A park like transformation for the study and the control of a nonsinusoidal brushless DC motor

By analogy to the case of sinusoidal flux distribution B X M , a Park-like transformation is defmed This transformation allows us to elaborate high performance control laws in "pseu

A Park-like Transformation for the Study and the Control

of a Non-Sinusoidal Brushless DC Motor

D Grenier 1>2, L.-A Dessaint ', 0 Akhrif J.-P Louis Groupe de Recherche en Electronique,de Puissance

et Commande Industrielle (GREXI)

Ecole de Technologie Sup6rieure

4750, Avenue Henri-Julien

MONTRJhL (Qukbec) H2T 2C8 CANADA

Abstrmt - The actual proposed techniques to cancel the torque

ripple in the non-sinussidal flux &ti+bwtion Brttfelefs BC

motors (BDCM) being not entirely satisfactory, a novel

approach for the control of this kind of motors is proposed By

analogy to the case of sinusoidal flux distribution B X M , a

Park-like transformation is defmed This transformation

allows us to elaborate high performance control laws in

"pseudo-dq" frame Thus, a nonlinear state feedback control

scheme is proposed and simulation results are presented The

feasibility of the implementation of this control scheme is

discussed

Laboratoce d'Electricit6, Signaux et Robotique

(LESiR) U.R.A C.N.R.S D1375

Ecole Normale Su@rieure de Cachan

6 1, Avenue du Pdt Wilson

94235 CACHAN Cedex FRANCE

iity of the implementation of the non-linear control scheme

is considered in section V

11 STATE OF l7-E ART IN TORQUE RIPPLE REDUCTION Several control schemes for PWM inverter fed drives have been reported in the litterature They can be classified

in three groups :

A Two-phase feeding

I INTRODUCTION For many years, research in the field of brushless DC

motors (BDCM) with non sinusoidal flux distribution was

aimed at reducing torque ripple Indeed, although torque

ripple is filtered out at high speeds by system inertia, it

becomes particulary annoying at low speed and in direct-

drive applications Moreover, torque rippie can cause

acoustic noise and mechanical vibrations Various tech-

niques have been proposed to minimize torque ripple but

none of them is entirely satisfactory Actually, the electroma-

gnetic torque can be efficiently controlled only for BDCM

with sinusoidal flux distribution These motors require how-

ever more complex mechanical design and are therefore

more expensive

In this work, we will show how the well known control

schemes for sinusoidal flux dstribution motors can be

extended to the case of a non sinusoidal flux distribution

This extension is based on a Park-like transformation that

allows to represent the non sinusoidal flux distribution motor

equations in the rotor field reference frame ("dq"-fkame)

Futhermore, a state feedback linearization control law will

be applied to the BDCM model in order to control precisely

the electromagnetic torque

The paper 1s orgamed as follows In section 11, a review

of the state of the art in torque ripple reduction is presented

In section 111, our Park-like transformation for non sinu-

soidal BDCM is introduced In section IV, the application of

the nonlinear feedback linearization technique is described

and simulation results are compared to those obtained with a

classical current vector control scheme Finally, the feasib-

In this feeding scheme, the inverter operates as commu- tator feeding the DC current into two phases of the motor, the third phase being in open circuit The rotating field is

created by switching the DC current from phase to phase at intervals equivalent to 60 electrical degrees 111 The syn-

chronous motor and its electrOnic commutator is analog to a classical DC motor and similarly it can be associated to a

DC chopper and a DC current regulator This current is used

as control variable for the motor torque

Although this current feeding scheme is economically

attractive, it suffers &om inherent torque pulsation due to

commutations between motor phase currents In addition, it

supposes that the motor's emfs are constant over 120 electrical degrees which requires specially designed trapezoidal motor [2] The torque pulsation due to the commutations can be reduced only by using "smooth"

comutations [3],[4], which leads to the adoption of a three- phase feeding control scheme [5]

B, Three-phase feeding

In this feeding scheme, arbitrary feed current waveforms

are imposed to the motor Various methods have k n pro-

posed to calculate the appropriate current profiles to drive a given BDCM without torque ripple These methods based on

the selective elimination of torque harmonics by the injection

of feed current harmonics using Fourier series decomposition of the emf's [6],[7], finite element analysis [81,[91, etc

However, since motor windings are inductive, the motor drive electronics has limited ability to produce the required

current waveforms Several current control schemes for

PWM inverters feeding BDCM have been studied and

reported in the litterature, mainly: the hysteris controller

and the linear controller [ 101

In the hysteresis current control scheme, the switching

frequency is variable over a wide range If the frequency

becomes too high, the switching losses become unacceptable

wheras if the frequency falls in the audible bandwith,

acoustic noise occurs

In the linear control scheme, the motor currents are com-

pared to the references and the errors are processed by PI

controllers to provide a control signal for a PWM modulator

Good performances can be obtained at low and medium

speeds At high speed, the phase shift introduced by the

controller may become unacceptable A large bandwith

controller is thus needed to minimize the phase shift

C Use of State Transformations

In this case, however, a Park-like transformation can be

defined We propose therefore in this paper a new transformation which preserves the same advantages as the

Park transformation

A Mathematical Model of the BDCM

We suppose that the motor has the following typical

- The airgap length is constant and large since the magnets are surface mounted and have the same permeability as

air As a result, the armature reaction is negligible

- The magnetic circuit has an important air part so that the

effects of the saturation in the iron parts are negligible

- In order to simplify the study, only the electromagnetic torque is considered Nevertheless, the cogging torque can be easily taken into account as shown in the appendix

features :

As seen previously, the tracking of imposed phase

current profiles in inductive windings requires power viour can be described by its electrical equations :

circuitry with high capabilities This costly requirement can

be avoided by replacing the variable phase current references

by "dq" current component references These component

have constant values in steady state Hence, the problem of

variable reference tracking simplifies into a problem of

constant reference regulation Unfortunately, the Park trans-

formation allowing to obtain the "dq" components is only

applicable to sinusoidal flux distribution BDCM The Park

distribution BDCM In fact, the Park transformation can be

considered as a state transformation such as the one required

Under these assummons, the synchronous motor beha-

(1)

where 9 and 9 are the stator ~y~~ voltage and current, and ag is the total flux induced in this yll stator phase, The total fluxes a can be split in fluxes self-induced by

= a, b or

cannot be to non sinsusoidal flux

the stator currents andifluxes due to the permanent magnets

of the rotor :

when applying state feedback linearization control to non-

linear systems [ 111, Indeed, the nonlinear feedback lineari-

zation scheme is based on a coordinate transformation and

an input transformation as well But the main advantage of

the Park transformation is to define an internal state variable

which is physically meaningful : that is the current vector

"6' component which has to be regulated to zero in order to

optimize the efficiency of the drive

In this work, we will show how an extension of the Park

transformation will lead to the 'jxeudo-dq" components of

the electric variables The electric equations of the BDCM in

the rotor-field reference frame will be obtained as a function

of the "pseudo-&" variables Futhermore, a nonlinear

control law will be applied to these equations and an input-

output linearization of the closed-loop system will result

111 PARK-LIKE TRANSFORMATION FOR

NON-SINUSOIDAL BDCM

Although an appropriate nonlinear coordinate transfor-

mation for motors with sinusoidal emf has been found (the

so-called Park transformation), it is not yet the case for

motors with non-sinusoidal emf's

where L, and M, are the self-inductance and the mutual- inductance of the stator coils Since we assume a constant airgap and no saturation, L, and M, are constant ara , arb

and arc are the rotor fluxes induced in the stator phases The electrical equations of the machine can therefore be written as follows :

are the backemf (p is the number of pairs of poles of the machine, 0 is its instantaneous position and Cl is the rotor

Through an analysis of the consumed power by the machine, we can deduce the electromagnetic torque expression, assuming constant airgap :

T =p.[@',,.i, + @';b.ib + (3)

This "abc" modeling of the synchonous drive works

Qrectly with measurable data But the electrical equations

are totally coupled Each stator current can be altered by a

modification of one of the stator phase voltages Finally, the

torque depends on the rotor position and on each stator

phase current

Our objective is therefore to find a nonlinear coordinate

transformation which, like the Park transformation, allows

to decompose the current into two parts, with only one

linked to the torque If the other one is k t to zero, the copper

losses will then be minimized

B Three-Phases / Two-Phases transfopmation

The classical Park transformation is, in fact, the succe-

sion of two transformations The first trandormation (the so-

called Concordials transformation) reduces a three-phase

system to an equivalent two-phase system plus the homo-

polar component A vector x is written as follows :

(4)

This transformation is first applied to the voltage, current

and flux vectors The torque, in the new coordinates can be

expressed as :

For a motor with sinusoidal flux distribution, we have :

0

which leads to : E'$ = [-@".sin@))

= 0, the homopolar current io does not take

part in the torque generation, although it contributes to

copper losses This kind of motor is often star connected, so

t ha t : i ,+ ib +i, =O n ; , i o = O

Using Concor&a's transformation, the three-phase syn-

chronous drive with sinusoidal electromotive force is thus

reduced to a two-phase system in the "ap" frame

@".cos(@)

Since

For a non-sinusoidal machine, @>o might not equal zero, and a homopolar current can be useful The trans- formed system is not a two-phase system anymore Never- theless, since the Concordia's Fansformation diagonalizes the inductance matrix L, the three-phase system obtained is totally decoupled Indeed, through this transformation, the electrical equations can be written as follows :

Vo = R,io + ( Z S + 2 A 4 J ~ +pn.@.',,

di,

The machine can therefore be decomposed into two motors having Werent time constants : a single-phase motor only able to provide a pulsating torque (the homopolar

part) and a two-phase motor with a rotating field (the "ap"

Part)

In order to simplirl the study, only a star-connected motor (without neutral point connection) will be hereafter

mnsidered With this assumption, the machine is reduced to

a two-phase motor

C Park-like Rotation

For a two-phase sinusoidal motor or equivalent, the clas- sical Park's transfonmation allows working in the rotor's reference frame, through a rotation of an angle @ Using the new "dq" variables, a vector expressed in "ap" frame, can be written as follows :

The torque in the "dq" coordinates is then given by :

For a drive with sinusoidal flux distribution, @>d = 0

Assuming constant airgap, the equation of the torque is

(9)

simplified to :

T = p.'.p>qiq, with GD>q = am as a constant The copper losses can be written as follows :

Pc = R,(i& + iq2),

and are mimimized if id = 0

Although the voltage equations of the obtained system are

still nonlinear, high-performance control schemes, such as vector control [12] can be elaborated, provided that the time scales of the electrical and mechanical subsystems are signi- ficantly separated Alternatively, state feedback linearization

in "&"-frame can be ]performed

For a non-sinusoidal BDCM, by analogy to the previous

case, an angle p0+p(B) has to be found, defining "pseudo-

dg" axes, so that @';.d = 0 Since we consider here a star-

connected motor, the homopolar current io is still zero and

the expression of the electromagnetic torque will be nearly

the same as in the sinusoidal case :

The first control sheme studied here is a very classical current control scheme in which off-line computed optimal currents are imposed in stator phases [2],[7],[9]

except that

rotor position 8

following :

is no longer constant, it depends on the

To obtain @ > d = 0 , it is necessary to have the

where: @',,(e) = &;a(8)z+@;p(e)z

p is therefore a function of 8 It can be verified that, in the

sinusoidal case, p = 0

The obtained transformation can be considered as an

extension of Park's transformation for the BDCM with any

emf pattern With this new transformation a vector x is

written as follows :

In the new "pseudo-dg" frame, the voltage equations are

written as follows :

IV CONTROL SCHEME FOR BDCM

In this paper, two different control schemes are studied

They are simulated using data corresponding to a motor with

a very simple design (Fig 1)

No-load fluxes have been computed using finite element

code Fig 2a shows the emf shape and the corresponding

evolution of (Fig 2.b) and p (Fig 2.c) with respect to

the electrical position pB

A "abc"j?ame vector control

Rotor

Magnets

stator

Figure 1 : GeommTf the studied motor (one quadrant represented)

".tJ

6.00

4.00

2.00

0.00

-2.m

-4.m

-6.00

-8.W 1

Figure 2 : Characterization ofthe studied motor

The extended Park modeling has been used to compute these optimal currents The drive being star connected, the homopolar current (io) equals always zero In order to

minimize copper losses, it is advisable to impose a "6'

current component reference equal to zero The reference torque will be assured if the "g" component of the stator current equals :

Trey

9 ref - p q q

By reversing the Park-like transformation, we obtain the

components of the current vector in the stator reference

("abc" frame) :

ia ref

h b c ref red = T32me+P(e)).( iq re ) (15)

The control strategy attempts to impose the reference

currents in the stator phases There is mainly two possibi-

lities for the current controller [lo] but in order to avoid

variable switching frequency due to the use of an hysteresis

current controller, a linear current control scheme has been

chosen Emf compensations have been added to improve

performance This scheme is presented in figure 3

=.-.%+J

8 p E B ) , a -

Figure 3: "ald-fiame torque control scheme

With high current loop gain, the current is close to the

optimal current shape A good quality torque is then

observed (fig 4.a.) with reduced copper losses, id being

negligible in relation with iq (fig 4.b)

Nevertheless, if high gains can not be chosen (to avoid

instability due to sampling effects for example), the perfor-

mance of the control scheme can rapidly degrade To velrfy

this, a second simulation has been performed with a current

loop gain 10 times lower than the first simulation The

obtained current deviates from the optimal shape with non

negligible magnitude and phase errors (fig 5.a) For sinu-

soidal motors, the effects of such errors are an attenuation of

the average torque value and an increase of the copper

losses But for a non-sinusoidal BDCM, small torque ripple

can be observed in addition (fig 5.b)

B Nonlinear feedback linearization in "dq"frame

In the following, we will show how the Park-like

transformation defined in this paper can allow us to

elaborate control laws in "pseudo-dq"-frame The main

advantage of this strategy is that the torque can directly be

defined as one of the controlled variables The "pseudo-d"

(b>

0 Bo 120 180 240 300 360 Figure 4: Simulatim results for high-gain current vector control

Torque (Nm)

040

om

~~~~~, 0 Bo 120 180 Electncal 240 posltim 330 (-) 360

om

id

-1003 1

0 Bo 120 180 240 300 380 Figure 5: Simulation resub for low-gain current vector oontrol

component of the current will be chosen as a second output

in order to minimize copper losses No tracking errors are

expected then and low-gain controllers can be used

For sinusoidal synchronous motors, "dq" frame vector

control is usually performed with linear controllers [ 121 In

the non-sinusoidal case, since the coupling terms, the emfs

and the open-loop gain depend on the rotor position, non-

linear compensators have to be used [ 131, [ 14 ]

Next, we proceed to apply the state feedback input-output

linearization technique [ 151 to the non-sinusoidal BDCM of

interest

The system is described by the state equations :

where : f&) =

and : g(x) =

'0

(TL is the total load torque, including damping effects; J is

the mechanical inertia)

a y I = T = p @ > i

The outputs are written as follows :

4 ' 4

a y 2 = i d

The outputs are differentiated with respect to time

repeatedly until at least one of the components of the input

vector U = (vd V d t appears For both outputs, one W e -

rentiation is enough and we obtain :

and

*Be) =-( 1 O p-?)

s- s I

(we write : @r>q =

B(x) is a matrix which is non singular for every operating

point and it is therefore possible to proceed to a linearization

of the system

It is then possible to impose to the outputs any desired dynamic behaviour, for example an exponential convergence

to the reference values :

dyi dT

a - - - - dt - dt - -h ( T-Tref) = V I t

dyz did

a dt - dt - - h i d = v 2 ,

This leads to :

Note that the resulting zero dynamics coincide with the mechanical dynamics and are therefore asymptotically stable due to damping effects

Using this new control scheme (fig 6), torque ripple is cancelled even when choosing low gains, while efficiency is

kept at its optimal value (fig 7)

D.C

in

I

Figure 6: "dq"-ftame nonlinear torque m o l scheme

0.40

m m

2 o W

i o m

000

-10 m

0 Bo 1m 180 240 300 JBO

Figure 7: Simulation results for low-gain "pseudo-dq"-fiame

nonlinear torque m o l

V IMPLEMENTATION OF THE NONLMAR

FEEDBACK CONTROL SCHEME

The implementation of the nonlinear feedback control

scheme in "pseudo-dq" frame described above requires

numerous control and computation tasks

The control tasks include current, position and speed

acquisitions, and PWM signal generation The computation

tasks include the Park-like coordinate transformation, the

torque and the id current component estimations, the calcu-

lation of the linear control outputs (vI and v2) and the non-

linear feedback total compensation A p o w e m micro-

processor-based control system appears to be imperative

These tasks have to be executed at every sampling period

In the ideal case, this sampling period equals the inverter

commutation rate, i.e typically 20k& in order to avoid the

acoustic bandwidth

These features can be reached by the use of a digital

signal processor (DSP)

In addition, the variables to be computed have known

dynamics Due to the measurement noises, more than 12 bit

resolution for analogic value acquisitions is unprofitable As

the algorithm does not use more than quadratic form of these

data, a 24 bit fixed-point DSP (for example the DSP56001)

is then sufkient

Using this microprocessor, the execution time for the

control and computation tasks of the proposed nonlinear

feedback control sheme is estimated to be about 25-35p

This estimation supposes that the most important part of the

control tasks is driven by an appropiate logic hardware (in

particular data acquisition synchronisation, peripheral

timing adaptation .) [ 161 It forsees likewise that the amount

of computations will be reduced by the use of look-up tables

These tables will contain the values with respect to the rotor position of @';d @'';d, sin(j+p), cos(P0+p,,, d@&

Hence, it is also necessary to allow some memory capacity

Finally, the computation time has to be majored because it

is well known that nonlinear feedback control is not robust

In order to improve the robustness, the addition of inte-

grators in the linear control part algorithm [ 10,171 or, better,

the use of an adaptwe nonlinear feedback control version

{lS] couldbe necessary

VI CONCLUSION The aim of this study was to reach an efficient control of the instantaneous torque of a non-sinusoidal brushless DC

motor The classical ways of traclang off-line computed optimal current shape (the so-called "abc" frame current control) is not totally satisfactory

By analogy with the sinusoidal case, a Park-like transfor- mation has been proposed This transformation allows to split the current in two parts : one linked to the torque (the

iq-like current), the other one (the idlike current) being regulated to zero in order to optimize the efficiency of the drive

A state feedback linearization of the system, with torque

and id-like current as outputs, has been performed A totally

compensated control scheme is then proposed which leads to

the cancellation of torque ripples while ensuring the mini-

mization of copper losses

This control scheme has been studied in simulation and

results are hopeful The total compensation algorithm could

be implemented using a digital signal processor (DSP)

VII APPENDIX : Compensation of the cogging torque The cogging torque T, results of the interaction between the rotor magnets and the stator slots It depends only on the rotor position and has to be compensated by the

electromagnetic torque

For a given torque reference Tre$ the electromagnetic torque reference becomes :

TeF ref is no longer constant and depends on the rotor posihon In order for the torque error to converge exponen- tially to zero, we should impose :

3 -h (T-Tref+Tc(0)) + p.f.2.Z = V I (A-2) With this new value of V I the new "dq" voltage components

can easily be computed as in section IV-B

[31

[41

[91

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