Use of an extension of the park's transformation to determine control laws applied to a non sinusoidal permanent magnet synchronous motor

USE OF AN EXTENSION OF THE PARK'S TRANSFORMATION TO DETERMINE CONTROL LAWS APPLIED TO A NON-SINUSOIDAL PERMANENT MAGNET SYNCHRONOUS MOTOR M i e n GRENIER, Prof.. This modelling permits

USE OF AN EXTENSION OF THE PARK'S TRANSFORMATION TO DETERMINE CONTROL LAWS APPLIED TO A NON-SINUSOIDAL PERMANENT MAGNET SYNCHRONOUS MOTOR

M i e n GRENIER, Prof Tx Ing Jean-Paul LOUE

1,aboratoire dElectricit6 SIgnaux et Robotique (URA CNRS D1375) - ENS de Cachan -FRANCE

VAI.EO Systkmes Electriques - R & D Moteurs Electriques - C R E E L - FRANCll

Abstract An extension of the analyt~c Park's modelling to the case of non-sinusoidal permanent magnet

synchronous motor is proposed This modelling permits us to analyse the classical 120' voltage control

law and to deduce new vector control laws in "abc" and "dq" fiames, which allow best dynamic and steady state behaviours

Keywords Non-Sinusoidal Permanent Magnet Synchronous Motor Extended Park's Transformation

Vector Control

INTRODUCTION

Actually, permanent magnet synchronous motors can be

divided in two types of drives

To those which have a sinusoidal pattern back-electromotive

force, it can be applied a vector control, using a modelling of

this motor based on Park's transformation This is the most

powerful method for dynamic control

The others, which are less expensive to build, can just be

controlled in steady state, using a supply by square-wave

currents, if the backelectromotive force is supposed to have a

trapezoidal pattem 'They can be so compared to d.c drives

with an electronic collector [I] More sophisticated supplies

can be computed using the real pattern of the back

electromotive force in order to optimise the steady state

[2,3,4,5]] These control laws are design in steady state

We propose here an extension of the Park's transformation

which allows to get a dynamic model of these motors and

thus to perform a vector control of permanent magnet

synchronous motors with backelectromotive force of any

pattem

We assume that these drives have a constant air gap large

enough to be able to neglect the effects of the saturation

Thus we obtain various results conceming the steady state

and the transients We can analyse the properties of classical

controls (120" voltage control) or propose new structures

which use all potentialities of the motor and which have

better properties

EQUATIONS OF THE PERMANENT-MAGNET

SYNCHRONOUS DRIVE

with: (J!)=( ks F: :) where Ls and M, are

M M L

the self-inductance &d &e mhual-inductance of the stator coils As we suppose a constant air gap and no saturation, L,

and M, are constant Qra , and QrC are the rotor fluxes induced in the stator phases

The electrical equations of the machine cm be written under

the form:

where: PI Q = p.!2 P) a'* are the back-electromotive forces

of the machine, 0 is its instantaneous position and C2 is the angular speed)

Through an analysis of the consumed power by the machine,

we can deduced the expression of the torque:

T = p ( ~ r ~ i ~ ~ ' ~ i ~ + ~ ' ~ , i ~ ) (3)

The fluxes in stator phases obey to the equation:

PROPOSAL OF AN EXTENSION OF THE PARK'S The torque is then given by:

For a drive with sinusoidal distribution, @lo = Wrd = 0 , so the equation of the torqm is simplified to:

Classical Park's transformation is in fact the succession of

two transfomations

Tc = p.@Irq.iq with @'rq is a constant (9)

Concordia's transformation This expression of the instantaneous torque of the machine is

enough simple to pennit to write, for these motors, control

laws with very good dynamic performances, as vector control

for example

This first transformation allows to reduce a three-phase

system to an equivalent two-phase (xa,xp) system plus the

zero-sequence component xo A vector g is written under the

With these new variables, the electromagnetic torque is now

equal to:

T, = p ( @ ' ~ i ~ + @ ' ~ ~ i ~ + @ ' ~ i ~ ) (5)

For a motor with sinusoidal back-electromotive force, @Iro= 0,

thus the zero-sequence current io does not take part to the

generation of the torque, but Contributes to copper losses

This kind of motor is then often star connected, so ia+ib+ic=O

=> io* Using Concordia's transformation, the three-phase

synchronous drive with sinusoidal back-electromotive force is

so reduce to a two-phase system in the "a-p" kame

For a non-sinusoidal machine can be not equal to zero,

and then a zero-sequence current can be useful The

Conmrdia's transfonnation has no more such interest, but

allows however to get, for th~s kind of drive, an other three-

phase system totally decoupled Indeed, through the

Concordia's transformation, the electrical equations can be

written under the form:

Park's transformation

AAer this first transfoxmation, the Pa& transfomation

allows us to work in the rotor's reference, through a rotation

of an angle PO Using the new variables "c-dq", a vector

can be written:

1

where: P(pe) = ( cos(pe) -sin(@)

sin(pe) CO@)

For a star ~ o ~ e c t e d non-sinusoidal motor, we have tried to fmd an angle pC"0) which defmes "pseudo d q " axes, so

that @Ird = 0 As io = 0 , the expression of the torque of the motor will be nearly the same as the expression for a sinusoidal drive:

T, = p.@',(0) i, (10)

except that @Ir can be no more constant It depends of the rotor's position%

To get @Ird = 0 , we must have:

p is a function of 0 We can verify that, in the sinusoidal case,

p=O p depends only of the value of the variations of rotor fluxes, witch can be easily known through a measuring of back electromotive forces, the motor being not supply

We get so a transformation that we can consider as an

extension of the Park's transformation for the synchronous machines with backelectromotive forces of any pattem With this new transformation a vector g is written under the form:

We can also write the dynamic model of a star connected non- sinusoidal permanent magnet synchronous motor with constant air gap (figure 1)

Figure 1: dynamic model of P.M synchronous motor

The electrical equations (14) are:

vo= pQ.P,,

di V,= Rs.id+(L;MJ

V,= R*.iQ+(L,-M) $+ pQ(L;M).( I +

because io = 0

- pR(L;M,).( I + *) iq

id+ pQQq

and the torque is given by ( 1 0): l e = p.Wrq(0) iq

DYNAMIC BEHAVIOUR OF NON-SINUSOIDAL

PERMANENT-MAGNET SYNCHRONOUS DRIVES

A four poles pmanent-magnet synchronous motor has been

modelled through a finite element code 161 Stator phases

being not supplied, the fluxes induced in this stator phases

have-been computed (figure 2)

I t%

Figure 2 Rotor fluxes in the stator phases and their variations

v s the position of tlie rotor

The Variations of these fluxes give the shape of the back-

electromotive forces e,, e,, , ec

Using the extension of the Park's transformation, @Ira , @Irp and the variations of the angle p and of versus the rsition 0 have been computed (figure 3)

I

(mrr *e: 13.14O)

I

:

;

;

"

;

Figure 3: @ I m , @Irp , p and @Irq versus the position of the rotor Several control laws have been simulated with this motor and their performance compared

1200 Voltage control with d.c current loop

When we look at the pattern of the variation of the rotor fluxes, we see that they are nearly constant during almost 120"/p As torque is given by:

(3)

're = p ( P ~ ~ i ~ + @ ' ~ ~ i h ~ ~ , i , )

a simple control law can be applied Stator phase (a) is supplied with a positive voltage E during the 120" inkrval of electrical angle where Wra is near of its maximal value @', supplied with a negative voltage -E during the 120" interval

of eltvtrical angle where @Im is near of its minimal value

-@I, and not supplied during the 2 x 60" other interval of the electrical position [ 11 Only two phases are then supplied in the same time and the torque can be estimated to:

vb

v:

-

Figure 4a: Scheme for 120" voltage control with

compensation of back electromotive force

IP controller has been computed to get successively a slow

and a fast response time The speed being constant, we study

the response of the system to a torque step

7 Torque I T ref

t

Figure 4b: 120" voltage control: torque and currents vs time

for a controller with slow response lime ( 15 ms)

With a controller with a slow response time, we observe

torque ripples due to the commutations (figure 4b) With a

fast response time, we are able to decrease the duration of the

torque ripples (figure 4c) but we can't, at high speed, reduce

their amplitude because of the saturation of the supply

voltage (figure 4d) [1,2]

1 Torque /Tref

t

~~ ~ _

Figure 4c: 120' voltage control: torque vs time for a controller

with fast response time (0.5 ms)

Instantaneous current variations are not possible with finite

supply voltage But, when we look at the d and q components

of the current, we see that the variations of the phases

currents are linked to the variations of the id currcnt

However, the id current don't produce torque, only copper

losses Square wave currents are not the best shape of

currents for this kind of motor

~

L-e+*,

1 L z - ~

t B

t

If .

Figure 4d: 120" voltage control: currents, supply voltage, id and iq vs time, for a controller with fast

r e q " time (0.5 ms)

"abc" frame vector control

Thanks to the extended Parks model, we can compute the

shape of the optimal currents to get the desired torque T ref :

Tref

- i, must be equal to: i,, ,cf = 7

- as the copper losses are given by: P, fl 'q = Rs( id%;), to minimise these losses, we must have id, = 0

We have tried to impose theses computed phases currents

(figure sa) and have note the results with two different

controllers

Torque I 'I' ref

t ia i ia ref

Figure 5b: "abc" frame vector control: torque and currents

versus time for a controller wth slow response time (1 5 ms)

Torque I T ref

t

ib

ic

Figure 5c: "abc" frame vector control: torque and currents

versus time for a controller with fast response time (0.5 ms)

With the controller with slow response time the currents are

not able to follow the reference currents The angle between

the rotor and the stator induction is no more 90", like for a

classical sinusoidal synchronous motor the average value of

the torque is reduced

With a faster time response, steady state torque is almost

constant but the implementation of such system present

"dq" frame vector control

To get best performance when response time has to be limited to avoid instability, we propose a "dq" frame vector control (figure 6a) where the controlled variahle 'r ref and id ref

are independent of the rotor position

Figure 6a: Scheme for "dq" vector control 'Tested "dq" vector control includes compensations of back- electromotive forces but do not take into account the coupling

of d and q phases

t

ia / ia ref

t

ib I ib ref

ic i ic ref

7

Figure 6b: "dq" frame vector control: torque, iq , iq ref,

id and phase currents for a controller with slow response time (IS ms)

For the controller with slow response time, we get best

performance with the "dq" h evector control as with the

"abc" frame vector control There is no more phase angle

m r between the phase currents i, , ib and i, and the ideal

currents i, rd, ib nf and i, nf (figure 6b)

" vc

I

1 id

voltage during the s W y state The "dq" frame is better than

" a h " frame It need more complex implementation, but the

band width may be smaller

CONCLUSION

The proposed extension of the Park's transfonnation permits

us to have a best knowledge of the non sinusoidal permanent

magnet synchronous motor We are so able as well to analyse the classical control laws such as 120' voltage control for

example, as to deduce of this new model, new control laws witch allow best dynamic and steady state behaviours, such

as vector controls in "a@ or "dq" frames

References

1

2

3

4

5

6

T.J.E Miller, "Brushless Permanent magnet and Reluctance Motor Drives", Clarendon Press, oxford 19x9

C.-S Berendsen G Champenoiq J Davoine, "Commutation strategy for k s h l e s u D.C Motors: Influence on Instant Torque", APEC 90 Los Angeles 11-16 Mars 90

Concept for the Veriical Axis of a Selective Compliance Arm

Robot", 3rd International Conference on Power Electronics and Variable-Speed Drives

C Marchand, A Razek, "Electromagnetic Modelling to

Optimise Low Speed and Position Control in Servo Motor", Intamtianal Workshop on Electric and Magnetic Fields from

Numerical Models to Industrial Applications, liege 28-30

C Menu "Machines Synchrones a FEM Trapezoidales:

autopilotage et c o n t d e de couple nunhique Etude et

simulation de dfikents strat6gies de " m a n & " Thise de

I'INF'G June 1989

Sinewave Permanent Magnel Synchronous Drive by an

Extension of the Park's Transformation", IMACS TCI-93,

Montreal 7-9 July 1993