Student Report
Department of Energy Technology

Pontopp
i
d
anstraede 101
Aalborg University, Denmark
Investigation of
F
ield
O
riented
C
ontrol
o
f
SPMSM w
ith Respect
t
o
Motor Parameter
Variation
a
nd Measurement Error
s
CONDUCTED BY GROUP PED 830
SPRING SEMESTER, 2011
Department of Energy Technology

Pontopp
i
d
anstraede 101
Aalborg University, Denmark
Investigation of
F
ield
O
riented
C
ontrol
of
SPMSM
w
ith Respect
t
o Motor Parameter
Variation
a
nd Measurement Errors
PED
8
3
0
CONDUCTED BY GROUP
PED
8
3
0
SPRING
SEMESTER, 201
1
Student report
DEPARTMENT OF ENERGY
TECHNOLOGY, AALBORG UNIVERSITY, DENMARK
Title:
Investigation of
F
ield Oriented
C
ontrol
of
SPMSM
with Respect t
o Motor
Parameter Variation
and Measurement
Errors
Semester:
8
th
Semester theme:
Control of
Converter

fed AC Drives
Project period:
01
.
02
.1
1
to
25
.
0
5
.1
1
ECTS:
1
5
Supervisors:
Kaiyuan Lu
Mehmet Sertug Basar
Project group:
PED 8
3
0
SYNOPSYS:
The purpose of the project is to design and
implement a Field Oriented Control for a
SPMSM
con
side
r
ing different levels of
performance
for
current sensors.
T
he machine
was
modelled then
validate
d
through
V/f open loop control.
Next the
rotor
FOC
was
implemented in Simulink
using
t
wo
and
three
current sensors
.
The developed model
w
as
implemented in laboratory,
using a DSpace
digital controller.
Study
cases were performed
in
simulation and
validated through
laboratory.
For
each case, different current sensor errors (gain
and offset) were introduced.
The most important
study cases
were presented and analysed.
A
n
important conclusion is that for high speed
applications the offset error has the highest
impact and for low speed applications the highest
impact is
due to
the gain error.
Ciprian Biris
Razvan Gheorghies
Copies:
3
Pages:
96
Appendix:
15
pages
Supplements:
1 DVD
B
y signing this document, each member of the group confirms that all participated in the project work and
thereby all members are collectively liable for the content of the report.
DEPARTMENT OF ENERGY
TECHNOLOGY, AALBORG UNIVERSITY, DENMARK
PREFACE
Preface
The present project report entitled ‘
Investigation of
F
ield
O
riented
C
ontrol of
P
ermanent
M
agnet
m
achine with
R
espect to
M
otor
P
arameter
V
ariation and
M
easurement
E
rrors
’
was
implemented by the group na
med ‘
PED 8
3
0
’ during the Master
s
8
th
semester at Aalborg University,
Department of Energy Technology, between
1
st
of
February
201
1
and
25
th
of
May
201
1
.
Reading instructions
In order
to simplify the reading,
some details about the structure are presented. References
are shown as a number in
round
brackets
(
x
)
, while figures are numbered X.Y, where X is the chapter
number and Y is the figure number. Equations are represented like (X

Y), where X is the chapter
number and Y is the equation number. Tables are numbered like X

Y, where X is the chapter number
and Y is the table number
.
.
The authors of the project would like to express their special thanks to the supervisor
Kaiyuan Lu
and PhD. Cristian Busca
for
thei
r
support and
valuable information
provided
throughout
the development of the project.
Ciprian Valeriu Biris
Razvan Gheorghies
PREFACE
iii
TABLE OF CONTENTS
Table of contents
Preface
................................
................................
................................
.............................
i
Table of figures
................................
................................
................................
...............
6
1.
Introduction
................................
................................
................................
.............
1
1.1.
Background
................................
................................
................................
..................
1
1.2.
Problem f
ormulation
................................
................................
................................
...
2
1.3.
Project limitation
................................
................................
................................
.........
2
1.4.
Project goals
................................
................................
................................
................
2
1.5.
Project outline
................................
................................
................................
.............
3
1.6.
Summary
................................
................................
................................
.....................
3
2.
PMSM Model
................................
................................
................................
...........
4
2.1.
Mathematical Model
................................
................................
................................
...
4
2.2.
Simulink Model
................................
................................
................................
............
6
2.3.
Reference frame transformations
................................
................................
...............
7
2.3.1.
“abc”
to “
αβ”
reference frame transformation
................................
..............................
7
2.3.2.
“αβ”
to
abc
reference frame transformation
................................
................................
.
9
2.3.3.
“abc”
to
dq0
reference frame transformation
................................
.............................
10
2.3.4.
“
dq0” to “abc” reference frame transformation
................................
..........................
11
2.3.5.
“αβ”
to “
dq0”
reference frame transformation
................................
............................
13
2.3.6.
“dq0”
to
“αβ”
reference frame transformation
................................
............................
14
2.4.
Validation of SPMSM Model
–
V/f Control
................................
...............................
15
2.5.
Summary
................................
................................
................................
...................
17
3.
Control Strategies
................................
................................
................................
..
18
3.1.
Control Strategies Overview
................................
................................
.....................
18
3.2.
Field Oriented Control for SPMSM Overview
................................
...........................
20
3.3.
Sensored FOC
................................
................................
................................
............
21
3.3.1.
PI Controllers Overview
................................
................................
................................
22
3.3.2.
Machine Parameters
................................
................................
................................
.....
23
iv
TABLE OF CONTENTS
3.3.
3.
Design of PI Current Controllers
................................
................................
...................
23
3.3.4.
Current PI Response in Simulation
................................
................................
................
24
3.3.5.
Current PI Response in Experiment
................................
................................
..............
26
3.3.6.
Interpretation
–
simulation vs. experiment
................................
................................
..
27
3.3.7.
Design of PI Speed Controller
................................
................................
.......................
27
3.3.8.
Speed PI Response in Simulation
................................
................................
..................
28
3.3.9.
Speed PI Respnse in Experiment
................................
................................
...................
30
3.3.10.
Interpretation
–
simulation vs. experiment
................................
................................
..
30
3.3.11.
Anti

windup system
................................
................................
................................
......
30
3.4.
Summary
................................
................................
................................
...................
31
4.
Laboratory work
................................
................................
................................
.....
32
4.1.
DSpace and ControlDesk overview
................................
................................
...........
32
4.2.
SPMSM rotor position alignment
................................
................................
..............
33
4.3.
Sensor calibration
................................
................................
................................
......
35
4.4.
Study Cases
................................
................................
................................
................
35
4.5.
Summary
................................
................................
................................
...................
37
5.
Sensored control of SPMSM
................................
................................
..................
38
5.1.
Extra PI for Torque
................................
................................
................................
....
38
5.2.
Speed response observations
................................
................................
...................
39
5.3.
Interpretation
–
simulation vs experiment
................................
...............................
40
5.4.
Torque response observations
................................
................................
..................
40
5.5.
Interpretation
–
simulation vs experiment
................................
...............................
41
5.6.
Three phase current observations
................................
................................
............
42
5.7.
Interpretation
–
simulation vs experiment
................................
...............................
43
5.8.
Introduced error effect
................................
................................
.............................
44
5.9.
Error effect in speed
–
observations
................................
................................
.........
45
5.10.
Error effect in torque
–
observations
................................
................................
....
47
5.11.
Summary
................................
................................
................................
................
48
6.
Sensorless control of SPMSM
................................
................................
................
4
9
v
TABLE OF CONTENTS
6.1.
Position estimator design (23)
................................
................................
..................
50
6.2.
Speed calculator
................................
................................
................................
........
51
6.3.
Error effect in speed
–
observations
................................
................................
.........
52
6.4.
Error effect in torque
–
observations
................................
................................
........
54
6.5.
Differences from sensored control
................................
................................
...........
55
6.6.
Interpretation of results
................................
................................
............................
59
6.7.
Summary
................................
................................
................................
...................
60
7.
Conclusions and future work
................................
................................
.................
61
7.1.
Conclusions regarding objectives
................................
................................
..............
61
7.2.
Conclusions regarding SMPMSM investigation
................................
........................
61
7.3.
Future work
................................
................................
................................
...............
62
7.4.
Summary
................................
................................
................................
...................
62
Bibliography
................................
................................
................................
..................
63
Appendix A: DVD content
................................
................................
.............................
65
6
CHAPTER 3:
CONTROL STRATEGIES
Table of figures
Figure 1.1 Basic configurations of SPMSM (4)
................................
................................
........................
1
Figure 2.1 Simulink Model
................................
................................
................................
......................
6
Figure 2.2 PMSM Layout
................................
................................
................................
.........................
7
Figure
2.3 “αβ” and “abc” reference frame
................................
................................
...........................
8
Figure 2.4 Simulink model of abc to αβ reference frame transformation
................................
.............
8
Figure 2.5 Simulink model of αβ to abc reference frame transformation
................................
.............
9
Figure 2.6
abc to dq0
reference frame
................................
................................
................................
.
10
Figure 2.7
abc
to
dq0
reference frame transformation blocks.
................................
............................
11
Figure 2.8 Simulink model of the
dq0
to
abc
reference frame transformation
................................
...
12
Figure 2.9
αβ
to
dq0
transformation reference frame
................................
................................
.........
13
Figure 2.10 Simulink model of the
αβ
to
dq0
reference frame transformation.
................................
..
14
Figure 2.11 V/f scalar control
................................
................................
................................
................
16
Figure 2.12 Output currents of SPMSM model with V/f control
................................
..........................
16
Figure 2.13 Electromagnetic torque
................................
................................
................................
.....
17
Figure 2.14 Output speed
................................
................................
................................
.....................
17
Figure 3.1 Field oriented control of SPMSM (14)
................................
................................
.................
21
Figure 3.2 PI controller layout
................................
................................
................................
...............
22
Figure 3.3 Simulation setup for PI current test
................................
................................
.....................
24
Figure 3.4 PI current controller response at 20% of the rated current
................................
................
25
Figure 3.5
Iq
current response with 1.85% overshoot
................................
................................
..........
26
Figure 3.6
Iq
current step input response for current PI
................................
................................
......
26
Figure 3.7
Speed PI response for step input
................................
................................
.........................
29
Figure 3.8
Speed response for Kp=0.0745
................................
................................
............................
29
Figure 3.9 Speed response for 1000[rpm] step input.
................................
................................
..........
30
Figure 3.10
Anti

windup scheme
................................
................................
................................
..........
31
Figure 4.1 Laboratory
DSpace setup
................................
................................
................................
.....
33
Figure 4.2
Alignment of flux axis
d
with phase axis
a
of the machine
................................
..................
34
Figure 4.3
SPMSM stator currents during alignment
................................
................................
...........
34
Figure 4.4 Current data acquisition an
d sensor calibration subsystem
................................
................
35
Figure 5.1

Speed response for ramp input

simulation
................................
................................
......
39
Figure 5.2

Speed response for ramp input

experiment
................................
................................
....
39
Figure 5.3

Torque response for ramp input

simulation
................................
................................
....
41
Figure 5.4

Torque response for ramp input

e
xperiment
................................
................................
...
41
Figure 5.5
–
Ia, Ic, Ic

simulation
................................
................................
................................
...........
43
Figure 5.6
–
Ia, Ib, Ic

experiment
................................
................................
................................
........
43
Figure 5.7

Current error generator for three current sensors
................................
............................
44
Figure 5.8

Current error generator for two current sensors
................................
...............................
45
7
CHAPTER 3:
CONTROL STRATEGIES
Figure 5.9
–
Error effect on speed fo
r three current sensors (sensored)
................................
..............
46
Figure 5.10
–
Error effect on speed for two current sensors (sensored)
................................
.............
46
Figure 5.11
–
Error effect on speed for two current sensors (sensored)
................................
..............
47
Figure 5.12
–
Error effect on speed for two current sensors (sensored)
................................
..............
48
Figure 6.1
–
Sensorless FOC Structure
................................
................................
................................
...
50
Figure 6.2
–
Rotor to stator flux
................................
................................
................................
.............
50
Figure 6.3
–
Position Estimator
................................
................................
................................
..............
51
Figure 6.4
–
Speed Estimator
................................
................................
................................
.................
52
Figure 6.5
–
Error effect on speed for three current sensors (sensorless)
................................
............
53
Figure 6.6
–
Error effect on speed for two current sensors (sensorless)
................................
..............
53
Figure 6.7
–
Error effect on speed for two current sensors (sensorless)
................................
..............
54
Figure 6.8
–
Error effect on speed for two current sensors (sensorless)
................................
..............
55
Figure 6.9
–
Speed response for ramp input

simulation
................................
................................
....
56
Figure 6.10
–
Speed response for ramp input

experiment
................................
................................
.
56
Figure 6.11
–
Torque response for step input

simulation
................................
................................
..
57
Figure 6.12
–
Torque response for step input

experiment
................................
................................
.
57
Figure 6.13
–
Three phase currents

simul
ation
................................
................................
..................
58
Figure 6.14
–
Three phase currents

experiment
................................
................................
.................
58
1
CHAPTER 3:
CONTROL STRATEGIES
1.
Introduction
In the present chapter a short introduction of the Surface Mounted Permanent Magnet
Synchronous Machine (
SPMSM
) is presented, followed by the problem formulation,
project
limitations, go
als and outline.
1.1.
Background
Due to high performance, the Permanent Magnet Synchronous Machines is used
worldwide
.
This type of machine combines the advantages of DC machines and
induction machines
. In
consequence, it eliminates disadvantages such as brushes for DC machines
and
slip

rings for
induction machines. The rotor copper losses can be translated into a high t
orque

to

inertia ratio and
a high efficiency.
(1)
,
(2)
In order to control the machine, information about the rotor position is
necessary. This is
provided
by position sensors such as encoders.
(3)
Permanent
M
agnet
S
ynchronous
M
achines can be
built
with either embedded magnets or
surface magnets on the rotor, and the location of the magnets can have a significant effect on the
mechanical and electrical characteristics
of
the motor.
(4)
(5)
Three basic configurations of
PMSM
s are shown in
Figure
1
.
1
:
Figure
1
.
1
Basic configurations of
SPMSM
(4)
a)
Non

salient surface magnet rotor.
b)
Salient pole surface magnet r
otor with inset magnets
.
c)
Embedded magnets in the rotor
.
Classical scalar control methods for variable speed drives
,
offer
s
simple implementation
with
limited performance.
With scalar control, algorithm
limitations encountering dynamic response
specifications require choosing a larger motor and drive. Field oriented control (FOC) overcomes
this problem by improving the performance of the same motor. This allows designers and engineers
to properly size motors and drives, lowering cost; thus the
result is a
n
efficient overall system.
(5)
2
CHAPTER 3:
CONTROL STRATEGIES
Different control methods can be applied to
SPMSM
, using different types of sensors, with
variable cost and volume. The control method implemented in this project is
F
ield
Orien
ted
C
on
trol
.
1.2.
Problem
formulation
Many industrial applications require variable speed drive
systems. In order to obtain
such a
system
, rotor position and/or
speed must be known. These v
ariables are determined using
an
encoder or position transducer
. T
he
machine can be controlled
so it can
provide
nominal
torque at
zero speed by upholding an appropriate angle between the magnetic fields of stator and rotor.
Nevertheless, the use of position encoder adds some disadvantages to the
SPMSM
drive
:
(6)

Increased drive system cost

Increased complexity and maintenance

Reduced reliability

Increased size of the drive system
In order to solve the problems presented above
,
this project proposes
the analysis of
SPMSM
Field Orien
ted Control
with current sensors that have measurement errors up to 5%. The
main
Id
ea is to use sensors with poor accuracy in order to reduce the cost of
the drive
and keep
similar
performance for the drive system.
1.3.
Project limitation
During the project ela
boration
,
some limitations had to be imposed.
The most important
are
the following
:

Sensored experiments at
2.3% (100 [rpm]) and 23% (1000 [rp
m]
)
o
f the
SPMSM
rated
speed
due to the fact that the IM rated speed is 1430 rpm

Sensorless experiments at 1400
[rpm]

Experiments at 20% (4 [Nm]) of the
rated torque
in order to pay respect to motor
parameter variation by keeping the SPMSM as cold as possible.

Torque
wa
s calculated based
on
Iq current
1.4.
Project goals
The project was
divided
into goals that needed to b
e achieved during the semester. These
goals can be summarized as:

Simulink model of
SPMSM

Design and implement
in Simulink
a
FOC for
SPMSM

Implement
the
developed FOC in DSpace

T
est the
sensored
controller performance with respect to measurement errors
in
troduced by the current sensors
3
CHAPTER 3:
CONTROL STRATEGIES

Analytical analysis between experiments and simulations o
f sensored control

Simulink model of the
SPMSM
sensorless
control

Implement
the
sensorless
model in DSpace

Test the
sensorless
controller performance with respect to
measurement errors
introduced by the current sensors

Analytical analysis between simulations and experiments on
sensorless
control
1.5.
Project outline
This project is divided in 7 chapters
as follows:

The first chapter consists on the problem formulation and
project goals with project
limitations

The second chapter consists on the mathematical and Simulink model of PMSM.
Reference frame transformations
are described and the PMSM is validated through V/f
control

The third chapter shows a quick briefing on the c
ontrol strategies and then focuses on
sensored Field Oriented Control. Machine parameters are presented and the PI
controllers for current and speed are tuned ma
thematically and then tested in
the
simulation and in the laboratory.

The fourth chapter focuse
s on laboratory setup showing the setup and presenting the
rotor position alignment algorithm and the current sensor calibration

The fifth chapter
focus is on sensored control of the PMSM.
Speed, torque and current
observations are made for both simulation
s and experiments. The effect of introduced
error is studied and then the effects of those errors in speed and torque are observed.

The sixth chapter focuses on sensorless control. The position estimator is studied and
the error effect in speed and torque
are observed. Differences from sensored control are
shown

The seventh chapter does an interpretation of results regarding introduced error effect
for both sensored and sensorless control.

In the eight chapter conclusions regarding objectives and PMSM inves
tigation are
drawn, followed by
directions regarding future work
1.6.
Summary
T
his chapter is an introduction for the work which was carried out during the project. Project
goals
and limitations were defined from the beginning with the problem formulation. The
chapter
also presents an outline for all the chapter
s
in this project.
4
CHAPTER 3:
CONTROL STRATEGIES
2.
PMSM Model
In this chapter
the Simulink model of
SPMSM
and an open loop control
is presented. The
machine model is developed in dq reference frame.
Next a V/f control is implemented in
order to
validate
the model
of the machine.
2.1.
Mathematical Model
Synchronous motor
s
are very popular AC machine
s
. The major components are the rotor
with the permanent magnets and the stator which carries the armature windings; stator windings
creates a
rotating magnetic field, by the AC current flow; while this phenomena is produced, the
magnetic field from the magnets locks to the magnetic field produced in the stator and rotates along
with it. Once the rotor is in synchronous speed, the spinning proces
s depends on the supplied
frequency.
Recently, SPMSM have increased popularity in industrial applications. Advantages such as
high torque density, efficiency and power factor make permanent magnet motors more competitive
over induction motors.
(7)
The rotor flux of the SPMSM is generated by surface permanent magnets, thus the need for
external power source is eliminated. In DC machines the rotor flux is created by the rotor windings
which are supplied by an external power source
. The use of permanent magnets in the rotor
structure of SPMSM allows the machines to be lighter and more compact sized compared to classical
DC machines. One important disadvantage of this machine is that the rotor flux cannot be controlled
in such an eas
y manner.
(8)
The general advantages of the PMSM are
:
(9)

High efficiency and power density

No field windings

Compact design (magnets are smaller than windings)

It can run at high speeds
–
up to
100.000 [rpm]

Low maintenance and long life
The general disadvantages are:
(9)

Permanent Magnets are susceptible to loss of magnetization abos Currie temperature

High production cost

Need for rotor position
The SPMSM
is modelle
d in
based on dq reference frame equations. An important statement
of this representation is that voltages and currents seen from stator
side
are
sinusoidal
, but
observed from the rotor side in steady state they become DC values.
On Laplace domain, the di
fferentiator gain is directly proportional to frequency, whereas
that of an integrator is inverse proportional to frequency. In other words a differentiator will be
more susceptible to high

frequency noise than an integrator. For this reason, integration i
s preferred
over differentiation, and is used all over this project in Simulink simulations
5
CHAPTER 3:
CONTROL STRATEGIES
Voltage equations in the d

q reference frame are shown in equations 2

1 and 2

2.
(8)
2

1
2

2
Where:

represent the voltage on the d

axis on rotor reference frame
[V]

represent the voltage o
n the q

axis on rotor reference frame
[V]

represent the resistance of the stator phase
[
Ω
]

represent the current on the d

axis in rotor reference frame
[A]

represent the current on the q

axis in rotor reference frame
[A]

represe
nt the flux linkage on the d

axis in rotor reference frame
[Wb]

represent the
flux linkage on the q

axis in rotor reference frame
[Wb]

represent
the electrical speed of the rotor
[rad/sec]
The flux linkage equations are described as in 2

3 and 2

4:
(8)
2

3
2

4
Where:

represent
the inductance on d

axis in rotor reference frame
[H]

represent
the inductance on q

axis in rotor reference frame
[H]

represent
the flux linkage of permanent magnet
[Wb]
If equations stated in (2

3; 2

4) are substituted in voltag
e equations (2

1; 2

2), the new
voltage equations will have the form of (2

5; 2

6):
2

5
2

6
The instantaneous electromagnetic torque can be expressed as equation 2

7:
(
)
2

7
Where:

represent
the electromagneti
c torque
[N
∙
m]

represent
the number of pole pairs of the motor
[

]
6
CHAPTER 3:
CONTROL STRATEGIES
Inductances on axis d and q used in SPMSM are equal as value, thus the electromagnetic
torque can be written as follows:
2

8
The mechanical equation can be described as equation 2

9:
)
2

9
Where:

represent
the electrical speed of the rotor
[rad/sec]

rep
resent
the inertia of the electrical machine
[kg
∙
m
2
]

represent
the electromagnetic torque
[N
∙
m]

represent
the load torque [N
∙
m]
2.2.
Simulink Model
The general Simulink layout is presented in
Figure 2.1. The three phase voltages together
with the
theta angle information enter the
Clarke transformation block in order to provide the
voltages in dq reference frame.
The Vd, Vq and the load torque Tl are the inputs of the SMPMSM.
Dq reference frame currents, Id and Iq, the electromagnetic torque Te
, the rotor angle
information theta_r and the rotor speed
are the outputs of the PMSM.
The dq cur
r
ents with the
theta angle are transformed from dq reference frame back to abc coordinate system.
The rotor
speed is transformed from [rad/s] in [rpm] with
equation 2

10
2

10
Figure
2
.
1
Simulink Model
The layout of the PMSM block is presented in Figure 2.2
. From dq voltages and with
the rotor angular speed information, the Id and Iq currents are calculated based on formulas
from
equation
2

1
to
2

4.
From the Iq current the electrical torque is computed based on equation 2

8.
The load torque Tl and the elect
romagnetic torque Te are the inputs for the block build based on
mechanical equation 2

9. The output of this block is the
angular speed
. The manual switch is used
in order to simulate a locked rotor test when tuning the current regulators.
After, the
rotor speed is
integrated and wrapped between 0 and
, providing the rotor position information theta_r.
7
CHAPTER 3:
CONTROL STRATEGIES
Figure
2
.
2
PMSM Layout
2.3.
Reference frame transformations
2.3.1.
“abc”
to “
αβ”
reference frame transformat
ion
The transformation is known also as “The Clarke Transformation” and basically have two
fixed reference frame components “
αβ”
as output and three, time varying, components “
abc”
as
input. In
Figure
2
.
3
can be seen an illustration of “
αβ”
reference frame related to the “
abc”
reference frame. Can be observed that “α” vector is aligned with “a” axis and “
α”
and “
β”
are
perpendicular vectors.
(10)
The space vector
can be described by the two components
and
. The relation
between these vectors is shown in the equation
2

11
.
)
)
2

11
Components “
” and “
” are calculated using equations
2

1
2 and
2

1
3.
(
)
)
2

12
(
√
√
)
√
)
2

13
The transformation of equations (
2

12
;
2

13
) can be written in matrix form,
2

14
.
[
]
[
√
√
]
[
]
2

14
8
CHAPTER 3:
CONTROL STRATEGIES
Figure
2
.
3
“αβ” and “abc” reference frame
The Simulink block representing the transformation from Iabc to Iαβ, with the internal connection, is
shown in
Figure
2
.
4
Figure
2
.
4
Simulink model of abc to αβ reference frame transformation
9
CHAPTER 3:
CONTROL STRATEGIES
2.3.2.
“αβ”
to
abc
reference frame transformation
The "Inverse Clarke Transformation", does exactly the opposite of the "Clarke
Transformation", transform from αβ to abc component. The equations are shown below.
(10)
2

15
√
(
√
)
2

16
√
(
√
)
2

17
The matrix is shown in equation
2

1
8:
[
]
[
√
√
]
[
]
2

18
Figure
2
.
5
shows the transformation block, from Vαβ to Vabc, with the inte
rnal connections.
Figure
2
.
5
Simulink model of αβ to abc reference frame transformation
10
CHAPTER 3:
CONTROL STRATEGIES
2.3.3.
“abc”
to
dq0
reference frame transformation
To transform voltage, from “abc” to “dq0” reference frame, are used "The Park
Transformations". The input element of the equations is a constant reference frame, while the
output is a variable component, in time. The connexions between, “dq” and “abc” ref
erence frame,
are shown in
Figure
2
.
6
also the θ (theta) angle is present between "d" vector, and "a" axis. (3) The
simulated blocks for the equations
(
2

1
9;
2

2
0) are shown in
Figure
2
.
7
The two components required to describe the
dq0
reference frame are shown below in
equations
2

1
9 and
2

2
0:
[
(
)
(
)
]
2

19
[
(
)
(
)
]
2

20
The matrix form of the transformation is shown in eq
uation
2

2
1:
[
]
[
(
)
(
)
(
)
(
)
]
[
]
2

21
Figure
2
.
6
abc to dq0
reference frame
11
CHAPTER 3:
CONTROL STRATEGIES
2.3.4.
“
dq0” to “abc” reference frame transformation
The simulation parameters does not correspond with the real machine, because different
factors. To correct this, "The Inverse Park Transformation", is used. The input elements in the
equation are the constant, θ (theta) angle and "dq", and the output is de
termined by the "abc"
components.
(10)
The equations used for the inverse transformation are shown below.
2

22
(
)
(
)
2

23
Figure
2
.
7
abc
to
dq0
reference frame transformation blocks.
12
CHAPTER 3:
CONTROL STRATEGIES
(
)
(
)
2

24
And the transformation matrix is shown in equation
2

2
5.
[
]
[
(
)
(
)
(
)
(
)
]
[
]
2

25
In
Figure
2
.
8
can be seen the Simulink block of the transformation with
Id
q
and the angle
θ
(theta)
as input and I
abc
as output as well as the internal connections.
Figure
2
.
8
Simulink model of the
dq0
to
abc
reference frame transformation
13
CHAPTER 3:
CONTROL STRATEGIES
2.3.5.
“αβ”
to “
dq0”
reference frame transformation
Figure
2
.
9
shows the relationship between
“αβ”
and “
dq0”
reference frames.
(10)
Figure
2
.
9
αβ
to
dq0
transformation reference frame
The equations and the matrix used in the transformation are shown below.
2

26
2

27
[
]
[
]
[
]
2

28
In
Figure
2
.
10
is shown the Simulink block of the transformation with I
αβ
and the angle
θ
(theta)
as input and
Id
q
as output as well as the internal connections.
14
CHAPTER 3:
CONTROL STRATEGIES
Figure
2
.
10
Simulink model of the
αβ
to
dq0
reference frame transformation.
2.3.6.
“dq0”
to
“αβ”
reference frame transformation
This transformation is used to convert two rotating axis frames into two stationary axis
frames in order to get back on the 3

phase voltage required to power the machine. The e
quations
used for this transformation are shown below.
(10)
The matri
x form is shown in equation 2

31
:
2

29
2

30
[
]
[
]
[
]
2

31
15
CHAPTER 3:
CONTROL STRATEGIES
2.4.
Validation
of
SPMSM
Model
–
V/f Control
Constant volt per hertz control in open loop technique for permanent magnet synchronous
machine, offers great advantages. Information about the angular speed can be estimated from the
frequency of the supply voltage, according to formula (2

32). To validat
e the motor model, V/f scalar
control strategy is used to test if the model works properly. A proper model should provide
sinusoidal current output as shown in figure 2

12.


2

32
2

33
2

34
Where:


–
represents
output generated voltage
[V]
–
represents
Rated voltage [V]
–
represents
given reference speed [rpm]
–
represents
reference speed [rpm]
–
represents
electrical speed [rad/s]
–
represents
rated electrical speed [rad/s]
From
equation
(2

32

2

34
),
voltage
is
obtained
. It is f
urther
connected to a simplified
PWM
Voltage Source Inverter
to obtain the
“
abc
”
voltages
,
required by
the machine.
The Simulink model for volt per hertz control is presented in Figure
2
.
11
; based on
theoretical model,
the prescribed speed measured in
[
rpm
]
, is transformed in angular speed. Angular
speed is transformed into voltage. With Laplace integrator angular speed is converted to theta
angle. The PWM uses as i
nput voltages from equation 2

32
and theta angle, and pr
ovides as output
3

phase voltage for the machine phases.
16
CHAPTER 3:
CONTROL STRATEGIES
Figure
2
.
11
V/f scalar control
To validate the machine model, a 75 rpm speed is prescribed, and the machine
3

phase
currents are measured (
Figure
2
.1
2
)
. The prescribed torque is set to 0, and based on v/f
simulation
the machine response in torque is
shown in Figure
2.13
. Nevertheless the speed response is shown
in Figure
2.14
Figure 2.12, shows that between 0 and 25 ms, the
currents amplitude is 4 times higher than
the normal values. Also the oscillation period is less than 250 ms; after this period the 3

phase
currents shape in sinusoidal wave

forms. Based on these results, the machine currents behave in
normal parameters.
Figure
2
.
12
Output currents of SPMSM model
with V/f control
Figure 2.13 shows, a torque oscillation from 0 to 250 ms; these oscillations are closely
related to current oscillation shown in figure 2.12. After 250 ms the electromagnetic output torque,
T
e
, is zero in steady state.
17
CHAPTER 3:
CONTROL STRATEGIES
Figure
2
.
13
Electromagnetic torque
The output speed follows the reference and demonstrates that the machine works properly.
This is shown
in Figure 2.14
, where reference speed is reached after 300 ms. Based on figu
res 2.12

2.14
, the machine model works well and the simulation is validated.
The machine was teste
d for
different
references, on a controlled range, in
order to
avoid
the overexcited phenomena.
Figure
2
.
14
Output speed
2.5.
Summary
A Simulink model for SPMSM was developed in
dq
axis reference frame
, due to the fact that
voltages and currents seen from the stator side are sinusoidal, but observed from the rotor side,
they become DC values
.
Furthermore t
he model was validated
through
open loop scalar control
,
known as V/f control.
Reference transformation
s
are also presented.
18
CHAPTER 3:
CONTROL STRATEGIES
3.
Control Strategies
In this chapter
an overview of the Field Oriented Control is presented, followed by the
implementation of the se
nsored control for the
SPMSM
.
The design
and response
of the current and
speed PI controllers
are
presented
.
3.1.
Control Strategies Overview
Permanent magnet synchronous machine, support vector and scalar control.
Scalar control was used to validate the machi
nes model functionality in sub

chapter 2.4;
simplicity and low cost makes volt per hertz control a very popular method. It is widely used to
control induction drives. The V/Hz ratio must be held constant to the whole operation speed range
of the motor. The
control simplicity costs in manipulating only the voltage amplitude and frequency;
these parameters are calculated from the PMSM nominal frequency and voltage values. If the V/Hz
ratio is increased, the machine may become overexcited, and if the ratio dec
reases it may become
under excited. The ratio for V/Hz is computed based on the application and rotor speed. Say, for low
rotor speeds a higher V/Hz is needed in order to compensate the stator voltage drop, throw the
resistance. Another important feature o
f scalar control is represented by the lack of a position
estimator/encoder. The speed is estimated just by looking to the supplied voltage frequency. This
great feature lowers the price considerably; another thing that lowers the price is the fact that sc
alar
control doesn’t require high performance DSP as in vector control cases.
An interesting observation is that some permanent magnet synchronous machines are built
up to self

start. It has the feature to start as an induction motor and when the synchron
ous speed is
almost reached, it locks in. These self

start PMSM are in a manner more suitable for scalar control.
This damper is similar to the induction machine rotor cage. The induction machine produces
asynchronous torque; this is similar to a PMSM when
a damper is used. It produces some amount of
asynchronous torque when needed. This phenomenon disappear when the machine reaches
synchronous speed, because there is no current induced in the damper windings when the slip is
zero, also no async
hronous torq
ue is produced.
(11)
Scalar control for PMSM can become unstable when sudden changes are made in load
torque. Because scalar control is an open loop method, it has the disadvantage of a low dynamic
performance. It can be used i
n application were speed and torque accuracy are not required, like
fans, pumps, blowers, etc. Considering the disadvantages that scalar control cannot overcome, it is
not suitable for permanent magnet synchronous drives in high dynamic applications.
On
the other hand, there is vector control technology which can be implemented in multiple
control strategies, like Direct Torque Control or Field Oriented Control. Compared with scalar
control, the system offers a better response in study and transient stat
e, the overall system
performance is much better. Accurate speed and torque control is possible when vector control is
used, but the cost is also higher than scalar control.
19
CHAPTER 3:
CONTROL STRATEGIES
Direct torque control had been introduced on the marked by ABB, and has the advan
tages of
simple control schematic, good dynamic response, and because it does not need a rotor speed or a
position feedback is conside
red a sensorless technique.
(11)
The basic DTC schematic, consist in torque and flux estimato
rs, torque and flux hysteresis
comparators, witching table and a voltage source inverter; the schematic is simpler than FOC. The
principle of DTC is to choose a voltage vector in order to control both stator flux and torque in the
same time. The torque and
flux estimation uses sampled stator currents and DC

link voltages.
DTC works in a few basic steps, first the estimated torque and flux amplitude is compared
with the references values, in the hysteresis comparators. Then the output of the comparators is
feed to the switching table in order to select the best values for voltage vector in each sampling
period.
As any control method, DTC has its disadvantages. Classical DTC have high torque and
current ripple during steady state. This problem can be elimina
ted by operating at high frequency
switching.
The last big category of vector control is represented by Field Oriented Control. The
objective of FOC is to control PMSM as a separately excited DC machine, meaning that flux and
torque can be controlled separately.
The stator currents are transformed fr
om the 3

phase, in dq rotating reference frame, with
mathematical algorithms. Rotor position is needed in order to complete the mathematical
transformations. The flux is controlled through the d

axis current, while torque is controlled
through
the q

axis c
urrent.
(11)
Two types of field oriented control are possible:

Rotor oriented FOC

Stator oriented FOC
Field Oriented Control needs a minimum of two current sensors, and for the sensored
method a shaft encoder. The mathematical
algorithms need software implementation, and a strong
hardware structure. Performance DSPs are quite expensive, and all of the great advantages of field
oriented control is translated as supplementary cost, but based on application this cost might be
justi
fied. Due to its advantages, like low current and torque ripple, constant VSI switching, low
audible noise Field Oriented Control is described more detailed during this project, i
n chapters that
come. Figure 3.0
shows the control tree, for permanent magnet
synchronous drives
20
CHAPTER 3:
CONTROL STRATEGIES
Figure 3.0
Co
ntrol strategies displacement
(10)
3.2.
Field Oriented Control
for
SPMSM
Overview
FOC
controls
the stator currents that are represented as a vector. In order to implement this
control the three phase currents need to be projected into a two coordinate (d and q) system. These
projections lead to a structure similar to the control of DC machines. The
main purpose of Field
Oriented Control is to separate the control of produced torque and flux of magnetization, thus
reproduce the operation of DC machines. FOC allows decoupling the components of stator current
into torque and magnetizing flux, thus the
torque and flux control are independent from each other.
The inputs of machine controlled through FOC are speed and
Id
current.
(12)
The FOC control strategy used is constant torque angle.
This is the most simple because the
reference current for the
d
axis is set to zero.
The control method is also called
α
=
π
/2
.
A constant
angle of 90 electrical degrees has to be kept between
d
and
q
axis.
(13)
Different contro
l methods can be applied to
SPMSM
, using different types of sensors, with
variable cost and volume. The method proposed, for controlling the machine is called
F
ield
Orien
ted
C
ontrol and a b
rief schematic is presented in
Figure
3
.
1
:
Field
O
rien
ted
C
ontrol is a mathematical method of controlling the
SPMSM
.
T
he
mathematical architecture of the structure, the motor size, cost and power can
be dramatically
minimized. [3]
The three
acquired
currents
are transformed
from
abc
to
dq
. The
Id
current is
subtracted
from the
Id
reference and the error is feed to the PI controller. The output of the two PI controllers
is the two component space vector control voltage that
control
s
the SVM.
The position obtained from the shaft encoder is converted into speed
.
This signal is
subtracted
from the prescribed speed
and fed to the speed PI.
The
output
provide
s
Iq
reference
current.
21
CHAPTER 3:
CONTROL STRATEGIES
3.3.
Sensor
e
d FOC
In Field Oriented Control, stator currents
are represented as a vector, and the strategy
system consist in control process of these vectors. In order to control currents as a vector, the 3

phase natural system must be converted into a two invariant co

ordinate system (d and q

coordinates). FOC stra
tegy needs a constant torque component (aligned with q
–
axis) and a flux
constant (aligned with d
–
axis), because FOC is based on projections, the control handles
instantaneous electrical values, and is accurate in steady and transient operations. The basi
c
schematic for Field Oriented Control is presented in Figure 3.1:
Figure
3
.
1
Field oriented control of SPMSM
(14)
As seen in figure 3.1, the difference
between the provided encoder speed an reference
represents the error for the speed PI. The output of the speed PI regulator is the Iq current
reference. The difference between this reference and the machine feedback represents the error for
the current PI
controller. The output of the current PI controller is the Vq voltage. The reference of
the Id current is 0 because the flus in the rotor is not controlled. The difference between this
reference and the Id feedback from the machine represents the error for
the Id current PI. The
output of the Id PI regulator is the Vd voltage. Both current PIs have the same parameters.
The
speed regulator is tuned separately.
The
voltages in the dq reference frame are converted to abs reference frame and fed to the
machine.
The machine feeds back the three phase currents Ia, Ib and Ic which are transformed from
abc coordinate system to alpha

beta coordinate system and then to the dq coordinate system
because the control is made in dq reference frame. The angle theta and the
speed represent the
position encoder
feedback. Position theta is fed to the reference frame transformation blocks.
22
CHAPTER 3:
CONTROL STRATEGIES
3.3.1.
PI Controllers Overview
C
lassical control theory
implies
that some specifications must be
stated
in order to describe
the characteristics
and
behaviour of the
system. The specifications are rise time
T
r
, time to reach the
peak value
T
p
, the overshoot percentage, settling time
T
s
–
time required for the settling of output
within a percentage of the final value.
(15)
The most common values for overshoot are between 2
%
and 5%. The settling time can be
described as:
3

1
Where:
k
–
Is
determined by the percentage used [

]. In this book
(15)
the value used for k is 4.
–
represent
the damping ratio [

]
The damping ratio can be:
嬰ⰱ)
1
(1, ∞)
獴慴uV
unT敲

T慭p敤
捲楴楣慬汹慭p敤
ov敲

T慭p敤
The
value of the damping ratio has been chosen
:
√
3

2
The general form of the PI controller in Laplace domain
is
:
3

3
Where:
–
represents the gain of the controller
–
represents
the g
ain of the integrator
The layout of the PI
is
shown
in
Figure
3
.
2
.
Figure
3
.
2
PI controller layout
23
CHAPTER 3:
CONTROL STRATEGIES
3.3.2.
Machine Parameters
For the simulation and laboratory work
,
the following parameters were used:
Parameter
Symbol
Value
Measuring
unit
Rated phase voltage
U
n
185
V
Rated current
I
n
19.5
A
Rated power
P
n
9.42
kW
Rated torque
T
n
20
Nm
Rated speed
n
rated
4500
Rpm
Stator resistance
r
s
0.18
Ω
Synchronous inductances
L
d
, L
q
0.002
H
Number of pole pairs
n
pp
4

Permanent magnet flux linkage
λ
m
0.123
Wb
Rotor inertia
J
0.00
48
Kgm
2
3.3.3.
Design of PI Current Controller
s
The S
PMSM
is modelled with the equations described in chapter
II
, thus it
results
that
voltage equation of
q

axis
are
influ
enced by the flux
linkage on
d

axis
.
3

4
In
order to
determine
the PI controller parameters
,
the rotor
must be blocked,
therefore
:
3

5
The voltage equation on
q

axis
become
s
:
3

6
Transferring equation (3

6) into
Laplace domain
,
results in
:
3

7
The
controller
transfer function is a first order system:
3

8
I
n order to determin
e the system response
, the closed loop
transfer function must be taken
into con
side
ration
as
stated by
classical
control theory
.
(15)
The closed loop transfer function of the
system containing the plant and the PI controller becomes:
24
CHAPTER 3:
CONTROL STRATEGIES
)
(
)
3

9
The characteristic equation of the system
i
s:
)
)
3

10
The canonical form of the
characteristic equation is:
)
3

11
By u
sing
equation (3

9; 3

10)
,
and
are
computed.
In order to have
increase
d
stability
,
the system
bandwidth
is 700 rad/sec (D
Space
work
s
with
a frequency of 5 kHz), thus the settling time is
.
Comparing the canonical
characteristic equation with the one of the system,
and
are
computed:
3

12
3

13
3.3.4.
Current
PI Response in Simulation
The simulated system for testing the current PI is shown in Figure 3.3. It consists in
the PI controller and
the plant.
Figure
3
.
3
Simulation setup for PI current test
25
CHAPTER 3:
CONTROL STRATEGIES
With the resulted values of
Kp and Ki, t
he system response at 20%
of
reference current
is
shown
in
Figure
3
.
4
. As it can be seen, the step is applied at 3.52 seconds. The rise tim
e
T
r
of the
current is 1 [ms] and the time to peak
T
p
is 3 [ms]. The settling time
T
S
is 10 [ms]. The overshoot is
17.2%
Figure
3
.
4
PI current controller response at 20% of the rated current
Classic
al
control theory states that the overshoot
interval
need
s
to be
between 2% and 5%.
(15)
. In order to reduce the overshoot at 2% is ne
cessary
to increase the proportional gain of the
controller (the DC gain of the system needs to
be
adjusted). The new
Kp
value
is 5.9387. The
response of
Iq
current
is shown
in
Figure
3
.
5
The step is applied again at 3.52 [s].
The
rise time is still 1 [ms] and the settling time
increased to 12 [ms]. The time to peak is 2 [ms]. A huge improvement is the overshoot which is now
1
.85%.
The current controller for
d

axis
is the same as the one for
q

axis
, because
L
d
and
L
q
inductance has the same value.
26
CHAPTER 3:
CONTROL STRATEGIES
Figure
3
.
5
Iq
current response with 1.85
% overshoot
3.3.5.
Current PI
R
esponse in
Experiment
The current PI response in the laboratory setup is shown in
Figure
3
.
6
.
In order to conduct
the experiment at zero speed, the rotor had to b
e locked mechanically. For this, a rotor locking
device was used. After this, a step was applied at 3.527 [s] to see the PI current response. The rise
time is 2 [ms].
No overshoot is observed
. The settling time is 14 [ms].
A
ripple is 1.2
%.
is present.
Figure
3
.
6
Iq
current step input response for current PI
27
CHAPTER 3:
CONTROL STRATEGIES
3.3.6.
Interpretation
–
simulation vs. experiment
One major difference between the simulation and laboratory response is the ripple. This
phenomenon is mor
e obvious in
the laboratory case. Figure 3.8
shows a ripple right from the start in
the measured current even though the reference step response is not yet applied. The ripple is
caused by many factors as, electromagnetic interferences, inverter switching,
parasitic signals,
current sensor accuracy and nonlinearities present in the system.
Settling time for the simulation and the experiment are close, 12 [ms] versus 14 [ms]. The
simulation shows
a overshoot of 1.85% but there seems to be none in the experim
ents. This is due
to the fact that having a ripple of 1.2%, the overshoot may not be observed as the values are close.
In the experiment, the rise time is 2 [ms] and 1 [ms] in the simulation.
The values are pretty close as
a large scale view, but the exper
iment rise time is twice as much as in the simulation.
This difference
occurs because the system in the laboratory is more complex than system of a PI and a plant in the
simulation.
As a conclusion, the PI response for the dq currents meets the requirement
s both in the
simulation and in the laboratory.
3.3.7.
Design of PI Speed Controller
The speed controller is
designed
based on the same theory used for current controller. The
parameters for this controller are determined at no load and based on the mechanical equation of
the electric machine.
The electr
ical equation
is presented
in
3

14
:
)
3

14
In order to calculate the speed controller parameters, n
o load torque is
need
ed
.
The
electrical
e
quation
3

14
becomes
:
3

15
Laplace form of
equation 3

15
is:
3

16
The PI needs the machine angular speed as a negative feedback, thus
equation
3

16
becomes
:
3

17
If
mechanical
speed
is used
instead of angular
one
, the equation
3

17
become
s
:
3

18
After simplification, the equation
form
is:
28
CHAPTER 3:
CONTROL STRATEGIES
3

19
The
speed controller
transfer function:
3

20
The
plant
transfer function and the PI controller
become
:
)
3

21
The
closed loop
transfer function is
:
)
)
3

22
The
system
characteristic equation is:
)
3

23
S
peed controller
bandwidth
needs to be between 20% and 40% of the
current controller
bandwidth
.
(16)
It was
chosen to be 30%, respectively 210 rad/sec
. The
resulting
settling time is:
3

24
The controller coefficients (
and
) are computed by c
omparing the
canonical
characteristic equation with the one of the system.
The controller proportional gain is
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