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Dynamic Average-Value Modeling of the 120° VSI-Commutated
Brushless DC Motors with Non-Sinusoidal Back EMF
by
Kamran Tabarraee
B.Sc., The Amirkabir University of Technology, 2008
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF APPLIED SCIENCE
in
THE FACULTY OF GRADUATE STUDIES
(ELECTRICAL & COMPUTER ENGINEERING)
THE UNIVERSITY OF BRITISH COLUMBIA
(Vancouver)
June 2011
© Kamran Tabarraee, 2011
Abstract
For large and small signal analysis of electromechanical systems with power electronic
devices such as Brushless DC (BLDC) motor-inverter drives, average-value models (AVMs)
are indisputable. Average-value models are typically orders of magnitude faster than the
corresponding detailed models. This advantage makes AVMs ideal for representing motordrive components in system level studies. Derivation of accurate dynamic average-value
model of BLDC motor-drive system is generally challenging and requires careful averaging
of the stator phase voltages and currents over a prototypical switching interval (SI) to find the
corresponding average-value relationships for the state variables and the resulting
electromagnetic torque.
The so-called 120° voltage source inverter (VSI) driven brushless dc (BLDC) motors are
very common in many commercial and industrial applications. This thesis extends the
previous work and presents a new and improved dynamic average-value model (AVM) for
such BLDC motor-drive systems. The new model is explicit and uses a proper qd model of
the permanent magnet synchronous machine with non-sinusoidal rotor flux. The model
utilizes multiple reference frame theory to properly include the back EMF harmonics as well
as commutation and conduction intervals into the averaged voltage and torque relationships.
The commutation angle is readily obtained from the detailed simulation.
The proposed model is then demonstrated on two typical industrial BLDC motors with
differently-shaped back EMF waveforms (i.e. trapezoidal and close to sinusoidal). The
results of studies are compared with experimental measurements as well as previously
established state-of-the-art models, whereas the new model is shown to provide appreciable
improvement especially for machines with pronounced trapezoidal back EMF.
ii
Preface
A version of Chapter 2 has been published in the following manuscript: Kamran Tabarraee,
Jaishankar Iyer, Sina Chiniforoosh, and Juri Jatskevich, “Comparison of Brushless DC
Motors with Trapezoidal and Sinusoidal Back EMF,” In proc. IEEE Canadian Conference on
Electrical and Computer Engineering, May 2011, Niagara Falls, Canada. I developed the
models, performed the tests and wrote most of the manuscript, while the conducted research
was supervised by Dr. Juri Jatskevich, and revised and assisted by my supervisor and
Jaishankar Iyer, and Sina Chiniforoosh.
A version of Chapter 3 has also been published in the following manuscript: K. Tabarraee, J.
Iyer, and J. Jatskevich, “Average-Value Modeling of Brushless DC Motors with Trapezoidal
Back EMF,” In proc. IEEE International Symposium on Industrial Electronics, June 2011,
Gdansk, Poland. I developed the models, performed the tests and wrote most of the
manuscript, while the conducted research was supervised by Dr. Juri Jatskevich, and revised
and assisted by my supervisor and Jaishankar Iyer.
A version of Chapter 4 has been submitted for publication: K. Tabarraee, J. Iyer, H.
Atighechi and J. Jatskevich, “Dynamic Average-Value Modeling of 120° VSI-Commutated
Brushless DC Motors with Trapezoidal Back EMF”. I developed and implemented the
model, performed and designed the tests and wrote most of the manuscript. The research has
been supervised by Dr. Juri Jatskevich, and assisted by Jaishankar Iyer and Hamid Atighechi.
iii
Table of Contents
Abstract .................................................................................................................................... ii
Preface ..................................................................................................................................... iii
Table of Contents ................................................................................................................... iv
List of Figures ......................................................................................................................... vi
Acknowledgements .............................................................................................................. viii
Dedication ............................................................................................................................... ix
1
2
Introduction ..................................................................................................................... 1
1.1
Why Average-Value Modeling? ........................................................................................... 1
1.2
Literature Review.................................................................................................................. 3
1.3
Brushless DC Motor-Inverter System ................................................................................... 4
1.4
Contributions......................................................................................................................... 5
1.5
Thesis Composition............................................................................................................... 7
Detailed Modeling of Brushless DC Motors with Non-Sinusoidal Back EMF .......... 8
2.1
Model Description................................................................................................................. 8
2.2
Effect of Back EMF Harmonics in Detailed Model ............................................................ 13
2.2.1
Model Verification in Steady-State ................................................................................ 13
2.2.2
Model Verification in Transient ..................................................................................... 17
2.3
3
4
Case studies ......................................................................................................................... 18
2.3.1
ASMG vs. SIMPOWER Model ...................................................................................... 18
2.3.2
Torque-Speed Characteristic .......................................................................................... 19
Average-Value Modeling of Brushless DC Motors with Trapezoidal Back EMF .. 21
3.1
Model Description............................................................................................................... 21
3.2
Case Studies ........................................................................................................................ 29
3.2.1
Start-Up Transient .......................................................................................................... 30
3.2.2
Steady-State .................................................................................................................... 31
Dynamic Average-Value Modeling of 120° VSI-Commutated Brushless DC Motors
with Trapezoidal Back EMF ................................................................................................ 35
4.1
Model Description............................................................................................................... 35
iv
5
4.2
Model Implementation ........................................................................................................ 40
4.3
Case Studies ........................................................................................................................ 43
4.3.1
Steady-State .................................................................................................................... 44
4.3.2
Transient Response to Mechanical Load Change ........................................................... 47
4.3.3
Start-Up Transient .......................................................................................................... 48
4.3.4
Transient Response to Input Voltage Change................................................................. 50
Conclusion ..................................................................................................................... 52
5.1
Summary ............................................................................................................................. 52
5.2
Future Research Topics....................................................................................................... 53
References .............................................................................................................................. 54
Appendices ............................................................................................................................. 58
Appendix A : Prototype Parameters ................................................................................................. 58
A.1
Motor-A Parameters ....................................................................................................... 58
A.2
Motor-B Parameters........................................................................................................ 58
Appendix B : BLDC Motor Controller ............................................................................................ 59
v
List of Figures
Figure 1.1
A typical power-electronic-based electro-mechanical system ............................. 1
Figure 1.2
Prototype Motor A Set-up Including Drive Circuit and Mechanical Load. ......... 6
Figure 1.3
Prototype Motor B Set-up Including Drive Circuit and Mechanical Load. ......... 6
Figure 2.1
Schematic diagram of a typical VSI-driven BLDC motor-drive system. ............ 9
Figure 2.2
Switching sequence of the inverter according to the 120° switching logic. ....... 11
Figure 2.3
Steady state waveforms of phase back EMF, phase current and electromagnetic
torque predicted by various models for Motor A.................................................................... 15
Figure 2.4
Steady state waveforms of phase back EMF, phase current and electromagnetic
torque predicted by various models for Motor B. ................................................................... 15
Figure 2.5
Measured and simulated waveforms of phase back EMF, phase current, and
phase voltage for Motor A. ..................................................................................................... 16
Figure 2.6
Measured and simulated waveforms of phase back EMF, phase current, and
phase voltage for Motor B. ..................................................................................................... 16
Figure 2.7
Measured and simulated source current and stator phase current waveforms for
Motor A................................................................................................................................... 17
Figure 2.8
Measured and simulated source current and stator phase current waveforms for
Motor B. .................................................................................................................................. 18
Figure 2.9
Steady-state waveforms of phase current, back-EMF, and electromagnetic
torque as predicted by models with trapezoidal back-EMF implemented in different
simulation packages. ............................................................................................................... 19
Figure 2.10
Steady-state torque-speed characteristic predicted by various models and
simulation packages. ............................................................................................................... 20
Figure 3.1
Start-up transient response as predicted by various models for the Motor A. ... 30
Figure 3.2
Start-up transient response as predicted by various models for the Motor B. .... 31
Figure 3.3
Steady-state torque as predicted by various models for the Motor A. ............... 32
Figure 3.4
Steady-state torque as predicted by various models for the Motor B................. 32
Figure 3.5
Steady-state torque-speed characteristic as predicted by various models for the
Motor A. .................................................................................................................................. 34
Figure 3.6
Steady-state torque-speed characteristic as predicted by various models .......... 34
vi
Figure 4.1
Commutation angle look-up table function for the Motor A. ............................ 41
Figure 4.2
Commutation angle look-up table function for the Motor B. ............................. 42
Figure 4.3
Block diagram of the AVM implementation. ..................................................... 43
Figure 4.4
Steady-state torque as predicted by various models for Motor A. ..................... 45
Figure 4.5
Steady-state torque as predicted by various models for Motor B. ..................... 45
Figure 4.6
Steady state torque-speed characteristic as predicted by various models for
Motor A. .................................................................................................................................. 46
Figure 4.7
Steady state torque-speed characteristic as predicted by various models for
Motor B. .................................................................................................................................. 46
Figure 4.8
System response to sudden load change for the Motor A. ................................. 47
Figure 4.9
System response to sudden load change for the Motor B. ................................. 48
Figure 4.10
Start-up transient of Motor A as predicted by various models. ........................ 49
Figure 4.11
Start-up transient of Motor B as predicted by various models. ........................ 49
Figure 4.12
Measured and simulated response to the input voltage change as predicted by
the detailed and proposed average-value models for Motor A. .............................................. 50
Figure 4.13
Measured and simulated response to the input voltage change as predicted by
the detailed and proposed average-value models for Motor B. .............................................. 51
Figure B.1
Motor controller schematic. ............................................................................... 59
Figure B.2
Controller PCB. ................................................................................................. 60
Figure B.3
BLDC motor controller box. .............................................................................. 61
vii
Acknowledgements
First of all, I would like to express my deepest appreciations to my research supervisor, Dr.
Juri Jatskevich, whose strong academic support and dedication to his students have been the
most precious assets to my studies and research during the last two years at UBC. I am also
very grateful for the Research Assistantship that has been made available to me through the
NSERC Discovery Grant lead by Dr. Jatskevich.
The availability of research equipment and resources in the Alpha technology Lab has also
been an important asset for my research. In this regard, I would like to express my special
thanks to our close colleague and collaborator, Dr. S. D. Pekarek at Purdue University, who
has generously donated to our group the high-power BLDC machine with trapezoidal back
EMF which has been extensively utilized in my research.
I also like to thank Dr. Jose Marti and Dr. K.D. Srivastava, who have accepted to be the
committee members and dedicated their time and effort for reading this thesis and providing
their constructive and valuable comments.
My special thanks go to my friends and colleagues in the UBC‟s Power Lab and Electrical
Power and Energy Systems Group, particularly to Jaishankar Iyer, Mehrdad Chapariha,
Hamid Atighechi, Sina Chiniforoosh, Milad Gougani, and all the members of research group
who have always been a supportive and helpful friend to me.
I also owe a debt of thanks to my loving parents and my brother who have supported me
morally and financially throughout these intense years of my graduate studies.
viii
Dedication
To My Family
ix
1
1.1
Introduction
Why Average-Value Modeling?
Modeling and simulation are indisputable steps in design and development of electromechanical systems with power electronic drives which are widely used in industrial
automation, robotics, automotive products, ships, and aircraft. Figure 1.1 represents the block
diagram of such an electro-mechanical system. The Electrical Subsystem often consists of an
electrical power distribution that may also contain energy source and/or storage (i.e. battery
in the case of vehicles). The Machine-Drive subsystem is basically composed of „Inverter‟,
and „Electrical Motor/Generator‟ modules. The inverter controls the flow of energy from
source to the electrical machine which is responsible for the electro-mechanical energy
conversion. The „Mechanical Subsystem‟ may also represent a mechanical drive train (i.e. in
vehicles) or an assembly actuating system (i.e. in industrial manufacturing/automation). In
general, if the inverter can operate in all four quadrants, the energy may flow in either
direction; from the Electrical Source to the Mechanical Subsystem, or from the Mechanical
Subsystem back to the Electrical Source. In the former case, the machine operates as a motor
where in the latter it functions as a generator. The output of the Mechanical Subsystem which
can be position or speed of the rotor, or any other mechanical variables might be used
directly or indirectly, to control the inverter. The control signal may also consist of external
variables such as duty cycle, voltage amplitude and etc.
Figure 1.1
A typical power-electronic-based electro-mechanical system
1
For the purposes of stability analysis and design of respective controllers, it is often desirable
to investigate both the large-signal time-domain transients as well as the small-signal
frequency-domain characteristics of such systems. Since the experimental tests with the
hardware is not always possible and/or cost-effective, in actual industrial practice most of the
studies are carried out using appropriate models, simulations, and mathematical apparatus.
Such computer-bases studies are usually carried out many times for tuning the system and
achieving the desired performance while satisfying the design specifications. This requires
the simulation speed and accuracy to be as high as possible. In particular, modeling and
simulation of Inverter Module is not often trivial, since this module includes switching
components such as MOSFETs, IGBTs and diodes, which make the respective models
discontinuous and time-variant. There are various simulation software packages such as [1]–
[5], which can be used to develop and implement the models where the switching of all
transistors and diodes is represented in full detail. On one hand, there is a need to run the
simulation for a sufficiently long time in order to capture the electromechanical transients
that may have relatively long time constants (on the order of several seconds); but on the
other hand, the presence of power electronic components and fast switching requires using
very small time-steps. Therefore, this type of detailed models requires excessively long CPU
(computing) times, especially for large systems that include many switching components and
consist of several subsystems.
However, since the fast switching of the transistors and diodes has only an average effect on
the system‟s slow dynamic behavior, it is advantageous to construct a simplified model that
matches the original detailed switching model in the low-frequency range. The approach of
establishing such simplified models is known as average-value modeling (AVM), wherein
the effects of fast switching are neglected or averaged within a prototypical switching
interval. Unlike the detailed models, average-value modes (AVMs) are continuous and the
respective state variables are constant in steady states. Therefore AVMs can be linearized
about a desired operating point; thereafter, obtaining a local transfer function and/or
frequency-domain characteristics becomes a straightforward and almost instantaneous
procedure. Many simulation programs offer linearization and frequency domain analysis
tools [4], [5]. In addition, since there is no switching, the AVMs typically execute, by orders
2
of magnitude, faster than their corresponding detailed models, making them ideal for
representing respective components in system-level time-domain transient studies. Such
dynamic average models have been very successfully used for modeling of distributed DC
power systems of spacecraft [6]–[8] and aircraft [9], [10], naval electrical systems [11], [12]
and vehicular electric power systems [13]. Average-value modeling has also been often
applied to variable speed wind energy systems [14]–[21], where the machines are typically
interfaced with the grid using the power electronic converters.
1.2
Literature Review
Average-value modeling of electro-mechanical systems with power electronic drives has
attracted attention of many researchers during the past few decades. R. Krishnan at the
Virginia Polytechnic who developed a basis for modeling of the PMSM with power
electronic drive [22], and P. C. Krause and S. D. Sudhoff at Purdue University who
established the detailed and average-value modeling of the power electronic driven motors
including the BLDC [23], [24] were among the pioneers of research in this area. Later on, P.
L. Chapman at the University of Illinois at Urbana Champaign, K. A. Corzine at the
University of Missouri-Rolla, and H. A. Toliyat at the Texas A& M University made
contributions to this field too by developing the models for continuous current operation.
Development of the current-regulated BLDC motor-drive system [25], [26] and analysis of
the BLDC motor-drive system in hybrid sliding mode observer [27] is often noted as the
outcome of their research in this field. In addition, most recently, there have been significant
contributions to this area lead by J. Jatskevich and his graduate students at The University of
British Columbia, which resulted in several state-of-the-art models including the numerical
state-space AVM for the DC-DC converters [28], parametric AVM for the synchronous
machine-rectifier system [29], and the most relevant AVM for the sinusoidal BLDC motorinverter system [30].
Prior to the work presented in this thesis, the best known to us average-value model of a
BLDC motor with a 120-degree inverter remains the dynamic AVM presented in [31]. That
work represents a significant contribution to the area but considers only the sinusoidal back
EMF of the machine.
3
1.3
Brushless DC Motor-Inverter System
A brushless dc (BLDC) motor-inerter system consists of a permanent magnet synchronous
machine (PMSM) that is driven by a voltage source inverter (VSI). Typically such motors
provide good torque-speed characteristics, fast dynamic response, high efficiency, long life,
etc., which make them favorable in wide range of applications including industrial
automation, instrumentation, and many other equipment and servo applications. This paper
considers typical voltage-source inverter driven (VSI-driven) BLDC motors wherein the
inverter operates using 120 commutation method [32], [33]. In this switching logic, each
phase is allowed to be open-circuited for a fraction of revolution, giving rise to complicated
commutation-conduction patterns of the stator currents [32]. In general, derivation of
dynamic average-value modeling of such BLDC systems requires careful averaging of the
stator phase voltages and currents over a prototypical switching interval (SI) to find the
corresponding average-value relationships for the state variables and electromagnetic torque.
A pioneering step in this approach has been the AVM for the BLDC motor-inverter system
with sinusoidal back EMF [31]. This approach has been extended to include both conduction
and commutation sub-intervals [31]. Another average-value model was proposed in [34] for
the non-sinusoidal back EMF PMSM driven by a three phase H-bridge inverter that can only
operate in continuous-current voltage-control mode by adjusting the duty cycle. However,
the average-value modeling of the 120 ° VSI driven BLDC motors with trapezoidal back
EMF becomes more challenging due to the discontinuous current and harmonics in the
voltage and torque equations; and to the best of our knowledge this has not been addressed in
the literature.
4
1.4
Contributions
This thesis extends the previous work in this area and presents a new AVM for the 120
VSI-driven BLDC motor with non-sinusoidal back EMF. The contributions of this thesis and
the property of the proposed model can be summarized as follows:
1) We show that there is a need for including the back EMF harmonics for modeling the
BLDC motors with pronounced non-sinusoidal back EMF. The new AVM is proposed
that simultaneously includes the back EMF harmonics and the commutation and
conduction subintervals.
2) The multiple reference frame theory [34] is utilized to properly include the effect of back
EMF harmonics into the average-value relationships of the AVM.
3) Since it is not practical to analytically derive a closed form solution for the commutation
angle, the solution is obtained numerically using detailed simulation. This method
reduces the complexity of analytical derivations and has been shown to provide accurate
results [31], [28], [29], [35].
4) The conducted studies are based on two typical industrial BLDC motors with various
back EMF harmonics content and parameters summarized in the Appendix A. Figures
1.2, and 1.3 show the prototype motors‟ test set-up arranged for the measurement
purposes in the laboratory. The details of the BLDC Motor Controller Box are shown in
Appendix B. The results of studies are compared with the experimental measurements as
well as previously established models [24], [31] whereas the new model is shown to
provide appreciable improvement.
5
Figure 1.2
Prototype Motor A Set-up Including Drive Circuit and Mechanical Load.
Figure 1.3
Prototype Motor B Set-up Including Drive Circuit and Mechanical Load.
6
1.5
Thesis Composition
This thesis is comprised of the following chapters:

Chapter 2 presents an improved approach for detailed modeling of the BLDC motor-drive
system where the effects of back EMF harmonics are properly incorporated into the
model. The proposed model is then verified against the hardware measurement from the
actual machine. It is also shown that including the back EMF harmonics often
significantly increases the accuracy of the detailed model in predicting the behavior of
the system and hence, it is necessary to take these harmonics into consideration for
further studies on the inverter-driven BLDC systems.

Chapter 3 describes a new average-value model for the BLDC machine-drive system in
which the multiple reference frame theory [34] is used to properly include the back EMF
harmonics into the AVM. The developed model is then implemented in Matlab/Simulink
[4] along with the previously developed AVMs in which the back EMF waveform is
assumed to be sinusoidal, and it is shown that the proposed AVM is more accurate.
However, since the commutation interval is neglected, the new AVM may still result in
some error in prediction of the system performance.

Chapter 4 completes the presented AVM in Chapter 3 by simultaneously including the
effects of the back EMF harmonics and the commutation and conduction subinterval. In
particular, this process is very challenging due to presence of both the higher harmonics
and the commutation angle in the voltage and torque relationships. The proposed AVM is
then shown to be considerably more accurate in comparison with the previously
developed AVMs and compensates the error arising due to neglecting the back EMF
harmonics and/or the commutation subinterval.

Chapter 5 concludes the thesis by summarizing the conducted research.

The parameters of the motors and the motor controller are summarized in Appendix.
7
2
Detailed Modeling of Brushless DC Motors with Non-Sinusoidal
Back EMF
The detailed modeling of the BLDC motor-inverter system has been described in literature
quite well [22], [23], [32], [33], [36]–[38] and can be easily carried out using a number of
simulation packages [1]–[5]. In many available detailed models, it is often assumed that the
induced back EMF waveform of the machine is sinusoidal [23], [24], [31], [32]. However,
the actual back EMF waveform might quite non-sinusoidal. Including the back EMF
harmonics into the voltage and torque equations increases the accuracy of the model. In
addition, to develop a detailed model that precisely predicts the performance of the BLDC
motor-drive system with trapezoidal back EMF, an appropriate simulation package must be
used such that the back EMF waveform can be modified to include the desired amount of
harmonics [2], [3]. Herein the typical voltage-source-inverter-driven (VSI-driven) BLDC
motors are considered where the inverter operates using the 120 commutation method [32].
The steady state analysis of such motors has been carried out by several researchers [22],
[23], [36].
In this chapter an improved detailed model of the typical 120 BLDC motor-drive system is
proposed in which the trapezoidal back EMF harmonics are properly included into the model.
It is also shown that an accurate model may be only obtained using simulation packages that
allow making proper changes in the model such that the effects of back-EMF harmonics are
appropriately included in the current and torque relationships.
2.1
Model Description
A schematic of the considered BLDC motor-inverter system is shown in Figure 1.2, in which
the logical signals from hall sensors are used to control the inverter switches-transistors S1 –
S 6 . Here, as previously described, the motor is driven according to the 120 switching logic.
In this method, switching signals are of the sequence shown in Figure 2.1 [31]. As a result,
each phase carries current for 120 two times during one electrical revolution which delays
8
the fundamental component of the voltage by 30 electrical degrees. To align the fundamental
component of the voltage with the back EMF, the advance firing angle of   30 is applied
[33], [37], [38].
Figure 2.1
Schematic diagram of a typical VSI-driven BLDC motor-drive system.
Although some BLDC machines are specifically designed to have low cogging torque and
consequently close to a sinusoidal back EMF waveform [39], [40], in practice, BLDC motors
often have trapezoidal back EMF. Including the back EMF harmonics into the voltage and
torque equations increases the accuracy of the model. The presented model is expressed in
physical variables and coordinates [38]. In particular, the electrical dynamics of stator shall
be described by the well-known voltage equation
9
v abcs  rs i abcs 
dλ abcs
.
dt
(1)
Here, the variable are represented in vectors such that f abcs   f as
f bs
f cs T , where
f may be voltage, current, or flux linkage. The stator resistance matrix is
rs  diag rs , rs , rs .
(2)
The flux linkages are then given by
λ abcs  L s i abcs  λ m
(3)
where the inductance matrix is defined by
 Lls  Lm
L s    0.5Lm
  0.5Lm
 0.5Lm
Lls  Lm
 0.5Lm
 0.5Lm 
 0.5Lm 
Lls  Lm 
(4)
in which Lls and Lm are the stator leakage and magnetizing inductances, and λ m is the
vector of flux linkages.
Assuming that stator windings are wye-connected, the three phase currents add up to zero.
Thus, (3) may be simplified as
λ abcs  Ls i abcs  λ m
where Ls  Lls 
(5)
3
Lm .
2
10
Figure 2.2
Switching sequence of the inverter according to the 120° switching logic.
Equations (1)–(5) hold true regardless of shape of the back EMF waveform. In general, the
flux linkages vector can be expressed as


sin 2n  1 r 


 
2

λ m   m  K 2 n 1 sin  2n  1 r 
3

n 1
 
2
 

sin  2n  1 r  3

 




 
 
 
 
 
 
(6)
where  r is the rotor‟s electrical position, and  m is the magnitude of the fundamental
component of the permanent magnet flux linkage. The coefficient K n denotes the
normalized magnitude of n th flux harmonic relative to the fundamental, i.e. K1  1 . Also, the
index 2n  1 explains that only odd harmonics may be present since the rotor is assumed to
be symmetrical.
11
The developed electromagnetic torque in presence of back EMF harmonics may then be
presented as [34],


T
cos 2n  1 r 
i as  

 
P
2

Te   m  2n  1K 2 n 1 ibs  cos 2n  1 r 
2
3

n 1
i cs   
2
 




cos
2
n

1



r


3

 



 
  .
 
 
 
 
(7)
The phase back EMF voltages can be measured at the stator terminals when the machine is
rotated by a prime mover and terminals are open-circuited. They can be also calculated based
on (1)–(6) as
e abcs


T
cos 2n  1 r 
i as  

 
2

  r  m  2n  1K 2 n 1 ibs  cos 2n  1 r 
3

n 1
i cs   
2
 

cos 2n  1 r  3

 



 
 
 
 
 
 
(8)
The mechanical subsystem is considered to be a single rigid body, for which the dynamics
shall be modeled by
d  P  1
   Te  Tm 
dt  2  J
(9)
where  r is the rotor‟s electrical angular speed, J is the combined moment of inertia of the
load and the rotor, P is the number of magnetic poles, and Tm denotes the combined
mechanical torque. Herein, a fan type load is used for which
Tm  T1n  To
(10)
12
where n represents the mechanical speed in revolution per minute (rpm), and the terms T1n
and To describe the dynamometer torque and the torque due to mechanical losses and friction
respectively.
Equations (1)–(10) form the detailed model of the BLDC motor driven by a 120 voltagesource inverter, where the back EMF waveform may be modified to possess the desired
amount of harmonics.
2.2
Effect of Back EMF Harmonics in Detailed Model
To demonstrate the importance of properly including the back EMF harmonics, the detailed
model described in previous sub-section is compared against the commonly-used model [22],
[31], [38] that considers sinusoidal back EMF. The considered detailed switching models
have been implemented in Matlab/Simulink using toolbox [3]. The conducted studies are
based on two typical industrial BLDC motors whose parameters summarized in the Appendix
A. As can be seen in the Appendix A, Motor A has a typical trapezoidal back EMF that
includes significant amount of 3rd , 5th , and 7th harmonics; whereas the back EMF
waveform of Motor B is much closer to sinusoidal.
2.2.1
Model Verification in Steady-State
The simulated steady state waveforms predicted by the models with sinusoidal and nonsinusoidal back EMF are superimposed in Figures 2.3 and 2.4 for Motor A and Motor B,
respectively. These waveforms correspond to a steady state operation when the inverter is
supplied with Vdc  26V , and a mechanical load of 330W at 2140 rpm applied for Motor A,
and 90W at 1650 rpm is applied for Motor B, respectively. The first subplot in Figure 2.3 and
2.4 shows the back EMF with and without the harmonics. As can be seen in Figure 2.3 (first
subplot), the Motor A has a strongly-pronounced trapezoidal back EMF, unlike the back
EMF of the Motor B in Figure 2.4 (first subplot), which is visibly close to sinusoidal. Figures
2.3 and 2.4 (second subplot) show that the back EMF harmonics also have effect on the
shape of the phase current during the conduction interval, which is also more pronounced for
the Motor A than Motor B. The simulated electromagnetic torque waveforms are shown in
13
Figures 2.3 and 2.4 (third subplot), where the effect of back EMF harmonics is also clearly
observed. According to (7), the electromagnetic torque is expected to have a larger average
value in the presence of harmonics. As a result, the difference in the torque ripple and its
average value when the harmonics are included or not for Motor A is more significant than
for Motor B.
Next, the detailed models that include the back EMF harmonics are compared to the actual
Motor A and Motor B. The measured and simulated waveforms corresponding to the same
steady state operating condition are superimposed in Figures 2.5 and 2.6. The first subplot
shows the measured back EMFs that have been recorded under the open-circuit condition
corresponding to the same speeds for the Motor A and Motor B, respectively. This
measurement was also used to extract the back EMF harmonics for each of the motors (with
the results summarized in Appendix A). Furthermore, Figures 2.5 and 2.6 (see second and
third subplots) also show that by including the back EMF harmonics into the detailed models,
an excellent match between the measured and simulated phase currents and voltages for both
motors is achieved. Therefore, these models can be considered as the reference for the future
studies.
14
Figure 2.3
Steady state waveforms of phase back EMF, phase current and electromagnetic torque
predicted by various models for Motor A.
Figure 2.4
Steady state waveforms of phase back EMF, phase current and electromagnetic torque
predicted by various models for Motor B.
15
Figure 2.5 Measured and simulated waveforms of phase back EMF, phase current, and phase voltage
for Motor A.
Figure 2.6 Measured and simulated waveforms of phase back EMF, phase current, and phase voltage
for Motor B.
16
2.2.2
Model Verification in Transient
The considered detailed model has been further verified in a transient study against the actual
Motor A and Motor B. In the following study, the motors are initially supplied with a voltage
Vdc1  20V , driving a fan–type load (emulated by a dynamometer machine). The
corresponding load characteristics for both machines (coefficients T1 and To ) are also
summarized in Appendix A. Then, at t  1s , the input voltage is stepped to Vdc2  23V . For
better comparison, the measured and simulated currents have been carefully aligned in time
axis and superimposed in Figure 2.7 and 2.8 for the Motor A and Motor B, respectively. As
can be seen in Figure 2.7 and 2.8, the two motors have different inertia and currents
corresponding to their respective loading conditions. However, the predicted dc current idc
(see Figure 2.1) and the phase current ibs are in good agreement with the experimental results
obtained for both motors, which also supports the use of these detailed models as the
reference in the future transient studies.
Figure 2.7 Measured and simulated source current and stator phase current waveforms for Motor A.
17
Figure 2.8 Measured and simulated source current and stator phase current waveforms for Motor B.
2.3
Case studies
The same industrial BLDC prototypes with parameters summarized in the Appendix A are
used to show the improvement of the proposed model when implemented in Matlab/ASMG
[3] against models for the BLDC motor that are implemented in other simulation packages
such as SIMPOWERSYSTEMS [1].
2.3.1
ASMG vs. SIMPOWER Model
The BLDC motor-inverter system modeling may be carried out using various simulation
packages [1]–[3]. Experimenting with these packages, it has been found that implementing
the BLDC motor that considers only the sinusoidal back EMF will result in the same output
waveforms of the currents and electromagnetic torque that are consistent among all the
mentioned tools. However, when the model includes the back EMF harmonics, the
simulation results of some packages might not be consistent anymore. To demonstrate this
point, the prototype BLDC Motor A with trapezoidal back EMF (see Appendix A.1) has been
also implemented in SIMPOWERSYSTEMS [1], wherein the user has a choice of using
either sinusoidal or trapezoidal back EMF. For comparison, the simulated waveforms of the
phase current, phase back EMF, and electromagnetic torque predicted by ASMG [3] and
18
SIMPOWERSYSTEMS [1] for the same operating point of Motor A are shown in Figure 2.8.
As can be seen in this figure, SIMPOWERSYSTEMS uses an ideal trapezoid for
representing the back EMF, which can be well matched with the measured/simulated
waveform of the back EMF shown in Figure 2.3 with the specified 3rd, 5th and 7th harmonics.
However, as it can be seen in Figure 2.9, when the trapezoid parameters are selected to match
the back EMF waveform (see top subplot), quite noticeable error will appear in the phase
current (see middle subplot) and electromagnetic torque (see bottom subplot).
Figure 2.9
Steady-state waveforms of phase current, back-EMF, and electromagnetic torque as
predicted by models with trapezoidal back-EMF implemented in different simulation packages.
2.3.2
Torque-Speed Characteristic
To further show the impact of accurately including the back EMF harmonics on the system
performance, the steady-state torque-speed characteristic of the prototype Motor A predicted
by various models is shown in Figure 2.10. As expected, the implemented model in ASMG
[3] that includes the effect of back EMF harmonics represents an improvement and predicts
higher torque that also confirms the results shown in Figure 2.3 and 2.9. Hence, it can be
again implied that neither the model that considers sinusoidal back EMF nor the
19
implemented model in SIMPOWERSYSTEMS can be used as a reference detailed model for
further investigation on the BLDC systems with trapezoidal back EMF.
Figure 2.10
Steady-state torque-speed characteristic predicted by various models and simulation
packages.
20
3
Average-Value Modeling of Brushless DC Motors with Trapezoidal
Back EMF
This chapter focuses on average-value modeling of typical voltage-source-inverter-driven
(VSI-driven) BLDC motors, where the inverter operates using the 120o commutation method
[24], [31], [32]. The AVM for the BLDC with sinusoidal back EMF has been proposed in the
literature [31]. Another average-value model was also proposed in [34] for the non-sinusoidal
back EMF PMSM with a 3-phase H-bridge inverter that can operate in continuous-current
voltage-control mode only by adjusting the duty cycle. However, average-value modeling of
the 120-degree BLDC with trapezoidal back EMF is more challenging due to the
discontinuous current and harmonics in the voltage and torque equations [34] and, to the best
of our knowledge, has not been addressed in the literature.
In this chapter a new and improved AVM for the typical 120-degree BLDC motor-drive
systems is proposed which makes a contribution by properly including the harmonics of the
trapezoidal back EMF into the average-value relationships of the model. It is also shown that
by utilizing the multiple reference frame theory and properly including the contributions of
harmonics, a more accurate AVM can be derived.
3.1
Model Description
For the purpose of deriving the AVM, the back EMF waveform is assumed to include only
5th and 7th harmonics because: i) the higher harmonics are negligible due to their relatively
smaller magnitude; and ii) the 3rd harmonic will have no effect in the averaging process since
the stator windings are wye-connected. Furthermore, the AVM is derived in a reference
frame in which the state variables are constant in steady-state. Therefore, the stator variables
are transformed into the qd -rotor reference frame using Park‟s Transformation [38]
f qd 0 s  K rs f abc
(11)
where
21

cos  r

2
r
Ks 
sin  r
3
 1

 2
2 
2 


cos r 
 cos r 

3 
3 


2 
2  


sin  r 
 sin  r 
 .
3 
3 



1
1

2
2

(12)
Applying transformation (11) to (1)–(9), the stator phase voltages in qd –rotor reference
frame may be described by
r
vqs
r
vds
r
diqs
r
  r Ls ids
  r  ' m 1  5K 5  7 K 7 cos6 r 

r
rs iqs
 Ls

r
rs ids
r
dids
r
 Ls
  r Ls ids
  r  ' m 5K 5  7 K 7 sin 6 r  .
dt
dt
(13)
(14)
The electromagnetic torque in qd -rotor reference frame may also be found by applying (11)
to (7) as


 3  P 
r
r
.
Te    m  1  5K 5  7 K 7  cos6 r iqs
 5K 5  7 K 7 sin 6 r ids
 2  2 
(15)
It is noticed that in (13)–(15), the addition of 5th and 7th harmonics into the model results in
extra harmonic terms corresponding to 6 r . This can be explained considering that (13)–(15)
are expressed in the reference frame rotating at rotor‟s electrical speed. In general, for a
round rotor PMSM, the n th harmonic term of the induced voltage in physical coordinates can
be expressed by an equivalent vector rotating at n times the rotor‟s electrical speed. Here,
only 5th and 7th harmonics of the back EMF are considered where the 5th travels in the
negative direction and 7th in the positive direction with respect to the fundamental harmonic
resultant vector [41]. However, in the qd -rotor reference frame, the transformed voltage
harmonics are equivalent to the vectors rotating at the relative speed of their respective
vectors in physical coordinates with respect to the fundamental harmonic which rotates with
22
the rotor. Therefore, considering the direction, both the 5th and 7th voltage harmonics result in
6 r dependant terms in (13)–(15).
r
r
The state variables iqs
and ids
must now be averaged with respect to a prototypical switching
interval, Ts (see Figures 2.5 and 2.6, second subplot), using
f 
1
Ts
t
t T
f ( )d
(16)
s
where f may represent voltages or currents, and the bar symbol is used to denote the
corresponding average-value. For the six-pulse converter, Ts   / 3 r  [7]. However, (15)
r
r
cannot be simply averaged using (16) since iqs
and ids
are both functions of rotor‟s electrical
position,  r .
To establish correct average-value relationship for torque, the multiple reference frame
theory is used [8], [9]. This step requires the state variables to be transformed from qd -rotor
reference frame to another reference frame rotating at some multiple of the rotor‟s electrical
speed. In particular, this transformation is described by
xr
r
f qd
 xr K rs f qd
(17)
where
xr
K rs 

r
K sxr

1
cosx r   r   sin x r   r 

.
 sin x r   r  cosx r   r  
(18)
Transformation (18) can be used to transform the variables from the rotor reference frame,
‗r‘, into the ‗xr‘ reference frame, rotating at ‗x‘ times the electrical speed of the rotor. After
23
algebraic manipulations, (15) can be expressed considering the multiple reference frame
theory as [34]

 3  P 
5r
7 r
r
Te     m  i qs
 5K 5 i qs
 7 K 7 i qs
2
2
  

(19)
5 r
r
where, for example, iqs
represents the transformation of iqs
into a reference frame rotating
at ‗5‘ times the rotor electrical speed and should not be misunderstood with the power sign.
Since there are no  r -dependent terms in (19), it can now be averaged using (16) resulting in
the following:


 3  P 
Te     m  i qsr  5K 5 i qs5r  7 K 7 i qs7 r .
 2  2 
(20)
To make use of (20), it is necessary to establish the state equations where the averaged
currents are the state variables. Since the state equations are derived from the voltage
equations, the next step is to find the transformed voltages in '5r ' and '7r ' reference frames
by applying (17) to (13)–(14), as
5 r
5 r
vqs
 rs iqs
 Ls
5 r
5 r
vds
 rs ids
 Ls
7 r
vqs
7 r
 rs iqs  Ls
7 r
7 r
vds
 rs ids
 Ls
5 r
diqs
dt
5 r
 5 r Ls ids
 5K 5 r m   r  ' m cos6 r   7 K 7 cos12 r  (21)
5 r
dids
5 r
 5 r Ls iqs
  r  ' m sin 6 r   7 K 7 sin 12 r 
dt
7 r
diqs
dt
(22)
7 r
 7 r Ls ids
 7 K 7 r m   r  ' m cos6 r   5K 5 cos12 r  (23)
7 r
dids
7 r
 7 r Ls iqs
  r  ' m sin 6 r   5K 5 sin 12 r .
dt
(24)
The transformed voltages are then averaged by applying (16) to (13)–(14) and (21)–(24),
which results in the following:
24
v qsr
v dsr
diqsr
 Ls

rs idsr
didsr
 Ls
  r Ls iqsr .
dt
dt
v qs5r  rs i qs5r  Ls
5 r
v ds
7 r
v qs
  r Ls i dsr   r  ' m

rs iqsr
5 r
 rs i ds
7 r
 rs i qs
 Ls
 Ls
v ds7 r  rs i ds7 r  Ls
di qs5r
dt
di ds5r
dt
di qs7 r
dt
di ds7 r
dt
(25)
(26)
 5 r Ls i ds5r  5K 5 r  m
(27)
 5 r Ls i qs5r
(28)
 7 r Ls i ds7 r  7 K 7  r  m
(29)
 7 r Ls i qs7 r .
(30)
To provide the input into the state model formed by (25)–(30), the averaged stator voltages
have to be established from the instantaneous voltages over a prototypical switching interval,
Ts , which consists of commutation and conduction subintervals [31]. The commutation
subinterval is often neglected [23] in order to simplify the averaging process. This
assumption is basis of the dynamic AVM proposed in [31] and also in this paper. In
particular, considering the switching interval II (see [24], [31], and Figures 2.5 and 2.6,
second subplot), which starts when the phase b transistor S 5 is being switched “off”, the
average voltages may be expressed as

 3   xr
vqsxr    2 vqs
,cond  r d r
   6 
(31)

 3   xr
vdsxr    2 vds
,cond  r d r
   6 
(32)
xr
xr
where vqs
,cond and v ds ,cond are the instantaneous voltages in the conduction subinterval
transformed into the ‗xr‘ reference frame. To obtain these voltages, the phase voltages in
25
direct abc coordinates should be known first. Based on analysis of inverter circuit [32], [38]
we have
v abc ,cond
1
 1

 2 Vdc  2 V B 


VB


 1 Vdc  1 V B 
2 
 2
(33)
where
 
2 
10

VB  m  r cos r 
  5K 5 cos 5 r 
3 
3

 
14 

 7 K 7 cos 7 r 

3 




.
(34)
Applying (11) to (33) and then transforming the result using (17), the conduction voltages in
the ' r ' , '5r ' and '7r ' reference frames are found as

1
2 

r
v qs

,cond  Vdc cos  r  cos   r 
3
3 



2  5
4 


 'm r cos 2  r 
  K 5 cos 4 r 

3  2
3 



(35)
7
7
4 
5


  K 5  K 7  cos 6 r   K 7 cos 8 r 

2
2
3 
2



1
2 

r
v ds

,cond  Vdc sin  r  sin   r 
3
3 


 
2  
2 
 'm r cos r 
 sin  r 

3  
3 
 

5
4   5
7
7
4



K 5 sin  4 r 
  K 5  K 7  sin 6 r   K 7 sin  8 r 
2
3  2
2
2
3



(36)



26

1
2 

 5 r 
v qs

,cond  Vdc cos 6 r cos  r  cos   r 
3
3 



2  5
4 


 'm r cos 6 r cos 2  r 
  K 5 cos 4 r 

3  2
3 




7
7
4  1
2 
5



  K 5  K 7  cos 6 r   K 7 cos 8 r 
  Vdc sin 6 r sin  r  sin  r 

2
2
3  3
3 
2




 
2  
2  5
4 

 'm r sin 6 r cos r 
 sin  r 
  K 5 sin  4 r 

3  
3  2
3 

 
7
7
4
5


  K 5  K 7  sin 6 r   K 7 sin  8 r 
2
2
3
2





(37)

1
2 

 5 r 
v ds

,cond   Vdc sin 6 r cos  r  cos   r 
3
3 



2  5
4 


 'm r sin 6 r cos 2  r 
  K 5 cos 4 r 

3  2
3 



7
7
4
5


  K 5  K 7  cos 6 r   K 7 cos 8 r 
2
2
3
2



2
 1

  Vdc cos 6 r sin  r  sin  r 
3
 3





 
2  
2  5
4 

 'm r cos 6 r cos r 
 sin  r 
  K 5 sin  4 r 

3  
3  2
3 

 
7
7
4 
5


  K 5  K 7  sin 6 r   K 7 sin  8 r 

2
2
3 
2


(38)

1
2 

7 r 
v qs

,cond  Vdc cos 6 r cos  r  cos   r 
3
3 



2

 'm r cos 6 r cos 2  r 
3


4 
 5

  K 5 cos 4 r 

3 
 2


7
7
4  1
2 
5



  K 5  K 7  cos 6 r   K 7 cos 8 r 
  Vdc sin 6 r sin  r  sin  r 

2
2
3  3
3 
2




 
2  
2  5
4 

 'm r sin 6 r cos r 
 sin  r 
  K 5 sin  4 r 

3  
3  2
3 

 
7
7
4 
5


  K 5  K 7  sin 6 r   K 7 sin  8 r 

2
2
3 
2


(39)
27

1
2 

7 r 
v ds

,cond   Vdc sin 6 r cos  r  cos   r 
3
3 



2  5
4 


 'm r sin 6 r cos 2  r 
  K 5 cos 4 r 

3  2
3 




7
7
4  1
2 
5



  K 5  K 7  cos 6 r   K 7 cos 8 r 
  Vdc cos 6 r sin  r  sin  r 

2
2
3  3
3 
2




 
2  
2  5
4 

 'm r cos 6 r cos r 
 sin  r 
  K 5 sin  4 r 

3  
3  2
3 

 
7
7
4
5


  K 5  K 7  sin 6 r   K 7 sin  8 r 
2
2
3
2





(40)
The averaged voltages over the conduction subintervals may now be obtained by substituting
(35)–(40) into the equations (31)–(32), respectively. The results in the ' r ' , '5r ' and '7r '
reference frames can be expressed as
v qsr  A r  Vdc  B r  m  r
(41)
vdsr  C r  Vdc  D r  m  r
(42)
v qs5r  A 5r  Vdc  B 5r  m  r
(43)
v ds5r  C 5r  Vdc  D 5r  m  r
(44)
v qs7 r  A 7 r  Vdc  B 7 r  m  r
(45)
v ds7 r  C 7 r  Vdc  D 7 r  m  r
(46)
where the coefficients A r   , B r   , C r   , D r   , A5r   , B 5r   , C 5r   ,
D 5r   , and A7 r   , B 7 r   , C 7 r   , D 7 r   are
A r   


cos   

6

(47)
B r   
1 3 3
  15 3
2  21 3





cos 2   
K 5 cos 4 
K 7 cos 8  

2 4
3  8
3  16
3



(48)
3
28
C r    
2 

cos  


3 

(49)
D r    
3 3
2  15 3
  21 3




cos 2 
K 5 cos 4   
K 7 cos 8  

4
6  8
6  16
6



(50)
3

2

cos5   cos 5  3





A 5r   
1
5
B 5r   
5
21 3
2  3 3
2  15
2 



K5 
K 7 cos 2 
cos 4 
K 5 cos10 



2
4
3  8
3  20
3 



(52)
C 5r   
1
5
D 5r    
A 7 r    

2

sin 5   sin 5  3


(51)



(53)
21 3
2  3 3 
2  15
2 


K 7 sin 2 
sin 4 
K 5 sin10 



4
3  8
3  20
3 



1 
 

cos7   cos 7  

7 
3 

(54)
(55)
B 7 r   
7
15 3
2  3 3
2  21 3
2 



K7 
K 5 cos 2 
cos 8 
K 7 cos14 



2
4
3  16
3  28
3 



(56)
C 7 r   
1
7
D 7 r    
3.2

 

sin 7   sin 7  3 



15 3
2

K 5 sin 2 
4
3

2
 3 3 
sin 8 

3
 16

(57)

 21 3

K 7 sin14  

3
 28

(58)
Case Studies
The proposed AVM has been implemented in Matlab/Simulink together with the detailed
model. The same prototype motors with parameters summarized in the Appendix A are used
here. It is important to recall that since we have neglected the commutation time, the model
accuracy depends on how large or small the commutation interval is, which in turn depends
on the motor parameters.
29
3.2.1
Start-Up Transient
Figures 3.1 and 3.2 depict the typical start-up transients of the prototype motors as predicted
by the detailed model, the average-value model with sinusoidal back EMF, and the proposed
AVM that takes the back EMF harmonics into account. As can be observed, the developed
AVM provides more accurate results in prediction of the transient response. However,
neglecting the commutation interval affects the accuracy of the proposed AVM despite the
improvement against the previous AVMs which consider sinusoidal back EMF [31].
Figure 3.1
Start-up transient response as predicted by various models for the Motor A.
30
Figure 3.2
3.2.2
Start-up transient response as predicted by various models for the Motor B.
Steady-State
To further explore the improvement of the proposed model against the previously established
AVM that considers a sinusoidal back EMF [31], all three models have been implemented
and compared. Without loss of generality, and to have consistent studies with chapter 2, the
same steady-state operating point which were defined by 330W mechanical load at 2140 rpm
for motor A, and 90W mechanical load at 1650 rpm for motor B, supplied with Vdc  26V ,
are used here again. The electromagnetic torque predicted by the three models is
superimposed in Figure 3.3 and Figure 3.4 corresponding to Motor A and Motor B
respectively. As shown in these figures, including the effect of back EMF harmonics
appreciably improves the accuracy of the new AVM, especially in the case of Motor A which
possess more significant back EMF harmonics in comparison with Motor B. As expected,
since the commutation angle is ignored, the proposed AVM which is noticeably more
accurate than the sinusoidal AVMs, might still result in some error if commutation time is
not negligible compared to the length of conduction period.
31
Figure 3.3
Steady-state torque as predicted by various models for the Motor A.
Figure 3.4
Steady-state torque as predicted by various models for the Motor B.
32
To examine the model performance in a wider operating range, the calculated steady state
torque-speed characteristics for all considered models are plotted in Figure 3.5 and 3.6 for the
prototype motors A and B, respectively. As expected, the AVM that includes the effect of
back EMF harmonics represents an improvement and overall predicts higher average torque
that is also closer to the torque predicted by the detailed model. This improvement is more
pronounced at light loads, which is also expected, because the commutation interval is
smaller in this operating region. In addition, it may be again noticed that neglecting back
EMF harmonics results in more significant error in the case of Motor A which has
trapezoidal back EMF.
33
Figure 3.5
Steady-state torque-speed characteristic as predicted by various models for the Motor A.
Figure 3.6
Steady-state torque-speed characteristic as predicted by various models
34
4
Dynamic Average-Value Modeling of 120° VSI-Commutated
Brushless DC Motors with Trapezoidal Back EMF
Dynamic Average-value modeling of 120° VSI-Commutated brushless DC motors with
sinusoidal back EMF has been well investigated in [31] where the challenge of including the
commutation interval into the voltage equations, was conventionally manipulated by utilizing
the data from detailed simulation in the form of a numerical look-up table. However, as
discussed in the previous section, the error arising due to neglecting the back EMF harmonics
might be quite significant in some cases. The proposed AVM in this chapter complements
the presented model in the previous chapter such that the effect of back EMF harmonics and
the commutation subinterval are taken into consideration simultaneously.
4.1
Model Description
To develop the average-value model of the BLDC motor-inverter system with trapezoidal
back EMF which properly considers the commutation time, the proposed procedure in
chapter 3 can be followed where equation (11)–(30) still hold true. However, the
commutation time is not ignored anymore meaning that the switching interval Ts , consists of
commutation and conduction subintervals. Therefore, the averaged instantaneous voltages
can be represented as [23], [32]
vqsxr  vqsxr,com  vqsxr,cond
(59)
vdsxr  vdsxr,com  vdsxr,cond .
(60)
xr
xr
where vqds
,com and vqds ,cond are the instantaneous voltages in the commutation and
conduction subintervals respectively, in the ‗xr‘ reference frame.
Considering the switching interval II (see [32]) which starts when the phase b transistor S 5
is being switched “OFF”, the averaged commutation and conduction voltages in the ‗xr‘
reference frame are
35

 3     xr
vqsxr,com    6
vqs ,com  r d r
   6 
(61)

 3     xr
vdsxr,com    6
vds ,com  r d r
   6 
(62)

 3   xr
vqsxr,cond    2
vqs ,cond  r d r
   6   
(63)

 3   xr
vdsxr,cond    2
vds ,cond  r d r
   6   
(64)
where  is the commutation angle, in electrical degrees, and  , is the advance in firing
angle that is assumed to be 30 .
Equations (61)–(64) require finding the instantaneous voltages during the commutation and
conduction subintervals prior to the averaging process. The commutation time (and angle)
depends on the stator winding electrical time constant and operating conditions, but in
general cannot be zero since the current in the inductor cannot be switched “OFF”
instantaneously. In the commutation subinterval of the SI II, the phase current ibs is negative
and going to zero through the upper diode. The stator phase voltages in direct abc
coordinates can then be readily established based on the inverter topology. In particular, after
some algebraic manipulation [23], the phase voltages during the commutation time may be
expressed as
v abc,com 
Vdc
1 1  2T .
3
(65)
Hence, (65) must also be transformed to the ' r ' , '5r ' and '7r ' reference frames according to
the multiple reference frame theory [34]. This is achieved by first applying the
transformation (11) to (65) and then transforming the result using (17), which results in the
following
36
2
2 

r
v qs

,com   Vdc cos r 
3
3 

(66)
2
2 

r
v ds
.
,com   Vdc sin  r 
3
3 

(67)
 

2
2 
2 

5r 
vqs
 cos6 r  sin r 
 sin 6 r 
,com   Vdc cos r 
3
3 
3 

 

(68)
 

2
2 
2 

5r 
vds
 sin 6 r  sin r 
 cos6 r  .
,com   Vdc cos r 
3
3 
3 

 

(69)
 

2
2 
2 

7 r 
vqs
 cos6 r  sin r 
 sin 6 r 
,com   Vdc cos r 
3
3 
3 

 

(70)

2
2

7 r 
vds
,com   Vdc  cos r 
3
3


(71)
2


 sin 6 r  sin r 
3




 cos6 r  .


However, the instantaneous voltage harmonics during the conduction subinterval remain the
same as (35)–(40).
The total averaged voltages over commutation and conduction subintervals can now be
obtained by substituting (66)–(71) and (35)–(40) into (61)–(64), respectively. The results in
the 'r ' , '5r ' and '7r ' reference frames are:
vqsr  A r  ,  Vdc  B r  ,  m  r
(72)
vdsr  C r  ,  Vdc  D r  ,  m  r
(73)
vqs5r  A5r  ,  Vdc  B 5r  ,  m  r
(74)
vds5r  C 5r  ,  Vdc  D 5r  ,  m  r
(75)
vqs7 r  A7 r  ,  Vdc  B 7 r  ,  m  r
(76)
vds7 r  C 7 r  ,  Vdc  D 7 r  ,  m  r
(77)
37
where the coefficients A r  ,   , B r  ,   , C r  ,   , D r  ,   , A5r  ,   , B 5r  ,   ,
C 5r  ,   , D 5r  ,   , A7 r  ,   , B 7 r  ,   , C 7 r  ,   , D 7 r  ,   are
A r   
2 
     
cos     sin   

 
3 2 6 2
    
5     


 cos    sin     2 cos  
  sin  
2 6 2
6 2   2 


(78)
3 

2

 
 15

  2

   cos 2     sin     
K 5 cos 4 
 2  sin 
 2 

2  3
3
3

 3
 4

  3

1
5K 5  7 K 7 cos6  3 sin 3   21 K 7 cos 8    4  sin 2  4 

2
8
3

  3

(79)
B r   
C r   
2 
     
    
5     


sin    sin   sin    sin    2 sin 
  sin 

 
3 2 6 2
2 6 2
6 2   2 


(80)
3
2
2

 
 15

  2

sin  2 
   sin     
K 5 sin  4 
 2  sin 
 2 
2
3
3

 3
 4

  3

1
5K 5  7 K 7 sin 6  3 sin 3   21 K 7 sin 8  2  4  sin 2  4 

2
8
3

  3

D r    
A5 r   
2
5
 
5    5 
cos 5  2  sin 6  2 
 

 
2 5   5 5 
 5   5  


 cos 5 


 sin
 2 cos 5  
 sin

3
2   6
2 
6 2   2 


1
3
2

  2

cos 6  3 sin 3  
cos 4 
 2   sin 
 2 
2
4
3

  3

3
2

 
 3 


K 5 cos10 
 5  sin   5  
   5K 5 
2
3

 3
 2  3

(81)
(82)
B 5 r   

(83)
7
21
2

 

K 7 cos 12  6 sin 6   
K 7 cos 2 
   sin    
4
2
3

 3

38
C 5 r   
 
5   5 5 
sin 5  2  sin 6  2 
 

 
2
5
2 5   5 5 
 5   5  


 sin 5 


 sin
 2 sin 5  
 sin

3
2   6
2 
6 2   2 


1
3
2

  2

sin 6  3 sin 3  
sin  4 
 2  sin 
 2 
2
4
3

  3

2

  2

K 5 sin 10 
 5  sin 
 5 
3

  3

21
2

 

K 7 sin 12  6  sin 6   
K 7 sin  2 
   sin    
2
3

 3

(84)
D 5 r   

3
2

7
4
A7 r    
2
7
 
7    7 
cos 7  2  sin 6  2 
 

 
 7    7 
 7    7  


 cos 7  
 sin 
 2 cos 7  
 sin

3 2  6 2 
6 2   2 


1
3
2

  2

cos 6  3 sin 3  
cos 8 
 4  sin 
 4 
2
8
3

  3

15
2

 
 3 


K 5 cos 2 
   sin     
   7 K 7 
2
3

 3
 2  3

(85)
(86)
B 7 r   

(87)
5
3
2

 

K 5 cos 12  6  sin 6   
K 7 cos14 
 7   sin   7  
4
2
3

 3

C 7 r   
2
7
 
7    7 
sin  7  2  sin  6  2 
 

 
 7    7 
 7    7  


 sin  7  
 sin  
 2 sin  7  
 sin 

3 2  6 2 
6 2   2 


(88)
39
1
3
2

  2

sin 6  3 sin 3  
sin  4 
 2   sin 
 2 
2
4
3

  3

2

  2

K 5 sin 10 
 5  sin 
 5 
3

  3

21
2

 

K 7 sin 12  6  sin 6   
K 7 sin  2 
   sin    
2
3

 3

D 7 r   

3
2

7
4
(89)
To complete the AVM, the commutation angle  has to be defined. An implicit
transcendental equation was proposed in [28] that considers steady-state operation and
sinusoidal back EMF, and requires iterative numerical solution. However, due to complexity
of equations in our case, it is impractical (and even impossible) to derive a closed-form
explicit analytical expression for the commutation time/angle.
4.2
Model Implementation
r
In general, the commutation angle depends on the motor speed  r , the phase currents iqs
r
and ids
, the supply voltage Vdc , and the machine parameters. Following the approach
established in [31],  may be expressed as an algebraic function of the state, and input
r
variables  r , iqds
, and Vdc , respectively, and can be numerically established based on the
results from the detailed simulation. Similar approach was also used for establishing the duty
ratio constraint for the analysis of dc-dc converters [28]–[29]. Thus, a function


r
 r ,Vdc , iqds
may be established by running the detailed simulation in a loop spanning a
desired range of operating conditions, where the currents are averaged according to (16), and
 r , Vdc , and  are recorded in a look-up table. The results can be further simplified by
defining the dynamic impedance of the inverter switching cell as
z
Vdc
r
iqds
(90)
which combines two parameters into one, and consequently reduces the dimension of the
developed look-up table. As a result, the commutation angle can be defined by a two40
dimensional look-up table   r , z  . The corresponding results for Motor A and Motor B are
plotted in Figures 4.1 and 4.2, respectively. Here, it may be noticed that Motor B, has slightly
higher commutation angle compared to Motor A, which is also consistent with their
respective stator electrical time constants. The considered range for each motor has been
chosen based on the rated values for each motor and the available measuring equipment.
Figure 4.1
Commutation angle look-up table function for the Motor A.
41
Figure 4.2
Commutation angle look-up table function for the Motor B.
Finally, the implementation of the proposed AVM is established according to the block
diagram depicted in Figure 4.3. The input into model is the inverter instantaneous dc voltage
v dc . Based on the commutation angle   r , z  , the combined average voltages in each
reference frame are calculated using (72)–(77) which provides the input into the state
equations of the electrical subsystem defined by (25)–(30). The outputs of the six-order state
model are the averaged currents (for each reference frame) that are used to calculate the
torque according to (20). The mechanical subsystem is defined by (9)–(10), and it calculates
the (mechanical and electrical) speed of the motor.
42
Figure 4.3
4.3
Block diagram of the AVM implementation.
Case Studies
To demonstrate the improvement introduced by the proposed average-value model the
proposed model has been implemented in Matlab/Simulink [4] together with the previously
established models and the detailed switching model (which is also used as the reference). To
demonstrate the generality of the new model, all studies are carried out for the two
considered motors with different back EMF shapes.
43
4.3.1
Steady-State
Without loss of generality, the same steady-state operating points described in Chapter 3, –
subsection 3.2.2– are considered here. The instantaneous and averaged electromagnetic
torque predicted by different models, are superimposed in Figures 4.4 and 4.5 for Motor A
and Motor B, respectively. It may be noticed from the figures that as expected, the relative
error due to neglecting back EMF harmonics is more significant for Motor A, where ignoring
the commutation angle has more pronounced impact on the performance of Motor B. The
reason can be explored considering that Motor A possesses more considerable amount of
back EMF harmonics in comparison with Motor B in which the commutation angle is
relatively larger.
To examine the model performance in a broader operating range, the torque-speed
characteristics predicted by all considered models are superimposed in Figures 4.6 and 4.7
for Motor A and Motor B, respectively. For convenience of comparing the two motors, here
the characteristics are plotted in terms of mechanical speed,  m  30  2 P  r , that has
units of rpm. As can be seen in Figure 4.6, the AVM that assumes sinusoidal back EMF and
neglect the commutation predicts the lowers torque. Including the back EMF harmonics
increases the torque and improves the AVM accuracy. The most accurate AVM is the one
that includes both the back EMF harmonics and the commutation. The characteristic
predicted by this AVM matches the detailed model very well. Similar observations can be
made in Figure 4.7 with regard to Motor B. However, since this motor has close to sinusoidal
back EMF, the impact of including the harmonics is not very significant. At the same time,
this motor has generally larger commutation angle (see Figure 4.2). Therefore, including the
commutation subinterval in this case has more pronounced improvement, which is also
achieved by the proposed AVM.
44
Figure 4.4
Steady-state torque as predicted by various models for Motor A.
Figure 4.5
Steady-state torque as predicted by various models for Motor B.
45
Figure 4.6
Steady state torque-speed characteristic as predicted by various models for Motor A.
Figure 4.7
Steady state torque-speed characteristic as predicted by various models for Motor B.
46
4.3.2
Transient Response to Mechanical Load Change
To further verify the proposed AVM, the following transient study is considered next. The
motor is assumed to initially operate in steady state defined by a certain mechanical load
(Motor A: 150 W at 1820 rpm; and Motor B: 90 W at 1650 rpm). The corresponding
mechanical torque is defined by (10) for each motor. At t  1s , the load torque To is changed
from 0.2 to 1N.m . The system response predicted by various models is plotted in Figures
4.8 and 4.9 corresponding to Motor A and Motor B, respectively. As expected, the increase
in load is followed by the decrease in motor speed  r and increase of the dc current idc . It
can be further seen in Figures 4.8 and 4.9, that the AVMs that do not include the
commutation effect are noticeably off in predicting the qd -axis current, ids . This is more
noticeable for Motor B, which has larger commutation angle (see Figure 4.9). At the same
time, including the commutation has less of an effect for Motor A (see Figure 4.8), wherein
including the back EMF harmonics has more pronounced improvement in accuracy.
Figure 4.8
System response to sudden load change for the Motor A.
47
Figure 4.9
4.3.3
System response to sudden load change for the Motor B.
Start-Up Transient
Next, the proposed AVM is next against the detailed model based on the prototypical start-up
transients of the prototype motors as shown in Figures 4.10 and 4.11, corresponding to Motor
A and Motor B, respectively. As can be observed in these figures, the developed AVM
precisely predicts the start-up transient of the motors, regardless of the back EMF harmonics
content and the length of the commutation interval.
48
Figure 4.10
Start-up transient of Motor A as predicted by various models.
Figure 4.11
Start-up transient of Motor B as predicted by various models.
49
4.3.4
Transient Response to Input Voltage Change
The accuracy of the detailed switching model has been established in Section II. For
consistency, the same transient study of Chapter 2, subsection 2.2.2 is considered again;
wherein the motors are subjected to the dc supply voltage step increase from 20V to 23V.
The resulted waveforms of idc (measured and simulated) and Te predicted by the AVM and
the detailed model are superimposed in Figures 4.12 and 4.13, for Motor A and Motor B,
respectively. Due to the limited space and for better clarity, only the final proposed AVM
that includes back EMF harmonics and commutation subinterval is considered in this study.
As can be seen in Figures 4.12 and 4.13, the increase in the applied voltage causes the
respective increase in the electromagnetic torque Te and the drawn dc current idc , and the
motors go through the transient that is determined by their respective electromechanical
parameters and inertia. Furthermore, the AVM predicts the transient responses for both
Motor A and Motor B with very good agreement with the measurement and the detailed
switching model.
Figure 4.12 Measured and simulated response to the input voltage change as predicted by the detailed
and proposed average-value models for Motor A.
50
Figure 4.13 Measured and simulated response to the input voltage change as predicted by the detailed
and proposed average-value models for Motor B.
51
5
Conclusion
Average-value modeling is indisputable for analysis of power-electronic-based electromechanical systems where the simulation speed and accuracy must be as high as possible.
Nevertheless, the AVM is particularly valuable for analysis of the so-called voltage-source
inverter driven brushless dc motors which are widely used in various industrial applications.
Several AVMs were previously proposed in the literature for the BLDC motor-drive system
in which the effects of the back EMF harmonics and/or the commutation subinterval were
ignored resulting in significant amount of error in predicting the machine‟s performance.
However, this thesis presents a new and improved AVM for the 120° VSI-driven BLDC
which is capable of precisely predicting the actual motor‟s behavior, regardless of the shape
of the back EMF waveform and/or the length of the commutation subinterval.
5.1
Summary
This thesis presented a new and improved average-value model for the commonly used 120degree VSI-controlled BLDC motors. The challenges in establishing dynamic average
models for such motor-drive systems include the commutation/conduction pattern of the
stator phase currents as well as the back EMF harmonics which may be quite significant in
trapezoidal BLDC machines. The approach considered in this paper is based on utilizing the
multiple reference frame theory for including the contribution from each significant
harmonic over conduction and commutation subintervals into the average-value relationships
of the voltages, currents, and developed electromagnetic torque, which are all combined into
the final state model.
The presented studies are based on two typical industrial BLDC motors with different back
EMF waveforms and electrical time constants. The results are compared with experimental
measurements as well as previously established reference models, whereas the proposed
average-value model is shown to provide appreciable improvement for either trapezoidal or
sinusoidal motors.
52
5.2
Future Research Topics
In this thesis, only one operating mode of the BLDC motor-inverter system, the so-called
negative-zero (NZ), is considered in which the overall switching interval is divided into two
subintervals. Average-value modeling of the BLDC motors with non-sinusoidal back EMF in
other operating modes has not been done can be further pursued by the UBC research group.
It has also been assumed that the BLDC operates with advance firing angle fixed at 30
degrees. However, in some applications, the BLDC machine can operate with different firing
angles and more complicated switching patterns of the stator currents with up to three
subintervals within a single switching interval may be taken into consideration for future
studies. In addition, the hall sensors are considered to be exactly 120° apart, which may not
be true in some cases. Including the effect of misalignment of the hall sensors on the
presented AVM can be another potential topic for the future research. Parameter
identification, which provides the model with the online dynamic changes of the system
parameters, is another potential subject for extending this research on the BLDC motorinverter system with the non-sinusoidal back EMF. These and other topics will be considered
by other graduate students who are joining the UBC power group.
53
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[3]
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[4]
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Appendices
Appendix A : Prototype Parameters
A.1 Motor-A Parameters
ECycle Inc., Model MGA-1-13, 4.5kW, 12poles, rs  0.2, Ls  0.025mH,
m  10.9mV.s , back-EMF harmonics: K1  1, 3K 3  0.20, 5K 5  0.047,
7 K 7  0.0067, J  5  10 3 kg.m 2 , J  5 10 3 kg.m 2 , T1  2.7 10 3 N.m
rpm
,
To  0.2N.m
A.2 Motor-B Parameters
Arrow Precision Motor Co., LTD., Model 86EMB3S98F-B1, 210W, 8poles, rs  0.125,
Ls  0.4mH,  m  21.8mV.s , back-EMF harmonics: K1  1, 3K 3  0.0035, 5K 5  0.039,
7 K 7  0.017, J  5 10 4 kg.m 2 , T1  7 10 4 N.m
rpm
, To  0.1N.m
58
Appendix B : BLDC Motor Controller
Figure B.1 Motor controller schematic.
59
Figure B.2
Controller PCB.
60
Figure B.3
BLDC motor controller box.
61