95 CHAPTER 5 ANALYSIS OF COGGING TORQUE 5.1 INTRODUCTION In modern era of technology, permanent magnet AC and DC motors are widely used in many industrial applications. For such motors, it has been a challenge for the designers to minimize the torque fluctuations which may cause vibrations, noise and speed fluctuations. These factors have two key components: torque ripple and cogging torque (Zhu & Howe 2000). Torque ripple is caused by the fluctuations generated in the field distribution and the armature MMF. Cogging torque is caused by the interaction between the stator air-gap permeance and permanent magnet MMF. Cogging torque is the main source of creating the torque ripple in PM machines. When the motor runs at high speed, the torque ripple is usually filtered out by the system inertia. In the case of lower speeds, torque ripple may result in undesirable speed variations, vibrations, and acoustic noise. Due to this, the machine performance is affected significantly. Reducing cogging torque is often a major concern during the design of PM machines, since it is one of the main sources of speed fluctuations. Figure 5.1 shows the variation of cogging torque of the proposed PMSRM with respect to different rotor positions. It is observed that the cogging torque of 1.2 N-m was generated by the proposed PMSRM which is to be reduced. If the cogging torque is reduced, the overall torque generated by the machine and its performance could improve to satisfy the optimum performance.
96 Figure 5.1 Cogging torque of the proposed PMSRM In order to eliminate the cogging torque theoretically, it is very important to investigate the equations which define it. In practice, however, the cogging torque cannot actually be eliminated, but it can be reduced (Zhu et al 2003). In the most fundamental form cogging torque can be represented as (Hanselman 1994): T cog 1 2 2 g dr d (5.1) where, g is the air-gap flux and R is the air-gap reluctance.
97 From Equation (5.1), cogging torque can be reduced either by dr forcing the air-gap flux g, or the rate of change of air-gap reluctance, to d be zero. It is not possible to make g as zero because some amount of air-gap flux is needed for the alignment and reluctance torque components for driving the machine. Therefore, the better option for reducing the cogging torque is to force the air-gap reluctance to be a constant with respect to the position of rotor. Cogging torque can be represented in terms of Fourier series as T cog k 1 T mk sin( mk c ) (5.2) where, m kc is the least common multiple of the number of stator slots and the number of poles is an integer T mk is a Fourier coefficient and is angular position of rotor It is because, cogging torque is a summation of sinusoidal harmonic component. In case of traditional machines, where there are no cogging torque reduction techniques, the rotor magnets will contribute an additive effect to cogging torque. It is because each magnet has the same relative position with respect to the stator slots. The torque generated from each magnet is in phase with the others, and as a result, the harmonic components of each are added. By properly designing a machine in such a way that the magnets are out of phase with each other the effect of cogging can be minimized.
98 5.2 METHODS OF REDUCING COGGING TORQUE Various techniques used for reducing the cogging torque are discussed in this chapter. Most of the techniques used are successful in reducing the undesired cogging torque, but they also reduce the desired mutual torque (Islam et al 2004; Ishikawa 1993). Conventional methods to reduce cogging torque are (Li & Slemon 1988): Using increased length of the air-gap Using fractional slots/pole Decreasing the width of the slot openings Using the increased air-gap length to decrease the cogging will increase the amount of the PM material, because low permeability of the air will rapidly increase the required MMF (Hwang et al 2001). Fractional slot/pole design will make the machine design more complicated, and it also leads to a higher harmonic content of the air-gap flux. It was shown by (Li & Slemon 1988), however, that with a proper design, the torque ripple of a fractional slot machine can be kept small. If the slot openings with decreased widths or even with semi-magnetic slot wedges are applied, the tooth-tip and the slot-leakage inductance will increase thus decreasing the torque production capability of the motor. It is also possible to decrease the cogging torque not only by the proper machine design, but also by modulating the inverter current waveform. Numerous papers have been written on this topic, as well as on the other control-based methods (Jahns & Soong 1996; Bianchi & Bolognani 2002; DeLaRee & Boules 1989). With the increased interest of the researchers, many new methods of reducing the cogging torque has been introduced in recent years (Dosiek & Pillay 2007). The most common methods of reducing cogging torque is discussed in this chapter and a better method is opted for the design of proposed PMSRM.
99 5.2.1 Introduction of Shoes in Stator Teeth The simplest and easiest way to eliminate cogging torque is to design a slot-less stator so that the saliency is eliminated. This type of design enables the generation of a constant air-gap reluctance that would minimize the cogging torque. However, in practice, this is not a feasible method due to the requirement of opening for winding. The alternate solution of reducing the cogging torque is by the addition of shoes to the stator teeth as shown in Figure 5.2. Figure 5.2 Air-gap reluctance with shoes in stator teeth The addition of shoes allow the stator inner surface to be mostly steel, thereby decreasing the air-gap reluctance variation. This type of arrangement also allows space for the insertion of the stator windings. In general, it is established that the variation in the magnitude of the cogging torque decreases with increased shoe size. Apart from reducing cogging torque, the main advantage with the addition of shoes is that the performance of the machine is not affected by this method. The disadvantage is that winding inductance is increased. The value of cogging torque with respect to
100 different slot openings is illustrated in Figure 5.3. It is observed that the value of cogging torque is minimum with no slot opening. It is practically not possible because the stator windings needed to get into the slots. The actual size of the shoes is selected so that the slot opening would be just large enough to allow the stator windings to fit. This value created a slot opening of 2mm. Also, it is seen that the value of cogging torque is nominal when the slot opening is at 2mm. Figure 5.3 Cogging torque as a function of slot opening 5.2.2 Optimal Magnet Arc Magnet pole-arc is a well established technique that can have a large effect on the amplitude of cogging torque. For minimizing the value of cogging torque, there is an optimum value for pole-arc that can be found using m n N sm 1, 0 1 (5.3)
101 where, m is the ratio of pole-arc to pole-pitch n N sm is an integer is the number of slots per pole and is the parameter that is varied to minimize cogging For any given value of n, there exist i values of that can minimize the i th harmonic of the cogging torque. Determining the proper values of is not trivial and requires the use of FEM. Finite element method determines the magnet pole-arc that can minimize cogging torque by changing several values of magnet arc shown in Figure. 5.4. Figure 5.4 Cogging torque as a function of magnet arc
102 The angle of the arc was varied for values of n between 4 and 5 to satisfy Equation (5.3). It was observed that an arc of 105 degree electrical produces lower cogging torque of 0.43N-m. 5.2.3 Magnet Edge Shaping By properly shaping the edges of the permanent magnets the rate of change of the air-gap flux density can be varied significantly. The variation of air-gap flux caused by the magnet edges, as one moves from one magnet pole to the next minimizes the cogging torque. In case of the conventional magnet shapes, the transition from magnet to non-magnet is immediate, and hence the rate of change is very high. This can be overcome by shaping the magnets, so that the thickness of magnet is smaller near the magnet edges as shown in Figure 5.5. As a result, the transition from rotor material or air to the magnet is more gradual, thereby reducing the rate of change of the air-gap flux density and the cogging torque. This effectively reduced the cogging torque to 0.4 N-m based on FEM as shown in Figure 5.6. (a) Flux barrier with normal magnet (b) Flux barrier with edges reshaped Figure 5.5 Magnet shaping of PMSRM
103 Figure 5.6 Comparison of cogging torque of original machine and magnet reshaped machine 5.2.4 Skewing Generally skewing is performed either in the stator or in the rotor along the axial length of the machine as shown in Figure 5.7. Skewing is one dr of effective method to eliminate. For an unskewed machine, the value of d instantaneous air-gap reluctance is uniform along the axial length. The reluctance varies with the period as the rotor rotates, thus generating cogging torque. The stator or rotor of a machine is skewed in such a way that the total circumferential angle of skewing is equal to one period of the air-gap reluctance variation. Each permanent magnet is instantaneously subjected to the variation of reluctance and thus the value of the cogging torque is varied. s 1 T T sin mk d (5.4) cog s k 1 0 mk
104 Figure 5.7 Machine with skewed magnet Where, s is one period of cogging torque which is obtained from: s 2 m (5.5) Figure 5.8 Cogging torque with skewing effect
105 The required angle of skewing to eliminate cogging torque was chosen to be 15 degrees from Equation (5.5). It was found in results shown in Figure 5.8 that the cogging torque is reduced to 0.35 N-m. Although continuous skewing theoretically reduces cogging torque, some residual will still remain in practice due to end effects and rotor eccentricity. Skewing along with reducing the cogging, also removes most of the harmonics of the back EMF. The problem of skewing is that it adds torque ripple in the machines fed with trapezoidal currents. Also skewing includes the difficulty in manufacturing and an increased winding resistance in the case of the skewed stator. 5.2.5 Fractional Pitch Winding In machines that have integral pitch windings, it is common that the poles will have a whole number which is in multiple of stator teeth. Thus the cogging effects of each magnet are in phase and added. The cogging torque generated by each magnet is given by: T T sin N k (5.6) s cog k 1 N k sl The fundamental frequency of Equation (5.6) is N sl (number of slot) times one mechanical rotation. Hence in the case of integral pitch wound machines, the least common multiple of the number slots and the number of poles, is equal to N s. It is because for such type of machines the number of slots present is an integer multiple of the number of poles. By using a fractional pitch winding, each magnet pole is subjected to a fractional number of slots and therefore, the cogging torque contributed by the magnets is out of phase with each other. As a result, the overall cogging torque is minimized. For fractional pitch windings, the least common multiple of the number of slots, number of poles, and the fundamental frequency of the overall cogging
106 torque, is always an integer multiple of the number of slots. Figure 5.9 shows the cogging torque for the fractional pitch winding. The value of cogging torque is observed as 0.45 N-m. Figure 5.9 Cogging torque of fraction pitch winding design 5.2.6 Auxiliary Slotting By adding some dummy slots in the stator tooth cogging torque can be minimized. The dummy slots vary the permeance of the stator to reduce the cogging torque. Dummy slots with the opening of 2mm are shown in Figure 5.10. The variation of cogging torque for 2mm and 4mm slot opening is illustrated in Figure 5.11. It is clear from the results that lower the width of slot opening, the cogging torque is reduced. The effect of including dummy slot is almost equivalent to doubling the number of slots which will increase the cost of production and complexity of design. Hence such methods are not advisable.
107 Figure 5.10 Introduction of dummy slot in stator Figure 5.11 Cogging torque Vs rotor position for different slot opening
108 5.3 SUMMARY The proposed machine is found to have a cogging torque of 1.2N-m which is 12.5% of the rated torque of the machine. Various techniques for reducing this cogging torque are discussed. The first method analyzed for reducing cogging torque is the addition of shoe to the stator teeth. The value of cogging torque is estimated for different slot openings. It is found that when the slot opening is at 2mm the cogging torque is reduced to 25% with the value of 0.9N-m. The lower value of slot opening can further reduce the cogging torque, but stator winding cannot be easily fixed. This method does not reduce the cogging torque significantly. In optimal magnet arc technique, the angle of arc is varied in steps of 5 degrees. It is observed that the cogging torque is reduced to 65% with the value of 0.42 N-m at 105 degrees. The problem with this technique is that the value of back emf is increased due to increase in arc angle. Magnetic shaping is one of the easiest techniques to implement without changing the stator or rotor construction. Only the edges of the magnets are reshaped to evenly distribute the air-gap flux. The cogging torque is effectively reduced to 67% with the value of 0.4 N-m. On skewing the rotor, cogging torque is reduced to a minimum value of 0.35N-m. However this method increases the manufacturing cost and complexity in design. Use of fractional pitch winding reduces the cogging torque to a value of 0.45N-m. This method is not attractive due to the complexity involved in winding construction.
109 On adding auxiliary slotting in stator, cogging torque can be reduced to 70% with the value of 0.35N-m. Due to the introduction of dummy slots, the stator area is to be increased which increases the size of the machine as well as cost. On analyzing various techniques it is proposed to implement the magnetic shaping method because, it is easy to implement and effectively reduces the cogging torque up to 67% with the value of 0.4N-m.