Comparison of cartesian vector control and polar

vector control for induction motor drives

Alain Bouscayrol

1,*

, 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

Mathematics and Computers in Simulation 46 (1998) 325±337

ÐÐÐÐ

* Corresponding author. Alain.Bouscayrol@univ-lillel.fr

new affectatiion in L2EP de Lille, USTL CiteÂ Scientifique, 59 655, Villeneuve d'Ascq (France)

PII S 0 3 7 8 - 4 7 5 4 ( 9 7 ) 0 0 1 4 5 - 6

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 !

) of the direct-axis d with respect of the

reference stator fixed frame (a, b)

. 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

Fig. 1. Control of an induction motor drive.

326

A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325±337

rotor frame,

(1)

The slip relation can also be defined with the angular frequencies (where

is the shaft rotation

speed):

(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:

ÿ !

0 R

ÿ !

ÿ p

0 R

ÿ p

<
>

(3)

with

<
>

(4)

The mechanical model is specified by the classical speed Eq. (5) in function of the electromagnetic

torque, T

, and of the load torque, T

load

ÿ T

load

(5)

Fig. 2. (a, b) and (d, q) axes.

A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325±337

327

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:

(6)

(7)

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

;

0. So, the general Eqs. (3)±(6), lead to the decoupling model of the flux

and of the rotation speed by using the stator currents in Laplace representation (with s Laplace

operator):

<
>

(8)

with

<
>

and

<
>

Then, these intermediary variables are expressed with the supply stator voltages by using non-linear

decoupling terms, E

and E

ÿ E

<
>

(9)

328

A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325±337

with

1 ÿ

<
>

and

ÿL

<
>

C. Fixed stator frame with the stator flux

This frame is a static frame where !

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):

ÿ E

<
>

(10)

with

<
>

and

3. Control vector structures

The control in the (a, b)

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

A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325±337

329

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

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

regulated separately with specific controllers (C

, C

,) associated to the current controllers (C

, C

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

with the slip relation (1). The third domain is associated to the rotating frame and contains the different

Fig. 3. Polar vector control structure.

330

A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325±337

controllers to define the (d, q) stator voltage references. The (d, q) model leads to the decoupling terms

, E

(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)

, 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)

domain

contains the flux controllers. Moreover, it uses an (a, b) model which gives the flux component

estimations and the decoupling terms, E

, E

(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.

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.

Fig. 4. Cartesian vector control structure.

A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325±337

331

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)

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 (R

L 10%), and by a load torque step (50% of the nominal

torque) at t 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.

332

A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325±337

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.

Fig. 5. Robustness test of the polar vector control. (R

L 10%, T

load

50%T

nom

Fig. 6. Robustness test of the cartesian vector control. (R

L 10%, T

load

50%T

nom

A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325±337

333

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.

Fig. 7. Phase influence on the polar control.

334

A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325±337

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.

Fig. 8. Phase influence on the cartesian control.

Fig. 9. Experimental results of cartesian control.

A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325±337

335

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

controller of the X variable

decoupling term

damping friction constant

, V

stator currents and voltages

inertia constant of the motor

, L

stator and rotor self inductances

mutual inductance

number of pole-pairs

PWM

pulse-width modulation

, R

stator and rotor resistances

time derivated operator

, T

stator and rotor time constants (T

)

, T

load

electromagnetic and load torque

stator and rotor flux

i components of the fluxes

phase of the d-axis with the a-stator-axis

phase of the d-axis with the a-rotor-axis

phase of the a-rotor-axis with the a-stator-axis

resultant leakage constant, 1 ÿ M

angular frequency associated to

(! d=dt)

angular frequency associated to

rotor shaft frequency (

p d

=dt)

, X

(a, b) components of X

, X

(d, q) components of X

^X:

estimated value of X

(a, b)

fixed stator frame

(d, q)

rotating frame

336

A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325±337

References

[1] W. Leonhard, 30 years space vectors, 20 years field orientation, 10 years digital signal processing with controlled ac

drives ± a review, EPE Journal 1(1) (1991) 13±20.

[2] M. Pietrzak-David, Algorithmes de commande d'un variateur eÂlectrique asynchrone: controÃle dynamique du flux,

commande optimale en position, TheÁse de Docteur es-sciences, LEEI-INP Toulouse, 1988.

[3] J.P. Caron, J.P. Hautier, ModeÂlisation et commande de la machine asynchrone, Editions technip, Paris, 1995.

[4] V.R. Stefanovic, Industrial ac drives status of technology, EPE Journal 2(1) (1992) 7±24.

[5] W. Leonhard, Adjustable-speed ac drives, Proceeding of the IEEE 76(4) (1988) 455±471.

[6] B.K. Bose, Power Electronics and AC Drives, Prentice-Hall, Englewood Cliffs, New-Jersey, 1986.

[7] X. Xu, D. Novotny, Implementation of direct stator flux orientation control on a versatile DSP based system, IEEE Trans.

on Industry Applications 27(4) (1991) 694±700.

[8] E. Ho, P. Sen, Decoupling control of induction motor drives, IEEE Trans. on Industrial Electronics 35(2) (1988) 253±

262.

[9] R. De Doncker, D. Novotny, The universal field oriented controller, IEEE Trans. on Industry Application 30(1) (1994)

92±100.

[10] P. Vas, Vector Control of AC Machines, Clarendon Press, Oxford, 1990.

[11] R. Lorenz, T. Lipo, D. Novotny, Motion control with induction motors, Proceeding of the IEEE 82(2) (1994) 1215±1240.

[12] S. Siala, B. de Fornel, M. Pietrzak-David, Comparative study of continuous and sampled control performances in

asynchronous drives, IMACS ± Modelling and control of electrical machine, Elsevier Science, North-Holland, 1991, pp.

179±183.

[13] A. Bouscayrol, Structures d'alimentation et strateÂgies de commande pour des systeÁmes multimachines asynchrones ±

application aÁ la motorisation d'un robot mobile, TheÁse de doctorat, LEEI-INP Toulouse, 1995.

[14] A. Bouscayrol, M. Pietrzak-David, B. de Fornel, Master-slave strategy for asynchronous robot propulsion set fed by four-

legged PWM inverter, Proceeding of ICEM Conference, Paris, 1994, pp. 209±214.

[15] P. Nandam, P. Sen, Analog and digital speed control of dc drives using proportional-integral and integral-proportional

control techniques, IEEE Trans. on Industrial Electronics, vol. IE-34, No. 2, 1987.

A. Bouscayrol et al. / Mathematics and Computers in Simulation 46 (1998) 325±337

337

Wyszukiwarka

Podobne podstrony:
A Comparison Of Dividend Cash Flow And Earnings Approaches To Equity Valuation
2002 Sensorless vector control of induction motors at very low speed using a nonlinear inverter mode
Comparison of Human Language and Animal Communication
Marijuana is one of the most discussed and controversial topics around the world
Comparison of the Russians and Bosnians
Comparative eco toxicity of nanoscale TiO2, SiO2, and ZnO
Issue of Gun Control and Violence As Seen in the U S and
CONTROL AND THE MECHANICS OF START CHANGE AND STOP
A Comparison of Plato and Aristotle
A Comparison of the Status of Women in Classical Athens and E
Comparison of the U S Japan and German British Trade Rivalr
Comparison of Plato and Aristotle's Political Theories
Comparison Of Judaism And Christianity
Comparison of theoretical and experimental free vibrations of high industrial chimney interacting
Alta J LaDage Occult Psychology, A Comparison of Jungian Psychology and the Modern Qabalah

więcej podobnych podstron