1. Introduction
Permanent magnet motors use permanent magnets with higher energy density than induction
motors, so they have a higher output density and many other advantageous characteristics,
such as efficiency, power factor, and controllability. If the shape of the permanent
magnet is appropriate, it can satisfy specifications required by the load or system,
so the degree of design freedom is very high. Because of these advantages, permanent
magent motors are widely used in fields that require high-quality motors, and their
application as industrial motors is increasing. However, they are more expensive than
induction motors and require a controller as they are not self-driving. Also, they
produces more noise and vibration than an induction motor.
Permanent magnet motors have cogging torque because they use permanent magnets. Because
of the presence of inserted permanent magnets and the slot opening width of the rotor,
cogging torque occurs and the back EMF waveform includes space harmonics. Cogging
torque deteriorates the controllability of the motor and causes noise/vibration, so
reducing cogging torque is a very important objective when designing a drive motor
for an electric vehicle or an EPS motor.
Methods to reduce cogging torque include rotor or stator skew, the use of fractional
slots, increasing the number of slots, minimizing the slot opening width, shoe notch,
shoe shape imbalance, and increasing the air gap length. The most common way to reduce
cogging torque is to apply skew to the rotor. In the case of SPM (Surface Permanent
Magnet) where the permanent magnet is attached to the surface of the rotor, general
skew is applied. However, in the case of IPM (Interior Permanent Magnet) where a permanent
magnet is inserted inside the rotor, general skew cannot be applied, so step skew
is used[1].
The 8-pole 12-slot permanent magnet motor designed in this paper is an IPM with integer
slots and is used as a steering motor, so reducing cogging torque is vital. This paper
describes methods for reducing cogging torque and discusses the characteristics of
each method. In addition, step skew was applied to reduce the cogging torque of the
8-pole 12-slot model, and the harmonic cogging torque waveform was anlyzed to study
the optimal number and angle of step skews to reduce cogging torque. Additionally,
the reasons why the design value and the measured value of cogging torque do not match
were studied and allowed for the identification of things to keep in mind when measuring
cogging torque.
2. Main Discourse
2.1 Method for reducing cogging torque
Cogging torque is a fundamental component of motors using permanent magnets, and it
causes torque ripple and noise/vibration, which can reduce motor performance. Therefore,
it must be considered when designing a permanent magnet motor. Equation (1) shows the general equation for cogging torque.
∵ Here, n=kS, K=1, 2, 3, …
S: Least common multiple of number of slots and number of poles
D: Rotor outer diameter
Lsk: Stator stack length
σ: Skew angle
$\dfrac{\sin(n\sigma L_{sk})}{n\sigma L_{sk}}$: The effect of skew
Λn: nth spatial harmonic of the gap permeance
fn: nth spatial harmonic of magnetic flux distribution by a magnet
ζ: Rotation angle of the rotor
A way to drastically reduce cogging torque is to use fractional slots for the number
of poles and slots. In the case of fractional slots, additional design is not necessary
to reduce cogging torque and torque ripple compared to integer slots, so the power
density can be increased. However, the impact of noise/vibration is considerably greater
than motors using integer slots due to rotor eccentricity. Minimizing the slot opening
width to reduce cogging torque makes it difficult to wind, and the method of eliminating
the slot opening width requires an additional process of fixing the back yoke. In
addition, the method of making the shape of the shoe uneven or applying a notch to
the shoe increases the effective gap, which reduces the output and does not have a
large effect on reducing cogging torque. Additionally, skewing the stator has the
disadvantage of lowering the space factor as there is a limitation on the winding
nozzle. Also, reducing the rotor outer diameter and lamination length in Equation
(1) cannot be considered as a design variable because it is directly related to the performance
of the motor. In this paper, step skew is applied to effectively reduce cogging torque.
Fig. 1 shows the 8-pole 12-slot motor designed in this paper.
Fig. 1. Designed 8-pole 12-slot PMSM
2.2 8 pole 12 slot step skew design
General skew applies skew equal to the slot pitch, but when a permanent magnet is
inserted into the rotor core, diagonal skew is impossible, so step skew is applied.
The angle of step skew varies depending on the number of steps. The equation for calculating
each step skew angle is expressed as equation (2).
$N_{step}\times 2 :(N_{step}\times 2)-2 =\alpha :\beta$
∵ Here, Nstep: Number of steps
⍺: One cycle of cogging torque
By applying equation (2), the angle of each step skew is calculated; the 2-stage step skew is 7.5⁰, the 3-stage
step skew is 5⁰, the 4-stage step skew is 3.75⁰, and the 5-stage step skew is 3⁰.
In this section, the cogging torque of the motor rotor model was compared and analyzed
without step skew and the model with step skew was applied from the 2nd to the 5th
stage. Fig. 2 shows the rotor with step skew applied.
In order to interpret the characteristics according to the step skew, three-dimensional
analysis is required. Two-dimensional analysis assumes that a physical phenomenon
is constant in the Z-axis, but when step skew is applied, the magnetic flux pattern
is not constant in the Z-axis, so 3D FEM design is required. Fig. 3 shows the cogging torque according to the step skew. Compared to the model without
the step skew, the cogging torque of the two-step skew model decreased by 56.4%, and
the cogging torque of the three-step skew model decreased by 76.7%. In addition, the
cogging torque did not decrease further when the four-step and five-step skews were
applied. This phenomenon can be understood through the results of the analysis of
cogging torque harmonic according to the step skew in Fig. 4. The analysis results show that most of the cogging torque is composed of first and
second harmonic components. When applying a 2-stage step skew, the fundamental wave
of the cogging torque generated from the upper magnet and the fundamental wave of
the cogging torque generated from the lower magnet have an electrical phase difference
of 180⁰, so the fundamental wave components are canceled out. However, the 2nd, 4th,
and 6th orders are not reduced. In addition, when applying a 3-stage step skew, a
phase difference of 120⁰ occurs, so the 1st and 2nd components are canceled out, but
the 3rd, 6th, and 9th components are not canceled out. Additionally, in the case
of a 4-stage step skew, a phase difference of 90⁰ occurs, so the 1st, 2nd, and 3rd
components are canceled out, but the 4th, 8th, and 12th components are not canceled
out. Therefore, a 3-stage step skew, which reduces the 1st and 2nd components, is
the most effective design for reducing cogging torque. If the third harmonic of the
cogging torque is large, a four-stage step skew design is required.
Fig. 2. The rotor of the motor with step skew
Fig. 3. Cogging torque according to step skew
Fig. 4. Analysis of cogging torque harmonics by step skew
Fig. 5. Torque according to step skew
Fig. 5 shows the torque according to the step skew. When the step skew is applied to the
rotor, torque is reduced because the direction of the magnetic flux generated from
the permanent magnet is divided, so the total magnetic flux is slightly reduced. This
is a similar concept to the difference in the distribution of the magnetomotive force
between the concentrated winding method and the distributed winding method. In addition,
the existence of axial leakage of the permanent magnet is also one of the causes.
When step skew is applied, torque is somewhat reduced, however the torque ripple rate
is significantly reduced. Compared to the model without the step skew, the torque
ripple rate is reduced by 50.1% for the 2-step step skew, 73.4% for the 3-step step
skew, 76.7% for the 4-step step skew, and 77.0% for the 5-step step skew. Therefore,
considering cogging torque, output, and torque ripple rate, applying the 3-step step
skew is the most optimal method. When applying the 3-step step skew, the cogging torque
was 0.04 Nm, the average torque at maximum output was 6.3 Nm, and the torque ripple
rate was 1.7%, which was the intended result.
2.3 Characteristics of cogging torque considering iron loss
In general, when comparing analysis results of cogging torque with the measured results,
the cogging torque waveform and values often do not match well. There are various
reasons for the difference in cogging torque results, but this paper analyzes the
effects of stator and rotor iron losses on cogging torque. Cogging torque is analyzed
as no-load without considering iron loss. However, if iron loss is considered, the
waveform and values of cogging torque change depending on the speed. Fig. 6 shows the distribution of no-load iron losses occurring in the stator and rotor depending
on the speed.
Iron loss is the sum of hysteresis loss and eddy current loss, and is proportional
to the magnetic flux and the square of the frequency. Iron loss occurs mostly in the
stator; theoretically, iron loss does not exist in the rotor, but in reality it occurs,
as seen in Fig. 6. This iron loss affects the waveform and values of cogging torque. Fig. 7 shows cogging torque according to the speed. Although cogging torque is not a function
of the speed, it can be seen that cogging torque
Fig. 6. Distribution of no-load iron loss by speed
Fig. 7. Cogging torque considering iron loss
increases as speed increases and the waveform changes. In particular, when the speed
increases, the average of the cogging torque does not become 0, but is pushed to one
side, and as can be seen at point A, a phenomenon occurs where the rotor does not
have periodicity and droops at the initial position. It is thought that this phenomenon
is due not only to the magnetic flux by the permanent magnet but also to the iron
loss component generated in the stator.
2.4 Cogging torque measurement method
Cogging torque is proportional to the square of the permanent magnet flux and is proportional
to the rate of change of in magnetic resistance. Cogging torque does not change with
the speed[3-6]. However, when cogging torque is measured with respect to speed, cogging torque often
changes. This is often because the accuracy of the torque sensor decreases at high
speeds or the cogging torque equipment is not set properly. In addition, as mentioned
in Section 2.3, cogging torque changes due to iron loss. In addition, if the cogging
torque analysis and measurement results are different, rotor eccentricity may have
introduced during manufacturing or noise generated from other equipment may have been
transmitted to the test motor through coupling. Also, it may vary depending on the
coupling clamping chain. Fig. 8 shows the steering motor designed in this paper, and Fig. 9 shows the 3D analysis and measurement results with iron loss not reflected and a
3-step step skew applied. Although the interpretation and measurement results differ
by about 2 times, the cogging torque value is quite small, so it is judged that the
difference is caused by various factors mentioned in this paper. Fig. 10 shows cogging torque measurement results according to speed. From the measurement
results, two things to keep in mind when measuring cogging torque can be seen. First,
filter adjustment according to speed is necessary. The low pass filter is the main
filter and is set to the cogging torque frequency when measuring cogging torque. Usually,
cogging torque equipment manufacturers often set it to 30Hz. A smooth filter is used
to remove noise that is not removed by the low pass filter. As can be seen from the
measurement results, if the filter value according to speed is not adjusted, cogging
torque changes according to speed. Second, it is desirable to measure cogging torque
at as low a speed as possible. From the measurement results, it can be seen that cogging
torque values at high speed change rapidly. This result is because, as explained in
Fig. 7, iron loss increases according to speed, and the waveform and value change of cogging
torque. Therefore, cogging torque should be measured at the lowest possible speed.
The general motor is measured at 1rpm, and the steering motor is measured at 30rpm.
Fig. 11 shows the results of cogging torque measurements according to the tigthening the
coupling chain. The coupling chain tighteness was adjusted using a torque wrench.
In the case of ultra-small motors, cogging torque values often vary depending on the
coupling chain tighteness, but it did not vary significantly in the motor developed
in this study.
Fig. 8. Cogging torque measurement
Fig. 9. Cogging torque analysis and measurement result
Fig. 10. Cogging torque measurement results according to speed
Fig. 11. Cogging torque measurement result by coupling tightening