Mathematics and Computers in Simulation 46 (1998) 325Ä…337
Comparison of cartesian vector control and polar
vector control for induction motor drives
Alain Bouscayrol1,*, Maria Pietrzak-David, Bernard de Fornel
Laboratoire d'Electrotechnique et d'Electronique Industrielle, UPRESA au CNRS n8 5004, INP Toulouse, ENSEEIHT,
2 rue Camichel, BP 7122, 31071 Toulouse cedex 07, France
Abstract
The performances of a vector control in the stator fixed frame are studied. This so called cartesian structure is compared
with a classical vector control, which uses the flux polar coordinates. This cartesian control avoids the phase determination
problem, but needs more complex controllers. # 1998 IMACS/Elsevier Science B.V.
1. Introduction
During the last 10 years, the induction motor drives are often controlled by vector control strategies
[1]. So, the static laws (as the classical V/f law) have been progressively replaced by these dynamic
methods, in which the transients are really better [2]. Moreover, the lower speed control can be realized
with a good behaviour with theses new techniques [3]. This evolution allows to obtain dynamic
performances for the induction motor drives as dc machines but with a lower cost [4].
The vector control techniques are based on the flux vector control (magnitude and phase) [5,6]. The
accurate knowledge of the flux magnitude and of the flux phase is absolutely necessary. In an
appropriated quadrature frame, the flux and the electromagnetic torque can be really decoupled. So a
separated control of them can be easier realized with some simple structures as in dc machines.
Many different schemes have been proposed in function of diverse criteria. The first one is the used
supply: current source (high power and high speed) or voltage source (lower or medium power) [5]. In a
second step, the application involves the choice of the mechanical control variable: torque, speed, or
position. So control loops can be defined. Then, the control flux and the associated quadrature frame
must be chosen: stator flux (for high speed and weakening flux operation [7] and for a simpler control
structure with voltage source [8]), rotor flux (for lower speed [9] and for a simpler control structure
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* Corresponding author. [e-mail]
1
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new affectatiion in L2EP de Lille, USTL Cite Scientifique, 59 655, Villeneuve d'Ascq (France)
0378-4754/98/$19.00 # 1998 IMACS/Elsevier Science B.V. All rights reserved
PII S 0 3 7 8 - 4 7 5 4 ( 9 7 ) 0 0 1 4 5 - 6
326 A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325Ä…337
with current source [8,10]), magnetizing flux (which involves complex structures [10]) or stator and
rotor flux (for a large speed range [9,11]). Moreover, the vector control structure is depending on the
determination technique of the flux phase [10,11]: flux measurement or flux estimation (so called direct
control), flux model using the speed measurement by the slip relation (so called indirect control). Thus,
the chosen flux can be obtained with an estimator (open loop observer which leads to a simpler
structure) or with a closed-loop observer (which leads to more robust behaviour) [11].
These different structures are based on the same approach. A rotating frame associated to the
controlled flux is used to simplify the induction machine model and to obtain decoupled controllers for
the flux and for the torque. But another strategy has been proposed by using the fixed stator frame
[12,13]. In this case, the cartesian components of the flux are controlled instead of its polar coordinates
(phase and magnitude). This new vector control structure has been verified in a mobile robot traction
application [14]. But its performances are not yet pointed out. This paper proposes a comparison
between the cartesian structure and the most popular structure of the polar vector control. The
drawbacks and the advantages of each one are discussed.
2. Induction machine models
The system to be controlled is composed by a three-phased voltage inverter associated to an
induction machine. The rotation speed of the motor shaft and the stator flux have been chosen as
control variables for a mobile robot propulsion application [14]. While the speed is directly measured,
the flux is estimated by a determinist observer through the stator current measurements. So, the speed
regulation and the flux control lead to the three-phased voltages which are imposed on the machine
(Fig. 1). A classical Pulse Width Modulation (PWM) is chosen to define the commutation orders of the
voltage inverter.
A. General model of induction machine
The electrical model can be defined in appropriate quadrature frames by considering simplifying
assumptions of the machine linearity. A general rotating frame (d, q) can be considered (Fig. 2). It
depends on its rotating phase (or by its angular frequency !ds) of the direct-axis d with respect of the
ds
reference stator fixed frame (a, b)s. This specific phase can be expressed with the electric phase
between the stator and the rotor, , and with the slip phase between the direct-axis d and the rotating
R
Fig. 1. Control of an induction motor drive.
A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325Ä…337 327
Fig. 2. (a, b) and (d, q) axes.
rotor frame, :
dr
ˆ ‡ (1)
ds dr R
The slip relation can also be defined with the angular frequencies (where is the shaft rotation
R
speed):
!ds ˆ !dr ‡ p R (2)
In this general frame, the Park's transformation leads to the electrical equations of the induction
machine in function of the machine parameters and of the angular frequencies:
8
d
>
> Vsd ˆ RsIsd ‡ !ds
sd sq
>
>
dt
>
>
>
>
> d
>
>
Vsq ˆ RsIsq ‡ ‡ !ds
sq sd
<
dt
(3)
>
d
>
>
0 ˆ RrIrd ‡ !ds p Ä…
> rd R rq
>
> dt
>
>
>
>
> d
:
0 ˆ RrIrq ‡ ‡ !ds p Ä…
rq R rd
dt
with
8
ˆ LsIsd ‡ MsrIrd
>
sd
>
<
ˆ LsIsq ‡ MsrIrq
sq
(4)
ˆ LrIrd ‡ MsrIsd
>
rd
>
:
ˆ LrIrq ‡ MsrIsq
rq
The mechanical model is specified by the classical speed Eq. (5) in function of the electromagnetic
torque, Tem, and of the load torque, Tload:
d
Tem Tload ˆ J ‡ f (5)
R R
dt
328 A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325Ä…337
Different equations can lead to the electromagnetic torque [3], in function of different fluxes and
currents. They represent the electro-magnetical energy conversion. Two of them are used:
Msr
Tem ˆ p Isq IsdÄ… (6)
Lr rd rq
2
r
Tem ˆ p !dr (7)
Rr
All of these equations have to be mixed in function of the chosen control variables and of the chosen
frame. They have to lead to the reference stator voltages for the inverter modulation.
B. Rotating frame associated to the rotor flux
In this case, the rotating frame (d, q) is oriented with the rotor flux. This method leads to the so called
oriented field control, and is the most popular choice [5]. Even if there are no simplification with the
angular frequencies, an easier model is obtained because the rotor flux has just one component on the
direct axis ˆ ; ˆ 0Ä…. So, the general Eqs. (3)Ä…(6), lead to the decoupling model of the flux
r rd rq
and of the rotation speed by using the stator currents in Laplace representation (with s Laplace
operator):
8
K
>
>
ˆ ˆ Isd
< r rd
1 ‡ s
(8)
> K
>
:
ˆ Isq
R
1 ‡ s
with
8
K ˆ Msr
>
<
Lr
>
:
ˆ
Rr
and
8
pMsr r
>
>
K ˆ
<
fLr
> J
>
:
ˆ
f
Then, these intermediary variables are expressed with the supply stator voltages by using non-linear
decoupling terms, Esd and Esq:
8
Kd
>
> Isd ˆ Vsd EsdÄ…
>
<
1 ‡ s
d
(9)
>
> Kq
>
:
Isq ˆ Vsq EsqÄ…
1 ‡ s
q
A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325Ä…337 329
with
8
TsTr
>
>
Kd ˆ Kq ˆ
>
<
LsTr ‡ LsTs1 Ä…
>
TsTr
>
>
: ˆ ˆ
d q
Tr ‡ Ts1 Ä…
and
8
Msr
>
>
Esd ˆ Ls!dsIsq
>
<
LrTr r
>
> Msr
>
:
Esq ˆ Ls!dsIsd ‡ p
R r
Lr
C. Fixed stator frame with the stator flux
This frame is a static frame where !ds ˆ 0. The electrical equations are simplified but contain all flux
components. The stator flux is chosen as control variable because it leads to a simpler structure in the
case of a voltage source. So the stator flux components and the rotation speed can be expressed in
function of the stator voltages and of the stator flux phase. A complete study leads to the specific
transfer functions by using Eqs. (3), (4), (5) and (7):
8
1
>
>
ˆ Vsa EsaÄ…
sa
>
>
s
>
>
>
<
1
(10)
ˆ Vsb EsbÄ…
sb
>
> s
>
>
>
> K
>
:
ˆ !dr
R
1 ‡ s
with
8
p 2
>
r
>
K ˆ
<
fRr
>
J
>
:
ˆ
f
and
Esa ˆ RsIsa
Esb ˆ RsIsb
3. Control vector structures
The control in the (a, b)s frame has been called cartesian vector control, because of the cartesian
components to be regulated. The control on the (d, q) frame associated to is well known as field
r
330 A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325Ä…337
Fig. 3. Polar vector control structure.
oriented control, because of the orientation of direct axis, d, on the rotor flux vector. In this paper we
use the name of polar vector control in this case, because of the flux polar coordinates. In the two
methods, the flux and the rotation speed have to be measured or estimated. In this paragraph, a classical
case is studied, where is obtained by a speed sensor, and the fluxes are obtained by an observer
R
through the current and voltage measurements. Moreover, you can remember, the chosen supply source
needs the stator voltages as modulation references.
A. Polar vector control structure
This control structure (Fig. 3) is involved by the transfer functions (8) and (9). In a first step,
measurements and estimations lead to the flux phase, the flux magnitude and the real rotation speed. In
this frame, the flux (d component, ) and the shaft rotation speed, , are independent, and can be
rd R
regulated separately with specific controllers (C , C ,) associated to the current controllers (CId, CIq).
In a last step, the (d, q) voltages are transformed in the (a, b) voltages, then, in the three-phased voltage
references.
This structure can be decomposed in four domains. The first one is the real electrical domain where
the electrical variables are composed by real three-phased components. The second one manages
cartesian components in the (a, b) fixed frame. The used (a, b) model involves relations in this stator
frame (including the slip relation) and leads to the estimation of the cartesian flux components and of
the rotor flux phase, . This transform angle, which is needed for the frame changing, can be obtained
dr
with the slip relation (1). The third domain is associated to the rotating frame and contains the different
A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325Ä…337 331
Fig. 4. Cartesian vector control structure.
controllers to define the (d, q) stator voltage references. The (d, q) model leads to the decoupling terms
Esd, Esq (9). The last domain is associated to the mechanical model.
B. Cartesian vector control structure
In this structure (Fig. 4), the flux phase reference is firstly determined by the speed controller and the
slip relation (10) (1). This phase leads to the flux cartesian references with the help of the flux
magnitude reference. Then, a double flux controller (one for each component) can define the stator
voltage references with respect of the transfer functions (10).
Even if this control is associated to the fixed stator frame (a, b)s, the four domain can be found also.
Of course, the real electrical domain is the same as the polar vector control one. The (a, b)s domain
contains the flux controllers. Moreover, it uses an (a, b) model which gives the flux component
estimations and the decoupling terms, Esa, Esb (10). The (d, q) domain is fictive in this case and owns
no operations. But it contains the stator flux magnitude reference, which can be represented by a direct
component reference in the (d, q) frame associated to the stator flux. Thus, the mechanical domain
yields to the flux phase, , with the speed controller and the slip relation.
ds
C. Comparisons of the both structures
Of course the both studied controls are different because of the chosen flux variables (stator or rotor
flux). But the phase determinations which are used, are the same for the both structures. The flux phase
is obtained by using the slip relation with the help of the speed measurement (2). So, they are both
indirect controls with regulated voltage source. The differences which are shown is this sub-paragraph,
point out the structure changing between the polar and the cartesian methods.
332 A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325Ä…337
The flux phase utilization is different. In the polar vector control, it is needed four time (see Fig. 4).
The flux phase determination is very important, because an error involves the non-orientation of the (d,
q) frame with the real flux magnitude, and so, the operating model does not allow a good decoupling
between flux and electromagnetic torque. In the cartesian structure, the flux phase is used only once, to
determine the flux references. So, this second structure can be less sensitive to a phase determination
error.
In another hand, the polar structure have to control continuous variables, which are separated in two
independent loops. The cartesian structure owns sine wave forms fluxes, and needs faster controllers to
yield the same performances. Moreover, the flux control and the speed control are mixed in this case.
The decoupled character is present, but is less evident.
Thus, the cartesian structure is simpler because its owns less function blocks and because the
coupling terms are not complex and not depending on variable estimations.
4. Comparison of the vector controls
A. Comparison criteria
All of the used discrete controllers are Integral-Proportional regulators because they avoid to have a
zero in the closed-loop transfer function [15]. In the both structures, a determinist observer is used to
define the fluxes in the (a, b)s frame.
A specific speed trajectory is used to study the induction machine behaviour on the global speed
range. In a another hand, the flux reference is kept to a constant value, and flux weakening operation is
not tested.
A robustness test is employed in a first study. It allows to compare the dynamic behaviour of the both
controls. But for this simulation test, the flux observers are not connected, because they are more
sensitive to perturbations than the control himself. Moreover, the observer robustness is not the same
for the stator flux and for the rotor flux. So, the performances of the both structures can be pointed out
without considering the chosen control variable difference. This robustness test is composed by an
electrical parameter change ( Rr ˆ Rs ˆ L ˆ 10%), and by a load torque step (50% of the nominal
torque) at t ˆ 3 s.
In a second time, the influence of the phase determination error is tested. The simulations use the
quantification of the phase flux with only 8 bits as in the experimental testing bench. Only the flux
phase is quantified because this study have to verify its influence without considering the flux control
variable difference. Indeed, the flux phase determinations do not use the same variables and model
parameters. In our case, these precautions are not justified, and a simulation with a global quantification
has shown smaller differences than with the only phase quantification.
B. Robustness studies
If the controllers are well determined, the polar vector control leads to a good dynamic behaviour
on the global speed range (Fig. 5). The cartesian structure yields also good performances except
in lower speeds without load torque (Fig. 6). Moreover, the torque ripple is a little more important
for the cartesian vector control, and the stator flux owns a smaller wave too. Indeed, the harmonics of
the inverter modulation (PWM) are less filtered on the stator flux than on the rotor flux. A
second reason can explain this problem. The flux reference must be controlled in transient operation.
A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325Ä…337 333
Fig. 5. Robustness test of the polar vector control. ( Rr ˆ Rs ˆ L ˆ 10%, Tload ˆ 50%Tnom).
Fig. 6. Robustness test of the cartesian vector control. ( Rr ˆ Rs ˆ L ˆ 10%, Tload ˆ 50%Tnom).
But the chosen controller involves a training error and so the sine wave form flux components
have delay with the references. More complex controllers have to be chosen to reduce these
disturbances.
334 A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325Ä…337
Fig. 7. Phase influence on the polar control.
In conclusion on the robustness test, the both structures have the same performances. The same
perturbations are obtained with respect of a test without parameter variation. The differences between
the results are function of the chosen flux as control variable, and of the chosen controllers.
C. Precision phase influence
As shown in the structure descriptions, the polar vector control is very dependent on the flux phase.
But in an discrete-time realization with microprocessor, its determination can involve some error. With
a classical 8 bits coding for this important variable, the dynamic behaviour of the polar control leads to
flux disturbances with lower speeds coupled with an important load torque (Fig. 7). In another hand,
the torque ripple is larger than without the flux phase coding, and so small perturbations are suitable on
the shaft rotation speed. These results confirm, the phase error involves a coupling between torque and
flux, specially for a smaller phase variation (or lower speed). Indeed, the (d, q) model used for the
control is not really oriented on the rotor flux with a flux phase error.
For the cartesian vector control, the phase influence is not preponderant (Fig. 8). Even if it amplifies
the torque ripple, the flux keeps a constant value in the global speed range. Of course the lower speed
problem is also amplified. But as discussed in the precedent sub-paragraph, these perturbations are
independent on the phase determination.
The phase error study shows a more sensitive influence for the polar vector control. Simulations
with a 16 bits coding of the flux phase, show this problem is reduced when increasing the flux
phase precision. But if the slip relation is used to define the flux phase (as in indirect control), the
precision of the speed sensor must be also increased. In an another hand, if a direct control is used, the
flux phase determination is sensitive to the flux estimation coding, which needs an important coding bit
number.
A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325Ä…337 335
Fig. 8. Phase influence on the cartesian control.
Fig. 9. Experimental results of cartesian control.
D. Experimental results of the cartesian control
The cartesian control have been realized in an experimental testing bench. The shaft rotation speed is
obtained by an incremental speed sensor, and the stator flux components are calculated with a discrete-
time observer through the stator current measurements. The global control is realized with a standard
68 000 Motorola microprocessor (16 MHz). The flux phase is coded with 8 bits, and is obtained with
the slip relation in a specific electronic card. The experimental results verify the good behaviour of this
cartesian vector control on the global speed range (Fig. 9). But problems of low speed measurements do
not allow to characterise the lower speed disturbances as indicated in the simulation results.
336 A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325Ä…337
5. Conclusion
The both methods are really vector control techniques. The first one uses the flux cartesian
components on the fixed stator frame, and the second one the flux polar coordinates in an appropriated
quadrature rotating frame.
The cartesian vector control is an interesting alternative of the classical dynamic controls of an
induction machine. Its stator flux structure is easier than the classical vector control one. If its allows
good dynamics behaviour, the cartesian control is less dependent on the flux phase determination,
which is often a sensitive point in discrete realization.
In another way, this cartesian strategy involves more complex controllers, because they have sine
wave form references, instead of continuous values with the polar control. Moreover the flux and speed
control are mixed, and not separated in independent loops. At least, the lower speed behaviour is more
disturbed than the polar control one.
6. Nomenclature
CX controller of the X variable
Es decoupling term
f damping friction constant
Is, Vs stator currents and voltages
J inertia constant of the motor
Ls, Lr stator and rotor self inductances
Msr mutual inductance
p number of pole-pairs
PWM pulse-width modulation
Rs, Rr stator and rotor resistances
s time derivated operator
Ts, Tr stator and rotor time constants (Ti ˆ Li=Ri)
Tem, Tload electromagnetic and load torque
, stator and rotor flux
s r
, i components of the fluxes
si ri
phase of the d-axis with the a-stator-axis
ds
phase of the d-axis with the a-rotor-axis
dr
phase of the a-rotor-axis with the a-stator-axis
R
2
resultant leakage constant, ˆ 1 Msr=LsLrÄ…
!ds angular frequency associated to (! ˆ d =dt)
ds
!dr angular frequency associated to dr
rotor shaft frequency ( ˆ p d =dt)
R R R
Xa, Xb (a, b) components of X
Xd, Xq (d, q) components of X
^
X: estimated value of X
(a, b)s fixed stator frame
(d, q) rotating frame
A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325Ä…337 337
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