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Alternating current (ac) windings in electrical machines basically have a twofold purpose. In electrical generators ac windings are place where electromotive force (EMF) should be induced. In electrical motors, primarily ac windings goal is to produce rotating magneto-motive force (MMF) wave. In any case, ac windings should be designed in such a manner that induced EMF or generated rotating MMF wave consists predominantly of the fundamental sinusoidal component. Therefore, the starting point of an induction machine study is the analysis of MMF waves in the air gap of such machines. This assumes knowledge of winding distribution along the air gap from stator as well as from rotor side.
The magnetic flux is analogous to the electric current. The MMF which sets up the magnetic flux is analogous to the EMF. The MMF is equivalent to a number of turns of wire carrying an electric current. If either the current through a coil (as in an electromagnet) or the number of turns of wire in the coil is increased, the MMF will be higher; and if the rest of the magnetic circuit remains the same, the magnetic flux increases proportionally.
At least two different approaches exist for describing winding distribution.
1. Current sheet concept
Fig. 1 (a) Cross-section view and (b) current sheet and MMF waveform around stator circumference in case of on full-pitch coil
Fig. 2 (a) Cross-section view and (b) current sheet and MMF waveform along
stator circumference for three full-pitch coils (one of the phase windings)
connected in series
Suppose that exact winding distribution in stator and rotor slots can be taken into account and current sheet is defined. Let us analyze the simplest case where just one full-pitch coil with N turns exists on the stator as shown in Fig. 1(a). A full-pitch coil has pitch that is equal to the pole pitch, y t. Let us assume that current I flows through the coil. MMF in one slot is the product of number of conductors in that slot and current that is equal in all conductors. A current sheet, as the name implies, is a plane current flowing in one direction of the plane. By definition, current sheet A is the ratio of the MMF of one slot, FS NSI, and slot mouth opening b:
(4.1) where NS is the number of conductors in a slot (NS N in this case). Current sheet distribution is depicted in Fig. 1(b), as two rectangles of different signs as the current in two slots is of different direction. The height of these two rectangles is given by (4.1). The MMF in the air gap due to the space distribution of windings (or current sheet) is also space-dependent. By definition, MMF on the place x along the circumference of the machine is:
Fx Z x 0 Ax dx (4.2)
…where position x 0 is defined as a place where current sheet distribution has maximal value. Previous expression was directly extracted from Ampere's circuital law. Trapezoidal MMF waveform along the machine circumference is also given in Fig. 1(b). The peak value of MMF is obtained by substituting x b/2 in (4.2):
F NI 2 (4.3)
In the case where three series-connected identical coils (one of the phase windings) are placed in neighboring slots as illustrated in Fig. 2(a), the current sheet and MMF waves have somehow more complex shape ( Fig. 2(b)). Now, N is the total number of turns per phase connected in series.
Current sheet could be alternatively defined as a ratio of slot MMF and slot pitch, and not slot mouth width. In this case, current sheet is...
(4.4)
...where S is the number of stator slots and tS is the slot pitch or tooth pitch which are the same as displayed in Fig. 3. Now the peak value of MMF wave for a machine with p poles pair is as follows:
(4.5)
Fig. 3 Current sheet and MMF waveform along the stator circumference for
three full-pitch coils connected in series (one of the phase windings)
This is identical with (4.3), for p 1, N in (4.5) is the number of series turns per phase.
Obviously, current sheet and MMF waveform in Fig. 3 are periodical.
So, these functions can be expressed as the sum of a Fourier series elements.
The fundamental harmonic approximation of both functions is given in Fig. 3.
These two harmonics are always shifted in space-MMF wave always leads one-half of pole pitch. Therefore, in general case, for fundamental harmonic, we have
(4.6) (4.7) (4.8) (4.9) (4.10)
...equation (4.6) can be given as a function of angular coordinate, electrical angle qel pqmeh:
(4.11)
(4.12)
Remark 1: Fourier series consists of the sum of harmonic functions with different amplitudes, periods and phase positions. Amplitudes of those harmonic functions are coefficients of Fourier series. For functions that are periodic with fundamental period 2p and integrable over the range of [ p,p ] that coefficients are (n is harmonic order; for n 0 DC offset or mean value of function on that range is obtained):
Often the exponential form of Fourier series is used which has a more concise form.
Bearing in mind Euler's identity,
Fourier series can be displayed in the following form,
.... where complex coefficient is:
The correlation between the previously introduced coefficients are given as follows:
In this section, the Fourier coefficients are obtained by utilizing the discrete Fourier transformation-FFT (Fast Fourier Transform) operator incorporated in software packet MATLAB_.
Fig. E4.1.1 Original function
2. Winding function concept
The winding function theory can be traced back to [2 ]. Contrary to the classical d-q model, this theory can take into account all winding MMF space harmonics in machines with small and uniform air gap. Now consider the simplest form of the stator phase winding which is a single concentrated coil with N turns. Positions of coil's sides along the stator circumference are described by mechanical angles q1 and q2, i.e., a q2 q1 which is coil pitch in mechanical radians as depicted in Fig. 4. Assuming uniform small air gap length, neglecting the stator and rotor slots, assuming infinitely permeable iron as well as adopting point conductor approximation, Ampere's circuital law in common shape is as the following:
(4.13)
...and it's much simpler form is...
(4.14) where H(q) is the radial component of magnetic field intensity in the air gap at position described by angle q, g is the air gap length, i is the coil current and n(q)is the turn function [3 ]. The assumption of small air gap length means that the rotor radius is significantly larger than air gap length. On the other hand, this also means that tangential component of the magnetic field strength in the air gap is negligible compared with its radial component. In expression (4.13), C is contour 1-2-3-4 and S is the area spanned by that contour.
Fig. 4 Turn function definition.
From Fig. 4, it is clear that by continuously changing the position q of side 2 of the closed-loop C, turn function could be defined as follows:
4.15) 4.16) 4.17) 4.18) 4.19) 4.20) 4.21) 4.22) 4.23) 4.24) 4.25)
…is winding function. Obviously, winding function is MMF per unit current.
Henceforth, the terms ''winding function'' and ''MMF per unit current'' will be used interchangeably. Winding function could be alternatively written as follows:
(4.26)
Since infinite permeable iron is assumed, the principle of superposition could be adopted. So turn function or winding function of one-phase winding is simply the sum of adequate functions of coils that are constituents of the phase winding.
2.1 Concentrated full-pitch coil MMF
For an N-turns full-pitch coil in a two-pole machine (p 1) and for the adopted system of reference, based on Fig. 5(a), turns and winding functions are as the following:
(4.27)
(4.28)
Fig. 5 illustrates the placement of the coil along the stator circumference, turn function profile, winding function profile as well as coefficients of the Fourier's series of winding function that are amplitudes of constituent space harmonics.
Obviously, only odd higher space-harmonics are present in Fourier series of winding function. The fundamental space-harmonic amplitude is 6.37 turns ( Fig. 5(b)), i.e., 4/p higher than the winding function value, so the fundamental MMF wave amplitude is as follows:
(4.29)
… where Imax is the amplitude of current passing the coil. Higher space-harmonic of order n has n times smaller amplitude than the fundamental one:
(4.30)
Fig. 5 (a) Crosssection view and (b) turn function, winding function and
Fourier coefficients of winding function for a full-pitch coil in a two-pole
machine (p 1). Coil has N 10 turns.
Turns and winding function for full-pitch coil as well as Fourier transform of winding function given in Fig. 5 could be easily defined using the following short MATLAB script:
-----------
In this section, the following two facts should be considered: (1) in order to obtain the exact discrete Fourier transform, the number of samples in 2p rad must be H 2n where n is an integer and (2) the first element in Fourier trans form is the coefficient equal to the mean value of the winding function. However, in this case, the number is already equal to zero, this element is not shown in Fig. 1(b).
Fig. 6 (a) Cross-section view and (b) turns function, winding function
and Fourier expansion of winding function of two series-connected full pitch
coils (1-1 and 2-2 ) in a four-pole machine (p 2). Every single coil has 10
turns
From Figures 1(b) and 5(b), it is apparent that the waveforms of MMF that are the result of the integration of current sheet and winding function are similar.
The main difference is in the fact that the winding function has step rise in the center of the slot which is the result of earlier adopted point conductor approximation. On the other hand, the MMF has linear rise across the slot. However, definition of winding function could be easily adopted in order to take the linear rise of the MMF across the slot into account.
Fig. 6(a) displays the crosssection view of the four-pole machine as well as turns and winding function for a full-pitch coil winding. Two full-pitched coils (in a four-pole machine, the full-pitch coil has a pitch amech p/2 rad) are connected in series. The total turns and winding function of phase winding is simply the sum of turns and winding functions of every single coil in the winding. Now, the fundamental harmonic is second-order harmonic (p), the third harmonic is sixth order harmonic (3p), etc. The amplitude of the fundamental MMF wave now is (4.31) where N is the total number of turns in the phase winding. The amplitude of the higher order space harmonic components of n is (4.32) In the above cases, turn and winding functions have been dealt with for the stator windings. All derived conclusions, however, are valid for the coils or windings in the rotor slots.
2.2 Distributed full-pitch phase winding
Fourier analysis showed that the air gap MMF produced by a full-pitch coil(s) consists of fundamental space-harmonic components as well as a series of all odd higher order space harmonic components. In the design of the ac windings, serious attempts are made to distribute the coils of the windings in such a manner as to minimize the higher order harmonic components and to produce an air-gap MMF wave consisting predominantly of the space-fundamental sinusoidal component.
In this way, the machine is better utilized. Therefore, in practice, ac winding of induction machine is always designed as a distributed winding. Distributed phase winding of a three-phase machine occupies one-third of the stator slots along the circumference of the machine.
Fig. 7(a) depicts such a one-phase winding in a two-pole machine when the stator has S 18 slots. It means that one-phase winding (three series-connected full-pitch coils) occupies six slots, three under one and three under the other pole.
The turn and winding function of phase A winding could be easily obtained by summing turns and winding functions of individual coils. As Fig. 7(b) illustrates, the resultant MMF profile of the distributed full-pitch winding has trapezoid shape which is much closer to the sinusoid than before (compare Figures 5(b) and 7(b)). This can also be concluded from the ratio of space harmonics amplitudes and fundamental harmonic amplitude. However, the amplitude of the resultant winding function is not three times greater than before, but somewhat smaller as it could be easily concluded observing the coefficients of the Fourier series. The reason for that is in the space displacement of individual coils.
The winding distribution factor kd, also known as a breadth factor, is ratio of the resultant MMF space-harmonic amplitude and sum of the MMF space-harmonics amplitudes of the constituent coils (ratio of vector and algebraic sum of constituent coil's MMFs):
(4.33)
Referring to Figures 5(b) and 7(b), this factor for the fundamental space harmonic (n 1) is
Fig. 7 (a) Cross-section view and (b) turns function, winding function
and the Fourier expansion of distributed winding consist of three series connected
full-pitch coils (1-1 , 2-2 and 3-3 ) in a two-pole (p 1) machine. Every single
coil has Ncoil 10 turns, so N 30
This is in good agreement with the factor obtained from the well-known equation for distribution factor:
(4.34)
where z is the number of slots per pole per phase and qel is the electrical angle between two slots, qel pqmech. For the analyzed phase winding, kd is as follows:
For higher space harmonic of order n, the distribution factor is
(4.35)
Considering the distribution factor, the distributed phase winding MMF amplitude for nth space harmonic could be defined as follows:
(4.36)
From Figures 7(b) and 2(b), it can be seen that the waveforms of MMF calculated by integration of current sheet and winding function are similar. Again, the main difference is that winding function has step rise in the center of the slot which is the result of earlier adopted point conductor approximation. On the other hand, MMF obtained from the integration of current sheet has linear rise across the slot.
2.3 Pulsating MMF
Winding function of N turns full-pitch coil in two-pole machine p 1 shown in Fig. 5(b), could be resolved in Fourier series. As it is illustrated before, Fourier series is a sum of all odd harmonics
(4.37)
(4.38)
Neglecting all higher harmonics, this winding function could be approximated with the following expression:
(4.39)
... where N1 max is the amplitude of the fundamental harmonic:
(4.40)
The same is true for Fig. 7(b), namely (4.37) is valid also for this case, but the amplitudes of the constituent harmonics are something different,
(4.41)
... where N is the series turns number per phase (in all three coils) and kdn is the distribution factor. As it has been already mentioned, the winding function is MMF per unit current. So the waveforms of winding function in Figures 5(b), 6(b) and 7(b) is simultaneously the waveforms of MMFs if a dc of 1A passes the winding. If the dc rises to 2 A, MMF waveforms will be the same as winding function waveforms, but with double higher stairs. If ac with amplitude of 2A and frequency of 50 Hz passes the winding, the MMF will pulsate. When the current is maximal, MMF has the shape as sketched for winding functions. When the current is zero, MMF is also zero. If the current is maximal in other direction, MMF waveform is the mirror image of winding function. From the above explanation, it is clear that MMF in this case is pulsating. Fig. 8 illustrates the waveforms of the pulsating MMF for the case of 3 turns full-pitch winding and ac of 2A amplitude. Therefore, MMF depends on two variables: angular coordinate and time.
Fig. 8 Pulsating MMF: N 3 turns full-pitch coil with ac of 2A amplitude
Taking only the fundamental harmonic of winding function into account, pulsating MMF is as follows:
(4.42)
Using trigonometric transformation (4.42) can be written as sum of two sinusoidal functions:
(4.43)
(4.44)
It means that the pulsating MMF can be resolved into two rotating MMF waves. One of them, called forward wave or direct wave (index d), rotates in anticlockwise direction and the other one, called backward or inverse wave (index i), rotates in opposite direction, as illustrated in Figures 4.9 and 4.10.
What is the speed of these waves in space? Let us look at forward wave. In different times, this wave will have fixed maximal value in different points along the stator circumference:
(4.45)
Therefore, the mechanical angular speed of direct MMF wave, also known as synchronous speed, is:
(4.46)
That is also true of the speed of backward wave. In general case, for a machine with p pair of poles, the synchronous speed is:
(4.47)
Fig. 9 Pulsating MMF at any point in space and any time is the sum of two rotating MMF waves with equal amplitudes: forward and backward rotating MMF waves
Fig. 10 Alternative view on pulsating MMF: pulsating MMF as a vector sum
of two rotating waves also given in vector notation
(4.48)
…where f is the frequency of ac through the phase winding, i.e., frequency of the voltage supply.
Fig. 11 (a) Cross-section view and (b) winding functions of phase windings.
Every phase winding consists of three series-connected full-pitch coils. Two-pole machine, p 1. Every single coil has 10 turns Fig. 12 Three-phase, 50Hz, unit currents: rms value of phase current are 1/ 2 0.707A
2.4 Three-phase full-pitch (single-layer) winding.
A three-phase machine has three distributed windings on the stator. Phase windings are identical, but shifted in space by 2p/3p mechanical radians (2p/3 electrical radians). One-phase winding in a three-phase machine occupies one-third of the stator slots under one pole as shown in Fig. 11(a). Winding functions for such a machine are provided in Fig. 11(b). In order to obtain the resultant MMF waveform, instantaneous value of phase currents must be taken into account. The phase-shifted current passes through the three-phase windings. At any time, for example t1 5 ms ( Fig. 12), the phase current a is ia 1 A, while the other two-phase currents are ib ic 0.5 A. The resultant MMF waveform could be obtained by multiplying phase winding functions with adequate instant values of the phase currents:
(4.49) (4.50) (4.51) (4.52)
... because ia(t2 10 ms) 0, ib(t2 10 ms) 3/2 and ic(t2 10 ms) 3/2, Fig. 12.
Fig. 13 displays the resultant MMF waveforms at these time instances.
Obviously, the positions of the maximal values of resultant MMFs are shifted in space, i.e., the resultant MMF waveform is not fixed in space; actually it is rotating MMF.
The rotating speed could be easily found from the following considerations: at t1 5 ms maximal value of the rotating MMF is in a position described by the axes of phase winding A, i.e., qt1 20 .At t2 10 ms, the maximal value of the rotating MMF is at position qt2 110 . Rotating MMF speed, i.e., synchronous speed, is:
(4.53)
More generally, in case of machine with p pole pairs, the synchronous speed is as the following:
(4.54)
Hence, the resultant MMF wave rotates with the synchronous speed, having different waveforms in different times. However, regardless of the shape of MMF waveform in different times, Fourier transform of the MMF is always the same.
Spectral content of the presented MMF waveforms is also shown in Fig. 13.
The main difference in comparison with the phase windings MMF is that the resultant rotating MMF wave does not contain harmonic components with the order of three, six, nine, . . . times the fundamental one. In other words, all MMF space harmonics belong to the following series:
(4.55)
Fig. 13 Resultant rotating MMF wave in two different times, t1 5
ms and t2 10 ms and Fourier transform of MMF wave.
Two-pole machine
... where k 0, 1, 2, . . . . It can be seen from (4.55) that the symmetrical three phase winding supplied with the symmetrical three-phase voltage supply contains fundamental n 1, fifth n 5, seventh n 7, eleventh n 11, etc., space harmonics. The minus sign means inverse rotating MMF wave. The synchronous speed of nth space-harmonic is n times smaller than the synchronous speed of the fundamental one:
(4.56)
As seen in Fig. 13, the higher space harmonics are fifth and seventh, the so-called phase belt harmonics, which are direct consequences of trapezoidal shape of the phase-winding MMF. However, the most significant higher space harmonics are slot harmonics of order S/p 1. In the analyzed case, it is 17th and 19th space harmonics. They are direct consequences of the discrete nature of the winding, i.e., conductor placement in the slots. These harmonics in the best manner fill the ''gaps'' in the stepwise shape of the resultant MMF wave. Additionally, the rotating MMF space-harmonic amplitude is 1.5 times higher than the amplitude of the phase-winding MMF space harmonics:
(4.57)
More generally, for a m-phase symmetrical winding distribution:
(4.58)
2.5 Three-phase shorted-pitch coil (double-layer) winding
An additional measure taken in order to further upgrade the rotating MMF wave form closer to the sinusoidal shape is the use of the short-pitch coils. However, in order to use short-pitch coils, the stator phase winding must be placed in two layers along the stator circumference, i.e., short-pitch coil must have one of its sides placed in the bottom layer in one slot but the other side should be placed in the top layer of other slot. In this way, double-layer winding is obtained. Using short-pitched coils, with adequate coil pitch, some of the higher space-harmonics in the resultant rotating MMF waveform could be canceled out or significantly attenuated.
Let us look at the following example. Fig. 14 depicts double-layer three phase stator winding. Every single coil in the phase windings is a short-pitch coil.
Shortening the coil is one stator slot pitch. Now, instead of three coils in one-phase winding as shown in Fig. 11(a), six shorted-pitch coils make one-phase winding. To compare with the previously analyzed case, every single coil should have one-half of the number of turns as before.
Fig. 14 Double-layer three-phase winding. Two-pole machine, p 1.
Coils are shortened for one stator slot pitch.
From phase winding in Fig. 14 one could observe the following manner: the phase winding is organized in two layers, and every layer, observed individually, is a layer with full-pitch coils, but the two layers are shifted in space for one stator slot pitch-shortening the coil.
Therefore, the resultant phase MMF, on the harmonic basis, could be seen as a vector sum of the layer's MMF. It is easy to introduce the other winding factor, chord or pitch factor, as ratio of vector and algebraic sum of layer's fundamental MMF wave:
(4.59)
...or for any other higher space-harmonic n:
(4.60)
In previous expressions, y is the coil pitch and t is the pole pitch, both commonly given in the number of slots. By taking just the defined pitch factor into account, the amplitude of the rotating MMF space-harmonic in a distributed double-layer three-phase winding is as follows:
(4.61)
where kn is the winding factor which includes distribution and pitch factor, kn kpn kdn. In many textbooks of electrical machines, the above expression is given in the following form:
(4.62)
where Neff kn Nphase is the so-called number of effective turns per phase and where I is the rms value of phase current. As kp1 < 1 for the shorted-pitch winding, it is implied that this measure has an impact on the fundamental MMF wave amplitude in such a way that this amplitude is somehow smaller than in full-pitch winding. However, the main result of the use of short-pitch coils is the attenuation of fifth and seventh space harmonics in the rotating MMF wave [8 ]. In fact, in order to cancel fifth harmonic the rotating MMF wave, the following condition must be satisfied:
(4.63)
where k is an integer, k 0,1,2, . . . and k must be chosen in order that the coil pitch y will be smaller than and close to the pole pitch expressed in the number of stator slots. For analyzed winding, t 9 slots, so the reasonable solution for y for…
As the coil-pitch must be an integer, possible solutions are y 7or y 8.
Obviously, for any choice, fifth space harmonic could not be eliminated, but for y 7 it will be significantly reduced. Similarly, for canceling out seventh space harmonic, the following condition must be satisfied:
(4.64)
Fig. 15 Rotating MMF wave in two different instants of time and the corresponding
Fourier's coefficients for machine from Fig. 14.
Two-pole machine. Every single coil has 5 turns.
Fig. 16 Comparison of MMF wave spectrum for full and shorted-pitch winding
from Figures 4.11 and 4.14
Again, y could be 7 or 8, and seventh space harmonic could be only attenuated, especially for y 8, but not absolutely eliminated from the MMF waveform. If coils with 7 stator slot pitch are chosen, fifth harmonic will be attenuated more than seventh. Inversely, if coils with 8 stator slot pitch are chosen, seventh space harmonic will be attenuated more than fifth. In the analyzed case study, coils with 8 stator slot pitch are used (one stator slot pitch shortening), so seventh space harmonic will be attenuated much more than fifth, as could be easily concluded by comparison of the rotating MMF wave spectrum from Figures 13 and 15, which is given again in Fig. 16. Usually, the coil pitch is chosen in order to attenuate simultaneously fifth and seventh space harmonic as much as possible, and the aim is fulfilled for the following coil pitch:
(4.65)
Along with the additional condition that k must be chosen in such a way that coil pitch y is slightly shorter than the pole pitch, condition (4.65) could be alternatively defined as follows:
(4.66)
As illustrated in Fig. 16, shortening the coils has also positive impact on 11th and 13th harmonics. However, from Fig. 16 it is observed that shortening the coils has no impact on the intensity of the slot harmonics. Why? Coil, regardless of its pitch, must begin in one slot and end in the other; therefore, the coil's pitch is an integral multiple of the slot pitch causing slot harmonics in the first place. It can be easily concluded from pitch factor for slot harmonics:
(4.68)
Obviously, this factor is the same as for the fundamental harmonic.
3 Rotating MMF wave-analytical approach
A general term for the rotational MMF of symmetrical three-phase machine fed by symmetrical three-phase voltage supply is easily reached also by following analytical procedures. Ac flowing through the phase A winding produces the following nth pulsating MMF harmonic:
(4.69) --(4.75)
... while the sum of backward waveforms gives the backward waveform of the rotating MMF:
(4.76)
The fundamental MMF harmonic of phase windings, i.e., n 1, results in forward rotating MMF waveform in which its amplitude is 3/2 times higher than the amplitude of constituent MMF waves while backward rotating MMF waveform does not exist:
(4.77)
(4.78)
A similar situation could be analyzed for higher space harmonics. For example, for n 3, there is no forward or backward MMF waveform.
(4.79) (4.80) (4.81) ... while backward rotating MMF exists:
(4.82) (4.83)
For n 7, there is forward rotating MMF as follows:
(4.84) (4.85) (4.86)
Remark 2: The current in phase A is given by iA(t) Imsin(wt). If the current in phase B lags by 2p/3 and the current in phase C lags by 4p/3, there is a direct system of three-phase ac:
It should be noted that lagging of currents is described by minus sign in analytical expressions of currents. On the other hand, phase A MMF in p pair poles machine is NA(qmeh) Nmsin(pqmeh). If phase B winding leads to space by 2p/3p mechanical radians and phase C winding leads to space by 4p/3p mechanical radians; then analytical expressions for MMFs are:
Now, the negative sign corresponds to leading of phases in space.
Described order of currents and windings gives forward rotating MMF wave.
So the conclusion is as follows:
The symmetrically constructed (identical phase windings with identical series number of turns per phase) and the symmetrically distributed three phase winding (phase windings space shifted by 2p/3 electrical radians) fed by symmetrical system of three-phase ac voltages (ac phase currents phase shifted by 2p/3) produce symmetrical rotating MMF waves. The angular speed of this wave is known as synchronous speed.
The described rotating MMF wave is not the result of mechanical rotation of any part of the electrical machine; rather, it is produced by the stationary phase windings powered by ac.
In addition to the fundamental MMF wave, there exist higher space harmonics of MMF that rotate forward or backward by angular speed which is nth times smaller than the synchronous speed. The order of harmonics are given by:
... where k 0, 1, 2, . . .
Noteworthy higher space harmonics are phase belt harmonics, i.e., fifth and seventh, as well as slot harmonics of order S/p 1.
In case of any asymmetry in the machine as different number of turns in phase windings, different phase shift of phase windings, asymmetry in phase currents in magnitude or in phase shift, in addition to the direct rotating MMF wave, the backward rotating MMF wave always exists, i.e., n could have value n 1. As a result, a resultant elliptical MMF wave exists in the air gap as shown in Fig. 17.
Fig. 17 The result of summing the forward and backward MMF waveforms of
the same order is the rotational MMF wave of elliptical shape
4 Fractional slot winding
The case in which only winding is embedded in the whole number of slots per pole per phase has been already analyzed. This winding is known as integral slot winding. However, there is also fractional slot winding. This is the case where the number of slots per pole per phase is not an integer. The manner of embodiment of this type of winding is numerous and so complex that it is beyond the scope of this guide. Their common characteristics are unsymmetrical phase as well as rotating MMF wave, so the MMF spectral content could not be described by introduced rule n 6k 1, where k 0, 1, 2, . . .
Fig. 18 Single-layer fractional slot winding (p 2, S 30, q 2.5)
Fig. 19 Fractional slot winding rotational MMF waveform at any time and
the corresponding spectral content.
As an example of such fractional slot winding in a three-phase four-pole machine inserted in S 30 stator slots in a single layer is provided in Fig. 18.
Here, the number of slots per pole per phase is obviously a fractional number:
The waveform of the rotating MMF at instant of time when ia 1A and ib ic 0.5A is given in Fig. 19, and it is asymmetrical. The same figure displays the spectral content of the MMF wave.
The noteworthy point is the existence of one sub-harmonic. Namely, the fundamental harmonic is second one of circa 24 A-turns amplitude. However, the first harmonic now exists with the amplitude of circa 5 A-turns. Additionally, there are now some even harmonics. However, the slot harmonics of order S p, i.e., 28th and 32nd harmonics, which are rather prominent, exist in this case, too.
5 Wound rotor MMF space harmonics
In wound-rotor induction motor, the rotor carries three-phase winding with identical number of pole pairs as stator winding. Therefore, all the previously derived conclusions for the three-phase stator winding are also true for rotor winding. The main difference is in the fact that rotor currents are induced currents of frequency that is different from voltage supply frequency. Fundamental rotor current frequency is,
(4.87) (4.88) (4.89)
if of interest are rotor currents induced by nth MMF space harmonic wave from stator side. Hence, as it was the case for three-phase stator winding, rotor winding produces MMF waves of the same order, m 6q 1 (4.90) where q 0, 1, 2, . . . The most significant harmonics are phase belt harmonics and slot harmonics. The rotor slot harmonics of order R/p 1 exist where R is the number of rotor slots. If rotor winding is of fractional slot type then some additional harmonics could appear as it was explained previously.
6 Cage rotor MMF space harmonics
The MMF waves of symmetrical, integral slot three-phase stator winding of an induction machine has been already derived as follows:
(4.91) where p is the number of pole pairs and Fsm is the amplitude of the mth space harmonic. It is clear that besides the fundamental MMF wave (k 0, m 1), there exist waves with 5p,7p,11p, . . . pair of poles even in the case of symmetrical machine. By assuming uniform air gap length, the flux density waves in the air gap produced by the stator windings have the same waveforms.
The winding function of multi-turn coil in slots described by angular coordinates q1 and q2 has been already defined by ...
(4.92)
... where N is the number of turns and a is a coil pitch, a q2 q1. In case of cage rotor, two nearby bars and ring segments between them form the rotor loop.
Therefore, the rotor loop can be observed as a one-turn coil with pitch a 2p/R, where R is the number of rotor bars. Hence, the winding function of the rotor loop 1, whose magnetic axis is in the center of the reference frame fixed to the rotor, is given by:
(4.93) (4.94) (4.95) (4.96) (4.97) (4.98)
Summing the MMFs of all rotor loops, the resultant MMF of the cage rotor is:
(4.99)
Let us analyze the last expression in the case where rotor currents are due to the mth stator flux density harmonic. By inspection of (4.99), the following rotor MMF waves exist:
for n mp, the first member of the sum is different from zero;
for n mp, the second member of the sum is different from zero;
In the healthy induction machine, there are only stator MMF waves of order m 6k 1, k 0, 1, 2,....
It means that the rotor cage produces MMF waves that are armature reaction to these MMF waves from the stator side. It should be noted that n can be only positive integer, n 1, 2, 3, . . . .
However, there are cage rotor MMF waves in the following two cases:
for n mp lR or n lR mp, where l 1, 2, 3 . . . the first member of the sum is nonzero.
for n mp lR or n lR mp, where l 1, 2, 3 . . . the second member of the sum is nonzero.
Therefore, in addition to the fundamental cage rotor MMF wave which is the armature reaction to the stator MMF wave, there is also the so-called rotor slot harmonics of order lR mp, l 1, 2, 3, . . . .
These harmonics are the direct con sequences of the space distribution of rotor bars, i.e., placement of rotor cage in slots. Consequently, the rotor currents caused by the mth stator flux density space harmonics generate the following MMF waves:
(4.100) (4.101) (4.102) (4.103) (4.104)
... these MMF waves, now observed from the stator side are ...
(4.105) (4.106) (4.107)
Fig. 20 shows the waveform of the fundamental cage rotor MMF wave (m 1) at an instant of time assuming that the amplitude of the rotor loop current is 1 A.
The rotor has R 40 bars and the machine has one pair of poles, p 1.
Fig. 21 illustrates the spectral contents of MMF wave extracted from Fig. 20. It is obvious that besides fundamental harmonic (mp 1st), there exist first-order rotor slot harmonics, for l 1, R p and R p, second-order rotor slot harmonics, for l 2, 2R p and 2R p, etc. (i.e., 39th, 41st, 79th, 81st). This directly follows from (4.106) and (4.107).
Fig. 20 Cage rotor MMF at an instant of time (p 1, R 40, m 1)
Fig. 21 Spectral content of cage rotor MMF given in Fig. 20
Fig. 22 Cage rotor MMF in an instant of time (p 1, R 40, m 5)
Fig. 22 displays the waveform of the cage rotor MMF, produced by rotor currents that are the result of m 5 under the assumption that the amplitude of rotor loop current is 1 A. In this case, the number of rotor bars per one pole of harmonic field is less than before; as a result, more rugged waveform is obtained.
The spectral content of the waveform in Fig. 22 is shown in Fig. 23.
It is obvious that in addition to the ''fundamental'' harmonic (now, mp fifth), the first-order rotor slot harmonics of the order (R 5p), (R 5p), (2R 5p), (2R 5p), i.e., 35th, 45th, 75th, 85th will appear. This result is consistent with (4.106) and (4.107) as well. It is now clear that the amplitude of rotor slot harmonics is more pronounced in relation to the ''fundamental'' harmonic when compared to the situation in Fig. 21.
However, the main conclusion can be derived from (4.105)-(4.107):
The cage rotor of induction machine reflects all MMF space harmonics from stator side at the fundamental frequency f1 and at frequencies (1 lR(1 s)/p)f1.
Fig. 23 Spectral content of cage rotor MMF given in Fig. 22