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Electrical Drives Juha Pyrhnen, LUT, Department of Electrical Engineering 5.1

5. CONTROL OF AN ELECTRICAL MACHINE ......................................................................... 15.1 Control of a DC Machine ..................................................................................................... 15.2 Vector Control of AC Motors .............................................................................................. 25.3 Direct Flux Linkage Control (DFLC) .................................................................................. 5

5.3.1 The Basis of Direct Torque Control (DTC) ................................................................. 55.3.2 Implementation of Direct Flux Linkage Control ......................................................... 8

5.3.3 Errors of Direct Flux Linkage Control ....................................................................... 125.4 Direct Torque Control (DTC) ............................................................................................ 19

5.4.1 Current Model Correction .......................................................................................... 195.4.2 Correction of the Eccentricity of the Stator Flux Linkage ......................................... 20

5. CONTROL OF AN ELECTRICAL MACHINE

In the following, the control section of an electrical drive is discussed in very general terms.

5.1 Control of a DC Machine

A DC machine has always been easy to control; further, the development of thyristors in the late1950s and in the 1960s brought new control applications that were adapted for really demandingdrives. The characteristics of a fully compensated DC machine in particular are excellent when con-sidering the aspects of control engineering.

In a separately magnetized, fully compensated DC machine, the electrical torque T e and the fluxlinkage F can be controlled independently. The flux linkage of a fully compensated DC machinedepends only on the magnetizing current I F

)( FF I f , (5.1)

since the effect of the armature current I A on the magnetization state, that is, the armature reaction,has been compensated.

The electrical torque depends on the controlled flux linkage and the armature current, since thestructure of the machine guarantees the perpendicularity of the flux linkage and the armature cur-rent ( = 90

). We may write

FAFAe sin I C I C T . (5.2)

C is a factor depending on the structure of the machine.

The flux linkage is controlled as follows: below the rated speed, the flux linkage is kept at the ratedvalue, and at higher speeds, the flux linkage is decreased inversely proportional to the speed. Thelatter area is known as the field weakening area. Since the armature reaction has been compensated,and depending on the speed, the flux linkage is controlled by the above method, the armature cur-rent is the only element having impact on the torque. This enables a rapid torque control, since theflux does not have to be altered. Figure 5.1 illustrates the block diagram for the control of a DC ma-chine.


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Figure 5.1 The block diagram of the control of a separately magnetized, fully compensated DC machine. Figure illus-trates clearly that the controls of the armature current iA and the magnetizing current iF are separate. The controller re-ceives the speed reference nref as its input. Also the field weakening control at high speeds is easy to implement for aDC machine by controlling the magnetizing current.

In this context, it should be borne in mind that the electrical torque required of an electrical ma-chine satisfies the differential equation for the rotation speed familiar from mechanics, the terms of

this equation being the torque T L, the inertia J , the mechanical angular speed , and the rotationfriction coefficient B

Bt

J T T dd

Le . (5.3)

However, a DC machine is an expensive apparatus, and its brushes require plenty of maintenance.Therefore, a long-time expectation has been to be able to completely change over to brushless ACdrives. For this purpose, a vector control has been developed, the core of which is the control of theDC machine presented above.

5.2 Vector Control of AC Motors

In general, AC motors require less maintenance, they have a better endurance for high speeds, andthey can be constructed for notably higher powers than the DC machines. Therefore, it is sensible toaim at selecting AC drives instead of DC ones. However, in the AC machines, the factors linked tothe magnetizing and the torque are interrelated, since there is, unlike in the DC machines, no com-

pensating winding to cancel the armature reaction.

In the 1960s, a German engineer Felix Blaschke invented a method that brought the control of rotat-ing-field machines close to the properties of DC drives. The Siemens transvector control was de-veloped; the vector control aims to control the flux linkage of the AC machine and the electricaltorque separately.

POWER ELECTR.

MEASUR.DEVICES

LOAD

POWER SOURCE

DCMOTOR

TORQUECONTROL

FLUXCONTROL

uAref

uFref

iF (~ olo)iA (~ T olo)

n

n ref

CONTROLLER

SIGNALPOWER


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In AC machines, the electrical torque can be expressed for instance as the cross product of the stator current and the stator flux linkage vector

T i e s s

32

p (5.4)

that is, the product of the absolute values times the sine of the angle between the above two vectors.The torque is a vector parallel to the shaft of the machine. The torque can be expressed as a scalar quantity as follows

sin23

sse i pT , (5.5)

whereT e the electrical torque,T e the absolute value of the electrical torque,

p the number of pole pairs,

i s the stator current vector,s the stator flux linkage vector, and

the angle between the above two vectors.

This expression obtains a positive value, when the current rotates in the positive direction ahead theflux linkage ( > 0). Thus, we have a motor drive. Instead, if the flux linkage is ahead of the current( < 0), we have a generator drive.

According to the principle of cross-field, the torque reaches its maximum, when the current vector and flux linkage vector are perpendicular to each other, in other words, the angle between thesevectors is 90 electrical degrees. As we have seen, in fully compensated DC machines, this conditionis always met thanks to the compensating and commutating-pole windings, whereas in AC ma-chines, the angle between these two quantities varies depending on the situation. The basic idea of the vector control is to bring the current and the flux linkage perpendicular to each other by meansof control technology. In that case, the quantities have to be treated as vectors with a direction and amagnitude, hence the term vector control.

In the traditional vector control, the target is to create the current references such that the magneti-zation and the electrical torque can be controlled independently also during transients. This is pos-sible if we manage to calculate a reference separately for the current component causing the electri-cal torque and for the magnetizing current component. The currents can be represented as compo-

nents for instance in the T reference frame fixed to the air gap flux linkage, or in a dq referenceframe fixed to the rotor as shown in Fig. 5.2.


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q

d

m

r

T

x

y

m

m

Figure 5.2 A dq reference frame fixed to the rotor, a T reference frame fixed to the air gap flux linkage, and an xyreference frame fixed to the stator. The rotation angle between the dq and T reference frames is the pole angle m of the air gap flux linkage. The rotation angle between dq and xy reference frames is the rotor position angle r . The rota-tion angle m between the T and xy reference frames is the position angle of the air gap flux linkage vector in the xyreference frame.

Figure 5.3 represents a principal block diagram of the vector control of an AC motor. When com- paring Fig. 5.3 with Fig. 5.1, we can see that the principle is the same in both controls. The aim, asstated before, is to control the flux linkage and the torque separately, yet there are two basic differ-ences: first, there is a difference between the treatment of currents. While in the DC control, thecurrents causing flux linkage and torque are obtained simply by measuring, in the vector control,coordinate transformations are required for the measured currents in order to determine the currentcomponents causing flux linkage and torque. Correspondingly, the inverse coordinate transforma-tions have to be carried out, when the phase voltages supplied to the motor are constructed. Another difference is that while in the control of a DC machine there is a direct link to the actual values of the torque and the flux linkage, in the vector control, the actual values of the flux linkage and thetorque have to be calculated with the motor model instead.

POWER ELECTRONICS

LOAD

AC-MOTOR

MEASURINGEQUIPMENT

FLUXCONTROL

TORQUE

CONTROL

CO-ORDINATETRANFORMATIONS

isaisb

isc

nref

n

isT

isT

usaref usbref uscref

m

CONTROLLER

usTref

us ref

SIGNAL

POWER

POWER SOURCE

MOTOR MODELANDCO-ORDINATETRANFORMATIONS


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Figure 5.3 The connection block diagram of the traditional vector control for an AC machine. In the vector control, thetarget is to control the flux linkage and the torque separately by applying the current vectors constructed by coordinatetransformations.

Nowadays, vector control is a method commonly applied to the control of various rotating-fieldmachines. Most of the manufacturers of frequency converters apply some vector control method intheir products. A challenge of the vector control is the correctness of the parameters of the motor model; if the inductance parameters are erroneous, the implementation of the vector control is natu-rally imperfect.

5.3 Direct Flux Linkage Control (DFLC)

In the mid-1980s, Manfred Depenbrock from Germany and Takahashi and Noguchi from Japan in-troduced a method based on Faradays induction law; in the method, the stator voltage vector is in-tegrated to determine the stator flux linkage. The researchers did not give a specific name to their method, and thus we have adopted the concept Direct Flux Linkage Control (DFLC), based on the

nature of the method. The principle differs form the previous vector control methods by the fact thatthe stator current fed to the motor is not controlled directly, but the target is to influence on the fluxlinkage of the motor as directly as possible, and thus on the torque produced by the machine.

From the control engineering perspective, the motor current in the DFLC is an output variable of the system, not an input variable.

5.3.1 The Basis of Direct Torque Control (DTC)

Since the direct flux linkage control was originally introduced for induction machines, the funda-mental philosophy of this control technique is investigated here with the induction machine. This,however, does not limit the application of the principle in any way, but the method can be appliedto any rotating-field machine.

As the equivalent circuit of the induction motor, a vector equivalent circuit according to Fig. 5.4,determined already in Ch. 3 is used. This equivalent circuit holds generally for a motor also duringtransients. The equivalent circuit does not take iron losses, harmonics, or the saturation of the ma-chine into account, yet it is accurate enough from the control-engineering point of view. The para-meters of the equivalent circuit are usually updated according to the operation mode of the machine.However, the time derivatives of the inductances are not usually taken into account.

ms r Lm

Ls Lr Rs Rr

j g s j( g-p ) r

u s

i s

i r

i m


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Figure 5.4 The equivalent circuit of an induction motor according to the space vector theory in a reference frame rotat-ing at a general speed g. The currents i and the voltages u are vectors, similarly as the flux linkages . In the stator reference frame, the angular frequency g disappears.

The equations of the asynchronous machine according to the space vector theory are recapitulatedand the bases for the direct flux linkage control are determined. Based on the equivalent circuit, we

may write the following representations for the voltages and flux linkages:

sgs

sss jdd

i u t

R . (5.6)

r gr r r r jdd

i u pt

R . (5.7)

r msssi i L L . (5.8)

smr r r i i L L . (5.9)

mmr smm i i i L L . (5.10)

Here the so-called stator inductance is the sum of the leakage inductance of the stator and the mag-netizing inductance

L L Ls s m . (5.11)

We determine the rotor inductance correspondingly

L L Lr r m . (5.12)

The instantaneous torque of the machine can be expressed by applying the familiar cross-field prin-ciple

r r sse 23

23

i i T p p . (5.13)

By substituting Eq. (5.8) to Eq. (5.13) we obtain

r msmsr msmse 232323i i i i i T p p L L L p . (5.14)

By applying air gap flux linkage, it is possible to express the torque both with the stator and the ro-tor current. We can also derive an expression for the torque, in which the determining terms are theflux linkages of the stator and the rotor and the leakage factor. By applying Eqs. (5.8), (5.9), and(5.13) we obtain


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2m

r sssm

r sse

m

r

m

ssss

m

r se

m

r r r ssse

23

23

23

23

23

L L L

L p p

L L

L L

L p

L L

p p

s

i

i T

i

T

i i T

. (5.15)

By taking32

p s si as a common factor in the latter equation, we may rewrite the torque equation

in the form

r s

m2m

r s

e

1

123

T

L L

L L p . (5.16)

By adopting the flux leakage factors for the stator leakage flux, the rotor leakage flux, and the totalleakage flux

r smr

r m

ss 11

11,,

L L

L L

(5.17)

we may write the denominator in Eq. (5.16) in the form

1

1

11 1

1

11

1

2

L L

L L

L L L

Ls r

mm

m s r mm

m

, (5.18)

consequently, the torque equation is now simplified into the form

r sm

e1

23

T L

p

. (5.19)

This equation justifies the direct flux linkage control. Namely, we know that the time constant of the rotor r = Lr / Rr is rather large in all induction machines, on the scale of 100 1500 ms. The tor-que of the induction motor can thus be adjusted very efficiently by controlling the angle betweenthe stator and rotor flux linkage. As the flux linkages also include the leakage components, we maystate that only the leakage flux of the machine constrains the rate of change. Hence the direct fluxlinkage control produces the fastest possible torque changes in induction motors.

Instead of the rotor flux linkage, we could also apply the air gap flux linkage in Eq. (5.19), in whichcase the torque equation is written as

mss

e

1

2

3 T

L p . (5.20)


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Also the air gap flux linkage behaves stably in the induction motor. This equation is applicable alsoto the direct flux linkage control of synchronous machines.

The switching of the modern IGBT switches typically takes place in less than a hundred nanose-conds. If the processor capacity is sufficient, switching frequencies up to tens of kilohertzes can beused. Thus, the time constants of the rotor flux linkage and the air gap flux linkage are easily even a

couple of decades longer than the time required for the computation of the motor model, making theswitching decisions, and the implementation. Figure 5.5 depicts how the air gap flux linkage andthe rotor flux linkage behave quite steadily irrespective of the somewhat unstable stator flux lin-kage.

Figure 5.5 Principles of the DFLC. As an effect of various voltage vectors, there occurs some ripple inthe stator flux linkage. The different time constantsof the machine filter the effect of ripple almostcompletely when observing the loci plotted both bythe point of the air gap flux linkage vector and bythe point of the rotor flux linkage vector. Contraryto the rippled locus plotted by the point of the sta-tor flux linkage vector, the locus drawn by the

point of the air gap flux linkage vector, and partic-ularly the locus plotted by the point of the rotor flux linkage vector is almost a perfect circle. In thesteady state, the angular speed of the rotor flux lin-kage r is very even, whereas the angular speed

s of the stator flux linkage vector is uneven;however, its average is equal to the angular speedof the rotor flux linkage vector or of the air gapflux linkage vector.

i r

i s

m

r

s

Equations (5.19) and (5.20) as well as Figure 5.5 indicate clearly that the torque of an rotating-fieldmachine can be controlled rapidly by making as quick adjustments to the stator flux linkage as

possible. When we accept this principle, it is easy to move on to investigate the direct flux linkagecontrol.

5.3.2 Implementation of Direct Flux Linkage Control

A fast torque control has a focal role in the realization of a good control. As it was shown previous-ly, the torque of an AC motor can be adjusted quickly by changing the angle between the stator androtor flux linkage. Also the absolute value of the stator flux linkage can be changed fairly easily dueto its leakage component. However, changing the main flux itself is a time-consuming task. Hence,the entire control can be based on a single equation. The stator flux linkage estimate is calculatedfrom the voltage by integrating

s,est s s s d u i R t ; (5.21)

no other machine parameters than the stator resistance Rs

are required. The estimate for the electric-al torque is obtained by the estimated stator flux linkage and the measured stator current vector


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sests,este, 23

i T p . (5.22)

In the case of a rotating-field machine, the FLC control provides thus the simplest equations. Here itis worth bearing in mind that the method is applicable to the control of all rotating-field machines.

For the implementation of the direct flux linkage control, in addition to the above data, also the po-sition information on the flux linkage vector on the flux circle is required. The flux circle is dividedinto six sectors in such a way that the borders of the sectors bisect the angles between the voltagevectos of the two-level inverter, Fig. 5.6.

u3

s

u7

u 0u

4

u6

u1

u2

u5

0

1 2

3

4 5

6 6

, 6 2

, 2

56

, 56

76

,

76

32

,

36 6

,

position vec-tor of the fluxlinkage s

= 0 = 1 = 2 = 3in the figure

= 4 = 5

Figure 5.6. The division of the flux circle into sectors determined by the voltage vectors.

In the direct flux linkage control, the stator flux linkage and the torque are kept within the set hyste-

resis limits (hysteresis band). The controls of the power switches are adjusted only when the torqueor the absolute value of the flux linkage deviate too much from their setpoint values. When the hys-teresis limit is reached, the next suitable voltage vector is selected to bring the stator flux linkageinto the right direction. When the stator flux linkage is selected as the target of regulation, the con-trol can be carried out by directly controlling the switches of the inverter. In each sector, two vol-tage vectors can be used in both rotation directions. One vector increases and the other decreasesthe stator flux linkage. The absolute value of the stator flux linkage is regulated according to thenormal two-point control, in other words, the flux is increased until it reaches the upper hysteresislimit, after which it is decreased until the lower limit is reached. The hysteresis limits are deter-mined based on the allowed switching frequency. If the switching frequency tries to exceed the al-lowed upper limit, the hysteresis limits have to be extended. On the other hand, by the correct set-ting of the hysteresis, it is possible to optimize the energy efficiency etc. The voltage vectors per-

pendicular to the stator flux linkage vector ( u 3 and u 6 for the two-level inverter of Fig. 5.7) have a particularly strong influence on the torque especially when operating clearly below the rated


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speed of the machine. By using a perpendicular voltage vector, a very rapid torque response isachieved.

s

m

i s

u6u 5

u4

u3

u2

u1

u 7

u0

x

y

'

min max

T emax

T emin

Figure 5.7 The voltage vectors and the shaded area illustrating the hysteresis control of a two-level inverter. This is acontrol of a synchronous machine, since the stator current is perpendicular to the stator flux linkage. Again, we areclose to the field weakening. If the absolute value of the stator flux linkage is driven to the upper limit of the hysteresisso that the angle with air gap flux linkage m is simultaneously at the maximum, the upper limit of the torque T emax isreached. Note that if the voltage vector shifts s away from the above location, also the position of the current changes,and the perpendicularity remains basically unchanged.

When combining the hysteresis controls of the flux linkage and the torque, as well as the positioninformation of the flux linkage vector, we obtain the so-called optimal switching table (Takahashiand Noguchi), Table. 5.1. Since the target is to keep the flux linkage within certain limits, it is al-ways either increased by = 1 or decreased by = 0. There are three possibilities with respect tothe torque control:

= 0 zero voltage is applied,= 1 the flux linkage is brought to the positive rotation direction,= -1 the flux linkage is brought to the negative rotation direction

Table 5.1. The selection of the voltage vector based on the location and the desired change in the flux linkage. Flux linkage sector

torque bit flux linkage bit

= 0 = 1 = 2 = 3 = 4 = 5

-1 0 u 5 u 6 u 1 u 2 u 3 u 4 -1 1 u 6 u 1 u 2 u 3 u 4 u 5 1 0 u 3 u 4 u 5 u 6 u 1 u 2 1 1 u 2 u 3 u 4 u 5 u 6 u 1

0 u 0 u 0 u 0 u 0 u 0 u 0

This core function of the DFLC can be illustrated by Figure 5.8, in which the torque and flux lin-kage control, as well as the position information of the flux linkage are combined.


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Figure 5.8 Block diagram of the combined torque and flux linkage control.

The behaviour of the hysteresis-controlled torque is depicted by Fig. 5.9. For instance at the nota-tion t the torque is let to decrease, when the upper limit of the torque has been reached. Here, a ze-ro voltage vector can be used, in which case the stator flux linkage remains stationary and isslightly shortened by the influence of the resistances. The instantaneous value of the torque T e di-minishes, until its value is below the lowest torque determined by the hysteresis band. Now again, avoltage vector is selected that keeps the length of the stator flux linkage appropriate and increasesthe angle between the flux linkage vectors thus increasing the torque ( t +).

Figure 5.9 Hysteresis control of air gap torque. Theabsolute value of the electrical torque oscillatesaround the reference value. Torque ripple is notharmful, since the switching frequency is typicallyat the scale of kilohertzes, and thus the mechanicalsystem filters the ripple out. The asymmetry of thetorque pattern at the position indicated byt t t is

based on a situation, in which the flux linkage con-troller reduces the flux linkage by selecting a newvoltage vector. Thus also the incrase rate of thetorque changes. The hysteresis T e1 is regulated bya PI controller, which thus also determines the av-

erage switching frequency.

0.9

0.85

0.8

T e,est

t

t t

t+

T e,ref + T e1

T e,ref T e1

T e,ref

Additional hysteresis of the stator FLC

In the basic application of the direct flux linkage control, the stator flux linkage control is subordi-nate to the torque control. If in this kind of a control the reference value of the torque becomes zero,the stator flux linkage control is not regulated at all. The situation may occur quite often for in-stance in the control of an induction machine when operating at a low speed and torque. If the tor-que control does not require an active voltage vector, the absolute value of the stator flux linkage isreduced due to the resistive losses according to Fig. 5.10 and is driven outside the hysteresis limits.A situation of this kind can be avoided by applying separate hysteresis limits min2 and max2 for the stator flux linkage. When the torque reference is other than zero, the hysteresis limits min1 and max1 are employed, in which case the control of the stator flux linkage is subordinate to the torque

s,est

+10

-1

T e,ref - T e,act

optimalswitchingtable

ASIC

SA

SB

SC

+10


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control. In that case, we apply the voltage vectors of the optimal switching table in the regulation.As the torque reference approaches zero, the stator flux linkage control subordinate to the torquecontrol does not function, and therefore, the stator flux linkage control comprising the hysteresislimits min2 and max2 have to be applied to. In such a situation, the voltage vectors located in themiddle of the stator flux linkage sector are selected, since in that case the effects of the stator fluxlinkage control on the torque remain lower than by using the vectors of the optimal switching table.

As the torque bit increases, the voltage vectors according to the optimal switching table are appliedagain.

s

i s

u6u 5

u4

u3

u2

u1

u7

u0

x

y

min2 max2

m

-i s Rs t

min1 max1

Figure 5.10 Additional hysteresis: The control of the stator flux linkage at low values of torque and low rotation speeds.In the torque-oriented control, the stator flux linkage can reduce due to the influence of resistive losses. Some of the points of influence are indicated by arrows in the figure. Such behaviour has to be taken into account by setting hyste-resis limits for the stator flux linkage that are independent of the torque control. In that case, voltage vectors that bestcorrespond to the direction of the stator flux linkage are employed, and thus the absolute value of the stator flux linkagecan be increased so that the angle between the stator and air gap flux linkage remains unaltered.

5.3.3 Errors of Direct Flux Linkage Control

An observant reader may have noticed that the DFLC involves few feedback elements typical of or-dinary control systems. This is also a reason why the method has been subject to criticism; theDFLC has been called only an advanced scalar control without an actual feedback, the only feed-

back information being the measuring of motor currents. If the integration of the flux linkage can be perfectly accomplished, the current measurement alone suffices, since the torque can now be exact-ly calculated. However, the voltage integration according to Eq. (5.21) alone is unfortunately notalways useful as such particularly when operating at low frequencies. The flux linkage estimation

by integration involves a problem, since even small errors may have a significant cumulative effect.The motor voltage is not usually measured (this is not profitable in commercial applications), anddifferent voltage losses , such as non-linear voltage losses in power switches and in the resistancesmay account for 50 70 % of the total voltage when operating at slow or zero speed. So far, the es-timation of these losses has been impossible by the present computation capacity provided by thecommercial processors. In particular, the vague voltage losses in the power switches that also vary

with the production lot, are very difficult to model. Therefore, if the integral of the stator flux lin-kage fails, the error easily accumulates in the flux linkage of the motor as time progresses; the mostcrucial problem in the DFLC is that due to the errors in the stator flux linkage estimate, the progres-


13/21


sion of the flux linkage of the motor does not remain origin-centered. Figure 5.10 illustrates howthe trajectory plotted by the point of the real stator flux linkage of the motor drifts gradually out of the origin-centered trajectory.

s,est s,motor

Figure 5.10 Due to the drifting of the stator flux linkage, the trajectory plotted by the point of the vector is no longer origin-centered. The DFL controller keeps the estimate origin-centered, however, the simultaneous integration errorscause the actual stator flux linkage to drift out of the origin.

The only feedback information the stator current involves information about the drifting of thestator flux linkage, since in practice, this kind of drifting produces a DC component in the currents.Furthermore, in the integration equation (5.21) of the stator flux linkage we observe a negativefeedback of the current. In principle, the term Rsi s stabilizes the situation, but since the stator resis-tance is in large machines in particular very small, the flux linkage of the motor will drift se-riously eccentric before the effect of Rsi s is high enough to prevent further eccentricity. This driftingto an eccentric trajectory nevertheless takes place rather slowly, and thus in practice we have sometime to correct the progression of the flux linkage by applying various methods. The effect of dif-ferent errors has been investigated for instance in the doctoral dissertation by Jukka Kaukonen; thefollowing discussion quotes his work.

The stator flux linkage estimate is thus obtained by integrating the stator voltage vectors in the sta-tor reference frame. The stator reference frame is a natural choice, and no extra coordinate trans-formations are required in the case of the DFLC, unlike in the current vector control. The integra-tion is carried out digitally by applying an efficient signal processor. Only the stator current andvoltage vectors u s and i s are required in addition to Rs.

As the voltage source inverter supplies voltage pulses into the windings of the motor, voltage vec-tors are formed. The phase currents of the stator are measured, and they are used to construct thecurrent vector of the stator. Additionally, the intermediate circuit voltage U DC is typically measured,as well as the switching states S A, S B, S C of the inverter; furthermore, a model for the voltage lossesin the switches is developed. Based on the above, the states receive the values "1" or "0" , depend-ing on the switching state. The stator voltage vector is thus written as

u s A B C DC A j B j

23

C

j43

a,hav j

b,hav

j23

c,hav

j43, ,S S S U S e S e S e u e u e u e

23

23

0 0

. (5.23)


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Here ua,hav , u b,hav , and uc,hav aim to model the voltage drops in the power switches both during theswitching state and when the switch is conductive. We may now write for the stator flux linkage es-timate

sest,1 s A B C s s s est,0

s A B C s s s est,0

d

d

u i

u i

S S S R t

S S S t R t

, ,

, ,. (5.24)

The equation shows that when the effect of the stator resistance is low, there is a small stator fluxlinkage moving into the direction of the instantaneous voltage vector. In the case of a non-zero vec-tor, the velocity is high. When applying zero voltage phasors, the velocity is small and proportionalto the loss i s Rs. In principle, the estimation of flux linkage is easy to implement, if we ignore the

problems caused by cumulative errors. The method is independent of the state of the machine in-ductances, which, in particular during the fast transients, makes the method superior to the tradi-tional current controls. Only Rs is a critical machine parameter, the measuring of which is quite easyto arrange also in practice.

Equations (5.23) show (5.24) that integration error is caused by four different factors:

1) an intermediate voltage measuring error,2) a stator current measuring error,3) a power switch voltage loss estimation error, and4) a stator resistance estimation error.

Measuring errors can be divided into two categories depending on their nature:

1) a gain error and2) an offset error.

Gain errors cause a constant error in the integration of the stator flux linkage. It can be shown thatthe torque error is directly proportional to the intermediate voltage measuring error and to thesquare of the current measurement. In the case of stator resistance estimation error, the torque error is directly proportional to the stator resistance estimation error. It is also worth noticing that the tor-que estimate error is inversely proportional to the frequency, and is thus insignificant at high fre-quencies. At the zero speed, the estimation error results in extremely large torque estimation error.

Offset error causes DC components to phase quantities, and the stator flux linkage vector rotateseccentrically at a certain distance from the origin. A pulsating error proportional to the rotationspeed is generated to the torque.

Thus, two types of errors are involved in the voltage determination: we obtain either a constant stable error or an unstable drifting error that drives the stator flux linkage out of the origin-centeredtrajectory. The latter is a serious problem.

Intermediate voltage measuring error

With respect to the intermediate voltage, the gain error and offset error produce a similar error,since the measurement signal is a DC quantity. Two different types of errors can be detected, both

of which result from the erroneous measuring of the intermediate voltage. One is a stable DC error and the other the unstable drifting of the flux linkage. Figure 5.11 illustrates these errors. Themeasured intermediate voltage U DC meas can be expressed as


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DCDCoffsDCDCgainmeasDC 1 U U U U k U , (5.25)

where U DC is the actual intermediate voltage and k gain is the gain coefficient, and U DC offs is the off-set voltage. U DC is the total measuring error at a certain instant of time: Kaukonen (1999) hasshown that if U DC 0 (U DC meas U DC), a stable flux linkage error occurs, and if U DC < 0 ( U DCmeas < U DC), unstable drifting of the stator flux linkage occurs.

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-0.9

-0.6

-0.3

0

0.3

0.6

0.9

1.2

-1.2 -0.9 -0.6 -0.3 0 0.3 0.6 0.9 1.2

s

s,est

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-0.9

-0.6

-0.3

0

0.3

0.6

0.9

1.2

1.2 -0.9 -0.6 -0.3 0 0.3 0.6 0.9 1.2

s

s,est

1) U DC 0 (U DC meas U DC) 2) U DC < 0 ( U DC meas < U DC)

Figure 5.11 Examples of the behaviour of the flux linkage integral caused by the intermediate voltage error.

Phase current errors

Gain and offset errors in the measured phase currents cause similar phenomena as the errors pre-sented above. Two different types of behaviour can be observed: the first is the constant stable error in the flux linkage, and the second is the alternating unstable drifting of the flux linkage. The condi-tions for these two errors are the same as for the intermediate voltage error. The gain error the cur-rent measuring can be considered a voltage loss calculation error, and eventually a voltage error.The measured stator voltage u s meas can be expressed as

ssssgainCBAcalcsmeass 1,, uui uu Rk S S S , (5.26)

where u s is the actual stator voltage, u s calc is the voltage calculated from the intermediate voltageand the switch states, and k gain is the gain coefficient. u s is the total measuring error at a certaintime instant. It has been shown by simulations that if us 0 (| u s meas |u s|) a stable stator flux lin-kage occurs, and if us < 0 (| u s meas | < |u s|), unstable drifting of the stator flux linkage occurs, Fig.5.12. The offset error in the measured phase currents causes a DC component, which leads inevita-

bly to the drifting of the integral.


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Voltage loss estimation error in power switches

Due to the present computation capacity, there is no accurate model for power switches, and there-fore, they are usually described by a simple resistance model ( Rd), supplemented by the tresholdvoltage u th.

u u i hv th s d R . (5.27)

The voltage loss estimation error in switch components can be considered similar to the stator resis-tance error.

Stator resistance error

The stability of the DFLC can be considered by estimating the stator resistance, since it is the mostsignificant variable in the voltage integration, and the integration of the flux linkage is very sensi-tive to this error. In the case of the induction motor, it has been shown both analytically and empiri-cally that if the stator resistance is estimated too large R

s est> R

s act, the error term of the stator flux

linkage increases in the direction of the flux linkage, and the DFL control is unstable (positivefeedback). If the stator resistance is estimated too small Rs est Rs act , the flux linkage error tends todecrease and the DFL control is stable (negative feedback) (Pohjalainen 1987). The same holds alsofor other machine types. The integral of the stator voltage can be written as

s s s sd d u i t R t , (5.28)

The stator resistance estimate can be expressed as

R R Rsest s s

. (5.29)

Rs est is the estimated resistance and Rs is its error. The error term of the stator flux linkage may bewritten as

s err s s est s act s act s act s act s act s s act

s s act s sx act s sy act sx err sy err sd err sq err j

= d d d d

d d j d j j r

u i u i

i

t R t t R R t

R t R i t R i t e . (5.30)

Figure 5.12 illustrates the drifting of the stator flux in the case of voltage loss estimation error. Thevoltage losses have been estimated to be larger than they actually are. As a result, the actual stator

voltage is higher than the estimate. The selected voltage vector is supplied until the stator flux lin-kage estimate exceeds its hysteresis limit. During this time, the real flux linkage exceeds its hysterislimit considerably and the control becomes unstable.


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s (t 0) =

u6u 5

u4

u 3u

2

u1

u7

u0

x

y

min

max

s (t 9)

s est (t 0)

s est (t 9)

Figure 5.12 Drifting of the stator flux linkage as a result of erroneous voltage loss estimation.

Let us assume that the stator flux linkage drifts from the origin due to the instantaneous error in thestator resistance estimate at time t 1 (Fig. 5.12). This error can be seen in the current vector of themotor

i s sd sq

sdsd md F D

sqsq mq Q

sdsd est sd err md f D

sqsq est sq err mq Q

sdsd est md F D

sd err

sd sqsq est mq Q

sq err

sq

j j

j

j

i i L

L i i L

L i

L L i i

L L i

L L i i

L L L i

L

1 1

1 1

1 1

(5.31)

As expected, the stator current vector moves in the same direction as the flux linkage error terms err . The error accumulating during one electric cycle can be calculated by assuming the angular

speed of the stator current vector constant

s err s err x s err y j s sd sq j j jr r e R i i e2 . (5.32)

Based on Eq. (5.31), we may write for the current components isd and isq caused by the estima-tion error

isd = s err d / Lsd and isq = s err q / Lsq.(5.33)

Now we can define the difference of the estimation error

s err sx err sy err s sd err sd

sq err

sq

j j j r

R L L

e2 2 (5.34)

We see that when estimating Rs est > Rs act , we obtain the unstable drifting as the flux linkage in-creases into the direction of the error. If the stator resistance is estimated to be smaller than the ac-


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tual resistance Rs est Rs act , we obtain the stable behaviour. This is easy to understand also on the basis of the energy principle a system, the losses of which are larger than the controller expects,tends to be more stable than an opposite system. Figure 5.13 depicts a situation in which the esti-mate is larger than the actual one Rs est > Rs act , and the opposite case Rs est < Rs act .

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0

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s

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0

0.3

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-0.9 -0.6 -0.3 0 0.3 0.6 0.9 1.2

ssest

sest

Rs est < Rs act Rs est > Rs act

Figure 5.13 Estimated and actual flux linkages when there is an estimation error in the stator resistance.

Further, Figure 5.14 illustrates a typical non-sinusoidal waveform of the stator current caused by thedrifting of the stator flux linkage.

1.5

1

0.5

0

-0.5

-1

-1.50 5 10 15 20 25 30 35 40 45 50 55

t [ms]

i, pu

ia i b ic

Figure 5.14 Non-sinusoidal waveforms of the stator currents caused by the eccentricity of the stator flux linkage (Nie-mel 1999).

We have shown that in practice, the direct flux linkage control is not very usable method. There-fore, in addition to the FLC, we need a method that corrects the problems of instability caused bythe non-idealities. The primary task of the correction method is to prevent the flux linkage fromdrifting out of the origin. A natural and very efficient method is to apply the current model based onthe motor parameters. Hence, when the direct flux linkage control is supplemented by the currentmodel, a method commonly known as the direct torque control (DTC) is achieved.


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5.4 Direct Torque Control (DTC)

Direct torque control (DTC) refers to a method, in which there is, in addition to the above-mentioned DFLC, also some method stabilizing the integration of the stator flux linkage , the

primary function of which is to keep the actual flux linkage of the motor origin-centered. In the fol-lowing, the current model correction and the correction of the eccentricity (drift correction) of thestator flux linkage are introduced and discussed in brief.

5.4.1 Current Model Correction

In the DTC, the voltage model is calculated very frequently; for instance in the DTC inverter byABB, the computation of the voltage model is repeated every 25 s. The drifting of the flux linkageout of the origin caused by the errors in the voltage model typically takes tens of milliseconds.Thus, the eccentricity of the flux linkage calculated by the voltage model has to be corrected for in-stance at every millisecond.

In the current model correction, the stator flux linkage integrated from the voltage is corrected bythe error vector. The error vector is constructed as a difference of the stator flux linkage vector in-tegrated from the voltage and the stator flux linkage vector calculated by the current model

susis . (5.35)

The flux linkage difference of Eq. (5.35), weighted suitably, is used to correct the stator flux lin-kage for instance at every 100 th microsecond.

s j

cm)su(1)su( e t

nn k . (5.36)

k cm is the weigthing coefficient of the current model. The correction has to be turned appropriately by the term t je that depends on the electric angular frequency and describes the progression of the flux linkage; the reason for this is that the correction term comprises information that is a milli-second old at maximum. The time t of the correction term is calculated from the instant of the de-termination of the error term onwards. The magnitude of the error is clearly dependent on the fre-quency, and therefore, if necessary, by suitably adjusting the weighting coefficient, the definition of the error vector can also be modified to be frequency-dependent. In practice however, a fixedweigthing coefficient is usually employed.

In particular, the superiority of the DTC when compared with the traditional current vector controlis based on combining the advantages of the voltage model and the current model. When applying alarge weigthing coefficient typical of the voltage model during fast transient phenomena, outstand-ing dynamic characteristics can be achieved for a drive. The current model tends to fail especiallyin dynamic states. In the DTC, the corrections given by the current model are weighted only slightlyin the case of a fast transient, and therefore, it is chiefly the DFLC that takes care of the transients.When the transient has stabilized, the weight of the current value increases in correcting the eccen-tricity of the stator flux linkage of the motor controlled by the DFLC.

In the context of the control of magnetically asymmetric machines, also the direct rotor positionmeasurement is required in the current model correction in order to enable coordinate transforma-

tions. This is clearly a defect that tempts to search for alternative methods for the stabilization of thestator flux linkage. The current model of the induction motor instead does not necessarily requirethe rotor position information, and thus a high-performance DTC drive can be achieved without po-


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sition angle information. The correction of the eccentricity of the stator flux linkage (drift correc-tion) functions quite well without the rotor position feedback information.

5.4.2 Correction of the Eccentricity of the Stator Flux Linkage

Based on the doctoral dissertations of M. Niemel and J. Luukko, the possible eccentricity of the

stator flux linkage s can be corrected, when operating at moderate frequencies, also by applyingthe scalar product of the stator flux linkage estimate s,est and the stator current i s, and by the com-

ponents s,x,est and s,y,est of the stator flux linkage estimate represented in the x and y directions inthe stator reference frame.

The scalar product of the stator flux linkage estimate s,est and the stator current i s is constructed

ys,esty,s,xs,estx,s,sests, iii . (5.37)

Next, the scalar product is lowpass filtered so that the filtering time constant is sufficiently longer than the periodic time of the supply frequency. The correction terms of the stator flux linkage esti-mate s,x,corr , s,y,corr are formed as a product of the difference of the calculated and filtered scalar

product and the components of the stator flux linkage estimate. The principle of the method is illu-strated in Figure 5.15.

calculation of correct. termsof stator fluxlinkage

LPFys,es ty,s,xs,es tx,s,

ests,

ii s

i

is,x

i s ,y

s,x s,y

s,x,corr

s,y,corr

+

Figure 5.15 Formation of the correction terms of the stator flux linkage.

The correction terms s,x,corr , s,y,corr of the stator flux linkage estimate are obtained from equations

s,x,corr corr s,est s s,est s filt s,x,est K i i (5.38)and

s,y,corr corr s,est s s,est s filt s,y,est K i i . (5.39)

The method thus enables the full utilization of the direct torque control without the rotor positionfeedback information. By applying this correction method, it is for the very first time possible toimplement the control of a rotating-field machine at a wide rotation speed range even though themotor parameter information is quite incomplete. Only the value of the stator resistance, which can

be easily measured, has always to be known. In principle, the method enables the application of theDTC to the power transmission of all rotating-field machines, and in the DC-to-AC inversion to thenetwork; the method is thus universal. Due to the filtering time required in determining the averageof the scalar product, the method is not applicable to the zero speed, and therefore the correctionmethod based on the current model is required in demanding drives around the zero speed.


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Figure 5.16 depicts the influence of applying the drift correction on the test drive when operating atthe supply frequency of 1 Hz.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5

sy,est sy,cm

pu

,

sx,est sx,cm pu,

s,est

driftcorrection

s,cm

Figure 5.16 Drift correction brings the flux linkage rapidly back to the origin (Niemel)

The drift correction can thus function very efficiently already when operating at the supply frequen-cy of 1 Hz.

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