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IEEE TRANSACTIONS ON EDUCATION, VOL. 44, NO. 2, MAY 2001
Vector Control Methods for Induction Machines:
An Overview
José Andrés Santisteban, Member, IEEE, and Richard M. Stephan, Member, IEEE
Abstract—In the last three decades, different vector control
methods [field-oriented control (FOC), field acceleration method
(FAM), universal field orientation (UFO), direct self control
(DSC) and Takahashi method among others) have been proposed.
It is difficult for students and nonspecialists to understand the
drawbacks and advantages of each one. With this in mind, the
objective of this paper is to propose a clear classification and
comparison of them.
Index Terms—Control of electric drives, induction machines,
vector control.
I. INTRODUCTION
I
N THE last three decades, the field oriented method (FOC)
for control of AC drives has been analyzed and discussed in
the literature, e.g., [1], [10], [17]–[19]. The current control loop
and the determination of the reference frame, necessary in this
control method, can be implemented in different ways. Moreover, the FOC method is sensitive to parameter variations, principally the rotor time constant. Therefore, different vector control strategies have been proposed trying to achieve the best performance and/or ease of implementation. Among these vector
control methods, the following can be mentioned: field acceleration method (FAM) [7], the proposal of Takahashi [6], direct self
control (DSC) [3], and universal field orientation (UFO) [2]. As
different mathematical notations have been used in these studies
(e.g., spatial vectors, phase segregation, spiral vectors, matrices,
…) it is difficult to introduce and evaluate the basic relationship
among them. In this paper, one methodology of presentation is
proposed in order to give clear concepts about the theme including an analytical comparison of the most important methods
found in the literature.
II. THE BASIC IDEA OF VECTOR CONTROL
In the so called “scalar control methods” for induction
machines, the motor model is considered just for steady
state. Therefore, it is expected that a controller based on
these methods can not achieve the best performance during
transients. This is the basic drawback of scalar control methods
for induction machines.
In “vector control methods,” the motor model considered is
valid for transient conditions [10]–[12]. The idea of the field
oriented control (FOC) proposed by Blaschke [1] can be understood from the “reference frame theory” [14]. The reference
frame used in the FOC is one whose real axis coincides with the
rotor flux vector. This frame is not static and does not have a constant speed during transients. Actually, it was not a commonly
used reference frame for the analysis of electric machines. The
great advantage of this “noninertial” frame is that for impressed
stator currents, this method allows independent flux and torque
controls as in a separately excited dc machine. Impressed stator
currents, that is currents controlled by a fast current loop that
can be implemented using cheap hall effect current sensors and
power electronics, are usual in the industry practice. Furthermore, the control proposed by Blaschke is the well-known control used for separately excited dc machines, and his theory
could be named “field oriented modeling.”
III. THE NOMENCLATURE
The first proposal of a vector control method has been presented by Blaschke [1] using the space vector notation suggested by Kovacs [11], [12]. Since the 1980s new approaches for
vector control methods have appeared in the literature: Yamamura (phase segregation, spiral vectors); Takahashi ( – comcomponents); De Donker (generponents); Depenbrock (
alized – components). Other proposals are basically a direct
consequence of these.
For the present paper, the – notation used by De Doncker
and Novotny [2] was chosen since it allows the machine representation in arbitrary reference frames (e.g., stator flux, rotor
flux, air-gap flux, etc.) with relative simplicity. This notation is a
generalization of the classical – notation [14] where the reference frame speed is no longer restricted to constant values. The
relationship between space vector, spiral vector, and – notations is presented during classes.
With the De Donker’s notation [2], it can be shown that the
induction motor model, assuming impressed stator currents, is
given by
(1)
Manuscript received August 13, 1997; revised March 27, 2000. This work
was supported by CNPq and GTZ.
J. A. Santisteban is with the Fluminense Federal University,
UFF/TEE/PGMEC, Niteroi, 24210-240, Brazil (e-mail: jasl@mec.uff.br).
R. M. Stephan is with the Federal University of Rio de Janeiro,
UFRJ/COPPE/DEE, Rio de Janeiro 21945-970, Brazil (e-mail:
richard@coe.ufrj.br).
Publisher Item Identifier S 0018-9359(01)03863-8.
0018–9359/01$10.00 © 2001 IEEE
(2)
(3)
SANTISTEBAN AND STEPHAN: VECTOR CONTROL METHODS FOR INDUCTION MACHINES: AN OVERVIEW
171
Fig. 2. Induction motor model into the rotor flux reference frame.
As can be seen from this block diagram, the flux can be controlled independently by ; whereas, the torque is controlled
by changing , maintaining the rotor flux constant.
Fig. 1.
General induction motor model with impressed currents.
IV. THE TORQUE EQUATIONS
where
dispersion factor;
arbitrary turn ratio, which physical
meaning is given below for three different values;
pole pair number;
flux slip frequency;
component of the arbitrary flux
based on the reference frame linked
to the arbitrary flux vector;
stator current component;
stator current component;
rotor inductance;
air-gap inductance;
stator inductance;
rotor resistor;
electromagnetic torque.
The reader should pay attention to the character “ ” that is
used as a subscript and superscript to identify the arbitrary reference frame and also as a letter that represents turns ratio. This
choice may appear confusing, but it is a good way to point out
that the arbitrary reference frame depends just on the choice of
a turn ratio.
These equations are presented as a block diagram in Fig. 1
where the coupling effect of the current components on the flux
and torque can be observed.
The “stator voltage/stator current” model of an induction machine allows different equivalent models with different internal
variables. This is a point stressed in good books on electrical
machines, e.g., [7], [10], [14].
, which is an
Additionally, according to the value of ,
internal variable, assumes different meanings:
then
(stator-flux);
then
(air-gap flux);
then
(rotor-flux).
Fig. 2 shows the case where the rotor-flux is chosen as the ref). The influence of
over the flux is
erence (therefore,
eliminated in this case.
This approach corresponds to the field orientation model proposed by Blaschke [1].
Torque is the most important variable in the control of electric-mechanical systems. It makes the connection of the mechanical and electrical parts. Equation (3) presents the electrical
torque as the product of a reference flux and the stator current
component orthogonal to it. This equation is similar to the one
for a separately excited DC machine. In this paper the control
methods based on this equation will be called “quadrature control methods.”
On the other hand, using (1) and (2), it is possible to write
as a function of
. Substituting this expression for
in (3),
it follows:
(4)
This expression presents the torque as a nonlinear and dynamic
. The control methods based
function of the slip frequency
on this equation will be called “slip control methods.”
If the derivative terms are neglected, i.e., steady-state or
quasi-steady-state conditions are assumed, the torque equation,
for constant flux, shows an approximately direct relationship
between torque and slip. As can be expected, with this approach
a linear control can not produce the best dynamic behavior.
However, when the rotor flux is chosen as the reference, then
and (5) below can be obtained from (4). In this case (5)
is always exact even under transient conditions. These observations are also in accordance with the comments presented by
Ueda et al. in [5]
(5)
V. QUADRATURE CONTROL METHODS
As mentioned, these control methods are based on (3). They
can be further subdivided into direct and indirect methods. Di-
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Fig. 3.
IEEE TRANSACTIONS ON EDUCATION, VOL. 44, NO. 2, MAY 2001
Velocity control using the general direct quadrature control method.
rect methods have a control loop for the flux and, therefore, a
flux estimator or flux sensors are necessary. Indirect methods
assume that the flux amplitude is constant, and its spacial position can be obtained with a feedforward block that has as an
input of speed. Clearly, indirect methods are easier to implement
but less accurate.
A. Rotor Flux Slip Control
As already mentioned, in this case
and (4) can be
written exactly as (5). Therefore the control scheme is valid
under transient and steady state conditions. Moreover, from (1),
it can be seen that there exists a direct relationship between
(the torque current component) and the slip, as shown in (7)
A. Direct Quadrature Control Methods
The general control scheme is presented in Fig. 3. The PI regulators shown in this figure could be substituted by other classical or modern controllers. To decouple the torque and flux conshown
trol loops, it is necessary to compensate for the term
in Fig. 1. As already shown with Fig. 2, this decouple occurs
naturally with the rotor-flux reference. In the general case, the
at the output
decouple can be achieved by adding the term
of the flux controller, as shown in Fig. 3, where
(6)
This term is obtained through the compensating block in
Fig. 3, using the values from the estimator block. An example
of this method can be found in [8].
Again choosing the rotor-flux reference, the compensating
.
term given in (6) will be zero since
(7)
Then, substituting (7) in (5), (8) below can be obtained. This is
the equivalent quadrature form of (5)
(8)
This means that the rotor-flux based slip control and the
rotor-flux based quadrature control are equivalent. For example, the FAM control applied to the T-I model [7] can be
mentioned. Moreover, reference [4] also shows the equivalence
of this control scheme and the indirect FOC.
B. Air-Gap and Stator-Flux Slip Control
In this case, the derivative terms in (4) are generally neglected, i.e., steady-state or quasisteady-state conditions are
assumed. Equation (4) then becomes (9)
B. Indirect Quadrature Control Methods
in Fig. 3
In this case, there is no flux control loop, and
is an imposed constant. Some examples of this control method
are given with the ideal indirect UFO scheme proposed by De
Doncker and Novotny [2] which is based on the inverse model
of the induction motor; and another is the well-known indirect
FOC [13].
VI. SLIP CONTROL METHODS
These methods are based on (4) and are the basic difference
from the quadrature control methods that are based on (3).
(9)
After some algebraic work, it can be shown that this equation is equivalent to the torque equation used in Yamamura’s
, where is
work, considering that in steady-state
the slip and the synchronous speed. FAM applied to T and
T-II models [7] are examples of air-gap and stator-flux slip control, respectively. Takahashi [6] and Depenbrock [3] are other
examples of stator-flux slip control methods, using nonlinear
SANTISTEBAN AND STEPHAN: VECTOR CONTROL METHODS FOR INDUCTION MACHINES: AN OVERVIEW
173
Fig. 4. Vector control methods classification.
controllers. Additionally, other alternatives for the slip control
methods were proposed by Ribeiro et al. [9].
The results discussed previously are summarized with the
general classification shown in Fig. 4. Here, the shaded blocks
show that quadrature and slip control methods are identical in
this case.
VII. SIMULATIONS
This analytical study will be supported in the following by
simulations using an induction motor with the following data:
kg-m ,
220 V, 3 , 60 Hz, 1472 W, 1725 r/min,
4 poles. Results will be presented in a normalized scale with a
speed base 900 r/min, a flux base 0.476 Wb, a current base 5.463
A, and a torque base 7.809 N-m.
Since the indirect control methods are structurally poor (they
do not have a flux control loop), the comparison focuses on
the direct ones. Moreover, slip control methods are based on
an approximated equation [see (5)], and the quadrature ones are
the more accurate. Therefore, to avoid a great number of simulations that would be confusing, only the direct SUFO (stator
UFO) and RUFO (rotor UFO) methods were selected from a
great number of performed simulations. Nevertheless, the reader
may find comparisons of some methods that are based on the
simplified equations (4) and (5) in [15] and [16].
For the RUFO control, the flux/torque estimator considers the
stator currents and the shaft speed as inputs (Fig. 5). For the
SUFO control the estimator has the stator currents and voltages
as inputs (Fig. 6).
From the stator currents and voltages it is possible to determine directly the stator flux reference frame. On the other hand,
from the stator currents and rotor speed it is possible to determine the rotor flux reference frame. Therefore, this choice allows the determination of the reference frames with the lowest
number of arithmetic operations.
It can be seen in Figs. 7 and 8 the results of the direct quadrature control method based on the rotor flux and the stator flux
Fig. 5. Rotor flux estimator.
Fig. 6. Stator flux estimator.
respectively. For the airgap flux-based case, the results are very
similar to the ones based on the stator flux.
Figs. 7(a) and 8(a) show the starting and speed reversal for the
induction motor when there are tuned conditions (i.e., parameter
values in the flux/torque estimator are equal to the ones on the
machine). The performance in both cases is the same.
Fig. 7(b) shows the result for 50% of variation in the rotor
time constant (doubling the rotor resistor). Fig. 8(b) shows the
results for 20% of variation (positive) in the stator resistor.
Comparing Figs. 7(b) and 8(b), the negative influence of nontuned estimators of the flux and on the torque can be seen in
both figures. It is particularly obvious during the starting of the
stator-flux-based case.
All simulations performed with detuned stator and rotor resistances have always shown some degradation. Some kind of
174
IEEE TRANSACTIONS ON EDUCATION, VOL. 44, NO. 2, MAY 2001
The content of this paper is the spine of the graduate-level
course “Control of Electrical Drives” regularly offered at
COPPE/UFRJ since 1995.
From the analysis, the following conclusions were deduced.
The quadrature control methods are based on exact
torque equations. Therefore, if the motor parameters
necessary for the control algorithm are known, the
transient performance can be optimal.
The slip control methods are based on approximate
torque equations, except when the rotor-flux is
chosen as reference. Therefore, the stator-flux and the
air-gap-flux slip control methods can not produce optimal transient responses for all operating conditions.
The rotor-flux based slip control is equivalent to the
rotor-flux based quadrature control method.
All direct quadrature control methods (stator-based,
air-gap-based or rotor-based) can present equivalent
optimal transient responses.
All direct quadrature control methods are sensitive to
some parameter changes. The flux/torque estimator
plays an important role on the control performance.
Fig. 7. Rotor-flux oriented direct quadrature control time scale: 0.05 sec/div.
(a) Tuned unload starting and speed reversal. (b) Detuned unload starting and
speed reversal.
REFERENCES
Fig. 8. Stator-flux oriented direct quadrature control. (a) Time scale: 0.05
sec/div. (b) Time scale: 0.1 sec/div. (a) Tuned unload starting and speed
reversal. (b) Detuned unload starting and speed reversal.
parameter adjustment is necessary in these cases, for instance,
adaptive control, neural, or fuzzy control.
VIII. CONCLUSION
A global approach of the vector control of induction machines
has been presented. Analytical comparisons were presented and
a general classification proposed.
[1] F. Blaschke, “The principles of field orientation as applied to the
new TRANSVEKTOR closed-loop control system for rotating field
machines,” Siemens Review, pp. 217–220, 1972.
[2] R. W. De Doncker and D. W. Novotny, “The universal field oriented
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[3] M. Depenbrock, “Direct Self-Control (DSC) of inverter-fed inductionmachine,” IEEE Trans. Power Electron., vol. 3, pp. 420–429, 1988.
[4] R. Stephan, “Field oriented and field acceleration control for induction
motors. Is there a difference?,” in Proc. IECON, 1991, pp. 567–572.
[5] R. Ueda, T. Sonoda, and K. Koga, “Uniqueness for constant control of
torque transfer in induction motor,” in Proc. APEC, 1990, pp. 809–816.
[6] I. Takahashi and T. Noguchi, “A new quick response and high efficiency
control strategy of an induction motor,” IEEE-IA, vol. 22, no. 5, pp.
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[7] S. Yamamura, AC Motors for High Performance Applications. New
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[8] X. Xu, R. W. De Doncker, and D. W. Novotny, “A stator flux oriented
induction machine drive,” in Proc. PESC, 1988, pp. 870–876.
[9] R. Ribeiro, C. Jacobina, and A. Lima, “Vector control on induction motor
drive systems,” in 9th CBA, Vitoria, Brasil, 1992, pp. 1119–1124.
[10] W. Leonhard, Control of Electrical Drives. Berlin, Germany:
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[11] K. P. Kovacs, Transiente Vorgänge in Wechselstrommachinen Verein der
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[12]
, Transient Penomena in Electrical Machines, Amsterdam: Elsevier, 1984.
[13] J. Murphy and F. Turnbull, Power Electronic Control of AC motors. Oxford, U.K.: Pergamon Press, 1988.
[14] P. Krause, Analysis of Electric Machinery. New York: McGraw-Hill,
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[15] G. Heinemann, “Comparison of several control schemes for AC induction motors under steady state state and dynamic conditions,” in Proc.
EPE, 1989, pp. 843–848.
[16] G. Garcia, R. Stephan, and E. Watanabe, “Comparing the indirect field
oriented control with a scalar method,” IEEE Trans. Ind. Electron., vol.
41, pp. 201–207, 1994.
[17] M. Kasmierkowski and H. Tunia, “Automatic control of converter fed
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[18] D. Novotny and T. Lipo, Vector Control and Dynamics of AC
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[19] B. Bose, P. Lataire, and J. Steinke, “Remarks on paper ABB Medium
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SANTISTEBAN AND STEPHAN: VECTOR CONTROL METHODS FOR INDUCTION MACHINES: AN OVERVIEW
José Andrés Santisteban (S’83–M’86) was born in Lima, Perú. He received the
B.Sc. degree in electronic engineering from the Universidad Nacional de Ingeniería (UNI), Lima, Perú, in 1986 and the M.Sc. and D.Sc. degrees in electrical
engineering from the Universidade Federal do Rio de Janeiro (UFRJ), Brazil, in
1993 and 1999, respectively.
From 1988 to 1991, he worked as a Researcher and as an Assistant Professor
in the Electrical and Electronic Department of UNI. From 1993 to 1995, he
worked as a Researcher in UFRJ. Since 1999 he has been with the Universidade
Federal Fluminense (UFF), Niterói, Brazil, where he is currently an Associate
Professor in the Electrical Engineering Department. His current research activities include power electronics, electrical drives, and bearingless machines.
Dr. Santisteban is a member of the Brazilian Power Electronics Society (SOBRAEP).
175
Richard M. Stephan (M’89) was born in Rio de Janeiro, Brazil. He received the
B.Sc. degree in electrical engineering from the Instituto Militar de Engenharia
(IME), Rio de Janeiro, in 1976, the M.Sc. degree in electrical engineering from
the Universidade Federal do Rio de Janeiro (UFRJ) in 1980, and the Dr.-Ing.
degree in electrical engineering from Ruhr Universität Bochum, Germany, in
1985.
During 1977, he worked as an engineer at Furnas Centrais Elétricas, Rio de
Janeiro. Since 1978, he has been with the Department of Electrical Engineering
UFRJ, Brazil. From 1982 to 1985, he was on a leave of absence from UFRJ as a
DAAD (Deutscher Akademischer Austauschdienst) scholar at the Ruhr Universität Bochum. He spent his sabbatical leave at CEPEL, the Research Center of
ELETROBRAS in 1993. His main interests are in the fields of control of electrical drives and power electronics.
Dr. Stephan is a member of the Brazilian Society for Automatic Control
(SBA), and of the Brazilian Power Electronics Society (SOBRAEP).