POLITECHNIKA
WARSZAWSKA
WARSAW UNIVERSITY OF TECHNOLOGY
Faculty of Electrical Engineering
ROZPRAWA DOKTORSKA
Ph.D. Thesis
Dariusz Świerczyński, M. Sc.
Direct Torque Control with
Space Vector Modulation (DTC-SVM) of Inverter-Fed
Permanent Magnet Synchronous Motor Drive
WARSZAWA
2005
WARSAW UNIVERSITY OF TECHNOLOGY
Faculty of Electrical Engineering
Institute of Control and Industrial Electronics
Ph.D. Thesis
M. Sc. Dariusz Świerczyński
Direct Torque Control with Space Vector
Modulation (DTC-SVM) of Inverter-Fed
Permanent Magnet Synchronous Motor Drive
Thesis supervisor
Prof. Dr Sc. Marian P. Kaźmierkowski
Warsaw, Poland - 2005
Contents
Table of Contents
Chapter 1
INTRODUCTION
1
Chapter 2
MODELING AND CONTROL MODES OF PM SYNCHRONOUS DRIVES
2.1 Mathematical model of PM synchronous motor
2.1.1 Voltage and flux-current equations
2.1.2 Instantaneous power and electromagnetic torque
2.1.3 Mechanical motion equation
2.2 Static characteristic under different control modes
2.3 Summary
8
Chapter 3
VOLTAGE SOURCE PWM INVERTER FOR PMSM SUPPLY
3.1 Introduction
3.2 Voltage source inverter (VSI)
3.3 Space vector based pulse width modulation (PWM) methods
3.4 Summary
Chapter 4
CONTROL METHODS OF PM SYNCHRONOUS MOTOR
4.1 Introduction
4.2 Field oriented control (FOC)
4.3 Direct torque control (DTC)
4.4 Summary
Chapter 5
DIRECT TORQUE CONTROL WITH SPACE VECTOR MODULATION
(DTC-SVM)
5.1 Introduction
5.2 Cascade structure of DTC–SVM scheme
5.2.1 Digital flux control loop
5.2.2 Digital torque control loop
5.3 Parallel structure of DTC–SVM scheme
5.3.1 Digital flux control loop
5.3.2 Digital torque control loop
5.4 Speed control loop for DTC–SVM structure control
5.5 Summary
8
9
17
22
25
33
34
34
35
46
52
53
53
54
57
64
65
65
66
68
82
91
92
102
113
122
Chapter 6
121
DIRECT TORQUE CONTROL WITH SPACE VECTOR MODULATION (DTCSVM) OF PMSM DRIVE WITHOUT MOTION SENSOR
6.1 Introduction
121
6.2 Initial rotor position estimation method
123
6.3 Stator flux estimation methods
127
6.3.1 Overview
127
6.3.2 Current model based flux estimator
127
6.3.3 Voltage model based flux estimator with ideal integrator
128
Contents
6.3.4 Voltage model based flux estimator with low pas filter
6.3.5 Improved voltage model based flux estimator
6.4 Electromagnetic torque estimation
6.5 Rotor speed estimation methods
6.5.1 Overview
6.5.2 Back electromotive force (BEMF) technique
6.5.3 Stator flux based technique
6.6 Summary
Chapter 7
DSP IMPLEMENTATION OF DTC-SVM CONTROL
7.1 Description of the laboratory test-stand
7.2 Steady state behaviour
7.3 Dynamic behaviour
7.3.1 Flux and torque control loop
7.3.2 Speed control loop
129
130
132
132
132
133
133
136
137
Chapter 8
SUMMARY AND CLOSING CONCLUSIONS
161
Appendices
163
137
140
143
143
151
Picture of rotor and stator of PMSM machine
Basic transformation
Model of PM synchronous motor- SABER
Parameters of PMSM machine
Parameters of voltage source inverter
PI speed controller
PWM technique - overmodulation
List of Symbols
170
References
172
Introduction
Chapter 1 INTRODUCTION
Recently, an increased interest in application of permanent magnet synchronous motors
(PMSM) in speed controlled drives has been observed. This is stimulated mainly by:
•
development of modern high switching frequency semiconductor power devices (as
for example IGBT modules of 5-th generation),
•
new rare earth magnetic materials as samarium-cobalt (Sm-Co) or neodymium-ironboron (Nd-Fe-B),
•
specialized digital signal processor (DSP) for AC drive applications with integrated
PWM function, A/D converters as well as processing of encoder signals (e.g
ADMC401, TMS320FL24XX, TMS320FL28XX).
Synchronous motors with an electrically excited rotor winding have a conventional threephase stator winding (called armature) and an electrically excited field winding on the rotor,
which carries a DC current. The armature winding is similar to the stator of induction motor.
The electrically excited field winding can be replaced by permanent magnet (PM) [1]. The use
of permanent magnets has many advantages including the elimination of brushes, slip rings,
and rotor copper losses in the field winding. It leads to higher efficiency. Additionally since
the copper and iron losses are concentrated in the stator, cooling of machines through the
stator is more effective. The lack of field winding and higher efficiency results in reduction of
the machine frame size and higher power/weight ratio.
Figure. 1.1. General classification of AC synchronous motors.
1
Introduction
Generally, the permanent magnet AC machines can be classified into two types (Fig.1.1):
trapezoidal type called “brushless DC machine” (BLDCM) and sinusoidal type called
permanent magnet synchronous machine (PMSM). The BLDC machines operate with
trapezoidal back electromagnetic force (EMF) and require rectangular stator phase current.
The PMSM’s generate sinusoidal EMF and operate with sinusoidal stator phase current.
The PMSM can be further divided into two main groups in respect how the magnet bars have
mounted in the rotor [6,7]. In the first group magnets are mounted in the rotor (Fig. 1.2 c-d)
and this type is called interior permanent magnet synchronous motors (IPMSM). The second
group is represented by surface permanent magnet synchronous motors (SPMSM). In the
SPMSM magnet bars are mounted on the rotor surface (Fig. 1.2 a-b).
q
SPMSM
S
N
a)
N
S
q
d
b)
S
N
q
IPMSM
S
N
S
N
S
N
d
q
d
d
c)
d)
Fig. 1.2. The cross section of the PMSM rotor shaft and the magnet bars placements:
a),b),c) axial field direction, d) radial field direction.
The magnets can be placed in many ways on the rotor (Fig. 1.2). In radial field fashion the
magnet bars are along the radius of the machine and this arrangement provides the highest air
gap flux density, but it has the drawback of lower structural integrity and mechanical
robustness. Machines with this arrangement of magnets are not preferred for high-speed
applications (higher than 3000 rpm). In axial field manner the magnets are placed parallel to
the rotor shaft. This arrangement of magnets is much more robust mechanically as compared
2
Introduction
to surface-mounted machine. It makes possible to use IPMSM for higher-speed applications
(contrary to SPMSM’s).
Regardless of the fashion of mounting the PM, the basic principle of motor control is the same
and the differences are only in particularities. An important consequence of the method of
mounting the rotor magnets is the difference in direct and quadrature axes inductance values.
The direct axis reluctance is greater than the quadrature axis reluctance, because the effective
air gap of the direct axis is multiple times that of the actual air gap seen by the quadrature
axis. As consequence of such an unequal reluctance, the quadrature inductance is higher than
direct inductance Lq > Ld . It produces reluctance torque in addition to the mutual torque.
Reluctance torque is produced due to the magnet saliency in the quadrature and the direct axis
magnetic paths. Mutual torque is produced due to the interaction of the magnet field and the
stator current. In case where the magnets bars are mounted on the rotor surface the quadrature
inductance is equal direct inductance Lq = Ld , because of the same flux paths in d and q axis.
As result the reluctance torque disappears.
Among the main advantage of PM machines are [12]:
•
high air gap flux density,
•
higher power/weight ratio,
•
large torque/inertia ratio,
•
small torque ripples,
•
high speed operation,
•
high torque capability (quick acceleration and deceleration),
•
high efficiency and high cos φ (low expense for the power supply),
•
compact design.
Thanks to this advantages the PMSM’s are usually used in high performance servo drives, in
special applications as computer peripheral equipment, robotics, ect. However, recently the
PMSM are also used as adjustable–speed drives in variety of application such as fans, pumps,
compressors, blowers. Another area is automotive application as an alternative drive in hybrid
mode with classical engine. The power of offered synchronous motors is in the range several
kW to MW.
3
Introduction
The main requirements for high performance PWM inverter-fed PMSM drive can be
formulated as follows:
•
operation with and without mechanical motion sensor,
•
fast flux and torque response,
•
available maximum output torque in wide range of speed operation region,
•
constant switching frequency,
•
uni-polar voltage PWM,
•
low flux and torque ripples,
•
robustness to parameters variation,
•
four quadrant operation.
To meet the above requirements, different control methods can be used [3,4,10].
Variable
Frequency
Control
Scalar based
controllers
V/Hz=const
with stabilization
loop
Vector based
controllers
Direct Torque
Control
(DTC)
Field
Oriented
(FOC)
PM (rotor)
Flux Oriented
(RFOC)
Stator Flux
Oriented
(SFOC)
Direct Torque
Control with Space
Vector Modulation
(DTC-SVM)
Circular flux
trajectory
(Takahashi)
Figure 1.3 Classification of PMSM control methods.
The general classification of the variable frequency control for PMSM is presented in Fig. 1.3.
The PMSM control methods can be divided into scalar and vector control. According to [3],
in scalar control, which based on a relation valid for steady states, only the magnitude and
frequency (angular speed) of voltage, currents, and flux linkage space vectors are controlled.
Thus, the control system does not act on space vector position during transient. Therefore, this
control is dedicated for application, where high dynamics is not demanded. Contrary, in
4
Introduction
vector control, which is based on relation valid for dynamics states, not just magnitude and
frequency (angular speed), but also instantaneous position of voltage, current and flux space
vectors are controlled. Thus, the control system adjust the position of the space vectors and
guarantee their correct orientation for both steady states and transients.
The scalar constant V/Hz control for PMSM without damper winding (squire cage) is not
simple as for induction motor. It requires additional stabilization control loop, which can be
provide by feedback from: rotor velocity perturbation, active power or DC-link current
perturbation [9].
The most popular vector control method developed in 70s, known as field oriented control
(FOC) [31] gives the permanent magnet synchronous motor high performance. In this method
the motor equation are transformed in a coordinate system that rotates in synchronism with
permanent magnet flux. It allows separately and indirectly control flux and torque quantities
by using current control loop with PI controllers like in well known DC machine control [3].
In search of a simpler and more robust high performance control system in 80s new vector
control called direct torque control (DTC) was developed [50]. It was innovative studies at
this time and completely different approach which depart from the idea of coordinate
transformation and the analogy with DC motor control. It allows direct control flux and torque
quantities without inner current control loops. Using bang-bang hysteresis controllers for flux
and torque control loops made this control concept very fast and not complicated. However,
the main disadvantage of DTC is fast sampling time required and variable switching
frequency, because of hysteresis based control loops. In order to eliminate above
disadvantages and kept basic control rules of classical DTC, at the beginning of 90’s a new
developed control technique called direct torque control with space vector modulator (DTCSVM) has been introduced [54,55]. However, from the formal consideration this method can
also be viewed as stator flux oriented control (SFOC). This control employed instead of
hysteresis controller as for classical DTC, the PI controllers and space vector modulator
(SVM). It allows to achieve fixed switching frequency, what considerably reduce switching
losses as well as torque and current ripples. Also requirement of very fast sampling time is
eliminated [113,115,117]. Therefore, this new method is subject of this thesis. In spite of
many control strategies there is no one which may be considered as standard solution.
5
Introduction
Therefore, the following thesis can be formulated:
“In the view of commercial manufacturing process the most convenient control scheme
for voltage source inverter-fed permanent magnet synchronous motor (PMSM) drives is
direct torque control with space vector modulator (DTC-SVM)”.
To prove the above thesis, the author used methodology based on an analyze and simulation
as well as experimental verification on the laboratory setup with 3kW PMSM motor.
Moreover, the presented control algorithm DTC-SVM has been introduced and used in serial
commercial product of Polish manufacture TWERD, Toruń.
In the author’s opinion the following results of the thesis are his original achievements:
•
development of a simulation algorithm in SABER package for the investigation
of PWM inverter-fed PMSM control,
•
elaboration and experimental verification of digital flux and torque controller
design based on the Z-transform approach for series (cascade) and parallel
structure of DTC-SVM schemes,
•
implementation and verification of series (cascade) and parallel DTC-SVM
schemes on experimental laboratory setup with 3kW PM synchronous motor
drive controlled by floating point DS1103 board.
•
bringing into production and testing of developed DTC-SVM algorithm in Polish
industry.
The thesis consists of eight chapters. Chapter 1 is an introduction. In Chapter 2 mathematical
model of PM synchronous motor and his basic control modes are presented. Chapter 3 is
devoted to voltage source inverter, his nonlinear characteristics and different PWM
techniques. Chapter 4 gives brief review of PM synchronous motor control method such as
FOC and classical DTC. In Chapter 5 two kind of DTC-SVM control schemes are presented.
Also, the analysis and synthesis of digital flux, torque and speed controllers based on Z
transform approach are given. Chapter 6 is devoted to initial rotor detection methods, stator
flux vector and rotor speed estimation algorithms. In Chapter 7 experimental results are
6
Introduction
presented and studied. Chapter 8 includes the finally conclusions. Description of the SABER
based control algorithm, basic coordinate transformations and parameters of used PM
synchronous machine as well as inverter are given in Appendices.
7
Modeling and control modes of PM synchronous motor drives
Chapter 2
MODELING AND CONTROL MODES OF PM SYNCHRONOUS DRIVES
2.1 Mathematical model of PM synchronous motor
Development of the machine model through the understanding of physics of the
machine is the key requirement for any type of electrical machine control. Since in this
project a Surface type Permanent Magnet Synchronous Motor (SPMSM) is used for the
investigation [9,13,14,15,16]. The development of those models is under bellow
assumptions as [3]:
•
three-phase motor is symmetrical,
•
only a fundamental harmonic of the magneto motive force (MMF) is taking in to
account,
•
the spatially distributed stator and rotor winding are replaced by a concentrated
coil,
•
an anisotropy effects, magnetic saturation, iron loses and eddy currents are not
taking into considerations,
•
the coil resistances and reactances are taking to be constant,
•
in many cases, especially when is considered steady state, the currents and
voltages are assumed to be sinusoidal,
•
thermal effect for permanent magnets is omitted.
The synchronous motor model will be presented in space vector notation. Space vector
form of the machine equations has many advantages such as compact notation, easy
algebraic manipulation, and very simple graphical interpretation. Specially, this notation
is very useful when analyzing the vector control based technique of the AC machines.
The space vector representation of AC machine equations has been discussed in detail
in number of text books ([3,4,12]).
The instantaneous value of a three-phase system K A , K B , KC (such as currents, voltages
and flux linkages) can be replaced by one resultant vector called the space vector,
K=
2
⎡1 ⋅ K A + a ⋅ K B + a 2 K C ⎤
⎦
3⎣
8
(2.1)
Modeling and control modes of PM synchronous motor drives
where: 1 , a = e
j
2π
3
−j
1
3 2
=− + j
,a =e
2
2
2π
3
=e
j
4π
3
1
3
=− − j
- complex vectors, 2/3 –
2
2
normalization factor (guarantee that for balanced sinusoidal waveforms the magnitude
of the space vector is equal to the amplitude of that phase waveforms).
The elements of this space vector satisfy the condition:
K A + K B + KC = 0
(2.2)
and it means that we have three-phase system without neutral wire.
2.1.1 Voltage and current equations
For idealized motor (Fig. 2.1), the following equations of the instantaneous stator phase
voltages can be written [3]:
B
Z sB
b
a
I sB
U sB
S
S
N
N
U sA
Z sA
γm
A
I sA
N
I sC
Z sC
S
U sC
C
c
Figure 2.1. Layout and symbols for three-phase PMSM electric motor windings.
U sA = I sA RsA +
d Ψ sA
dt
(2.3a)
U sB = I sB RsB +
d Ψ sB
dt
(2.3b)
U sC = I sC RsC +
dΨ sC
dt
(2.3c)
9
Modeling and control modes of PM synchronous motor drives
where U sA ,U sB ,U sC are the instantaneous stator voltage values, I sA , I sB , I sC are
instantaneous values of the current, Rs = RsA = RsB = RsC is the resistance of the stator
windings, and Ψ sA , Ψ sB and Ψ sC are magnetic flux linkages stator windings A, B and
C , respectively.
Using the space vector theory to voltage equations we can written in vector form
U sABC = Rs I sABC +
d Ψ sABC
dt
(2.4)
where:
2
2
2
2
2
U sABC = (1U sA + aU sB + a U sC ) , I sABC = (1I sA + aI sB + a 2 I sC ) , Ψ sABC = (1Ψ sA + aΨ sB + a Ψ sC )
3
3
3
are the
stator voltage, current and flux space vectors, respectively.
The stator winding flux consist of rotor flux and stator flux linkages:
Ψ sABC = Ψ ABC ( s ) + Ψ ABC ( r )
(2.5)
where,
Ψ ABC ( s )
⎡ LsA
= ⎢⎢ M sBA
⎢⎣ M sCA
Ψ ABC ( r ) = Ψ PM
M sAB
LsB
M sCB
M sAC ⎤ ⎡ I sA ⎤
M sBC ⎥⎥ ⎢⎢ I sB ⎥⎥
LsC ⎥⎦ ⎢⎣ I sC ⎥⎦
⎡
⎤
⎢ cos θ r
⎥
⎢
⎥
⎢ cos(θ − 2π ) ⎥
r
⎢
3 ⎥
⎢
⎥
⎢cos(θ r + 2π ) ⎥
3 ⎦⎥
⎣⎢
(2.6)
(2.7)
and, θ r is electrical rotor position. Mechanical rotor position is defined as:
θ r = pbγ m
(2.8)
where: pb - number of pole pairs, γ m - mechanical position.
In equation (2.6) LsA is the self-inductance of phase A winding, M sAB and M sAC are the
mutual inductances between A and B phase, A and C phase, respectively. For self and
mutual inductances of B and C phase the same notations used. In (2.7), ΨPM is the
10
Modeling and control modes of PM synchronous motor drives
amplitude of the flux linkages established by the permanent magnet on the rotor. The
inductances are described below.
Due to the rotor saliency in IPMSM the air gap is not uniform and, therefore, the self
and mutual inductances of stator windings are a function of the rotor position.
The derivation of these rotor position dependent inductances is available in details in
[5]. The results are summarized here as follows:
The stator winding self-inductances are
LsA = Lls + LA − LB cos 2θ r
(2.9a)
LsB = Lls + LA − LB cos 2(θ r −
2π
4π
) = Lls + LA − LB cos(2θ r −
)
3
3
(2.9b)
LsC = Lls + LA − LB cos 2(θ r +
2π
4π
) = Lls + LA − LB cos(2θ r +
)
3
3
(2.9c)
where, Lls is stator-winding leakage inductance and LA , LB are given by
2
⎛m ⎞
LA = ⎜ s ⎟ πµ 0 rlε1
⎝ 2 ⎠
(2.10a)
2
1⎛m ⎞
LB = ⎜ s ⎟ πµ 0 rlε 2
2⎝ 2 ⎠
(2.10b)
where, ms is number of turns of each phase winding, r is radius, which is from center
of machine to the inside circumference of the stator, and l is the axial length of the air
gap of the machine, µ 0 is permeability of the air, ε1 and ε 2 are defined as s:
1 1
1
)
+
2 g min g max
(2.11a)
1 1
1
)
−
2 g min g max
(2.11b)
ε1 = (
ε2 = (
where, g min is minimum air gap length and g max is maximum air gap length.
The mutual inductances between stator phase are:
1
π
1
2π
M sAB = M sBA = − LA − LB cos 2(θ r − ) = − LA − LB cos(2θ r −
)
2
3
2
3
11
(2.12a)
Modeling and control modes of PM synchronous motor drives
1
π
1
2π
)
M sAC = M sCA = − LA − LB cos 2(θ r + ) = − LA − LB cos(2θ r +
2
3
2
3
(2.12b)
1
1
M sBC = M sCB = − LA − LB cos 2(θ r + π ) = − LA − LB cos(2θ r + 2π )
2
2
1
= − LA − LB cos 2θ r
2
(2.12c)
Using the space vector theory, the flux linkage Ψ sABC space vector can be written as:
Ψ sABC = ( Lls +
3
3
∗
LA ) I sABC − LB I sABC e j 2θr + Ψ PM e jθr
2
2
2
3
(2.13)
2
3
where, I sABC = (1I sA + aI sB + a 2 I sC ) , I sABC ∗ = (1I sA + a 2 I sB + aI sC ) are the stator current
space vector and conjugate stator current space vector.
Taking into account that:
Ld = Lls + Lmd
(2.14a)
Lq = Lls + Lmq
(2.14b)
3
3
where, Lmd = ( LA + LB ) , Lmq = ( LA − LB ) are d and q magnetizing inductances and
2
2
are defined as [5].
Finally, equations (2.13) comes as:
Ψ sABC = (
Ld + Lq
2
) I sABC − (
Lq − Ld
2
∗
) I sABC e j 2θr + Ψ PM e jθr
(2.15)
where, Ld , Lq are d and q inductances.
Space vector form of machine equations (2.4, 2.15) becomes more compact, but the
rotor position dependent parameters still exist in that form of expressions for the stator
flux linkage space vector. Therefore, the space vector model is still not simple to use for
the analysis. A simplification can be made if the space vector model is referred to a
suitably selected rotating frame.
12
Modeling and control modes of PM synchronous motor drives
Figure 2.2 shows axes of reference for the three-stator phase A, B, C . It also shows a
rotating set of x, y axes, where the angle θ K is position of x -axis in respect to the stator
A phase axis. Variables along the A, B and C axes can be referred to the x − and
y − axes by the expression:
⎡ K x ⎤ 2 ⎡ cos θ K
⎢K ⎥ = ⎢
⎣ y ⎦ 3 ⎣ − sin θ K
⎡K ⎤
cos(θ K + 2π / 3) ⎤ ⎢ A ⎥
KB
− sin(θ K − 2π / 3) − sin(θ K + 2π / 3) ⎥⎦ ⎢ ⎥
⎢⎣ K C ⎥⎦
cos(θ K − 2π / 3)
(2.16)
y
KB
K ABC
ΩK
x
Ky
Kx
θK
KA
KC
Figure 2.2. Stator fixed three phase axes (A,B,C) and general rotating reference frame ( x, y ).
Finally, the space vector in general rotating frame can be written as:
K ABCs = K K (cos Θ K + j sin Θ K ) = K K e jθ K
(2.17)
In this case the voltage equation (2.4) using (2.17) can written as:
U sK e jθ K = Rs I sK e jθ K +
d
(Ψ sK e jθ K )
dt
(2.18)
Using chain rule, equation. (2.17) and divided by term e jθ K can be written as:
U sK = Rs I sK +
d Ψ sK
+ jΩ K Ψ sK
dt
(2.19)
where U sK , I sK , Ψ sK is the stator voltage, current and flux space vector in general
rotating frame.
Making similar arrangement like for the voltage equation the flux linkage vector in
general reference frame can be expressed as:
13
Modeling and control modes of PM synchronous motor drives
Ψ sK = (
Ld + Lq
2
) I sK − (
Lq − Ld
∗
) I sK e j 2(θr −θ K ) + Ψ PM e j (θr −θ K )
2
(2.20)
Stator fixed system ( α , β )
Taking the angular speed of the reference frame to be Ω K = 0 and θ K = 0 , the set of
synchronous machine vector equations (2.19) and (2.20) my be written as:
U sαβ = Rs I sαβ +
Ψ sαβ = (
d Ψ sαβ
Ld + Lq
2
(2.21)
dt
) I sαβ − (
Lq − Ld
2
∗
) I sαβ e j 2θr + Ψ PM e jθr
(2.22)
Substituting to above equations the following expressions for complex vectors
U sαβ = U sα + jU s β ,
I sαβ = I sα + jI s β , Ψ sαβ = Ψ sα + j Ψ s β
and splitting into real and
imaginary parts one can obtain the scalar form of the machine equations in stationary
α , β reference frame:
d Ψ sα
dt
U sα = Rs I sα +
d Ψ sβ
U s β = Rs I s β +
Ψ sα = (
Ld + Lq
Ψ s β = −(
2
(2.23b)
dt
−
Lq − Ld
2
(2.23a)
Lq − Ld
2
cos 2θ r ) I sα − (
)sin 2θ r I sα + [(
Ld + Lq
2
Lq − Ld
2
)+(
)sin 2θ r I sβ + Ψ PM cosθ r
Lq − Ld
2
)cos 2θ r ]I sβ + Ψ PM sin θ r
(2.24a)
(2.24b)
Note, that in the flux-current equations (2.24a and b) still we can observe that value of
inductances depends on rotor position θ r .
Stator flux fixed system ( x, y )
In order to take advantage of the set of equations (2.19) and (2.20) in rotating coordinate
system, one assumes that the coordinate system rotates with the stator flux linkage
angular speed Ω K = Ω Ψ and θ K = θ Ψ . As a Ψ sx = Ψ s , δ Ψ = −(θ r − θ Ψ )
U sxy = Rs I sxy +
Ψ sxy = (
d Ψ sxy
Ld + Lq
2
dt
+ jΩ Ψs Ψ sxy
) I sxy − (
Lq − Ld
2
14
(2.25)
∗
) I sxy e − j 2δ Ψ + Ψ PM e − jδ Ψ
(2.26)
Modeling and control modes of PM synchronous motor drives
Substituting to above equations the following expressions for complex vectors
U sxy = U sx + jU sy , I sxy = I sx + jI sy , Ψ sxy = Ψ s and splitting into real and imaginary parts
one can obtain the scalar form of the machine equations in stationary x, y reference
frame:
U sx = Rs I sx +
dΨs
dt
(2.27a)
U sy = Rs I sy + Ω Ψ s Ψ s
(2.27b)
1
1
Ψ s = [( Ld + Lq ) − ( Lq − Ld )cos 2δ Ψ ]I sx + ( Lq − Ld )sin 2δ Ψ I sy + Ψ PM cos δ Ψ
2
2
(2.28a)
1
1
0 = ( Lq − Ld )sin 2δ Ψ I sx + [( Ld + Lq ) + ( Lq − Ld ) cos 2δ Ψ ]I sy − Ψ PM sin δ Ψ
2
2
(2.28b)
The current-flux equations can be expressed also in simplest form as:
Ψ s = ( Ld cos 2 δ Ψ + Lq sin 2 δ Ψ ) I sx + ( Lq − Ld )sin δ Ψ cosn δ Ψ I sy + Ψ PM cos δ Ψ
(2.29a)
0 = ( Lq − Ld )sin δ Ψ cos δ Ψ I sx + ( Ld sin 2 δ Ψ + Lq cos 2 δ Ψ ) I sy − Ψ PM sin δ Ψ
(2.29b)
Rotor flux fixed system ( d , q )
In order to take advantage of the set of equations (2.19) and (2.20) in rotating coordinate
system, one assumes that the coordinate system rotates with the rotor flux angular speed
Ω K = pb Ω m and θ K = pbγ m = θ r
U sdq = Rs I sdq +
Ψ sdq = (
d Ψ sdq
Ld + Lq
2
dt
+ jpb Ω m Ψ sdq
) I sdq − (
Lq − Ld
2
(2.30)
∗
) I sdq + Ψ PM
(2.31)
Substituting the following expressions for complex vectors U sdq = U sd + jU sq ,
I sdq = I sd + jI sq , Ψ sdq = Ψ sd + j Ψ sq to (2.30) and (2.31), and splitting for real and
imaginary parts the scalar form of the machine equations in rotational fixed reference
frame can be obtained:
U sd = Rs I sd +
d Ψ sd
− pb Ω m Ψ sq
dt
15
(2.32a)
Modeling and control modes of PM synchronous motor drives
U sq = Rs I sq +
d Ψ sq
dt
+ pb Ω m Ψ sd
(2.32b)
where,
Ψ sd = Ld I sd + Ψ PM
(2.33a)
Ψ sq = Lq I sq
(2.33b)
It should be noted that when transforming the flux linkage vector Ψ s to the d , q
reference frame the rotor position θ r dependent terms disappear it can be seen from
equation (2.31). This is the main advantage of rotor-oriented representation.
Substituting the relationship of (2.33a-b) into (2.32a-b), and also considering
d Ψ PM
= 0 , the most common scalar form of the machine voltage equations in the rotor
dt
reference frame can be obtained as:
U sd = Rs I sd + Ld
U sq = Rs I sq + Lq
dI sd
− pb Ω m Lq I sq
dt
dI sq
dt
(2.34a)
+ pb Ω m Ψ PM + pb Ω m Ld I sd
(2.34b)
Based on the above voltage-current equations it is possible to draw the equivalent
electrical circuit separately for d and q axes (Fig. 2.3).
pb Ω m Lq I sq
Rs
Rs
I sq
I sd
U sd
pb Ω m Ld I sd
Ld
U sq
pb Ω mΨ PM
Lq
Figure 2.3. Equivalent circuit model of PMSM in the rotor reference frame. (a) Rotor d-axis
equivalent circuit, (b) Rotor q-axis equivalent circuit.
16
Modeling and control modes of PM synchronous motor drives
2.1.2 Instantaneous power and electromagnetic torque
The three-phase star-connection system without neutral wire is shown in Fig. 2.4. This
is classical configuration for AC motor windings connections.
A
U sAB
U sAC
U sA
B
U sBC
I sC
I sA
Z sA
Z sC
C
U sAB
Z sB
U sC U sB
I sB
Figure 2.4. Three-phase star connection system without neutral wire.
For this configuration the expression for instantaneous active power supplied to load
can be expressed as:
P = U sA I sA + U sB I sB + U sC I sC
(2.35)
Introducing space vector definition, after some arrangement and taking into account the
relation: I sA + I sB + I sC = 0 , the equation (2.35) can be written as:
3
∗
P = Re[U sABC I sABC ]
2
(2.36)
For d , q frame, the equation (2.35) for the active power can be expressed as:
3
P = (U sd I sd + U sq I sq )
2
(2.37)
Substituting voltage equation (2.4) into (2.36), and adopting Ω K = pb Ω m one obtains
d Ψ sABC
3
∗
∗
∗
P = [Re( Rs I sABC I sABC +
I sABC − jpb Ω m Ψ sABC I sABC )]
dt
2
(2.38)
Note that I sABC I sABC ∗ = Is and:
2
d Ψ sABC
3
2
∗
∗
P = [ Rs Is + Re(
I sABC ) + Re(− jpb Ω m Ψ sABC I sABC )]
dt
2
(2.39)
Hence, neglecting the losses in stator resistance Rs and assuming that
d Ψ sABC
= 0 , the
dt
electromagnetic power is expressed:
17
Modeling and control modes of PM synchronous motor drives
Pe =
3
∗
pb Ω m Im(Ψ sABC I sABC )
2
(2.40)
In d , q frame the active power can be written:
Pe =
3
pb Ω m (Ψ sd I sq − Ψ sq I sd )
2
(2.41)
For the presented system (Fig. 2.4) the expression for instantaneous reactive power
supplied to the three-phase load system without neutral wire can be calculated as:
Q=
1
3
( I sAU sBC + I sBU sCA + I sCU sAB )
(2.42)
Introducing the space vector definition into equation (2.42), after some arrangement,
and taking into account the relation: I sA + I sB + I sC = 0 , one obtains:
Q=
3
∗
Im[U sABC I sABC ]
2
(2.43)
In d , q frame the reactive power is expressed as:
3
Q = (U sq I sd − U sd I sq )
2
(2.44)
Substituting voltage equation (2.4) into (2.43), adopting Ω K = pb Ω m and made similar
arrangements like for active power calculation, the final expression for reactive power
is:
Q=
3
∗
pb Ω m Re(Ψ sABC I sABC )
2
(2.45)
In d , q frame the expression (2.45) for the reactive power becomes:
Q=
3
pb Ω m ( Ψ sd I sd + Ψ sq I sq )
2
(2.46)
The important quantity of the drive is the power factor cos φ , which can be calculated
as:
cos φ =
Q
S
(2.47)
where S is module of apparent power vector S = P + jQ :
S = P2 + Q2
(2.48)
18
Modeling and control modes of PM synchronous motor drives
The instantaneous electromagnetic torque developed by an electric motor can be defined
as:
Me =
Pe
Ωm
(2.49)
where, Pe is the electromagnetic power and Ωm is the mechanical angular rotor speed.
Finally, taking into account equation (2.49) the expression for electromagnetic torque
can be obtained as:
Me =
3
∗
pb Im( Ψ sABC I sABC ) ,
2
(2.50)
Me =
3
pb (Ψ sd I sq − Ψ sq I sd )
2
(2.51)
and in d , q frame:
Substituting Ψ sd , Ψ sq from (2.33a-b), the torque expression of equations (2.47)
becomes:
Me =
3
pb (Ψ PM I sq − ( Lq − Ld ) I sd I sq )
2
(2.52)
It can be seen from (2.52), that developed torque consist of two parts, one produced by
the permanent magnet flux called synchronous torque ( M es ) and the second called
reluctance torque ( M er ), which is produced by the difference of the inductance in rotor
d- and q-axes. Expressions for those two torque components are:
M es =
3
pb Ψ PM I sq
2
M er = −
(2.53a)
3
pb ( Lq − Ld ) I sd I sq
2
(2.53b)
It should be mentioned that for SPMSM ( Ld = Lq ) the reluctance torque does not exist
due to the same inductance paths in rotor d- and q-axes.
The torque expression (2.52) can also be written in polar form using the current vector
amplitude Is and the torque angle δ I , i.e. angle between rotor d-axis and current
vector (Fig. 2.5.).
19
Modeling and control modes of PM synchronous motor drives
q − axis
Is
I sq
Ωs
d − axis
δI
I sd
Ψ PM
Figure 2.5. Stator current vector in rotor reference frame.
For two current components using trigonometrical rules we can write:
I sd = I s cos δ I
(2.54a)
I sq = I s sin δ I
(2.54b)
Substituting I sd , I sq into equation (2.52), the torque expression can be obtain as:
Me =
3
1
2
pb [ Ψ PM I s sin δ I − ( Lq − Ld ) I s sin 2δ I ]
2
2
M es
(2.55)
M er
For given current amplitude the synchronous and reluctances torque varies according to
the sine of torque angle δ I . The variation of M es and M er and resultant torque M e with
torque angle are illustrated in Fig. 2.6. The IPMSM parameters used for this calculation
M e [ Nm]
M er [ Nm ]
M es [ Nm ]
are given in the Appendices.
δ I [deg]
Figure 2.6. Variation of synchronous torque M es , reluctance torque M er and resultant
torque M e as a function of torque angle (for rated current amplitude).
20
Modeling and control modes of PM synchronous motor drives
Referring to Fig. 2.7, stator flux components in rotor reference frame can be written as:
Ψ sd = Ψ s cos δ Ψ = Ld I sd + Ψ PM
(2.56a)
Ψ sq = Ψ s sin δ Ψ = Lq I sq
(2.56b)
where: Ψ s is stator flux linkage amplitude, Ψ PM is rotor permanent magnet and δ Ψ is
torque angle (angle between stator flux linkage vector and rotor permanent magnets flux
vector).
q − axis
Is
Ψ sd
Ψ PM
I sq
Ωs
Ψs
Ψ sq = Lq I sq
δΨ
d − axis
I sd
Ld I sd
Ψ sd = Ld I sd + Ψ PM
Figure 2.7. Rotor permanent magnet flux vector and stator flux linkage vector in rotor reference
frame.
From (2.56a) and (2.56b) the I sd and I sq can be obtained as:
I sd =
I sd =
Ψ s cos δ Ψ − Ψ PM
(2.57a)
Ld
Ψ s sin δ Ψ
(2.57b)
Lq
Substituting current components (2.57a), (2.57b) into equation (2.51), one can obtain
another useful torque expressions:
Ψ Ψ sin δ Ψ Ψ s ( Lq − Ld )sin 2δ Ψ
3
M e = pb [ s PM
−
]
Ld
2
2 Ld Lq
2
M es
(2.58)
M er
where: Ψ s stator flux linkage amplitude, and Ψ PM rotor flux, δ Ψ is torque angle, M es synchronous torque, M er - reluctance torque.
21
Modeling and control modes of PM synchronous motor drives
For the PM synchronous motor the amplitude of stator flux Ψ s is established by
permanent magnet. Operation with stator flux amplitude belong the nominal value of
rotor flux amplitude Ψ PM increases the amplitude of stator phase current. Please note
that maximum amplitude of the stator current vector is calculated as: I s ≤
Ψ PM
, and
Ld
higher value may damage the PM (complete demagnetization).
From the Fig. 2.8 it can be observed that rated torque is achieved for torque angle
M e [ Nm ]
M er [ Nm ]
M es [ Nm ]
0 < δ Ψ < 25D electrical degree.
δ Ψ [deg]
Figure 2.8. Variation of synchronous torque M es , reluctance torque M er and resultant
torque M e as a function of torque angle (for constant stator flux equal value of PM).
2.1.3 Mechanical motion equation
The equation of rotor motion dynamics describes the mechanical equilibrium of a drive
system. Taking the moment of inertia to be constant ( J = const. ) and neglecting friction
and elastic torque we can write:
Me = Ml + Md
(2.59)
where, M l is the external torque on the motor shaft, and M d the dynamic torque
Md = J
d Ωm
dt
(2.60)
where: J is total moment of inertia, Ω m angular speed of the rotor.
22
Modeling and control modes of PM synchronous motor drives
In general, for a drive system,
J = Jm + Jl
(2.61)
where: J m - motor inertia, J l load moments of inertia.
From equation (2.55) and (2.56) one can write:
d Ωm 1
= (M e − M l )
dt
J
(2.62)
d Ωm 1 3
∗
= ( pb Im(Ψ s I s ) − M l )
dt
J 2
(2.63)
and, with (2.50),
Finally, the full mathematical model of PM synchronous machine which is used in
simulation studies [Appendices] is described in d , q reference frame as:
U sd = Rs I sd +
U sq = Rs I sq +
d Ψ sd
− pb Ω m Ψ sq
dt
d Ψ sq
dt
+ pb Ω m Ψ sd
(2.64a)
(2.64b)
Ψ sd = Lsd I sd + Ψ PM
(2.65a)
Ψ sq = Lsq I sq
(2.65b)
d Ωm 1
= (M e − M l )
dt
J
Me =
(2.66)
3
3
∗
pb Im( Ψ s I s ) = pb ( Ψ sd I sq − Ψ sq I sd )
2
2
(2.67)
Based on above equations we can create the block scheme of the PMSM machine (Fig.
2.9), where the input signals are the voltage components in d , q reference frame
U sd ,U sq and the output signal is the mechanical speed of the rotor Ω m . As the external
load torque M l is disturbance.
23
Modeling and control modes of PM synchronous motor drives
Ml
ΨPM
pb Ω m Ψ sd
U sd
+
−
∫
pb
−
Ψsd
I sd
1
Ld
Ψ sq I sd
Rs
−
3
pb
2
Me
−
Rs
−
U sq
−
∫
Ψsq
∫
1
J
Ωm
Ψ sd I sq
1
Lq
I sq
pb
pb Ω m Ψ sq
Figure 2.9. Block scheme of PM synchronous machine in rotating d , q frame.
Based on equations (2.64-2.67) we can also draw the vector diagram of PM
synchronous motor (Fig. 2.10). From this vector representation it can see the positions
of the vectors (currents, voltages and fluxes). Especially, power angle φ (angle between
voltage and current vectors) and torque angle defined in two manners: as an angle
between current and rotor flux vectors - δ I , or as angle between stator flux and rotor
flux vectors - δ Ψ .
q− axis
β
Us
Is
Rs I s
Isq
Ld Isd
Ψsq
Ωs Ψs
Ψs
φ
δI
δΨ
θr
Isd
θ
Lq Isq
Ψsd Ψs
d − axis
ΨPM
rotor
α (A)
stator
Figure 2.10. Vector diagram of PM synchronous motor in rotor reference frame d , q .
24
Modeling and control modes of PM synchronous motor drives
2.2 Static characteristic under different control modes
In this section, basic steady state properties of the PMSM under different control mode
strategies will be study [6,9]. The key control strategies for the PMSM can be listed as
follows:
•
Constant torque angle control (CTAC).
•
Maximum torque per ampere control (MTPAC)
•
Unity power factor control (UPFC)
•
Constant stator flux control (CSFC)
Constant torque angle (CTA) control
This control strategy for PMSM keeps the torque angle δ I (angle between stator current
vector and rotor permanent magnet flux) at constant value 90D .
q − axis
I sq = I s
δ I = 90D
d − axis
Ψ PM
Figure 2.11. Current vector and permanent magnet flux vector for constant torque angle
operation (CTAC)
Hence, this control can be achieved by controlling the d-axis current components to
zero leaving the current vector on the rotor q-axis (see Fig. 2.11). Therefore, this
strategy is also referred to as I sd = 0 control. The amplitude of rotor flux vector is
constant and also the torque angle is constant. So, the torque depends only on the value
of stator current amplitude. Therefore, this control strategy is not recommended for
IPMSM with high saliency ratio. However, for SPMSM, this strategy is commonly
used.
The torque equation in this mode of operation becomes:
Me =
3
3
pb Ψ PM I sq = pb Ψ PM I s
2
2
25
(2.68)
Modeling and control modes of PM synchronous motor drives
The steady state voltage components based on the equations (2.34a) and (2.34b) are:
U sd = − pb Ωm Lq I qs = − pb Ω m Lq I s
(2.69a)
U sq = Rs I qs + pb Ωm Ψ PM = Rs I s + pb Ω m Ψ PM
(2.69b)
The amplitude of stator voltage vector can be calculated as:
U s = U 2 sd + U 2 sq
(2.70)
The stator flux vector amplitude can be calculated from equations (2.65a-b) as:
Ψ s = Ψ 2 sd + Ψ 2 sq
(2.71)
The active and reactive power and also the power factor can be obtained from equations
(2.41),(2.46), (2.47).
Maximum torque per ampere (MTPA) control
The main idea of this control is develop the torque using minimum value of stator
current amplitude. In this case the I sd components is not equal zero, and may cancel the
reluctance torque produced by high saliency ratio. Therefore, this control strategy is
recommended for IPMSM.
q − axis
Is
I sq
δ I >= 90D
I sd
d − axis
Ψ PM
Figure 2.12. Current vector I s and permanent magnet flux vector Ψ PM for maximum torque
per ampere operation (MTPAC).
In order to obtain the maximum torque per ampere we should solve the derivative of
torque equations (2.55) in respect to torque angle. Solving for torque angle α and taking
into account that only negative sign should be considered for the solution, we can
calculate torque angle as:
δ I = cos −1[
−1
1
1
)2 ]
−
+(
4( Ld − Lq ) I s
2 4( Ld − Lq ) I s
26
(2.72)
Modeling and control modes of PM synchronous motor drives
From Fig. 2.8, it can be seen that M e is maximum when torque angle is 90D < δ I < 180D .
The relevant torque equation in this mode of operation becomes from (2.55).
The steady state voltage equations can be written using the current vector amplitude I s
and the torque angle δ I as:
U sd = Rs I s cos δ I + pb Ωm Lq I s sin δ I
(2.73a)
U sq = Rs I s sin δ I − pb Ω m Ld I s cos δ I + pb Ωm Ψ PM
(2.73b)
The amplitude of stator voltage vector can be calculated from equation (2.70) and
amplitude of stator flux vector from (2.71). The active and reactive power and also the
power factor can be obtained from equations (2.41),(2.46), (2.47).
Unity power factor (UPF) control
Under this control strategy there is no phase different between the current vector and the
voltage vector. Hence, power factor angle φ (see Fig. 2.13) becomes zero. Since only
active power is supplied to the machine under unity power factor operation, the VA
rating requirement of the inverter can be reduced.
q − axis
Us
Is
φ =0
δI
d − axis
ΨPM
Figure 2.13. Current vector and permanent magnet flux vector under unity power factor
operation (UPFC).
In this case when φ = 0 we have the relationship:
U sq
U sd
=
I sq
I sq
= tan δ I
(2.74)
Substituting the voltage equations (2.69a-b) into (2.71) and made some simplifying, we
can obtain:
I s ( Ld − Lq )cos 2 δ I − Ψ PM cos δ I + Lq I s = 0
27
(2.75)
Modeling and control modes of PM synchronous motor drives
Solving for the torque angle δ I :
2
−1
δ I = cos [
Ψ PM − Ψ PM 2 − 4 I s ( Ld − Lq ) Lq
2( Ld − Lq ) I s
]
(2.76)
only positive sign should be take into consideration.
After obtaining δ I the amplitude of stator voltage vector can be calculated from
equation (2.70) and amplitude of stator flux vector from (2.71). The active and reactive
power and also the power factor can be obtained from equations (2.41),(2.46), (2.47).
Constant stator flux (CSF) control
As it can be see from the torque expression (2.58) for a given stator flux amplitude Ψ s
the electromagnetic torque M e is a function of torque angle δ Ψ . The stator flux linkage
amplitude Ψ s is kept constant of the permanent magnet flux amplitude Ψ PM .
q − axis
Is
Ψs
δI
δΨ
d − axis
ΨPM
Figure 2.14. Flux vector and permanent magnet flux vector under constant stator flux operation
(CSFC).
The amplitude of the stator flux linkage vector is
Ψ s = Ψ sd 2 + Ψ sq 2 = ( Lq I sq ) 2 + ( Ld I sd + Ψ PM ) 2
(2.77)
Ψ s = Ψ PM
(2.78)
Equating
can be obtain the relationship for rotor frame currents as:
( Lq I sq ) 2 + ( Ld I sd ) 2 + 2 Ld Ψ PM I sd = 0
(2.79)
This condition is true if I sd < 0 , because expression ( Lq I sq ) 2 + ( Ld I sd ) 2 and Ld , Ψ PM are
always positive values.
28
Modeling and control modes of PM synchronous motor drives
( Ld 2 − Lq 2 ) I s cos δ I 2 + 2 Ld Ψ PM I s cos δ I + Lq 2 I s = 0
2
2
(2.80)
Solving for the torque angle δ I
2
−1
δ I = cos [
− Ld Ψ PM ± Ld 2 Ψ PM 2 − I s ( Ld 2 − Lq 2 ) Lq 2
( Ld 2 − Lq 2 ) I s
]
(2.81)
For given δ I , the amplitude of stator voltage vector we can be calculated from equation
(2.70) and amplitude of stator flux vector from (2.71). The active and reactive power
and also the power factor can be obtained from equations (2.41),(2.46),(2.47),
respectively for defined speed.
Comparison study
In order to compare the control strategies and to cancel dependence of machine power,
per unit values defined as shown in Table 2.1 below have been introduced [3,9].
The value of current vector:
I sN =
Is
Is
=
Ib
2 I srms ( rated )
(2.82)
The value of voltage vector:
U sN =
Us
Us
=
U b Ωb Ψ PM
(2.83)
where: Ωb = 2π fb and f b is rated frequency
of the PM motor.
The value of flux vector is:
Ψ sN =
The value of torque is:
M eN =
Ψs
Ψs
=
Ψ b Ψ PM
Me
Me
=
Mb 3 p Ψ I
b PM b
2
S
S
=
SN =
3
Sb
U b Ib
2
P
PN =
Sb
The value of apparent power vector
The value of active power
The value of reactive power
QN =
Q
Sb
(2.84)
(2.85)
(2.86)
(2.87)
(2.88)
Table 2.1. Per unit values definition.
In order to compare the steady state performance characteristic of the above discussed
control strategies, for each of the control strategy some important quantities of the
machine have been plotted as a function of the torque. The PMSM parameters, which
are used for the calculations are given in Appendices.
29
Modeling and control modes of PM synchronous motor drives
The current requirement versus torque is illustrated in Fig. 2.15 for the different control
strategies. It can be seen, that up to 1 pu torque, the requirement for current is lowest for
CSF control. Highest than 1 pu torque the low current needs MTPA control requirement
lowest current for a given torque.
3
2.5
I sN [ pu ]
2
CSF
CTA
UPF
MTPA
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
M eN [ pu ]
Figure 2.15. Stator current amplitude under different control strategies versus electromagnetic
torque.
The voltage requirement versus torque for the different control strategies is illustrated in
Fig. 2.16. It can be seen, that CSF requires the highest value of stator voltage.
2.5
CSF
CTA
2
1.5
U sN [ pu ]
MTPA
1
0.5
0
UPF
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
M eN [ pu ]
Figure 2.16. Stator voltage amplitude versus electromagnetic torque under different control
strategies (at 1 pu rotor speed).
3
CSF
CTA
MTPA
2
PN [ pu ]
UPF
1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
M eN [ pu ]
Figure 2.17. Active power versus electromagnetic torque under different control strategies.
30
Modeling and control modes of PM synchronous motor drives
The active power requirement as a function of torque is illustrated in Fig. 2.17 for the
different control strategies. It can be seen, that all control strategies require
approximately the same value of active power for a given torque. CSF control needs
less active power in the region up to 1.3 pu torque.
The reactive power requirement as a function of torque is illustrated in Fig. 2.18. It can
be seen, that CTA control requires the highest value of active power for a given torque
and the CSF control lowest.
4
CTA
MTPA
3
QN [ pu ]
2
CSF
1
0
UPF
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
M eN [ pu ]
Figure 2.18. Reactive power versus electromagnetic torque under different control strategies.
1.1
UPF
1
CSF
0.9
cos φ
0.8
MTPA
0.7
CTA
0.6
0.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
M eN [ pu ]
Figure 2.19. Power factor as a function of electromagnetic torque under different control
strategies.
The power factor as a function of torque is illustrated in Fig. 2.19. It can be seen, that as
it could be expected, UPF control requires constant power factor for a given torque.
CSF control is very close to the unity power factor up to 1 pu torque.
The above analysis can be summarized as shown in Table. 2.2.
31
Modeling and control modes of PM synchronous motor drives
Table. 2.2. Summary of voltage, power, power factor requirements under control modes.
Requirement
Control
method
Voltage
Current
Power
factor
CTA
middle
low
low
MTP
A
low
low
middle
UPF
low
high
1
CSF
high
lowes up to 1.1 Close to 1 up to
1 pu torque
pu torque
From this comparison study it can be concluded that CSF control appears to be superior
in terms of steady state performance characteristics compared to other methods under
discussion.
32
Modeling and control modes of PM synchronous motor drives
2.3 Summary
¾ There are different forms to express the PMSM equations, but the rotor
reference frame equations are the most widely used. The simplification in rotor
d , q reference frame equations results from the disappearance of position
dependent inductances.
¾ The electromagnetic torque of the IPMSM is not only produced by the
permanent magnet flux, but also by the reluctance difference in rotor d- and qaxes.
¾ Electromagnetic torque as cross vector product of the stator flux linkage and
current space vectors or rotor and stator flux linkages is independent of
coordinate system selected. Therefore, can be expressed in stationary ( α , β ) or
rotated ( d , q ) coordinates.
¾ For further control strategies consideration it is convenient to express the
electromagnetic torque of PMSM machine by:
•
vector product of stator current and rotor flux vectors. The rotor flux
vector in PMSM machine is constant, because of the PM. Therefore, to
increase and decrease the torque, the current amplitude and the torque
angle δ I should be changed (see Fig. 2.20a),
•
vector product of stator flux vector and rotor flux vector. Generally, the
value of the stator flux amplitude is kept constant at value of rotor flux
produced by permanent magnets. So, in this case to change the torque we
should adjust the torque angle δ Ψ (see Fig. 2.20b).
q β
q β
a)
Is
δI
γm
b)
Ψs
Ψ PM
d
Ψ PM
δΨ
α
γm
d
α
Fig. 2.20 Torque production: a) current control, b) flux control
¾ Taking into account discussion regarding static characteristic under different
control strategies it can be said that –depart from special requirements- the most
suited for general application PMSM drives is constant stator flux (CSF)
operation.
33
Voltage source PWM inverter for PMSM supply
Chapter 3 VOLTAGE SOURCE PWM INVERTER FOR PMSM SUPPLY
3.1 Introduction
The block scheme of an adjustable speed drive (ASD) commonly used in industrial
applications to supply three-phase AC motor is presented in Fig. 3.1.
Rectifier
Three-phase
grid
DC link
filter
Inverter
AC motor
Choke
Figure 3.1 Basic scheme of adjustable speed AC motor system.
An ASD is supplied from three or single phase grid. It consists of a diode rectifier, DC
link filter and an inverter. The rectifier converts supply AC voltage into DC voltage.
The DC voltage is filtered by a capacitor in the DC link. The inverter converts the DC
to an variable voltage, variable frequency AC for motor speed or (torque/current)
control.
The rectifier section of an ASD, called the front end, is responsible for generating
current harmonics into the power supply system. Therefore, to reduce the total harmonic
distortion (THD) of phase current it is necessary to add additional choke inductances.
There are generally to way how insert choke inductances (see Fig. 3.2).
b)
a)
D1
D3
D5
LF
LF
D1
D3
D5
CF
CF
LF
U DC
U DC
LF
D2
D4
D6
D2
D4
D6
LF
Fig. 3.2 Three phase diode rectifier with smoothing choke: a) at the input b) at the DC link side.
By adding a choke inductance at the input of rectifier gives the significant harmonic
reduction. Some drive manufactures are starting to include this choke inductance in the
DC link of the drive, providing the same harmonic current reduction benefit.
Regarding to power electronics standards IEEE Std 519 it is recommended that
production of harmonics should be less than 5%. It is a trend to replace the diode
34
Voltage source PWM inverter for PMSM supply
rectifier by fully controllable active rectifier [8] (see Fig. 3.3), which guaranties
following futures:
•
power flow from AC/DC or DC/AC side (there is no need of break resistor ),
•
significant reduction of phase current THD,
•
unity power factor (phase voltage is in phase with current),
•
reduction of DC link capacitor,
•
controllable DC link voltage.
Active
Rectifier
Three-phase
grid
DC link
filter
Inverter
AC motor
Choke
Fig. 3.3 Modern AC/DC/AC converter topology of adjustable speed drives.
In the next part of this thesis the author will be focus on voltage source inverter.
3.2 Voltage source inverter (VSI)
The made constant DC voltage by rectifier is delivered the to the input of inverter (Fig.
3.4), which thanks to controlled transistor switches, converts this voltage to three-phase
AC voltage signal with wide range variable voltage amplitude and frequency [3].
Voltage Source Inverter
T1
CF
D7
T3
T5
D9
D11
O
U DC
CF
T2
D8
T4
A
T6
D10
D12
C
B
U sAN
U sBN
U sCN
N
Three-phase motor windings
Figure 3.4 Basic scheme of voltage source inverter circuit.
35
Voltage source PWM inverter for PMSM supply
The one leg of inverter consists of two transistor switches. A simple transistor switch
consist of feedback diode connected in anti-parallel with transistor. Feedback diode
conducts current when the load current direction is opposite to the voltage direction.
Assuming that the power devices are ideal: when they are conducting the voltage across
them is zero and they present an open circuit in their blocking mode. Therefore, each
inverter leg can be represented as an ideal switch. Its gives possibility to connect each
of the three motor phase coils to a positive or negative voltage of the dc link ( U DC ).
Thus the equivalent scheme for three-phase inverter and possible eight combinations of
the switches in the inverter are shown in Fig. 3.5.
U 7 = 111
1
U DC
U 0 = 000
1
SA
SB
1
SC
SA
U DC
SB
0
A
C
B
A
U1 = 100
1
U DC
SA
SB
A
SC
1
U DC
U 3 = 010
1
SA
SB
C
B
SC
A
C
B
SB
1
SC
0
B
C
SB
SC
0
A
C
B
U 6 = 101
SA
U DC
1
0
U 5 = 001
1
SA
U DC
0
0
A
C
B
0
U 4 = 011
SA
SC
0
U 2 = 110
0
U DC
0
SB
1
SC
1
U DC
SA
0
A
SB
1
SC
0
B
C
A
B
C
Figure 3.5 Possible switches state in VSI.
The six positions of switches ( U1 − U 6 ) produce an output phase voltage equal ± 1/3 or
± 2/3 of the DC voltage. The last two ( U 0 ,U 7 ) give zero output voltage. The output
phase voltages produced by inverter are shown in Fig. 3.6a. and adequate line to line
voltage calculated in bellow formula also are presented in Fig. 3.6b.
U sAB = U sAN − U sBN
(3.1a)
U sBC = U sBN − U sCN
(3.1b)
U sCA = U sCN − U sAN
(3.1c)
36
Voltage source PWM inverter for PMSM supply
a)
b)
U sAN
U sAB
2
U DC
3
U DC
2π
Ωt
2
− U DC
3
Ωt
2π
Ωt
2π
Ωt
−U DC
U sBN
U sBC
2
U DC
3
U DC
2π
Ωt
2
− U DC
3
−U DC
U sCA
U sCN
2
U DC
3
U DC
2π
2
− U DC
3
2π
Ωt
−U DC
U1
U2
U3
U4
U5
U6
U1
a)
U2
U3
U4
U5
U6
b)
Figure. 3.6 Three voltage waveforms generated by the inverter: a) phase voltages, b) line to line
voltages.
Form the Fourier analysis for phase voltage produced by inverter (Fig. 3.7) the
maximum amplitude of fundamental phase voltage for a given DC link voltage is given
by:
U _ amp =
2
π
(3.2)
U DC
Uout
2
UDC
3
1
UDC
3
1
− UDC
3
2
− UDC
3
2
π
UDC
2π ωt
Figure 3.7 Inverter phase voltage generated during six step operation (solid line), corresponding
fundamental component of output voltage (dashed line) and harmonic spectrum of phase
voltage.
37
Voltage source PWM inverter for PMSM supply
The three-phase output voltage of the inverter can be described by space vector
definition as:
π
j ( k −1)
⎧2
3
⎪ U DC e
3
⎪⎪
Uk = ⎨
⎪0
⎪
⎪⎩
for k =1,2...,6.
,
(3.3)
for k=0,7.
where k denotes numbers vector.
Vectors from 1-6 are called active vectors, whereas vectors 0,7 are called zero vectors
or non active vectors. The voltage space vector U k in complex plane forms a regular
hexagon and divides in into six equal sectors (one sector takes 60 electrical degree) Fig.
3.8.
Im
U 3 (010)
U 2 (110)
2
U DC
3
sec tor 2
sec tor1
sec tor 3
U 4 (011)
U 0 (000)
U 1 (100)
U 7 (111)
Re
sec tor 4
sec tor 6
sec tor 5
U 5 (001)
U 6 (101)
Figure 3.8 Representation of the inverter states in the complex space.
In practice the real voltage source inverter has non-linear characteristic due to [19]:
•
the dead-time,
•
a voltage drop across the power switches,
•
pulsation of the DC link voltage.
Dead time effect [17,20,27,30]
Semiconductors power switches of voltage source inverter operate not ideally. They do
not turn-on or turn-off instantaneously. Therefore it is necessary to include a protection
time to avoid a short circuit in the DC link, when two switching devices are in the same
leg (see Fig. 3.9). This time Td is included in the control signals and it is called “dead
38
Voltage source PWM inverter for PMSM supply
time”. It guarantees safe operation of the inverter. The typical value is from 1µ s - 5µ s .
When the lowest value is for small power IGBT and is growing in respect to increasing
of IGBT power. More details about real IGBT module you can find in Appendices.
The effect of dead time can be examined from one phase of PWM inverter. The basic
configuration is shown in Fig. 3.9. Consist of upper and lower power devices T1 and T2 ,
and reverse recovery diodes D1 and D2 , connected between the positive and negative
rails of power supply. The gating signals S A and S Ai come from control block. Output
voltage terminal U 0 is connected to motor phase.
T1
SA
D1
I sA
U DC
S Ai
T2
LOAD
D2
U0
S
Dead time
Td
Figure 3.9 Circuit diagram of one inverter leg.
Fig. 3.10 shows the ideal control signals and real control signals with inserted dead time
Td . As can be observed the time duration of real drive signal for upper transistor is
shorted than ideal drive signal and for lower transistor is longer than ideal.
Ideal drive signals
Real drive signals
SA
SA
S Ai
S Ai
Td
Td
Figure 3.10 Gate signals control of one inverter leg.
39
Voltage source PWM inverter for PMSM supply
As a consequence when the phase current I sA is positive, the output voltage is reduced,
and when the current I sA is negative the output voltage is increased (see Fig. 3.11).
U0
ideal voltage
I sA > 0
U DC
2
U DC
2
−U DC
2
−U DC
2
deacresing
I sA < 0
U0
real voltage
increasing
Figure 3.11 Dead time effect on the inverter output voltage: (fat line real voltage, doted line
ideal voltage).
Voltage drop across power devices
In real voltage source inverter power switches do not conduct ideally. When they are
conducting the voltage across them is not zero and equal the voltage drop on the
conducted transistor VT . Also in blocking mode the power switches have voltage drop
on the conducted diode VD . More details about real IGBT module you can find in
Appendices.
The voltage drop across the power devices is dependent on the direction of the phase
current. It has influence on the output voltage, especially at low speed operation of
motor and high load current [17,20,27,30]. Fig. 3.12 shows the voltage drop influence
on the output voltage. Also shows that the output voltage is asymmetric (with offset)
and the voltage drop decreases the output voltage when the phase current is positive and
increases the output voltage when the phase current is negative.
I sA > 0
U0
VT
U DC
2
−U DC
2
VD
I sA < 0
U0
VD
U DC
2
−U DC
2
VT
Figure 3.12 Output voltage in voltage source inverter due to voltage drop across the power
devices a) for I sA > 0 , b) I sA < 0 (fat line real voltage, doted line ideal voltage).
The influence of dead time effect and voltage drop across power devices on the output
voltage from inverter is illustrated in block diagram (Fig. 3.13). The ideal reference
voltage components in stationary reference frame ( U *α _ ideal , U *β _ ideal ) are equal real
40
Voltage source PWM inverter for PMSM supply
( U *α _ real , U *β _ real ) and delivered to pulse width modulation (PWM) modulator block
with real non-linear inverter. As a result the output voltages ( Uα _ out , U β _ out ) are distorted
(green signals) and as consequence the phase currents ( Iα _ out , I β _ out ) in the load (red
signals) are also distorted.
U *α _ ideal
U
*
β _ ideal
U *α _ real
U
*
β _ real
SA
PWM
Modulator
SB
Inverter
SC
Uα _ out
Iα _ out
U β _ out
I β _ out
Load
Figure 3.13 Block diagram illustrated the dead time effect and voltage drop across power
devices in three phase motor supplied from non-ideal voltage source inverter.
Pulsation of the DC link voltage [19]
In practice it should be take into account that the real input dc-link voltage required for
supply VSI is not ideal. It has ripples and fluctuation, because of not ideal filtering and
disadvantages of diode rectifier. Therefore, the quality of dc-link voltage has impact on
the output voltage from inverter. If dc-link voltage will change we can observed the
changing at the output of inverter. In order to overcome this problem:
•
in PWM modulator we can not assume a constant dc-link voltage and we should
measured this voltage in order to calculate the modulation index (see subchapter
3.3),
41
Voltage source PWM inverter for PMSM supply
•
instate of
diode rectifier will be use the active rectifier, which provided
controllable DC-link voltage.
•
or used bigger capacitor in the DC-link side in order to increase possibility of
filtering.
Let us summarize, adding influence of non-linear VSI causes by:
•
serious distortion in the inverter output voltage,
•
distorted machine currents,
•
torque pulsation,
Additionally, also causes motor instability due to the interaction between motor and the
PWM inverter, or the choice of the PWM strategy [25].
Based on simulated and experimental observation one can say that the dead time effect
is:
•
more visible in low speed operation of the motor,
•
may become significantly in drives where high switching frequency is required
for good dynamics performances.
In some applications such as sensor-less vector control, the inverter output voltages are
needed to calculate the rotor or stator flux vectors. Unfortunately, it is very difficult to
measure the output voltage and requires additional hardware. The most desirable
method to obtain the output voltage feedback signal is to use the reference voltages
instead. However, the relation between the output and reference voltage is nonlinear due
to the dead-time effect and voltage drop across power devices. Thus, unless the properly
dead-time and voltage drop compensation will be applied, the reference voltage can not
be used instead of the inverter output voltage. Several compensation method were
proposed to overcome this problem. One of them will be present bellow.
Compensation based on modification of reference voltage waveform [17]
The compensation process of dead time effect and voltage drop across power devices on
the inverter output voltage from is illustrated in Fig. 3.14.
42
Voltage source PWM inverter for PMSM supply
Compensation
block
I sA I sB I sC
U DC
Compensation of
inverter
Tcomp
U α _ comp
U
U β _ comp
*
U *α _ real
α _ ideal
U
*
α _ real
U *β _ real
U *β _ ideal
SA
PWM
Modulator
U *β _ real
SB
Inverter
SC
Uα _ out
Iα _ out
U β _ out
I β _ out
Load
Figure 3.14 Block diagram illustrating the dead time and voltage drop across power devices
compensation method in three phase motor supplied from non-ideal voltage source inverter.
In order to compensate the inverter non-linearity to the ideal reference voltage
components in stationary reference frame ( U *α _ ideal , U *β _ ideal ) a compensation signal
( Uα _ comp , U β _ comp ) is added. As a consequence the real reference voltage components
( U *α _ real , U *β _ real ) are pre-distorted. Further those signals are delivered to PWM
modulator with non-ideal inverter. As a result the output voltages are not distorted in
Fig. 3.14 and thus phase currents in the load (red signals in Fig. 3.14) are almost
sinusoidal.
To calculate an average compensation voltages ( Uα _ comp , U β _ comp ), the parameters of
IGBT modules as:
•
dead time Td ,
•
turn on TON and turn off TOFF of IGBT transistors,
•
and also on a voltage drop on diode VD and transistor VT ,
should be know.
43
Voltage source PWM inverter for PMSM supply
The total compensation time to compensate the non-linearity of inverter can be
calculated as:
Tcomp = Td + TON − TOFF +
U dp
U DC
Ts
(3.4)
Where U dp = VD − ton (VD − VT ) / Ts and ton is conducting time of IGBT devices in one
sampling time.
The compensation voltage vector can be obtained as:
U comp = 2
Tcomp
Ts
U DC s ign( Is ) = 2U th s ign( Is )
(3.5)
2
3
where sign( I s ) = ( sign( I sA ) + asign( I sB ) + a 2 sign( I sC )) ,
and
⎧ sign( I sA ) = 1 if IsA > 0
sign( I sA ) = ⎨
⎩ sign( I sA ) = 0 if IsA < 0
(3.6)
The sign function for remain phase currents are calculated similarly.
Solving equations (3.5) for real and imagine part in stationary frame, one obtains:
1
Uα _ comp = 2U th (2sign( I sA ) − 0.5sign( I sB ) − 0.5sign( I sC ))
3
(3.7a)
1
( sign( I sB ) − sign( I sC ))
3
(3.7b)
U β _ comp = 2U th
The waveform of compensation voltages in stationary frame are shown in Fig. 3.15.
44
Voltage source PWM inverter for PMSM supply
8
U th
3
4
U th
3
2
U th
3
Figure 3.15 Voltage compensation components in stationary reference frame.
From the top reference α and β components.
In order to illustrate the effectiveness of the proposed compensation a simulation study
has been performed. Fig. 3.16a shows the phase current in α , β frame and their
hodograph without compensation and Fig. 3.16b with proposed compensation method.
a)
a)
b)
b)
Figure 3.16 Nonlinearity effect of voltage source inverter on phase current of AC machine:
a) without compensation, b) with compensation.
45
Voltage source PWM inverter for PMSM supply
3.3 Space vector based pulse width modulation (PWM) methods
In voltage source inverter the transistors are controlled in a on-off fashion. In order to
obtain a suitable duty cycle for each switches the technique pulse with modulation is
used. The modulation methods [18,21,22,23,24,26,28,29,30] have the influence on:
wide range of linear operation, low content of higher harmonics in voltage and current,
low frequency harmonics, minimal number of switching to decrease switching losses in
the power components.
The most important factor in PWM mode is modulation index defined as the ratio of the
reference voltage amplitude value to the maximum voltage amplitude value during sixstep operation (see Fig 3.17) and is given by:
M=
U ref
2
π
(3.8)
U DC
where the U DC is the DC link voltage (for three phase six diodes rectifier is 560 V ).
The modulation index varies between 0-1 and can be divided into two regions: the
linear ( 0 < M ≤ 0.907 ) and the nonlinear ( 0.907 < M ≤ 1 ) as is shown in Fig. 3.17.
End of overmodulation
region (six-step mode)
Im
U 3 (010)
U 2 (110)
U max =
2
π
U DC ⇒ M = 1
sec tor = 2
2
U dc
3
sec tor = 3
U max =
U ref
U 0 (000)
U 4 (011)
End of linear region
U 7 (111)
sec tor = 4
sec tor = 1
θ ref
T1
U1
Ts
sec tor = 6
sec tor = 5
U 5 (001)
U DC
3
⇒ M = 0.907
Re
U1 (100)
nonlinear
(overmodulation) region
linear region
U 6 (101)
Figure 3.17 Space vector diagram of the available switching vectors.
46
Voltage source PWM inverter for PMSM supply
Linear range of operation ( 0 > M <= 0.907 )
In linear region, shown in Fig. 3.17, the rotating reference voltage vector U ref remains
within the hexagon. It means that the maximum amplitude reference voltage is equal
U DC
3
and adequately modulation index is equal:
M=
U DC / 3
π
=
= 0.9068
2U DC
2 3
(3.9)
π
The active vectors ( U1 ,U 2 ,U 3 ,U 4 ,U 5 ,U 6 ) are used to change the position of voltage
vector. The zero vectors ( U 0 ,U 7 ) are using to increase or decrease the amplitude of
voltage vector in one sampling time.
The desired voltage U ref is approximated by a time average of selected voltage vectors.
These selected vectors are non-zero vectors which are adjacent to the U ref .
Reference voltage vector control section is sampled at the fixed clock frequency of 2 f s
(where f s is the switching frequency). After that, the reference stator voltage vector
magnitude and position are calculated.
The principle for the implemented space vector modulation is described below. First,
based on the position of reference voltage vector is calculated the sector with help of
accordance below equations.
sec tor = int(
θ ref
) +1
π /3
(3.10)
Where θ ref is position of reference voltage vector U ref in respect to real axis in complex
plane.
Next based on the sector information angle α ref in respect to adjacent vector is
calculated (see Fig 3.18).
α ref = θ ref − (sec tor − 1)
π
3
47
(3.11)
Voltage source PWM inverter for PMSM supply
Next from U ref , α ref it is necessary to calculate the time interval for particular vectors.
Uref
U2
120°- α
t2
U0,U7
t0
120°
αref
t1
U1
Figure 3.18 One sector in voltage plane.
Using the low of sine it is possible to write:
U ref
sin120°
=
U1
sin(60° − α ref )
=
U2
sin α ref
(3.12)
From this relations calculated value of vectors
U1 = U ref
U 2 = U ref
sin(60° − α ref )
=
sin120°
sin α ref
sin120°
=
2
3
2
3
U ref sin(60° − α ref )
U ref sin α ref
(3.13a)
(3.13b)
and respectively the normalized times are given:
t1 =
t2 =
U1
2
U DC
3
=
3 U ref
U DC
sin(60° − α ref )
3 U ref
U2
=
sin α ref
2
U
DC
U DC
3
(3.14a)
(3.14b)
Putting Eq. (3.13a-b) in to Eq. (3.14a-b) the normalized value can be presented as:
t1 =
2 3
t2 =
2 3
π
π
M sin(60° − α ref )
(3.15a)
M sin α ref
(3.15b)
or in other form:
48
Voltage source PWM inverter for PMSM supply
t2 =
2 3
t1 =
3
π
M sin α ref
(3.16a)
1
M cos α ref − t2
2
π
(3.16b)
After t1 and t2 calculation, the remaining normalized time is reserved for zero vectors
U 0 , U 7 with condition t1 + t2 ≤ 1 .Therefore, the normalized total time for zero vectors
becomes:
t07 = t0 + t7 = 1 − (t1 + t2 )
(3.17)
The equations (3.15a-b) for time interval of active vectors and equation (3.17) for total
time interval of zero vectors are identical for all variants of space vector modulation
(SVM) techniques.
The absence of neutral wire in star connected load provides a degree of freedom in
selecting the partitioning (zero sequence signals -ZSS) time of the two zero vectors. It is
equivalent to the freedom of injected signals in to phase signals. Therefore, it gives
different equations of t0 and t7 for each PWM method, but normalized duration time of
must fulfill condition in Eq. 3.17. As a consequence is only in different placement of
zero vectors U 0 , U 7 . Therefore, we can introduce the portioning factor of zero vectors,
which is defined as:
k=
t7
t
= 7
t0 + t7 t07
(3.18)
Please note that, the zero sequence signals does not change the inverter output line-toline voltage.
From knowledge of the neutral voltage U N 0 (see Fig. 3.4) and information what kind of
zero sequence signal (ZSS) will be injected in each phase of motor it is possible to
calculate normalized duration time of zero vectors t0 and t7 . In bellow Table 3.1 are
summarized different three-phase modulation techniques and remarks. Also graphical
interpretation are shown in Fig. 3.19.
49
Voltage source PWM inverter for PMSM supply
Zero Sequence Signal
Time interval of zero vectors
SPWM
for even sectors
ZSS =0
4
t0 = (1 − ( M cos θ )) / 2
π
M=
U DC / 2 π
= = 0.785
2U DC
4
π
for odd sectors
complicated
4
t0 = t07 − (1 − ( M cos θ )) / 2
π
for
time
calculation
interval
of
vectors
and t7 = t07 − t0
THIPWM
for even sectors
ZSS =sinusoidal signal
4
1
t0 = (1 − ( M cosθ ) − cos 3θ )) / 2
π
6
with triple harmonic
Remark
M=
for odd sectors
U DC / 3
π
=
= 0.907
2U DC
2 3
π
Increase the linearity of
inverter grater than 15%
4
1
t0 = t07 − (1 − ( M cos θ ) − cos 3θ )) / 2
6
π
of SPWM
and t7 = t07 − t0
SVPWM
for all sectors
M = 0.906
ZSS=triangle signal
t0 = t07 / 2
Simple
with triple harmonic
t7 = t07 − t0 = t07 / 2
time interval
k=0.5
calculation
for
the portioning
factor of zero vectors
Table 3.1. Variants of three-phase space vector modulation techniques.
a)
b)
c)
Fig. 3.19 PWM techniques with various zero signal sequence shape: a) SPWM, b) THIPWM,
c) SVPWM. The upper part of figure: phase voltage U AN (green), pole voltage U AO (blue),
voltage between neutral points U NO (black). The lower part of figure shows the portioning
factor of zero vectors for all PWM techniques.
50
Voltage source PWM inverter for PMSM supply
Except SPWM technique all PWM method guarantee that the ZSS extends the range of
modulation index from 0.78 to 0.906, i.e. 15% greater than that obtained with standard
version of SPWM.
The duty time cycles in sector 1 for each phase can be written:
d a = t1 + t2 + kt07
(3.19a)
db = t2 + kt07
(3.19b)
d c = kt07
(3.19c)
for all sectors the duty time calculations for each phase can be calculated:
⎡1
⎢
⎢ d ⎥ = ⎢0
⎢ b⎥ ⎢
⎢⎣ d c ⎥⎦ ⎢ 0
⎣
⎡da ⎤
sec tor 1
sec tor 2
sec tor 3
sec tor 4
sec tor 5
sec tor 6
1
k1
0
k0
0
k0
0
k0
1
k1
1
1
k1
1
k1
1
k1
0
k0
0
k0
0
0
k0
0
k0
1
k1
1
k1
1
k1
0
⎤
⎥
k⎥
⎥
k⎥
⎦
k
T
⎡ t1 ⎤
⎢ t ⎥ (3.20)
⎢ 2⎥
⎢⎣t07 ⎥⎦
Depending on the location of the space vector, the basic vectors must be chosen in order
to get the minimum number of changes in the switches of the converter. The switching
sequence for each sector and suitable pulse pattern for first sector are shown in Fig.
3.20.
Sector
TS
Three-phase Modulation
TS / 2
t1
t2
t7
0
1
1
1
1
1
1
0
0
0
1
1
1
1
0
0
0
0
0
1
1
0
0
0
U2
U7
t0
1
U 0 , U 1 , U 2 , U 7 ,U 2 ,U 1 ,U 0
2
U 0 , U 3 ,U 2 , U 7 ,U 2 , U 3 ,U 0
3
U 0 , U 3 ,U 4 , U 7 ,U 4 , U 3 ,U 0
4
U 0 , U 5 , U 4 ,U 7 , U 4 ,U 5 ,U 0
5
U 0 , U 5 ,U 6 ,U 7 , U 6 , U 5 , U 0
6
U 0 , U 1 , U 6 ,U 7 , U 6 , U 1 , U 0
U0
U1
dA
U2
U1
U0
dB
dC
Fig. 3.20 Switching sequence for three-phase PWM techniques (on the left ) and pulse pattern
of three-phase vector modulator in sector 1 (on the right).
51
Voltage source PWM inverter for PMSM supply
3.4 Summary
The general conclusions from this chapters can be summarized as follows:
¾ to supply the voltage source inverter can be used diode rectifier or active
rectifier with IGBT transistors,
¾ supplied PMSM machine from VSI do not required mechanical comutator. It is
thanks to electronic commutation. This overcome the problem with brushes and
periodical service,
¾ The voltage source inverter is non-linear power amplifier in respect to:
o dead time effect,
o voltage drop across the power devices,
o DC link voltage pulsation.
¾ Without appropriate dead-time and voltage drop compensation, the sensorless
operation in low speed range is not possible.
¾ The quality of DC link voltage has influence on proper operation of AC drive,
¾ Sinusoidal modulation technique (SPWM) guarantees the inverter output voltage
amplitude from 0 until
U DC
V. This correspond to changes modulation index
2
from 0-0.785.
¾ Modulation techniques with zero sequence signals not equal zero guarantees the
inverter output voltage amplitude from 0 until
U DC
3
V (it is 15% grater than
SPWM). This correspond to changes modulation index from 0-0.907.
¾ In this work the PWM modulator with zero sequence signal of triple harmonics
(SVPWM) will be used. Mainly because of simple calculation of zero vector
duration and placement.
52
Control methods of PM Synchronous motor
Chapter 4 CONTROL METHODS OF PM SYNCHRONOUS MOTOR
4.1 Introduction
The basic block scheme of adjustable speed drive with control block for PMSM is
presented in Fig. 4.2. It
consists of two parts: power (fat line) and control part
employed microprocessor (thin line). The first one previously have been explained in
chapter 3. The second one will be described bellow.
Figure 4.2. The basic block scheme of PMSM drive supplied voltage source inverter.
The main task of control block is follow demand reference speed by motor and provide
proper operation in static (insight of the limits) and dynamic states without any
instability. This is ensured through suitable generated gate signals for the IGBT
transistor inside of the inverter. Therefore, to make good decision how to control power
transistors in the inverter, the following feedback signals are measured and used:
•
DC link voltage,
•
motor phase currents,
•
speed or position of the rotor.
This significantly improve dynamic behavior of the system (good performance of the
torque and speed response, very fast dynamics response with fully controllable torque in
wide speed range).
The scalar control for PMSM without damper winding (squire cage) is not simple as for
induction motor [39,40]. It requires additional stabilization loop, which can be provide
by feedback loop from: rotor velocity perturbation, active power or DC-link current
perturbation [9].
The vector control method will be described bellow.
53
Control methods of PM Synchronous motor
4.2 Field oriented control (FOC)
During many years a DC motor has been mostly used. Because of simple control
method, which based on fact that flux and torque can be controlled separately using
current control loop with PI controllers. However the weak point of this drive was DC
motor, which could not worked in aggressive or volatile environment and required
cyclical maintenance. This disadvantages has been eliminated, when instead of DC
machine a three phase PMSM motor were used.
In searching new control method for induction machine in 1971 was developed vector
control method known as field oriented control (FOC) [31,38,49]. This method allows
control the flux and the torque in the AC machine in similar way as for DC motor. It
was achieved by transform current vector in stationary reference frame ( α , β ) into new
coordinate system ( d , q ) with respect to rotor (magnet) flux vector. So the flux
produced by permanent magnet is frozen to the direct axis of the rotor (see Fig. 4.4).
q − axis
β − axis
Is
Ωs
Isq
γI
δI
Isd
d − axis
ΨPM
γm
α − axis
Figure 4.4 Vector diagram illustrated the principle of FOC.
Further, stator current vector can be split into two current components: flux current I sd
and torque producing current I sq . In analogy to separate commutator motor, the flux
current components corresponds to excitation current and torque-producing current
corresponds to the armature current. Therefore, the goal of the control system is to
reference the I sd _ ref , I sq _ ref stator current components in respect to requirement of
references torque and flux. The flux and torque producing stator current references are
obtained on the output of the reference current generation block (see Fig. 4.5).
54
Control methods of PM Synchronous motor
Reference Current
Calculation
Reference Current
Calculation
Isd _ ref
FG1
reference
torque
Me_ ref
Is
reference
d-axis current
FG2
FG3
Isq _ ref
reference
torque
reference
q-axis current
Me_ ref
FG3
δI
γI
γm
Figure 4.5. Reference current generator block for FOC technique
a) in cartesian form, b) in polar form.
Function generation FG1 gives the relationship between the torque and the direct axis
stator current component I sd _ ref , and function generator FG2 gives the relationship
between the torque and the quadrature axis stator current I sq _ ref . His graphical
illustration in Fig. 4.6 are presented.
0.5
CTA
0
MTPA
0.5
I sdN [ p.u ]
1
CSF
1.5
UPF
2
2.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
M eN [ p.u ]
2.5
CTA
2
MTPA
1.5
I sqN [ p.u ]
CSF
1
UPF
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
M eN [ p.u ]
Figure 4.6. Generated current components I sd and I sq dependent on required electromagnetic
torque under difference control strategies. CTA-constant torque angle control, MTPA-maximum
torque per ampere control, CSF-constant stator flux control, UPF – unity power factor control.
55
Control methods of PM Synchronous motor
For many years the CTA control ( I sd _ ref = 0 ) method has been a popular technique for a
long time because of simple control. This method was dedicated for surface permanent
magnet synchronous motor (SPMSM), where the magnetic saliency does not exist
[126]. So the maximum torque per ampere is obtained, when stator current vector is
shifted in respect to rotor flux vector 90 degree. However, in IPMSM, maximum torque
per ampere is obtained with torque angle more than 90 degree. This is because of
existence of reluctance torque component due to magnetic saliency (see subchapter
2.2.2). Therefore, the I sd _ ref should be negative value [44].
The main question is how or in which manner produce the reference currents in d-q
frame. Its leads to many realization of current control structure. Among them generally
we can distinguish two structures of current control loop. One of them is hysteresis
based control (Fig.4.7a) [3,52] and the second one is PI based current controllers
(Fig.4.7b).
Hysteresis based current control has following disadvantage such as [3]:
•
measurement of three phase currents are required,
•
three independent hysteresis current controllers are required,
•
variable switching frequency is achieved,
•
fast sampling time is required.
All this listed above disadvantage can be eliminate, when the PI current control are
used. This structure are mostly used in industrial application (Fig. 4.7b).
UDC
a)
IA_ref
Id _ ref
Me_ref
Fig. 4.5
Iq _ref
d,q/ABC
SA
IB_ref −
IC_ref
SB
−
SC
−
current
feedback
IA current
IB sensors
IC
rotor
position
sensor
56
Inverter
PMSM
Control methods of PM Synchronous motor
UDC
b)
Me_ref
Fig. 4.5
Isd _ref
eIsd
Isq_ ref
eIsq
-
PI
PI
Usd
Usq
SA
Us _ref
Space S
Reference
B
VoltageVector
Vector
ϕUs _ref
S
Calculation
Modulator C
Inverter
γs
Isd
current
sensors
dq/ABC
Isq
Is
γm
rotor
position
sensor
PMSM
Figure 4.7 Vector control structure for PM synchronous motor with: a) hysteresis current
control, b) synchronous PI current control
4.3 Direct torque control (DTC)
The name direct torque control is deliver by the fact that, on the basis of the errors
between the reference and the estimated values of torque and flux, it is possible to
directly control the inverter states without inner current control loop as for FOC
[32,34,35,50] and [57-66].
The basic idea of this control rely on stator voltage vector equation of AC motor.
U s = Rs Is +
d Ψs
dt
(4.1)
Making the assumption that ohmic voltage drop on the stator resistor can be neglected
the equation for stator flux vector takes the form:
Ψ s = ∫ (U s )dt
(4.2)
It can be said that the stator voltage vector has directly influence on control stator flux
vector. Using a three phase voltage source inverter to supply the AC motor, there are six
non-zero vectors and two zero voltage vectors. The active vectors change the amplitude
and position of stator flux vector, while the zero vectors stop the stator flux vector as
shown in Fig. 4.8.
57
Control methods of PM Synchronous motor
moment with active
forward vector
y
stops with zero
vector
moment with active
backward vector
Ψs
γs
δΨ
Ψ r = Ψ PM
x
α
rotates
continuously
stator
Figure. 4.7 Stator flux vector Ψ s movement relative to rotor flux vector Ψ r = Ψ PM under the
influence of active and zero inverter voltage vectors.
Therefore, it is possible control the torque angle δ Ψ across control stator flux vector Ψ s
position in respect to rotor flux vector produced by permanent magnet Ψ r = Ψ PM , what
further allows to have impact on control the electromagnetic torque in accordance with
following formula:
2
Ψ Ψ sin δ Ψ Ψ s ( Lq − Ld )sin 2δ Ψ
3
M e = pb [ s PM
−
]
Ld
2
2 Ld Lq
(4.3)
Generally the DTC technique operate at constant stator flux amplitude Ψ s , what
correspond to CSF operation, because of simple reference stator flux amplitude equal
nominal value of permanent magnet. For DTC technique can be also apply all control
strategies discussed in Chapter 2.
Using the block generator for reference stator flux amplitude and electromagnetic
torque as is shown in Fig 4.8. it is possible to draw the relationship between required
Reference
Flux
Calculation
M e _ ref
reference
torque
reference
flux
Ψ s _ ref
reference
torque
M e _ ref
Figure 4.8. Reference flux and torque generator block for DTC technique.
reference flux and torque. His graphical illustration in Fig. 4.9 is presented.
58
Control methods of PM Synchronous motor
2
CTA
1.5
MTPA
Ψ sN [ pu ] 1
CSF
0.5
UPF
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
M eN [ pu ]
Figure 4.9. Generated stator flux amplitude Ψ s _ ref
dependent on required electromagnetic
torque under difference control strategies. CTA-constant torque angle control, MTPA-maximum
torque per ampere control, CSF-constant stator flux control, UPF – unity power factor control.
The basic structure of direct flux and torque control voltage-sourced PWM inverter-fed
permanent magnet synchronous motor is shown in Fig.4.10.[3,36,37,41,45]
U DC
Flux
hysteresis
H Ψs
Ψ s _ ref
−
M e _ ref
−
dΨs
dMe
Switching
Table
Hm
Torque
histeresis
sector
Ψs
Me
SA
SB
SC
Inverter
γ s( N )
Flux and
Torque
Estimation
Is
γm
PMSM
Figure. 4.10. Block diagram of switching table based direct torque control ST-DTC.
The command stator flux amplitude Ψ s _ ref and electromagnetic torque M e _ ref values
are compared with the actual Ψ s and M e values, in hysteresis flux and torque
controllers, respectively. The flux and the torque controllers are a two-level
comparators.
The digitized outputs signals of the flux controllers are defined as:
d Ψs = 1 (increase flux)
for Ψ s > Ψ s _ ref + H Ψ
(4.4a)
d Ψs = 0 (decrease flux)
for Ψ s < Ψ s _ ref − H Ψ
(4.4b)
59
Control methods of PM Synchronous motor
and those of the torque controller as
d M e = 1 (increase torque) for M e > M e _ ref + H m
(4.5a)
d M e = 0 (decrease torque) for M e < M e _ ref − H m
(4.5b)
where H m and H Ψ are hysteresis bands for torque and flux, respectively.
The digitized variables d Ψ s , d M e and the stator flux position sector γ s ( N ) information
create a digital word, which select appropriate voltage vector from the switching table.
Next, from the selection table the proper voltage vectors are selected and the gate pulses
S A , S B , SC to control the power switches in the inverter are generated.
The circular stator flux vector trajectory can be divided into six symmetrical sectors
(according to the non zero voltage vectors), which are defined as (see Fig. 4.11):
sector 1 −30° ≤ γ s < 30°
V3 (010)
sector 2 30° < γ s ≤ 90°
Sector 3
sector 3 90° < γ s ≤ 150°
sector 4 150° < γ s ≤ 210°
V4 (011)
Sector 4
V7 (111)
sector 5 210° < γ s ≤ 270°
V2 (110)
Sector 2
Sector 1
Vo (000)
V1 (100)
Sector 5 Sector 6
sector 6 270° < γ s < −30°
V5 (001)
V6 (101)
Figure. 4.11 Sector definition for DTC
In each region, two adjacent voltage vectors, which give the minimum switching
frequency, may be selected to increase or decrease the amplitude of stator flux and
electromagnetic torque.
The selection table is created accordance with following formula explained for first
sector as is illustrated in Fig. 4.12 .
When stator flux vector is located in first sector Ψ s and rotates clockwise in order to
increase the torque ( d M e = 1 ) the vectors V2,V3 can be used. In order to reduce torque
the opposite two vectors are used.
60
Control methods of PM Synchronous motor
β
Ψs ↓
Ψs ↑
Me ↑
Me ↑
Ψs ↓
Ψs ↑
Ψs
γs
γ
α
s
Ψs ↓
Ψs ↑
Me ↓
Me ↓
Figure. 4.12 Voltage vector effects in sector 1 on stator flux and torque.
The presented rule for first sector can be extended for other sectors, what further help
construct the switching Tables 4.1 and 4.2 as below [56]
Flux
dψs=1
dψs=0
Torque
sector 1
sector 2
sector 3
sector 4
sector 5
sector 6
dme.=1
V2
V3
V4
V5
V6
V1
dme=0
V6
V1
V2
V3
V4
V5
dme.=1
V3
V4
V5
V6
V1
V2
dme=0
V5
V6
V1
V2
V3
V4
Table 4.1. Switching table for DTC with active vectors.
Flux
dψs=1
dψs=0
Torque
sector 1
sector 2
sector 3
sector 4
sector 5
sector 6
dme.=1
V2
V3
V4
V5
V6
V1
dme=0
V7
V0
V7
V0
V7
V0
dme.=1
V3
V4
V5
V6
V1
V2
V0
V7
dme=0
V0
V7
V0
V7
Table 4.2. Switching table for DTC with zero and active vectors.
Tables 4.1 and 4.2 represent the eight and six voltage-vectors switching tables.
61
Control methods of PM Synchronous motor
The DTC has lesser parameter dependence and fast torque when compare with the
torque control via PWM current control.
Among the well-know advantages of the DTC scheme are the following:
•
Simple control,
•
Excellent torque dynamics,
•
Absence of coordinate transformations,
•
Absence of separate voltage modulation block,
•
Absence of voltage decoupling circuits,
•
There are no current control loops, hence, the current is not regulated directly,
•
Stator flux vector and torque estimation is required.
Among the well-know disadvantages of the DTC scheme are the following:
•
variable switching frequency (difficulties of LC input EMI filter design),
•
high sampling time is required (fast microprocessor and A/D converter ),
•
inverter switching frequency depending on: flux and torque hysteresis bands,
machine parameters, sampling frequency,
•
violence of polarity consistency rules (huge voltage stress for IGBT transistor),
•
current and torque distortion caused by sector changes,
•
start and low speed operation problems,
•
high sampling frequency needed for digital implementation of hysteresis
comparators,
•
high noisy level,
•
high current and torque ripple.
Many modifications of the basic switching table based direct torque control (ST-DTC)
scheme at improving starting, overload condition, very low speed operation, torque
ripple reduction, variable switching frequency functioning, and noise level attenuation
have been proposed during last decade.
In the last five years, many researcher have been carried out to try solve the above
mentioned problems of ST-DTC scheme. Therefore, the following solutions have been
developed in order to eliminated mentioned before problems:
62
Control methods of PM Synchronous motor
•
Use of improved switching table,
•
Use of comparators with and without hysteresis, at two or three levels,
•
Use of multi-level inverter,
•
Introduction of fuzzy or neuro-fuzzy techniques,
•
Use of sophisticated flux estimators to improve the low speed behavior,
•
Implementation of DTC schemes with constant switching frequency operation
In multi-level inverter there will be more voltage vectors available to control the flux
and torque. Therefore, a smoother torque can be expected. However, more power
switches are needed to achieved a lower ripple, which will increase the system cost and
complexity.
All this contributions allow the DTC performance to be improved, but at the same time
they lead to more complex schemes. As expected, conventional DTC is growing in
field-oriented control area and the so-called improved DTC with space vector
modulation (SVM). Let us call it DTC-SVM. This control concept will be deeply
discussed in the next Chapter.
63
Control methods of PM Synchronous motor
4.4 Summary
¾ In FOC drive flux linkage and electromagnetic torque are controlled indirectly
and independently by PI controllers with space vector modulator (SVM). In this
control concept current control loop is required,
¾ In DTC drive, flux linkage and electromagnetic torque are controlled directly
and independently by hysteresis controllers and selection of optimum inverter
switching modes. In this control concept flux and torque control loop is required,
¾ In DTC all switch changes of the inverter are based on the electromagnetic state
of the motor.
¾ The DTC technique is different from the traditional methods of controlling
torque, where the current controllers in the rotor reference frame are used. It is
completely different control concept (approach) from FOC. The new control
technology was characterized by simplicity, good performance and robustness,
because of bang-bang hysteresis control. Using DTC it is possible to obtain a
good dynamic control of the torque without current controllers and any
mechanical transducers on the machine shaft. Moreover, in this control structure
the PWM modulator is not required. Its is occupied by variable switching
frequency.
¾ The flux weakening control becomes easier because stator flux linkage can be
controlled directly in the DTC system of PMSM.
64
Direct Torque Control with Space Vector Modulation
Chapter 5
DIRECT TORQUE CONTROL WITH SPACE VECTOR MODULATION
(DTC-SVM)
5.1 Introduction
The DTC-SVM greatly improves torque and flux performance by:
•
Achieved fixed switching frequency,
•
Reduced torque and flux ripples.
The main idea of DTC-SVM is based on analyze of the torque equation
2
Ψ Ψ sin δ Ψ Ψ s ( Lq − Ld )sin 2δ Ψ
3
M e = pb [ s PM
−
]
2
Ld
2 Ld Lq
(5.1)
Assuming that the Ld = Lq = Ls
Me =
Ψ Ψ sin δ Ψ
3
pb [ s PM
]
Ls
2
(5.2)
From equation (5.2) we can see that for constant stator flux amplitude Ψ s and flux produced
by permanent magnet Ψ PM , the electromagnetic torque can be changed by control of the
torque angle δ Ψ . This is the angle between the stator and rotor flux linkage, when the stator
resistance is neglected. The torque angle, in turn, can be changed by changing position of the
stator flux vector θ Ψs in respect to PM vector using the actual voltage vector supplied by
PWM inverter.
In the steady state, δ Ψ is constant and corresponds to a load torque, whereas stator and rotor
flux rotate at synchronous speed. In transient operation, δ Ψ varies and the stator and rotor
flux rotate at different speeds (Fig. 5.1).
65
Direct Torque Control with Space Vector Modulation
q − axis
β
Ψ s _ ref
Ψs
∆δ Ψ
δΨ
θ Ψs ΨPM
d − axis
θr
α
Figure 5.1. Space vector diagram illustrating torque control conditions.
5.2 Cascade structure of DTC–SVM scheme
The structure of proposed control scheme is shown in the Fig. 5.2. [11,33,42,48,51,53,54]
U DC
Ψ s _ ref
M e _ ref
eM
∆δ Ψ
U sα _ ref
SA
U sβ _ ref
SC
SB
Ψ s θ Ψs I s
Me
Is
γm
Figure 5.2. Cascade structure of DTC-SVM scheme.
The error between reference and measured torque can be expressed as:
3 Ψs _ ref ΨPM sin(δΨ +∆δΨ ) Ψs ΨPM sin δΨ
eM = Me _ ref − Me = pb[
−
]
Ls
Ls
2
(5.3)
From equation (5.3) we can see that the relation between torque error and increment of load
angel ∆δ Ψ is nonlinear. Therefore, we used PI controller which generates the load angel
66
Direct Torque Control with Space Vector Modulation
increment required to minimize the instantaneous error between reference M e _ ref and actual
M e torque.
In control scheme of Fig. 5.2 the torque error signal eM is delivered to the PI controller,
which determines the increment of torque angle ∆δ Ψ . Based on this signal and reference
amplitude of stator flux Ψ s _ ref , the reference voltage vector in stator coordinates α , β is
calculated. The calculation block of reference voltage vector also uses information about the
actual stator flux vector (amplitude Ψ s and position θ Ψs ) as well as measured current vector
Is . The reference stator voltage vector is delivered to space vector modulator (SVM), which
generates the switching signals S A , S B , S C for power transistors of inverter.
The calculation block of reference voltage vector is shown in Fig. 5.3.
Rs I sα
Ψ s _ ref
Ψ sα _ ref
U sα _ ref
∆Ψ sα
−
∆δ Ψ
Ψ sβ _ ref
U sβ _ ref
∆Ψ sβ
Ψ sβ
Ψ sα
θ Ψs
−
Rs I sβ
Figure 5.3. Calculation block of reference voltage vector.
Based on ∆δ Ψ signal, reference of stator flux amplitude Ψ s _ ref and measured stator flux
vector position θ Ψs (Fig. 5.3.), the reference flux components Ψsα _ ref , Ψsβ _ ref in stator
coordinate system are calculated as:
Ψ sα _ ref = Ψ s _ ref cos(θ Ψs + ∆δ Ψ )
Ψ sβ _ ref = Ψ s _ ref sin(θ Ψs + ∆δ Ψ )
(5.4)
Pleas note that for constant flux operation region the reference value of stator flux amplitude
Ψ s _ ref is equal flux amplitude of permanent magnet ΨPM .
67
Direct Torque Control with Space Vector Modulation
The references of stator flux components (see Fig. 5.3) are compared with estimated value:
Ψ sα = Ψ s cos θ Ψs ,
(5.5)
Ψ sβ = Ψ s sin θ Ψs ,
The command voltage can be calculated from flux errors in α , β coordinate system as
follows:
U sα _ ref =
∆Ψ sα
+ Rs I sα
Ts
U sβ _ ref =
∆Ψ sβ
Ts
(5.6)
+ Rs I sβ
Where: Ts is sampling time, ∆Ψ sα = Ψ sα _ ref − Ψ sα , ∆Ψ sβ = Ψ sβ _ ref − Ψ sβ .
The presented bellow design methodology for flux and torque control loops based on the
approach presented in literature [11,43].
5.2.1 Digital flux control loop
The flux control loop is based on the voltage equations of PMSM machine in stator
coordinates.
U sα = Rs I sα +
U sβ = Rs I sβ +
d Ψ sα
dt
(5.7a)
d Ψ sβ
(5.7b)
dt
Using Laplace transformation the above equations can be written as:
U sα = sΨ sα + Rs I sα
(5.8a)
U sβ = sΨ sβ + Rs I sβ
(5.8b)
It corresponds to flux model of PMSM machine in α , β system presented in Fig. 5.4.
68
Direct Torque Control with Space Vector Modulation
Rs I sα
U sα
−
Ψ sα
1
s
U sβ
Ψ sβ
1
s
−
Rs I sβ
Figure 5.4. Flux model of PMSM in stator coordinates.
In order to control the flux components in α , β frame the bellow control structure can be
applied.
Flux control
part
Ψ sα _ ref
P block
∆Ψ sα
P = 1/ Ts
−
Ψ sβ _ ref
P block
∆Ψ sβ
P = 1/ Ts
Rs I sα
∆Ψ sα
Ts
U sα _ ref
Rs I sα
−
Flux PMSM
model
1
s
Ψ sα
∆Ψ sβ
U sβ _ ref
Ts
−
−
Rs I sβ
Ψ sα
1
s
Ψ sβ
Rs I sβ
Ψ sβ
Figure 5.5. Flux control loop with two P controller in α , β reference frame.
Pleas note that regarding to Fig. 5.1 the following rules are keeping:
Ψ sα _ ref = Ψ s _ ref cos(θ Ψs + ∆δ )
(5.9a)
Ψ sβ _ ref = Ψ s _ ref sin(θ Ψs + ∆δ )
(5.9b)
and
Ψ sα = Ψ s _ ref cos θ Ψs
(5.10a)
Ψ sβ = Ψ s _ ref sin θ Ψs
(5.10b)
In order to find the formula for tuning the P controllers in the flux loop, the following
assumptions should be made:
69
Direct Torque Control with Space Vector Modulation
•
increment of torque angle ∆δ Ψ coming from torque control loop (see Fig.5.2.) is equal
zero. It means that the torque is not produced,
•
stator flux vector position θ Ψs and rotor flux vector position θ r are equal zero. It
corresponds to situation, where the those two flux vectors lie along the α axis.
In this special case the reference stator flux amplitude Ψ s _ ref = Ψ sα _ ref can be controlled
trough the reference stator voltage component U sα _ ref = U s _ ref , when the voltage drop on the
stator resistances in α , β axes are neglected (see Fig. 5.5). Therefore, the simplified flux
control loop can be shown in Fig. 5.6.
Ψ s _ ref = Ψ sα _ ref
∆Ψ sα
U sα _ ref = U s _ ref
P
Ψ sα
PMSM
Ψ s = Ψ sα
−
controller
U sβ _ ref = 0
Ψ sβ
Figure 5.6. Simplified flux control loop in α , β coordinates.
Continuous s-domain
Simplified flux control loop in s domain is shown in Fig. 5.7, where CΨ ( s) is a transfer
function of the P controller given by:
CΨ ( s ) = K p Ψ
(5.11)
The transfer function between stator flux amplitude Ψ s = Ψ sα and stator voltage amplitude
Us
can be expressed as:
GΨ ( s ) =
Ψs 1
=
Us
s
(5.12)
70
Direct Torque Control with Space Vector Modulation
Control Plant
P controller
Ψ s_ref
Us
CΨ ( s )
Ψs
GΨ ( s )
Figure 5.7. Block diagram of flux controller in s domain.
Hence the transfer function of the closed stator flux amplitude control loop is obtained as:
GΨ _ closed ( s ) =
Ψs
_ ref
( s)
Ψ s ( s)
=
CΨ ( s )GΨ ( s )
1 + CΨ ( s )GΨ ( s )
(5.12)
Substituting transfer function for CΨ ( s) and GΨ ( s ) one becomes:
⎛1⎞
⎛1⎞
K pΨ ⎜ ⎟
K pΨ ⎜ ⎟
⎝s⎠ =
⎝ s ⎠ = K pΨ
GΨ _ closed ( s ) =
⎛ 1 ⎞ s + K p Ψ s + K pΨ
1 + K pΨ ⎜ ⎟
s
⎝s⎠
(5.13)
Discrete design
The transfer function for P controller in discrete system is expressed as:
CΨ ( z ) = K p Ψ
(5.14)
D( z )
Ψ s _ ref ( z )
CΨ ( z )
U sα = U s
z −1
G ( z)
}
Ψ
ZOH
1
s
Ψ s ( z)
Figure 5.8. Block diagram of flux controller in discrete domain.
Where, CΨ ( z ) is discrete transfer function for P controller, D( z ) z −1 is for one sampling time
delay for voltage generation from PWM .
71
Direct Torque Control with Space Vector Modulation
The GΨ ( z ) is discrete transfer function for voltage-flux relationship with zero hold order
(ZOH) can be calculated as:
GΨ ( z ) = (1 − z −1 ) Z [
GΨ ( s )
z −1 1
]=
Z[ 2 ]
s
z
s
(5.15)
Using table of Z transformation [2]. Finally, it gives
GΨ ( z ) =
( z − 1)
z
zTs
( z − 1)
2
=
AΨd
( z − 1)
(5.16)
Where AΨd = Ts and Ts is sampling time of the discrete system.
Hence, the closed loop transfer function between Ψ s
GΨ _ closed ( z ) =
Ψs
_ ref
( z)
Ψ s ( z)
_ ref
( z ) and Ψ s ( z ) is obtained as:
CΨ ( z )GΨ ( z ) D( z )
1 + CΨ ( z )GΨ ( z ) D( z )
=
K pΨ AΨd
K pΨ AΨd
z ( z − 1)
=
= 2
K A
z − z + K pΨ AΨd
1 + pΨ Ψd
z ( z − 1)
(5.16)
The flux step response depended on poles placement of closed flux control loop. The pole
placement can be selected by setting the K pΨ .
Assuming, that CΨd = K pΨ AΨd the GΨ _ closed ( z ) expressed by equations (5.16) will take the
following form:
GΨ _ closed ( z ) =
CΨ d
z − z + CΨ d
2
(5.17)
The nomogram of Fig. 5.9 shows the relationship between overshoot M p [%] , rise time tr and
settling time t s in respect to CΨd .
Please not that tr is time calculate from 10% to 90% of output signals and t s is the time it
takes the system transient to decay +-1%.
72
Direct Torque Control with Space Vector Modulation
Figure 5.9. The relationship between overshoot, rise time and settling time versus to CΨd for stator
flux amplitude control loop.
Now from few values of CΨd =[0.4046, 0.2688, 0.1720] we can choose CΨd =0.2688, which
guaranties overshoot about 0% and settling time about 10 times of sampling time.
1
2
3
1
Figure 5.10. Step flux response for different to CΨd : red line(1) CΨd =0.4046, blue line (2)
CΨd =0.2688, black line (3) CΨd =0.1720.
It corresponds to the transfer function of closed stator flux control loop as:
GΨ _ closed ( z ) =
C Ψd
0.2688
= 2
z − z + CΨd z − z + 0.2688
2
(5.18)
Using digitalized motor parameters AΨd = Ts and chosen CΨd value we can calculate the
parameters of P digital flux controller as:
73
Direct Torque Control with Space Vector Modulation
K pΨ =
C Ψ d CΨ d
=
AΨd
Ts
(5.19)
For example let us assume that sampling time in digital flux control loop is equal Ts = 200µ s .
The gain of P controller is:
K pΨ =
0.2688 0.2688
=
= 1344
0.0002
AΨd
(5.20)
In digital control when the sampling time changes the parameters of digitalized plant control
AΨd will also change. Therefore, to keep closed loop transfer function as close as possible to
GΨ _ closed ( z ) =
C Ψd
0.2688
= 2
, the gain of P flux controller should also be
z − z + CΨd z − z + 0.2688
2
changed (see Table 5.1.).
Keeping constant transfer function GΨ _ closed ( z ) the flux step response for different sampling
times Ts = 50µ s , 100 µ s , 200µ s , 400µ s , which correspond to switching frequency f s =
20kHz, 10kHz, 5kHz, 2.5kHz are presented in Fig. 5.11.
Figure 5.11. Flux tracking performance for different sampling times Ts = 50 µ s (blue line -1), 100 µ s
(green line -2), 200 µ s (red line -3), 400 µ s (light blue line -4).
74
Direct Torque Control with Space Vector Modulation
From Fig. 5.11 it can observed that overshoot is around 0% and the settling time took 10
times of microprocessor sampling time. So, it is possible control the flux amplitude in 10
samples
The settings of P flux controller for different sampling time Ts = 50µ s,100µ s, 200µ s, 400µ s
are summarized in Table 5.1.
75
Direct Torque Control with Space Vector Modulation
76
Direct Torque Control with Space Vector Modulation
The behavior of the flux control loop was tested using SABER simulation package. The
model created in SABER takes into account the whole control system, which include
real models of inverter and permanent magnet synchronous motor.
The flux step response is shown in Fig. 5.12., when parameters of P flux controller
designed for sampling time Ts = 200µ s were used for control plant for different
sampling times Ts = 50µ s , 100µ s , 200µ s , 400µ s , which correspond to switching
frequency f s = 20kHz, 10kHz, 5kHz, 2.5kHz.
Figure 5.12. Flux tracking performance for different sampling time Ts = 50µ s , 100 µ s ,
200µ s , 400µ s ( f s = 20kHz, 10kHz, 5kHz,2.5kHz) using gain of P controller designed
for Ts = 200µ s ( f s = 5kHz).
After modification of P flux controller gain according to Table 5.1 it is possible to
achieve better results as shown in Fig. 5.13, what confirms proper flux tracking
performance in steady and dynamics state.
77
Direct Torque Control with Space Vector Modulation
Figure 5.13. Flux tracking performance for different sampling time Ts = 50 µ s , 100 µ s ,
200µ s , 400µ s ( f s = 20kHz, 10kHz, 5kHz,2.5kHz) using designed gain of P controller
calculated individually (see table 5.1.)
The simulation results in SABER package for Ts=200us is presented in Fig. 5.14.
Figure 5.14. Simulated (SABER) flux tracking performance for step change from 70% - 100%
of nominal flux.
As we can observed from Fig. 5.14 that overshoot is around M p = 0% and settling time
took about 10 sampling time of microprocessor, what proved the design procedure of P
digital flux controller gain.
78
Direct Torque Control with Space Vector Modulation
5.2.2 Digital torque control loop
Ψ s _ ref
The considered torque control loop is shown in Fig. 5.15.
Torque
controller
∆δ Ψ
PI
−
Reference
flux generator
In stator frame
Ψ sα _ ref
∆Ψ sα
−
Ψ s β _ ref
∆Ψ sβ
θ Ψs
Me
Flux
controllers
In stator frame
U sα _ ref
U sβ _ ref
Ψ sβ
−
Ψ sα
M e _ ref
Figure 5.15. Torque control loop with PI controller.
Based on the equation (5.9a-b and 5.10a-b) the stator flux errors in α , β coordinates can
be calculated as:
∆Ψ sα = Ψ s _ ref cos(θ Ψs + ∆δ Ψ ) − Ψ s _ ref cosθ Ψs = Ψ s _ ref [cos(θ Ψs + ∆δ Ψ ) − cosθ Ψs ]
(5.21a)
∆Ψ sβ = Ψ s _ ref sin(θ Ψs + ∆δ Ψ ) − Ψ s _ ref sin θ Ψs = Ψ s _ ref [sin(θ Ψs + ∆δ Ψ ) − sin θ Ψs ]
(5.21b)
Assuming that for small changes of ∆δ Ψ the cos ∆δ Ψ ≅ 1 and sin ∆δ Ψ ≅ ∆δ Ψ , the equations
(5.21a) and (5.21b) are given by:
∆Ψ sα _ ref = − Ψ s _ ref ∆δ Ψ sin θ Ψs
(5.22a)
∆Ψ sβ _ ref = Ψ s _ ref ∆δ Ψ cos θ Ψs
(5.22b)
In order to design the PI torque controller the following assumption are made:
•
stator flux vector position θ Ψs and rotor flux vector position θ r are equal zero. It
correspond to situation, when those two flux vectors lie along the α axis,
•
the reference stator flux amplitude is equal value of permanent magnet flux
Ψ s _ ref = Ψ PM ,
•
stator resistance is neglected.
Therefore, the error stator fluxes in α , β coordinates are calculated as:
∆Ψ sα _ ref = 0 ,
(5.23a)
79
Direct Torque Control with Space Vector Modulation
∆Ψ sβ _ ref = Ψ PM ∆δ Ψ .
(5.23b)
Stator voltage components using equations (2.23a-b) can be expressed as:
U sα _ ref = 0
(5.24a)
U sβ _ ref = ∆Ψ sβ _ ref K pΨ
(5.24b)
And further because of U sβ _ ref = U s _ ref
U s = Ψ PM ∆δ K pΨ
(5.25)
So, the transfer function between stator voltage amplitude U s and increment of torque
angle ∆δ Ψ can be written as:
GM δ ( s ) =
Us
∆δ Ψ
= K pΨ Ψ PM
(5.26)
Where K pΨ is the gain of stator flux P controller.
For example for sampling time Ts = 200 µ s , calculated K pΨ =
0.2688
= 1344 (see
Ts
Table5.1.) and nominal value of Ψ PM = 0.264Wb the calculated GM δ ( s) = 354,82 V / rad .
The obtained transfer function between electromagnetic torque M e and stator voltage
amplitude U s is (see equation 5.72):
GM ( s ) =
M e ( s)
AM s
= 2
U s ( s ) s + BM s + CM
(5.27)
3 Ψ s Ψ PM pb 2
3 pb Ψ PM
Rs Ψ PM
Where AM =
and BM =
CM =
2 JLs
2 Ls
Ψ s Ls
Using the motor parameters (see Appendices), one obtains:
AM = 198 and BM = 115.3 CM = 9065
Continuous s-domain
The torque control loop is shown in Fig. 5.16, where CM ( s ) is a transfer function of the
PI controller given by [105]:
80
Direct Torque Control with Space Vector Modulation
⎛
K
K pM ⎜ s + iM
⎜
K pM
⎝
CM ( s ) =
s
where K iM =
⎞
⎟⎟
⎠
(5.28)
K pM
TiM
M e _ ref
CM ( s )
∆δ Ψ
GM δ ( s )
U sβ = U s
Me
GM ( s )
Figure 5.16. Block diagram of torque control loop represented in s-domain.
Hence, the transfer function between M e _ ref ( s ) and M e ( s ) is obtained as:
GM _ closed ( s ) =
M e _ ref ( s )
M e (s)
=
CM ( s )GM ( s )GM δ ( s )
1 + CM ( z )GM ( s )GM δ ( s )
(5.29)
Substituting transfer function for CM ( s) and GM ( s ) equation (5.29) becomes:
K pΨ Ψ PM AM K pM ( s +
GM _ closed ( s ) = =
K iM
)
K pM
(5.30)
s 2 + ( BM + K pM AM ) s + CM + K iM AM
Discrete design
The transfer function (equations 5.28) for PI controller in discrete system using
backward difference method for discretization process ( s =
(z −
CM ( z ) = ( K pM + K iM )
where: K iM =
K pM
TiM
K pM
K pM + K iM
( z − 1)
z −1
) [2] is expressed as:
Ts z
)
(5.31)
Ts , Ts - sampling time, CM ( z ) is the discrete transfer function of
torque PI controller, D( z ) z −1 is one sampling time delay for voltage generation from
PWM, and
81
Direct Torque Control with Space Vector Modulation
D( z )
M e _ ref ( z )
∆δ Ψ
CM ( z )
GM δ ( z )
Us
z −1
G ( z)
}
M
ZOH
M e ( z)
AM s
s + BM s + CM
2
Figure 5.17. Block diagram of torque control loop in discrete domain.
GM δ ( z ) = K pΨ Ψ PM
(5.32)
is discrete transfer function for relation between stator voltage amplitude U s and
increment of torque angle ∆δ Ψ (see Fig. 5.17)
The discrete transfer function GM ( z ) for voltage-torque relationship with zero order
hold (ZOH) can be calculated as:
GM ( z ) = (1 − z −1 ) Z [
GM ( s )
AM
z −1
]=
Z[ 2
]
s
z
s + BM s + CM
(5.33)
Finally, the discrete transfer function of controlled plant GM ( z ) can be written as:
⎛
⎞
AMd
GM ( z ) = ( z − 1) ⎜ 2
⎟
⎝ z − BMd z + CMd ⎠
Where: AM _ d =
BM _ d = 2e
−
BM
Ts
2
AM
B 2
CM − M
4
e
cos(Ts CM −
−
BM
Ts
2
(5.34)
BM 2
sin(Ts CM −
),
4
BM 2
) , CM _ d = e − BM Ts , and Ts is sampling time.
4
Hence, the transfer function of closed torque control loop is obtained in the following
form:
GM _ closed ( z ) =
M e _ ref ( z )
M e ( z)
=
CM ( z )GM ( z ) D( z )GM δ ( z )
=
1 + CM ( z )GM ( z ) D( z )GM δ ( z )
82
Direct Torque Control with Space Vector Modulation
K pΨ Ψ PM AM _ d ( K pM + K iM )( z −
=
K pM
K pM + K iM
)( z − 1)
[ z 3 − BM _ d z 2 + [ AM _ d ( K pM + K iM ) + CM _ d ]z − AM _ d K pM ]( z − 1)
K pΨ Ψ PM AM _ d ( K pM + K iM )( z −
K pM
K pM + K iM
=
(5.35)
)
z 3 − BM _ d z 2 + [ AM _ d ( K pM + K iM ) + CM _ d ]z − AM _ d K pM
Selecting K pΨ , K iΨ will influence poles placement of closed torque control loop and as
a consequence also torque step responses can be selected.
The transfer function of closed torque control loop is more complicated than flux
control loop (see design of P-flux controller – section 5.2.1). One possibility is use to
the SISO tools from Matlab package to tune parameters of PI torque controller [106].
a)
b)
Figure 5.18. a) Torque step response for sampling time Ts = 200 µ s , b) with denoted rise time,
overshoot and settling time.
As can be observed in (Fig. 5.18) torque response is characterized by overshoot about
40%, rise time 4 samples and settling time 17 samples.
To eliminate high overshoot it is recommended to insert at the input prefilter (see
Fig.5.19 ) with transfer function:
PM ( z ) = K
where K =
(z −
z −b
K pM
K pM + K iM
=K
)
z − 0.6878
z − 0.855
1
=0.466 is gain of the prefilter.
z − 0.6878
lim
z →1 z − 0.855
83
(5.36)
Direct Torque Control with Space Vector Modulation
GM ( z )
D( z )
M e _ ref ( z )
PM ( z )
CM ( z )
∆δ Ψ
GM δ ( z )
Us
z −1
ZOH
M e ( z)
AM s
s + BM s + CM
2
Figure 5.19. Block diagram of torque control loop with prefilter (discrete domain).
Finally, the step response of closed torque control loop with prefilter at the input is
presented bellow:
a)
b)
Figure 5.20. a) Torque step response for sampling time Ts = 200 µ s , b) with denoted rise time,
overshoot and settling time.
As it can be observed, the response is now characterized by overshoot about 2%, rise
time 5 samples and the settling time 15 samples.
In digital control system when the sampling time is changed the parameters of
digitalized control plant AMd , BMd , CMd will also change. Therefore, the parameters of PI
torque controller will change also (see Table 2).
Simulation results for digital torque control loop in SABER package for 5KHz with and
without prefilter are shown in Fig. 5.21. Also, the torque step response for different
level of reference torque are presented in Fig. 5.22.
The settings for PI torque controller for different sampling time Ts = 50µ s , 100 µ s ,
200µ s , 400µ s are summarized in Table 5.2.
84
Direct Torque Control with Space Vector Modulation
a)
b)
Figure 5.21. Torque step response: a) without prefilter, b) with prefilter.
Figure 5.22. Torque step response with prefilter (from 0 to 25%, 50%, 75% and 100% of
nominal torque).
85
Direct Torque Control with Space Vector Modulation
86
Direct Torque Control with Space Vector Modulation
The torque step response is shown in Fig. 5.23, when parameters of PI torque controller
designed for sampling time Ts = 200µ s were used for control plant for different sampling
times Ts = 50µ s , 100 µ s , 200µ s , 400µ s , which correspond to switching frequency f s =
20kHz, 10kHz, 5kHz, 2.5kHz.
Figure 5.23. Torque tracking performance for different sampling time Ts = 50 µ s , 100 µ s , 200 µ s ,
400µ s ( f s = 20kHz, 10kHz, 5kHz,2.5kHz) using parameters of PI controller designed for
Ts = 200µ s ( f s = 5kHz). (Please note that for 2.5kHz the system was unstable).
After modification of PI torque controller parameters according to Table 2 it is possible to
achieve better results as shown in Fig. 5.23, what confirms and proper torque tracking
performance in steady and dynamics state.
Figure 5.24. Torque tracking performance for different sampling time Ts = 50 µ s , 100 µ s , 200 µ s ,
400µ s ( f s = 20kHz, 10kHz, 5kHz,2.5kHz) using designed parameters of PI controller calculated
individually (see table 5.2)
87
Direct Torque Control with Space Vector Modulation
5.3 Parallel structure of DTC–SVM scheme
Block scheme of the control structure is shown in Fig. 5.25. Two PI controllers are used for
regulation torque and flux magnitude loops [11,55].
U DC
Ψ s _ ref
M e _ ref
eΨs
U sx _ ref
U sα _ ref
eM
U sy _ ref
U sβ _ ref
SA
SB
SC
θ Ψs
Ψs
Is
Me
γm
Figure 5.25. Parallel structure of DTC-SVM scheme.
In this control scheme the reference stator flux magnitude
Ψ s _ ref
and reference
electromagnetic torque M e _ ref are compared with estimated values, respectively. The flux and
torque errors eΨs , eM are delivered to PI controllers, which generate command value the stator
voltage components in stator flux coordinates U sx _ ref , U sy _ ref . This voltage signals are
transformed to stationary coordinates using the stator flux position angle θ Ψs . The reference
stator voltage vector ( U sα _ ref , U sβ _ ref ) is delivered to space vector modulator (SVM), which
generates the switching signals S A , S B , S C to control power transistors of the inverter.
The presented control strategy is based on simplified stator voltage equations described in
stator flux oriented x-y coordinates (equations 2.27a-b):
U sx = Rs I sx +
d Ψs
dt
(5.37)
U sy = Rs I sy + Ω Ψs Ψ s = Rs I sy + Esy = k s M e + Esy
88
(5.38)
Direct Torque Control with Space Vector Modulation
where: k s =
2
2 Rs
, M e = pb Ψ s I sy and Esy = Ω Ψs Ψ s .
3
3 pb Ψ s
The above equations show that the U sx component has influence only on the change of stator
flux magnitude Ψ s , and the component U sy – if the term Ω Ψs Ψ s is decoupled – can be
used for torque adjustment. Therefore, the flux and torque quantities can be controlled as
shown in Fig. 5.26.
Note, that this DTC-SVM scheme formally corresponds to the stator flux oriented voltage
source inverter-fed drive induction motor. The block diagram of the DTC-SVM scheme with
two PI controllers is shown in Fig. 5.26 The dashed line represents the PMSM part [124,125].
Ψ s _ ref
Rs I sx
Ψs
_
Rs I sx
U sx
∫
Ω Ψs
Rs I sy
M e _ ref
_
⊗
Ω Ψs Ψ s
U sy
Esy
Ψs
1 I sy 3
pb Ψs _ ref
Rs
2
Me
Me
Figure 5.26. Block diagram of the scheme presented in Fig. 5.27
5.3.1 Digital flux control loop
Putting the stator x-axis current expression from equation 2.28a under the assumption Ld = Lq
I sx =
Ψ s − Ψ PM cos δ Ψ
(5.39)
Ls
into equation (5.37) one can obtaines
U sx = Rs (
Ψ s − Ψ PM cos δ Ψ
Ls
)+
d Ψs
dt
=
d Ψ s Rs Ψ PM
Rs
cos δ Ψ
Ψs +
−
Ls
dt
Ls
(5.40)
Using Laplace transformation to equation 5.40 can be written as:
U sx = Ψ s ( s +
Rs
RΨ
) − s PM cos δ Ψ .
Ls
Ls
89
(5.41)
Direct Torque Control with Space Vector Modulation
Assuming small changes of δ Ψ , the cos δ Ψ ≅ 1 , and equations (5.41) reduces to:
U sx = Ψ s ( s +
Where WΨ =
Rs
) + WΨ
Ls
(5.42)
Rs Ψ PM
Ls
The transfer function between stator flux amplitude Ψ s and x-axis of stator voltage is:
GΨ ( s ) =
Where WΨ ≅
Ψs
Ls
1
1
=
=
=
U sx + WΨ sLs + Rs s + Rs s + AΨ
Ls
(5.43)
Rs
R
Ψ PM cos δ Ψ ≅ s Ψ PM
Ls
Ls
For motor parameters (see Appendices ): AΨ =
Rs
= 115.333 .
Ls
Continuous s-domain
The flux control loop is shown in Fig. 5.27, where CΨ ( s ) is a transfer function of the PI
controller given by [105]:
⎛
K
K pΨ ⎜ s + i Ψ
⎜
K pΨ
⎝
CΨ ( s ) =
s
where K iΨ =
⎞
⎟⎟
⎠
(5.45)
K pΨ
TiΨ
WΨ
Rs
Ψ PM cos δ Ψ
Ls
Ψ s_ref
CΨ ( s )
U sx +
GΨ ( s )
Ψs
Figure 5.27. Block diagram of flux control loop in s-domain.
Hence, the transfer function of the closed stator flux amplitude control loop is obtained as:
90
Direct Torque Control with Space Vector Modulation
GΨ _ closed ( s ) =
Ψs
_ ref
( s)
Ψ s ( s)
=
CΨ ( s )GΨ ( s )
1 + CΨ ( z )GΨ ( s )
(5.46)
Substituting transfer function for CΨ ( s) and GΨ ( s ) one becomes
⎛
K
K pΨ ⎜ s + i Ψ
⎜
K pΨ
⎝
s
⎞
⎟⎟
⎛
K ⎞
⎠⎛ 1 ⎞
K pΨ ⎜ s + iΨ ⎟
⎜
⎟
⎜
K pΨ ⎠⎟
⎝ s + AΨ ⎠ =
⎝
(5.47)
GΨ _ closed ( s ) =
s 2 + ( AΨ + K pΨ ) s + K iΨ
⎛
K iΨ ⎞
K pΨ ⎜ s +
⎟
⎜
K pΨ ⎟⎠ ⎛ 1 ⎞
⎝
1+
⎜
⎟
s
⎝ s + AΨ ⎠
Discrete design
Using backward difference method for discretization process ( s =
z −1
) [2] the transfer
Ts z
function of equation (5.45) for flux PI controller in discrete system is expressed as:
Ts z
CΨ ( z ) = K pΨ (1 +
)=
TiΨ ( z − 1)
Where: K iΨ =
K pΨ
TiΨ
( K pΨ + K iΨ )( z −
K pΨ
K pΨ + K iΨ
)
(5.48)
( z − 1)
Ts ; Ts - sampling time.
W ( z)
D( z )
Ψ s _ ref ( z )
CΨ ( z )
U sx
z −1
G ( z)
}
Ψ
ZOH
1
s + AΨ
Ψ s ( z)
Figure 5.28. Block diagram of flux control loop in discrete domain.
Where: CΨ ( z ) discrete transfer function of PI controller, D( z ) z −1 - one sampling time delay
for voltage generation from PWM, and W ( z ) - disturbance voltage due to cross coupling
between x-y axis (see Fig. 5.28).
The GΨ ( z ) is discrete transfer function of voltage-flux relationship with zero order hold
(ZOH) block can be calculated as:
91
Direct Torque Control with Space Vector Modulation
GΨ ( z ) = (1 − z −1 ) Z [
( z − 1)
z
GΨ ( s )
AΨ
z −1
]=
]=
Z[
s
z
AΨ s ( s + AΨ )
(5.49)
AΨ
1
]
Z[
AΨ s ( s + AΨ )
Using table of Z transformation [2] one can calculate:
GΨ ( z ) =
Where: AΨd =
( z − 1)
AΨd
1
z (1 − e − AΨTs )
=
− AΨTs
AΨ ( z − 1) ( z − e
) z − BΨd
z
(5.50)
(1 − e − AΨTs )
, BΨd = e − AΨTs and Ts is sampling time.
AΨ
Hence, the transfer function of closed stator flux control loop can be expressed in the
following form:
GΨ _ closed ( z ) =
Ψs
_ ref
( z)
Ψ s ( z)
=
CΨ ( z )GΨ ( z ) D( z )
1 + CΨ ( z )GΨ ( z ) D( z )
( K pΨ + K iΨ ) AΨd ( z −
=
K pΨ
K pΨ + K iΨ
z ( z − 1)( z − BΨd ) + ( K pΨ + K iΨ ) AΨd ( z −
(5.51)
)
K pΨ
K pΨ + K iΨ
)
Now selecting K pΨ , K iΨ is possible to obtain poles placement, which define the dynamic of
closed torque control loop.
Assuming, that BΨd =
K pΨ
K pΨ + K iΨ
⇒ K iΨ =
K pΨ − BΨd K pΨ
BΨd
=
K pΨ
BΨd
(1 − BΨd )
and the transfer function of closed stator flux control loop will take the following form:
GΨ _ closed ( z ) =
( K pΨ + K iΨ ) AΨd
z − z + ( K pΨ + K iΨ ) AΨd
2
Putting into above equation K pΨ + K iΨ =
K pΨ
GΨ _ closed ( z ) =
BΨd
z −z+
2
K pΨ
BΨd
one obtains:
AΨd
K pΨ
BΨd
(5.52)
=
AΨd
92
CΨ d
z − z + CΨ d
2
(5.53)
Direct Torque Control with Space Vector Modulation
where CΨd =
K pΨ
AΨd
BΨd
The bellow diagrams shows the relationship between overshoot M p , rise time tr and settling
time t s as function CΨd value.
Please note that tr is time calculated from 10% to 90% of output signals and ts is the time in
witch the system transient decay to +-1%.
Figure 5.29. The relationship between overshoot, rise time and settling time versus CΨd for stator flux
amplitude control loop.
From a few values of CΨd =[0.4046, 0.2688, 0.1720] we can selected CΨd =0.2688, which
guaranties overshoot 0% and settling time about 10 samples.
1
2
3
Figure 5.30. Flux step response for different values of CΨd : red line (1) CΨd =0.4046, blue line (2)
CΨd =0.2688, black line (3) CΨd =0.1720.
93
Direct Torque Control with Space Vector Modulation
It corresponds to the transfer function of closed stator flux control loop:
CΨd
0.2688
= 2
z − z + CΨd z − z + 0.2688
GΨ _ closed ( z ) =
2
Using digitalized motor parameters AΨd =
(5.54)
(1 − e − AΨTs )
, BΨd = e − AΨTs and chosen CΨd value, we
AΨ
can calculate the parameters of digital PI flux controller as:
K pΨ =
CΨd BΨd
AΨd
(5.55a)
K pΨTs
TiΨ =
(5.55b)
K iΨ
K pΨ
K iΨ =
(1 − BΨd )
BΨd
(5.55c)
For example with sampling time Ts = 200µ s , parameters of PI controller are:
K pΨ =
TiΨ =
0.2688BΨd 0.2688*0.9772
=
= 1328.64
AΨd
0.0001977
(5.56a)
1328.64* 200 µ s
= 8572µ s
30.999
(5. 56b)
K pΨTs
K iΨ =
K iΨ
K pΨ
BΨd
=
(1 − BΨd ) =
1328.64
*(1-0.9772) = 30.999
0.9772
(5. 56c)
For different sampling time the closed transfer function GΨ _ closed ( z ) of digital flux control
loop should be kept to:
GΨ _ closed ( z ) =
CΨd
0.2688
= 2
z − z + CΨd z − z + 0.2688
2
(5.57)
In order to find the original function of Z transfer function GΨ _ closed ( z ) using the Z properties
as (sum transformations) [2]:
n
z
z
F ( z )] = Z −1[
GΨ _ closed ( z )]
z −1
z −1
k =0
z
a
= Z −1[
]
2
z − 1 ( z − z + a)
∑ f (kT ) = Z
s
−1
[
94
(5.58)
Direct Torque Control with Space Vector Modulation
As an example the calculated response for 4 samples are given:
n
∑ f (kT ) = az
k =0
−2
s
+ 2az −3 − a(1 + a) + 2az −4 + .......
which gives:
f (0) = f (k ) = 0
f (1Ts ) = f (k + 1) = 0
f (2Ts ) = f (k + 2) = a = 0.2688
f (3Ts ) = f (k + 3) = a (b + 1) = 2a = 0.5376
f (4Ts ) = f (k + 4) = −a (b + c) + a(b + 1) 2 = − a(1 + a) + 2a = 0.7342
Keeping constant CΨd in equation (5.57) the flux step response for different sampling time
Ts = 50µ s, 100µ s, 200µ s, 400µ s , which correspond to switching frequency f s = 20kHz,
10kHz, 5kHz, 2.5kHz are presented.
1
2
3
4
Figure 5.31.Flux step response for different sampling time Ts = 50 µ s, 100 µ s, 200 µ s, 400 µ s
(switching frequency f s 20kHz (1 -blue line), 10kHz (2 -green line), 5kHz (3 -red line),2.5kHz
(4 -light blue line).
We may observe from Fig. 5.31 that overshoot is 0% and the settling time is 10 samples.
Selected parameters of PI flux controller for sampling time Ts = 50µ s, 100µ s, 200µ s,
400µ s are summarized in Table 5.3
95
Direct Torque Control with Space Vector Modulation
96
Direct Torque Control with Space Vector Modulation
The behavior of the flux control loop was tested using SABER simulation package. The
model created in SABER takes into account the whole control system, which include real
models of inverter and permanent magnet synchronous motor.
The flux step response is shown in Fig. 5.32., when parameters of PI flux controller designed
for sampling time Ts = 200µ s were used for control plant for different sampling times
Ts = 50µ s , 100 µ s , 200µ s , 400µ s , which correspond to switching frequency f s = 20kHz,
10kHz, 5kHz, 2.5kHz.
Figure 5.32. Flux tracking performance for different sampling time Ts = 50 µ s , 100 µ s , 200 µ s ,
400µ s ( f s = 20kHz, 10kHz, 5kHz,2.5kHz) using parameters of PI controller designed for
Ts = 200µ s ( f s = 5kHz).
After modification of PI flux controller parameters according to Table 5.31 it is possible to
achieve better results as shown in Fig. 5.33, what confirms proper flux tracking performance
in steady and dynamics state.
97
Direct Torque Control with Space Vector Modulation
Figure 5.33. Flux tracking performance for different sampling time Ts = 50 µ s , 100 µ s , 200 µ s ,
400µ s ( f s = 20kHz, 10kHz, 5kHz,2.5kHz) using parameters of PI controller calculated individually
(see table 5.3.)
The simulation results in SABER package for Ts=200us is presented in Fig. 5.34.
Figure 5.34. Simulated (SABER) flux tracking performance for step change from 70% - 100% of
nominal flux.
As we can observed from Fig. 5.34 that overshoot is around M p = 0% and settling time took
about 10 sampling time of microprocessor, what proved the design procedure of PI digital
flux controller parameters.
98
Direct Torque Control with Space Vector Modulation
5.3.2 Digital torque control loop
The PMSM equations (2.27a,b-2.28a,b) in stator flux coordinates under the assumption
Ld = Lq can be written as:
U sy = Rs I sy + Ω Ψ s Ψ s
(5.59)
0 = Ls I sy − Ψ PM sin δ Ψ
(5.60)
Me =
3
pb Ψ s I sy
2
(5.61)
d Ωm 1
= (M e − M l )
dt
J
(5.62)
The load angle can be expressed (Fig. 5.1):
δ Ψ = θ Ψs − pbγ m ,
(5.63)
Where: δ Ψ is torque angle, θ Ψs is stator flux vector position, and γ m is rotor position in stator
α , β coordinates, pb is number of pole pars.
After differentiation equation (5.63) can be written as:
dγ
dδ Ψ dθ Ψs
=
− pb m
dt
dt
dt
(5.64)
dδ Ψ
dδ
= Ω Ψ s − pb Ω m ⇒ Ω Ψ s = Ψ + pb Ω m
dt
dt
(5.65)
Putting equations (5.64) and (5.65) into voltage equation (5.59) one obtains:
U sy = Rs I sy + Ω Ψ s Ψ s = Rs I sy + Ψ s (
dδ Ψ
+ pb Ω m )
dt
(5.66)
From equation 0 = Ls I sy − Ψ PM sin δ Ψ with assumption that for small angle δ Ψ = sin δ Ψ , the
torque angle can be expressed as:
δΨ =
Ls I sy
(5.67)
Ψ PM
So, the voltage equation (5.59) becomes:
U sy = Rs I sy + Ω Ψ s Ψ s = Rs I sy + Ψ s (
99
Ls dI sy
+ pb Ω m )
Ψ PM dt
(5.68)
Direct Torque Control with Space Vector Modulation
After differentiating of the above equation one obtains:
dU sy
dt
= Rs
dI sy
dt
+ Ψs (
2
Ls d I sy
d Ωm
+ pb
)
Ψ PM dt
dt
Take into account that from equation (5.61) the y-axis current is equal I sy =
(5.69)
2M e
,
3 pb Ψ s
d Ωm 1
= ( M e − M l ) and under assumption that the motor is no loaded equation (5.69) takes
dt
J
form:
dU sy
dM e
Ls
d 2 M e pb
2
2
= Rs
+ Ψs (
+
Me )
dt
Ψ PM 3 pb Ψ s dt
J
3 pb Ψ s dt
(5.70)
Using Laplace transformation and after some arrangements the equation (5.70) can be written:
sU sy = M e (
Ψ p
2 Ls
2 Rs
s2 +
s+ s b)
3 pb Ψ PM
3 pb Ψ s
J
(5.71)
Hence, the transfer function between electromagnetic torque M e and y-axis voltage U sy can
be obtained as:
GM ( s ) =
Where: AM =
M e ( s)
AM s
= 2
U sy ( s ) s + BM s + CM
(5.72)
3 Ψ s Ψ PM pb 2
3 pb Ψ PM
RΨ
and BM = s PM CM =
2 JLs
2 Ls
Ψ s Ls
Using the motor parameters (see Appendices) we may calculates:
AM = 198 , BM = 115.3 and CM = 9065
Continuous s-domain
The torque control loop of the block scheme DTC-SVM from Fig. 5.25 is shown in Fig. 5.35,
where CM ( s ) is a transfer function of the PI controller given by equation 5.28:
100
Direct Torque Control with Space Vector Modulation
M e _ ref
U sy
CM ( s )
Me
GM ( s )
Figure 5.35. Block diagram of torque control loop in s-domain.
The transfer function of torque control loop is obtained as:
GM _ closed ( s ) =
M e _ ref ( s )
M e ( s)
=
CM ( s )GM ( s )
1 + CM ( z )GM ( s )
(5.73)
Substituting in equation (5.73) transfer function for CM ( s ) -Eq.5.28 and GM ( s ) - Eq.5.72 we
may calculate:
AM K pM ( s +
GM _ closed ( s ) = =
K iM
)
K pM
(5.74)
s 2 + ( BM + K pM AM ) s + CM + K iM AM
Discrete design
Using backward difference method for discretization process ( s =
z −1
) the transfer function
Ts z
for discrete PI controller is expressed as:
Ts z
CM ( z ) = K pM (1 +
) = ( K pM + K iM )
TiM ( z − 1)
Where: K iM =
K pM
TiM
Ts - integration gain; Ts - sampling time
101
(z −
K pM
K pM + K iM
( z − 1)
)
(5.75)
Direct Torque Control with Space Vector Modulation
Time delay
PI controller
M e _ ref ( z )
CM ( z )
D( z )
U sy
Plant G ( z )
}
z −1
M
ZOH
M e ( z)
AM s
s + BM s + CM
2
Figure 5.36. Block diagram of torque control loop in discrete domain.
Where: CM ( z ) - discrete transfer function for PI controller, D ( z ) z −1 - one sampling time
delay for voltage generation from PWM block (see Fig. 5.36).
The discrete transfer function GM ( z ) for voltage-torque relationship with zero order hold
(ZOH) can be calculated as:
GM ( z ) = (1 − z −1 ) Z [
GM ( s )
z −1
AM
Z[ 2
]=
]
s
z
s + BM s + CM
⎡
⎤
⎤
⎢
⎥
⎥
⎢
⎥
AM
A
−
1
z
M
⎥=
=
Z
⎢
2
2⎥
z
2 ⎞ ⎥
BM 2 ⎥
⎞
⎛
⎢
B 2
B
⎥
⎟ + CM −
⎢ ( s + M ) + ⎜ CM − M ⎟ ⎥
4 ⎦
⎠
⎜
2
4 ⎟⎠ ⎥
⎢⎣
⎝
⎦
.
⎡
⎤
⎢
⎥
BM 2
−
C
M
⎢
⎥
z −1
AM
4
=
Z⎢
2⎥
z
2 ⎞ ⎥
BM 2 ⎢
⎛
B 2
B
CM −
⎢ ( s + M ) + ⎜ CM − M ⎟ ⎥
4
⎜
2
4 ⎟⎠ ⎥
⎢⎣
⎝
⎦
⎡
⎢
z −1 ⎢
=
Z
⎢⎛
z
BM
⎢⎜ s +
2
⎣⎝
(5.76)
Assuming that a =
BM
B 2
and b = CM − M , and using table of Z transformation [2] we
2
4
have:
⎡
⎤
ze − aTs sin(bTs )
b
Z⎢
=
− aTs
−2 aTs
2
2⎥
2
⎣ ( s + a) + b ⎦ z − 2e (cos(bTs )) z + e
102
(5.77)
Direct Torque Control with Space Vector Modulation
Finally, the discrete transfer function of controlled plant GM ( z ) can be written as:
⎛
⎞
AMd
GM ( z ) = ( z − 1) ⎜ 2
⎟
⎝ z − BMd z + CMd ⎠
AM
Where: AM _ d =
e
−
BM
Ts
2
sin(Ts CM −
BM 2
4
and Ts is sampling time.
CM −
CM _ d = e − BM Ts
(5.78)
B
− M Ts
BM 2
B 2
) , BM _ d = 2e 2 cos(Ts CM − M )
4
4
Hence, the transfer function of closed torque control loop is obtained as:
GM _ closed ( z ) =
M e _ ref ( z )
M e ( z)
=
CM ( z )GM ( z ) D( z )
=
1 + CM ( z )GM ( z ) D( z )
AM _ d ( K pM + K iM )( z −
=
K pM
K pM + K iM
)
z 3 − BM _ d z 2 + [ AM _ d ( K pM + K iM ) + CM _ d ]z − AM _ d K pM
(5.79)
Selecting K pΨ , K iΨ will influence poles placement of closed torque control loop and as a
consequence also torque step responses can be selected.
The transfer function of closed torque control loop is more complicated than flux control loop
(see design of PI-flux controller – section 5.3.1). One possibility is use to the SISO tools from
Matlab package to tune parameters of PI torque controller [106].
a)
b)
Figure 5.37. a) Torque step response for sampling time Ts = 200 µ s , b) with denoted rise time,
overshoot and settling time.
As can be observed the response is characterized by overshoot about 40%, rise time 4 samples
and settling time 17 samples.
103
Direct Torque Control with Space Vector Modulation
The torque control loop should be as fast as possible even with some overshoot. This improve
response to disturbance (for example from flux control loop –see Fig. 5.38).
D( z )
M e _ ref ( z )
CM ( z )
U sy
G ( z)
}
M
ZOH
z −1
M e ( z)
AM s
s + BM s + CM
2
Figure 5.38. Block diagram of torque controller in discrete domain with disturbance.
a)
b)
Figure 5.39. Disturbance rejection in torque control loop: a) short voltage impulse, b) voltage
step.
To improve reference tracking performance (without any overshoot) it is recommended to
insert a input prefilter (see Fig. 5.40 ) described by transfer function:
PM ( z ) = K
Where: K =
(z −
z −b
K pM
K pM + K iM
=K
)
z −b
z − 0.6663
=K
( z − a)
z − 0.855
1
=0.43413 is gain of the prefilter.
z − 0.6663
lim
z →1 z − 0.855
104
(5.80)
Direct Torque Control with Space Vector Modulation
D( z )
M e _ ref ( z )
PM ( z )
CM ( z )
U sy
z −1
G ( z)
}
M
ZOH
M e ( z)
AM s
s + BM s + CM
2
Figure 5.40. Block diagram of torque control loop with prefilter in discrete domain.
Finally, the reference tracking performance of closed torque control loop with prefilter is
presented in Fig. 5.41.
Figure 5.41. a) Reference tracking performance of the torque control loop for sampling time
Ts = 200µ s , b) with denoted rise time, overshoot and settling time.
In digital control when the sampling time is changing the parameters of digitalized plant
control AMd , BMd , CMd will be also changes. It is normally that the parameters of PI torque
control will be changes also (see Table 5.4.).
Simulation for PI flux calculated parameters with and without prefilter are shown in Fig.
5.42a-b. Also torque step response for different level of reference torque are presented in Fig.
5.43.
105
Direct Torque Control with Space Vector Modulation
a)
b)
Figure 5.42. Torque step response: a) without prefilter, b) with prefilter.
Figure 5.43. Torque step response with prefilter (from 0 to 25%, 50%, 75% and 100% of nominal
torque).
106
Direct Torque Control with Space Vector Modulation
107
Direct Torque Control with Space Vector Modulation
The behavior of the torque control loop was tested using SABER simulation package.
The model created in SABER takes into account the whole control system, which
include real models of inverter and permanent magnet synchronous motor.
The torque step response is shown in Fig. 5.44, when parameters of PI torque controller
designed for sampling time Ts = 200µ s were used for control plant for different
sampling times Ts = 50µ s , 100µ s , 200µ s , 400µ s , which correspond to switching
frequency f s = 20kHz, 10kHz, 5kHz, 2.5kHz.
Figure 5.44. Torque tracking performance for different sampling time Ts = 50 µ s , 100 µ s ,
200µ s , 400µ s ( f s = 20kHz, 10kHz, 5kHz,2.5kHz) using parameters of PI controller
designed for Ts = 200 µ s ( f s = 5kHz). Pleas note that for 2.5kHz the system was unstable.
After modification of PI flux controller parameters according to Table 5.4 it is possible
to achieve better results as shown in Fig. 5.45, what confirms proper torque tracking
performance in steady and dynamics state.
108
Direct Torque Control with Space Vector Modulation
Figure 5.45. Torque tracking performance for different sampling time Ts = 50 µ s , 100 µ s ,
200µ s , 400µ s ( f s = 20kHz, 10kHz, 5kHz,2.5kHz) using parameters of PI controller
calculated individually (see Table 5.4.)
109
Direct Torque Control with Space Vector Modulation
5.4 Speed control loop for DTC-SVM structure
The structure of speed control loop for cascade (a) and parallel (b) DTC-SVM scheme
is shown in the Fig. 5.46.
U DC
a)
Ψ s _ ref
Ω m _ ref
M e _ ref
SA
SB
U sα _ ref
eM
U sβ _ ref
∆δ Ψ
SC
Ψ s θ Ψs I s
Me
Is
γm
d
dt
U DC
b)
Ω m _ ref
Ψ s _ ref
eΨs
U sx _ ref
U sα _ ref
M e _ ref
eM
U sy _ ref
U sβ _ ref
SA
SB
SC
θ Ψs
Ψs
Is
Me
γm
d
dt
Figure 5.46. Speed control loop for: a) cascade structure of DTC-SVM scheme, b) parallel
structure of DTC-SVM scheme.
The mechanical speed equation (2.66) for the PMSM is:
Me − ML = J
d Ωm
dt
(5.81)
Taking Laplace transformation to equation (5.81) one obtains:
M e ( s) − M L ( s) = JsΩ m ( s )
(5.82)
The transfer function between mechanical rotor speed Ω m and electromagnetic torque
M e can be expressed as:
110
Direct Torque Control with Space Vector Modulation
GΩ ( s ) =
Ωm 1
1
=
=
M e Js Js
(5.83)
Continuous s-domain
The block diagram of speed control loop is shown in Fig. 5.47, where CΩ ( s ) is a
transfer function of the PI speed controller given by:
CΩ ( s ) = K pΩ (1 +
1
TiΩ s
K pΩ ( s +
)=
K iΩ
)
K pΩ
(5.84)
s
and DΩ ( s ) is approximated transfer function of closed torque control loop for cascade
or parallel DTC-SVM structure.
ML
Ω m _ ref
CΩ ( s )
M e _ ref
DΩ ( s )
Me
GΩ ( s )
Ωm
Figure 5.47. Block diagram of speed control loop in s-domain.
Discrete design
The transfer function for PI controller in discrete system using backward difference
method for discretization process is expressed as:
(z −
CΩ ( z ) = ( K pΩ + K iΩ )
Where: K iΩ =
K pΩ
TiΩ
K pΩ
K pΩ + K iΩ
( z − 1)
)
(5.85)
Ts - integration and K pΩ proportional gain of speed controller, Ts -
sampling time.
111
Direct Torque Control with Space Vector Modulation
M L ( z)
G ( z)
}
Ω
Ω m _ ref ( z )
CΩ ( z )
Me_ref (z)
DΩ ( z )
M e ( z)
ZOH
1
Js
Ωm ( z )
Figure 5.48. Block diagram of speed control loop in discrete domain.
The closed torque control loop transfer function (see for cascade DTC-SVM -Table 5.2
or parallel DTC-SVM - Table 5.4) is:
DΩ ( z ) = GM _ closed ( z ) ≅
0.466*0.37267
a
= 2
z - 1.289z + 0.4633) z +bz + c
2
(5.86)
The GΩ ( z ) is discrete transfer function for torque-speed relationship with zero order
hold (ZOH). The GΩ ( z ) can be calculated as:
GΩ ( s )
1
] = (1 − z −1 ) Z [
]
s
( Js ) s
1 1
1 Ts
= (1 − z −1 ) Z [ 2 ] =
J s
J ( z − 1)
GΩ ( z ) = (1 − z −1 ) Z [
(5.87)
Finally, it can be expressed as:
GΩ ( z ) =
Where: AΩd =
AΩd
( z − 1)
(5.88)
Ts
, and Ts is sampling time.
J
Using the sampling time Ts = 200µ s and motor parameters J = 0.0173 the GΩ ( z ) can
be calculated as:
GΩ ( z ) =
0.01156
( z − 1)
(5.89)
Hence, the transfer function of closed speed control loop can be written:
GΩ _ closed ( z ) =
Ωm ( z)
CΩ ( z ) DΩ ( z )GΩ ( z )
=
Ω m _ ref ( z ) 1 + CΩ ( z ) DΩ ( z )GΩ ( z )
And finally
112
(5.90)
Direct Torque Control with Space Vector Modulation
GΩ _ closed ( z ) =
( K pΩ + K iΩ )( z −
K pΩ
K pΩ + K i Ω
)aAΩd
( z − 1)( z 2 − bz + c)( z − 1) + ( K pΩ + K iΩ )( z −
(5.91)
K pΩ
K p Ω + K iΩ
)aAΩd
In many practical cases the digital filter is used in speed measurement loop (see Fig.
5.49).
M L ( z ) Control Plant
GΩ ( z )
PI controller
Ω m _ ref ( z )
CΩ ( z )
Me_ref (z)
DΩ ( z )
}
M e ( z)
ZOH
1
Js
Ωm ( z )
Torque control loop
FΩ ( z )
Digital Filter
Figure 5.49. Block scheme of speed control with digital filter in speed measurement loop
(discrete domain).
The transfer function FΩ ( s ) of first order low pass filter in s domain is expressed as:
FΩ ( s ) =
1
s
ωi
(5.92)
+1
Where ωc = 2π f c and f c is cut off frequency and ωi =
ωT
2
tan( c s ) [2]. In practice f c
Ts
2
is selected in the range 20-250Hz
2( z − 1)
for discretization process, the
Ts ( z + 1)
discrete transfer function of first order low pass filter can be expressed as:
Using the Tutsins’s approximation method s =
Tsωi
( z + 1)
2 + Tsωi
a ( z + 1)
= 1
FΩ ( z ) =
2 − Tsωi
z − b1
z−
2 + Tsωi
Where: a1 =
Tsωi
2 − Tsωi
, b1 =
2 + Tsωi
2 + Tsωi
113
(5.93)
Direct Torque Control with Space Vector Modulation
The discrete transfer function of digital filter for Ts = 200µ s and f c = 25 Hz can be
calculated as:
FΩ ( z ) =
0.015466 (z+1)
(z-0.9691)
(5.94)
Hence, the transfer function speed control loop with digital filter:
GΩ _ closed ( z ) =
Ωm ( z)
CΩ ( z ) DΩ ( z )GΩ ( z )
=
Ω m _ ref ( z ) 1 + CΩ ( z ) DΩ ( z )GΩ ( z ) F (c)
(5.95)
And finally
GΩ _ closed ( z ) =
( K pΩ + K iΩ )( z − b1 )( z −
=
K pΩ
K pΩ + K i Ω
( z 2 − bz + c)( z − 1)( z − 1)( z − b1 ) + ( K pΩ + K iΩ )( z −
)aAΩd
K pΩ
K p Ω + K iΩ
)aAΩd a1 ( z + 1)
(5.96)
Selecting K pΩ , K iΩ will influence poles placement of closed speed control loop and as
a consequence also speed step responses can be selected.
In order to select the best value of PI speed controller it is recommended to use the
SISO tools from Matlab package to tune the parameter of PI speed controller.
The speed response with digital filter simulated in SIMULINK is shown in Fig. 5.50
and simulated in SABER in Fig. 5.51 is presented.
114
Direct Torque Control with Space Vector Modulation
Figure 5.50. Simulated (SIMULINK) speed response with digital filter
Figure 5.51. Simulated (SABER) speed response with digital filter in feedback. From the top
reference torque, measured speed.
However, the speed respond is characterized by large overshoot. Therefore, the prefilter
will be applied in order to reduce overshoot (see Fig. 5.52). The discrete transfer
function of prefilter P ( z ) can be expressed as”
P( z ) =
0.005
KK _ s
=
z − bb _ s z − 0.995
(5.97)
where KK _ s = lim( z − 0.995)=0.005 is gain of the prefilter.
z →1
115
Direct Torque Control with Space Vector Modulation
M L ( z)
G ( z)
}
Ω
Ω m _ ref ( z )
PΩ ( z )
CΩ ( z )
Me_ref (z)
DΩ ( z )
M e ( z)
ZOH
1
Js
Ωm ( z)
FΩ ( z )
Figure 5.52. Speed response with digital filter in feedback FΩ ( z ) and prefilter PΩ ( z ) at the
input.
The speed response with and without prefilter are shown in Fig. 5.53.
Without prefilter
With prefilter
Figure 5.53. Speed response: blue signal without prefilter and green signal with prefilter at the
input.
Design parameters of PI speed controller for sampling time Ts = 50µ s, 100µ s, 200µ s,
400µ s are summarized in Table 5.5.
116
Direct Torque Control with Space Vector Modulation
117
Direct Torque Control with Space Vector Modulation
Simulated results for speed tracking performance for different reference speed level in
Fig. 5.54 are shown.
Figure 5.54. Simulated speed tracking performance with prefilter at the input for 10%, 20%,
50%, 100% of nominal speed. From the top actual speed, reference torque.
Investigation for influence of load torque in Fig. 5.55 is presented.
Figure 5.55. Simulated disturbance rejection performance of speed control loop for step load
change 50% of nominal torque. From the top electromagnetic torque, measured speed
Simulation results for speed control loop in Saber package for sampling time
Ts = 200 µ s and PI speed parameters controller: K pΩ = 1.1940 , TiΩ = 0.0398 (see Table
5.5) in Fig. 5.56 is shown.
118
Direct Torque Control with Space Vector Modulation
Figure 5.56. Simulated speed tracking performance to step of speed from 0 to 1000rpm.
The presented simulation results confirm well proper operation and design methodology
for digital speed control loop.
119
Direct Torque Control with Space Vector Modulation
5.5 Summary
This Chapter presents design of discrete control loops for two DTC-SVM schemes:
series (cascade) and parallel structures of flux and torque controllers, Fig. 5.2 and
Fig. 5.25, respectively. The cascade structure operates with P-flux controller and PItorque controller whereas in parallel structure two PI controllers are used. In the first
step of design calculation of discrete Z- transfer function from continuous s- domain
transfer function using zero order hold (ZOH) method of discretization has been
performed. The continuous PI controller transfer function has been discretized using
backward difference approach. Secondly, a SISO tool from Matlab package for
digital controller parameter calculation has been applied. The results of design were
verified by Simulink (using simplified discrete transfer function) and Saber (using
full motor and inverter model) simulation. Also, the influence of sampling time
selection on controller parameters have been discussed. Finally, also the speed
control loop was synthesized using similar methodology.
120
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
Chapter 6
DIRECT TORQUE CONTROL WITH SPACE VECTOR MODULATION (DTCSVM) OF PMSM DRIVE WITHOUT MOTION SENSOR
6.1 Introduction
Many motion control applications, such as material handling, packaging and hydraulic
or pneumatic cylinder replacement, require the use of a position transducer for speed or
position feedback, such as an encoder or resolver. In addition, permanent magnet
synchronous motors require position feedback to perform commutation. Some of
systems utilize velocity transducer as well. These sensor add cost, weight, and reduce
the reliability of the system. Also, a special mechanical arrangement needs to be made
for mounting the position sensors. An extra signal wires are required from the sensor to
the controller. Additionally, some type of position sensors are temperature sensitive and
their accuracy degrades, when the system temperature exceed the limits. Therefore, the
research in the area of sensorless speed control of PMSM is beneficial because of the
elimination of the feedback wiring, reduced cost, and improved reliability.
Sensorless speed DTC-SVM control block scheme is presented in Fig. 6.1.
Figure 6.1. Block scheme of DTC-SVM for PMSM drive without motion sensor.
As we can see the operation of speed controlled PMSM drive without mechanical
motion sensor is based only on measurement of following signals, which are available
in every PWM inverter-fed drive system as:
•
DC link voltage,
•
motor phase currents,
121
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
Based on this signals, the state variable of the drive can be indirectly calculated or
estimated what further allow to achieve the estimated (actual) rotor speed of PMSM.
Two motion sensorless control schemes of DTC-SVM for PMSM drive are presented in
Fig. 6.2
U DC
a)
Ψ s _ ref
Ω m _ ref
M e _ ref
SA
SB
SC
U sα _ ref
eM e
U sβ _ ref
∆δ
θ Ψs Ψ s I s
Us
Me
Is
Ω m _ est
U DC
b)
Ω m _ ref
Ψ s _ ref
eΨs
U sx _ ref
M e _ ref
eM e
U sy _ ref
U sα _ ref
U sβ _ ref
SA
SB
SC
θ Ψs
Ψs
Us
Me
Is
Ω m _ est
Figure 6.2. The DTC-SVM block schemes of PMSM without motion sensor: a) cascade
structure and b) parallel structure.
In motion sensorless PMSM drives, as shown in Fig.6.2, the position or speed
transducer (see Fig. 5.52) is replaced by a speed estimation block, which generates the
speed feedback signal Ω m _ est into the control systems and stator flux model.
122
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
The problem associated with speed sensorless operation of PMSM drive sourced from
VSI is listed below:
•
initial rotor flux detection at start up of PMSM controlled drive,
•
stator flux estimation without measured speed or position signal from sensor,
•
rotor speed estimation based on state variable of the PMSM, especially in low
speed operation region.
Therefore, in this Section the initial rotor position detection method of permanent
magnets, as well as stator flux and rotor speed estimation techniques will be discussed.
6.2 Initial rotor detection method
In a PMSM drive the detection of initial flux position is an important task. The initial
position of the rotor must be detected correctly in order to initialize the flux estimation
procedure. In case of wrong detection the control algorithm has incorrect information
and the rotor shaft can be rotated through few second in positive or negative direction.
This situation is not acceptable in any drive system. Therefore, for the stable starting of
PMSM drive without the temporary reversal rotation, the initial rotor position
estimation is proposed.
The simplest method to achieved the initial rotor flux position is based on the following
rule. For short time the stator winding is supplied by the DC voltage. It impress the DC
current, which generates the magnetic field. The permanent magnet of PMSM sets
accordance with this field line. This position of PM flux is used to set initial values for
the stator flux estimation algorithm.
This method is very simple and not complicate. However, has disadvantage that during
this process the rotor can be moved in unknown direction depending on:
•
position of PM before initial detection procedure,
•
direction of DC voltage supply into the motor phase.
In order to make the initial rotor flux position correct without any movement (at
standstill) the following algorithms can be used in the literature [78,90,93,95,99,101]
123
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
To estimate the initial rotor position before starting DTC-SVM control, two kind of the
rectangular pulsewise voltages are applied from the inverter to the motor:
•
one is the short pulsewise voltage,
•
and another is the longer one.
Short pulsewise voltage test
This test based on general principle that the three-phase winding inductances of PMSM
are a function of the mechanical rotor position. Therefore, from the line current
responses in stator oriented coordinates α , β under the short pulse wise voltage (see
Fig. 6.3) the position of PM can be estimated.
Figure 6.3. Voltage pulse wise during short time voltage test.
During the short time (100 µ s ) the vector V1 =(100) and opposite V4 =(011) is
generated by voltage source inverter. The achieved current responses in α , β system for
two type of PMSM during this test in respect to mechanical rotor position are presented
in Fig. 6.4.
a)
b)
Figure 6.4. Current components in stator oriented coordinates α , β under supplied voltage
vector to the motor for very short time : a) for Ld = Lq and b) for Lq > Ld .
124
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
The stator current trajectory can be divided in to eight sectors (see Fig. 6.5a).
5π
8
3π
8
112 .5o
67 .5o
Sector 3
7π
8
Sector 4
157 .5
o
Sector 2
−−
−+
++
Sector 5
202 .5o
+−
−+
−−
Sector 6
247 .5o
π
8
++
9π
8
22 .5o
Is
+−
Sector 8
Sector 1
15π
8 π
−
8
337 .5 o
292 .5o
11π
8
13π
8
Sector 7
Figure 6.5. Stator current trajectory.
Based on the measured response of phase currents in α , β
coordinates, the
sign( Is ) + and sign( Is ) − is calculated from following formulas:
sign( Is ) + = I sα + I sβ − Is
(6.1)
sign( Is ) − = I sα − I sβ − Is
(6.2)
The possible combinations of sign( Is ) + and sign( Is ) − are shown in Fig. 6.6:
sign( Is ) +
+
+
−
−
sign( Is ) −
+
−
−
+
Figure 6.6. Possible combination of sign( Is ) + and sign( Is ) − under short pulse supply.
Let us assuming, for example, the case where the sign( Is ) + and sign( Is ) − have positive
sign. The position γ m exist in the domain of −
π
8
~
π
8
or
7π 9π
and two estimated
~
8
8
position can be obtained.
The mathematical analysis of I sα , I sβ waveforms leads to following equations:
I sα = Is + ∆ Is cos 2γ m
125
(6.3)
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
I sβ = ∆ Is sin 2γ m
(6.4)
where Is is DC component in I sα and ∆ Is amplitude of fluctuated component.
The current components in α , β can be modeled as an average value of Is plus some
offset value ∆ Is , as a function of the mechanical position γ m . Fig. 6.4 shows the
current components given in Eq. (6.1-2), which are function of the phase angle 2γ m .
Solving those Eqs. (6.1 and 6.2) in respect to mechanical rotor position, two domains of
mechanical rotor position can be obtained as:
γ m1 =
γ m2 =
I sβ
(6.5)
2( I sα − Is )
I sβ
2( I sα − Is )
+π
(6.6)
The estimated rotor positions for other combination of sign( Is ) + and sign( Is ) − are
summarized in Table 6.7.
15π π
−
8
8
7π 9π
−
8
8
π 3π
−
8
8
9π 11π
−
8
8
3π 5π
−
8
8
11π 13π
−
8
8
5π 7π
−
8
8
13π 15π
−
8
8
I s α − I s + I sβ
γm
I s α − I s − I sβ
I sβ
+
+
+
−
2 ( I sα − I s )
I sβ
+π
2 ( I sα − I s )
I sα + I s
π
−
+
2 I sβ
4
−
−
−
−
+
I sα + I s
5π
+
2 I sβ
4
I sβ
π
+
2 ( I sα − I s )
2
I sβ
3π
+
2 ( I sα − I s )
2
+ Is
I
3π
− sα
+
2 I sβ
4
−
I sα + I s
7π
+
2 I sβ
4
Table 6.7. Mechanical rotor position calculations.
Long pulsewise voltage test
This test help us to choose the proper estimated value of mechanical rotor position from
two values calculated during the short pulse wise voltage test.
126
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
This test is based on the saturation effect of magnetic circuit. If the long pulse will send
in direction of north pole of PM, the current response will be slower, and, if current
response will be faster that the previous, it means that it is south pole.
6.3 Stator flux estimation methods
Flux estimation is an important task in implementation of high-performance DTC-SVM
motor drives. Vector control method of PMSM drive needs knowledge about actual
value of the stator flux magnitude and position as well electromagnetic torque. Also, the
flux estimation is needed to calculate the actual rotor speed for sensorless operation.
6.3.1 Overview
Many different technique has been developed for PMSM flux estimation [107].
Generally, they may be divided into two groups: open loop estimators and closed loop
estimators/observers. Most of these method are based on so called “current model” or
“voltage model” [110,113]. In fact closed loop estimators/observers are based on the
current or voltage model with an error correction loop, which drives error between two
flux models to zero in steady state. However, an observer has its own dynamics, is
sensitive to parameter changes, and has to be carefully designed for individual drives.
Therefore, for commercially manufactured drives is to complicated and impractical.
This is the reason why in this Chapter only open loop flux estimators will be
considered.
6.3.2 Current model based flux estimator
The block scheme of the current based flux model is presented in Fig. 6.7. It requires:
•
knowledge of PMSM machine inductance Ld , Lq ,
•
speed or position signal,
•
PMSM phase currents.
This kind of flux estimator served in experimental test as a master (standard) to run the
DTC-SVM scheme with speed sensor.
127
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
ΨPM
I sA
ABC
I sB
I sα
I sβ
αβ
I sd
αβ
dq
I sq
Ld
Lq
Ψsd
Ψsq
dq
αβ
Ψsα
Ψs
Ψsβ
θ Ψs
γm
Figure 6.7. Current model based stator flux estimator.
6.3.3 Voltage model based flux estimator with ideal integrator
The stator flux linkage can be obtained by using terminal voltages and currents. It is the
integral of terminal voltages minus the resistance voltage drop:
d Ψ sα
= (U sα − Rs I sα )
dt
d Ψ sβ
dt
= (U sβ − Rs I sβ )
(6.7)
(6.8)
However, at low speed (frequencies) some problems arise, when this technique is
applied, since the stator voltage becomes very small and the resistive voltage drops
become dominant, requiring very accurate knowledge of the stator resistance Rs and
very accurate integration. The stator resistance can vary due to temperature changes.
This effect can also be taken into consideration by using the thermal model of the
machine. Drifts and offsets can greatly influence the precision of integration. The
overall accuracy of the estimated flux linkage vector will also depend on the accuracy
of the monitored voltages and currents.
The most know classical voltage model obtains the flux components in stator
coordinates ( α , β ) by integrating the motor back electromotive force Esα , Esβ (see Fig.
6.8). The method is sensitive for only one motor parameter, stator resistance Rs .
However, the application of pure integrator is difficult because of dc drift and initial
value problems.
128
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
I sA
ABC
I sB
I sα
I sβ
αβ
U sA
U sB
ABC
αβ
Rs
Rs
U sα
Esα
Es β
U sβ
∫
∫
Ψsα
Ψs
Ψsβ
θ Ψs
Figure 6.8. Voltage model based estimator with ideal integrator.
There are proposed many improvements of the classical voltage model. Some of them
are presented bellow.
6.3.4 Voltage model based flux estimator with low pas filter
A common way to improve the stator flux voltage based model is to use a first-order
low-pass filter (LP) instead of the pure integrator. The equations (6.7 and 6.8) are
transferred to the form:
d Ψ sα
= (U sα − Rs I sα ) + Fc Ψ sα
dt
d Ψ sβ
dt
= (U sβ − Rs I sβ ) + Fc Ψ sβ
(6.9)
(6.10)
The block diagram of the estimator is presented in Fig. 6.9. Discrete time
implementation of the integrator becomes:
z Ψ sα ( z ) = Ψ sα ( z ) + (U sα − Rs I sα )Ts
(6.11)
z Ψ sβ ( z ) = Ψ sβ ( z ) + (U sβ − Rs I sβ )Ts
(6.12)
A LP filter does not give high accuracy at frequencies lower than cutoff frequency
ωc = 2π Fc . There will be errors both in the magnitude and in the phase angle. As
results, the proposed voltage estimator with LP filter can be used successfully only in a
limited speed range above cutoff frequency
129
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
I sA
ABC
I sB
I sα
I sβ
αβ
U sA
ABC
U sB
αβ
Rs
U sα -
U sβ
Fc
Rs
Esα
-
Es β
-
-
Ψsα
∫
Ψsβ
∫
Cartesian
To
Polar
Ψs
θ Ψs
Fc
Stator flux estimator (improved voltage model)
Figure 6.9. Voltage model based estimator with low-pass filter.
Discrete time implementation of the LP filter becomes:
z Ψ sα ( z ) = Ψ sα ( z ) + (U sα − Rs I sα )Ts + Fc Ψ sα ( z )
(6.13)
z Ψ sβ ( z ) = Ψ sβ ( z ) + (U sβ − Rs I sβ )Ts − Fc Ψ sα ( z )
(6.14)
6.3.5 Improved voltage model based flux estimator
Many other methods were developed in order to eliminate dc-offset and initial values
problems [107]. In general, the output Y of these new integrators (Fig. 6.10) is
expressed as:
Y=
ωc
1
X+
Ylim
s + ωc
s + ωc
(6.15)
Where X is the input and Y is output of the integrator respectively. The Ylim is a
compensation signal used as a feedback and ωc is cutoff frequency.
1
s + ωc
Y
Compensation
signal
X
ωc
s + ωc
lim
Ylim
− lim
Figure 6.10. Improved integration method with saturation block.
130
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
The first part of the equation represents a LP filter. The second part realizes a feedback,
which is used to compensate the error in the output. The block diagram of new
integration algorithm with saturation block is shown in see Fig. 6.11.
I sA
ABC
I sB
I sα
U sA
U sB
ωc
I sβ
αβ
ABC
αβ
s + ωc
Rs
U sα
U sβ
Rs
Esα
Es β
1
s + ωc
lim
Ylim
− lim
Ψα _ comp
1
s + ωc
Ψsα
Ψs
Ψsβ
θ Ψs
Ψ sβ _ comp
ωc
s + ωc
lim
Ylim
− lim
Figure 6.11. Full block diagram of voltage model based estimator with saturation block on the
α , β components.
The main task of saturation block is to stop the integration when the output signal Ψ sα
or Ψ sβ exceed the reference value of stator flux amplitude. Please note that if the
compensation signal is set to zero, the improved integrator represents a first-order LPfilter. If the compensation signal Ψ sα _ comp or Ψ sβ _ comp is not zero the improved
integrator operates as a pure integrator.
Discrete time implementation of the improved integrator becomes:
z Ψ sα ( z ) = Ψ sα ( z ) + (U sα − Rs I sα )Ts +
z Ψ sβ ( z ) = Ψ sβ ( z ) + (U sβ − Rs I sβ )Ts +
ωc
Ts
ωc
Ts
(Ψ sα _ lim ( z ) − Ψ sα ( z ))
(6.16)
(Ψ sβ _ lim ( z ) − Ψ sβ ( z ))
(6.17)
The output of saturation block can be described as:
⎧Ψ sαβ ( z ) if (Ψ sαβ (z))<lim
Ψ sαβ _ lim ( z ) = ⎨
if (Ψ sαβ (z))>=lim
⎩ lim
131
(6.18)
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
Where lim is the limited value. Please note that lim should be set at reference stator
flux amplitude lim = Ψ s _ ref equal Ψ PM .
From Eq. 6.18 it can be observed that when one of the stator flux linkage components
Ψ sα or Ψ sβ exceeds the limit, it causes in distortion of the output waveform.
6.4 Electromagnetic torque estimation
The PMSM motor output torque is calculated based on the equations (2.51), (2.52),
(2.55), (2.58) presented in Chapter 2, which for stator oriented coordinate system can be
written as follows:
Me =
3
pb (Ψ sα I sβ − Ψ sβ I sα )
2
(6.19)
It can be seen that calculated torque is dependent on the current measurement accuracy
and stator flux estimation method. In practice current measurement is performed with
high accuracy ( ≤ 1% with 150kHz frequency bandwidth) using, for example, LEM
sensors.
6.5 Rotor speed estimation methods
6.5.1 Overview
High performance operation of motion sensorless PMSM drives depends mainly on
accurate knowledge of rotor PM flux magnitude, position and speed. The rotor position
estimation methods can be classified into two major groups:
•
motor model based,
•
rotor saliency based techniques.
The rotor saliency based approach is suitable only for the Interior PMSM (see Fig. 1.2 c
and d). Motor model based approach detect the back EMF vector, which includes
information about position and speed, using either open loop models/estimators
[81,85,86] or closed loop estimators/observers [70, 73,74,96,97,100]. Also adaptive
observers [72,92,98,83] and Extended Kalman Filters (EKF) [67,73] have been
proposed for motor position and speed estimation. These methods, however, are
computationally intensive and require careful design and proper initialization.
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Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
Therefore, for commercially manufactured drives are impractical and further a simple
open loop based techniques will be considered.
6.5.2 Back electromotive force (BEMF) technique
This technique uses the back electromotive force to estimate the rotor speed [70]. The
velocity signal could be integrated to generate a position estimate. However, this signal
is sensitive to parameter variations and tends to drift and have offset problem. Another
problem with using BEMF to estimate position is that at zero speed the BEMF goes to
zero and at low speed the signal to noise ratio can not be ignored.
6.5.3 Stator flux based technique
Generally, the calculation of rotor speed is based on the simple relationship:
θ r = θ Ψs − δ Ψs ,
(6.20)
where θ r is electrical position, θ Ψs is stator flux position and δ Ψs is torque angle.
After differentiation equation (6.20) and taking into account that θ r = pbγ m the
mechanical speed of PMSM rotor can be expressed as:
dδ ⎞
⎛ dθ
Ω m = ⎜ Ψs − Ψ ⎟ / pb ,
dt ⎠
⎝ dt
where Ω Ψs =
(6.21)
dθ Ψs
is angular speed of stator flux vector and δ Ψ is torque angle.
dt
As we can observe form equation (6.21) in order to calculate the mechanical rotor speed
it is necessary to calculate separately two components. One of them is angular speed of
stator flux vector Ω Ψs and the second one is change of the load angle
6.12).
133
dδ Ψ
(see Fig.
dt
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
dδ Ψ
dt
I sα
Ψ sα
I sβ
U sα
Ω Ψs =
Ψ sβ
dθ Ψ s
dt
U sβ
−
1
pb
Ωm
Figure 6.12. Block diagram of stator flux vector based rotor speed estimator.
The synchronous speed Ω Ψs is calculated based on the stator flux estimator:
θ Ψs = arctg (
Ψ sβ
Ψ sα
)
(6.22)
and the calculation of Ω Ψs can be done as:
Ω Ψs =
dθ Ψs θ Ψs ( k ) − θ Ψs ( k −1)
=
dt
Ts
(6.23)
The estimation of the synchronous speed Ω Ψs based on the derivative of the position of
stator flux space vector can be modified taking in to account equation (6.22), which
finally gives [12]:
Ω Ψs =
Ψ sα
d Ψ sβ
− Ψ sβ
d Ψ sα
dt
dt
Ψ sα 2 + Ψ sβ 2
(6.24)
Digital implementation of equation (6.24) can be written as:
Ω Ψs ( k ) =
Ψ sα ( k −1) Ψ sβ ( k ) − Ψ sβ ( k −1) Ψ sα ( k )
2
Ψ s Ts
(6.25)
Also from equation (2.27b) in stator flux coordinate system the synchronous speed Ω Ψs
can be obtained as:
Ω Ψs =
U sy − Rs I sy
Ψs
=
Esy
Ψs
134
(6.26)
Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
Calculation of
dδ Ψ
dt
Based on the flux-current (Eq. 2.29b) and torque (Eq. 2.58) equations in stator flux
coordinates under consideration that for surface PMSM machine Ld = Lq and making
assumption that for small changes of torque angle δ Ψ , the sin δ Ψ = δ Ψ , the equations can
be written as:
0 = Ls I sy − Ψ PM sin δ Ψ
Me =
(6.27)
3
pb Ψ PM I sy
2
(6.28)
the torque angle δ Ψ can be calculated as:
δΨ =
Further, the
Ls I sy
Ψ PM
=
2M e Ls
3 pb Ψ s Ψ PM
(6.29)
dδ Ψ
is calculated as:
dt
dδ Ψ δ Ψ ( k ) − δ Ψ ( k −1)
=
dt
Ts
(6.30)
Figure 6.13. Simulated oscillograms of rotor speed estimation according to block scheme from
Fig. 6.12 (in Saber). From the top: synchronous speed Ω Ψs , the
dδ Ψ
signal, the measured and
dt
estimated rotor speed, the measured and estimated electromagnetic torque.
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Sensorless Speed Direct Torque Control with Space Vector Modulation (DTC-SVM)
The speed estimation problem is still open, especially at low and zero speed operations.
The accuracy of presented method depends on accuracy of applied stator flux estimation
and differentiation algorithm. It allows, however, for robust start, closed loop operation
above 10% of nominal speed, and braking the drive to zero speed.
6.5 Summary
The main problems associated with PMSM sensorless speed operation are presented
in this Chapter. For robust start of PMSM without the temporary rotor reversal a
special initialization algorithm has been used. This algorithm performs two test:
short and the longer voltage generated by the PWM inverter. The used speed
estimation algorithm is based on stator flux vector and torque angle estimation and
does not operate accurately around zero speed region. However, it allows robust
start and closed speed operation in the speed range above 10% of nominal speed.
The effectiveness of DTC-SVM with and without motion sensor has been proved by
simulation and experimental results (see Chapter 7)
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DSP implementation of DTC-SVM control
Chapter 7 DSP IMPLEMENTATION OF DTC-SVM CONTROL
7.1 Description of the laboratory test-stand
The basic structure of laboratory setup is presented in Fig. 7.1 and the photo of
laboratory setup is shown in Fig. 7.2. The motor setup consists of 3kW permanent
magnet synchronous motor and DC motor, which is used as a load. The PMSM machine
is supplied by PWM inverter, which is controlled by digital signal processor (DSP)
based on DS1103 board. The voltage inverter is supplied from three-phase diodes
rectifier. The DSP interface is used in order to separate the high power from the low
power circuit (computer part). Please note that the DS1103 is inserted inside the PC
computer.
Figure 7.1. Block scheme of laboratory setup.
Figure 7.2. Laboratory setup. 1-voltage inverter, 2-control for DC motor, 3- PMSM machine, 4
– DSP interface.
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DSP implementation of DTC-SVM control
The detailed power circuit of the laboratory setup is shown in Fig. 7.3.
Rectifier
three-phase
supply
network U
A
D1
D3
UB
Inverter
T1
D5
D1
T3
D3
T5
CF
UC
D2
D4
T2
D6
D2
T4
D4
T6
D5 current
sensors
A
B
C
PMSM
D6
speed or position
sensor
SA SA
SC SC
SB SB
DTC-SVM
Reference speed
microprocessor
DS1103
Figure 7.3. Power circuit of the laboratory setup.
In presented system the actual two currents and DC link voltage are measured by LEM
sensors and processed by A/D converter. The rotor position and speed are obtained with
an encoder of 2500 pulse per revolution. All internal data of DSP can be sent through a
D/A converter and displayed in the scope. All data of the PMSM and inverter are given
in the Appendices.
The control algorithm for PMSM machine was written in C language and was
implemented inside the processor. Also, a simple dead-time compensation method and
voltage drop on the semiconductor elements are implemented.
The phase voltage of the motor are reconstructed inside the processor using the
measured DC-link voltage and duty cycles of PWM for each phases. Motor and PI
controller model are given in Appendices.
Various tests have been carried out in order to investigate the drive performance and to
characterize the steady-state and transient behavior.
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DSP implementation of DTC-SVM control
Control Desk experiment software run on the PC computer provides all function for
controlling, monitoring and automation of real-time experiments and makes the
development of controllers more effective. A Control Desk experiment layout for
control the PMSM machine using DTC-SVM control method is presented in Fig. 7.4.
Figure 7.4. Performed Control Desk experimental layout for control of PMSM drive.
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DSP implementation of DTC-SVM control
7.2 Steady state behaviour
The experimental steady state no load operation at 25Hz stator frequency for
conventional ST-DTC (Fig. 4.10) and DTC-SVM (Fig. 5.46b) control is presented in
Fig. 7.5. The sampling time has been set at Ts = 200 µ s for DTC-SVM and Ts = 25µ s for
hysteresis based ST-DTC method, respectively.
a)
b)
Figure 7.5. No load experimental steady state oscillograms at stator frequency 25Hz.
(a) ST-DTC for Ts = 25µ s (b) DTC-SVM for Ts = 200 µ s .
From the top: line to line voltage, phase current, amplitude of stator flux, motor torque.
As we can observed from Fig. 7.5a the motor phase current characterized by high
current ripples. This is mainly because the inductances of the PMSM is smaller than an
equivalent power induction motor IM. In order to reduce the current ripples the
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DSP implementation of DTC-SVM control
sampling time of microcontroller should be decrease. However there is hardware
limitation. The loaded program to microprocessor can not run faster.
Using the space vector modulation (SVM) based DTC much better results can be
obtained. Note, that in spite of lower switching frequency DTC-SVM guarantees less
current and torque ripple. This is mainly because contrary to hysteresis operation with
SVM operation, the inverter output voltage is unipolar (compare output voltage
waveform in Fig. 7.5a with Fig. 7.5b). This also reduces semiconductor device voltage
stress and instantaneous current reversal in DC link.
The presented experimental results (Fig. 7.6-7.9) are measured in the system with
measured speed taken to the feedback. These investigations have been performed to
show the behaviour of the DTC-SVM system without influence of the speed estimation.
In Fig. 7.6 and Fig. 7.7 steady state operation for different values of the mechanical
speed and load torque are shown.
Figure 7.6. Experimental steady state operation of PMSM controlled via DTC-SVM with the
encoder speed signal taken to the feedback ( Ω m = 300rpm , M l = 0 ).
From the top: 1) amplitude of stator flux (0.25Wb/div), 2) electromagnetic torque (10Nm/div),
3) line to line voltage (1000V/div), 4) stator phase current (10A/div).
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DSP implementation of DTC-SVM control
Figure 7.7. Experimental steady state operation of PMSM controlled via DTC-SVM with the
encoder speed signal taken to the feedback ( Ω m = 300rpm , M l = 10 Nm -50% of nominal
torque). From the top: 1) amplitude of stator flux (0.25Wb/div), 2) electromagnetic torque
(10Nm/div), 3) line to line voltage (1000V/div), 4) stator phase current (10A/div).
Figure 7.8. Experimental steady state operation of PMSM controlled via DTC-SVM with the
encoder speed signal taken to the feedback ( Ω m = 1500rpm , M l = 0 ). From the top: 1)
amplitude of stator flux (0.25Wb/div), 2) electromagnetic torque (10Nm/div), 3) line to line
voltage (1000V/div), 4) stator phase current (10A/div).
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DSP implementation of DTC-SVM control
Figure 7.9. Experimental steady state operation of PMSM controlled via DTC-SVM with the
encoder speed signal taken to the feedback ( Ω m = 1500rpm , M l = 10 Nm -50% of nominal
torque). From the top: 1) amplitude of stator flux (0.25Wb/div), 2) electromagnetic torque
(10Nm/div), 3) line to line voltage (1000V/div), 4) stator phase current (10A/div).
7.3 Dynamic behaviour
The experimental results of flux and torque control loop obtained in dynamic states for
PMSM machine controlled via two different DTC-SVM schemes are presented.
7.3.1 Flux and torque control loop
Cascade DTC–SVM control scheme (Fig. 5.46a)
In order to show behaviour of the system the dynamic testes for the flux and torque
controllers has been carried out for sampling time, Ts = 200 µ s . It corresponds to
switching frequency f s = 5kHz . Please note that the flux digital controller parameters
were selected according to Table 5.1 and the torque digital controller parameters were
selected from Table 5.2. (see Chapter 5.2).
In Fig. 7.10 stator flux tracking performance is presented. This result is comparable
with simulation results presented in Fig. 5.15.
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DSP implementation of DTC-SVM control
Figure 7.10. Experimental flux tracking performance of PM synchronous motor for zero speed
at sampling time Ts = 200 µ s . Reference flux from70% to 100% of nominal value . From the
top: 1-reference stator flux amplitude(0.05Wb/div), 2- stator flux amplitude (0.05Wb/div), 3electromagnetic torque (2Nm/div), 4- motor phase current (10A/div).
In Fig. 7.11 torque tracking performance is presented. The achieved result is
comparable with simulation results presented in Fig. 5.21b.
Figure 7.11. Experimental torque tracking performance of PM synchronous motor for zero
speed at sampling time Ts = 200 µ s . Reference torque from 0 to nominal value. From the top:1reference torque (10Nm/div), 2 - estimated torque (10Nm/div), 3 - stator flux amplitude
(0.1Wb/div), 4- motor phase current (10A/div)
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DSP implementation of DTC-SVM control
Figure 7.12. Experimental torque tracking performance of PM synchronous motor for zero
speed (zoom) at sampling time Ts = 200µ s . Reference torque from 0 to nominal value. From
the top:1- reference torque (10Nm/div), 2 - estimated torque (10Nm/div), 3 - stator flux
amplitude (0.1Wb/div), 4- motor phase current (10A/div).
All experimental results presented in Fig. 7.10-7.12 confirm very well proper and stable
operation of flux and torque control loops for cascade DTC-SVM structure.
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DSP implementation of DTC-SVM control
Influence of sampling time for torque control loop in cascade DTC-SVM
The influence of sampling time on experimental torque tracking performance is
illustrated in Fig. 7.13-7.15. The dynamic test has been carried out for the same
condition ( Ω m = 0rpm ) as for simulation shown in Fig. 5.24. The controller parameters
has been set according to Table 5.2. In all oscilograms we may see proper operation of
the torque control loop for different sampling time used in practice.
Figure 7.13. Experimental torque tracking performance of PM synchronous motor for zero
speed at sampling time Ts = 100µ s ( f s = 10kHz ). Reference torque from 0 to nominal value.
From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div)
Figure 7.14. Experimental torque tracking performance of PM synchronous motor for zero
speed at sampling time Ts = 200µ s ( f s = 5kHz ). Reference torque from 0 to nominal value.
From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div)
146
DSP implementation of DTC-SVM control
Figure 7.15. Experimental torque tracking performance of PM synchronous motor for zero
speed at sampling time Ts = 400 µ s ( f s = 2.5kHz ). Reference torque from 0 to nominal value.
From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div).
Parallel structure of DTC–SVM scheme (Fig. 5.46b)
Dynamic testes for the flux and torque controllers were carried out for sampling time
Ts = 200 µ s , which corresponds to switching frequency f s = 5kHz . Please not that the
digital flux controller parameters were selected according to Table 5.3 and the digital
torque controller parameters were selected from Table 5.4. (see Chapter 5.3).
In Fig. 7.16 stator flux tracking performance is presented. This result is comparable
with simulation results presented in Fig. 5.33 for flux and Fig. 5.45 for torque loop,
respectively.
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DSP implementation of DTC-SVM control
Figure 7.16. Experimental flux tracking performance of PM synchronous motor for zero speed
at sampling time Ts = 200 µ s . Reference flux from70% to 100% of nominal value . From the
top: 1-reference stator flux amplitude(0.05Wb/div), 2- stator flux amplitude (0.05Wb/div), 3electromagnetic torque (2Nm/div), 4- motor phase current (10A/div).
It can be observed that achieved result is comparable with simulation results presented
in Fig. 5.34.
Figure 7.17. Experimental torque tracking performance of PM synchronous motor for zero
speed at sampling time Ts = 200 µ s . Reference torque from 0 to nominal value. From the top:1reference torque (10Nm/div), 2 - estimated torque (10Nm/div), 3 - stator flux amplitude
(0.1Wb/div), 4- motor phase current (10A/div)
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DSP implementation of DTC-SVM control
Figure 7.18. Experimental torque tracking performance of PM synchronous motor for zero
speed (zoom) at sampling time Ts = 200µ s . Reference torque from 0 to nominal value. From
the top:1- reference torque (10Nm/div), 2 - estimated torque (10Nm/div), 3 - stator flux
amplitude (0.1Wb/div), 4- motor phase current (10A/div).
The achieved result is comparable with simulation results presented in Fig. 5.42b.
Experimental results presented in Fig. 7.16-7.18 confirm very well the effectiveness of
controller design and proper operation of flux and torque control loops for DTC-SVM
structure.
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DSP implementation of DTC-SVM control
Influence of sampling time for torque control loop in parallel DTC-SVM
The influence of sampling time on experimental torque tracking performance is
illustrated in Fig. 7.19-7.21. The dynamic test has been carried out for the same
condition ( Ω m = 0rpm ) as for simulation shown in Fig. 5.45. The controller parameters
has been set according to Table 5.4. In all oscilograms we may see proper operation of
the torque control loop for different sampling time used in practice.
Figure 7.19. Experimental torque tracking performance of PM synchronous motor for zero
speed at sampling time Ts = 100µ s . Reference torque from 0 to nominal value. From the top:1reference torque (4Nm/div), 2 - estimated torque (4Nm/div)
Figure 7.20. Experimental torque tracking performance of PM synchronous motor for zero
speed at sampling time Ts = 200µ s . Reference torque from 0 to nominal value. From the top:1reference torque (4Nm/div), 2 - estimated torque (4Nm/div).
150
DSP implementation of DTC-SVM control
Figure 7.21. Experimental torque tracking performance of PM synchronous motor for
zero speed at sampling time Ts = 400 µ s . Reference torque from 0 to nominal value.
From the top:1- reference torque (4Nm/div), 2 - estimated torque (4Nm/div).
In this Chapter the results of experimental verification 3kW PMSM drive with two
DTC-SVM schemes has been presented. As shown the drive performance confirms
applied design methodology. The performance of both cascade and parallel DTC-SVM
control structure are similar. However, parallel structure has been selected for industrial
manufacturing because of :
•
less noisy control algorithm (differentiation required in cascade structure –
equation (5.6) is eliminated),
•
stator flux control in closed loop,
•
the same structure can be used for IM and PMSM control (universal control for
AC motors).
7.3.2 Speed control loop
0peration with speed sensor
Dynamic testes for the speed control loop were measured for sampling time Ts = 200 µ s .
Please note that the digital speed controller parameters were selected according to Table
5.5 (see Chapter 5.4). In Fig. 7.22-7.26 rotor speed tracking performance are presented.
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DSP implementation of DTC-SVM control
Figure 7.22. Experimental start up and breaking to zero speed of PMSM motor controlled via
DTC-SVM with the encoder speed signal taken to the feedback ( Ω m = 0rpm → 300rpm ). From
the top: 1- reference speed (180rpm/div), 2- measured speed (180rpm/div), 3 - electromagnetic
torque (20Nm/div), 4- motor phase current (20A/div).
Figure 7.23. Experimental speed reversal for PMSM motor controlled via DTC-SVM with the
encoder speed signal taken to the feedback ( Ωm = −300rpm → 300rpm ). From the top: 1reference speed (360rpm/div), 2- measured speed (360rpm/div), 3 - electromagnetic torque
(20Nm/div), 4- motor phase current (20A/div).
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DSP implementation of DTC-SVM control
Figure 7.24. Experimental start up and breaking to zero speed of PMSM motor controlled via
DTC-SVM with the encoder speed signal taken to the feedback ( Ω m = 0rpm → 1500rpm ).From
the top: 1- reference speed (900rpm/div), 2- measured speed (900rpm/div), 3 - electromagnetic
torque (20Nm/div), 4- motor phase current (20A/div).
Figure 7.25. Experimental speed reversal for PMSM motor controlled via DTC-SVM with the
encoder speed signal taken to the feedback ( Ω m = −1500rpm → 1500rpm ). From the top: 1reference speed (1800rpm/div), 2- measured speed (1800rpm/div), 3 - electromagnetic torque
(20Nm/div), 4- motor phase current (20A/div).
153
DSP implementation of DTC-SVM control
Figure 7.26. Experimental speed reversal for PMSM motor controlled via DTC-SVM with the
encoder speed signal taken to the feedback ( Ωm = −1200rpm → 1200rpm ). From the top: 1stator flux component Ψ sα (0.25Wb/div), 2- measured speed (360rpm/div), 3 - electromagnetic
torque (20Nm/div), 4- motor phase current (20A/div).
Figure 7.27. Experimental response to load torque step change of PMSM motor controlled via
DTC-SVM with the encoder speed signal taken to the feedback at
Ωm = 0rpm .
( M l = 0 Nm → 10 Nm ) From the top: 1- reference speed (180rpm/div), 2- measured speed
(180rpm/div), 3 - electromagnetic torque (5Nm/div), 4- motor phase current (10A/div).
154
DSP implementation of DTC-SVM control
Figure 7.28. Experimental response to load torque step change of PMSM motor controlled via
DTC-SVM with the encoder speed signal taken to the feedback at Ω m = 300rpm .
( M l = 0 Nm → 10 Nm ).From the top: 1- reference speed (180rpm/div), 2- measured speed
(180rpm/div), 3 - electromagnetic torque (5Nm/div), 4- motor phase current (10A/div).
Figure 7.29. Experimental response to load torque step change of PMSM motor controlled via
DTC-SVM with the encoder speed signal taken to the feedback at Ω m = 1500rpm .
( M l = 0 Nm → 10 Nm ). From the top: 1- reference speed (900rpm/div), 2- measured speed
(900rpm/div), 3 - electromagnetic torque (5Nm/div), 4- motor phase current (10A/div).
155
DSP implementation of DTC-SVM control
Experimental results presented in Figures 7.22-7.29 well confirm the effectiveness of
developed controller synthesis methodology and proper operation of speed control loop
for parallel DTC-SVM structure.
Sensorless speed operation
Dynamic tests for the speed control loop without motion sensor were measured for
sampling time Ts = 200 µ s . Please note that the digital speed controller parameters were
selected exactly like for operation with speed sensor according to Table 5.5 (see
Chapter 5.4).
The results of speed estimator dynamic test are presented in Fig. 7.30. In this test speed
controller operates with the encoder signal in feedback and speed estimator works in
open loop fashion.
Figure 7.30. Experimental dynamic test of the speed estimation. Speed reversal Ω m = ±300rpm
From the top: 1- reference speed (180rpm/div), 2- measured speed (180rpm/div), 3 - estimated
speed (180rpm/div), 4- error of estimated speed (5%/div).
The typical dynamic performance tests of sensorless DTC-SVM drive has been
illustrated in Fig. 7.31-7.36. Start up and breaking to zero speed for different speed level
are shown in Fig. 7.31 and 7.33. Also, the speed reversal for low (Fig. 7.32) and
nominal (Fig. 7.34) speed are presented. The half load torque step change tests are
shown in Fig. 7.36 and 7.37.
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DSP implementation of DTC-SVM control
Figure 7.31. Experimental start up and breaking to zero speed of PMSM motor controlled via
DTC-SVM with the estimated speed signal taken to the feedback ( Ω m = 0rpm → 300rpm ).
From the top: 1- reference speed (180rpm/div), 2- measured speed (180rpm/div), 3 electromagnetic torque (20Nm/div), 4- motor phase current (20A/div).
Figure 7.32. Experimental speed reversal for PMSM motor controlled via DTC-SVM with the
estimated speed signal taken to the feedback ( Ω m = −300rpm → 300rpm ). From the top: 1reference speed (360rpm/div), 2- measured speed (360rpm/div), 3 - electromagnetic torque
(20Nm/div), 4- motor phase current (20A/div).
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DSP implementation of DTC-SVM control
Figure 7.33. Experimental speed step response for PMSM motor controlled via DTC-SVM with
the estimated speed signal taken to the feedback ( Ω m = 0rpm → 1500rpm ).From the top: 1reference speed (900rpm/div), 2- measured speed (900rpm/div), 3 - electromagnetic torque
(20Nm/div), 4- motor phase current (20A/div).
Figure 7.34. Experimental speed reversal for PMSM motor controlled via DTC-SVM with the
estimated speed signal taken to the feedback ( Ω m = −1500rpm → 1500rpm ). From the top: 1reference speed (1800rpm/div), 2- measured speed (1800rpm/div), 3 - electromagnetic torque
(20Nm/div), 4- motor phase current (20A/div).
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DSP implementation of DTC-SVM control
Figure 7.35. Experimental speed reversal for PMSM motor controlled via DTC-SVM with the
estimated speed signal taken to the feedback ( Ωm = −1200rpm → 1200rpm ). From the top: 1stator flux component Ψ sα (0.25Wb/div), 2- measured speed (360rpm/div), 3 - electromagnetic
torque (20Nm/div), 4- motor phase current (20A/div).
Figure 7.36. Experimental response to load torque step change of PMSM motor controlled via
DTC-SVM with the estimated speed signal taken to the feedback at Ωm = 300rpm .
( M l = 0 Nm → 10 Nm ). From the top: 1- reference speed (180rpm/div), 2- measured speed
(180rpm/div), 3 - electromagnetic torque (10Nm/div), 4- motor phase current (10A/div).
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DSP implementation of DTC-SVM control
Figure 7.37. Experimental response to load torque step change of PMSM motor controlled via
DTC-SVM with the estimated speed signal taken to the feedback at Ω m = 1500rpm .
( M l = 0 Nm → 10 Nm ). From the top: 1- reference speed (900rpm/div), 2- measured speed
(900rpm/div), 3 - electromagnetic torque (10Nm/div), 4- motor phase current (10A/div).
Experimental results presented in Figures 7.31-7.37 confirm very well the effectiveness
of speed estimation algorithm of speed control loop for parallel DTC-SVM structure.
160
Summary and closing conclusion
Chapter 8
SUMMARY AND CLOSING CONCLUSIONS
This thesis studied basic problems related to selection, investigation and implementation
of the PWM inverter-fed permanent magnet synchronous motor (PMSM) drives suitable
for serial manufacturing. The selected method should provide: robust start and operation
in wide speed control range including zero speed, with and without mechanical motion
sensor; guarantee good and repeatable parameters of PMSM drive for wide power range
(1-100kW). The control and protection algorithm should be implemented in simple and
cheap microprocessor.
To solve so formulated general task several related problems had to be solved. At first,
the space vector based mathematical description and static characteristic of PMSM
under different control modes were studied (Chapter 2). Secondly, the three phase
voltage source inverter model including nonlinearities (dead time, semiconductor
voltage drop, DC link pulsation) and pulse with modulation PWM techniques were
presented (Chapter 3). Next, based on the study of most important high performance
control methods as field oriented control (FOC), and direct torque control (DTC), the
method called direct torque control with space vector modulator (DTC-SVM) has been
selected for further consideration (Chapter 4). This methods combines main advantage
of FOC (space vector modulator and fixed switching frequency) and DTC (simple
structure, rotor parameter independent), as well as eliminates disadvantages like:
coordinate transformation, the need of internal current control loops, high sampling
time, high torque and current ripple at steady state operation, etc.
Consequently the most important contribution of this thesis is included in the Chapter 5,
where the two basic variants of DTC-SVM schemes: series (cascade) and parallel
structures of flux and torque controllers are presented (Fig. 5.2 and Fig. 5.25). Also, a
systematic methodology of digital controller design for these both DTC-SVM variants
have been given. This methodology has been verified by Simulink (using simplified
discrete transfer function) and Saber (using full motor and inverter model) simulation
studies. The influence of sampling time selection on controller design has also been
discussed.
The main problems associated with PMSM sensorless speed operation are presented in
the Chapter 6. It should be noted that PMSM differ from IM drives mainly in:
•
PMSM parameters strongly depend on construction ,
161
Summary and closing conclusion
•
position of PM flux has to be known prior to start up to achieve smooth
operation.
Therefore, for robust starting of PMSM without the temporary rotor reversal a simple
initialization algorithm has been used. This algorithm performs two tests: short and the
longer voltage pulses generated by the PWM inverter. The used speed estimation
algorithm is based on stator flux vector and torque angle estimation and does not
operate accurately in zero speed region. However, it allows robust start and closed
speed operation in the speed range above 10% of nominal speed. For application where
high performance operation around zero speed are required, the DTC-SVM drive with
motion sensor (encoder) is recommended. The effectiveness of DTC-SVM scheme with
and without motion sensor has been proved by simulation and experimental results
(Chapter 7).
Simulation study and experimental results have shown that from two variants of DTCSVM schemes the parallel structure is more flexible in torque and flux controller
design. Also, because of lack of the differentiation in the main control path (compare
Fig. 5.2 and Fig. 5.25), it is less sensitive to noise which inherently associates signal
processing in power electronic converters. Therefore, the parallel structure has been
selected for industrial manufacturing and implemented on digital signal processor
(DSP).
Thanks to direct torque and flux control structure the described control is suitable to
almost all – industrial applications including electrical vehicles (for example hybrid
cars).
Finally, it should be stressed that the developed system was brought into serial
production. Presented algorithm DTC-SVM has been used in new generation of inverter
drives produced by Polish company Power Electronic Manufacture – “TWERD”,
Toruń.
162
Appendices
APPENDICES
A1 Rotor and stator of PMSM machine
A1.1. View of rotor (on the left side) and stator armature (on the right side) of PMSM.
A2 Basic transformation
β
y
KB
K
θK Kβ
Ky
x
Kx
θK
Kα
α
KA
KC
Fig. A2.1. Space vector representation in stationary α , β coordinates and synchronous
rotating x, y coordinates.
A, B, C ⇒ x, y
2
K x = [ K A cos θ K + K B cos(2π / 3 − θ K ) + K C cos(2π / 3 + θ K )]
3
2
K y = [ K A cos(π / 2 + θ K ) + K B cos(2π / 3 − (π / 2 + θ K )) + K C cos(2π / 3 + π / 2 + θ K )]
3
163
Appendices
2
K x = [U A cosθ K + K B cos(θ K − 2π / 3) + K C cos(θ K + 2π / 3)]
3
2
K y = [− K A sin θ K − K B sin(θ K − 2π / 3) − K C sin(θ K + 2π / 3)]
3
⎡ K x ⎤ ⎡ cosθ K
⎢K ⎥ = ⎢
⎣ y ⎦ ⎣ − sin θ K
⎡KA ⎤
cos(θ K + 2π / 3) ⎤ ⎢ ⎥
KB
− sin(θ K − 2π / 3) − sin(θ K + 2π / 3) ⎥⎦ ⎢ ⎥
⎢⎣ K C ⎥⎦
cos(θ K − 2π / 3)
A, B, C ⇒ α , β
θK = 0
2
1
1
( K A − K B − KC )
3
2
2
2 3
3
1
( K B − KC )
Kβ = (
KB −
KC ) =
3 2
2
3
1
1 ⎤
⎡
⎡KA ⎤
−
−
1
⎡ Kα ⎤ 2 ⎢
2
2 ⎥⎢ ⎥
⎥ KB
⎢K ⎥ = ⎢
⎢ ⎥
3
3
3
⎢
⎥
β
⎣ ⎦
⎢⎣ K C ⎥⎦
−
0
⎢⎣
2
2 ⎥⎦
Kα =
x, y ⇒ A, B, C
K A = K x cos θ K + K y cos(π / 2 + θ K ) = K x cosθ K − K y sin θ K
K B = K x cos(2π / 3 − θ K ) + K y cos(2π / 3 − (π / 2 + θ K )) = K x cos(θ K − 2π / 3) − K y sin(θ K − 2π / 3)
K C = K x cos(2π / 3 + θ K ) + K y cos(2π / 3 + (π / 2 + θ K )) = K x cos(θ K + 2π / 3) − K y sin(θ K + 2π / 3)
cos θ K
− sin θ K
⎡KA ⎤ ⎡
⎤
⎢ K ⎥ = ⎢ cos(θ − 2π / 3) − sin(θ − 2π / 3) ⎥ ⎡ K x ⎤
K
K
⎢ B⎥ ⎢
⎥ ⎢K y ⎥
⎢⎣ K C ⎥⎦ ⎢⎣ cos(θ K + 2π / 3) − sin(θ K + 2π / 3) ⎥⎦ ⎣ ⎦
α , β ⇒ A, B, C
θK = 0
K A = Kα
1
3
K B = − Kα +
Kβ
2
2
1
3
K C = − Kα −
Kβ
2
2
164
Appendices
⎡
⎢ 1
⎡KA ⎤ ⎢
⎢ K ⎥ = ⎢− 1
⎢ B⎥ ⎢ 2
⎢⎣ K C ⎥⎦ ⎢
⎢− 1
⎢⎣ 2
⎤
0 ⎥
⎥
3 ⎥ ⎡ Kα ⎤
⎢ ⎥
2 ⎥ ⎣Kβ ⎦
⎥
3⎥
−
2 ⎥⎦
x, y ⇒ α , β
Kα = K x cos θ K + K y cos(π / 2 + θ K ) = K x cos θ K − K y sin θ K
K β = K x cos(π / 2 − θ K ) + K y cosθ K = K x sin θ K + K y cosθ K
⎡ Kα ⎤ ⎡cos θ K
⎢K ⎥ = ⎢
⎣ β ⎦ ⎣ sin θ K
− sin θ K ⎤ ⎡ K x ⎤
⎢ ⎥
cosθ K ⎥⎦ ⎣ K y ⎦
α , β ⇒ x, y
K x = Kα cos θ K + K β cos(π / 2 − θ K ) = Kα cos θ K + K β sin θ K
K y = Kα cos(π / 2 + θ K ) + K β cos θ K = − Kα sin θ K + K β cos θ K
⎡ K x ⎤ ⎡ cos θ K
⎢K ⎥ = ⎢
⎣ y ⎦ ⎣ − sin θ K
sin θ K ⎤ ⎡ Kα ⎤
⎢ ⎥
cosθ K ⎥⎦ ⎣ K β ⎦
A3 Model of PM synchronous motor
# This template models the permanent magnet synchronous motor(pmsm)
# t1,t2 and t3 are motor input terminals
# rotor speed (in rad/s) is the output
# rs-Stator windings' resistence per phase(in Ohms)
# ld-d_axis inductance(in H)
# lq-q_axis inductance(in H)
# pm-Rotor magnet flux(Wb)
# j-Moment of inertia(in kgm2)
# d- Damping constant (Nm/rad/s)
# tl-motor load (Nm)
# p-Number of pole pairs
# power-The total input power (W)
# Assumptions:No core losses,no saturation,thermal effects
#(rs,ld,lq and pm values are constants)
element template pmsm_dtc t1 t2 t3 t0 speed out_me out_psi out_thetam out_ualf out_ubet out_psia
out_psib out_ialf out_ibet out_theta
out_tl=rs,ld,lq,pm,d,tl,j,p,init_theta_m,init_omega_m,omega_m_const
electrical t1,t2,t3,t0
# motor input terminals and stator neutral point
output nu speed,
out_me,out_psi,out_thetam,out_ualf,out_ubet,out_psia,out_psib,out_ialf,out_ibet,out_theta,out_tl
number rs=0.692,ld=6m,lq=6m,pm=0.26379,d=0.002044,tl=0.0,j=0.003,p=3.0,
init_theta_m=0.0,init_omega_m=0.0,omega_m_const=0.0
{
165
Appendices
<consts.sin
val v vq,vd,va,vb
val v vt1,vt2,vt3,v0
val tq_Nm te
val f fd,fq
val i ialf,ibet
val f psia,psib, psi
val p power
val ang_rad theta
# Stator phase voltages
# Electro magnetic torque
# d and q axis fluxes
var w_radps omega_m
var ang_rad theta_m
var i iq,id
var i it1,it2,it3
# stator phase currents
number y
parameters {
y=2*math_pi/3
}
values
{
vt1=v(t1)-v(t0)
vt2=v(t2)-v(t0)
vt3=v(t3)-v(t0)
va=2.0*(vt1-0.5*(vt2+vt3))/3.0
vb=(vt2-vt3)/sqrt(3)
#ialf=2.0*(it1-0.5*(it2+it3))/3.0
#ibet=(it2-it3)/sqrt(3)
ialf=id*cos(p*theta_m)-iq*sin(p*theta_m)
ibet=id*sin(p*theta_m)+iq*cos(p*theta_m)
vd=2*(vt1*cos(p*theta_m)+vt2*cos(p*theta_m-y)+vt3*cos(p*theta_m+y))/3 #d_axis voltage
vq=2*(-vt1*sin(p*theta_m)-vt2*sin(p*theta_m-y)-vt3*sin(p*theta_m+y))/3 #q_axis voltage
fd=ld*id+pm
#d_axis flux
fq=lq*iq
#q_axis flux
psi=sqrt(fd*fd+fq*fq)
psia=fd*cos(p*theta_m)-fq*sin(p*theta_m)
psib=fd*sin(p*theta_m)+fq*cos(p*theta_m)
te=1.5*p*(pm*iq+(ld-lq)*id*iq)
#electromagnetic torque
power=3.0*(vd*id+vq*iq)/2.0
theta=p*theta_m
}
control_section{
initial_condition(theta_m,init_theta_m)
initial_condition(omega_m,init_omega_m*math_pi/30.0)
}
equations {
id: vd=rs*id + d_by_dt(fd)-p*omega_m*fq
iq: vq=rs*iq + d_by_dt(fq)+p*omega_m*fd
166
Appendices
i(t1->t0)+=it1
it1: it1=id*cos(p*theta_m)-iq*sin(p*theta_m)
i(t2->t0)+=it2
it2: it2=id*cos(p*theta_m-y)-iq*sin(p*theta_m-y)
i(t3->t0)+=it3
it3: it3=id*cos(p*theta_m+y)-iq*sin(p*theta_m+y)
omega_m: (te-tl-d*omega_m)/j=d_by_dt(omega_m)
#omega_m: omega_m=omega_m_const*math_pi/30.0
theta_m: omega_m=d_by_dt(theta_m)
speed: speed=omega_m
out_me: out_me=te
out_psi: out_psi=psi
out_thetam:out_thetam=theta_m
out_ualf: out_ualf=va
out_ubet: out_ubet=vb
out_psia: out_psia=psia
out_psib: out_psib=psib
out_ialf: out_ialf=ialf
out_ibet: out_ibet=ibet
out_theta: out_theta=theta
out_tl: out_tl=tl
}
}
A4 Motor parameters
Surface type motor
Power
Number of pole pairs
Phase current
Phase voltage
Magnetic flux-linkage
P
p
I(rms)
U(rms)
Ψ PM
3kW
3
6.9A
70V
0.264 Wb
Rotor speed
Ωm
Me
J
Rs
Ld
Lq
3000rpm
20Nm
0.0174kgm2
0.692 Ω
6mH
6mH
Power
Number of pole pairs
Phase current
Rated voltage
Magnetic flux-linkage
P
p
I(rms)
U(rms)
Ψ PM
2,2kW
3
4.1A
380V
0.4832 Wb
Rotor speed
Ωm
Me
J
Rs
Ld
Lq
1750rpm
Nominal torque
Moment of the inertia
Stator winding resistance
Stator d-axis inductance
Stator d-axis inductance
Interior type motor
Nominal torque
Moment of the inertia
Stator winding resistance
Stator d-axis inductance
Stator d-axis inductance
12Nm
0.010074kgm2
3.3 Ω
41.59mH
57.06mH
167
Appendices
A5 Voltage Source Inverter parameters
Detailed date of IGBT transistors (module TOSHIBA M675Q2YS50):
U CE = 1200V , I C = 75 A
U CEsat = 2.8 − 3.6V , forward diode voltage 2.4 − 3.5V
Turn on time tON = 0.2µ s , Turn off time tOFF = 0.6 µ s
Delay of IGBT drivers tONd = 0.5µ s TOFFd = 1µ s
TON = tON + tONd = 0.7 µ s total turn on time of IGBT
TOFF = tOFF + tOFFd = 1.6 µ s total turn off time of IGBT
Dead time Td = 2.5µ s
A6 PI speed controller
The commonly used in industrial application speed controller is a Proportional-Integral
PI controller thanks to possibility to reduce the speed error between the reference ( X ref )
and actual rotor speed ( X m ) to zero (see Fig. A6.1). The output signal of controller is a
reference torque, which has upper and lower limitation for this value equal the nominal
torque or more than 130% of nominal torque. The output of the speed controller acts as
a current reference command for the current controllers. This current command is
limited to a nominal current of the motor.
The speed controller demands produce proper electromagnetic torque.
X _ ref
error signal
−
Feedback signal
Reference signal
a)
Controller output
Kp
Ti
error signal
−
Feedback signal
b)
∫
X _m
X _ ref
Reference signal
YL
Kp
YNL
Kp
lim_max
Controller output
lim_ min
Kp
Ti
∫
X _m
YL
−
YE
1
Ti
A.6.1. General structure of Proportional- Integral controller without antiwindup (a) and with
antwindup (b).
168
Appendices
A7 PWM technique – six step mode
Six-stepped-voltage waveforms are rich in harmonics. These time harmonics produce
respective stator current harmonics, which in turn interact with fundamental air gap
flux, generating harmonics torque pulsations. The torque pulsations are undesirable:
they generate audible noise, speed pulsations, and losses. In case of supplied motor by
using only active vectors (six step mode) we can observed non sinusoidal current, which
generates torque ripples with frequency of six time fundamental frequency of supplied phase
voltages.
Fig. A7.1. Experimental operation in six step mode. From the left side stator voltages in α , β
coordinates, From right side voltage trajectory.
Fig. A7.2. Experimental operation in six step mode. From the left side stator currents in α , β
coordinates, From right side stator current trajectory.
169
Appendices
Fig. A7.3. Experimental operation in six step mode. From the top: α stator voltage,
electromagnetic torque in machine, phase current.
170
List of symbol
List of symbols
Symbol (general)
X - instantaneous value
X N - normalized value
X - vector
X ∗ - conjugate vector
X - amplitude of vector
Re( X ) – real part of X
Im( X ) – imaginary part of X
Symbol (special)
α , β - stator fixed system
d , q - rotor reference system
x, y - general reference system
Ls - stator inductance
Z s - stator impedance
M s - mutual inductance
I s - phase current value
U s - phase voltage value
Ψ s - phase flux value
P -active power
Q - reactive power
S - apparent power
Pe - electro-magnetic power
Ωm - mechanical rotor speed
Ω s - synchronous speed
cos φ - power factor
δ I , δ Ψ - torque angle
φ - power angle
Rs - stator resistance
Ld , Ld - direct and quadrature inductances
θ r - electrical rotor position
γ m - mechanical rotor position
pb - number of pole pairs
Ψ PM - rotor flux of permanent magnets
M e - electromagnetic torque
M es - synchronous torque
M er - reluctance torque
M l - load torque (external load torque)
M d - dynamic torque
J m - motor moment of inertia
J l - load moment of inertia
J - moment of inertia of total system (sum of J m and J l )
171
List of symbol
Subscripts
A , B , C - denote arbitrary phase quantities in a system of natural coordinate A, B, C .
d , q - arbitrary direct and quadrature components in a system of rotor coordinate d , q .
α , β - arbitrary alpha and beta components in a system of stator coordinate α , β .
x , y - denote arbitrary components in a system of general coordinate x, y .
.. r - denotes value of rotor
.. s - denotes value of stator
.. _max - maximum value
.. _min – minimum value
.. _ ref - reference value
.. _ est - estimated value
.. _ amp -amplitude value
.. _ rms - root mean square value
.. _ LL - line to line value
*
^
- reference value
- estimated value
Abbreviations
RSM – reluctance synchronous motor
BLDCM – blushless DC motor
PMSM – permanent magnet synchronous motor
IPMSM - interior permanent magnet synchronous motor
SPMSM - surface permanent magnet synchronous motor
EMF – electro-magnetic force
VSI - voltage source inverter
SVM – space vector modulator
PWM – pulse width modulation
PWM-VSI – voltage source inverter with PWM
DTC - direct torque control
DTC-SVM - direct torque control with space vector modulator
RFOC - rotor field oriented control
SFOC - stator field oriented control
CTAC - constant torque angle control
MTPAC - maximum torque per ampere control
UPFC - unity power factor control
CSFC - constant stator flux control
172
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