OPERATION OF INTERIOR PERMANENT MAGNET SYNCHRONOUS MACHINES
WITH FRACTIONAL SLOT CONCENTRATED WINDINGS UNDER BOTH
HEALTHY AND FAULTY CONDITIONS
By
Shanelle Nicole Foster

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Electrical Engineering - Doctor of Philosophy
2013

ABSTRACT
OPERATION OF INTERIOR PERMANENT MAGNET SYNCHRONOUS
MACHINES WITH FRACTIONAL SLOT CONCENTRATED WINDINGS
UNDER BOTH HEALTHY AND FAULTY CONDITIONS
By
Shanelle Nicole Foster
Design for fault tolerance and early detection of insulation failure are critical for automotive and aerospace applications to ensure passenger safety. Permanent magnet machines can
be designed to better withstand stator insulation failures. In this work, the performance of
three fault tolerant fractional slot concentrated winding machine designs experiencing stator
winding insulation failure are evaluated. Two of the machines are designed with double-layer
windings and one with single-layer. The single-layer fractional slot concentrated winding design is shown most reliable; however this design has the worst torque performance. A ripple
reduction control technique is developed based on an analytical description of torque. This
technique is shown to improve the torque performance of the single-layer fractional slot design.

Fault tolerant design alone does not provide high reliability since thermal stress from
aging, overloading, cycling or fast switching of the inverter causes most stator insulation
failures. Early detection of incipient stator winding faults could avoid catastrophic machine
failure, allow implementation of mitigation techniques to continue operation, reduce the
occurrence of secondary faults and allow adequate time to plan maintenance. In this work,
two of the machines designed were manufactured with windings that allow the introduction of
faults with three severity levels and varying degrees of incipient faults. Through a parametric

identification method, the characteristic flux linkages of the machines are extracted under
both healthy and faulty conditions. It is shown that incipient stator windings faults are
reflected in the machine’s characteristic parameters. These parametric changes are reflected
in the phase voltage for current-controlled applications. Incipient stator winding faults can
be detected online, if accurate knowledge of the healthy machine parameters is available.

To my husband, children, mother and brother

iv

ACKNOWLEDGMENTS

First and foremost, I thank my Lord and Savior Jesus Christ for without Him none of this
would have been possible. I’d also like to thank my husband, Kyle Foster, and my children,
Shayna, Kyle, Jr. and Kailen, for their patience and forgiveness as I attempted to juggle
being a wife, mother and researcher. Without your assistance, this balancing act would have
been impossible. I must thank my mother, Defoird Rochelle Hogan, for being an example of
strength and courage all of my life. Thank you for your prayers and encouragement over the
years. Also, a big thank you to my brother, Shannon Hogan, and my cousin, Dr. Marlon
Mitchell, for all the words of encouragement. I can’t forget my mother-in-law, Toni Lee,
my beautiful sisters-in-law, Carmen Hogan and Michelle Lee, and my God-sister, Shanequa
Dennie . There are too many family members to name them all, but I love each of you and
appreciate all of the support and encouragement over the years.

I’d like to thank Dr. Elias G. Strangas for his patience, guidance, support and encouragement. I’d also like to thank Dr. John Miller, Dr. Harold Schock, Dr. Bingsen Wang and
Dr. George Zhu for their time and effort as part of my guidance committee.

A big thank you to present and past members of the Electrical Machines and Drives Laboratory team for all of their support. I must also thank Mrs. Meagan Kroll, Mrs. Pauline
Vandyke, Mrs. Roxanne Peacock, Mr. Brian Wright, Mr. Gregg Mulder, Ms. Laurie Rashid,
Ms. Jennifer Woods and Ms. Joyce Foley for their assistance.

The assistance with child care that I’ve received over the years has made balancing mothv

erhood and research much easier. Thank you to Mrs. Lauren Mitchner Brown, Ms. Racheal
Mitchner, Ms. Ciera Magee-Wiggins, Mrs. Jacqueies Thomas, Ms. Deanna Anderson and
AppleTree Christian Learning Center for the magnificent care that you provided for my children. The peace of mind that I had when my precious children were in your care allowed
me to the time to focus on my research. I also have to thank a few people for their care
friendship, support and encouragement. Dr. Aurles U. Wiggins, Mrs. Jennifer Mitchner,
Mrs. Jamillah Gross-Caldwell, Mr. Theodore Caldwell and Mrs. Lisa Raths you are all very
special to me.

I’d also like to thank Mr. Rickey Caldwell, Ms. Lisa Henry and Mrs. Maggie BlairRamsey from the Diversity Programs Office, Ms. Audrey Smith and Mrs. Evette Chavez
Lockhart from the MSU Women’s Resource Center, Ms. Paulette Granberry Russell from
the Office of Inclusion and Intercultural Initiatives, Dr. Percy Pierre and the Alfred P. Sloan
Foundation for their support.

vi

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Objectives and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
4

Chapter 2 Stator Winding Insulation Failure . . . . . . . . . . . . . . . . . .
2.1 Analytical Model of PMSM with Stator Inter-turn Faults . . . . . . . . . . .
2.2 Finite Element Model of PMSM with Stator Inter-turn Faults . . . . . . . .

5
7
14

Chapter 3 PMSM Reliability . . . .
3.1 Fault Tolerant Design . . . . . . .
3.2 Stator Winding Configurations . .
3.3 Effects of Winding Configurations
3.4 Stator Winding Failure Detection

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35

Chapter 4 Design and Analysis of Fault Tolerant Machines . . . .
4.1 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Star of Slots Theory . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Double-layer PM Machine Design . . . . . . . . . . . . . . . . . . .
4.3 Single-layer PM Machine Design . . . . . . . . . . . . . . . . . . . .
4.4 Experimental Characterization . . . . . . . . . . . . . . . . . . . . .
4.4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Experimental Data Acquisition . . . . . . . . . . . . . . . .
4.4.3 Experimental Data Processing . . . . . . . . . . . . . . . . .
4.4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . .
4.5 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Performance with Stator Inter-turn Failure . . . . . . . . . . . . . .
4.6.1 Double-layer 21 SPP PM Machine . . . . . . . . . . . . . . .
4.6.2 Single-layer 52 SPP PM Machine . . . . . . . . . . . . . . . .
4.7 Reliability and Performance of PM Machines Designed with FSCW

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38
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42
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46
51
53
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57
63
69
69
75
80

Chapter 5 PMSM Torque Ripple . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Stator MMF Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83
83

vii

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5.2
5.3
5.4
5.5
5.6

Rotor MMF Distribution . . . . . . . . . . . . . . . .
Air-gap Permeance Variations Due to Rotor Slotting
Air-gap Permeance Variation Due to Stator Slotting .
Air-gap Flux Density . . . . . . . . . . . . . . . . . .
Analytical Torque Expression . . . . . . . . . . . . .

Chapter 6 Torque Ripple Reduction
6.1 PMSM Controller Development .
6.2 Torque Ripple Estimation . . . .
6.3 Ripple Reduction Technique . . .
6.4 Finite Element Results . . . . . .

Control
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86
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102
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110
111

Fault
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116
117
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126

Chapter 7 Detection of Stator Winding Insulation Degradation
Tolerant PM Machines . . . . . . . . . . . . . . . . . . .
7.1 Winding Insulation Degradation and Machine Performance . . . .
7.2 Experimental Setup and Results . . . . . . . . . . . . . . . . . . .
7.3 Online Detection of Winding Insulation Degradation . . . . . . .

in
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Chapter 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
BIBLIOGRAPHY

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viii

. . . . . . . . . . . . . . . . . 145

LIST OF TABLES

Table 2.1

Design Parameters of PMSM modeled in FEA . . . . . . . . . . . .

16

Table 2.2

Circulating current in a PMSM designed with sinusoidally distributed
windings under short-circuit faults. Torque ripple as a percentage of
the average torque in a healthy machine. . . . . . . . . . . . . . . .

23

MMF harmonic order and distribution factor for transformation of
double-layer winding to single-layer winding for various cases based
on machine periodicity and number of stator slots. . . . . . . . . . .

42

Table 4.2

Double-layer 12 SPP machine design parameters. . . . . . . . . . . .

44

Table 4.3

Single-layer 52 SPP machine design parameters. . . . . . . . . . . . .

49

Table 4.4

Table used for data acquisition. . . . . . . . . . . . . . . . . . . . .

56

Table 4.5

Comparison of Torque Ripple and Current Circulating in Windings
of Double-layer 12 SPP PM Machine Due to Short-Circuited Inter-turns. 70

Table 4.6

Comparison of Torque Ripple and Current Circulating in Windings
of Single-layer 52 SPP PM Machine Due to Short-Circuited Inter-turns. 80

Table 4.7

Circulating Current in PM Machines Designed with Fractional Slot
Concentrated Windings as a Factor of Rated Current. . . . . . . . .

82

Torque Ripple in PM Machines Designed with Fractional Slot Concentrated Windings as a Percentage of Average Torque. . . . . . . .

82

Table 4.1

Table 4.8

Table 6.1

Change in torque ripple due to implementation of the ripple compensation control on 25 SPP PM machine. . . . . . . . . . . . . . . . . . 112

Table 6.2

Change in machine losses due to implementation of the ripple reduction control on 25 SPP PM machine. . . . . . . . . . . . . . . . . . . 115

ix

Table 7.1

Torque Ripple in PM Machines Designed with Fractional Slot Concentrated Windings with Degraded Insulation (As a Percentage of the
Average Torque of a Healthy Machine). . . . . . . . . . . . . . . . . 121

Table 7.2

Resistance of a Single Turn for Both Machine Designs . . . . . . . . 124

x

LIST OF FIGURES

Figure 2.1

Stator Winding with Damaged Insulation. (For interpretation of the
references to color in this and all other figures, the reader is referred
to the electronic version of this disseration.) . . . . . . . . . . . . . .

6

Figure 2.2

Equivalent circuit model of winding with inter-turn fault . . . . . .

8

Figure 2.3

FEA geometry of PMSM with 2 slots per pole per phase:(a)healthy
(b)faulty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

FEA circuit model of PMSM with 2 slots per pole per phase:(a)healthy
(b)faulty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

Cogging Torque of Healthy PMSM with 2 slots per pole per phase
(finite element results) . . . . . . . . . . . . . . . . . . . . . . . . .

17

Healthy back-EMF of PMSM with 2 slots per pole per phase (finite
element results) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Back-EMF of PMSM with 2 slots per pole per phase and a single
turn in a slot short-circuited (finite element results) . . . . . . . . .

18

Back-EMF of PMSM with 2 slots per pole per phase and 10% of turns
in a slot short-circuited (finite element results) . . . . . . . . . . . .

19

Back-EMF of PMSM with 2 slots per pole per phase and 20% of turns
in a slot short-circuited (finite element results) . . . . . . . . . . . .

19

Circulating current due to a single turn in a slot involved in a shortcircuit (finite element results) . . . . . . . . . . . . . . . . . . . . . .

20

Circulating current due to a 10% of the turns in a slot involved in a
short-circuit (finite element results) . . . . . . . . . . . . . . . . . .

20

Circulating current due to 20% of the turns in a slot involved in a
short-circuit (finite element results) . . . . . . . . . . . . . . . . . .

21

Torque of healthy PMSM with 2 slots per pole per phase (finite element results) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 2.13

xi

Figure 2.14

Figure 2.15

Figure 2.16

Torque of 2 slots per pole per phase PMSM with a single turn in a
slot short-circuited (finite element results) . . . . . . . . . . . . . . .

22

Torque of 2 slots per pole per phase PMSM with 10% of turns in a
slot short-circuited (finite element results) . . . . . . . . . . . . . . .

22

Torque of 2 slots per pole per phase PMSM with 20% of turns in a
slot short-circuited (finite element results) . . . . . . . . . . . . . . .

23

Figure 3.1

Overlapping Winding Category - (a)Distributed and (b)Concentrated 27

Figure 3.2

Non-overlapping Winding Category - (a)Double-layer and (b)Singlelayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

Figure 3.3

MMF of different winding configurations - (a)Distributed windings
with 2 SPP and (b)Fractional Slot Concentrated Windings with 21 SP P 31

Figure 3.4

MMF harmonic spectrum of different winding configurations - (a)Distributed
windings with 2 SPP and (b)Fractional Slot Concentrated Windings
with 21 SP P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 4.1

Star of slots for double-layer 12 SP P PM machine . . . . . . . . . . .

43

Figure 4.2

Double-layer 21 SP P PM machine laminations . . . . . . . . . . . . .

45

Figure 4.3

Manufactured parts of 12 SPP PM machine - (a)Stator and (b)Rotor
with NdFeB magnets-N38EH nickel coated . . . . . . . . . . . . . .

46

Figure 4.4

Star of slots for double-layer 25 SP P PM machine . . . . . . . . . . .

47

Figure 4.5

Star of slots for single-layer 52 SP P PM machine . . . . . . . . . . .

49

Figure 4.6

Single-layer 52 SP P PM machine laminations . . . . . . . . . . . . .

50

Figure 4.7

Laminations for 25 SPP PM machine - (a)Stator and (b)Rotor . . . .

50

Figure 4.8

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Figure 4.9

Experimental setup for collecting data to extract the machine characteristic inductances. . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Experimental D-axis Flux Linkage of 21 SPP PM Machine . . . . . .

58

Figure 4.10

xii

Figure 4.11

Experimental Q-axis Flux Linkage of 21 SPP PM Machine . . . . . .

58

Figure 4.12

Experimental Torque-speed envelope for 12 SPP PM Machine . . . .

59

Figure 4.13

Experimental D-axis Flux Linkage of 52 SPP PM Machine . . . . . .

59

Figure 4.14

Experimental Q-axis Flux Linkage of 25 SPP PM Machine . . . . . .

60

Figure 4.15

Experimental Torque-speed envelope for 25 SPP PM Machine . . . .

60

Figure 4.16

Experimental D-axis Inductance of 21 SPP PM Machine . . . . . . .

61

Figure 4.17

Experimental Q-axis Inductance of 12 SPP PM Machine . . . . . . .

61

Figure 4.18

Experimental D-axis Inductance of 52 SPP PM Machine . . . . . . .

62

Figure 4.19

Experimental Q-axis Inductance of 25 SPP PM Machine . . . . . . .

62

Figure 4.20

Double-layer 21 SPP and Single-layer 25 SPP PM Machine Cogging
Torque (finite element results). . . . . . . . . . . . . . . . . . . . . .

64

Figure 4.21

Double-layer 21 SPP PM Machine BackEMF (finite element results).

65

Figure 4.22

Harmonic Content in Double-layer 12 SPP PM Machine. . . . . . . .

65

Figure 4.23

Single-layer 52 SPP PM Machine BackEMF (finite element results). .

66

Figure 4.24

Harmonic Content in Single-layer 52 SPP PM Machine. . . . . . . . .

66

Figure 4.25

Double-layer 12 SPP PM Machine Torque (finite element results). . .

67

Figure 4.26

Single-layer 52 SPP PM Machine Torque (finite element results). . .

67

Figure 4.27

Double-layer 12 SPP PM Machine Inductances (finite element results). 68

Figure 4.28

Single-layer 52 SPP PM Machine Inductances (finite element results).

68

Figure 4.29

Inductance of single-layer FSCW PMSM with iron having constant
permeability (finite element results) . . . . . . . . . . . . . . . . . .

69

Circulating Current in Double-layer 21 SPP PM Machine with SingleTurn Short-Circuited (finite element results). . . . . . . . . . . . . .

70

Figure 4.30

xiii

Figure 4.31

Figure 4.32

Figure 4.33

Figure 4.34

Figure 4.35

Figure 4.36

Figure 4.37

Figure 4.38

Figure 4.39

Figure 4.40

Figure 4.41

Figure 4.42

Figure 4.43

Figure 4.44

Figure 4.45

Circulating Current in Double-layer 12 SPP PM Machine with 10% of
Turns in Slot Short-Circuited (finite element results). . . . . . . . .

71

Circulating Current in Double-layer 12 SPP PM Machine with 20% of
Turns in Slot Short-Circuited (finite element results). . . . . . . . .

71

Open-Circuit Voltage in Double-layer 12 SPP PM Machine with SingleTurn Short-Circuited (finite element results). . . . . . . . . . . . . .

72

Open-Circuit Voltage in Double-layer 21 SPP PM Machine with 10%
of Turns in Slot Short-Circuited (finite element results). . . . . . . .

72

Open-Circuit Voltage in Double-layer 21 SPP PM Machine with 20%
of Turns in Slot Short-Circuited (finite element results). . . . . . . .

73

Torque of Double-layer 21 SPP PM Machine with Single-Turn ShortCircuited (finite element results). . . . . . . . . . . . . . . . . . . . .

73

Torque of Double-layer 12 SPP PM Machine with 10% of Turns in
Slot Short-Circuited (finite element results). . . . . . . . . . . . . .

74

Torque of Double-layer 12 SPP PM Machine with 20% of Turns in
Slot Short-Circuited (finite element results). . . . . . . . . . . . . .

74

Circulating Current in Single-layer 25 SPP PM Machine with SingleTurn Short-Circuited (finite element results). . . . . . . . . . . . . .

75

Circulating Current in Single-layer 25 SPP PM Machine with 10% of
Turns in Slot Short-Circuited (finite element results). . . . . . . . .

76

Circulating Current in Single-layer 25 SPP PM Machine with 20% of
Turns in Slot Short-Circuited (finite element results). . . . . . . . .

76

Open-Circuit Voltage in Single-layer 52 SPP PM Machine with SingleTurn Short-Circuited (finite element results). . . . . . . . . . . . . .

77

Open-Circuit Voltage in Single-layer 52 SPP PM Machine with 10%
of Turns in Slot Short-Circuited (finite element results). . . . . . . .

77

Open-Circuit Voltage in Single-layer 25 SPP PM Machine with 20%
of Turns in Slot Short-Circuited (finite element results). . . . . . . .

78

Torque of Single-layer 25 SPP PM Machine with Single-Turn ShortCircuited (finite element results). . . . . . . . . . . . . . . . . . . . .

78

xiv

Figure 4.46

Figure 4.47

Figure 4.48

Figure 4.49

Figure 5.1

Figure 5.2

Figure 5.3

Torque of Single-layer 52 SPP PM Machine with 10% of Turns in Slot
Short-Circuited (finite element results). . . . . . . . . . . . . . . . .

79

Torque of Single-layer 25 SPP PM Machine with 20% of Turns in Slot
Short-Circuited (finite element results). . . . . . . . . . . . . . . . .

79

Circulating current in short-circuited turns of double-layer 25 SPP PM
(finite element results). . . . . . . . . . . . . . . . . . . . . . . . . .

81

Torque for double-layer 52 SPP PM with short-circuited turns (finite
element results). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

Simplified phase A winding distribution of single-layer fractional slot
concentrated winding design . . . . . . . . . . . . . . . . . . . . . .

84

Square-wave approximation of the single-layer fractional slot concentrated winding design stator phase A MMF distribution . . . . . . .

84

Trigonometric Fourier series of the single-layer fractional slot concentrated winding design stator phase A MMF distribution . . . . . . .

85

Figure 5.4

Single-layer Fractional Slot Concentrated Winding Rotor Configuration 87

Figure 5.5

Square-wave Approximation of the single-layer fractional slot concentrated winding design rotor MMF distribution over one pole pair . .

88

Trigonometric Fourier series of the single-layer fractional slot concentrated winding design rotor MMF distribution . . . . . . . . . . . .

88

Square-wave Approximation of Reluctance Variation Due to Rotor
Slotting in Singlelayer Fractional Slot Concentrated Winding PMSM

90

Square-wave Approximation of Reluctance Variation Due to Rotor
Slotting in Singlelayer Fractional Slot Concentrated Winding PMSM

91

Square-wave approximation of Reluctance variations due to stator
slotting over one slot pitch . . . . . . . . . . . . . . . . . . . . . . .

93

Square-wave approximation of Reluctance variations due to stator
slotting over one slot pitch . . . . . . . . . . . . . . . . . . . . . . .

93

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Simulink model of analytical torque expression . . . . . . . . . . . . 100

Figure 5.12

Simulink model of analytical torque expression . . . . . . . . . . . . 101
xv

Figure 6.1

Permanent magnet synchronous machine regions of operation defined
by the torque speed curve. . . . . . . . . . . . . . . . . . . . . . . . 103

Figure 6.2

PMSM model-based controller . . . . . . . . . . . . . . . . . . . . . 105

Figure 6.3

Comparison of finite element single-layer 25 SPP PMSM inductances
to piece-wise approximation. . . . . . . . . . . . . . . . . . . . . . . 105

Figure 6.4

Magnitude of rotor MMF harmonic components in single-layer 25 SPP
PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 6.5

Estimated torque ripple for 35Nm command in single-layer 52 SPP
PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 6.6

Analytical Torque Ripple Estimation . . . . . . . . . . . . . . . . . 109

Figure 6.7

Finite element torque ripple at varying speeds and torques . . . . . 110

Figure 6.8

Torque ripple estimator . . . . . . . . . . . . . . . . . . . . . . . . . 110

Figure 6.9

Torque ripple compensation controller . . . . . . . . . . . . . . . . . 111

Figure 6.10

Simulink model of torque compensation . . . . . . . . . . . . . . . . 112

Figure 6.11

Torque for 10Nm command with and without the torque ripple compensation controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Figure 6.12

Finite element results 10Nm torque command with and without the
torque ripple compensation controller. . . . . . . . . . . . . . . . . . 113

Figure 6.13

Finite element results 20Nm torque command with and without the
torque ripple compensation controller. . . . . . . . . . . . . . . . . . 114

Figure 6.14

Finite element results 35Nm torque command with and without the
torque ripple compensation controller. . . . . . . . . . . . . . . . . . 114

Figure 7.1

Circulating Current in Single-layer 25 SPP Permanent Magnet Synchronous Machine with Single-turn Insulation Degraded at Different
Speeds - (a)200 rpm (b)300 rpm (c)400 rpm (finite element results) . 118

Figure 7.2

Circulating Current in Single-layer 25 SPP Permanent Magnet Synchronous Machine with Insulation Degraded in Different Number of
Turns at 300 rpm - (a)Single-turn in Slot (b)10% Turns in Slot (c)20%
Turns in Slot (finite element results) . . . . . . . . . . . . . . . . . . 118
xvi

Figure 7.3

Open-Circuit Voltage (at Different Speeds) in Single-layer 52 SPP Permanent Magnet Synchronous Machine with the Insulation on a Singleturn in a Slot Degraded - (a)200 rpm (b)300 rpm (c)400 rpm (finite
element results) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Figure 7.4

Open-Circuit Voltage (at Different Speeds) in Single-layer 52 SPP Permanent Magnet Synchronous Machine with the Insulation of 20% of
the Turns in a Slot Degraded - (a)200 rpm (b)300 rpm (c)400 rpm
(finite element results) . . . . . . . . . . . . . . . . . . . . . . . . . 119

Figure 7.5

Open-Circuit Voltage in Single-layer 25 SPP Permanent Magnet Synchronous Machine with Insulation Degraded in Different Number of
Turns at 300 rpm - (a)Single-turn in Slot (b)10% Turns in Slot (c)20%
Turns in Slot (finite element results) . . . . . . . . . . . . . . . . . . 120

Figure 7.6

Harmonic Spectrum of Open-Circuit Voltage in Single-layer 25 SPP
Permanent Magnet Synchronous Machine with Insulation Degraded
Across a Single-turn in a Slot . . . . . . . . . . . . . . . . . . . . . . 120

Figure 7.7

Harmonic Spectrum of Open-Circuit Voltage in Single-layer 25 SPP
Permanent Magnet Synchronous Machine with Insulation Degraded
Across 20% of Turns in a Slot . . . . . . . . . . . . . . . . . . . . . 121

Figure 7.8

Winding Schematic of Double-layer 21 SPP PM Machine Prototype . 122

Figure 7.9

Winding Schematic of Single-layer 52 SPP PM Machine Prototype. . 123

Figure 7.10

Circuit Diagram for Introducing Faults of Varying Severity . . . . . 125

Figure 7.11

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Figure 7.12

Double-layer 21 SPP PM Machine with Insulation on a Single-turn
Degraded( 21 Ω) (experimental results). . . . . . . . . . . . . . . . . . 127

Figure 7.13

Double-layer 21 SPP PM Machine with Insulation on a Single-turn
Degraded( 41 Ω) (experimental results). . . . . . . . . . . . . . . . . . 128

Figure 7.14

Double-layer 12 SPP PM Machine with Insulation on 10% of turns in
a slot Degraded( 12 Ω) (experimental results). . . . . . . . . . . . . . . 129

Figure 7.15

Double-layer 12 SPP PM Machine with Insulation on 10% of turns in
a slot Degraded( 14 Ω) (experimental results). . . . . . . . . . . . . . . 130

xvii

Figure 7.16

Double-layer 12 SPP PM Machine with Insulation on 20% of turns in
a slot Degraded( 12 Ω) (experimental results). . . . . . . . . . . . . . . 131

Figure 7.17

Double-layer 12 SPP PM Machine with Insulation on 20% of turns in
a slot Degraded( 14 Ω) (experimental results). . . . . . . . . . . . . . . 132

Figure 7.18

Single-layer 25 SPP PM Machine with Insulation on a Single-turn
Degraded( 21 Ω) (experimental results). . . . . . . . . . . . . . . . . . 133

Figure 7.19

Single-layer 25 SPP PM Machine with Insulation on a Single-turn
Degraded( 41 Ω) (experimental results). . . . . . . . . . . . . . . . . . 134

Figure 7.20

Single-layer 25 SPP PM Machine with Insulation on 10% of turns in
a slot Degraded( 12 Ω) (experimental results). . . . . . . . . . . . . . . 135

Figure 7.21

Single-layer 25 SPP PM Machine with Insulation on 10% of turns in
a slot Degraded( 14 Ω) (experimental results). . . . . . . . . . . . . . . 136

Figure 7.22

Single-layer 25 SPP PM Machine with Insulation on 20% of turns in
a slot Degraded( 12 Ω) (experimental results). . . . . . . . . . . . . . . 137

Figure 7.23

Single-layer 25 SPP PM Machine with Insulation on 20% of turns in
a slot Degraded( 14 Ω) (experimental results). . . . . . . . . . . . . . . 138

Figure 7.24

Phase voltage of Double-layer 12 SPP PM Machine with Varying Number of Turns Having Degraded Insulation from Finite Element Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 7.25

Phase voltage of Single-layer 25 SPP PM Machine with Varying Number of Turns Having Degraded Insulation from Finite Element Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 8.1

Experimental Voltage Estimate for Single-layer 25 SPP PM Machine
Applying a 15A@120◦ Current Command . . . . . . . . . . . . . . . 142

Figure 8.2

Finite Element Torque for Single-layer 25 SPP PM Machine with incipient faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

xviii

Chapter 1
Introduction
Permanent magnet synchronous machines are appealing for aerospace and hybrid/electric
transportation applications due to their many advantages over other machine designs, such
as higher efficiency, higher power density and faster dynamics in a compact package. However, permanent magnet machines are subject to electrical, magnetic and mechanical failures.
These failures are a challenge because the rotating magnets cannot be turned off.

In aerospace and automotive industries a very high importance is placed on passenger
safety. Machine failures in these industries put human lives at risk [1]. For this reason,
high reliability and robustness are required of permanent magnet synchronous machines.
Machines used in aerospace and automotive applications must have a low possibility of
failure as well as continue to operate or fail safely in the event of a failure. Technology has
advanced the reliability of electrical machines; however, machine failures still occur.

1.1

Objectives and Contributions

Stator winding failures are among the most common. About 35% of electrical machine
failures are stator related. 70% of the stator related failures are due to failures in the insulation [2]. Low probability of stator winding failure together with the ability to continue
operating if the winding insulation fails can be accomplished through fault tolerant machine
1

design. According to [3], the turn-to-turn short-circuit in a single phase is more severe than a
phase-to-phase short-circuit. For this reason, the authors conclude that a double-layer fractional slot concentrated winding permanent magnet synchronous machine design is as reliable
as a single-layer. In this work, the reliability of both a double- and single-layer fractional
slot concentrated winding interior permanent magnet synchronous machine is compared.

Both the double- and single-layer fractional slot concentrated winding designs options
offer advantages. The primary difference is two phases share a stator slot with double-layer
windings. The MMF sub-harmonics in double-layer windings typically have lower amplitude. However, in single-layer windings, the MMF consists of the main and sub-harmonics
which lead to unbalanced magnetic loading of the rotor resulting in saturation, decreased
average torque, unbalanced pull forces and high torque ripple. The advantage of selecting
the double-layer winding design is reduced sub-harmonics and losses. Single-layer fractional
slot concentrated windings provide the best isolation between phases; however, the inherent
characteristics of this design negatively impact the machine performance. In this work, a
control technique is developed that improves the machine performance by reducing torque
ripple.

Permanent magnet synchronous machines are typically modeled in the two-axes synchronous frame of reference. DQ theory assumes sinusoidal MMF; however, the arrangement
of fractional slot concentrated windings produces a non-sinusoidal MMF. According to [4],
the DQ model remains reliable. Self- and cross-saturation are non-linear effects that have
been proven important for development of high performance permanent magnet synchronous
machine controllers. In [5], it was determined that cross-coupling is not significant in perma2

nent magnet machines with fractional slot concentrated windings. The frozen permeability
method was utilized to prove that the classical d-q model of permanent magnet synchronous
machines is accurate when fractional slot concentrated windings are used. In this work,
the cross-coupling parameters of both double- and single-layer fractional slot concentrated
winding interior permanent magnet synchronous machines are evaluated.

The need for fail-safe operation of permanent magnet synchronous machines in applications requiring high reliability has prompted development of techniques for detecting incipient faults. Without knowledge of degrading stator winding insulation, a stator inter-turn
short could develop and lead to catastrophic failure. It is desired to detect the insulation
degradation during normal machine operation to avoid these failures. In this work, use of
extracted machine parameters to monitor the condition of the stator winding insulation is
evaluated. Degrading insulation leads to variations in the machine characteristic parameters;
the flux linkages will change as the fault progresses. It is well known that the use of accurate machine parameters for control improves a machine’s operating performance. Accurate
knowledge of the parameters is also useful for condition monitoring.

The contributions of this work are:
• Development of an analytical expression of torque developed by permanent magnet
synchronous machines with fractional slot concentrated windings, including permeance
variations.
• Development of a control technique for reducing torque ripple in permanent magnet
synchronous machines with fractional slot concentrated windings.

3

• Demonstrating that indicators of incipient stator winding faults are present in the dand q-axis flux linkages of permanent magnet synchronous machines with fractional
slot concentrated windings.

1.2

Organization

In Chapter 2 stator winding insulation stator winding insulation failure in permanent magnet synchronous machines with sinusoidally distributed windings is examined. An analytical
model of a permanent magnet synchronous machine with stator winding insulation failure
is presented. Additionally, a finite element model is used to evaluate the machine’s performance when a turn-to-turn short-circuit is present. Chapter 3 describes techniques that
are available for improving permanent magnet synchronous machine reliability. Designs for
two fractional slot concentrated winding interior permanent magnet synchronous machine
designs are presented in Chapter 4. One machine is designed with double-layer windings,
while the other with single-layer windings. Experimental setup and characterization results
are provided. Additionally, a finite element model of the fault tolerant machine designs
is used to evaluate the machine characteristic inductances and machine performance under
stator inter-turn failure. Chapter 5 presents an analytical description of torque including
the sources of torque ripple. A technique for reducing torque ripple is presented in Chapter
6. In Chapter 7 the finite element and experimental parameters of the two fault tolerant
design under incipient fault conditions. Finally, conclusions and anticipated future work are
presented in Chapter 8.

4

Chapter 2
Stator Winding Insulation Failure
Permanent magnet machine failures include core insulation degradation, stator winding or
insulation failures, electrical trips, demagnetization of the magnets, bearing failures and
loss of stator/rotor mechanical integrity. Root causes of these failures have been identified as defective design/manufacturing, defective materials/components, incorrect installation/operation, ambient conditions, fatigue/stress and debris/corrosion [6].

The stator of a permanent magnet machine can experience faults in both the winding
and core. Root cause analysis of failed stators has shown that thermal deterioration of
the insulation, short/long term electrical stress on the insulation, mechanical stress induced
by vibration of the winding, dirt or moisture contaminating the winding and manufacturing/design flaws (i.e., loose bracing for the end winding, slack in the core laminations, slot
wedges and joints) lead to faults in stators [6, 7].

Stator winding failures are among the most common in electric machines [1, 6–9]. Most
winding faults are caused by insulation deterioration, resulting from thermal stress due to
aging, overloading or cycling. Also, demagnetization of the magnets, even partially, causes
current to flow in the stator windings that exceed the ratings and degrade the winding insulation. Degradation of the insulation, as shown in Figure 2.1, could result in a turn-to-turn

5

Figure 2.1: Stator Winding with Damaged Insulation. (For interpretation of the references
to color in this and all other figures, the reader is referred to the electronic version of this
disseration.)
short-circuit which increases the current flowing in that phase and the amount of heat generated.

Turn-to-turn short-circuit faults are among the most common faults occurring in electrical
machines. This fault could permanently demagnetize the magnets by producing a magnetic
field intensity that exceeds the coercivity of the magnets. High temperature could increase
the fault severity, leading to phase-to-phase or phase-to-ground faults. Due to heat, shortcircuit faults progress quickly and can lead to a fire.

A turn-to-turn short-circuit generates a circulating current which increases temperature.
The heat can lead to insulation of other turns failing, a phase-to-phase short-circuit or even
phase-to-ground short circuit. The circulating current also affects the performance of the
permanent magnet synchronous machine. The machine model is useful for identifying how

6

stator turn-to-turn short circuits affect the permanent magnet synchronous machine performance.

In this chapter both an analytical model and a finite element model of permanent magnet
synchronous machines with stator inter-turn faults are examined.

2.1

Analytical Model of PMSM with Stator Inter-turn
Faults

Several authors have used equivalent circuit models to study the effects of inter-turn winding
faults [10–12]. The equivalent circuit model of a permanent magnet synchronous machine
with an inter-turn winding fault in one phase is shown in Figure 2.2. The fault is modeled
as a resistance, Rf , across a portion of the series connected phase a winding. The machine
equations provide an indication of how the fault affects the machine performance. Total
phase a voltage, Equation (2.1), is the sum of the voltages across the healthy, va1 , and faulty,
va2 , portions of the winding. The healthy and faulty portions of the phase are identified
as a1 and a2 , respectively. Their voltages are described in Equations (2.2) and (2.3) where
rs1,s2 are the resistances of the healthy and faulty portions of the phase a winding, ia is the
phase a current, if is the fault current and λas1,as2 are the flux linkages of the healthy and
faulty portions of the phase a winding. Voltages of phase b and c, vb and vc , are described
in Equations (2.4) and (2.5) where rs is the phase resistance, ib and ic are the phase currents
and λbs and λcs are the flux linkages. The phase back-EMFs are eas1 , eas2 , ebs and ecs .

va = va1 + va2
7

(2.1)

Figure 2.2: Equivalent circuit model of winding with inter-turn fault

va1 = rs1 ia +

va2 = rs2 (ia + if ) +

dλas1
+ eas1
dt

dλas2
+ eas2 = −Rf if
dt

(2.2)

(2.3)

vb = rs ib +

dλbs
+ ebs
dt

(2.4)

vc = rs ic +

dλcs
+ ecs
dt

(2.5)

Assuming that each phase has the same number of conductors, Ns and the windings are
sinusoidally distributed, the MMFs for all of the phases can be described, where Ns,a , Ns,b
and Ns,c are the number of conductors for each phase. The number of conductors involved

8

in the fault is Ns,a2 and Ns,a1 is the number of healthy phase a conductors, φs is the stator
angular displacement.
Ns,a = Ns,b = Ns,c = Ns

(2.6)

Ns,a = Ns,a1 + Ns,a2 = Ns

(2.7)

M M Fas = M M Fas1 + M M Fas2

(2.8)

Ns,a1
ia cos (φs )
2

(2.9)

Ns,a2
(ia + if ) cos (φs )
2

(2.10)

M M Fbs =

Ns
2π
ib cos (φs −
)
2
3

(2.11)

M M Fcs =

Ns
2π
ic cos (φs +
)
2
3

(2.12)

M M Fas1 =

M M Fas2 =

µ, the ratio of faulted turns to total turns, can be used to describe the number of turns
involved in the fault as shown in Equations (2.13) to (2.15).

µ=

Ns,a2
Ns

9

(2.13)

Ns,a2 = µNs

(2.14)

Ns,a1 = (1 − µ)Ns

(2.15)

The flux linkages are described in Equations (2.16) to (2.19) as functions of self and mutual
inductances.
λas1 = La1a1 ia + La1a2 (ia + if ) + La1b ib + La1c ic

(2.16)

λas2 = La2a1 ia + La2a2 (ia + if ) + La2b ib + La2c ic

(2.17)

λbs = Lba1 ia + Lba2 (ia + if ) + Lbb ib + Lbc ic

(2.18)

λcs = Lca1 ia + Lca2 (ia + if ) + Lcb ib + Lcc ic

(2.19)

The phase back-EMFs are described in Equations (2.20) to (2.22) where λpm is the permanent magnet flux linkage, the rotor speed is ωr and the rotor position is θr .

eas = eas1 + eas2 = ωr λpm cosθr

(2.20a)

eas1 = (1 − µ)ωr λpm cosθr

(2.20b)

eas2 = µωr λpm cosθr

(2.20c)

10

ebs = ωr λpm cos(θr −

2π
)
3

(2.21)

ecs = ωr λpm cos(θr +

2π
)
3

(2.22)

The machine inductances can be described from the airgap flux densities.

µ
B = 0 MMF
g

(2.23)

Self inductance of the healthy and faulty portions of phase a, La1a1 and La2a2 , are shown
in Equations (2.24) and (2.25), where Lls and Lm are the leakage and magnetizing inductances, respectively. The mutual inductances between the healthy and faulty portions,
La1a2 , of phase a is shown in Equation (2.26). Self inductance of phases b and c, Lbb and
Lcc , are shown in Equation (2.27). Mutual inductances between phases b and c, Lbc and
Lcb , are described in Equation (2.28). Mutual inductances between the healthy portion of
phase a and the other two phases, La1b and La1c , are described in Equation (2.29). Mutual
inductances between the faulty portion of phase a and the other two phases, La2b and La2c ,
are described in Equation (2.30).

La1a1 = (1 − µ)2 (Lls − Lm )

(2.24)

La2a2 = µ2 (Lls − Lm )

(2.25)

11

La1a2 = La2a1 = −µ(1 − µ)Lm

(2.26)

Lbb = Lcc = Lls + Lm

(2.27)

1
Lbc = Lcb = − Lm
2

(2.28)

1
La1b = Lba1 = La1c = Lca1 = − (1 − µ)Lm
2

(2.29)

1
La2b = Lba2 = La2c = Lca2 = − µLm
2

(2.30)

As shown in Equations (2.24) to (2.30), turn-to-turn faults in the stator winding will be
reflected in the machine characteristic inductances. Since the flux linkages are functions of
the inductances, they are also affected by turn-to-turn faults.

The open-circuit voltage of healthy permanent magnet synchronous machines is simply produced by the magnet flux.

Equations (2.2) to (2.5) can be used to evaluate the

open-circuit voltage for a permanent magnet synchronous machines with turn-to-turn short
circuits.
dif
va1 = La1a2
+ (1 − µ)ωr λpm cosθr
dt
dif
= −µ(1 − µ)Lm
+ (1 − µ)ωr λpm cosθr
dt

12

(2.31)

dif
va2 = La2a2
+ µωr λpm cosθr
dt
dif
+ µωr λpm cosθr
= µ2 (Lls − Lm )
dt

(2.32)

dif
2π
+ ωr λpm cos(θr −
)
vb = Lba2
dt
3
dif
1
2π
= − µLm
+ ωr λpm cos(θr −
)
2
dt
3

(2.33)

dif
2π
+ ωr λpm cos(θr +
)
dt
3
dif
2π
1
+ ωr λpm cos(θr +
)
= − µLm
2
dt
3

(2.34)

vc = Lca2

The circulating current, due to the turn-to-turn short-circuit, affects the open-circuit voltage
as shown in Equations (2.31) to (2.34). Evaluation of the equations indicates that this effect
can be minimized by reducing the mutual inductance.

The machine torque can be evaluated through the energy stored in the coupling field.
Assuming a linear magnetic system, the energy stored in the coupling field is described
in Equation (2.35).
1
1
Wf = La1a1 ia 2 + La1a2 ia (ia + if ) + La2a2 (ia + if )2
2
2
1
+ La1b ia ib + La2b (ia + if )ib + Lbb ib 2
2
1
+ La1c ia ic + La2c (ia + if )ic + Lbc ib ic + Lcc ic 2
2

13

(2.35)

The machine torque can be calculated by Equation (2.36). The circulating current also
affects the machine torque.
T =p

2.2

∂Wf
∂θr

(2.36)

Finite Element Model of PMSM with Stator Interturn Faults

The effects of stator inter-turn faults, described analytically in Section 2.1 can be verified through finite element analysis. A 2 slot per pole per phase PMSM with sinusoially
distributed windings was modeled in finite element. Design parameters for the model are
provided in Table 2.1. The geometry of both the healthy and faulty cases are shown in Figure 2.3. The red regions in the stator slots of Figure 2.3b, represent the turns involved in the
short-circuit. The primary difference between the two models is the circuit, shown in Figure 2.4. The circuit for the faulty model has two additional coil conductors in the phase
where the fault is introduced. One of the additional coil conductors has a resistor connected
in parallel to simulate the fault severity. The region associated with this coil conductor is
red in Figure 2.3b. The other coil conductor simulates the healthy turns of the faulty phase
that share the slot with faulted turns.

The torque due to the interaction between the permanent magnets in the rotor and the
stator slots, or cogging torque, was evaluated using finite element. The cogging torque is
shown in Figure 2.5.
The open-circuit voltage for the healthy machine model is shown in Figure 2.6. The
faulty machine model is used to evaluate the open-circuit voltage for different number of
14

(a) Healthy

(b) Faulty
Figure 2.3: FEA geometry of PMSM with 2 slots per pole per phase:(a)healthy (b)faulty.

15

Number of stator slots
Number of poles
Coil Pitch
Number of turns per coil
Number of turns per slot
Stack Length
Maximum Current
Stator Line Voltage
Maximum Power

24
4
◦
150 (5 slots)
32
64
72mm
18Arms
480V
10kW

Table 2.1: Design Parameters of PMSM modeled in FEA

(a) Healthy

(b) Faulty
Figure 2.4: FEA circuit model of PMSM with 2 slots per pole per phase:(a)healthy (b)faulty

16

Torque (Nm)

0.4
0.2
0
−0.2
−0.4
0

5
10
position (degrees)

15

Figure 2.5: Cogging Torque of Healthy PMSM with 2 slots per pole per phase (finite element
results)
turns involved in the slot. The effect of the short-circuited turns is apparent in the opencircuit voltage, as expected from Equations (2.31) to (2.34). The open-circuit voltage of the
faulty machine with a short-circuit involving a single-turn, 10% of the turns in the slot and
20% of the turns in the slot are shown in Figures 2.7 to 2.9. This effect is due to the current
circulating in the shorted turns, shown in Figures 2.10 to 2.12. The circulating current is
more than 9 times higher than the motor’s maximum current given in Table 2.1, in each case.
Torque ripple also increases as the number of turns involved in the short-circuit increases,
as summarized in Table 2.2.

17

60

Voltage (V)

40
20
0
Phase A
Phase B
Phase C

−20
−40
−60
0

50
100
150
position (degrees)

Figure 2.6: Healthy back-EMF of PMSM with 2 slots per pole per phase (finite element
results)

60
Voltage (V)

40
20
0
−20

Phase A
Phase B
Phase C

−40
−60
0

50
100
150
position (degrees)

Figure 2.7: Back-EMF of PMSM with 2 slots per pole per phase and a single turn in a slot
short-circuited (finite element results)

18

Voltage (V)

50

0
Phase A
Phase B
Phase C

−50
0

50
100
150
position (degrees)

Figure 2.8: Back-EMF of PMSM with 2 slots per pole per phase and 10% of turns in a slot
short-circuited (finite element results)

Voltage (V)

50

0
Phase A
Phase B
Phase C

−50
0

50
100
150
position (degrees)

Figure 2.9: Back-EMF of PMSM with 2 slots per pole per phase and 20% of turns in a slot
short-circuited (finite element results)

19

300

Current (A)

200
100
0
−100
−200
−300
0

50
100
150
position (degrees)

Figure 2.10: Circulating current due to a single turn in a slot involved in a short-circuit
(finite element results)

600
Current (A)

400
200
0
−200
−400
−600
0

50
100
150
position (degrees)

Figure 2.11: Circulating current due to a 10% of the turns in a slot involved in a short-circuit
(finite element results)

20

400

Current (A)

200
0
−200
−400
0

50
100
150
position (degrees)

Figure 2.12: Circulating current due to 20% of the turns in a slot involved in a short-circuit
(finite element results)

Torque (Nm)

70
65
60
55
50
0

50
100
position (degrees)

150

Figure 2.13: Torque of healthy PMSM with 2 slots per pole per phase (finite element results)

21

Torque (Nm)

70
65
60
55
50
0

50
100
position (degrees)

150

Figure 2.14: Torque of 2 slots per pole per phase PMSM with a single turn in a slot shortcircuited (finite element results)

70

Torque (Nm)

65
60
55
50
45
40
0

50
100
position (degrees)

150

Figure 2.15: Torque of 2 slots per pole per phase PMSM with 10% of turns in a slot shortcircuited (finite element results)

22

Torque (Nm)

70
60
50
40
30
0

50
100
position (degrees)

150

Figure 2.16: Torque of 2 slots per pole per phase PMSM with 20% of turns in a slot shortcircuited (finite element results)

Machine Condition Circulating Current Torque Ripple
Healthy
11.99%
Single-Turn
163.61Arms
14.33%
10% Turns
273.61Arms
32.21%
20% Turns
174.75Arms
52.65%
Table 2.2: Circulating current in a PMSM designed with sinusoidally distributed windings
under short-circuit faults. Torque ripple as a percentage of the average torque in a healthy
machine.

23

Chapter 3
PMSM Reliability
One approach to improving machine reliability is through design. Permanent magnet synchronous machines can be designed to better tolerate stator winding inter-turn short-circuits.
Machines designed to tolerate failures are called fault tolerant machines. However, designing
for fault tolerance alone will not provide the desired level of reliability for ensuring passenger
safety. Accurate detection and diagnosis of a pending fault would provide an additional level
of reliability.

This chapter introduces available techniques for improving reliability through design and
fault detection.

3.1

Fault Tolerant Design

Stator inter-turn short-circuits in permanent magnet machines can experience high values
of circulating currents in the shorted turns magnitudes above the machine rated current.
The circulating current results in higher temperatures. In [7] the authors state that the
winding degrades twice as fast as under normal operating conditions for every 10◦ C above
the insulation rated temperature. With increased winding temperature a turn-to-turn short
circuit may advance to include additional turns or even lead to a secondary fault, i.e., phaseto-phase short, phase-to-ground short or even demagnetization.
24

The authors of [13] identify requirements for fault tolerant permanent magnet synchronous machines. The requirements include considerations for:
1. Number of phases
2. Limiting of fault currents
3. Magnetic isolation between phases
4. Thermal isolation between phases
5. Physical isolation between phases
The first requirement listed, increasing the number of phases, reduces the fault current by
minimizing the coupling between the magnet flux and stator windings. Meeting the second
requirement reduces the effect that an inter-turn short-circuit has on the open-circuit voltage
and torque. According to [14], using additional phases provides a few benefits.
• Lower amplitude of torque pulsations
• Lower stator current per phase without increasing the voltage per phase
• Higher torque per amp for the same volume machine
However, machines with more than three phases are overrated to provide the desired
operation in case of failure, in which case the failed phase is disconnected. The tradeoff for
complexity, F , is a function of the number of phases, m, as shown in Equation (3.1). [15].

F =

m
m−1

25

(3.1)

The fault tolerance of permanent magnet machines with more than three phases has
been demonstrated by several researchers [14–20]; however, these machines are more complex to control and require additional power electronic components. Machine reliability is
high priority for aerospace and automotive applications; however, cost of manufacturing is
also a priority. The cost associated with the complexity may not be as appealing for mass
production in these industries.

In [13,21], inter-turn short-circuits sharing a stator slot were examined. It was determined
that the circulating current opposes the air-gap flux and increases the slot leakage flux in
[21]. The authors show that by modifying the stator tooth shape, the slot leakage inductance
increases resulting in reduced short-circuit current. Stator winding configuration also affects
the phase inductance and can be used to effectively limit the short-circuit current.

3.2

Stator Winding Configurations

There are two categories of stator winding configurations: overlapping and non-overlapping
end-windings [22], shown in Figures 3.1 and 3.2, respectively. The categories are distinguished by the end-windings. Windings in the overlapping end-windings category have
different phases in close proximity of each other; the configurations are distributed or concentrated. Distributed windings, shown in Figure 3.1a are arranged to produce a sinusoidal
MMF. Concentrated windings, shown in Figure 3.1b are wrapped around a tooth. In this
category, concentrated windings have 1 slot per pole per phase.

Windings in the non-overlapping end-windings category are arranged such that different

26

(a) Distributed

(b) Concentrated
Figure 3.1: Overlapping Winding Category - (a)Distributed and (b)Concentrated

27

phases do not touch. This winding configuration has less than one slot per pole per phase
and is called fractional slot concentrated windings. The configurations are concentrated,
wrapped around a single tooth, and are arranged as either double- or single-layer. In the
double-layer windings, every tooth is wound, as shown in Figure 3.2a. Single-layer windings
have every other tooth wound, as shown in Figure 3.2b.

The characteristics of fractional slot concentrated winding machines make them a viable
solution for applications requiring fault tolerance [23]. Designs with fractional slot concentrated windings provide magnetic, thermal and physical separation of phases.Single-layer
windings eliminate the possibility of overlapping between end windings of different phases.
This lends to low mutual coupling between phases and provide better thermal isolation since
different phases do not share stator slots. The low mutual coupling indicates that the short
circuit current in one phase will not affect the other phases.

3.3

Effects of Winding Configurations

The MMF of permanent magnet synchronous motors with different winding configurations
are shown in Figures 3.3a and 3.3b. Due to the arrangement of the windings, the overlapped distributed windings, shown in Figure 3.1a, produce a nearly sinusoidal MMF as
shown in Figure 3.3a. The arrangement of the non-overlapped double-layer windings, shown
in Figure 3.2a, produces a less sinusoidal MMF as shown in Figure 3.3b. The fractional
slot concentrated winding MMF contains extra harmonic components when compared to an
integral slot distributed winding, as shown in Figure 3.4. The MMF harmonics also create
flux in the machine that induce voltages at fundamental frequency in the windings.

28

(a) Double-layer

(b) Single-layer
Figure 3.2: Non-overlapping Winding Category - (a)Double-layer and (b)Single-layer

29

As with conventional machine analysis, the winding function can be used to calculate the
self and mutual inductances of fractional slot concentrated windings; however, the effect of
the harmonics must be considered. In [24] it is stated that in fractional slot concentrated
winding machines saturation influences both the d-and q-axis inductance more than in distributed windings. It is also noted that higher order harmonics in the mutual inductance
affect saliency. In [25], there are four components of leakage inductance, Lσ , presented that
should be considered when evaluating winding configurations. Equation (3.2) describes the
total leakage inductance due to the winding configuration.
1. Airgap leakage inductance, Lδ
2. Slot leakage inductance, Lu
3. Tooth-tip leakage inductance, Lt
4. End winding leakage inductance, Lw

Lσ = Lδ + Lu + Lt + Lw

(3.2)

The airgap leakage component is described in Equation (3.3) is a function of the machine
geometry. The permeability of air,

µ0 , the number of phases, m, the effective airgap

length, δ, the machine diameter, D, the effective core length, l , and the slot permeance
factor, λu are all used to calculate the airgap leakage component.

µ m
Lδ = 0 Dl
πδ

∞
N 2
p

ν=1

30

kw ν 2
ν

(3.3)

80
60
40
MMF

20

Actual
Sinusoidal Approximation

0
−20
−40
−60
−80
0

100

200
Position (degrees)

300

(a) Distributed Winding

100

MMF

50
Actual
Sinusoidal Approximation
0

−50

−100
0

50

100

150
200
250
Position (degrees)

300

350

(b) Fractional Slot Concentrated Winding
Figure 3.3: MMF of different winding configurations - (a)Distributed windings with 2 SPP
and (b)Fractional Slot Concentrated Windings with 12 SP P

31

80

Magnitude

60
40
20
0
0

10

20
30
Harmonic order

40

(a) Distributed Winding

Magnitude

80
60
40
20
0
0

10

20
30
Harmonic order

40

(b) Fractional Slot Concentrated Winding
Figure 3.4: MMF harmonic spectrum of different winding configurations - (a)Distributed
windings with 2 SPP and (b)Fractional Slot Concentrated Windings with 12 SP P

32

The slot leakage inductance, in Equation (3.4) is significantly higher in fractional slot
concentrated winding machines when compared to machines with distributed windings [26].
For this reason, an accurate estimation of the slot leakage inductance is suggested, including
mutual slot leakage for double-layer windings.

Lu =

4m
µ l N 2 λu
Q 0

(3.4)

According to [26], the tooth-tip, in Equation (3.5), and end leakage, in Equation (3.6)
inductances are negligible.

Lt =

Lw =

4m
µ l λ N2
Q 0 d

(3.5)

4m 2
qN µ0 lw λw
Q

(3.6)

As described in [27], joule loss for one phase, shown in Equation (3.7), is a function of the
average conductor length, Lc , the slot fill factor, cf , and the winding factor, kw . Fractional
slot concentrated windings can also have a low winding factor which leads to higher heat
generation, iron saturation and lower efficiency.

Pwinding =

Lc
cf K w 2

(3.7)

The MMF harmonics that are asynchronous with the rotor, ν = p, induce current in
the rotor. In [28], it is stated that due to the higher wavelength of subharmonics, the flux
lines enter deeply in the rotor, implying that subharmonics lead to higher rotor loss. The
subharmonics also lead to an increased torque ripple, as concluded in [29].
33

Several authors have presented design techniques for reducing losses and torque ripple.
The authors of [30] demonstrate in finite element that by using a different number of turns
for each coils side and adding flux barriers to the back iron of every other slot the magnet
and iron loss associated with fractional slot concentrated windings is reduced. In [31], two
stators with the same slot per pole per phase combination shifted by an electrical angle will
cancel harmonics that contribute to rotor loss. The authors of [32] develop an analytical
expression of torque neglecting permeance variations. They conclude that selecting stator
windings with an odd number of slots per pole pair reduces the torque ripple. In [33], the
torque expression developed in [32] is used to demonstrate the the number of stator and
rotor slots used in the machine design affect the losses and torque ripple. The authors of [34]
add slits on the q-axis of the rotor surface and modify the stator tooth face to reduce the
torque harmonics. The addition of flux barriers and stators provide for more complex manufacturing.

The MMF harmonics of fractional slot concentrated windings contribute extra stator
leakage inductance; this however, may be helpful in reducing current for field weakening and
reducing short-circuit current. As presented in [35], a fault tolerant PMSM design should
have high magnetizing inductance to limit short-circuit current. Fractional slot concentrated
windings offer other advantages.
• A reduction in the volume of copper in the end region
• A reduction of the joule loss in the end region due to the shorter end turns
• Reduced mutual coupling between the phases because no tooth carries coils for more
34

than one phase
• Higher slot fill factor
• Reduced cost because the manufacturing is simplified
• Lower cogging torque, due to the higher cogging torque frequency, fc
The cogging torque frequency is described as frequency, f , multiplied by the least common
multiple of the number of poles, where p is the number of pole pairs, and the number of
stator slots, Q, as shown in Equation (3.8). Higher cogging frequency yields lower cogging
amplitude.

fc = LCM (2p, Q)f

3.4

(3.8)

Stator Winding Failure Detection

Machine reliability can also be improved through accurate detection and diagnosis of pending
stator winding inter-turn faults. Stator winding inter-turn faults lead to unbalanced air gap
voltages and line currents, increased torque pulsations, decreased average torque, increased
losses, reduced efficiency and excessive heating. Stator inter-turn faults are effectively imbalances reflected in signals. However, these signals may be stationary, cyclostationary,
non-stationary or even buried in a machine-borne noise.

Condition monitoring is a means of measuring signals and through analysis extract a
detection and diagnosis of a fault. Techniques have been developed to diagnose faults from
various measured signals, including stray flux, air gap flux, rotor field voltage, torque and
35

stator current. These techniques can be separated into two categories: 1) intrusive and 2)
non-intrusive. Intrusive techniques require additional equipment whereas non-intrusive techniques do not. The goal of condition monitoring is to diagnose the machine with minimal
measurements. Inter-turn short-circuits can be detected in torque, air-gap flux and stator
currents, to name a few.

The voltage reference has been used as a fault indicator. A turn-to-turn fault in one
winding has been shown to reduce the number of effective turns in that winding. This parametric variation is reflected at the voltage references. The difference between the non-fault
and fault voltage references indicates a fault when the preset threshold is exceeded. The
fault estimation threshold may vary since the voltage reference is speed and load dependent
[36].

The negative-sequence voltage component has also been used to diagnose stator winding
faults. One disadvantage is the zero-sequence voltage component measurement requires access to the machine neutral [37].

Stator turn-to-turn faults can increase slot leakage flux and decrease the main air gap
flux [37]. The torque is also affected by the increased slot leakage flux [21]. In reference [38] the permeance network is used to identify the inductances of a surface mounted
permanent magnet machine, including the effects of saturation and leakage. The authors
validate their results through comparison to finite element analysis finding good accuracy
for use in a machine model. Under fault, the inductances are no longer only functions of the
number of turns and slot geometry but dependent on the location of the fault in the slot [39].
36

Machine parameters have been used in [40–42] to detect inter-turn short circuit faults.
In [40] the authors present a scheme for estimating the winding resistance and rotor flux
linkages by injecting a short pulse in the d-axis current. The authors validate that their
method can detect changes in the winding resistance due to heat; however, state that the
scheme is suitable for monitoring non-salient pole permanent magnet synchronous machines.
In [41] the recursive least squares algorithm and the Extended Park model are used to
estimate the machine parameters. The authors show that winding short circuit faults can be
detected through variations in the estimated parameters. The parametric variations indicate
the number of turns involved in the short circuit. In [42], a parametric model of a 5-phase
permanent magnet machine is used to detect winding short circuit faults from the secondorder harmonic of the q-axis current. The authors of [41, 42] both validate their methods
through simulation for short-circuit winding faults.

The time between a single inter-turn short circuit and catastrophic failure of the machine
is short. Catastrophic permanent magnet machine failure can be avoided if a degrading stator
winding insulation was detected during normal operation. Detecting the degradation prior
to the existence of a turn-to-turn short circuit would allow for timely maintenance and repair.

37

Chapter 4
Design and Analysis of Fault Tolerant
Machines
Challenges with fractional slot concentrated windings include high harmonic content in the
MMF distribution, the production of losses in the rotor, unbalanced saturation that leads
to torque ripple, high slot leakage inductance and high acoustic noise for some slot/pole
combinations. It is shown in [35] that through careful selection of the slot/pole combination
the rotor losses, torque ripple and acoustic noise can be reduced.

The most common slot per pole per phase selections are 12 and 25 . According to [11],
1
2

slot per pole per phase is a good selection for reduced rotor losses in both double and

single-layer windings. However, 12 slot per pole per phase machines may have high torque
ripple. 25 slot per pole per phase machines may have lower cogging torque and torque ripple.

In this chapter two fractional slot concentrated winding interior permanent magnet machines are designed with the most common slot per pole per phase selections, 12 and 25 . The
1
2

slot per pole per phase machine is designed with double-layer windings and the 52 slot

per pole per phase with single-layer windings. Experimental characterization and finite element results are presented for both machines. It is shown through finite element analysis

38

that single-layer fractional slot concentrated windings with null mutual inductance provide
highest reliability at the cost of machine torque performance.

4.1
4.1.1

Tools
Star of Slots Theory

The star of slots theory presented by Bianchi [35] is a tool used to design permanent magnet
machines with fractional slot concentrated windings and evaluate the machine performance.
The star of slots is a complex representation of the main EMF harmonic induced in the
coil side of each slot of the machine, drawn only for the left-hand coil sides. The machine
periodicity, t, is the greatest common divisor of the number of stator slots, Q, and the
number of pole pairs, p, as shown in Equation (4.1).

t = GCD(Q, p)

Q
t

(4.1)

spokes form the star of slots and each spoke contains t phasors. Equation (4.2) is used to

examine the feasibility of a certain slot/pole combination. The winding is considered feasible
if the number of spokes per phase, qph , is an integer. The number of spokes is determined
by the number of stator slots divided by the product of the number of phases, m, and the
machine periodicity.
qph =

Q
∈N
mt

(4.2)

The angle between the phasors of adjacent slots, αse , is the electrical angle in Equation (4.3).
The angle between adjacent spokes, αph , is the electrical angle in Equation (4.4). A phase

39

π radians, rotate both sectors by 2π
contains the phasors of opposite sectors each covering m
m

radians to identify other phases.
αse =

2πp
Q

(4.3)

αph =

2πt
Q

(4.4)

The star of slots can also be used to analyze the harmonic content of the EMF waveform
and the air-gap MMF distribution. If Qt is even, the main winding distribution will only
include harmonics of odd order; however, if Qt is odd, the main winding distribution will
include both even and odd order harmonics. Additionally, if the number of spokes per phase
is even, there is an even number of EMF phasors to sum and the main distribution factor,
kd , is found by Equation (4.5). If the number of spokes per phase is odd, the distribution is
given by Equation (4.6).
π )
sin ( 2m
kd = q
αph
ph
2 sin ( 2 )

kd =

π )
sin ( 2m

α
qph sin ( 4ph )

(4.5)

(4.6)

The coil span angle, σw , and pitch factor, kp , are described by Equations (4.7) and (4.8),
respectively.
σw =

2πpyq
Q

kp = sin (

40

σw
)
2

(4.7)

(4.8)

The main winding factor, kw , is given by Equation (4.9).

kw = kd kp

(4.9)

A high winding factor does not guarantee desirable machine performance because if all coil
sides of one phase are on the same side of the machine, the winding will produce unwanted
unbalanced magnetic pull resulting in noise.

The star of slots theory also provides a means of transforming a double-layer winding
to single-layer and evaluating its performance, simply by removing all of the even phasors
from the star of slots; however, there are a few constraints, both geometrical and electrical.
The two geometrical constraints are 1) Q must be even to maintain symmetry between the
phases and 2) the coil throw must be odd so that each slot contains only one coil side. The
electrical constraints can be divided into two categories: even and odd periodicity. If the
periodicity is even, the transformation is always possible; however, the harmonic content and
winding factor are determined by Qt . The evaluation of the harmonic content and winding
factor for the transformation to a single-layer can be described by several cases, as shown
in Table 4.1.
The star of slots theory provides a method for designing a machine with null mutual
inductance. In double-layer windings, the ratio of stator slots to periodicity must be an even
number. Single-layer windings will have null mutual inductance when the ratio of the slots
to twice the periodicity is even.

41

Case #
1
2
3
4
5

Periodicity
even
even
even
odd
odd

Q
t

Q
2t

odd
even even
even odd
even odd
even even

Distribution Harmonic Order
even & odd
odd
even & odd
odd
even & odd

Distribution Factor
same
increase
same
same
increase

Table 4.1: MMF harmonic order and distribution factor for transformation of double-layer
winding to single-layer winding for various cases based on machine periodicity and number
of stator slots.

4.1.2

Software

Two software packages were used to design and analysis the permanent magnet synchronous
machines with fractional slot concentrated windings: SPEED and Flux2D. SPEED, a lumpedparameter model, has a graphical interface that was used to design the stator laminations,
define the stator windings and analyze the resulting MMF harmonic content and winding
factors. Flux2D, time-dependent nonlinear finite element method with circuit coupling, was
used to analyze torque, voltage, current for each machine design. This finite element software was also used to analyze iron loss, eddy current loss in the magnets and copper loss.
The iron loss calculations, both eddy current and hysteresis, were computed from Bertotti
coefficients. Flux2D magneto-static simulations were used to calculate inductances.

4.2

Double-layer PM Machine Design

An available 4-pole neodymium iron boron permanent magnet rotor was used to design a
double-layer 12 slot per pole per phase machine. Star of slots theory was applied to select
the number of stator slots and examine the MMF harmonic content and the winding factor.

42

Figure 4.1: Star of slots for double-layer 12 SP P PM machine
The star of slots evaluation is shown in Equations (4.10) to (4.16) and Figure 4.1.

Q=6

(4.10)

p=2

(4.11)

m=3

(4.12)

t = GCD(Q, p) = 2
Q
= 1, f easible
mt
2πp
2π
αse =
=
Q
3
2π
2πt
αph =
=
Q
3
qph =

(4.13)
(4.14)
(4.15)
(4.16)

Since Qt is odd, the main winding distribution will contain both even and odd harmonic
orders. This also indicates that mutual inductance is a factor. It is observed that there
are no sub-harmonics present, since t = p. The winding factor calculations are shown

43

Design Parameter
Value
Stator Diameter
182.03mm
Rotor Diameter
98.34mm
Stack Length
72mm
Number of Stator Slots
6
Number of poles
4
Air gap length
1mm

Design Parameter
Value
Magnet Type
NdFeB N38EH
Winding Type
Double-layer Concentrated
Number of Turns
152 turns per coil
Stator Line Voltage
480V
Maximum Current
18Arms
Maximum Power
8kW

Table 4.2: Double-layer 12 SPP machine design parameters.
in Equations (4.17) to (4.20).

kd =

π )
sin( 2m

α
qph sin( 4ph )

σw =

=1

2πpyq
2π
=
Q
3

√
σw
3
kp = sin( ) =
2
2

(4.17)

(4.18)

(4.19)

√
kw = kp kd =

3
2

(4.20)

An outline of the machine is shown in Figure 4.2 and machine design parameters are
given in Table 4.2.

The stator and rotor of the manufactured prototype are pictured in

Figures 4.3a and 4.3b, respectively.

44

Figure 4.2: Double-layer 21 SP P PM machine laminations

45

(a) Stator

(b) Rotor

Figure 4.3: Manufactured parts of 12 SPP PM machine - (a)Stator and (b)Rotor with NdFeB
magnets-N38EH nickel coated

4.3

Single-layer PM Machine Design

A prototype single-layer 25 slot per pole per phase machine was designed. Star of slots
theory was applied to examine the MMF harmonic content and the winding factor. The star
of slots evaluation begins with the double-layer design as shown in Equations (4.21) to (4.26)

46

Figure 4.4: Star of slots for double-layer 25 SP P PM machine
and Figure 4.4.

Q = 12

(4.21)

p=5

(4.22)

t = GCD(Q, p) = 1

(4.23)

Q
= 4, f easible
mt
2πp
5π
αse =
=
Q
6
2πt
π
αph =
=
Q
6
qph =

(4.24)
(4.25)
(4.26)

Since Qt is even, the main winding distribution will contain only odd harmonic orders
and the design yields null mutual inductance. The winding factor calculations are shown

47

in Equations (4.27) to (4.30).
π )
sin( π6 )
sin( 2m
=
kd = q
αph
π ) = 0.966
ph
2 sin( 12
sin(
)
2
2

σw =

2πpyq
5π
=
Q
6

kp = sin(

σw
) = 0.966
2

kw = kp kd = 0.933

(4.27)

(4.28)

(4.29)

(4.30)

Q
t, Qt , 2t
are all even, according to Table 4.1 the harmonic orders remain unchanged how-

ever, the distribution factor increases which yields a higher winding factor, shown in Equations (4.31) and (4.32). The change in the distribution factor is a result of removing half of
the phasors, as shown in Figure 4.5. The winding harmonics include sub-harmonics, since
t = p, with magnitudes higher than the main harmonic.
π )
sin( 2m
sin( π6 )
kd = q
=
=1
αph
ph
sin( π6 )
sin(
)
2
2

(4.31)

kw = kp kd = 0.966

(4.32)

An outline of the machine is shown in Figure 4.6 and machine design parameters are given
in Table 4.3.

The stator and rotor laminations of the resulting prototype are pictured in

Figures 4.7a and 4.7b, respectively.
48

Figure 4.5: Star of slots for single-layer 25 SP P PM machine

Design Parameter
Value
Stator Diameter
220mm
Rotor Diameter
138mm
Stack Length
72mm
Number of Stator Slots
12
Number of poles
10
Air gap length
1mm

Design Parameter
Value
Magnet Type
NdFeB N38EH
Winding Type
Single-layer Concentrated
Number of Turns
150 turns per coil
Stator Line Voltage
480V
Maximum Current
18Arms
Maximum Power
3.8kW

Table 4.3: Single-layer 52 SPP machine design parameters.

49

Figure 4.6: Single-layer 52 SP P PM machine laminations

(a) Stator

(b) Rotor

Figure 4.7: Laminations for 52 SPP PM machine - (a)Stator and (b)Rotor
50

4.4

Experimental Characterization

Permanent magnet synchronous machines are typically modeled in the two axes synchronous
frame of reference. DQ theory assumes sinusoidal MMF; however, the arrangement of fractional slot concentrated windings produces a non-sinusoidal MMF. According to [4], the
DQ model remains reliable. Self- and cross-saturation are non-linear effects that have been
proven important for development of high performance permanent magnet synchronous machine controllers. The d- and q-axis model, including self- and cross-saturation, is given in
by Equations (4.35) to (4.37).

λd (id , iq ) = Ld (id )id + Ldq (id , iq )iq + λpm

(4.33)

λq (id , iq ) = Lq (iq )iq + Lqd (id , iq )id

(4.34)

diq
di
vd = Rs id + [Ld (id ) d + Ldq (id , iq ) ] − ωe [Lq (iq )iq + Lqd (id , iq )id ]
dt
dt
dλ (i , iq )
− ωe λq (id , iq )
= Rs id + d d
dt

(4.35)

diq
di
+ Lqd (id , iq ) d + ωe [Ld (id )id + Ldq (id , iq )iq + λpm ]
dt
dt
dλq (id , iq )
= Rs iq +
+ ωe λd (id , iq )
dt

(4.36)

vq = Rs iq + [Lq (iq )

T =

3p
[λ (i , iq )iq − λq (id , iq )id ]
2 d d

51

(4.37)

In [5], it was determined that cross-coupling is not significant in permanent magnet machines
with fractional slot concentrated windings. The frozen permeability method was utilized to
prove that the classical d-q model of permanent magnet synchronous machines is accurate
when fractional slot concentrated windings are used. It was concluded that cross-coupling
in fractional slot concentrated winding permanent magnet machines is negligible, implying
that the simplified model in Equations (4.40) to (4.42) provides sufficient accuracy.

λd (id ) = Ld (id )id + λpm

(4.38)

λq (iq ) = Lq (iq )iq

(4.39)

vd = Rs id − ωe Lq (iq )iq

(4.40)

= Rs id − ωe λq (iq )

vq = Rs iq + ωe [Ld (id )id + λpm ]

(4.41)

= Rs iq + ωe λd

T =

3p
[λ (i )iq − λq (iq )id ]
2 d d

52

(4.42)

Figure 4.8: Experimental setup

4.4.1

Experimental Setup

The permanent magnet synchronous machines with fractional slot concentrated windings
were individually mounted on an Emerson 20hp DC dynamometer to extract characteristic
parameters, as shown in Figure 4.8. A PCB torque transducer is connected between the
dynamometer and the machine under test. The machine under test is connected to an APS
inverter, equipped with current sensors for each phase and a voltage transducer for the DC
link. The inverter DC link is connected to an uncontrolled rectifier. The inverter gate driver
is connected to an ESI controller board equipped with a Texas Instrument DSP. The DSP’s
controller area network is used to transmit and receive data from a PC running LabVIEW.
An Agilent oscilloscope is used to monitor the phase current measured with an LEM current
transducer.

The machine characteristic inductances for both the double-layer 12 slot per pole per phase
and single-layer 25 slot per pole per phase designs are experimentally extracted to evaluate
53

the significance of the cross-coupling terms [43]. A two step process is used to acquire the
machine characteristic parameters.
1. Experimental Data Acquisition
2. Experimental Data Processing

4.4.2

Experimental Data Acquisition

The experimental setup is shown in Figure 4.9 is used to acquire the data required for extracting the machine’s characteristic inductances. The dynamometer controls the speed of the
motor under test. The AC currents and DC voltage are fed back for control. The Texas Instrument floating point DSP is flash programmed to control the machine current. Controller
area network (CAN) is used to transfer the machine variables required for characterization
from a DSP to a computer running a LabVIEW virtual instrument. The LabVIEW virtual
instrument is used to transmit the desired current magnitude and angle, as well as save data
to the computer hard disk. As shown in the permanent magnet synchronous machine model
including cross-coupling terms, shown in Equations (4.35) and (4.36), the variables required
to extract the machine characteristic inductances are:
• Phase Voltage
• Phase Current
• Rotor Position
• Machine Speed
The phase voltage is estimated instead of using voltage transducers. This is as accurate
as measuring, since DSP internal variables are used to perform the estimation [43]. Current
54

Figure 4.9: Experimental setup for collecting data to extract the machine characteristic
inductances.
transducers provide phase current measurement. An optical rotary position encoder is used
for position feedback and speed calculation. The encoder must be aligned to the rotor
magnet axis in software. The motor’s open circuit voltage is used for alignment [43, 44].
After verifying proper rotor alignment, the current command is set to 0 90◦ A and the
variables are recorded. This process is repeated for all of the Is δ ◦ A combinations shown
in Table 4.4.

4.4.3

Experimental Data Processing

The acquired data is processed off-line to extract the flux linkages, which inherently include
self- and cross-saturation effects. The acquired data during 0 90◦ A command is processed
first to extract the permanent magnet flux linkage, λpm . The fundamental component of the
55

δ
Is
5A
10A
15A
20A
25A

90◦

100◦

110◦

120◦

130◦

140◦

150◦

160◦

170◦

180◦

Table 4.4: Table used for data acquisition.
estimated voltage and the position are used to determine the d- and q-axis components, vd
and vq . Manipulating Equation (4.36) with id = 0 and iq = 0 results in Equation (4.43).

λpm =

vq
ωe

(4.43)

The d- and q-axis flux linkages are determined from the data acquired for each current
command. Phase resistance is measured using a multimeter. Solving Equations (4.35)
and (4.36) in steady-state for d- and q-axis flux linkages result in Equations (4.44) and (4.45).

vq − iq Rs
ωe

(4.44)

i Rs − vd
λq (id , iq ) = d
ωe

(4.45)

λd (id , iq ) =

The inductances in Equations (4.33) and (4.34) are calculated using Equations (4.46)
to (4.49).

λ (i , 0) − λpm
Ld (id ) = d d
id

56

(4.46)

Lq (iq ) =

(4.47)

λ (i , 0) − λd (id , iq )
Ldq (id , iq ) = d d
iq

(4.48)

λq (0, iq ) − λq (id , iq )
id

(4.49)

Lqd (id , iq ) =

4.4.4

λq (0, iq )
iq

Experimental Results

The dynamometer rotated the double-layer 12 slot per pole per phase machine at 700rpm
during the data acquisition process. The extracted flux linkages are shown in Figures 4.10
and 4.11. The single-layer 25 slot per pole per phase machine was rotated by the dynamometer
at 300rpm. The resulting flux linkages are shown in Figures 4.13 and 4.14. These experimental flux linkages were used to determine the torque-speed characteristics of each machine
at rated current and voltage. The resulting torque-speed envelopes are shown in Figure 4.12
for the double-layer 12 slot per pole per phase machine and Figure 4.15 for the single-layer 52
slot per pole per phase machine.

The extracted inductances of the double-layer 12 slot per pole per phase design are shown
in Figures 4.16 and 4.17. The single-layer 52 slot per pole per phase design extracted inductances are shown in Figures 4.18 and 4.19. It is noted that cross-coupling is significant and
should not be neglected.

57

λd−(V−s)

1
0
−1
10
120

20
Amps

180

160

100

140
Angle−(degs)

Figure 4.10: Experimental D-axis Flux Linkage of 21 SPP PM Machine

λq−(V−s)

1.5
1
0.5
180
160
140
120
100
Angle−(degs)

0
20
Amps

10

Figure 4.11: Experimental Q-axis Flux Linkage of 12 SPP PM Machine

58

70

Torque (Nm)

60
50
40
30
20
10
0
0

1000

2000
3000
Speed (RPM)

4000

Figure 4.12: Experimental Torque-speed envelope for 12 SPP PM Machine

λd−(V−s)

0.5
0
−0.5
5
10
15
20
Amps

120

100

140
25 180 160 Angle−(degs)

Figure 4.13: Experimental D-axis Flux Linkage of 52 SPP PM Machine

59

λq−(V−s)

1
0.5
0
25

180
160

20

140

15
Amps

120

10
5

100

Angle−(degs)

Figure 4.14: Experimental Q-axis Flux Linkage of 25 SPP PM Machine

50

Torque (Nm)

40
30
20
10
0
0

1000

2000
3000
Speed (RPM)

4000

Figure 4.15: Experimental Torque-speed envelope for 25 SPP PM Machine

60

Ld−(H)

0.1
0.05
0
180

25
20
160

15

140

120
Angle−(degs) 100

10
5

Amps

Figure 4.16: Experimental D-axis Inductance of 12 SPP PM Machine

Lq−(H)

0.4
0.2
0
5
10
15
20
Amps 25

160

180

140
120
100
Angle−(degs)

Figure 4.17: Experimental Q-axis Inductance of 12 SPP PM Machine

61

Ld−(H)

0.05
0
180
160

20

140
120
Angle−(degs) 100

10

Amps

Figure 4.18: Experimental D-axis Inductance of 25 SPP PM Machine

Lq−(H)

0.3
0.2
0.1
0
5
10
15
20
Amps 25

160

100

180

140
120
Angle−(degs)

Figure 4.19: Experimental Q-axis Inductance of 25 SPP PM Machine

62

4.5

Finite Element Analysis

Finite element analysis of both healthy machines was completed to analyze the cogging
torque, back-emf, torque ripple and q- and d-axis inductances. As expected, cogging is lower
in the 25 slot per pole per phase design, shown in Figure 4.20.

The open-circuit voltage of the double-layer 21 slot per pole per phase design and the
single-layer 25 slot per pole per phase design are shown in Figures 4.21 and 4.23, respectively.
Higher order harmonics are present in the open-circuit voltages of both designs as shown
in Figures 4.22 and 4.24.
The double-layer 12 slot per pole per phase permanent magnet design delivered more torque
than the single-layer 25 slot per pole per phase design on average. The 12 slot per pole per
phase design delivered an average 55Nm with a 10Nm ripple, shown in Figure 4.25, whereas
the 52 slot per pole per phase design delivered an average 34Nm with a 10Nm ripple, shown
in Figure 4.26.

The inductance of the double-layer design was also calculated from finite element data,
including self-saturation only. The results are shown in Figure 4.27. Inductances of the
single-layer 25 slot per pole per phase permanent magnet machine design are shown in Figure 4.28. Iron permeability was confirmed as an inductance factor using the single-layer
machine finite element model. A second model of the machine was developed describing
a constant permeability for both the stator and rotor iron. The resulting inductances are
shown in Figure 4.29.

63

Torque (Nm)

0.5

0

−0.5
0

10
20
position (degrees)

30

(a) 12 SPP

Torque (Nm)

0.2
0.1
0
−0.1
−0.2
0

2
4
position (degrees)

6

(b) 25 SPP
Figure 4.20: Double-layer 12 SPP and Single-layer 52 SPP PM Machine Cogging Torque (finite
element results).

64

50

Voltage (V)

Phase A
Phase B
Phase C
0

−50
0

100
200
300
position (degrees)

Figure 4.21: Double-layer 12 SPP PM Machine BackEMF (finite element results).

50

Magnitude

40
30
20
10
0
0

10

20
30
40
Harmonic order+1

50

Figure 4.22: Harmonic Content in Double-layer 21 SPP PM Machine.

65

40

Voltage (V)

20
0
Phase A
Phase B
Phase C

−20
−40
0

100
200
position(degrees)

300

Figure 4.23: Single-layer 52 SPP PM Machine BackEMF (finite element results).

40

Magnitude

30
20
10
0
0

10

20
30
40
Harmonic order+1

50

Figure 4.24: Harmonic Content in Single-layer 25 SPP PM Machine.

66

Torque (Nm)

60

55

50

45
0

50
100
position (degrees)

150

Figure 4.25: Double-layer 21 SPP PM Machine Torque (finite element results).

45

Torque − Nm

40
35
30
25
20
0

50
100
position (degrees)

150

Figure 4.26: Single-layer 52 SPP PM Machine Torque (finite element results).

67

Ld

Inductance (H)

0.15

Lq

0.1

0.05

0
0

5

10
15
Current (A)

20

25

Figure 4.27: Double-layer 12 SPP PM Machine Inductances (finite element results).

0.1
Ld
Inductance (H)

0.08

Lq

0.06
0.04
0.02
0
0

5

10
15
Current (A)

20

25

Figure 4.28: Single-layer 25 SPP PM Machine Inductances (finite element results).

68

Inductance (H)

0.095
0.0945
0.094
Ld

0.0935

Lq
0.093
0

5

10
15
Current (A)

20

25

Figure 4.29: Inductance of single-layer FSCW PMSM with iron having constant permeability
(finite element results)

4.6

Performance with Stator Inter-turn Failure

The addition of inter-turn short-circuits of varying number of turns to the finite element
models revealed that the circulating current due to a short circuit in the single-layer machine
design is highest for a single-turn fault, as reported in [45]. However, the circulating current
due to a short-circuit across 10% of turns in the double-layer machine design is highest.
The back-emf of the machine with inter-turn faults was also analyzed using finite element.
The back-emf of the non-faulted phases in double-layer winding machine was affected by the
inter-turn faults, however, not in the machine with single-layer windings. The torque ripple
was found to increase with number of turns involved for both designs.

4.6.1

Double-layer

1
2

SPP PM Machine

The open-circuit voltage of each phase is affected by the current circulating in the shorted
turns, shown in Figures 4.30 to 4.32. As the number of turns increases, the voltage of the

69

150
Current (A)

100
50
0
−50
−100
−150
0

100
200
300
position (degrees)

Figure 4.30: Circulating Current in Double-layer 12 SPP PM Machine with Single-Turn
Short-Circuited (finite element results).

Machine Condition
Healthy
Single-turn Short-Circuited
10% Turns in Slot Short-Circuited
20% Turns in Slot Short-Circuited

Torque Ripple
Fault Current
(Nm)
(Arms) Average (A)
10
11.13
106.88
1.26
46.11
125.79
72.38
50.59
72.42
45.82

Table 4.5: Comparison of Torque Ripple and Current Circulating in Windings of Doublelayer 12 SPP PM Machine Due to Short-Circuited Inter-turns.
faulted phase reduces, as shown in Figures 4.33 to 4.35. The torque ripple increased as
the number of turns involved in the short increased, as shown in Figures 4.36 to 4.38. The
average torque of a healthy machine is 55 Nm and the rated current is 18 Arms. The effects
of a shorted turn on the machine torque and the current flowing in the faulted phase is shown
in Table 4.5.

70

Current (A)

300
200
100
0
0

100
200
300
position (degrees)

Figure 4.31: Circulating Current in Double-layer 12 SPP PM Machine with 10% of Turns in
Slot Short-Circuited (finite element results).

Current (A)

150
100
50
0
0

100
200
300
position (degrees)

Figure 4.32: Circulating Current in Double-layer 12 SPP PM Machine with 20% of Turns in
Slot Short-Circuited (finite element results).

71

Voltage (V)

50
Phase A
Phase B
Phase C
0

−50
0

100
200
300
position (degrees)

Figure 4.33: Open-Circuit Voltage in Double-layer 12 SPP PM Machine with Single-Turn
Short-Circuited (finite element results).

Voltage (V)

50

0
Phase A
Phase B
Phase C
−50
0

100
200
300
position (degrees)

Figure 4.34: Open-Circuit Voltage in Double-layer 12 SPP PM Machine with 10% of Turns
in Slot Short-Circuited (finite element results).

72

50

Voltage (V)

Phase A
Phase B
Phase C
0

−50
0

100
200
300
position (degrees)

Figure 4.35: Open-Circuit Voltage in Double-layer 12 SPP PM Machine with 20% of Turns
in Slot Short-Circuited (finite element results).

60

Torque (Nm)

50
40
30
20
10
0

50
100
position (degrees)

150

Figure 4.36: Torque of Double-layer 21 SPP PM Machine with Single-Turn Short-Circuited
(finite element results).

73

60

Torque (Nm)

50
40
30
20
10
0

50
100
position (degrees)

150

Figure 4.37: Torque of Double-layer 12 SPP PM Machine with 10% of Turns in Slot ShortCircuited (finite element results).

60

Torque (Nm)

50
40
30
20
10
0

50
100
position (degrees)

150

Figure 4.38: Torque of Double-layer 12 SPP PM Machine with 20% of Turns in Slot ShortCircuited (finite element results).

74

150

Current (A)

100
50
0
−50
−100
−150
0

100
200
300
position (degrees)

Figure 4.39: Circulating Current in Single-layer 25 SPP PM Machine with Single-Turn ShortCircuited (finite element results).

4.6.2

Single-layer

2
5

SPP PM Machine

The open-circuit voltage of the faulted phase is affected by the current circulating in the
shorted turns, shown in Figures 4.39 to 4.41. The open-circuit voltage of the non-faulted
phases is not affected. As the number of turns increases, the voltage of the faulted phase
reduces, as shown in Figures 4.42 to 4.44. The torque ripple increased as the number of turns
involved in the short increased, as shown in Figures 4.45 to 4.47. The average torque of a
healthy machine is 35 Nm and the rated current is 18 Arms. The effects of a shorted turn
on the machine torque and the current flowing in the faulted phase is shown in Table 4.6.

75

80

Current (A)

60
40
20
0
−20
−40
0

100
200
300
position (degrees)

Figure 4.40: Circulating Current in Single-layer 52 SPP PM Machine with 10% of Turns in
Slot Short-Circuited (finite element results).

40

Current (A)

30
20
10
0
−10
−20
0

100
200
300
position (degrees)

Figure 4.41: Circulating Current in Single-layer 52 SPP PM Machine with 20% of Turns in
Slot Short-Circuited (finite element results).

76

40

Voltage (V)

20
0
Phase A
Phase B
Phase C

−20
−40
0

100
200
position(degrees)

300

Figure 4.42: Open-Circuit Voltage in Single-layer 25 SPP PM Machine with Single-Turn
Short-Circuited (finite element results).

40

Voltage (V)

20
0
Phase A
Phase B
Phase C

−20
−40
0

100
200
position(degrees)

300

Figure 4.43: Open-Circuit Voltage in Single-layer 52 SPP PM Machine with 10% of Turns in
Slot Short-Circuited (finite element results).

77

40

Voltage (V)

20
0
Phase A
Phase B
Phase C

−20
−40
0

100
200
position(degrees)

300

Figure 4.44: Open-Circuit Voltage in Single-layer 52 SPP PM Machine with 20% of Turns in
Slot Short-Circuited (finite element results).

45

Torque (Nm)

40
35
30
25
20
15
10
0

50
100
position (degrees)

150

Figure 4.45: Torque of Single-layer 25 SPP PM Machine with Single-Turn Short-Circuited
(finite element results).

78

45

Torque (Nm)

40
35
30
25
20
15
10
0

50
100
position (degrees)

150

Figure 4.46: Torque of Single-layer 52 SPP PM Machine with 10% of Turns in Slot ShortCircuited (finite element results).

45

Torque (Nm)

40
35
30
25
20
15
10
0

50
100
position (degrees)

150

Figure 4.47: Torque of Single-layer 52 SPP PM Machine with 20% of Turns in Slot ShortCircuited (finite element results).

79

Machine Condition
Healthy
Single-turn Short-Circuited
10% Turns in Slot Short-Circuited
20% Turns in Slot Short-Circuited

Torque Ripple
Fault Current
(Nm)
(Arms) Average (A)
10.66
11.42
93.75
1.72
27.56
30.18
12.88
27.75
15.85
7.65

Table 4.6: Comparison of Torque Ripple and Current Circulating in Windings of Single-layer
2 SPP PM Machine Due to Short-Circuited Inter-turns.
5

4.7

Reliability and Performance of PM Machines Designed with FSCW

A finite element model of a double-layer 25 slot per pole per phase PM machine, using the
same rotor and stator laminations as the single-layer described in Section 4.6.2, was analyzed
to determine if mutual inductance is the largest contributor to the magnitude of the circulating current and torque ripple. According to the star of slots theory the double-layer 25 slot
Q
per pole per phase design has null mutual inductance because 2t
is even. The circulating

currents in the double-layer design are shown in Figure 4.48. The torque ripple associated
with insulation failures in this design are shown in Figure 4.49.

Results indicate that mutual inductance affects the magnitude of circulating current
produced as a result of short-circuited inter-turns; however, the number of layers used in
the fractional slot concentrated winding is also a factor. Double-layer windings, even those
designed with zero mutual inductance according to the star of slots, yield higher circulating
current except in the case of a single-turn short, as shown in Table 4.7. Single-layer fractional
slot concentrated winding designed with null mutual inductance provide increased reliability.

80

150
100
Current (A)

50
0
Singleturn
10%
20%

−50
−100
−150
0

50
100
150
position (degrees)

Figure 4.48: Circulating current in short-circuited turns of double-layer 25 SPP PM (finite
element results).

45

Torque (Nm)

40
35
30
25
20
0

Healthy
Singleturn
10%
20%
50
100
position (degrees)

150

Figure 4.49: Torque for double-layer 25 SPP PM with short-circuited turns (finite element
results).

81

Number in Slot Short-Circuited
Machine Design

Single-Turn

10% Turns 20% Turns

Double-layer 12 SPP

5.94

6.99

4.02

Double-layer 25 SPP

4.8

3

1.58

Single-layer 25 SPP

5.21

1.68

.8

Table 4.7: Circulating Current in PM Machines Designed with Fractional Slot Concentrated
Windings as a Factor of Rated Current.
Machine Condition
Number in Slot Short-Circuited
Machine Design

Healthy Single-Turn

10% Turns 20% Turns

Double-layer 12 SPP

18.18%

20.55%

83.84%

91.98%

Double-layer 25 SPP

14.53%

19.78%

51.48%

56.4%

Single-layer 52 SPP

30.21%

32.36%

78.1%

78.63%

Table 4.8: Torque Ripple in PM Machines Designed with Fractional Slot Concentrated
Windings as a Percentage of Average Torque.
This increased reliability is achieved with a compromise in machine performance. The
torque ripple of the healthy single-layer machine with zero mutual inductance is 30% of
the average torque produced, as shown in Table 4.8. Additionally, higher average torque
is achieved with double-layer windings. Higher average torque and lower torque ripple are
achieved with double-layer windings. Double-layer windings with null mutual inductance
provide higher reliability and less torque ripple when more than one turn is involved in a
short.

82

Chapter 5
PMSM Torque Ripple
The high harmonic content in the stator MMF distribution is attributed to the torque ripple
associated with single-layer fractional slot concentrated windings permanent magnet synchronous machines; however, there are four sources of torque ripple in permanent magnet
synchronous machines. These sources are the harmonics in the stator MMF distribution, the
harmonics in the rotor MMF distribution, the variations in permeance due to stator slotting
and the variations in permeance due to rotor slotting.

In this chapter, analytical descriptions of each source are developed for a permanent magnet synchronous machine with a single-layer fractional slot concentrated winding. Finally,
an analytical description of the torque is presented for the single-layer design.

5.1

Stator MMF Distribution

The stator magnetomotive force (MMF) distribution is directly related to the stator winding
configuration. A simplified description of the single-layer fractional slot concentrated winding
is shown in Figure 5.1. In this figure the phase a coil conductors are represented as a single
conductor in a slot, neglecting the presence of teeth. The direction of the current flowing in
the conductor is denoted by

, out of the page, and ⊗, into the page. The resulting MMF

can be approximated as a square-wave, as shown in Figure 5.2 with amplitude equivalent to
83

Figure 5.1: Simplified phase A winding distribution of single-layer fractional slot concentrated winding design

Figure 5.2: Square-wave approximation of the single-layer fractional slot concentrated winding design stator phase A MMF distribution

the product of the number of turns per slot, Nt , and the phase current, ias . The trigonometric Fourier series of the square-wave function is shown in Figure 5.3 and Equation (5.1).
The MMF of phases B and C are offset by 120◦ , as shown in Equations (5.2) and (5.3).
Three phase balance currents are described in Equations (5.4) to (5.6). The resulting stator
MMF is shown in Equation (5.7).

∞

M M Fas (φs ) =
ν=1

∞

M M Fbs (φs ) =
ν=1

2Nt ias
π
sin ν
νπ
Q

2Nt ibs
π
sin ν
νπ
Q

84

[1 − cos (νπ)] cos (νφs )

[1 − cos (νπ)] cos ν φs −

2π
3

(5.1)

(5.2)

200
Square−wave
Fourier series

MMF

100
0
−100
−200
0

2
4
6
Stator Angular Displacement(rad)

Figure 5.3: Trigonometric Fourier series of the single-layer fractional slot concentrated winding design stator phase A MMF distribution

∞

M M Fcs (φs ) =
ν=1

π
2Nt ics
sin ν
νπ
Q

[1 − cos (νπ)] cos ν φs +

2π
3

(5.3)

ˆ
ˆ
ias (t) = Icos(ω
r t + δ) = Icos(pθ
r + δ)

(5.4)

2π
2π
ˆ
ˆ
) = Icos(pθ
)
ibs (t) = Icos(ω
rt + δ −
r+δ−
3
3

(5.5)

2π
2π
ˆ
ˆ
ics (t) = Icos(ω
) = Icos(pθ
)
rt + δ +
r+δ+
3
3

(5.6)

∞

M M Fs (φs ) =
ν=1

3Nt Iˆ
π
sin ν
νπ
Q

85

[1 − cos (νπ)] cos (νφs − pθr − δ)

(5.7)

5.2

Rotor MMF Distribution

The MMF of the rotor is directly related to the rotor configuration. The amplitude of the
magnetomotive force of permanent magnets is given by the product of the coercive field
intensity of the permanent magnets, Hc , and the magnet thickness, tm , as shown in Equation (5.8). In surface mounted designs, the flux density on the rotor surface is equivalent to
that on the magnet surface; however, this is not true for interior permanent magnet designs.

In [46], the open-circuit flux density on the rotor surface for interior permanent magnet
machines with multiple magnets per pole is described analytically with a lumped magnetic
circuit model. The reluctances in the model are calculated using the geometry of the rotor
configuration. It is desired to describe the MMF and describe the permeance, separately.
According to [47, 48], the flux density on the rotor surface, Br , for buried magnet designs is
proportional to the the flux density on the magnet surface, Bm , neglecting the magnet flux
leakage. The constant of proportionality is the ratio of the total magnet width used in one
pole, wmpp to the pole-arc in meters, τ , as shown in Equation (5.9). This same constant of
proportionality is used to describe the MMF on the rotor surface, Equation (5.10).

Fm = Hc tm

(5.8)

Br =

wmpp
Bm
τ

(5.9)

Fr =

wmpp
Fm
τ

(5.10)

86

Figure 5.4: Single-layer Fractional Slot Concentrated Winding Rotor Configuration

The amplitude of the MMF for the 52 SP P rotor configuration shown in Figure 5.4,
where rectangular magnets are used to populate the slots, can also be described with Equation (5.10). The rotor MMF can be approximated as a square-wave approximation. The
amplitude of the square-wave is determined by the angular position along the rotor. Above
the magnets in each pole-arc the full amplitude of the MMF is assumed, as shown in Figure 5.5 for one pole pair. The trigonometric Fourier series of the square-wave function is
shown in Figure 5.6 and Equation (5.11).

∞

M M Fr (φr ) =
η=1

2Fr
τ
sin ηp
[cos (ηπ) − 1] cos (ηpφr )
πη
2

87

(5.11)

MMF

Fr
0
−Fr

tau/2

taup−(tau/2) taup+(tau/2) 2*taup−(tau/2)
Rotor Angular Displacement (radians)

Figure 5.5: Square-wave Approximation of the single-layer fractional slot concentrated winding design rotor MMF distribution over one pole pair

4

2

x 10

MMF

1

Square−wave
Fourier Series

0
−1
−2
0

0.5
1
Rotor Angular Displacement (radians)

Figure 5.6: Trigonometric Fourier series of the single-layer fractional slot concentrated winding design rotor MMF distribution

88

5.3

Air-gap Permeance Variations Due to Rotor Slotting

The rotor configuration of the single-layer fractional slot concentrated winding design is
shown in Figure 5.4. The slots in the rotor laminations lead to variations in the permeance.
In [47, 48], the permeance function due to rotor slotting is described through the magnetic
reluctance with the following assumptions:
• the stator is a slotless
• the flux excited by the current sheet flows radially in the air-gap
• there are only two flux paths, through
1. the rotor iron only
2. the center of the rotor pole
Following the same assumptions, with the inclusion of the infinite permeability of the
stator and rotor iron, the reluctance variations due to the rotor slotting can be described
as a square-wave function of the rotor angular displacement, shown in Figure 5.7. The
reluctance over the magnet pole-arc, Rpole , is described in Equation (5.12), where hs is
the height of the slot. The reluctance over the remaining pole-pitch, Riron , is described
in Equation (5.13). The square-wave function is described using the trigonometric Fourier
series, shown in Figure 5.8 and Equation (5.14).

Rpole =

g
hs
+
µ0 µ0

89

(5.12)

Reluctance

R_pole

R_iron
tau/2
taup−(tau/2) taup
Rotor Angular Displacement (radians)
Figure 5.7: Square-wave Approximation of Reluctance Variation Due to Rotor Slotting in
Singlelayer Fractional Slot Concentrated Winding PMSM

Riron =

τ
Rr (φr ) = Riron +
Rpole − Riron +
τp

5.4

∞
κ=1

g
µ0

(5.13)

2
sin (κpτ ) Rpole − Riron cos (κ2pφr ) (5.14)
πκ

Air-gap Permeance Variation Due to Stator Slotting

It is well known that the air-gap permeance varies due to the presence of slots in the stator.
Carter’s coefficient is conventionally used to account for the reduction of flux due to the
stator slotting. As described in [25],Carter’s coefficient, kc , is multiplied by the physical
measure of the air-gap, g, to find an equivalent air-gap length, ge , which is constant,as
90

10000

Reluctance

8000
6000
4000

Square−wave
Fourier Series

2000
0
0
0.2
0.4
0.6
Rotor Angular Displacement (radians)
Figure 5.8: Square-wave Approximation of Reluctance Variation Due to Rotor Slotting in
Singlelayer Fractional Slot Concentrated Winding PMSM

shown in Equation (5.15).

ge = kc g

(5.15)

An analytical description of the variations in the air-gap permeance due to stator slotting
is desired. The authors of [47,48] describe a technique for including the effect of stator slotting
in the permeance function. The magnetic reluctance is used to form a permeance function
that captures the variations due to the stator slotting. In this technique the following
assumptions are made:
• the rotor is slotless
• the flux excited by the magnets flows radially in the air-gap
• the flux excited by the magnets flows along a circular path around the slot opening to
the stator tooth
91

The length of the path from the air-gap around the stator tooth is described in Equation (5.16), where r is the radius from the stator tooth.

1
ls = 2πr
4

(5.16)

The reluctance variations due to stator slotting are described for the 25 SP P machine
design following the same assumptions made in [47, 48], with the addition of infinite permeability of the stator and rotor iron. The radius around the stator tooth is assumed 21
the width of the stator slot opening, wso . The reluctance under a tooth is given in Equation (5.17). The reluctance under a slot opening is given in Equation (5.18). A square-wave
function is used to describe the reluctance variations around the stator periphery. This function is shown in Figure 5.9 for one slot pitch, τs . This square-wave function is described by
the trigonometric Fourier series is shown in Equation (5.19) and Figure 5.10.

Rt =

g
µ0

1
1
4 2π 2 wso
g
+
Rso =
µ0
µ0

1
Rs (φs ) =
(Rt wt + Rso wso ) +
τs

∞
κ=1

2
wt
sin κQ
[Rt − Rso ] cos (κQφs )
πκ
2

92

(5.17)

(5.18)

(5.19)

Reluctance

R_so

R_t
0

(wt/2) wso+(wt/2)
Stator Angular Displacement (radians)

Figure 5.9: Square-wave approximation of Reluctance variations due to stator slotting over
one slot pitch

4

4

x 10

Reluctance

3
2

Square−wave
Fourier Series

1
0
−1
0

0.1
0.2
0.3
0.4
0.5
Stator Angular Displacement (radians)

Figure 5.10: Square-wave approximation of Reluctance variations due to stator slotting over
one slot pitch

93

5.5

Air-gap Flux Density

Air-gap flux density is the product of the air-gap MMF, M M Fg (φr ), and the air-gap permeance, Λg (φr ), as shown in Equation (5.27). The total air-gap MMF is the sum of the
contributions from the stator, Equation (5.20), and rotor, Equation (5.21), as described
in Equation (5.22).

∞

M M Fs (φr ) =
ν=1
∞

3Nt Iˆ
π
sin ν
νπ
Q

[1 − cos (νπ)] cos (νφr + (ν − p)θr − δ)
(5.20)

M M Fsν cos (νφr + (ν − p) θr − δ)

=
ν=1

∞

M M Fr (φr ) =
η=1
∞

=

2Fr
τ
sin ηp
[cos (ηπ) − 1] cos (ηpφr )
πη
2
(5.21)
M M Frη cos (ηpφr )

η=1

M M Fg (φr ) = −M M Fs (φr ) + M M Fr (φr )
∞

(5.22)
−M M Fsη cos (ηφr + (η − p)θr − δ) + M M Frη cos (ηpφr )

=
η=1

The total air-gap permeance function is described as the inverse of the total air-gap reluctance variation, shown in Equation (5.26). The total air-gap reluctance variation is the
sum of the reluctance variations due to rotor and stator slotting. However, since the reluctance variation due to rotor slotting assumed a smooth stator, it accounts for the reluctance

94

below a stator tooth. For this reason, the reluctance under a stator tooth is subtracted from
the the stator slotting contribution to the air-gap reluctance variations. The total air-gap
reluctance is described in Equation (5.25).

1
Rs (φr ) =
(Rt wt + Rso wso ) +
τs

∞
κ=1

2
wt
sin κQ
[Rt − Rso ] cos (κQ (φr + θr ))
πκ
2
(5.23)

∞

= Rs0 +

Rsκ cos (κQ (φr + θr ))
κ=1

τ
Rr (φr ) = Riron +
Rpole − Riron +
τp

∞
κ=1

2
sin (κpτ ) Rpole − Riron cos (κ2pφr )
πκ

∞

= Rr0 +

Rrκ cos (κ2pφr )
κ=1

(5.24)

Rg (φr ) = (Rs (φr ) − Rt ) + Rr (φr )
∞

= Rs0 − Rt + Rr0 +

Rsκ cos (κQ (φr + θr )) + Rrκ cos (κ2pφr )

(5.25)

κ=1
∞

= Rg0 +

Rsκ cos (κQ (φr + θr )) + Rrκ cos (κ2pφr )
κ=1

Λg (φr ) =

1
Rg (φr )

(5.26)

∞

= Λg0 +

Λsκ cos (κQ (φr + θr )) + Λrκ cos (κ2pφr )
κ=1

95

Bg (φr ) = M M Fg (φr ) Λg (φr )
∞

∞

Λg0 M M Frη cos (ηpφr ) − Λg0 M M Fsη cos (ηφr + (η − p)θr − δ)

=
κ=1 η=1

Λsκ M M Frη
[cos ((κQ − ηp)φr + κQθr ) + cos ((κQ + ηp)φr + κQθr )]
2
Λsκ M M Fsη
−
cos ((κQ − η)φr + (κQ − η + p)θr + δ)
2

+

+ cos ((κQ + η)φr + (κQ + η − p)θr − δ)

(5.27)

Λrκ M M Frη
cos ((κ2 − η)pφr ) + cos ((κ2 + η)pφr )
2
Λrκ M M Fsη
cos ((κ2p − η)φr − (η − p)θr + δ)
−
2
+

+ cos ((κ2p + η)φr + (η − p)θr − δ)

5.6

Analytical Torque Expression

Torque is described in Equation (5.29) where L is the motor stack length and rs−id is the
inner radius of the stator. Total air-gap flux density, Bg (φr ) is shown in Equation (5.27).
The linear current density, Ks (φr ) is described in Equation (5.28). The integrand, shown
in Equation (5.30), is only non-zero when the coefficient of φr is equal to zero. The conditions
for non-zero torque are shown in Equation (5.31).

∞

Ks (φr ) =
ν=1

3Nt Iˆ
π
sin ν
νπrs−id
Q

2
T = Lrs−id

[1 − cos (νπ)] cos (νφr + (ν − p)θr − δ)

(5.28)

2π

Bg (φr ) Ks (φr ) dφr
0

96

(5.29)

∞

∞

∞

Bg (φr )Ks (φr ) =
κ=1 η=1 ν=1

Λg0 M M Frη Ksν
cos ((ηp − ν)φr − (ν − p)θr + δ)
2

+ cos ((ηp + ν)φr + (ν − p)θr − δ)
−

Λg0 M M Fsη Ksν
cos ((η − ν)φr + (η − ν)θr )
2
+ cos ((η + ν)φr + (η + ν − 2p)θr − 2δ)

+

Λsκ M M Frη Ksν
cos ((κQ − ηp − ν)φr + (κQ − ν + p)θr + δ)
4
+ cos ((κQ − ηp + ν)φr + (κQ + ν − p)θr − δ)
+ cos ((κQ + ηp − ν)φr + (κQ − ν + p)θr + δ)
+ cos ((κQ + ηp + ν)φr + (κQ + ν − p)θr − δ)

−

Λsκ M M Fsη Ksν
cos ((κQ − η − ν)φr + (κQ − η − ν + 2p)θr + 2δ)
4
+ cos ((κQ − η + ν)φr + (κQ − η + ν)θr )

(5.30)

+ cos ((κQ + η − ν)φr + (κQ + η − ν)θr )
+ cos ((κQ + η + ν)φr + (κQ + η + ν − 2p)θr − 2δ)
+

Λrκ M M Frη Ksν
cos ((κ2p − ηp − ν)φr − (ν − p)θr + δ)
4
+ cos ((κ2p − ηp + ν)φr + (ν − p)θr − δ)
+ cos ((κ2p + ηp − ν)φr − (ν − p)θr + δ)
+ cos ((κ2p + ηp + ν)φr + (ν − p)θr − δ)

−

Λrκ M M Fsη Ksν
cos ((κ2p − η − ν)φr + (2p − η − ν)θr + δ)
4
+ cos ((κ2p − η + ν)φr + (ν − η)θr − δ) + cos ((κ2p + η − ν)φr + (η − ν)θr )
+ cos ((κ2p + η + ν)φr + (ν + η − 2p)θr − 2δ)

97

T =0






ηp = ν









ηp = ν









ηp = ν










ηp = ν








ηp = ν




η









η








η








η









η

and κ = η
and κ =

η(p−1)
Q

and κ =

η(1+p)
2p

and κ =

η(p−1)
2p

=ν
=ν

and κ =

η(p−1)
Q

=ν

and κ =

η(p+1)
2p

=ν

and κ =

η(p−1)
2p

=ν

and κ = ηp

The resulting torque expression is shown in Equation (5.32).

98

(5.31)

∞

T =

2
πLrs−id

Λg0 M M Frη Ksηp cos (−(ηp − p)θr + δ)
η=1

− Λg0 M M Fsη Ksη
+
∞

+
η=3,9,15

Λrη M M Frη Ksηp
cos (−(ηp − p)θr + δ)
2

1
− Λ η(p−1) M M Fsη Ksηp
2
s Q
+ Λ η(p−1) M M Frη Ksη cos ((ηp − p)θr − δ)
s
Q

(5.32)

∞

1
+
− Λ η(p+1) M M Fsη Ksηp cos ((2p − η − ηp)θr + δ)
2
r 2p
η=5,15,25
+ Λ η(p+1) M M Frη Ksη cos (−(η − p)θr + δ)
r
2p

− Λ η(p−1) M M Fsη Ksηp cos ((η − ηp)θr )
r
2p

+ Λ η(p−1) M M Frη Ksη cos ((η − p)θr − δ)
r
2p

− Λr η M M Fsη Ksη cos ((2p − 2η)θr + δ)
p

The analytical expression given in Equation (5.32) was simulated with MATLAB. The
MATLAB FSCW machine model includes the first two non-zero harmonic interactions between the rotor and stator, η = 1&3, as shown in Equation (5.33) at 300rpm. The results
of the analytical model at two current levels are compared to the torque produced at the
same current levels in the finite element model of the single-layer 25 SP P machine. As shown
in Figures 5.11 and 5.12, Equation (5.32) closely approximates the finite element results.

99

Torque (Nm)

15

10

5
Analytical
Flux2D
0
0

10
20
Position (degrees)

30

Figure 5.11: Simulink model of analytical torque expression

3
2
T = πLrs−id

Λrη
M M Frη Ksηp cos (−(ηp − p)θr + δ) Λg0 +
2

η=1

(5.33)
1
+ Λ 3(p−1) M M Fr3 Ks3 cos ((3p − p)θr − δ)
2 s Q

100

45

Torque (Nm)

40
35
30
Analytical
Flux2D

25
20
0

50
100
Position (degrees)

150

Figure 5.12: Simulink model of analytical torque expression

101

Chapter 6
Torque Ripple Reduction Control
The single-layer fractional slot concentrated winding design provides the best isolation between phases. In the event of a stator winding inter-turn short circuit, the single-layer design
best limits circulating current. This winding configuration provides highest reliability; however, the torque ripple associated with this design is undesirable, nearly 30% of the average
torque.

In this chapter, a permanent magnet synchronous machine controller is modified to include torque ripple estimation. The estimation is used to decrease the torque ripple of
the single-layer fractional slot concentrated winding design. The losses associated with this
controller are also evaluated.

6.1

PMSM Controller Development

Two regions of operation can be defined by the torque speed curve of the permanent magnet
synchronous machine, shown in Figure 6.1:
1. the region below base speed or the constant torque region
2. the region above base speed or the field weakening region

102

40

Torque(Nm)

←Base Speed
30
Constant Torque Region
Field Weakening Region

20
10
0
0

500

1000
Speed(RPM)

1500

Figure 6.1: Permanent magnet synchronous machine regions of operation defined by the
torque speed curve.

In the constant torque region, the maximum current, Imax is the only restriction because
the induced voltage remains below the maximum. In this region, the angle at which the
current is applied, δ, remains constant. Equation (6.3) is applied to Equation (6.4) to
determine the current angle, δ, in this region, shown in Equation (6.5).

id = Imax cos (δ)

(6.1)

iq = Imax sin (δ)

(6.2)

d IT
max
dδ

T =

=0

3p
λpm + Ld − Lq Imax cos (δ) Imax sin (δ)
2
103

(6.3)

(6.4)


δ = cos−1 

−λpm +

λ2pm

+8

2 2
Ld − Lq Imax

4 Ld − Lq Imax




(6.5)

Once the machine exceeds base speed, entering the field weakening region, the induced
voltage reaches the maximum voltage, Vmax , requiring that the magnet flux is controlled to
restrict the voltage. Control of the magnet flux is achieved through adjustments in the angle
at which the current, Is , is applied. Substituting Equations (4.35) and (4.36) for maximum
current (Imax ) into the maximum steady-state voltage equation Equation (6.6), neglecting
the resistive drop. Applying the quadratic equation results in Equation (6.7)

2
Vmax
= vd2 + vq2


 −2Ld λpm Imax ±

−1
δ = cos 



2Ld λpm Imax

2

2
− 4Imax
L2d − L2q

(6.6)

2
L2q Imax
+ λ2pm −

2
Vmax
ωe2

2
2 L2d − L2q Imax

(6.7)
The model-based torque controller is shown in Figure 6.2. The performance of this control
algorithm is directly dependent on the accuracy of the parameters. The machine parameters
are included in the controller as piece-wise approximations. In this controller, the d- and qaxis inductances of the single-layer 25 SP P machine found in finite element are separated into
three sections to develop the piece-wise approximations, as shown in Figure 6.3. Inductances
in section I and II are described as linear functions of current. In section III, the inductances
are described as 2nd -order polynomials. As shown in Figure 6.3, these function closely
approximate the inductances.
104








Figure 6.2: PMSM model-based controller

0.1

Inductance (H)

Ld Finite Element
Ld Approximation

0.08

Lq Finite Element
0.06
0.04
0.02
0

Lq Approximation
Sect.I

Sect.III
Sect.II

5

10
15
Current (A)

20

25

Figure 6.3: Comparison of finite element single-layer 25 SPP PMSM inductances to piece-wise
approximation.

105

6.2

Torque Ripple Estimation

The analytical description of torque given in Equation (5.32) can be divided into two components:
• DC Torque Component
• Torque Ripple Component
The DC component of torque is shown in Equation (6.8). All other values of η, ν and κ
contribute to torque ripple as shown in Equation (6.9).

Λ
2
TDC = πLrs−id
M M Fr1 Ksp cos(δ) Λg0 + r1
2
∞

−

Λg0 M M Fsη Ksη
η=1
∞

−
η=3,9,15

106

Λ η(p−1) M M Fsη Ksηp
s
Q

(6.8)

∞

Tripple =

2
πLrs−id

Λg0 M M Frη Ksηp cos (−(ηp − p)θr + δ)
η=3

+
∞
η=3,9,15
∞

Λrη M M Frη Ksηp
cos (−(ηp − p)θr + δ)
2

1
Λ
M M Frη Ksη cos ((ηp − p)θr − δ)
2 s η(p−1)
Q

1
− Λ η(p+1) M M Fsη Ksηp cos ((2p − η − ηp)θr + δ)
2
r 2p
η=5,15,25

(6.9)

+ Λ η(p+1) M M Frη Ksη cos (−(η − p)θr + δ)
r
2p

− Λ η(p−1) M M Fsη Ksηp cos ((η − ηp)θr )
r
2p

+ Λ η(p−1) M M Frη Ksη cos ((η − p)θr − δ)
r
2p

− Λr η M M Fsη Ksη cos ((2p − 2η)θr + δ)
p

The largest rotor MMF component beyond the fundamental that contributes to the
torque ripple is the 3rd harmonic, as shown in Figure 6.4. This harmonic interacts with both
the 3rd and 15th harmonic of the linear current density to produce torque ripple. Only using
the 3rd harmonic of the rotor MMF in the ripple estimator, ripplec alc, shown in Figure 6.5,
the torque ripple is estimated for a 35Nm torque command, from the current command and
rotor position. The peak-to-peak amplitude is approximately 7Nm. This nearly 70% of the
torque ripple found with finite element for the same torque level. Finite element verified that
the frequency of the torque ripple is a spatial quantity, as shown in Figure 6.7, the period
of the torque ripple is not affected by machine speed.

107

10000

MMF

8000
6000
4000
2000
0
0

5

10
15
20
Harmonic Order

25

Figure 6.4: Magnitude of rotor MMF harmonic components in single-layer 52 SPP PMSM

Torque (Nm)

4
2
0
−2
−4
0

100
200
Position (degrees)

300

Figure 6.5: Estimated torque ripple for 35Nm command in single-layer 25 SPP PMSM

108

Figure 6.6: Analytical Torque Ripple Estimation

109

40

300rpm
500rpm
700rpm

Torque

30
20
10
0
0

10
20
position (degrees)

30

Figure 6.7: Finite element torque ripple at varying speeds and torques

Figure 6.8: Torque ripple estimator

6.3

Ripple Reduction Technique

A priori estimation of torque ripple can be used to reduce torque ripple. In the PMSM
controller presented in Section 6.1, id and iq are determined from the PMSM model in Equation (6.4). The estimation of torque ripple can be implemented on-line with current, rotor
position and the torque command, as shown in Figure 6.8. The PMSM model in Figure 6.8
is populated with ideal inductances.

The torque ripple estimation can be used to modify the torque command as shown

110

Figure 6.9: Torque ripple compensation controller

in Figure 6.9. This technique generates sinusoidal d- and q-axis current commands. This
technique was verified in Simulink using the model shown in Figure 6.10. The resulting
reduction is shown for a 10 Nm command in Figure 6.11

6.4

Finite Element Results

The ripple compensation controller, shown in Figure 6.9 was implemented with the singlelayer 25 SPP machine model in finite element. The torque ripple estimation only includes the
contribution from the 3rd harmonic of the rotor MMF. The results are shown for the 10Nm,
20Nm and 35Nm cases in Figures 6.12 to 6.14. A priori estimation of reduced torque ripple
in the single-layer design by as much as 70%, see Table 6.1. The frequency of the torque
ripple increased in some cases, however the amplitude is reduced. The increased frequency
is a result of remaining higher order harmonic content.

This technique increase the harmonic content of the current commands, which may result
111

Figure 6.10: Simulink model of torque compensation

Torque
Torque Ripple
Percent
Command
Without
With
Change
(Nm)
Compensation Compensation
(%)
10
1.53
.453
-70%
20
4.11
1.44
-64.96%
35
10.59
3.92
-62.98%
Table 6.1: Change in torque ripple due to implementation of the ripple compensation control
on 25 SPP PM machine.

112

11.5

Without
With

Torque (Nm)

11
10.5
10
9.5
9
0

0.05

0.1
time (sec)

0.15

0.2

Figure 6.11: Torque for 10Nm command with and without the torque ripple compensation
controller.

Torque (Nm)

11
Without
With

10.5
10
9.5
9
0

10
20
position (degrees)

30

Figure 6.12: Finite element results 10Nm torque command with and without the torque
ripple compensation controller.

113

23
Without
With

Torque (Nm)

22
21
20
19
18
0

10
20
position (degrees)

30

Figure 6.13: Finite element results 20Nm torque command with and without the torque
ripple compensation controller.

45

Torque (Nm)

Without
With
40

35

30
0

10
20
position (degrees)

30

Figure 6.14: Finite element results 35Nm torque command with and without the torque
ripple compensation controller.

114

Torque
Loss Type
Command
Iron
(Nm)
Magnet Copper Hysteresis Eddy Current
10
0%
+.5%
+3%
+2%
20
+2%
+1.5%
+.5%
+3%
35
+15%
+8%
0%
+4%
Table 6.2: Change in machine losses due to implementation of the ripple reduction control
on 25 SPP PM machine.
in higher losses. The losses associated with the decreased torque ripple were evaluated in
finite element. Magnet losses increase by as much as 15% at high torque, as shown in Table 6.2.; however, the ripple compensation control is effective for reducing torque ripple in
the single-layer 52 SPP machine design.

115

Chapter 7
Detection of Stator Winding
Insulation Degradation in Fault
Tolerant PM Machines
Most winding faults are caused by insulation deterioration, resulting from thermal stress due
to aging, overloading, cycling or fast switching of the inverter, which implies that designing
fault tolerant machines will not eliminate winding fault occurrences. Deteriorated winding
insulation will lead to an inter-turn short circuit overtime, without corrective measures. The
time between a single inter-turn short circuit and catastrophic failure of the machine is short.
Without knowledge of degrading stator winding insulation, a stator inter-turn short could
develop and lead to catastrophic failure. Catastrophic permanent magnet machine failure
can be avoided if degrading stator winding insulation is detected during normal operation.
Detecting the degradation prior to the existence of a turn-to-turn short circuit would allow
for timely maintenance and repair.

Off-line techniques for detecting insulation degradation are commonly implemented as
part of the machine’s preventive maintenance schedule. The surge test, DC conductivity
test, insulation resistance test, DC/AC HiPot test and the polarization index test are exam-

116

ples of off-line techniques used to check health of stator winding insulation [2].

In this chapter, the effect of degrading insulation on the machine performance is evaluated. Parametric changes due to degraded insulation is demonstrated. Extracted machine
parameters are used to identify a means to monitor the condition of the stator winding insulation.

7.1

Winding Insulation Degradation and Machine Performance

The open-circuit voltage is affected as winding insulation degrades, even before the presence
of a short-circuit. The changes in open-circuit voltage are directly related to the circulating current. Varying the fault resistance in the finite element model simulates insulation
degradation. Machine speed and the number of turns in which the insulation has degraded
affect the magnitude of the circulating current, as shown in Figures 7.1 and 7.2. The shape
of the open-circuit voltage deformation is consistent regardless of machine speed, as shown
in Figures 7.3 and 7.4. However, the number of turns with degraded insulation determines
the level of deformation, or the harmonic spectrum, as shown in Figures 7.5 to 7.7.

As with shorted inter-turns, the machine torque ripple increases with the number of turns
with degraded insulation, shown in Table 7.1. The ability to detect that the winding insulation is degrading during machine operation offers opportunity to mitigate the fault and
maintain decent machine performance.
117

0.4
200 rpm
300 rpm
400 rpm

Current (A)

0.2
0
−0.2
−0.4
0

50
100
position (degrees)

Figure 7.1: Circulating Current in Single-layer 52 SPP Permanent Magnet Synchronous Machine with Single-turn Insulation Degraded at Different Speeds - (a)200 rpm (b)300 rpm
(c)400 rpm (finite element results)

Single−turn
10%
20%

Current (A)

5

0

−5
0

50
100
position (degrees)

Figure 7.2: Circulating Current in Single-layer 52 SPP Permanent Magnet Synchronous Machine with Insulation Degraded in Different Number of Turns at 300 rpm - (a)Single-turn in
Slot (b)10% Turns in Slot (c)20% Turns in Slot (finite element results)

118

Voltage (V)

50
200 rpm
300 rpm
400 rpm
0

−50
0

50
100
position (degrees)

Figure 7.3: Open-Circuit Voltage (at Different Speeds) in Single-layer 25 SPP Permanent
Magnet Synchronous Machine with the Insulation on a Single-turn in a Slot Degraded (a)200 rpm (b)300 rpm (c)400 rpm (finite element results)

Figure 7.4: Open-Circuit Voltage (at Different Speeds) in Single-layer
Magnet Synchronous Machine with the Insulation of 20% of the Turns in
(a)200 rpm (b)300 rpm (c)400 rpm (finite element results)

Voltage (V)

50
200 rpm
300 rpm
400 rpm
0

−50
0

50
100
position (degrees)

119

2 SPP
5

Permanent
a Slot Degraded -

40
Single−turn
10%
20%

Voltage (V)

20
0
−20
−40
0

50
100
position (degrees)

Figure 7.5: Open-Circuit Voltage in Single-layer 25 SPP Permanent Magnet Synchronous
Machine with Insulation Degraded in Different Number of Turns at 300 rpm - (a)Single-turn
in Slot (b)10% Turns in Slot (c)20% Turns in Slot (finite element results)

50
200 rpm
300 rpm
400 rpm

Magnitude

40
30
20
10
0
0

5
10
Harmonic order+1

15

Figure 7.6: Harmonic Spectrum of Open-Circuit Voltage in Single-layer 25 SPP Permanent
Magnet Synchronous Machine with Insulation Degraded Across a Single-turn in a Slot

120

Magnitude

40

200 rpm
300 rpm
400 rpm

30
20
10
0
0

5
10
Harmonic order+1

15

Figure 7.7: Harmonic Spectrum of Open-Circuit Voltage in Single-layer 25 SPP Permanent
Magnet Synchronous Machine with Insulation Degraded Across 20% of Turns in a Slot

Machine Design

Machine Condition
Number in Slot Short-Circuited
Healthy Single-Turn 10% Turns 20% Turns

Double-layer 12 SPP

18.18%

19.5%

19.67%

20.55%

Single-layer 52 SPP

30.21%

31.29%

31.4%

32.17%

Table 7.1: Torque Ripple in PM Machines Designed with Fractional Slot Concentrated
Windings with Degraded Insulation (As a Percentage of the Average Torque of a Healthy
Machine).

7.2

Experimental Setup and Results

It is observed in [39] that the location of the shorted turn in the slot affects the magnitude
of the circulating current. It was shown that shorted turns nearest the slot opening develop
the highest short-circuit current. For this reason, the windings of the double-layer 12 and
single-layer 52 slot per pole per phase permanent magnet machines were designed to allow for
121

Figure 7.8: Winding Schematic of Double-layer 21 SPP PM Machine Prototype
introducing faults of varying severity in a slot nearest the air gap, as shown in Figures 7.8
and 7.9,respectively. The leads of several turns are separate coils brought out of the housing
to allow the introduction of a single-turn fault, a fault involving 10% of the turns in the slot
and a fault involving 20% of the turns in the slot.

The resistance of a single turn of the winding for each design was estimated using Equations (7.1) and (7.2) where ρ is the resistivity of copper, L is the length of a turn (estimated
using the machine geometry) and A is the estimated cross-section of the wire. Both stators
were wound with 4 parallel 22AWG wires. The resistance of a turn for both machine designs
is shown in Table 7.2.

122

Figure 7.9: Winding Schematic of Single-layer 25 SPP PM Machine Prototype.

123

Machine Design
Double-layer 12 SPP
Single-layer 25 SPP

Turn Resistance
3.7mΩ
3.06mΩ

Table 7.2: Resistance of a Single Turn for Both Machine Designs

ρL
A

(7.1)

Rwire
4

(7.2)

Rwire =

Rturn =

Varying the number of turns involved in the fault is accomplished by connecting a resistor,
as shown in Figure 7.10. The level of insulation degradation is controlled by the value of the
resistor connected. The experimental setup used to characterize the machines with insulation
degradation is shown in Figure 7.11. The insulation degradation connector includes a LEM
current transducer for monitoring the circulating current due to the fault on the oscilloscope.

The experimental procedure described in Section 4.4.2 is repeated for both machines
under various inter-turn fault conditions. Both a 12 Ω and 41 Ω resistor are used to simulate
insulation degradation. The experimental data processing method described in Section 4.4.3
is modified slightly. It was shown in Sections 4.6 and 7.1 that the open-circuit voltage is affected by circulating current due to the fault so the permanent magnet flux is not calculated
from the open-circuit voltage in the faulty cases. The permanent magnet flux calculated
with the healthy machine acquired data instead.

124

Figure 7.10: Circuit Diagram for Introducing Faults of Varying Severity

Figure 7.11: Experimental Setup

125

The circulating current and resulting flux linkages for the double-layer 12 and single-layer
2
5

slot per pole per phase machine varying the number of turns with degraded insulation

were evaluated. As indicated by finite element analysis, circulating current is a function of
the number of turns with degraded insulation and the machine speed.

It is shown in Figures 7.12 to 7.23 that the insulation degradation was reflected in the
extracted flux linkages. The effects of saturation in the d- and q-axis flux linkages of the
double-layer 21 slot per pole per phase machine change as the insulation of more turns degrade. Insulation degradation in the single-layer 25 slot per pole per phase machine also
affects saturation in both the d- and q-axis flux linkages. The resulting flux linkages reveal
a means for early detection of incipient stator inter-turn faults.

7.3

Online Detection of Winding Insulation Degradation

The parametric changes due to degrading winding insulation result in variations in the estimated voltage. The 12 slot per pole per phase design voltage variations, in an electrical cycle,
observed in finite element are shown in Figure 7.24. Voltage variations in an electrical cycle
for the 52 slot per pole per phase design are shown in Figure 7.25. The changes due to the
insulation on a single-turn degrading doesn’t vary much from the healthy voltage; however,
this is a result of the low value of circulating current.

Degrading insulation leads to variations in the machine characteristic parameters; the flux
126

λd−(V−s)

0.5
0
−0.5
5
10

100

15
Amps
20

150 Angle−(degs)

λq−(V−s)

(a) d-axis

1.5
1
0.5
0
20
150

15
10
Amps

5

100
Angle−(degs)

(b) q-axis
Figure 7.12: Double-layer 12 SPP PM Machine with Insulation on a Single-turn
Degraded( 21 Ω) (experimental results).

127

λd−(V−s)

0.5
0
−0.5
5
10
15
Amps
20 160

120

100

140
Angle−(degs)

(a) d-axis

λq−(V−s)

1.5
1
0.5
0
20

160
140

15

120

10
Amps

5

100
Angle−(degs)

(b) q-axis
Figure 7.13: Double-layer 12 SPP PM Machine with Insulation on a Single-turn
Degraded( 41 Ω) (experimental results).

128

λd−(V−s)

0.5
0
−0.5
5

100
10

120

15
Amps

140
Angle−(degs)
20 160

λq−(V−s)

(a) d-axis

1.5
1
0.5
0
20

160
15
10
Amps

140
120
5

100
Angle−(degs)

(b) q-axis
Figure 7.14: Double-layer 12 SPP PM Machine with Insulation on 10% of turns in a slot
Degraded( 12 Ω) (experimental results).

129

λd−(V−s)

0.5
0
−0.5
5
10

100

15
Amps

150 Angle−(degs)

20

λq−(V−s)

(a) d-axis

1.5
1
0.5
0
20

160
140

15
10
Amps

120
5

100
Angle−(degs)

(b) q-axis
Figure 7.15: Double-layer 12 SPP PM Machine with Insulation on 10% of turns in a slot
Degraded( 14 Ω) (experimental results).

130

λd−(V−s)

0.5
0
−0.5
5

100

10
15
Amps

20

150

Angle−(degs)

(a) d-axis

λq−(V−s)

1.5
1
0.5
0
20

160
140

15
10
Amps

120
5

100
Angle−(degs)

(b) q-axis
Figure 7.16: Double-layer 12 SPP PM Machine with Insulation on 20% of turns in a slot
Degraded( 12 Ω) (experimental results).

131

λd−(V−s)

0.5
0
−0.5
2
4

100
6

Amps

8
10

150 Angle−(degs)

λq−(V−s)

(a) d-axis

1.5
1
0.5
0
10

160
8

140
6
Amps

120

4
2

100
Angle−(degs)

(b) q-axis
Figure 7.17: Double-layer 12 SPP PM Machine with Insulation on 20% of turns in a slot
Degraded( 14 Ω) (experimental results).

132

λd−(V−s)

0.4
−0.2
−0.8
5
10
Amps

120

100

140
15 180 160 Angle−(degs)
(a) d-axis

λq−(V−s)

1
0.5
0
15

180
160
140

10

120

Amps

5

100

Angle−(degs)

(b) q-axis
Figure 7.18: Single-layer 25 SPP PM Machine with Insulation on a Single-turn Degraded( 21 Ω)
(experimental results).

133

λd−(V−s)

0.4
−0.2
−0.8
5
10
120

100

140
15 180 160 Angle−(degs)

Amps

λq−(V−s)

(a) d-axis

1
0.5
0
15

180
160
140

10
Amps

120
5

100

Angle−(degs)

(b) q-axis
Figure 7.19: Single-layer 25 SPP PM Machine with Insulation on a Single-turn Degraded( 41 Ω)
(experimental results).

134

λd−(V−s)

0.4
−0.2
−0.8
5
10
Amps

120

15 180

160

100

140
Angle−(degs)

λq−(V−s)

(a) d-axis

1
0.5
0
15

180
160
140

10
Amps

120
5

100

Angle−(degs)

(b) q-axis
Figure 7.20: Single-layer 25 SPP PM Machine with Insulation on 10% of turns in a slot
Degraded( 12 Ω) (experimental results).

135

λd−(V−s)

0.4
−0.2
−0.8
6
8
Amps

120 100
140
10 180 160 Angle−(degs)

λq−(V−s)

(a) d-axis

1
0.5
0
10

180
160

8
Amps

140
120

6

100

Angle−(degs)

(b) q-axis
Figure 7.21: Single-layer 25 SPP PM Machine with Insulation on 10% of turns in a slot
Degraded( 14 Ω) (experimental results).

136

λd−(V−s)

0.4
−0.2
−0.8
6
8
Amps

120

10 180

160

100

140
Angle−(degs)

λq−(V−s)

(a) d-axis

1
0.5
0
10

180
160
8
Amps

140
120

6

100

Angle−(degs)

(b) q-axis
Figure 7.22: Single-layer 25 SPP PM Machine with Insulation on 20% of turns in a slot
Degraded( 12 Ω) (experimental results).

137

λd−(V−s)

0.4
−0.2
−0.8
2
4

120

Amps

6 180

160

100

140
Angle−(degs)

(a) d-axis

λq−(V−s)

1
0.5
0
6

180
160
140

4

120

Amps

2

100
Angle−(degs)

(b) q-axis
Figure 7.23: Single-layer 25 SPP PM Machine with Insulation on 20% of turns in a slot
Degraded( 14 Ω) (experimental results).

138

100
Healthy
Singleturn
10%
20%

Voltage (V)

50
0
−50
−100
0

50
100
150
position (degrees)

Figure 7.24: Phase voltage of Double-layer 12 SPP PM Machine with Varying Number of
Turns Having Degraded Insulation from Finite Element Simulation.

Voltage (V)

200

Healthy
Singleturn
10%
20%

100
0
−100
−200
0

20
40
position (degrees)

60

Figure 7.25: Phase voltage of Single-layer 52 SPP PM Machine with Varying Number of
Turns Having Degraded Insulation from Finite Element Simulation.

139

linkages will change as the fault progresses. It is well known that the use of accurate machine
parameters for control improves a machine’s operating performance. Accurate knowledge of
the parameters is useful for more than control. Monitoring the condition of the winding
insulation can be accomplished online, if accurate knowledge of the machine parameters is
available. The machine model populated with accurate parameters together with measured
phase currents and machine speed provide voltage estimates, Equations (4.35) and (4.36).
Circulating current in windings with degraded insulation affect the machine parameters,
which will be reflected in the voltage estimations. The ability to detect that the winding
insulation is degrading during machine operation offers opportunity to mitigate the fault
and avoid machine failure, thereby increase machine reliability. Additionally, early detection
of winding insulation deterioration should reduce the occurrence of secondary faults and
provide opportunity for planned maintenance.

140

Chapter 8
Conclusion
Fractional slot concentrated windings can be designed to provide thermal, magnetic and
physical isolation between phases; however, machine performance is a direct trade-off. Singlelayer fractional slot concentrated windings provide the best isolation between phases; however, the inherent characteristics of this design negatively impacts the torque performance,
due to the high ripple. In this work, a torque ripple compensation technique was developed
to improve the torque performance of this reliable design.

An analytical torque expression was developed in this work. The resulting torque expression was divided into DC and ripple components. The ripple component was used as an
a priori estimator of torque ripple calculated from the current command and rotor position.
The estimated torque ripple was used to modify the torque command effectively decreasing
the ripple torque contributed by the harmonic of the rotor MMF with highest magnitude.
Finite element results demonstrated that the torque ripple can be decreased by as much
as 70% with the technique. It was shown that the ripple compensation technique increases
the machine losses, affecting efficiency. At high torque, the magnet losses increase by as
much as 15%. The model-based controller used in this work was populated with parameters
that only include the self-saturation effect. Additionally, both self- and cross-saturation were
neglected in the torque ripple estimator.

141

150
Healthy
1/2Ω−singleturn
1/2Ω−10%
1/2Ω−20%

Voltage (V)

100
50
0
−50
−100
−150
0

0.05
0.1
0.15
Time (seconds)

0.2

Figure 8.1: Experimental Voltage Estimate for Single-layer 25 SPP PM Machine Applying a
15A@120◦ Current Command

Designing a machine with magnetic, thermal and physical isolation of the phases does not
guarantee that the winding insulation will not wear. Increased machine reliability is gained
by monitoring the condition of the winding insulation during operation. Circulating current
begins to flow in the windings with worn insulation before the presence of a short-circuit.
The circulating current affects the machine characteristic parameters. In this work, insulation degradation was introduced experimentally with a resistor to two permanent magnet
machines with fractional slot concentrated windings. Experimental results demonstrated
changes in the characteristic flux linkages due to winding degradation. Finite element results showed that these changes are reflected in the machine voltage. The number of turns
with degraded insulation affects the shape of the machine voltage. Voltage variations, due to
degraded winding insulation, are not easily observed in the experimental voltage estimates,
as demonstrated in Figure 8.1.

142

Torque (Nm)

45
Singleturn
10%
20%

40
35
30
25
0

10
20
position (degrees)

30

Figure 8.2: Finite Element Torque for Single-layer 25 SPP PM Machine with incipient faults
The incipient fault in finite element is placed in phase a and physically located in the slot
between 4.87◦ and 25.12◦ . The machine torque reduces near the fault location, as shown
in Figure 8.2.

Permanent magnet synchronous machines equipped with single-layer fractional slot windings provide low possibility of failure. Of the machine designs evaluated in this work, the
single-layer design has the lowest value of circulating current in the event of stator winding
failure. The torque performance of this machine design is improved with the ripple compensation technique presented in this work. Prior to stator winding failure, the machine
characteristic parameters change as a result of incipient faults. These parametric changes
are reflected in the machine voltages and can be used to detect pending failures online, allowing for timely maintenance and repair.

Future work will demonstrate the use of experimental voltage estimates to detect incipi-

143

ent faults in the fault tolerant machines designed with fractional slot concentrated windings.
Additionally, methods for accurately identifying the fault location and improving the performance of the torque ripple compensation technique over the machine’s entire range of
operation.

Self- and cross-saturation are non-linear effects that have been proven important for development of high performance permanent magnet synchronous machine controllers. These
effects will be included in both the model-based controller and the torque ripple estimator.

During the parametric identification process for the 12 slot per pole per phase permanent
magnet machine, acoustic noise was noted when applying high d-axis current, angles close to
180◦ . Once the insulation of the winding was degraded, the acoustic noise was noted earlier,
as the number of turns with degraded insulation increased. Acoustic noise was also noted
when applying high d-axis current to the 25 slot per pole per phase machine with incipient
stator faults. Future work will explore the relationship between the angle of the applied
current and the acoustic noise, as well as its relation to insulation degradation.

144

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145

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