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Darmstadt University of Technology

Department of Power Electronics and Control of Drives

Manual S3 Field oriented control

Version 2010

Ingo Müller

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1. Generalities For a long period, the technology of electric drives was dominated by DC motors, as their speed could be easily controlled by the armature voltage and the field current. In spite of their robustness, induction motors were used only in special cases, e.g. if the environment forbad using a dc motor because of its commutator, or if it was not necessary to control the speed, because processes were controlled by other actors like indexing gears, restrictors and bypasses. With the development of power semiconductors it became possible to change the speed of induction motors continuously and with low losses, because they could be supplied with voltages varying in amplitude and frequency. Due to the invention of the field oriented control it has been possible to control the torque and the flux of induction motors independently and in consequence to make the induction motors act like dc motors.

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2. Symbols

UN rated voltage (rms-value, linked) IN rated current (rms-value) PN rated power, assumption: mechanical power ƒN rated frequency nN rated speed M mutual inductance Lh magnetizing inductance O Blondel’s dispersion coefficient L1 stator inductance L2 rotor inductance R1 stator resistance R2 rotor resistance ƒA sampling rate TA sampling period T1 time constant, which can be compensated, general T2 time constant, which cannot be compensated, general T1 dead time TTP time constant of a low-pass filter, general TR time constant of a PI controller, general KS gain of a control system, general KR gain of the controller, general h a parameter of the angular position controller φ mechanical angular position of the rotor αmax maximum angular acceleration of the drive s slip ω syn synchronous angular speed ωm angular speed of the rotor ω2 angular speed of the slip ωmech angular position of the rotor, mechanical βm angular position of the rotor, converted to electrical value ßk angular position of the rotor flux linkage ß2 slip angle i 1 vector of the rotor current (length ? amplitude i 2 vector of the rotor current (length ? amplitude

1ψ vector of the state flux linkage 2ψ vector of the rotor flux linkage

1u vector of the stator voltage (length ? amplitude Tel torque produced by the induction motor

2e e.m.f. (due to the voltage inducted in the stator windings by the rotor flux) 1 ,A Bx∗ setpoint, general

1x∗ value of a complex number or vector, mirrored at the real axis, general J inertia of the drive

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3. The set-up

3.1 Overview An induction motor (so-called ”servo motor”) is placed on the wagon of the set-up. In order to investigate its load response, it is equipped with a disc brake. An angular position sensor is mounted on the other end of the shaft. The motor is fed by an inverter with a d.c. voltage-link, which is placed in the lower area of the wagon. A computer is placed close to the inverter. It contains bus cards which receive the signals from the angular position sensor and from the current sensors, to generate exact time bases and to control the inverter. The BNC sockets on the operator panel can be used to display measured values with an oscilloscope.

Figure 1: Set-up

3.1.1 Induction motor The induction motor used for the set-up is a so-called servo motor. Its dispersion inductance is rather small and in consequence the motor can be controlled very quickly. The type plate contains the following data: UN,eff = 350V at star connection IN,eff = 4A PN = 1,5kW assumption mechanical power ƒN = 50Hz nN = 2840min-1

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The following parameters were approximately identified by some measurements: M=Lh = 200...300mH mutual and magnetizing inductance 1Lσ⋅ = '

2Lσ⋅ = 9mH dispersion inductance R1 = 3Ω stator resistance R2 = 3,4Ω rotor resistance 3.1.2 D.c. voltage-link inverter The inverter is connected to the 400V power line. The control features which are normally used in industry are disabled. Instead, the power semiconductors (IGBTs) are directly controlled by the computer via optical waveguides.

3.1.3 Computer The whole control algorithm and editing of the measurements is realized by the computer. That is why it is a discrete control or a sampling control. As the computer is not able to provide a reliable time pattern, a timer card is used for generating the time pattern. The timer card executes the following tasks:

• At the beginning of each sample period it generates an interrupt request for the computer in order to start the control algorithm.

• It sends trigger signals in order to sample the currents and the position at exactly

defined instants. • It reads the switching times for the next period from the computer, generates the

signals and sends them to the inverter by optical waveguides. • It contains a watchdog function which observes, whether the computer sends the

voltage vectors at least once during a sampling period. A missing of the voltage vectors during one sampling period indicates that the computer has got stuck. In this case, the inverter has to be stopped immediately.

The sampling rate in this experiment is 10Af kHz= . This is equal to a sampling period of

100AT s= μ . The computer contains two additional cards for the reception of the measured currents and for the generation of 4 analog values, and a card for the reception of the angular position. The latter edits the signals received from the angular position sensor. It is described in detail in [1].

4. Field oriented control

4.1 Representation of the space vectors, reference frames

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In the three-phase system, the state variables – voltages, currents, fluxes – are usually represented as space vectors. That are vectors in a two-dimensional reference frame. For the description of the states of an induction motor, usually the following three reference frames are used, see figure 2:

• (αβ-) reference frame fixed to the stator. This is a reference frame which does not move. Its real axis is called α-axis, the imaginary axis is called β-axis. The quantities which are represented in the α- reference frame are marked by a ”s”, like 1

sΨ .. • (dq-) reference frame fixed to the rotor. The position of the dq-reference frame is

identical to the mechanical position of the rotor, converted to the electrical one1 . The quantities which are represented in the dq- reference frame are marked by a ”m”, like

1sΨ .

. • (AB-) reference frame fixed to the rotor flux linkage = field oriented. The

position of this reference frame is identical to the position of the rotor flux linkage. In consequence, the rotor flux linkage has a component only in the A-axis (real axis), but it has no component in the B-axis (imaginary axis). The quantities which are represented in the AB- reference frame are marked by a ”k”, like 1

kΨ .

α

d

AqB

ψ 2

reference frame fixed to stator

reference frame fixed to rotor flux

reference frame fixed to rotorβk

β

β2

βm

Figure 2: Reference frames

The torque of an induction motor is usually controlled by a field oriented control, because of the good dynamic behaviour achievable by this method. The equations of the model are first set up in the different reference frames according to the parts of the motor they belong to. After that, they are all converted to the AB-reference frame. By this means, the model of the induction motor becomes much smaller. The space vector representation is converted between the reference frames, like shown in the following example of the rotor flux linkage vector, see figure 2: The space vector 2Ψ is placed on the real axis of the AB- reference frame. Relative to this reference frame, its representation is written as 2

kΨ and it contains a real component only.

1The electrical position (and the electrical angular speed) is Pm Nß = , or

PmechPm NN ⋅⋅= ωω NP is the number of pole pairs.

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Representing 2Ψ relative to the αβ-reference frame, it is obviously placed at an angle of kβ to the real axis. Hence, the conversion is:

2

sΨ = 2kΨ jßke⋅ (1)

The conversions of the other space vectors are performed in the same manner. As both the space vector and the angle are time variant, the product rule has to be used for derivating. The length of the space vector is equal to the amplitude of the corresponding quantity. E.g. the stator current is 1 1| | 2ri I= ⋅ in steady state.

4.2 Basic equations of the induction motor An induction motor is – on principle – a three-phase transformator with its secondary windings being revolving and short-circuited. That is why the equations are rather similar. For simplification, equal numbers of stator and rotor windings are assumed. Due to this assumption, the rotor parameters become '

2 2L L= and '2 2R R= . In the progress, only 2R and

2L are written. At first, the stator is analysed. Therefore, the quantities are written relative to the αβ- reference frame. A current is flowing into the windings with the resistance 1R . Additionally, the windings are placed in a magnetic field with the flux linkage 1

sΨ . Then the voltage is:

11 1 1

ss s du R i

dtΨ

= ⋅ + (2)

Accordingly, the rotor voltage is:

22 2 2 0

mm m du R i dt

Ψ= ⋅ + =

as the rotor windings are short-circuited.

22 2

mmd R idt

Ψ = − ⋅ (3)

Like in a transformator, the stator and the rotor are coupled by a mutual inductance M. The stator flux linkage, relative to the αβ-reference frame, is: 1 1 1 2

s s sL i M iΨ = ⋅ + ⋅ (4) Accordingly, the rotor flux linkage, relative to the AB- reference frame, is: 2 1 2 2

m m mM i L iΨ = ⋅ + ⋅ (5) As algebraic equations are invariant to coordinate transformations, (5) looks quite similar relative to the αβ-reference frame: 2 1 2 2

s s sM i L iΨ = ⋅ + ⋅ (6)

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By setting up equations (4) and (6) as a matrix,

1 1 1

2 22

s s

ss

L M iM L i

⎡ ⎤Ψ ⎡ ⎤⎡ ⎤= ⋅⎢ ⎥ ⎢ ⎥⎢ ⎥Ψ ⎣ ⎦ ⎣ ⎦⎣ ⎦

(7)

the currents can easily be calculated:

1 122

11 22 2

1s s

s s

i L MM LL L Mi

⎡ ⎤⎡ ⎤ Ψ−⎡ ⎤= ⋅ ⎢ ⎥⎢ ⎥ ⎢ ⎥−− Ψ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

(8)

By introducing Blondel’s leakage factor

2 2

1 2

1 2 1 21 M L L M

L L L L−σ = − = (9)

The equations can be written as follows:

1 1 2 11

22

1 2 2

1

1

ss

ss

ML L LiMi

L L L

−⎡ ⎤⎢ ⎥σ σ ψ⎡ ⎤ ⎡ ⎤

= ⋅⎢ ⎥⎢ ⎥ ⎢ ⎥− ψ⎣ ⎦⎣ ⎦ ⎢ ⎥σ σ⎢ ⎥⎣ ⎦

(10)

or

1 1 21 1 2

1s s sMi L L L= ⋅Ψ − ⋅Ψσ σ

(11)

2 1 21 2 2

1s s sMi L L L−= ⋅Ψ + ⋅Ψ

σ σ (12)

4.3 Derivation of the time constants of the induction motor The purpose of the field oriented control is to control two components, 1Ai parallel to the rotor flux linkage vector and 1Bi orthogonal to it, independently. Therefore, it makes sense to represent the space vectors relative to the AB-reference frame. In this representation, the rotor flux linkage vector consists of a real component only, which simplifies the equations. Setting up the voltage equations for the stator (2) and for the rotor (3) relative to the AB-reference frame results in the following differential equations for the fluxes:

11 1 1 1

kk k k

kd u R i jdt

Ψ = − ⋅ − ω ⋅Ψ (13)

22 2 2 2

kk kd R i jdt

Ψ= − ⋅ − ω ⋅Ψ (14)

The equations (11) and (12), concerning the dependency between the currents and the fluxes, don’t change due to their invariance to coordinate transformations:

1 1 21 1 2

1k k kMi L L L= ⋅Ψ − ⋅Ψσ σ

(15)

2 1 21 2 2

1k k kMi L L L−

= ⋅Ψ + ⋅Ψσ σ

(16)

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In order to analyse the time behaviour of the induction motor, a differential equation is derived, which primarily contains only the stator voltage and the rotor flux linkage. Inserting (16) in (14) results in the following differential equation for 2

kΨ :

2 2 21 2 2 2

1 2 2

kk k kd R M R jdt L L L

Ψ = ⋅ Ψ − ⋅ Ψ − ω ⋅ Ψσ σ

(17)

Derivating (15) and inserting (13) and (17) results:

( )1 2 21 1 1 1 1 2 2 2

1 1 2 1 2 2

1kk k k k k k

kdi M R M Ru R i j jdt L L L L L L

⎛ ⎞= − ⋅ − ω ⋅ Ψ − ⋅ Ψ − ⋅ Ψ − ω ⋅ Ψ⎜ ⎟σ σ σ σ⎝ ⎠ (18)

Resolving (15) to 1kψ and inserting results in:

( )2 3

21 1 1 2 2 21 2 1 22 2 2 3 2

1 1 1 21 2 1 2 1 2

21 2 2

1 1 2 1 22 21 1 1 21 2 1 2

1

k kkk k k k

k

mk k k k kk

Mdi u R R M R M R Mi j idt L L L LL L L L L L

R R M R M Mu i j iL L L LL L L L

ω − ω⎛ ⎞⎛ ⎞ ⎛ ⎞= − + ⋅ − − ⋅Ψ − ω ⋅ + ⋅ Ψ⎜ ⎟⎜ ⎟ ⎜ ⎟σ σ σσ σ σ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ω⎛ ⎞= ⋅ − + ⋅ + ⋅ Ψ − ω ⋅ + ⋅ Ψ⎜ ⎟ ⎜ ⎟σ σ σσ σ ⎝ ⎠⎝ ⎠

(19)

The first two items describe the principle time behaviour if the voltage is fixed. Obviously, the stator current changes with the short time constant 2

1 22

1 1 2

1R R ML L L

+σ σ

. The knowledge of this

time constant is necessary for the design of the current controller. The remaining items describe additional coupling. The influence of this coupling is eliminated by decoupling methods, described in detail on page 18. By inserting (3) in (5) and by this way eliminating 2

mi , the following differential equation is obtained:

2 22 1

2

mm mL d M iR dt

Ψ⋅ + Ψ = ⋅ (20)

Representing (20) in the AB-reference frame by ( )222

jßkm e⋅Ψ=Ψ

2 2 22 2 2 1

2 2

kk k kL d L j M iR dt R

Ψ⋅ + ⋅ ω ⋅ Ψ + Ψ = ⋅ (21)

is obtained.

2kΨ contains only a real component 2 AΨ (because the AB-reference frame has been placed

accordingly); the imaginary component 2BΨ is zero. Regarding the real component of (21):

2 22 1

2

AA A

L d M iR dtΨ⋅ + Ψ = ⋅ (22)

it is obvious, that the rotor flux linkage follows the component of the current in rotor flux

linkage direction by the large time constant 2

2

LR . This time constant has to be considered

only during the startup of the control by increasing the flux slowly. E.g. the rotor flux linkage should be increased from zero to the rated flux with constant speed within some

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tenths of a second. During operation, this time constant is no longer important, because the amplitude of the rotor flux linkage is generally held constant. If the amplitude of the rotor flux linkage is constant, it is linked to the stator current accordingly to (22) by 2 1A AM iΨ = ⋅ (23) Regarding the imaginary component of (21):

2 2 22 2 2 1

2 2

BA B B

L d L M iR dt RΨ⋅ + ⋅ω ⋅Ψ + Ψ = ⋅ (24)

22 2 1

2A B

L M iR ⋅ω ⋅ Ψ = ⋅ (25)

as 2BΨ and 2 0BddtΨ = , an equation for the calculation of the slip frequency is obtained:

2 12

2 2

B

A

M R iLω = ⋅

Ψ (26)

(25) and (26) can be made clear as follows: The rotor ”sees” a flux linkage 2Ψ which rotates with slip frequency. A voltage 2 2ω ⋅Ψ is induced in the rotor. At a slip equal to zero, no voltage is induced, because in this case, the

rotor ”sees” a constant flux linkage. The current 2 22

2i R

ω ⋅Ψ= is transformed to the stator

windings by 2LM .

4.4 Generation of the torque The torque generated by an induction motor is calculated by (without derivation):

1 13 Im2

s sel PT N i∗= ⋅ Ψ ⋅ (27)

Eliminating 1sΨ by resolving (11) and inserting it, results in

1 1 1 2 12

3 Im2s s s s

el PMT N L i i iL

∗ ∗= ⋅ σ ⋅ + Ψ (28)

The first item is zero, because 1 1

s si i∗ ⋅ is real only. Representing the equation relative to the AB-reference frame, it is reduced to

2 12

32el P A B

MT N iL= ⋅ Ψ (29)

because the rotor flux linkage consists only of a real component, when it is represented relative to the AB-reference frame. Obviously, the torque depends on the component of the stator current orthogonal to the rotor flux linkage. The rotor flux linkage itself depends on

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the component of the stator current, which has the same direction as the rotor flux linkage, see (22). In consequence, 1i A is called the flux forming component and 1i B is called the torque forming component of the current. This is shown in the following figure.

M

ML2

T = L / R2 2 2

3N2

P 1J

Tload

Tel

i1A

i1B

2A

mech+-

Figure 3: Model, if the current is controlled

Therefore, it is necessary to control the stator current and to know the position of the rotor flux linkage. The stator current of this set-up is controlled by an inner current controller. The torque resulting from the motor torque elT and the load torque LastT accelerate the drive, which is assumed to have an inertia J.

4.5 Behaviour at stationary use If the induction motor is fed by a three-phase system with constant amplitude and frequency and if the load is constant, too, this is called stationary use. In an induction motor with a number of pole pairs 1PN = , the three phases of the stator windings are distributed around one whole circuit. One period of the three-phase system is equal to one mechanical revolution of the stator flux and hence to approximately one revolution of the rotor. If 1PN > , the three phases of the stator windings are distributed around 1

PN of the

revolution and repeat PN times. As a result, one period of the three-phase system is equal

to 1PN

of one mechanical revolution only.

A voltage has to be induced in the rotor windings in order to be able to produce a torque. This is only possible if the rotor rotates at a lower (or higher) speed than the stator flux linkage. Therefore, the slip s is introduced:

21 m

syn syns ω ω= − =

ω ω (30)

Here, synω denotes the electrical angular speed of the stator flux linkage and mω denotes the electrical angular speed of the rotor. At rated usage, the value of the slip is some percent. The following equivalent circuit diagram can be used for stationary use:

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R1

R2 sLh

σ L1 σ L2

U1

I1

Ih

Figure 4: T circuit diagram

First, no load (s=0) is assumed. No current is flowing through the rotor windings. As a result, the stator current is a pure magnetizing current hI . According to the equivalent circuit diagram, figure 4, its effective value is

( ) ( )

12 22

1 2h

UIR f L M

=+ π σ +

(31)

The magnetizing current is responsible for the generation of the flux. Hence, it is the flux forming component of the stator current.

4.6 Obtaining the angular position of the rotor flux linkage The angular position of the rotor flux linkage can be obtained by calculating it from the stator current. This is done by the so-called current model of the induction motor. It is derived from (26).

MR

2

N P

ω 2i1B

*

Ψ2A*

2

L β2

βmech

βm

βk

Figure 5: Obtaining the position of the rotor flux by the current model

As the rotor flux linkage cannot be measured with reasonnable efforts, this model is usually fed with the setpoint of the flux. Moreover, the setpoint of the current is used because it is assumed that the inner current controller controls the current fast enough. The angular position mechß is measured by the angular position sensor. The quantities 2ω (angular

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frequency of the slip), ´2ß (slip angle) and kß (angular position of the rotor flux linkage) are calculated in the model and can differ from the real quantities. This method is used at standstill and low speed. The calculated slip can be wrong due to the inaccurate knowledge of the inductances and of the rotor resistance 2R (the latter is even temperature dependent). Unfortunately, the error is integrated to the slip angle and is only limited by the feedback in the motor, resulting in a lack of performance. At high speed, the angular position of the rotor flux linkage can be obtained more accurately by the voltage model. Derivating (11) and inserting it into the voltage equation for the stator (2) results in an equation for the voltage 2e induced in the stator windings due to the rotation of the rotor flux linkage (electromagnetic force – e.m.f.):

2 2 1 2 1 2 12 1 1:

s ss s sd L R L L L die u idt M M M dt

Ψ σ= = ⋅ − ⋅ − ⋅ (32)

The e.m.f. can be calculated from the measured quantities during operation with this equation. Afterwards, it can be separated into the components and rotated into the AB-reference frame, see figure 6. The e.m.f. is just the variation of the rotor flux linkage relative to the αβ-reference frame. In stationary operation, the rotor flux linkage rotates at synchronous angular speed

k synω = ω , hence it is 2 2

synj tsA e ωΨ = Ψ ⋅ (33)

and in consequence 2 2

synj tssyn Ae j e ω= ω ⋅Ψ ⋅ (34)

2se is 90° ahead of the rotor flux linkage.

In stationary operation, the e.m.f. contains only a B-component, but no A-component. If in consequence the e.m.f., calculated by the voltage model, contains an A-component, then the angular position of the rotor flux linkage has not been calculated correctly and has to be corrected. An advantage of this method is, that the error does not sum up. First, the angular position of the rotor flux linkage is calculated by the current model as described above, see figure 5. Additionally, the A-component of the e.m.f. is obtained by the voltage model, see figure 6, and it is made zero by a PI controller. This is done by correcting the angular speed of the slip 2iω (this is the angular speed of the slip calculated by the current model – here it is marked by the index i) by the output of the controller.

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N P

ω 2ii1B

*

Ψ2A*

MR 2

2L β2

βmech

βm

βkω 2

K mech ω ( )

e -2A

contr.αβAB

e2A

e2B

e2A* = 0 correction

e2α

e2β

e.m.f-calcul-ation

u1α

u1β

i 1α

i 1β

βk Figure 6: Calculation of the flux angle by the voltage model

The parameters of the controller have been set to the following values:

Gain: 2 2

1201400korrsK

s V

ω −= ⋅ ⋅

ω

It has been obtained by experiments. In order to keep the angular position of the rotor flux stetic during acceleration of the motor, the correction is activated continuously, starting at 120

sω = , that is approximately 1200

min. As the gain of the system is proportional to the

speed, the gain of the controller has to be set proportional to the inverse of the speed, in order to keep the system stable.

Time constant: 2

2korr

LT R=

4.7 Realisation of the control The field oriented control can control the rotor flux linkage and the torque independently. The setpoints 1Ai∗ of the flux-forming current and 1Bi∗ of the torque-forming current are given. Each of these two components of the current is controlled by a PI controller. The controller output – that is the voltage vector to be reached –, which is represented relative to the AB-reference frame, is transformed into the αβ-reference frame by a vector rotator. Then, the inverter impresses the voltage vector on the motor. Another vector rotator transforms the measured currents into the AB- reference frame. The resulting components are sent to the current controllers. The angular position of the rotor flux linkage kß , which is required by the vector rotators, is calculated from measured currents and the rotor position mß by the model described in 4.6, see figure 6.

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The field-oriented control is primarily only a current control. Usually, it is used together with a higher-level speed control or also a position control. The speed controller calculates a setpoint of the torque-forming current as a function of the speed. This setpoint is evaluated by the current controller.

4.7.1 Controller basics The behaviour of a controlled drive depends substantially on the structure and the setting of the control loops. Usually, a cascade control is used, see figure 7. In this application, that means that the outermost control loop is the position control. It controls the angular position to a given setpoint. The output of the position controller is the setpoint for the angular speed. The inner speed controller controls the angular speed to that setpoint. Its output is the setpoint for the torque-forming current. The inner current control loop controls the torque-forming current to its reference value coming from the speed controller (and the flux- forming current to a constant value).

1J

i1*

i1

speed-controller

ωmfilt*

TLast

Tel ωm ωmωm* setpoint-filter

filter

inverter+ motor

current-controller

u1*position-controller

ϕ*

Figure 7: Cascade control

Cascade control provides high dynamics because it evaluates additional measured values in the inner control loops2 . It compensates interferences in the inner control loops, before they affect the higher-level control loops. In the following, some usual rules for designing control loops are presented. They serve as guide values only. The parameters of the control are, based on these values, optimized by tests. A more accurate derivation of the rules would go beyond the scope of this laboratory course. Interested students can read the derivation in [3] and [9]. The time constants which are relevant for the design of control are indicated subsequently. PI controller The transfer function of a PI controller can be represented e.g. as follows:

( ) 1 1R RR R R

R

s T s TG s K Vs s T+ ⋅ + ⋅= ⋅ = ⋅

⋅ (35)

2If the position control would not be realised as a cascade control, then the position controller evaluated the measured and the reference value of the position only and calculated the voltage vector. If it is realised as cascade control, it also evaluates the speed and the stator current.

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Amplitude optimum The controller setting according to the amplitude optimum is frequently used for systems with a PT2-behaviour, e.g. the control of the stator current. The transfer function of a PT2-system is

( ) ( )( )1 21 1S

SKG s

sT sT=

+ + (36)

Here, 1T means the larger time constant and can be compensated. 2T means the smaller time constant and cannot be compensated. For such a system, a PI controller is normally used. If 2T is much smaller than 1T ( ( )2 14T T< ), then the controller is set according to the amplitude optimum. Otherwise, it is set according to the symmetrical optimum, see below. The setting rule for the amplitude optimum is:

12

12R R

ST T K T K= =

⋅ ⋅ (37)

The response function to setpoint changes3 of a system, which is controlled by a controller designed according to the amplitude optimum, can roughly be approximated by the alternative transfer function

( ),2

11 2w ersG s T s=

+ (38)

It still contains the time constant 2T (which cannot be compensated) of the system. The equivalent transfer function is necessary for the design of the higher-level control loops. Symmetrical optimum The controller setting according to the symmetrical optimum is often applied to systems with a non-compensatable PT1-behaviour and an integral-action component, if the stationary offset has to be controlled to zero even at the presence of interferences. This is e.g. the case for speed control loops with an inner current controller. Assume a system with the transfer function

( )2

11

SS

KG s s s T= ⋅+ ⋅

(39)

Assume that its time constant 2T cannot be compensated.

3It describes the time behaviour to a setpoint change

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The setting rule for a PI controller according to the symmetrical optimum is: 24RT T= ⋅ (40)

22 22

1 1 12 8R

S R SS R R

K K T T K TK T T T= = =

⋅ ⋅ ⋅ ⋅⋅ (41)

This setting provides a good response to interferences. However, the response to setpoint changes is more weakly damped than if the controller is set according to the amplitude optimum. In order to avoid an overshoot of the system at a step change of the setpoint, while ensuring a dynamic response to interferences, the setpoint has to be smoothed by a low-pass filter of first order. The transfer function of such a low-pass filter is:

11TP

TPG T s=

+ (42)

Usually, the time constant of the filter is set as a function of the time constant of the system: 24TPT T= ⋅ (43) In drive control the speed setpoint is usually filtered. The time constant 2T to be considered is equal to the equivalent transfer function of the inner current control loop. The guide transfer function of a system, which is controlled by a controller designed according to the symmetrical optimum, can very roughly be approximated by the equivalent transfer function

( ),2

11 4w ersG s T s=

+ (44)

4.7.2 Behaviour of the systems and design of the controllers The inner control loop is the stator current controller. The stator current controller compensates the time constant of the stator circuit. In consequence, the higher-level speed controller does not see anymore the time constant of the stator, but only the time constant of the current- controlled motor which is much smaller. A possibly higher-level position controller sees the equivalent time constant of the speed-controlled loop. Inverter and sampling interval One sampling period passes from the sampling of the measured values up to the beginning of the output of the calculated voltage vector to the inverter. The output vector from the controller is always averaged over one sampling period. In the following sampling period, the inverter approximates this vector by a series of up to three out of seven creatable voltage vectors. This causes an additional dead time between 0 and one sampling period.

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The dead time tT of the inverter and the control algorithm cannot directly be used for the controller design, but it can be approximated by a PT1-transfer function with the time constant

232

ATT = (45)

This time constant is not compensatable by the control4 and limits the gain of the current controller. Behaviour of the stator circuit as the stator current controller sees it The differential equations of the stator circuit are coupled, see (19), i.e. the equation for the A-component also contains values of the B-component and vice versa: 5

2 2

211 1 1 1 1 1 22 2

2 2

AA A K B A

M Rdi M RL R i L idt L L⎛ ⎞σ = υ − + ⋅ + ω ⋅σ ⋅ + ⋅ Ψ⎜ ⎟⎝ ⎠

(46)

1 2 21 1 1 1 1 1 22

22

BB B K A m A

di M R ML R i L idt LL⎛ ⎞σ = υ − + ⋅ − ω ⋅σ ⋅ − ⋅ω ⋅ Ψ⎜ ⎟⎝ ⎠

(47)

Because of that, the voltage vectors 1Ac

∗υ and 1Bc∗υ , which are put out by the current

controller, are not directly impressed into the motor, but an decoupling is done first by substracting from each component those terms which derive from the other component, see also (19):

21 1 1 1 22

2A Ac K B A

M RL iL

∗ ∗ ∗ ∗υ = υ − σ ω − Ψ (48)

21 1 1 1 2

2B Bc K A m A

M RL iL

∗ ∗ ∗ ∗υ = υ + σ ω − ω Ψ (49)

By this method the two last terms in (46) and (47) are compensated and the transfer function of the stator circuit is reduced as follows:

( )

1

12

12121 2

221 2

2

1 1

1

s

Ac

A

K

T

i sLM R sR M RL R

L

∗ = ⋅++

+

συ (50)

( )1

11

11

Bcs

B

i sK T s∗ = ⋅

+ ⋅υ (51)

4otherwise the controller would have to be acausal, thus ”clearvoyant”. Maybe this is possible in magic, however not in control. 5When derivating, please consider that 022 jA

k +Ψ=Ψ , because 02 =Ψ B .

19

The equations (Fehler! Verweisquelle konnte nicht gefunden werden.) and (51) don’t yet contain the dead time tT due to the sampling and the actuator. The current control consists of two individual current controllers – one for the component

1Ai and one for the component 1Bi . As the transfer function of the stator circuit is equal in both components, see (Fehler! Verweisquelle konnte nicht gefunden werden.) and (51), both current controllers are designed identically.

de-coup-ling

Tt

i1Bi1B-contr.

i1B*

system

Figure 8: System ”stator circuit”

Mathematically, the decoupling can also be assigned to the system. Even a digital control offers different possibilities do decide, which parts are treated as the output of the controller and which gain is assigned to the actuator:

• The numerical value of the controller output is equal to the duty factor λ , i.e. the controller output is equal to the length of the voltage vector related to the biggest one possible. Thus, a gain is assigned to the inverter (actuator), by which λ is multiplied in the inverter in order to convert it to the voltage to be impressed.

• The numerical value of the controller output is equal to the voltage to be impressed to

the motor. The gain of the actuator is equal to 1. The gain of the system is thus equal to that of the motor.

In this practical course experiment, the second version is used. In this way, the reinforcement of the inverter is void for the design of the controller. The block diagram can thus be redrawn as follows:

1

R1+M2R2L2

2

v1B,Ci1B

T1= M2R2L2

2R1+

σL1

i1B-contr.

i1B*

KS=

motor with decoupling

Tt

approximation

T2=2

3TA

Figure 9: System ”stator circuit”, redrawn

20

This transfer function of the stator circuit inclusive dead time due to the sampling, see page 17, is seen by the current controller. The current controller is designed as a PI controller. It is set according to the amplitude optimum in order to compensate the time constant of the stator circuit. Behaviour of the current-controlled motor as the speed controller sees it The speed controller controls the system ”current-controlled motor”. It thus sees a PT1-transfer function with a following integrator.

3M NP 1J

i1B* i1Bspeed-controller 2L2

ψ2A

ωmfilt*

TLast

Tel ωm ωm

current-controlled motor

T2T2

ωm*

setpoint-filter

filter

Figure 10: Block diagram speed control

The time constant 2T of the PT1-transfer function is equal to the equivalent time constant of the current-controlled system. It must not be compensated by the controller, because it contains the non-compensatible time constant of the system (dead time of the inverter). The gain

1S

BK

iω

= is, according to figure 10:

2

2

32

P AS

M NK L J⋅ ⋅ ⋅Ψ=

⋅ ⋅ (52)

The speed signal contains high-frequency interferences due to the limited accuracy of the incremental encoder and the sampling. If these are not filtered, the speed controller tries to compensate them and becomes very nervous. Therefore, the speed signal has to be filtered by a low-pass filter. The low-pass filter has a PT1-behaviour with a time constant fT which is not compensatible6 . fT has to be considered for the design of the speed controller. It is added to the equivalent time constant of the current-controlled system. In order to prevent a permanent offset due to the load torque, a PI controller has to be used. It is designed according to the symmetrical optimum. Behaviour of the speed-controlled motor as the position controller sees it From the point of view of the position controller, the speed-controlled system looks like a PT1-system (with the equivalent time constant from the speed-controller) and an additional integrator (speed → position).

6A compensation of the time constant of the filter would cancel the effect of the filtering.

21

PT1ϕposition-

controllerωω∗ϕ∗

Figure 11: Block diagram position control

A P-controller is sufficient for the position control because there are no interferences which would lead to a permanent offset. Its gain RK is set according to the amplitude optimum. Large-signal operation The controller outputs stay within the allowed or physically possible limits (e.g. the maximum voltage which can be impressed by the inverter or the maximum allowable stator current), if the interferences and the changing of the setpoints are small. This case is called small-signal operation. The design rules described above are valid in that small-signal operation. E.g. on a large step change of the setpoint the controller output reaches its limit. This case is called large-signal operation. In drive control, the position controller and the integral-action components of the speed- and the current controllers are affected by the limitation of the outputs. If an offset is present at the input of a PI controller during a longer period, because the actuator cannot follow the controller output, the integral-action component of the controller increases to very large values. If the offset disappears after some time, the controller cannot adjust its output immediately, because the integral-action component has to be reduced. The system will overshoot with high amplitude and for a rather long time. Therefore, a PI controller must contain a so-called anti-windup. This feature ensures a limitation of the integral-action component, if the controller output is at its limit. If a P controller RK∗ω = − ⋅ Δϕ with a constant gain RK is used for position control, then the following problem exists: On the one hand, the gain has to be rather small in order to avoid oscillations of the system when the reference value changes. On the other hand, large changes of the position are followed very slowly if the gain is small. In order to solve this problem, the controller is modified according to the following considerations: The fastest method to change the position is to accelerate the drive by maximum torque up to the maximum speed and to decelerate it from a certain position, so that it comes to a stop exactly at the reference position. At large offsets the controller has to control the angular speed ( ) max2sig∗ω = − Δϕ ⋅ α ⋅ Δϕ (53)

22

max2

RK

∗ αω = − ⋅ΔϕΔϕ

(54)

in principle. There, maxα is the maximum possible angular acceleration7 . Regarding (54), it is stated that the controller gain RK tends to infinitive for small deviations Δϕ. This leads to instability due to the the finite response time of the current and the angular speed. By modifying (54) to

max2

RK

h∗ αω = − ⋅Δϕ

+ Δϕ (55)

the parameter h limits the maximum gain for small deviations.

4.7.3 Sampling of the measured values Inverters with d.c. voltage link cannot impress continuous voltage vectors. They can switch between a few discrete voltage vectors only. The other, not directly impressable, voltage vectors are approximated from the discrete ones by modulation. Assuming e.g. a stationary operating point of the induction motor (speed and torque held constant), the shape of the currents is sinusoidal (fundamental wave) in the ideal case. If the motor is fed by an inverter, the current oscillates at operating frequency around this fundamental wave. If the instantaneous value of the current is sampled thoughtlessly at any fixed time within the sampling interval,8 the sampled value will rarely be on the fundamental wave, but mostly be above or below, depending on the actual pulse pattern. See also figure 12. You see the shape of iα in the upper area and the shape of uα in the lower area, each during 2 sampling periods. uα is displayed in order to show when a zero voltage vector is impressed. If the zero voltage vector is present, then uα is zero. If the current is not sampled in the centre point of the zero voltage vector – in the figure at -90μs and +160μs –, but at another time, e.g. at -210μs and +40μs, then the fundamental wave is nearly never sampled.

7At the set-up the current controller limits the stator current to its rated value and thus the torque to its rated value. 8Here, the sampling rate of the control is equal to the operating frequency of the inverter.

23

Figure 12: Modulation and shape of the current

Sampling beyond the centre point of the zero voltage vector causes another problem. Directly after a switching between two voltage vectors, the current signal is disturbed by a transient response. Such a transient response is shown in figure 13. If the current is sampled during this time, then any point in the transient response signal may be sampled.

Figure 13: Transient response due to switching

In consequence signal components beyond the nyquist frequency ( 2fA ) are added to the

fundamental wave of the current signal. This causes the aliasing-effect. Those signal

24

components are mirrored into the area 0... 2fA due to the sampling. A dynamic control tries

to compensate these components. But as they don’t exist in reality, the control becomes agitated. The following methods are possible against the effects described above:

• Analog anti-aliasing filter, placed before the sampling in order to filter the components above the nyquist frequency. This filter must be of high order and causes a time delay (phase shift). In order to avoid instability, the controller has to be designed slow, resulting in a loss of dynamics. For drive control, usually no anti-aliasing filter is used.

• Sampling at instants when the analog signal is exactly on the fundamental wave.

Thus the components above the nyquist frequency are not sampled and it can be done without an anti-aliasing filter (which causes a phase-shift as we know). The dynamics of the controller then depend only on the dead times due to the sampling, the inverter and the measured-data acquisition.

At the set-up, the current is always sampled in the centre point of the zero-voltage vector. In consequence, only the fundamental wave of the current is sampled and an anti-aliasing filter is not necessary.

4.8 Operation

4.8.1 Turning on Turn on the set-up by the main switch and turn on the computer. Actually, the computer is equipped with the operating systems DOS and Linux. Simply press ”Enter” when the prompt of the boot manager (display lilo boot: ) appears. Answer the question whether the networking software should be loaded, with ”n”. Switch into the folder c: \for. Start the control software by typing the command tv_for. A user interface appears, enabling you to change parameters and to input reference values for the control. Turn on the inverter and reset it by pressing the buttons ”Umrichter Reset” and ”Umrichter Freigabe”.

4.8.2 User interface The user interface consists of two pages.

25

One page serves the settings of the controller and the reference values. On the second page you can choose, which values to assign to the outputs DA-1 to DA-4. You can change between the two pages by pressing <F6>. Within a page you can jump from one to the next element by pressing the tabulator key. Shortcuts are assigned to some of the elements. They are activated by pressing <Alt>+<marked character>. Inputs of numbers must be terminated by the enter button. The decimal point character must be a point. Alternative options (radio buttons) are selected by the arrow keys. The selection and deselection is done by the space key. In order to take over the changes, the button <<Uebernehmen>> or the enter key has to be pressed. You can start or stop the drive with the buttons <<Start>> and <<Stop>> or by the corresponding shortcuts <Alt><S> and <Alt><P>. Starting the drive resets all setpoints. Page: Controller settings and setpoints

26

Figure 14: User interface: Controller settings and reference values At the top of the right half of the screen, 3 radio buttons are placed to select the controller. Changing the selection of the controller stops the drive. If Stromregelung is selected, the current iB is controlled to the setpoint I_B_SOLL. iA is controlled to its rated value. The speed- and the position-controllers are not running. If Drehzahlregelung is selected, the speed is controlled to the setpoint N_SOLL. The speed controller is running together with the inner current controller, forming a cascade control. The setpoint I_B_SOLL is not used. If Lageregelung is selected, the position is controlled to the setpoint PHI_SOLL. The position controller is running together with the inner speed controller and the current controller, forming a cascade control. The setpoints I_B_SOLL and N_SOLL are not used. The meanings of the variables shown on the user interface are: T_R_I RT of the current controller K_R_I RK of the current controller I_B_SOLL setpoint, if the current control is selected TP_Filter WM_IST activate low-pass filter for ω TP_Filter WM_SOLL activate low-pass filter for ∗ω T_F_WM_IST time constant of the low-pass filter for ω T_F_WM_SOLL time constant of the low-pass filter for ∗ω T_R_WM RT of the speed controller K_R_WM RK of the speed controller N_SOLL setpoint of the speed for speed control R2 resistance of the rotor Lh magnetizing inductance a_max maximum angular acceleration Lageregler h Parameter for the position controller, see (55)PHI_SOLL setpoint of the position for position control Page: DA-outputs The indicated values can be assigned to the 4 DA-outputs.

27

Figure 15: User interface: DA-outputs

4.8.3 Use of the buttons and the switches Some buttons and switches are placed on the operator panel. They are assigned to the following functions:

• Button ”Umrichter Reset”: Resets error flags, e.g. overcurrent. • Button ”Umrichter Freigabe”: Activates the inverter. • Button ”Lageerf. Reset”: Resets the position acquisition. • Switch ”Impuls Freigabe”: Disables the pulses for the igbts independently even if the

control software is running. However, it does not influence the pulses sent to the operator panel for measuring purposes.

Moreover, the state of the set-up is displayed by the following LEDs:

• LED ”Überstrom”: Indicates an overcurrent, detected by the inverter. • LED ”Übertemperatur”: Indicates an overheating of the inverter.

28

• LED ”Freigabe”: Indicates, that the inverter receives an enabling signal from the computer.

• LEDs ”Timerkarte”: If both LEDs are alight, the watchdog function is activated.

After turning on or after an error of the inverter (overcurrent or overheating) the buttons ”Umrichter Reset” and ”Umrichter Freigabe” have to be pressed successively. Take care that the control is not active when you press those buttons.

4.8.4 Program flow for speed- and position-control Due to the missing absolute angular position of the motor the controller is not able to adjust a reference speed or position instantly after startup. At startup, the controller performs some revolutions in order to find the zero track of the incremental encoder [1]. Therefore the motor is fed by a volts per hertz characteristic [10]. After that, the flux forming current 1i A is slowly (remember the large time constant of the rotor, (22)) increased from zero to the rated value. During this time, the rotor flux linkage of the model is already calculated. Finally, the speed control (or the position control) starts. The control parameters can be modified during operation.

4.8.5 Error messages from the control software In order to avoid damage of the inverter, the control software stops in the following situations:

• ”Zwischenkreisspannung zu klein” (low d.c. link voltage). This error occurs, if the inverter has not been turned on when the control is started.

• ”Überstrom” (overcurrent). This error can occur if the control parameters are set very

badly. An extremly high overcurrent is also detected by the inverter (LED ”Überstrom”).

• ”Zu großer Spannungsvektor” (voltage vector too large). Due to the tiny leakage

inductance σ⋅L the current can vary very fast if a unsuitable voltage vector is impressed. Thus it can increase far beyond the maximum current allowed by the inverter. In order to avoid this by software, the calculated voltages vector is examined for plausibility before it is sent to the inverter.

29

5. Problem section Process the problems of the part ”preparing tasks” at home and bring the results to the laboratory course afternoon. The protocol must contain the calculation methods and the results of the numbered problems.

5.1 Preparing tasks

1. Compute the stator inductance 1L with help of (9). 2. Quote the number of pole pairs PN and calculate the flux-forming current 1Ai , the

torque-forming current 1Bi , the rated flux NΨ and the rated torque NT with help of (23), (26), (29), (Fehler! Verweisquelle konnte nicht gefunden werden.) and (52). Assume an average magnetizing inductance of 250hL mH= for the calculation.

3. Calculate the time constant 1T and the amplification SK of the stator circuit from the

given parameters of the induction motor. 4. Design the PI-controller of the inner current control according to the amplitude

optimum. 5. Design the PI-controller of the speed control according to the symmetrical optimum.

Assume a total inertia 3 25 10J Nm s−= ⋅ ⋅ for the drive. Also assume the low-pass filter for smoothing WM_IST set to T=1ms.

6. Design the low-pass filter for smoothing WM_SOLL. 7. Deduce the relation between h and the controller gain for 0 from (55). 8. Calculate the controller gain RK of the position controller according to the

amplitude optimum. Then calculate h. 9. Try to approximate roughly, how much the stator current can vary during one sample

period. Why is it not sufficient sometimes to use the sampled current only for the detection of an overcurrent?

10. Reflect on a possibility how to create a criterion ”voltage vector too large”. It is

sufficient to make the criterion react, if the current is twice as much as the rated current. Don’t forget the counter-e.m.f. due to the speed.

30

5.2 Problems for the laboratory course afternoon First, the current controller has to be optimised.

• Turn on the inverter and run the control software. • Select the current control. • Set the control parameters to the calculated values. • Make the DA-converters output the currents 1 ,B solli and 1Bi . 11. Keep the motor at standstill by using the brake. Perform some step changes of the

current (max. 400mA) and try to optimise the current controller by modifying the gain.

12. How can you recognise that the controller gain is set too high? 13. Try to identify the time constant of the controlled current loop by evaluating some

step responses. Now, the speed controller has to be adjusted.

• Select the DA-outputs in a manner that you can watch the values of N_SOLL,

N_SOLL_FILT, N_IST and I_B with the oscilloscope. • Set the trigger of the oscilloscope to the channel of N_SOLL. • Select the speed control. • Set the time constant of the low-pass filter for WM_IST to 1ms. • Deactivate the filtering of the setpoint of the speed. • Set the calculation of the flux to ”Strom- und Spannungsmodell” (current and voltage

model). 14. Redesign the speed controller with the measured time constant of the current control

loop and adjust the parameters on the user interface. 15. Perform some step changes of the speed and modify the gain in order to make the

control stable. 16. Perform large step changes of the speed and identify the maximum angular

acceleration of the drive and the gain 1s

BK i

ω= of the current-controlled system.

Don’t forget to transform the speed into radian. 17. Design the speed controller and the filter for the setpoint of the speed according to

the symmetrical optimum and activate the filter.

31

18. Perform some step changes of the speed and try to optimise the controller. At high speed, the position of the rotor flux linkage is calculated both with the current model and the voltage model. Now the difference in behaviour should be examined, if the position of the rotor flux linkage is calculated only by the current model. Moreover, the rotor resistance has to be adjusted.

• Set the DA-outputs and the oscilloscope in a way to display the space vector of the

e.m.f. relative to the AB-reference frame (E2A_MOD, E2B_MOD). • Set the calculation of the rotor flux linkage to ”nur Strommodell” (current model

only). 19. How should the space vector of the e.m.f. ideally look like? 20. Why can a deviation of the assumed rotor resistance from the real value be detected

only at operation with load? • Run the motor at approximately 1500min− and add some load by carefully using the

brake. 21. Modify the values of the rotor resistance and the magnetizing inductance, until the

rotor flux linkage is calculated correctly. How do the different settings of 2R and hL affect the calculation of the rotor flux linkage?

22. Run the motor at a higher speed, e.g. 11500min− , and verify the setting of the rotor

resistance by comparing the space vector of the e.m.f. with and without using the voltage model.

Now the position control has to be adjusted.

• Select the position control. • Display the values PHI_SOLL, PHI_IST, N_SOLL and N_IST with the oscilloscope. • Set the maximum angular acceleration to the measured value (radian! ). 23. Redesign the gain of the position controller according to the amplitude optimum for

small deviations and set the value h. 24. Optimise the position controller. E.g. maxα might be reduced, so that the drive can

follow the reference speed during deceleration.

32

References [1] Bähr, A., Manual S4 - Encoder Signal Processing, Institut für Stromrichtertechnik und

Antriebsregelung, 2003 [2] Budig, P.-K., Drehzahlvariable Drehstromantriebe mit Asynchronmotoren, vde-

Verlag, 1988 [3] Föllinger, O., Regelungstechnik, Hüthig Verlag [4] Hasse, K., Regelung in der Antriebstechnik, Skript, Institut für Stromrichtertechnik und

Antriebsregelung, TU Darmstadt [5] Mutschler, P., Control of Drives, Skript, Institut für Stromrichtertechnik und

Antriebsregelung, TU Darmstadt [6] Mutschler, P., Antriebe in der Automatisierungstechnik, Skript, Institut für

Stromrichtertechnik und Antriebsregelung, TU Darmstadt [7] Pahlke, D., Implementierung einer direkten Stromregelung, Studienarbeit, Institut für

Stromrichtertechnik und Antriebsregelung, TU Darmstadt, 2001 [8] Schröder, D., Elektrische Antriebe 1, Springer, 1995 [9] Schröder, D., Elektrische Antriebe 2, Springer, 1995 [10] Weigel, J., Manual S2 - Three-Phase Drive with d.c. Voltage Link Inverter, Institut für

Stromrichtertechnik und Antriebsregelung, TU Darmstadt, 2003