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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.

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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):

ddRT gcog

2

21 (5.1)

where,

g is the air-gap flux and

R is the air-gap reluctance.

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From Equation (5.1), cogging torque can be reduced either by

forcing the air-gap flux g , or the rate of change of air-gap reluctanceddR , to

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

1)sin(

kcmkcog mkTT (5.2)

where,

m is the least common multiple of the number of stator slots and

the number of poles

ck is an integer

Tmk 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.

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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.

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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

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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

,1sm

m Nn

10 (5.3)

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where,

m is the ratio of pole-arc to pole-pitch

n is an integer

Nsm 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 ith 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

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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

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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

of effective method to eliminateddR . For an unskewed machine, the value of

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.

1 0

sin1k

mks

cog

s

dmkTT (5.4)

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Figure 5.7 Machine with skewed magnet

Where,

s is one period of cogging torque which is obtained from:

ms2 (5.5)

Figure 5.8 Cogging torque with skewing effect

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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:

1sin

kslkNcog kNTT

s (5.6)

The fundamental frequency of Equation (5.6) is Nsl (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 Ns. 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

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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.

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Figure 5.10 Introduction of dummy slot in stator

Figure 5.11 Cogging torque Vs rotor position for different slot opening

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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.

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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.