Tài liệu High Performance Driver P2 pdf

HIGH PERFORMANCE DRIVES
---------------------------------------------------------------------------------------------------------------------------------------
 E Levi, 2001
7
2. MATHEMATICAL MODELLING OF AN INDUCTION MACHINE
AND THE SUPPLY
2.1. INTRODUCTION
As far as the AC machines are concerned, simple speed control systems are not capable of providing
decoupled (independent) flux and torque control. All the so called scalar speed control methods
(constant volts/hertz control, slip frequency control, voltage control etc.) are able of controlling the
steady-state behaviour of the machine only. All these methods rely on controlling the rms values of AC
voltage and/or current while instantaneous torque depends on instantaneous values of currents.
Therefore torque developed by the machine exactly corresponds to the commanded torque in steady state

only, while the dynamic response is generally sluggish and slow. Transition from one steady-state to
another is not controllable and follows internal dynamics of the machine. The idea of field orientation,
or vector control as it is called as well, can be briefly stated as a ‘control method that converts an AC
machine into its DC machine equivalent from the control point of view and thus enables
instantaneous decoupled control of flux and torque’. Instantaneous decoupled flux and torque control
is made possible by control of instantaneous current values rather than rms values. Extremely fast
response, that fully corresponds to the one obtainable from a DC machine, is enabled by this method of
speed control. However, the control system capable of realising such a good quality speed control is,
due to AC nature of all the variables in the machine, much more complicated. Due to significantly more
complex structure of AC machines, compared to DC machines, application of field orientation as a
practical speed control method has become possible only by microprocessors. Field oriented control is
nowadays applied in variety of manners in conjunction with both induction and synchronous machines

(sinusoidal and trapezoidal permanent magnet synchronous machines, wound rotor synchronous
machines, synchronous reluctance machines). The emphasis here is on the two most frequent types of
AC machines that are utilised in vector controlled drives, namely three phase squirrel cage (singly fed)
induction machine (IM) and three phase sinusoidal permanent magnet synchronous machine (SPMSM).
As shown shortly, electro-magnetic torque of a three-phase induction motor can be expressed in terms of
phase currents of stator and rotor as
()()()
T L ii ii ii ii ii ii ii ii ii
e aA aAbBcC aCbAcB aBbCcA
=− + + + −
æ
è

ç
ö
ø
÷
++
ì
í
î
++
æ
è
ç

ö
ø
÷
++
ü
ý
þ
sin sin sin
θθ
π
θ
π

2
3
2
3
(2.1)
where
θ
denotes instantaneous position of the rotating rotor phase ‘A’ magnetic axis with respect to
stationary stator phase ‘a’ magnetic axis and L
aA
is the peak value of the mutual inductance between
stator and rotor windings of the machine. This torque expression holds true in both steady-state and

transient operation of the induction machine. The angle
θ
is determined with the speed of rotation, that
is
θω
=
ò
dt
(2.2)
Note that rotor currents are induced in rotor windings and they are thus governed by feeding conditions
at stator side (and load). Hence both flux and torque component of the current stem from stator (there
are no independent windings for separate flux current and torque current control, in contrast to DC

machines). The question then arises: is it possible somehow to express the torque of the induction
machine in terms of some other, fictitious currents in such a way that it resembles torque expression for
a separately excited DC machine? In other words, can the torque be somehow transformed into the form
TCi
edqs
=
ψ
(2.3)
HIGH PERFORMANCE DRIVES
---------------------------------------------------------------------------------------------------------------------------------------
 E Levi, 2001
8

where flux
ψ
may be stator flux, air gap flux or rotor flux linkage, and i
qs
is a certain fictitious
component of the stator current. If such a transformation is possible, then induction machine may be
made to behave from the control point of view as separately excited DC machine.
FIELD ORIENTED
CONTROL (VECTOR CONTROL)
is a theory which enables achievement of the stated goal, not only with
respect to induction machines but for all the other listed types of AC machines.
Field oriented control may be therefore shortly defined as a set of control methods which, with respect

to control of the machine, enable conversion of an ac machine into an equivalent separately excited
DC machine. Thus field oriented control enables decoupled (independent) control of flux and torque in
an AC machine by means of two independently controlled (fictitious) currents, as the case is in a
separately excited DC machine.
It has to be noted that, as instantaneous time-domain variables are under consideration at all times and
the subject of analysis is dynamic (transient) behaviour of an AC machine, it is not possible to use in
analysis approach with phasor representation of sinusoidal quantities. The variables are not sinusoidal
(except in steady-state) nor are the regimes under consideration steady-states. The whole theory of field
oriented control relies on machine modelling in time-domain.
Vector control requires existence of the current control, in very much the same way as it was explained
in conjunction with a separately excited DC machine. However, in the case of a DC machine
instantaneous change of torque requires only instantaneous change of the current amplitude, since the

armature current is a DC current. In the case of an AC machine requirement of instantaneous change of
current is much more involved. To illustrate this, consider a steady state operation of an induction
machine. Let us assume that the supply source is capable of providing purely sinusoidal currents of any
amplitude and any frequency. Instantaneous stator phase currents are then given with:
()
()
()
iI t
iI t
iI t
ae
be

ce
=−
=−−
=−−
sin
sin /
sin /
ωϕ
ωϕπ
ωϕπ
23
43

(2.4)
Suppose that a step speed command increase takes place, that asks for instantaneous stepping of the
torque. The problem of stepping the torque in the machine from the appropriate steady state value to the
maximum permissible value in order to achieve the fastest possible acceleration of the drive may be
understood in terms of three-phase stator currents as a problem of providing new set of currents:
()
()
()
iI t
iI t
iI t
ae

be
ce
11 1 1
11 1 1
11 1 1
23
43
=−
=−−
=−−
sin
sin /

sin /
ωϕ
ωϕπ
ωϕπ
(2.5)
such, that a transient-free torque response is obtained. In other words, it is necessary to provide control
of stator current amplitude, frequency and phase in an appropriate manner. Field-oriented control
actually explains how these parameters have to be changed in order to obtain a transient-free torque
response.
In order to further examine behaviour of an induction machine torque response, Fig. 2.1 illustrates no-
load acceleration from standstill, with 50 Hz, rated sinusoidal voltage supply. Note that this is not the
case of a variable speed drive and that there is not any control of the motor. It is simply an acceleration

transient with mains supply. At time instant zero the motor is connected to the mains. There is no load
connected to the shaft. The motor accelerates from standstill to the steady-state no-load speed. In final
steady state operation the motor torque is zero, while the speed is constant no-load speed. As witnessed
by the torque trace in Fig. 2.1, torque developed by the motor is highly oscillatory during the transient.
It even takes negative values in some instants, during which the speed reduces rather than increases.
Recall that in a high performance drive torque response is required to be instantaneous and equal to the
maximum allowed torque during the transient. Complex nature of an induction machine makes such a
torque response rather difficult to achieve and that is why vector control is widely used.
HIGH PERFORMANCE DRIVES
---------------------------------------------------------------------------------------------------------------------------------------
 E Levi, 2001
9

In a standard induction motor drive with open loop or closed loop V/f speed control torque transient
during transition from one operating speed to the other behaves similarly to the trace of Fig. 2.1. Hence
more dedicated control has to be used if high performance is to be achieved.
-10
0
10
20
30
40
Torque (Nm)
0 0.05 0.1 0.15 0.2 0.25 0.3
Time (s)

Torque for no-load acceleration
Sinusoidal, 50 Hz supply
Fig. 2.1 - Variation of the induction motor torque during no-load acceleration from standstill with 50
Hz sinusoidal supply.
On the basis of the considerations of this sub-section and discussion of high performance DC motor
drives, the following statements can be made:
• High performance operation requires that the electro-magnetic torque of the motor is controllable in
real time;
• What the commutator does in a DC machine physically (i.e. enables decoupled flux and torque
control), has to be done in an AC machine mathematically (theory of vector control or field oriented
control);
• Instantaneous flux and torque control require that the machine windings are fed from current

controlled AC sources;
• Current and speed sensing is necessary in order to obtain the feedback signals for real time control
(current and speed are controlled in closed loop manner, with current control loop embedded within
the speed control loop).
Compared to the statements given at the end of discussion of high performance DC drives, one notes
that the first and the last two are the same. However, the second statement replaces the second and the
third in the list for DC drives and is the subject that will be discussed shortly.
2.2. HISTORY AND APPLICATIONS OF FIELD ORIENTED CONTROL
Rapid development in industry automation asks for permanent improvement of different types of electric
drives. The imposed requirements are increased reliability, decrease in electric energy consumption,
minimisation of the maintenance costs and improved capability of dealing with complicated and precise
tasks required by the given technological process. About 50% of the generated electric energy in

developed countries is converted into mechanical energy by means of electrical drives, and about 20 %
HIGH PERFORMANCE DRIVES
---------------------------------------------------------------------------------------------------------------------------------------
 E Levi, 2001
10
of drives are variable speed drives. Variable speed drives which were for an extended period of time
based on standard DC machines are more and more being substituted with appropriate variable speed
AC drives. The annual rate of substitution varies in different areas of applications, but attains even
such value as 15 % per year in the field of servo drives. The main reason for such a widespread
utilisation of DC drives in the past is the capability of decoupled flux and torque control in DC
machines, which asks for just a moderate investment in appropriate power electronics source. It was
not until the fundamentals of field oriented control were set forth, that such a decoupled control of flux

and torque in AC machines became feasible.
Basic principles of field oriented control show that it is possible to realise theoretically perfectly
decoupled control of flux and torque in AC machines. The idea of field oriented control requires that
instantaneous values of magnitude and position of the stator current space vector with respect to the
appropriate flux space vector in the machine, in relation to which the orientation is performed, can be
controlled. The way of obtaining field oriented control is to orientate stator current space vector with
respect to rotor flux space vector. The notion of "space vector" stems from the general theory of
electric machines and the other popular name for field oriented control which is widely used, namely
"vector control", has its origin in the fact that field oriented control is frequently dealt with in terms of
space vectors, which are commonly applied in analysis and modelling of AC machines. Realisation of
decoupled control of flux and torque is possible with both induction and synchronous machines and they
can be fed from a converter which is either of the voltage or current source type .

The original realisations of rotor flux oriented control from early seventies employ analogue techniques.
Due to the complexity of the control part of the system, which is caused mainly by necessity to perform
co-ordinate transformation, analogue versions of the field oriented control did not find wider
application.
Development in microprocessors in the late seventies made however realisation of vector controlled
induction motor drives both attractive and achievable. During the last fifteen years, research in the area
of field oriented control has become subject of wide interest in the whole world. Superiority of
dynamics of vector controlled induction machines in relation to classic control algorithms represents the
fundamental reason for such a trend in development of controlled AC drives. On the other hand, the
complexity of the control system inevitably forces researchers to look for simpler control schemes which
should still be able to retain dynamic behaviour comparable to vector controlled drives. However, for
high performance drives, where the most severe constraints are imposed on dynamics, simplified control

methods can not be expected to replace field oriented control due to poorer dynamic behaviour. As a
conclusion to this discussion it can be stated that field oriented control remains the best available choice
for the applications where decoupled control of flux and torque is an absolute "must" in order to obtain
the highest possible accuracy and speed of the drive response.
Application areas of vector controlled induction machines in industry are numerous. One of the most
frequent applications is in machine tools, were usually induction machines with rated speed of 1500 rpm
are utilised and field weakening feature extends the range of operating speeds up to 4500-6000 rpm.
Completely digital versions of field oriented controllers for machine tools, manufactured by Bosch, are
capable of operating at as high speeds as 10.000 rpm, thus providing for speed range in the field
weakening region of up to 1:6. The advantage of vector controlled induction machines in relation to
field oriented permanent magnet synchronous machines, in the domain of machine tools, is simple
provision for field weakening feature. Another important area of application are servo drives where

either permanent magnet synchronous machines or induction machines are used, depending on operating
requirements.
If operation in the field weakening region is needed, induction machines are advantageous and they are
used in servo drives for positioning. The applications discussed so far comprise low and medium power
range. Induction motor drives with field oriented control are however used in high power range as well.
This type of application was initiated in Japan in late seventies. The complete automation of a
production line in paper industry, where requirement on speed control accuracy is 0.02% and speed has
to be varied in the range 180:1, is performed by five induction machines with power ratings 340-500
HIGH PERFORMANCE DRIVES
---------------------------------------------------------------------------------------------------------------------------------------
 E Levi, 2001
11

kW in 1979. A number of vector controlled induction machines with power ratings of the order 100-
300 kW have been installed in the period 1980-1983 in steel industry. Two complete production lines
were introduced in 1979 in Japan in steel rolling mills, each containing 40 vector controlled induction
machines in the power range 5.5-11 kW.
The research in the area of field oriented control, due to the complexity of the overall system, runs in
parallel in a number of different sub-areas, namely VLSI design, power electronic converters, modern
control techniques and parameter variation effects and parameter identification (as will be shown later,
vector control schemes require accurate knowledge of induction machine parameters). New laboratory
prototypes utilise single chip for all the control functions, or are alternatively based on application
specific integrated circuits. Topologies of power electronic converters which ask for semiconductor
switches with bi-directional current flow and bi-directional voltage blocking capabilities are gaining
more and more attention recently, because it is expected that such switches will become available in the

near future. At this stage, instead of bi-directional switches, appropriate combinations of unidirectional
switches are utilised for experimental purposes. As the bi-directional switches are still not available,
converter topology with resonant DC link at the moment seems to be more prosperous solution.
Development of high-speed low-cost microprocessors and signal processors enables implementations of
more and more sophisticated control algorithms in vector controlled drives. Different methods based on
modern control theory are being proposed with ultimate goal to further improve the drive performance.
Among the large variety of the methods, the most important seem to be application of state observers,
model reference adaptive control and state-space controllers. The need for application of modern
control theory stems from the complexity of an AC machine as a control object, whose parameters are
variable. The ideal decoupled control of flux and torque can be obtained by means of standard vector
control approach only if the parameters of an AC machine are exactly known and constant. This is
unfortunately not the case in reality. The parameters of machines are subject to variation due to their

dependency on operating state of the machine. A discrepancy between parameter values assumed at the
stage of the control system design and actual parameter values in the machine results, causing loss of
decoupled torque and flux control and deterioration in quality of dynamic response.
2.3. PHASE-DOMAIN MODEL OF AN INDUCTION MACHINE AND ITS
TRANSFORMATION
2.3.1. Model of the machine in terms of physical phase variables
As the field oriented control asks for instantaneous control of machine flux and torque via instantaneous
current control, it is not possible to deal with induction machine representation in terms of equivalent
circuit and phasors. Instead, time domain mathematical model in the original phase reference frame has
to be utilised as a starting point. Furthermore, this model has to be mathematically transformed into
new fictitious reference frame by suitably chosen mathematical transformation. It becomes obvious
even from this short discussion that the process of designing and achieving decoupled flux and torque

control in an induction machine is much more tedious than with DC machines.
The procedure of mathematical modelling of an induction machine is subject to a number of different
common assumptions and idealisations. More specifically, it is assumed that stator phase windings are
identical with mutual space displacement of exactly 120 degrees, that magneto-motive force of a
winding is sinusoidally distributed along the air gap circumference, that air gap is constant, that rotor
cage winding can be substituted with a balanced three-phase winding, that winding resistances and
leakage inductances are constant parameters, eddy-currents and iron losses are neglected as well as all
the parasitic capacitances, and finally, it is assumed that magnetising curve can be treated as a linear
function, i.e. that the main flux saturation can be neglected.
Voltage equilibrium equations of a three-phase induction machine in original phase domain, if the above
listed assumptions are adopted, are given with the following expressions (underlined symbols denote
matrices) in terms of time domain instantaneous variables:

HIGH PERFORMANCE DRIVES
---------------------------------------------------------------------------------------------------------------------------------------
E Levi, 2001
12
vRi
d
dt
vRi
d
dt
abc
s

abc
abc
ABC
s
ABC
ABC
=+
=+


(2.6)

abc
s abc sr ABC
ABC
r ABC sr
t
abc
Li L i
Li L i
=+
=+
(2.7)

where lower case indices apply to stator quantities, while upper case indices denote rotor quantities, and
vvvv iiii
v vvv i iii
abc
abc
t
abc
abc
t
ABC
ABC
t

ABC
ABC
t
==
==
(2.8.a)
LL
sr
aA
=
+
ổ

ố
ỗ
ử
ứ
ữ

ổ
ố
ỗ
ử
ứ
ữ

ổ
ố
ỗ
ử
ứ
ữ
+
ổ
ố
ỗ
ử

ứ
ữ
+
ổ
ố
ỗ
ử
ứ
ữ

ổ
ố

ỗ
ử
ứ
ữ
ộ
ở
ờ
ờ
ờ
ờ
ờ
ờ

ờ
ự
ỷ
ỳ
ỳ
ỳ
ỳ
ỳ
ỳ
ỳ
cos cos cos
cos cos cos

cos cos cos

2
3
2
3
2
3
2
3
2
3

2
3
(2.8.b)
L
LLL
LLL
LLL
s
aa ab ac
ba bb bc
ca cb cc
=

ộ
ở
ờ
ờ
ờ
ự
ỷ
ỳ
ỳ
ỳ
L
LLL

LLL
LLL
r
AA AB AC
BA BB BC
CA CB CC
=
ộ
ở
ờ
ờ
ờ

ự
ỷ
ỳ
ỳ
ỳ
(2.8.c)
Each matrix equation in (2.6)-(2.7) is therefore an abbreviated way of writing three equations, one per
phase. The angle , as already discussed, denotes instantaneous position of magnetic axis of rotor phase
A winding with respect to stationary magnetic axis of stator phase a winding and is correlated with
rotor (electrical) speed of rotation through dependence

=

ũ
dt
(2.2)
The mechanical equation of motion is the same as for a DC or any other motor (a torque that describes
mechanical losses is now included and the equation is given in terms of electrical speed of rotor rotation)
TT
J
P
d
dt P
k
eL

= +


1
(2.9)
where is once more electrical speed of rotation of rotor, mechanical power is taken as positive when it
leaves the machine (for motoring) and electromagnetic torque can be expressed in terms of
instantaneous phase currents as
()()()
T L ii ii ii ii ii ii ii ii ii
e aA aAbBcC aCbAcB aBbCcA
= + + +

ổ
ố
ỗ
ử
ứ
ữ
++
ỡ
ớ
ợ
++
ổ

ố
ỗ
ử
ứ
ữ
++
ỹ
ý
ỵ
sin sin sin

2
3
2
3
(2.1)
Equation (2.1) for electromagnetic torque is, as already discussed, significantly more complicated than
the corresponding equation for a DC machine (2.3). The main reasons are that the machine under
consideration is an AC, three phase machine, and, additionally, currents in rotor windings are induced
current (i.e. rotor and stator windings are not fed from separate supplies, as in a DC machine). Hence
the excitation current and armature current in the case of an induction machine stem from the same

supply.
Schematic representation of a three-phase induction machine in original phase domain is given in Fig.
2.2. Model described with (2.1), (2.9) is very inconvenient for any type of analysis and it has to be
transformed by applying an appropriate mathematical transformation. The main shortcomings of this
model are time dependent coefficients of differential equations (all the mutual inductances between
stator and rotor phases are indirectly time dependent through dependence on rotor angular position ),
HIGH PERFORMANCE DRIVES
---------------------------------------------------------------------------------------------------------------------------------------
 E Levi, 2001
13
and full inductance matrix with 36 non-zero inductance terms. The system of differential equations that
describe the machine is said to be non-linear, with time-varying coefficients. There are all together seven

first-order differential equations, six for the electrical sub-system (voltage equilibrium equations) and
one for the mechanical sub-system (equation of rotor motion).
b
ω
CB
a
θ
A
c
Fig. 2.2 - Schematic representation of a three-phase induction machine: all the windings are placed
on magnetic axes (windings are illustrated for phases a and A); rotor windings A,B,C rotate with
rotor, while stator windings a,b,c are stationary.

2.3.2. Transformation of the model into a common reference frame, rotating at an arbitrary
angular speed
Mathematical model of an induction motor, expressed in terms of phase variables and parameters, can
be transformed into a corresponding model in the so-called common reference frame, by means of
appropriate mathematical transformations. In general, two approaches are possible. The first one
utilises the model given in the preceding section as the starting point and relies on the use of matrix
transformations. The second approach at first defines so-called space vectors and then applies
appropriate transformation without resorting to the use of matrices. The approaches lead to the same
final result. In what follows, the matrix transformation approach is used. Space vectors are defined in
the following sub-section.
Regardless of which approach is used, the idea is to replace the physically existing machine with its
three-phase stator and rotor windings with a fictitious machine whose all windings are in the common

reference frame. This means that stationary stator windings and rotor windings that rotate at rotor speed
are all replaced with new fictitious stator and rotor windings that all have the same speed. This speed of
the common reference frame can be arbitrarily selected for an induction machines, due to the uniform
air gap.
In order to transform the model of the machine from the original phase variables into new variables, it is
necessary to apply appropriate transformation matrices on stator and rotor variables. If the stator
equations and rotor equations are transformed by means of A
s
and A
r
transformation matrices
respectively,

HIGH PERFORMANCE DRIVES
---------------------------------------------------------------------------------------------------------------------------------------
 E Levi, 2001
14
A
s
ss s
ss s
=
−
æ
è

ç
ö
ø
÷
+
æ
è
ç
ö
ø
÷
−−−

æ
è
ç
ö
ø
÷
−+
æ
è
ç
ö
ø

÷
é
ë
ê
ê
ê
ê
ê
ê
ê
ù
û

ú
ú
ú
ú
ú
ú
ú
2
3
2
3
2

3
2
3
2
3
1
2
1
2
1
2
cos cos cos

sin sin sin
θθ
π
θ
π
θθ
π
θ
π
(2.10.a)
A
r

rr r
rr r
=
−
æ
è
ç
ö
ø
÷
+
æ

è
ç
ö
ø
÷
−−−
æ
è
ç
ö
ø
÷

−+
æ
è
ç
ö
ø
÷
é
ë
ê
ê
ê

ê
ê
ê
ê
ù
û
ú
ú
ú
ú
ú
ú

ú
2
3
2
3
2
3
2
3
2
3
1

2
1
2
1
2
cos cos cos
sin sin sin
θθ
π
θ
π
θθ

π
θ
π
(2.10.b)
equations of an induction machine in arbitrary common reference frame result. Note that the
transformation matrices for stator and rotor windings differ in the sense that different angles are met in
sin and cos terms in these two matrices. The procedure of transforming equations of an induction
machine from original phase domain into so called arbitrary reference frame may be viewed, as already
pointed out, as substitution of actual phase windings with new fictitious windings. These new windings
are all, in general, rotating; it is important to realise that both original stator (stationary) and rotor
(rotating) windings are substituted with new windings that have the same arbitrary speed of rotation
(hence the name "common reference frame"). The model obtained after the application of the

transformation may be given as follows (indices "s" and "r" denote stator and rotor variables and
parameters, respectively):
vRi
d
dt
vRi
d
dt
vRi
d
dt
ds s ds

ds
aqs
qs s qs
qs
ads
os s os
os
=+ −
=+ +
=+
ψ
ωψ

ψ
ωψ
ψ
(2.11)
vRi
d
dt
vRi
d
dt
vRi
d

dt
dr r dr
dr
aqr
qr r qr
qr
adr
or r or
or
=+ −−
=+ +−
=+

ψ
ωωψ
ψ
ωωψ
ψ
()
()
(2.12)
where d-q-o axis flux linkages are given with
ψ
ψ
ψ

ds s ds m dr
qs s qs m qr
os os os
Li L i
Li L i
Li
=+
=+
=
(2.13)
ψ
ψ

ψ
dr r dr m ds
qr r qr m qs
or or or
Li L i
Li L i
Li
=+
=+
=
(2.14)
for stator and for rotor equivalent windings, respectively. In equations (2.13)-(2.14) the inductance

terms are correlated with phase domain inductances through the following expressions
LL L LL
LL L LL L L L
LL LLL
saaab sm
r AA AB r m s aa ab
m aA r AA AB
=−=+
=−=+ =+
==+
γ
γσ

σ
2
32 2(/)
(2.15)