Graduate Theses and Dissertations
Graduate College
2014
Advanced wound-rotor machine model with
saturation and high-frequency effects
Yuanzhen Xu
Iowa State University
Follow this and additional works at: http://lib.dr.iastate.edu/etd
Part of the Electrical and Electronics Commons
Recommended Citation
Xu, Yuanzhen, "Advanced wound-rotor machine model with saturation and high-frequency effects" (2014). Graduate Theses and
Dissertations. Paper 13720.
This Thesis is brought to you for free and open access by the Graduate College at Digital Repository @ Iowa State University. It has been accepted for
inclusion in Graduate Theses and Dissertations by an authorized administrator of Digital Repository @ Iowa State University. For more information,
please contact digirep@iastate.edu.
Advanced wound-rotor machine model with saturation and high-frequency effects
by
Yuanzhen Xu
A thesis submitted to the graduate faculty
in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
Major: Electrical Engineering
Program of Study Committee:
Dionysios Aliprantis, Major Professor
Venkataramana Ajjarapu
Jiming Song
Iowa State University
Ames, Iowa
2014
c Yuanzhen Xu, 2014. All rights reserved.
Copyright ii
DEDICATION
I would like to dedicate this dissertation to my parents Rugang Xu and Chunyun Wang
without whose encouragement and support I would not have been able to complete this work.
iii
TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . .
1
CHAPTER 2. LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . .
3
2.1
High-frequency Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.1.1
Distributed Parameters Models . . . . . . . . . . . . . . . . . . . . . . .
4
2.1.2
Lumped Parameters Models . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Parameter Identification of High-frequency Models . . . . . . . . . . . . . . . .
9
2.3
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
CHAPTER 3. METHODS, PROCEDURES AND RESULTS . . . . . . . . .
16
3.1
The Proposed Machine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.2
Parameter Identification Procedure . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2.1
Stator and Rotor DC Resistances . . . . . . . . . . . . . . . . . . . . . .
20
3.2.2
Magnetizing Characteristic and Turns Ratio Identification . . . . . . . .
20
3.2.3
Indefinite Admittance Matrix . . . . . . . . . . . . . . . . . . . . . . . .
23
3.2.4
Standstill Frequency Response Tests . . . . . . . . . . . . . . . . . . . .
24
3.2.5
Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
Fitting Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.3.1
33
3.3
Fitting Results of Proposed Model . . . . . . . . . . . . . . . . . . . . .
iv
3.4
3.3.2
Verifications at Different Rotor Positions
. . . . . . . . . . . . . . . . .
34
3.3.3
Comparison to Classical Model . . . . . . . . . . . . . . . . . . . . . . .
41
Dynamic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3.4.1
Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3.4.2
Steady State Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3.4.3
Dynamic Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
CHAPTER 4. CONCLUSION AND FUTURE WORK . . . . . . . . . . . . .
81
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
v
LIST OF TABLES
Table 2.1
Summary of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Table 2.2
Summary of models (Cont.) . . . . . . . . . . . . . . . . . . . . . . . .
15
Table 3.1
List of GA genes and identified values . . . . . . . . . . . . . . . . . .
34
vi
LIST OF FIGURES
Figure 2.1
Models proposed by [Zhong and Lipo (1995)] and [Ran et al. (1998a)] .
4
Figure 2.2
Models proposed by [Consoli et al. (1996)] and [Grandi et al. (1997b)]
6
Figure 2.3
Models proposed by [Boglietti and Carpaneto (1999)] and [Boglietti and
Carpaneto (2001)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 2.4
7
Models proposed by [Moreira et al. (2002)] and [Gubı́a-Villabona et al.
(2002)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Figure 2.5
Models proposed by [Weber et al. (2004)] and [Schinkel et al. (2006)] .
9
Figure 2.6
Model proposed by [Mirafzal et al. (2007)] . . . . . . . . . . . . . . . .
10
Figure 2.7
Models proposed by [Magdun and Binder (2012)] and [Wang et al. (2010)] 11
Figure 2.8
Model proposed by [Degano et al. (2010)]
. . . . . . . . . . . . . . . .
12
Figure 2.9
CM impedance measurement connections . . . . . . . . . . . . . . . . .
13
Figure 2.10
DM impedance measurement connections . . . . . . . . . . . . . . . . .
13
Figure 3.1
Classic machine model in the stationary abc reference frame . . . . . .
17
Figure 3.2
as winding configuration . . . . . . . . . . . . . . . . . . . . . . . . . .
17
Figure 3.3
ar winding configuration . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Figure 3.4
Complete proposed three-phase model . . . . . . . . . . . . . . . . . .
19
Figure 3.5
Experimental setups for magnetizing characteristic identification procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Figure 3.6
Measured hysteresis curves . . . . . . . . . . . . . . . . . . . . . . . . .
22
Figure 3.7
Terminal variables
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Figure 3.8
SSFR test configuration . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Figure 3.9
Error compensation experiment configuration . . . . . . . . . . . . . .
31
vii
Figure 3.10
Error compensation results . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 3.11
Measured data and fitted curves for diagonal elements of first quarter
of Y using the proposed model at θr = 0◦ . . . . . . . . . . . . . . . .
Figure 3.12
45
Measured data and fitted curves for off-diagonal elements of second and
third quarters of Y using the proposed model at θr = 30◦ . . . . . . .
Figure 3.24
44
Measured data and fitted curves for diagonal elements of second and
third quarters of Y using the proposed model at θr = 30◦ . . . . . . .
Figure 3.23
43
Measured data and fitted curves for off-diagonal elements of forth quarter of Y using the proposed model at θr = 30◦ . . . . . . . . . . . . . .
Figure 3.22
42
Measured data and fitted curves for off-diagonal elements of first quarter
of Y using the proposed model at θr = 30◦ . . . . . . . . . . . . . . . .
Figure 3.21
41
Measured data and fitted curves for diagonal elements of forth quarter
of Y using the proposed model at θr = 30◦ . . . . . . . . . . . . . . . .
Figure 3.20
40
Measured data and fitted curves for diagonal elements of first quarter
of Y using the proposed model at θr = 30◦ . . . . . . . . . . . . . . . .
Figure 3.19
39
Measured data and fitted curves for off-diagonal elements of second and
third quarters of Y using the proposed model at θr = 0◦ (Cont.) . . . .
Figure 3.18
38
Measured data and fitted curves for off-diagonal elements of second and
third quarters of Y using the proposed model at θr = 0◦ . . . . . . . .
Figure 3.17
37
Measured data and fitted curves for diagonal elements of second and
third quarters of Y using the proposed model at θr = 0◦ . . . . . . . .
Figure 3.16
36
Measured data and fitted curves for off-diagonal elements of forth quarter of Y using the proposed model at θr = 0◦ . . . . . . . . . . . . . .
Figure 3.15
35
Measured data and fitted curves for off-diagonal elements of first quarter
of Y using the proposed model at θr = 0◦ . . . . . . . . . . . . . . . .
Figure 3.14
35
Measured data and fitted curves for diagonal elements of forth quarter
of Y using the proposed model at θr = 0◦ . . . . . . . . . . . . . . . .
Figure 3.13
32
46
Measured data and fitted curves for off-diagonal elements of second and
third quarters of Y using the proposed model at θr = 30◦ (Cont.) . . .
47
viii
Figure 3.25
Measured data and fitted curves for diagonal elements of first quarter
of Y using the classical model at θr = 0◦ . . . . . . . . . . . . . . . . .
Figure 3.26
Measured data and fitted curves for diagonal elements of forth quarter
of Y using the classical model at θr = 0◦ . . . . . . . . . . . . . . . . .
Figure 3.27
. . .
57
Measured data and fitted curves for off-diagonal elements of second and
third quarters of Y using the classical model plus Res at θr = 0◦
Figure 3.38
56
Measured data and fitted curves for diagonal elements of second and
third quarters of Y using the classical model plus Res at θr = 0◦
Figure 3.37
55
Measured data and fitted curves for off-diagonal elements of forth quarter of Y using the classical model plus Res at θr = 0◦ . . . . . . . . . .
Figure 3.36
54
Measured data and fitted curves for off-diagonal elements of first quarter
of Y using the classical model plus Res at θr = 0◦ . . . . . . . . . . . .
Figure 3.35
54
Measured data and fitted curves for diagonal elements of forth quarter
of Y using the classical model plus Res at θr = 0◦ . . . . . . . . . . . .
Figure 3.34
53
Measured data and fitted curves for diagonal elements of first quarter
of Y using the classical model plus Res at θr = 0◦ . . . . . . . . . . . .
Figure 3.33
52
Measured data and fitted curves for off-diagonal elements of second and
third quarters of Y using the classical model at θr = 0◦ (Cont.) . . . .
Figure 3.32
51
Measured data and fitted curves for off-diagonal elements of second and
third quarters of Y using the classical model at θr = 0◦ . . . . . . . . .
Figure 3.31
50
Measured data and fitted curves for diagonal elements of second and
third quarters of Y using the classical model at θr = 0◦ . . . . . . . . .
Figure 3.30
49
Measured data and fitted curves for off-diagonal elements of forth quarter of Y using the classical model at θr = 0◦ . . . . . . . . . . . . . . .
Figure 3.29
48
Measured data and fitted curves for off-diagonal elements of first quarter
of Y using the classical model at θr = 0◦ . . . . . . . . . . . . . . . . .
Figure 3.28
48
. . .
58
Measured data and fitted curves for off-diagonal elements of second and
third quarters of Y using the classical model plus Res at θr = 0◦ (Cont.) 59
ix
Figure 3.39
Measured data and fitted curves for diagonal elements of first quarter
of Y using the classical model plus Res and Cgs at θr = 0◦ . . . . . . .
Figure 3.40
Measured data and fitted curves for diagonal elements of forth quarter
of Y using the classical model plus Res and Cgs at θr = 0◦ . . . . . . .
Figure 3.41
61
Measured data and fitted curves for off-diagonal elements of forth quarter of Y using the classical model plus Res and Cgs at θr = 0◦ . . . . .
Figure 3.43
60
Measured data and fitted curves for off-diagonal elements of first quarter
of Y using the classical model plus Res and Cgs at θr = 0◦ . . . . . . .
Figure 3.42
60
62
Measured data and fitted curves for diagonal elements of second and
third quarters of Y using the classical model plus Res and Cgs at θr = 0◦ 63
Figure 3.44
Measured data and fitted curves for off-diagonal elements of second and
third quarters of Y using the classical model plus Res and Cgs at θr = 0◦ 64
Figure 3.45
Measured data and fitted curves for off-diagonal elements of second and
third quarters of Y using the classical model plus Res and Cgs at θr = 0◦
(Cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 3.46
Measured data and fitted curves for diagonal elements of first quarter
of Y using the classical model plus Res and Cgr at θr = 0◦ . . . . . . .
Figure 3.47
67
Measured data and fitted curves for off-diagonal elements of forth quarter of Y using the classical model plus Res and Cgr at θr = 0◦ . . . . .
Figure 3.50
66
Measured data and fitted curves for off-diagonal elements of first quarter
of Y using the classical model plus Res and Cgr at θr = 0◦ . . . . . . .
Figure 3.49
66
Measured data and fitted curves for diagonal elements of forth quarter
of Y using the classical model plus Res and Cgr at θr = 0◦ . . . . . . .
Figure 3.48
65
68
Measured data and fitted curves for diagonal elements of second and
third quarters of Y using the classical model plus Res and Cgr at θr = 0◦ 69
Figure 3.51
Measured data and fitted curves for off-diagonal elements of second and
third quarters of Y using the classical model plus Res and Cgr at θr = 0◦ 70
x
Figure 3.52
Measured data and fitted curves for off-diagonal elements of second and
third quarters of Y using the classical model plus Res and Cgr at θr = 0◦
(Cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Figure 3.53
Proposed qd model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
Figure 3.54
Simulation block diagram . . . . . . . . . . . . . . . . . . . . . . . . .
77
Figure 3.55
Comparison of rotor line currents at steady state . . . . . . . . . . . .
78
Figure 3.56
Comparison of rotor line currents at steady state . . . . . . . . . . . .
78
Figure 3.57
Comparison of stator line currents at steady state . . . . . . . . . . . .
79
Figure 3.58
Comparison of electromagnetic torque at steady state . . . . . . . . . .
79
Figure 3.59
Comparison of electromagnetic torque during a no-load startup . . . .
80
Figure 3.60
Comparison of rotor speed during a no-load startup . . . . . . . . . . .
80
xi
ACKNOWLEDGEMENTS
I would like to take this opportunity to express my thanks to those who helped me with
various aspects of conducting research and the writing of this thesis. First and foremost, I
greatly appreciate Dr. Aliprantis’ guidance, patience and support. His insights, experience,
and knowledge on electric machines and related fields have often inspired me in the research.
His words of encouragement have renewed my hopes for completing my graduate education.
I would also like to thank my committee members, Dr. Venkataramana Ajjarapu and Dr.
Jiming Song, for their support with my research. I would additionally like to thank my friend
and partner Nicholas David, for his generous help and indelible contributions throughout our
master studies.
xii
ABSTRACT
Nowadays, the doubly-fed induction generator (DFIG) is perhaps the most common type
of generator used in onshore wind turbines. With the thriving development of wind energy,
there is a high demand for precisely predicting the dynamic performance of the DFIG. Due to
the involvement of power electronics, massive high-frequency harmonics are injected into the
machine leading to high-frequency effects such as parasitic currents. As the core of DFIGs, the
wound-rotor induction machine must be well modeled to capture these significant phenomena
in the operation of wind turbines. However, the T-equivalent model, which is currently the
most widely used model in machine dynamic studies and controller designs, is incapable to
simulate machine behaviors in the high-frequency range.
The aim of this research is to develop a novel model of a wound-rotor induction machine,
which incorporates magnetic saturation of the main flux path and high-frequency effects. The
model’s experimental parameterization procedure is described in detail. This consists of standstill frequency response tests, and a test for determining the machine’s magnetizing characteristic and turns ratio. Time-domain simulations are used to highlight the capabilities of the
proposed model, and to compare its predictions with those of a classical model at both transient
and steady states. The results show that the proposed model can better capture the dynamic
performance with the consideration of magnetic saturation. What is more, the high-frequency
current ripples, which are caused by common-mode ground currents can also be simulated in
the proposed model.
1
CHAPTER 1.
INTRODUCTION
The wound-rotor induction machine, which has been traditionally encountered in a rather
limited domain of industrial applications, is nowadays being widely used by the wind industry.
The doubly-fed induction generator (DFIG), which is perhaps the most common type of generator used in onshore wind turbines today [Müller et al. (2002); Liserre et al. (2011)], employs a
wound-rotor induction generator. The generator’s rotor is connected to a partially-rated converter that is driving the currents in the rotor windings, thereby controlling real and reactive
power output from the machine. The classical T-equivalent model has been used in the vast
majority of studies involving wind turbine dynamics and control design.
Newer semiconductor devices, such as gallium nitride high electron mobility transistors, are
being considered as possible replacements of traditional silicon-based devices in motor drives
due to their improved efficiency and switching characteristics [Morita et al. (2011); Mitova et al.
(2014); Rodrı́guez et al. (2014)]. Increasing the switching frequency in a motor drive is advantageous because it reduces current and torque ripple, but it also inevitably leads to higher parasitic current injections into the machine, which may flow through either common-mode (CM)
or differential-mode (DM) paths, exacerbating conducted electromagnetic interference (EMI)
issues and circulating bearing currents through the frame and shaft [Grandi et al. (1997a);
Mäki-Ontto and Luomi (2005)]. Usually, the turn-to-turn capacitance in the winding and the
parasitic capacitance between winding and frame are used to capture these effects [Boglietti
and Carpaneto (2001)], whereas phase-to-phase, stator-to-rotor, and rotor-to-frame capacitance
may also be taken into account. Furthermore, eddy currents could play an important role at
higher frequencies [Boglietti and Carpaneto (1999)].
Several types of advanced squirrel-cage induction machine models can be found in the
literature, featuring representation of main flux path and/or leakage magnetic saturation and
2
deep bar effects [Moreira and Lipo (1992); Smith et al. (1996); Sudhoff et al. (2002); Seman et al.
(2003); Guha and Kar (2006); Ranta et al. (2009)]; however, similar models for wound-rotor
induction machines are missing. Most models considering high-frequency effects are typically
used for EMI filter design in the frequency range between 1 kHz and 100 MHz, and are generally
not suitable for dynamic analysis of electromechanical transients. In squirrel-cage induction
machines, where power electronics are connected to the stator terminals, the flux penetration
into the rotor circuit at high frequencies is minimal [Mirafzal et al. (2007)], hence existing
high-frequency models for EMI studies only take stator windings into account. In contrast,
in DFIGs, the power electronics are connected to the rotor terminals, and the rotor has an
entirely different structure. To fill this gap, a wound-rotor induction machine model that can
represent both low-frequency and high-frequency dynamics is proposed. Moreover, a suitable
“wide-bandwidth” parameter identification procedure, which is based on standstill frequency
response (SFFR) testing, is devised.
The dissertation is structured as follows: The parameter identification methods applied by
other authors along with the models they proposed are summarized in chapter 2. Chapter 3
comes with the detailed description of the proposed model, including the experimental procedure for parameter identification and dynamic simulations. Lastly, chapter 4 concludes and
briefly introduces the future work.
3
CHAPTER 2.
2.1
LITERATURE REVIEW
High-frequency Models
Conducted EMI refers to the undesirable effects in the machine due to the electromagnetic
conduction. The high dv/dt due to power device switching within a voltage-type inverter causes
significant CM currents [Ran et al. (1998a)]. Meanwhile, the high-frequency currents flowing in
the winding generate fast changing magnetic fields, which induce large eddy currents inside the
magnetic core and the motor frame [Boglietti et al. (2007)]. Various induction machine models
considering the conducted EMI caused by high switching frequency from power electronics have
been proposed in the last few decades.
In general, machine models can be classified into two types: distributed parameters models
and lumped parameters model. Distributed parameters models divide the winding into multiple sections by assuming that attributes such as resistance, inductance and capacitance are
distributed continuously throughout the winding. This enables the variation of current penetration into the winding with frequency to be modeled [Ran et al. (1998a)], but it may require
machine dimensions and it is difficult for parameter identifications. In contrast, lumped parameters models concentrate all attributes into ideal electrical components. Although distributed
parameters models are more accurate than lumped parameters models especially in modeling
long distance cables, lumped parameters models are more popular in machine modeling with
the acceptable accuracy and the simple structure. Both lumped parameters and distributed
parameters models are adopted by different authors.
4
Cwe
Rwe
Lwe
Rw1
Cce
Cce
Lw
Lw
Cw
Cw
Ct
Rce
Rce
(a) Zhong and Lipo (1995)
Figure 2.1
2.1.1
Rw1
Ct
Rw2
Rw2
(b) Ran et al. (1998a)
Models proposed by [Zhong and Lipo (1995)] and [Ran et al. (1998a)]
Distributed Parameters Models
Some high-frequency distributed parameters models of induction motors have been proposed
by other authors, and they are briefly introduced and compared here.
[Zhong and Lipo (1995)] (Fig. 2.1(a)) proposed a distributed parameter high-frequency
model considering the turn-to-turn capacitance Cwe , the capacitance between phases (not drawn
in the single phase-belt winding model shown in Fig. 2.1(a)), and the capacitance Cce between
winding and iron core. However, the turn-to-turn capacitance can only be measured by separating the coils form the motor. A frequency-dependent resistor Rce is added in the path
of winding to iron core representing the loss along the iron, and Rwe which accounts for the
copper loss and skin effects is connected in series with leakage inductance Lwe .
[Ran et al. (1998a,b)] (Fig. 2.1(b)) proposed a distributed parameter high-frequency machine model with ten sections. The model includes the winding-to-ground capacitors Ct and
Cw , frequency-dependent resistors Rw1 and Rw2 , and leakage inductor Lw . The phase-to-phase
capacitance and mutual inductance are ignored in this model. The authors represented the
low-frequency behaviors and high-frequency behaviors separately using two models. Parallel
resonant circuits are added before the high-frequency model to block fundamental frequency
components.
The model proposed in [Llaquet et al. (2002)] is similar to the one in [Ran et al. (1998a)],
and the structure of each section is similar to the lumped model in [Grandi et al. (1997b)]. The
5
difference is that the authors considered the phase-to-phase capacitance in their model. This
paper also showed that the motor speed does not affect the high-frequency characteristic.
A model suitable for the analysis of bearing voltages and currents is presented in [MäkiOntto and Luomi (2005)]. This model focuses on modeling the shaft voltages which include
capacitive and induced shaft voltage. The capacitive shaft voltage is caused by the CM voltage
between the parasitic capacitances which are responsible for capacitive couplings between the
stator winding, the rotor core, and the stator core. Meanwhile, the CM current will generate
a circumferential magnetic flux in the stator core, which causes an induced shaft voltage.
[Henze et al. (2008); Boucenna et al. (2012)] applied finite element method (FEM) on their
distributed models. [Henze et al. (2008)] used a transmission-line-based model. [Boucenna
et al. (2012)] adopted [Mäki-Ontto and Luomi (2005)]’s method but took more metallic parts
into account such as stator core, frame, and end shields.
2.1.2
Lumped Parameters Models
More authors adopted lumped parameters models for modeling high-frequency effects of
induction motors. Some of them are summarized and compared in this section.
The contribution of [Consoli et al. (1996)] (Fig. 2.2(a)) is to decouple the CM and DM
components using qd0 transformation. This allows the separate evaluations of CM and DM
currents. After the transformation, the DM current is only related to the qd -axes circuits,
and CM current is only related to the 0 -axis circuit. The parasitic capacitor Cg is only added
between the neutral and ground. The resistor R is frequency dependent due to skin effect in
the qd -axes circuits but constant in 0 -axis circuit. The author did not illustrate the physical
meanings of L and Cw in the proposed model. They are supposed to represent the leakage
inductance and turn-to-turn capacitance. Different parameter values are used for common
mode and different mode models.
A detailed single-coil stator winding high-frequency model has been given in [Grandi et al.
(1997b)] (Fig. 2.2(b)). A air-core coil model was built at first, and then developed to a twoterminal iron-core coil model. Two circuits with same topology are connected in series to
capture the two measured resonant points. For each circuit, the resistor RL in the inductive
6
Rp1
Cw
R
L
Rp2
C1
Rc1
C2
Rc2
RL1
L1
RL2
L2
Rg
Rg
Cg
Ctg
2
(a) Consoli et al. (1996)
Figure 2.2
Ctg
2
(b) Grandi et al. (1997b)
Models proposed by [Consoli et al. (1996)] and [Grandi et al. (1997b)]
path is responsible for the ac wire resistance. Other two resistors Rp and Rc represent the
dissipative phenomena due to HF capacitive currents and dielectric losses. The capacitor C
accounts for the turn-to-turn capacitance. Furthermore, the coil-to-iron capacitive coupling
is considered by adding two equal capacitors
Ctg
2
at two terminals along with resistors Rg
representing the iron loss.
Grandi developed his own single-coil model into a three-phase model in [Grandi et al.
(1997a, 2004)]. The three-phase model takes inductive couplings among phases into account,
but neglects the coil-to-coil capacitive couplings since the corresponding capacitances are much
lower then the phase-to-ground capacitances.
[Boglietti and Carpaneto (1999, 2001)] (Fig. 2.3(a) and Fig. 2.3(b)) proposed a high-frequency
model with considerations of phase resistance R, eddy current loss resistance Re , phase leakage
inductance Ld /Ls , turn-to-turn capacitance Ct , and winding-to-ground capacitance Cg . The
experiment showed that winding-to-ground capacitance is usually much larger than turn-toturn capacitance. In addition, the phase resistance is much smaller than the leakage reactance
at high frequency. So two components turn-to-turn capacitance and phase resistance can be
neglected in the simplified model. The frame resistance Rg taking the skin effect into account
was modeled in the front end of the winding. In the paper, the effects of magnetic saturation
and rotor presence for leakage inductance under CM supply were discussed. It also showed a
combination with low-frequency qd -axes model at the end, but not revealed in details.
7
Ct
Re
R
Ls
Re
Ld
Cg
Cg
Cg
Cg
Rg
(a) Boglietti and Carpaneto (1999)
Figure 2.3
(b) Boglietti and Carpaneto (2001)
Models proposed by [Boglietti and Carpaneto (1999)] and [Boglietti and Carpaneto
(2001)]
[Boglietti et al. (2007)] further applied the high-frequency model introduced in [Boglietti
and Carpaneto (1999, 2001)] on other types of machines such as synchronous reluctance and
brushless motors, and it indicated that the model is applicable to other types of machines
as well. Furthermore, the author adopted least squares method for parameter identification
instead of analytical calculation based on resonant points.
[Moreira et al. (2002)] (Fig. 2.4(a)) developed the high-frequency model based on [Boglietti
and Carpaneto (1999)]’s and [Grandi et al. (1997a)]’s work. Instead of using pure capacitor
to represent the turn-to-turn capacitance, Moreira chose a RLC circuit (LT , RT and CT ) to
capture the second resonance in the frequency response, which is related to the turn-to-turn
capacitance.
[Gubı́a-Villabona et al. (2002)] (Fig. 2.4(b)) used a non-symmetric π-model in which the capacitance Ci between winding terminal and ground is smaller than the capacitance Cn between
neutral and ground.
[Weber et al. (2004)] (Fig. 2.5(a)) combined [Zhong and Lipo (1995)]’s model with [Grandi
et al. (1997a)]’s model to make it fit a wider frequency range. The authors additionally considered the coupling between two series-connected sections in one phase.
[Schinkel et al. (2006)] (Fig. 2.5(b)) proposed a simple model which can be used for fast
8
Cs
LT
RT
CT
Rd
Re
Cg
Rs
L
Cg
Ci
Rg
Rg
(a) Moreira et al. (2002)
Figure 2.4
Cng
(b) Gubı́a-Villabona et al. (2002)
Models proposed by [Moreira et al. (2002)] and [Gubı́a-Villabona et al. (2002)]
time domain simulation. A voltage source which represented the back-EMF is connected in
series with high-frequency model to model the low-frequency dynamic behavior of the motor.
This voltage source model is a function of load and speed, and it includes the number of turns
and the coupling between rotor and stator windings. The authors also adopted non-symmetric
π-model structure similar to [Gubı́a-Villabona et al. (2002)].
[Mirafzal et al. (2007)] (Fig. 2.6) developed a low-to-high frequency model based on IEEE
standard 112 circuit. This model combined CM, DM, and bearing models into one circuit. All
parameters are identified from machine dimensions. In [Mirafzal et al. (2009)], the analytical
approach for parameter identification is set forth. The proposed model will be similar to this
one, but further consider the high-frequency effects on rotor side, main flux saturation and
dynamics.
In [Idir et al. (2009)], authors did the CM test with windings shorted. They used RLC
circuits to capture CM and DM impedance characteristic. However, this kind of model is
lack of physical significance, and it involves massive components which will highly increase the
simulation time.
Magdun proposed a new lumped model by modifying Schinkel’s model in [Magdun and
Binder (2012)] (Fig. 2.7(a)). He used different values for parasitic capacitances Cg1 and Cg2 but
kept same value for iron resistance Rg . This modification keeps both simplicity and accuracy
9
Cwk
Rwk1
Rwk2
Ls
Re
Lwk
Cg
Cg
Cg1
Cg2
Rg
Rg
Rg1
Rg2
(a) Weber et al. (2004)
Figure 2.5
(b) Schinkel et al. (2006)
Models proposed by [Weber et al. (2004)] and [Schinkel et al. (2006)]
of the model.
[Wang et al. (2010)]’s model (Fig. 2.7(b)) is based on [Schinkel et al. (2006)]’s work. Furthermore, they added a RLC branch which is similar to the one in [Moreira et al. (2002)] to
capture the second resonance in the motor impedance characteristic, which may be caused by
the skin effect and turn-to-turn capacitance of the stator windings.
[Degano et al. (2010)] (Fig. 2.8(a)) further developed [Grandi et al. (1997a)]’s model. To
get better accuracy, the winding-to-frame capacitor is replaced by a more complicated RLC
network. This will increase the time cost by using trial-and-error method. Therefore, authors
used genetic algorithm (GA) to identify the parameters. Degano modified his own model later
in [Degano et al. (2012)]. This model includes three sections connected in series. Two sections
at each end are used to represent two winding ends, and the middle one models the winding
in the slot. The complicated parasitic network used in [Degano et al. (2010)] is replaced by a
single RC circuit.
2.2
Parameter Identification of High-frequency Models
The analytical calculation with impedance measurements is the most common method
used in the parameter identification process of high-frequency models. Taking advantage of the
variation of impedance at different frequencies, the parameters of the machine can be identified.
10
Rsw
Csw
Phase
Rs
ηLls
Lls
Llr
µRs
Ground
Csfeffective
Lm
Rcore
Rr
s
Neutral
Figure 2.6
Model proposed by [Mirafzal et al. (2007)]
Frequency response tests are conducted on the machine to derive two types of impedance: CM
impedance and DM impedance.
CM impedance refers to the input impedance between the windings and grounded machine
frame. CM currents flow from windings (or shorted wires), through parasitic capacitance, then
go to the grounded machine frame, and back to the ac power supply through the ground path.
Based on whether the neutral point is accessible or not, there may be two different connections
which are illustrated in Fig. 2.9. For either connection, the windings are connected in parallel,
and the input impedance is measured between the winding terminals and the grounded machine
frame. The difference is that the windings are short circuited at the same time in Fig. 2.9(b).
This allows the simplification of transfer functions.
DM impedance refers to the input impedance between windings. DM currents flow into
windings and return to the source through another winding or the neutral point. Based on
whether the neutral point is accessible or not, there may be two different connections which are
illustrated in Fig. 2.10. When the neutral is inaccessible, two windings are connected in parallel.
The input impedance is measured between the common terminal of parallel connected windings
and the terminal of the third winding as shown in Fig. 2.10(a). When the neutral is accessible,
three windings can be connected in parallel. The input impedance is measured between the
11
Ld
Ls
LT
Re
Lc
Cg1
CT
Re
Cg2
Rg
(a) Magdun and Binder (2012)
Figure 2.7
RT
Cg1
Cg2
Rg1
Rg2
(b) Wang et al. (2010)
Models proposed by [Magdun and Binder (2012)] and [Wang et al. (2010)]
common terminal of three windings and the neutral point. This is shown in Fig. 2.10(b).
The CM/DM impedances vary with frequency. When the frequency increases, they can
reach some resonant points (usually one or two points, based on the frequency range of interest)
where the impedance has a minimum amplitude and some anti-resonant points where the
impedance has a maximum amplitude. By utilizing the resonant behavior of the machine
impedance, all circuit parameters can be calculated. Other advanced fitting methods such as
least squares and genetic algorithm (GA) are also used to fit the measured data [Boglietti et al.
(2007); Degano et al. (2010, 2012)]. Besides that, FEM has been applied on some distributed
models with knowledge of machine dimensions and electromagnetic field analysis [Henze et al.
(2008); Boucenna et al. (2012)].
2.3
Summary
Tables 2.1 and 2.2 summarize the features of all models, and compare them to the proposed
model. In the tables, “Y” means the feature is considered in proposed model, “N” means it is
not. “N/A” represents that the feature is not applicable to this type of model. Most features
have been clarified. The only one that may be confusing is the iron resistance. This refers to
the resistance in the CM ground path.
From the literature review, we can see that there is not a comprehensive model including
12
Rp1
Rp2
C1
Rc1
C2
Rc2
RL1
L1
RL2
L2
Rcg
RLg
Rpg
Rcg
RLg
Rpg
C2g
Lg
C1g
C2g
Lg
C1g
(a) Degano et al. (2010)
Figure 2.8
Model proposed by [Degano et al. (2010)]
both high-frequency effects and mechanical dynamics. The proposed model in this dissertation
solved this issue with a “lumped parameters” structure since it is capable to capture the
frequency response in the range of interest and easier to identify all parameters.
13
W
W
+
+
ZCM (s)
ZCM (s)
–
–
G
G
(a) Neutral inaccessible
Figure 2.9
(b) Neutral accessible
CM impedance measurement connections
W
W
+
+
ZDM (s)
–
ZDM (s)
W
–
N
(a) Neutral inaccessible
Figure 2.10
(b) Neutral accessible
DM impedance measurement connections
Distributed
EMI
1K–500K
DM
Model type
Use of model
Frequency
range (Hz)
Type of data
Fitting
method
Variables
DC resistance
Eddy current
loss
Stator leakage
inductance
Phase-tophase
inductive
coupling
Phase-tophase
capacitive
coupling
Turn-to-turn
capacitance
Phase-toground
capacitance
Iron
resistance
Main-flux
saturation
Low-frequency
behavior
Rotor
components
Induction
motor
Induction
motor
Machine type
Analytical
qd0
N
Y
Y
N
N
Y
Y
N
N
N
N
N/A
abc
Y
Y
Y
Y
Y
Y
Y
Y
N
N
N
CM/DM
10K–2M
EMI
Lumped
Consoli et al.
(1996)
Zhong and
Lipo (1995)
Literature
N
N
N
Y
Y
Y
N
Y
Y
Y
abc
N
Analytical
CM/DM
10K–1M
EMI
Lumped
Induction
motor
Grandi et al.
(1997b,a,
2004)
N
Y
N
Y
Y
N
N
N
Y
Y
abc
N
Analytical
CM/DM
1K–3M
EMI
Distributed
Induction
motor
Ran et al.
(1998a,b)
N
Y
N
Y
Y
Y
N
N
Y
Y
abc
Y
Least squares
CM/DM
1K–1M
EMI
Lumped
Induction
motor
N
Y
N
Y
Y
Y
N
N
Y
Y
abc
N
Analytical
CM/DM
1K–2M
EMI
dual-voltage
ac induction
motor
Lumped
Moreira
et al. (2002)
N
N
N
Y
Y
Y
Y
N
Y
Y
abc
N
Analytical
CM
10K–1M
EMI
Distributed
Induction
motor
Llaquet et al.
(2002)
Summary of models
Boglietti et
al. (1999,
2001, 2007)
Table 2.1
N
N
N
N
Y
Y
N
N
Y
Y
abc
Y
Analytical
CM
N/A
EMI
Lumped
ac motor
Gubı́aVillabona
et al. (2002)
N
N
N
Y
Y
Y
Y
Y
Y
Y
abc
N
Analytical
CM/DM
1K–100M
EMI
Lumped
Induction
motor
Weber et al.
(2004)
N
N
N
Y
Y
N
N
Y
Y
N
abc
Y
Y
Y
Y
Y
Y
N
N
Y
Y
Y
abc/qd
Y
GA
Admittance
matrix
CM, machine
dimensions
Analytical
0.01–51.2K
Lumped
Dynamic
simulation
wound-rotor
Proposed
model
10K–10M
Distributed
Bearing
current
Induction
motor
Mäki-Ontto
and Luomi
(2005)
14
CM/DM,
machine
dimensions
Analytical
abc
Y
Y
Y
Y
N
Y
Y
Y
N
Y
Y
CM/DM
Analytical
abc
Y
Y
Y
Y
N
N
Y
Y
N
Y
N
Type of data
Fitting method
Variables
DC resistance
Eddy current
loss
Stator leakage
inductance
Phase-to-phase
inductive
coupling
Phase-to-phase
capacitive
coupling
Turn-to-turn
capacitance
Phase-toground
capacitance
Iron resistance
Main-flux
saturation
Low-frequency
behavior
Rotor
components
10–10M
10K–30M
Frequency
range (Hz)
Mirafzal
et al. (2007,
2009)
Induction
motor
Lumped
EMI and
bearing
current
EMI
Induction
motor
Lumped
Schinkel
et al. (2006)
Use of model
Model type
Machine type
Literature
N
N
N
N
Y
Y
N
N
Y
Y
FEM
N/A
Y
Machine
dimensions
100K–10M
EMI
Induction
motor
Distributed
Henze et al.
(2008)
N
N
N
N/A
N/A
N/A
N/A
N/A
N/A
N/A
Analytical
N/A
N/A
CM/DM
100K–40M
EMI
Induction
motor
Lumped
N
N
N
Y
Y
N
N
N
Y
Y
CM/DM,
machine
dimensions
Analytical
abc
N
1K–100M
EMI
Lumped
squirrel-cage
Magdun and
Binder
(2012)
N
N
N
Y
Y
Y
N
Y
Y
Y
Analytical
abc
N
CM/DM
100–10M
EMI
Lumped
squirrel-cage
Wang et al.
(2010)
Summary of models (Cont.)
Idir et al.
(2009)
Table 2.2
N
N
N
Y
Y
Y
N
Y
Y
Y
GA
abc
N
CM/DM
10K–30M
EMI
induction
motor
Lumped
Degano et al.
(2010)
N
N
N
Y
Y
Y
N
Y
Y
Y
CM/DM,
machine
dimensions
GA
abc
N
10K–30M
EMI
Induction
motor
Lumped
Degano et al.
(2012)
N
N
N
N/A
N/A
N/A
N/A
N/A
N/A
N/A
FEM
N/A
N/A
Machine
dimensions
Y
Y
Y
Y
Y
N
N
Y
Y
Y
GA
abc/qd
Y
Admittance
matrix
0.01–51.2K
Dynamic
simulation
Bearing
current
10K–200K
Lumped
wound-rotor
Proposed
model
Induction
motor
Distributed
Boucenna
et al. (2012)
15
16
CHAPTER 3.
METHODS, PROCEDURES AND RESULTS
3.1
The Proposed Machine Model
The classical abc induction machine model is shown in Fig. 3.1. It generally includes the
stator and rotor resistances (rs and rr ), stator and rotor leakage inductances (Lls and Llr ),
stator and rotor magnetizing inductances (Lms and Lmr ) and the mutual inductances between
stator and rotor windings (Lsr ). All variables are assumed to be constant. It needs to be
noticed that the mutual couplings between stator and rotor windings are not plotted.
The proposed 3-phase model is developed based on the classical abc model of a symmetric
machine, with the following modifications:
1. Resistors representing eddy current loss are connected in parallel with the stator leakage
inductances. The eddy current resistance is a frequency dependent parameter. In order
to run time-domain dynamic simulation, it can be assumed a constant value for a certain
frequency range.
2. Parasitic capacitors connected between terminal and frame ground are added at each
terminal. (The iron resistors that are in series with the parasitic capacitors are not used
in the SSFR fitting process, because they can be safely ignored in the measured frequency
range. Nevertheless, they are necessary in the dynamic simulation to avoid large current
ripples caused by high dv/dt of the supplied PWM voltage.)
3. Saturation in the magnetizing path is considered. (This is a large-signal phenomenon,
and thus is ignored in small-signal SSFR tests.)
These modifications were made in order to fit the observed behavior of the machine under
test. The proposed as-winding configuration is shown in Fig. 3.2. bs- and cs-windings are
17
rs
rr
Lls
Llr
bs
Lms
br
Lmr
Lms
Lls
rs
Lmr
as
rr
ar
Lms
Lmr
cs
cr
Lls
Llr
rs
Figure 3.1
Llr
rr
Classic machine model in the stationary abc reference frame
Res
I1
Ias
rs
+
Igas
Lls
Lms
Vas
−
Cgs
Figure 3.2
as winding configuration
assumed identical to as-winding due to symmetry. Similarly, the proposed ar-winding configuration is shown in Fig. 3.3, and br- and cr-windings are assumed identical to ar-winding.
The complete three-phase model is shown in Fig. 3.4. Furthermore, the fitting procedure indicates that it is not necessary to have a capacitor representing the turn-to-turn capacitance in
measured frequency range. After modifications, setting s = jω, the frequency-domain voltage
equations for the proposed model at standstill can be written as:
1
1
Vas = (Zws + sLms )Ias − sLms Ibs − sLms Ics +
2
2
sLsr
2π
2π
)Ibr + cos(θr −
)Icr
cos(θr )Iar + cos(θr +
3
3
(3.1)
18
I4
Iar
rr
Llr
+
Igar
Lmr
Var
−
Cgr
Figure 3.3
ar winding configuration
1
1
Vbs = (Zws + sLms )Ibs − sLms Ias − sLms Ics +
2
2
2π
2π
)Iar + cos(θr )Ibr + cos(θr +
)Icr (3.2)
sLsr cos(θr −
3
3
1
1
Vcs = (Zws + sLms )Ics − sLms Ias − sLms Ibs +
2
2
2π
2π
sLsr cos(θr +
)Iar + cos(θr −
)Ibr + cos(θr )Icr (3.3)
3
3
1
1
Var = (Zwr + sLmr )Iar − sLmr Ibr − sLmr Icr +
2
2
sLsr
2π
2π
cos(θr )Ias + cos(θr −
)Ibs + cos(θr +
)Ics
3
3
(3.4)
1
1
Vbr = (Zwr + sLmr )Ibr − sLmr Iar − sLmr Icr +
2
2
2π
2π
sLsr cos(θr +
)Ias + cos(θr )Ibs + cos(θr −
)Ics (3.5)
3
3
1
1
Vcr = (Zwr + sLmr )Icr − sLmr Iar − sLmr Ibr +
2
2
2π
2π
sLsr cos(θr −
)Ias + cos(θr +
)Ibs + cos(θr )Ics (3.6)
3
3
where θr is electrical rotor angle,
Zws = rs +
1
Res
1
+
1
sLls
= rs +
sLls Res
s(Lls Res + Lls rs ) + rs Res
=
Res + sLls
Res + sLls
(3.7)
and
Zwr = rr + sLlr .
(3.8)
19
Res
as
rs
Lls
Lms
Lmr
Llr
Cgs
Cgr
Rgs
bs
Rgr
Res
rs
Lls
Lms
Lmr
Llr
br
rr
Cgs
Cgr
Rgs
cs
ar
rr
Rgr
Res
rs
Lls
Lms
Lmr
Llr
cr
rr
Cgs
Cgr
Rgs
Rgr
Figure 3.4
Complete proposed three-phase model
All currents in equations (3.1)-(3.6) are not current injections at terminals. Instead, they
are currents in the series branches as shown in Fig. 3.2 and 3.3. Also, voltages are those across
the series branches.
3.2
Parameter Identification Procedure
The experimental procedure is applied to a YZR 160M2-6 wound-rotor induction motor.
This is a six-pole Y-Y connected machine, rated for 7.5 kW, 195 V (rotor), 360 V (stator), at
1200 rpm.
20
3.2.1
Stator and Rotor DC Resistances
Taking advantage of the accessibility of both stator and rotor windings of the wound-rotor
induction machine, the stator and rotor dc resistance can be easily measured by a multimeter.
The measured values are rs = 0.53 Ω and rr = 0.17 Ω at room temperature. These values
are also verifiable by the SSFR tests. The variation of resistance with temperature is not
incorporated in the model.
3.2.2
Magnetizing Characteristic and Turns Ratio Identification
In [Aliprantis et al. (2005a)], a turns ratio identification method for synchronous machines
was proposed, which takes advantage of the magnetic nonlinearity properties of the machine’s
iron. The same technique is applied here, with slight modification. Specifically, two magnetization curves are measured by exciting the machine from the stator and rotor side, respectively,
leaving the opposite side open-circuited. The connection when the machine is excited from the
stator side is illustrated in Fig. 3.5. The other connection is similar to this one but the source
is connected to the rotor side and the voltage is measured on the stator side. At the rotor
side, measurements were taken that included the brush/slip-ring configuration. The excitation
comes from a controllable dc supply that can generate the positive half of a sine wave at low
frequency, which was 50 mHz in this experiment. First, the machine is fully saturated using a
sufficiently high current (ca. 120% of rated in this case), then the current is allowed to drop
to zero as slowly as possible to minimize other effects. The rotor needs to be carefully aligned
for these tests, as shown in Fig. 3.5. Conventionally, this rotor angle is defined as θr = 0◦ ,
where the current ics or icr produces a magnetomotive force aligned with the d -axis[Krause
et al. (2002)].
Same to synchronous machine, the main flux path saturation can be given by
[Aliprantis et al. (2005a)]:
imq = Γmq (λ̂m )λmq
(3.9)
imd = Γmd (λ̂m )λmd
(3.10)
21
Oscilloscope
PC
Function Generator
Ch1
Ch2
Current Probe
Voltage Probe
Agilent 33220A
DC Power Supply
Sorensen SGI 330/45
+
−
ics
−
bs
br
as
ar
vcbr
cs
cr
+
Figure 3.5
Experimental setups for magnetizing characteristic identification procedure
Due to the non-salient rotor construction, q- and d -axis should have the same magnetizing
characteristic. i.e.,
Γmq (λ̂m ) = Γmd (λ̂m )
(3.11)
The magnitude of the effective magnetizing flux vector λ̂m is defined by
λ̂m =
q
λ2md + λ2mq
(3.12)
We can write the following sets of equations that relate the magnetizing currents and flux
linkages with measured currents and voltages. We use standard machine notation and the
reference frame transformation defined in [Krause et al. (2002)]. When exciting the stator side,
we have:
2
imd ≈ ids = √ ics
3
λ̂m = λmd ≈ λ0dr = √
(3.13)
1
1
λcbr = √
3Nrs
3Nrs
Z
vcbr dt
(3.14)
22
Magnetizing flux linkage (Vs)
Magnetizing flux linkage (Vs)
1
1
0.8
0.6
Nrs=0.45
0.4
0.2
0.8
0.6
Nrs=0.65
0.4
0.2
rotor excited
stator excited
0
0
5
10
15
20
25
30
35
rotor excited
stator excited
0
0
40
5
10
Magnetizing current (A)
15
20
25
30
35
40
Magnetizing current (A)
Magnetizing flux linkage (Vs)
1
0.8
0.6
Nrs=0.56
0.4
0.2
0
0
rotor excited
stator excited
fitted magnetizing characteristic curve
5
10
15
20
25
30
35
40
Magnetizing current (A)
Figure 3.6
Measured hysteresis curves
When exciting the rotor side,
2
imd ≈ i0dr = √ Nrs icr
3
Z
1
1
λ̂m = λmd ≈ λds = √ λcbs = √
vcbs dt
3
3
(3.15)
(3.16)
The flux linkage is calculated by performing numerical integration using Simpson’s rule. The
rotor-to-stator turns ratio Nrs is a free parameter that is manually adjusted until a value is
found for which the upper parts (“extrados”) of the two hysteresis curves coincide. Integration
constants are removed by artificially forcing the extrados to return to the origin, so that the
two curves can be compared to each other. The result is shown in Fig. 3.6, and the turns ratio
is found to be Nrs = 0.56. The turns ratio is identified first, because it is needed in the SSFR
testing procedure that follows.
A nonlinear function for the (single-valued, nonhysteretic) magnetizing characteristic can
23
be obtained as well. For example, the d -axis inverse magnetizing inductance, which relates flux
linkage and current by equation (3.10), and is often used in saturable dynamic models [Aliprantis et al. (2005b)], was derived by a least-squares fitting technique as:
Γmd1 (λ̂m ) =
1614λ̂4m − 3214λ̂3m + 2129λ̂2m − 1038λ̂m + 686
λ̂2m − 50.65λ̂m + 51.86
(3.17)
This expression is valid for λ̂m ≤ λ̂m1 = 0.92 V·s, because the rational function diverges after
this point. An appropriate continuation is defined so that the flux linkage increases linearly
with the same slope as at λ̂m1 . This results in
Γmd2 (λ̂m ) =
"
1
Lmd,sat
+ Γmd1 (λ̂m1 ) −
1
Lmd,sat
#
λ̂m1
λ̂m
(3.18)
for λ̂m > λ̂m1 , with Lmd,sat = 3.1 mH.
3.2.3
Indefinite Admittance Matrix
When the wound-rotor induction machine is at standstill, and saturation is not considered
(in SSFR tests, the excitation is of a small-signal nature), it can be treated as a seven-terminal
linear network [Balabanian and Bickart (1981)]. The seven terminals include three stator terminals, three rotor terminals, plus the frame ground. Choosing an arbitrary point of reference,
and denoting terminal voltages V1 to V7 and terminal current injections I1 to I7 as shown in
Fig. 3.7, the network can be expressed as a linear combination of the terminal voltages to yield
 

I1  y11
  
  
I2  y21
  
.= .
.  .
.  .
  
  
I7


y12 · · · y17  V1 
y22 · · ·
..
.
 
 
 
y27 
 V2 


..  
 .. 
 . 
. 
 
 
y71 y72 · · · y77
(3.19)
V7
The coefficient matrix in (3.19) is called the indefinite admittance matrix and is designated
Yi . According to Kirchhoff’s current law, the sum of elements in each column equals zero. Rows
must also sum to zero. Hence, if the frame ground is chosen as the common terminal, then
the last row and column of Yi can be removed, and the system becomes a common-terminal
six-port network.
24
I1
as
+
I2
bs
+
cs
+
I4
ar
+
I5
V1
br
+
V2
I6
Induction Machine
I3
cr
V3
+
V4
I7
V5
V6
Frame
+
V7
−
−
−
−
−
−
Figure 3.7
3.2.4
−
Terminal variables
Standstill Frequency Response Tests
Standstill frequency response tests are conducted to derive the elements of the admittance
matrix Y of the common-terminal six-port network. The elements are transfer functions and
can be calculated by
yjk =
Ij
Vk
(3.20)
where j = 1, 2, · · · , 6 and k = 1, 2, · · · , 6. When a voltage source is connected to the kth
terminal, all other terminals are grounded to the frame ground (cable effects are neglected).
Then the elements of kth column of Y can be obtained by measuring the voltage of the source
and current injections at each terminal. An example of test configuration when the voltage
source is connected to as winding is shown in Fig. 3.8.
3.2.4.1
Theoretical Calculation in Frequency Domain
With the setup shown in Fig. 3.8 and winding models shown in Fig. 3.2 and 3.3, there are
Ias = −2Ibs = −2Ics
(3.21)
25
HP 35670A Signal Analyzer
Src
+
Ch1
Ch2
Voltage Probe
Current Probe
−
+
−
+
+
I2
I5
V2
bs
V5
Stator Winding
br
Rotor Winding
as
ar
−
I1
+
−
cs
cr
+
V3
V1
+
V6
I3
−
−
+
I4
V4
−
I6
−
Machine Frame
Figure 3.8
SSFR test configuration
Iar + Ibr + Icr = 0
(3.22)
Vbs = Vcs
(3.23)
Var = Vbr
(3.24)
26
Substituting equations (3.2), (3.3), (3.21) into (3.23) yields
1
1
(Zws + sLms )Ibs − sLms Ias − sLms Ics +
2 2
2π
2π
sLsr cos(θr −
)Iar + cos(θr )Ibr + cos(θr +
)Icr
3
3
1
1
= (Zws + sLms )Ics − sLms Ias − sLms Ibs +
2
2
2π
2π
sLsr cos(θr +
)Iar + cos(θr −
)Ibr + cos(θr )Icr (3.25)
3
3
and rearranging, yields
cos(θr −
2π
2π
)Iar + cos(θr )Ibr + cos(θr +
)Icr
3
3
= cos(θr +
2π
2π
)Iar + cos(θr −
)Ibr + cos(θr )Icr (3.26)
3
3
then simplifiy to
sin(θr )Iar + cos(θr +
π
π
)Ibr − cos(θr − )Icr = 0
6
6
(3.27)
Substituting equation (3.22) into (3.27) yields
sin(θr )(−Ibr − Icr ) + cos(θr +
π
π
)Ibr − cos(θr − )Icr = 0
6
6
2π
2π
)Ibr = cos(θr +
)Icr
cos(θr −
3
3
(3.28)
(3.29)
Similar relationships can also be derived as follows
cos(θr +
2π
)Iar = cos(θr )Ibr
3
(3.30)
cos(θr −
2π
)Iar = cos(θr )Icr
3
(3.31)
Substituting equations (3.21), (3.29)-(3.31) into (3.4) yields
1
1
Var = (Zwr + sLmr )Iar − sLmr Ibr − sLmr Icr
2
2
2π
2π
+ sLsr cos(θr )Ias + cos(θr −
)Ibs + cos(θr +
)Ics
3
3
!
2π
cos(θr + 3 )Iar
1
1
= (Zwr + sLmr )Iar − sLmr
− sLmr
2
cos(θr )
2
cos(θr − 2π
3 )Iar
cos(θr )
2π
1
2π
1
+ sLsr cos(θr )Ias + cos(θr −
) − Ias + cos(θr +
) − Ias
3
2
3
2
3
3
cos(θr ) Ias + Zwr + sLmr Iar
= sLsr
2
2
!
(3.32)
27
Substituting equations (3.21), (3.29)-(3.31) into (3.5) yields
1
1
Vbr = (Zwr + sLmr )Ibr − sLmr Iar − sLmr Icr
2
2
2π
2π
+ sLsr cos(θr +
)Ias + cos(θr )Ibs + cos(θr −
)Ics
3
3
!
2π
cos(θr + 3 )Iar
1
1
= (Zwr + sLmr )
− sLmr Iar − sLmr
cos(θr )
2
2
cos(θr − 2π
3 )Iar
cos(θr )
2π
2π
1
1
)Ias + cos(θr ) − Ias + cos(θr −
) − Ias
3
2
3
2
"
#
√
3
3 3
= sLsr − cos(θr ) −
sin θr Ias
4
4
"
!
!
#
√
√
1
3
3 3 3
+ − −
tan(θr ) Zwr −
+
tan(θr ) sLmr Iar
2
2
4
4
!
+ sLsr cos(θr +
(3.33)
Substituting equations (3.32) and (3.33) into (3.24) yields
sLsr
"
#
√
3
3 3
3
3
cos(θr ) Ias + Zwr + sLmr Iar = sLsr − cos(θr ) −
sin θr Ias
2
2
4
4
!
!
#
"
√
√
3 3 3
3
1
tan(θr ) Zwr −
+
tan(θr ) sLmr Iar (3.34)
+ − −
2
2
4
4
Then simplify to
sLsr
!
√
3 3
9
cos(θr ) +
sin(θr ) Ias
4
4
!
!#
"
√
√
3
9 3 3
3
tan(θr ) Zwr + − −
tan(θr )sLmr Iar (3.35)
=
− −
2
2
4
4
Further simplify to
3
3
sLsr cos(θr )Ias = −(Zwr + sLmr )Iar
2
2
(3.36)
Substituting equations (3.30) and (3.31) into (3.36) respectively, yields
3
2π
3
sLsr cos(θr +
)Ias = −(Zwr + sLmr )Ibr
2
3
2
(3.37)
3
2π
3
sLsr cos(θr −
)Ias = −(Zwr + sLmr )Icr
2
3
2
(3.38)
and
The current injection at each terminal under the test setup shown in Figure 3.8 is
I1 =
V1
+ Ias
Zgs
(3.39)
28
I2 = Ibs
(3.40)
I3 = Ics
(3.41)
I4 = Iar
(3.42)
I5 = Ibr
(3.43)
I6 = Icr
(3.44)
Using equations (3.1), (3.2), (3.21), and (3.36)-(3.38),
V1 = Vas − Vbs
3
1
3
− Ias
= Zws + sLms Ias − Zws + sLms
2
2
2
√
3sLsr cos(θr )
π
+ 3sLsr cos(θr + ) −
Ias
6
2Zwr + 3sLmr
!
√
3sLsr cos(θr + 2π
π
3 )
− 3sLsr cos(θr − ) −
Ias
6
2Zwr + 3sLmr
+
√
3sLsr cos(θr − 2π
3 )
3sLsr sin(θr ) −
Ias
2Zwr + 3sLmr
!
3
= Zs Ias
2
(3.45)
where
9
s2 L2sr
3
Zs = Zws + sLms −
2
4 Zwr + 32 sLmr
(3.46)
then substituting equation (3.39) into (3.45) yields
V1 =
3Zs Zgs
I1
2Zgs + 3Zs
(3.47)
1
sCgs
(3.48)
where
Zgs =
Hence, the transfer functions of the first column of Y can be expressed as
y11 =
I1
2Zgs + 3Zs
=
V1
3Zs Zgs
(3.49)
Substituting equations (3.40) and (3.21) into (3.49) yields
y21 =
I2
1
=−
V1
3Zs
(3.50)
29
Substituting equations (3.41) and (3.21) into (3.49) yields
y31 =
1
I3
=−
V1
3Zs
(3.51)
Substituting equations (3.42) and (3.36) into (3.49) yields
y41 =
I4
sL cos(θr )
sr
= −
3
V1
Zws + 2 sLms Zwr + 32 sLmr − 49 s2 L2sr
(3.52)
Substituting equations (3.43) and (3.37) into (3.49) yields
y51 =
sLsr cos(θr + 2π
I5
3 ) = −
3
3
V1
Zws + 2 sLms Zwr + 2 sLmr − 49 s2 L2sr
(3.53)
Substituting equations (3.44) and (3.38) into (3.49) yields
y61 =
sLsr cos(θr − 2π
I6
3 ) = −
V1
Zws + 32 sLms Zwr + 32 sLmr − 49 s2 L2sr
(3.54)
Due to the symmetric construction, the expressions of elements for other columns of Y can
be derived with similar manipulations, and the results show that
y44 =
2Zgr + 3Zr
I4
=
V4
3Zr Zgr
(3.55)
where
3
9
s2 L2sr
Zr = Zwr + sLmr −
2
4 Zws + 23 sLms
(3.56)
and
1
sCgr
(3.57)
I5
1
=−
V4
3Zr
(3.58)
Zgr =
In addition,
y54 =
30
and the following relationships hold:
3.2.4.2
y11 = y22 = y33
(3.59)
y44 = y55 = y66
(3.60)
y21 = y31 = y12 = y13 = y23 = y32
(3.61)
y54 = y64 = y45 = y46 = y56 = y65
(3.62)
y41 = y52 = y63 = y14 = y25 = y36
(3.63)
y51 = y43 = y62 = y15 = y26 = y34
(3.64)
y61 = y42 = y53 = y16 = y24 = y35
(3.65)
Error Compensation of measuring instruments
During the experimental measurements, all instruments (signal analyzer, voltage probe,
current probe, etc.) may lead to a bias of measured results in both magnitude and phase.
Therefore, a frequency response test is conducted at first to compensate the measurement error
as much as possible. A 25 Ω non-inductive resistor is connected to the voltage source of the
signal analyzer. Assume this resistor is ideally resistive. The experiment setup is shown in
Fig. 3.9.
Based on the assumption, the magnitude should keep at 25 Ω and the phase should be
at zero degree. However, the test results show that both magnitude and phase vary when
the frequency increases as shown in Fig. 3.10. So a second-order polynomial and a first-order
polynomial are used to fit the test results. For the magnitude, there is
Emag (f ) = −1.876 × 10−9 f 2 + 3.97 × 10−5 f + 25.17
(3.66)
For the phase, there is
Ephase (f ) = 2.011 × 10−5 f + 0.008412
(3.67)
Then the data from measurements after the compensation become
Mag(f ) = Magraw (f )
25
Emag (f )
(3.68)
31
HP 35670A Signal Analyzer
Ch2
Current Probe
+
Ch1
Voltage Probe
Src
−
+
−
25 Ω
Non-inductive Resistor
Figure 3.9
Error compensation experiment configuration
and
Phase(f ) = Phaseraw (f ) − Ephase (f )
3.2.4.3
(3.69)
Experiment Process
First, move the rotor to zero position where θr = 0◦ . (For instance, the rotor position may
be obtained using an encoder.) Then measure the voltages and currents at each configuration
to derive all elements of the 6 by 6 admittance matrix Y at θr = 0◦ . Afterwards, one may rotate
the rotor to other arbitrarily chosen positions to verify the validity of the proposed model. The
SSFR tests were conducted in the frequency range from 15.625 mHz to 51.2 kHz, limited by
32
Magnitude
Phase
30
120
Experimental results
Fitted curve
Angle (degree)
Impedance (Ω)
100
25
20
80
60
40
15
10 1
10
20
Experimental results
Fitted curve
2
10
3
4
10
10
Frequency (Hz)
Figure 3.10
5
10
0 1
10
2
10
3
4
10
10
Frequency (Hz)
5
10
Error compensation results
the capability of the measuring instrument that was available. According to the theoretical
analysis, there are only five independent elements in Y. Other elements can be related to these
five independent elements by algebraic relationships. Therefore, tests of dependent elements
may only be conducted in high frequency range in order to save testing time. Measurements
were taken excluding the brush/slip-ring configuration.
3.2.5
Genetic Algorithm
The parameters in proposed model are identified using a genetic algorithm (GA). Here, we
use error functions adapted from [Aliprantis et al. (2005a)]. Let x denote a transfer function
for which {x̂k } is the set of measured data, and {xk } the corresponding set of predicted data
for k = 1, ..., N points. A normalized magnitude error is defined by
E
mag
(ρk ) =



min{ρk ,10}−1


,

9
if ρk ≥ 1



k ,0.1}

 1−max{ρ
,
0.9
if ρk < 1
(3.70)
where ρk = |x̂k /xk |. A normalized angle difference error is defined by
E ang (∆φk ) =


π

k, 2 }
 min{∆φ

,
π

2


min{2π−∆φk , π2 }



,
π
2
if ∆φk ∈ [0, π)
if ∆φk ∈ [π, 2π)
(3.71)
33
where ∆φk = |6 x̂k − 6 xk |. The total error of a transfer function is calculated by averaging the
magnitude and angle errors:
PN
E(x) =
k=1 [E
mag (ρ )
k
+ E ang (∆φk )]
.
2N
(3.72)
Although there are 36 transfer functions in total in an admittance matrix, only five independent elements need to be fitted. The total error is defined as the mean error of the five
transfer functions ,
E=
1
[Ey11 + Ey21 + Ey41 + Ey44 + Ey54 ]
5
(3.73)
In the GA, each unknown variable (gene) belongs to a specified range. There are two types
of genes: linear and exponential. For linear genes, the GA operates on their actual value. For
exponential genes, the GA operates on the logarithm of the gene, which allows the search to
cover a wider range. Different from least-square method, the results obtained from GA may
not be the best mathematic solution. We should run GA several times to make sure every gene
can converge to a certain value and doesn’t approach any bound of the specified range. More
details about the GA implementation can be found in [Aliprantis et al. (2005a)].
3.3
3.3.1
Fitting Results
Fitting Results of Proposed Model
In the proposed model, the following 10 unknown parameters need to be identified: stator
and rotor dc resistances rs and rr , stator and rotor leakage inductances Lls and Llr , stator
resistance due to eddy current loss Res , stator and rotor self-inductance Lms and Lmr , statorto-rotor mutual-inductance Lsr , stator and rotor winding-to-ground capacitances Cgs and Cgr .
Since Lmr and Lsr can be related to Lms by
2
Lmr = Lms × Nrs
(3.74)
Lsr = Lms × Nrs
(3.75)
and
where Nrs is already found in section 3.2.2, the number of unknowns reduces to 8.
34
Table 3.1
Variable
rs
rr
Lls
Llr
Res
Lms
Cgs
Cgr
Unit
Ω
Ω
H
H
Ω
H
F
F
List of GA genes and identified values
Gene Type
lin.
lin.
exp.
exp.
exp.
lin.
exp.
exp.
Min.
0.4
0.1
1 × 10−10
1 × 10−10
1
0.001
1 × 10−10
1 × 10−10
Max.
0.6
0.3
1
1
1 × 1010
0.1
1 × 10−5
1 × 10−5
Result
0.5457
0.1776
2.4279 × 10−3
9.7461 × 10−4
4.2422 × 102
0.03155
4.1778 × 10−9
5.4423 × 10−9
The GA settings and identified parameter values are shown in Table 3.1. The identified
stator and rotor resistances are very close to the values obtained in section 3.2.1. The identified
value for stator self-inductance Lms will not be further used in dynamic simulation, because it
is related to minor hysteresis loops that are traversed in SSFR small-signal tests. Instead, the
dynamic model will use (3.17) and (3.18). The fitted transfer function curves along with the
measured data are plotted in Fig. 3.11-Fig. 3.17.
It can be seen from figures that the proposed model fits well in the test frequency range.
The total error as defined in (3.73) is E = 0.014. In Fig. 3.13, data points at high frequency
(over 40 kHz) were noisy and thus were disregarded. According to (3.50) and (3.61), the real
part of the transfer functions must be negative. So this admittance must have an angle greater
than 90◦ . The measured data and fitted curves verified the transfer functions of the proposed
model and the relationships concluded in equations (3.59)-(3.65).
3.3.2
Verifications at Different Rotor Positions
To verify the correctness of the proposed model, same SSFR tests are repeated at randomly
different rotor positions. For instance, the fitted transfer function curves along with the measured data at θr = 30◦ are plotted in Fig. 3.18 - Fig. 3.24.
From the expressions of transfer functions in admittance matrix, we can see that parameter θr only appears in the transfer functions which are in second and third quarters of the
admittance matrix. The tests verified that the transfer functions in first and forth quarters
did not change when θr is changed. For elements in second and third quarters, the change of
35
MAGNITUDE (mho)
ANGLE (deg)
100
0
y11
10
0
−5
10
−100
100
0
y22
10
0
−5
10
−100
100
0
y33
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.11
frequency (Hz)
Measured data and fitted curves for diagonal elements of first quarter of Y using
the proposed model at θr = 0◦
MAGNITUDE (mho)
ANGLE (deg)
100
0
y44
10
0
−5
10
−100
100
0
y55
10
0
−5
10
−100
100
0
y66
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.12
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for diagonal elements of forth quarter of Y using
the proposed model at θr = 0◦
36
0
MAGNITUDE (mho)
y12
10
200
−2
10
100
−4
10
0
0
y13
10
200
−2
10
100
−4
10
0
0
y21
10
200
−2
10
100
−4
10
0
0
200
y23
10
100
−5
10
0
0
y31
10
200
−2
10
100
−4
10
0
0
y32
10
200
−2
10
100
−4
10
Figure 3.13
ANGLE (deg)
10 10 10 10 10 10 10 10
0 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
frequency (Hz)
−2
−1
0
1
2
3
4
5
Measured data and fitted curves for off-diagonal elements of first quarter of Y
using the proposed model at θr = 0◦
37
MAGNITUDE (mho)
ANGLE (deg)
200
0
y45
10
100
−5
10
0
200
0
y46
10
100
−5
10
0
200
0
y54
10
100
−5
10
0
200
0
y56
10
100
−5
10
0
200
0
y64
10
100
−5
10
0
200
0
y65
10
100
−5
10
Figure 3.14
10 10 10 10 10 10 10 10
0 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
frequency (Hz)
−2
−1
0
1
2
3
4
5
Measured data and fitted curves for off-diagonal elements of forth quarter of Y
using the proposed model at θr = 0◦
38
MAGNITUDE (mho)
ANGLE (deg)
200
0
y14
10
0
−5
10
−200
200
0
y25
10
0
−5
10
−200
200
0
y36
10
0
−5
10
−200
200
0
y41
10
0
−5
10
−200
200
0
y52
10
0
−5
10
−200
200
0
y63
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.15
−200 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for diagonal elements of second and third quarters
of Y using the proposed model at θr = 0◦
39
0
MAGNITUDE (mho)
y15
10
100
−2
10
0
−4
10
−100
0
y26
10
100
−2
10
0
−4
10
−100
0
y34
10
100
−2
10
0
−4
10
−100
0
y43
10
100
−2
10
0
−4
10
−100
0
y51
10
100
−2
10
0
−4
10
−100
0
y62
10
100
−2
10
0
−4
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.16
ANGLE (deg)
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for off-diagonal elements of second and third
quarters of Y using the proposed model at θr = 0◦
40
0
MAGNITUDE (mho)
y16
10
200
−2
10
0
−4
10
−200
0
y24
10
200
−2
10
0
−4
10
−200
0
y35
10
200
−2
10
0
−4
10
−200
0
y42
10
200
−2
10
0
−4
10
−200
0
y53
10
200
−2
10
0
−4
10
−200
0
y61
10
200
−2
10
0
−4
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.17
ANGLE (deg)
−200 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for off-diagonal elements of second and third
quarters of Y using the proposed model at θr = 0◦ (Cont.)
41
MAGNITUDE (mho)
ANGLE (deg)
100
0
y11
10
0
−5
10
−100
100
0
y22
10
0
−5
10
−100
100
0
y33
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.18
frequency (Hz)
Measured data and fitted curves for diagonal elements of first quarter of Y using
the proposed model at θr = 30◦
position only changes the magnitude by scale and phase by signs. It does not affect the shape
of curves along with frequency. As for the measured data and fitted curves plotted in Fig. 3.24,
the theoretical expressions equal zero when θr is set to 30 degree. So there is no fitted curves
in magnitude because the figure is plotted in logarithmic scale. The measured data with small
values are caused by noises and slight angular misalignment.
3.3.3
Comparison to Classical Model
To see the advantages of the proposed model, the classical induction machine model is used
to fit the SSFR test data. We used the same values of stator and rotor resistances, stator and
rotor leakage inductances, and magnetizing inductance as identified using GA in the proposed
model. The results are plotted in Fig. 3.25 - Fig. 3.31.
It can be observed that the classical model fits well at low frequency range, but it cannot
capture the high-frequency phenomena. The differences at high frequency are circled in the
figures. The mismatch is easier to see in phase. There is a obvious trend to go capacitive in
Fig. 3.25 and Fig. 3.26. In other figures, there is a trend from inductive to resistive at high
42
MAGNITUDE (mho)
ANGLE (deg)
100
0
y44
10
0
−5
10
−100
100
0
y55
10
0
−5
10
−100
100
0
y66
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.19
frequency (Hz)
Measured data and fitted curves for diagonal elements of forth quarter of Y using
the proposed model at θr = 30◦
frequency.
By adding the resistor parallel connected with the stator leakage inductor, the model can
capture the high-frequency phenomena in all off-diagonal elements of the admittance matrix.
Diagonal elements of the admittance matrix are improved at the same time, but they are not
perfect yet. The results are shown in Fig. 3.32 - Fig. 3.38. All mismatches between measured
data and fitted curves are circled out.
By further adding the capacitor between stator terminal and frame ground, the model can
well capture the high-frequency phenomena in y11 , y22 , and y33 . All other admittances are not
affected by Cgs at the same time. The results are shown in Fig. 3.39 - Fig. 3.45. All mismatches
between measured data and fitted curves are circled out.
By adding the capacitor between rotor terminal and frame ground along with Res , the
model can capture the high-frequency phenomena in y44 , y55 , and y66 . All other admittances
are not affected by Cgr at the same time. The results are shown in Fig. 3.46 - Fig. 3.52. All
mismatches between measured data and fitted curves are circled out.
43
0
MAGNITUDE (mho)
y12
10
200
−2
10
100
−4
10
0
0
y13
10
200
−2
10
100
−4
10
0
0
y21
10
200
−2
10
100
−4
10
0
0
y23
10
200
−2
10
100
−4
10
0
0
y31
10
200
−2
10
100
−4
10
0
0
y32
10
200
−2
10
100
−4
10
Figure 3.20
ANGLE (deg)
10 10 10 10 10 10 10 10
0 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
frequency (Hz)
−2
−1
0
1
2
3
4
5
Measured data and fitted curves for off-diagonal elements of first quarter of Y
using the proposed model at θr = 30◦
44
MAGNITUDE (mho)
ANGLE (deg)
200
0
y45
10
100
−5
10
0
200
0
y46
10
100
−5
10
0
200
0
y54
10
100
−5
10
0
200
0
y56
10
100
−5
10
0
200
0
y64
10
100
−5
10
0
200
0
y65
10
100
−5
10
Figure 3.21
10 10 10 10 10 10 10 10
0 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
frequency (Hz)
−2
−1
0
1
2
3
4
5
Measured data and fitted curves for off-diagonal elements of forth quarter of Y
using the proposed model at θr = 30◦
45
MAGNITUDE (mho)
ANGLE (deg)
200
0
y14
10
0
−5
10
−200
200
0
y25
10
0
−5
10
−200
200
0
y36
10
0
−5
10
−200
200
0
y41
10
0
−5
10
−200
200
0
y52
10
0
−5
10
−200
200
0
y63
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.22
−200 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for diagonal elements of second and third quarters
of Y using the proposed model at θr = 30◦
46
0
MAGNITUDE (mho)
y15
10
100
−2
10
0
−4
10
−100
0
y26
10
100
−2
10
0
−4
10
−100
0
y34
10
100
−2
10
0
−4
10
−100
0
y43
10
100
−2
10
0
−4
10
−100
0
y51
10
100
−2
10
0
−4
10
−100
0
y62
10
100
−2
10
0
−4
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.23
ANGLE (deg)
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for off-diagonal elements of second and third
quarters of Y using the proposed model at θr = 30◦
47
MAGNITUDE (mho)
−2
y16
10
200
−4
10
0
−6
10
−200
−2
y24
10
200
−4
10
0
−6
10
−200
−2
y35
10
200
−4
10
0
−6
10
−200
−2
y42
10
200
−4
10
0
−6
10
−200
−2
y53
10
200
−4
10
0
−6
10
−200
−2
y61
10
200
−4
10
0
−6
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.24
ANGLE (deg)
−200 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for off-diagonal elements of second and third
quarters of Y using the proposed model at θr = 30◦ (Cont.)
48
MAGNITUDE (mho)
ANGLE (deg)
100
0
y11
10
0
−5
10
−100
100
0
y22
10
0
−5
10
−100
100
0
y33
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.25
frequency (Hz)
Measured data and fitted curves for diagonal elements of first quarter of Y using
the classical model at θr = 0◦
MAGNITUDE (mho)
ANGLE (deg)
0
0
y44
10
−50
−5
10
−100
0
0
y55
10
−50
−5
10
−100
0
0
y66
10
−50
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.26
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for diagonal elements of forth quarter of Y using
the classical model at θr = 0◦
49
0
MAGNITUDE (mho)
200
y12
10
100
−5
10
0
0
200
y13
10
100
−5
10
0
0
200
y21
10
100
−5
10
0
0
200
y23
10
100
−5
10
0
0
200
y31
10
100
−5
10
0
0
200
y32
10
100
−5
10
Figure 3.27
ANGLE (deg)
10 10 10 10 10 10 10 10
0 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
frequency (Hz)
−2
−1
0
1
2
3
4
5
Measured data and fitted curves for off-diagonal elements of first quarter of Y
using the classical model at θr = 0◦
50
MAGNITUDE (mho)
ANGLE (deg)
200
0
y45
10
100
−5
10
0
200
0
y46
10
100
−5
10
0
200
0
y54
10
100
−5
10
0
200
0
y56
10
100
−5
10
0
200
0
y64
10
100
−5
10
0
200
0
y65
10
100
−5
10
Figure 3.28
10 10 10 10 10 10 10 10
0 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
frequency (Hz)
−2
−1
0
1
2
3
4
5
Measured data and fitted curves for off-diagonal elements of forth quarter of Y
using the classical model at θr = 0◦
51
MAGNITUDE (mho)
ANGLE (deg)
200
0
y14
10
0
−5
10
−200
200
0
y25
10
0
−5
10
−200
200
0
y36
10
0
−5
10
−200
200
0
y41
10
0
−5
10
−200
200
0
y52
10
0
−5
10
−200
200
0
y63
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.29
−200 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for diagonal elements of second and third quarters
of Y using the classical model at θr = 0◦
52
0
MAGNITUDE (mho)
y15
10
100
−2
10
0
−4
10
−100
0
y26
10
100
−2
10
0
−4
10
−100
0
y34
10
100
−2
10
0
−4
10
−100
0
y43
10
100
−2
10
0
−4
10
−100
0
y51
10
100
−2
10
0
−4
10
−100
0
y62
10
100
−2
10
0
−4
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.30
ANGLE (deg)
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for off-diagonal elements of second and third
quarters of Y using the classical model at θr = 0◦
53
0
MAGNITUDE (mho)
y16
10
200
−2
10
0
−4
10
−200
0
y24
10
200
−2
10
0
−4
10
−200
0
y35
10
200
−2
10
0
−4
10
−200
0
y42
10
200
−2
10
0
−4
10
−200
0
y53
10
200
−2
10
0
−4
10
−200
0
y61
10
200
−2
10
0
−4
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.31
ANGLE (deg)
−200 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for off-diagonal elements of second and third
quarters of Y using the classical model at θr = 0◦ (Cont.)
54
MAGNITUDE (mho)
ANGLE (deg)
100
0
y11
10
0
−5
10
−100
100
0
y22
10
0
−5
10
−100
100
0
y33
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.32
frequency (Hz)
Measured data and fitted curves for diagonal elements of first quarter of Y using
the classical model plus Res at θr = 0◦
MAGNITUDE (mho)
ANGLE (deg)
0
0
y44
10
−50
−5
10
−100
0
0
y55
10
−50
−5
10
−100
0
0
y66
10
−50
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.33
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for diagonal elements of forth quarter of Y using
the classical model plus Res at θr = 0◦
55
0
MAGNITUDE (mho)
y12
10
200
−2
10
100
−4
10
0
0
y13
10
200
−2
10
100
−4
10
0
0
y21
10
200
−2
10
100
−4
10
0
0
200
y23
10
100
−5
10
0
0
y31
10
200
−2
10
100
−4
10
0
0
y32
10
200
−2
10
100
−4
10
Figure 3.34
ANGLE (deg)
10 10 10 10 10 10 10 10
0 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
frequency (Hz)
−2
−1
0
1
2
3
4
5
Measured data and fitted curves for off-diagonal elements of first quarter of Y
using the classical model plus Res at θr = 0◦
56
MAGNITUDE (mho)
ANGLE (deg)
200
0
y45
10
100
−5
10
0
200
0
y46
10
100
−5
10
0
200
0
y54
10
100
−5
10
0
200
0
y56
10
100
−5
10
0
200
0
y64
10
100
−5
10
0
200
0
y65
10
100
−5
10
Figure 3.35
10 10 10 10 10 10 10 10
0 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
frequency (Hz)
−2
−1
0
1
2
3
4
5
Measured data and fitted curves for off-diagonal elements of forth quarter of Y
using the classical model plus Res at θr = 0◦
57
MAGNITUDE (mho)
ANGLE (deg)
200
0
y14
10
0
−5
10
−200
200
0
y25
10
0
−5
10
−200
200
0
y36
10
0
−5
10
−200
200
0
y41
10
0
−5
10
−200
200
0
y52
10
0
−5
10
−200
200
0
y63
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.36
−200 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for diagonal elements of second and third quarters
of Y using the classical model plus Res at θr = 0◦
58
0
MAGNITUDE (mho)
y15
10
100
−2
10
0
−4
10
−100
0
y26
10
100
−2
10
0
−4
10
−100
0
y34
10
100
−2
10
0
−4
10
−100
0
y43
10
100
−2
10
0
−4
10
−100
0
y51
10
100
−2
10
0
−4
10
−100
0
y62
10
100
−2
10
0
−4
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.37
ANGLE (deg)
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for off-diagonal elements of second and third
quarters of Y using the classical model plus Res at θr = 0◦
59
0
MAGNITUDE (mho)
y16
10
200
−2
10
0
−4
10
−200
0
y24
10
200
−2
10
0
−4
10
−200
0
y35
10
200
−2
10
0
−4
10
−200
0
y42
10
200
−2
10
0
−4
10
−200
0
y53
10
200
−2
10
0
−4
10
−200
0
y61
10
200
−2
10
0
−4
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.38
ANGLE (deg)
−200 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for off-diagonal elements of second and third
quarters of Y using the classical model plus Res at θr = 0◦ (Cont.)
60
MAGNITUDE (mho)
ANGLE (deg)
100
0
y11
10
0
−5
10
−100
100
0
y22
10
0
−5
10
−100
100
0
y33
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.39
frequency (Hz)
Measured data and fitted curves for diagonal elements of first quarter of Y using
the classical model plus Res and Cgs at θr = 0◦
MAGNITUDE (mho)
ANGLE (deg)
0
0
y44
10
−50
−5
10
−100
0
0
y55
10
−50
−5
10
−100
0
0
y66
10
−50
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.40
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for diagonal elements of forth quarter of Y using
the classical model plus Res and Cgs at θr = 0◦
61
0
MAGNITUDE (mho)
y12
10
200
−2
10
100
−4
10
0
0
y13
10
200
−2
10
100
−4
10
0
0
y21
10
200
−2
10
100
−4
10
0
0
200
y23
10
100
−5
10
0
0
y31
10
200
−2
10
100
−4
10
0
0
y32
10
200
−2
10
100
−4
10
Figure 3.41
ANGLE (deg)
10 10 10 10 10 10 10 10
0 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
frequency (Hz)
−2
−1
0
1
2
3
4
5
Measured data and fitted curves for off-diagonal elements of first quarter of Y
using the classical model plus Res and Cgs at θr = 0◦
62
MAGNITUDE (mho)
ANGLE (deg)
200
0
y45
10
100
−5
10
0
200
0
y46
10
100
−5
10
0
200
0
y54
10
100
−5
10
0
200
0
y56
10
100
−5
10
0
200
0
y64
10
100
−5
10
0
200
0
y65
10
100
−5
10
Figure 3.42
10 10 10 10 10 10 10 10
0 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
frequency (Hz)
−2
−1
0
1
2
3
4
5
Measured data and fitted curves for off-diagonal elements of forth quarter of Y
using the classical model plus Res and Cgs at θr = 0◦
63
MAGNITUDE (mho)
ANGLE (deg)
200
0
y14
10
0
−5
10
−200
200
0
y25
10
0
−5
10
−200
200
0
y36
10
0
−5
10
−200
200
0
y41
10
0
−5
10
−200
200
0
y52
10
0
−5
10
−200
200
0
y63
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.43
−200 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for diagonal elements of second and third quarters
of Y using the classical model plus Res and Cgs at θr = 0◦
64
0
MAGNITUDE (mho)
y15
10
100
−2
10
0
−4
10
−100
0
y26
10
100
−2
10
0
−4
10
−100
0
y34
10
100
−2
10
0
−4
10
−100
0
y43
10
100
−2
10
0
−4
10
−100
0
y51
10
100
−2
10
0
−4
10
−100
0
y62
10
100
−2
10
0
−4
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.44
ANGLE (deg)
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for off-diagonal elements of second and third
quarters of Y using the classical model plus Res and Cgs at θr = 0◦
65
0
MAGNITUDE (mho)
y16
10
200
−2
10
0
−4
10
−200
0
y24
10
200
−2
10
0
−4
10
−200
0
y35
10
200
−2
10
0
−4
10
−200
0
y42
10
200
−2
10
0
−4
10
−200
0
y53
10
200
−2
10
0
−4
10
−200
0
y61
10
200
−2
10
0
−4
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.45
ANGLE (deg)
−200 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for off-diagonal elements of second and third
quarters of Y using the classical model plus Res and Cgs at θr = 0◦ (Cont.)
66
MAGNITUDE (mho)
ANGLE (deg)
100
0
y11
10
0
−5
10
−100
100
0
y22
10
0
−5
10
−100
100
0
y33
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.46
frequency (Hz)
Measured data and fitted curves for diagonal elements of first quarter of Y using
the classical model plus Res and Cgr at θr = 0◦
MAGNITUDE (mho)
ANGLE (deg)
100
0
y44
10
0
−5
10
−100
100
0
y55
10
0
−5
10
−100
100
0
y66
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.47
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for diagonal elements of forth quarter of Y using
the classical model plus Res and Cgr at θr = 0◦
67
0
MAGNITUDE (mho)
y12
10
200
−2
10
100
−4
10
0
0
y13
10
200
−2
10
100
−4
10
0
0
y21
10
200
−2
10
100
−4
10
0
0
200
y23
10
100
−5
10
0
0
y31
10
200
−2
10
100
−4
10
0
0
y32
10
200
−2
10
100
−4
10
Figure 3.48
ANGLE (deg)
10 10 10 10 10 10 10 10
0 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
frequency (Hz)
−2
−1
0
1
2
3
4
5
Measured data and fitted curves for off-diagonal elements of first quarter of Y
using the classical model plus Res and Cgr at θr = 0◦
68
MAGNITUDE (mho)
ANGLE (deg)
200
0
y45
10
100
−5
10
0
200
0
y46
10
100
−5
10
0
200
0
y54
10
100
−5
10
0
200
0
y56
10
100
−5
10
0
200
0
y64
10
100
−5
10
0
200
0
y65
10
100
−5
10
Figure 3.49
10 10 10 10 10 10 10 10
0 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
frequency (Hz)
−2
−1
0
1
2
3
4
5
Measured data and fitted curves for off-diagonal elements of forth quarter of Y
using the classical model plus Res and Cgr at θr = 0◦
69
MAGNITUDE (mho)
ANGLE (deg)
200
0
y14
10
0
−5
10
−200
200
0
y25
10
0
−5
10
−200
200
0
y36
10
0
−5
10
−200
200
0
y41
10
0
−5
10
−200
200
0
y52
10
0
−5
10
−200
200
0
y63
10
0
−5
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.50
−200 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for diagonal elements of second and third quarters
of Y using the classical model plus Res and Cgr at θr = 0◦
70
0
MAGNITUDE (mho)
y15
10
100
−2
10
0
−4
10
−100
0
y26
10
100
−2
10
0
−4
10
−100
0
y34
10
100
−2
10
0
−4
10
−100
0
y43
10
100
−2
10
0
−4
10
−100
0
y51
10
100
−2
10
0
−4
10
−100
0
y62
10
100
−2
10
0
−4
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.51
ANGLE (deg)
−100 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for off-diagonal elements of second and third
quarters of Y using the classical model plus Res and Cgr at θr = 0◦
71
0
MAGNITUDE (mho)
y16
10
200
−2
10
0
−4
10
−200
0
y24
10
200
−2
10
0
−4
10
−200
0
y35
10
200
−2
10
0
−4
10
−200
0
y42
10
200
−2
10
0
−4
10
−200
0
y53
10
200
−2
10
0
−4
10
−200
0
y61
10
200
−2
10
0
−4
10
−2
−1
0
1
2
3
4
5
10 10 10 10 10 10 10 10
frequency (Hz)
Figure 3.52
ANGLE (deg)
−200 −2 −1 0 1 2 3 4 5
10 10 10 10 10 10 10 10
frequency (Hz)
Measured data and fitted curves for off-diagonal elements of second and third
quarters of Y using the classical model plus Res and Cgr at θr = 0◦ (Cont.)
72
Res
iqs
rs
ωλld
Lls
−
+
+
ωλmd
+
−
vqs
(ω − ωr )λ′dr
−
L′lr
rr′
i′qr
+
+
′
vqr
LM
−
−
Res
ids
rs
ωλlq
Lls
−
+
ωλmq
+
−
vds
+
(ω − ωr )λ′qr
+
−
LM
−
rr′
i′dr
+
′
vdr
−
Figure 3.53
3.4
3.4.1
L′lr
Proposed qd model
Dynamic Simulation
Simulation Model
Dynamic simulation is implemented using Matlab/Simulink and the Automated State Model
Generator (ASMG) toolbox [PC Krause and Associates (2014)]. For the simulations, the machine is connected as ∆-Y. This means that the stator windings shown in figure Fig. 3.4 are
now connected in ∆. This does not affect the parameters identified in the previous section. In
order to take main-flux saturation into account, the portion of the proposed model within the
parasitic capacitors and iron resistors is transferred to the arbitrary reference frame Krause
et al. (2002). The detailed qd -axes model is shown in Fig. 3.53.
73
The state space equations of the proposed qd model are
d
dt iqls
d
dt idls
d 0
dt iqr
1
[(iqs − iqls )Res − ωidls Lls ]
Lls
1
[(ids − idls )Res + ωiqls Lls ]
=
Lls
1
= − 0 [vqs − rs iqs − (iqs − iqls ) Res − ωλmd
Llr
=
0
+ (ω − ωr ) λ0dr + rr0 i0qr − vqr
d 0
dt idr
=−
i
(3.76)
(3.77)
(3.78)
1
[vds − rs ids − (ids − idls ) Res + ωλmq
L0lr
0
− (ω − ωr ) λ0qr + rr0 i0dr − vdr
i
(3.79)
d
dt λmq
= vqs − rs iqs − (iqs − iqls ) Res − ωλmd
(3.80)
d
dt λmd
= vds − rs ids − (ids − idls ) Res + ωλmq
(3.81)
using the following algebraic equations
λ0qr = λmq + i0qr L0lr
(3.82)
λ0dr = λmd + i0dr L0lr
(3.83)
iqs = imq − i0qr
(3.84)
ids = imd − i0dr
(3.85)
The magnetizing currents depend on the magnetizing fluxes, based on the saturation equations that have been presented in section 3.2.2. The electromagnetic torque can be expressed
as [Aliprantis et al. (2005b)]
Te =
3P
(iqs λmd − ids λmq )
22
(3.86)
where P is the number of poles. The mechanical dynamics are modeled by [Krause et al. (2002)]
d
dt ωr
=
P
(Te − Tm )
2J
(3.87)
where J is the moment of inertia, and Tm is the mechanical torque.
Since the qd -axes model operates using the transformed line-to-neutral voltages, and is
agnostic with respect to line-to-ground voltages, appropriate interfacing with the abc model is
necessary. The rotor is connected in Y . It can be assumed that the line-to-neutral voltages
74
sum to zero (var + vbr + vcr = 0) because the rotor currents add to zero. we can derive the
line-to-line voltages from line-to-ground voltages by




vagr 
vabr 
 1
 



 



 vbcr  = A  vbgr  =  0

 


 



Then there exist




var 
vabr 
0  vagr 
1


−1
  vbgr 
0
1
1 −1




vcgr
(3.88)

0  var 






(3.89)
 
−1
  vbr 
1
1
vcr




  


  


 vbcr  = B  vbr  = 0
  


  


0
−1
−1
vcgr
vcar



1
1
vcr
Hence,




 2
vabr 
var 


 

1


 

 vbr  = B−1  vbcr  = −1


 
3


 

1
1 vabr 
1


1
  vbcr 
−1 −2 1
0
vcr









0
(3.90)
All rotor variables are referred to stator side through the turns ratio Nrs . Then the qd0
transformation is applied [Krause et al. (2002)].



0
vqr 


0
var 
cos β
 
 
2
 0 
 0 

vdr  = Kr  vbr  =  sin β
 
 
3
 
 

0
v0r

0
vcr

cos β −
2π
3
2π
3
sin β −
1
2

0
iar 

1
2
2π
3
cos β
i00r
cos β +
sin β −
2π
3
sin β +








1 i0qr 
sin β

 
  
 0 

−1  0 
 ibr  = Kr idr  = cos β − 2π
3
 
  
  
 
(3.91)
0
vcr
1
2

0
iqr 
i0cr
  
0
cos β +
 var 
 

 0 
 
sin β + 2π
3   vbr 
 
2π
3
2π
3
0 
1
 idr 
1
i00r
(3.92)
where β = θ − θr .
The stator is connected in ∆. So the stator voltages and current injections into the qd -axes
model have following relationships:





vags   1
vabs 
 






 
 vbcs  = A  vbgs  =  0
 






 
vcas
vcgs
−1


−1
0  vags 
1


−1
  vbgs 
0
1






vcgs
(3.93)
75




iabs   1
ias 
 

 
 

 
 ibs  = C  ibcs  = −1
 

 
 

 
0
−1 iabs 
1


0
  ibcs 
−1
0
icas
ics



1






icas
(3.94)
Applying transformation [Krause et al. (2002)],





vabs 
vqs 
cos θ


 

2


 

vds  = Ks  vbcs  =  sin θ


 

3


 



cos θ −
2π
3
2π
3
sin θ −
1
2
vcas
v0s


1
2

 
cos θ +
 vabs 





2π 


sin θ + 3   vbcs 



2π
3
1
2

icas
i0s
vcas


cos θ
sin θ
iqs  
iabs 
  







 ibcs  = Ks −1 ids  = cos θ − 2π
sin θ − 2π
3
3
  


  


cos θ +
2π
3
sin θ +
2π
3
(3.95)
1 iqs 






 
1
 ids 
1
i0s
(3.96)
In the proposed model, all 0-axis variables are assumed to be zero. The proposed model (with
ideal capacitors in the ground path) can capture features well in the frequency range up to
approximately 50 kHz. However, higher-frequency harmonics that are present in the dynamic
simulation cause unrealistically high current ripples in the simulation of the drive system. To
eliminate these, a series resistor is added in the ground path. Its value is assumed to be 30 Ω.
The dynamic equations of the parasitic capacitors are
d
dt vcgas
=
d
dt vcgbs
=
d
dt vcgcs
=
d
dt vcgar
=
d
dt vcgbr
=
d
dt vcgcr
=
1
Rgs Cgs
1
Rgs Cgs
1
Rgs Cgs
1
Rgr Cgr
1
Rgr Cgr
1
Rgr Cgr
(vags − vcgas )
(3.97)
(vbgs − vcgbs )
(3.98)
(vcgs − vcgcs )
(3.99)
(vagr − vcgar )
(3.100)
(vbgr − vcgbr )
(3.101)
(vcgr − vcgcr )
(3.102)
76
Then the currents flowing through the parasitic capacitors can be expressed as
iags =
ibgs =
icgs =
iagr =
ibgr =
icgr =
1
Rgs
1
Rgs
1
Rgs
1
Rgr
1
Rgr
1
Rgr
(vags − vcgas )
(3.103)
(vbgs − vcgbs )
(3.104)
(vcgs − vcgcs )
(3.105)
(vagr − vcgar )
(3.106)
(vbgr − vcgbr )
(3.107)
(vcgr − vcgcr )
(3.108)
The total line currents in the cables are
ias c = ias + iags
(3.109)
ibs c = ibs + ibgs
(3.110)
ics c = ics + icgs
(3.111)
iar c = iar + iagr
(3.112)
ibr c = ibr + ibgr
(3.113)
icr c = icr + icgr
(3.114)
A simulation block diagram is shown in Fig. 3.54.
3.4.2
Steady State Study
A steady-state simulation is run first to highlight the capability of the proposed model to
predict high-frequency effects. The rotor is connected to a power electronic converter, using
the high-frequency model described in [Zhong and Lipo (1995)]. The stator is connected to
a three-phase voltage source with rated line-to-line rms value of 208 V. The proposed model
is compared to the classical model. The parameters of the classical model are the same as
those in the proposed model, except the magnetizing inductance whose value is derived using
a standard test procedure. In this study, the machine is spinning at a constant speed of
1800 rpm. A control system is regulating electromagnetic torque and stator reactive power.
77
(3.97)-(3.99)
(3.103)-(3.105)
iabcgs
vabcgs
iabcs c
iabcs
+
′
vqd0r
vqd0s
vabs /vbcs /vcas
Ks
A
iabs /ibcs /icas
(Ks )−1
C
iqd0s
(3.9)-(3.12)
(3.17)-(3.18)
(3.76)-(3.87)
Kr
i′qd0r
(Kr )−1
′
vabcr
i′abcr
ωr
TL
ω
(3.100)-(3.102)
(3.106)-(3.108)
iabcgr
iabcr
c
+
iabcr
Nrs
vabcgr
vabr /vbcr /vcar
A
Figure 3.54
(B)−1
vabcr
Nrs
Simulation block diagram
The same torque (45 N-m) and reactive power (0 VAr) commands are used for both models. The
R
simulation is implemented in MATLAB/Simulink
using an ode23s solver. The maximum step
size is limited to 0.1 ms to ensure all high-frequency transients can be captured. The simulation
time to run the classical model for two seconds is approximately 5 minutes in real time. In
contrast, the simulation time to run the proposed model for one second is about 22 minutes in
real time.
The rotor currents are plotted in Figs. 3.55 and 3.56. In Fig. 3.55, we can observe the
appearance of high-frequency current ripples that are not predicted by the classical model.
It should be noted that the amplitude of this CM ripple is sensitive to the value of iron
resistance Rgr . Fig. 3.56 shows the waveform of rotor currents at different time scale. The
stator currents are shown in Figs. 3.57 containing low-frequency harmonics, but they can be
captured adequately by both models. This implies that high-frequency currents do not travel
through the air gap, and that CM currents are flowing through the ground path on the rotor
50
50
40
40
30
20
10
0
−10
−20
−30
iar
−40
ibr
Rotor line currents (A)
Rotor line currents (A)
78
30
20
10
0
−10
−20
iar
−30
i
br
−40
icr
−50
1.7
1.702
1.704
1.706
1.708
icr
−50
0.7
1.71
0.701
Time (s)
0.704
Comparison of rotor line currents at steady state
50
50
40
40
30
20
10
0
−10
−20
−30
iar
−40
ibr
30
20
10
0
−10
−20
−30
iar
−40
ibr
icr
−50
1.7
1.71
1.72
1.73
1.74
1.75
Time (s)
icr
−50
0.7
0.71
0.72
0.73
0.74
0.75
Time (s)
(a) Classical model
Figure 3.56
0.705
(b) Proposed model
Rotor line currents (A)
Rotor line currents (A)
0.703
Time (s)
(a) Classical model
Figure 3.55
0.702
(b) Proposed model
Comparison of rotor line currents at steady state
side where the converter is connected. The torques from two models are almost the same in
Fig. 3.58. Since there exists only one ground path between the winding terminal and the frame,
and under the assumption that neutral points and the ground are isolated, CM current cannot
flow in the winding itself. Hence, the electromagnetic torque is not affected by high-frequency
CM components (in the proposed model).
79
30
Stator line currents (A)
Stator line currents (A)
30
20
10
0
−10
ias
−20
ibs
20
10
0
−10
ias
−20
ibs
ics
−30
1.7
1.71
1.72
1.73
1.74
ics
−30
0.7
1.75
0.71
Time (s)
Electromagnetic torque (N.m.)
Electromagnetic torque (N.m.)
48
47
46
45
44
43
42
41
1.994
1.996
1.998
2
50
49
48
47
46
45
44
43
42
41
40
0.99
0.992
Time (s)
(a) Classical model
Figure 3.58
3.4.3
0.75
Comparison of stator line currents at steady state
49
1.992
0.74
(b) Proposed model
50
40
1.99
0.73
Time (s)
(a) Classical model
Figure 3.57
0.72
0.994
0.996
0.998
1
Time (s)
(b) Proposed model
Comparison of electromagnetic torque at steady state
Dynamic Study
To observe the electromechanical dynamics of the proposed model, a no-load startup test
is simulated with an inertia value of J = 0.15 kg-m2 . The stator is connected to the grid with
rated line-to-line rms value of 208 V, and the rotor is connected to a three-phase Y-connected
resistor bank with R = 4.9 Ω. The results are shown in Fig. 3.59. The torque converged to
zero after t = 2 s. Two models have markedly different dynamic performance. This is mainly
due to the varying magnetizing inductance in the proposed model.
The rotor speed dynamic performance of two models are shown in Fig. 3.60. They are
100
Electromagnetic torque (N.m.)
Electromagnetic torque (N.m.)
80
90
80
70
60
50
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
100
90
80
70
60
50
40
30
20
10
0
0
0.1
0.2
Time (s)
(a) Classical model
0.5
Comparison of electromagnetic torque during a no-load startup
1400
1400
1200
1200
1000
800
600
400
200
0
0
0.4
(b) Proposed model
Rotor speed (rpm)
Rotor speed (rpm)
Figure 3.59
0.3
Time (s)
1000
800
600
400
200
0.5
1
1.5
Time (s)
(a) Classical model
Figure 3.60
2
0
0
0.5
1
Time (s)
(b) Proposed model
Comparison of rotor speed during a no-load startup
slightly different before t = 0.2 s.
1.5
2
81
CHAPTER 4.
CONCLUSION AND FUTURE WORK
A new model and corresponding experimental parameterization procedure for a wound-rotor
induction machine with magnetizing saturation and high-frequency effects has been introduced.
The model parameters are obtained with a combination of low- and high-frequency tests at
standstill. A new method is adopted to implement the SSFR tests. The machine’s turns ratio
was also derived during the measurement of the main flux path magnetization characteristic.
The dynamic machine model is built to study both dynamic and steady state performance of
the machine with incorporation of power electronics.
A possible improvement to the model might be to allow the leakage inductances to vary,
for capturing the saturation of the leakage flux paths. Alternatively, circuit elements could be
added (e.g., capacitors to model turn-to-turn parasitics), to better match the observed behavior
of a machine under test. The main challenge in developing such integrated, wide-bandwidth
models is to maintain the physical significance of the magnetizing path, so that electromagnetic
torque can be predicted. Moreover, in order to model accurately the CM impedance, it is
important to conduct the SSFR tests with an instrument that can measure the input impedance
at frequencies up to several hundreds of kHz. Further experiments are needed to verify the
accuracy of the proposed model under a range of operating conditions. Validation of the model
is challenging since current probes that can measure high-amplitude/high-frequency currents
(e.g., higher than 100 kHz) are not readily available. These aspects are part of ongoing and
future work.
82
BIBLIOGRAPHY
Aliprantis, D. C., Sudhoff, S. D., and Kuhn, B. T. (2005a). Experimental characterization
procedure for a synchronous machine model with saturation and arbitrary rotor network
representation. IEEE Trans. Energy Convers., 20(3):595–603.
Aliprantis, D. C., Sudhoff, S. D., and Kuhn, B. T. (2005b). A synchronous machine model
with saturation and arbitrary rotor network representation. IEEE Trans. Energy Convers.,
20(3):584–594.
Balabanian, N. and Bickart, T. A. (1981). Linear network theory: analysis, properties, design
and synthesis. Matrix series in circuits and systems. Matrix.
Boglietti, A. and Carpaneto, E. (1999). Induction motor high frequency model. In Proc. IAS
Ann. Meet., volume 3, pages 1551–1558. IEEE.
Boglietti, A. and Carpaneto, E. (2001). An accurate induction motor high-frequency model for
electromagnetic compatibility analysis. Electr. Power Comp. Syst., 29(3):191–209.
Boglietti, A., Cavagnino, A., and Lazzari, M. (2007). Experimental high-frequency parameter
identification of AC electrical motors. IEEE Trans. Ind. Appl., 43(1):23–29.
Boucenna, N., Hlioui, S., Revol, B., and Costa, F. (2012). Modeling of the propagation of
high-frequency currents in AC motors. In Proc. Intern. Symp. Electro. Comp., pages 1–6.
Consoli, A., Oriti, G., Testa, A., and Julian, A. (1996). Induction motor modeling for common
mode and differential mode emission evaluation. In Proc. IAS Ann. Meet., volume 1, pages
595–599. IEEE.
83
Degano, M., Zanchetta, P., Clare, J., and Empringham, L. (2010). HF induction motor modeling using genetic algorithms and experimental impedance measurement. In Proc. Intern.
Symp. Industr. Electron., pages 1296–1301. IEEE.
Degano, M., Zanchetta, P., Empringham, L., Lavopa, E., and Clare, J. (2012). HF induction
motor modeling using automated experimental impedance measurement matching. IEEE
Trans. Ind. Electron., 59(10):3789–3796.
Grandi, G., Casadei, D., and Massarini, A. (1997a). High frequency lumped parameter model
for AC motor windings. In Proc. Euro. Conf. Power Electron. Appl., volume 2, pages 578–
583.
Grandi, G., Casadei, D., and Reggiani, U. (1997b). Equivalent circuit of mush wound AC
windings for high frequency analysis. In Proc. IEEE Intern. Symp. Ind. Electr., volume 1,
pages 201–206.
Grandi, G., Casadei, D., and Reggiani, U. (2004). Common- and differential-mode HF current
components in AC motors supplied by voltage source inverters. IEEE Trans. Power Electron.,
19(1):16–24.
Gubı́a-Villabona, E., Sanchis-Gúrpide, P., Alonso-Sádaba, Ó., Lumbreras-Azanza, A., and
Marroyo-Palomo, L. (2002). Simplified high-frequency model for AC drives. In Proc. Ind.
Electron. Society, Ann. Conf., volume 2, pages 1144–1149.
Guha, S. and Kar, N. C. (2006). A new method of modeling magnetic saturation in electrical
machines. In Proc. Canadian Conf. Electr. Comp. Eng., pages 1094–1097.
Henze, O., Cay, Z., Magdun, O., De Gersem, H., Weiland, T., and Binder, A. (2008). A stator
coil model for studying high-frequency effects in induction motors. In Proc. Intern. Symp.
Power Electron. Electr. Drives, Automation and Motion, pages 609–613.
Idir, N., Weens, Y., Moreau, M., and Franchaud, J. J. (2009). High-frequency behavior models
of AC motors. IEEE Trans. Magn., 45(1):133–138.
84
Krause, P. C., Wasynczuk, O., and Sudhoff, S. D. (2002). Analysis of electric machinery and
drive systems. IEEE Press, 2 edition.
Liserre, M., Cárdenas, R., Molinas, M., and Rodrı́guez, J. (2011). Overview of multi-MW wind
turbines and wind parks. IEEE Trans. Ind. Electron., 58(4):1081–1095.
Llaquet, J., Gonzalez, D., Aldabas, E., and Romeral, L. (2002). Improvements in high frequency
modelling of induction motors for diagnostics and EMI prediction. In Proc. Ind. Electron.
Society, Ann. Conf., volume 1, pages 68–71.
Magdun, O. and Binder, A. (2012). The high-frequency induction machine parameters and
their influence on the common mode stator ground current. In Proc. Intern. Conf. Electr.
Mach., pages 505–511.
Mäki-Ontto, P. and Luomi, J. (2005). Induction motor model for the analysis of capacitive and
induced shaft voltages. In Proc. Intern. Conf. Electr. Mach. Drives, pages 1653–1660. IEEE.
Mirafzal, B., Skibinski, G., and Tallam, R. (2009). Determination of parameters in the universal
induction motor model. IEEE Trans. Ind. Appl., 45(1):142–151.
Mirafzal, B., Skibinski, G., Tallam, R., Schlegel, D., and Lukaszewski, R. (2007). Universal
induction motor model with low-to-high frequency-response characteristics. IEEE Trans.
Ind. Appl., 43(5):1233–1246.
Mitova, R., Ghosh, R., Mhaskar, U., Klikic, D., Wang, M., and Dentella, A. (2014). Investigations of 600-V GaN HEMT and GaN diode for power converter applications. IEEE Trans.
Power Electron., 29(5):2441–2452.
Moreira, A., Lipo, T., Venkataramanan, G., and Bernet, S. (2002). High-frequency modeling
for cable and induction motor overvoltage studies in long cable drives. IEEE Trans. Ind.
Appl., 38(5):1297–1306.
Moreira, J. C. and Lipo, T. A. (1992). Modeling of saturated AC machines including air gap
flux harmonic components. IEEE Trans. Ind. Appl., 28(2):343–349.
85
Morita, T., Tamura, S., Anda, Y., Ishida, M., Uemoto, Y., Ueda, T., Tanaka, T., and Ueda,
D. (2011). 99.3% efficiency of three-phase inverter for motor drive using GaN-based gate
injection transistors. In Proc. Appl. Power Electron. Conf. Expos. (APEC), pages 481–484.
IEEE.
Müller, S., Deicke, M., and de Doncker, R. W. (2002). Doubly fed induction generator systems
for wind turbines. IEEE Ind. Appl. Mag., 8(3):26–33.
PC Krause and Associates (2014). Automated State Model Generator (version 1.1.4).
Ran, L., Gokani, S., Clare, J., Bradley, K., and Christopoulos, C. (1998a). Conducted electromagnetic emissions in induction motor drive systems. I. Time domain analysis and identification of dominant modes. IEEE Trans. Power Electron., 13(4):757–767.
Ran, L., Gokani, S., Clare, J., Bradley, K., and Christopoulos, C. (1998b). Conducted electromagnetic emissions in induction motor drive systems. II. frequency domain models. IEEE
Trans. Power Electron., 13(4):768–776.
Ranta, M., Hinkkanen, M., and Luomi, J. (2009). Rotor parameter identification of saturated
induction machines. In Proc. Energy Convers. Congr. Expos. (ECCE), pages 1524–1531.
IEEE.
Rodrı́guez, M., Zhang, Y., and Maksimović, D. (2014). High-frequency PWM Buck converters
using GaN-on-SiC HEMTs. IEEE Trans. Power Electron., 29(5):2462–2473.
Schinkel, M., Weber, S., Guttowski, S., John, W., and Reichl, H. (2006). Efficient HF modeling
and model parameterization of induction machines for time and frequency domain simulations. In Proc. Appl. Power Electron. Conf. Expos. (APEC), pages 1181–1186.
Seman, S., Saitz, J., and Arkkio, A. (2003). Dynamic model of cage induction motor considering
saturation and skin effect. In Proc. Intern. Conf. Electr. Mach. Syst., volume 2, pages 710–
713.
Smith, A. C., Healey, R. C., and Williamson, S. (1996). A transient induction motor model
including saturation and deep bar effect. IEEE Trans. Energy Convers., 11(1):8–15.
86
Sudhoff, S., Aliprantis, D., Kuhn, B., and Chapman, P. (2002). An induction machine model
for predicting inverter-machine interaction. IEEE Trans. Energy Convers., 17(2):203–210.
Wang, L., Ho, C., Canales, F., and Jatskevich, J. (2010). High-frequency modeling of the
long-cable-fed induction motor drive system using TLM approach for predicting overvoltage
transients. IEEE Trans. Power Electron., 25(10):2653–2664.
Weber, S., Hoene, E., Guttowski, S., John, W., and Reichl, H. (2004). Modeling induction
machines for EMC-analysis. In Proc. Power Electron. Spec., Ann. Conf., volume 1, pages
94–98.
Zhong, E. and Lipo, T. (1995). Improvements in EMC performance of inverter-fed motor drives.
IEEE Trans. Ind. Appl., 31(6):1247–1256.