wedges themselves are used as the bars of the damer winding. In synchronous generators, the function of damer windings is to dam for instance counter-rotating fields. To minimize losses, the resistance is ket to a minimum in damer windings. The crosssectional area of the damer bars is selected to be 0 30 % of the cross-sectional bar area of the armature winding. The windings are made of coer. In single-hase generators, damer bar crosssectional areas larger than 30 % of the stator coer area are emloyed. The frequency of the voltages induced by counter-rotating fields to the damer bars is double when comared with the network frequency. Therefore, it has to be considered whether secial actions are required with resect to the skin effect of the damer windings (for instance the utilization of the Roebel bars (braided conductors) to avoid skin effect). The cross-sectional area of the short-circuit rings is selected to be aroximately % of the cross-sectional area of the damer bars er ole. The damer bars have to dam the fluctuations of the rotation seed caused by the ulsating torque loads. They also have to guarantee a good starting torque when the machine is starting as an asynchronous machine. Thus, brass bars or small-diameter coer damer bars are emloyed to increase the rotor resistance. The cross-sectional area of coer bars is tyically only 0 % of the cross-sectional area of the coer of the armature winding. In ermanent magnet synchronous machines, in axial flux machines in articular, the damer winding may be easily constructed by lacing a suitable coer or aluminium late on the surface of the rotor, on to of the magnets. However, achieving a total conducting surface in the range of 0 30 % of the stator coer surface may be somewhat difficult as the late thickness easily increases too large and limits the air ga flux density created by the magnets. References and Further Reading Aho, T Electromagnetic design of a solid steel rotor motor for demanding oerational environments. Dissertation. Acta Universitatis Laeenrantaensis 34. Laeenranta University of Technology. (htts://oa.doria.fi/) Heikkilä, T. 00. Permanent magnet synchronous motor for industrial inverter alications analysis and design. Dissertation. Acta Universitatis Laeenrantaensis 34. Laeenranta University of Technology. (htts://oa.doria.fi/) Hindmarsh, J Electrical Machines and Drives. Worked Examles. Second edition. Oxford: Pergamon Press. Huunen, J High-seed solid-rotor induction machine electromagnetic calculation and design. Acta Universitatis Laeenrantaensis 97. Laeenranta University of Technology. (htts://oa.doria.fi/) IEC International electrotechnical vocabulary (IEC). Rotating machines. Geneva: International Electrotechnical Commission. Pyrhönen, J. 99. The high-seed induction motor: calculating the effects of solid-rotor material on machine characteristics. Acta Polytechnica Scandinavica, Electrical Engineering Series 68. (htts://oa.doria.fi/) Richter, R Electrical machines. (Elektrische Maschinen.) Volume IV. Induction machines. (Die Induktionsmaschinen.) Second edition. Basel and Stuttgart: Birkhäuser Verlag.

97 J. Pyrhönen, T. Jokinen, V. Hrabovcová.97 Richter, R Electrical machines. (Elektrische Maschinen.) Volume II. Synchronous machines and rotary converters. (Synchronmaschinen und Einankerumformer.) Third edition. Basel and Stuttgart: Birkhäuser Verlag. Richter, R Electrical machines. (Elektrische Maschinen.) Volume I. General calculation elements. DC machines. (Allgemeine Berechnungselemente. Die Gleichstrommaschinen.) Third edition. Basel and Stuttgart: Birkhäuser Verlag. Salminen, P Fractional slot ermanent magnet synchronous motors for low seed alications. Dissertation. Acta Universitatis Laeenrantaensis 98. Laeenranta University of Technology. Vogt, K Design of electrical machines. (Berechnung elektrischer Maschinen.) Weinheim: VCH Verlag.