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IEA Industrial Electrical Engineering andAutomationLund Institute of Technology

September 1991

Oscillations in Inverter Fed

Induction Motor Drives

Bo Peterson

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Lund 1991

Oscillations in Inverter Fed

Induction Motor Drives

Bo Peterson

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Department of

Industrial Electrical Engineering andAutomation (IEA)Lund Institute of TechnologyBox 118S-221 00 LUNDSweden

Copyright 1991 by Bo Peterson

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Issuing organization

Department of Industrial

Document name

Tekn. Lic. Dissertation

Electrical Engineering andAutomation (IEA),

Date of issue

Sep 1991

Lund Institute of Technology Document numberCODEN: LUTEDX/(TEIE-1001)/1-

Author(s)

Bo PetersonSupervisor

Gustaf Olsson, Jaroslav ValisSponsoring organization

Title and subtitle

Oscillations in Inverter Fed Induction Motor Drives

Abstract

Severe oscillations in the range of 1 to 100 Hz have been encountered ininverter fed induction motor drive systems, especially where there are noexternal damping loads, such as fan drives. These oscillations may damagethe drive system or generate noise. It is found that the induction machinehas two resonance frequencies. The damping of the first resonance isdecreased with increased stator resistance, while the damping of the secondresonance is increased with increased stator resistance. Simple mechanicalmodels are presented which give physical insight into the reason for theoscillations, as well as suggestions of how to suppress them.

Key words

Instability, Resonance, Induction machine, Inverter, Mechanical model

Classification system or index terms

Supplementary bibliographical information

ISSN and key title ISBN

Language

EnglishNumber of pages

67Recipient's notes

Security classification

The document may be ordered from: IEA, Box 118, S-221 00 LUND, Sweden

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Abstract

Severe oscillations in the range of 1 to 100 Hz have beenencountered in inverter fed induction motor drive systems,especially where there are no external damping loads, such asfan drives. These oscillations may damage the drive system orgenerate noise. It is found that the induction machine has tworesonance frequencies. The damping of the first resonance isdecreased with increased stator resistance, while the damping ofthe second resonance is increased with increased statorresistance. Simple mechanical models are presented which givephysical insight into the reason for the oscillations, as well assuggestions of how to suppress them.

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Contents

1 Introduction......................................................................................................9

Oscillations in Induction Machines.............................................9Observations of Oscillations ............................................................9Induction Machines Connected to Inverters.......................10

Analysis of the Oscillations...........................................................10Damping of the Oscillations.........................................................11Outline of the Thesis........................................................................12

2 Dynamic Behaviour...................................................................................133 Mechanical Model.......................................................................................15

Mechanical Model of the Stator Resistance ......................... 17Mechanical Model with Fewer Parameters..........................18

4 Experimental Set-up.................................................................................235 Measurements ..............................................................................................266 Series Resonance.........................................................................................29

Comparison to Earlier Expressions ..........................................33Series Resonance with DC Link Capacitor...........................35

7 Parallel Resonance.....................................................................................418 Linearization.................................................................................................489 Feedback.......................................................................................................... 5310 Conclusions ....................................................................................................5511 References....................................................................................................... 56

Appendix .........................................................................................................57A Per Unit Notation ..............................................................................57B Motor Data............................................................................................58C Vector Equations................................................................................59D Coordinate Transformations........................................................61E Mechanical Analogy.........................................................................62F Roots, Natural Frequency and Damping.............................. 63G Simulation File....................................................................................65H List of Symbols....................................................................................67

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Acknowledgements

I was introduced to the problem of oscillating induction machinesin 1990 by Professor Jaroslav Valis and Tommy Magnusson,both at the Department of Industrial Electrical Engineering and

Automation, together with Claus Holm Hansen, at that timeworking for Speed Control A/S, Denmark. I am very thankful fortheir support during my work and I also thank ProfessorFrantisek Bernt at the Czechoslovak Academy of Sciences forsupplying me with background material. Last but not least Ithank my supervisor Professor Gustaf Olsson who has helpedand encouraged me through my work.

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Introduction 9

1 Introduction

Oscillatory behaviour of electrical machines in certainapplications - also known as instability or hunting - is a well-known topic. The conditions under which a DC motor exhibits anoscillatory response to a sudden change of its supply voltage orload is a common issue in any textbook on electrical machines(Fitzgerald et al 1985). The oscillatory behaviour of synchronousmachines is also well known and understood (Wagner 1930).Stepper motors are ill-famed for their poorly damped oscillatingresponse.

Oscillations in Induction Machines

The oscillatory behaviour of line-fed induction motors in constantspeed applications has also been noticed long time ago, howeverless well understood. The reason is probably that the inductionmotor is an intricate electro-magnetic-mechanical system,difficult to analyze and describe in purely electrical terms. Theclassical T-shaped per-phase equivalent diagram describes theinduction motor at steady state, giving no hint on the dynamicbehaviour.

Only recently, using the space vector method, a profoundanalysis of the oscillations is found (Kovcs 1984). Kovcs hasanalyzed overshoot after acceleration, response to a sudden loadchange, forced oscillations excited by load fluctuations, and hasderived expressions for the natural frequency and damping ofoscillations. His expressions are not valid at all times however,due to some limiting assumptions, for example that the statorresistance is equal to zero.

Kovcs' formulas exposed the fact that the typical resonancefrequencies were in the range of 1 to 100 Hz, i.e. within theusual range of motor speeds in inverter-fed variable speed drives.This is the reason why problems caused by instability has becomemore accentuated in variable speed drives.

Observations of Oscillations

Excessive oscillations at certain motor speeds are actuallyencountered in many applications, especially when the load itself

can not provide sufficient mechanical damping, as fan blowers,or when the load torque contains a periodic component, as a

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10 Introduction

piston compressor, which can excite the oscillations. Theoscillations may also be excited by fluctuations of the air gaptorque due to imperfections in the motor, as eccentricity andslotting.

Induction Machines Connected to Inverters

It has been observed that large motors (> 10 kW) oscillate morethan small ones (

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Introduction 11

Unfortunately, it is an almost hopeless task to derive algebraicexpressions for the eigenvalues of a matrix of this size. Anumerical solution does not give any physical insight into theoscillatory behaviour.

Another way to simplify the problem is to use an electro-mechanical analogy to describe the induction machine.Interpreting the mathematical model of the induction motor asan equivalent mechanical system reveals that an induction itselfis a poorly damped oscillating system, whose two resonancefrequencies depend on the motor flux, the leakage and themagnetizing inductance and rotor moment of inertia. Thedamping is only due to the winding resistances.

When the motor is fed from an inverter the DC link capacitor

becomes part of the oscillating system and can also be included inthe mechanical model.

Damping of the Oscillations

Methods for suppressing the oscillations using electronics havebeen presented (Mutoh et al 1990) but they are often morecomplicated than necessary. However, with a good under-standing of the physical reasons for the resonance, and someadditional feedback, the resonance problems can be solved in a

better way.

This thesis suggests an efficient yet simple way of dampingtorque oscillations disrupting the function of induction motorsthat are fed and controlled by inverters.

The new idea is to introduce an electronic damping by a fastacting control of the angular frequency of the stator flux vector.

A control law based on feedback of the current in the DC link isderived and its properties are verified by simulations.

Outline of the Thesis

In section 2 the fundamental dynamic equations of the inductionmachine are described. Section 3 presents a mechanical analogyto the dynamic equations. Sections 4 and 5 describe how todetermine the resonance frequencies experimentally. Section 6and 7 examine the physical reasons for the oscillations. In section8 the resonance is discussed in terms of eigenvalues of alinearized system. Finally it is shown how feedback can damp the

oscillations.

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12 Dynamic Behaviour

2 Dynamic Behaviour

The dynamic behaviour of an induction machine can bedescribed by three vector differential equations in a referenceframe attached to the stator (Kovcs 1984):

dYs

dt=

us

isRs (2.1)

dYr

dt=jw

Yr

i rRr (2.2)

Jdwdt

=Im(Ys*

is) Tm (2.3)

with the two algebraic conditions

Ys =

is(Lsl +Lm) +

i rLm =

isLs +

i rLm (2.4)

Yr =

isLm +

i r(Lrl +Lm) =

isLm +

i rLr (2.5)

The equations are in per unit (p.u.) notation. (The terminologyand the derivation of the p.u.-values are found in App. A). Thestator current vector and stator voltage vector are

is =

23

(ia +aib +

a2ic) (2.6)

us =

23

(ua +aub +

a2uc) (2.7)

where

a = ej 2p/3 (2.8)

ua, ub and uc are instantaneous values of phase-to-neutralvoltages, ia, ib and ic are instantaneous values of line currents.

The system of equations is non-linear and it is difficult to see thereasons for the oscillations in the equations. If the equationswere linear, the eigenvalues could be calculated, but they

wouldn't give any clue to why the oscillations occur.

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14 Mechanical Model

3 Mechanical Model

A mechanical model of equations (2.4-2.5) is shown in Fig. 3.1,where inductances have become springs and currents are forcespulling the springs (Trk et al 1985). The inductance is theinverse of the stiffness of the spring. (App. E gives a completedescription of corresponding parameters in the electrical andmechanical models.)

Ys Yr

Lm

Lsl Lrl

is ir

Figure 3.1 Mechanical model of equations (2.4-2.5)

The linked fluxes are vectors with a length and a directioncorresponding to the original fluxes. It is seen that the torque on

the springs developed by the currentis is

Im (Ys*

is) = T (3.1)

which is the driving torque in equation (2.3). The point in Fig. 3.1where the force

i r is attached to the spring Lrl follows the

restrictions of equation (2.2). The resistanceRr is represented asa viscous damper which can be a drag-pad on an oily surface.The velocity difference between the drag-pad and the surface is

proportional to the force applied to the drag-pad, and the velocityand the force have the same direction. The time derivative of theflux in the mechanical model is the velocity of the tip of the fluxvector.

If the oily surface is a rotating disc with the angular speed w(Fig. 3.2) then the relative velocity difference between the drag-pad and the disc is

vdiff=

dYr

dt jwYr (3.2)

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Mechanical Model 15

The force on the drag-pad is i r and then the velocity difference

between the disc and the drag-pad can be expressed as

vdiff=

i rRr (3.3)

Yr

is

w

Rr

Ys

drag-pa

rotor dis

Figure 3.2 Mechanical model of equations (2.2-2.5)

Combining equations (3.2) and (3.3) gives equation (3.4),

dYr

dt=jw

Yr

i rRr (3.4)

The torque transferred from the drag-pad to the disc is given byequation (3.1),

Im (Ys*

is) = T. (3.5)

If the rotating disc represents the rotor and has the actual rotormoment of inertia, we have an exact mechanical analogy ofequations (2.2-2.5). To get a complete analogy of equations (2.1-2.5), the stator resistance must be modelled.

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16 Mechanical Model

Mechanical Model of the Stator Resistance

Equation (2.1) can also be represented by a mechanicalequivalent. The term

isRs is represented as a drag-pad on a

viscous surface. The drag-pad is transferring the forceis to the

springs. The viscous surface is moving with the velocity usindependently of the stator flux. All points of this surface aremoving in the same direction at the same speed. Contrary to therotor disc, this surface can not rotate. The velocity of the tip ofthe stator flux will be the difference between the stator voltageand the resistive voltage-drop in the stator. Figure 3.3 shows themechanical model with the stator resistance included.

Yr

is

w

RrRs

us

Ys

rotor dis

stator pan

us us

shaft

stator pan

rotor dis

RrRs

Figure 3.3 Mechanical model with the stator resistance included

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Mechanical Model 17

Mechanical Model with Fewer Parameters

It is possible to get an equivalent mechanical model of Fig. 3.2,with fewer parameters than the original one. A few simpletransformations have to be done. The same transformations can

be done to the traditional T-shaped per-phase equivalent circuitof the stationary motor, to get a model with fewer parameters(Slemon et al 1980).

In a squirrel-cage induction motor it is impossible to separate thestator leakage inductance from the rotor leakage inductance. Oneof the parameters is redundant and models with one totalleakage inductance instead of two separate ones can be obtained.

Various models can be derived. In the model described below, thevalues of the inductances correspond to the values that are

obtained from no-load and locked-rotor tests.

The original model is shown in Fig. 3.4 (a), with the stator androtor omitted. The two springsLsl andLm in Fig. 3.4 (b) can bereplaced by one spring, LT, and the flux Ys is transformed intoYT. This is shown in Fig. 3.4 (c) where

YT=Lm

Lm+ LslYs (3.6)

LT= LmLm+ LslLsl (3.7)

Now let

LmLm+ Lsl

=k (3.8)

This transformation is analogous to using Thvenin's theorem inthe electrical circuit of Fig. 3.4 (f). The resistorsR1 andR2 can be

replaced byRTif the voltage Uis replaced by UT, where

RT=R2

R1+ R2R1 and UT=

R2R1+ R2

U

From Fig. 3.4 (d) it is seen that the rotor current vector is

i r =

YT+

Yr

LT+ Lrl=

kYs

kLsl+ Lrl+

Yr

kLsl+ Lrl(3.9)

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18 Mechanical Model

(a)

Ys Yr

Lsl Lrl

is ir

Rr

Lm

(b)

Ys

Lm

Lsl

ir

Ym

is

(c)

ir

YmYT

LT

(d)

Ym

YT

LTLrl

ir

Yr

(e)

is

LL

LM

RR

YR

iR

Ys

(f)

UR1

R2

UT

RT

Figure 3.4 Transformations of the mechanical model.

The stator currentis in Fig. 3.4 (b) may be expressed as the sum

of two components, one due to the stator fluxYs, the other due to

the rotor currenti r. If the current

i r is set to zero, the first

component is

is1 =

Ys

Lm+ Lsl=

kYs

Lm(3.10)

When the stator flux is set to zero, the second component is

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Mechanical Model 19

is2 =

LmLm+ Lsl

i r = k

i r (3.11)

Thus by superposition,

is =

is1 +

is2 =

kYsLsl

ki r (3.12)

By substitution from equation (3.9),

is =

Ys

Lm/k+

Ys

Lsl/k+ Lrl/k2

Yr/k

Lsl/k+ Lrl/k2=

=

YsLM+

YsLL

YRLL(3.13)

where

LM=Lmk

(3.14)

LL =Lslk

+Lrlk2

(3.15)

YR =

Yrk

(3.16)

Equation (3.13) describes the model in Fig. 3.4 (e). The rotorresistance must also be transformed. Equation (2.2),

dYr

dt=jw

Yr

i rRr

must be replaced by

dYRdt

=jwYR

iRRR (3.17)

where

iR =

is2 =k

i r (3.18)

Equation (3.16) implies that

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20 Mechanical Model

dYRdt

=1

kdYr

dt=

1k

jwYr

1k

i rRr =jw

YR

1k2

iRRr (3.19)

Now the last parameter is found to be

RR =Rrk2

(3.20)

and the model in Fig. 3.4 (e) is an exact equivalent of the modelin Fig. 3.4 (a). The complete improved model with the rotor discincluded is shown in Fig. 3.5. Due to the resemblance to theGreek letter G, this model is sometimes referred to as the G-model.

is

w

LL

LM

RR

YRYs

Rs

Figure 3.5. Mechanical G-model

The stator equation is equal to the that of the original model,which leads to the final set of equations (compare equations (2.1-2.5)):

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Mechanical Model 21

dYs

dt=

us

isRs (3.21)

dYRdt =jw

YR

iRRR (3.22)

Jdw

dt=Im(

Ys*

is) Tm (3.23)

Ys = (

is +

iR)LM (3.24)

YR =

Ys +

iRLL (3.25)

The relations between the models are:

G-model original model

LM =Lmk

(3.26)

LL =Lslk

+Lrlk2

(3.27)

Rs = Rs (3.28)

RR =Rrk2

(3.29)

Ys =

Ys (3.30)

YR =

Yrk

(3.31)

is =

is (3.32)

iR = k

i r (3.33)

k =Lm

Lm+ Lsl(3.34)

Note that the vectorsYr and

YR have the same arguments, only

the magnitudes are different. This is also true for

i r and

iR.

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22 Experimental Set-up


With a simple experimental set-up, the oscillatory behaviour ofthe induction machine can be investigated. With this test, tworesonance frequencies can be distinguished. The test is performedwith a non-rotating rotor. However, the rotor is not locked, butfree to oscillate.

The machine is magnetized in one direction to rated flux by aDC-current, flowing through two of the phases of the statorwinding. A voltage source with variable frequency is applied tothe third phase, exciting the machine in a directionperpendicular to the DC-flux (Fig. 4.1). The voltage source maybe a function generator connected to an audio power amplifier.

ua

Idc

ia

ib

ic

Figure 4.1 DC-excited induction machine

With vector representation of the induction machine, the DC-current and voltage become the components of the stator currentand voltage in the imaginary axis direction. The stator currentvector and stator voltage vector are

is =

23

(ia +aib +

a2ic) (4.1)

us =

23

(ua +aub +

a2uc) (4.2)

where

a = ej 2p/3 (4.3)

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Experimental Set-up 23

ua, ub and uc are instantaneous values of phase-to-neutralvoltages, ia, ib and ic are instantaneous values of line currents,all in p.u.-notation.

If the line currents are ib = Idc and ic =Idc then the imaginary

parts of the stator current and voltage vectors are

isy = 2

`3Idc (4.4)

usy =Rsisy (4.5)

The current needed for rated flux Y= 1 p.u. is

|isy| =

Y

LM (4.6)

The frequency of the voltage ua is varied and the the current iais measured. The rotor is free to move and will oscillate with thesame frequency as the voltage.

The voltage ua in vector representation becomes the voltage inthe real axis direction

usx =2

3ua (4.7)

The current Idc needed for rated flux can be calculated in twoways, either by combining equations (4.4) and (4.6) which gives

Idc =3

2

Y

LM(4.8)

or by using information on the rating plate of the machine. Therated current consists of two parts, one magnetizing current and

one torque generating current. The torque generating current isequal to the rated current multiplied by rated cosj and themagnetizing current is the rated current multiplied by ratedsinj,

imagn=In sinj= |isy| (4.9)

where In is thepeak-value of the rated current. Equation (4.4)

and (4.9) together withIn,rms =In/2 give

Idc =32 In,rms sinj (4.10)

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At low frequencies, the induction machine will behave as aninductive load i.e. the current ia lags the voltage ua. As thefrequency is increased, the phase angle is reduced, and themachine will behave as a capacitive load i.e. the currentleads thevoltage. This is due to the moment of inertia of the rotor, actingas a capacitance. When the frequency is further increased, themotor will behave as an inductive load again.

The frequencies where the current changes from lagging toleading, and from leading to lagging are the two resonancefrequencies. The current is low at the first frequency, as in thecase of parallel resonance. The current is high at the secondfrequency as in series resonance.

In the following sections the two kinds of resonance will be

referred to asparallel resonance andseries resonance.

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Measurements 25

5 Measurements

The measurements described in section 4 have been performedon a Y-connected 1.1 kW two-pole motor from Grundfos. The dataof the motor are found in App. B.

Table 5.1 shows the measured data, where fis the frequency ofthe feeding voltage ua and the line current ia shown in Fig. 4.1.The values of ua and ia are peak values in natural units.

According to equation (4.7), usx = 2/3 ua, and the p.u.-value isobtained if this is divided by the nominal voltage Un. Thus,

usx =23

uaUn (5.1)

Similarly, isx can be calculated,

isx =23

ia

In(5.2)

The nominal values are Un = 310 V and In = 3.82 A. q is thephase angle between ua and ia (q > 0 ifia leads ua). The lastcolumn shows the gain from u

sxto i

sx.

The value of the magnetizing current was (equation (4.10))

Idc =32In,rms sin j=`32 2.7 ```````1cos 2j= 1.8 APlots of the gain, 20 log(isx/usx), versus the frequency fin p.u.,and the phase angle qversus the frequency are found in Fig. 5.1.The two resonance frequencies can be clearly distinguished

where the phase curve crosses the zero line, the first at aboutf=0.2 p.u. (10 Hz) and the second at f= 0.7 p.u. (35 Hz). It is alsoseen that the gain curve has a minimum at the parallelresonance frequency, and a maximum at the series resonancefrequency.

In the following sections, models for the two kinds of resonancewill be developed, as well as expressions for calculation of theresonance frequencies.

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26 Measurements

f [p.u.

f [p.u.

q

[deg]

i

/

u

[dB]

sx

sx

Figure 5.1 Measured gain and phase angle between voltage and current

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Measurements 27

f[Hz] f[p.u.] ua [V] usx [p.u.] ia [A] isx [p.u.] q[deg] isx/usx5 0.10 7.0 0.015 0.378 0.0660 69 4.40

6 0.12 7.0 0.015 0.281 0.0490 71 3.27

7 0.14 7.0 0.015 0.201 0.0350 72 2.33

8 0.16 7.0 0.015 0.126 0.0220 72 1.47

9 0.18 7.0 0.015 0.083 0.0145 55 0.97

10 0.20 7.0 0.015 0.055 0.0096 20 0.64

11 0.22 7.0 0.015 0.070 0.0122 24 0.81

12 0.24 7.0 0.015 0.105 0.0184 41 1.23

15 0.30 7.0 0.015 0.218 0.0380 48 2.53

20 0.40 7.0 0.015 0.372 0.0650 38 4.33

25 0.50 7.0 0.015 0.487 0.0850 28 5.67

30 0.60 7.0 0.015 0.562 0.0980 16 6.53

35 0.70 7.0 0.015 0.596 0.1040 3 6.93

40 0.80 7.0 0.015 0.596 0.1040 8 6.93

45 0.90 7.0 0.015 0.590 0.1030 15 6.8750 1.00 7.0 0.015 0.567 0.0990 20 6.60

Table 5.1 Measured data from the Grundfos motor.

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28 Series Resonance

6 Series Resonance

A clear mechanical model can be obtained for the seriesresonance. The leakage inductance of the machine plays anessential role in this oscillation. The magnetizing inductanceLMdoes not effect this oscillation and can be considered infinite.

At the nominal flux the mechanical model is examined in thetangential direction, and the stator voltage vector is assumed tobe perpendicular to the flux. This leads to a model with a seriesconnection between a spring, a mass, and two viscous dampers,as in Fig. 6.1 (a).

JRs RRLL

usw

J

R LL

usw

(a) (b)

Figure 6.1 Mechanical representation of the series resonance

The spring represents the total leakage inductanceLL, the mass

represents the rotor moment of inertia, and the dampers thestator and rotor resistances. The two dampers can be replaced bya single damperRs +RR =R as in Fig. 6.1 (b). If the flux differsfrom the nominal flux, the mass J can be replaced by a hoistdrum with variable radius Y and moment of inertia J(Fig. 6.2(a)), or by a lever with the mass Jattached to its end (Fig. 6.2(b)).

R LLY

us wY

w

i s

w

Y

1

Jus

i s R LL

(a) (b)

Figure 6.2 Alternative representations of the series resonance allowingvarying flux

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Series Resonance 29

The dynamic equations for the system in Fig. 6.2 can be derivedeither directly from the mechanical system or from the vectorequations. The force acting on the mechanical system isrepresented by the stator current is. It is seen that

disdt

=us Ris wY

LL(6.1)

dw

dt=

Yi sJ

(6.2)

Another way of deriving these relations is to start with the vectorequations in a coordinate system rotating with the angularvelocity wk = 1. The differential equations (3.21-3.25) have to bechanged according to App. D. The real and imaginary parts ofthe equations become

dYsxdt

= usx isxRs + Ysy (6.3)

dYsydt

= usy isyRs Ysx (6.4)

dYRxdt

= wYRy iRxRR + YRy (6.5)

dYRydt

= wYRx iRyRR YRx (6.6)

dw

dt=

Ysxisy Ysyisx

J(6.7)

IfLM then is = iR according to (3.24). At steady state, theflux, voltage and current vectors might appear as in Fig. 6.3.

usisi

YsYR

LLR

Figure 6.3 Flux, current and voltage vectors at steady state

It is seen in the figure that

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30 Series Resonance

isy = iRy = 0 (6.8)

iRx = isx (6.9)

YRx = 0 (6.10)

Ysy = YRy (6.11)

isx = (Ysx YRx)/LL (6.12)

If (6.5) is subtracted from (6.3) and relations (6.8-6.12) areinserted, (6.13) is obtained,

LLdisxdt

= usx isx (Rs +RR) + wYsy (6.13)

With the following denotation, equation (6.13) is identical toequation (6.1),

is = isx (6.14)

R =Rs +RR (6.15)

Y= Ysy (6.16)

us = usx (6.17)

Equation (6.7) becomes identical to (6.2) with isy = 0,

dw

dt=

Yi sJ

(6.18)

The system described by (6.1-6.2) can also be described by anequation of the second order,

w+R

LL.w+

Y2

JL Lw=

usY

JL L(6.19)

If the characteristic equation

s2 +RLL

s +Y2

JL L= 0 (6.20)

is compared to a standard oscillatory characteristic equation(App. F),

s2 +2Wzs + W2 = 0 (6.21)

the natural frequency Wof (6.19) is found to be

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Series Resonance 31

W=Y

```JLL(6.22)

and the relative damping z

z=R2Y

`JLL (6.23)It is seen that the damping is decreased with decreasing statorresistance. This is typically a problem for large motors with smallrelative stator and rotor resistances. In large motors, there ismore space for the windings, resulting in a larger copper area,reducing the resistances.

If the parameters of the Grundfos motor described in App. B areinserted into equation (6.22), the natural frequency is

W=Y

```JLL=

1

`````````13.50.138= 0.733 p.u.

and the damping is from (6.23)

z=R2Y

JLL

=0.146

2

``

13.50.138

= 0.72

The frequency in Hz is equal to the frequency in p.u. multipliedby 50 (rated electrical frequency) which gives f=37Hz, veryclose to the measured value of 35 Hz in the previous section.

To compare the system described by equation (6.19) with themeasured data for other frequencies than the resonancefrequency, a Bode plot can be used. As the measured variablesare ia and ua, a transfer function from voltage to current issuitable for comparison. The Laplace transforms of equations

(6.18) and (6.19) are

s W=YI sJ

(6.24)

s2W+sRLL

W+Y2

JL LW=

UsY

JL L(6.25)

where Wis the Laplace transform ofw.

Combining these equations give the transfer functionHs from UstoIs, for the series resonance model:

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32 Series Resonance

Is =Hs(s) Us=s/LL

s2+ sRLL

+Y2

JL L

Us (6.26)

Amplitude and argument ofHs(jw) together with the measureddata is found in Fig. 6.4. It is seen that the measured curves(dashed) lie close to the curves of the transfer function Hs (solid)for frequencies greater than 0.4 p.u. (20 Hz).

Comparison to Earlier Expressions

Oscillations are also discussed in (Kovcs 1984). The followingcharacteristic equation is presented:

s2 +sps + 2Tph = 0 (6.27)

wheresp denotes the pull-out slip, Tp the pull-out torque and h isthe moment of inertia in p.u. Note the following differences in thenotation used by Kovcs and in this thesis:

Kovcs Here

p.u. moment of inertia h J

p.u. inductance X LThe pull-out torque is defined as

Tp =us

2k2

2Xr'(6.28)

and the pull-out slip

sp =RrXr'

(6.29)

where

Xr' =Xrl +XmXsl

Xm+ Xsl=Xrl +kXsl (6.30)

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Series Resonance 33

f [p.u.

f [p.u.

q

[deg]

i

/

u

[dB]

sx

sx

H Measured datas

Figure 6.4 Amplitude and argument of transfer functionHs

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34 Series Resonance

Substituting h by J and X by L and using equations (3.25),(6.28) and (6.30), the term 2Tp/h can be written

2TpJ

=2us

2k2

J2 Xr'=

us2

J

k2

Lrl+ kLsl=

us2

JL L(6.31)

Using (3.29), the termsp can be written

RrXr'

=Rr/ k2

Xr'/ k2=

RRLL

(6.32)

Inserting (6.31) and (6.32), the characteristic equation becomes

s2 +RRLL

s +us

2

JL L= 0 (6.33)

which only shows two differences to the characteristic equation(6.20).

In (6.33) the stator voltage us is found instead of the flux Y. Thisis of no practical importance as the flux is proportional to thestator voltage. Thus, at nominal voltage (us = 1), the flux alsoattains its nominal value (Y= 1), and if the voltage is increasedor decreased, the magnitude of the flux will follow that of thevoltage. Therefore, either the flux or the voltage can be used

with exactly the same result.

The other difference is of greatest importance. The sum of statorand rotor resistance in found in (6.20) while only the rotorresistance is found in (6.33). In (Kovcs 1984) it is assumed thatthe stator resistance is zero, and then (6.20) and (6.33) areidentical. However, as will be seen later, the influence of thestator resistance can not be neglected. A large stator resistancewill reduce the damping of other resonance phenomena.

Series Resonance with DC Link Capacitor

If the induction motor is connected to an inverter the DC linkcapacitor becomes part of the oscillating system. The inverter can,as well as the induction machine, be represented by a mechanicalmodel.

The inverter can produce voltage vectors in six directions, andthe zero voltage vector, according to Fig. 6.5. The length of thevectors are

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Series Resonance 35

|u| =23

Ud (6.34)

If the switching frequency is high enough, a voltage vector inany direction can be generated by taking a time average of

vectors in two directions and the zero voltage vector.

u1

u2u3

u4

u5 u6

u0

t 1u 1

t 2u 2

t0u 0

Tu

Figure 6.5 Output voltage vectors of an inverter

The zero voltage vector is used during time t0, then u1 is usedduring t1 and finally u2 during time t2. Ift0+ t1+ t2= Tthen theaverage amplitude of the voltage vector during time T is

|u| =``````````(t1+ t2)2 t1t2

T23

Ud (6.35)

and the average argument is

a= arccost1+0.5 t2

``````````(t1+ t2)2 t1t2; 0 a 60 (6.36)

By choosing another pair of voltage vectors, an arbitraryargument, and an arbitrary amplitude

|u| =KUd32

23

Ud =Ud

`3(6.37)

can be obtained, where K is the amplitude factor (almost thesame as duty cycle),

K=``````````(t1+ t2)2 t1t2

T23

1

`3(6.38)

Let the average current from the inverter be denoted byId. Theaverage power from the inverter during time Tis

Pinverter = UdId (6.39)

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36 Series Resonance

and the power to the motor is

Pmotor =32

(usxisx + usyisy) (6.40)

The power to the motor must equal the power from the inverter,and it is assumed that usx= us= KUd, isx= is and isy =0. Thisleads to

Id =32

Kis (6.41)

A mechanical model can now be obtained for the inverter and itsDC link capacitor. The capacitor is modelled by a mass, m = 2C/3.This mass is attached to a lever as in Fig. 6.6 (a).

J

R LL

us

w

Y

11 i s

Ud

K

23C

us

1 i s

Ud

K

23C

(a) (b)

Figure 6.6 (a) Mechanical model of DC link capacitor (b) DC link capacitorconnected to induction machine

A model of the inverter connected to the motor model of Fig. 6.2(b) is seen in Fig. 6.6 (b). The flux is proportional to the the stator

voltage us and the inverse of the stator frequency ws,

Y=usws

=KUdws

(6.42)

If the flux is supposed to be constant, Yo, the dynamic equationsof the system become

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Series Resonance 37

disdt

=us Ris wYo

LL=

KUd Ris wYoLL

(6.43)

dw

dt=

isYoJ

(6.44)

dUddt

= Kis

2

3C(6.45)

The equations for this system are non-linear, but can belinearized around an operating point (iso, wo, Udo), which resultsin a system with the state variables

x1 = dis= is iso (6.46)

x2 = dw= w wo (6.47)

x3 = dUd= Ud Udo (6.48)

and the input signal

u = dK=KKo (6.49)

At steady state the state, we have the following relations for thestator current, angular velocity and flux,

iso = 0 (6.50)

wo = ws (6.51)

Yo =KoUdo

ws=

KoUdowo

(6.52)

The system can now be described by

.x =Ax +Bu (6.53)

where

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38 Series Resonance

A =

R

LL

YoLL

KoLL

YoJ

0 0

Ko23C

0 0

(6.54)

and

B =

Udo

LL

00

(6.55)

if the non-linear terms are neglected.

It is seen in (6.44) and (6.45) that

dUddt

=3 JK2CYo

dw

dt(6.56)

and it can be supposed that

dUd =3 JK2CYo

dw (6.57)

The system can now be reduced to a second order system withx =(disdw)T

.x =Ax +Bu =

=

RLL

YoLL

3JKo2

2LLCYo

YoJ

0

x +

Udo

LL

0

u (6.58)

The characteristic polynomial of matrixA is

s2 +RLL

s +Yo

2

JL L+

Ko2

2

3CL L

(6.59)

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Series Resonance 39

Compare this to equation (6.20). In equation (6.59) a termincluding the DC link capacitor is added.

The natural frequency and damping are

W=````Yo2

JL L+

Ko2

2/3CL L(6.60)

z=R

2`````LLYo2

J+

LLKo2

2/3C

(6.61)

If the DC link capacitor C , the expressions for damping andnatural frequency become identical to (6.22) and (6.23).

The smallest value ofC andJdetermines basically the resonancefrequency and damping. The damping can not be increased byincreasing C ifJis small compared to C.

One way to increase the damping seems to be to increase thestator resistance, but this is an unfavourable alternative due totwo reasons: the losses are increased with additional resistances,and worse, other resonance phenomena might appear. Thesephenomena are described in the following sections.

To solve the resonance problems some kind of feedback must beused, which is described in a later section.

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40 Parallel Resonance


A clear mechanical model can be obtained also for the parallelresonance. Computer simulations of the measuring setup showthat the amplitudes of the stator, rotor and leakage fluxes arenearly constant at the parallel resonance frequency. To derive auseful model which describes the oscillations, starting with themechanical model of Fig. 3.5, a few simplifying assumptions aremade, based on the results from simulations:

The rotor resistance is zero in the tangential direction of therotor. This means that if the rotor flux is turned an angle j,

the rotor will rotate the same angle j (the slip is zero).However, the rotor drag-pad can move in the radial direction,allowing the motor to be magnetized.

The leakage inductanceLL is zero. The amplitudes of the fluxes are constant usy is constant.

It follows from the assumptionLL = 0 that

Ys =

Yr =

Y (7.1)

This can be described by a mechanical equivalent where thestator and rotor drag-pads are stuck together as if they were one,and this single drag-pad moves in a radial slot i.e.RR = 0 in thetangential direction (Fig. 7.1).

The amplitudes of the fluxes are constant,

Y= Ys = Yr =usyRs

LM (7.2)

First we make an analysis of this model to determine the naturalfrequency. Let j denote the angle between the flux andimaginary axis (Fig. 7.2). As the rotor resistance is assumed to bezero, jis also equal to the displacement of the rotor. In this firstanalysis it is assumed that

Y isyLM (7.3)

The torque is approximately

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Parallel Resonance 41

TisyYsinj Y2

LMsinj

Y2

LMj (7.4)

dw

dt

= j=T

J

= Y2

JLMj (7.5)

The natural frequency is

W=Y

````JLM(7.6)

rotor dis

is

LM

Y

im

Figure 7.1. Mechanical representation of the parallel resonance

The behaviour of this oscillation can be compared to thebehaviour of a pendulum according to Fig. 7.2.

isy

Y

LM

j jm

l

F = mg

(a) (b)

Figure 7.2 (a) Mechanical representation of parallel resonance (b)Oscillatingpendulum

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Compare the frequency in equation (7.6) to that of equation(6.22). The only difference between the frequencies is that theleakage inductance in (6.22) is replaced by the magnetizinginductance in (7.6). Physically, the difference is much moresignificant. The magnetizing inductance LM influences theresonance frequency, but it is not oscillating in the radialdirection, which means that the energy stored in LM is notoscillating. In the case of the series resonance, the leakageinductance does not only effect the resonance frequency but isalso part of the oscillating system, exchanging energy with therotor of the machine.

This is a system without damping. To find the dampingcharacteristics, the model must be more extensive. The statorequation has to be included. The stator equation is the same asfor the complete model of the machine,

dY

dt=

dYs

dt=

us

isRs (7.7)

As the amplitude ofY is constant, the flux can only change in a

direction perpendicular to itself,

d

Ydt=jwY (7.8)

(7.7) and (7.8) imply

us

isRs =jw

Y= w(Ysy +jYsx) (7.9)

If (7.9) is split into real and imaginary parts, (7.10) and (7.11)are obtained,

usx =Rsisx wYsy (7.10)

usy =Rsisy + wYsx (7.11)

Equations (7.2), (7.10) and (7.11) give

isx =w

RsYsy +

usxRs

(7.12)

isy = w

RsYsx

Y

LM(7.13)

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The real and imaginary parts ofYbecome, according to Fig. 7.2,

Ysx = Ysinj (7.14)

Ysy = Ycosj (7.15)The torque can now be expressed as

T=Im (Ys*

is) = Ysxisy Ysyisx =

= YcosjusxRs

Y2cos2jw

RsY2sin2j

w

Rs

Y2

LMsinj (7.16)

Ifj is small, then cosj 1, cos2j + sin2j= 1 and sinjj, and

the torque is

T= YusxRs

Y2w

Rs

Y2

LMj (7.17)

The dynamic equations for the system in Fig. 7.1 are

dw

dt=

TJ

=1

J

Yusx

Rs Y2

w

Rs

Y2

LMj (7.18)

djdt

= w (7.19)

and can also be written as a second-order differential equation,

j+Y2

RsJ.j+

Y2

JLMj= Y

usxJRs

(7.20)

We have the same natural frequency as in (7.6)

W=Y

````JLM(7.21)

and the relative damping

z=Y

2RsLMJ (7.22)

The parameters of the Grundfos motor (App. B) are inserted intoequation (7.21) and (7.22), which give the natural frequency

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W=Y

````JLM=

1

```````13.51.6= 0.215 p.u. = 10.8 Hz

and the relative damping

z=Y

2Rs`LMJ = 120.07 `1.613.5 = 2.5

Note that a value of damping greater than 1 means that thereare no complex poles.

As with the series resonance model, a transfer function fromvoltage to current can be used to compare the model withmeasured data. If cosj 1, then (7.12) and (7.15) give

isx =usxRs

wY

Rs(7.23)

The Laplace transforms of (7.19), (7.20) and (7.23) are

sF= W (7.24)

s2F+Y2

RsJsF+

Y2

JLMF= Y

UsxJRs

(7.25)

Isx =UsxRs

WY

Rs(7.26)

and they give

Isx =Hp(s) Usx =1

Rs

s2+Y2

JLM

s2+ sY2

RsJ+

Y2

JLM

Usx (7.27)

The amplitude and argument of the transfer functionHp for theparallel resonance model is found in Fig. 7.3 (solid). Theaccordance to the measured data (dashed) is good for fre-quencies up to the resonance frequency, 0.22 p.u. (10.8 Hz).

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f [p.u.

f [p.u.

q

[deg]

i

/

u

[dB]

sx

sx

H Measured datap

Figure 7.3 Amplitude and argument of transfer functionHp

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It must be pointed out that since some simplifications are made,the expressions for the resonance frequency do not give exactvalues, but they describe how the different parameters influencethe resonance properties.

The two mechanical models for series resonance (Fig. 6.2) andparallel resonance (Fig. 7.1) can be used to understand thebehaviour of the oscillations of the induction machine.

It is important to notice that an increasing stator resistancereduces the damping of the parallel resonance but increases thedamping of the series resonance. If MOSFETs are used in theinverter, with a relatively high on-state resistance, thisresistance will act as an increased stator resistance, reducing thedamping of the parallel resonance.

Another case with high stator resistance is if the motor is fedfrom long wires. The resistances of the wires in series with thestator windings have the same effect as stator windings withhigher resistance.

Trying to solve the series resonance problem by increasing thestator resistance will not only increase the losses but also makethe parallel resonance problem worse.

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Linearization 47

8 Linearization

Resonance frequencies of linear systems can be found bycomputing the eigenvalues of the system matrix. The inductionmachine is non-linear, and this method can not be used directly.However, the equations describing the machine can belinearized, and the eigenvalues of the linear system can becomputed.

This method is in many ways inferior to the methods describedearlier in this thesis, as it only results in numerical values butgives no physical insight into the problem. As a comparison, theresults of linearization are presented.

To facilitate the linearization, the following time constants aredefined:

tls =LL/Rs (8.1)

tlr =LL/RR (8.2)

tms =

LMLL

LM+ LL

Rs =

1

Rs(1/ LM+1/ LL) (8.3)

tmech =JLL (8.4)

The dynamic equations are rewritten so that the deviation froman operating point, Yso, YRo, wo, is investigated,

Ys =

Yso + D

Ys (8.5)

YR =

YRo + D

YR (8.6)

w= wo + Dw (8.7)

us =

uo + D

u (8.8)

At steady state, the stator and rotor fluxes are equal,

Yso =

YRo =

Yo (8.9)

The voltage needed at steady state for the flux Yo is

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48 Linearization

uo =

RsLM

Yo (8.10)

With the equations (8.1-10) inserted into the equations (3.21-3.25), the following equations for the fluxes are obtained,

dDYs

dt= D

u

1tms

DYs +

1tls

DYR (8.11)

dDYR

dt=

1tlr

DYs

1tlr

DYR +jDwD

YR +jDw

Yo (8.12)

The value ofYo andwo in the tests of section 5 is

Yox = 0 and Yoy = 1 (8.13)

wo = 0 (8.14)

The flux equations, split into real (x) and imaginary (y) partsbecome

dDYsxdt

= Dux 1

tmsDYsx +

1tls

DYRx (8.15)

dDYsydt

= Duy 1tmsDYsy + 1tls

DYRy (8.16)

dDYRxdt

=1tlr

DYsx 1tlr

DYRx DwDYRy+ Dw (8.17)

dDYRydt

=1tlr

DYsy 1tlr

DYRy + DwDYRx (8.18)

and the mechanical equation (3.23) can be written

dDw

dt=

DYsxDYsxDYry DYrx+DYrxDYsy Tm

JL L(8.19)

If the non-linear terms, underlined above, are neglected,equations (8.15-8.19) can be replaced by the matrix equation

.x =Ax +Bu

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Linearization 49

or

.x =

1tms 0

1tls

0 0

0 1

tms 01

tls 0

1tlr

0 1

tlr0 1

01

tlr0

1tlr

0

1tmech

0 1

tmech0 0

x +

1 0 0

0 1 0

0 0 0

0 0 0

0 01

tmech

u

(8.20)

where

x =

DYsxDYsyDYRxDYRy

Dw

(8.21)

and

u =

usx

usyTm

(8.22)

The Grundfos motor described in App. B has the following timeconstants:

tls =LL/Rs = 0.138/0.07 = 1.97

tlr =LL/RR = 0.138/0.076 = 1.82

tms =1

Rs(1/ LM+1/ LL)=

10.07(1/1.66+1/0.138)

= 1.82

tmech =JLL = 13.5 0.138 = 1.86

which give the matrixA of equation (8.20),

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50 Linearization

A =

0.55 0 0.51 0 0

0 0.55 0 0.51 0

0.55 0 0.55 0 1

0 0.55 0 0.55 0

0.54 0 0.54 0 0

The eigenvalues of this matrix are shown in Fig. 8.1 (a). In (b)are shown the poles of the transfer function Hs from equation(6.26) and in (c) the poles ofHp from equation (7.27),

Hs =s/LL

s2+ sRLL

+Y2

JL L

=7.25s

s2+1.06 s+0.54

Hp =1

Rs

s2+Y2

JLM

s2+ sY2

RsJ+

Y2

JLM

=14.3s2+0.64

s2+1.06 s+0.045

It is seen in the figure that the two poles ofHs lie very close to

two of the eigenvalues ofA, and the two poles ofHp lie close totwo other eigenvalues ofA. This justifies thatHs andHp give agood description of the oscillatory behaviour of the inductionmachine.

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Linearization 51

(a)

(b)

(c)

p1,2 = 0.53 0.4j

p3 = 1.08

p4 = 0.04

p5 = 0.02

p1,2= 0.53 0.51j

p1 = 1.01

p2 = 0.04

Figure 8.1 (a) Eigenvalues of matrixA (b)Poles of Hs (c)Poles of Hp

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52 Feedback

9 Feedback

Damping of the series resonance described in section 6 can beobtained by feedback. One way is to measure the DC link currentId and let it control the voltage and stator frequency.

The feasibility of feedback as a way of damping the oscillations isillustrated by a standard proportional controller with gain G. Thedamping of the system described by equation (6.58) can now bechosen arbitrarily. With the feedback law

dK= G dis (9.1)

the damping of the closed-loop system becomes

z=R+ GUdo

2`````LLYo2

J+

LLKo2

2/3C

(9.2)

The gain can now be calculated,

G =2z

`````LLYo

2

J +

LLKo2

2/3C RUdo

(9.3)

If for example the damping z = 1 is desired for the Grundfosmotor at nominal flux with Ko = 0.58 and Udo = 1.73, the gainwill be

G = 0.0325

It is important that the stator frequency ws is adjusted in

accordance with the voltage amplitude factor K, to get thevoltage to frequency ratio constant,

ws = wsoKKo

(9.4)

A simulation of this (App. G) is shown in Fig. 9.1, where (a)shows a start of the motor without feedback, and (b) shows astart with G=0.0325. Note that the oscillation is damped, butthat the start is slower with feedback.

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Feedback 53

time [p.u.

time [p.u.

w

[p.u.]

w ws

w

[p.u.]

(a)

(b)

Figure 9.1 Start of induction machine (a) without and (b) with feedback

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54 Conclusions

10 Conclusions

The induction machine itself is an oscillating system withdifferent oscillating modes. If the induction machine is connectedto an inverter, a torque ripple is introduced that can excite theoscillations. The DC link capacitor of the inverter will also becomea part of the oscillating system.

In this thesis, mechanical models are presented which give anintuitive understanding of the different oscillating modes. Thisunderstanding is necessary for the design of regulators that cansuppress the oscillations. One such regulator is presented whichmeasures only current from the DC link.

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References 55

11 References

1) Alexandrovitz, A. (1987), Determination of InstabilityRange of a Frequency Controlled Three-Phase InductionMotor Fed Through Series Capacitors, ETZ Arch .(Germany), vol. 9, no. 7, July 1987, 235-9.

2) Fitzgerald, A. E., Charles Kingsley, Jr., Stephen D. Umans(1985).Electric Machinery, McGraw-Hill, New York.

3) Kovcs, P. K. (1984) Transient phenomena in electricalmachines, Elsevier, Amsterdam.

4) Leonhard, W. (1985). Control of electrical drives, Springer,Berlin.

5) Mutoh, N., A. Ueda, K. Sakai, M. Hattori and K. Nandoh(1990). Stabilizing Control Method for SuppressingOscillations of Induction Motors Driven by PWMInverters, IEEE Transactions on Industrial Electronics,vol. 37, no 1, 48-56

6) Palit, B. B. (1978), Eigenvalue Investigations of a 3-phase

Asynchronous Machine with Self-Excited Hunting, Bull.Assoc. Suisse Electr. (in German), vol .69, no. 8, 29 April1978, 371-6.

7) Slemon, G. and A. Straughen (1980). Electric Machines,Addison-Wesley, Reading, Mass..

8) Trk, V., and J. Valis (1985).Elektroniska drivsystem (inSwedish), Tutext, Stockholm.

9) Wagner, C. F. (1930). Effects of Armature Resistance Upon

Hunting of Synchronous Machines, Transactions AIEE,July 1930.

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56 Appendix

Appendix

A Per Unit Notation

The basic quantities are:

- rated phase voltage Un (peak value)- rated phase currentIn (peak value)- rated electrical angular velocity w1- rated mechanical angular velocity wn = w1/zp

wherezp is the number of pole pairs- rated phase flux Yn = Un/w1- rated apparent powerPn = 3/2 UnIn- rated torque Tn =Pn/wn- base impedanceZn = Un/In- rated start timeH=Jwn

2/Pn

The p.u. values of parameters and variables can now becalculated (voltage in p.u. is equal to the voltage in natural unitsdivided by the rated voltage Un etc):

voltage u/Uncurrent i/Inresistance R/Zninductance w1L/Zncapacitance w1CZnflux Y/Ynmoment of inertia w1H

torque T/Tntime w1t

electrical angular velocity wel /w1mechanical angular velocity wmech /wn

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Appendix 57

B Motor Data

The measurements described in section 4 and 5 have beenperformed on a Y-connected Grundfos MG 80 B with thefollowing characteristics:

rated phase voltage, Un (peak value) 310 V rated phase current,In (peak value) 3.82 A rated electrical angular velocity, w1 314 rad/s

number of pole pairs,zp 1rated power,P 1.1 kWrated power factor, cos j 0.84base impedance,Zn 81.2 W

stator resistance,Rs 6 Wrotor resistance,Rr 6 Wstator leakage inductance,Lsl 0.0173 Hrotor leakage inductance,Lrl 0.0173 Hmain inductance,Lm 0.414 Hmoment of inertia,J 0.00077 kg m2

in p.u. notation:

Rs 0.07Rr 0.07Lsl 0.065Lrl 0.065Lm 1.6J 13.5

k 0.96LM 1.66

LL 0.138RR 0.076

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58 Appendix

C Vector Equations

For simulation and other purposes, the vector equations (3.21-3.25) can be split up. One way is to separate the real and ima-ginary parts, another is to separate amplitude and argument ofthe flux vectors (Fig. C.1).

The following system of five differential equations is obtained if(3.21-3.23) are split into real (x) and imaginary (y) parts :

dYsxdt

= usx isxRs

dYsy

dt = usy isyRs

dYRxdt

= wYRy iRxRR

dYRydt

= wYRx iRyRR

dw

dt=

Ysxisy Ysyisx

J

Note that these equations are written in the stator referenceframe.

YsYsy

Ysx

istan

israd

is

s

Im

Re

YR YRy

YRx

iRtan

iRrad

iR

Im

Re

r

(a) (b)

Figure C.1 Flux and current vectors and their components(a) Stator (b)Rotor

The stator and rotor flux vectors can be expressed withamplitudes and arguments,

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Appendix 59

Ys = Ys e

js

YR = YR e

jr

Let the index rad represent components pointing in the radialdirection, parallel to the flux, and the index tan representcomponents in the tangential direction, perpendicular to the flux.This means for example that israd is the x-component of thestator current expressed in a reference frame attached to thestator flux, and istan is the corresponding y-component. Theequations (3.21-3.25) can be written as five differential equationsand some algebraic relations,

dYsdt = usrad Rsisrad

ds

dt= (ustan Rsistan)/Ys

dYRdt

= iRradRR

dr

dt= (YRwiRtan)/YR

dw

dt= Ysistan /J

israd = Ys/LMYs/LL YR cosd/LL

istan = YR sind/LL

iRrad = YR/LL Ys cosd/LL

iRtan = Ys sind/LL

d= sr

The following two relations transform the stator voltage fromcoordinates in the stator reference frame to the stator fluxreference frame,

usrad = usx coss+ usy sins

ustan = usx sins+ usy coss

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60 Appendix

D Coordinate Transformations

The vector equations can be transformed to a coordinate systemrotating with the angular velocity wk. The quantities in therotating system are denoted by a superscript-k. The anglebetween the rotating system, and the coordinate system fixed tothe stator is denoted by j. It follows that

dj

dt= wk

Ys =

Ys

k e jj,is =

isk e jjetc.

dYsdt

=dYsk

dte jj+j

djdt

Ys

k e jj

dYr

dt=

dYr

k

dte jj+j

dj

dtYr

k e jj

If these relations are inserted into the vector equations in statorcoordinates, the following equations are obtained,

dYskdt

=usk is

kRs jwkYs

k

dYr

k

dt=jw

Yr

k i rkRr jwk

Yr

k

The torque equation is invariant against coordinate transfor-mations.

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62 Appendix

F Roots, Natural Frequency and Damping

The equation

s2 +2Wzs + W2 = 0

has the roots

z1: s = Wz`````z21

In the case of complex roots, they can be demonstrated by Fig.F.1. Wis the naturalfrequency and zis the damping.

W

j

cosj= z

Figure F.1 Relation between location of roots, natural frequency and damping

Figure F.2 shows the time variation of the state variable x of thesystem

x +2Wz.

x + Wx= 0

with the initial value xo = 1. The natural frequency W = 1, thedamping z= 0.1 for the solid line, and z= 1 for the dashed line.The dotted line shows

ezWt; z= 0.1, W= 1

The frequency of oscillation for z= 0.1 is

w= W`````1 z2 = 0.995 rad/s

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Appendix 63

time

x

Figure F.2 Simulation of x + 2Wz

.

x + Wx= 0

ezWt

z= 0.1

z= 1

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64 Appendix

G Simulation File

The simulations have been performed with the programSimnon. The following file is used to simulate the start of aninduction motor with and without feedback.

continuous system feedback"gamma model of induction machine in p.u. notation"stator coordinates"DC link current feedback

"stator flux, rotor flux, angular velocity, rotor angle,"arg of stator voltagestate psisx psisy psirx psiry w theta arguder dpsisx dpsisy dpsirx dpsiry dw dtheta dargu

time t

"Grundfos parameters in p.u.LM:1.6LL:0.138Rs:0.07Rr:0.076J:13.5Tload:0zero:0psin:1 "nominal flux

Ud:1.73K0:0.58ws0:1 "stator frequency

"feedbackdeltaK=-gain*deltaId*2/3/K0K=K0+deltaKu=K*Udws=ws0*K/K0Id0:0.0273Id=is*cos(argis-argu)deltaId=Id-Id0gain:0.0325

"feeding voltagedargu=wsusx=u*cos(argu)usy=u*sin(argu)

us=sqrt(sqr(usx)+sqr(usy))

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Appendix 65

"dynamic equationsdpsisx=usx-Rs*isxdpsisy=usy-Rs*isydpsirx=-w*psiry-Rr*irxdpsiry= w*psirx-Rr*irydw=(Torque-Tload)/J

dtheta=w

imx=psisx/Lmimy=psisy/Lmim=sqrt(imx*imx+imy*imy)argim=atan2(imy,imx)irx=(psirx-psisx)/LLiry=(psiry-psisy)/LLargir=atan2(iry,irx)ir=sqrt(irx*irx+iry*iry)isx=imx-irxisy=imy-iry

is=sqrt(isx*isx+isy*isy)argis=atan2(isy,isx)Torque=(psisx*isy-psisy*isx)

P2=w*TorqueP1=(3/2)*(usx*isx+usy*isy)

"rotating reference framefi0:-0.17fi=ws*t-fi0cfi=cos(fi)sfi=sin(fi)psisxr= psisx*cfi+psisy*sfi

psisyr=-psisx*sfi+psisy*cfipsirxr= psirx*cfi+psiry*sfipsiryr=-psirx*sfi+psiry*cfi

end

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66 Appendix

H List of Symbols

amplitude factor K

angular velocity wcapacitance Ccurrent i

rotor current ir, iRstator current isline current ia, ib, ic

frequency fstator frequency fs, ws

inductance Lmutual inductance Lm,LM

rotor leakage inductance Lrlstator leakage inductance Lsl

Laplace operator slinked flux Y

rotor flux Yr, YRstator flux Ys

moment of inertia Jnatural frequency Wpower factor cosj

relative damping zresistance Rrotor resistance Rr,RRstator resistance Rs

slip spull-out slip sp

time constant ttime ttorque T

load torque Tm

pull-out torque Tpvoltage u

stator voltage usphase-to-neutral voltage ua, ub, uc

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CODEN: LUTEDX/(TEIE-1001)/1-67/(1991)

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