Cogging Torque Measurement,
Moment of Inertia
Determination and Sensitivity
Analysis of an Axial Flux
Permanent Magnet AC motor
P.W. Poels
DCT 2007.147
Traineeship report
Traineeship performed at the Charles Darwin University, Darwin, Australia
Coach(es):
dr. ir. F. de Boer
G. Heins
Supervisor:
dr.ir. M. Steinbuch
Technische Universiteit Eindhoven
Department Mechanical Engineering
Control Systems Technology Group
Eindhoven, June, 2008
ii
Abstract
Many technical applications require a smooth torque. An axial flux permanent magnet
(PM) AC motor is used to achieve this with control methods. Required are motor parameters, such as the moment of inertia. This parameter is determined by calculation with
help of the CAD drawings. To verify the result, an experimental setup is designed. The
resulting difference of 6.6% between the calculated and experimental determinded value
of the moment of inertia is explained with the help of a sensitivity analysis.
One of the properties of the type of motor used to achieve smooth torque is the presence of
cogging torque. To compensate for cogging torque, this parameter needs to be measured.
To be able to do this, a measurement method is designed.
To explain the resulting RM Serror a sensitivity analysis of the calibration method is made.
This is done theoretically and verified experimentally. The remaining RM Serror of 0.8 −
1.5% is caused by the current sensor and control errors.
iii
iv
Samenvatting
Voor vele technische toepassingen is een constant koppel vereist. Door enkele regel methodes toe te passen op een axiale flux permanente magneet (PM) AC motor wordt dit
bereikt. Hiervoor is het nodig om de motor parameters te weten, zoals het massatraagheidsmoment. Met behulp van de CAD tekeningen wordt deze parameter berekend. Met een
experiment wordt de berekende waarde geverifieerd. Het verschil van 6.6% tussen deze
twee waarden wordt verklaard aan de hand van een foutenanalyse.
Een van de eigenschappen van het type motor dat gebruikt wordt is cogging. Door deze te
meten kan hiervoor gecompenseerd worden. Voor deze meting is een opstelling bedacht.
Om de resulterende RM Sf out te verklaren wordt een foutenanalyse van de kalibratie
methode gemaakt. Allereerst gebeurt dit theoretisch. Hierna zijn de antwoorden experimenteel geverifieerd. De overgebleven RM Sf out van 0.8 − 1.5% wordt veroorzaakt door
de stroomsensor en regelfouten.
v
vi
Contents
Abstract
iii
Samenvatting
v
Table of contents
vii
Nomenclature
ix
1
Introduction
1
1.1
Motor Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Report overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2 Determination of the Moment of Inertia
2.1
Determination of the moment from the CAD drawings . . . . . . . . . . .
5
2.2 Experimental determination of the Moment of Inertia . . . . . . . . . . .
7
2.3
3
5
2.2.1
Method of determining the Moment of Inertia experimentally . . .
7
2.2.2
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2.3
Results of the experiment . . . . . . . . . . . . . . . . . . . . . . .
10
Comparison and sensitivity analysis of the results . . . . . . . . . . . . . .
13
2.3.1
Sensitivity analysis of the experiment . . . . . . . . . . . . . . . .
13
2.3.2
Improvements of the experiment . . . . . . . . . . . . . . . . . . .
18
Measurement of the Cogging Torque
19
3.1
Defining problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2
Measurement method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
vii
Contents
4 Sensitivity Analysis of the Calibration
4.1
5
25
Calibration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4.1.1
Determination of y
~ and ~
z over operating range . . . . . . . . . . .
28
4.1.2
∗
~ ∗ and ~
Determination of α
~ ∗, β
τcog
. . . . . . . . . . . . . . . . . .
28
4.1.3
~ and ~
Compensation for α
~, β
τcog . . . . . . . . . . . . . . . . . . .
28
4.2 Theoretical Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . .
29
4.3
Measurements for verification of the calibration method . . . . . . . . . .
30
4.4 Comparison of the Theoretical and practical sensitivity . . . . . . . . . . .
34
Conclusion
39
Bibliography
39
viii
Nomenclature
Symbols from chapter "Determination of the Moment of Inertia"
δ
= Logarithmic increment
θ
= Rotation of the electric motor (rad)
ζ
= Damping ratio
F
= Force (N m)
J
= Moment of inertia (kg · m2 )
K
= Spring stiffness (N m)
L
= Distance to the rotation point (m)
M
= Mass attached to a sping (kg)
m
= Mass of a spring (kg)
T
= Driving torque of the electric motor (N m)
t
= (Period) time (s)
x
= Elongation of the spring (m)
x1
= Maximum of the first oscillation
x5
= Maximum of the fifth oscillation
Symbols from chapter "Measurement of the Cogging Torque"
2p
= Number of motor poles
τcog
= Cogging torque (N m)
ix
Nomenclature
HCF = Highest Common Factor
Np
= Number of periods of the cogging torque in a slot pitch rotation
Q
= Number of stator slots
Symbols from chapter "Sensitivity Analysis of the Calibration"
τ̄m
= Φ × 1 vector of the mean torque
δ
= torque sensor scaling error
= current inverter scaling error
τrated = Rated torque of the electric motor (N m)
N
= Number of measurements points
RM Serror = Root Mean Square error of the motor torque
α
~∗
= 3 × 1 vector of current scaling error estimate
αp
= current scaling error in phase p
α
~
= 3 × 1 vector of current scaling error
~∗
β
= 3 × 1 vector of current offset error estimate
βp
= current offset error in phase p
~
β
= 3 × 1 vector of current offset error
•
= element-wise multiplication operator
I∗
= Φ × 3 matrix of the current estimate (A)
I
= Φ × 3 matrix of the current (A) (NOT the identity matrix)
iφ,p
= current in phase p at encoder point φ (A)
K
= Φ × 3 matrix of the back EMF (V s/rad)
kφ,p
= normalised back EMF for phase p at encoder point φ (V s/rad)
x
Nomenclature
∗
~
τcog
= Φ × 1 vector of the cogging torque estimate (N m)
τφcog
= cogging torque at encoder point φ (N m)
~
τcog
= Φ × 1 vector of the cogging torque (N m)
~
τem
= Φ × 1 vector of the electro-magnetic torque (N m)
∗
~
τm
= Φ × 1 vector of the estimate of motor torque (N m)
∗
τm,φ
= estimate of motor torque at encoder point φ (N m)
~
τp∗
= Φ × 1 vector of the pulsating torque estimate (N m)
~
τr∗
= Φ × 1 vector of the estimated mean torque (N m)
X
= Φ × 6 matrix of torque and back EMF
y
~
= 9 × 1 vector of scaling and offset errors
~
z
= Φ × 1 vector of residuals
xi
Nomenclature
xii
Chapter 1
Introduction
Many electric motor applications require a constant torque, especially applications that
require precise tracking. These processes are for instance laser cutting and numerically
controlled machining. Pulsating torque (any kind of variation in the torque output of the
motor) can have a negative effect on, for example, the surface finish when using rotary
machine tools. Also pulsating torque can excite resonances in the drive-train of the application. This produces acoustic noise as well.
A smooth torque output can be achieved by using a programmed reference current waveform. This method has the ability to work at different speed and torque set points. This
minimizes restrictions on motor design and manufacture.
When limiting pulsating torque mechanically, accurate manufacturing is required. This
limits the practicality for low-cost, high volume production.
Research on this subject is performed at the Charles Darwin University (CDU) in Darwin,
Australia. The goal of the CDU electric motor research program is to create an output
torque with a maximum RM Serror of 1%. The contribution to this research explained in
this report consists of three parts:
• For control purposes, the moment of inertia of the motor has to be known. With the
help of CAD drawings of the electric motor, the moment of inertia is calculated. To
verify this result, the moment of inertia is determined experimentally.
• One of the properties of the electric motor used, is the presence of cogging torque.
To achieve a smooth output torque, compensation for the cogging torque is added
to the control scheme. Therefore the cogging torque is measured.
• To improve the result of a programmed reference current waveform, a calibration
1
Chapter 1. Introduction
method is designed. The sensitivity of this calibration method is analyzed theoretically and verified practically. In this analysis also the cause for the remaining torque
ripple is explained.
1.1
Motor Setup
The type of motor used for this research is a Permanent Magnet Synchronous AC motor.
The first natural frequency of the motor mounted on a force table is 700 Hz. The stator
consists of 48 slots and the rotor has 16 poles. The motor has a rated torque of 6N m and
a rated voltage of 24V .
The position of the axle is measured with a 12-bit (4096 states), gray code, absolute encoder. The torque is measured with piezo electric reaction torque sensors. An eddy current
brake is installed to apply different torque set points.
The defined set points are 1,2,3,4 and 5 N m and 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0 Hz. This
provides a mesh which consists of 30 measurements. Data acquisition is done with the
help of Labview.
In figure 1.1 a picture of the electric motor can be seen. In this picture the magnets of the
eddy current brake have been removed.
1.2
Report overview
The contribution to the research consist of three parts, which are divided into three chapters. The determination of the moment of inertia is described in chapter 2. In chapter
3 the cogging torque is measured. The sensitivity analysis of the calibration method is
explained in chapter 4. The conclusions and recommendations of the three parts are combined in chapter 5.
2
1.2. Report overview
Figure 1.1: The electric motor with the different components named. The magnets of
the Eddy Current brake have been removed.
3
Chapter 1. Introduction
4
Chapter 2
Determination of the Moment of
Inertia
The goal of this experiment is to determine the moment of inertia of the electric motor.
This motor parameter is necessary in the control scheme. First of all the moment of inertia
is calculated based on the CAD drawings. To verify this calculation, a experimental setup
is designed to determine the moment of inertia experimentally. A sensitivity analysis of
this experiment is made.
2.1
Determination of the moment from the CAD
drawings
A solid model of the motor is available in Pro/ENGINEER. The moment of inertia of
the assembly of the rotating parts (figure 2.1) can be calculated in Pro/ENGINEER. The
rotation frequency of the bearing cage assembly is 42% of the inner race and the attached
parts [12]. The acceleration is proportional to the moment of inertia. Only 42% of the
moment of inertia of the cage assembly contributes to the total moment of inertia. The
calculated moment of inertia is 0.01056kg · m2 .
5
Chapter 2. Determination of the Moment of Inertia
Figure 2.1: CAD assembly of the rotating components of the electric motor.
6
2.2. Experimental determination of the Moment of Inertia
2.2
Experimental determination of the Moment of Inertia
2.2.1
Method of determining the Moment of Inertia experimentally
The moment of inertia can be determined experimentally by an acceleration or an oscillation method. Both methods are discussed for this particular case. This discussion is based
on work done by Genta en Delprete [4].
Acceleration Method
The rotating parts of the electric motor are constrained. Rotating motion is only possible
about one axis. Somehow, the body is subject to a driving torque T . During the test the
time t and the accompanying rotation θ are measured. The moment of inertia can be
calculated with equation (2.1). The presence of damping is neglected.
J=
T t2
2θ
(2.1)
The acceleration method has a non-periodic motion. To decrease measurement errors in
the time and position, long tests need to be performed. This however increases the error
caused by neglecting damping. Due to the presence of the bearings and the eddy current
brake, the influence of damping on the experiment is rather large.
Oscillation Method
Rotating motion is only possible about one axis. To create an oscillating motion, an elastic
spring with stiffness K is attached. Measuring the period time T of an oscillation, in
combination with the damping ratio, the moment of inertia can be calculated:
J=
KT 2
1 − ζ2
2
4π
(2.2)
The oscillation method is periodic. Measuring a number of oscillations reduces measurement errors.
Comparison Acceleration and Oscillation Method
Because of the following reasons there is chosen to use the oscillation method to determine the moment of inertia:
• By measuring a large number of oscillations, the relative error for time and position
measurements is reduced.
7
Chapter 2. Determination of the Moment of Inertia
• With the oscillation method, the measurement can be started after a slowly decaying
motion has been reached. Errors due to initial transients are not present in the
measurements.
• From a number of oscillations the damping ratio can be calculated. This is possible
because the position can be measured with the encoder.
2.2.2
Experimental Setup
To get an oscillating motion around the rotation point of the electric motor a lever (an
aluminum bar) is attached to the rotor. This is done with two, already available, screws.
On the end of the aluminum bar the springs are fixed. The other end of the springs is attached to the solid world. The springs are prestressed. Only the linear part of the springs
is used. To determine the position of the rotating parts of the electric motor during the
oscillation, the encoder is used.
During the oscillation experiment the stator of the electric motor is removed. Otherwise
oscillating motion is not possible due to the cogging torque. The magnets of the eddy
current break are positioned in a way in which they minimize the force exerted on the disc
attached to the axle. Removing the eddy current brake is difficult.
Equation (2.2) is valid for one spring attached to the bar. To get an oscillating motion
two springs, in opposite direction, are attached to the bar. Also the springs are placed at a
distance L of the rotation point. Adapting equation (2.2) gives:
J=
2KL2 2
T 1 − ζ2
2
4π
(2.3)
The length of de bar (L) and the stiffness of the springs (K) has to be chosen. The chosen
length of the bar is 0.5m. An estimation of the J can be made based on the CAD answer
(chapter 2.1) with added the J of the bar. The last one can be calculated with standard
formulas. When the damping is neglected, the period time can be plot as a function of the
spring stiffness, figure 2.2. A spring constant of about 100N/m is chosen. The influence
of a small error in the spring constant on the oscillation time is small. Nevertheless the
oscillation time is reasonable. Due to limitations of the available springs, on each side of
the bar three springs with a total constant of about 100N/m are used. The constant of
every spring is determined experimentally by using F = −Kx. Deriving the equation for
8
2.2. Experimental determination of the Moment of Inertia
Oscillation time for varying spring constants, L = 0.5m
0.5
0.45
Oscillation time t [s]
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0
20
40
60
80
100
120
Spring constant K [N/m]
140
160
Figure 2.2: Oscillation time for various spring constants with L = 0.5m.
9
180
Chapter 2. Determination of the Moment of Inertia
the case with three springs on both sides gives:
J=
(K1 + K4 ) L21 + (K2 + K5 ) L22 + (K3 + K6 ) L23 2
T 1 − ζ2
2
4π
(2.4)
A schematic drawing of the setup with the variables can be seen in figure 2.3.
Figure 2.3: Overview test setup.
2.2.3
Results of the experiment
The result of 30 oscillation measurements (and the steady state) is shown in figure 2.4. As
can be seen, the zero crossings for each measurement is nearly identical, only the amplitude varies. The period time is determined by averaging the time needed for 5 oscillations.
10
2.2. Experimental determination of the Moment of Inertia
30 Oscillation experiments
2450
Encoder counts φ
2400
2350
2300
2250
2200
0
0.1
0.2
0.3
0.4
Time [s]
0.5
0.6
0.7
Figure 2.4: The oscillation of 30 measurements and the steady state
The logarithmic increment [13] is used to determine the factor ζ :
1 x1 δ = ln 4
x5
δ
ζ=√
2
4π + δ 2
(2.5)
(2.6)
The maximum of the first and fifth oscillation are used to minimize errors. In figure 2.5
a measurement for the last oscillation is shown. There are sufficient measurement points
for an accurate determination of the maximum. The moment of inertia determined from
the measurements includes the moment of inertia of the motor, bar and the springs. The
moment of inertia of the springs can be neglected. This is not possible for the mass of
the accelerated part of the springs. According to Thomson [13] the mass which should be
included (also called "effective mass") is
1
3
of the spring mass. The springs are assumed
to be point masses. Due to the large distance to the rotation point, the influence is significant.
Final answer for the experimentally determined moment of inertia is 0.009902kg · m2 .
The standard deviation of the 30 moments of inertia determined, is 0.0038%.
11
Chapter 2. Determination of the Moment of Inertia
Measurements last oscillation of one experiment
2340
Encoder counts φ
2330
2320
2310
2300
2290
2280
0.6
0.62
0.64
0.66
Time [s]
0.68
0.7
0.72
Figure 2.5: Measurement points of last oscillation of a measurement. Also the steady
state is shown.
12
2.3. Comparison and sensitivity analysis of the results
2.3
Comparison and sensitivity analysis of the results
The difference between the moment of inertia calculated from the CAD drawings and
the moment of inertia experimentally determined is 6.6%. Assuming that the calculation
performed in Pro/ENGINEER is perfect, the difference is caused by errors in the experimental method. Therefore a sensitivity analysis of the experimental method is made.
2.3.1
Sensitivity analysis of the experiment
Errors can be caused by the lengths (L1 ,L2 and L3 ), the determined spring constants (K1
- K6 ) and a different weight of the springs. The effective mass of the springs as taken into
account is incorrect.
The percentage errors, as showed in the figures, are related to a realistic error in the
determined variable. On the y-axis the percentage error in the moment of inertia is shown.
An error of 6.6% explains the difference between the calculation and the experiment.
13
Chapter 2. Determination of the Moment of Inertia
Deviation in the spring constants
In figure 2.6 the resulting error in the moment of inertia of an error in the spring constant
can be seen. An error of ±3.0% is equal to a change in the spring constant of ±1N m. The
six lines correspond to an error in one or more (up to 6) spring constants. An error in all
springs can result in an error in the moment of inertia of 6.6%, however this is not very
likely.
Moment of inertia error caused by an error in the spring constants
8
Moment of inertia error [%]
6
4
2
0
K1
K1,2
−2
K1,2,3
−4
K1,2,3,4
K1,2,3,4,5
−6
K1,2,3,4,5,6
−8
−3
−2
−1
0
1
2
Percentage error in the spring constants [%]
3
Figure 2.6: Moment of inertia error for a percentage error in the measured spring constants.
14
2.3. Comparison and sensitivity analysis of the results
Deviation in the measured lengths
The lengths (L1 ,L2 and L3 ) in figure 2.3 are measured. A percentage error of ±0.5% is
a measurement error of ±2mm. The resulting change in the moment of inertia of this
error can be seen in 2.7. An error in all three distances (red line) explains max 2% of the
total difference between calculation and experiment. This particular case, an error in all
three distances, can be caused by an incorrect determination of the rotation point
Moment of inertia error caused by an error in the lengths
2.5
2
Moment of inertia error [%]
1.5
1
0.5
0
−0.5
−1
L1
−1.5
L1,2
L1,2,3
−2
−2.5
−0.5
0
Percentage error in the lengths [%]
0.5
Figure 2.7: Moment of inertia error for a percentage error in the measured lengths.
15
Chapter 2. Determination of the Moment of Inertia
Mass of the springs
The accuracy of the balance used to weigh the springs is ±1g. The mass of the springs is
determined by weighing all of the springs and dividing by six. An error of 1g in the total
mass, is about 0.2g in the mass of one spring. This is ±3%. The result of this error on the
error in the moment of inertia can be seen in figure 2.8.
Moment of inertia error caused by an error in the weighted spring mass
1
0.8
Moment of inertia error [%]
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−3
−2
−1
0
1
Percentage error in the spring mass [%]
2
3
Figure 2.8: Moment of inertia error for a percentage error in the weighted spring mass.
16
2.3. Comparison and sensitivity analysis of the results
Effective mass factor
According to Thomson [13], the effective mass of a spring is
1
3
of its total mass. However
Fox and Mahanty [3] claim that this is wrong. The effective mass depends on the mass
attached to the spring. Also Armstrong [1] and Sears [11] claim this. Speaking of a mass
M attached to a spring with mass m. When
proaches
m
3.
M
m
→ ∞, the effective mass of the spring ap-
On the other hand, when there is no mass attached to the spring (M = 0) the
effective mass is
4m
.
π2
The influence of this varying factor on the moment of inertia error is
shown in figure 2.9. Calculation of the exact effective mass in this case is rather difficult.
This is caused by the nonlinearity of the equation and the difficulty of determining the
attached mass M .
Moment of inertia error for different effective mass factors
0
Moment of inertia error [%]
−1
−2
−3
−4
−5
−6
0.34
0.35
0.36
0.37
0.38
Effective spring mass factor
0.39
0.4
Figure 2.9: Moment of inertia error for different factors for the effective mass of the
springs.
17
Chapter 2. Determination of the Moment of Inertia
2.3.2
Improvements of the experiment
A combination of the uncertainties in the lengths, spring constants, spring mass and effective mass factor can cause the moment of inertia error of 6.6%. Therefore the accuracy
of the experiment should be improved to get a more precise answer. The largest uncertainties are caused by the spring constants and the measured lengths. By taking only one
spring on each side, instead of three springs, the possible error is lowered. Only 1 length
has to be determined instead of 3. This also reduces the importance of the effective mass
factor and the weight of the springs. The length of the bar, currently 0.5m, should be reduced. The error caused by a measurement mistake is reduced quadratically. By reducing
the length of the bar, the measured moment of inertia is dominated by the electric motor
and not by the bar used.
Finally the Eddy Current brake should be removed. This reduces the effect of damping on
the experiment.
18
Chapter 3
Measurement of the Cogging
Torque
One of the properties of the electric motor used in this research is cogging torque. This is
the variation in torque due to the permanent magnets on the rotor having a much greater
attraction to the steel cores than to the copper windings between the steel cores.
To achieve a smooth output torque compensation for the cogging torque is necessary.
Therefore good knowledge (a measurement) of the cogging torque is necessary. A measurement method is designed. The acquired results are analyzed and compared with the
theory.
3.1
Defining problem
It is difficult to measure cogging torque. If measured incorrectly dynamic effects of speed
variations of the stator are present. Determination of a correct measurement method is
difficult. Many publications ([7],[14],[9] and [10] ) present measurement results. However
the explanation about how the measurement is performed is very summary.
Cogging torque measurements can be performed by static, quasi static and dynamic methods. More explanation about these methods can be found in Heins [5]. All measurement
equipment of the electric motor is optimized for measuring during rotation. For this reason a dynamic cogging torque measurement method is chosen.
19
Chapter 3. Measurement of the Cogging Torque
3.2
Measurement method
To measure the cogging torque the rotor should turn without any input current. To accomplish this, the rotor is turned with the help of an external motor. On the rotor a pulley
is mounted, which is connected to the pulley of the external motor with a flexible rubber
belt. The force which is caused by the rotation of the rotor is measured by the reaction
torque sensor which is mounted underneath the housing of the motor. The position is
measured by the encoder. The measurement setup can be seen in figure 3.1.
At the defined speeds (see page 2) data from 24 revolutions is saved. The number of revolutions is bounded by the maximum amount of data which can be sampled in one trial.
Due to the reaction torque sensor compensation for inertial forces is not necessary [5].
Figure 3.1: Setup for measuring the cogging torque.
20
3.3. Results
3.3
Results
The measurement is started at an arbitrary point of the rotation. To get complete revolutions starting at the "zero" point, the 24 revolutions are reduced to 23. By averaging
the noise is reduced. The Discrete Fourier transform (DFT) of the averaged revolution is
showed in figure 3.2. All speed set points are shown.
DFT of 1 revolution (averaged)
0.5 Hz
0.6 Hz
0.7 Hz
0.8 Hz
0.9 Hz
1.0 Hz
3
10
2
Amplitude
10
1
10
0
10
−1
10
−2
10
−250 −200 −150 −100
−50
0
50
Harmonics
100
150
200
250
Figure 3.2: DFT of the averaged revolution for all speeds
The number of τcog periods in a slot pitch rotation (Np ) can be calculated [2]:
Np =
2p
HCF {Q, 2p}
(3.1)
In this equation, 2p and Q are the number of motor poles and the number of stator slots
respectively. The Highest Common Factor (HCF) of Q and 2p is the denominator. Calculating the Np with the motor data from page 2 gives 1. The stator consists of 48 slots. The
main harmonic of the cogging torque is therefore the 48th. This corresponds with the
DFT shown. Other harmonics are also present. The significant harmonics are multiples
of 16. The cause of these harmonics is probably a dislocated slot. In combination with the
16 permanent magnets on the rotor this dislocated slot gives a 16th harmonic. The only
exception to this is the 0th harmonic. This harmonic is caused by the torque sensor.
21
Chapter 3. Measurement of the Cogging Torque
By extracting the multiples of 16 and converting the signal from the frequency domain
back to the time domain, the cogging torque is constructed. Averaging over the speed
range gives figure 3.3.
Measured cogging torque
Torque sensor output voltage (V)
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0
500
1000
1500
2000
2500
Encoder position φ
3000
3500
4000
Figure 3.3: Measured cogging torque averaged over the speed range
22
3.3. Results
The percentage error of the averaged cogging torque can be seen in figure 3.4. The
averaged cogging torque is compared to the cogging torque of every speed. As can be
seen, the amplitude of the error is speed dependent. This is caused by the flexibility of
the rubber belt used to drive the rotor of the motor. This causes the errors for the lower
speeds. In the higher speeds slip did occur. The cogging torque measurements could be
improved by using a stiffer transmission between the two pullies.
Percentage error of the averaged cogging torque
2.5
0.5 Hz
0.6 Hz
0.7 Hz
0.8 Hz
0.9 Hz
1.0 Hz
2
1.5
% Error
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
0
500
1000
1500
2000
2500
Encoder position φ
3000
3500
Figure 3.4: Percentage error of the measured cogging torque
23
4000
Chapter 3. Measurement of the Cogging Torque
24
Chapter 4
Sensitivity Analysis of the
Calibration
A smooth torque output can be achieved by using a programmed reference current waveform. If the calculation is performed perfect, the resulting pulsating torque can be caused
by an unbalance in the current or an error in the cogging torque. The unbalance can be
divided in an offset and a gain error.
To compensate for these errors, a calibration method is designed. From this method a theoretical and practical sensitivity analysis is made. Also the cause of the resulting RM Serror
is explained.
4.1
Calibration Method
The calibration method is designed by Heins [5]. A global approach is shown here.
If the "ideal" currents have been correctly calculated, pulsating torque will only come from
an error in either:
1. the cogging torque; and/or
2. an unbalance in the current, caused by an offset or gain error in the current sensors.
These are shown in pink in figure 4.1, and their determination will be the focus of this
section.
By decoupling the pulsating torque into the components created from each of these errors, it is possible to determine where the errors lie and compensate accordingly. To do
25
Chapter 4. Sensitivity Analysis of the Calibration
Figure 4.1: Block diagram of parameters to be determined (simplified)
this, it is important to note that the cogging torque will be independent of current input.
The cogging torque is redefined as the residual resulting from a least squares minimisation matching the electro-magnetic torque to the measured torque.
The torque of the electric motor with the current offset and scaling factors is:


X
(iφ,p αp + βp ) kφ,p + τ cog 
τ∗ = δ 
m,φ
φ
(4.1)
p=a,b,c
where:
∗
τm,φ
= estimate of motor torque at encoder point φ (N m)
δ = torque sensor scaling error
iφ,p = current in phase p at encoder point φ (A)
αp = current scaling error in phase p
= current inverter scaling error
βp = current offset error in phase p
kφ,p = normalised back EMF for phase p at encoder point φ (V s/rad)
τφcog = cogging torque at encoder point φ (N m)
Equation (4.1) can also be expressed in matrix notation:
∗ = δ (I∗ •K)~
~+~
~
τm
α + (K)β
τcog
where:
∗ = Φ × 1 vector of the estimate of motor torque (N m)
~
τm
I∗ = Φ × 3 matrix of the current estimate(A)
26
(4.2)
4.1. Calibration Method
α
~ = 3 × 1 vector of current scaling error
~ = 3 × 1 vector of current offset error
β
K = Φ × 3 matrix of the back EMF (V s/rad)
~
τcog = Φ × 1 vector of the cogging torque (N m)
A dynamic torque sensor is used which does not measure the average component of the
torque. The estimate of the motor torque is therefore:
∗ =~
~
τm
τp∗ + ~
τr∗
(4.3)
where:
~
τp∗ = Φ × 1 vector of the pulsating torque estimate (N m)
~
τr∗ = Φ × 1 vector of the estimated mean torque (N m)
~ and ~
~
τr∗ is effectively the torque created without α
~, β
τcog :
~
τr∗ = (I∗ •K)δ
Substituting Equations 4.3 and 4.4 into equation 4.2:
~+~
~
τp∗ = δ (I∗ •K)(~
α − 1) + (K)β
τcog
(4.4)
(4.5)
If we concatenate the matrices to let:
X = I•K K
(4.6)
and concatenate the vectors to let:

y
~=
(~
α − 1)δ

~
βδ

(4.7)
and let:
∗
~
z = δ~
τcog = ~
τcog
(4.8)
∗ = X~
~
τm
y+~
z
(4.9)
then:
where y
~ and ~
z are unknown.
Using the Moore-Penrose pseudo inverse is a convenient way conducting a least squares
minimisation. By using this inverse and assuming that ~
z will be the residual, y
~ can be
found.
The residual (~
z ) can then be found by rearranging equation 4.9:
∗ − X~
~
z=~
τm
y
27
(4.10)
Chapter 4. Sensitivity Analysis of the Calibration
4.1.1
Determination of y
~ and ~
z over operating range
y
~ is attributed to errors in the current sensor system, so regardless of speed and torque
set-point it should be constant. ~
z is attributed to cogging torque so should also be independent of operating point. This method will only be valid if y
~ and ~
z are the same
over the entire operating range. One method to ensure that the same y
~ is determined for
all operating points is to combine all tests at different operating points into one long X
(6 × Φ × number of trials). Though this gives only one y
~ for all trials, it does give a
different ~
z for every trial.
4.1.2
∗
~ ∗ and ~
Determination of α
~ ∗, β
τcog
Consideration of y
~ and ~
z and Equation 4.8 suggests the best estimates for the parameters
responsible for pulsating torque are:
α
~∗ = α
~
y
~ 1,2,3
+1
δ
~∗ = δβ
~
β
=
∗ = δ~
~
τcog
τem
(4.11)
(4.12)
(4.13)
(4.14)
Overall system gains
~ and ~
These expressions suggest that while an estimate of β
τcog is possible, an estimate of
α
~ requires a knowledge of the product of δ and .
A method for finding this product and compensating to ensure that the estimate of the
overall system gain is correct is available. Assuming this is possible, analysis will continue
with:
δ = 1
(4.15)
~ ∗ and ~
If δ = 1 then α
~∗ = α
~, β
τem however are still effected by the unknown δ.
4.1.3
~ and ~
Compensation for α
~, β
τcog
∗
~ ∗ and ~
Once α
~ ∗, β
τcog
have been determined from an uncompensated set of measure-
ments over the operating range, they can be used to pre-compensate I∗ to cancel their
affect for future operation. This is done as shown in figure 4.2.
28
4.2. Theoretical Sensitivity Analysis
Figure 4.2: Block diagram of parameters to be determined (simplified)
4.2
Theoretical Sensitivity Analysis
~ and ~τcog can be found from a set of measurements. Some variation
The values for α
~, β
exists in these measurements, therefore the determined vectors are not perfect. The goal
of this research is minimizing the torque ripple(page 2). Therefore only the RM Serror
(equation (4.16)) of the motor torque caused by an error in one, or a combination, of these
three vectors is considered.
The performance of the motor is expressed in a root-mean-square (RM S) value of the
motor torque according to:
k~
τm − τ̄m k2
RM Serror = √
· 100%
N · τrated
(4.16)
~ = 0) and ~
If there are no scaling errors (~
α = 1), no offset errors (β
τcog is known perfectly,
~ one will
the RM Serror (equation (4.16) ) is zero. Following the definition of α
~ and β
find:
αa · αb · αc = 1
(4.17)
βa + βb + βc = 0
(4.18)
From formula (4.17) follows the value of αb and αc after inducing an error in αa , if
defined that αb and αc are equal . In a similar way the values of βb and βc are calculated
after inducing an error in βa .
To induce an error in ~
τcog , the measured cogging torque (chapter 3) is multiplied with a
~ and ~
factor to change the magnitude. The intervals in which the values of α
~, β
τcog are
varied:
The resulting RM Serror due to variation of the variables in their interval for torque
set point 3 is shown in figure 4.3. The first row of three figures is the change of the
29
Chapter 4. Sensitivity Analysis of the Calibration
Variable
Interval
αa
0.8 - 1.2
βa
-1.0 - 1.0
factor of ~
τcog
0.65 - 1.35
Table 4.1: Intervals of the variables
RM Serror which is caused by a change in βa and τcog for three values (the two extremes
and the mean) of the interval of αa . The two other possible combinations are shown in
the second and third row.
As can be seen in the first row, a change of αa with 20% positive, results in a different
RM Serror than a change of 20% negative. This is caused by the nonlinearity in the
determination of αb (equation(4.17)) after αa has been defined.
The change of βa and τcog is symmetric. The influence of a 35% change in τcog is much
smaller than a 20% change in αa .
The chosen intervals give an error which is larger than the maximum allowed RM Serror
of 1%. The maximum variation of the variables αa , βa and τcog to stay below a maximum
RM Serror of 1% is shown in figure 4.4. After the calibration all the variables should
have their ideal value, the RM Serror is equal to zero. Due to fluctuations in temperature
the current offset and scaling have small variations.
4.3
Measurements for verification of the calibration
method
If the proposed method of compensating for scaling and offset errors and cogging torque
works, after calibration, the RM Serror should be minimized. Inducing an error in the
scaling, this induced value should be found after a new calibration. This should also be
valid for an induced offset error. The results for these two experiments are shown in table
4.2. The error is the difference between the calculated,and induced value, from a new set
of measurements. Also the resulting RM Serror is calculated. As can be seen, the error
between the induced and calculated α
~ is small. Besides the error in phase c, this is also
the case for the offset compensations. The large error in phase c can be caused by an error
in the first calibration. Though the error is relatively large, the resulting RM Serror is
still within the allowed interval.
30
4.3. Measurements for verification of the calibration method
Figure 4.3: Resulting RM Serror due to the variation of the variables for torque set
point 3 N m.
Induced value
Calibrated value
Error
RM Serror
αa = 0.8000
αa = 0.7717
−3.5%
0.54%
αb = 1.1180
αb = 1.1035
−1.3%
0.27%
αc = 1.1180
αc = 1.0884
−2.6%
0.54%
βa = 1.0
βa = 0.9994
0.1%
0.002%
βb = −0.5
βb = −0.5078
1.6%
0.028%
βc = −0.5
βc = −0.6205
24.1%
0.439%
Table 4.2: Induced scaling and offset errors and the found compensating values
31
Chapter 4. Sensitivity Analysis of the Calibration
Figure 4.4: Limiting interval for αa , βa and τcog with maximum 1% RM Serror .
32
4.3. Measurements for verification of the calibration method
A similar approach for cogging torque is not possible. The induced change in magnitude
is not found after a new calibration. This is caused by small changes in phase. Instead
the cogging torque found by calibrating is compared with the cogging torque which has
been measured (chapter 3). The error in the time domain is showed in figure 4.5. The
maximum error is 8%, which is still within the allowed interval (figure 4.4).
Percentage error of the cogging torque based on the time signal
8
6
4
% Error
2
0
−2
−4
−6
−8
500
1000
1500
2000
2500
Encoder point φ
3000
3500
4000
Figure 4.5: Percentage error between the cogging torque measured and the cogging
torque determined by calibration. The error shown is in the time signal.
Besides the comparison of the time signals, the DFTs of the measured and calibrated
cogging torque are compared. Only the harmonics which are multiples of 16 are available
in the signals. The error in the magnitude of the signals is approximately 0.5%. The
phase causes the largest part of the error, but never excedes 0.5 of an encoder point.
33
Chapter 4. Sensitivity Analysis of the Calibration
4.4
Comparison of the Theoretical and practical
sensitivity
To check the theoretical sensitivity, errors are induced with values which are in the interval
which has also been used for the theoretical analysis (table 4.1). These errors are induced
after the motor has been calibrated. This has been done for torque set point 3 with speed
0.7 Hz. These set points are chosen because they are in the middle of the operating range.
The resulting practical RM Serror caused by an induced error in αa is showed in figure
4.6. In the same figure the theoretical error is shown. To compare the measurements
and the theory, two lines are fitted by using the measurements in a way as shown in the
legend. The 5th measurement point is not used because, as can be seen from the lines, it
~ ~
is not part of the two slopes. In a similar way the theoretical models for β,
τcog and δ are
verified. The deflection of the middle measurement points from the slopes is caused by
~ should compensate for current offset
an error in the current sensors. As per definition, β
values.
The difference in height between the theoretical an practical slopes is caused by the part
from the signal from which no information can be extracted, the so called "residual". This
signal consists mainly of measurement noise. Some averaging errors remain in this signal
as well.
The slopes of the theoretical and practical induced α
~ error are similar and therefore there
can be concluded that the theoretical model is correct.
The ideal βa is higher than the actual βa calculated from the calibration. This is also the
case for ~
τcog . The calibration method finds the best value for all set points and speeds.
For the used set point the calibrated values are not the best.
34
4.4. Comparison of the Theoretical and practical sensitivity
RMS
error
due to change in α
a
4.5
1
Measurements
Fit on measurement 1−4
Fit on measurement 6−9
Theoretical RMSerror
4
3.5
2
9
8
RMSerror
3
2.5
3
7
2
6
4
1.5
5
1
0.5
0
0.8
0.85
0.9
0.95
1
αa
1.05
1.1
1.15
1.2
Figure 4.6: The RM Serror caused by an error in αa practical and theoretical
RMSerror due to change in βa
4.5
1
Measurements
Fit on measurement 1−5
Fit on measurement 7−11
Theoretical RMSerror
4
2
3.5
RMSerror
3
11
10
3
9
2.5
4
2
8
1.5
5
6
1
7
0.5
0
−1
−0.5
0
β
0.5
1
a
Figure 4.7: The rmserror caused by an error in βa practical and theoretical
35
Chapter 4. Sensitivity Analysis of the Calibration
RMSerror due to change in τcog
3
1
Measurements
Fit on measurement 1−8
Fit on measurement 10−15
Theoretical RMSerror
2
2.5
3
4
RMSerror
2
15
14
13
5
12
6
1.5
11
7
9
8
1
10
0.5
0
0.7
0.8
0.9
1
factor of τ
1.1
1.2
1.3
cog
Figure 4.8: The rmserror caused by an error in ~
τcog practical and theoretical
36
4.4. Comparison of the Theoretical and practical sensitivity
Performing a theoretical analysis on the sensitivity of δ is complicated due to the
interactions with other variables. The results of the practical influence is showed in figure
4.9. To find the cause of the remaining measured RM Serror , as shown in figure 4.10,
RMSerror due to change in δε
1.4
1.2
RMSerror
1
0.8
0.6
0.4
0.2
0
0.6
0.7
0.8
0.9
1
δε
1.1
1.2
1.3
1.4
Figure 4.9: Practical sensitivity due to an error in δ
the reconstructed torque is calculated. This is done by multiplying the feedback current
with the back EMF. The reconstructed torque consist of the error in the current sensors
and the error in the control loop. The result can be seen in figure 4.11. As can be seen the
remaining measured RM Serror of 0.8−1.5% is in the same range as the reconstructed
torque. To reduce the RM Serror more accurate current sensors are neccessary. The
current sensors used have an error of ±0.7% [8] in the current. Also the control loop
needs to be more accurate.
37
Chapter 4. Sensitivity Analysis of the Calibration
RMSerror of τm
RMSerror of τm
1.4
1.2
1
0.8
5
4
3
2
1
Torque [Nm]
0.5
0.6
0.7
0.8
0.9
1
Speed [Hz]
Figure 4.10: Remaining measured RM Serror in τm
Error of the current sensors and control loop
1.3
RMSerror of Ifb
1.2
1.1
1
0.9
0.8
5
4
3
2
Torque [Nm]
1
0.5
0.6
0.7
0.8
0.9
Speed [Hz]
Figure 4.11: Error due to the current sensor and control loop
38
1
Chapter 5
Conclusion
The report consists of three parts, moment of inertia determination, cogging torque measurement and a sensitivity analysis of the calibration method of the electric motor.
The moment of inertia is calculated in Pro/ENGINEER with help of the CAD drawings.
To verify this calculation the moment of inertia is determined experimentally. The experimentally determined moment of inertia is 6.6% lower than the value calculated. The
calculated value is assumed to be correct. To find a cause for the lower experimentally determined value a sensitivity analysis from the experiment is made. There are determined
4 parameters of the experimental setup which can cause conjointly the deflection in the
determined moment of inertia. Based on these results, recommendations for an improved
experimental setup are made.
The measurement devices of the electric motor are optimized for measuring during rotation. Therefore the cogging torque is measured dynamically. The measurement result
is satisfying; nevertheless some improvements to the experimental setup are suggested.
This should result in a more accurate answer.
To improve the performance, a calibration method for the electric motor is designed. To
verify the calibration method and to explain the remaining RM Serror of 0.8 − 1.5%
a sensitivity analysis is made. This is done theoretically and verified experimentally. The
resulting differences are explained. The remaining RM Serror is a combination of the
error in the current sensors and a control error.
39
Chapter 5. Conclusion
40
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42