3. fundamentals of space-vector theory · 3. fundamentals of space-vector theory ... electrical...

Electrical drives Juha Pyrhönen, LUT, Department of Electrical Engineering

3.1

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................................................................................................................8 3.3 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 the context 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.


3.2

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 CiB

iC

Figure 3.1 Magnetic axes of the stator phase windings. We assume here that each winding alone produces a sinusoidal flux 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 γ on the periphery of the machine. The field strength HA(γ) oriented radially in the air gap of the machine, caused by the current of the phase A, is calculated by applying Ampère’s law. Next, we consider the integration path of Fig. 3.3, which passes from the angle γ to γ + π on the periphery of the ma-chine. The winding density of the machine per radian is Ns/2 sinγ. The equation for the magnetic field strength of the air gap is obtained by applying the instantaneous current iA of the phase A

HN i

As A=δ

γcos . (3.1)


3.3

γ

υdυ

γ = 0γ = π

iA

HA

-HA

Figure 3.3 Applying Ampère’s 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 γ is

B HN i

A As A= =μ μδ

γ0 0 cos . (3.2)

Consequently, the magnetomotive force is written as

ΘA A s A= =δ γH 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

ΘB s B= −⎛⎝⎜

⎞⎠⎟N i cos γ

23

π , (3.4)

and

ΘC s C= −⎛⎝⎜

⎞⎠⎟N i cos γ

43

π . (3.5)

Equations (2.3−2.5) 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


3.4

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 θA1, θB1, and θC1 of the phase-specific sinusoidal current linkage corresponding to these currents. For instance

θA1= ξ1NAiA. (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 observation1 2

θC1

θA1

θB1

Σθ1

θA2θB2

θC2

Σθ2

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

3π2j

e=a , (3.7) which can be applied to construct the space vector ( )t'sθ 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' sC2

sB1

sA0

s θθθ aaaθ ++= (3.8) Assume that there is a constant air gap δ 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


3.5

determined completely analogically; this space vector can be represented as a function of time in the stator reference frame by the equation

( ) ( ) ( ) ( )( )tititit' sC2

sB1

sA0

s 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 is reduced by the factor of 2/3 is often employed.

( ) ( ) ( ) ( )( )⋅++= tititit sC2

sB1

sA0

s 32 aaai (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 with the 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 sC2

sB1

sA0

s 32 aaau (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 sC2

sB1

sA0

s 32 ψψψ aaaψ (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 iC in 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.


3.6

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 current is created as an effect of one phase current (isx' or isy') only.

t

1

-1

iB iCiA

moment of observation

A

B

CiA

iB

iC

1.5

1

-1

-1.5

0.5

-0.5t

isx' isy'

moment of observation

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 the xy-axes. Both the reference frames are fixed, and there is a phase shift of the constant angle κ between the A-axis and x-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

1240sin240cos1120sin120cos1sincos

iii

iii

κκκκ

κκ, (3.13)


3.7

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

( ) ( )( ) ( )

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++++

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

sC

sB

sAoo

oo

ss0

ssy

ssx

½½½240sin120sinsin240cos120coscos

32

iii

iii

κκκκκκ

. (3.14)

In a symmetrical case, there is no zero component in a three-phase system. Usually the angle κ 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 reference frame. 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 xy axes fixed to the stator to the reference frame rotating at the general angular speed ωg , the axes of this system being denoted dg and qg. Consider the components of the stator current vector is in different reference frames

titii gsygsxgsd sincos ωω += , (3.15)

titii gsygsxg´sq cossin ωω +−= . (3.16)

qg

ωg

iqisx

is

x

y

dg

isy

idωgt

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 ωg. 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.


3.8

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 xy frame of reference to the dg−qg frame of reference is obtained by the equation

( ) tt eiieii gg jsysx

jss

gsq

gsd

gs jj ωω −− +==+= 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

γθr = ωrt

Figure 3.8 The vector i in different frames of reference: dq reference frame is the rotor reference frame xy reference frame is the stator reference frame i ir j= e γ

( )i i is j r jr r= =+e eγ θ θ the coordinate transformation of the vector is thus from the rotor reference frame to the stator reference frame i is r j r= e θ , and from the stator reference frame to the rotor reference frame i i is j r j j rr r re e e− −= =θ θ θ , i ir s j r= −e θ

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

t

Rd

d sss

ssss

ψiu += . (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 r

dd

= = +e R et

eθ θ θψ . (3.19)

By deriving the latter term of the equation we obtain

u u iss

sr j

s sr j s

rj

r sr jr r r r

dd

j= = + +e R et

e eθ θ θ θωψ

ψ . (3.20)

The sides of the above equation are multiplied by e− j rθ , which yields the equation of the stator volt-age in the rotor reference frame


3.9

u u iss j

sr

s sr s

r

r srd

dje R

tr− = = + +θ ω

ψψ . (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 term r

srj ψω caused by the speed difference of the 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

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

rule, yields the term rr d/d ωθ =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 ωg. 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 + jωg s

gψ is caused to the equations, which are

u isg

s sg

g

g sgd

dj= + +R

tsψ

ψω , (3.22)

( )u irg

r rg r

g

g r rgd

dj= + + −R

tψ

ψω ω . (3.23)

In Eq. (3.23), ωr is the rotor electrical angular speed (ωr = pΩ). The stator and rotor flux linkages occurring in Eqs. (3.22) and (3.23) can be written as

ψ sg

s sg

m rg= +L Li i , (3.24)

ψ rg

m sg

r rg= +L Li i . (3.25)

In Eqs. (3.24) and (3.25) Lm is the magnetizing inductance, Ls = Lm + Lsσ is the total inductance of the stator, and Lr = Lm + Lrσ is the total inductance of the rotor. Lsσ and Lrσ are the leakage induct-ances of the stator and the rotor. By using Eqs. (3.22) − (3.25), an equivalent circuit based on the space 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.

ψsg

jωsψsg

usg Lm ψr

gj(ωg-ωr)ψr

g

Rs

isg

Lsσ Lrσ

irg

Rr

urg

+ −

ψsσ ψrσ

ψmg

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.


3.10

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

u iss

s ss s

sdd

= +Rt

ψ , (3.25)

srr

srs

rrsr j

dd ψψ ω−+=

tR iu , (3.26)

srm

sss

ss ii LL +=ψ , (3.27)

srr

ssm

sr ii LL +=ψ . (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 direct x-axis and quadrature y-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 + jiy and ψ = ψ x + jψy to the equations; thus we obtain

( ) ( )tψψ

uuRuud

jdjj sysx

sysxssysx

+++=+

( ) ( ) ( )ryrxrryrx

ryrxrryrx jjd

jψdj ψψ

tψ

jiiRuu +−+

++=+ ω

tiRu

dd sx

sxssxψ

+=

tiRu

dd sy

syssyψ

+=

ryrrx

rxrrx dd ψωψ

++=t

iRu (3.29)

rxrry

ryrry dd

ψωψ

−+=t

iRu

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

( ) srxm

ssxs

srx

ssxm

ssxsσ

ssx iLiLiiLiL +=++=ψ

( ) ssxm

srxr

ssx

srxm

srxrσ

srx iLiLiiLiL +=++=ψ

( ) srym

ssys

sry

ssym

ssysσ

ssy iLiLiiLiL +=++=ψ

( ) ssym

sryr

ssy

srym

sryrσ

sry iLiLiiLiL +=++=ψ (3.30)


3.11

usxs Lm ωrψry

s

Rs

isxs

Lsσ Lrσ

irxs

Rr

usys Lm

Rs

isys

Lsσ Lrσ

irys

Rr

+

−

−

+ωrψrx

s

ψsxs ψrx

s

ψrys

ψmxs

ψmys

ψsys

Figure 3.10 The equivalent circuits corresponding to the two-axis model fixed to the stator. Lm magnetizing inductance, Lrσ = Lr − Lm the leakage inductance of the rotor, and Lsσ = Ls − Lm the 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 for instance 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 i1 rotating in the positive direction at the electrical angular speed ωe and the complex conjugate i2* of the negative-sequence phasor rotating in the negative direction at angular speed −ωe. During a steady state, the lengths of the positive-sequence and the negative-sequence components are constant ( 21 ˆ andˆ 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 ˆˆ ii − . When we write the phasors of the positive-sequence and

negative sequence components in the form

( )i1 11= +$i e tj eω α and (3.31)

( )i2 22= +$i e tj eω α , (3.32)

the angle of the major axis is α α1 2

2−

.


3.12

ω

y

x

ei 1i2

i ss

*

i2

−α 2

α 2

ωe

α1

ωe

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 the xy-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 + Lσ

Ms

θr

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 θr. The main inductance of the stator phase is Lp and the mutual inductance between the phases is Ms. The maximum mutual inductance between the stator and the rotor, when the magnetic axes of the phases coincide, is Msr. Next, we construct the space vector s

sψ 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 θr, the instantaneous flux linkages of different stator phases can be represented by the equations


3.13

( ) ( )ψ θ

θ θ

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

σ

π π

cos

cos cos23

43

(3.33)

( )

( )

ψ θ

θ θ

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

σ

π

π

cos

cos cos

43

23

(3.34)

( )

( )

ψ θ

θ θ

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

σ

π

π

cos

cos cos ,

23

43

(3.35)

where Lsσ is the leakage inductance of the stator, Lp is the main inductance of the stator, Ms is the mutual inductance between the stator windings, Msr is the maximum value of the mutual inductance between the stator

and rotor circuits, θr is the angle between the magnetic axes of the rotor and stator, and irA, irB, and irC are 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 pp=

⎛⎝⎜

⎞⎠⎟ = −cos

23 2π

, (3.36)

since there is a phase difference of 23π 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 sAp

sB sC s p sA

s m sA s sA

σ σ σ

σ

+ + + = + − + = +⎛⎝⎜

⎞⎠⎟

= + =

232

,(3.37)

where Lm is the magnetizing inductance and Ls is the total inductance of the stator. The maximum value Msr of 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


3.14

M Lsr p= . (3.38)

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

s of the stator flux linkage

ψ ss = 2

3⋅( a0ψsA + a1ψ sB + a2ψ sC) . (3.39)

We obtain by simplification

ψ ss = Ls is

s + Lm irr te ejω−

= Ls iss + Lm ir

s . (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 ψsA, ψsB, and ψsC caused by the instantaneous phase currents. The space vector ψ s

s created by the flux linkage components ψsA, ψsB, and ψsC can 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 ixs and iys, we obtain the sum flux linkageψ s

s from the equation

ψ ss = ψsx

s + j ψsys . (3.41)

When simulating or constructing the digital control of machines, the machines are often modelled according 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-


3.15

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' =3/2 iCejγ

x

y y

xγ

C

C'B' B

A'

A

x

y is'= 3/2iC

iA

iBiC

x

a) b)

c)

y

A

B

C

κ

Figure 3.13 a) The current vector components isA, isB, and isC of the windings of different phases (A−A', B−B', and C−C') represented at the same instant as the winding currents isA, isB, and isC of Fig. 2.4. b) The corresponding space vector 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 κ = 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 pole pairs, 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.


3.16

$Θ

Ω = ω/3

$Θ

Ω = ω0 − 6π

π

2π,

3π4π

5π

0 − 2π

π

κΦsΦs/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.

3. fundamentals of space-vector theory · 3. fundamentals of space-vector theory ... electrical...

Documents