on inter-bar currents in induction motors with cast aluminium …325097/fulltext… · ·...

On Inter-bar Currents in Induction Motors with Cast

Aluminium and Cast Copper Rotors

ALEXANDER STENING

Licentiate Thesis

Stockholm, Sweden 2010

TRITA-EE 2010:027ISSN 1653-5146ISBN 978-91-7415-682-9

Electrical Machines and Power ElectronicsSchool of Electrical Engineering, KTH

SE-100 44 StockholmSWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie licentiatexamen tisdagen den15 Juni 2010 klockan 10.00 i E2, Kungl Tekniska högskolan, Lindstedtsvägen 3,Stockholm.

© Alexander Stening, May 2010

Tryck: Universitetsservice US AB

iii

Abstract

This thesis presents a study of the effects of inter-bar currents on induction

motor starting performance and stray-load losses. The work is focused on the

performance differences between aluminium and copper casted rotors.

A method to predict the stator current when starting direct-on-line is

developed. This includes modelling of skin-effect, saturation of the leakage

flux paths and additional iron losses. The results are verified by measure-

ments. An analytical model accounting for inter-bar currents is derived, and

the dependency of the harmonic rotor currents on the inter-bar resistivity is

investigated. It is found that the inter-bar currents can have considerable

effect on motor starting performance and stray-load losses, the amount being

strongly dependent on the harmonic content of the primary MMF.

Based on measurements of inter-bar resistivity, the starting performance

of an aluminium and a copper casted rotor is simulated. The results indicate

a higher pull-out torque of the aluminium rotor than for the equivalent copper

rotor. This is rather due to an increase of the fundamental starting torque of

the aluminium rotor, than due to braking torques from the space harmonics

in the copper rotor. The results are verified by measurements. It is found

that the difference between the pull-out torques is even larger than calculated

from the model. Thereby, it can be concluded that the inter-bar currents have

a considerable effect on motor starting performance.

At rated speed the braking torques are larger in the aluminium rotor than

in the copper rotor. This is seen as increased harmonic joule losses in the ro-

tor cage. Simulations have shown, that these losses can be as large as 1% of

the output power for the studied machine.

Keywords: Induction motors, Inter-bar currents, Copper rotors, Aluminium

rotors, Starting torque, Asynchronous torques, Starting current, Stray losses.

v

Sammanfattning

Denna licentiatavhandling presenterar en studie av tvärströmmars påver-

kan på startegenskaper och tillsatsförluster för asynkronmaskiner med gjutna

aluminium- och kopparrotorer.

En metod för estimering av startströmmen i asynkronmaskiner vid direk-

tstart mot nätet utvecklas. Metoden inkluderar strömförträngning, järnmät-

tning av läckflödesvägar samt järnförluster på grund av läckflöden. Resultaten

verifieras med mätningar. En analytisk modell för beräkning av tvärström-

mar härleds, med vilken beroendet av rotorns övertonsströmmar på kontak-

tresistivitet mellan rotorledare och rotorplåt utreds. Simuleringar visar att

tvärströmmar kan ha stor inverkan på asynkronmaskinens startmoment och

dess tillsatsförluster. Effekten av tvärströmmars inverkan är direkt kopplad

till övervågsinnehållet i den av statorlindningen skapade MMK:n.

Baserat på mätningar av kontaktresistivitet mellan rotorledare och rotor-

plåt, beräknas startprestanda för en gjuten aluminium- respektive koppar-

rotor. Resultaten indikerar att aluminiumrotorn har ett högre kippmoment

än motsvarande kopparrotor. Enligt simuleringar beror detta mer på ett

ökat grundtonsmoment i aluminiumrotorn än på ett reducerat totalmoment

i kopparrotorn. Mätningar visar att denna skillnad existerar och att den

dessutom är större än beräknat från modellen. Det kan således konstateras

att tvärströmmar har en betydande effekt på asynkronmaskinens startegen-

skaper.

Moment av högre ordning än grundtonsmomentet skapar vid nominell

drift ett resulterande bromsande moment, vilket visar sig vara större i alu-

miniumrotorn än i kopparrotorn. Detta leder till en ökning av högfrekventa

resistiva förluster i rotorkretsen. Simuleringar av den studerade asynkron-

maskinen visar att dessa förluster kan vara så stora som 1% av märkeffekten.

Sökord: Asynkronmotorer, Tvärströmmar, Kopparrotorer, Aluminiumro-

torer, Startmoment, Asynkrona moment, Startström, Tillsatsförluster.

Acknowledgements

This work has been carried out within the High Performance Drives program of theCenter of Excellence in Electric Power Engineering at the department of ElectricalMachines and Power Electronics. Since the start of this project, several people havebeen involved and have contributed to this thesis in different ways; I am gratefulto them all.

First of all, I would like to thank my supervisor Prof. Chandur Sadarangani forhis help throughout this project and for sharing his knowledge in our conversations.I am also grateful to Assoc. Prof. Juliette Soulard, for being available and inspiringme whenever I needed it. I would like to thank the personnel at ABB LV Motorsand ITT Flygt for giving me a rewarding stay outside KTH. A special thanks goesto Bo Malmros and Jörgen Engström, for their useful inputs and for helping mewith supplies of prototype motors.

I would like to thank Jan Timmerman and Olle Brännvall, for always helpingme to find the best solutions to my problems in the laboratory, and also for thenice moments we have had discussing hunting and boating. Further, I would liketo thank Dr Stephan Meier for his help during the time he worked in the labora-tory. Thanks to Prof. Hans-Peter Nee for reading the thesis and for his valuablecomments.

I am very grateful to all employees at EME, for contributing to a pleasantatmosphere. In the same way I would like to thank the former employees at EME,for keeping up the good spirit with different kinds of Roebel-activities. A specialthanks to my colleges and friends, Henrik Grop and Dmitry Svechkarenko, for thenice times we have had and for the times to come. To my former college RathnaChitroju, I would like to say that I am glad you are still in Sweden, thanks forthe joy you are bringing. Further, a thank goes to my office room mate AlijaCosic, with whom I have shared many laughs. Thanks to Eva Pettersson and PeterLönn for helping me with administration and computers, things I sometimes do notunderstand.

Finally, I would like to express my deepest gratitude to my family, for theirsupport and understanding. This truly means a lot to me. To my beloved cohabiteeIda Axelsson; thanks for your patience during the late hours when I have beenworking with this thesis, you mean everything to me.

Alexander SteningStockholm, May 2010

vii

Contents

Contents ix

1 Introduction 1

1.1 The need of accurate induction motor models . . . . . . . . . . . . . 11.1.1 Improved efficiency . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Starting performance . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Die cast aluminium and copper rotors . . . . . . . . . . . . . . . . . 31.3 Rotor skewing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Scientific contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.7 Studied motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.8 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Measurements of inter-bar resistance 7

2.1 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Modelling of the rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Results from measurements . . . . . . . . . . . . . . . . . . . . . . . 112.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Model for the analysis of inter-bar currents 17

3.1 Rotor circuit taking inter-bar currents into account . . . . . . . . . . 173.2 Stator flux linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 Airgap flux density due to stator current . . . . . . . . . . . . 203.2.2 Stator flux linked by the rotor circuit . . . . . . . . . . . . . 24

3.3 Rotor flux linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.1 Airgap flux density due to rotor current . . . . . . . . . . . . 263.3.2 Flux caused by the phase belt harmonics . . . . . . . . . . . 283.3.3 Flux caused by slot harmonics . . . . . . . . . . . . . . . . . 29

3.4 General set of equations . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Effects of a finite inter-bar resistance on rotor current distribution . 31

3.5.1 Rotor without skew . . . . . . . . . . . . . . . . . . . . . . . 323.5.2 Rotor with skew . . . . . . . . . . . . . . . . . . . . . . . . . 36

ix

x CONTENTS

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Effects during a direct-on-line start 41

4.1 Skin effect in the rotor bars . . . . . . . . . . . . . . . . . . . . . . . 414.1.1 Numerical method used to account for skin effect . . . . . . . 424.1.2 Verification with FEM . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Saturation of the leakage paths . . . . . . . . . . . . . . . . . . . . . 484.2.1 Model used to account for saturation . . . . . . . . . . . . . . 494.2.2 Iron losses due to leakage flux . . . . . . . . . . . . . . . . . . 52

4.3 Rotor losses and starting torque . . . . . . . . . . . . . . . . . . . . . 524.3.1 Rotor losses during a start . . . . . . . . . . . . . . . . . . . . 524.3.2 Effects of a finite inter-bar resistance on starting torque . . . 53

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Simulation results and measurements 59

5.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.1.1 Starting torque . . . . . . . . . . . . . . . . . . . . . . . . . . 595.1.2 Rotor losses at rated speed . . . . . . . . . . . . . . . . . . . 63

5.2 Measurements of starting torque . . . . . . . . . . . . . . . . . . . . 645.2.1 Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . 645.2.2 Results from measurements . . . . . . . . . . . . . . . . . . . 66

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6 Conclusions and Future work 71

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.2 Future work guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Bibliography 73

List of Figures 76

Chapter 1

Introduction

Electrical machines are widely used for high efficient conversion between electricaland mechanical energy. Ever since the induction motor was invented in 1886, ithas been serving the ever growing industry. Due to the simplicity and the robustdesign of the induction motor, it has become the most commonly used electricalmachine.

1.1 The need of accurate induction motor models

Despite the simple working principle, the designer requires extensive knowledge ofthe induction motor, in order to construct an efficient motor. During the years, theinduction motor designs have been refined, which increases the need for accuratemotor models.

1.1.1 Improved efficiency

In order to reduce the consumption of electricity, the manufacturing of high efficientmotors has become a topic of current interest. New efficiency standards make theinduction motor design even more challenging. This requires not only the mini-mization of the well known stator and rotor copper losses, iron losses and frictionlosses, but also the reduction of the additional losses. These losses are defined asthe additional losses that occur in the machine over the normal losses that are con-sidered in usual induction motor performance calculations.

At rated load the additional losses are referred to as stray-load losses. For smallto medium sized induction motors these losses vary typically within the range 0,5%- 3% of the motor input power [1]. Measurements have, however, shown that theselosses can be even larger [2]. Measuring these losses with a reasonable accuracyis a difficult task. As there are different efficiency standards defining this measur-ing procedure, the amount of stray-load losses depend on the standard used [3, 4].

1

2 CHAPTER 1. INTRODUCTION

Investigations of the different stray-load loss components have been performed,among others by Nishizawa in [2]. Figure 1.1 shows the obtained stray-load losscomponents for a set of small to medium-size induction motors.

3%

10%

17%

30%

40%

Leakage flux losses

High-frequency losses

Pulsation losses

Inter-bar current losses

Surface losses

Figure 1.1: Stray-load loss components (0,2-37 kW induction motors) [2].

According to this study, the largest portion of the stray-load losses is composedof surface losses caused by high frequency flux. However, these losses can be suffi-ciently suppressed by the use of non-machined rotors [2], reducing the eddy currentsat the rotor surface. The second largest portion of the stray-load losses, accordingto [2], are the losses caused by inter-bar currents. These currents, flowing betweenthe rotor bars through the iron lamination, can cause considerable losses in the in-duction motor unless the rotor bars are insulated [5]. These losses, primary causedby the stator slot harmonics, result in increased torque dips during a direct-on-linestart, which can reduce the pull-out torque [6]. Therefore, the torque-speed curvecontains information regarding these losses.

1.1.2 Starting performance

The starting characteristic of the induction motor is an important design factor.For the motor to start, the starting torque must be larger than the load torque.Furthermore, to ensure a reasonable margin of overload capability, it is usual torequire that an induction motor is able to deliver momentarily at least twice itsrated torque at rated voltage [7]. In traction applications, the pull-out torque is

1.2. DIE CAST ALUMINIUM AND COPPER ROTORS 3

an important design criteria since the speed range of the motor is limited by itspull-out torque [8]. This reinforces the need of inter-bar current models.

1.2 Die cast aluminium and copper rotors

Large motors are manufactured with fabricated aluminium or copper bar rotors.The prefabricated bars are inserted into rotor slots that are punched around theperiphery of the rotor lamination and the short circuit rings are welded or brazed tothe bars. However, this choice is unattractive for small to medium sized motors dueto cost reasons. These machines are generally equipped with die cast aluminiumrotors. The casting process results in a low resistive path between the rotor barsand the iron core. This is referred to as inter-bar resistance. Resistance, however,is not an appropriate unit to use for this contact region as it depends on the stacklength. Therefore, this resistance is usually multiplied with the stack length, defin-ing the inter-bar resistivity.

Due to advancements in casting technology, it is possible to manufacture die castcopper rotors. Thanks to the higher conductivity of copper, the motor efficiencycan be increased. Measurements have shown, however that the inter-bar resistivityin copper rotors can be as much as 10 times lower than in casted aluminium rotors[9], this promotes the flow of inter-bar currents.

1.3 Rotor skewing

The stator, creating the primary MMF, is usually equipped with semi-closed slots.These openings create a non-uniform distribution of the air-gap permeance, dis-torting the fundamental MMF. This give rise to airgap space harmonics referredto as slot harmonics, the order depending on the number of stator slots. In anunskewed rotor, these harmonics induce high frequency currents in the rotor cage,the resulting cage losses can form a considerable part of the stray losses [10].

If the rotor is skewed by one stator slot pitch and the rotor bars are insulated,these currents are efficiently suppressed, improving the motor efficiency. However,by the introduction of casted rotors, the inter-bar resistivity being low, inter-barcurrents start to flow. The magnitude of these currents are highly dependent onrotor skew and inter-bar resistivity.

1.4 Objectives

The objective of this thesis is to study the effects of inter-bar currents on aluminiumand copper casted rotors. This is achieved by developing analytical models tosimulate the starting performance and to calculate additional rotor losses. Themodels should be verified by measurements. The objectives can be summarized asfollows:


• Measure the inter-bar resistance on a set of aluminium and copper rotors.

• Develop a computer program for the calculation of the effects of inter-barcurrents on motor performance.

• Develop an analytical model that that can account for saturation of the leak-age paths during a direct-on-line start.

• Verify the analytical models by measuring the starting performance of theinduction motor equipped with either an aluminium or a copper rotor.

1.5 Scientific contribution

This work has resulted in the following contributions:

• Measurements have shown that the inter-bar resistivity in the studied castedrotors is lower in the copper rotors than in the aluminum rotors.

• Measurements have shown results indicating an uneven distribution of theinter-bar resistivity in the studied aluminium rotors, while in the copper ro-tors, the inter-bar resistivity is evenly distributed.

• A numerical method to account for skin-effect has been verified by finiteelement simulations.

• A method to estimate the starting current of induction motors has been de-veloped and verified by measurements.

• An analytical model to predict the effects of inter-bar currents on startingperformance has been verified by measurements.

• The starting torques of one aluminium- and one copper rotor skewed by onestator slot pitch have been measured. The results show that the pull-outtorque is lower for the copper rotor than for the equivalent aluminum rotor.This is verified by simulations.

1.6 Publications

The work presented in this thesis has resulted in two international conference paperslisted below:

• A. Stening and C. Sadarangani, The effects of inter-bar currents in cast alu-minium and cast copper rotors, In Proc. International Conference on Elec-

trical Machines, Vilamoura, Portugal, September 2008.

1.7. STUDIED MOTORS 5

• A. Stening and C. Sadarangani, Starting performance of induction motorswith cast aluminium and copper rotors including the effects of saturation andinter-bar currents, In Proc. International Conference on Electrical Machines

and Systems, Tokyo, Japan, November 2009.

1.7 Studied motors

In this thesis the models developed are used to simulate the performance of twodifferent motors, referred to as Motor A and Motor B.

Motor A 11 kw, 4-pole, 36 stator slots and 44 rotor slots with aluminum castedbars.

Motor B 11 kW, 4-pole, 36 stator slots and 28 rotor slots with both aluminiumand copper casted bars.

Measurements have been performed on Motor B, this motor is therefore used toverify the analytical models. The aluminum and the copper rotors have the samegeometry, except for a small difference in the short-circuit ring design. The samestator is used when measuring the performance of the two rotor concepts.

Motor A is used for analytical studies to demonstrate the dependency of theinter-bar resistivity on motor performance.

1.8 Outline of the thesis

Chapter 1: This chapter introduces the thesis, gives a brief introduction to thetopic and presents the objectives.

Chapter 2: A method for measurements of inter-bar resistance is presented. Anequivalent circuit of the rotor is used to calculate the resistivity from measuredvoltages. Results are presented from the measurements on five different rotors.

Chapter 3: In this chapter the main model used to account for inter-bar currentsis derived. Motor A is simulated, results are presented showing the distribution ofthe inter-bar currents in a skewed and an unskewed rotor at different inter-bar re-sistivities.

Chapter 4: Models used to account for inter-bar effects during a direct-on-linestart are presented. A numerical method to account for skin effect is verified withFEM-simulations. A method to include saturation of the leakage flux paths is de-rived. The rotor losses and the produced torque is derived from the rotor currents.Simulation results are presented showing the dependency of the starting torque onthe inter-bar resistivity for Motor A.


Chapter 5: This chapter highlights the differences in starting performance be-tween the aluminum and the copper rotor used in Motor B. The additional lossescreated in the rotor cage at rated speed are simulated as a function of inter-barresistivity. The simulated starting performance is verified by measurements.

Chapter 6: This chapter concludes the thesis, the results are summarized andsuggested future work is presented.

Chapter 2

Measurements of inter-bar resistance

The casting process results in a distributed low resistive path between the rotorcage and core. To determine this resistance accurately is a difficult task. It is notpossible to measure the inter-bar resistivity directly; it has to be calculated frommeasurements. The methods known can be categorized as non-destructive and de-structive methods.

In 1958, Odok presented a method to measure inter-bar resistance on castedrotors [6]. A direct current is fed into one short-circuit ring and taken out throughthe shaft on the opposite side. The voltage drop between the ring and iron core ismeasured along the axial direction. Based on the average value of this voltage, theinter-bar resistivity is calculated. Odok also came to the important conclusion thatthe inter-bar impedance can be assumed to be purely resistive. Odok’s method issimple but not so accurate since it does not take the distribution of the bar currentsinto account. Odok’s method was further developed, among others by Dabala in[11]. Assuming an equally distributed inter-bar resistivity, this method takes thedistribution of the bar currents into account.

When casted copper rotors were introduced the measurements became evenmore challenging. Dabala suggested an improved method for measurements oncasted copper rotors [9]. The improved method is not only taking the distributionof the bar currents into account, it also considers the resistivity of the iron sheets.

The method in [9] is used to determine the inter-bar resistivity for a set ofaluminium and copper rotors. All rotors are made for the same 4-pole stator, ratedat 11 kW. The geometries of the rotors are the same for both concepts, except fora minor difference in the short-circuit ring design.

7

8 CHAPTER 2. MEASUREMENTS OF INTER-BAR RESISTANCE

2.1 Test setup

A test-rig was built with the intention to avoid unnecessarily destruction of therotor. This setup, shown in Figure 2.1, makes it possible to measure the inter-barresistance with a negligible impact on the rotor construction. The top of the rig,on which the rotor is standing, consist of a smooth aluminium plate. One end ofthe rotor shaft is insulated with a thin plastic film and inserted into a hole in thecenter of this plate. Due to the rotor weight a conducting path is created betweenthe plate and the short-circuit ring. In order to create a uniform distribution of thecurrent in this contact path, the aluminium plate is machined as well as the surfaceof the short-circuit ring. To complete the current path a copper ring is mountedon the other side of the shaft. With a potential difference between this copper ringand the aluminium plate, a current will flow from one short-circuit ring to the shafton the opposite side via the bar to core region.

By the use of an equivalent circuit of the rotor assuming that the current isevenly distributed between the rotor bars, it is possible to determine the ring toring voltage UAB and the ring to shaft voltages UAD and UBC , as a function of theinter-bar resistivity.

It is, however, appropriate to note some important issues regarding these mea-surements. As this setup basically is a short-circuit, it requires a relatively highcurrent in order to obtain voltage levels that are possible to measure. And it is ofgreat importance to exclude the connection points of the rotor to the test-rig fromthe voltage measuring circuit. It turned out, during the development of the test-rig,that the currents where not evenly distributed between the rotor bars. Especiallyfor the copper rotors which where incidently manufactured without any fins on theshort-circuit rings. One reason for this is of course that the inter-bar resistivitymight be unevenly distributed. But an improvement was obtained by placing aconducting washer between the aluminium plate and the rotor short-circuit ring,according to Figure 2.2. This washer, being quite soft, distributes the force moreequally around the short-circuit ring, resulting in a smoother distribution of thecurrent in this contact region.

2.2 Modelling of the rotor

The rotor is modelled as proposed by Dabala [9], with a parameter network dis-tributed in the axial direction x. As indicated by Odok in [6], the inter-barimpedance is assumed to be purely resistive. However, the equivalent circuit inFigure 2.3 has been further developed taking full consideration of the voltage dropalong the shaft. This is obtained by the parallel connection of all the rotor barsinstead of representing one bar. With the total current flowing through the shaft,the corresponding voltage drop is then modelled correctly.In the equivalent circuit the following notations are used:

2.2. MODELLING OF THE ROTOR 9

I

A B

CD

x

(a) Circuit.

(b) Setup in the lab.

Figure 2.1: Rotor test setup for measurements of inter-bar resistance.


Figure 2.2: Conducting washer between test-plate and rotor short-circuit ring.

U : Bar to shaft voltage [V].

Ib, Is: Sum of all bar currents, and shaft current, respectively [A].

Rb, Rs: Bar and shaft resistance per unit length, respectively[

Ωm

]

.

ℓ: Length of the rotor bars [m].

Qr: Number of rotor bars.

gtn: Inter-bar conductivity[

1Ωm

]

.

gFe: Iron core conductivity[

1Ωm

]

.

From Kirchhoff’s voltage law the first differential equation is obtained as:

dU(x)

dx= Is(x)Rs − Ib(x)

Rb

Qr(2.1)

And the second equation is obtained from Kirchhoff’s current law, resulting in:

dIb(x)

dx= −U(x)

gtngFe

gtn + gFe(2.2)

By realizing that the change in the shaft current is caused by the current flowingthrough the iron core, the third and the last equation within the system becomes:

dIs(x)

dx= U(x)

gtngFe

gtn + gFe(2.3)

The boundary conditions are expressed with the total current flowing through therotor I, as;

Ib(0) = Is(ℓ) = I (2.4)

2.3. RESULTS FROM MEASUREMENTS 11

++

U(x)

Ib(x)RbQr

∆x Ib(x + ∆x)

gtn∆x

gFe∆x

Rs∆xIs(x) Is(x + ∆x)

U(x+ ∆x)

x x+ ∆x

A B

CD

0 ℓ

Figure 2.3: Equivalent circuit of the rotor used for the calculation of the inter-barresistivity.

and

Ib(ℓ) = Is(0) = 0. (2.5)

Based on the solution of the equations presented above, the ring to ring voltageUAB and the ring to shaft voltages UAD and UBC are determined as a function ofthe inter-bar resistivity. Figure 2.4 shows these voltages for the studied aluminiumrotor at a total current of 200 A. The corresponding results for the equivalent copperrotor is shown in Figure 2.5.

These results show that the inter-bar resistivity can be determined, based onthe assumptions made in the model, through measurements of the correspondingvoltages on the considered rotor.

2.3 Results from measurements

A direct current power supply, Delta Elektronika SM15-400, was used to supply therotors with a total rotor current of 200 A. The voltages UAB, UAD and UBC werethen measured using sharp probes connected to the multimeter Agilent 34410A.Based on these measured voltages, the inter-bar resistivities were calculated fromthe model described in the previous section. It could be noted from the measure-ments, that the accuracy for calculating the inter-bar resistivity was poor when thederivative of voltage to inter-bar resistivity was low, measurements within theseareas were therefore avoided in the analysis. The results obtained for the studied


Vol

tage

[µV

]

Inter-bar resistivity [Ωm]10−8 10−7 10−6 10−5 10−4

120

140

160

180

200

220

240

260

(a) Ring to ring voltage UAB .

Inter-bar resistivity [Ωm]

Vol

tage

[mV

]

UAD

UBC

10−8 10−7 10−6 10−5 10−40

20

40

60

80

100

120

(b) Ring to shaft voltages UAD and UBC .

Figure 2.4: Calculated voltages for the aluminium rotor at a total current of 200A.

Vol

tage

[µV

]

Inter-bar resistivity [Ωm]10−8 10−7 10−6 10−5 10−470

80

90

100

110

120

130

140

(a) Ring to ring voltage UAB .

Inter-bar resistivity [Ωm]

Vol

tage

[mV

]

UAD

UBC

10−8 10−7 10−6 10−5 10−40

20

40

60

80

100

120

(b) Ring to shaft voltages UAD and UBC .

Figure 2.5: Calculated voltages for the copper rotor at a total current of 200 A.

rotors are shown in Table 2.1.

From these results it can be concluded that the inter-bar resistivity is higherin cast copper rotors than in cast aluminium rotors. For the studied rotors, thedifference is at least a factor of ten. These results are however consistent with thefindings in [9]. Measurements of inter-bar resistivity on aluminium rotors with thesame slot shape as the rotors studied in this thesis have shown very similar results[12]. Even though, in that work, the author removed the short-circuit rings andmeasured directly between the rotor bars.

It shall be noted that, depending on the casted rotor material, some of the

2.3. RESULTS FROM MEASUREMENTS 13

Figure 2.6: Two of the studied aluminium and copper rotors.

Rotor Al 1 Al 2 Cu 1 Cu 2 Cu 3

UAB [µV] 134 130 125 124 123Rtn [µΩm] - - 0,3 0,4 0,4

UBC [mV] 15,15 12,22 2,87 2,60 3,49Rtn [µΩm] 9,0 7,0 0,4 0,3 0,6

UAD [mV] 5,35 3,04 20·10−3 22·10−3 15·10−3

Rtn [µΩm] 7,0 4,0 - - -

Mean valueRtn [µΩm] 8,0 5,5 0,35 0,35 0,5

Table 2.1: Measured voltages at a total rotor current of 200 A and the resultinginter-bar resistivities Rtn.


voltages in Table 2.1 are unsuitable for the determination of the inter-bar resistivity.Regarding the aluminium rotors, the measured ring to ring voltage UAB results inan infinitely high inter-bar resistivity. This could be due to the fact that the modelassumes an equally distributed inter-bar resistivity along the rotor bars, and thering to ring voltage is strongly dependent on this distribution. One can concludethat the inter-bar resistivity is unevenly distributed in the aluminium rotors. Thiseffect is studied further by measuring the voltage drop along the rotor bars withreference to one short-circuit ring, referred to as UAX .

UAX =

∫ ℓ

0

Rb

QrIb(x,Rtn)dx (2.6)

The measurements were performed along one fourth of the total number of rotorbars. The results are presented in Figure 2.7 together with the calculated valuesusing the inter-bar resistivity from Table 2.1. According to the measured voltageprofile, it can be concluded that the bar currents have decreased to zero in the lastthird of the rotor. This implies, for the studied aluminum rotor, that the inter-bar resistivity might be unevenly distributed. Probably due to the existence ofaluminium oxide along the rotor bar surface, naturally created through the reactionwith oxygen. This process is enhanced by the high casting temperature [13].

The corresponding results for an equivalent copper rotor is shown in Figure 2.8.In this case, the measured and the calculated voltage shows good correlation. For

Vol

tage

[µV

]

Axial position x [m]

Rtn = 5, 5 · 10−6 [Ωm]

Measured bar 1

Measured bar 2

Measured bar 3

Measured bar 4

Measured bar 5

Measured bar 6

Measured bar 7

Analytical using

0 0.05 0.1 0.150

20

40

60

80

100

120

140

160

Figure 2.7: Measured and calculated voltage UAX for rotor Al 2 at a total currentof 200 A

2.4. SUMMARY 15

Vol

tage

[µV

]

Axial position x [m]

Rtn = 0, 35 · 10−6 [Ωm]

Measured bar 1

Measured bar 2

Measured bar 3

Measured bar 4

Measured bar 5

Measured bar 6

Measured bar 7

Analytical using

0 0.05 0.1 0.150

10

20

30

40

50

60

70

Figure 2.8: Measured and calculated voltage UAX for rotor Cu 2 at a total currentof 100 A

the studied copper rotor, the theory of an evenly distributed inter-bar resistivityseems to hold, indicating an important difference between aluminium and coppercasted rotors.

2.4 Summary

A test-rig has been built for the measurement of rotor voltages, from which theinter-bar resistivity can be calculated through an equivalent circuit of the rotor.Measurements have shown that the inter-bar resistivity is as much as ten timeshigher in cast aluminum- than in cast copper rotors. An important difference wasnoted between the two rotor concepts. Aluminium rotors show results indicatingan unequal distribution of the inter-bar resistivity, while the copper rotors seem tohave a more equal distribution of this resistivity.

Chapter 3

Model for the analysis of inter-bar currents

The analytical model used to include the effects of inter-bar currents is derived fromBehdashti´s work in [14]. The inter-bar currents are taken into account by intro-ducing a transverse bar to bar resistivity distributed along the rotor bars. Basedon Behdashti´s proposed equivalent circuit of the rotor the inter-bar current dis-tribution along the rotor core can be obtained. Apart from the machine geometry,this model requires the fundamental stator current as an input parameter.

3.1 Rotor circuit taking inter-bar currents into account

The equations describing the distribution of the inter-bar currents along the ironcore is derived in the rotor reference frame by studying a small element of the rotorcircuit. In Figure 3.1, the bar current of order n in bar number k, at time t andaxial position x is denoted ibn,k(t, x). The inter-bar current distribution at thecorresponding time and position is referred to as Jtn,k(t, x), given in [A/m]. Atpoint A in the rotor circuit Kirchhoff’s current law gives:

ibn,k(t, x)− ibn,k(t, x− dx) + (Jtn,k−1(t, x) − Jtn,k(t, x)) dx = 0 (3.1)

In the rotor reference frame the fundamental component of these currents are vary-ing with slip frequency. Assuming sinusoidal rotor bar currents and using peak-value scaling, Equation 3.1 is written with complex notion as:

∂

∂xIbn,k(t, x) = Jtn,k(t, x) − Jtn,k−1(t, x) (3.2)

In the following text, bold symbols represent complex quantities. The phasor ofthe inter-bar currents at a certain time can be illustrated as in Figure 3.2, wherethe phase displacement of the inter-bar currents of order n between two adjacentslots, is determined from the number of poles p and the number of rotor bars Qr,i.e.

17

18 CHAPTER 3. MODEL FOR THE ANALYSIS OF INTER-BAR CURRENTS

ibn,k+1(t, x)ibn,k(t, x)

Jtn,k(t, x+ dx)

Jtn,k(t, x)

α

x+ dx

xA B

CD

k k + 1

Figure 3.1: Definition of bar- and inter-bar currents in a small element of the rotorcircuit.

Jtn,k−1(t, x) = Jtn,k(t, x)ejnpπQr (3.3)

Combining Equation 3.2 and Equation 3.3 gives the relation between the inter-barcurrent distribution and the bar current as;

Jtn,k(t, x) = − e−jnpπ

2Qr

2j sin(

npπ2Qr

)

∂

∂xIbn,k(t, x). (3.4)

Jtn,k(x)

Jtn,k−1(x)

n p πQr

Figure 3.2: Phase displacement between inter-bar currents.

3.1. ROTOR CIRCUIT TAKING INTER-BAR CURRENTS INTO ACCOUNT 19

The currents in the rotor element ABCD in Figure 3.1 are linked by both theflux from the stator current φsn, and the flux produced by the rotor current φrn.The voltage equation for this current loop is therefore given by;

0 = ZbnIbn,k(t, x)dx +RnJtn,k(t, x) −ZbnIbn,k+1(t, x)dx

−RnJtn,k(t, x+ dx) +∂

∂t

(

φsn(t, x) + φrn(t, x))

,(3.5)

where Zbn is the bar impedance per unit length and Rn is the inter-bar resistivity.

In Figure 3.3(a), Rn is defined as the resistance between two adjacent barsmultiplied with the stack length. The inter-bar current path can also be definedvia the rotor shaft, according to Figure 3.3(b). The induced voltages in the rotorbars are the same for these two cases. If the inter-bar current losses should be thesame for these two models, the following equation must be satisfied [14]:

Rn = 4Rtn sin2

(

npπ

2Qr

)

(3.6)

As the measurements of the inter-bar resistance resulted in the bar to shaft resistiv-ity Rtn, Equation 3.6 is used for the conversion into bar to bar resistivity. In ordersolve the voltage equation of the rotor circuit, the phase angle of the bar currentsare expressed in a similar way as for the inter-bar currents.

Ibn,k+1(t, x) = Ibn,k(t, x)e−jnpπQr . (3.7)

Combining the Equations; 3.4, 3.5, 3.6, and 3.7 gives the following differential

(a) Resistance between rotor bars.

RnRn

(b) Resistance between rotor bars and shaft.

RtnRtnRtn

Figure 3.3: Definitions of inter-bar resistances.


equation for the rotor bar currents:

0 =ZbnIbn,k(t, x) −Rtn∂2

∂x2Ibn,k(t, x)

+1

dx

ejnpπ2Qr

2j sin(npπ2Qr)

∂

∂t

(

φsn(t, x) + φrn(t, x))

(3.8)

The solution gives the axial variation of the bar currents. The inter-bar currentsfor the solution of Equation 3.8 can then easily be obtained through Equation 3.4.Required are the stator and rotor fluxes linked by the rotor element ABCD.

3.2 Stator flux linkage

In this section the stator flux linked by the rotor element is calculated in therotor reference frame. The stator flux of order n is expressed as a function of thefundamental stator current.

3.2.1 Airgap flux density due to stator current

The time varying current in the distributed three phase stator winding results inin a rotating MMF-wave in the airgap. The magnitude of the resulting flux densityis depending on the airgap permeance. Due to the stator and rotor slotting therotating MMF-wave sees a permeance that is varying in both time and space.

The airgap flux density is conventionally determined as the product of the MMF-wave and the permeance function. This method is described further among othersin [7, 15]. However, in this thesis the airgap flux density caused by the stator cur-rent is calculated by the use of a different method. The results are verified withthe finite element method (FEM).

The method used is based on the fourier analysis of a simplified flux densitydistribution along the airgap, resulting from the current in one phase. The totalairgap flux density is then obtained by superposition of the other two phases. Twodifferent shapes of possible flux density distributions in the air gap are studied,referred to as Model A and Model B. It turns out that the two models give verysimilar results.First the calculation is simplified by introducing the following assumptions:

• The stator and rotor iron is assumed to have infinite permeability.

• The rotor slotting do not contribute to the permeance variation along theairgap circumference.

• The current in phase a is assumed to have the following variation in time:

ia = i cos(ω t). (3.9)

3.2. STATOR FLUX LINKAGE 21

Model A

The distribution of the airgap flux density in space due to the current in phase ais approximated according to Figure 3.4. By applying Amperes law along the lineC at time t = 0 the following is obtained;

∮

C

−→H · −→dl =

qNs1

Csi (3.10)

which gives:

Bδ =µ0

2δ

qNs1

Csi (3.11)

Where q is the number of slots per pole per phase, Ns1 is the number of conductorsper slot and Cs is the connection factor. It is assumed that the airgap flux densitycreated by the current in one phase is zero beneath the stator slot openings definedby the distance d in Figure 3.4, given in [16] as:

d =2π

Qs

(

1− 1

Cfs

)

. (3.12)

Where Qs is the number of stator slots and Cfs is the Carter factor due to the statorslotting. Based on this distribution of the flux density, the harmonic componentsare calculated by means of Fourier analysis. The amplitude of these harmoniccomponents of order n are obtained as:

Bnδ =2

nπ

µ0

δ

qNs1

Cs

sin(

npπ2CfsQs

)

sin(

npπ2Qs

) i (3.13)

Bδ

C

d

θ1

stator

rotor

Figure 3.4: Airgap flux density due to current in phase a, Model A.


With the contribution from the other two phases, the peak value of the total airgapflux density becomes;

Bnm =3

2BnδKrn (3.14)

which is equal to:

Bnm =3

nπ

µ0

δ

qNs1

CsKrn

sin(

npπ2CfsQs

)

sin(

npπ2Qs

) i (3.15)

Where Krn is the winding factor for the wave of order n. With known Fouriercoefficients the resulting airgap flux density can be expressed in both time andspace as:

Bnm(t, θ1) =

∞∑

n=1

Bnm cos(

ωt− np2θ1

)

(3.16)

Where θ1 is the mechanical angular position given in stator coordinates.

Model B

The distribution of the airgap flux density in space due to the current in phase ais approximated according to Figure 3.5. The resulting airgap flux density of ordern is then determined by Fourier analysis to [17];

Bnm =3

nπ

µ0

δ

qNs1

CsKrnKfni (3.17)

Bδ

σbsys2β

θ1

stator

rotor

Figure 3.5: Airgap flux density due to current in phase a, Model B.


where the form factor is given by

Kfn = 1− 2 β + 2βsin(

ksnπ2Qs

p2

)

ksnπ2Qs

p2

sin(

nπQs

p2

(

1− ks2)

)

sin(

nπQs

p2

) . (3.18)

Where ks is defined as:

ks =σbsys

τs. (3.19)

The distance bsys is the slot opening and τs is the slot pitch. The coefficients σ andβ are geometry dependent coefficients which can be found in [15, 17].

Verification with FEM

The two models are slightly different when it comes to the modeling of the perme-ance change beneath a stator slot. This will affect the harmonic components of theairgap flux density. It is believed that Model B is closer to the actual flux densitydistribution in the machine.

In order to verify the analytical models, a comparison is made with the resultsfrom a FEM simulation. The FEM results are obtained from a no-load test per-formed with the simulation software Flux2d. As the electrical steel is assumed tohave infinite permeability in the analytical models, the same assumption is adaptedto the FEM model, i.e. saturation is not taken into account. The stator currentobtained from this simulation is used to calculate the corresponding flux densitieswith the analytical models.

θ1 [mek o]

Air

gap

flux

den

sity

[T]

0 20 40 60 80 100 120 140 160 180-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

(a) FEM simulated airgap flux density at no-load.

Harmonic order

Air

gap

flux

den

sity

[T]

Model AModel BFlux2d

1 5 7 11 13 17 190

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(b) Harmonic spectrum of the airgap flux den-sity at no load.

Figure 3.6: Comparison between analytical and FEM-simulated airgap flux densityat no-load.


The FEM simulated airgap flux density distribution at no load is shown in Fig-ure 3.6(a). From this simulation it is obvious that the stator slotting has a largeimpact on the flux density distribution, and hence, also on the harmonic compo-nents describing this distribution. It can be seen that the rotor slotting affects theairgap flux density, but to a lesser extent than the stator slotting. In Figure 3.6(b)the harmonic spectrum is compared with the corresponding analytical values. Inthe analysis, harmonic components up to the order of the first pair of slot harmonicsare considered.

The fundamental component calculated with Model A is 3.4 % higher thanthe FEM simulated value, the corresponding value for Model B is 2.4 %. Apartfrom the leakage flux, the main difference between the models is believed to bedescended from the modeling of the airgap permeance. The overall harmonic spec-trum obtained from the two analytical models, for the harmonics considered, showsacceptable correlation with the FEM results. Based on the values of the fundamen-tal components, Model B will be used for the calculation of the airgap flux densitycreated by the stator current.

3.2.2 Stator flux linked by the rotor circuit

In the previous section the airgap flux density produced by the stator current wasderived in stator coordinates. Based on these results this section presents the cal-culation of the stator flux linked by the rotor circuit.

As the calculations are performed in the rotor circuit, the airgap flux densityproduced by the stator is expressed in rotor coordinates. This requires informationregarding the position of the rotor relative to the stator flux waves. Therefore, itis assumed that a wave of order n at the time t = 0, has a position relative to therotor as shown in Figure 3.7. Based on these conditions, the relation between thestator and the rotor positions is given by:

θ1 = θ2 +2

p(1− s1)ωt (3.20)

The airgap flux density of order n seen by the rotor can then be expressed withrotor coordinates in time and space as:

Brn = Bnm cos(

snωt− np

2θ2

)

(3.21)

Where sn is the slip of a wave of order n given by;

sn = 1− n(1− s1), (3.22)

n should be positive or negative according to the direction of rotation of the wave.The flux linked by the element ABCD in Figure 3.7 is equal to;


2πQr

(k − 1) αxℓ

αxℓ

2πQrk

ǫ

x

x+ dx

θ2

A B

CD

Figure 3.7: Position of a wave of order n at time t = 0 in the rotor reference frame.

φsn(x, t) =

∫ 2πQrk+αx

ℓ−ǫ2

2πQr

(k−1)+αxℓ

+ ǫ2

rdxBnm cos(

snωt− np

2θ2

)

dθ2 (3.23)

where ǫ is given by [16].

ǫ =2π

Qr

(

1− 1

Cfr

)

(3.24)

The rotor skewing is taken into account by introducing the mechanical skewingangle α, given by:


α =2π

Qsαs (3.25)

Where αs is the skewing in number of stator slots. This gives the linked stator fluxexpressed with complex notation as:

φsn(x, t) =2rdx

np2Bnm sin

(

npπ

2CfrQr

)

· ej(

snωt−np

2

(

π(2k−1)Qr

+αxℓ

))

(3.26)

3.3 Rotor flux linkage

The stator flux linked by the rotor circuit induce currents in the rotor bars, themagnitude and frequency of these currents are depending on the rotor speed. Asa result, a rotor flux is created which counteracts the stator flux. In this section,the rotor flux linked by the small rotor element is derived as a function of the barcurrent of order n.

3.3.1 Airgap flux density due to rotor current

The calculation is simplified by introducing the following assumptions:

• The stator and rotor iron is assumed to have infinite permeability.

• The rotor slotting do not contribute to the permeance variation in the airgap.

• The rotor bar currents are assumed to vary sinusoidally in time.

The current of order n in rotor bar k, ibn,k, and the corresponding MMF, Mn,k, isshown in Figure 3.8. Applying amperes law along the line C gives:

Mn,k(x, t) −Mn,k−1(x, t) = ibn,k(x, t) (3.27)

The phase displacement between the MMF over two adjacent rotor teeth is givenby:

Mn,k−1(x, t) = Mn,k(x, t)ejnpπ

Qr (3.28)

The rotor bar current ibn,k is replaced with the complex bar current Ibn,k introducedto define the rotor circuit equation in Section 3.1. This gives the following expressionfor the complex airgap MMF created by the rotor current.

Mn,k(x, t) = − e−jnpπ

2Qr

2j sin(

npπ2Qr

)Ibn,k(x)ejsnωt (3.29)

3.3. ROTOR FLUX LINKAGE 27

r

xθ2

ibn,k+1ibn,kibn,k−1

Mn,k−1 Mn,k

C

A B

CD

Figure 3.8: Current in bar number k and the corresponding MMF in the airgap.

The corresponding airgap flux density at the point θ1, is obtained by multiplyingthe MMF with the airgap magnetic permeance per unit area.

Bn,k(x, θ1, t) = Mn,k(x, t)Λ(θ1) (3.30)

According to the previously stated assumption the airgap permeance variationalong the circumference is only created by the stator slotting, i.e. the rotor is as-sumed to have closed slots. This simplifies the forthcoming analysis considerably.As the effect of the rotor is neglected, the airgap permeance function can be ex-pressed independent of time in stator coordinates. In analogy with the calculationof the airgap flux density due to the stator current in Section 3.2.1, the airgappermeance function is approximated, Model A is used for simplicity reasons. Bythe use of Fourier analysis, the distributed permeance defined by Figure 3.9, canbe expressed in stator coordinates as;

Λ(θ1) =µ0

δCfs+

2µ0

πδ

∞∑

γ=1

sin(

γπCfs

)

γcos (γQsθ1) (3.31)

where γ is the order of the permeance harmonic considered. In this thesis, onlythe permeance harmonics of the first order are considered. With complex notation,this simplifies Equation 3.31 to;

Λ(θ1) = Λ0 + Λs1ejγQsθ1 (3.32)

whereΛ0 =

µ0

δCfs(3.33)


µ0

δ

Λ

dτs

θ1

Figure 3.9: Permeance variation along the airgap circumference as defined by ModelA.

and

Λs1 =2µ0

γπδsin

(

γπ

Cfs

)

. (3.34)

The airgap flux density is now calculated as the product of the MMF and thepermeance function. Note that the permeance function has to be expressed inrotor coordinates by the use of Equation 3.20.

Bn,k(x, θ2, t) = Λ(θ2, t)

∞∑

n=1

Mn,k(x, t) (3.35)

From this definition a series of harmonic components is obtained, they are treatedseparately in the following.

Bn,k(x, θ2, t) = (M1,k + M5,k + M7,k ...)(

Λ0 + Λs1ejQsγ(θ2+ 2

p(1−s1)ωt)

)

(3.36)

3.3.2 Flux caused by the phase belt harmonics

Phase belt harmonics are due to the concentration of the MMF in slots, in a threephase machine they fulfil the condition:

• n = 1± 6k

The dominating terms of the airgap flux densities caused by rotor currents in-duced by the phase belt harmonics are those acting through the average permeanceΛ0. The interaction with the harmonic permeance Λs is neglected in this study.Thus, the airgap flux densities due to these harmonics are simply calculated as:

Bn,k(x, t) = Mn,k(x, t)Λ0 (3.37)

3.3. ROTOR FLUX LINKAGE 29

The corresponding flux is obtained by integrating the flux density over the areaABCD in Figure 3.7.

φrn(x, t) = rdx

∫ 2πQrk+αx

ℓ−ǫ2

2πQr

(k−1)+αxℓ

+ ǫ2

Bn,k(x, t)dθ2 (3.38)

Which gives the same result as in [17]:

φrn(x, t) = jrπdx

CfrCfsQr

µ0

δIbn,k(x)

e−jnpπ2Qr

sin(

npπ2Qr

)ejsnωt (3.39)

3.3.3 Flux caused by slot harmonics

The slot harmonics are partly made up of the MMF slot harmonics of order:

• n = 1 + γ 2Qsp

These rotor fields are caused by the harmonic rotor current of order n = 1 + γ 2Qsp

,acting through the average permeance. The corresponding flux densities are deter-mined by Equation 3.37.

The slot harmonics are also generated through the interaction with the perme-ance harmonics of order γ, having the same number of poles. These harmonicsarise from the interaction between the fundamental rotor field and the harmonicpermeance. In the rotor reference frame this is seen as a wave, that can be sepa-rated into a wave rotating in the positive direction and one in the negative directionrespectively, i.e. γ = ±1. Each of the waves produces a flux density given by:

Bγ,k(x, θ2, t) = M1,k(x, t)Λs12ejγQs(θ2+ 2

p(1−s1)ωt) (3.40)

The backward rotating permeance field corresponding to γ = −1, produces to-gether with the fundamental MMF, the same rotor frequency as the MMF harmonicof order n = 1 + 2Qs

p, rotating in the forward direction. Therefore, the resulting

rotor flux of order n, linked by the element ABCD in Figure 3.7 is calculated as:

φrn(x, t) = rdx

∫ 2πQrk+αx

ℓ−ǫ2

2πQr

(k−1)+αxℓ

+ ǫ2

(Bn,k(x, t) + B−γ,k(x, θ2, t))dθ2 (3.41)

Which gives similar results as in [17]:

φrn =jrdxµ0

δejsnωt

( π

QrCfsCfr

In,k(x)e−j

npπ

2Qr

sin(

npπ2Qr

) +

1

γπQssin( γπ

Cfs

)I1,k(x)e−j

pπ2Qr

sin(

pπ2Qr

) sin( Qsπ

QrCfr

)

e−jγ(QsQr

2π(k− 12 )+Qs

αxℓ

))

(3.42)


3.4 General set of equations

For a given stator current the rotor Equation 3.8 can be solved by inserting Equa-tions; 3.39, 3.42 and 3.26. The solution gives the corresponding rotor bar currentas a function of the axial position x. In the following, the rotor current of order nin bar number k = 1, is denoted Ibn. The solution is depending on the order of theconsidered current.

In case of a phase-belt harmonic the differential equation for the rotor circuitbecomes:

• If n 6= 1 + γ 2Qsp

, γ = ±1

Z2nIbn(x)− ρtn∂2

∂x2Ibn(x) + En0e

−jnpαx

2ℓ = 0 (3.43)

Where:

En0 =2rℓBnmω

np

sin(

npπ2CfrQr

)

sin(

npπ2Qr

) (3.44)

Z2n =ℓ

snZbn + jXn (3.45)

Xn =2πrℓ

QrCfsCfr

µ0

δ

ω

4 sin2(

npπ2Qr

) (3.46)

ρtn =ℓ

snRtn (3.47)

If the considered harmonic is a slot harmonic of the first order, the correspondingequation becomes:

• If n = 1 + γ 2Qsp

, γ = ±1

Z2nIbn(x)− ρtn∂2

∂x2Ibn(x) + En0e

−jnpαx

2ℓ + jX1Ib1(x)e−jγQsαxℓ = 0 (3.48)

Where:

X1 =rℓω

2γπQs

µ0

δ

sin(

γπCfs

)

sin(

πQsCfrQr

)

sin(

npπ2Qr

)

sin(

pπ2Qr

) (3.49)

Due to the interaction with the permeance harmonics, the rotor bar current of or-der n is now also a function of the fundamental bar current Ib1, obtained from the

3.5. EFFECTS OF A FINITE INTER-BAR RESISTANCE ON ROTOR CURRENT

DISTRIBUTION 31

solution of Equation 3.43.

The boundary conditions for these two differential equations are obtained bystudying the rotor circuit at the point where it connects to the adjoining short-circuit ring, as in Figure 3.10(a). At this point the ring voltage and the bar to barvoltage are equal, i.e.

Ian,kZan = Jtn,kRn (3.50)

Where Zan is the impedance of the short-circuit ring segment with respect to thefrequency of the nth harmonic. Using the phasor relation between the bar- andthe short-circuit ring currents given in Figure 3.10(b), together with Equation 3.4,gives the boundary conditions as:

∂∂x

Ibn(

− ℓ2)

− Zan

RnIbn(

− ℓ2)

= 0∂∂x

Ibn(

ℓ2

)

+ Zan

RnIbn(

ℓ2

)

= 0(3.51)

From this equation it is obvious that the inter-bar current density at the rotorboundary is determined by the impedance of the short-circuit ring, regardless therotor skew. Thereby, it can be concluded that inter-bar currents are present alsoin unskewed machines, unless the impedance of the short-circuit ring is zero.

Rn

Zan

Ibn,k+1Ibn,k

Ian,kIan,k−1

Jtn,kJtn,k−1 − ℓ2

x

(a) Rotor circuit.

Ian,k−1

Ian,k

Ibn,k

npπQr

(b) Ring- and bar current phasors.

Figure 3.10: Rotor currents at the boundary x = − ℓ2 .

3.5 Effects of a finite inter-bar resistance on rotor current

distribution

According to the previous section, a finite inter-bar resistance introduces the inter-bar current density Jtn,k. With the contribution of the inter-bar current density inthe adjacent rotor tooth Jtn,k−1, a resulting bar current is obtained dIbn,k.

dIbn,k(x) = (Jtn,k(x) − Jtn,k−1(x)) dx (3.52)


This current defines the change in the bar current when moving a small distancedx in the positive x-direction, according to Figure 3.11. In this figure, the airgapflux density caused by the stator current Bnm is used as reference phasor.

From this figure one can see that the inter-bar currents causes a non uniformdistribution of the rotor bar currents. This effect is enhanced by the rotor skew, butthe dependency of the size of the inter-bar resistance, or the required bar insulationto cancel out this effect, is not that obvious.With the intention to get a better understanding of this effect, a case study isperformed on an 11 kW 4-pole machine. The machine having 36 stator slots anda full pitch single layer winding is equipped with a cast aluminium rotor with 44rotor slots, referred to as Motor A. The machine is simulated both with and withoutrotor skew. The rotor bar- and inter-bar currents are then studied along the coreat different values of inter-bar resistance.

Ibn(x)

Ibn(x+ dx)dIbn

βn

Bnm

Brn(x)

γn

Figure 3.11: Change in rotor bar current due to the interaction with inter-barcurrents.

3.5.1 Rotor without skew

The inter-bar current density in a rotor without skewing is focused towards therotor ends, and its magnitude is directly determined by the ratio of short-circuitring impedance to inter-bar impedance. This analysis is somewhat simplified by,as stated in Chapter 2, assuming that the inter-bar impedance is purely resistive.


DISTRIBUTION 33

The magnitude of the fundamental locked-rotor inter-bar current density alongthe rotor core is shown in Figure 3.12(a). Due to the low impedance of the short-circuit ring, it requires a quite low inter-bar resistivity to introduce fundamentalinter-bar currents. These currents, having their maximum values at the rotor endsdecreases towards zero in the middle of the rotor.

Figure 3.12(b) shows the angle β1 defined in Figure 3.11, giving the phase angleof the inter-bar currents flowing along the rotor bar, contributing to the bar current.At the rotor ends, this angle is set by the voltage over the corresponding short-circuit ring segment. In the middle of the rotor, where the inter-bar currents arezero, a phase shift of 180 degrees occurs. In other words; the voltage over thering segments at each end of the rotor introduces two circumferential current pathsthrough the rotor teeth, opposing each other. In a symmetrical rotor, assuming anequally distributed inter-bar resistivity, these currents cancel each other out in themiddle.

As a result the bar current is reduced towards the ends of the rotor. This canbe seen in Figure 3.13(a) which shows the fundamental locked rotor bar current.But the effect is of minor importance, as well as for the phase of the correspondingcurrent phasor shown in Figure 3.13(b).

In the following, the variation of the inter-bar currents with rotor speed, andthe influence on the rotor currents of higher order is studied. The analysis is per-formed with the inter-bar resistivity Rtn = 5 ·10−5 Ωm. Figures 3.14(a) and 3.14(b)show the fundamental bar and inter-bar current as a function of both slip and axialposition in the bar. It can be seen that the inter-bar currents have their maximum

Axial position x [cm]

Inte

r-bar

curr

ent

den

sity[

A mm

] Rtn = 5 · 10−4 Ωm

Rtn = 5 · 10−5 Ωm

Rtn = 5 · 10−6 Ωm

-10 -8 -6 -4 -2 0 2 4 6 8 100

5

10

15

20

25

30

35

40

45

50

(a) Inter-bar current density.


β1[

]

Rtn = 5 · 10−4 Ωm

Rtn = 5 · 10−5 Ωm

-10 -8 -6 -4 -2 0 2 4 6 8 1010-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

8080

(b) Resulting fundamental inter-bar current an-gle β1.

Figure 3.12: Magnitude of the fundamental locked rotor inter-bar current densityand the resulting angle β1, for Motor A with unskewed rotor.



Rot

orbar

curr

ent

[kA

]

Rtn = 5 · 10−4 Ωm

Rtn = 5 · 10−5 Ωm

Rtn = 5 · 10−6 Ωm

-10 -8 -6 -4 -2 0 2 4 6 8 103.44

3.46

3.48

3.50

3.52

3.54

3.56

(a) Fundamental bar current.


Phas

e[

]

Rtn = 5 · 10−4 Ωm

Rtn = 5 · 10−5 Ωm

Rtn = 5 · 10−6 Ωm

-10 -8 -6 -4 -2 0 2 4 6 8 1089.4

89.6

89.8

90

90.2

90.4

90.6

90.8

91

91.2

(b) Fundamental bar current angle γ1.

Figure 3.13: Magnitude of fundamental locked rotor bar current and the corre-sponding angle γ1, for Motor A with unskewed rotor.

value at the speed corresponding to the largest slip. However, their influence onthe fundamental rotor current is of minor importance.

The cases when the currents are caused by higher order space harmonics aredifferent. As these fields have low synchronous speeds, the corresponding rotorcurrents create torques that counteract the fundamental torque at nominal speed.The rotor currents of order higher than the fundamental are therefore sources ofstray losses. Figure 3.14(c) and Figure 3.14(e) shows the rotor currents caused bythe first pair of slot harmonics, having the order n = 1 − 2Qs

pand n = 1 + 2Qs

p,

respectively.These currents, caused by the slot harmonics, are large in the unskewed rotor.

Especially at low speeds when the high fundamental current creates large slot MMFharmonics. Thus, large asynchronous torques are expected during a start.

Furthermore, at nominal speed the magnitude of these harmonic currents arestill quite large. This is also true at no-load, when the magnitude of the slotspace harmonics are determined by the fundamental no-load current. The resultingcage losses can form a considerable part of the no-load losses [10]. This effect isenhanced by main flux saturation increasing the fundamental current, which hasbeen neglected in this thesis.

Rotor skewing by one stator slot pitch is a common practice to reduce the rotorcurrents caused by slot harmonics. But this promotes the flow of inter-bar currents.


DISTRIBUTION 35

Slip

Rot

orbar

curr

ent

[kA

]

Axial position x [cm] 00.5

11.5

-10-5

05

100

1.0

2.0

3.0

4.0

(a) Rotor bar current, n = 1.

Slip

Inte

r-bar

curr

ent

den

sity

[A mm

]


11.5

-10-5

05

100

2

4

6

8

(b) Inter-bar current density, n = 1.

Slip

Rot

orbar

curr

ent

[A]


11.5

-10-5

05

100

100

200

300

400

500

(c) Rotor bar current, n = 1− 2Qsp

.

Slip

Inte

r-bar

curr

ent

den

sity

[A mm

]


11.5

-10-5

05

100

0.02

0.04

0.06

0.08

(d) Inter-bar current density, n = 1− 2Qsp

.

Slip

Rot

orbar

curr

ent

[A]


11.5

-10-5

05

100

50

100

150

200

250

300

(e) Rotor bar current, n = 1 + 2Qsp

.

Slip

Inte

r-bar

curr

ent

den

sity

[A mm

]


11.5

-10-5

05

100

0.05

0.1

0.15

0.2

(f) Inter-bar current density, n = 1 + 2Qsp

.

Figure 3.14: Magnitude of the currents in the unskewed rotor caused by the fun-damental and the first pair of slot space harmonics when Rtn = 5 · 10−5 Ωm.


3.5.2 Rotor with skew

The analysis continues with the same motor as in the previous section but witha rotor skew of one stator slot pitch, i.e. αs = 1. When the rotor is skewedthe voltages induced in the skewed conductors are reduced in comparison to theunskewed conductors. The ratio of these voltages defines the skewing factor for thenth rotor voltage as:

ksk =sin(

npπαs2Qs

)

npπαs2Qs

≤ 1 (3.53)

This expression becomes zero if the skewing in number of stator slots is equal to:

αs0 =Qsnp2

(3.54)

For the first pair of slot harmonics ksk is close to zero when αs0 is close to one. Asa result, when skewing insulated rotor bars by one stator slot pitch, a large reduc-tion of the corresponding rotor currents can be expected, improving the machineperformance, both in terms of reduced asynchronous torques and rated efficiency.However, in casted rotors the rotor skewing might have an opposite effect.

Figure 3.15 shows the fundamental locked- rotor inter-bar current density andthe resulting angle β1 for different values of inter-bar resistivity. The inter-barcurrent densities at the rotor ends are approximately the same as for the unskewedrotor. The important difference is that the inter-bar currents now are focusedtowards the middle of the rotor, somewhat depending on the inter-bar resistivity.By studying the angle β1, one can expect a continuous increasing bar current in


Inte

r-bar

curr

ent

den

sity[

A mm

]

Rtn = 5 · 10−2 Ωm

Rtn = 5 · 10−3 Ωm

Rtn = 5 · 10−4 Ωm

Rtn = 5 · 10−5 Ωm

Rtn = 5 · 10−6 Ωm

-10 -8 -6 -4 -2 0 2 4 6 8 100

5

10

15

20

25

30

35

40

45

(a) Inter-bar current density.


Phas

e[

]

Rtn = 5 · 10−4 Ωm

Rtn = 5 · 10−5 Ωm

Rtn = 5 · 10−6 Ωm

-10 -8 -6 -4 -2 0 2 4 6 8 10-120

-100

-80

-60

-40

-20

0

20

40

60

80

(b) Angle β1.

Figure 3.15: Magnitude of the fundamental locked-rotor inter-bar current densityand the resulting angle β1 for Motor A with skewed rotor.


DISTRIBUTION 37


Rot

orbar

curr

ent

[kA

]

Rtn = 5 · 10−2 Ωm

Rtn = 5 · 10−3 Ωm

Rtn = 5 · 10−4 Ωm

Rtn = 5 · 10−5 Ωm

Rtn = 5 · 10−6 Ωm

-10 -8 -6 -4 -2 0 2 4 6 8 10

0.8

3.05

3.10

3.15

3.20

3.25

3.30

3.35

3.40

3.45

3.50

(a) Rotor bar current.

Axial position x [cm]P

has

e[

]

Rtn = 5 · 10−4 Ωm

Rtn = 5 · 10−5 Ωm

Rtn = 5 · 10−6 Ωm

-10 -8 -6 -4 -2 0 2 4 6 8 1080

82

84

86

88

90

92

94

96

98

100

(b) Bar current angle γ1.

Figure 3.16: Magnitude of fundamental locked-rotor bar current and the corre-sponding angle γ1, for Motor A with skewed rotor.

the positive x-direction, except for very low values of inter-bar resistivity. This canbe seen in Figure 3.16(a), showing the fundamental rotor bar currents.

An interesting result can be found in Figure 3.16(b), showing the angle of thefundamental bar current phasor with respect to the stator flux density. For lowvalues of inter-bar resistivity the phase change between the rotor ends is equal tothe electrical skewing angle, indicating that the skewing is ineffective. These resultsare consistent with what was found in [18].

Figure 3.17 shows the rotor currents caused by the fundamental and the firstpair of slot harmonics as a fuction of speed, when the inter-bar resistivity Rtn =5 · 10−2 Ωm. The overall inter-bar current density is low, the value of the inter-bar resistivity is thereby to be considered as high. In comparison to the unskewedcase the currents caused by the slot harmonics are drastically reduced. The corre-sponding asynchronous torques will vanish and an improved motor efficiency can beexpected. With high values of inter-bar resistivity the theory, assuming negligibleinter-bar current flow, holds. i.e. the skewing is effective.

Measurements have shown that the inter-bar resistivity in cast rotors is muchlower than the value referred to as high in the previous case. In Figure 3.18 thecorresponding results are shown with an inter-bar resistivity of Rtn = 5 · 10−5 Ωm.Inter-bar currents now appears for all the considered harmonics. High fundamentalinter-bar currents at start indicate an increased locked rotor torque. But mostnotable is the large increase of inter-bar currents caused by the slot harmonics,resulting in a huge increase and a distortion of the corresponding bar currents. Inthis case the skewing is not effective. As the inter-bar resistivity is much higherthan the bar resistivity, large additional losses are created in the bar to core region.Large asynchronous torques and increased stray losses are expected.


Slip

Rot

orbar

curr

ent

[kA

]


11.5

-10-5

05

100

1.0

2.0

3.0

4.0


Slip

Inte

r-bar

curr

ent

den

sity

[A mm

]


11.5

-10-5

05

100

0.05

0.1

0.15

0.2


Slip

Rot

orbar

curr

ent

[A]


11.5

-10-5

05

100

10

20

30

40


.

Slip

Inte

r-bar

curr

ent

den

sity

[A mm

]


11.5

-10-5

05

100

0.005

0.01

0.015

0.02


.

Slip

Rot

orbar

curr

ent

[A]


11.5

-10-5

05

100

2

4

6

8


.

Slip

Inte

r-bar

curr

ent

den

sity

[A mm

]


11.5

-10-5

05

100

0.005

0.01

0.015

0.02

0.025

0.03


.

Figure 3.17: Magnitude of the currents in the rotor skewed by one stator slotpitch, caused by the fundamental and the first pair of slot space harmonics whenRtn = 5 · 10−2 Ωm.


DISTRIBUTION 39

Slip

Rot

orbar

curr

ent

[kA

]


11.5

-10-5

05

100

1.0

2.0

3.0

4.0


Slip

Inte

r-bar

curr

ent

den

sity

[A mm

]


11.5

-10-5

05

100

5

10

15

20

25

30


Slip

Rot

orbar

curr

ent

[A]


11.5

-10-5

05

100

100

200

300

400

500


.

Slip

Inte

r-bar

curr

ent

den

sity

[A mm

]


11.5

-10-5

05

100

2

4

6

8


.

Slip

Rot

orbar

curr

ent

[A]


11.5

-10-5

05

100

100

200

300

400


.

Slip

Inte

r-bar

curr

ent

den

sity

[A mm

]


11.5

-10-5

05

100

2

4

6

8

10


.

Figure 3.18: Magnitude of the currents in the rotor skewed by one stator slotpitch, caused by the fundamental and the first pair of slot space harmonics whenRtn = 5 · 10−5 Ωm.


3.6 Summary

A model to calculate the inter-bar currents in cage induction motors has been de-rived. This model requires the stator current and the inter-bar resistivity as inputparameters. Simulations have shown that the inter-bar current density is increasingrapidly with decreasing inter-bar resistivity. This affects the bar current distribu-tion and phase angle.

In unskewed rotors the inter-bar currents are focused towards the rotor ends, andtheir magnitude is directly determined by the ratio of short-circuit ring impedanceto inter-bar impedance. Unless this ratio is very high, inter-bar currents are smallin unskewed rotors and can therefore be neglected.

When the rotor is skewed by one stator slot pitch inter-bar currents increase.This effect is most significant for rotor currents that are caused by the slot spaceharmonics. For the studied machine, the inter-bar currents are reduced to a negli-gible level when the bar to shaft resistivity is larger than 5 · 10−2Ωm. In this caserotor skewing becomes effective.

Chapter 4

Effects during a direct-on-line start

When a cage induction motor is started directly against the grid, the presence ofskin effect and leakage path saturation has large impact on the machine perfor-mance. The high frequency of the fundamental slot leakage flux gives rise to highcurrent density in the upper parts of the rotor bars. As a result, the effective barresistance is increased and the effective bar inductance is decreased. The start-ing currents will generate large differential and slot leakage fluxes. This saturatesthe differential and slot leakage flux paths, which will reduce the correspondingleakage inductances. It also creates additional iron losses in the stator and rotorteeth. These factors have to be taken into to account when calculating the startingperformance of the machine.

4.1 Skin effect in the rotor bars

A good starting performance usually implies high starting torque and low startingcurrent, requiring a high rotor resistance and a high rotor leakage inductance. Onthe other hand, to obtain a high full load efficiency and to ensure a reasonablemargin of overload capability, low rotor resistance and leakage inductance is prefer-able. This trade-off between starting performance and rated efficiency is a classicalproblem for the induction motor designer [19].

However, during an induction motor start when the rotor frequency equals themains frequency, advantage is taken of the skin effect. In the deep bar rotor shownin Figure 4.1(a), the self inductance is highest at the bottom of the bar. Duringa start when the rotor frequency is high, the rotor current is focused towards theupper parts of the bar. As a result, the rotor bar losses are increasing and a largertorque is produced. This is normally modeled by an increased equivalent bar re-sistance and a decreased bar inductance. When the rotor has accelerated up tonominal speed the rotor frequency is very low, resulting in an evenly distributedbar current. Utilizing the whole rotor bar area ensures a low slip and therefore

41

42 CHAPTER 4. EFFECTS DURING A DIRECT-ON-LINE START

(a) Deep bar. (b) Double cage. (c) Bar in the studiedmachine.

Figure 4.1: Different types of rotor bars.

reduced losses. Analytical expressions for the frequency dependent impedances ofdifferent deep bar rotors have been derived by Liwschitz-Garik in [20].

With the introduction of the double cage rotor shown in Figure 4.1(b), theAC-resistance of the bar can be increased even further. The idea is to create twoparallel current paths in the rotor bar, one with low resistance and high inductanceto conduct most of the current at rated speed, and one with high resistance andlow inductance which will conduct the largest part of the current during start. Ananalytical method for performance calculations of multiple squirrel cage rotors hasbeen presented by Alger in [19]. Finite element modeling of a double cage rotor hasshown good correlation with analytical methods [21]. In this work Williamson andGersh highlight the effects of magnetic saturation, showing that the phenomena hasa considerable effect on the leakage inductance, while the effects on the frequencydependent resistance is of minor importance.

The drawback with the analytical models is that they are restricted to a cer-tain slot geometry, which is often quite simple. In the case of more complicatedgeometries numerical methods are preferable.

4.1.1 Numerical method used to account for skin effect

In this thesis the skin effect is taken into account by the use of a one-dimensionalnumerical model described in [22]. Besides the accuracy, the main advantage ofusing this method is that it can easily be adapted for different slot geometries. Thedrawback might be that it requires a software with a numerical solver. However,the flexibility of the method definitely makes it worth the effort. In the followinga brief description of the model is presented, giving the assumptions used and themain equations describing the theory.

4.1. SKIN EFFECT IN THE ROTOR BARS 43

The following assumptions are made:

• The magnetic field lines are crossing the rotor bar perpendicular to the slotsides.

• The rotor iron reluctance is negligible.

• The current density is axial-symmetrical in the slot.

• The rotor bar current density J varies sinusoidally in time with angular fre-quency ω.

The bar is modeled with a large number of rectangular segments of equal height.Figure 4.2 shows the dimensions of the vth segment. Within this section Maxwell’sinduction law defines the induced electric field ~Ev, produced by the flux densityfield ~Bv.

∇× ~Ev = −∂~Bv

∂t(4.1)

With rectangular slot sections defined in a cartesian coordinate system according toFigure 4.2, and with a bar current defined in the positive y-direction, the inductionlaw can be rewritten as:

−ex∂Ey(z)v∂z

= −ex∂Bx(z)v∂t

(4.2)

With section resistivity ρv and a current density ~Jv varying sinusoidally in time,the first equation within the system can be defined as:

ρvdJy(z)vdz

= jωµ0Hx(z)v (4.3)

Neglecting the displacement current, Ampere’s circuit law states that the mag-netic field ~Hv is generated by an electrical current according to:

∇× ~Hv = ~Jv (4.4)

Assuming that the bar width is equal to the slot width, which is reasonable in acasted rotor, the following is obtained:

ey∂Hx(z)v∂z

= eyJy (4.5)

Which defines the second and the last equation within the system as:

dHx(z)vdz

= Jy (4.6)


z

x

y

bv

hv

Figure 4.2: The vth section of the rotor slot.

If section i carries the current Ii, the first boundary condition can be obtained fromAmpere’s law.

bvHx(z = hv)v = bv+1Hx(z = 0)v+1 =

v∑

i=0

Ii (4.7)

As the sections are short-circuited at the bar ends the voltages over all sections areequal, giving the final boundary condition as:

ρv+1Jy(z = 0)v+1 = ρvJy(z = hv)v (4.8)

These equations are part of the iterative process described in Figure 4.3. Fora known bar current Ibar with the angular frequency ω, this method calculatesthe bar current distribution and phase. The corresponding impedance correctionfactors are then obtained from:

kr =PAC

PDC(4.9)

kx =WAC

WDC(4.10)

Where the active power P and the stored magnetic energyW are simply calculatedas:

P =

nsec∑

v=0

|Iv|2 ρvℓ

bvhv(4.11)

W =1

2

nsec∑

v=0

µ0ℓhv

bv

∣

∣

∣

∣

∣

v∑

i=0

Ii

∣

∣

∣

∣

∣

2

(4.12)


Iv=0 = Ibarnsec

Calculate Jv+1

Error =|∑nsec

i=0Ii|−IbarIbar

OK?

Iv=0 = Iv=0 (1− Error) Calculate kr and kx

For: nsec

NO YES

Figure 4.3: Procedure for the calculation of the impedance correction factors krand kx that accounts for skin- effect.

4.1.2 Verification with FEM

In order to verify the model, the radial distribution of the locked rotor bar currentwas calculated using harmonic analysis in Flux2D. The studied 4-pole machine isequipped with a rotor having 28 rotor slots, casted with either aluminium or copper.Figure 4.4(a) shows the magnitude of the locked rotor bar current density for anumber of rotor bars corresponding to one pole. The result is somewhat dependingon the relative position between the bar and the stator core. The correspondingcurrent distribution obtained by the proposed model, for same current, is shown


Mag

nit

ude

ofcu

rren

tden

sity[

Am

m2

]

Radial position in rotor bar [mm]

bar 1bar 2bar 3bar 4bar 5bar 6bar 7

0 5 10 15 20 25 300

20

40

60

80

100

120

140

(a) FEM-simulation.


Mag

nit

ude

ofcu

rren

tden

sity[

Am

m2

] One-dimensional numerical methodFEM - average

0 5 10 15 20 25 300

20

40

60

80

100

120

140

(b) Proposed numerical method.

Figure 4.4: Locked rotor bar current density for the aluminium rotor.


Phas

e[

]

bar 1

bar 2

bar 3

bar 4

bar 5

bar 6

bar 7

0 5 10 15 20 25 300

50

100

150

(a) FEM-simulation.


Phas

e[

]

One-dimensional numerical methodFEM - average

0 5 10 15 20 25 300

50

100

150


Figure 4.5: Locked rotor bar current phase angle relative to the bottom of the barfor the aluminium rotor.

in Figure 4.4(b). The method used shows good correlation with the FEM results,except in the upper parts of the bar. This is due to saturation of the rotor toothtips. As a result, the current density in the top of the bar is overestimated by 22%.

Figure 4.5(a) shows the phase angle of the bar current density with referenceto the current flowing at the bottom of the slot. The result obtained from theproposed model is shown in Figure 4.5(b), which indicates a slightly overestimatedinductance in the top of the bar.

As part of the goal of this thesis is to study the differences between aluminiumand copper casted rotors, the corresponding simulation is performed with a copper


rotor. The results are shown in Figures 4.6 and 4.7. Due to the lower resistivity ofcopper the skin depth is smaller, resulting in a more pronounced skin-effect. Thisis seen in the results as a slightly increased saturation of the tooth tips. As a con-sequence, the current density at the top of the bar is overestimated by 28 %.


Mag

nit

ude

ofcu

rren

tden

sity[

Am

m2

] bar 1bar 2bar 3bar 4bar 5bar 6bar 7

0 5 10 15 20 25 30

0

0

50

100

150

200

250

(a) FEM-simulation.


Mag

nit

ude

ofcu

rren

tden

sity[

Am

m2

]

Analytical

FEM - average

0 5 10 15 20 25 300

50

100

150

200

250


Figure 4.6: Locked rotor bar current density for the copper rotor.


Phas

e[

]

bar 1

bar 2

bar 3

bar 4

bar 5

bar 6

bar 7

0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

200

(a) FEM-simulation.


Phas

e[

]

Analytical

FEM - average

0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

200


Figure 4.7: Locked rotor bar current phase relative to the bottom of the bar forthe copper rotor.

When simulating the machine performance the skin effect correction factors aremultiplied with the corresponding DC-values, giving the effective AC-values. Basedon the presented theory these factors vary with the rotor frequency according toFigure 4.8. The more pronounced skin-effect in the copper rotor will somewhatcompensate for the lower resistivity. If the locked rotor torque should be main-


tained when changing from aluminium rotor to copper rotor, the rotor slot must beredesigned. In this case, it could be obtained by the use of a double cage conductor,increasing the skin-effect even further, but at the expense of reduced power factorat rated speed.

Frequency [Hz]

Skin

effec

tco

rrec

tion

fact

ors

kr - Al

kx - Al

kr - Cu

kx - Cu

0 5 10 15 20 25 30 35 40 45 50

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Figure 4.8: Skin-effect correction factors for the studied rotor slot.

4.2 Saturation of the leakage paths

At low rotor speeds when the current is high, the large differential and slot leakagefluxes saturate the stator and rotor tooth tips. Regarding the rotor slots, this wasalready indicated in the previous section. The slot design has a large impact onthe saturation level during a start, especially the design of the tooth tips. In [23],Agarwal and Alger show that if the tooth tips are designed correctly, a reductionof the starting current can be obtained with a negligible change of the power fac-tor at rated speed. There are different techniques to account for these saturationeffects. One idea is to adjust the length of the slot opening depending on the levelof estimated saturation [24]. In [25], Chalmers introduce saturation factors similarto those commonly used to account for skin-effect, showing good agreement withmeasurements.

The tooth tip saturation will affect the distribution of all space harmonics.However, the complete analysis of these effects is beyond the scope of this thesis.In this work the influence of the leakage path saturation on the fundamental currentis investigated. In order to simplify the analysis a combined analytical and finiteelement model is used. The method defines impedance correction factors dependingon the fundamental current. These equations are then solved with initial valuesfrom a FEM-simulated locked rotor test.

4.2. SATURATION OF THE LEAKAGE PATHS 49

4.2.1 Model used to account for saturation

During an online start the rotor impedance is much smaller than the magnetizingimpedance. The magnetizing branch in the equivalent circuit can therefore beneglected at low rotor speeds. The resulting circuit is shown in Figure 4.9. Thestator and rotor resistances are usually quite small. The starting current is thereforemainly determined by the value of the leakage inductances. The saturated reactanceis defined as:

Xsat =1

ksatXunsat (4.13)

The large fundamental current is the reason for the leakage path saturation. Themodel is developed based on the assumption that the level of leakage paths satu-ration is proportional to the stator current. If the saturation factor is assumed tohave the following variation with the rotor slip;

ksat(s) = 1 +A

B + 1s

(4.14)

the validity of the model will depend on the definition of the constants A and B.The first condition used for the calculation of these constants is obtained from a

finite element simulation of a locked-rotor test. From this simulation the saturationfactors can be obtained giving the boundary condition as zero speed.

ksat(s = 1) = kFEM (s = 1) (4.15)

Based on the results from the FEM-simulation, the stator current is studied asfunction of the terminal voltage. Figures 4.10(a) and 4.10(b) show the normalizedcurrent for the aluminum rotor and the copper rotor, respectively. The red linesshowing the starting current obtained if saturation is neglected. Du to saturationeffects the starting current is increased by as much as 37 % for the copper motor

U1

+

I1 R1 jX1

R21

s

jX21

Figure 4.9: Equivalent circuit during online start.


Normalized stator voltage[

UUn

]

Nor

mal

ized

stat

orcu

rren

t[

I I n

]

FEM - IFEMLinear extrapolation - Ik

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

(a) Aluminium rotor.

Normalized stator voltage[

UUn

]

Nor

mal

ised

stat

orcu

rren

t[

I I n

]

FEM - IFEMLinear extrapolation - Ik

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

(b) Copper rotor.

Figure 4.10: FEM-simulated locked rotor test.

and 34 % for the aluminium motor. The saturation factor at zero speed is definedas:

kFEM (s = 1) =IFEM

Ik(4.16)

The stator current remains quite high during the rotor acceleration, up to thespeed where the peak-torque occurs, above this speed it decreases rapidly. Thevalue of the current at this point is therefore used when defining the second condi-tion for the calculation of the constants A and B. The stator current at peak-torqueIp, can be estimated from the locked rotor current by the use of a circle diagram.According to Alm in [26], when neglecting the magnetizing current and the ironlosses, this current is obtained as shown in Figure 4.11.

The phase displacement between the stator voltage and current φk is, for thestudied machine, close to 60, for both the aluminium rotor and the copper rotor. Inthis case, the graphical solution of the problem gives the following relation betweenthe starting current and the current at peak torque:

Ip =1√3Ik (4.17)

Thus, the saturation factor is decreased by the same amount giving:

ksat(s = sp) = 1 +kFEM (s = 1)− 1√

3(4.18)

Neglecting the stator resistance, the corresponding speed is found as:

sp =R21

X21(4.19)

4.2. SATURATION OF THE LEAKAGE PATHS 51

ℜ

ℑ

Ip

Ik

ϕk

Figure 4.11: Simplified circle diagram for the induction motor defining the startingcurrent and the current at break-down torque.

Based on Equation 4.14, with the conditions defined by Equations 4.16 and 4.18,the saturation factor can be obtained as a function of fundamental rotor slip. Themethod ensures that the saturation factor is roughly proportional to the statorcurrent during a start. Figure 4.12 shows the saturation factors for the studiedmachine as a function of rotor speed. As the peak-torque occurs at a higher speedfor the copper rotor, the two saturation curves becomes a bit different. In the

Rotor speed [× 100 rpm]

Sat

ura

tion

fact

or

Al-rotor

Cu-rotor

-4 -2 0 2 4 6 8 10 12 14 161

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Figure 4.12: Saturation factors as a function of rotor speed.


forthcoming analysis, this method is used when calculating the fundamental statorcurrent during a direct-online-start.

4.2.2 Iron losses due to leakage flux

The large leakage flux during start gives rise to additional iron losses in the statorand rotor teeth. In the equivalent circuit, these losses can be modeled by introduc-ing additional resistances in parallel with the stator and rotor leakage reactances[22], as shown in Figure 4.13.

As a result, the equivalent leakage reactance is reduced and the equivalentresistance is increased. Referred to the stator side, the increase in resistance isgiven by;

∆R(s) = kLR(X1 +X21)√s (4.20)

and the reduction of the leakage reactance is given by;

∆X(s) = kLX(X1 +X21)√s (4.21)

Where X1 is the stator leakage reactance and X21 is the rotor leakage reactance,referred to the stator side. The coefficients kLR and kLR are empirical constantsthat are obtained from short-circuit tests on a large number of rotors.

These values can be used when correcting the analytical starting current ac-cording to measurements.

⇔X

RFe

+∆R −∆X X

Figure 4.13: Additional resistance taking iron losses into account.

4.3 Rotor losses and starting torque

In Chapter 3, the rotor bar currents were derived as a function of stator current andinter-bar resistivity. The case study of the 4-pole machine having 36 stator slotsand 44 rotor slots showed the effects of a finite inter-bar resistance on bar currentdistribution. In this section the corresponding torque is calculated, showing theeffects of inter-bar currents on the starting performance.

4.3.1 Rotor losses during a start

According to the model used, the losses created within the rotor cage is derivedfrom three different regions. The rotor bars, the short-circuit rings and the contact

4.3. ROTOR LOSSES AND STARTING TORQUE 53

region between bar and core. This defines the total loss created by a bar current oforder n as:

P2n = Pbn + Prn + Ptn (4.22)

Where the bar losses are;

Pbn = QrRbn

∫ ℓ2

−ℓ2

Ibn(x)2dx (4.23)

and the end ring losses, derived through the ring currents becomes:

Prn = QrRanIbn(

ℓ2 )2 + Ibn(−ℓ2 )2

(

1− ejnpπQr

)2 (4.24)

Finally the inter-bar current losses created in the contact region.

Ptn = QrRn

∫ ℓ2

−ℓ2

Jtn(x)2dx (4.25)

The airgap power, which is the power transferred through the airgap, is givenby:

Pδn =ωTnnp2

(4.26)

Together with the equation describing the relation between airgap power and cagelosses,

P2n = snPδn (4.27)

the developed shaft torque is obtained as:

Tn =P2n

snω

np

2(4.28)

The total shaft torque is given by the sum of all the harmonic torques.

T =∞∑

n=1

Tn (4.29)

If the torque of order n at slip sn is positive, it contributes to the acceleration. Ifit is negative it counteracts the accelerating torque, reducing the acceleration.

4.3.2 Effects of a finite inter-bar resistance on starting torque

In Chapter 3 it was indicated that the inter-bar currents are of minor importancein the unskewed rotor. This is further reinforced by Figure 4.14(a), showing theobtained starting torque. The total torque is more or less the same for all thestudied inter-bar resistivities. Although fundamental inter-bar currents are present


Speed [rpm]

Tor

qu

e[N

m]

5 · 10−6 ≤ Rtn ≤ 5 · 10−2

-400 -200 0 200 400 600 800 1000 1200 14000

50

100

150

200

250

300

350

(a) Total torque.

Speed [rpm]

Tor

qu

e[N

m]

n = 1

n = −5

n = 7

n = −11

n = 13

n = −17

n = 19

-400 -200 0 200 400 600 800 1000 1200 1400-150

-100

-50

0

50

100

150

200

250

(b) Torque components.

Figure 4.14: Starting torque of Motor A with unskewed rotor.

4.3. ROTOR LOSSES AND STARTING TORQUE 55

at the rotor ends at zero speed, the inter-bar resistivity is too small to create anytorque.

As expected for an unskewed rotor, there are large asynchronous torques duringstart-up. The torque components considered are shown separately in Figure 4.14(b).Apart from the fundamental component, the dominating torques are those createdby the first order slot harmonics. These torques reduce the rotor acceleration duringthe start, and at rated speed they create additional losses. It should be noted thatthese torques do not become zero at fundamental synchronous speed. A smallbreaking torque at this high speed can create considerable additional losses. Again,at no-load when the main flux path usually is saturated, these braking torques areincreased even further due to the increased fundamental current. This effect is notincluded in this simulation.

However, these effects are one of the reasons why rotors generally are skewed.Figure 4.15 shows the starting torque when the rotor is skewed by one stator slotpitch. When the inter-bar resistivity is high the asynchronous torques caused bythe slot harmonics are efficiently suppressed. When the inter-bar resistivity getslower the starting torque decreases rapidly. And at a certain level the machinemight not even be able to start.

Speed [rpm]

Tor

qu

e[N

m]

Rtn = 5 · 10−2 Ωm

Rtn = 5 · 10−3 Ωm

Rtn = 5 · 10−4 Ωm

Rtn = 5 · 10−5 Ωm

Rtn = 5 · 10−6 Ωm

-400 -200 0 200 400 600 800 1000 1200 1400

0

50

100

150

200

250

300

350

400

Figure 4.15: Starting torque of Motor A with a rotor skewed by one stator slotpitch.


Speed [rpm]

Tor

qu

e[N

m]

Rtn = 5 · 10−2 Ωm

Rtn = 5 · 10−3 Ωm

Rtn = 5 · 10−4 Ωm

Rtn = 5 · 10−5 Ωm

Rtn = 5 · 10−6 Ωm

-400 -200 0 200 400 600 800 1000 1200 14000

25

50

75

100

125

150

175

200

225

250

275275

(a) Fundamental torque.

Speed [rpm]

Tor

qu

e[N

m]

Rtn = 5 · 10−2 Ωm

Rtn = 5 · 10−3 Ωm

Rtn = 5 · 10−4 Ωm

Rtn = 5 · 10−5 Ωm

Rtn = 5 · 10−6 Ωm

-400 -200 0 200 400 600 800 1000 1200 1400-100

-80

-60

-40

-20

0

20

40

60

80

100

120

(b) Torque caused by slot harmonics, dashed lines are representing n = 1− 2Qsp

and solid lines are representing n = 1 + 2Qsp

.

Figure 4.16: Main torque components for Motor A with a rotor skewed by onestator slot pitch.

4.4. SUMMARY 57

As there are inter-bar currents having fundamental rotor frequency, the fun-damental torque is increased. This can be seen in Figure 4.16(a), showing thefundamental components at different inter-bar resistivities, indicating that a con-siderable increase of the fundamental torque can be obtained. However, for thismachine, in the motoring region the braking torques are obviously larger than thisincrease. The torques created by the first pair of slot harmonics are the maincause of this effect, shown in Figure 4.16(b). For low inter-bar resistivities, theasynchronous torques are much larger than in the unskewed rotor, and their mag-nitudes remain large even at speeds well above their synchronous speed. This canresult in a considerable decrease of the pull-out torque. This machine will mostlikely have large stray-load losses unless the cage is insulated.

4.4 Summary

When a cage induction motor is started directly against the grid, difficulties mayarise in the calculation of the starting current. FEM-simulations of the studied ma-chine have shown that the starting current is increased by 34% with an aluminumrotor and 37% with a copper rotor due to the presence of leakage path saturation.

A numerical model used to account for the skin-effect has been verified withFEM-simulations. A combined analytical and finite element model has been devel-oped for the calculation of the fundamental starting current, taking saturation ofthe slot- and differential leakage paths into account. A method to introduce theeffects of additional iron losses during a start has been introduced.

It have been shown that the effects of the inter-bar currents on the startingtorque in unskewed rotors are of minor importance. While in skewed rotors, theinter-bar currents can have a considerable effect on the motor starting performance.The inter-bar currents having fundamental frequency, contribute to the fundamen-tal torque, i.e. they create useful torque and should not be considered as a sourceof losses. It have been shown that the inter-bar currents created by the slot har-monics can cause large asynchronous torques and are still very large even at speedswell above their synchronous speed, resulting in a reduced pull-out torque. In somecases the machine might not even be able to start.

Chapter 5

Simulation results and measurements

In this chapter, the 11 kW machine having 36 stator slots and 28 rotor slots issimulated with either an aluminium- or a copper cage rotor, skewed by one statorslot pitch. The differences in starting performance between the copper and the alu-minium rotor is studied. In the analysis, the starting torque is calculated using thevalues of inter-bar resistivities obtained from measurements on the correspondingrotors. The additional rotor losses at rated speed, caused by the inter-bar currents,are also calculated as a function of the inter-bar resistivity.

In order to verify the results, measurements of the starting current and torquehave been performed.

5.1 Simulation results

Based on the models described in the previous chapters, the machine performanceis calculated at different rotor slips s and inter-bar resistivities Rtn. In the anal-ysis, space harmonics of order n, up to the first pair of stator slot harmonics areconsidered. The calculation procedure is described in the flow-chart in Figure 5.1.

5.1.1 Starting torque

The torque speed characteristics for the two rotor concepts are calculated whenstarting direct-on-line at rated voltage, neglecting the line impedance.

First the torque is calculated assuming insulated rotor bars. Secondly the mea-sured inter-bar resistivities are used for the study of inter-bar effects. Figure 5.2shows the torque speed characteristics in the case of insulated and uninsulated rotorbars for the two motor concepts.

In the case of insulated rotor bars, the stator slot harmonics are sufficiently sup-pressed by the rotor skew. Since the 5th and the 7th space harmonics have largerwave lengths than the slot harmonics, they still cause asynchronous torques. As

59

60 CHAPTER 5. SIMULATION RESULTS AND MEASUREMENTS

Start

For: Rtn

For: n

For: s

If n = 1 If n 6= 1

Calculate coefficients accounting for;

skin-effect, saturation and additional iron losses.

Calculate stator current

from equivalent circuit.

Calculate the rotor- ring, bar

and inter-bar currents.

Calculate the terminal voltage from

the stator and rotor fluxes.

Is the terminal voltage correct?

Yes

No

Calculate the rotor losses and torque.

Calculate skin-effect coefficients.

From the fundamental stator

current, calculate; rotor- ring,

bar and inter-bar currents.

Estimate a new value

of the stator current.

End

Figure 5.1: Procedure for the calculation of motor performance at different speedsand inter-bar resistivities.

5.1. SIMULATION RESULTS 61

Speed [rpm]

Tor

qu

e[N

m]

Al

Cu

-400 -200 0 200 400 600 800 1000 1200 1400 16000

50

100

150

200

250

(a) Calculated for insulated rotor bars.

Speed [rpm]

Tor

qu

e[N

m]

Al with Rtn = 8, 0µΩm

Cu with Rtn = 0, 35µΩm

-400 -200 0 200 400 600 800 1000 1200 1400 16000

50

100

150

200

250

(b) Calculated at measured values of inter-bar resistivity.

Figure 5.2: Simulated starting torque for the studied aluminium and copper rotorsskewed by one stator slot pitch.


Speed [rpm]

Tor

qu

e[N

m]

n = 1

n = −5

n = 7

n = −11

n = 13

n = −17

n = 19

0 500 1000 1500-50

-25

0

25

50

75

100

125

150

175

200

225

250

275275

(a) Aluminium rotor with Rtn = 8, 0µΩm.

Speed [rpm]

Tor

qu

e[N

m]

n = 1

n = −5

n = 7

n = −11

n = 13

n = −17

n = 19

0 500 1000 1500-50

-25

0

25

50

75

100

125

150

175

200

225

250

275275

(b) Copper rotor with Rtn = 0, 35µΩm.

Figure 5.3: Starting torque components for the studied aluminium and copperrotors calculated with measured values of inter-bar resistivity.

5.1. SIMULATION RESULTS 63

there are no inter-bar currents, the pull-out torques were expected to be the samefor the two motor concepts. However, the short-circuit ring of the copper rotor issomewhat smaller than the short-circuit ring of the aluminium rotor. This, com-bined with the fact that the copper motor is slightly more saturated in the leakagepaths during a start, results in a somewhat higher pull-out torque of the copperrotor.

With uninsulated rotor bars, the skewing is no longer as effective. Large asyn-chronous torques occurs due to the first order slot harmonics, indicating inter-barcurrent flow. Contrary to the results obtained for the motor having 44 rotor slotssimulated in chapter 4, the inter-bar currents seem to increase the pull-out torqueof the aluminum rotor. The pull-out torque is now 4,5 % higher in the aluminiumrotor than in the copper rotor. This is most likely due to a more suitable slot num-ber combination, reducing the influence of the slot harmonics on the starting torque.

The different torque components contributing to the starting torque in the alu-minium and the copper rotor are shown in Figure 5.3(a) and Figure 5.3(b), re-spectively. Due to the very low inter-bar resistivity in the copper rotor, the slotharmonic torques becomes quite narrow, resulting in a lower braking torque at highspeeds than the the corresponding torques in the aluminium rotor. From these re-sults it can be concluded that, for the studied machine, the higher pull-out torqueof the aluminium rotor is rather due to an increase of the fundamental torque thandue to the braking torques from the space harmonics in the copper rotor.

5.1.2 Rotor losses at rated speed

At rated speed the airgap space harmonics create high frequency rotor cage losses.If the rotor bars are insulated these losses can be suppressed by rotor skewing. In-vestigation of the starting torque characteristics has however indicated that unin-sulated rotor bars increase the high frequency currents in the rotor circuit. Thedependency of the rotor stray-load losses is therefore studied as a function of theinter-bar resistivity at rated power.

As the torque depends on the inter-bar resistivity the slip is recalculated tomaintain the shaft power. In order to get a realistic value of the rotor slip, thesimulation is performed at a motor temperature of 75 , and measured valuesof friction- and iron losses are included in the analysis. Since the temperaturedependency of the inter-bar resistivity is unknown, no correction is performed onthis value. Figure 5.4 shows the high frequency cage losses obtained in the studiedmachines, the rotors being either skewed by one stator slot pitch or unskewed.

The rotor without skew is not affected by inter-bar currents, as long as theimpedance of the short circuit ring to inter-bar resistance ratio is low. When therotor is skewed by one stator slot pitch and the rotor bars are insulated, the volt-ages induced by the stator slot harmonics are efficiently suppressed. The resulting


Inter-bar resistivity Rtn [Ωm]

Addit

ional

roto

rlo

sses

[W]

αs = 1

αs = 0

Rtn Al-1

Rtn Al-2

10−8 10−6 10−4 10−2 1000

20

40

60

80

100

120

(a) Aluminium rotor

Inter-bar resistivity Rtn [Ωm]

Addit

ional

roto

rlo

sses

[W]

αs = 1

αs = 0

Rtn Cu-1

Rtn Cu-2

Rtn Cu-3

10−8 10−6 10−4 10−2 1000

20

40

60

80

100

120

(b) Copper rotor.

Figure 5.4: High frequency cage losses as a function of inter-bar resistivity Rtn at75 .

losses are mainly due to the lower order phase belt harmonics.

As the bar to core resistance decreases from a very large value, the high fre-quency losses start to increase rapidly. The maximum value of these losses isstrongly influenced by the number of stator and rotor slots [15]. According to themodel used, for the studied machine, these losses can be as large as 1 % of theoutput power. If the bar to core resistivity is reduced even further, the additionallosses starts to decrease, and reaches the value of losses equal to the losses of theunskewed rotor. The higher bar to core resistivity in the aluminium rotor is, in thiscase, resulting in higher inter-bar current losses than the equivalent copper rotor.

5.2 Measurements of starting torque

The torque during start-up was measured on the rotors having 28 slots skewed byone stator slot pitch. One aluminum rotor and one copper rotor have been tested.The torque was measured dynamically during start when the machine was loadedwith a flywheel. The unfiltered torque signal was sampled at a high frequencyand the signal noise in the sampled data was suppressed by the use of a low-passbutterworth filter of the 5th order with a cut-off frequency selected sufficiently high.

5.2.1 Measurement setup

A rotating torque transducer was used to measure the torque. The torque trans-ducer, Magtrol TM-312 rated at 200 Nm, was mounted between the motor shaftand the flywheel, according to Figure 5.2. The flywheel, having a moment of inertia

5.2. MEASUREMENTS OF STARTING TORQUE 65

FlywheelTorque

transducer Motor

Figure 5.5: Setup for measurements of starting torque.

of 1.6 kgm2, reduces the rotor acceleration making it possible to capture the torquesignal.

First, the 4-pole motor was accelerated in reverse direction to 1000 rpm. Then,the torque was measured after shifting two phases of the sinusoidal supply voltage,forcing the machine to accelerate in the opposite direction. By doing this, the in-fluence of the asynchronous torques could be measured with minimum distortionfrom the switching transient.

Time [s]

Tor

que

[Nm

]

Measured

Filtered

0.4

0 0.5 1 1.5 2 2.5 3-50

0

50

100

150

200

250

300

(a) Torque

Time [s]

Sp

eed

[rpm

]

Measured

Filtered

0 0.5 1 1.5 2 2.5 3-800

-400

0

400

800

1200

16001600

(b) Speed.

Figure 5.6: Measured and filtered torque and speed as a function of time.


Using this measuring procedure, resulted in an acceleration time of approxi-mately 3 seconds for the copper rotor at rated voltage. Figure 5.6 shows a typicalresult of the measured torque and speed as a function of time. The red lines showthe corresponding values obtained after the filtering process. One can see that thethere are huge fluctuations in the torque when starting direct-on-line. Therefore,the use of a low-pass filter is necessary in order to validate the static model usedin the analysis.

5.2.2 Results from measurements

The RMS-value of the starting current, obtained after postprocessing of the mea-sured sinusoidal current, is shown in Figure 5.7(a) and Figure 5.7(b) for the alu-minium and the copper rotor, respectively. The current has been adjusted linearlyto account for the voltage drop during start. For a comparison, the correspondinganalytical values are calculated in three different ways. Firstly, they are calculatedby only considering the skin effect in the rotor bars. Secondly, the effect of satura-tion of the leakage paths is introduced. And finally, the effect of iron losses in theleakage paths is included.

From these results it can be concluded that the high level of leakage flux duringa start has considerable impact on the starting current due to saturation effectsand iron losses. These effects have to be considered when calculating a machinesstarting performance.

Figure 5.8 shows the measured starting torques for the two rotor concepts. Thetorque has been adjusted in quadratic relation to the voltage to account for thevoltage drop during the start. Large asynchronous torques are caused by the firstorder stator slot harmonics. This verifies the prediction from the analytical model,

Speed [rpm]

Curr

ent

[A]

Measured

Skin-effect

Skin-effect + saturation

Skin-effect + saturation + iron-losses

-400 -200 0 200 400 600 800 1000 1200 1400 16000

20

40

60

80

100

120

140

160

180

200

(a) Aluminium rotor

Speed [rpm]

Curr

ent

[A]

Measured

Skin-effect

Skin-effect + saturation

Skin-effect + saturation + iron-losses

-400 -200 0 200 400 600 800 1000 1200 1400 16000

20

40

60

80

100

120

140

160

180

200

220

(b) Copper rotor.

Figure 5.7: Simulated and measured starting currents.

5.2. MEASUREMENTS OF STARTING TORQUE 67

Speed [rpm]

Tor

qu

e[N

m]

Al

Cu

-400 -200 0 200 400 600 800 1000 1200 1400 16000

50

100

150

200

250

Figure 5.8: Measured torque for the aluminium and the copper rotor when startingdirect-on-line at rated voltage.

which showed that the inter-bar currents are counteracting the effect of rotor skew,and that the pull-out torque of the aluminum rotor is higher than for the copperrotor. The measured pull-out torque of the aluminium rotor is 7 % higher than thepull-out torque of the copper rotor. This is even larger than expected theoretically.

After the introduction of the coefficients accounting for additional iron losses inthe leakage paths, the simulated starting current correlated well with the measure-ments. By comparing the corresponding starting torques, one can get an idea ofthe accuracy of the analytical model used to calculate the rotor losses. Figure 5.9shows a comparison between the measured and the simulated starting torques forthe two rotor concepts.

It should be mentioned that, since the simulations are based on a static model,it is difficult to model rapid changes in the rotor acceleration. That is probablythe reason for the overestimated pull-out torques. However, given that the machineis heavily saturated during start, the overall torque speed characteristics show ac-ceptable correlation, except for the asynchronous torque caused by the 5th spaceharmonic, which is overestimated by the analytical model. Regarding the asyn-chronous torque caused by the 7th space harmonic, it is difficult to draw conclusionsas a synchronous torque is present at the same speed.

As a result of the calculated rotor losses caused by the inter-bar currents, the


Speed [rpm]

Tor

qu

e[N

m]

Simulated

Measured

-400 -200 0 200 400 600 800 1000 1200 1400 16000

50

100

150

200

250

(a) Aluminium rotor

Speed [rpm]

Tor

qu

e[N

m]

Simulated

Measured

-400 -200 0 200 400 600 800 1000 1200 1400 16000

50

100

150

200

250

(b) Copper rotor.

Figure 5.9: Simulated and measured torques for the aluminium and the copperrotor when starting direct-on-line at rated voltage.

5.3. SUMMARY 69

torque speed characteristic of the slot harmonic torques was predicted to be dif-ferent for the copper- and the aluminium rotors. This result is verified by themeasurements, showing a similar behavior, indicating that the analytical modelseems to give a reasonable estimate of the inter-bar current losses.

Note that the rotor slot harmonics are neglected in the analysis, which resultsin less distortion of the simulated torque, especially since the rotor slots are semi-closed and not closed.

5.3 Summary

Good agreement has been demonstrated between simulated and measured startingcharacteristics for Motor B, for both aluminium and copper rotors.

Simulations have shown that the pull-out torque of the studied aluminium rotoris higher than that for the equivalent copper rotor. This is rather due to an increaseof the fundamental starting torque of the aluminium rotor, than due to the brakingtorques from the space harmonics in the copper rotor.

Measurements have shown that the difference between the pull-out torques iseven larger than calculated from the model. The measured pull-out torque of thestudied aluminium rotor was 7 % higher than for the equivalent copper rotor.Thereby, it can be concluded that the inter-bar currents have a considerable effecton motor starting performance.

At rated speed the braking torques are larger in the aluminium rotor than inthe copper rotor. This is seen as increased harmonic joule losses in the rotor cage.Simulations have shown, that these losses can be as large as 1 % of the outputpower for the studied machine.

Chapter 6

Conclusions and Future work

6.1 Conclusions

A numerical model used to account for the skin-effect has been verified with FEM-simulations. A combined analytical and finite element model has been developedfor the calculation of the fundamental starting current, taking saturation of theslot- and differential leakage paths into account. A method to introduce the effectsof additional iron losses during a start has been introduced. FEM-simulations ofthe studied machine have shown that the starting current is increased with approx-imately 35 % due to the presence of leakage path saturation and skin-effect.

A test-rig has been built for the measuring of rotor voltages, from which theinter-bar resistivity has been calculated. Measurements have shown that the inter-bar resistivity is as much as 10 times higher in cast aluminum than in cast copperrotors. The aluminium rotors showed results indicating an unevenly distributionof the inter-bar resistivity, while the copper rotors where indicating a more evenlydistributed inter-bar resistivity.

A model to include the effects of inter-bar currents in cage induction motorshas been derived. Simulations have shown that the inter-bar current density isincreasing rapidly with decreasing inter-bar resistivity in skewed machines. Thisaffects the bar current distribution and phase angle in such a way that the skewingis made ineffective, these results are consistent with the findings in [18].

It have been shown that in skewed rotors, the inter-bar currents can have aconsiderable effect on the motor starting performance. The inter-bar currents hav-ing fundamental frequency, contribute to the fundamental torque, i.e. they createuseful torque and should not be considered as a source of losses. It have beenshown that the inter-bar currents created by the slot harmonics can cause hugeasynchronous torques and are still very large even at speeds well above their syn-

71

72 CHAPTER 6. CONCLUSIONS AND FUTURE WORK

chronous speed, resulting in a reduced pull-out torque. For Motor A, in some casesthe machine might not even be able to start. These effects are strongly dependenton the combination of number of stator and rotor slots.

Good agreement has been demonstrated between simulated and measured start-ing characteristics for Motor B with both aluminium and copper casted rotors.Simulations have shown that the pull-out torque is 4,5% higher for the aluminiumrotor than for the equivalent copper rotor. This is rather due to an increase of thefundamental starting torque of the aluminium rotor, than due to braking torquesfrom the space harmonics in the copper rotor. Measurements have, however, shownthat the difference between the pull-out torques is even larger than calculated fromthe model. The measured pull-out torque of the studied aluminium rotor was 7%higher than for the equivalent copper rotor. Thereby, it can be concluded that theinter-bar currents have a considerable effect on motor starting performance.

At rated speed the braking torques are larger in the aluminium rotor than inthe copper rotor. This is seen as increased harmonic joule losses in the rotor cage.Simulations have shown, that these losses can be as large as 1% of the output powerfor the studied machine.

6.2 Future work guidelines

Based on the results obtained in this thesis, for a cast copper rotor having a suitablenumber of slots skewed by one stator slot pitch, one can expect similar performanceas for an equivalent rotor without skewing. This could be verified by measurementson prototype machines. In case of a cast aluminium rotor, suitable measurementsshould be made to verify if the stray-load losses are larger than in cast copper rotorsor not. It could also be favorably to study the starting torque and efficiency of amachine having an unsuitable slot number combination, increasing the inter-barcurrent flow, making the verification of the analytical model much easier.

In general, unskewed rotors are said to create larger noise levels than skewedrotors. As the inter-bar currents seem to counteract the rotor skew, it is of interestto measure and compare the noise levels of skewed and unskewed rotors havinglow inter-bar resistivities. Further, in order to suppress both noise and inter-barcurrents, the concept of asymmetrical rotor slots studied among others in [27],should be evaluated by measurements on a prototype machine.

Bibliography

[1] A.R. Hagen, A. Binder, M. Aoulkadi, T. Knopik, and K. Bradley. Comparisonof measured and analytically calculated stray load losses in standard cageinduction machines. 18th International Conference on Electrical Machines,pages 1–6, 2008.

[2] H. Nishizawa, K. Itomi, S. Hibino, and F. Ishibashi. Study on reliable reductionof stray load losses in three-phase induction motor for mass production. IEEE

Transactions on Energy Conversion, EC-2:489–495, 1987.

[3] A. Boglietli, A. Cavagnino, M. Lazzari, and A. Pastorelli. Induction motorefficiency measurements in accordance to ieee 112-b, iec 34-2 and jec 37 inter-national standards. IEEE International Electric Machines and Drives Confer-

ence, 3:1599–1605, 2003.

[4] A.A. Jimoh, R.D. Findlay, and M. Poloujadoff. Stray losses in induction ma-chines: Part i, definition, origin and measurement. IEEE Transactions on

Power Apparatus and Systems, PAS-104:1500–1505, 1985.

[5] Y.N. Feng, J. Apsley, S. Williamson, A.C. Smith, and D.M. Ionel. Reducedlosses in die-cast machines with insulated rotors. IEEE International Electric

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[6] A. M. Odok. Stray-load losses and stray torques in induction machines. Power

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[7] P. L. Alger. Induction Machines: Their Behavior and Uses. Taylor & Francis,1995.

[8] A. Harson, P.H. Mellor, and D. Howe. Design considerations for inductionmachines for electric vehicle drives. Seventh International Conference on Elec-

trical Machines and Drives, pages 16–20, 1995.

[9] K. Dabala. Modified method to determine rotor bar-iron resistance in three-phase copper casted squirrel-cage induction motors. Proceedings of ICEM,pages 231–234, 2006.

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[10] P. L. Alger. Induced high-frequency currents in squirrel-cage windings. Power

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[11] K. Dabala. A new experimental-computational method to determine rotorbar-iron resistance. Proceedings of ICEM, 2:69–72, 1996.

[12] O. Aglén. Calorimetric Measurements of Losses in Induction Motors. Licen-tiate thesis, Royal Institute of Technology, Stockholm, Sweden, 1995.

[13] A.H. Bonnett and T. Albers. Squirrel-cage rotor options for ac inductionmotors. IEEE Transactions on Industry Applications, pages 1197–1209, 2001.

[14] A. Behdashti and M. Poloujadoff. A new method for the study of inter-barcurrents in polyphase squirrel-cage induction motors. IEEE Transactions on

Power Apparatus and Systems, PAS-98(3):902–911, 1979.

[15] B. Heller and V. Hamata. Harmonic Field Effects in Induction Machines.Elsevier Science Ltd, 1977.

[16] M. Ivanes and M. Bourmault. Etudes des pertes supplementaires dans lesmoteurs asynchrones. Technical report, Cie Electro-Mecanique, October 1968.

[17] A. Behdashti. Contribution a l’etude des pertes supplementaires des machines

asynchrones dans une tres large zone de fonctionnement. PhD thesis, L’Institutnational polytechnique de Grenoble, June 1975.

[18] A.C. Smith, S. Williamson, and C.Y. Poh. Distribution of inter-bar currentsin cage induction machines. Second International Conference on Power Elec-

tronics, Machines and Drives, 1:297–302, 2004.

[19] P. L. Alger and J. H. Wray. Double and triple squirrel cages for polyphase in-duction motors. Power Apparatus and Systems, Transactions of the American

Institute of Electrical Engineers, 72(2):637 – 645, 1953.

[20] M. Liwschitz-Garik. Skin-effect bars of squirrel-cage rotors. Power Apparatus

and Systems, Transactions of the American Institute of Electrical Engineers,73(1):255 – 258, 1954.

[21] S. Williamson and D. R. Gersh. Finite element calculation of double-cagerotor equivalent circuit parameters. IEEE Transactions on Energy Conversion,11(1):41–48, 1996.

[22] C. Sadarangani. Electrical machines - design and analysis of induction and

permanent magnet motors. KTH Hogskoletryckeriet, 2000.

[23] P. D. Agarwal and P. L. Alger. Saturation factors for leakage reactance of in-duction motors. Power Apparatus and Systems, Transactions of the American

Institute of Electrical Engineers, 79(3):1037–1042, 1960.

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[24] H. M. Norman. Induction motor locked saturation curves. Transactions of the

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[25] B. J. Chalmers and R. Dodgson. Saturated leakage reactances of cage inductionmotors. Proceedings IEE, 116(8):1395–1404, 1969.

[26] E. Alm. Elektroteknik, Band 3, Del 2B, Elektromaskinlära - Asynkronmaski-

nens teori, driftegenskaper och beräkning. Alb. Bonniers boktryckeri, 1931.

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with Non-Skewed Asymmetrical Rotor Slots. Licentiate thesis, Royal Instituteof Technology, Stockholm, Sweden, 2009.

List of Figures

1.1 Stray-load loss components (0,2-37 kW induction motors) [2]. . . . . . . 2

2.1 Rotor test setup for measurements of inter-bar resistance. . . . . . . . . 9

2.2 Conducting washer between test-plate and rotor short-circuit ring. . . . 10

2.3 Equivalent circuit of the rotor used for the calculation of the inter-barresistivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Calculated voltages for the aluminium rotor at a total current of 200 A. 12

2.5 Calculated voltages for the copper rotor at a total current of 200 A. . . 12

2.6 Two of the studied aluminium and copper rotors. . . . . . . . . . . . . . 13

2.7 Measured and calculated voltage UAX for rotor Al 2 at a total currentof 200 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.8 Measured and calculated voltage UAX for rotor Cu 2 at a total currentof 100 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Definition of bar- and inter-bar currents in a small element of the rotorcircuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Phase displacement between inter-bar currents. . . . . . . . . . . . . . . 18

3.3 Definitions of inter-bar resistances. . . . . . . . . . . . . . . . . . . . . . 193.4 Airgap flux density due to current in phase a, Model A. . . . . . . . . . 21

3.5 Airgap flux density due to current in phase a, Model B. . . . . . . . . . 22

3.6 Comparison between analytical and FEM-simulated airgap flux densityat no-load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.7 Position of a wave of order n at time t = 0 in the rotor reference frame. 25

3.8 Current in bar number k and the corresponding MMF in the airgap. . . 27

3.9 Permeance variation along the airgap circumference as defined by ModelA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.10 Rotor currents at the boundary x = − ℓ2 . . . . . . . . . . . . . . . . . . . 31

3.11 Change in rotor bar current due to the interaction with inter-bar currents. 32

3.12 Magnitude of the fundamental locked rotor inter-bar current density andthe resulting angle β1, for Motor A with unskewed rotor. . . . . . . . . 33

3.13 Magnitude of fundamental locked rotor bar current and the correspond-ing angle γ1, for Motor A with unskewed rotor. . . . . . . . . . . . . . . 34

76

List of Figures 77

3.14 Magnitude of the currents in the unskewed rotor caused by the funda-mental and the first pair of slot space harmonics when Rtn = 5 · 10−5 Ωm. 35

3.15 Magnitude of the fundamental locked-rotor inter-bar current density andthe resulting angle β1 for Motor A with skewed rotor. . . . . . . . . . . 36

3.16 Magnitude of fundamental locked-rotor bar current and the correspond-ing angle γ1, for Motor A with skewed rotor. . . . . . . . . . . . . . . . 37

3.17 Magnitude of the currents in the rotor skewed by one stator slot pitch,caused by the fundamental and the first pair of slot space harmonicswhen Rtn = 5 · 10−2 Ωm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.18 Magnitude of the currents in the rotor skewed by one stator slot pitch,caused by the fundamental and the first pair of slot space harmonicswhen Rtn = 5 · 10−5 Ωm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 Different types of rotor bars. . . . . . . . . . . . . . . . . . . . . . . . . 424.2 The vth section of the rotor slot. . . . . . . . . . . . . . . . . . . . . . . 444.3 Procedure for the calculation of the impedance correction factors kr and

kx that accounts for skin- effect. . . . . . . . . . . . . . . . . . . . . . . 454.4 Locked rotor bar current density for the aluminium rotor. . . . . . . . . 464.5 Locked rotor bar current phase angle relative to the bottom of the bar

for the aluminium rotor. . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.6 Locked rotor bar current density for the copper rotor. . . . . . . . . . . 474.7 Locked rotor bar current phase relative to the bottom of the bar for the

copper rotor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.8 Skin-effect correction factors for the studied rotor slot. . . . . . . . . . . 484.9 Equivalent circuit during online start. . . . . . . . . . . . . . . . . . . . 494.10 FEM-simulated locked rotor test. . . . . . . . . . . . . . . . . . . . . . . 504.11 Simplified circle diagram for the induction motor defining the starting

current and the current at break-down torque. . . . . . . . . . . . . . . 514.12 Saturation factors as a function of rotor speed. . . . . . . . . . . . . . . 514.13 Additional resistance taking iron losses into account. . . . . . . . . . . . 524.14 Starting torque of Motor A with unskewed rotor. . . . . . . . . . . . . . 544.15 Starting torque of Motor A with a rotor skewed by one stator slot pitch. 554.16 Main torque components for Motor A with a rotor skewed by one stator

slot pitch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.1 Procedure for the calculation of motor performance at different speedsand inter-bar resistivities. . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2 Simulated starting torque for the studied aluminium and copper rotorsskewed by one stator slot pitch. . . . . . . . . . . . . . . . . . . . . . . . 61

5.3 Starting torque components for the studied aluminium and copper rotorscalculated with measured values of inter-bar resistivity. . . . . . . . . . 62

5.4 High frequency cage losses as a function of inter-bar resistivity Rtn at75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.5 Setup for measurements of starting torque. . . . . . . . . . . . . . . . . 65

78 List of Figures

5.6 Measured and filtered torque and speed as a function of time. . . . . . . 655.7 Simulated and measured starting currents. . . . . . . . . . . . . . . . . . 665.8 Measured torque for the aluminium and the copper rotor when starting

direct-on-line at rated voltage. . . . . . . . . . . . . . . . . . . . . . . . 675.9 Simulated and measured torques for the aluminium and the copper rotor

when starting direct-on-line at rated voltage. . . . . . . . . . . . . . . . 68

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