DESIGN AND CONTROL OF A HIGH-EFFICIENCY DOUBLY-FED
BRUSHLESS MACHINE FOR POWER GENERATION APPLICATIONS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
in the Graduate School of The Ohio State University
By
Bo Guan, M.S.
Graduate Program in Electrical and Computer Engineering
The Ohio State University
2014
Dissertation Committee:
Dr. Longya Xu, Advisor
Dr. Jin Wang
Dr. Mahesh Illindala
Copyright by
Bo Guan
2014
Abstract
Due to its similar terminal characteristics like a wound-rotor Doubly-Fed Induction Machine
(DFIM) and rugged rotor structure without brushes and slip rings, the Doubly-Fed Brushless
Machine (DFBM) has attracted much attention for years, especially in the applications of
variable-speed constant-frequency power generation and in adjustable speed drive system.
However, the theory and modeling of DFBM still have not been sufficiently developed.
Suffering from inherent deficiencies such as low mutual coupling between two windings,
difficulties in rotor structure design and non-optimized control, the DFBM made an impression of
low torque density and lower energy efficiency.
The research objectives of this dissertation are to: (1) give a systematic and quantitative
analysis of the mechanism, modeling and control of DFBM; and (2) design and build a highefficiency DFBM based doubly-fed power generation system.
Firstly, a detailed nonlinear analysis of DFBM by using the Finite Element Analysis (FEA)
method is given. The mechanism of electromechanical energy conversion of the DFBM is
investigated through the analysis of internal magnetic flux distribution, and of terminal
characteristics of winding flux linkage and induced speed voltage (Back-EMF). The proposed
harmonic decomposition method is utilized for the quantitative analysis of the asymmetric, nonsinusoidal and pulsating air-gap flux density, and assists in the evaluation of the “modulation”
capability of the DFBM.
Secondly, the mathematic models of DFBM in both dynamic and steady state conditions are
systematically examined. The results of the study confirm the previous findings by the FEA and
contribute to a quantitative analysis of the mathematic model of DFBM. The deduced dynamic
ii
and steady state equivalent circuits suggest that the DFBM has the same form and similar
expression of mathematical model as those of a conventional DFIM. After the modeling of
DFBM, the basic concept of field orientation control of torque and flux (and decouple control of
active and reactive power) is introduced and analyzed in the steady state conditions. Then, the
dynamic responses of the field orientation are discussed with the highlights of transient
characteristics and expression of the flux linkage and torque. The implementation of the field
orientation is also investigated.
Finally, the latest investigation of optimal design and advanced control results of DFBM are
presented. The challenges of designing a high-efficiency DFBM system are discussed in detail
with the identified solutions of the optimal design and control. Using design principles assisted
by FEA, an original design of a 200kW radially-laminated reluctance DFBM system is achieved,
capable of 2,000 N.m in the speed range of 400 - 1,200 rpm, with a frame size comparable to that
of a brush-type DFIM. The designed machine is built and tested in both a steady state and in
dynamic conditions, and the experimental results are suitably agreed with the design objectives.
The most successful results include the DFBM power delivery with more than 25% overload
capability and energy efficiency higher than 90%, occupying 75% of the designed torque-speed
regime. The theoretical and experimental results represent breakthroughs in doubly-fed brushless
technology. The feasibility of DFBM technology for practical application is fully established.
iii
Dedicated to my family and people that I love.
iv
Acknowledgments
I would like to express deepest gratitude to my advisor, Dr. Longya Xu, who has been
consistently providing academic guidance, funding support, opportunities and encouragement
throughout my graduate study.
His profound knowledge, creative way of thinking and
perspective of future have inspired me to explore in the academic world, keep thinking and put
thoughts into practice. His guidance not only helps me in the graduate study but also will deeply
and positively influence me a whole life.
I also want to thank Dr. Jin Wang, Dr. Donald Kasten, Dr. Steven Sebo and Dr. Mahesh S.
Illinda for helping me understand the essence of the power electronics and power systems. I
would like to give my thanks to Dr. Vadim Utkin for his education of the control theory.
Additionally, I would like to thank Dr. Hao Huang and Dr. Abbas Mohamed from GE
Aviation for providing attractive research projects, financial support, and also the opportunity for
me to gain high-level industry experience in USA.
My special thanks go to Dr. Yifan Zhao, Dr. Li Zhen and Dr. Xingyi Xu from Kinway Tech.
Inc. and Dr. Yi Ruan from Shanghai University for initially bringing me into the area of power
electronics and motor drives.
I would like to thank my senior fellows Dr. Jiangang Hu, Dr. Yuan Zhang, Dr. Wenzhe Lu
and Dr. Song Chi for their help to my study and life in Columbus. I also want to thank my junior
group members Dr. Kaichien Tsai and Dr. Zhendong Zhang for the team works on many fantastic
projects, and spending days and nights together. I have to give thanks to Yazan Alsmadi for his
contribution to my work especially when I was out of the campus. I also want to thanks my other
v
junior graduate students Dr. Ernesto Inoa, Mr. Yu Liu, Mr. Haiwei Cai, Mr. Dakai Hu, Dr. Ke
Zhou, Mr. Cong Li, Mr. Feng Guo and Mr. Mark Scott for their friendship and support. I have to
give special thanks to Hui Li and Zifei Dai for their friendship and support.
Finally, I would like to thank my parents Yunya Guan and Weirong Guan, and my girl friend
Anqi Zhang, for their endless love and consistent efforts to support me. I am truly grateful for
having such a happy and supportive family.
vi
Vita
July 2003……………………………………………………
B.S. Electrical Engineering,
Shanghai University, Shanghai, China
October 2004 – September 2006…………………………..Research and Development Engineer,
Kinway Technologies, Inc., Shanghai,
China
March 2006…………………………………………………
M.S. Electrical Engineering,
Shanghai University, Shanghai, China
March 2007 – February 2009……………………………...Project Research Associate,
GE Aviation, Dayton, Ohio
September 2006 – present…………………………………Graduate Research Associate,
The Ohio State University, Columbus,
Ohio
Publications
B. Guan and L. Xu, “A Novel Adaptive Algorithm for Rotor-Flux and Slip Estimation of FieldOriented Induction Machine Drives,” in IEEE Energy Conversion Congress and Exposition,
(ECCE) 2009, pp.1547,1552, Sept. 20-24, 2009.
B. Guan, Y. Zhao and Y. Ruan, “Torque Ripple Minimization in Interior PM Machines using
FEM and Multiple Reference Frames,” in 1st IEEE Conference on Industrial Electronics and
Application, Singapore, May 24-26, 2006.
H. Cai, B. Guan and L. Xu, “Low Cost PM Assisted Synchronous Reluctance Machine for
Electrical Vehicles,” in IEEE Transactions on Industry Electronics, vol.61, no.10, pp.5741-5748,
Oct. 2014.
vii
L. Xu, E. Inoa., Y. Liu and B. Guan, “A New High-Frequency Injection Method for Sensorless
Control of Doubly-Fed Induction Machines,” in IEEE Transactions on Industry Applications,
vol.48, no.5, pp.1556-1564, Sept. - Oct. 2012
L. Xu, B. Guan and J. Hu, “A Robust Sensorless Control Algorithm for Induction Generator
Operating in Deep Flux Weakening Region,” in Conf. Rec. of IEEE IAS Annual Meeting 2008,
pp.1-8, Oct. 5-9, 2008.
L. Xu, B. Guan, H. Liu, L. Gao and K. Tsai, “Design and Control of a High-efficiency DoublyFed Brushless Machine for Wind Power Generator Application,” in IEEE Energy Conversion
Congress and Exposition (ECCE) 2010, pp.2409-2416, Sept. 12-16, 2010.
H. Cai, B. Guan and L. Xu, “Optimal Design of Synchronous Reluctance Machine – a Feasible
Solution to Eliminating Rare Earth Permanent Magnets for Vehicle Traction Applications,” in
The International Journal for Computation and Mathematics in Electrical and Electronic
Engineering, Vol. 33 Iss: 5, 2014
Z. Zhang, L. Xu, Y. Zhang and B. Guan, “Novel Rotor-side Control Scheme for Doubly- Fed
Induction Generator to Ride through Grid Faults,” in IEEE Energy Conversion Congress and
Exposition (ECCE) 2010, pp.3084-3090, Sept. 12-16, 2010.
L. Xu, E. Inoa., Y. Liu and B. Guan, “A New High-Frequency Injection Method for Sensorless
Control of Doubly-Fed Induction Machines,” in IEEE Energy Conversion Congress and
Exposition (ECCE) 2011, pp.1758-1764, Sept. 17-22, 2011.
L. Gao, B. Guan, Y. Zhou and L. Xu, “Model Reference Adaptive System Observer Based
Sensorless Control of Doubly-Fed Induction Machine,” in IEEE International Conference
on Electrical Machines and Systems (ICEMS) 2010, pp.931- 936, Oct. 10-13, 2010.
H. Cai, B. Guan, L. Xu and W. Choi, “Optimal Design of Synchronous Reluctance Machine – A
Feasible Solution to Eliminating Rare Earth Permanent Magnets for Vehicle Traction
Applications,” in Ninth International Conference on Ecological Vehicles and Renewable Energies
(EVER) 2012, March 22-24, 2012
Fields of Study
Major Field: Electrical and Computer Engineering
viii
Table of Contents
Abstract ............................................................................................................................................ ii
Dedication. ...................................................................................................................................... iv
Acknowledgments............................................................................................................................ v
Vita................................................................................................................................................. vii
List of Tables ................................................................................................................................. xii
List of Figures ............................................................................................................................... xiii
Chapter 1: Introduction .................................................................................................................. 1
1.1 History of Doubly-Fed Brushless Machines ....................................................................... 1
1.2 DFBM Structure and System Configuration....................................................................... 4
1.3 Dissertation Overview ........................................................................................................ 7
Chapter 2: Analysis of DFBM by Finite Element Analysis (FEA) Method .................................. 9
2.1 DFBM Geometry and Winding Structure ........................................................................... 9
2.2 Air-gap Flux Density Distribution .................................................................................... 11
2.3 Winding Flux Linkage and Back-EMF............................................................................. 24
2.4 Comparison of 4/6 and 4/8 Pole Combination .................................................................. 41
2.5 Conclusions....................................................................................................................... 49
Chapter 3: Modeling and Equivalent Circuit of DFBM .............................................................. 51
3.1 Dynamic Equations of DFBM in a Stationary a,b,c Reference Frame ............................. 51
3.1.1 Voltage Equations of DFBM ................................................................................. 52
ix
3.1.2 Inductances of DFBM ............................................................................................ 53
3.2 Complex Variable Model of DFBM ................................................................................. 57
3.3 Equations of DFBM in a Rotating d-q Reference Frame.................................................. 59
3.4 Operational Equivalent Circuits of DFBM ....................................................................... 63
3.5 Power and Torque Equations ............................................................................................ 68
3.6 Conclusions....................................................................................................................... 71
Chapter 4: Field Orientation Control of DFBM for Doubly-Fed Power Generation Applications
....................................................................................................................................................... 72
4.1 Steady State Field Orientation Control of DFBM ............................................................ 72
4.1.1 Steady State Operation of DFBM in the Stator Flux 2 Reference ......................... 75
4.1.2 Active and Reactive Power Control ....................................................................... 84
4.2 Dynamic Field Orientation Control of DFBM.................................................................. 87
4.3 Conclusions....................................................................................................................... 94
Chapter 5: Design, Construction and Experimental Study of the Prototype DFBM System........ 96
5.1 Energy Efficiency of a DFBM .......................................................................................... 96
5.2 Torque Density of a DFBM .............................................................................................. 97
5.3 Sizing of the Prototype DFBM ....................................................................................... 100
5.4 Segmental Rotor Design of the Prototype DFBM .......................................................... 101
5.5 Performance Prediction of the Prototype DFBM by the FEA Method ........................... 103
5.6 Experimental Results of the Prototype DFBM ............................................................... 106
5.6.1 Steady State Testing of the Prototype DFBM ...................................................... 111
x
5.6.2 Dynamic Testing of the Prototype DFBM ........................................................... 117
5.7 Conclusions..................................................................................................................... 124
Chapter 6: Conclusions and Future Work .................................................................................. 126
6.1 Conclusions..................................................................................................................... 126
6.2 Future Work .................................................................................................................... 128
Bibliography ................................................................................................................................ 129
xi
List of Tables
Table 1 Comparison of Effective Air-Gap Flux Density and Rotor Modulation Capability
between the 4/6 and 4/8 Pole Combination ................................................................................... 45
Table 2 Specifications and Main Dimensions of the Prototype DFBM..................................... 100
xii
List of Figures
Figure 1.1 Two Induction Machines in Cascade Connection ........................................................ 2
Figure 1.2 Rotor Structures of the DFBM ..................................................................................... 5
Figure 1.3 Main Configuration of a DFBM system ....................................................................... 5
Figure 2.1 DFBM Stator and Rotor Assembly (4/6 Pole Combination) ...................................... 10
Figure 2.2 Dual Stator Windings of the DFBM (4/6 Pole Combination) .................................... 10
Figure 2.3 Discretized Field Model of the DFBM (Mesh Plot in FEA) ...................................... 11
Figure 2.4 Cross Section Flux Distribution with DC Excitation of 4-Pole Winding ................... 15
Figure 2.5 Air-Gap Flux Density with DC Excitation of 4-Pole Winding (4/6 Pole Combination)
....................................................................................................................................................... 16
Figure 2.6 Details of Cross Section Flux Distribution and Air-Gap Flux Density
with 4-Pole MMF Excitation ......................................................................................................... 17
Figure 2.7 Distribution of the Magnetic Poles with the 4-Pole MMF Excitation (Case I) .......... 18
Figure 2.8 Distribution of the Magnetic Poles with the 4-Pole MMF Excitation (Case II) ......... 19
Figure 2.9 Harmonics of Air-Gap Flux Density with 4-Pole MMF Excitation (4/6 Pole
Combination) ................................................................................................................................. 20
Figure 2.10 Effective Harmonics of Air-Gap Flux Density with 4-Pole MMF Excitation ......... 20
Figure 2.11 Actual Waveform vs. Effective Harmonics of Air-Gap Flux Density with 4-Pole
MMF Excitation ............................................................................................................................. 21
Figure 2.12 Cross Section Flux Distribution with DC Excitation of 6-Pole Winding ................. 22
Figure 2.13 Air-Gap Flux Density with DC Excitation of 6-Pole Winding (4/6 Pole
Combination) ................................................................................................................................. 23
xiii
Figure 2.14 Harmonics of Air-Gap Flux Density with 6-Pole MMF Excitation (4/6 Pole
Combination) ................................................................................................................................. 24
Figure 2.15 Self Flux Linkage of 4-Pole Winding Versus Rotor Positions with DC Excitation . 26
Figure 2.16 Self Flux Linkage of 6-Pole Winding Versus Rotor Positions with DC Excitation . 27
Figure 2.17 Self Flux Linkage of 4-Pole Winding at 0 rpm with AC, 40Hz Excitation .............. 28
Figure 2.18 Self Flux Linkage of 4-Pole Winding at 1,200 rpm with AC, 40Hz Excitation ....... 28
Figure 2.19 Self Flux Linkage of 4-Pole Winding at -1,200 rpm with AC, 40Hz Excitation ..... 29
Figure 2.20 Mutual Flux Linkage of 4-Pole Winding with DC Excitation of 6-Pole Winding ... 31
Figure 2.21 Back-EMF of 4-Pole Winding with DC Excitation of 6-Pole Winding at 1,200 rpm
....................................................................................................................................................... 32
Figure 2.22 Mutual Flux Linkage of 6-Pole Winding with DC Excitation of 4-Pole Winding ... 33
Figure 2.23 Mutual Flux Linkage of 4-Pole Winding with AC, -60Hz Excitation of 6-Pole
Winding ......................................................................................................................................... 35
Figure 2.24 Back-EMF of 4-Pole Winding with AC, -60Hz Excitation of 6-Pole Winding ....... 37
Figure 2.25 Mutual Flux Linkage of 6-Pole Winding with AC, 40Hz Excitation of 4-Pole
Winding ......................................................................................................................................... 38
Figure 2.26 Cross Section Flux Distribution with DC Excitation of 4-Pole Winding (4/8 Pole
Combination) ................................................................................................................................. 42
Figure 2.27 Air-Gap Flux Density with DC Excitation of 4-Pole Winding (4/8 Pole
Combination) ................................................................................................................................. 43
Figure 2.28 Harmonics of Air-Gap Flux Density with 4-Pole MMF Excitation (4/8 Pole
Combination) ................................................................................................................................. 44
Figure 2.29 Harmonics of Air-Gap Flux Density with 8-Pole MMF Excitation (4/8 Pole
Combination) ................................................................................................................................. 44
xiv
Figure 2.30 Mutual Flux Linkage of 8-Pole Winding with AC, 40Hz Excitation of 4-Pole
Winding ......................................................................................................................................... 46
Figure 2.31 Mutual Flux Linkage of 4-Pole Winding with AC, 80Hz Excitation of 8-Pole
Winding ......................................................................................................................................... 47
Figure 3.1 Magnetic Axes of a DFBM ........................................................................................ 52
Figure 3.2 d-q Axes and - Axes Relative to Magnetic Axes of a DFBM ............................... 59
Figure 3.3 Complex Vector Equivalent Circuit of a DFBM in the d-q Reference Frame ........... 61
Figure 3.4 Scalar Form Equivalent Circuits of a DFBM in the d-q Reference Frame................. 63
Figure 3.5 Operational Complex Vector Equivalent Circuit of the Dynamic Model of a DFBM64
Figure 3.6 Positive Sequence Steady State Equivalent Circuit of a DFBM ................................ 66
Figure 3.7 Negative Sequence Steady State Equivalent Circuit of a DFBM ............................... 67
Figure 4.1 General Steady State Equivalent Circuit of a DFBM (per phase) with Referral Ratio a
....................................................................................................................................................... 73
Figure 4.2 Steady State Equivalent Circuit of a DFBM (per phase) without Leakage Inductance
on the Side of Stator 2.................................................................................................................... 74
Figure 4.3 Steady State Equivalent Circuit of a DFBM in the Stator Flux 2 Reference – When
Magnetizing Current is Supplied Only by Stator Winding 1 ......................................................... 75
Figure 4.4 Phasor Diagram of a DFBM in the Stator Flux 2 Reference – When Magnetizing
Current is Supplied Only by Stator Winding 1 .............................................................................. 77
Figure 4.5 Steady State Equivalent Circuit of a DFBM in the Stator Flux 2 Reference – When
Magnetizing Current is Supplied Only by Stator Winding 2 ......................................................... 79
Figure 4.6 Phasor Diagram of a DFBM in the Stator Flux 2 Reference – When Magnetizing
Current is Supplied Only by Stator Winding 2 .............................................................................. 80
xv
Figure 4.7 Steady State Equivalent Circuit of a DFBM in the Stator Flux 2 Reference – When
Magnetizing Current is Supplied by Both Stator Windings 1 and 2 .............................................. 82
Figure 4.8 Phasor Diagrams of a DFBM in the Stator Flux 2 Reference – When Magnetizing
Current is Supplied by Both Stator Windings 1 and 2 ................................................................... 83
Figure 4.9 Active Power Balance of a Brushless Doubly-Fed Power Generation System .......... 85
Figure 4.10 Phasor Diagram of a DFBM in d-q Axes – Stator Flux 2 Reference ....................... 89
Figure 4.11 Dynamic Torque Diagram of a DFBM Represented by Oriented Flux and q-Axis
Current – Stator Flux 2 Reference ................................................................................................. 90
Figure 4.12 Dynamic Torque Diagram of a DFBM Represented by Currents – Stator Flux 2
Reference ....................................................................................................................................... 92
Figure 4.13 Control Block Diagram for Field Orientation Control of a DFBM .......................... 94
Figure 5.1 Stator Lamination of the Prototype DFBM .............................................................. 101
Figure 5.2 Rotor Cross-sectional View of the Prototype DFBM ............................................... 102
Figure 5.3 Lamination for Rotor Segments ............................................................................... 103
Figure 5.4 Asymmetrical Rotor Magnetic Structure of the Prototype DFBM in a No-Load
Condition ..................................................................................................................................... 104
Figure 5.5 Magnetizing Curve of the Prototype DFBM by the FEA ......................................... 104
Figure 5.6 Torque Capability of the Prototype DFBM with Rated Currents by the FEA ......... 105
Figure 5.7 Magnitude of Air-Gap Flux Density Distribution of the Prototype DFBM by the FEA
..................................................................................................................................................... 106
Figure 5.8 Stator, Rotor and Total Assembly of the Prototype DFBM ..................................... 108
Figure 5.9 Main Circuit Configuration of the Developed Converter Module ........................... 109
Figure 5.10 Photos of the Experimental Testing Setup.............................................................. 109
Figure 5.11 v-i Waveforms in the No-Load Condition of the Prototype DFBM ....................... 111
xvi
Figure 5.12 Measured Magnetizing Curve of the Prototype DFBM ......................................... 112
Figure 5.13 Measured Iron Losses in the No-Load Conditions of the Prototype DFBM .......... 113
Figure 5.14 4-Pole Stator Winding Current and Grid Voltage Waveforms in Steady State Loaded
Conditions .................................................................................................................................... 114
Figure 5.15 Grid/ 6-Pole Winding Current Voltage Waveforms in Steady State Loaded
Conditions .................................................................................................................................... 115
Figure 5.16 Efficiency Contours in Loaded Conditions Based on Measurements .................... 116
Figure 5.17 Voltages and Current Waveforms of the Grid-Side Converter during the Process of
its Grid-Friendly Integration ........................................................................................................ 118
Figure 5.18 Grid, Induced 6-Pole Winding Voltages and 4-Pole Winding Current Waveforms of
the Prototype DFBM System before Stator-Side Grid Integration .............................................. 119
Figure 5.19 Grid, Induced 6-Pole Winding Voltages and4-Pole Winding Current Waveforms of
the Prototype DFBM System during the Process of Stator-Side Grid-Friendly Integration ........ 120
Figure 5.20 Dynamic Performance of the Prototype DFBM System Active Power Control .... 121
Figure 5.21 Dynamic Performance of the Prototype DFBM System Reactive Power Control . 122
Figure 5.22 Continuous Full Load Operation of the Prototype DFBM in a Wide Speed Range
..................................................................................................................................................... 123
xvii
Chapter 1: Introduction
1.1 History of Doubly-Fed Brushless Machines
Doubly-Fed Brushless Machines (DFBMs) have attracted attention in recent years in the
application of variable-speed constant-frequency generating and adjustable speed drive systems
[1-9]. With the advances of power electronics technologies, a DFBM appears very attractive
because of its rugged structure (absence of slip rings and brushes), decent compatibility with a
power converter, and flexible operational modes for various application needs. In particular, a
DFBM resembles the terminal characteristics of a wound-rotor Doubly-Fed Induction Machine
(DFIM), which has made DFBM a very competitive candidate in applications where doubly-fed
operational modes are preferred but slip rings and brushes are not allowed.
The basic concept of DFBM can be traced back nearly 100 years and has experienced three
major development waves. In the very early and first wave, around the 1910-1920s, Hunt and
Creedy researched the concept of the “self-cascaded induction motor”. Stemming from the
cascaded induction machine, the “self-cascaded induction motor” essentially is two wound
induction motors in a special arrangement. Mechanically, the two induction motors share a
common shaft and, at the same time, the two sets of rotor windings are electrically self-cascaded
[1, 2]. As a result, the overall system was left with two sets of terminals for the two sets of stator
windings. Since the two sets of rotor windings are self-cascaded, no brushes and slip rings are
needed, as shown in Figure 1.1.
1
Stator winding1
VABC
Common shaft
DFIM1
Stator winding2
DFIM2
Vabc
Connected
rotor windings
Figure 1.1 Two Induction Machines in Cascade Connection
High starting torque and speed regulation were obtained with certain success. With the help
of power electronics control, this early version of DFBM, with minor variations, still finds
application today [3].
Fifty years later was the second wave (around the 1970s). Broadway,
Thomas, Kusko, Somuah, and others conducted further research and published their in-depth
understanding in “self-cascaded induction motor” [4-8]. It was proposed to merge the two sets of
stator windings by a dual-tapped stator windings wound into a common stator core. Also, it was
proposed to design a special rotor common to the dual-tapped stator windings. In this way, the
early version of DFBM evolved from the “self-cascaded induction motor” with a common shaft,
to a single electric machine in a common stator house. However, in terms of modeling and
analysis of the machine, all researchers during that period still treated the machine as two separate
machines but built into a single frame. Along the way, creative ideas sprouted: of particular
mention is Broadway’s rotor in two styles, the nested cage rotor and the salient reluctance rotor
that could be equally effective for the single-frame “self-cascaded induction motor” [4-5]. With
somewhat complicated derivations, Broadway gave steady-state equations and an equivalent
circuit for the self-cascaded machine, and named it “Brushless Stator-Controlled SynchronousInduction Machines” [6-7]. At the beginning of the 1980s one attempt was made by Heyne and
El-Antably to prototype Broadway’s version of DFBM [9]. This prototype is perhaps the firstever detailed experimental investigation, which, to a certain degree, verified the basic principles
2
but did not produce meaningful torque density and energy efficiency for practical applications. In
addition, control principles were developed and algorithms implemented.
The renewed interest in DFBM in the 1990s, regarded as the third wave with much strength,
was driven by the rapid advances in modern power electronics, which, supposedly, took full
advantage of possible potentials of the DFBMs in variable-speed drives and variable-speed
constant-frequency generators. Many papers have been published continuing the discussions of
the DFBM principles, modeling, operational characteristics, and possible advanced control
methods and applications [10-23].
The most important development in this stage is that
researchers have formally established the identity of DFBM as a special machine, instead of
treating it as two machines in “self-cascade”, that is, a combination of two induction machines
built on a common shaft or single stator house [23-32]. Their research established that, regardless
of rotor types, the DFBM terminal characteristics resemble those of a wound-rotor DFIM.
Detailed field analysis of DFBM further verified that the two sets of stator windings in a DFBM
resemble the mutual coupling characteristics of the stator and rotor windings in a conventional
wound-rotor DFIM. However, from the structure robustness point of view, the attractiveness of
the DFBM machine is evident – the machine can be operated as a conventional DFIM but
eliminate all the headaches associated with brushes and slip rings, from high costs of building and
maintaining the machine to the serious reliability issues. Attracted by the features of brushless
and doubly-fed operation modes, several prototype DFBMs were built.
Unfortunately, the
experiment results did not show great promise, with efficiency only about 75% and inability to
reach the designed full power [32-34].
Nevertheless, the research results during the last twenty years did reveal clearly a series of
fundamental issues and challenges with respect to the design and control of a DFBM. Compared
to its counterpart with brushes/slip-rings, the issues of DFBM to the researchers are serious,
including: a) What are the rules for DFBM optimal electromagnetic design to maximize torque
3
and power density? b) What are the suitable control algorithms for a DFBM system? c) How can
the energy efficiency be substantially improved? d) What are the ultimate limits on design and
control of such a machine?
Evidently, these challenges are of practical significance and
demanding application. Without innovative breakthroughs to the issues, the dreamed DFBM
technology would remain as a dream on academic papers.
1.2 DFBM Structure and System Configuration
The rotor structures of the DFBM (shown in Figure 1.2), are categorized into two major types:
nested-loop cage rotor and reluctance rotor [32, 33]. Compared to the cage rotor structure, the
reluctance rotor with flux barriers produces stronger mutual coupling characteristics of the two
stator windings, resulting in higher torque capability and efficiency [27-29].
The axially-
laminated reluctance rotor has been well investigated in the literature so that its pros and cons are
quite clear [35, 36]. To further reduce the eddy current losses of the rotor, a radially-laminated
reluctance rotor recently has been chosen in many research studies for applications in which high
efficiency is required [37, 38]. In this thesis, the study has concentrated on the design and control
of a radially-laminated reluctance rotor based DFBM with the aim of high efficiency performance
in doubly-fed power generation applications.
A typical radially-laminated reluctance DFBM based variable speed drive (or power
generation) system as shown in Figure 1.3 consists of three main components: a DFBM, a backto-back converter and an associated controller [39-42]. The DFBM is a controlled electric
machine, meaning that for its practical applications, it is necessary to mate the DFBM with a bidirectional power flow converter.
4
(a) Nested-Loop Cage Rotor
(b) Salient Pole Rotor without Flux Barrier
(c) Axially-Laminated Reluctance Rotor
(d) Radially-Laminated Reluctance Rotor
with Flux Barrier
with Flux Barrier
Figure 1.2 Rotor Structures of the DFBM
3-phase Power Grid
2q Stator
Winding 2
DFBM
2p Stator
Winding 1
Converter
Mechanical
Feedback
Controller
Electrical
Feedback
Figure 1.3 Main Configuration of a DFBM system
5
As shown, the stator of a DFBM has two sets of 3-phase sinusoidally distributed windings.
One set of windings, the primary, is fed with variable voltages at variable frequencies from a
converter, which is also connected to the power grid. The other set of windings, the secondary, is
directly connected to the power grid. When both sets of stator windings are fed from a set of 3phase symmetric currents, two rotating MMFs are produced along the DFBM air-gap. Since the
two sets of windings differ in pole numbers, one of 2p and another 2q, and the excitation currents
are of different frequencies, one of 1 and another 2, the rotating MMFs differ from each other.
According to electric machine fundamentals, under normal conditions, the two stator MMFs have
no useful interaction for electromechanical energy conversion, except for torque and force
oscillations. However, in the structure as described, with the magnetic modulation of the rotor,
the two MMFs do have useful interaction for electromechanical energy conversion [41, 49].
The rotor of the DFBM can be built in one of the two styles: the rotor with nested cages of pr
circuits, or rotor with reluctance segments of pr pieces where:
(1.1)
Eq. (1.1) indicates that pr, the number of rotor nested cage circuits or reluctance segments, is
constrained by the pole numbers of the two sets of stator windings. Eq. (1.1) simply states that
among the three numbers (pr, p and q) we can independently choose two (two degrees of
freedom) and the third is determined by the equation. The operation of the DFBM relies on the
interaction of the two stator MMFs through modulation action of the rotor. When one set of
symmetrical sine-wave currents of frequency 1 are flowing in the primary windings, threephases of back EMFs will be induced with a frequency of 2 in the secondary windings and vice
versa. The two electrical frequencies, 1 and 2, in the primary and secondary are related to the
rotor mechanical speed rm by
6
(1.2)
Electromechanical energy conversion will occur in the DFBM if Eq. (1.2) is satisfied. Eq. (1.2)
clearly shows that, among the three speeds (1, 2 and rm), we can independently control two
(two degrees of freedom) and the third is determined by the equation. In motor operation for
speed control, a predetermined rotor speed rm is achievable if 1 and 2 are controlled. As in
generator operation, variable-speed constant-frequency generation is achievable by the DFBM if
1 or rm are predetermined, with 2 controlled conforming to Eq. (1.2).
Depending on the sequence and value of the controlled frequency 2, a DFBM can operate in
different modes [40, 41]. In particular, in the doubly-fed mode with 2=0, the DFBM operates as
a synchronous machine. On the other hand, with 20 (positive sequence) or 2<0 (negative
sequence), a DFBM can operate as an induction machine below or above synchronous speed. In
terms of power flow, regardless at the sub-, super-, or synchronous rotor speed, a DFBM always
can be operated as a motor or generator. For the above arguments, it is more reasonable just to
name this machine “doubly-fed brushless machine” or DFBM, rather than to call it doubly fed
brushless “cage induction” or “reluctance” machine, as in the past literature. The machine is not
called a “cage” or “reluctance” machine because it is substantially different from both
conventional cage induction and synchronous reluctance machines in the traditional and classical
sense of cage or reluctance machines.
1.3 Dissertation Overview
This dissertation presents the latest investigation of modeling, optimal design and advanced
control of a radially-laminated reluctance DFBM with an original design of 200kW/1,200rpm for
7
high-efficiency and high-reliability doubly-fed power generation applications. The dissertation is
organized as follows.
Chapter 2 presents the detailed nonlinear analysis of a DFBM by using the Finite Element
Analysis (FEA) method. Through the investigation of the air-gap flux distribution, winding flux
linkage and back-EMF, the magic and mechanism of DFBM have been studied.
Based on the information in Chapter 2, Chapter 3 examines the dynamic and steady state
modeling (equations and equivalent circuits) of the DFBM in both stationary (a,b,c) and rotating
(d-q) reference frames. The concept of the complex vector is used to represent the machine
equations.
Chapter 4 studies the principles of field orientation control of the DFBM. Firstly, the basic
concept of decoupling control of torque and flux (or active and reactive power) is introduced and
analyzed in the steady state conditions.
Next, the dynamic response characteristics and
implementation of the field orientation are discussed.
In Chapter 5, the challenges of designing a high-efficiency DFBM and system are
highlighted. Following the challenge description are the identified solutions to the optimal
design. Using Finite Element Analysis (FEA), the thesis presents the original design of DFBM
and system in a power rating of 200kW for a speed range of 400-1,200rpm. The designed
machine is built and tested in the laboratory and both of the steady-state and dynamic
experimental results are presented and analyzed.
Chapter 6 concludes the research of the dissertation and renders an outline for possible future
works.
8
Chapter 2: Analysis of DFBM by Finite Element Analysis (FEA) Method
This chapter presents the detailed nonlinear analysis of DFBM by using the Finite Element
Analysis (FEA) method. The results are considered accurate because the complicated geometry
of the DFBM and nonlinearity of the materials are taken into full account [27]. Through the
investigations of the internal magnetic field analysis of flux distribution, terminal characteristics
of winding flux linkage and back-EMF, the magic and mechanism of DFBM are studied. Lastly,
a comparison study of two cases of pole number combinations is presented to show various
advantages and disadvantages of different pole number combinations with an odd number and
even number of rotor segments.
2.1 DFBM Geometry and Winding Structure
Firstly, a DFBM with 4/6 pole combination in the stator winding and five rotor segments is
chosen as the prototype for the following analysis. The cross-section of the DFBM is shown in
Figure 2.1. The dual 4-pole and 6-pole three-phase sinusoidally distributed stator windings are
illustrated in Figure 2.2.
As shown in Figure 2.3, to achieve an accurate computation of FEA, the cross-section are
well discretized especially for the area of air-gap where the Magnetomotive Force (MMF) drop is
dominant in the whole magnetic path.
9
Figure 2.1 DFBM Stator and Rotor Assembly (4/6 Pole Combination)
(a) 4-pole Stator Winding
(b) 6-pole Stator Winding
Figure 2.2 Dual Stator Windings of the DFBM (4/6 Pole Combination)
10
Figure 2.3 Discretized Field Model of the DFBM (Mesh Plot in FEA)
2.2 Air-gap Flux Density Distribution
For electrical machines, the total flux of the core is indeed dependent on the sum of the air-gap
flux and leakage flux [54]. As known, the torque capability, induced speed voltage, core and
teeth saturation of an electrical machine are closely related to the air-gap flux density. Therefore,
the air-gap flux density is chosen as a key parameter to investigate the main magnetic flux
characteristics of the DFBM.
Since the core flux density is the spatial integral of the air-gap flux density, if the core flux
density is sinusoidal, then the air-gap flux density also should be co-sinusoidal with the same
fundamental frequency as that of the core flux density [54]. Compared to conventional AC
machines, the pole number of the two windings and the number of rotor segment are different for
the DFBM. It is interesting to determine the unique characteristics of the flux distribution and
particular air-gap flux pattern.
As indicated in Eq. (1.2), when the 2p pole and 2q pole windings are doubly excited by two
sets of three-phase sinusoidal currents with different frequencies of ω1 and ω2, two sinusoidally
11
distributed MMFs are created with frequencies of
and
. The combined MMF will interact
with the rotor permeance in conditions of p+q (could be an odd number) pieces of rotor segments
and rotor speed of
, and essentially produce a complicated and inexplicit air-gap flux
distribution [28].
For an electrical machine, the flux pattern is dependent solely on the relative position between
the MMF and rotor permeance, no matter what the frequency of the applied current excitation is,
and no matter whether the rotor is rotating or not. Therefore, to simplify the analysis of flux
distribution and air-gap flux density, the DFBM is singly excited by a three-phase DC current
[27].
In Figure 2.4, when the 4-pole winding is excited by a three-phase DC current, the flux
distribution and air-gap flux density are plotted at six different rotor positions spatially spanning
72 degrees. Since the rotor has five pieces of rotor segment, the air-gap flux pattern should be
repetitive for every 72 degrees in space. Note that, although a 4-pole sinusoidally distributed
MMF is produced in the stator winding, a non-sinusoidal and even asymmetric flux distribution is
created by the effect of the rotor permeance of five pieces of rotor segment. The asymmetric
magnetic field will produce an unbalanced magnetic pull, resulting in noises and vibrations [49,
50]. Some interesting observations are listed as follows:
 When the 4-pole MMF is applied in the stator, with the rotor permeance, the flux distribution
is unlike the conventional 4-pole electrical machine with four symmetric curls of flux line.
Due to the effect of the magnetic barrier in the rotor structure, the flux follows the path
provided by the laminations, resulting in curls of flux line over each segment. The entire flux
distribution over the cross section of this machine is asymmetric. For instance, in Figure 2.4
(a), the flux density over segments 1and 4 are obviously higher than those of segments 2, 3
and 5. Furthermore, each curl of flux line over each segment also is not exactly symmetric
12
about its axis. As shown in Figure 2.5, the characteristic of this asymmetric flux distribution
spanning each segment is more clearly present by the air-gap flux density.
 A special feature is in one of the five segments (e.g., Segment #3 in Figure 2.4 (a) and
Segment #4 in Figure 2.4 (c)) over which there are more than one curl of flux line. Meanwhile,
the flux pattern and corresponding air-gap flux density are extremely different from those of
the other four segments. For this segment, as shown in Figure. 2.6, the waveform of the airgap flux density is even more asymmetric and full of harmonics, and its magnitude is much
lower than the others.
It means that the magnetic capability of 1/5 (20%) of the iron
lamination has not been fully utilized, and potentially resulting in degraded torque capability
and efficiency to some extent.
 To describe the magnetic flux characteristics of the DFBM more straightforwardly, the
distribution of the magnetic polarities over the cross section of the machine is pointed out.
Figure 2.7 illustrates the distribution of the magnetic poles with the 4-pole MMF excitation.
Unlike the conventional 4-pole machine with the normal flux pattern:“N-S”-“S-N”-“N-S”-“SN”, the flux pattern of the DFBM is “N-S”-“S-N”-“N-S”-“S-N”-“S-N”. It seems that one
more pair of “S-N” is created, since there are five pieces of rotor segments with special flux
barrier design. As shown in Figure 2.8, the distribution of the magnetic polarities becomes
“N-S”-“S-N”-“N-S”-“N-S-N”-“S-N” at some of the rotor positions.
 In previous literature, due to the asymmetric, non-sinusoidal and pulsating features of the airgap flux density, it was difficult to perform some quantitative analysis and find an effective
method to evaluate the characteristics of the air-gap flux density. In this study, the harmonic
decomposition method has been utilized. Since there are two sinusoidally distributed MMFs
with frequencies of
and
when the DFBM are doubly excited by two sets of three-phase
sinusoidal currents, not only the “fundamental component” but also the corresponding
13
“effective harmonics” of the flux need to be taken into account. If the Fast Fourier Transform
(FFT) algorithm is applied to the waveform of the air-gap flux density in Figure 2.4, the
amplitude of the harmonic components across the frequency spectrum are achieved and shown
in Figure 2.9, where the 2nd harmonic component indicates the 4-pole magnetic flux
distribution, and the 3rd harmonic component represents the 6-pole magnetic flux distribution.
It is noted that, although only a 4-pole sinusoidally distributed MMF is produced in the stator
winding, both the 4-pole and 6-pole flux distributions are dominantly created along the air-gap
by the effect of the rotor permeance. The magnitude of the 6-pole air-gap flux density (B6) is
96.1% of that of the 4-pole air-gap flux density (B4). The equation of
could be
used to represent the mutual coupling between the 4 and 6-pole windings in terms of flux
density. As can be shown in Figures 2.10 and 2.11, the main characteristics of the air-gap flux
density are clearly demonstrated by the combination of the 4- and 6-pole air-gap flux densities.
This special function of the rotor structure could be interpreted as the “modulation” capability
of modulating the 4-pole MMF into the 6-pole flux along the air-gap. If this 6-pole flux
(generated by the rotor modulation) interacts with the 6-pole MMF (generated by the current
excitation in the 6-pole stator winding), the electromagnetic torque is possibly created.
Therefore, the “effective” harmonic component of the air-gap flux density is the 6-pole flux by
the rotor modulation when the 4-pole MMF is applied, and vice versa. To achieve higher
torque, the higher mutual coupled flux density resulting from the rotor modulation is desired.
The design of the rotor structure of the DFBM has been extensively studied by researchers to
enhance the modulation function. The modulation ability of the rotor is a key feature affecting
the torque capability of this machine and could be represented by the amplitude of the
effective harmonic component of the air-gap flux density.
14
#2
#3
#1
#4
#5
(a) θr
(b) θr = 12
#2
#3
#4
#1
#5
(c) θr = 24
(d) θr = 36
(e) θr = 48
(f) θr = 60
Figure 2.4 Cross Section Flux Distribution with DC Excitation of 4-Pole Winding
15
Air-Gap Flux
Density (T)
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
θr = 0
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
Air-Gap Flux
Density (T)
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
Air-Gap Flux
Density (T)
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
Air-Gap Flux
Density (T)
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
Air-Gap Flux
Density (T)
Air-Gap Flux
Density (T)
θr = 12
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
θr = 24
θr = 36
θr = 48
θr = 60
0
90
180
270
360
Locations in Air-Gap (Degree)
Figure 2.5 Air-Gap Flux Density with DC Excitation of 4-Pole Winding (4/6 Pole Combination)
16
Air-Gap Flux Density (T)
1.6
θr = 24
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
0
90
180
270
Locations in Air-Gap (Degree)
Figure 2.6 Details of Cross Section Flux Distribution and Air-Gap Flux Density
with 4-Pole MMF Excitation
17
360
S
N
N
N
S
N
S
N
Air-Gap Flux Density (T)
1.6
S
S
S
N
N
S
N
S
S
N
S
N
θr = 36
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
0
90
180
270
360
Locations in Air-Gap (Degree)
Figure 2.7 Distribution of the Magnetic Poles with the 4-Pole MMF Excitation (Case I)
18
S
N
S
N
N
S
N
N
S
N S
Air-Gap Flux Density (T)
1.6
N
S
S
N
S N
N
S
N S
N
θr = 24
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
0
90
180
270
360
Locations in Air-Gap (Degree)
Figure 2.8 Distribution of the Magnetic Poles with the 4-Pole MMF Excitation (Case II)
19
Amplitude of Air-Gap Flux Density (T)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
Harmonic Order
Figure 2.9 Harmonics of Air-Gap Flux Density with 4-Pole MMF Excitation (4/6 Pole
Combination)
1.5
Air-Gap Flux Density (T)
1
4 Pole Air-Gap
Flux Density
0.5
6 Pole Air-Gap
Flux Density
0
Total Air-Gap Flux
Density
-0.5
-1
-1.5
0
90
180
270
360
Locations in Air-Gap (Degree)
Figure 2.10 Effective Harmonics of Air-Gap Flux Density with 4-Pole MMF Excitation
20
S S N N S N S S N N
1.5
Air-Gap Flux Density (T)
1
Effective Air-Gap
Flux Density
0.5
0
Actual Air-Gap
Flux Density
-0.5
-1
-1.5
0
90
180
270
360
Locations in Air-Gap (Degree)
Figure 2.11 Actual Waveform vs. Effective Harmonics of Air-Gap Flux Density with 4-Pole
MMF Excitation
Figures 2.12 and 2.13 show the similar characteristics of the flux distribution and air-gap flux
density at six different rotor positions spatially spanning 72 degrees, while the 6-pole winding is
singly excited by a set of three-phase DC current. The harmonics of air-gap flux density under
the excitation of 6-pole MMF are illustrated in Figure 2.14, where the 4-pole air-gap flux is
generated by the rotor modulation and treated as the effective flux component. The magnitude of
the 4-pole air-gap flux density (B4) is 62.4% of that of the 6-pole air-gap flux density (B6).
21
#2
#3
#1
#4
#5
(a) θr
(b) θr = 12
#2
#1
#3
#4
#5
(c) θr = 24
(d) θr = 36
(e) θr = 48
(f) θr = 60
Figure 2.12 Cross Section Flux Distribution with DC Excitation of 6-Pole Winding
22
Air-Gap Flux
Density (T)
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
Air-Gap Flux
Density (T)
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
Air-Gap Flux
Density (T)
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
Air-Gap Flux
Density (T)
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
Air-Gap Flux
Density (T)
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
Air-Gap Flux
Density (T)
1.6
1.2
0.8
0.4
0
-0.4
-0.8
-1.2
-1.6
θr = 0
θr = 12
θr = 24
θr = 36
θr = 48
θr = 60
0
90
180
270
360
Locations in Air-Gap (Degree)
Figure 2.13 Air-Gap Flux Density with DC Excitation of 6-Pole Winding (4/6 Pole Combination)
23
Amplitude of Air-Gap Flux Density (T)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
6E-16
0
2
4
6
8
-0.1
10 12 14 16 18 20 22 24 26 28 30
Harmonic Order
Figure 2.14 Harmonics of Air-Gap Flux Density with 6-Pole MMF Excitation (4/6 Pole
Combination)
2.3 Winding Flux Linkage and Back-EMF
To verify the electromechanical energy conversion of an electrical machine or an
electromechanical system, in general, the winding flux linkage and induced voltage due to the
movement of the rotor are investigated. If the stator winding flux linkage varies as the rotor is
rotating, a voltage will be induced according to Faraday’s Law, and will be referred to as the
“speed voltage” or “Back-EMF”. Simultaneously, if a current is injected into the stator winding
as well, the electrical power is generated depending on the phase angle between the induced
voltage and current, thus the electromechanical energy conversion is accomplished [51].
For either a field winding or a permanent magnet synchronous machine, if the rotor flux
linking the stator winding changes due to the rotation of the rotor, then a speed voltage is induced.
An induction machine has a similar mechanism of electromechanical energy conversion. Under
24
normal conditions of an induction machine, a rotor current is induced by the slip between the
rotor speed and rotating field for a squirrel cage rotor, while a current is intentionally injected into
the rotor winding for a wound-rotor DFIM [51]. Therefore, a rotor flux resulting from its current
links the stator winding and varies according to the rotation of the rotor; thus, a speed voltage is
produced.
In a reluctance machine, although there is no effective current in the rotor, a speed voltage is
also induced and proves the electromechanical energy conversion. When the stator current enters
into the armature winding, the corresponding MMF is established along the air-gap and produces
flux linking of the rotor side. The paths of the resulting flux are different in terms of permeance
according to the different positions of the reluctance type rotor [51]. As a result, the stator flux
linkage changes due to the movement of the rotor and then a speed voltage is induced in the stator.
In this chapter, the mechanism of electromechanical energy conversion of the DFBM is
investigated by the analysis of the winding flux linkage and induced speed voltage (Back-EMF).
As known, there are two sets of three-phase windings with different pole numbers in the stator
of DFBM. Two types of winding flux linkage are particularly defined as follows [27]:

Self flux linkage: the flux linked by one phase of the windings when this set of three-phase
windings with the same pole number is excited by a three-phase sinusoidal current.

Mutual flux linkage: the flux linked by one phase of the windings when the other set of threephase windings with a different pole number is excited by a three-phase sinusoidal current.
(a) Self Flux Linkage
Flux linkage is a property of a winding (two-terminal element), which is not only related to
flux distribution but also to winding distribution.
The defined self flux linkage could be
explained as the flux weighted by one phase of the windings in a set of same pole number, when
an air-gap flux is created by this set of three-phase windings excited by a three-phase sinusoidal
25
current. Thus, the self flux linkage is a lumped description or terminal characteristic of the
magnetic flux in an electrical machine from a single phase point of view [27].
There are two ways to evaluate the characteristics of the self flux linkage, by a DC or AC
current excitation alternatively.
In Figure 2.15, the three-phase self flux linkages of the 4-pole winding versus the rotor
positions are plotted, when the 4-pole winding is singly excited by a three-phase DC current. As
shown, the self flux linkage is mainly a DC quantity with small AC pulsations. For the DC
component, it could be explained as the self flux linkages are independent of rotor positions, and
there are mainly no flux linkage variations and so as no induced speed voltages. On the other
hand, the small AC pulsation of flux linkage will cause a small AC speed voltage. However, the
DC current excitation and AC induced speed voltage will not produce average electromagnetic
power and torque. The similar results of the self flux linkages of the 6-pole winding with DC
excitations are shown in Figure 2.16.
XY Plot 2
2.0
2.00
150kwdc
Curve Info
FluxLinkage(PhaseA_4)
Setup1 : Transient
FluxLinkage(PhaseB_4)
Setup1 : Transient
FluxLinkage(PhaseC_4)
Setup1 : Transient
1.5
1.0
0
1.00
Y1 [Wb]
Self Flux Linkage (T)
1.50
0.5
0.50
0
0.00
-0.5
-0.50
-1.0
-1.00
0.00
0
10.00
20.00
90
Time [ms]
30.00
180
40.00
270
50.00
360
Rotor Position (Degree)
Figure 2.15 Self Flux Linkage of 4-Pole Winding Versus Rotor Positions with DC Excitation
26
XY Plot 1
1.50
1.50
150kw17(dc)
Curve Info
FluxLinkage(PhaseA_6)
Setup1 : Transient
FluxLinkage(PhaseB_6)
Setup1 : Transient
1.25
FluxLinkage(PhaseC_6)
Setup1 : Transient
1.00
1.00
0.75
0.75
0.50
0.50
Y1 [Wb]
Self Flux Linkage (Wb)
1.25
0.25
0.25
0.00
0.00
-0.25
-0.25
-0.50
-0.50
-0.75
-0.75
0.00
0
10.00
20.00
90
Time [ms]
30.00
180
40.00
270
50.00
360
Rotor Position (Degree)
Figure 2.16 Self Flux Linkage of 6-Pole Winding Versus Rotor Positions with DC Excitation
As shown in Figures 2.17, 2.18 and 2.19, when the 4-pole winding is singly excited by a 40Hz
three-phase AC current, the corresponding self flux linkages are also sinusoidal in 40Hz and have
a 120 degree phase angle difference in each phase. By comparing the characteristics at 0 rpm,
1,200 rpm and -1,200rpm, it has been proven that the fundamental components of the self flux
linkage in the three speeds are the same in terms of magnitude and phase angle. The self flux
linkage with AC current excitation is not related to the rotor speed and position, if the effect of
the slight harmonic components is neglected. The harmonic induced speed voltage interacting
with the fundamental current will not deliver average real power but energy oscillation. Similar
characteristics may be achieved in the conditions of the 6-pole winding with AC excitation.
27
XY Plot 3
2.00
2.00
150kwjixie
Curve Info
FluxLinkage(PhaseA_4)
Setup1 : Transient
FluxLinkage(PhaseB_4)
Setup1 : Transient
1.50
FluxLinkage(PhaseC_4)
Setup1 : Transient
1.00
1.00
0.50
0.50
Y1 [Wb]
Self Flux Linkage (Wb)
1.50
0.00
0.00
-0.50
-0.50
-1.00
-1.00
-1.50
-1.50
-2.00
-2.00
0.00
0
10.00
10
20.00
20
Time [ms]
30.00
30
40.00
50.00
40
50
Time (ms)
Figure 2.17 Self Flux Linkage of 4-Pole Winding at 0 rpm with AC, 40Hz Excitation
XY Plot 2
2.00
2.00
150kwjixie
Curve Info
FluxLinkage(PhaseA_4)
Setup1 : Transient
FluxLinkage(PhaseB_4)
Setup1 : Transient
1.50
FluxLinkage(PhaseC_4)
Setup1 : Transient
1.00
1.00
0.50
0.50
Y1 [Wb]
Self Flux Linkage (Wb)
1.50
0.00
0.00
-0.50
-0.50
-1.00
-1.00
-1.50
-1.50
-2.00
-2.00
0.00
10.00
0
10
550
20.00
20
Time [ms]
30.00
40.00
50.00
30
40
50
Time (ms)
Figure 2.18 Self Flux Linkage of 4-Pole Winding at 1,200 rpm with AC, 40Hz Excitation
28
XY Plot 3
2.00
2.00
150kwjixie
Curve Info
FluxLinkage(PhaseA_4)
Setup1 : Transient
FluxLinkage(PhaseB_4)
Setup1 : Transient
1.50
FluxLinkage(PhaseC_4)
Setup1 : Transient
1.00
1.00
0.50
0.50
Y1 [Wb]
Self Flux Linkage (Wb)
1.50
0.00
0.00
-0.50
-0.50
-1.00
-1.00
-1.50
-1.50
-2.00
-2.00
0.00
0
550
10.00
10
20.00
20
Time [ms]
30.00
30
40.00
40
50.00
50
Time (ms)
Figure 2.19 Self Flux Linkage of 4-Pole Winding at -1,200 rpm with AC, 40Hz Excitation
Through the investigation of the characteristics in both DC and AC current excitations, the
fundamental component of the defined self flux linkage is verified to be not a function of rotor
position and speed, and so cannot produce the speed voltage which is necessary for the
electromechanical energy conversion. It is concluded that the singly excited DFBM is certainly
in a no-load condition without average power production.
(b) Mutual Flux Linkage
The mutual flux linkage is defined as the flux linked by one phase of the windings when the
other set of three-phase windings with a different pole number is excited by a three-phase
sinusoidal current. The defined mutual flux linkage could be explained as the flux weighted by
one phase of the windings in a set of same pole number, when an air-gap flux is created by the
other set of three-phase windings with a different pole number excited by a three-phase sinusoidal
current [27, 28]. By intuitional thinking, the mutual flux linkage should be substantially different
29
with the self flux linkage due to the different pole numbers and pole pitches of these two
windings and the special structure of the rotor.
There are also two methods to evaluate the characteristics of the mutual flux linkage, by a DC
or AC current excitation alternatively.
In Figure 2.20, the defined three-phase mutual flux linkages of the 4-pole winding versus the
rotor positions at two rotor speeds of 1,200 rpm and -1,200 rpm are plotted, when the 6-pole
winding is singly excited by a three-phase DC current. As shown, the mutual flux linkage is
mainly a sinusoidal AC quantity with high frequency harmonics. For the fundamental component
of the waveforms, it should be noted that, unlike the self flux linkage, the mutual flux linkages are
definitely related to the rotor positions. These fundamental flux linkage variations will produce
induced Back-EMF in the 4-pole windings (shown in Figure 2.21) which are needed for
electromechanical energy conversion.
If a three-phase sinusoidal current with the same
frequency as that of the induced speed voltage is injected into the 4-pole winding, an average
electromagnetic power and torque could be produced.
The mechanism of the electromechanical energy conversion of DFBM could be explained as:
when the stator current enters the 6-pole winding, the corresponding MMF is created along the
air-gap, producing flux which links both the rotor side and the 4-pole winding. The paths of the
resulting flux are different in terms of permeance, according to the different positions of the
reluctance type rotor [27]. Moreover, the rotor related flux variation could only be weighted by
the 4-pole winding instead of the excited 6-pole winding. As a result, the defined mutual flux
linkage of the 4-pole winding varies due to the movement of rotor and then a speed voltage is
induced. In other words, the mutual coupling between the two windings is implemented by the
effect of the designed reluctance type rotor.
30
XY Plot 2
1.25
1.25
150kw17(dc)
Curve Info
FluxLinkage(PhaseA_4)
Setup1 : Transient
FluxLinkage(PhaseB_4)
Setup1 : Transient
0.63
0.63
Y1 [Wb]
Mutual Flux Linkage (Wb)
FluxLinkage(PhaseC_4)
Setup1 : Transient
0.00
0.00
-0.63
-0.63
-1.25
-1.25
0.00
0
10.00
20.00
10
20
Time [ms]
30.00
40.00
30
40
50.00
50
Time (ms)
(a) Rotor Speed at 1,200 rpm
XY Plot 2
1.25
1.25
150kw17(dc)
Curve Info
FluxLinkage(PhaseA_4)
Setup1 : Transient
FluxLinkage(PhaseB_4)
Setup1 : Transient
0.63
0.63
Y1 [Wb]
Mutual Flux Linkage (Wb)
FluxLinkage(PhaseC_4)
Setup1 : Transient
0.00
0.00
-0.63
-0.63
-1.25
-1.25
0.00
10.00
20.00
0
10
20
Time [ms]
30.00
30
40.00
40
50.00
50
Time (ms)
(b) Rotor Speed at -1,200 rpm
Figure 2.20 Mutual Flux Linkage of 4-Pole Winding with DC Excitation of 6-Pole Winding
31
XY Plot 3
1.25
1.25
150kw17(dc)
Curve Info
InducedVoltage(PhaseA_4)
Setup1 : Transient
InducedVoltage(PhaseB_4)
Setup1 : Transient
InducedVoltage(PhaseC_4)
Setup1 : Transient
0.63
Y1 [kV]
Back-EMF (kV)
0.63
0.00
0.00
-0.63
-0.63
-1.25
-1.25
0.00
0
10.00
10
20.00
Time [ms]
20
30.00
30
40.00
40
50.00
50
Time (ms)
Figure 2.21 Back-EMF of 4-Pole Winding with DC Excitation of 6-Pole Winding at 1,200 rpm
On the other hand, although the flux distribution along the air-gap is asymmetrical, nonsinusoidal and full of harmonics, it is observed that the defined mutual flux linkage is quite
sinusoidal while the MMF, winding and rotor are of different pole numbers. It proves the fact
that the flux linkage is related not only to flux distribution but also to the winding distribution.
As observed, the frequency relation among the 4-pole, 6-pole windings and rotor speed is
following the constraint indicated in Eq. (1.2).
Similar results of the mutual flux linkages of the 6-pole winding with DC excitations in the 4pole winding are shown in Figure 2.22.
32
XY Plot 1
1.25
1.25
150kwdc
Curve Info
FluxLinkage(PhaseA_6)
Setup1 : Transient
FluxLinkage(PhaseB_6)
Setup1 : Transient
0.63
0.63
Y1 [Wb]
Mutual Flux Linkage (Wb)
FluxLinkage(PhaseC_6)
Setup1 : Transient
0.00
0.00
-0.63
-0.63
-1.25
-1.25
0.00
0
10.00
10
20.00
20
Time [ms]
30.00
40.00
50.00
30
40
50
Time (ms)
(a) Rotor Speed at 1,200 rpm
XY Plot 2
1.25
1.25
150kwdc
Curve Info
FluxLinkage(PhaseA_6)
Setup1 : Transient
FluxLinkage(PhaseB_6)
Setup1 : Transient
0.63
0.63
Y1 [Wb]
Mutual Flux Linkage (Wb)
FluxLinkage(PhaseC_6)
Setup1 : Transient
0.00
0.00
-0.63
-0.63
-1.25
-1.25
0.00
10.00
20.00
0
10
20
Time [ms]
30.00
30
40.00
40
50.00
50
Time (ms)
(b) Rotor Speed at -1,200 rpm
Figure 2.22 Mutual Flux Linkage of 6-Pole Winding with DC Excitation of 4-Pole Winding
33
As shown in Figure 2.23, when the 6-pole winding is singly excited by a 60Hz three-phase AC
current, the corresponding mutual flux linkages of the 4-pole winding are also mainly sinusoidal
AC waveforms and have a 120 degree phase angle difference in each phase. By comparing with
the characteristics at 0 rpm, 1,200 rpm and -1,200 rpm, it has been validated that the mutual flux
linkage is a function of the rotor position. The magnitudes of the fundamental component of the
mutual flux linkages in the three speeds are the same, due to the same current excitation.
However, the frequencies of the waveforms are different, while the frequency relation among the
4-pole, 6-pole windings and rotor speed is also following the constraint in Eq. (1.2). Similar
characteristics could be observed in Figure 2.25 under the conditions of the 6-pole winding with
AC excitation in the 4-pole winding. It is noted that the mutual flux linkage varies in a sinusoidal
(with harmonics) profile with respect to the rotor position. Therefore, as shown in Figure 2.24,
the induced Back-EMF will be mainly sinusoidal as the rotor is rotating. The induced speed
voltage interacting with the fundamental current will deliver average torque as real power. The
rotor position related mutual flux linkage proves the effective mutual coupling of the two
windings with different pole numbers through the modulation function of the designed rotor.
It should be noted that, in many cases, the three-phase mutual flux linkages of one set of
windings are not identical in terms of profile by each phase, which is unlike conventional AC
electrical machines. The reasons are listed as follows:

The number of the rotor segment is odd.

The pole numbers and pole pitches of the two sets of windings are different.

There exists unbalanced air-gap flux distribution.
However, Figures 2.23 (b) and 2.25 (b) show the cases of the mutual flux linkages with the
three-phase balanced waveforms and same profile, when the frequencies of the 4-pole winding, 6pole winding and rotor are 40Hz, -60Hz, and 20Hz respectively. It is interesting to note that the
34
three-phase mutual flux linkages in one set of winding have the same profile with a 120 degree
phase angle difference in each phase, when the frequencies are following the equation:
(2.1)
With the constraints of both Eqs. (1.2) and (2.1), the generated flux will be felt and weighted
evenly by each phase of windings in one set with the same pole number , and therefore create
three-phase balanced flux linkage. This is an important hint to control this machine in an optimal
way with three-phase balanced flux linkages, and so as balanced voltages and currents. The
specific control method of this type of electrical machine will be discussed in details in the later
chapter.
XY Plot 3
1.25
1.25
150kwjixie1
Curve Info
FluxLinkage(PhaseA_4)
Setup1 : Transient
FluxLinkage(PhaseB_4)
Setup1 : Transient
0.63
0.63
Y1 [Wb]
Mutual Flux Linkage (Wb)
FluxLinkage(PhaseC_4)
Setup1 : Transient
0.00
0.00
-0.63
-0.63
-1.25
-1.25
0.00
10.00
20.00
0
10
20
Time [ms]
30.00
30
40.00
40
50.00
50
Time (ms)
(a) Rotor Speed at 0 rpm
continued
Figure 2.23 Mutual Flux Linkage of 4-Pole Winding with AC, -60Hz Excitation of 6-Pole
Winding
35
Figure 2.23 continued
XY Plot 3
1.25
1.25
150kwjixie1
Curve Info
FluxLinkage(PhaseA_4)
Setup1 : Transient
FluxLinkage(PhaseB_4)
Setup1 : Transient
0.63
0.63
Y1 [Wb]
Mutual Flux Linkage (Wb)
FluxLinkage(PhaseC_4)
Setup1 : Transient
0.00
0.00
-0.63
-0.63
-1.25
-1.25
0.00
10.00
20.00
0
10
20
550
Time [ms]
30.00
40.00
50.00
30
40
50
Time (ms)
(b) Rotor Speed at 1,200 rpm
XY Plot 3
1.25
1.25
150kwjixie1000
Curve Info
FluxLinkage(PhaseA_4)
Setup1 : Transient
FluxLinkage(PhaseB_4)
Setup1 : Transient
0.63
0.63
Y1 [Wb]
Mutual Flux Linkage (Wb)
FluxLinkage(PhaseC_4)
Setup1 : Transient
0.00
0.00
-0.63
-0.63
-1.25
-1.25
0.00
10.00
20.00
0
10
20
Time [ms]
30.00
30
Time (ms)
(c) Rotor Speed at -1,200 rpm
36
40.00
50.00
40
50
XY Plot 4
600
600.00
150kwjixie1
Curve Info
InducedVoltage(PhaseA_4)
Setup1 : Transient
InducedVoltage(PhaseB_4)
Setup1 : Transient
InducedVoltage(PhaseC_4)
Setup1 : Transient
400
200
200.00
Y1 [V]
Back-EMF (V)
400.00
0
0.00
-200
-200.00
-400
-400.00
-600
-600.00
0.00
10.00
0
10
20.00
20
Time [ms]
30.00
30
40.00
50.00
40
50
Time (ms)
(a) Rotor Speed at 0 rpm
XY Plot 4
750
750.00
150kwjixie1
Curve Info
InducedVoltage(PhaseA_4)
Setup1 : Transient
InducedVoltage(PhaseB_4)
Setup1 : Transient
InducedVoltage(PhaseC_4)
Setup1 : Transient
500
250
250.00
Y1 [V]
Back-EMF (V)
500.00
0
0.00
-250
-250.00
-500
-500.00
-750
-750.00
0.00
0
10.00
10
20.00
20
Time [ms]
30.00
30
40.00
50.00
40
50
Time (ms)
(b) Rotor Speed at 1,200 rpm
continued
Figure 2.24 Back-EMF of 4-Pole Winding with AC, -60Hz Excitation of 6-Pole Winding
37
Figure 2.24 continued
XY Plot 4
2.0
2.00
150kwjixie1000
Curve Info
InducedVoltage(PhaseA_4)
Setup1 : Transient
InducedVoltage(PhaseB_4)
Setup1 : Transient
1.5
1.50
InducedVoltage(PhaseC_4)
Setup1 : Transient
1.0
0.5
0.50
Y1 [kV]
Back-EMF (kV)
1.00
0
0.00
-0.5
-0.50
-1.0
-1.00
-1.5
-1.50
-2.0
-2.00
0.00
0
10.00
10
20.00
20
Time [ms]
30.00
30
40.00
50.00
40
50
Time (ms)
(c) Rotor Speed at -1,200 rpm
-------------------------------------------------------------------------------------------------------------------------------XY Plot 2
1.25
1.25
150kwjixie
Curve Info
FluxLinkage(PhaseA_6)
Setup1 : Transient
FluxLinkage(PhaseB_6)
Setup1 : Transient
0.63
0.63
Y1 [Wb]
Mutual Flux Linkage (Wb)
FluxLinkage(PhaseC_6)
Setup1 : Transient
0.00
0.00
-0.63
-0.63
-1.25
-1.25
0.00
10.00
20.00
0
10
20
Time [ms]
30.00
30
40.00
40
50.00
50
Time (ms)
(a) Rotor Speed at 0 rpm
continued
Figure 2.25 Mutual Flux Linkage of 6-Pole Winding with AC, 40Hz Excitation of 4-Pole
Winding
38
Figure 2.25 continued
XY Plot 1
1.25
1.25
150kwjixie
Curve Info
FluxLinkage(PhaseA_6)
Setup1 : Transient
FluxLinkage(PhaseB_6)
Setup1 : Transient
0.63
0.63
Y1 [Wb]
Mutual Flux Linkage (Wb)
FluxLinkage(PhaseC_6)
Setup1 : Transient
0.00
0.00
-0.63
-0.63
-1.25
-1.25
0.00
0
10.00
10
20.00
20
Time [ms]
30.00
40.00
50.00
30
40
50
Time (ms)
(b) Rotor Speed at 1,200 rpm
XY Plot 2
1.25
1.25
150kwjixie
Curve Info
FluxLinkage(PhaseA_6)
Setup1 : Transient
FluxLinkage(PhaseB_6)
Setup1 : Transient
0.63
0.63
Y1 [Wb]
Mutual Flux Linkage (Wb)
FluxLinkage(PhaseC_6)
Setup1 : Transient
0.00
0.00
-0.63
-0.63
-1.25
-1.25
0.00
10.00
20.00
0
10
20
Time [ms]
30.00
30
Time (ms)
(c) Rotor Speed at -1,200 rpm
39
40.00
50.00
40
50
By means of the FEA of the flux linkages and induced voltages in various excitations, the
mechanism of the electromechanical energy conversion of DFBM is investigated. The important
conclusions could be summarized as follows:
 As DFBM is singly excited in one set of winding with the same pole number, only the defined
self flux linkage will be produced in this winding and not related to the rotor position. In this
condition, there is no directly mutual coupling between the singly excited winding and rotor.
Therefore, the singly excited DFBM is certainly in a no-load condition without average torque
delivery.
 Through the modulation effect of the designed reluctance type rotor, these two windings of
different pole numbers are mutual coupled. Thus, the electromechanical energy conversion is
feasible when the two windings are properly and doubly excited.
 The corresponding flux linkages and induced voltages are primarily in a sinusoidal profile
with harmonics, if a sinusoidal current is injected into the winding. The frequency relation
among the two windings and rotor speed is always following the constraint in Eq. (1.2).
 Unlike the conventional AC machine, in many cases, the three-phase mutual flux linkages of
one set of windings are not identical in terms of profile by each phase. It is also identified that,
with the constraint of Eq. (2.1), the flux could be evenly weighted by each phase of winding in
one set, consequently producing a three-phase balanced flux linkage with the exactly same
profile. The frequency relation in Eq. (2.1) could be implemented to control this machine and
so as to achieve three-phase balanced performances.
40
2.4 Comparison of 4/6 and 4/8 Pole Combination
According to the FEA results of the DFBM with the 4/6-pole combination, the asymmetric
magnetic flux distribution due to the rotor peameance of the five pieces of rotor segment is an
issue resulting in unbalanced magnetic pull, noise, vibration and also a three-phase unbalanced
flux linkage.
To avoid these undesirable characteristics of the asymmetric magnetic flux
distribution, another pole combination of the 4/8 with the 6 rotor segments is chosen for further
investigation.
The DFBM of the 4/8 pole combination consists of the same stator core, air-gap length and 4pole winding distribution as those of the 4/6-pole combination version. Obviously, there is a set
of 8-pole, three-phase windings in the stator of the machine with the 4/8 pole combination. The
major distinction between these two machines is the number of rotor segments, since the 4/8 pole
combination will have six (an even number) rotor segments in contrast to the case of five (an odd
number) rotor segments for the 4/6-pole combination.
To assess the flux distribution of the 4/8 pole combination, the 4-pole winding is also excited
by a three-phase current, which is the same case as the 4/6-pole combination.
From the
information shown in Figure 2.26, it is apparent that the flux follows the path of the iron
lamination, and results in six curls of flux line over the cross section. Interestingly, unlike the
results of the 4/6-pole combination, the entire flux distribution is centro-symmetric about the
center of the circle through the effect of the rotor permeance of the six pieces of rotor segments.
Therefore, the serious issues of unbalanced magnetic pull, noise and vibration are avoided for the
4/8 pole combination with six rotor segments.
41
#3
#2
#4
#1
#5
#6
Figure 2.26 Cross Section Flux Distribution with DC Excitation of 4-Pole Winding (4/8 Pole
Combination)
As can be seen from Figure 2.27, the air-gap flux density also clearly indicates the
characteristic of this centro-symmetric flux distribution, although the waveform is non-sinusoidal
and full of harmonics. Furthermore, special findings are in two of the six segments (e.g.,
Segment #2 and 5 in Figure 2.28) in which the magnitudes of the air-gap flux density are much
lower than the other four segments. These results indicate that the magnetic capability of 2/6
(33.3%) of the iron lamination has not been fully utilized. It is important to note that, compared
to the 4/6-pole combination, the utilization of the magnetic core for the 4/8 pole arrangement is
less effective, and possibly results in degraded torque capability and efficiency.
42
Air-Gap Flux Density (T)
1.8
θr =
1.4
1
0.6
0.2
-0.2
-0.6
-1
-1.4
-1.8
0
90
180
270
360
Locations in Air-Gap (Degree)
Figure 2.27 Air-Gap Flux Density with DC Excitation of 4-Pole Winding (4/8 Pole
Combination)
For the purpose of quantitative analysis, the FFT algorithm is also applied to the waveform of
the air-gap flux density in Figure 2.27. The results, as shown in Figures 2.28 and 2.29, indicate
that the 4-pole and 8-pole flux components are dominantly generated through the 4-pole or 6-pole
MMF excitation and modulation effect of the rotor permeance. The comparisons of the effective
air-gap flux density and rotor modulation capabilities between the 4/6 and 4/8 pole combination
are highlighted in Table 1.
The data of this comparative study indicate that the 4/6-pole
combination motor has higher air-gap flux density no matter which stator winding is excited.
Another important finding is that the mutual coupling between the 4- and 6-pole windings in
terms of flux density is more effective than that of the 4/8 pole combination. Therefore, this
study confirms that the rotor modulation capability of the 4/6 pole combination is stronger than
that of the 4/8 pole combination; as a result, higher torque production and efficiency are expected
for the 4/6-pole combination.
43
Amplitude of Air-Gap Flux Density (T)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
Harmonic Order
Figure 2.28 Harmonics of Air-Gap Flux Density with 4-Pole MMF Excitation (4/8 Pole
Amplitude of Air-Gap Flux Density (T)
Combination)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
Harmonic Order
Figure 2.29 Harmonics of Air-Gap Flux Density with 8-Pole MMF Excitation (4/8 Pole
Combination)
44
4/6 Pole Combination
4-Pole MMF Excitation:
4-Pole Flux
Density (B4, T)
0.509
6-Pole Flux
Density (B6, T)
0.489
Mutual
Coupling (%)
B6/B4= 96.1%
6-Pole MMF Excitation:
0.377
0.604
B4/B6 = 62.4%
4/8 Pole Combination
4-Pole Flux
8-Pole Flux
Mutual
Density (B4, T)
Density (B8, T)
Coupling (%)
4-Pole MMF Excitation:
0.508
0.475
B8/B4= 93.5%
8-Pole MMF Excitation:
0.302
0.580
B4/B8= 52.1%
Table 1 Comparison of Effective Air-Gap Flux Density and Rotor Modulation Capability
between the 4/6 and 4/8 Pole Combination
As discussed, for the 4/6-pole combination, mutual flux linkage is three-phase unbalanced in
many cases, mainly due to the effect of the rotor structure having an odd number of pieces of
segments.
The corresponding mutual flux linkage of the 4/8 pole combination under the
conditions of various current excitations and speeds are calculated by the FEA and exhibited in
Figures 2.30 and 2.31. From the graphs, interestingly, it can be seen that the three-phase mutual
flux linkages of one set of winding are exactly identical in terms of profile with a 120 degree
phase angle difference in each phase, despite their different winding excitations and rotor speeds.
It indicates that the generated flux could always be felt and weighted evenly by each phase of the
windings in one set with the same pole number, and thus creates a three-phase balanced flux
linkage. This is a significant difference of electric magnetic characteristic between the 4/6 and
4/8 pole combination, which could be mainly interpreted by the parity of the number of the rotor
segment. The results of this study show that the rotor permeance with an even number of
segments will produce a centro-symmetric flux (density) distribution and accordingly three-phase
balanced flux linkages, while the even number pieces of rotor segments could create asymmetric
flux distribution, resulting in unbalanced three-phase flux linkages. The three-phase balanced
45
flux linkage property (winding terminal characteristic) of the machine makes it more suitable for
grid-tied and smooth torque required applications.
1
Mutual Flux Linkage (Wb)
0.8
0.6
0.4
0.2
Phase A
0
Phase B
-0.2 0
10
20
30
40
50
Phase C
-0.4
-0.6
-0.8
-1
Time (ms)
(a) Rotor Speed at 0 rpm
Mutual Flux Linkage (Wb)
1.2
0.7
Phase A
0.2
Phase B
-0.3
0
10
20
30
40
50
Phase C
-0.8
-1.3
Time (ms)
(b) Rotor Speed at 1,200 rpm
Figure 2.30 Mutual Flux Linkage of 8-Pole Winding with AC, 40Hz Excitation of 4-Pole
Winding
46
Mutual Flux Linkage (Wb)
1.2
0.7
Phase A
0.2
-0.3
0
10
20
30
40
50
Phase B
Phase C
-0.8
-1.3
Time (ms)
(a) Rotor Speed at 0 rpm
Mutual Flux Linkage (Wb)
1.2
0.7
Phase A
0.2
Phase B
-0.3
0
10
20
30
40
50
Phase C
-0.8
-1.3
Time (ms)
(b) Rotor Speed at 1,200 rpm
Figure 2.31 Mutual Flux Linkage of 4-Pole Winding with AC, 80Hz Excitation of 8-Pole
Winding
47
Taken together, the findings of this comparative study between the 4/6 and 4/8 pole
combination could be extended to represent the different features between the odd number and
even number of rotor segments. The following conclusions can be drawn from the present study:
 When an even number of rotor segments is used, the flux distribution is centro-symmetric
about the center of the circle without the issues of the unbalanced magnetic pull, noise and
vibration due to the asymmetric flux distribution existing in the rotor structure with an even
number of segments.
 As the difference between the pole numbers of the two sets of windings becomes smaller, the
higher is the air-gap flux density and the more effective rotor modulation capability is
achieved.
In other words, when
(e.g., the 4/6 pole combination), the torque
capability is potentially optimized. However, at the same time, the number of rotor segments
is odd with the issue of unbalanced flux distribution.
 The flux linkage as a winding terminal characteristic is also affected by the pole combination.
Evidence shows that the flux linkage is three-phase balanced when the number of rotor
segments is even, whereas the three-phase mutual flux linkages of one set of windings are not
always identical in terms of profile.
 For the pole combination selection, to achieve higher torque production and higher efficiency
of the motor, an odd number of rotor segments with the smallest difference between the pole
numbers of the two windings is the best choice. On the other hand, an even number of rotor
segments could be considered for the grid-tied and smooth torque output applications where
the three-phase balanced terminal characteristics and magnetic properties are important.
48
2.5 Conclusions
In this chapter, magnetic field analysis of flux distribution, terminal characteristics of winding
flux linkage and back-EMF are examined by using the FEA to illustrate the characteristics of the
DFBM. These findings are summarized as follows:
The main magnetic field characteristic of the DFBM is its non-sinusoidal and even asymmetric
flux distribution by the effect of the rotor permeance of rotor segments. In this study, the
harmonic decomposition method has been utilized for the quantitative analysis of the asymmetric,
non-sinusoidal and pulsating air-gap flux density, and aids in the evaluation of the “modulation”
capability of the DFBM.
The mechanism of electromechanical energy conversion of the DFBM is investigated by the
analysis of the winding flux linkage and induced speed voltage (Back-EMF). By means of both
DC and AC current excitations alternatively, the characteristics of the defined self- and mutualflux linkages are evaluated. Although the flux distribution along the air-gap is asymmetrical,
non-sinusoidal and full of harmonics, it is observed that the defined mutual flux linkage is rotor
position dependent and quite sinusoidal while the MMF, winding and rotor are of different pole
numbers. Compared to conventional AC machines, in many cases, the three-phase mutual flux
linkages of one set of windings are not identical in terms of profile by each phase. It is also
identified that, with the constraint of Eq. (2.1), the flux could be evenly weighted by each phase
of winding in one set, consequently producing a three-phase balanced flux linkage with the
exactly same profile.
Through the comparative study between the 4/6 and 4/8 pole combination, it is concluded that,
to achieve high torque production and high efficiency of the motor, an odd number of rotor
segments with the smallest difference between the pole numbers of the two windings is the best
choice. On the other hand, an even number of rotor segments could be considered for the grid49
tied and smooth torque output applications when the three-phase balanced terminal characteristics
and magnetic properties are important.
50
Chapter 3: Modeling and Equivalent Circuit of DFBM
After the investigation of the mechanism of the DFBM by using the FEA method in the
previous chapter, Chapter 3 examines the mathematic models of the DFBM in both dynamic and
steady state conditions. Following the logic of analysis of electrical machines proposed by Lipo
and Krause [48, 52, 53], the dynamic model of the DFBM is first developed in the stationary a,b,c
reference frame. The complex vector approach is used to express the machine’s equation in a
more compact form [48]. Then, to eliminate the coupling among the two stator windings and
rotor with different frequencies and positions, the mathematic equations are transformed to a
common rotating d-q reference frame. Based on the machine’s equations, the corresponding
equivalent circuits are developed to give a clear illustration of the DFBM.
3.1 Dynamic Equations of DFBM in a Stationary a,b,c Reference Frame
For a DFBM, there are two separate three-phase windings placed in the stator with different
pole numbers and without direct electromagnetic coupling between each other.
The rotor
permeance plays a dominant role in the indirect interaction or mutual coupling of the two stator
windings.
The corresponding winding configuration and magnetic axes of a DFBM are
conceptually shown in Figure 3.1.
51
𝑏1-axis
𝑏2-axis
𝑣𝑏1
𝑖𝑏1
𝑎2-axis
𝑣𝑎2
𝑖𝑏2
𝑖𝑎2
𝑣𝑏2
𝜃𝑟
𝑣𝑎1
𝑖𝑎1
𝑖𝑐1
𝑣𝑐1 𝑣𝑐2
𝑐1-axis
𝑎1-axis
𝑖𝑐1
𝑐2-axis
Figure 3.1 Magnetic Axes of a DFBM
3.1.1 Voltage Equations of DFBM
The stator voltage equations of a DFBM could be expressed as [48]:
𝑣
𝑟𝑖
(3.1)
𝑣
𝑟𝑖
(3.2)
where the vectors are defined as
𝑣
𝑣
𝑣
𝑣
𝑣
𝑣
𝑣
𝑣
𝑖
𝑖
𝑖
𝑖
(3.3)
𝑖
𝑖
𝑖
𝑖
(3.4)
The flux linkages are functions of inductance and current [48]:
(3.5)
52
(3.6)
where
𝑖
𝑖
𝑖
(3.7)
𝑖
𝑖
(3.8)
𝑖
𝑖
(3.9)
𝑖
𝑖
(3.10)
3.1.2 Inductances of DFBM
Applying the winding function theory, the mutual inductance between any two windings “i”
and “j” of the DFBM could be expressed as [24, 29, 52 and 53]:
𝜃
𝑟
𝑐
𝜃
𝑖
𝑖
(3.11)
where pi and pj are the pole pair numbers of the windings “i” and “j”, pr is the number of the
rotor segments which is also called the rotor pole number here. The
difference between phases in winding “j”.
53
represents the angle

Inductances of one stator winding
By setting pr= pi+pj, pi pj, i = j= 1 and
= 0, then the self inductances of the stator winding
1 are obtained:
𝑟
(3.12)
The subscript “m” here denotes the effective magnetizing inductance.
The leakage inductance
is added to express the total self inductances of the stator
winding 1:
𝑟
𝑟
(3.13)
𝑟
By denoting the magnetizing inductance of the stator winding 1 as:
𝑟
(3.14)
then the total self inductance of the stator winding 1could be written as:
(3.15)
By setting
=
, the mutual inductances between the phases a1, b1 and c1 in stator winding 1
are obtained:
𝑟
As a result, the flux linkage of stator winding 1 due to the three-phase current 𝑖
54
(3.16)
is expressed:
𝑖
𝑖
Similarly, the flux linkage of stator winding 2 due to the three-phase current 𝑖
(3.17)
is:
𝑖
(3.18)
𝑖
where

is the leakage inductance of the stator winding 2.
Inductances between two stator windings
By setting pr = pi+pj, pi pj, i = 1, j= 2, and
= 0 in Eq. (3.11), then the mutual inductances
between the stator winding 1 and 2 are obtained:
𝑟
By setting
=
𝜃
𝜃
(3.19)
, the mutual inductances between the phases a1 and b2, phase b1 and c2, phase
c1 and a2, are obtained:
𝑟
𝜃
(3.20)
𝜃
55
By setting
=
, the mutual inductances between the phases a1 and c2, phase b1 and a2,
phase c1 and b2, are obtained:
𝑟
𝜃
(3.21)
𝜃
As a result, the mutual flux linkage of stator winding 1 due to the three-phase current 𝑖
is
expressed as:
𝑖
𝜃
𝜃
𝜃
𝜃
𝜃
𝜃
(3.22))
𝑖
𝜃
𝜃
𝜃
Similarly, the mutual flux linkage of stator winding 2 due to the three-phase current 𝑖
is
expressed as:
𝑖
𝜃
𝜃
𝜃
𝜃
𝜃
𝜃
𝜃
(3.23)
𝑖
𝜃
𝜃
Through the calculations of the inductances based on the principle of winding function, it is
concluded that the self inductances and mutual inductances between the phases in the same set of
stator winding are constants, while the mutual inductances between the two windings are
sinusoidal functions of the rotor position. The results of this study confirm the previous findings
by the Finite Element Analysis of the flux linkage and contribute a quantitative analysis of the
math model of the DFBM.
56
)
3.2 Complex Variable Model of DFBM
To express the machine equations in a more compact form, the complex vector approach is
utilized [48]. Defining
𝑎
(3.24)
and complex variable
𝑎
(3.25)
𝑎
where the symbol f could represent any of the three-phase variables like voltage, current, flux
linkage, etc.
𝑣
𝑟𝑖
(3.26)
where
𝑣
𝑣
𝑖
𝑖
𝑎𝑣
(3.27)
𝑎 𝑣
𝑎𝑖
𝑎 𝑖
𝑎
𝑎
(3.28)
(3.29)
From Eq. (3.17):
𝑎
𝑎
𝑖
𝑖
(3.30)
From Eq. (3.22):
𝑖
𝑎
𝑎
𝑎
where 𝜃
𝜃
𝑎
𝑎
𝑎
𝑎
𝑎
𝑎
.
57
𝑎
𝑎
𝑎
(3.31)
𝑖
𝑖
(3.32)
So the total flux linkage of stator winding 1 in complex variable form is
𝑖
𝑖
(3.33)
Similarly, the total flux linkage of stator winding 2 in complex variable form is
𝑖
𝑖
(3.34)
The turns ration transformation has been done by referring the stator winding 2 to winding 1
as utilized for a transformer and defining the variables [48]:
𝑣
𝑣
(3.35)
𝑖
𝑖
(3.36)
(3.37)
𝑟
(3.38)
𝑟
(3.39)
Denoting the effective magnetizing inductance
𝑟
(3.40)
The voltage equations could be written as:
𝑣
𝑟𝑖
𝑖
𝑖
(3.41)
𝑣
𝑟𝑖
𝑖
𝑖
(3.42)
or
58
𝑣
𝑟𝑖
𝑖
𝑖
𝑣
𝑟𝑖
𝑖
𝑖
where
𝜃
𝑖
(3.43)
𝑖
(3.44)
.
3.3 Equations of DFBM in a Rotating d-q Reference Frame
As known, the theory of reference frame has been widely used for electrical machine analysis,
where the time-varying and position-varying parameters and variables become constants [48, 55
and 56]. A rotating d-q reference frame is defined and illustrated in Figure 3.2, where 𝜃 is the
angle by which the q-axis leads phase a of the stator winding 1. For a DFBM, the d-q axes could
be arbitrary, but normally are set to rotate with the same frequency as one set of the stator
windings.
𝑏1-axis
𝑏2-axis
β-axis
d-axis
𝑎2-axis
𝜃
q-axis
𝜃𝑟
α-axis
𝑎1-axis
𝑐1-axis
𝑐2-axis
Figure 3.2 d-q Axes and - Axes Relative to Magnetic Axes of a DFBM
59
The equations express the transformation of variables from the stationary a,b,c to the rotating
d-q reference frame [48].
𝜃
𝜃
𝜃
𝜃
𝜃
𝜃
(3.45)
(3.46)
(3.47)
So the transformation matrix is:
𝜃
𝜃
𝜃
𝜃
𝜃
𝜃
(3.48)
Due to the winding (wye or delta) connection of DFBM, the zero sequence component is
neglected here.
𝑎
(3.49)
𝑎
Similarly,
𝑎
𝑎
(3.50)
The voltage equations in a rotating d-q reference frame could be written as:
𝑣
𝑟𝑖
𝑖
𝑖
(3.51)
𝑖
𝑣
𝑟𝑖
𝑖
𝑖
𝑖
(3.52)
𝑖
where
𝜃
𝑖
.
60
The d-q equations in complex variable form could be rewritten as:
𝑣
𝑟𝑖
𝑖
𝑖
𝑖
(3.53)
𝑣
𝑟𝑖
𝑖
𝑖
𝑖
(3.54)
and where the flux linkages are:
𝑖
𝑖
𝑖
𝑖
𝑖
(3.55)
𝑖
𝑖
𝑖
𝑖
𝑖
(3.56)
So the complex vector equivalent circuit of a DFBM is shown in Figure 3.3:
𝑟1
+
𝑣
1
𝑖
1
+3
′
𝑚
+3
𝑖′
1
1
2
𝑟2′
𝑚
𝑟
2
𝑚
′
2
+
𝑣′
2
-
-
Figure 3.3 Complex Vector Equivalent Circuit of a DFBM in the d-q Reference Frame
Eqs. (3.51) and (3.52) could also be transformed to be in scalar form as:
𝑣
𝑟𝑖
(3.57)
𝑣
𝑟𝑖
(3.58)
where
𝑖
𝑖
𝑖
𝑖
𝑖
(3.59)
𝑖
𝑖
𝑖
𝑖
𝑖
(3.60)
For the stator winding 2:
61
𝑣
𝑟𝑖
(3.61)
𝑣
𝑟𝑖
(3.62)
where
𝑖
𝑖
𝑖
𝑖
𝑖
(3.63)
𝑖
𝑖
𝑖
𝑖
𝑖
(3.64)
The real variable d-q voltage equations in a matrix form could be rewritten as:
𝑣
𝑣
𝑣
𝑣
𝑟
𝑟
𝑟
𝑟
𝑖
𝑖
𝑖
𝑖
(3.65)
The d-q equivalent circuit of a DFBM in scalar form is in exactly the same form as the
complex vector equivalent circuit shown in Figure 3.4.
62
𝑟1
+
𝑣
1
1
+3
𝑖
1
′
𝑚
2
+3
𝑟2′
𝑚
𝑖′ 2
1
𝑟
′
+
2
𝑣
𝑚
-
2
𝑟1
+
𝑣
′
1
𝑖
1
+3
′
𝑚
2
+3
𝑟2′
𝑚
𝑖′ 2
1
1
𝑟
′
2
𝑚
+
𝑣′ 2
-
-
Figure 3.4 Scalar Form Equivalent Circuits of a DFBM in the d-q Reference Frame
3.4 Operational Equivalent Circuits of DFBM
By combining Eqs. (3.53) and (3.54) with Eqs. (3.55) and (3.56), the voltage equations of the
stator winding 1 and 2 in complex variable form could be reconstructed and expressed as:
𝑣
𝑟𝑖
𝑖
𝑖
(3.66)
𝑟𝑖
𝑣
𝑖
𝑖
𝑟𝑖
𝑖
𝑟𝑖
𝑖
𝑖
𝑖
𝑖
(3.67)
𝑖
To obtain a more straightforward expression, the voltage equation of the stator winding 2 is
multiplied by the operator
𝑣
[48]. Then,
𝑟
𝑖
𝑖
(3.68)
𝑖
𝑖
63
As shown in Figure 3.5, the resultant equivalent circuit is a clear and operational explanation
of the dynamic model of a DFBM [48].
𝑟1
+
𝑣
1
+3
1
+
𝑚
𝑖
′
2
+3
𝑚
+
+
𝑚
𝑖′
1
𝑟2′
2
𝑣′
+
+
𝑟
+
+
2
+
-
𝑟
-
Figure 3.5 Operational Complex Vector Equivalent Circuit of the Dynamic Model of a DFBM
The analysis of the steady state operation is convenient and effective for understanding the
basic characteristics and performance of the electrical machine. Generally, the steady state
characteristics are represented by using equivalent circuits and phasor diagrams.
These constraints of steady state operation are normally summarized as [57]:

Three-phase sinusoidal voltage excitations which result in three-phase sinusoidal currents
in two sets of stator windings

Constant rotor speed operation

No saturation effect in the iron core

Constant resistances and inductances except for variable mutual inductances due to rotor
motion
For a general three-phase voltage excitation,
𝑣
𝑐
(3.69)
𝑣
𝑐
(3.70)
𝑣
𝑐
(3.71)
64
the voltage could be expressed as the sum of two complex exponentials by using the Euler
relation [48]:
(3.72)
𝑣
Denote the peak values of the voltages as
𝑣
𝑣
𝑣
(3.73)
then the voltages could be written as [48]:
𝑣
(3.74)
𝑣
(3.75)
𝑣
(3.76)
The defined voltage in complex variable form in Eq. (3.27) becomes
𝑣
𝑣
𝑎𝑣
𝑎 𝑣
(3.77)
𝑎
𝑎
𝑎
𝑎
By defining the positive sequence voltage
𝑎
𝑎
(3.78)
and the negative sequence voltage
𝑎
𝑎
(3.79)
then the voltage in complex variable form is rewritten as
𝑣
(3.80)
For a balanced three-phase voltage, the negative sequence component equals zero and so
65
𝑣
where
(3.81)
is the peak value of the phase voltage.
The voltage equations in a rotating d-q reference frame could be written as [48]:
𝑣
𝑣
(3.82)
To obtain the steady state equivalent circuit of a DFBM, the complex vector voltage equations
in Eq. (3. 66) and (3.68), and the corresponding dynamic equivalent circuit shown in Figure 3.5
could be utilized. If the model is referred to the stator winding 1, then
(3.83)
For the positive sequence which is an exponential function of
, the operators
(3.84)
and
(3.85)
Then, the positive sequence steady state equivalent circuit of a DFBM is achieved and shown
in Figure 3.6.
𝑟1
+
1
1
𝐼
1
+3
1
𝑚
′
2
+3
𝑟2′
𝑚
𝐼 ′2
1
+
′
2
1 𝑚
-
-
Figure 3.6 Positive Sequence Steady State Equivalent Circuit of a DFBM
Similarly, for the negative sequence that is an exponential function of
66
, the operators
(3.86)
and
(3.87)
Then, the negative sequence steady state equivalent circuit of a DFBM is obtained and shown
in Figure 3.7.
𝑟1
1
+
1
1
+3
𝐼
1
1
𝑚
′
2
+3
𝑟2′
𝑚
2
+
𝐼 ′2
1 𝑚
′
2
2
-
-
Figure 3.7 Negative Sequence Steady State Equivalent Circuit of a DFBM
Taken together, these results of both dynamic and steady state equivalent circuits suggest that
the DFBM has the same form and similar expression of a mathematical model as those of a
conventional DFIM.
For a DFBM, the magnitude of the defined
is 50% of the magnetizing inductance of a
conventional DFIM with the same stator winding design, stator iron shape, air-gap dimension and
stack length. However, it does not mean that the magnetizing inductance of a DFBM is only half
that of the DFIM. The reason is that these mathematical results of DFBM are all based on the
assumption of the inverse gap function as [29, 53 and 54]:
𝜃
𝑐
𝜃
(3.88)
which is approximately derived by utilizing a rotor of a conventional salient pole machine. As
mentioned previously, various researches are being done - such as special electromagnetic and
67
structure design of rotor, pole combination choosing, etc. - in order to improve the mutual
coupling between the two windings and to enlarge the magnetizing inductance of a DFBM.
The leakage inductances of a DFBM could be redefined as
and
for two
stator windings which are definitely larger than those of the stator and rotor windings of a DFIM.
As indicated, for a DFBM based power generation system, the leakage inductance of one stator
winding could be effective to smoothing out the harmonics of the current from the inverter, while
the leakage inductance of the other winding becomes part of an LC or LCL filter tied to the grid.
Thus, the leakage inductances of a DFBM are useful to enhance the performance and reduce the
cost of the extra inductance based filter in doubly-fed power generation applications.
3.5 Power and Torque Equations
The electrical power flowing into the two windings of the DFBM is the sum of products of the
voltages and currents [48]:
𝑣 𝑖
𝑣 𝑖
𝑣 𝑖
𝑣 𝑖
𝑣 𝑖
𝑣 𝑖
(3.89)
𝑣
𝑖
𝑣
𝑖
The power equation in a rotating d-q reference frame could be written as [48]:
𝑣
𝑖
𝑣
𝑖
(3.90)
𝑣
𝑖
𝑣
𝑖
which could also be expressed in a scalar form
𝑣 𝑖
𝑣 𝑖
𝑣 𝑖
𝑣 𝑖
By use of Eqs. (3.66) and (3.67), Eq. (3.90) is expressed as
68
(3.91)
𝑟𝑖
𝑖
𝑖
𝑟𝑖
𝑖
(3.92)
𝑖
𝑖
𝑖
which could be reorganized as
𝑟 𝑖
𝑟 𝑖
𝑖
𝑖
𝑖
𝑖
(3.93)
𝑖
𝑖
𝑖
𝑖
where the first term 𝑟 𝑖
𝑟 𝑖
𝑖
is the copper loss due to winding resistance; the
𝑖
second term
𝑖
𝑖
𝑖
𝑖
represents the
change rate of the magnetic energy of the leakage and mutual inductances of the DFBM; Thus
only the third term is related to the electromechanical power [48]:
𝑖
𝑖
𝑖
(3.94)
𝑖
𝑖
𝑖
Through a process of mathematical derivation, the electromechanical power is developed as
𝐼𝑚
𝑖
𝑖
𝑖 𝑖
(3.95)
𝑖 𝑖
Then, the electromagnetic torque of the DFBM could be expressed as
𝐼𝑚
𝑖
𝑖
𝑖 𝑖
Note that the flux linkage of stator 1 is
69
𝑖 𝑖
(3.96)
𝑖
𝑖
(3.97)
Then the electromagnetic torque can be expressed by the flux linkage and current of stator 1(
and 𝑖
) as
𝐼𝑚 𝑖
𝑖
𝑖
(3.98)
𝐼𝑚 𝑖
Similarly, the electromagnetic torque can be expressed by the flux linkage and current of stator 2
(
and 𝑖
) as
𝐼𝑚
𝑖
(3.99)
𝑖
𝑖
(3.100)
Denote the air-gap flux linkage as
The electromagnetic torque could also be represented by the air-gap flux linkage and current of
stator 1 or 2 (
and 𝑖
or𝑖
) as
𝐼𝑚 𝑖
(3.101)
𝐼𝑚
𝑖
Considering the torque control of the DFBM, the electromagnetic torque is reasonably derived
from the interaction of flux linkage of one stator winding and current of the other winding [48].
By inserting the following stator current equations into Eq. (5.85)
𝑖
𝑖
(3.102)
𝑖
𝑖
(3.103)
the corresponding torque expressions become
70
𝐼𝑚 𝑖
(3.104)
𝐼𝑚
𝑖
3.6 Conclusions
By means of the FEA of the electromagnetic field, the mechanism of a DFBM was addressed
in the previous chapter.
In this chapter, mathematic models of DFBM are systematically
examined.
Based on the principle of winding function, the inductances in the stationary a,b,c reference
frame are calculated. In this investigation, it is shown that the self inductances and mutual
inductances between the phases in one set of stator winding are constants, while the mutual
inductances between the two windings are sinusoidal functions of the rotor position. The results
of this study confirm previous findings by the FEA of the flux linkage and contribute a
quantitative analysis of the mathematic model of the DFBM.
The equations of the DFBM in the rotating d-q reference frame are deduced and expressed in a
complex vector form. The analysis of both dynamic and steady state equivalent circuits suggest
that the DFBM has the same form and similar expression of mathematical model as those of a
conventional DFIM.
For a doubly-fed power generation application, the comparatively larger leakage inductances
of stator winding of a DFBM are useful in smoothing out the harmonics of the current and could
be treated as part of an inductance based filter.
71
Chapter 4: Field Orientation Control of DFBM for Doubly-Fed Power
Generation Applications
In this chapter, similar to the analysis of the conventional DFIM [41-48], the principles of
field orientation control of a DFBM especially for doubly-fed power generation applications are
investigated. First, the basic concept of decoupling control of torque and flux (or active and
reactive power) has been introduced and analyzed in the steady state conditions. The dynamic
response characteristics and implementation of the field orientation are then discussed.
4.1 Steady State Field Orientation Control of DFBM
As known, the basic principle of the field orientation control is to control the electromagnetic
field flux and armature MMF (torque) independently [48]. Usually, the flux and MMF are
perpendicularly oriented.
Denote stator inductances:
(4.1)
(4.2)
By introducing a referral ratio a, the general dynamic model of a DFBM in complex form
could be transformed as [48]:
𝑣
𝑟𝑖
𝑟𝑖
𝑖
𝑎
𝑖
𝑎
𝑎
𝑖
and
72
𝑎
𝑖
(4.3)
𝑖
𝑎
𝑎𝑣
𝑖
𝑎 𝑟
𝑎 𝑟
𝑎
𝑎
𝑖
𝑎
𝑎
𝑖
𝑖
𝑎
(4.4)
𝑎
𝑎
𝑖
𝑖
𝑎
or
𝑎𝑣
𝑖
𝑎 𝑟
𝑎
𝑎
If 𝑎
𝑎
𝑖
𝑎
(4.5)
𝑎
𝑖
𝑖
𝑎
, the dynamic equations of a DFBM become the expressions shown in Eqs. (3.66),
(3.67) and (3.68). Therefore, the general voltage equations (per phase) in steady state conditions
(only the positive sequence is considered) could be represented as [48]:
𝑟𝐼
𝑎 𝑟 𝐼
𝑎
𝑎
𝑎
𝐼
𝑎
𝑎
𝐼
𝑎
𝑎
𝐼
𝑎
𝐼
𝑎
(4.6)
𝐼
𝑎
𝐼
(4.7)
The general equivalent circuit of a DFBM in steady state conditions is shown in Figure 4.1.
𝑟1
+
1
𝑎
1
𝐼1
𝑎
𝑚
𝑎
𝑚
-
2
𝑎2 𝑟2
2
𝐼2
𝑎
𝑎
𝑚
+
𝑎𝐸2
-
+
𝑎
2
-
Figure 4.1 General Steady State Equivalent Circuit of a DFBM (per phase) with Referral Ratio a
73
Clearly, by setting 𝑎
, the conventional equivalent circuit previously shown in Figure
3.6 has been achieved.
From the equivalent circuit, the torque could be represented by using the air-gap power
𝑟
𝐼
𝐸 𝐼
𝐼
(4.8)
where 𝐸 is the induced voltage due to the flux of stator winding 2, and
is the angle between the
phasors of 𝐸 and 𝐼 . Similar to a synchronous machine, the torque control of the DFBM could
be achieved through the independent control of the induced voltage 𝐸 and stator current 𝐼 ,
where 𝐼 is related to the air-gap flux.
To further simplify the torque control of a DFBM to be like a DC machine, the turns referral
ratio a is set as [48]:
𝑎
(4.9)
Then, the leakage inductance on the side of stator 2 becomes zero. The corresponding
equivalent circuit is observed in Figure 4.2.
𝑟1
+
1
-
1
1
2
𝑚
2
𝑚 𝑟2
2
2
2
𝐼1
1
2
2
𝑚
𝑚
+
+
𝐼2
2
𝑚
2
-
𝐸2
𝑚
2
2
-
Figure 4.2 Steady State Equivalent Circuit of a DFBM (per phase) without Leakage Inductance
on the Side of Stator 2
74
It is important to note that the voltage across the magnetizing reactance equals the induced
𝐸 . It means that the magnetizing current is directly in charge of the flux of stator
voltage
winding 2
rather than the defined air-gap (mutual) flux
. Thus, it could be called “stator
flux 2 orientation” [48].
4.1.1 Steady State Operation of DFBM in the Stator Flux 2 Reference
There are three cases to explain the basic principles of the steady state operation of a DFBM in
the stator flux 2 reference.

Case I: when the magnetizing current is supplied only by stator winding 1
In the Case I, the current of stator winding 1 𝐼 could be straightforwardly divided into the
magnetizing component 𝐼
reactance and 𝐼
and torque component 𝐼 , where the 𝐼
into the induced voltage
goes into the magnetizing
𝐸 due to the stator flux 2. So the equivalent circuit
could be redrawn and shown in Figure 4.3.
𝑟1
+
1
-
1
2
𝑚
1
2
𝑚 𝑟2
2
2
2
+
+
𝐼1
1
2
𝑚
𝐼1𝑀
𝐼1 =
2
𝑚
𝐼2
𝑚
2
2
𝐸2
-
𝑚
2
2
-
Figure 4.3 Steady State Equivalent Circuit of a DFBM in the Stator Flux 2 Reference – When
Magnetizing Current is Supplied Only by Stator Winding 1
75
From the equivalent circuit, the magnetizing current 𝐼
𝐸
𝐼
is calculated by
𝐸
(4.10)
As known, the induced voltage is defined as the derivative of flux linkage with respect to time.
In other words, the induced voltage is the time rate change of flux linkage. So the induced
voltage 𝐸 due to the flux of stator winding 2 is
𝐸
(4.11)
Substituting the induced voltage equation Eq. (6.11) into Eq. (6.10), yields the stator 2 flux
linkage
(4.12)
𝐼
Therefore, the flux linkage of stator winding 2 could be directly controlled by the magnetizing
current component 𝐼
of stator winding 1.
In the stator flux 2 reference, when all the magnetizing current is provided by the stator
winding 1, the power factor angle between the phasors 𝐸 and 𝐼 is zero. As a result, the torque is
expressed as
𝐸 𝐼
𝐸
𝐼
(4.13)
𝐼
where 𝐼
𝐼
𝐼
𝐼 .
As indicated in Eqs. (4.12) and (4.13), the flux of stator winding 2 and torque of a DFBM
could be respectively controlled by the magnetizing and torque component of current of the stator
winding 1 in the defined stator flux 2 reference, when the magnetizing current is only provided by
the stator winding 1.
76
The phasor diagram of a DFBM in the case I based on the equivalent circuit of Figure 4.3 is
shown in Figure 4.4. It is interesting to note, if the stator winding 2 is short circuit, the equivalent
circuit of DFBM becomes the same form as that of an induction machine.
𝑟1 𝐼1
1
2
𝑚
1
𝐼1
𝐼2
𝑚
2
𝐼1𝑀
2
2
𝑚
2
1
2
𝑚
2
𝐼1
𝐸2
𝐼1
=
𝑚 𝐼1𝑀
Figure 4.4 Phasor Diagram of a DFBM in the Stator Flux 2 Reference – When Magnetizing
Current is Supplied Only by Stator Winding 1

Case II: when the magnetizing current is supplied only by stator winding 2 (
In the Case II, the current of stator winding 1 𝐼 goes through only the induced voltage
)
𝐸 .
The flux linkage of stator winding 2 is solely produced by the magnetizing current in stator
winding 2.
The current of stator winding 2, 𝐼 , has been divided into a magnetizing component 𝐼
and a
torque component 𝐼 . As shown in Figure 4.5 (a), the equivalent circuit is redrawn with
𝐼
𝐼
The voltage equation becomes
77
(4.14)
𝑟𝐼
𝐼
𝐼
(4.15)
𝑟𝐼
𝐼
𝐸
Then, the induced voltage is represented as a voltage source in Figure 4.5 (b) without
magnetizing reactance in the circuit. As known, the induced voltage is provided by the flux of
stator winding 2, only due to the magnetizing excitation in stator winding 2, and is proportional to
the sum of the rotor electrical speed (electrical radians per second) and the electrical angular
frequency of stator winding 2.
𝐸
𝐼
(4.16)
Since there is no magnetizing current from the stator winding 1, the phasor angle
and 𝐼 is zero, so
between 𝐸
. Therefore, the torque is
𝐸
𝐼
𝐼
(4.17)
Another interpretation of a DFBM is obtained by using a synchronous model in Figure 4.5 (c),
where the voltage is deduced as
𝑟𝐼
𝑟𝐼
𝐼
𝐼
𝐼
𝐼
𝐼
(4.18)
𝑟𝐼
𝐼
𝑟𝐼
𝐼
𝑟𝐼
𝐼
𝐼
𝐼
𝐼
Here the current of stator winding 2, 𝐼 , is similar to the field current of a synchronous
machine, where
𝐼 is the Back-EMF in this machine.
78
The phasor diagram of a DFBM in the case II, based on the equivalent circuit of Figure 4.5, is
shown in Figure 4.6. As indicated, when
, the terminal power factor of stator winding 1 is
always lagging.
1
𝑟1
2
𝑚
1
2
𝑚 𝑟2
2
2
2
+
𝐼2 =
1
𝐼1
1
-
2
𝑚
𝐼2𝑀
2
𝑚
+
+
𝐼2
𝑚
2
2
𝑚
𝐸2
2
2
-
-
(a) Induction Machine-Type” Equivalent Circuit
𝑟1
+
1
2
𝑚
1
2
𝐼1
𝑚
1
2
𝐸2 =
1
𝑚
2
2
(b) Equivalent Circuit without Magnetizing Reactance
𝑟1
+
1 1
𝐼1
𝑏
1
=
1 𝑚 𝐼2
(c) Synchronous Machine-Type Equivalent Circuit
Figure 4.5 Steady State Equivalent Circuit of a DFBM in the Stator Flux 2 Reference – When
Magnetizing Current is Supplied Only by Stator Winding 2
79
𝑟1 𝐼1
1
𝐼2
1
𝐼1
𝑚
2
𝑚
2
𝑚
𝐼2𝑀
𝐼2
2
=
𝑚 𝑟2
2
2
2
1
2
𝑚
2
𝐼1
𝐸2
𝐼2
𝑚 𝐼2𝑀
Figure 4.6 Phasor Diagram of a DFBM in the Stator Flux 2 Reference – When Magnetizing
Current is Supplied Only by Stator Winding 2
In the study of this case, the equivalent circuit of DFBM is found to be very similar to a
wound-rotor synchronous machine.
Compared to the DC field current excitation of a
conventional AC synchronous machine, the magnetizing current of a DFBM could be an AC
variable.

Case III: when the magnetizing current is supplied by both stator windings 1 and 2
(
)
In the Case III, both of stator windings1 and 2 provide magnetizing currents for the flux of
stator winding 2. Therefore,
𝐼
𝐼
𝐼
𝐼
(4.19)
The corresponding “induction machine-type” equivalent circuit is shown in Figure 4.7 (a),
where the voltage equation is
80
𝑟𝐼
𝐼
𝑟𝐼
𝐼
𝑟𝐼
𝐼
𝐼
𝐼
(4.20)
𝐸
Similarly, the voltage equation in a synchronous machine model is deduced
𝑟𝐼
𝐼
𝑟𝐼
𝐼
𝑟𝐼
𝐼
𝐼
𝐼
(4.21)
𝐼
𝐸
𝐼
The resultant “synchronous machine-type” equivalent circuit is presented in Figure 4.7 (b).
Comparing Case III to Cases I and II, the power factor angle
could be no longer zero, and
between the phasors 𝐸 and 𝐼
. The phasor diagram of a DFBM in Case III (
) is
shown in Figure 4.8. Through the vector control of the current of stator winding 1, the terminal
power factor of the stator windings could be intentionally regulated to be unity, lagging or leading.
The torque is calculated from the real power in the circuit
𝐸 𝐼
𝐼
81
(4.22)
𝑟1
+
1
1
2
𝑚
2
𝐼1
1
1
-
2
𝑚 𝑟2
2
2
𝐼𝑀
2
𝑚
2
𝑚
+
+
𝐼2
𝑚
𝐸2
2
2
-
𝑚
2
2
-
(a) Induction Machine-Type” Equivalent Circuit
𝑟1
+
1 1
𝐼1
𝑏
=
1
=
-
1 𝑚 𝐼2
𝑚
2
𝐸2
1
2
𝑚
2
𝐼1
(b) Synchronous Machine-Type Equivalent Circuit
Figure 4.7 Steady State Equivalent Circuit of a DFBM in the Stator Flux 2 Reference – When
Magnetizing Current is Supplied by Both Stator Windings 1 and 2
82
2
𝑚
𝑟1 𝐼1
𝐼2
1
𝑚
𝑚
𝐼1
𝐼2
𝑚
2
2
𝐼𝑀
2
=
𝑚
𝐼1 +
(a) 𝐼 Lagging
2
𝑚
2
𝑚
𝑚 𝑟2
2
𝐼2
𝐼2
𝑟1 𝐼1
1
𝑚
2
𝑚
𝑚
𝐼𝑀
2
𝐼1
𝐼2
1
𝐼2
2
, and 𝐼 Leading
𝐼1
2
1
𝐸2
2
2
2
𝑚
1
2
2
2
=
𝑚
(b) 𝐼 Leading
𝐼1 +
2
𝑚
𝑚 𝑟2
1
2
𝑚
2
𝐼1
𝐸2
𝐼2
𝐼2
, and 𝐼 Leading
continued
Figure 4.8 Phasor Diagrams of a DFBM in the Stator Flux 2 Reference – When Magnetizing
Current is Supplied by Both Stator Windings 1 and 2
83
Figure 4.8 continued
2
𝑚
𝐼2
𝑚
𝑚 𝑟2
2
2
2
𝑚
2
𝑚
𝐼𝑀
2
𝐸2
𝐼2
1
𝑟1 𝐼1
1
2
𝑚
2
1
𝐼1
𝐼2
𝐼1
2
=
𝑚
𝐼1 +
(c) 𝐼 Lagging
2
𝑚
𝐼2
, and 𝐼 Lagging
4.1.2 Active and Reactive Power Control
In a doubly-fed power generation system as shown in Figure 1.3, if the mechanical loss in the
shaft is neglected, then mechanical power equals the electromagnetic power:
(4.23)
where
and
are the electromagnetic powers of stator windings 1 and 2, respectively.
From Eq. (4.22), the total electromagnetic power is expressed as:
𝐼
(4.24)
and for each stator windings
𝐼
(4.25)
(4.26)
𝐼
The total active power generated from the machine to the grid is
84
(4.27)
where
and
are the active powers generated from stator windings 1 and 2 to the grid,
respectively.
(4.28)
(4.29)
where
and
are the losses within the two windings, which are normally neglected for
convenient analysis.
Taken together, it is concluded that the torque and active powers in these two windings could
be effectively controlled in the stator flux 2-oriented reference through the vector control of the
current of stator winding 1. The active power flow of a brushless doubly-fed power generation
system at different speed conditions is illustrated in Figure 4.9.
1
1
2
1
2
𝑀 𝑐ℎ
2
3-phase Power Grid
2
Stator
Winding 2
Stator
Winding 1
Converter
1
𝑀 𝑐ℎ
DFBM
(a) Sub-synchronous Speed (
)
continued
Figure 4.9 Active Power Balance of a Brushless Doubly-Fed Power Generation System
85
Figure 4.9 continued
1
2
2
1
2
𝑀 𝑐ℎ
3-phase Power Grid
2
Stator
Winding 2
Stator
Winding 1
Converter
𝑀 𝑐ℎ
DFBM
(b) Synchronous Speed (
1
1
)
2
1
2
2
𝑀 𝑐ℎ
3-phase Power Grid
2
Stator
Winding 2
Stator
Winding 1
Converter
1
𝑀 𝑐ℎ
DFBM
(c) Super-synchronous Speed (
86
)
For the reactive power, since the stator winding 1 is connected to the back-to-back converter
not directly tied to the grid, the reactive power from the generator in steady state operation is
defined as the reactive power in the stator winding 2:
𝐼𝑚
𝐼
𝐼𝑚
𝐼
𝐼
(4.30)
Since the reactive power is aimed to be controlled by the current of stator winding 1, the reactive
power is expressed by using the following equations:
𝐼
𝐼
(4.31)
𝐼
𝐼
(4.32)
and
Therefore,
𝐼
𝐼
(4.33)
𝐼
4.2 Dynamic Field Orientation Control of DFBM
For a high performance motor drive or generation system, the dynamic response
characteristics are important [48]. The decoupling control of torque and flux of a DFBM in
steady state operations is investigated in Section 4.1, based on complex vector equivalent circuits.
In this section, with the same basic concepts of the steady state field orientation control, concerns
about the dynamics are discussed and analyzed in the rotating d-q reference frame.
The control strategy of the designed DFBM is developed based on the features of a doubly-fed
generation system illustrated in Figure 1.3. One set of the stator windings is constantly connected
87
to the power grid with a fixed voltage of constant frequency, and the other to a converter of
variable voltage and frequency, adapted to the rotor speed. Instructed by the torque equations in
the previous analysis, we can improve current utilization of the DFBM by locking the phase angle
of the two sets of stator current to a special angle, to realize the so-called vector control principle;
that is, the phase angle of the current in the two sets of stator windings needs to be maintained at
an optimal value for maximum torque production [41 and 49].
The dynamic model of a DFBM in the arbitrary rotating d-q reference frame at the speed of
is [41]:
𝑣
𝑟𝑖
(4.34)
𝑣
𝑟𝑖
(4.35)
𝑣
𝑟𝑖
(4.36)
𝑣
𝑟𝑖
(4.37)
𝑖
𝑖
(4.38)
𝑖
𝑖
(4.39)
𝑖
𝑖
(4.40)
𝑖
𝑖
(4.41)
𝑖
𝑖
(4.42)
As described in the previous chapter, to implement the decouple control of the torque and flux,
field orientation along the second winding field flux
is utilized. It can be realized by
selecting the second winding flux as the reference frame and orienting the d axis to
result,
is achieved with the following important relations [41 and 49]:
88
. As a
𝑣
𝑟𝑖
(4.43)
𝑣
𝑟𝑖
(4.44)
𝑖
𝑖
𝑖
(4.45)
𝑖
(4.46)
𝑖
(4.47)
Similar to the expression of the (stator flux 2) field orientation in the steady state, the phasor
diagram of a DFBM in the d-q axes is shown in Figure 4.10.
2
𝑚
𝑖
𝑚
𝑖
𝑖
1
1
𝑟𝑚 )
2
(
2
𝑟1 𝑖
1
𝑣
1
𝑟2 𝑖
2
𝑣
1
1
2
q-axis
2
2
=0
2
2
=
2𝑖 2
+
𝑚𝑖 1
d-axis
Figure 4.10 Phasor Diagram of a DFBM in d-q Axes – Stator Flux 2 Reference
The major difference between the steady state and dynamic behaviors concerns the oriented
flux. Using Eq. (4.45), the current 𝑖
could be expressed by
𝑖
𝑖
and be inserted into Eq. (4.43) yielding
89
and 𝑖
(4.48)
𝑟
𝑟
𝑣
𝑖
(4.49)
Therefore, the oriented stator flux 2 becomes
𝑟
𝑣
𝑟
𝑟
𝑖
(4.50)
𝑣
𝑖
As indicated in Eq. (4.50), in a transient state, the oriented flux linkage
first order transfer function (with a time constant
is the output of a
) of two inputs of the current 𝑖
and
voltage 𝑣 . It means that there is a lag in the response of the flux to the corresponding
magnetizing current and voltage [48]. Combining the information of the flux linkage from Eq.
(4.50) and torque expression from Eq. (4.47) in the stator flux 2 reference, the torque diagram in
terms of the oriented flux linkage in stator winding 2 and q-axis current of stator winding 1 has
been illustrated in Figure 4.11.
𝑖
𝑚
1
1+
𝑣
2
2
𝑟2
+
2
+
2
𝑟2
1 + 𝑟2
2
𝑖
×
3
2
𝑟
𝑚
2
1
Figure 4.11 Dynamic Torque Diagram of a DFBM Represented by Oriented Flux and q-Axis
Current – Stator Flux 2 Reference
90
Alternatively, in the same reference frame, the torque could be represented in totality by
currents instead of flux and current. Inserting the flux linkage expression of Eq. (4.40) into the
voltage Eq. (4.43) to suppress the oriented flux, the d-axis current of stator winding 2 is obtained.
𝑖
𝑣
The investigation of current 𝑖

In the steady state, 𝑖
(4.51)
has shown that
is the current only due to the voltage 𝑣
across the resistance 𝑟
𝑣
𝑟
𝑖

𝑖
(4.52)
In the dynamic state, as shown in Eq. (4.51), changes in current 𝑖
additionally create induced current 𝑖
and voltage 𝑣
will
and result in a transient in the torque production.
As a result, the torque is calculated by
𝑖
𝑖
𝑖
𝑖
(4.53)
𝑣
𝑖
𝑖
𝑖
The corresponding torque diagram in terms of the current in stator windings 1 and 2 is
illustrated in Figure 4.12.
To implement the field orientation control, the spatial phase angle and magnitude of the
oriented stator flux must be investigated. Through the so-called Clark transformation in Eq.
(4.54), the voltage components 𝑣
𝑣
and current components 𝑖
𝑖
in the stationary -
reference are obtained [41].
(4.54)
91
𝑚
𝑖
1
𝑚𝑖 1
+
𝑚
𝑟2
1 + 𝑟2
2
+
-
𝑖
2𝑖
2
2
2
𝑣
2
+
1
𝑟2
1+𝑟
×
3
2
𝑟
𝑚
2
2
2
𝑖
1
Figure 4.12 Dynamic Torque Diagram of a DFBM Represented by Currents – Stator Flux 2
Reference
The corresponding magnitude and phase angle of the stator flux 2 could be calculated by using
the voltage and current components in the stationary - reference [41]:
𝑣
𝑟𝑖
(4.55)
𝑣
𝑟𝑖
(4.56)
Therefore:
(4.57)
𝜃
𝑎
(4.58)
In the practical applications, the pure integrator is replaced by a designed low-pass filter to avoid
the DC bias issue of an integral due to the initial values and noises of the input signals.
As mentioned, in a DFBM based doubly-fed generation system, the stator winding 2 is directly
connected to the grid with the mostly constant magnitude and frequency of the terminal voltage.
92
Therefore, the corresponding flux linkage
is approximately unchanged and can be derived
from Eqs. (4.55) and (4.56).
Consequently, as indicated in Eq. (4.47), in the (stator flux 2) field orientation reference frame,
the instantaneous torque control could be implemented by controlling the q axis current
component of the stator winding 1: 𝑖 . If the voltage drops of the winding resistance are
neglected, active and reactive power at the terminal of the stator winding 2 can be derived as [41]:
𝑖
𝑣 𝑖
(4.59)
𝑣 𝑖
𝑖
(4.60)
𝑖
The less obvious control strategy for the DFBM is that the two sets of stator windings are of
different pole-pitch and excited with currents of different frequencies, implying that the same
space angle in the DFBM air-gap means different electrical angles to the two sets of stator
windings [41 and 49]. The angle equivalence transformation, however, can be obtained by
integrating both sides of Eq. (1.2), resulting in
𝜃
𝜃
𝜃
(4.61)
Evidently, Eq. (4.61) plays a key role to translate the three different phase angles into a
uniform scale and then field orientation or the vector control principle is applied to the DFBM.
According to Eq. (4.59), if the current component, 𝑖
, which is in phase with respect to the
induced speed voltage is controlled, the torque production and real power of the DFBM is
proportionally controlled. On the other hand, if the current component 𝑖
orthogonal to the
induced speed voltage, due to the mutual flux linkage variation is controlled, the reactive power
and, thus, the terminal power factor will be proportionally controlled according to Eq. (4.60) [41
and 49]. The control block diagram for the field-orientation control of the DFBM is shown in
93
Figure 4.13.
It is not surprising that this DFBM, like its brush-type counterpart, can be
conveniently used to achieve decoupled control of active and reactive power, a very useful
feature for variable-speed constant-frequency in wind turbine generator applications, in which the
active power often needs to be controlled, due to the wind speed, reactive power and reactive
power demands from the utility grids [41 and 49].
A B C
Grid Side
Converter
Q2*+
id2*
P2* +
ωrm*
id1* +
PI
-
PI
-
+
vdc
-
Ssw
iq1*
+
vα1*
j(θ2+prθrm) v *
β1
e
vq1*
PI
-
PI
-
vd1*
PI
id1
θ2
ia1
ib1
Encoder
Flux linkage
Observer
vd2
vq2
P&Q
Caculation
3Φ/2Φ
θrm, ωrm
+
iα1
iβ1
-j(θ2+prθrm)
e
iq1
Q2
SVPWM
θ2, θrm
ωrm
P2
va1*
vb1*
vc1*
2Φ/3Φ
e -jθ2
vα2
vβ2
3Φ/2Φ
va2
vb2
e -jθ2
iα2
iβ2
3Φ/2Φ
ia2
ib2
Cal.
DFBM
vab2
vbc2
θ2
id2
iq2
Grid
Figure 4.13 Control Block Diagram for Field Orientation Control of a DFBM
4.3 Conclusions
In this chapter, the principles of field orientation control of a DFBM, especially for doublyfed power generation applications, are investigated.
94
Based on the analysis of steady state operation of the DFBM in the field orientation control,
in the three types of cases, the equivalent circuit of a DFBM could be transformed into the same
form as an induction machine, a wound-rotor synchronous machine, or a conventional DFIM,
respectively. Through the proper vector control of the current of one stator winding, the active
and reactive power could be independently controlled, while the terminal power factor of the
stator windings could be intentionally regulated to be unity, lagging or leading.
With the same basic concepts of the steady state field orientation control, concerns about the
dynamics are discussed and analyzed in the rotating d-q reference frame. The major difference
between the steady state and dynamic behaviors concerns the oriented flux.
characteristics and expression of the flux linkage and torque are discussed.
Finally, the implementation of the field orientation of a DFBM is investigated.
95
The transient
Chapter 5: Design, Construction and Experimental Study of the Prototype DFBM
System
To achieve a design of a DFBM with high efficiency, many factors need to be considered. In
this chapter, the challenges of designing a high-efficiency DFBM system are highlighted.
Following the challenge description are the identified solutions to the optimal design. Using
Finite Element Analysis (FEA), the thesis presents the original design of a DFBM and system in a
power rating of 200kW for a speed range of 400 - 1,200 rpm. The designed machine is built and
tested in the laboratory and both of the steady state and dynamic experimental results are
presented and analyzed [49].
5.1 Energy Efficiency of a DFBM
To maximize energy efficiency of a DFBM is equivalent to minimizing various losses, mainly
the iron and copper losses for the machine operated in various load conditions [49]. First,
considerations of minimizing copper losses immediately exclude the selection of a rotor with
nested cage circuits for the DFBM design. This is because for the DFBM with the nested cage
rotor, the modulation function of the rotor is obtained at the costs of currents in the nested cages,
inherently causing significant conduction losses. If choosing the current-free reluctance rotor,
there will be no rotor copper losses while achieving the goal of modulating the two stator MMFs.
While minimizing copper losses favors a current-free reluctance rotor, there are still choices
among the various reluctance structures: simple saliency, radially-laminated reluctant segments,
axially-laminated and others. As discussed before, the simple salient rotor is easily eliminated for
96
its poor modulation capability. If turning to the axially-laminated rotor for greater modulation
capability, the eddy current losses become unacceptable. It is logical that a radially-laminated
rotor in segments is chosen over the axially-laminated rotor, if equal or stronger modulation
capability is achievable. The design of a radially-laminated segmental rotor to achieve large
modulation capability and minimized eddy-current losses seems the only hopeful solution to a
high efficiency DFBM. A detailed design approach of a radially-laminated rotor is discussed in
the next section [49].
5.2 Torque Density of a DFBM
In the previous publication, it was made clear by both the linear and non-linear models that
for a DFBM, when one set of stator windings is excited, back EMFs will be induced in the other
set of stator windings of different pole numbers, due to mechanical rotation of the rotor [33]. The
induced EMFs under the influence of a moving rotor are the keys for a DFBM to accomplish
electromechanical energy conversion because the induced voltage is termed “induced speed
voltage”. Assuming that a set of currents with the same frequency as that of the induced speed
voltage are injected into the second set of stator windings, then the electromechanical power and
electromagnetic torque occur. The general form of the torque production in DFBM is:
𝐸𝐼
𝐸 𝐼
𝐼
𝐼
(5.1)
𝐼
where the E1 and E2, are the induced back EMFs, the speed voltages associated with the
variations of the mutual flux linkages; I1 and I2 are the phase currents; isthe angle between the
induced speed voltage and the current; and m1 and m2 are the mutual flux linkages under the
97
given currents. Note that in the torque derivation, the relation m1/m2=I1/I2 is used and the
frequency constraint given in Eq. (1.2) is observed [49].
Examining Eq. (5.1), the torque production is related to the DFBM magnetic structure. Note
that in the equation, the currents, I1 and I2, and phase angle  are externally controlled variables
and only the mutual flux linkages, m1 and m2 are linked to the DFBM parameter (mutual
inductance) or its magnetic structure.
According to the electric machine fundamentals, the
mutual inductance of any electric machine between two windings is proportional to the winding
turn numbers and the magnetic permeance of the magnetic flux paths. If the DFBM stator
windings and iron laminations are of a traditional design as those in any traditional machine, we
are left with choices only in the DFBM rotor design. It is important to note that the function of
the DFBM rotor is to modulate one MMF (for example a 6-pole MMF) to create the largest
possible number of flux lines in another pole number (for example, the 4-pole flux lines). It is
clear that the larger the rotor modulation capability, the better the rotor design and the mutual
inductance between two sets of stator windings [49].
Previous investigations on rotor styles indicate that the rotor style has strong effects on the
rotor modulation capabilities. For the rotor with nested cage circuit, the modulation effects are
medium but rotor currents in the cage circuits are necessary and, hence, conduction losses are
inevitable. On the other hand, for the current-free reluctance segmental rotor, the modulation
capabilities are diverse, depending on the reluctance structures. Three types of current-free
reluctance rotors were investigated in detail, including simple salient reluctance rotor, axiallylaminated reluctance rotor and radially-laminated reluctance segmental rotors [27]. It is discovered
that the simple salient rotor, through robust and low cost, has a poor modulation capability and
has parasitic pole numbers in addition to the intent segmental number pr. Therefore, the simple
salient pole rotor is of little use in DFBM design. The axially-laminated rotor showed very strong
98
magnetic modulation, creating the largest mutual coupling between the two stator windings. One
major concern with the axially-laminated rotor, however, is its inability to block eddy-current
circulation in the planes vertical to the magnetic flux lines that continuously and radically change
due to the rotor’s magnetic modulation. The resultant eddy currents can cause serious problems
for the DFBM: a) heavy eddy current losses; and b) distorted flux line distribution pushed by the
large eddy currents.
Based on the above considerations, innovative approaches are needed to
design the rotor that achieves both strong rotor modulation capability and is immune to eddy
current losses [49].
In addition to the rotor style selection and lamination design, it is also found that the
combination of the DFBM pole numbers are of great influence to the rotor modulation and, thus,
to the mutual coupling of the two stator windings.
In [27], three cases of pole number
combinations are investigated using the same stator frame and lamination structures: a) p=1, q=2
and pr=3; b) p=1, q=3 and pr=4; and c) p=2, q=3 and pr=5. With finite element analysis for the
conditions of the same currents, the torque production for the 4/6-pole combination (pr=5) is 4050% more than what is achieved in the other two cases. Returning to the study posted in Chapter
2.4, it is known that the mutual coupling between the 4 and 6-pole windings in terms of flux
density is more effective than that of the 4/8 pole combination. Therefore, the investigation
confirms that the rotor modulation capability of the 4/6-pole combination is stronger than that of
the 4/8 combination, resulting in higher torque production. To achieve a motor design with high
efficiency performance, the 4/6-pole combination is chosen for the prototype motor design and
construction [49].
The strong impact of the DFBM pole number combination on DEBRM is obvious. The
results of torque capability investigation are consistent with those from the investigation of the
induced speed voltages and the effective air-gap flux density of the DFBM.
99
5.3 Sizing of the Prototype DFBM
The sizing of the 200kW/1,200rpm DFBM begins with a comparable and conventional
wound-rotor DFIM. Since the DFBM to be designed relies on the rotor modulation to create
mutual coupling between the two sets of stator windings with different pole numbers, the frame
size has been purposely enlarged by 25%, compared to that of the conventional doubly fed
induction machine, to take its inherent lower mutual coupling into consideration. By trial-anderror, the DFBM machine’s main dimensions are determined as listed in Table 2, together with
the DFBM specifications.
Power (kW):
200
Rated rpm:
1,200
Stator OD (mm):
740
Vline (volts, rms):
380
Power Grid Hz:
50
Stator ID (mm):
501
I1 (amps, rms):
125.5
1st Winding:
3-, 4-pole
Rotor OD (mm):
499.4
I2 (amps, rms):
125.5
2nd Winding:
3-, 6-pole
Rotor ID (mm):
200
Power Factor:
variable
Rotor Segmts:
5
Length (mm):
600
Cooling:
water
Insulation Class:
F
Air-Gap (mm)
0.8
Table 2 Specifications and Main Dimensions of the Prototype DFBM
As summarized in the table, the DFBM is designed with two sets of stator windings, one for
the 6-pole and another for the 4-pole. The synchronous speed of the designed DFBM is set at 600
rpm and, in order to deliver the rated power of 200kW, the DFBM needs to run at a super
synchronous speed of 1,200rpm. In operation, the second stator winding of the 6-pole will be
connected to the power grid of 50Hz while the primary stator winding of the 4-pole will be
controlled by a bi-directional converter, consisting of two inverters in back-to-back connection.
The stator lamination of the prototype DFBM is shown in Figure 5.1.
100
Figure 5.1 Stator Lamination of the Prototype DFBM
5.4 Segmental Rotor Design of the Prototype DFBM
The emphasis of the DFBM design is placed in its current-free rotor since it significantly
affects its power density and energy efficiency. As indicated in Figure 5.2, the rotor is designed
with five laminated rotor segments while each segment is formed by a stack of radial laminations
with a proper pressure. A single sheet of rotor lamination for one rotor segment is shown in
Figure 5.3. The designed laminations in the radial direction serves dual purposes: a) to force the
magnetic flux lines travelling along the five paths imposed by the designed magnetic paths; and b)
to minimize the eddy-current losses by orienting the lamination parallel to the magnetic flux lines.
101
The ratio of air space to the width of the magnetic path is carefully maintained at 2:3, so that
when the full load currents apply, the iron materials do not saturate to guarantee strong
modulation capabilities of the rotor.
It is noticeable that the rotor design ensures that the entire rotor body is non-electrical and nonmagnetically conductive except for the magnetic paths designed on purpose. This is because any
magnetic and electrical short circuits are not allowed, so as not to misguide the flux lines and
degrade the DFBM power density, and therefore reduce the energy efficiency. Also evident is
that, if any heat generation caused by losses on the rotor is excessive, the heat dissipation
becomes formidable. Precautions have been taken in the rotor design to minimize all possible
losses. Additionally, non-electrical and non–magnetic conductive epoxy bonding materials are
chosen to hold all laminated segments together for rotor mechanical integrity and proper heat
dissipation.
Figure 5.2 Rotor Cross-sectional View of the Prototype DFBM
102
Figure 5.3 Lamination for Rotor Segments
5.5 Performance Prediction of the Prototype DFBM by the FEA Method
As the final step in the design, finite element analysis has been utilized to verify three design
objectives: a) mutual coupling and induced back EMFs; b) torque production under various
controlled conditions; and c) magnetic field distribution and magnetic loading under no-load and
loaded conditions. As shown in Figure 5.4, the magnetic field of the prototype DFBM in a noload condition is asymmetrical, and calculation of the entire magnetic field for the DFBM is
needed.
Obviously, the simulation result of the magnetic flux distribution agrees with the
previous investigation in Chapter 2.
For the first objective verification, only one of the two sets of stator windings is excited with
a three-phase sinusoidal current; the induced back EMF voltages are computed for another set of
the stator windings based on the winding flux linkage variations as the rotor positions change.
This calculation is repeated at many current levels all the way to close iron core saturation. All
calculation results of induced voltages are summarized in the magnetizing curves shown in Figure
5.5. The results show that the two sets of stator windings do emulate the functions of the stator
and rotor windings, respectively, in a conventional doubly fed induction machine; that is, when
one set of windings is excited, an induced speed voltage occurs in another set of windings. In
103
addition, the relationship among the three frequencies constrained by Eq. (1.2) is fully verified.
Figure 5.4 Asymmetrical Rotor Magnetic Structure of the Prototype DFBM in a No-Load
Voltage/V
Induced Voltage (V)
Condition
500
450
400
350
300
250
200
150
100
50
0
600rpm
0
50
100
150
Current/A
Excitation
Current (A)
Figure 5.5 Magnetizing Curve of the Prototype DFBM by the FEA
For the second design objective verification, finite element analysis is used to calculate
torque production of the designed DFBM. In the calculation, both sets of the stator windings are
excited with the frequencies conforming to Eq. (1.2). At the same time the relative phase angles
104
between the two stator current vectors are changed as a parameter of the torque production
computation. The torque computation by finite element analysis is directly based on the magnetic
flux density and field intensity on each element. The results are considered accurate because the
complicated geometry of the DFBM and nonlinearity of the materials are taken into full account.
The torque production of the DFBM as a function of the phase angle between the currents in the
two sets of stator windings is shown in Figure 5.6 for the rated current levels listed in Table 2.
The results clearly show that the rated torque is fully achievable and the designed DFBM
power capability is verified. Another result we can derive at the rated power level is the copper
losses for the proportional amount of torque. For the rated torque condition (1,600 N.m), the
copper losses of both stator windings are summed to be about 3kW, less than 2% of the rated
output power.
Torque-Angle(600rpm)
3000
Torqur/Nm
Torque (N.m)
2000
1000
0
-1000
0
50
100
150
200
250
300
350
400
-2000
-3000
Phase Angle Angle/°
of Currents (Degree)
Figure 5.6 Torque Capability of the Prototype DFBM with Rated Currents by the FEA
For the third objective verification, finite element analysis is used to plot the magnetic field
distributions for typical no-load and loaded conditions. It is our experience that difficulties in
predicting iron core losses are substantial since both the magnetic field distributions in space and
variations in time of the DFBM are not sinusoidal as compared to those in a conventional AC
105
machine.
We have investigated the typical locations across all iron core areas to identify
magnetic saturation points and magnetic field variation patterns. The most important observation
is the asymmetrical distribution of air-gap flux density as shown in Figure 5.7. In the figure, the
five segments conduct magnetic flux in different amounts and an asymmetrical magnetic
distribution forms. The asymmetrical magnetic distribution implies an unbalanced pulling force
in the radial direction and non-uniformed torque force distribution along the DFBM air-gap. The
finite element field plot suggests special attention: unbalanced radial forces have to be
investigated by experimental testing of the designed DFBM.
XY Plot 1
1.50
150kw17
Curve Info
Mag_B
Setup1 : Transient
Time='0s'
1.25
1.25
1.00
1.00
Mag_B [tesla]
Magnitude of Air-Gap Flux Density (T)
1.50
0.75
0.75
0.50
0.50
0.25
0.25
0.00
0.00
0.00
0
0.20
0.40
90
0.60
0.80
Distance [meter]
1.00
180
1.20
270
1.40
1.60
360
Rotor Position (Degree)
Figure 5.7 Magnitude of Air-Gap Flux Density Distribution of the Prototype DFBM by the FEA
5.6 Experimental Results of the Prototype DFBM
The designed 200kW/1,200rpm DFBM was built and tested in the laboratory. Photos of the
DFBM stator, rotor and total assembly are shown in Figure 5.8. As observed, the stator terminals
106
are doubled because of the dual stator windings, while the rotor is current-free with no windings,
brushes, and slip rings. The stator cooling is achieved by the water jacket built into the stator
frame and the rotor cooling is greatly simplified, solely relying on natural ventilation. The simple
rotor cooling method is made possible because of greatly reduced losses and heat generation: no
copper losses and only about half of the total iron losses.
A high performance water cooled 150kW back-to-back converter has been designed and built
to drive the DFBM. Figure 5.9 shows the main circuit configuration of the developed converter
module. The specification is as follows:
 Input Voltage: 380 ~ 560 Vac
Output Voltage: 0 ~ 560 Vac
 Converter efficiency: 98.84% (rated power)
 Water cool flow rate: 1.5 GPM
 DC-bus Voltage: 800 Vdc
 Heatsink temp. rise: 30.64 deg. (rated power)
 Rated output current: 250 A
 Switching device (IGBT) rating: 1200V, 300 A
 Max output current: 300A
 Switching frequency: 5 ~ 10 kHz
The overall experimental testing setup is shown in Figure 5.10 and all important system
components are labeled.
107
a) Stator
b) Rotor
c) Assembled DFBM
Figure 5.8 Stator, Rotor and Total Assembly of the Prototype DFBM
108
Converter Module for PWM Rectifier
Converter Module for PWM inverter
3-Phase
D/Y Transformer
DFBM
Dynamic
Braking
Figure 5.9 Main Circuit Configuration of the Developed Converter Module
(a) 200kW Dynamometer Testing Bed
continued
Figure 5.10 Photos of the Experimental Testing Setup
109
Figure 5.10 continued
(b) 150kW Back-to-Back Converter
(c) DSP Based Control System
(d) 3-, 690V/300A LCL Filter
110
5.6.1 Steady State Testing of the Prototype DFBM
The steady state testing of the DFBM focuses on five major aspects: i) no-load induced
voltages, ii) iron losses, iii) grid connection with doubly fed control modes, iv) torque-power
capabilities, and v) energy efficiency over the designed speed-torque region.
For the steady state experimental testing concerning i), the purpose is to experimentally
verify the magnetizing characteristics of the designed DFBM. The test results are shown in
Figure 5.11 (a) for the two stator winding voltages, and (b) excitation current in one and induced
voltage in another set of stator windings.
(a) Yellow Trace: Line to Line Voltage of Stator Winding 1 𝑣
Blue Trace: Line to Line Voltage of Stator Winding 2 𝑣
(b) Purple Trace: Line to Line Voltage of Stator Winding 2 𝑣
Green Trace: Phase Current of Stator Winding 1 𝑖
= 405V(peak)
= 228V (peak)
= 244V(peak)
= 55.2A (peak)
Figure 5.11 v-i Waveforms in the No-Load Condition of the Prototype DFBM
111
Also, the measured magnetizing curve of the DFBM is shown in Figure 5.12. As indicated in
the figures, the induced voltages are in good sine waveforms and the induced speed voltage
levels, as a function of excitation current, in very good agreement with what were predicted by
the design and finite element analysis. The results clearly confirm that the design principles and
calculation approaches used here are reliable, and the designed DFBM itself is capable of meeting
the design expectations.
Induced Voltage (V)
500
400
300
200
100
00
0
20
30
60
80
100
120
140
Excitation Current (A)
Figure 5.12 Measured Magnetizing Curve of the Prototype DFBM
For the experimental testing concerning iron losses in ii), first the DFBM is driven by the
dynamometer to the synchronous speed of 600 rpm without exciting any stator windings, and
input power to the shaft measured. Then, maintaining the same speed as in the first step, one set
of stator windings are excited with a controllable current over a predetermined range,
corresponding to the specified induced voltage range. For this voltage range, both input power
and induced voltages are recorded. Based on the difference of input power in the first and second
steps, we obtain the iron losses as a function of operation voltages.
Figure 5.13 shows,
corresponding to the operation voltage range, the iron losses of the DFBM in a range from several
112
hundred watts to as high as 3.0-3.5kW, depending on the operation voltages.
3500
Iron Losses (W)
3000
2500
2000
2000
1000
500
0
0
140
100
200
300
400
500
600
Operating Voltage (V)
Figure 5.13 Measured Iron Losses in the No-Load Conditions of the Prototype DFBM
For the steady state experimental testing concerning iii), the grid connection characteristics
and control algorithms, the 6-pole stator windings of the DFBM are fed directly from the power
grid and the 4-pole to the controlled power converter. The recorded steady state current and grid
voltage waveforms of both the 4-pole and 6-pole winding are shown in Figures 5.14 and 5.15,
respectively. As demonstrated by the waveforms, the currents in those two windings are well
sinusoidally controlled, even though there are substantial harmonics existing in the grid voltages.
Moreover, the DFBM can continuously achieve decoupled reactive and active power through the
vector control of current components, 𝑖
and 𝑖 , via the converter for lagging, leading, and
unity power factor operations. As featured by any doubly-fed machines, the controlled active
power through the back-to-back converter set is only a fraction of the total rating of the system,
resulting in a low-cost DFBM drive system.
113
Time (10.0ms/div)
Green Trace: Grid/6-Pole Stator Winding Voltage (250V/div)
Yellow Trace: DC-bus Voltage of Converter (100V/div)
Purple Trace: 4-Pole Stator Winding Phase Current (100A/div)
Figure 5.14 4-Pole Stator Winding Current and Grid Voltage Waveforms in Steady State Loaded
Conditions
114
(a) Lagging Power Factor
(b) Leading Power Factor
(c) Unity Power Factor
(d) 50% Loaded Condition
Purple Traces: Grid/ 6-Pole Stator Winding Phase Current (50A/div)
Yellow Trace: Grid/ 6-Pole Stator Winding Voltage (250V/div)
Figure 5.15 Grid/ 6-Pole Winding Current Voltage Waveforms in Steady State Loaded
Conditions
For the steady state experimental testing concerning iv) and v), the power capabilities and
energy efficiency, the DFBM is loaded with active power on the two stator windings, and input
and output power recorded for efficiency evaluation. The power capability testing is conducted
in the neighborhood of five levels of torque (400, 800, 1,200, 1,600, and 2,000 N.m), combined
with four levels of speed (400, 600, 900, and 1,200 rpm). In this way, the total tested sample
points are twenty. At each operating point, the energy efficiency is examined. As indicated by
115
the results, the designed DFBM machine is fully capable of rated power (200kW) and can be 25%
above the rated power (250kW).
The energy efficiency of the DFBM is plotted in Figure 5.16 by the equal-efficiency contours.
Clearly shown by the contour plot, for the tested torque-speed (power) region, 75% of operating
points in the regime have efficiencies higher than 90%, 50% operating points higher than 92%,
and 35% operating points higher than 94%, including the rated and over-rated power operating
points. The torque and power capability of the designed DFBM is very satisfactory and energy
efficiency is record-breakingly high in the electric machine family of similar ratings.
1200
96.00%-98.00%
Speed (rpm)
94.00%-96.00%
900
92.00%-94.00%
90.00%-92.00%
600
88.00%-90.00%
400
400
86.00%-88.00%
800
1200
1600
2000
Torque (NM)
Figure 5.16 Efficiency Contours in Loaded Conditions Based on Measurements
Acoustic noise and vibration have been found to be a problem for the DFBM built. At a fixed
rotor speed, the noise and vibration intensities are proportional to the voltage levels applied to the
DFBM stator windings, instead of load levels. The relationship between the noise intensity and
voltage levels can be attributed to the magnetic flux levels in the DFBM field. This is a direct
indication that the asymmetrical magnetic field distribution might be responsible for the acoustic
noises and mechanical vibrations. Furthermore, it can be derived that, if the magnetic flux
116
distribution of the field can be improved to symmetric around the rotating structures, the noise
and vibration problems may be alleviated.
5.6.2 Dynamic Testing of the Prototype DFBM
The dynamic testing of the DFBM system and its grid integration in both the grid-side and
stator-side converter focuses on these major aspects: i) grid-friendly integration of the grid-side
converter, ii) grid-friendly integration of the stator-side converter, iii) active and reactive power
control, and iv) a full load operation in variable speeds.
For the dynamic experimental testing concerning i), the grid-friendly integration of the gridside converter, an algorithm of grid voltage-oriented vector control, is utilized to realize the gridfriendly integration and the regulation of the DC bus voltage of the Back-to-Back converter. As
shown in Figure 5.17, through the proper control, there is no inrush current flowing from the gridside converter to the grid once the power circuit starts to operate as a PWM rectifier, and so
implements the smooth grid-friendly integration. The DC bus voltage of the converter could be
well regulated by the control of the active power current component of the grid-side converter.
For the dynamic experimental testing concerning ii), the grid-friendly integration of the
stator-side converter, the stator flux oriented vector control discussed in the previous chapter
remains the key principle of this technology. It is known that the grid voltage waveforms are
sampled by the voltage sensors and fed back to the real time controller. Firstly, based on the
information of grid and the principle of the proposed field orientation control, by means of
regulation of the reactive power current component 𝑖 , the induced 6-pole stator voltage could
be controlled with the same magnitude, frequency and in phase to the grid voltage, regardless of
the rotor speed (as indicated in Figure 5.18). After this, as can be seen in Figure 5.19, the grid
117
integration of the stator winding is expected to be friendly without any unacceptable current or
voltage transients to the grid.
Time (20.0ms/div)
Yellow Trace: DC-bus Voltage of Converter (100V/div)
Purple Trace: Grid-Side Converter Voltage (250V/div)
Blue Trace: Grid Voltage (250V/div)
Green Trace: Grid-Side Converter Current (100A/div)
Figure 5.17 Voltages and Current Waveforms of the Grid-Side Converter during the Process of
its Grid-Friendly Integration
118
Time (10.0ms/div)
Purple Trace: Induced 6-Pole Winding Voltage before Grid Integration (250V/div)
Blue Trace: Grid Voltage (250V/div)
Yellow Trace: DC-bus Voltage of Converter (200V/div)
Green Trace: 4-Pole Stator Winding Phase Current (100A/div)
Figure 5.18 Grid, Induced 6-Pole Winding Voltages and 4-Pole Winding Current Waveforms of
the Prototype DFBM System before Stator-Side Grid Integration
119
Time (40.0ms/div)
Yellow Trace: DC-bus Voltage of Converter (100V/div)
Blue Trace: 6-Pole Stator Winding/ Grid Voltage (250V/div)
Green Trace: 6-Pole Stator Winding Phase Current (100A/div)
Purple Trace: 4-Pole Stator Winding Phase Current (200A/div)
Figure 5.19 Grid, Induced 6-Pole Winding Voltages and4-Pole Winding Current Waveforms of
the Prototype DFBM System during the Process of Stator-Side Grid-Friendly Integration
For the dynamic experimental testing concerning iii), it is known that the active and reactive
power control is equivalent to the vector control of the current components, 𝑖
and 𝑖
,
respectively. The dynamic responding of the active power generation during the process of the
35kW load on and off is shown in Figure 5.20. The generated active power follows the command
well with acceptable transients in the controlled phase current of the 4-pole winding and also the
DC bus voltage of the converter. Similarly, the experimental results of the dynamic reactive
power control under the step on and off commands are shown in Figure 5.21, where the reactive
power and corresponding current components are effectively regulated to follow the commands.
120
As observed, during the transient of reactive power, there is no influence to the DC bus voltage of
the converter.
Time (100.0ms/div)
Yellow Trace: DC-bus Voltage of Converter (100V/div)
Blue Trace: Active Power Command (50kW/div)
Green Trace: Active Power Response (50kW/div)
Purple Trace: 4-Pole Stator Winding Phase Current (100A/div)
Figure 5.20 Dynamic Performance of the Prototype DFBM System Active Power Control
121
Time (100.0ms/div)
Time (40.0ms/div)
Yellow Trace: DC-bus Voltage of Converter (100V/div)
Blue Trace: Reactive Power Command (50kVA/div)
Green Trace: Reactive Power Response (50kVA/div)
Purple Trace: 4-Pole Stator Winding Phase Current (100A/div)
Figure 5.21 Dynamic Performance of the Prototype DFBM System Reactive Power Control
122
For the dynamic experimental testing concerning iv), the full load operation in variable
speeds, as shown in Figure 5.22, the prototype DFBM generation system is stably and
continuously operated with constant torque speeding up from 200 rpm to 1,000 rpm and also
down from 1,000 rpm to 200 rpm. As can be seen, due to the constant torque and grid connection,
the active power and current of the 6-pole stator winding is remaining constant under variable
speed operations. On the other hand, for the converter controlled 4-pole stator winding, the active
power depends upon the rotor speed and is proportional to its changeable electrical frequency.
Time (1.0s/div)
(a) Rotor Speed from 200 rpm to 1,000 rpm
Yellow Trace: DC-bus Voltage of Converter (250V/div)
Green Trace: 6-Pole Stator Winding Phase Current (200A/div)
Purple Trace: 4-Pole Stator Winding Phase Current (200A/div)
continued
Figure 5.22 Continuous Full Load Operation of the Prototype DFBM in a Wide Speed Range
123
Figure 5.22 continued
Time (1.0s/div)
(b) Rotor Speed from 1,000 rpm to 200 rpm
5.7 Conclusions
This chapter presents the latest investigation of optimal design and advanced control results
of the DFBM. The challenges of designing a high-efficiency DFBM and system are discussed in
detail and the effective solutions to the optimal design and control are identified. Using design
principles assisted by the FEA, an original design of a 200kW radially-laminated reluctance
DFBM system is achieved, capable of 2,000 N.m in the speed range of 400 - 1,200 rpm with a
frame size comparable to that of a brush type doubly fed induction machine. The designed
machine is built and tested in both the steady state and dynamic conditions, and the experimental
results are in excellent agreement with the design objectives. The most successful results include
the DFBM power capability of more than 25% over the rated value and energy efficiency higher
than 90%, occupying 75% of the designed torque-speed regime.
124
These theoretical and
experimental results represent breakthroughs of doubly-fed brushless technology. The feasibility
of DFBM technology for practical applications is fully established.
The following can be concluded from the original design of the DFBM: a) the rotor design
and the selection of the pole number combination are keys to the power density and efficiency,
competing with its brush-type counterpart; b) in addition to the features of robustness and
maintenance-free, the DFBM is a controllable machine and can achieve decoupled control of
active and reactive power by implementation of vector control theory; and c) research attention is
needed to improve the magnetic field distribution to reduce noise and vibrations in DFBM
machines. Based on the achieved results, it is concluded that DFBM technology, with its intrinsic
advantages of brushless and doubly-fed operation modes, has high potential in today’s green
energy economy, especially for wind turbine generators.
125
Chapter 6: Conclusions and Future Work
6.1 Conclusions
The dissertation presents a systematic and quantitative analysis of the mechanism, modeling
and control of a DFBM. After this, a 200kW prototype of a high-efficiency DFBM based doublyfed power generation system has been successfully designed, built and tested. The conclusions
are summarized as:

The main magnetic field characteristic of a DFBM is its non-sinusoidal and even
asymmetric flux distribution due to the effect of the rotor permeance of rotor segments.
In this study, the harmonic decomposition method has been proposed for the quantitative
analysis of the asymmetric, non-sinusoidal and pulsating air-gap flux density, and assists
in the evaluation of the modulation capability of the DFBM.

Although the flux distribution along the air-gap is asymmetrical, non-sinusoidal and full
of harmonics, it is observed that the defined mutual flux linkage is rotor position
dependent and quite sinusoidal.

In many cases, the three-phase mutual flux linkages of one set of winding are not
identical in terms of profile by each phase. In this study, it is also identified that, with a
frequency constraint, the flux could be evenly weighted by each phase of winding in one
set, and consequently produces a three-phase balanced flux linkage with the exactly same
profile.

Through the comparative study between the 4/6 and 4/8 pole combinations, it is
concluded that to achieve high torque production and high efficiency of the motor, an odd
126
number of rotor segments with the smallest difference between the pole numbers of the
two windings is the optimal choice.
On the other hand, an even number of rotor
segments could be considered for the grid-tied and smooth torque output applications
where the three-phase balanced terminal characteristics and magnetic properties are
important.

The analysis of both dynamic and steady state equivalent circuits suggest that the DFBM
has the same form and similar expression of the mathematical model as those of a
conventional DFIM.

In the three types of cases, the equivalent circuit of a DFBM could be transformed to be
in the same form as an induction machine, a wound-rotor synchronous machine or a
conventional DFIM, respectively. Regardless of its steady state and dynamic conditions,
through the proper vector control of the current of one stator winding, the active and
reactive power may be independently controlled while the terminal power factor of the
stator windings can be intentionally regulated to be unity, lagging or leading.

For a doubly-fed power generation application, the comparatively larger leakage
inductances of stator winding of DFBM are useful to smooth out the harmonics of the
current and could be treated as part of an inductance based filter.

The latest investigation of optimal design and advanced control results of a DFBM are
herein presented. A 200kW radially-laminated reluctance DFBM system is designed,
built and tested in both a steady state and dynamic conditions, and the experimental
results are in agreement with the design objectives. The most successful result is the
energy efficiency higher than 90% occupying 75% of the designed torque-speed regime.
The feasibility of DFBM technology for practical applications is established.
127
6.2 Future Work
Considerably more work is required before the DFBM system becomes commercially
practical. Future research is as follows:

Investigate the quantitative relationship between the asymmetric magnetic flux level in
the DFBM field and in the noise/vibration intensities.

Investigate methods to alleviate the asymmetrical magnetic flux level in the DFBM field.

Investigate methods for iron loss prediction and reduction in design procedures.

Further study the influence of different stator/rotor pole number combinations to the
DFBM performance and torque capabilities.

Examine the air-gap sensitivities of this type of DFBM.

Develop low voltage ride-through (LVRT) technology for a DFBM to meet the power
transmission reliability standard, especially for wind power applications.

Optimally design a MW level DFBM system with high performance for industry
applications.
128
Bibliography
[1]
L. J. Hunt, “The Cascade Induction Motor,” Proc. IEE, Vol. 52, pp. 406 - 426, 1914.
[2]
F. Creedy, “Some Developments in Multi-Speed Cascade Induction Motors,” J. IEEE, Vol.
59, pp. 551 - 552, 1921.
[3]
G. Boardman, J. G. Zhu and Q. Ha, “Dynamic and Steady State Modeling of Brushless
Doubly Fed Induction Machines,” Proceedings of the Fifth International Conference on
ICEMS 2001, Vol. 1, pp. 412 - 416, 2001.
[4]
A. R. W. Broadway and G. Thomas, “Single-Unit PAM Induction Frequency Converters,”
Proc. IEE, Vol. 114, pp. 958 - 964, 1967.
[5]
A. R. W. Broadway, “Cageless Induction Motor,” Proc. IEE, Vol. 18, pp. 1593 - 1600,
1971.
[6]
A. R. W. Broadway, “Brushless Stator-Controlled Synchronous-Induction Machine,” Proc.
IEE, Vol. 120, pp. 860 - 866, 1973.
[7]
A. R. W. Broadway and G. Thomas, “Brushless Cascade Alternator,” Proc. IEE, Vol. 121,
pp. 1529 - 1535, 1974.
[8]
A. Kusko and C. B. Somuah, “Speed Control of a Single-Frame Cascade Induction Motor
with Slip-Power Pump Back,” IEEE Trans. on Industry Applications, Vol. IA, pp. 97 - 105,
1978.
[9]
C. J. Heyne and A. M. El-Antably, “Reluctance and Doubly-Excited Reluctance Motors,”
Final Report, Oak Ridge National Laboratories, Report ORNUSUB/81-95013, 1981.
[10] R. Li, A. Wallace, and R. Spée, “Dynamic Simulation of Brushless Doubly-Fed Machines,”
IEEE Transactions on Energy Conversion, Vol. 6, No.3, pp. 445 - 452, 1991.
[11] R. Spée, A. K. Wallace, and H. K. Lauw, “Performance Simulation of Brushless DoublyFed Adjustable Speed Drives,” In Conference Record of the IEEE Industry Applications
Society Annual Meeting, San Diego, CA, 1989.
[12] A. K. Wallace, R. Spée, and H. K. Lauw, “Dynamic Modeling of Brushless Doubly-Fed
Machines,” IEEE Industry Applications Society Annual Meeting, San Diego, CA, 1989.
129
[13] A. K. Wallace, P. Rochelle, and R. Spée, “Rotor Modeling and Development for Brushless
Doubly-Fed Machines,” International Conference on Electrical Machines, Vol. 1,
Cambridge, MA, 1990.
[14] R. Li, A. Wallace, and R. Spée, “Determination of Converter Control Algorithms for
Brushless Doubly-Fed Induction Motor Drives using Floquet and Lyapunov Techniques,”
IEEE Transactions on Energy Conversion, Vol. 10, No. 1, pp. 78 - 85, 1995.
[15] A. Ramchandran and G. C. Alexander, “Frequency-Domain Parameter Estimations for the
Brushless Doubly-Fed Machine,” Power Conversion Conference, Yokohama, pp. 346 - 351,
1993.
[16] A. Ramchandran, G. C. Alexander, and R. Spée, “Off-Line Parameter Estimation for the
Doubly-Fed Machine,” Proceedings of Conference on Industrial Electronics, Control,
Instrumentation, and Automation, Vol. 3, IEEE, pp. 1294 - 1298, 1992.
[17] S. Williamson, A. C. Ferreira, and A. K. Wallace, “Generalized Theory of the Brushless
Doubly Fed Machine. Part 1: Analysis,” IEE Proceedings - Electric Power Applications,
Vol. 144, No. 2, pp. 111 - 122, 1997.
[18] S. Williamson and A. C. Ferreira, “Generalized Theory of the Brushless Doubly-Fed
Machine. Part 2: Model Verification and Performance,” IEE Proceedings - Electric Power
Applications, Vol. 144, No. 2, pp. 123 - 129, 1997.
[19] A. C. Ferreira, “Analysis of the Brushless Doubly-Fed Induction Machine,” Ph.D.
Dissertation, University of Cambridge, 1996.
[20] M. S. Boger, “Aspects of Brushless Doubly-Fed Induction Machines,” Ph.D. Dissertation,
University of Cambridge, 1997.
[21] S. Williamson and M. Boger, “Impact of Inter-Bar Current on the Performance of the
Brushless Doubly Fed Motor,” IEEE Transactions Industry Applications, Vol. 35 (2), pp.
453 - 460, 1999.
[22] A. C. Ferreira and S. Williamson, “Time-Stepping Finite-Element Analysis of Brushless
Doubly Fed Machine Taking Iron Loss and Saturation into Account,” IEEE Transactions
on Industry Applications, Vol. 35 (3), pp. 583 - 588, 1999.
[23] L. Xu, F. Liang and T. A. Lipo, “Analysis of a New Variable Speed Doubly Excited
Reluctance Motor,” Proceedings of International Conference on Electric Machines,
presented at MIT Boston, 1990.
[24] L. Xu, F. Liang, T. A. Lipo, “Transient Model of a Doubly Excited Reluctance Motor”,
IEEE Transaction on Energy Conversion, Vol. 6, Issue 1, pp. 126 - 133, 1991.
130
[25] F. Liang, L. Xu, and T. A. Lipo, “Analysis of a New Variable Speed Doubly Excited
Reluctance Motor,” Electric Machines and Power System, Vol. 19, No. 2, pp. 125 - 138,
1991.
[26] F. Liang, L. Xu, T. A. Lipo, “d-q Analysis of a Variable Speed Doubly AC Excited
Reluctance Motor,” Electric Machines and Power Systems, Vol. 19, No. 2, pp. 849 - 855
1991.
[27] L. Xu, “Analysis of a Doubly-Excited Brushless Reluctance Machine by Finite Element
Method,” IEEE/IAS Annual Meeting, Houston, pp. 171 - 177, 1992.
[28] L. Xu, Y. Tang, L. Ye, “Comparison Study of Rotor Structures of Doubly Excited
Brushless Reluctance Machine by Finite Element Analysis,” IEEE Transaction on Energy
Conversion, Vol. 9, Issue 1, pp.165 - 172, 1994.
[29] Y. Tang, “High Performance Variable Speed Drive System and Generating System with
Doubly Fed Machine,” Ph.D. Dissertation, Columbus: The Ohio State University, 1994.
[30] R. Li, A. Wallace, R. Spee, Y. Wang, “Two-Axis Model Development of Cage-Rotor
Brushless Doubly-Fed Machines,” IEEE Transactions on Energy Conversion, Vol. 6, No. 3,
p. 453 - 460, 1991.
[31] R. E. Betz, M. G. Jovanovic, “The Brushless Doubly Fed Reluctance Machine and the
Synchronous Reluctance Machine - a Comparison,” IEEE Transactions on Industry
Applications, Vol. 36, Issue 4, pp. 1103 - 1110, 2000.
[32] F. Wang, F. Zhang and L. Xu, “Parameter and Performance Comparison of Doubly-Fed
Brushless Machine with Cage and Reluctance Rotors,” IEEE Transactions on Industry
Applications, Vol. 38, Issue 5, pp. 1237 - 1243, 2002.
[33] L. Xu and F. Wang, “Comparative Study of Magnetic Coupling for a Doubly Fed
Brushless Machine with Reluctance and Cage Rotors,” Thirty-Second IAS Annual Meeting,
IAS '97, Conference Record of the 1997, IEEE, Vol. 1, pp. 326 - 332. 1997.
[34] L. Xu and H. Liu, “Rotor Pole Number Studies for Doubly Excited Brushless Machines,”
IEEE Energy Conversion Congress and Exposition (ECCE2009), San Jose, California,
Proceedings, pp. 208 - 213, 2009.
[35] A. J. O. Cruickshank, A. F. Anderson, and R. W. Menziers, “Theory and Performance of
Reluctance Motors with Axially Laminated Anisotropic Rotors,” Proc. IEE, Vol. 118, No.
7, pp. 887 - 894, 1971.
131
[36] I. Boldea, Z. X. Fu, S. A. Nasar, “Performance Evaluation of Axially-Laminated
Anisotropic (ALA) Rotor Reluctance Synchronous Motors,” IEEE Transactions on
Industry Applications, Vol. 30, No. 4, pp. 977 - 985, 1994.
[37] E.M. Schulz and R. E. Betz, “Impact of Winding and Gap Flux Density Harmonics on
Brushless Doubly Fed Reluctance Machines,” Second International Conference on Power
Electronics, Machines and Drives, 2004. Vol. 2, pp. 487 - 491, 2004.
[38] M. G. Jovanovic, R. E. Betz, Y. Jian, “The Use of Doubly Fed Reluctance Machines for
Large Pumps and Wind Turbines,” IEEE Transactions on Industry Applications, Vol. 38,
No. 6, pp. 1508 - 1516, 2002.
[39] Y. Liao and L. Xu, “Design of a Doubly Fed Reluctance Motor for Adjustable-Speed
Drives,” IEEE Transactions on Industry Applications, Vol. 32, No. 5, pp. 1195 - 1203,
1996.
[40] Y. Tang and L. Xu, “A Flexible Active and Reactive Power Control Strategy for a Variable
Speed Constant Frequency Generating System,” IEEE Transactions on Power Electronics
Vol. 10, Issue 4, pp. 472 - 478, 1995.
[41] L. Xu, L. Zhen, and E. Kim, “Field-Orientation Control of a Doubly Excited Brushless
Reluctance Machine,” IEEE Transactions on Industry Applications, Vol. 34, No. 1, , pp.
148 - 155, 1998.
[42] L. Xu and Y. Tang, “Vector Control and Fuzzy Logic Control of Doubly Fed Variable
Speed Drives with DSP Implementation,” IEEE Transactions on Energy Conversion,
Vol.10, No.4, pp. 661 - 668, 1995.
[43] L. Xu and Y. Tang, “A High Efficiency Variable Speed Wind-Power Generating System
Using Doubly-Excited Brushless Reluctance Machine,” American Wind Energy
Association, 23rd Annual Conference, San Francisco, 1993.
[44] Y. Tang and L. Xu, “Adaptive Fuzzy Control of a Variable Speed Power Generating
System with Doubly Excited Reluctance Machine,” IEEE Power Electronics Specialist
Conference Proceeding, Taipei, pp. 377 - 384, June, 1994.
[45] L. Xu and Y. Tang, “A Novel Variable Speed Constant Frequency Wind-Power Generating
System Using Doubly-Excited Brushless Reluctance Machine,” IEEE/IAS Annual Meeting,
Houston, 1992.
[46] Y. Tang and L. Xu, “Stator Field Oriented Control of Doubly-Excited Induction Machine
in Wind Power Generating System,” 35th Mid West Symposium on Circuit and System,
Washington D.C., 1992.
132
[47] L. Xu and W. Cheng, “Torque and Reactive Power Control of a Doubly-Fed Induction
Machine by Position Sensorless Scheme,” IEEE Transactions on Industry Applications,
Vol. 31, No. 3, pp. 636-642, 1995.
[48] D. W. Novotny and T. A. Lipo, Vector Control and Dynamics of AC Drives, Oxford
University Press, 1996.
[49] L. Xu, B. Guan, H. Liu, L. Gao, and K. Tsai, “Design and Control of a High-Efficiency
Doubly-Fed Brushless Machine for Wind Power Generator Application,” in Proc. IEEE
ECCE, pp. 2409 - 2416, 2010.
[50] D. G. Dorrell, A. M. Knight, and R. E. Betz, “Issues With the Design of Brushless DoublyFed Reluctance Machines: Unbalance Pull, Skew and Iron Losses,” in Proc. IEEE IEMDC,
Niagara Falls, ON, Canada, May 2011, pp. 663 - 668, 1991.
[51] L. Xu, “Design and Evaluation of a Converter Optimized Synchronous Reluctance Motor
Drive,” Ph.D. Dissertation, University of Wisconsin-Madison, 1990.
[52] T.A. Lipo, Analysis of Synchronous Machines, University of Wisconsin, 1987.
[53] P. C. Krause, Analysis of Electric Machinery, McGraw Hill, 1986.
[54] T. A. Lipo, Introduction to AC Machine Design, WPERC University of Wisconsin, 2007
[55] P. C. Krause, O. Wasynczuk, and Scott D. Sudhoff, Analysis of Electric Machinery and
Drive System, John Wiley Sons, 2004.
[56] B. K. Bose, Modern Power Electronics and AC Drives, Prentice Hall PTR, 2002.
[57] D. W. Novotny, T. A. Lipo and T. M. Jahns, Introduction to Electrical Machine and Drives,
WPERC University of Wisconsin, 2009
133