drives vector contr

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Electrical drives Juha Pyrhnen, LUT, Department of Electrical Engineering3.1

3. FUNDAMENTALS OF SPACE-VECTOR THEORY...........................................................1

3.1 Introduction to the Space Vector of the Current Linkage....................................................2

3.1.1 Mathematical Representation of the Space Vector......................................................4

3.1.2 Two-Axis Representation of the Space Vector............................................................5

3.1.3 Coordinate Transformation of the Space Vector .........................................................7

3.2 Voltage Equations................................................................................................................83.3 Space Vector Model.............................................................................................................9

3.4 Two-Axis Model................................................................................................................10

3.5 Application of Space Vector Theory .................................................................................11

3. FUNDAMENTALS OF SPACE-VECTOR THEORY

The treatment of the transient states of electrical machines has gained significance as the develop-

ment of electrical drives has progressed. It is nowadays increasingly common to feed a motor for

instance by a frequency converter, in which case the voltage is far from sinusoidal. Even in a sinu-

soidal supply, the electrical machines experience transients for example in the start-up and in thecontext of process control. The traditional single-phase equivalent circuit that can well be applied to

the sinusoidal quantities in the stationary state, is not applicable to transient states. The space-vector

theory has therefore been developed for the treatment of the transients occurring in electrical ma-

chines.

In space-vector theory, the following assumptions are usually made to simplify the analysis:

1. the flux density distribution is assumed sinusoidal in the air gap,

2. the saturation of the magnetizing circuit is assumed constant,

3. there are no iron losses, and

4. the resistances and inductances are independent of the temperature and

frequency

The assumption of sinusoidal flux density distribution usually brings good results, since the target

in designing the structures of a rotating-field machine is that the flux density is as sinusoidal as pos-

sible. In teaching the fundamentals of space-vector theory, each phase winding is assumed to be

constructed such that the single phase winding alone produces a sinusoidal flux density. This as-

sumption is naturally wrong, yet the result is not poor, since for three-phase machines in particular,

the curvature form of the flux density created together by the windings is quite beautiful, although

the flux density curve created by a single winding is far from sinusoidal.

First, a space vector constructed of three-phase quantities is considered; this space vector can be

employed in the analysis of the actual space-vector model. The mathematical treatment of space

vectors requires certain coordinate transformations of the quantities; the bases of these transforma-

tions are also discussed in brief.


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3.1 Introduction to the Space Vector of the Current Linkage

Figure 3.1 illustrates the magnetic axes A, B, and C of a two-pole three-phase machine. We assume

here that each winding alone produces a sinusoidal flux density distribution in the air gap.

Magnetic axis of phase A

iA

Magnetic axis of phase B

Magnetic axis of phase C

iB

iC

Figure 3.1 Magnetic axes of the stator phase windings. We assume here that each winding alone produces a sinusoidalflux density distribution in the air gap.

When we adopt the space-vector theory approach to the electrical machines, the phase windings of

the stator divided into slots are usually illustrated by windings now concentrated on the magnetic

axes, as shown by Fig. 3.2.

A+

A-

B+

B-

C+

C-

A

B

C

A

B

C

Figure 3.2 In space-vector theory, the phase windings of the stator divided into slots are usually illustrated by windings

concentrated on the magnetic axes.

Consider the field strengths caused by the winding of the phase A in Fig. 3.3 at the point on theperiphery of the machine. The field strengthHA() oriented radially in the air gap of the machine,

caused by the current of the phase A, is calculated by applying Ampres law. Next, we consider

the integration path of Fig. 3.3, which passes from the angle to + on the periphery of the ma-chine. The winding density of the machine per radian is Ns/2 sin. The equation for the magnetic

field strength of the air gap is obtained by applying the instantaneous current iAof the phase A

HN i

A

s A=

cos . (3.1)


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d

= 0 =

iA

HA

-HA

Figure 3.3 Applying Ampres law to a certain integration path. The conductor density of the slots is illustrated by the

size of the diameters of the conductors.

The subscript A indicates that the field strength is created by the phase A alone. We also agree here

that the field strength from the rotor to the stator is positive, and the field strength in the opposite

direction is negative. The respective air gap flux density at the point is

B HN i

A A

s A= =

0 0 cos . (3.2)

Consequently, the magnetomotive force is written as

A A s A

= = H N i cos . (3.3)

Neglecting the magnetic saturation, the magnetomotive force, the magnetic field strength, and the

magnetic flux density in the air gap are linearly dependent on each other. By treating the two other

phases similarly, we obtain

B s B=

N i cos

2

3 , (3.4)

and

C s C=

N i cos

4

3 . (3.5)

Equations (2.32.5) show that at any instant of time, the peak value of the magnetomotive force of

each phase coincides with the magnetic axis of the phase. Further, due to the sinusoidal conductor

density, the magnetomotive forces are sinusoidal. This sinusoidal form remains unaltered irrespec-

tive of the instantaneous values of the currents, for instance, if the current of one phase is direct

current and the currents of the other two phases follow an arbitrary curve as a function of time.

Figure 3.4 illustrates the instantaneous phase currents iA, iB, and iC of the different phases of a

symmetric three-phase system, fixed to the stator reference frame, at time 1, when the current of the


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phase A is at the maximum, and the currents of both the phases B and C are a half of the

negative maximum value. The figure also depicts the amplitudes A1, B1, and C1 of the phase-

specific sinusoidal current linkage corresponding to these currents. For instance

A1= 1NAiA. (3.6)

The amplitudes are indicated at the peaks of each sinusoidal current linkage distribution. The peaks

coincide with the magnetic axes. The total current linkage is the sum of the phase-specific current

linkages.

iA

iB

iC

moment of observation

1 2

C1

A1

B1

1

A2

B2

C2

2

Figure 3.4 The formation of the total current linkage of the phase-specific current linkage. The phase-specific current

linkage are created as a result of the phase currents.

The space vector representation enables the representation of a three-phase quantity by a single

space vector rotating in the complex plane; this intensifies the mathematical treatment of electrical

machines.

3.1.1 Mathematical Representation of the Space Vector

Next, we consider the mathematical representation of the sinusoidal distributions created by the

windings. Since in a three-phase machine, there is a local phase shift of 120 electrical degrees be-

tween the different phases, we have to introduce the phase-shift operator

3

2j

e=a , (3.7)

which can be applied to construct the space vector ( )t's of the magnetomotive force of the stator(subscript s) from the instantaneous values of the magnetomotive forces of the different stator

phases. This space vector is created together by all three phase windings and their currents.

( ) ( ) ( ) ( )( )tttt' sC2sB1sA0s aaa ++= (3.8)

Assume that there is a constant air gap in the machine, and the permeability of the iron parts is

very high. Now the magnetic field strength in the machine with a constant air gap varies in relation

to the magnitude of the current linkage. When the current linkages of Eq. (3.8) are divided by the

effective number of turns, we see that the space vector ( )t'si formed by the phase currents can be


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determined completely analogically; this space vector can be represented as a function of time in

the stator reference frame by the equation

( ) ( ) ( ) ( )( )tititit' sC2sB1sA0s aaai ++= . (3.9)

This current vector thus illustrates the effect of the current linkage created together by the windings.It has the same direction as the current linkage. We can show that in the case of Eq. (3.9), the mag-

nitude of the space vector of the stator current equals to 3/2 of the peak value of the sinusoidal cur-

rent.

Instead of the stator current vector is, in literature, the current vector isreduced by the factor of 2/3

is often employed.

( ) ( ) ( ) ( )( )++= tititit sC2sB1sA0s3

2aaai (3.10)

The reduction of the vector by the factor 2/3 facilitates the use of the space vectors: if the current

vector is defined this way, the parameters according to the real equivalent circuit of the machine the resistances and inductances can be employed.

Similarly as the current vector, the voltage vector ( )tsu of the stator is determined analogically withthe stator current vector, although it does not have as clear physical importance as the current vec-

tor, which has the same direction as the current linkage of the current of the winding

( ) ( ) ( ) ( )( )++= tututut sC2sB1sA0s3

2aaau (3.11)

Here, the same directions of the magnetic axes are used as in the definition for the current.

Respectively, the flux linkage vector is formed analogically from the phase quantities

( ) ( ) ( ) ( )( )++= tttt sC2sB1sA0s3

2 aaa (3.12)

The current, voltage, and flux linkage vectors of the rotor can be determined accordingly.

When we change over to vector representation, all the current, voltage, and flux linkage compo-

nents are to be represented in the vector form. Now, all the ordinary relations originating from the

circuit analysis and three-phase system occur between the vectors.

3.1.2 Two-Axis Representation of the Space Vector

A three-phase winding can now be replaced by an equivalent two-phase winding, and thus we can

easily move to the two-axis representation of the space vector.

Figure 3.5 illustrates the symmetrical three-phase currents iA, iB, and iCin the time plane. The figure

also indicates the instant when the current of the phase C has reached its negative peak value, and

the current of the phases A and B is a half of the positive peak value.


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Figure 3.6 depicts the symmetric two-phase current components isx'and isy'in the time plane

corresponding to the space vector is'of the stator current. The zero points of the time axes of Figs.

3.5 and 3.6 are set at the same instant of time. If there is an equal number of turns in the two-phase

system, the peak values of the currents isx'and isy'of the symmetrical two-phase system have to be

1.5 times the symmetrical three-phase currents in order to produce an equal magnetomotive force.

The space vector is' is created in a three-phase system always from at least two phase currents,

however, in the case of a two-phase system, there are instants when the space vector of the currentis created as an effect of one phase current (isx'or isy') only.

t

1

-1

iB iCiA


A

B

CiA

iB

iC

1.5

1

-1

-1.5

0.5

-0.5

t

isx' isy'


x

y isy'

isx'

Figure 3.5. Three-phase currents in the time plane Figure 3.6. Two-phase currents causing the respective current

linkage

Next, we determine the mathematical transformations of the instantaneous values of the three-phase

currents presented on the A, B, and C-axes into the space vector components of the stator current on

thexy-axes. Both the reference frames are fixed, and there is a phase shift of the constant angle

between theA-axis andx-axis.

Further, when we take the possible current component i0s of the zero-sequence network into ac-count, the following matrix notation can be introduced for the instantaneous values of the three-

phase currents

( ) ( )( ) ( )

++

++=

ss0

ssy

ssx

oo

oo

sC

sB

sA

1240sin240cos

1120sin120cos

1sincos

i

i

i

i

i

i

, (3.13)


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thus we may write for the transformation from the three-phase current components into two-

phase current components

( ) ( )( ) ( )

++

++

=

sC

sB

sA

oo

oo

s

s0

s

sy

s

sx

240sin120sinsin

240cos120coscos

3

2

i

i

i

i

i

i

. (3.14)

In a symmetrical case, there is no zero component in a three-phase system. Usually the angle is

set zero and the zero component of the phase currents is neglected.

3.1.3 Coordinate Transformation of the Space Vector

In the implementation of vector controls, coordinate transformations are often necessary. This need

can be justified for instance by magnetic asymmetry or by the fact that thanks to the transformation,

the control becomes easier to implement. For instance, the traditional vector control of an induction

motor has usually been implemented in the rotor flux linkage-oriented reference frame (coordinate

system). Respectively, in the case of synchronous machines, it is natural to employ a rotor referenceframe. Figuratively speaking, this transfer to the rotor reference frame corresponds to a situation in

which we view the events from the rotor perspective; the stator seems now to rotate around us at

high speed.

By using Fig. 3.7, it is possible to show the validity of the following equations when we want to

transfer from the xyaxes fixed to the stator to the reference frame rotating at the general angular

speed g, the axes of this system being denoted dgand qg. Consider the components of the stator

current vector isin different reference frames

titii gsygsxgsd sincos += , (3.15)

titii gsygsxgsq cossin += . (3.16)

qg

g

iqisx

is

x

y

dg

isy

idgt

Figure 3.7 The stator current can be represented by two components in the stator reference frame (the axes xy) or in a

general reference frame (the axes dg and qg). The transfer from the reference frame fixed to the stator to the reference

frame rotating at the general angular speed g. The figure illustrates the current vector corresponding to the constant

state of a symmetric system and the locus plotted by the point of the current vector.


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The simplest representation for the coordinate transformation is achieved by using polar complex

representation for the space vectors. Now the transformation of the stator current from the xyframe

of reference to the dg

qg

frame of reference is obtained by the equation

( )tt

eiieii

gg j

sysx

js

s

g

sq

g

sd

g

s jj

+==+= ii

.

(3.17)

The complex vector has thus only to be turned to the required direction. The angular connections

required in the coordinate transformation are recapitulated by Fig. 3.8.

d

q

x

y

i

r= rt

Figure 3.8 The vector iin different frames of reference:

dqreference frame is the rotor reference frame

xyreference frame is the stator reference frame

i ir j= e

( )i i i

s j r jr r= =+e e

the coordinate transformation of the vector is thus

from the rotor reference frame to the stator reference frame

i is r j r= e , and

from the stator reference frame to the rotor reference frame

i i is j r j j r r r re e e

= = ,i i

r s j r= e

3.2 Voltage Equations

The general voltage equation of the stator of rotating-field machines in the stator reference frame is

given in the familiar form

tR

d

d sssss

s

s

iu += . (3.18)

The same equation is given by using the previous stator quantities presented in the rotor reference

frame transformed into the stator reference frame

( )u u iss sr j s sr j sr jr r rd

d= = +e R e

te . (3.19)

By deriving the latter term of the equation we obtain

u u iss

s

r j

s s

r j s

r

j

r s

r jr r r r d

dj= = + +e R e

te e . (3.20)

The sides of the above equation are multiplied by ej r , which yields the equation of the stator volt-

age in the rotor reference frame


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u u iss j

s

r

s s

r s

r

r s

rd

dje R

tr = = + + . (3.21)

Here it is worth noticing that when the voltage equation of the winding is given in a frame of refer-

ence foreign to the winding itself (the voltage equation of the stator winding e.g. in the rotor refer-

ence frame), there occurs a rotation induced voltage termr

srj caused by the speed difference ofthe reference frames. This in fact is the most demanding detail in the coordinate transformations,

which, however, can be justified both mathematically and physically. Mathematically, this can be

seen when deriving the product rjr

s

e in Eq. (3.19), which, according to the product derivation

rule, yields the term rr d/d =t for the multiplier.

3.3 Space Vector Model

Let us next investigate how the space vector model of an asynchronous machine is generated. The

voltage equations of an asynchronous machine should first be represented in a general frame of ref-

erence rotating at an angular speed g. The reason for this is that it is often necessary to change

over to some other than a stator reference frame. When the observation frame of reference rotates,

an additional motion voltage term +jg sg

is caused to the equations, which are

u isg

s s

g

g

g s

gd

dj= + +R

t

s , (3.22)

( )u irg r rg rg

g r r

gd

dj= + + R

t

. (3.23)

In Eq. (3.23), ris the rotor electrical angular speed (r=p). The stator and rotor flux linkages

occurring in Eqs. (3.22) and (3.23) can be written as

s

g

s s

g

m r

g= +L Li i , (3.24)

r

g

m s

g

r r

g= +L Li i . (3.25)

In Eqs. (3.24) and (3.25) Lm is the magnetizing inductance,Ls= Lm+ Lsis the total inductance of

the stator, andLr= Lm+ Lris the total inductance of the rotor.LsandLrare the leakage induct-

ances of the stator and the rotor. By using Eqs. (3.22) (3.25), an equivalent circuit based on thespace vector theory can be constructed according to Fig. 3.9, this equivalent circuit holding also for

transient states. In the equivalent circuit, iron losses are neglected.

sg

jssg

usg Lm

rg

j(g-r)rg

Rs

isg

Ls

Lr

irg

Rr

urg

+

s r

mg

Figure 3.9 The equivalent circuit of an asynchronous machine according to space vector theory. The equivalent circuit

functions at any three-phase voltage or current of the machine. The equivalent circuit does not require sinusoidal quan-

tities unlike the effective-value equivalent circuit.


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In a reference frame fixed to the stator (g= 0), Eqs. (3.22) (3.25) can be written as

u iss

s s

s s

sd

d= +R

t

, (3.25)

srr

s

rsrr

sr j

dd +=

tRiu , (3.26)

srm

sss

ss ii LL += , (3.27)

srr

ssm

sr ii LL += . (3.28)

Note that in the case of an induction motor, the rotor voltage vector is defined zero.

3.4 Two-Axis Model

As shown above, a two-axis model can be employed. For the directx-axis and quadraturey-axis the

equivalent circuits of Fig. 3.10 are valid. These equivalent circuits are obtained by separating the

real and imaginary parts from Eqs. (2.43) (2.46). We substitute u= ux+ juy, i= ix+ jiyand = x+ jyto the equations; thus we obtain

( ) )

t

uuRuu

d

jdjj

sysx

sysxssysx

+++=+

( ) ( ) ( )ryrxrryrxryrxrryrx jjd

jdj

t

jiiRuu +

+++=+

tiRu d

d sxsxssx

+=

tiRu

d

d sysyssy

+=

ryrrx

rxrrxd

d

++=

tiRu (3.29)

rxrry

ryrryd

d

+=

tiRu

We may write for the flux linkages shown in Fig. 3.10

( ) srxmssxssrxssxmssxsssx iLiLiiLiL +=++= ( ) ssxmsrxrssxsrxmsrxrsrx iLiLiiLiL +=++=

) srymssyssryssymssysssy iLiLiiLiL +=++= ( ) ssymsryrssysrymsryrsry iLiLiiLiL +=++= (3.30)


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usxs Lm rry

s

Rs

isxs

Ls Lr

irxs

Rr

usys Lm

Rs

isys

Ls Lr

irys

Rr

+

+rrx

s

sxs

rxs

rys

mxs

mys

sys

Figure 3.10 The equivalent circuits corresponding to the two-axis model fixed to the stator.Lmmagnetizing inductance,

Lr=LrLmthe leakage inductance of the rotor, andLs=LsLmthe leakage inductance of the stator.

3.5 Application of Space Vector Theory

Space vectors can be applied for instance to the representation of all kinds of asymmetric or dis-torted three-phase currents. For example, the space vector representation is applicable to a case, in

which the motor is fed by a frequency converter based on pulse width modulation. We see that the

space vector representation simplifies the machine representation considerably when compared forinstance with a case in which we try to write equations individually for each winding.

As an example, let us first consider the representation of symmetrical sinusoidal states. In the case

of an asymmetric three-phase system, the point of a three-phase system plots an ellipse. If there is

no zero component, the current vector is created from the positive-sequence phasor i1rotating in thepositive direction at the electrical angular speed eand the complex conjugate i2

*of the negative-

sequence phasor rotating in the negative direction at angular speed e. During a steady state, the

lengths of the positive-sequence and the negative-sequence components are constant ( 21and ii ).

According to Fig. 3.11, the major (transverse) axis of the created ellipse is $ $i i1 2+ , and the length

of the minor (conjugate) axis is 21 ii . When we write the phasors of the positive-sequence and

negative sequence components in the form

( )i1 1

1= +$i e tj e and (3.31)( )

i2 22= +$i e tj e , (3.32)

the angle of the major axis is 1 2

2

.


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y

x

e

i 1i2

i ss

*

i2

2

2

e

1

e

Figure 3.11 The locus plotted by the point of the space vector of the current of an asymmetric three-phase system in a

steady state, when the phase currents have no zero component.

In practice, this kind of an asymmetric system can easily be represented by including the suitable

temporal components to the definition of the current vector, which will automatically result in the

above elliptical orbit. If for instance one of the phase currents is lower than the others, the current

circle starts to change into an ellipse. If one phase current is completely missing, the result is a sin-

gle-phase system, which plots a line into thexy-plane.

Previously, the transformation of the three-phase system into a two-phase one and the representa-

tion in the two-axis reference frame were presented. Figure 3.12 illustrates the stator and the rotor

of an asynchronous machine with a three-phase winding. Let us investigate how difficult the model-

ling of an asynchronous machine would be without the space vector representation.

statorrotor

Lp+L

Ms

r

Msr

Figure 3.12 The stator and the rotor of an asynchronous machine with a three-phase winding, the rotor being short-

circuited. The rotation angle of the rotor with respect to the stator is r. The main inductance of the stator phase is L

p

and the mutual inductance between the phases isMs. The maximum mutual inductance between the stator and the rotor,

when the magnetic axes of the phases coincide, isMsr.

Next, we construct the space vectors

s of the stator flux linkage in a reference frame fixed to the

stator. All the six phase currents of the stator and rotor have an impact on the generation of the total

flux linkage of the machine.

The angle between the rotor and stator being r, the instantaneous flux linkages of different stator

phases can be represented by the equations


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( ) ( )

sA s p sA s sB s sC sr r rA

sr r rB sr r rC

= + + + +

+ +

+ +

L L i M i M i M i

M i M i

cos

cos cos2

3

4

3

(3.33)

( )( )

sB s p sB s sA s sC sr r rA

sr r rB sr r rC

= + + + + +

+ + +

L L i M i M i M i

M i M i

cos

cos cos

4

3

2

3

(3.34)

( )

( )

sC s p sC s sA s sB sr r rA

sr r rB sr r rC

= + + + + +

+ +

+

L L i M i M i M i

M i M i

cos

cos cos ,

2

3

4

3

(3.35)

where Lsis the leakage inductance of the stator,

Lpis the main inductance of the stator,Msis the mutual inductance between the stator windings,

Msris the maximum value of the mutual inductance between the stator

and rotor circuits,

ris the angle between the magnetic axes of the rotor and stator, and

irA, irB, and irCare the instantaneous phase currents of the rotor.

When the flux density is sinusoidally distributed in the air gap, the mutual inductance between the

phases of the stator windings is

M LL

s p

p=

= cos

2

3 2

, (3.36)

since there is a phase difference of2

3

between the magnetic axes of the stator windings. Due to

this phase difference, for instance the current flowing in the phase A creates a flux linkage in the

phase B, which is a half of the flux linkage created by the A-phase current to the phase A. If there is

no zero component in the stator currents, the first three components in Eq. (3.33) are replaced by

the expression

( ) ( ) ( )

( )

L L i M i M i L L iL

i i L L i

L L i L i

s p sA s sB s sC s p sA

p

sB sC s p sA

s m sA s sA

+ + + = + + = +

= + =

2

3

2

,

(3.37)

where Lmis the magnetizing inductance and

Lsis the total inductance of the stator.

The maximum valueMsrof the mutual inductance between the stator and rotor circuits is obtained,

when the magnetic axes of the stator and rotor coincide. When treating the main flux, the coupling

factor between the stator and rotor circuits is k= 1. When the rotor currents are referred to the stator

side, we may write


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M Lsr p= . (3.38)

The instantaneous values of the flux linkages of the different phases are substituted to the equation

of the space vector ssof the stator flux linkage

ss = 2

3(a0sA+ a1sB+ a2sC) . (3.39)

We obtain by simplification

s

s=Lsis

s+Lmirr te e

j =Lsis

s+Lmirs . (3.40)

Instead of Eqs. (3.33 3.35), we can thus apply a simple space vector representation as presented

by Eq. (3.40). The result is the same as the one that we employed already in Eq. (3.27). Hence we

have confirmed that we have found a really simple and efficient presentation method for the motor

when considering the control model. Eq. (3.40) is given in the complex plane; the currents and flux

linkages are complex vectors.

Figure 3.13 a) depicts the instantaneous flux linkage components sA, sB, and sCcaused by the

instantaneous phase currents. The space vector ss created by the flux linkage components sA,

sB, and sCcan be represented as shown in Fig. 3.13 b). In the figure, the windings are located in

the stator in the direction of the direct and quadrature axes. By feeding the windings with the ap-

propriate instantaneous current components ixsand iys, we obtain the sum flux linkage ssfrom the

equation

s

s= sx

s+ j sys . (3.41)

When simulating or constructing the digital control of machines, the machines are often modelledaccording to the above two-phase system. For instance when using a microprocessor-based control,

the equations of the motor have to be represented in a two-axis system, since the processors are

incapable of handling the vectors in a polar coordinate system, which would mathematically be the

best method.

The two-axis model is particularly useful when modelling magnetically asymmetric machines, such

as salient-pole synchronous machines. When applying the two-axis model, the rotor reference frame

is often employed, since synchronous machines are genuinely asymmetric either with respect to the

magnetic circuit and the electric circuit, or with respect to the electric circuit alone, which leads

naturally to the selection of the rotor reference frame.

Here we remark that so far, we have used three representation methods for the machine quantities:

the first method is a three-phase one; in the three-phase method, the equations become quite heavy,

since they include plenty of terms. The second alternative is the two-phase method, which simpli-

fies the equations considerably, since the windings perpendicular to each other have in principle no

mutual inductance, which makes the representation easier. Another benefit of the two-axis model is

that the quantities can be treated in the complex plane. Figure 3.13 c illustrates the generation of the

space vector and its three representation methods. The space vector is a single-phase complex quan-

tity, which can naturally be used as such; however, it is usually decomposed into its components,

and thus we have returned to the representation method according to the two-axis model. The treat-


15/16


ment of complex numbers, for instance in signal processors, is at least nowadays still easier to be

carried out in the component form than in the form of polar representation, and therefore, returning

to the two-phase representation makes the computation easier.

iA

iB iC

ix

iy

is'=3/2iCej

x

y y

x

C

C'

B' B

A'

A

x

yis'= 3/2iC

iA

iB iC

x

a) b)

c)

y

A

B

C

Figure 3.13 a) The current vector components isA, isB, and isCof the windings of different phases (AA', BB', and CC') represented at the same instant as the winding currents isA, isB, and isCof Fig. 2.4. b) The corresponding spacevector is of the stator flux linkage represented by the direct and quadrature windings. c) The three representation meth-

ods of the space vector the three-phase, the two-phase, and the single-phase, complex method. In practice, the angle

between the three-phase system and the two-phase system is usually set zero = 0.

The purpose of Fig. 3.14 is to illustrate the connection between the physical flux distribution and

the flux space vector. The connection is clear in a two-pole machine, however, the situation is less

easy to comprehend in a poly-phase machine, in which the definition of the total flux depends on

the structure and the connection of the winding of the machine. Nevertheless, in the case of a multi-

pole machine, the illustration of the flux distribution itself is easier than in the case of a two-pole

machine.

When applying the space vector theory, the mathematical treatment of the machines reverts in prac-

tice to the two-pole representation. A motor controller attached to the terminals of the motor cannot

have any information on the number of the pole pairs of the motor, unless the operator of the ma-

chine gives this information to the frequency converter. Only the angular data provided by the pos-

sible angle sensor attached to the motor has to be handled in connection with the number of polepairs, since a motor control based on the space vector theory operates on the basis of the electrical

angle. Furthermore, the number of pole pairs impacts on the magnitude of the torque of the ma-

chine.


16/16


$

= /3

$

=

0 6

2,

34

5

0 2

ss/3

Figure 3.14 The generation of the flux space vector with different numbers of pole pairs. In a six-pole machine, outlin-ing the flux vector is somewhat unclear. The total vector comprises the vectors of the different poles. We have to note

in the figure that all the vectors point in the same electrical direction.

drives vector contr

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