ABSTRACT
Title of Thesis: CHARACTERIZATION AND MODELING
OF BRUSHLESS DC MOTORS AND
ELECTRONIC SPEED CONTROLLERS WITH
A DYNAMOMETER
Robert H. Brown
Masters of Science, 2019
Thesis directed by: Professor Inderjit Chopra
Professor Anubhav Datta
Department of Aerospace Engineering
The global drone market is expected to grow from $4.9 billion to $14.3 billion
within the next decade, indicating a heavy demand for high performance electric
aircraft. Modern drones are propelled with brushless DC (BLDC) motors and elec-
tronic speed controllers (ESCs). However, a current lack of information concerning
the performance and efficiency of BLDC motors and ESCs prevents their use in
rigorous aircraft design. Low cost hobby ESCs and BLDCs are typically used in
research aircraft, but few technical details are released by their manufacturers.
To better understand these devices, a custom dynamometer was constructed
to study the performance of ESCs and BLDC motors. By properly recording the
DC, AC, and mechanical power, information on peak efficiency and performance
for the ESCs and BLDC motors are determined experimentally. Motors between
920 KV to 2500 KV were tested with 18 A, 30 A, and 40 A ESCs. A combination
of these tests were carried out at 7.2 V, 11.1 V, and 14.8 V DC to explore trade
offs in the design process. While typically neglected in formal analysis, this work
seeks to better understand the power loss mechanisms in ESCs, as it was found
that ESCs could have efficiencies as low as 65%, reducing the overall efficiency of
the system considerably. This custom dynamometer features a load varying device,
power analyzers, and a unique two DAQ setup to properly capture the high frequency
electrical signals of BLDC motors.
From the sets of experimentally recorded motor and ESC tests, a novel ana-
lytical model is developed to predict the performance of ESCs and BLDC motors.
At the heart of this modeling effort is describing the 3 phase AC circuit as a sin-
gle equivalent circuit, which encapsulating the motor?s performance. This work is
critical in the design process, as properly sizing ESCs, motors, and rotors for an
electric aircraft can improve aircraft endurance and range. Performance metrics are
extracted from experimental results and are fit into the analytical model. Predic-
tions for the system?s mechanical power, AC power, and DC power agree well with
experimental results, demonstrating applicability of the robust model.
CHARACTERIZATION AND MODELING OF BRUSHLESS DC
MOTORS AND ELECTRONIC SPEED CONTROLLERS WITH A
DYNAMOMETER
by
Robert H. Brown
Thesis submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Master of Science
2019
Advisory Committee:
Professor Inderjit Chopra, Co-Chair
Professor Anubhav Datta, Co-Chair
Professor Christopher Cadou
?c Copyright by
Robert H. Brown
2019

Preface
No motors, ESCs, or electric components were shorted in the making of this thesis.
ii
Dedication
To my Mother and Father
I am forever grateful for the help you gave me during this journey.
iii
Acknowledgments
Words cannot express the gratitude I have to the many people who have helped
me over the years.
First and foremost is Dr. Chopra, as my advisor he has always managed to
bring out the best problem solving skills in me. You have taught me something that
you will not find in any textbook: to never stop thinking about the problem. For
someone so important and on so many committees, I have always been amazed at
how every time I come by your office you were able to meet with me and answer my
questions. I will never forget the day when I sat in your office and you invited me
to join the Rotorcraft Center, at first I did not believe it because it was too good to
be true! It is an honor to be your student, and I look forward to working together
in the future.
Dr. Datta, I am lucky to have you as my co-advisor. Your ability to ask the
hard, direct questions has instilled a mentality in me to always be improving and
refining my work. Due to your key insights into motor theory, I was able to progress
my modeling work considerably. Your feedback allowed me to craft a physics based
motor theory. Without your help this thesis would have been 100 pages shorter.
Dr. Cadou, thank you for lending me the critical pieces of equipment needed
to make this dynamometer possible. I greatly appreciate the assistance you have
given me over the past year to get this work started. Thank you for your help and
for being on my committee.
Dr. VT, thank you for helping me relate my work back to the world of rotors.
iv
You were able to ask fundamental questions, which allowed me to think critically
about how my work fits into the grand scheme. Through our conversations you were
able to help me get a glimpse of the bigger picture, and for that I am thankful.
Dr. Maqbool, you are the mentor that has known me the longest. The expe-
rience I gained working for you has always helped me stand out, and it never would
have been possible if our paths did not cross. I still have the quadcopter we built
sitting next to me on my desk, as it is a constant reminder of the fond memories.
You were the first one who convinced me to go to graduate school and I am grateful
for following your advice. Thank you for your encouragement and friendship.
Brent Mills, thank you for your help in getting me up to speed with motor
theory. When I was first getting started the world of motors seemed like a jungle,
but you were able to guide me to the most important points. I would often show
you my data and analysis first because I valued your opinion and you weren?t afraid
to tell me some hard truths. It was difficult to find someone to talk about the
finer points of my model, but you were always available to meet and ask meaningful
questions that lead me to new discoveries.
Thank you to the various friends I have made here at the Rotorcraft Center:
Ilya, Ehis, Vikram, and Brandyn. I appreciate your feedback on my work and
keeping me sane through the hard times. Finally, I would like to thank my family
members for their support, especially my fiance? Leah who has always been my
biggest supporter.
v
Table of Contents
Preface ii
Dedication iii
Acknowledgements iv
Table of Contents vi
Commonly Used Definitions viii
1 Introduction 1
1.1 Electric Aviation Demands . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Commercial Off the Shelf Components . . . . . . . . . . . . . . . . . 5
1.3 The Need for a Dynamometer . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Outline of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 DC Motors 12
2.1 Basic Motor Performance - the Brushed Motor Model . . . . . . . . . 12
2.1.1 User Input Throttle . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.2 KT , KE, and KV . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Motor Component Theory . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 Brushed DC Motors . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.2 Brushless DC Motors . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Electronic Speed Controllers . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Commutation Logic . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.2 MOSFET Operation . . . . . . . . . . . . . . . . . . . . . . . 45
3 Experimental Setup 56
3.1 Dynamometer Overview . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Two DAQ Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3 DC Power Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4 Mechanical Power Analyzer . . . . . . . . . . . . . . . . . . . . . . . 67
3.5 AC Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5.1 Single Phase AC Relations . . . . . . . . . . . . . . . . . . . . 69
3.5.2 Polyphase AC Power Relations . . . . . . . . . . . . . . . . . 76
3.5.3 2-Wattmeter Proof for Y-Circuits . . . . . . . . . . . . . . . . 83
3.5.4 2-Wattmeter Proof for ?-Circuits . . . . . . . . . . . . . . . 88
vi
3.5.5 AC Power Analyzer . . . . . . . . . . . . . . . . . . . . . . . . 93
4 BLDC Motor and ESC Modeling 99
4.1 Brushless DC Motors . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1.1 Waveform RMS Evaluation . . . . . . . . . . . . . . . . . . . 101
4.1.2 Active Power from RMS Waveforms . . . . . . . . . . . . . . 112
4.1.3 Power Factor for a BLDC Motor . . . . . . . . . . . . . . . . 116
4.1.4 Equivalent Motor Circuit . . . . . . . . . . . . . . . . . . . . 118
4.2 Electronic Speed Controller . . . . . . . . . . . . . . . . . . . . . . . 129
4.2.1 MOSFET Power Losses . . . . . . . . . . . . . . . . . . . . . 133
4.2.2 Ideal DC to AC Transformer . . . . . . . . . . . . . . . . . . . 137
4.2.3 DC Current Draw for an ESC . . . . . . . . . . . . . . . . . . 141
4.2.4 ESC Power Losses . . . . . . . . . . . . . . . . . . . . . . . . 145
5 Experimental Results 152
5.1 Tests at 7.2V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.2 Test at 11.1V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.3 Tests at 14.8V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
5.4 Determining Motor Parameters from Experimental Data . . . . . . . 227
5.4.1 Common Trends . . . . . . . . . . . . . . . . . . . . . . . . . 236
5.5 Motor Variation Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 244
5.6 Motor and Rotor Pairing . . . . . . . . . . . . . . . . . . . . . . . . 254
5.6.1 Hover Test Stand . . . . . . . . . . . . . . . . . . . . . . . . . 256
5.6.2 Example Helicopter Propulsive Trim . . . . . . . . . . . . . . 262
6 Conclusions 272
6.1 Summary of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
6.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Bibliography 280
vii
Commonly Used Definitions
A Rotor Area, ?R2
C0 ESC transformation offset
C1 ESC transformation slope
CP Rotor shaft power coefficient,
PR
?A(?R)3
C QRQ Rotor torque coefficient, ?A(?R)2R
CT Rotor thrust coefficient,
TR
?A(?R)2
?ESC ESC efficiency,
PAC
PDC
?m Motor efficiency,
Pm
PAC
?sys Overall system efficiency,
Pm
PDC
fsw MOSFET switching frequency
GR Gear ratio
i(t) Instantaneous current
v(t) Instantaneous voltage
IDC DC Current
Io Current required to overcome static motor friction
Ipk Peak current
Irms Root mean square value of current, rms{i(t)}
KE Motor back voltage constant, VE = KE?
KT Motor torque constant, Q = KT I
KV Manufacturer specified RPM/VDC for a BLDC motor, includes ESC
N DC Ground
N? Neutral point of AC system
Np Number of pole pairs, (# of magnets)/2
? Motor RPM
?e Electrical driving frequency, Np?
? Rotor RPM
?NL Motor no-load speed? T
P Active AC Power, 1AC v(t)i(t)dtT 0
Pcond Conduction power loss, I
2R
PDC DC Power, VDCIDC
PESC ESC Power, PDC ? PAC
pf Power factor, PAC
S
pinst Instantaneous power, v(t)i(t)
Pm Motor output power, Q?
Psw Switching power loss
Q Mechanical torque
Q1?6 MOSFET numbering
Qstall Motor torque at 0 RPM
R Rotor radius
viii
RESC ESC equivalent resistance
Rm Motor resistance
S Apparent power in AC system, VrmsIrms
TR Throttle input to ESC, normalized between 0 ? TR ? 1
? Motor shaft angle
?e Electrical angle, Np?
vAN(t) Instantaneous voltage of line A wrt DC Ground
vAN?(t) Instantaneous voltage of phase A wrt motor neutral point
VDC DC Voltage
VLL,rms Line-to-line rms voltage, ex. VAC , VBC , etc.
Vph,rms Phase rms voltage, ex. VAN? , VBN? , etc.
Vpk Peak value of phase voltage
Ipk Peak value of line current
2S, 3S, 4S Battery cell count (nominally 7.4 V, 11.1 V, and 14.8 V, respectively)
AC Alternating current
BLDC Motor Brushless DC motor
DC Direct current
ESC Electronic Speed Controller
Line-to-line Voltage of one line measured wrt another line
MOSFET Metal Oxide Field Effect Transistor
Phase Voltage of one line measured wrt common point, typically N?
rms Root mean square
RPM Revolutions per minute
ix
Chapter 1: Introduction
1.1 Electric Aviation Demands
The world of Aerospace is changing fast. Electrification of aircraft has allowed
for flying to become an everyday occurrence, with millions of worldwide drone users
taking to the skies. Emergence of light weight composite structures, high density
lithium polymer batteries, and reliable low-cost sensors have allowed for an increased
interest into electric aviation. One of the largest advancements in recent decades are
high power to weight ratio brushless DC (BLDC) motors. Increased efficiency and
peak power allow for small, unmanned autonomous aircraft to complete useful tasks
such as surveillance, infrastructure inspections, and package delivery. The promise
of a new electric aircraft market has excited investors, as the global drone market
is expected to grow from $4.9 billion in 2019 to $14.3 billion by 2028 [1].
Inexpensive drones have opened the door to a transportation revolution - ur-
ban air mobility. Recent interest in urban air mobility has placed a great deal of
importance on electric aircraft. Prominent companies such as Uber have taken in-
terests in air taxis, and promise to deliver full scale 4,000 lbs autonomous electric
aircraft by 2023 [2, 3]. Numerous startups have emerged recently to provide low
cost mobility and shipping solutions to Uber and other customers. Most of these
1
companies aim for ?on demand aviation?, where users are able to summon aircraft
directly to their location and carry them a short 20 minutes to their destination.
Demand for electric motors comes from efficiency, economic, and reliability
standpoints. An hour?s flight time in a single engine airplane costs $3 in electricity
and $40 in gasoline [4]. This difference can be associated with increased electric
motor efficiency and increased aircraft performance associated with reduced sys-
tem mass. Additionally, electric motors can be sized to directly drive propellers,
eliminating heavy and expensive gearboxes and further reduces costs.
Figure 1.1: Left: E-volo?s VoloCopter. Right: NASA?s X-57.
The first manned e-VTOL personal air vehicle, E-volo?s VoloCopter, flew on
March 30th, 2016 [5]. Following this, many companies have begun testing radical
designs, featuring distrusted electric propulsion systems, with the goal of carrying
passengers. Introduction of manned electric aircraft is first expected in niche mar-
kets, such as electrifying 3 hour flight time trainer aircraft [4]. Within the last couple
of years, the Vertical Flight Society?s eVTOL Aircraft Directory has expanded to
include over 200 electric aircraft, with designs ranging from vectored thrust config-
urations to personal hover bikes [6].
2
NASA?s X-57 Maxwell is an experimental aircraft right on the front lines of
electric aviation. An Italian Tecnam P2006T gas driven airplane has been retrofitted
to become an all electric aircraft with the intent of increasing cruise performance
by 2.5 times the base model [7]. A disturbed electric power and propulsion system
allows for the X-57 to replace it?s 74.5 kW, 275 kg gas motors and propeller with
a 60 kW, 125 kg electric motor and rotor. The reduction in mass shows a 76.6%
increase in propulsor specific power density, 0.27 kW/kg for a gas propulsors, and
0.48 kW/kg for electric propulsors [8].
For small unmanned autonomous air vehicles (> 10 lbs), physically small sized
motors are required. For a 160 W power requirement, a small gas powered engine
such as the AP Yellowjacket was found to have a specific power density of 1.1
kW/kg [9]. For comparison at 160 W peak power, brushless DC electric motors
tested in this thesis contain a specific power density up to 4.9 kW/kg, whereas
commercially available electric brushed DC motors have specific power densities up
to 1.0 kW/kg. Electric motors scale well with size, making them excellent candidates
for powering unmanned air vehicles.
Modern electric motors use advanced electromagnetic, thermal, and structural
design practices to improve performance. Both the Maxwell?s engines and future
NASA designs are using finite element modeling approaches to accurate represent
the electromagnetic interactions between the motor?s rotor and stator. In an effort to
increase performance and eliminate weight, designs seek to optimize air gap length,
minimize rotor temperature, and reduce the number of heavy iron components. The
most ambitious of these programs targets >13 kW/kg with 95% efficient electric
3
motors for use in hybrid-electric propulsion schemes, outlined in [10, 11]. Thermal
management and a system?s focus effort seeks to properly incorporate electric motors
into future aircraft designs [7,11]. Design of the motor is dependent upon the mission
requirements.
To better understand the motor and rotor interaction, consider the following
example of a typical propulsion design. High fidelity aerodynamic models exist to
allow for blade design and optimization. Typically, the aerodynamic design deter-
mines a required rotor torque and RPM such that propulsion and lift requirements
are satisfied. In this example suppose the rotor has a 2500 RPM requirement and
that two brushless motors are under consideration, with masses of 30 grams and 50
grams respectively. Ideally, the least massive motor should be selected. However,
the efficiencies curves of the motors must be considered, and are overlaid with the
rotor?s requirement in figure 1.2.
Figure 1.2: Components considered for equivalent circuit modeling.
4
Once the motor?s performance is factored into the design, it is clear that motor
2 should be selected, as this design requires less power to operate. A design tradeoff
exists between motor efficiency and motor mass, as more efficient motors are heavier
but require less power to operate. Exchanging mass and power can change the
aircraft?s performance, endurance, payload capability, and range. An unaddressed
question in research is considering which motor controller to use, as again heavier
controllers tend to be more efficient. This example demonstrates a key point, that
performance parameters of commercial off the shelf components must be available
for use in detailed electric aircraft design.
1.2 Commercial Off the Shelf Components
Improving range and endurance for electric aircraft relies on knowing key per-
formance parameters related to their propulsion system. An accurate knowledge of
motor performance, design trade offs, and power losses are needed for comprehensive
aircraft design. Knowing how the motor?s efficiency changes with RPM and torque
allows gearing to be designed such that both the rotor and the motor operate at
peak performance. More efficient propulsor designs require less power, translating
into longer flight times, increased range, and a more flexible mission profile.
Unfortunately, many BLDC motor suppliers do not provide detailed informa-
tion on their motor?s performance. A typical brochure on BLDC motor performance
is shown in figure 1.3, and shows the motor?s DC current draw at various DC volt-
ages, with different rotors attached to the motor. While this information is useful
5
to hobbyist, it becomes largely irrelevant when considering designing an electric air-
craft. The reason is that it does not infer how the motor performance will change
when a different load is applied, or when a different motor controller is used. For ex-
ample, using a more efficient motor controller may deliver more power to the motor,
thereby changing its performance for the same amount of input power.
Figure 1.3: Information typically given for BLDC performance.
Given the lack of information from manufacturers, several design and engi-
neering questions remain unaddressed:
1. What operating conditions produce the maximum efficiency for the motor and
the ESC?
6
2. How does motor performance change when different rotors are used?
3. How does the change in throttle setting impact the output torque and speed
of the motor?
4. What is the peak power output of the motor? Max continuous?
5. How does changing motor controller effect the motor?s performance?
Due to the coupled nature of the controller and the motor, these are quite involved
questions to answer. Additionally, from manufacturer specified details, designers
are unaware where the power loss in the system is occurring, either in the motor
controller, the wiring, or the motor itself.
When designing electric aircraft, it is critical to the designer to know which
motor and motor controller to use. To do this, key performance information about
the motor is required, however mostly not provided by the manufacturer. Even less
understood are the Electronic Speed Controllers (ESCs), which are responsible for
open loop control of the motor. Exact knowledge of the maximum efficiency points of
an ESCs and BLDC motors allow designers to evaluate tradeoffs between selecting
different motors with specific ESCs. This problem is unexplored in research, as
only moderate attempts have been made to better understand power loss for BLDC
motors and ESCs. A clear need is established - to develop an apparatus for testing
commercial off the shelf components to determine their performance criteria.
7
1.3 The Need for a Dynamometer
A dynamometer was created to study BLDC motor and ESC characteristics.
A dynamometer is a device that is used to vary the load on a motor, while simul-
taneously record the electrical and mechanical power. With this information, an
analytical model was developed to help understand propulsion system performance.
A key point in this work is to study both the ESC and BLDC motor performance,
with ESCs being a special focus of this work. It was found that ESC peak efficiency
points are different than BLDC motors, which is critical knowledge to designers.
Figure 1.4: Components considered for equivalent circuit modeling.
A typical propulsion system used in a University research setting is shown in
figure 1.4. It contains a DC power supply, the ESC, and a BLDC motor. There are
three types of power in this setup: the DC power, the AC power, and mechanical
power. The DC power supply emits a steady voltage and current to the ESC, the
ESC modulates this power into the AC voltage and current signals for the motor,
and the motor converts this electrical power into mechanical power. To isolate the
performance of the ESC and BLDC motor separately, both a DC and AC power
analyzer are used. As a result, there are three different efficiencies one may discuss
8
for this system: the ESC efficiency ?ESC , the motor efficiency ?m, and the overall
system efficiency ?sys. These definitions can be expanded below:
PAC PAC
?ESC = = (1.1)
PDC VDCIDC
Pm Q?
?m = = (1.2)
PAC PAC
PAC Pm Q?
?sys = ?ESC?m = = (1.3)
PDC PAC VDCIDC
where PDC , PAC , and Pm are the DC, AC, and mechanical powers, ?ESC , ?m, and
?sys are the ESC, motor, and system efficiencies, and VDC , IDC , Q, and ? are the DC
voltage, DC current, motor output torque, and motor rotational speed, respectively.
A diagram relating the different powers and efficiencies can be found below:
Figure 1.5: Propulsion system power and efficency breakdown for an ESC and BLDC
motor.
Each efficiency relates to a particular transformation of power in the propul-
sion system. ESCs transform DC power into AC power to run the motor, whereas
the motors transform AC power into mechanical power. Increasing the ESC effi-
9
ciency allows for more power to be available to the motor, meaning that a different
performance is expected for the same amount of input power. Typically researchers
discuss equation 1.3 when discussing propulsion system performance, only relating
the DC input power to the mechanical output power. This is a high-level view, as
it does not distinguish between motor and controller inefficiencies during the power
conversion process. By properly recording the AC power of the system, researchers
can use this information to search for high efficiency ESCs and BLDC motors.
As a result, testing motors is coupled with testing ESCs. Without the motor,
the ESC does not output any AC signals and without the ESC, the motor cannot
spin. It is not possible to test one without the other. The work in this thesis
focuses on developing an experimental apparatus to study motor performance. This
information is then used to develop an analytical, first principles based model for
both the ESC and the BLDC motor.
1.4 Outline of This Thesis
Each Chapter in this thesis is self contained and independent. However, sub-
sequent chapters build off the foundations of previous Chapters, which is critical for
the modeling attempt. The organization of this thesis is as follows:
Chapter 2: Introduction to motor performance and component breakdown. Before model-
ing attempts can be made, it is essential to understand the critical components
in DC motors and ESCs. Key concepts such as the brushed motor model are
covered to give additional context into motor operation.
10
Chapter 3: A detailed explanation of dynamometer components, sensors, and power an-
alyzers is provided. Here the DC, AC, and mechanical power analyzers are
explained, with special attention is given to measure the 3 phase AC power.
The final 2 DAQ configuration is described in detail.
Chapter 4: An analytical model is derived for both the BLDC motor and ESC. The goal
of this section is to identify parameters that do not change with different
operating conditions. Both the BLDC motor and the ESC are modeled using
equivalent circuit diagrams.
Chapter 5: Experimental results for 7 BLDC motors and 3 ESCs are given at DC voltages
of 7.2 V, 11.1 V, and 14.8 V. A description of extracting performance variables
from experimental data is given. Additional topics, such as motor variation
testing and rotor/motor/ESC pairing are addressed at the end of the chapter.
Chapter 6: Conclusions about this work and areas for future work are described. Future
work includes enlarging the dynamometer to handle larger motors, as well as
closed loop control of a motor, ESC, and rotor.
11
Chapter 2: DC Motors
2.1 Basic Motor Performance - the Brushed Motor Model
Motors convert electrical power into mechanical power. Before describing the
different types of motors, a brief discussion on motor performance is required. An
equivalent circuit can be drawn, which encapsulates both the electrical and mechan-
ical properties of the system.
A motor is a series of coils and magnets, which operate under the principles
of electromagnetic induction. The core concept of any electric motor is that passing
current through a wire creates a magnetic field around the wire and this is used
to spin the rotor. If a separate magnetic field is created and is brought close to
the wire, the interaction of these magnetic fields causes an electromagnetic force to
develop on the coil according to Flemin?s left hand rule [12]:
FB = BlI (2.1)
Now consider if the wire is bent into a loop like figure 2.1.
The current on the left side of the centerline passing through the magnetic
field B causes a force to be generated at a distance r on segment l. The result is a
12
Figure 2.1: Generation of a magnetic torque for a current.
magnetic force which is generated out of the page. However, since that wire is bent
in a loop, the same current must now pass through the right hand side of the wire l,
meaning the result of this interaction is a magnetic force into the page. This results
in a torque Q to be generated and is equal to:
Q = 2rFB = 2rBlI (2.2)
This forms the very basic relation that the output torque of the motor is proportional
to the current, by some constant KT :
Q = KT I (2.3)
Now consider a stationary magnetic field and the wire is moving perpendicular
to the field with some speed v, shown in figure 2.2. Farraday?s law of electromagnetic
induction dictates that a voltage differential must be created to oppose the change
13
Figure 2.2: Generation of back-EMF in a moving wire.
in magnetic flux:
VE = vBl (2.4)
This shows that a voltage is generated across the wire and is proportional to the
speed of the moving wire. If the wire was moving around in a rotor, the speed would
be proportional to the RPM of the motor:
v
? =
r
Thus, for a motor, the back electromagnetic force, or back EMF, generated in the
coil windings can be written as:
VE = KE?
Where KE is some proportaionality constant, relating voltage induced in the coil to
RPM.
Both KT and KE are fundamental constants in motor theory. They are func-
tions of the magnetic field strength, material design of the motor, number of turns
in the inductors, and overall electrical design of the motor. A commonly employed
14
tool for steady-state analysis is to simplify the motor into an equivalent electrical
model. To demonstrate the key concepts of motor performance, a sufficient motor
model [12?14] is shown in figure 2.3.
Figure 2.3: Simplified motor electrical circuit, the arrows indicate component voltage
drop.
An equivalent circuit is developed, that contains a motor resistance Rm and
an induced back voltage KE? from the rotating magnets. Based on figure 2.3, the
sum of voltages around the loop is equal to 0:
VDC = VRm + VE
Substituting terms yields the following:
VDC = IDCRm +KE? (2.5)
Output torque is related to the DC current, from equation 2.3, but must be aug-
mented with an additional torque to overcome the static friction of the motor:
Q = QEM ?Qstatic
15
Q = KT (IDC ? Io) (2.6)
Where the term KT Io is the torque required to overcome the internal friction of the
motor. Equations 2.5 and 2.6 are considered to be the fundamental equations of
an electric motor. These two equations can be combined together to determine the
linear relationship between RPM and torque:
KT KTKE
Q = [VDC ? IoRm]? ? (2.7)
Rm Rm
which is plotted in figure 2.4. Equation 2.7 states that the Q vs RPM relationship is
linear in nature, with a slope related to the parameters of the motor and a y-intercept
related to the DC supply voltage VDC .
Figure 2.4: Left: Torque vs RPM relation. Right: Torque vs current relation.
There are two key points along the Q vs RPM line: the stall torque and the
no-load speed. Stall torque is defined as the torque generated from the motor when
16
the shaft is no longer rotating, and is the maximum torque the motor can produce.
Such a torque is useful to hold positions of shafts at a constant angle, for example
in a robot arm. The other point is the no-load speed, which is the RPM that the
motor shaft spins when no load is applied to the motor. This is the maximum RPM
that the motor shaft spins.
The no load speed can be found by setting torque equal to 0 in equation 2.7:
VDC ? IoRm
?NL = (2.8)
KE
Similarly, the stall torque can be found by setting RPM to 0 in equation 2.7:
KT
Qstall = [VDC ? IoRm] (2.9)
Rm
Both of these are significant as they relate to the maximum performance of
the motor: either maximum rotational speed or maximum load. The torque-RPM
line can be separated into three regions, which relates to the load on the motor and
is depicted in figure 2.5.
17
Figure 2.5: Motor operation regions.
Region 1: Region of continuous use. High RPM and low torque. This region contains
the location of the maximum efficiency of the motor. Rotors should be sized
to operate in this region during trim.
Region 2: Intermittent use region. Region contains the location of maximum power
output for the motor. Motor operation should only enter this region during
maneuvers.
Region 3: Limited use region. High torque and and low RPM. Region contains the cur-
rent limit for either the motor or the controlling electronics. Motor operation
should only enter this region during protracted maneuvers. Extended time in
this region could cause thermal effects to degrade the motor?s performance,
both electrically and/or physically.
A properly sized motor operation should be contained in region I, with only short
18
term use in the other regions during take-off, landing, or maneuvers. Here the load
on the motor is lowest, and the motor operates close to its maximum efficiency point.
Power output from the motor is the product of the mechanical output torque
with the mechanical RPM:
Pm = Q? (2.10)
and can be reformulated as:
KT KTKE
Pm = ( [VDC ? IoRm]? ?)?
Rm Rm
KT KTKE
Pm = [V
2
DC ? IoRm]? ? ? (2.11)
Rm Rm
The maximum power rotational speed ?max P can be found by differentiating equa-
tion 2.11 with respect to RPM:
dPm KT KTKE
= [VDC ? IoRm]? 2 ?max P = 0
d? Rm Rm
1 VDC ? IoRm ?NL
?max P = ( ) = (2.12)
2 KE 2
Maximum power lies at half of the no-load speed ?NL. Maximum power occurs
when the distrubition between torque and RPM is balanced at half of no load speed
and half of stall torque. Equation 2.11 can be plotted and is shown below:
Note that at ?NL and Qs the mechanical output power is 0. Again, this
corresponds to the limits of the motor, at maximum RPM there is no load and
19
Figure 2.6: Motor output power vs RPM.
thus no output power. Since the input and output powers have been identified, the
overall efficiency can be evaluated.
The ratio of mechanical output power to DC input power is the system effi-
ciency:
Pm Q?
?sys = =
PDC VDCIDC
K VDC?KE?T ( ? Io)
? = Rmsys ? (2.13)V (VDC KE?DC )Rm
Maximum efficiency of the system occurs at ?opt, and can be found by differentiating
equation 2.13 with respect to the RPM, to find:
d?sys KT (K
2?2E ? 2KEVDC? + (V 2DC ? IoRmVDC)= = 0
d? VDC(VDC ?Rm?)
?
VDC ? IoRmVDC
?opt = (2.14)
KE
20
Figure 2.7: System powers and efficiency vs RPM.
The optimal RPM for system performance is typically between 75% to 95% of
the motor?s no-load speed, and can have a maximum efficiency of 95% for brushless
motors and 75% for brushed motors. A typical plot of a motor?s efficiency curve is
shown in figure 2.7. Motor performance is augmented by changing the voltage of
the motor, which is done using user throttle.
2.1.1 User Input Throttle
The motor equations shown so far assumed that the motor was operating at
the full supply voltage VDC . A more realistic model for motor operation is shown
in figure 2.8. User throttle TR is a value between 0-1, and has been normalized
appropriately. The throttle is analogous to the gas pedal on a car, depending on
how far down the gas pedal is pressed, more or less fuel enters the engine. With
more fuel, the engine produces more power, and the car responds appropriately.
A voltage regulator is a device that has an output voltage proportional to
21
Figure 2.8: Motor efficiency vs RPM.
some input signal, called the throttle TR. This means that the output voltage of the
regulator, which is the voltage acting across the simplified motor circuit, is VDCTR.
Meaning that at 60% throttle, the motor has 0.6VDC acting across the circuit. The
regulation and modulation schemes employed for brushed and brushless motors are
discussed in section 2.3.
Changing the voltage seen by the motor is accomplished by changing the throt-
tle TR into the voltage regulator. This changes the voltage acting across the motor,
and thereby changes the performance of the motor. Applying Kirchhoff?s voltage
law to the right side of figure 2.8 yields:
VDCTR = IDCRm +KE? (2.15)
the torque-current relation, equation 2.6, remains unchanged for a given VDC . How-
ever, the user throttle TR appears when evaluating the motor?s performance:
KT KTKE
Q = [VDCTR ? IoRm]? ? (2.16)
Rm Rm
22
KT ? ? KTKEP = [V T I R ]? ?2m DC R o m (2.17)
Rm Rm
K (VDCTR?KE?T ? Io)
? Rmsys =
V (VDCTR?
(2.18)
KE?
DC )Rm
changing the input throttle to the motor shifts the plots in a predictable manner.
The change in performance for different throttles is shown in figures 2.9 and 2.10.
Figure 2.9: Torque-RPM curves at different throttles.
A reduction in throttle changes the no-load speed and stall torque of the motor
by:
VDCTR ? IoRm
?NL(TR) =
KE
KT
Qstall(TR) = [VDCTR ? IoRm]
Rm
Throttle has a linear relationship with the torque-RPM curves, and can be
though of as shifting the plots left or right. Increasing throttle allows for a higher
no-load speed and higher value of stall torque to be achieved, which consequently
23
shifts the maximum power point higher up the power axis and to the right in RPM.
This is because the maximum power point is still half of the no-load speed, and
since no-load speed increases, so must the maximum power point.
Figure 2.10: System performance at different throttles.
Figure 2.11: Constant power curves, solid lines indicate torque-RPM relationship
whereas dotted lines indicate constant power curves.
The user or closed loop controller, changes the throttle value to the voltage
regulator. This regulator acts like a value, and changes the voltage available for the
24
motor. Changing the motor voltage allows more or less power to be delivered to the
motor, and has 2 effects: faster RPM for the same load, or more current (and thus
more torque) for the same RPM.
Certain motor applications require constant power, and so it is important to
know how these power curves affect throttle, RPM, and torque requirements. For a
given power P , the torque required to produce a constant power is Q = P/?. An
example is shown in figure 2.11, with the left plot showing the required torque to
generate peak power at 70% throttle Pmax|T =0.7. For 70% throttle clearly only oneR
torque and RPM value satisfies this power requirement. However the figure on the
right shows how the peak powers curves of a given throttle setting only intersect the
torque-RPM lines for throttle settings above the original setting. Although the peak
power is only reached once for a given throttle, it is reached an infinite number of
times for the throttle settings above it. However, as the peak power increases, fewer
throttle settings are available to meet the requested torque and RPM requirements.
Now that the basics have been discussed, key terms for motor performance
must be identified.
2.1.2 KT , KE, and KV
Precise definitions of KT , KE, and KV are required for discussing brushless
motors. These quantities are often misquoted, so it is important to remember the
context that are provided. Incorrectly using one quantity for another can lead one
to arrive at incorrect conclusions. These quantities are defined as:
25
1. KT [Nm/A] - the amount of torque produced for each amp of current
2. KE [Vs/rad] - the amount of back EMF (voltage) produced for each rad/s of
mechanical rotation
3. KV [RPM/V] - Manufacturer specified ?NL per volt of DC voltage applied to
the ESC (?NL =KV VDC). This quantity is not motor specific as it includes
the ESC.
Typically manufactures provide only the KV rating for a BLDC motor, but
this is not useful for engineering applications. Use of KV only tells the motors
no-load speed at full throttle, but does not describe how the torque-RPM evolves.
For this, KT and KE, as well as additional motor parameters, are required. Due
to a lack of manufacturer specified modeling values, only general statements can
be made. Typically, a lower KV motor translates to a motor with a higher torque
constant KT . For example, if choosing between a 935 KV and 2500 KV BLDC
motor, the 935 KV motor would be selected for operation with larger/slower rotors.
However, even this simplification may not always be adequate, as manufactur-
ers typically do not specify which ESC was used. During testing for this work, it
was found that a 920 KV motor has KT = 13.20 mNm/A, whereas a 935 KV motor
had a KT = 13.85 mNm/A when tested at VDC = 7.2V with a MultiStar 30A ESC.
The reason for this dependency between KV and KT or KE is that KV involves
the motor and the ESC performance, whereas KT and KE are only related to the
motor performance. In terms of practical engineering application, KV value does
not accurately describe the motor?s performance.
26
There are multiple ways of defining KT and KE. References [12,13,15] include
definitions of BLDC motors that rely on the peak voltage Vpk and peak current Ipk.
The reason for this is that BLDC motors operate on AC waveforms, meaning there
are multiple ways to define KT and KE. These values of KT and KE are given in the
context of a model, and so usually are easy to interpret. The ideal ratio of KT/KE
is of importance when discussing BLDC motors. Based on the waveform properties
and the motor type, the following chart can be obtained and is taken from [13]:
Table 2.1: Ideal KT and KE based on motor type.
DC Squarewave 3 Phase Sinewave 3 Phase
KT [Nm/A] Q/IDC Q/Ipk Q/Ipk
KE [Vs/rad] VE/? Vpk,LL/? Vpk,LL/?
?
KT /KE 1 1 3/2
For this work, the ratios of voltage and torque are related using the rms value
of voltage and current, as this is easy to calculate during a test. Additionally,
determining the peak voltage and current usually is difficult due to the presence of
electrical noise. Further information on the model used in this work can be found
in Chapter 4. It is up to the reader to infer the proper use of the quoted values in
any manual or data sheet. For now, a detailed component breakdown of brushed
motors, brushless motors, and ESCs are given.
2.2 Motor Component Theory
Brushed and brushless DC motors operate with a fixed DC supply voltage,
hence the name DC. There are two major differences between brushed and brushless
27
motors:
1. Commutation of a brushed motor is built into the device, whereas in a brush-
less motor it is contained in the ESC
2. Commutation methods for a brushed motor are mechanical, whereas brushless
motors are commutated with electronic methods
Next sections describe the construction and operation of the brushed motors, brush-
less motors, and ESCs.
2.2.1 Brushed DC Motors
Brushed motors are the simplest types of motors. They are cheap, easy to use,
and available in all ratings. The disadvantages of these motors is that they have a
low power-to-weight ratio and they are not suited to long term operation [16].
Every motor contains fundamentally two parts: the rotor and the stator. The
rotor is the component that rotates and transmits the mechanical torque to the
load, whereas the stator remains stationary and houses the internal components of
the motor. Commutation is the process of reversing the direction of current into the
motor to keep it spinning, and can be accomplished either mechanically (brushed)
or electronically (brushless).
Use of a mechanical commutator means that physical contact with the rotor
must be made to continue reversing the direction of current within a motor. A
commutator is a device which ensures an electrical contact from the voltage supply
VDC and the rotating coil of wire is kept constant.
28
Figure 2.12: Components in a brushed motor.
Figure 2.13: Commutator and brushes for a brushed motor.
Coil wires are connected to the commutator so the commutator rotates with
the rotor. Carbon brushes connect the rotating wire to the non-rotating stator, and
are used to electrically complete the circuit. Construction of a brushed motor is
relatively simple, as shown in figure 2.12.
These brushes are a key distinction in a brushed motor, hence the name. The
brushes ensure consistent current flow through the motor. To better illustrate how
29
a brushed motor is commutated, consider figure 2.14. Here a wire is wrapped to
form a loop within a permanent magnetic field, similar to figure 2.1. The flow of
current generates a torque as a result of equation 2.3. If there was no way to reverse
Figure 2.14: Generation of a magnetic torque for a current.
the direction of current, then the area of the coil becomes perpendicular to the field
at the top of the rotation. At this point, no torque will be generated because the
moment arm is 0, however the rotor has some angular velocity and so it continues
to spin. If the direction of current does not change, then the resulting magnetic
force generated after the midpoint will start to slow down, and eventually reverse
the rotation of the rotor. To prevent this, the direction of current must be reversed
with a commutator, depicted in figures 2.15 and 2.16.
Figure 2.15: Before midpoint.
30
Figure 2.16: After midpoint.
By constantly changing the direction of current in the coil, it ensures contin-
uous rotation of the rotor. Additional coils of wires are used to ensure the torque
output remains constant. Use of springs, brushes, and rotating components means
that considerable wear and tear takes place during motor operation. Sparking can
also occur during a commutation event, resulting in a loss of power.
Control and simplicity for a brushed motor is not to be overstated. The model
outlined in section 2.1.1 is exceptionally well suited to describe a brushed motor.
To control the speed of a brushed motor, a modulated signal called an analog duty
cycle signal is used.
A voltage modulation board is placed between the DC power supply and the
brushed motor. The microcontroller communicates with this board and passes the
user throttle TR to the voltage modulation board. The modulation board typically
consists of MOSFETs, which can either be in an on or off state. By changing the
amount of time the MOSFET is open, or allowing current to flow, the effective
output voltage can be controlled. The throttle TR commands how much voltage the
motor sees by controlling the duty cycle of the motor. The effective output voltage
31
is a function of the throttle command TR:
ton
VDC,eff = VDCTR = VDC (2.19)
ton + toff
In this context, the user throttle is the same as the analog duty cycle ratio. Reversing
the direction of motor rotation can be achieved by reversing the polarity of the
motor. By flipping the hot and ground terminals, current will flow in the opposite
direction as before and thus, the motor spins in the opposite direction. Typically the
control of motor speed and direction is accomplished within a single device, called
an H-bridge, shown in figure 2.17.
Figure 2.17: Left: Positive brushed motor rotation. Right: Reverse brushed motor
rotation.
H-bridges consists of 4 MOSFETS, arranged in a 2 per pair fashion, which
allows current to enter the motor in either a positive or negative direction. By
determining which pair of MOSFETs are conducting or nonconducting, the direction
of motor current entering the motor can be controlled. Regulating the length of time
that the MOSFETs are conducting controls the current entering the motor. When
the positive direction signal is sent, MOSFETs Q1 and Q4 are activated, allowing
32
current to flow in the positive direction and results in the motor spinning in the
positive direction. When the reverse direction signal is sent, MOSFETS Q2 and
Q3 are active and the direction of current is reversed, resulting in a reversed RPM.
A short will occur if Q1 and Q3 or Q2 and Q4 are opened at the same time. The
basic components to build and control a brushed motor are used in brushless motor
operation.
2.2.2 Brushless DC Motors
Brushless motors operate on the same principles as brushed motors, but are
augmented to allow for electronic commutation. Allowing for electronic commuta-
tion greatly enhances the life of a DC motor, as the physical brushes in a brushed
motors degrade over time. The construction of a brushless motor is almost identical
to that of a brushed motor, with two major differences:
1. No mechanical carbon brushes
2. Additional motor phases to allow for electronic commutation
Electronic commutation of a BLDC motor will be explained in section 2.3.
Construction of the BLDC motor is shown in figure 2.18. Just like a brushed motor,
BLDC motors consist of a rotor and a stator. The stator has three phases of wires
wrapped around a central core to enhance the magnetic field of the motor.
33
Figure 2.18: Components in a brushless motor.
Figure 2.19: Left: 3 pole pair motor. Right: 7 pole pair motor.
34
These phases can be combined into either a wye (Y) or delta (?) configura-
tion, and are discussed in more detail in section 3.5.2. The stator has mounting
holes on the bottom to allow for bolts to attach the motor to the aircraft?s struc-
ture. Conducting wire is wrapped in a loop to form an inductor, and then placed
around the core?s ferromagnetic material to enhance the magnetic properties of the
electromagnets. These loops are referred to as slots, and are part of the stator.
The rotor consists of magnets with alternating polarity facing towards the
center of the motor. An important concept with BLDC motors is the concept of
pole pairs. Each north and south pole facing the center of the rotor makes a single
pole pair, meaning that:
number of magnets
Np = number of pole pairs = (2.20)
2
The electrical and magnetic frequencies are related to one another by this ratio Np.
If there were only 2 magnets, then for every 1 mechanical revolution, the induced
back-EMF would experience 1 electrical cycle. If the number of magnets are doubled
to 4, then for half a mechanical revolution, the back-EMF experiences 1 electrical
cycle. The electrical angle ?e is necessary for electronic commutation, and is related
to the mechanical angle ? by [15]:
?e = Np? (2.21)
?e = Np?
35
where Np is the number of pole pairs of the motor, and ? is the mechanical RPM.
This relationship for a 3 pole pair and 7 pole pair motor is shown in 2.19. The
takeaway is that for a given electrical frequency ?e, doubling the mechanical rotation
speed ? has the effect as doubling the number of poles Np (or quadruplicating the
number of magnets). All motors used in this study are 7 pole pair motors, as this
arrangement is most common for commercially avialble motors.
As mentioned previously, the stator houses the slots for the motor, which are
the inductors inside a brushless motor. The ratio of poles to slots considered the
gearing for the motor, and can augment the motor?s KT and KE value. A high
number of pole pairs is indicative of a motor that has a higher KT , whereas fewer
pole pairs means that the motor has a lower KT and thus a higher RPM. Pole pairs
are related to the electrical frequency of the signals for a brushless motor, and are
interpreted by the ESC.
2.3 Electronic Speed Controllers
ESCs commutate, or spin, the BLDC motor by electronically regulating the
direction of current into the motor by alternating which phases of the motor are
energized. By changing the direction and strength of the magnetic field, a push-pull
mechanism is developed that causes the rotor to chase the magnetic field. Metal
Oxide Silicon Field Effect Transistors (MOSFETs) are used to regulate the direction
of current into the BLDC, by changing which MOSFETs are conducting at a time.
A typical ESC and MOSFET configuration is shown in figure 2.20.
36
Figure 2.20: 6 MOSFET configuration for an ESC.
Six MOSFETs are used to commutate a BLDC motor, with 2 conducting at
any time. If Q1 and Q5 are conducting, then current forms the DC supply, through
Q1, into the motor by line A, and then out of the motor from line B, through Q5, and
then returns to the DC supply. During this time, iA > 0 and iB < 0, which changes
the orientation of the magnetic field inside the motor. One half of the coils are
energized with positive current, which pulls on the rotor via magnetic forces. The
second half of the coils are energized with negative current, resulting in a magnetic
repulsion which pushes the rotor.
Most low cost commercial ESCs and BLDC motors use 120o, 6-step commuta-
tion methods. The 120o refers to the conduction interval of the MOSFET, meaning
each MOSFET is on for 1/3rd of an electrical revolution, with 2 on at any one time.
Coils are connected together, in an either a Y or a ? configuration, to allow the same
current to pass through the both positively and negatively charged coils. Six step
refers to six commutation intervals, required for one electrical revolution, ?e = 0
o to
37
360o.
Figure 2.21: Detailed electrical schematic of a sensorless ESC.
The primary component in an ESC are the 6 MOSFETS, which are responsible
for commutating the BLDC motor. Each MOSFET is controlled by a gate driver,
which in turn is controlled by the motor controller IC. The motor controller chip is
responsible for sensing the voltage of the motor phases and comparing this to its own
commutating logic. The commutating logic is the set of instructions that dictates
which MOSFETs, and therefore which phases, are active at one time. Commutating
logic timing is heavily dependent on the location of the rotor. Chaning the active
MOSFETs changes which lines are pushing and pulling the rotor?s magnets, and
hence mistiming commutation intervals results in a loss in performance and at worst
an electrical short.
38
To detect the rotor?s position, two different methods are used: sensored and
sensorless ESCs. A sensored ESC uses hall effect sensors to detect the rotor?s posi-
tion. As the alternating magnetic fields of the motor pass by the hall effect sensor,
a series of digital 1s and 0s are read into the ESC?s microcontroller. The exact
sequence of 1s and 0s from the three hall effect sensors are used to determine the
rotor?s exact location [17,18]. The other type of ESC is a sensorless ESC.
Sensorless refers to an ESC which detects the rotor?s position by the use of
back-EMF. The motor?s voltage and current waveform are well defined, and thus by
tracking the back-EMF voltage, the electrical angle ?e can be deduced. To sense the
voltage of each phase, a resistor network is employed to reduce the voltage of the
phase to within the range of the microcontroller?s analog-to-digital converter. As
the rotor spins, it induces an EMF in the coil wingdings of the phases of the motor.
Sensorless ESCs are referred to as sensorless in that they do not have hall effect
sensors to measure the rotor?s location. Instead they sense the voltage waveforms
of the motor to infer the rotor?s location. The use of sensorless or sensored ESC
depends upon the application.
Sensorless ESCs are the more popular choice for today?s electric aircraft,
whereas sensored ESCs are used more in land based applications. Sensorless ESCs
operate best when the rotor is spinning at high RPMs, since the back-EMF is pro-
portional to the RPM of the rotor. When at low RPM, the maximum value of the
back-EMF (VE = KE?) is not large enough for accurate detection by the microcon-
troller, and thus the ESC can not properly commutate a BLDC motor. This is not
a problem for high RPM rotors for electric aircraft, but is a problem for accurate
39
small angle control or low RPM. Since the sensored ESC can accurately detect the
rotor?s location for all RPMs, they work well at low speeds. Typical applications
involve ground vehicles or devices which use high torque, low RPM operation.
Figure 2.22: Detailed electrical schematic of an ESC.
Figure 2.23: Component identification of a sensorless ESC.
A detailed component breakdown of a sensorless ESC is given in figures 2.23
and 2.22. Controlling the MOSFETs is accomplished with a gate driver, which is
an electrical component that interprets the microcontrollers signal and augments it
to drive the gate terminal of the MOSFET. A series of additional components, like
40
diodes, capacitors, and resistors are used to increase the performance of an ESC.
From the aforementioned list, the most important components are resistors, as they
are essential in determining the phase voltages and thus commutating the BLDC
motor.
2.3.1 Commutation Logic
With the ability to sense the voltage wavforms of the BLDC motor, the com-
mutation logic can be stated. This logic presents a mapping from the voltage states
to the active MOSFETs. The delicate dance between transistors and current keeps
the motor spinning. Although the method for determing the rotor?s position can
vary, the commutation logic is the same for both sensored and sensorless ESCs and
is given by table 2.2.
Table 2.2: 6-step 120o commutation intervals in an ESC and BLDC motor.
?e [deg] Active Switches vAN? iA vBN? iB vCN? iC
0 ? 60 Q1, Q5 + + ? ? floating 0
60 ? 120 Q1, Q6 + + floating 0 ? ?
120 ? 180 Q2, Q6 floating 0 + + ? ?
180 ? 240 Q2, Q4 ? ? + + floating 0
240 ? 300 Q3, Q4 ? ? floating 0 + +
300 ? 360 Q3, Q5 floating 0 ? ? + +
Commutating the rotor depends on the electrical angle ?e of the motor, and is
related to the mechanical rotation by equation 2.21. Commutation logic follows a
very simple rule - when the voltage is high the current is high, and when the voltage
is low the current is low. High or low refer to the values of the voltages and currents
being either positive or negative.
41
Figure 2.24: Commutating a BLDC motor with an ESC, ?e = 0
o to 60o.
Consider section ? = 0oe to 60
o of table 2.2 and the ESC diagram of figure
2.20. Having switch Q1 open means positive voltage and current flows from the
high side of the DC circuit into the motor. A second switch is required to connect
the circuit to ground, which is Q5 in this case. Positive current is defined when
current is flowing into the motor, so for phase B this means a negative current is
passing through the wire, since it is exiting the motor. A similar case can be said
about the voltage of vBN?(t), making this value negative. No switches are active for
phase C, meaning no current flows in or out of this phase. The voltage is deemed
floating, since the back-EMF is inducing a change in voltage from either positive to
negative or vice versa. The direction of the change in back-EMF can be found by
referencing the next sector. Switches Q1 and Q6 are active, meaning that current
is exiting the motor through line C. This means that the voltage of C must be low,
and that it would have needed to transition from positive to negative in the previous
commutation interval. This sequence continues for the remaining intervals, and is
represented pictorially in figures 2.25-2.29.
42
Figure 2.25: Commutating a BLDC motor with an ESC, ? = 60o to 120oe .
Figure 2.26: Commutating a BLDC motor with an ESC, ?e = 120
o to 180o.
Figure 2.27: Commutating a BLDC motor with an ESC, ? = 180oe to 240
o.
43
Figure 2.28: Commutating a BLDC motor with an ESC, ?e = 240
o to 300o.
Figure 2.29: Commutating a BLDC motor with an ESC, ? = 300o to 360oe .
This cycle repeats itself as long as the motor is spinning. Senorless feedback
works well when the motor is spinning, but is ineffective at lower RPMs. Commuta-
tion timing is dependent on the rotor position, and mistiming commutation events
results in a loss in performance. For sensorless ESCs, precise commutation timing
is achieved using zero-crossing point (ZCP) detection schemes [18?20]. The exact
moment of when to switch from one set of switches to the next is imperative for
proper motor commutation. ZCP based methods work to detect when the floating
44
phase of the motor changes sign value. By recording the phase voltage of the float-
ing signal, the microcontroller can detect when a ZCP occurs. By knowing the time
between 2 ZCPs, and knowing they must have been 120o electrical degrees apart,
the speed of the electrical signal ?e can be determined.
Operation of the ESC relies on accurate knowledge of the electrical angle to
commutate the motor ?e. Figures 2.25 to 2.29 depict the necessary steps to keep
the rotation of the motor continuous. Additionally circuitry is required to sense
the phase voltages, filter noise ripples, and freewheel voltage spikes. ESCs are
fundamentally 6 switches, known as MOSFETs, which are the focus of the next
section.
2.3.2 MOSFET Operation
The primary means of controlling induction motors are the MOSFETs. MOS-
FETs, function as electronic switches, that open or close rapidly based on a control
signal. These MOSFETs can be combined into useful arrangements, to allow for
precise control of the motor, such as an a H bidge or ESC configuration. To under-
stand how MOSFETs work, we must first understand the basic working principle
behind MOSFETs. A MOSFET consists of three terminals, shown in figure 2.30:
1. Drain - terminal connected to +VDC
2. Source - terminal connected to load
3. Gate - terminal which controls if current is flowing between terminals
45
Figure 2.30: MOSFET model.
These names are based on the electron current definition, because when the MOS-
FET allows current to flow, electrons enter the source terminal and exit out through
the drain terminal. However, this work discusses MOSFET operation using conven-
tional current. For a standard NPN channel MOSFET, when no voltage is applied
to the gate, no current is allowed to flow from the drain to the source.
To control current flow between MOSFET terminals, a voltage (typically +5V )
is applied to the gate. The transistor allows current to flow as long as a voltage
is applied to the gate, which is above some threshold value. To stop current flow,
simply drop the gate voltage to 0. The use of a MOSFET is analogous to a switch,
Figure 2.31: MOSFET model.
46
and can be used to augment the physical circuit into multiple equivalent circuits
based on the state of the MOSFET, with a simple example shown in figure 2.31.
MOSFETs are useful when combined with additional circuitry that enables
closed loop control of a particular system. A microcontroller, or some other type of
MOSFET driver, is paired with a set of MOSFETs, and the controller regulates how
much time the MOSFET is active. The controlling signal is typically a pulse width
modulated (PWM) signal, with the length of time regulating the output voltage of
the device.
Now consider if a microcontroller ?C is added to the previously examined
circuit, shown in figure 2.32. This microcontroller controls how long the MOSFET
is open or closed, and sends a 50% duty cycle of 5V to the gate of the MOSFET. This
Figure 2.32: MOSFET circuit with a microcontroller ?C.
input duty cycle is mimiced on the output of the MOSFET, where the same duty
cycle waveform appears except with an augmeted amplitude. During on operation,
the gate and drain are connected, meaning they share the same value of +VDC .
47
During off operation, no voltage is dropped across the load z, meaning that the
source is connected to ground and has a value of 0V . The result is that the input
duty cycle with an amplitude of 5V has been turned into a duty cycle with an
amplitude of VDC . This means that powerful circuits can be created.
An effective modeling tool is to consider the output voltage of the MOSFET as
the average value of the source voltage over one waveform. The duty cycle ratio, or
throttle, sets the effective output voltage of the circuit. In reality, the output of the
MOSFET are voltage pulses between 0 and VDC , however by modeling the output
as the average value of voltage signal, a single simplified model can be determined.
Figure 2.33: Output voltage of VSN based off duty cycle duration.
At no throttle, the MOSFET is always off, meaning that no voltage is seen at
the source terminal. At full throttle, the MOSFET is always on, meaning the source
voltage is equal to the supply voltage. When the duty cycle is neither 0 or 100%,
the output voltage is the combination of ?on? voltage and ?off? voltage. The average
voltage of VSN is given by equation 2.19. Common circuits involving MOSFETs
48
are switch mode power supplies, such as: DC-DC step up boost converters, DC-DC
step down buck converters, and H-bridge topologies [21].
Combining MOSFETs with BLDC motors creates a complicated and excit-
ing circuit to analyze. The voltage and current waveforms of a BLDC motor are
trapezoidal and rectangular, respectively. The nonlinearity of the voltage waveform
must be taken into account when designing BLDC motors. User throttle changes
the duty cycle of the MOSFET, effectively scaling the output voltage in a controlled
way. However, instead of seeing 0 to VDC on the output of the MOSFET, one sees
the back-EMF of the motor superimposed onto the MOSFET signal. An example of
this is shown on the next pages in figures 2.34 and 2.35, which shows experimentally
recorded waveforms of vAN(t) and iA(t). Recall that N is the DC ground of the
system.
Consider the experimentally obtained voltage and current waveforms for a
BLDC motor, figure 2.34. A snapshot of the instantaneous waveform over approxi-
mately 2 electrical cycles is shown in figure 2.35, for throttle settings of 40%, 70%,
and 100%. When comparing individual waveforms, the most obvious result is there
appears to be less noise in the waveforms as the throttle increases. For example,
40% throttle contains a large amount of electrical noise, resulting in a disorganized
appearing waveforms. On the other hand, the 100% throttle waveforms look smooth
and continuous.
49
Figure 2.34: Instantaneous voltage and current waveforms over 1ms for a BLDC
motor at VDC = 7.2V .
50
Figure 2.35: Instantaneous voltage and current waveforms over 2 cycles for a BLDC
motor at VDC = 7.2V . Note the difference in time between waveforms of different
throttles.
51
The difference in appearance of these signals relates to the amount of time the
MOSFETs are open. At a lower throttle setting, more switching occurs and thus
the signal appears to be noisy in comparison to the 100% throttle setting. From
this, 3 conclusions can be drawn:
1. Increase in throttle results in increased on time for a MOSFET, and thus fewer
switching events for the same amount of time.
2. Increase in throttle increases effective motor voltage, which increases motor
rotation speed
3. Waveforms are ?ideal-like? at 100% throttle
These results will be useful when determining the motor model in Chapter 4,
and the ESC power loss model of section 4.2.4.
Careful examination of figure 2.35 shows large unaccounted spikes in voltage
and current, which are most noticeable at 100% throttle. Additionally, the voltage
spikes are larger than the supply voltage of 7.2V, meaning they must result from a
different source. Also, these spikes appear to come in a repeatable fashion, indicat-
ing that it is not simply random noise. When examining the current waveform, a
step response-like behavior appears. The voltage appears to change instantaneous,
but the current contains some time-lag component before the current reaches its
maximum value.
These voltage spikes are actually attributed to commutation events, when the
rotor reaches a certain location to trigger a change in active MOSFETs. During this
time, one MOSFET is turned off whereas a separate MOSFET turns on, and occurs
52
every 60 electrical degrees. A spike in voltage occurs whenever a commutation event
occurs, due to the inductor discharging the energy stored in its magnetic field. The
voltage across an inductor is [22]:
VL(t) = LI?(t) (2.22)
Just before commutation, current is flowing, whereas just after commutation, cur-
rent no longer flows through that phase. This results in a large change in current
I?, and thus a large spike in voltage acting across the inductor. Diodes are used to
create a complete circuit path for the inductors to discharge, and so the resulting
voltage spike is quickly dissipated. This process is referred to as freewheeling [13],
and can be assisted by using additional MOSFETs to make the energy dissipation
process more efficient.
The time-lag response of current can be explained by modeling a simple
resistor-inductor circuit, shown in figure 2.36. At time t = 0, a change in voltage
Figure 2.36: Simple RL circuit, switch closes at time = 0.
?V occurs which can be modeled as a step input. Kirchhoff?s voltage law sums
the voltages around the loop and can be used to expressed the set of differential
53
equations:
?V = I(t)R + LI?(t) (2.23)
This equation can be solved to find the current time response:
?V R
I(t) = (1? e? tL ) (2.24)
R
For a given step input of voltage V , the current will reach a steady state value of V/R
after some transient response occurs, and has a time constant of L/R. The larger
the inductor value L, the longer it takes for the current to reach its steady-state
value, shown in figure 2.37.
Figure 2.37: Output voltage of VSN based off duty cycle duration.
Interestingly, this has a noticeable impact on the current waveforms for BLDC
motors. For a large torque constant KT , it is desirable to increase the number of
turns in a coil to increase the strength of the magnetic field, from equation 2.2.
54
This large number of turns however, increases the inductance of each phase, which
causes larger time delays. Additionally, more turns represents increased resistance,
so the final value of steady state current V/R will be lower. This is consistent with
needing less current to generate the same torque, as compared to a motor with a
smaller KT .
With the basic operation of DC motors and ESCs covered, the experimental
setup to record motor and ESC performance can be explained.
55
Chapter 3: Experimental Setup
3.1 Dynamometer Overview
A dynamometer consists of a load varying device and sensors to monitor the
motor?s performance. Performance of commercial off the shelf motors and ESC is
studied by recording the power at various points for a range of operating conditions.
Figure 3.1: Basic components in a dynamometer for electric motors.
The centerpiece of this work was designing and fabricating a custom dy-
namometer using in-house resources. This involved creating custom hardware for
56
the power analyzers, constructing a versatile frame, and implementing a unique
2 data acquisition device (DAQ) LabView setup to properly record the system?s
performance. The final result is a highly customizable dynamometer, with a large
amount of testing flexibility, in terms of ESCs, motors, and DC voltages.
To study BLDC motors and ESCs, the following components were used: a
DC power supply, a DC power analyzer, an AC power analyzer, and a mechanical
power analyzer, shown in figure 3.1. A breakdown of these items and their use can
be found in table 3.1:
Table 3.1: Dynamometer component overview.
Item Purpose
DC Power Supply Provides DC electrical power to the ESC
DC Power Analyzer Measures the power content of the DC signal
ESC Modulates DC power to AC power for the BLDCM
AC Power Analyzer Measures the power content of the AC signal
BLDCM Converts AC power to mechanical power
Torque Sensor Records the shaft torque of the motor
RPM Sensor Records the shaft RPM of the motor
Brake Varies load on the BLDCM
Brake Power Supply Allows user to control operation of the brake
The critical component of any dynamometer is the brake, which is responsible
for absorbing the motors mechanical output and should be controllable. The brake
is physically connected to the output shaft of the motor and thus any change in
load of the brake must be produced by the motor. Several types of brakes exist for
dynamomemters [9]:
57
1. Friction Brakes: Mechanical devices impeded the rotation of the shaft, similar
to an automobile?s brakepad
2. Hydraulic Brakes: A working fluid, typically water, is compressed and ex-
panded, which changes the load on the motor
3. Magnetic Brakes: Current flowing through a non rotating coil causes a mag-
netic interaction with a rotating magnet that produces a magnetic torque
For each of these types of brakes, the adsorbed energy must be dissipated as
heat, and thus typically requires cooling for extended use. Additionally, the pres-
ence of rotating components inevitably causes wear on the system. For extended
use, the number of these components must be minimized. A typical friction or hy-
draulic brake requires pistons, levers, pulleys or other moving components that are
placed under cyclical stress. Both friction and hydraulic brakes consist of moving
parts, whereas the primary action in the magnetic brake is a non-contacting mag-
netic interaction. For magnetic brakes, the only component subject to wear are the
bearings. In addition, there are several other factors that make a magnetic brake
desirable [23]:
1. Mechanical torque is only a function of the input current to the brake, Q =
f(Ibrake), and is easily controlled
2. Mechanical torque is independent of operational RPM
3. High degree of repeatability and controllability associated with ease of use
4. Long life cycle associated with low wear components
58
5. Smooth and quiet operation
The ability to control torque independent of RPM makes the magnetic hysteresis
an attractive option for studying electric motor performance.
For this purpose, a Magtrol magnetic hysteresis brake (HB-10) was acquired
to vary the load on the motor. The brake is rated for 10 in-oz at up to 20,000
RPM, however it was found that the load on the brake could go as high as 20 in-oz,
for short term operation. The RPM limit is set by the maximum rated speed of
the internal bearings used in the HB-10. Construction of the brake consists of a
rotating pole structure and a coil, shown in figure 3.2. Energizing the stationary
coil with DC current causes a magnetic interaction between the stationary coil and
the moving magnets. This magnetic interaction causes a braking torque to develop,
and thus the torque of the motor is changed. Controlling the DC current into the
brake allows one to control the braking torque.
Figure 3.2: Exploded view of a magnetic hysteresis brake. Image on the left is
from [24], image on the right is taken from [23].
As previously mentioned, the operation of the brake is only a function of the
input current and is independent of RPM. Changing the output voltage of the brake?s
DC power supply allows the input current to the brake to be regulated, allowing for
59
control of the motor?s torque. The input current to the brake and output current
relationship can be found in figure 3.3.
Figure 3.3: Input current vs output torque for the Magtrol HB-10 magnetic hystersis
brake. Image taken from [25].
With the ability to regulate the motor?s torque, the brake is the primary opera-
tional component in the dynamometer. Once the torque is varied, additional sensors
record the performance of the motor at the various operating points. Once enough
points at different throttles have been recorded, the motor and ESC parameters can
be deduced.
Various sensors are used in the system?s DC, AC, and mechanical power an-
alyzers. A breakdown of the different sensors used is included in table 3.2. Each
power analyzer uses a set of sensors to monitor the performance of the ESC and
BLDC motor, each targeting a specific domain.
60
Table 3.2: Sensors used on the dynamometer.
Analyzer Sensor Quantity Measurement Rating
DC Current Sensor 1 IDC +30A
DC
Voltage Sensor 1 VDC +15V
AC Current Sensor 2 iA, iB +/?30A
AC
Voltage Sensors 3 vAN , vBN , vCN +15V
Torque Sensor 1 Q +/?50 in-oz
Mechanical
RPM Sensor 1 ? 30,000 RPM
The remainder of this Chapter shall be devoted to exploring the 2 DAQ setup
used to record the motor?s performance and the power analyzers themselves. Special
attention is given to the AC power analyzer as this requires an understanding of 3
phase electricity to implement. Derivations for the active power, apparent power,
and power factor will be critical in the modeling efforts of Chapter 4.
3.2 Two DAQ Setup
To record DC, AC, and mechanical power, a DAQ system must have 2 fea-
tures: a high number of input channels, and a data acquisition read rate high
enough to properly sample the AC waveforms. Between the three analyzers, up to
10 analog input channels are required to record all necessary metrics. The National
Instrument?s (NI)-USB 6251 had a maximum channel capacity of 16 analog inputs.
However running all inputs to 1 DAQ reduced the read rate to be 100 kHz.
Due to the number of input channels, and speed of the electrical signals, 2
Data Acquisition Devices (DAQs) were required to properly analyze the system?s
61
performance. While originally using 1 DAQ, it was found that a slow sample rate
had a detrimental effect on calculating the AC power. An insufficient sample rate
resulted in nonphysical values for PAC . To eliminate this issue, the maximum data
rate for the system was increased by using a second DAQ. Each DAQ has a data
limit, which is a function of the number of samples/sec and the number of samples
collected.
The final sample rate of setup was 250 kHz, which appears adequate for sam-
pling the electrical signals. The maximum RPM of the brake is 20,000 RPM and
all motors studied contain 7 pole pairs, meaning that at 250 kHz sampling, each
waveform contained a minimum of 107 samples. Such a high resolution is necessary
to detect the large amount of MOSFET switching, shown in figures 2.34 and 2.35,
which occurs during normal operation. Additionally, the ideal voltage and current
waveforms contain six distinct steps over one cycle and can only be captured by
rapid data acquisition. Note that at lower motor speeds, the number of samples per
waveform is higher.
62
Figure 3.4: DAQ and sensor arrangement.
Use of two DAQs with in house power analyzers produced a dynamometer
system that was completely customizable, and allowed for testing numerous motors
and ESCs at different voltages. Figure 3.4 shows which signals were read into which
DAQ. Due to the time sensitive nature of the AC power analyzer, all signals related
the AC system were read into DAQ-2. Each DAQ contains 2 components: the
NI-USB 6251 is the actual data sampling and collection device, and the SC-2345
functions as a breakout board for the NI-USB 6251. Measurements from motor
were connected to the SC-2345 and transmitted to the labview environment via the
NI-USB 6251. Both DAQs were read into the same labview program, streamlining
data collection.
63
Figure 3.5: Screenshot of labview front panel with 2 DAQs.
All parameters related to system performance are monitored directly from the
front panel of the labview setup, shown in figure 3.5, whereas the power calculations
are done in post processing. Redcuing the number of calculations to a minimum
keeps the labview setup as bare bones as possible, allowing for a 250 kHz read rate.
The labivew program had certain custom additional features such as PWM signal
generation, a robust file data saving network, and automatic zeroing of sensors before
tests. The final two DAQ configuration is shown in figure 3.6.
64
Figure 3.6: Final dynamometer configuration.
With an understanding of the dynamometer components, the focus shifts to exam-
ining the various power analyzers used to measure performance.
65
3.3 DC Power Analyzer
The DC power supply provides a constant DC voltage VDC and a constant DC
current IDC to ESC. The DC power is [21,22,26]:
PDC = IDCVDC (3.1)
A noninvasive hall effect based current sensor was used to measure the value of
the DC current, and deduces the strength of the DC current based off the current?s
magnetic field strength. The output of this sensor was a 0-5 V analog voltage signal,
which was converted to the DC current value based of a simple factor of 133 mV/A.
This sensor showed good agreement during use with the DC power supply?s LED
display, which showed the value of the instantaneous current while the motor was
running. The voltage measurement was read into the DAQ directly for VDC < 10V .
Figure 3.7: DC power occurs between the DC power supply and the ESC.
The DAQ has a 10 V hardware limit, meaning that for higher voltages a simple
voltage divider was used to reduce the value to within 10 V. By sampling the voltage
in the middle of two 500 ? resistors in series, the signal?s voltage is reduced by a
66
factor of 2, meaning that at the maximum VDC = 14.8 V, only 7.4 V was being
read into the DAQ. Calibration of the voltage divider is accomplished by reading in
voltages between 0-10 V into the DAQ both before and after the voltage divider and
then deducing the gain. Finally, a low pass filter was applied to both the voltage
and current signals to smooth out any spikes from the power supply. Both sensors
were placed before the ESC.
3.4 Mechanical Power Analyzer
A mechanical power analyzer was developed to measure the output perfor-
mance of the motor. The mechanical power is [13,27,28]:
Pm = Q? (3.2)
An infrared (IR) optical sensor was used to monitor RPM by counting the
time between pulses. Mounted on the rotating shaft is a disk with slots, so as the
shaft rotates the IR beam is interrupted periodically and the RPM can be deduced.
For low speeds (less than 10,000 RPM) a 20 slot disk was used, whereas for high
speeds (up to brake limit of 20,000 RPM) a 4 slot disk was used. Accuracy of the
RPM measurement was verified with a hand held IR tachometer.
Torque measurements were made with a transducer techniques 50 in-oz torque
sensor, consisting of a full strain gauge bridge. When a load is applied, the shaft
deflects and outputs a voltage proportional to the deflection of the device. This
voltage is amplified to between 0-2 V and then read into the SC-2345. To calibrate
67
this sensor, calibration weights were hung off the device at a known distance. This
test was repeated several times before arriving on the calibration plot, shown in
figure 3.8.
Figure 3.8: Calibration curve for the 50 in-oz torque cell.
During motor testing, it was found that the torque sensor is sensitive enough
to pick up minor changes in the mass of the motor. Additionally, changes in the
position of the sensor wire were found to augment the offset of the torque cell. As
a result, a zeroing program was implemented in the LabView environment, which
would zero out the sensors readings before the start of a test. This provided an
effective strategy, as the brake required a repeatable 3.1 in-oz torque to overcome
the cogging friction of the brake. The torque sensor was connected the BLDC motor
via a short adapter piece. The sensor was then mounted to the remainder of the
physical structure shown in figure 3.6.
DC and mechanical power measurements are straight forward to achieve, how-
68
ever AC power measurement requires additional circuit theory to implement.
3.5 AC Power
Before describing the custom AC power analyzer, an understanding of AC
power is required. Basic power relations for AC systems, such as active power,
apparent power, and the power factor, shall be expressed for a single phase sinusoidal
AC system. Next these relations will be expanded to include polyphase and non-
sinusoidal waveforms.
3.5.1 Single Phase AC Relations
When discussing AC circuits, it is common to explain the differences between
DC and AC circuits by first considering a single phase AC circuit. Eventually, this
analysis will be expanded to include additional phases. To start, consider a single
phase AC system with sinusoidal voltage and current waveform:
v(t) = Vpk cos(?et+ ?v) (3.3)
i(t) = Ipk cos(?et+ ?i) (3.4)
where pk represents the peak value of the waveform and ? represents the initial
phase shift of the signal. The user sets ?v and the electrical frequency ?e, and the
load impedance z determines the amount of current and the current phase angle
?i. Changing the impedance z can have two effects: changing the magnitude of the
69
current Ipk, or changing the phase lag between voltage and current ?v ? ?i. This
second point will be important when considering power transmission.
Figure 3.9: Left: Single phase AC system. Right: Voltage and current in a sinusoidal
system.
The instantaneous power in any signal is simply the product of the instanta-
neous voltage v(t) and the instantaneous current i(t) [21, 22,26]:
pinst(t) = v(t)i(t) (3.5)
pinst(t) = [Vpk cos(?et+ ?v)][Ipk cos(?et+ ?i)]
?VpkIpk VpkIpkpinst(t) = co?s?(?v ? ?i?) + ? cos(2???e + ?v + ?i?) (3.6)2 2
non-ossicilating ossicilating
An important distinction of instantaneous power in an AC system is that it consists
of an oscillating component and non-oscillating component. The steady component
is found by the cosine difference of the phase angles whereas the oscillating compo-
nent has a frequency of twice the driving frequency ?e. Notice how the instantaneous
70
Figure 3.10: Left: Voltage and current in an AC system. Right: Instantaneous
power in an AC system.
power pinst oscillates between a maximum and minimum, and is offset by some av-
erage value in figure 3.10. To find the average power in the signal, it is simply the
instantaneous power over a waveform:
?
1 T
PAC = v(t)i(t)dt (3.7)
T 0
?
1 T VpkIpk
PAC = [ [cos(?v ? ?i) + cos(2?e + ?v + ?i)]]dt
T 0 2
Taking the integral over one waveform (2? radians) eliminates the oscillating com-
ponent of the instantaneous power. This is a result of the fact that this component
does no work on the system, as it both adds and subtracts power in an unusable
manner. Hence, the term active power describes the mean power in the system, as
this represents the net power transfer of the system. Active power is the power to
spin motors, heat up wires, and do useful work. For efficiency metrics of the ESC
71
and motor, it is only proper to use the active power PAC . This makes the active
power for a single-phase sinusoidal system:
VpkIpk
PAC = cos(?v ? ?i) (3.8)
2
where term cos(?v ? ?i) is referred to as the power factor, pf . Power factor re-
lates the leading or lagging of the voltage and current signals in the system. The
important takeaway is that if the current and voltage signals are out of phase with
one another, less real power can be transfered. This phase shift between current
and voltage comes from the load impedance z. When inductors and capicators are
present, a leading or lagging of the current signal occurs [22]. Some key notes about
power factor and power transmission are:
1. Power factor has a range between ?1 ? pf ? 1
2. ?v > ?i relates to a lagging power factor, which is indicative of an inductive
load. An example is a RL circuit.
3. ?v < ?i relates to a leading power factor, which is indicative of a capacitive
load. An example is a RC circuit.
4. ?v = ?i has a power factor of unity, which is indicative of a purely resistive
load. An example is a R or RLC circuit.
5. Higher power factor allows more work to be done on the system.
6. Increasing the peak current or voltage increases the active power.
72
To demonstrate the need for a high power factor, consider if voltage and current
are completely out of phase with one another such that ?v ? ?i = ?/2. Here, the
power factor is equal to 0, meaning even though voltage and current are flowing
through the system, no power is transfered. An example of such a circuit is a source
connected to only either a capacitor or a source connected to only an inductor. For
maximum power transfer, a power factor of 1 is needed. The power factor can still
be unity even with the presence of inductors capacitors are present in a load, as long
as the values of the capacitors, inductors, and driving frequency are such that the
net phase lag is 0.
The oscillating component of the active power is still useful for power analysis,
as wires must be sized according to the maximum power in the system. To describe
the ?equivalent DC magnitude? of the AC voltage and current, the root-mean-square
(rms) function is used. This is required since the average value of both the voltage
and current is 0 in an AC system, by definition. The rms of any signal is defined as:
? ? T
rms{ 1x(t)} = x2(t)dt (3.9)
T 0
Thus the rms values for voltage and current are:
? ? T
Vrms = rms{
1
v(t)} = v2(t)dt
T 0
V?pkVrms = (3.10)
2
73
? ?
1 T
Irms = rms{i(t)} = i2(t)dt
T 0
?IpkIrms = (3.11)
2
?
The ratio of the peak value to the rms value is the 2 for all single phase sinusoidal
AC signals. Using equations 3.10 and 3.11 along with the active power equation 3.8
allows one to conclude:
PAC = IrmsVrms cos(?v ? ?i) (3.12)
The power factor relates the distortion from the IrmsVrms power to the active power.
Consider again if the power factor is 0, meaning no active power is transferred.
Voltage and current still flows through the system, and so IrmsVrms is nonzero. This
quantity is important for determining how much effective power is in the line. The
IrmsVrms power is defined as the apparent power S in the line:
S = IrmsVrms (3.13)
The ratio of active power to apparent power is an alternative way of defining the
power factor:
PAC = S cos(?v ? ?i) = S pf
PAC
pf = (3.14)
S
Apparent power S represents the maximum amount of power that can be used for
work, whereas the power factor relates how much of the maximum power does work.
74
Typically the power factor angle is used to describe the triangle of power, figure
3.11, which are used to relate the active, reactive, and apparent power in a system.
Figure 3.11: The Triangle of Power.
The final leg of the power triangle is the reactive power QAC , and is used as
a modeling tool to describe the effect of the inductors and capacitors on a load.
In an AC system, current and voltage are always changing which causes voltage
and current to always be flowing across inductors and capacitors. The product of
this current and voltage over these components would indicate that power is being
lost. However, careful examination will show that this power is actually oscillating
in nature. That is, it describes the second component of the instantaneous power
relation from equation 3.6. Ideal inductors and capacitors are referred to as loseless
components, in that they do no work. Their presence changes the distribution of
power in a system, and is represented by QAC . Since PAC , QAC , and S relate to legs
of the triangle of power, the following relationship can be obtained:
S = PAC + jQAC
S2 = P 2 2AC +QAC (3.15)
75
?
where j is the complex quantity ?1, which is useful for determining the amount
of reactive load in a circuit. Since voltage and current can be monitored, both the
active and apparent power can be recorded by:
? ?
1 T
QAC = (I 2rmsVrms) ? ( v(t)i(t)dt)2 = IrmsVrms sin(?v ? ?i) (3.16)
T 0
The following relations were derived in the context of a single phase, sinusoidal
voltage and current waveforms. The basic ideas of active, reactive, and apparent
power hold even when the waveform is non-sinusoidal. The only term which is
altered significantly is the definition of the power factor. The power factor still
relates the lagging or leading of the voltage signal to the current signal, but is no
longer defined in this manner when the signals are sinusoidal. Instead, the general
definition for power factor, stemming from the triangle of power, as the ratio of
active to apparent power must be used:
???? PAC? = cos(?v ? ?i) for sinusoidal voltage and currentSpf = ?? PAC 6= cos(?v ? ?i) for non-sinusoidal voltage or currentS
With the basics covered, these ideas can be expanded to include circuits with mul-
tiple phases and eventually brushless motors.
3.5.2 Polyphase AC Power Relations
The study of AC power analysis becomes more interesting when additional
phases are added to the circuit. Typical power generation and distribution happens
76
using 3 phase electricity, as it will be shown in the following sections that the
transfer of 3 phase power is constant. This is in stark contrast to the single phase
system where power contains both a constant and oscillating term, equation 3.6.
The remainder of this section discusses polyphase circuits, and the concepts can be
applied to either sinusoidal or non-sinusoidal waveforms. Three phase non-sinusoidal
waveforms are of importance for a brushless motors, as the voltage and current
waveforms are trapezoidal and rectangular, respectively.
Figure 3.12: Left: Generic 3-phase AC system. Right: Phasor diagram for a 3 phase
system. Each waveform is 120o electrical degrees apart.
In an ideal three phase system, the voltage waveforms are 120o apart from
one another and is referred to as a balanced system. For a sinusoidal voltage and
current, these waveforms take the following form:
vAN?(t) = Vpk cos(?et+ ?v) (3.17)
v(t)BN?(t) = Vpk cos(?et+ ?v ? 2?/3) (3.18)
77
v(t)CN?(t) = Vpk cos(?et+ ?v ? 4?/3) (3.19)
iA(t) = Ipk cos(?et+ ?i) (3.20)
iB(t) = Ipk cos(?et+ ?i ? 2?/3) (3.21)
iC(t) = Ipk cos(?et+ ?i ? 4?/3) (3.22)
The goal is now to define the rms values for polyphase voltages and current, as this
will be necessary when describing the motor model in Chapter 4. From figure 3.12,
and the definition of a balanced load, the following relationship can be drawn [21]:
vAN?(t) + vBN?(t) + vCN?(t) = 0 (3.23)
Which is equivalent to its phaser representation:
V AN? + V BN? + V CN? = 0
When discussing polyphase circuits, it is important to bookmark the voltages. A
voltage is described as a difference in a potential field, which means both the starting
and final potentials must be taken into account. A phase voltage is a voltage
measured with respect to the neutral point of the AC system (ex. vAN? , figure
3.12), whereas a line? to? line voltage is a voltage from one phase measured with
respect to another phase (ex. vAC , figure 3.15). A neutral point N
? acts like an ?AC
ground? for the system, and is used as a common measuring points for all phases.
78
This is similar to the DC ground, which is used as a common reference point for DC
systems. It is important to note that the DC ground and the neutral point do not
share the same voltage, and have a time varying value of vN?N(t) from one another.
A safe assumption is that all 3 phases have the same peak value and thus has the
same rms value:
VAN? = VBN? = VCN? = Vph,rms (3.24)
The three phases are identical, only having been rotated 120o electrical degrees from
one another. With this in mind, voltages can be treated like positional vectors or
phasers. Assuming a balanced voltage set, the line-to-line voltages can be calculated
using [22,26]:
V AB = V AN? ? V BN?
?
? ?1 ? 3VAB = VAN? [1 + j0] VBN? [ j ]
2 2
?
3 3
V AB = V [ + j ]2 2
?
VLL,rms = 3Vph,rms (3.25)
which is also shown in figure 3.12. Circuit analysis depends on defining a complete
circuit, and for 3-phase systems there are two ways to connect the circuit together.
AC circuits are connected in either a delta (?) or wye (Y ) connection, de-
pending on the power application. The derivation for the active power and apparent
power depend upon the configuration.
79
Figure 3.13: Y and ? connected 3 phase circuits.
Y connected circuits have each input line connected together at a central point,
referred to as the neutral point N?. The ? connection has each leg connected to
the other 2 in a triangular formulation. The most notable difference between the 2
configurations is that a ? connection does not have a neutral point.
When studying BLDC motors, the type of connection configuration can be
thought of as the gearing. Y connected motors are favored for low speed/high
torque applications, whereas the ? motors are favored for high speed low torque
conditions [29]. The relation for this can be stated as:
?
MY = 3M? (3.26)
1
?Y = ? ?? (3.27)
3
In fact, since most research and commercial electric aircraft use high RPM
motors for propulsion, the majority of the commercial off the shelf products are ?
wound motors.
80
For an AC system, there are three principal ways of recording power in a 3
phase system: the 1-wattmeter, 2-wattmeter, and 3-wattmeter method [22, 30?32].
A wattmeter is a device that measures the voltage differential between two points
and the current passing through the device. This is done for a DC system by
measuring the DC current IDC and DC voltage VDC , with the DC power being
simply PDC = VDCIDC . With a DC system, the voltage differential between ?high?
and ?low? volts makes sense, but how can this be applied to a polyphase time-
varying system? To answer this, we shall first consider the 1-wattmeter method.
Figure 3.14: 1-wattmeter method for AC power measurement.
The 1-wattmeter method assumes that the neutral point of the system can be
accessed. With this, then the definition of three phase power in the AC system is:
pinst = pA + pB + pC (3.28)
pA = vAN?(t)iA(t) (3.29)
pB = vBN?(t)iB(t) (3.30)
pC = vCN?(t)iC(t) (3.31)
81
?
1 T
PAC = pinst (3.32)
T 0
The 1-wattmeter method also assumes that each phase of the circuit carries the
same power and so pA = pB = pC . This allows for the following simplification:
pinst = 3pA = 3vAN?(t)iA(t) (3.33)
?
3 T
PAC = vAN?(t)iA(t)dt (3.34)
T 0
Which is simply the result of 3 times the single phase power, found from equation
3.7. An advantage of this method is that it is easy to implement, as only one phase
voltage and current must be recorded. The disadvantage is that a neutral wire
must be available to access, which is not possible as that point is either inside the
motor (Y -wound motors) or does not exist (?-wound motors). To circumvent this,
additional meters are required.
An alternative way to measure 3-phase power is with the 3-wattmeter method,
shown in figure 3.12. Here the phase voltages are measured with respect to neutral,
and each line current recorded. The instantaneous power is the v(t)i(t) for each
phase. Using equation 3.28 and the expressions for 3 phase voltage and current,
equations 3.17 to 3.22, an expression for the instantaneous power using 3-wattmeters
can be deduced. Combining terms and using trigonometric identities allows for the
following expression for instantaneous power [22]:
VpkIpk
pinst = 3 cos(?v ? ?i) = constant (3.35)
2
82
Here, for a 3 phase system the instantaneous power is constant, whereas for a
single phase system it contained an oscillating component. Also note that the 3
phase active power PAC,3p is simply 3 times the single phase power PAC,1p:
PAC,3p = 3PAC,1p
Power factor returns for 3 phase systems, and has the same meaning as it did for
single phase systems. Recall, one wants to increase power factor for greater power
transmission.
As with the 1-wattmeter method, the disadvantage of the 3-wattmeter method
is that a neutral wire must be present. The major difficulty when dealing with 3
phase systems is that, in most cases, is that accessing the neutral point of the system
is not possible. Why not measure the phases relative to the DC ground and repeat
the analysis with vAN? This is because there is some voltage differential vN?N(t)
between the DC ground and the neutral point. To calculate the 3 phase power when
no neutral is available, the 2-wattmeter method is used.
3.5.3 2-Wattmeter Proof for Y-Circuits
Proof of the 2-wattmeter method depends upon the connection scheme, and
so first the Y connection is analyzed. The goal is to calculate active and apparent
power in this configuration, and then compare the results for a ? connection in
section 3.5.4 To begin, consider the Y connected circuit shown below:
83
Figure 3.15: Y-connected circuit.
The AC power in a 3-phase circuit is given by equation 3.28, and is stated
here again:
pinst = vAN?(t)iA(t) + vBN?(t)iB(t) + vCN?(t)iC(t) (3.36)
In a Y configuration, the sum of currents at the neutral point N? is 0 by Kirchoff?s
voltage law [22]:
iA(t) + iB(t) + iC(t) = 0 (3.37)
In this method, only iA(t) and iB(t) are measured, meaning the third current iC(t)
can be deduced by:
iC(t) = ?iA(t)? iB(t) (3.38)
Plugging equation 3.38 into equation 3.37 yields:
pinst,Y = vAN?(t)iA(t) + vBN?(t)iB(t) + vCN?(t)[?iA(t)? iB(t)]
84
pinst,Y = [vAN?(t)? vCN?(t)]iA(t) + [vBN?(t)? vCN?(t)]iC(t) (3.39)
The advantage of the 2-wattmeter method is by referencing voltage differen-
tials with respect to another phase, and taking advantage of equation 3.37, only
2-wattmeters are required. Both voltages for line A and B are referenced with
respect to line C.
vAC(t) = vAN?(t)? vCN?(t) (3.40)
vBC(t) = vBN?(t)? vCN?(t) (3.41)
Plugging into equation 3.39 yields:
pinst,Y = vAC(t)iA(t) + vBC(t)iB(t) (3.42)
Note that because the same circuit was analyzed for the 3-wattmeter method, equa-
tion 3.35 and equation 3.39 are equal to one another, such that:
VpkIpk
pinst = vAC(t)iA(t) + vBC(t)iB(t) = 3 cos(?v ? ?i) = constant (3.43)
2
indicating that AC power transmission is constant with the 2-wattmeter method
as well. Integrating over a number of cycles yields the AC power for the Y-
configuration: ?
1 T
PAC,Y = (vAC(t)iA(t) + vBC(t)iB(t))dt (3.44)
T 0
To evaluate the power factor, the total apparent power in all three phases must
85
be evaluated. Recall that the power factor is the ratio of the active power to the
apparent power in the circuit, and is expressed in equation 3.14.
The apparent power for a single phase AC circuit is given by equation 3.13,
and must be expanded to include 3 phase circuits. Just like instantaneous power,
apparent power is merely the sum of all single phase apparent powers:
S = SA + SB + SC (3.45)
with
SA = rms{vAN?(t)}rms{iA(t)} = VAN?IA (3.46)
SB = VBN?IB (3.47)
SC = VCN?IC (3.48)
Assume a balance loading condition, such that:
1
VAN? = VBN? = VCN? = Vph = ? VLL,rms (3.49)
3
where the rightmost equation is given by equation 3.25. By definition, VAC and VBC
are line-to-line rms voltages VLL,rms, wheras VAN? and VBN? are rms phase voltages
Vph,rms. The line currents are all balanced as well, meaning:
IA = IB = IC = IL,rms (3.50)
86
with IL,rms meaning rms line current. Plugging into equation 3.45 yields:
S = VAN?IA + VBN?IB + VCN?IC = Vph,rms(IA + IB + IC)
1
S = 3Vph,rmsIL,rms = 3(? VLL,rms)IL,rms
3
?
S = 3VLL,rmsIL,rms (3.51)
In reality, both line-to-line voltages VAC and VBC are not perfectly equal, and
thus the average from the recorded values of SA and SB is used to compute the total
apparent power:
?
3
SY = (VACIA + VBCIB) (3.52)
2
The ratio of equation 3.44 to equation 3.52 gives the power factor pf for the system:
PAC,Y
pfY =
SY
With the active and apparent power deduced for the Y configuration, the ? config-
uration can be analyzed.
87
3.5.4 2-Wattmeter Proof for ?-Circuits
Like the Y connection, the objective for the ? connection analysis is to eval-
uate expressions for the active power, apparent power, and power factor. A ?
connected circuit is shown below:
Figure 3.16: ?-connected circuit.
The key difference between a ? circuit and a Y circuit is that a ? circuit does
not have a neutral point N?. The concept of a phase becomes nebulous, as there
is no common reference point for the AC system. Instead, the voltages must be
referenced to one another. For a ? connection [22, 26,32], the instantaneous power
in the circuit is:
pinst = vab(t)i1(t) + vbc(t)i2(t) + vca(t)i3(t) (3.53)
where points a, b, and c represent different vertices of the ? triangle. Kirkoff?s node
88
law at points a, b, and c yield the following current relations:
iA(t) = i1(t)? i3(t) @ point a (3.54)
iB(t) = i2(t)? i1(t) @ point b (3.55)
iC(t) = i3(t)? i2(t) @ point c (3.56)
It can also be stated that the voltages of A and a, B and b, as well as C and c are
the same since they are physically connected. For a balanced, 3 phase ? connection
the following voltage relation can be stated:
vAB(t) + vBC(t) + vCA(t) = 0 (3.57)
which is pictorial represented in figure 3.12. With this, one can deduce:
vAB(t) = ?vBC(t)? vCA(t)
vAB(t) = ?vBC(t) + vAC(t) (3.58)
From this, equation 3.53 can be expanded to:
pinst,? = [?vBC(t) + vAC(t)]i1(t) + vBC(t)i2(t) + vCA(t)i3(t) (3.59)
= ?vBC(t)i1(t) + vAC(t)i1(t) + vBC(t)i2(t)? vAC(t)i3(t)
89
= vAC(t)[i1(t)? i3(t)] + vBC(t)[i2(t)? i1(t)]
pinst,? = vAC(t)iA(t) + vBC(t)iB(t) (3.60)
?
1 T
PAC,? = (vAC(t)iA(t) + vBC(t)iB(t))dt (3.61)
T 0
which is the same as active power as the Y connection, equation 3.44.
The expression for apparent power is made on similar grounds as the instan-
taneous power. The total apparent power of the system is the sum of all S for each
leg of the ?:
S? = VabI1 + VbcI2 + VcaI3 (3.62)
Note that Vab, Vbc, and Vca are the line-to-line rms voltages and can be replaced
with VAB = VBC = VCA = VAC = VLL,rms. A relationship between the rms current
and line currents can be found via an application of Kickoff?s voltage law around
the different legs of the ?. The results are [22, 32]:
1
I1 = I2 = I3 = I? = ? IL,rms (3.63)
3
where IL,rms represents the rms current in each line (IL,rms = IA = IB = IC). Thus
the following relation can be found:
S? = VLL(I1 + I2 + I3) = 3VLL,rmsI?
1
S? = 3VLL,rms(? IL,rms)
3
90
?
S? = 3VLL,rmsIL,rms (3.64)
In reality, both line-to-line voltages VAC and VBC are not perfectly equal, and thus
are averaged from the recorded values of SA and SB is used to compute the total
apparent power:
?
3
S? = (VACIA + VBCIB) (3.65)
2
which is the same relation for the Y connection, equation 3.52.
Regardless of the connection scheme, the expressions for active power, appar-
ent power, and power factor are the same. The strength of the 2-wattmeter method
is that it does not require a neutral wire and is invariant of the connection type.
These results can be summarized as:
?
1 T
PAC = PAC,Y = PAC,? = (vAC(t)iA(t) + vBC(t)iB(t))dt (3.66)
T 0
?
S = SY = S? = 3VLL,rmsIL (3.67)
PAC
pf = pfY = pf? = (3.68)
S
These results show another important proerty of Y and ? circuits: that they can
be transformed into one another. This transformation can write a ? connection
as an equivalent Y connection [21, 22, 26, 32], shown in figure 3.17. This is useful
for motors, because this means a ? connected motor, which has no neutral point
N?, can be represented as an equivalent Y circuit containing a neutral point N?.
The phase waveforms of a BLDC motor are most naturally defined in reference to
91
a neutral point, and are typically expressed in that manner.
Figure 3.17: Conversion of a ? connected circuit to an equivalent Y connected
circuit.
To transform a 3 phase connected circuit into its evil twin, the following rela-
tion is used [26]:
zY,1 = zY,2 = zY,3 = zY
z?,1 = z?,2 = z?,3 = z?
1
zY = z? (3.69)
3
where zY and z? are the impedances of each leg of the circuit. In the event that the
impedances for each leg of the connection are not equal to one another, additional
relations are used and are found in [26]. However, most models assume balanced
voltages and impedances and therefore equation 3.69 is used.
Even though most commercial BLDC motors are ? connected motors [29], the
theory describing BLDC motors is typically given for Y connected motors. Y con-
nected motors, and circuits, are less complicated to analyze, and contain a neutral
92
point, which is a logical point to reference phase voltages. However, it is important
to remember that 3 phase active and apparent power are equal to one another, and
hence makes this transformation valid.
Now that the proofs are completed, the system to record the performance of
the AC system can be discussed.
3.5.5 AC Power Analyzer
The AC power analyzer uses the 2-wattmeter method for determining both
the real and apparent power entering the motor. Because the DAQ had a 10 V
limit, two separate setups were used to study the motors at different voltages.
Two types of AC power analyzers were employed to study the motors and
ESCs; a low voltage (VDC < 10 V ) and a high voltage setup (10 V < VDC < 15 V ).
The NI-USB-6251 and SC-2345 DAQs used in the setups had a maximum voltage
rating of 10 V, and so recording voltages higher than 10 V required special attention.
First the low voltage setup shall be described and then the high voltage setup.
Figure 3.18: AC Power Analyzer connection scheme for VDC < 10 V.
93
The low voltage setup was the first AC power analyzer developed, shown in
figure 3.18. Phase voltages below the DAQ?s 10 V rating were read directly into the
DAQ. For current measurements, a shunt resistor method was used and operates
by measuring the voltage drop across a small, high tolerance, resistor. In this case
RS = 0.005 ? with a tolerance of 0.25% shown in figure 3.19. Once the voltage and
resistance are known, ohms law is used to determine the current i(t) = v(t)/RS.
Accuracy of this measurement depends on two things: the accuracy of the resistor?s
known value, and the accuracy of the voltage measurement. The shunt resistor was
shipped with a calibration certificate, indicating the calibrated resistor value and
the test date.
Figure 3.19: 0.005 ? shunt resistor and calibration certificate, used to sense current
in the low voltage AC power analyzer.
The NI-USB-6251 DAQ has a high accuracy feature that allowed for all 16-
bits of the DAQ to be focused into a specific region, which is ideal for differential
voltage measurements. Accurate voltage measurements across the shunt resistors
was accomplished by ensuring the voltage differential was less than +/ ? 100 mV.
This high-accuracy mode was sufficient for recording differential voltages of the
94
shunt resistor accurately, with the cost of running the DAQ channel in differential
mode. This meant that 2 additional voltage measurements, 1 for line A and 1 for
line B, were required to accurately measure the line currents, reducing the maximum
read rate of the system to 250 kHz.
A high voltage setup was used to study systems between 10 V and 15 V. The
10 V lower limit comes from the DAQ?s hardware upper limit and the 15 V comes
from the maximum voltage of the DC power supply. In the dynamometer?s current
state, the setup employs two +/- 30 A bi-directional non-invasive current sensors
(NICS) to measure the current pass through lines A and B. The voltage of each
phase must be reduced by a voltage divider before being read into the DAQ.
Figure 3.20: AC Power Analyzer connection scheme for VDC > 10 V.
Two 500 ? resistors are used in series to drop the voltage of the phases to
within the 10 V range of the DAQ. The estimate of power loss from the 3 voltage
dividers at maximum voltage is 0.04 W, which is negligible compared to the 200 W
DC input power. Two equally sized resistors means the phase voltage is dropped
by half, meaning the sensed value VAN,sens must be multiplied by 2 to arrive at
95
the correct phase voltage VAN . The bi-directional current sensors output an analog
voltage between 0-5 V , proportional to the amount of current, similar to the DC
current sensor described in section 3.3. The supplier provided the following relation
between output voltage of the sensor vNICS(t) and current:
1
i(t) = [vNICS(t)? 2.5]
0.066
The sensor is centered at 2.5 V , meaning that if VNCIS?A > 2.5 V current is flowing
in the positive direction. If VNCIS?A < 2.5 V, then current is flowing in the opposite
direction. A key difference between this sensor and the shunt resistor is that it
naturally contains a small amount of low pass filtering to the signals, with a cutoff
frequency of 90kHz. This could be problematic with high pole count motors spinning
at high RPM. Additionally the higher harmonic content of the current signal is
attenuated by this filter.
The presence of a low pass filter in the non-invasive current sensor means
that the AC current signals will be attenuated. To compensate, less filtering of the
signals is done in post processing. A noticeable drop in the power factor is seen when
studying motors at higher voltages, which is a result of this filtering. Additionally,
the sensed AC power could be slightly lower than its actual value as a result of this
sensor. For future setups, it is required that the DAQs are sized to handle the DC
operating voltage so that the more accurate shunt method can be used exclusively.
To ensure that both setups are giving the same results for a given test, the same
motor, ESC, and voltage were tested on both the low-voltage and high-voltage setup.
96
An EMAX 935 KV motor and a MultiStar 30A ESC were tested at VDC = 7.2V .
Results of the calibration test are shown on the next page in figure 3.21
The results of the calibration test show that the two setups are in agreement
and produce the same results. Minor deviations in the x-crossing point of the lines
are related to the fact that during the test not all throttle settings are the same (ex
49% vs 51% throttle). The important parameter is the slope of each line relative to
the other setup, which seem to agree well. This test also determined the accuracy
of the bi-directional current sensors, as it gave results that agreeme with the low
voltage setup. However, when testing additional motors at higher voltages and
faster RPMs, it was noticed that the noninvasive current sensor?s filtering did have
an impact on the recorded AC power values.
With the setup and power analyzers covered, we proceed to modeling the
system in Chapter 4 and experiential results in Chapter 5.
97
Figure 3.21: Calibration test for same conditions on both low and high voltage
setup.
98
Chapter 4: BLDC Motor and ESC Modeling
A first principles based analytical model will be derived from the experimental
results to represent BLDC motors and ESCs. The motor theory shall be presented
as it was discovered, originally studying systems at VDC = 7.2 V and then later
expanded to cover higher voltage systems. Primary objectives for this model are:
1. Determination of an equivalent AC motor model, which relates the inputs
VLL,rms, Irms, and TR to the both the ESC?s and BLDC motor?s performance.
2. Prediction of the torque-RPM curve and mechanical output power.
3. Required AC power to run the motor.
4. ESC?s DC to AC power conversion characteristics, ending with the steady
state DC current draw out of the DC power supply.
First the BLDC motor shall be modeled in section 4.1, which describes the equivalent
circuit model for the motor alone. Secondly, the ESC is modeled in section 4.2,
with an emphasis being placed on MOSFET modeling and power transformation
relations. The BLDC motor relies on an enhanced version of the DC motor model,
presented in Chapter 2. However, in the BLDC model of section 4.1, the relevant
motor parameters have been augmented into an equivalent AC circuit. For the ESC
99
model, a literature review of MOSFETs is given to identify the types of power losses
that occur within an ESC.
Alternative models for BLDC motors exist, however, they typically are used
with a time domain analysis [33, 34] and can include even a rotating mutual phase
inductance [35]. This work seeks to model the steady state behavior of ESCs and
BLDC motors, as they are determined with the dynamometer. A unique equivalent
circuit model is to represent both the ESC and the BLDC motor.
By modeling the ESCs and the BLDC motors separately, a better understand-
ing of power loss can be gained, with the goal of identifying unique parameters
to describe different ESCs and BLDC motors. For example, in the DC model of
Chapter 2, the ESC and BLDC motor are lumped into a single black box model.
If the ESC was replaced with a different ESC, the system would have to be re-
characterized before use. Characterizing the ESC and the BLDC motor separately
allows for a more programmatic and discipled approach to aircraft design.
4.1 Brushless DC Motors
Brushless DC motors transform AC electrical power into mechanical power,
and this section models the electromechanical interaction by using an equivalent
circuit approach. The underlying equations for calculating AC power from instan-
taneous voltage and current measurements was presented in section 3.5 and are
applied to the trapezoidal and rectangular waveforms of a BLDC motor. During
motor testing, the active power as well as other key parameters related to AC power
100
are recorded:
VAB + VBC + VCA
VLL,rms = (4.1)
3
IA + IB + IC
Irms =
3
IA + IB
Irms = (4.2)
2
Equation 4.2 stems from the fact that only IA and IB are recorded for the 2-
wattmeter method, and IC = IA = IB = IL,rms. It will be shown that VLL,rms
maps to the motor?s mechanical rotation ?, whereas Irms relates to the motor?s
output torque. Both of these observations are consistent with the model presented
in Chapter 2. By referring to the DC motor model of Chapter 2, this serves as a
?sanity check?, to verify that predictions with the presented AC motor model agree
with previous work. An important distinction between the DC and AC motor model
for BLDC motors is the evaluation of AC power in a 3-phase system. In the DC
model, power to run the ESC and motor system is only DC, whereas in the pre-
sented AC motor model, only AC power is considered to run the motor. To start, let
us consider evaluating AC power directly from the rms values of VLL,rms and Irms,
which requires evaluating the rms of the trapezoidal and rectangular waveforms.
4.1.1 Waveform RMS Evaluation
Waveforms entering a BLDC motor are time varying in nature, and are shown
in figure 4.1. The derivation of the AC power entering a BLDC is built upon the
following assumptions:
101
1. The voltage waveform is trapezoidal [12,13,15]
2. The current waveforms are rectangular [12,13,15]
3. Power, for any waveforms, can be calculated using the 2 wattmeter method [22]
By leveraging the fact that the exact waveform shapes are known, the AC power
can be calculated from recorded rms values for VLL,rms and Irms. Rms values are
critical to the AC motor model as they will replace the DC values in the model from
Chapter 2.
Figure 4.1: Expected voltage and current waveforms entering a BLDC motor.
A distinction must be made: during experiments AC power is calculated using
102
the 2-wattmeter method directly from time varying AC signals (ex. vAN(t), iB(t), etc.),
whereas in the AC motor model predicts VLL,rms and Irms values to determine AC
power. It will be shown that determining PAC from predicted VLL,rms and Irms values
is accomplished with the 2-wattmeter method as well. This relies upon subtracting
waveforms from one another to arrive at an equation for the total instantaneous
power that is similar to equation 3.66.
From figure 4.1, it can be seen that the waveforms oscillate back and forth
between a peak voltage Vpk and a peak current Ipk value. To determine the AC
power in these waveforms, first the waveforms for vAN? , vBN? , and vCN? will be
mathematically defined and then subtracted from one another to find vAC and vBC .
In a similar manner, the current waveforms iA and iB are explicity stated and then
used to compute the AC power in the signal via the 2-wattmeter method.
All voltage waveforms follow the same trend, but are shifted by 2?/3 radians
from one another. Inspection of waveform vAN? from figure 4.1 shows the following
intervals:
1. +Vpk for 2?/3 radians
2. A decreasing transition from +Vpk to ?Vpk for ?/3 radians
3. ?Vpk for 2?/3 radians
4. An increasing transition from ?Vpk to +Vpk for ?/3 radians
With this, the piecewise waveform function for vAN? can be mathematically de-
103
scribed as: ? ?
??? ??? ????? 1 for 0 ? ? < ?/3 ?
????
e ??
???
?
??
?
? 1 for ?/3 ? ?e < 2?/3 ????????
?
?
? 5? 6?e/? for 2?/3 ? ?e < ?
??
vAN? = Vpk???? ? (4.3)???? ?1 for ? ? ?e < 4?/3 ?
???
??? ?? ?? ???
??
? ?1 for 4?/3 ? ?e < 5?/3 ???? ??????6?e/? ? 11 for 5?/3 ? ? < 2? ?e
Although there are only 4 unique regions, the waveform has been split into 6 regions
in order to assist with determining vAC and vBC . Note that at the end of each 60
o
interval, a commutation occurs and is shown in figures 2.25 to 2.29. A similar set
of equations can be found for vBN? and vCN? , as they have the same waveform but
are shifted 2?/3 radians:
??? ???
??????? ?1 for 0 ? ?e < ?/3 ?????
?????
??
???
??
6?e/? ? 3 for ?/3 ? ?e < 2?/3 ????????
? 1 for 2?/3 ? ?e < ?
??
vBN? = Vpk???? ? (4.4)???? 1 for ? ? ?e < 4?/3 ??
???
? ?? ????? ??? 9?
?
6?e/? for 4?/3 ? ? < 5?/3 ?? e? ? ???
????
1 for 5?/3 ? ? ?e < 2?
104
???????? ?
?
?? 1? 6?e/? for 0 ? ?e < ?/3
??
? ????????????? ?1 for ?/3 ? ?e < 2?/3 ????
?
???? ?? ?1 for 2?/3 ? ?
?
? e
< ? ?
vCN? = Vpk??? ????? (4.5)???? 6?e/? ? 7 for ? ? ?e < 4?/3
???????? 1 for 4?/3 ? ? < 5?/3 ?
???
? e? ?
????????
1 for 5?/3 ? ? < 2? ?e
Rms values for any piecewise waveform can be found using the following formula [21]:
????6
VAN? = rms{v ?} = ? I2AN i (4.6)
0
????e,i+1I2i = rms{vAN?} (4.7)
?e,i
where Ii represents the rms integral over the piecewise interval i from ?e,i to ?e,i+1.
Equation 4.6 states that sections of a piecewise function can be evaluated indepen-
dently and then combined to find the total rms value over the entire interval. When
evaluating VAN? , the expression can be simplified, after noting that only half of the
waveform needs to be evaluated due to symmetry, and then multiplied by 2 to arrive
at the final expression.
?
VAN? = I21 + I
2 2
2 + I3 + I
2
4 + I
2 2
5 + I6
I1 = I2 I1 = I4 I2 = I5 I3 = I6
105
?
V ? = 4I2 + 2I2AN 1 3 (4.8)
Likewise, integrals I1, I3 are evaluated using equation 3.9:
? ?
3
I2
1
1 = V
2
2? pk
d?e
0
V 2
2 pkI1 = (4.9)
? 6?
2 1 2 ? 6?eI3 = Vpk(5 )2d?e2? 2? ?
3
V 2
I2
pk
3 = (4.10)? 18
V 2pk V
2
pk
VAN? = 4( ) + 2( )
6 ? 18
7
VAN? = Vpk (4.11)
9
due to the balanced loading condition between vAN? , vBN? , and vCN? , the rms values
for all phase are equal to one another:
?
7
VAN? = VBN? = VCN? = Vph,rms = Vpk (4.12)
9
Note the similarities between trapezoidal and sinusodial waveform rms values:
????? Vpk??
1 = 0.7071Vpk for sinusodial waveforms2
Vph,rms = ?? (4.13)V 7pk = 0.8819Vpk for trapezodial waveforms9
106
which is a 24.7% increase in rms value when compared to a sinusoidal waveform.
Line-to-line rms value VLL,rms can be found by evaluating the rms value of
vAC . However, first the analytical expression of vAC is found by subracting equation
4.3 from equation 4.5:
? vAC = vAN?? ? vCN?? ?
????? ?????? 1 ?? ?? ???? ????? ?
????????
?
1? 6? /? ??e ???
????? 1 ?????
???? ????1 ???? 5? 6? /? ?? ??????
??????
e ?1 ??
vAC = Vpk??????
?? ??1 ???????
? Vpk??
? ? ???????
?
6?e/? ? 7 ???
????? ????? ????? ??
???
?1
????? ????? ?
?
???
1 ??
6? /? ? 11 ? 1 ?
??????
e ?? ?????????
?
6? /? ??
???????
e ??????
? 2?? ?????
???
? 6? 6? /? ?
?
vAC = Vpk??????
e ? (4.14)
?? 6? 6? /? ???
?
? e
????????
?
? ?2 ??????? ?????6?e/? ? 12 ?
By subracting two phase waveforms, the Vpk?s combine to yield 2Vpk, but now
occur with an interval of ?/3 instead of 2?/3. This can be visualized in figure 4.2.
107
Figure 4.2: Voltage waveforms vAN?, vCN? and vAC .
The rms value of vAC can be computed in a similar manner to vAN? by using
equation 4.6 and identifying symmetrical patterns in the waveform:
?
VAC = I21 + I
2
2 + I
2
3 + I
2 2
4 + I5 + I
2
6
I1 = I4 I2 = I5 I3 = I6
?
VAC = 2(I2 2 21 + I2 + I3 ) (4.15)
Integrals I1, I2, and I3 are evaluated using equation 3.9:
? ?
3
I2
1
= V 2
6?e
( )21 d?e2? pk0 ?
2V 2pk
I21 = (4.16)
? 92?
1 3
I2 = V 22 pk(2)
2d?e
2? ?
3
108
2V 2
2 pkI2 = (4.17)
? 3
1 ?2 2 6?eI3 = Vpk(6? )2d?e2? 2? ?
3
2V 2
2 pkI3 = (4.18)? 9
2V 2pk 2V
2 2
pk 2Vpk
VAC = 2(( ) + ( ) + ( ))
9 ?3 9
20
VAC = Vpk (4.19)
9
Due to the balanced loading condition between vAB, vBC , and vCA, the rms values
for all phase are equal to one another:
?
20
VAB = VBC = VCA = VLL,rms = Vpk (4.20)
9
Equation 4.20 can be extended to include any phase voltage subtracted from one
another, the resulting rms value is the same as VLL,rms. This is most notable for:
VLL,rms = VAC = VBC (4.21)
and will be useful when determining the AC motor model. Note the similarities
between trapezoidal and sinusodial waveform rms values:
???? ? ?? V3V = 3( ?pk? ph ) = 1.2247Vpk for sinusodial waveformsVLL,rms = ?? 2 (4.22)20V
9 pk
= 1.4907Vpk for trapezodial waveforms
109
which is a 21.7% increase in rms value when compared to a sinusoidal waveform.
Additionally, the ratio of VLL,rms to Vph,rms for trapezoidal waveforms can be com-
puted: ?
?20 ?V V ?LL,rms 9 pk 20= = ? 1.6903 =6 3 (4.23)
Vph 7V 7
9 pk
Interestingly, use of trapezoidal waveforms reduces the ratio of VLL,rms to Vph by
2.4%, when compared to sinusoidal waveforms.
To continue the derivation of the AC power, we must produce the rms values
for iA and iB. Inspection of waveform iA from figure 4.1 shows the following intervals:
1. +Ipk for 2?/3 radians
2. 0 for ?/3 radians
3. ?Ipk for 2?/3 radians
4. 0 for ?/3 radians
Only rms of iA will be computed, since the procedure is the same for the other line
currents. To start, consider the piecewise expression for the line current iA:
????? ??????
?
1 for 0 ? ?e < 2?/3 ?????????
?? 0 for 2?/3 ? ?e < ?
??
iA = Ipk???? ? (4.24)???? ?1 for ? ? ?e < 5?/3 ???? ????
?
? ?0 for 5?/3 ? ? ?e < 2?
The rms value of iA can be computed in a similar manner to vAN? , by using equation
110
4.6 and identifying symmetrical patterns in the waveform:
?
I 2 2 2 2A = rms{iA} = I1 + I2 + I3 + I4
I1 = I3 I2 = I4 = 0
?
IA = 2I21 (4.25)
Integrals I1 is evaluated using equation 3.9:
? 2?
2 1 3I1 = I
2
pkd?e2? 0
1
I2 21 = Ipk (4.26)? 3
1
IA = 2(I2 )
?pk 3
2
IA = Ipk (4.27)
3
Note the similarities between trapezoidal and sinusodial waveform rms values:
???? I ??1? pk = 0.7071I for sinusodial waveforms?? 2
pk
Irms = (4.28)
I 2pk = 0.8165Ipk for trapezodial waveforms3
which is a 15.4% increase when compared to a sinusoidal waveform. Table 4.1 con-
tains the complied information on the rms differences between the relevant wave-
forms.
A great deal of effort has been made to identify the relevant sinusoidal electrical
111
Table 4.1: Relation between rms values for motor vs sinusoidal waveforms.
?BLDC motor Pure Sinusoid Difference [%]
V 7 1ph,rms ? Vpk = 0.88Vpk ?? Vpk = 0.70Vpk +24.79 2
V 20LL,rms ? Vpk = 1.49V 3pk Vpk = 1.22V +21.89 2 pk?
V 20LL,rms ?Vph,rms = 1.69Vph,rms 3Vph,rms = 1.73Vph,rms ?2.47
I 2 1rms Ipk = 0.82Ipk ? Ipk = 0.70Ipk +15.43 2
circuit theory, and then augment this model for BLDC motors. This is necessary
as it properly encapsulates the correct waveforms for these devices. Now that a
relation between peak values and rms values has been established for BLDC motors,
a formulation for active and apparent power can be made.
4.1.2 Active Power from RMS Waveforms
Relation from AC power PAC to rms motor values VLL,rms and Irms is built
upon the 2-wattmeter method, which states that the instantaneous power for a
3-phase AC system is given by equations 3.42 and 3.60, and is repeated here for
convenience:
pinst = vACiA + vBCiB = pAC + pBC
where pAC and pBC are the wattmeter readings for lines A and B, respectively.
It is important to emphasize that this is not the power flowing through line A
and line B, since they are referenced to line C. It will be shown that the power
entering the BLDC motor is constant, which is consistent with 3-phase electrical
power transmission and equation 3.35.
112
Section 4.1.1 determined the relationship between peak values and rms val-
ues, whereas this section focuses on relating peak values to 3-phase active power.
Overlaying the waveform equations on top on one another shows the instantaneous
power of the system, shown in figure 4.3. To emphasize the connection between
voltage and current waveforms to AC power, the relevant plots are shown on top of
one another to visually aid in determining 3-phase active power.
113
Figure 4.3: Total instantaneous power entering a BLDC motor.
114
From figure 4.3, the total instantaneous 3-phase power entering the BLDC
motor from the ESC is:
pinst = 2VpkIpk (4.29)
and can be verified by combining the various mathematically defined waveforms
from section 4.1.1. Using equation 3.66, the instantaneous power is integrated over
one waveform cycle to determine the active power in the load:
?
1 2?
PAC = pinstd?e
2? 0
?
1 2?
PAC = 2VpkIpkd?e
2? 0
PAC = 2VpkIpk (4.30)
Now that PAC is written in terms of peak voltage and current values, table 4.1 from
section 4.1.1 can be used to relate these quantities to VLL,rms and Irms values.
? ?
9 3
Vpk = VLL,rms Ipk = Irms
20 2
? ?
9 3
PAC = 2( VLL,rms)( Irms)
? 20 2
27
PAC = VLL,rmsIrms ? 1.64VLL,rmsIrms (4.31)
10
Equation 4.31 is similar in nature to DC power PDC = VDCIDC . For AC?power,
voltage and current must be evaluated as rms quantities, and the factor of 27/10
can be thought of as an exchange rate between rms values and active power. In DC
115
power, the peak value is equal to the rms value, meaning no conversion is required.
AC power analysis has additional tools to evaluate the power content of a 3-phase
system such as apparent power and the power factor.
4.1.3 Power Factor for a BLDC Motor
A question now arises if one can calculate the power factor for these waveforms.
Recall that power factor is simply the ratio of the active power to the apparent power,
expressed by:
PAC
pf =
S
The expressions for apparent power S, derived in section 3.5, must be modified for
non-sinusoidal systems. In a standard AC system, changing the driving frequency
?e changes the impedances of the line, and thus the balance of power in the circuit
[22, 26, 31]. Augmenting the driving frequency for a sinusoidal circuit implies that
the power factor is not constant, but a function of the components in a circuit.
For BLDC motors, the waveforms are actively controlled by the ESC, mean-
ing that ?e does not change the power distribution for a circuit, as evidenced in
figure 2.35. Actively controlling the waveforms indicates that the apparent and ac-
tive powers in a circuit are set by the commutation steps and the user throttle.
Definition of 3-phase apparent power, equation 3.67, must be corrected for BLDC
waveforms. Previously, total apparent power was found under the assumption of
?
3-phase sinusoidal system and required VLL,rms = 3Vph,rms, but must be revised
for BLDC motors. This modification can be made by using the expressions in table
116
4.1 for an equivalent Y connected circuit:
S = VAN?IA + VBN?IB + VCN?IC
S = 3V
? ph,rms
Irms
7
S = 3( VLL,rms)Irms
? 20
63
S = VLL,rmsIrms (4.32)
20
With apparent power S defined, and active power given by equation 4.31, the ideal
power factor for a BLDC motor is found to be:
?
?27V10 LL,rmsIrmspf =
63V
20 LL,rms
Irms
pf = 92.6% (4.33)
Surprisingly, this result is independent of operating voltage, input throttle,
rotational speed, and all other physical parameters. A constant power factor stems
from the fact that the waveform shapes are set and controlled by the ESC. Passive
RLC circuits do not have the ability to hold the power factor constant, and so
the driving frequency ?e effects the power factor. Figure 4.4 shows experimentally
obtained data points for a BLDC motor?s power factor vs the predicted 92.6%.
117
Figure 4.4: Power factor for the EMAX 1900KV motor. Black represents the pre-
dicted 92.6% value whereas each other color represents a different throttle.
Evaluation of active power from AC quantities, as well as the power factor,
is useful as it allows one to relate rms values to active power. Next, an equivalent
circuit is determined that includes the motor?s performance.
4.1.4 Equivalent Motor Circuit
An equivalent motor circuit for BLDC motors and the ESCs can be found
by modifying the existing DC motor model to include AC rms quantities. This
augmented model serves to predict the performance metrics of the motor and the
required AC and DC powers of the system. In the section 4.2, this motor model
is expanded to include the ESC. Key points of the motor model include predicting
the torque-RPM relation, properly capturing the torque-current relationship, and
accurate prediction of the AC power required to run the motor. The model outlined
in this work will be described as it was discovered, in a logical step-by-step manner.
118
To start, an actual motor and ESC are examined:
Figure 4.5: Components considered for equivalent circuit modeling.
Proper characterization of the system relies on 3 sets of equations:
1. AC motor equations, which relate input quantities such as Vrms, Irms, and t to
the motor?s performance.
2. AC power equations, which determines the required amount of power drawn
by the motor.
3. ESC transformer equations, relating the required AC power to the required
DC current draw.
Equations relating DC, AC, and mechanical power will be derived in this
Chapter, and then overlaid with the experimental tests in Chapter 5. To start, a
component-by-component analysis of figures 4.5 and 2.21, is drawn:
Figure 4.6: Components considered for equivalent circuit modeling.
119
An actual circuit for a BLDC motor requires 3-phases to commutate the motor.
However, it has been shown that the AC power to run the motor is only related to
VLL,rms and Irms, meaning these values can be used to make a simplified ?quasi-
single phase? circuit, like a DC circuit. With this, the AC motor model for brushless
motors uses the following equivalent model for the ESC and motor:
Figure 4.7: Equivalent BLDC and ESC circuit.
The equivalent model has the following features added: a motor resistance
Rm, a back constant EMF KE, and a motor torque constant KT . By reducing the
three phases system into an equivalent DC type circuit, the motor?s complexity is
reduced, and evaluating its performance can be referenced against the Chapter 2
model. A discussion on how AC electrical parameters influence the motor torque-
RPM line starts by referring to the equivalent DC motor model that was presented
in Chapter 2:
VDCTR = IDCRm +KE?
Q = KT (IDC ? IDC,o)
To work with solely the brushless motor, AC quantities are needed. This includes
120
the terms VLL,rms and Irms, since they describe the power entering the motor. With
this, the DC model is augmented to:
VLL,rms = IrmsRm +KE? (4.34)
Q = KT (Irms ? Io) (4.35)
where VLL,rms is the line-to-line rms voltage entering the motor, Irms is the input
current to the motor, and Io is the required rms current to overcome the static
friction of the motor, all AC quantities. Equations 4.34 and 4.35 relate motor
voltage to mechanical rotation, and rms current to output torque. Equation 4.35 is
shown in figure 4.8 and is quite easy to verify with experimental data.
Figure 4.8: Q vs Irms for an EMAX 1900KV motor and MultiStar 30A ESC at VDC
= 7.2 V.
Rms values for voltages and currents can be easily measured in an experiment,
but how do these values relate to the DC voltage and the input throttle? One
121
expects a higher VLL,rms to be associated with a higher DC voltage and a higher
input throttle, but by how much? To determine the link between rms voltage and
input DC voltage and throttle, consider the following experimentally retrieved data
on motor voltages, shown in figure 4.9.
Figure 4.9: VLL,rms vs Irms for an EMAX 1900KV motor and MultiStar 30A ESC
at VDC = 7.2 V. Note how all voltages are below VDC .
From figure 4.9, it can be seen that for a given throttle setting t, all voltages
following the same linear trend with Irms:
y = b?mx
??
VLL,rms = VLL,rms? ?RESCIrms (4.36)
Irms=0
where RESC is the equivalent ESC resistance. Phase voltages are measured directly
at the output of the ESC, indicating that some voltage drop occurs across the ESC?s
internal resistance. Additionally changing the length of wire to the motor was found
122
to have no influence on RESC . These qualitative relations between VLL,rms vs Irms
can be summarized:
? Motor rms voltage VLL,rms is a function of throttle TR, rms current Irms, and
the DC voltage VDC .
?
? ?Initial offset rms voltage VLL,rms? is only a function of the DC Voltage
Irms=0
VDC and the input throttle TR, and has a value less than VDC .
? Equivalent ESC resistance RESC is the same for all throttles.
It was found from [13, 21], that the maximum rms output voltage depends on the
connection and commutation setup, and the results and copied into table 4.2 for
convenience.
Table 4.2: Relation between maximum rms values for BLDC, depedning on control
scheme. Taken from [13].
Control Strategy VLL,rms
?
Six step operation, with 180o conduction in each transistor 6VDC 0.780V? DC
Six step operation, with 120o conduction in each transistor ?3 V 0.675V
2? DC DC
?
Limit of linear modulation in ?sine/triangle? PWM ?3 VDC 0.612V2 2 DC
The ESCs studied in this work are of six step, 120o conduction commutation
type, as displayed by figures 2.25 to 2.29. Therefore, the effective rms voltage offset
of the motor is: ?? 3
VLL,rms? = ? VDCTR (4.37)
Irms=0 2?
123
with this, the rms motor voltage can be stated:
VLL,rms = ?
3
VDCTR ?RESCIrms (4.38)
2?
which satisfies the original stipulations for rms motor voltage. Overlaying the pre-
dicted VLL,rms values via equation 4.38 with the original experimentally obtained
VLL,rms values from figure 4.9 results in the plot shown in figure 4.10.
Figure 4.10: Predicted and experimentally obtained VLL,rms vs Irms for an EMAX
1900KV motor and MultiStar 30A ESC at VDC = 7.2 V.
Predicted values for VLL,rms agree well with experiment up to 90% throttle.
Somewhere between 90% and 100% throttle, the ESC begins to saturate and can no
longer provide the required 0.675VDCTR voltage, and is believed to have happened
because table 4.2 predicts the maximum theoretical rms voltage value. Saturation
is a phenomenon that occurs at high control inputs, and takes place in this system
between 90% and 100% throttle. Physically, this means that some DC voltage
124
cannot be converted into AC voltage due to inherit hardware limitations in the
ESC.
As a result of this upper saturation limit, the model presented in this Chapter
is only valid up to 90% throttle. In this region, no saturation occurs - meaning
that equation 4.37 is valid. However, at 100% throttle a nonlinearity occurs, which
means that equation 4.37 over predicts the amount of voltage available to the motor.
An area of future work is to further explore the region between 90% and 100%
throttle, because at 100% the absolute limits of the system?s performance can be
evaluated. However, within the non-saturated region, we can determine the torque-
RPM relationship.
Throttle ranges between 0 and 90% are predicted well with this model, and
are shown in Chapter 5. To determine the relationship between voltage, throttle,
and RPM, equation 4.34 can be combined with equation 4.38:
3
(? VDCTR ?RESCIrms) = IrmsRm +KE?
2?
?3 VDCTR = (RESC +Rm)Irms +KE? (4.39)
2?
which yields a more useful form of the AC motor equation. Here the influence of
user throttle and DC voltage is linked to the physical parameters of the motor, and
is similar to the brushed motor model, equation 2.15, in Chapter 2. This modified
equivalent motor model is shown in figure 4.11, and has been updated to include
the equivalent RESC .
125
Figure 4.11: Equivalent BLDC and ESC circuit, updated to include RESC .
With the voltage-RPM relation and the torque-current relation defined, an
expression relating torque to RPM can be found by combining equations 4.39 and
4.35:
KT 3 KTKE
Q = [? VDCTR ? Io(Rm +RESC)]? ? (4.40)
Rm +RESC 2? Rm +RESC
which is the same result as equation 2.7 for the equivalent brushed model. From
equation 4.40, determining the motor?s output power can be found:
Pm = Q?
KT? ?3 KTKEP = [ V 2m DCTR ? Io(Rm +RESC)]? ? (4.41)
Rm +RESC 2? Rm +RESC
Equations 4.40 and 4.41 are plotted against experimentally obtained data in figure
4.12. Again, the motor?s output power is a concave down parabola, and has a peak
power point at half the no-load speed of the motor. Without the output power of
the motor defined, an expression for the required AC power to drive the motor can
126
be made. However, first an observation about the torque vs RPM line.
Figure 4.12: Predicted and experimentally obtained Q vs RPM curve for an EMAX
1900KV motor and MultiStar 30A ESC at VDC = 7.2 V.
Careful examination of the torque vs RPM relation for a BLDC motor fur-
ther demonstrates ESC saturation at high throttle. Consider the required RPM to
generate 3.1 in-oz of torque, depicted as the lowest points for each throttle in figure
4.12. Between 40% and 90% throttle, each 10% increase in throttle has a consistent
1230 RPM increase in motor speed. However, the points between 90% throttle and
100% throttle only have 750 RPM increase, and stems from the fact that the ESC
cannot produce the required rms voltage. Reduced voltage is attributed to slower
rotational speeds by equations 2.7 and 4.40.
The strength of this equivalent motor model is that it combines easily with the
AC power relations from section 4.1.2. Given a load Q and speed ?, the required
voltage VLL,rms and current Irms can be found, and are related to the active power
PAC in the circuit via their peak values. With the insight of figure 4.11 and equation
127
4.38, a modification to the AC power entering the motor is required to account for
the impact of RESC . Power output from the ESC has an IrmsRESC voltage drop
associated with the ESC, and reduces the effective motor rms voltage. Therefore,
power entering the BLDC motor sees an I2rmsRESC power loss:
?
27
PAC = VLL,rmsIrms
10
?
27 3
PAC = (? VDCTR ?RESCIrms)Irms (4.42)
?10 2? ?
3 27 27
PAC = VDCIrmsTR ? I2rmsRESC? 20 10
AC power for a BLDC motor contains 2 components: a positive VLL,rmsIrms and
a negative I2rmsRESC component. In reality, the I
2
rmsRESC is a conductive power
loss to the ESC, which will be explained further in section 4.2. As the rms current
Figure 4.13: Predicted and experimentally obtained Q vs RPM curve for an EMAX
1900KV motor and MultiStar 30A ESC at VDC = 7.2 V.
increases, less power is available to the motor. Predicted and experimental results
128
for the rms current and AC power are shown in figure 4.13.
Representing the BLDC motor as an equivalent AC circuit predicts the motor?s
performance well. As mentioned previously, this work was originally developed for
7.2 V motors, but later expanded to include 11.1 V and 14.8 V motors. Looking
ahead: Chapter 4 continues with modeling the ESC, whereas Chapter 5 shows
experimental results of motor and ESC testing overlaid with predicted results. The
next section of work seeks to model the remainder of the propulsion system: the
ESC.
4.2 Electronic Speed Controller
An ESC is responsible for properly modulating the DC power into the proper
AC waveforms for a BLDC motor. A series of MOSFETs control the commutation
interval, based on the rotor?s position. The length of time that the MOSFETs are
conducting controls the rms voltage to the motor. Objectives for ESC modeling are
to determine:
1. DC current IDC draw out of the DC power supply
2. Physical characteristics of the ESC which are invariant of the motor
3. Power loss of the ESC PESC
Traditionally, the ESC is neglected in formal analysis. However, this work seeks to
determine a physics based model of the ESC, and to identify physical parameters
that uniquely describe the ESC. Testing ESC models is complicated because different
129
motors bring about different test environments for the ESC, regarding RPM and
current usage. To simplify this work, the following observations have been made
about ESCs:
1. Power loss increases with DC Voltage VDC .
2. Power loss increases with AC current Irms.
3. Power loss decreases with throttle TR.
Power losses for an EMAX 935 KV and MultiStar 30A ESC are shown in figures
4.14 and 4.15.
130
Figure 4.14: ESC power loss vs RPM for a MultiStar 30A ESC and EMAX 935 KV
motor.
131
Figure 4.15: ESC power loss vs Irms for a MultiStar 30A ESC and EMAX 935 KV
motor.
ESC power loss shows a strong correlation with DC voltage, rms current, and throt-
tle. Before modeling an ESC, first MOSFET power loss literature is reviewed.
132
4.2.1 MOSFET Power Losses
As was explained in section 2.3 and 2.3.2, an ESC is comprised of 6 MOSFETs
that are responsible for commutating the brushless motor. Two MOSFETs control
a commutation event and alternating the two out of six MOSFETs, the rotor is kept
spinning. Both the act of switching the MOSFET on and off, as well as the act of
allowing current to flow results in power losses. Results show that the power loss of
a MOSFET contains two primary components [36?40]:
1. A linear Irms component - related to the rapid switching on and off of the
MOSFET. Referred to as switching losses
2. A quadratic I2rms component - related to the conductive power loss of any
physical component. Referred to as conduction losses
Ideally, these losses can be written as:
PMOSFET = Psw + Pcond (4.43)
1
Psw = VDCIrms(ton + toff )fsw (4.44)
2
Pcond = I
2
rmsR (4.45)
where ton, toff , and fsw are the time to switch on in seconds, time to switch off
in seconds, and switching frequency in Hz, respectively. This relates how long
each MOSFET switches to how many switches occur each second. Switching losses
originate with a changing voltage and current acting across the MOSFET. Resistive
133
losses are evident in all physical conducting elements. Diode losses and capacitive
effects have also been included in some models [37, 40]. Favorable qualities to have
in a MOSFET are fast swtiching times, low forward voltage drop, low on resistance,
and high breakdown voltage [41].
To better understand the physical nature of switching losses, consider the
following diagram of a MOSFET:
Figure 4.16: MOSFET model.
As explained in section 2.3.2, MOSFETs have three terminals: a drain, a gate,
and a source. When a sufficient voltage is applied between the source and gate, the
MOSFET will allow current Irms to flow from the drain to the source. However, this
transition from no current flowing to current flowing takes some time. Changes in
waveforms during this switching event is what causes the switching power loss. A
typical MOSFET waveform during switching is shown in figure 4.17.
134
Figure 4.17: MOSFET switching waveforms for on and off operation.
Switching, as the name applies, occurs when the MOSFET is switched from
on (i.e. non-conducting to conducting) and off (i.e. conducting to non-conducting).
First, switching on losses shall be considered [37], which occurs between the initial
on time ti,on and the voltage falling time tv,f . At ti,on, VGS is pulled high, which
allows current to start flowing through the MOSFET. A linear transition between
0 to Irms concludes at the time of maximum current rise ti,r. After this, the voltage
difference between drain and source VDS drops from it?s maximum value to 0. The
MOSFET is fully on at tv,f , when there is no voltage difference between the drain
and source. Energy consumed during the on transition is the area under the VDSIrms
curve:
1
Esw,on = VDSIrms(tv,f ? ti,on)
2
?
ton = tv,f ? ti,on
135
1
Esw,on = VDSIrmston (4.46)
2
Energy lost during the on transition per second Psw,on can be found by:
Psw,on = Esw,on ? (number of switches each second)
1
Psw,on = VDSIrmstonfsw (4.47)
2
where fsw is the switching frequency, or the number of switches each second.
Similar logic is used to determine the switching off losses. When VGS changes
from high to low, the MOSFET begins the off transition at ti,off . At this time, VDS
linearly rises to block the flow of current and reaches its maximum value at time tv,r.
After this, the current Irms linearly decreases to 0. The MOSFET is fully off at ti,f ,
when there is no current flowing from the drain to the source. Energy consumed
during the off transition is the area under the VDSIrms curve:
1
Esw,off = VDSIrms(ti,f ? ti,off )
2
?
toff = ti,f ? ti,off
1
Esw,off = VDSIrmstoff (4.48)
2
Energy lost during the off transition per second Psw,off can be found by:
1
Psw,off = VDSIrmstofffsw (4.49)
2
136
With the switching on and off power modeled, the total switching loss can be com-
puted:
1
Psw = VDSIrms(ton + toff )fsw (4.50)
2
Conduction losses occur between tv,f to ti,off , that is: when the MOSFET is con-
ducting. Total power loss during MOSFET operation is the sum of the switching
and conductive losses, and is stated as:
1
PMOSFET = V
2
DSIrms(ton + toff )fsw + IrmsR (4.51)2
A single MOSFET contains both a linear and quadratic component of power loss
with the rms current. If there were only one MOSFET to block the entire supply
voltage, then VDS = VDC . Recall from figure 2.21, that a micro controller controls
the rate at which switching occurs, and is by user throttle. That is, the user controls
the amount of switching. From equation 4.51, it is evident that minimizing switching
losses occurs when fsw = 0, indicating that the MOSFET only has conductive losses.
In an ESC, the switching frequency fsw shows a strong correlation to the input
throttle. Before determining a power loss model for an ESC, first an ideal DC to
AC transformer is considered.
4.2.2 Ideal DC to AC Transformer
ESCs invert the steady DC waveforms into AC waveforms for the brushless mo-
tor, acting as transformers, changing DC power into AC power by rapidly switching
137
MOSFETs in a specific sequence. It was shown in the previous section that MOS-
FETs contain some power loss related to switching and conduction, however this
section presents an experiment to evaluate ESC behavior. It will be shown that by
identifying ideal ESC behavior, and tracking actual ESC behavior, the difference
between these models relates to the ESC power.
Figure 4.18 shows a diagram that relates the input power to the output power.
Here VDC and IDC are augmented into VLL,rms and Irms. Based on the equivalent
motor circuit: VLL,rms, Irms, and VDC are known, whereas the DC current draw IDC
is unknown.
Figure 4.18: ESC transformer diagram.
Converting DC power PDC into AC power PAC requires some power PESC , which is
lost to the environment as heat. A power balance relates these quantities:
PDC = PESC + PAC (4.52)
which states that the DC power is lost to either the ESC or is converted into AC
power to run the motor. From equation 4.52 and 1.1, the efficiency of the ESC can
138
be reformulated as:
PAC
?ESC =
PDC
PESC
?ESC = 1? (4.53)
PDC
Increasing ESC efficiency indicates that power lost to the ESC PESC should be
minimized. However, what if the power lost to the ESC PESC was exactly 0? Here,
an ESC is operating in an ideal state with an efficiency of 100%. An ideal ESC
implies that DC and AC powers are equal to one another:
PDC = PAC (4.54)
To explore how DC current is converted into AC current, we will first examine
an ideal transformer and then generalize this work to an non-ideal transformer.
A key concept in the transmission of electrical power, or any power, is that the
input and output power must remain accounted for. For electrical systems [41], a
reduction in voltage requires an increase in current such that if Vout < Vin then it
is guaranteed that Iout > Iin. Exchanging voltage for current stems from the fact
that power must remained balanced, and the electromagnetic system will work to
achieve this. Power cannot be created or destroyed, but merely changes form. This
logic can be extended for an ideal ESC, by using the relations for DC and AC power
from equations 3.1 and 4.42 respectively:
?
27 3 0
VDCIDC = (? V T ?R:DC R ESCIrms)Irms
10 2?
139
?
3 27
VDCIDC = VDCTRIrms (4.55)
? 20
For an ideal ESC, the equivalent resistance RESC = 0, meaning that no voltage is
dropped across the inverter. Equation 4.55 can be further simplified into a more
useful form: ?
3 27
IDC = TRIrms
?? 20
IDC 3 27 10
= TR ? TR (4.56)
Irms ? 20 9
which states that for a given amount AC current Irms, the required DC current
draw IDC out of the power supply is dependent upon the throttle alone and is
independent of the DC voltage. It is also worth noting that at 90% throttle, the
ratio IDC/Irms = 1. Equation 4.56, referred to as the ideal transformer relation, is
plotted in figure 4.19 for all throttles.
Figure 4.19: Ideal IDC and Irms relationship.
Note that this IDC/Irms relationship is a straight line with regions above and
below the line. If an actual ESC produced a IDC/Irms curve above the ideal line, then
140
more DC current is required for the same amount of rms current when compared
to the ideal relationship. Deviating above the ideal line demonstrates a nonideal
transformer, with ?ESC < 1. It is nonphysical for an ESC to occupy the region
below the ideal line, as that would suggest ?ESC > 1. Determination of the exact
location of an ESC transformation curve will allow a designer to determine the exact
amount of power lost to the ESC as heat and the DC current draw out of the battery.
4.2.3 DC Current Draw for an ESC
The concept of an ideal transformer must be extended to include losses. Ex-
perimentally obtained plots for rms current vs DC current is shown in figure 4.20,
and reveals plots similar to the ideal transformer plots shown in figure 4.19. In
both figures, for a given throttle, it appears all DC currents line on the same line,
originating from 0. When the throttle changes, the slope of the line changes and
all IDC follow the same trend. The higher the throttle, the more the ratio of DC
current to rms current becomes closer to 1.
141
Figure 4.20: IDC vs Irms for a MultiStar 30A ESC and EMAX 935 KV motor.
Figure 4.20 implies that the ratio of DC current to rms current does not depend
on voltage, which was first shown in the ideal transformer relation, by equation 4.56.
Experimentally observed ratio of IDC/Irms is the same regardless of the operating
voltage and are located on the same line for a given throttle. The goal now is to
determine an expression for the current ratio, and then deduce an expression for
ESC power. By plotting the ratio of IDC/Irms vs throttle at different voltages for
this motor/ESC combination in figure 4.21, it is apparent that these points lie above
the ideal transformer relation.
142
Figure 4.21: ESC transformer diagram for the MultiStar 30A ESC.
For all voltages, the grouping of the points is consistent. To describe the actual
ratio of points, a simple linear expression is required:
IDC
= C1TR + C0 (4.57)
Irms
IDC = (C1TR + C0)Irms (4.58)
where C1 and C0 are coefficients that describe the transformation from DC to AC
current. Note that C1 = 10/9 and C0 = 0 for an ideal transformer. Both C1 > 0
and C0 > 0, to make physical sense. One expects a more efficient ESC to have a
smaller value of C0, meaning the actual ESC transformer curve lies close to the ideal
curve. Transformation coefficients C1 and C0 are easy to find in post processing by
using a best fit approximation. In section 4.2.4, it will be shown that C1 and C0 are
related to the MOSFET switching frequency fsw. Combining the motor model with
143
equation 4.57 is shown in figure 4.22.
Figure 4.22: Predicted and experimentally obtained Q vs RPM curve for an EMAX
1900KV motor and MultiStar 30A ESC at VDC = 7.2 V.
Combining the ESC transformer relations with the equivalent motor model
allows one to determine the required DC current/power. Using the rms current,
throttle, and equation 4.57, the DC power can be evaluated:
PDC = VDCIDC
PDC = VDC(C1TR + C0)Irms (4.59)
which shows that the DC power is correlated to both the throttle and the rms
current. Recall that throttle sets the speed of the motor and the rms current sets
the torque of the motor. Therefore, it makes sense that both variables appear in
the final DC power equation. With AC and DC power determined, an expression
for ESC power is made.
144
4.2.4 ESC Power Losses
An ESC is a non ideal component, indicating that power must be lost as current
conducts across the device. Section 4.2.1 showed the literature review of MOSFET
power losses, revealing that MOSFETs contain both conductive and switching power
losses. The goal of this section is to arrive at an ESC power loss equation which
contain both a switching loss and a conduction loss, similar to equation 4.51. From
section 4.1.4, the motor model has already identified an I2rmsRESC conductive power
loss across the ESC. The remaining item to quantify is the ESC?s switching losses.
Considering again the diagram in figure 4.11, a power balanced can be used
to relate the DC and AC power to the power lost to the ESC PESC :
PDC = PESC + PAC
It was shown in the previous sections, that the AC power can be found from the
motor parameters and the operating point, whereas the DC power is determined
based on the ESC transformer relations. With PDC and PAC defined, the difference
in power is the ESC power:
PESC = PDC ? PAC (4.60)
Combining equations 4.42 and 4.59 with equation 4.60, the ESC power can be
145
determined:
?
? 27 ?3PESC = VDC(C1TR + C0)Irms ( VDCTR ? IrmsRESC)Irms
10 2?
? ?
3 27 27
PESC = ?VDCIrms((C1 ??? )TR + C0) +? 20 ? ? R10 ?E? 2SCIrm?s (4.61)
switching losses conduction losses
Equation 4.61 contains both switching losses and conduction losses, mimicking
the power loss form of a single MOSFET, equation 4.51. The result of this analysis
is that each ESC can be represented as a single equivalent MOSFET, with some
modifications. In an ESC, two MOSFETs are active at any one time, meaning that
each MOSFET VDS = VDC/2. Having two connected in series means that the total
voltage of both MOSFETs is VDS,1 +VDS,2 = VDC . The equivalent MOSFET model
contains this, as 2VDS,equiv = VDC . Additionally, the switching frequency decreases
as the throttle increases, and is consistent with the results displayed in figure 4.15,
which shows lower power poss PESC for the same Irms. Mathematically, this result
can be represented by:
2PMOSFET,sw = PESC,sw ?
3 27
VDCIrms(ton + toff )fsw = VDCIrms((C1 ? )TR + C0)
? ? 20
3 27
(ton + toff )fsw = (C1 ? )TR + C0 (4.62)
? 20
Recall that for a MOSFET, ton and toff are set by the physical characteristics
of the device, whereas fsw is controlled via the duty cycle to the gate of the MOS-
FET. The microcontroller of the ESC, shown in figure 2.21, regulates the amount of
146
switching to keep the motor commutating based of the phase voltages and the user
throttle.
Figure 4.23: Left: ESC transformer diagram for a MultiStar 30A ESC (C1 =
1.02, C0 = 0.16). Right: Predicted and experimentally obtained PESC vs Irms
curve for an EMAX 1900KV motor and MultiStar 30A ESC at VDC = 7.2 V.
Plotting equation 4.62 vs throttle shows how the C1 and C0 describe a de-
creasing switching frequency with an increased throttle. It was shown in figures
2.34 and 2.35 in section 2.3.2, that increasing the throttle linearly decreases the
number of switching events. S?imilarly, the conduction?losses are modeled as having
an equivalent resistance R = 27RESC . The factor
27 is consistent with having
10 10
defined the RESC in the previous motor modeling section and needing to convert to
active AC power.
A linear transformation relation, specifically equation 4.57, seems well suited
to describe the conversion between DC and AC current and is critical to model cor-
rectly, as time of flight estimates for electric aircraft are dependent upon the total
DC current drawn from the battery. With the AC motor model and the transfor-
147
mation relations, the designer is well equipped to answer many design challanges.
Equations 4.57 and the AC motor equations match experimental results well for all
voltages, but further work is required for the ESC model.
Experimental results show that this model for ESC works well for VDC = 7.2 V,
but contains some inconsistencies at higher voltages, which are not easily explained.
At higher voltages, the expression for ESC power, equation 4.61, both under-predicts
and over-predicts the power loss of the ESC. Additionally, there is a noticeable offset
in PESC , which remains unaccounted for in the current model, shown in figure 4.24.
It appears that conduction losses are overly pessimistic, as there is an widening gap
between experimental and expected results, even at moderate Irms.
This result is surprising, considering both the DC power and AC power are
accurately predicted, and that the final form of the ESC power equation agrees,
in theory, with the MOSFET power loss equations 4.61 and 4.51. The ESC power
loss discrepancy results in an over-estimation of cooling requirements for an ESC on
board an electric aircraft. However, the important prediction of DC current draw
remains matches with experimental results, which is important as this determines
an aircraft?s endurance.
148
Figure 4.24: ESC modeling results for a MultiStar 30A ESC and EMAX 935 KV
motor with the high voltage setup.
149
It is believed that some of this inaccuracy between model and experimental
results is attributed to the high voltage experimental setup. In section 3.5.5, it was
described how a different current measurement technique was used, and contains
some inherent low pass filtering. At higher RPMs, the effect of this filtering is
believed to augment the power content of the signals, indicating that experimental
results may contain some inaccuracies.
Additionally, it will be shown in the next Chapter that using equation 4.61
to predict ESC efficency reveals a linear decrease in ESC efficiency, something that
was first observed in [42]. Figure 4.25 shows on the left a plot of efficiency vs torque
Figure 4.25: System efficiency vs load, image on the left was taken from [42] and
image on the right was produced by this thesis. Note the linear decrease in ESC
efficiency.
from [42], and on the right information collected and predicted on ESC, motor,
and system performance from this thesis. It is believed that the PESC prediction is
correct because it matches observations from external authors. This further supports
the idea that the high voltage setup contains some error in recording AC power.
Future work consists of further refining the AC power analyzers to allow for
150
only current shunt resistors to be used for all voltages. Shunt resistors are favorable
to non-invasive hall effect sensors because shunts allow for nearly instantaneous
capturing of the current signal, and do not contain any filtering. Nevertheless, this
model is applied to all BLDC motor and ESC tests, and the results are shown in
Chapter 5.
151
Chapter 5: Experimental Results
Experimental tests were conducted with the dynamometer to study the per-
formance of different combinations of BLDC motors and ESCs under different DC
voltages. Motors between 920 KV to 2500 KV, 7 pole pairs, and at voltages of 7.2
V (2S), 11.1 V (3S), and 14.8 V (4S) were tested. Basic parameters of the motors
are outlined in table 5.1 and the motors are shown in figure 5.1. For each BLDC
motor, LAB represents the total non-rotating inductance measured between lines A
and B with a precision inductance meter in mili-henries, and Rec. Prop. is the
manufacturer recommended rotor diameter in inches.
Table 5.1: BLDC motors tested using the custom dynamometer.
KV Mass Np VDC Rec. Prop LAB
Motor
[RPM/V] [g] [-] [V] [in] [?H]
DJI 2212 920 50 7 7.2, 11.1, 14.8 8-12 29.5
EMAX 2213 935 55 7 7.2, 11.1, 14.8 8-12 30.6
EMAX ECO2306 1700 32 7 7.2, 11.1 3-5 17.6
EMAX MT2206 1900 35 7 7.2, 11.1 3-5 16.0
EMAX RS2205 2300 31 7 7.2, 11.1 3-5 10.4
Samguk Series 2500 36 7 7.2 3-5 8.2
Not all motors could be tested at higher voltages due to the magtrol magnetic
152
hysteresis brake?s 20,000 RPM limit. This effects higher KV motors, as using 14.8
V on a 2500 KV (or RPM/V) motor results in a no-load speed of 37,000 RPM,
nearly double the brake?s limit. Additionally, motors with low KV rating quickly
saturated the brake?s torque rating, and so motors with rating below 800 KV could
not be tested.
Figure 5.1: BLDC motors tested using the custom dynamometer.
ESC selection was restricted by the inherent nature of using a magnetic brake,
as it contained a repeatable static 3.1 in-oz cogging torque [24]. All motors and ESCs
must produce a sufficient initial torque to overcome this cogging torque before the
motor could start spinning. Senorless ESCs use the back EMF from the motor?s
phases to commutate the motor. As discussed in section 2.3, this is a problem for
slowly spinning motors as the back EMF (VE = KE?) is not large enough for the
ESC to detect.
153
Table 5.2: ESCs tested using the custom dynamometer.
ESC Max Cont. IDC [A] Voltage Range Mass [g]
SpiderLite Series 18 2-4S 11
Turnigy MultiStar 30 2-4S 23
DYS BlHeliOpto 40 2-6S 27
In the event of a stalled rotation, modern ESCs stop commutating the motor
in order to prevent an electrical short. Therefore, not all ESCs could begin spinning
when there was already a substantial load on the motor. It is important to note
that this is not a defect of the ESC, but rather a result of using a sensorless ESC in
an environment for which it was not designed.
Figure 5.2: ESCs tested using the custom dynamometer.
ESCs that were able to start the motor under the 3.1 in-oz static cogging
torque were tested, and shown in table 5.2 and figure 5.2. These ESCs have a max
continuous DC current rating of 18 A, 30 A, and 40 A, a voltage range of 2S - 6S, and
a mass between 11 g to 27 g. Note how manufacturers provide few details related to
the physical parameters of ESCs. Identifying key performance parameters for ESCs
is a main focus point of this work.
154
Sections 5.1 to 5.3 shows the experimental results at different voltages. Section
5.4 describes how system parameters (KT , Rm, etc.) can be extracted directly from
experimental results, and displays these results for all test in tables 5.4 to 5.9. Over
the course of experimental testing, additional questions arose related to variation in
motors as well as pairing motors to rotors. These topics are covered in sections 5.5
and 5.6.
When plotting the motor performance data, dots represent experimentally
obtained data points using the dynamometer, whereas solid lines represent predicted
performance curves using the model outlined in the previous Chapter.
155
156
5.1 Tests at 7.2V
Figure 5.3: Performance plots - DJI 920 KV - SpiderLite 18A - 7.2V
157
Figure 5.4: Efficiency plots - DJI 920 KV - SpiderLite 18A - 7.2V
158
Figure 5.5: Performance plots - DJI 920 KV - MultiStar 30A - 7.2V
159
Figure 5.6: Efficiency plots - DJI 920 KV - MultiStar 30A - 7.2V
160
Figure 5.7: Performance plots - DJI 920 KV - DYS BlHeliOpto 40A - 7.2V
161
Figure 5.8: Efficiency plots - DJI 920 KV - DYS BlHeliOpto 40A - 7.2V
162
Figure 5.9: Performance plots
163
Figure 5.10: Efficiency plots - EMAX 935 KV - SpiderLite 18A - 7.2V
164
Figure 5.11: Performance plots - EMAX 935 KV - MultiStar 30A - 7.2V
165
Figure 5.12: Efficiency plots - EMAX 935 KV - MultiStar 30A - 7.2V
166
Figure 5.13: Performance plots - EMAX 935 KV - DYS BlHeliOpto 40A - 7.2V
167
Figure 5.14: Efficiency plots - EMAX 935 KV - DYS BlHeliOpto 40A - 7.2V
168
Figure 5.15: Performance plots - EMAX 1700 KV - SpiderLite 18A - 7.2V
169
Figure 5.16: Efficiency plots - EMAX 1700 KV - SpiderLite 18A - 7.2V
170
Figure 5.17: Performance plots - EMAX 1700 KV - MultiStar 30A - 7.2V
171
Figure 5.18: Efficiency plots - EMAX 1700 KV - MultiStar 30A - 7.2V
172
Figure 5.19: Performance plots - EMAX 1700 KV - DYS BlHeliOpto 40A - 7.2V
173
Figure 5.20: Efficiency plots - EMAX 1700 KV - DYS BlHeliOpto 40A - 7.2V
174
Figure 5.21: Performance plots - EMAX 1900 KV - SpiderLite 18A - 7.2V
175
Figure 5.22: Efficiency plots - EMAX 1900 KV - SpiderLite 18A - 7.2V
176
Figure 5.23: Performance plots - EMAX 1900 KV - MultiStar 30A - 7.2V
177
Figure 5.24: Efficiency plots - EMAX 1900 KV - MultiStar 30A - 7.2V
178
Figure 5.25: Performance plots - EMAX 1900 KV - DYS BlHeliOpto 40A - 7.2V
179
Figure 5.26: Efficiency plots - EMAX 1900 KV - DYS BlHeliOpto 40A - 7.2V
180
Figure 5.27: Performance plots - EMAX 2300 KV - SpiderLite 18A - 7.2V
181
Figure 5.28: Efficiency plots - EMAX 2300 KV - SpiderLite 18A - 7.2V
182
Figure 5.29: Performance plots - EMAX 2300 KV - MultiStar 30A - 7.2V
183
Figure 5.30: Efficiency plots - EMAX 2300 KV - MultiStar 30A - 7.2V
184
Figure 5.31: Efficiency plots - EMAX 2300 KV - DYS BlHeliOpto 40A - 7.2V
185
Figure 5.32: Efficiency plots - EMAX 2300 KV - DYS BlHeliOpto 40A - 7.2V
186
Figure 5.33: Performance plots - Samguk 2500 KV - SpiderLite 18A - 7.2V
187
Figure 5.34: Efficiency plots - Samguk 2500 KV - SpiderLite 18A - 7.2V
188
Figure 5.35: Performance plots - Samguk 2500 KV - MultiStar 30A - 7.2V
189
Figure 5.36: Efficiency plots - Samguk 2500 KV - MultiStar 30A - 7.2V
190
Figure 5.37: Performance plots - Samguk 2500 KV - DYS BlHeliOpto 40A - 7.2V
191
Figure 5.38: Efficiency plots - Samguk 2500 KV - DYS BlHeliOpto 40A - 7.2V
192
193
5.2 Test at 11.1V
Figure 5.39: Performance plots - DJI 920 KV - SpiderLite 18A - 11.1V
194
Figure 5.40: Efficiency plots - DJI 920 KV - SpiderLite 18A - 11.1V
195
Figure 5.41: Performance plots - DJI 920 KV - MultiStar 30A - 11.1V
196
Figure 5.42: Efficiency plots - DJI 920 KV - MultiStar 30A - 11.1V
197
Figure 5.43: Performance plots - DJI 920 KV - DYS BlHeliOpto 40A - 11.1V
198
Figure 5.44: Efficiency plots - DJI 920 KV - DYS BlHeliOpto 40A - 11.1V
199
Figure 5.45: Performance plots - EMAX 935 KV - SpiderLite 18A - 11.1V
200
Figure 5.46: Efficiency plots - EMAX 935 KV - SpiderLite 18A - 11.1V
201
Figure 5.47: Performance plots - EMAX 935 KV - MultiStar 30A - 11.1V
202
Figure 5.48: Efficiency plots - EMAX 935 KV - MultiStar 30A - 11.1V
203
Figure 5.49: Performance plots - EMAX 935 KV - DYS BlHeliOpto 40A - 11.1V
204
Figure 5.50: Efficiency plots - EMAX 935 KV - DYS BlHeliOpto 40A - 11.1V
205
Figure 5.51: Performance plots - EMAX 1700 KV - SpiderLite 18A - 11.1V
206
Figure 5.52: Efficiency plots - EMAX 1700 KV - SpiderLite 18A - 11.1V
207
Figure 5.53: Performance plots - EMAX 1700 KV - MultiStar 30A - 11.1V
208
Figure 5.54: Efficiency plots - EMAX 1700 KV - MultiStar 30A - 11.1V
209
Figure 5.55: Performance plots - EMAX 1700 KV - DYS BlHeliOpto 40A - 11.1V
210
Figure 5.56: Efficiency plots - EMAX 1700 KV - DYS BlHeliOpto 40A - 11.1V
211
Figure 5.57: Performance plots - EMAX 1900 KV - MultiStar 30A - 11.1V
212
Figure 5.58: Efficiency plots - EMAX 1900 KV - MultiStar 30A - 11.1V
213
Figure 5.59: Performance plots - EMAX 1900 KV - DYS BlHeliOpto 40A - 11.1V
214
Figure 5.60: Efficiency plots - EMAX 1900 KV - DYS BlHeliOpto 40A - 11.1V
215
216
5.3 Tests at 14.8V
Figure 5.61: Performance plots - DJI 920 KV - SpiderLite 18A - 14.8V
217
Figure 5.62: Efficiency plots - DJI 920 KV - SpiderLite 18A - 14.8V
218
Figure 5.63: Performance plots - DJI 920 KV - DYS BlHeliOpto 40A - 14.8V
219
Figure 5.64: Efficiency plots - DJI 920 KV - DYS BlHeliOpto 40A - 14.8V
220
Figure 5.65: Performance plots - EMAX 935 KV - SpiderLite 18A - 14.8V
221
Figure 5.66: Efficiency plots - EMAX 935 KV - SpiderLite 18A - 14.8V
222
Figure 5.67: Performance plots - EMAX 935 KV - MultiStar 30A - 14.8V
223
Figure 5.68: Efficiency plots - EMAX 935 KV - MultiStar 30A - 14.8V
224
Figure 5.69: Performance plots - EMAX 935 KV - DYS BlHeliOpto 40A - 14.8V
225
Figure 5.70: Efficiency plots - EMAX 935 KV - DYS BlHeliOpto 40A - 14.8V
226
5.4 Determining Motor Parameters from Experimental Data
Once a BLDC motor, ESC, and VDC are tested, the next step is to extract the
physical parameters of the model. It is important to have enough points to properly
cover the motor?s expected operating range, otherwise insufficient data points could
result in a poorly fit model (as was with motors tested below 800 KV). This section
describes a procedure to minimize the error between experimentally recorded points
and model predictions. A summary and description of modeling parameters from
Chapter 4 is given in table 5.3.
Table 5.3: ESCs tested using the custom dynamometer.
Constant Description
KT [mNm/A] Q = KT Irms, Motor torque constant
KE [mVs/rad] VE = KE?, Motor back EMF constant
Io [A] Motor profile rms current
Rm [?] Motor resistance
C1 [-] ESC switching slope
C0 [-] ESC switching offset
RESC [?] ESC Resistance
Key importance is placed on matching the Q-rms current, Q-RPM curve,
AC power curves, and the DC current draw curves. First KT , Io, and RESC are
found, then using these values KE and Rm are estimated. Once the motor has been
described, the ESC transformer coefficients C0 and C1 are deduced. This process
can be outlined as:
1. Determine KT and Io from Q vs Irms plot
227
2. Estimate RESC from VLL,rms vs Irms relation
3. Using the estimated motor values, predict KE and Rm from the Q vs ? plot
4. Evaluate ESC coefficients C0 and C1 from IDC/Irms vs TR
Together, these 6 values predict the performance of the motor and the ESC for all
throttle ranges and loads.
Starting with the Q-rms current plot, shown in figure 5.71, the equation relat-
ing rms current to output torque is a straight line described by equation 4.35 and
is copied below:
Q = KT (Irms ? Io)
Figure 5.71: Experimentally observed Q-RPM relation.
Figure 5.71 shows that all of the points line on the same line, for all throttles.
228
To find KT and Io a linear relation (y = mx+ b) is needed with:
y = Q x = Irms
m = KT b = ?KT Io
For determining phyiscal parameters from experimental data, 2 strategies exist.
The first is to determine parameters for each throttle setting and then average the
individual estimates together, resulting in a single estimate for all throttle settings.
The second method involves formulating a linear least squares problem to minimize
the total error for all throttles using a single estimate. Which method is chosen
depends on the amount of allowed error.
For KT and Io, determining KT for each throttle setting alone could produce
KT estimates which can change up to 10% between throttle settings. As such, the
second method is used, which involves considering each test at a constant throttle
containing a vector of information and combining them in a specific manner. Each
test at constant throttle produces a vector of experimental obtained points, an
example for torque at 40% throttle is:
Q = [Q T
T =0.4 1,T
. . . Q
R=0.4 k,TR=0.4]
R
where TR is the throttle setting, and k is the number of data points collected during
that test. Using this notation, y and x are redefined to make vectors containing all
sets of data:
229
?? ???Q ?
? ?
?? I? TR=0.4???Q ???? ?
??? rms,TR=0.4????I
y = ? TR=0.5? ?? ?? . ? ?
? rms,TR=0.5?
? ..? ?
x = ???
?
. ?
?? ?
.. ????
?Q ???? ????I ?T =0.8 rms,TR=0.8?R ??
Q I
T =0.9 rms,TR=0.9R
Next a linear regression is required to find the values for the slope m? and the y-
intercept b?, which minimizes the error between the experimental values and the
estimated regression line. Matlab?s polyfit function returned estimates for m? and
b?, allowing KT and Io to be estimated.
Before determining the remaining motor parameters, RESC must be estimated.
As current flows across the ESC, a voltage drop of IrmsRESC occurs, meaning less
voltage acts on the motor and is given by equation 4.38:
3
VLL,rms = ? VDCTR ? IrmsRESC
2?
RESC is found by examining figure 5.71, which shows a linear drop in line-to-line
motor voltage with rms current. A linear estimation is applied to each VLL,rms vs
Irms relation separately, and then the results were averaged to find a single RESC to
describe the system. This was found to be effective, as RESC changed less than 5%
for each throttle setting. Now that Io, KT , and RESC have been determined, the
remaining motor parameters can be extracted.
To determine the motor resistance Rm and back-EMF constant KE, an ap-
230
proach similar to finding KT and Io is taken. Equation 4.40 relates Q to RPM:
KT 3 KTKE
Q = [? VDCTR ? Io(Rm +RESC)]? ?
Rm +RESC 2? Rm +RESC
which depends on throttle. It is more accurate to minimize the error for the entire
system at once, rather than for each throttle setting individually and then averaging
the results into a single estimate for KE and Rm. A linear least squares fit is used,
which minimizes error between all Q-RPM points and the expected model. However,
first equation 4.40 must be written in the following form:
y = Ax (5.1)
with:
?? ? ? ???Q ??? ???? ?3T =0.4 VDC(0.4)? TR=0.4? R 2? ?????Q? TR=0.5?? ???? ?
?????? ?3TR=0.5 V2? DC(0.5)??
??? ?? ?KE? . . . R +R ?m ESCy = ? .. ??+KT Io1 A = K ?T .. .. ?? ? ?
x = ?? ??
???Q ???? ?
??? ? 1
TR=0.8 ??
? Rm+RESC
?? ?3 ?TR=0.8 V2? DC(0.8)???
Q ?? ?3 V
T =0.9 TR=0.9 2? DC
(0.9)
R
where
1 = [1 1 . . . 1 length(y)]T
The vector 1 contains only 1s and has as many elements as y. Once A and y
are populated with experimental data, a linear least squares regression is applied to
231
determine a vector x? which has minimal error between the model and experimentally
obtained points, found via [43]:
x? = (ATA)?1ATy (5.2)
from here, the model parameters for Rm and KE can be found. At this point, the en-
tire motor (KT , KE, Io, and Rm) and RESC have been estimated from experimental
observations. With the motor modeled, attention turns to the ESC.
ESC transformer relation, equation 4.57, is used to find suitable values for C0
and C1 by taking the average ratio of IDC/Irms for each throttle. Matlab?s polyfit
function is used to find a linear fit for C0 and C1. Now that the motor and ESC
performance parameters have been identified, this model is applied to all of the
motors and ESCs studied, with the results listed in tables 5.4 to 5.9.
Table 5.4: Parameters for the DJI 920 KV motor.
BLDC Motor ESC
VDC ESC KT KE Io Rm C1 C0 RESC
[V] Used [A] [mNm/A] [mVs/rad] [A] [?] [-] [-] [?]
18 13.5796 6.9699 0.2918 0.1408 0.9975 0.2049 0.0725
7.2 30 13.2212 7.1323 0.2721 0.1683 0.9692 0.1774 0.0997
40 13.5531 7.2492 0.2993 0.1549 1.0078 0.1586 0.0763
18 12.7273 6.9709 0.3783 0.1557 1.0713 0.1503 0.0802
11.1 30 14.0378 7.2008 0.6722 0.1563 1.0298 0.1084 0.1254
40 14.1674 7.3004 0.7046 0.1786 1.0540 0.0982 0.0596
18 14.3306 6.8977 0.7770 0.1686 1.0682 0.1325 0.0983
14.8
40 14.3459 7.3194 0.9183 0.1722 1.0541 0.0661 0.0837
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Table 5.5: Parameters for the EMAX 935 KV motor.
BLDC Motor ESC
VDC ESC KT KE Io Rm C1 C0 RESC
[V] Used [A] [mNm/A] [mVs/rad] [A] [?] [-] [-] [?]
18 14.3088 7.0255 0.3158 0.1522 1.0322 0.2022 0.0777
7.2 30 13.8519 7.1497 0.2838 0.1638 0.9873 0.1596 0.1221
40 13.9595 7.3758 0.2861 0.1496 1.0178 0.1536 0.0860
18 13.8310 7.0950 0.4988 0.1692 1.0497 0.2155 0.0642
11.1 30 13.5964 7.2888 0.4528 0.1680 0.9868 0.1989 0.1033
40 13.9678 7.4391 0.5571 0.1735 1.0522 0.1503 0.0580
18 14.0402 6.8776 0.5699 0.1956 1.0059 0.2450 0.0943
14.8 30 13.7628 7.0752 0.5981 0.1814 0.9446 0.2102 0.1415
40 14.1937 7.4209 0.7428 0.1999 1.0561 0.1340 0.0669
Table 5.6: Parameters for the EMAX 1700 KV motor.
BLDC Motor ESC
VDC ESC KT KE Io Rm C1 C0 RESC
[V] Used [A] [mNm/A] [mVs/rad] [A] [?] [-] [-] [?]
18 7.1635 4.0680 0.4001 0.0839 1.0016 0.2108 0.0578
7.2 30 7.0592 4.1228 0.5337 0.1039 0.9124 0.1763 0.0421
40 7.1554 4.1339 0.4759 0.0945 0.9619 0.1589 0.0353
18 7.4288 3.8686 0.8052 0.0831 1.0274 0.1714 0.0565
11.1 30 7.0592 4.0982 0.7585 0.1098 0.9524 0.1658 0.0473
40 7.2421 4.0784 0.7451 0.1096 0.9822 0.1534 0.0301
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Table 5.7: Parameters for the EMAX 1900 KV motor.
BLDC Motor ESC
VDC ESC KT KE Io Rm C1 C0 RESC
[V] Used [A] [mNm/A] [mVs/rad] [A] [?] [-] [-] [?]
18 6.2241 3.3632 0.2669 0.1222 0.9954 0.2111 0.0428
7.2 30 6.2417 3.6411 0.6699 0.1419 0.9439 0.1605 0.0301
40 6.2456 3.6113 0.5176 0.1355 0.9588 0.1638 0.0244
30 6.2710 3.7178 0.6012 0.1357 0.9352 0.1710 0.0433
11.1
40 6.3790 3.6104 0.5685 0.1524 0.9970 0.1471 0.0229
Table 5.8: Parameters for the EMAX 2300 KV motor.
BLDC Motor ESC
VDC ESC KT KE Io Rm C1 C0 RESC
[V] Used [A] [mNm/A] [mVs/rad] [A] [?] [-] [-] [?]
18 4.9924 2.7274 0.7198 0.0654 0.9638 0.2605 0.0443
7.2 30 4.9681 3.0137 0.7269 0.0743 0.9183 0.1908 0.0366
40 5.0273 2.9522 0.7179 0.0644 0.9541 0.1868 0.0426
Table 5.9: Parameters for the Samguk 2500 KV motor.
BLDC Motor ESC
VDC ESC KT KE Io Rm C1 C0 RESC
[V] Used [A] [mNm/A] [mVs/rad] [A] [?] [-] [-] [?]
18 4.6697 2.5155 0.5709 0.0579 0.9707 0.2765 0.0460
7.2 30 4.5483 2.7451 0.9172 0.0688 0.9253 0.1954 0.0387
40 4.7499 2.7572 1.2073 0.0652 0.9667 0.1840 0.0313
Predicted motor and ESC performance using these parameters are overlaid
with experimental results in sections 5.1 to 5.3. The AC motor model predicts
mechanical power, AC power, and DC power quite well. However, when estimating
234
the efficiencies of the motor and ESC, the results are suboptimal. It appears that
the motor efficiency is overestimated, whereas the ESC efficiency is underestimated.
The product of ?ESC and ?m is the system efficiency, which is predicted well by the
model.
ESC modeling is a future area of work. It is believed that the type of current
sensors used in the high voltage setup causes the apparent ?curving? of the ESC
efficiency and power factor plots. Recall these non invasive hall effect sensors contain
low pass filtering, which attenuates the higher harmonic content of the signal. A
lower current draw would explain the lower AC power, power factor, and ESC
efficiency. These results will be further verified in the future with more accurate AC
power testing equipment. Next, some common trends in the data are discussed.
235
5.4.1 Common Trends
Experimental observations reveal 3 trends: a 43% difference in experimental
KE and manufacturer KV values, specific masses of BLDC motors ranging from 0.4
kW/kg to 4.9kW/kg, and experimentally observed values for KT/KE > 1.64.
Figure 5.72: Experimentally determined KE vs manufacturer provided KV for all
tests, in the same units. Circles, crosses, and squares indicate the 18 A, 30 A, and
40 A ESCs respectively.
Figure 5.72 shows experimentally obtained values for KE vs manufacturer
provided values for KV, both in terms of RPM/Volt. From the trend, it is clear that
experimental KE values are 43% higher than the manufacturer provided KV value.
Section 2.1.2 discussed the differences between KT , KE, and KV. Manufacturer
provided KV value relates the DC voltage applied to the ESC to the maximum
motor no-load speed ?NL:
?NL = KV VDC
236
This definition includes the ESC, and thus is not specific to the motor alone. Based
on the results from Chapter 4, the motor no load speed is:
?3 V t? (R +R
2? DC m ESC
)Io
?NL = (5.3)
KE
It is clear that these definitions of KV and KE are not related, and that providing
KV alone does not describe the motor?s performance.
Of key importance with motor models is the ratio of KT/KE, with some
common examples listed in table 2.1. Most brushed motor models assume that
KE = KT [12, 13], however, experimental results for the model used in this work
show KT > KE. This discrepancy comes from the fact that all three phases of the
brushless motor have been replaced with a single, equivalent circuit. Figure 5.73
shows experimentally obtained KE values vs KT values.
Figure 5.73: KT vs KE for all tests. Circles, crosses, and squares indicate the 18 A,
30 A, and 40 A ESCs respectively.
237
It appears that motors with smaller values of KT and KE have a ratio of
KT/KE ? 1.64. Referencing the motor parameters from tables 5.4 to 5.9 reveals
that motors with ratios KT/KE closer to 1.64 are physically smaller motors with
higher KV ratings. For the model presented in this work, the ideal ratio of KT/KE
can be found by using an ideal power relation between mechanical and AC power:
Pm = PAC
?
27
Q? = VLL,rmsIrms
10
In an ideal motor:
Q = KT Irms
VLL,rms = KE?
because Io = 0 and Rm = 0. Therefore:
?
27
(KT Irms)? = (KE?)Irms
10
?
KT 27
= ? 1.64 (5.4)
KE 10
This explains why some motors appear to approach an ideal ratio of 1.64. However,
why do some motors have KT/KE ratios close to 1.9? The answer can be found
by augmenting the previous KT/KE analysis to include some tiny motor resistance
238
such that Rm > 0.
VLL,rms = IrmsRm +KE?
Additionally, the presence of torque requires Irms > 0:
Rm > 0 and Irms > 0 =? IrmsRm > 0
therefore:
IrmsRm +KE? > KE?
Returning to the original ideal power equation:
Pm = PAC
?
27
Q? = VLL,rmsIrms
? 10 ?
27 27
(KT Irms)? = (IrmsRm +KE?)Irms > (KE?)Irms
10 10
for a nonideal motor: ?
KT 27
>
KE 10
this is supported by plotting KT/KE vs Rm in figure 5.73. For moto?rs with larger
resistance Rm, the ratio of KT/KE is larger than the ideal value of 27/10. It is
important to note that having a ratio of KT/KE = 1 does not imply a 100% efficient
motor. There is a clear upwards trend: increasing Rm results in a larger ratio of
KT/KE. Motors with higher Rm are physically larger motors, with higher KT
239
and KE ratings. Power losses associated with the motor places the experimentally
recorded value at 1.90, indicating a 15.6% difference from the ideal ratio.
Finally, power to mass ratios for several motors are considered. BLDC motors
used for this thesis are commercially available devices, so how do they compare
against commercially available brushed motors? To answer this, brushed motors
from an online vendor [44] were examined, and their results summarized below:
Table 5.10: Commercially available Brushed Motors at VDC = 12 V.
mm [gr] Qs [in-oz] ?NL [RPM] Pmax [W] Pmax/mm [kW/kg] Max ?m [%]
215 23.0 6100 25.9 0.12 69.4
220 62.5 19300 222.4 1.01 74.2
295 48.0 5600 49.5 0.17 78.7
350 76.1 7000 98.3 0.28 72.0
350 125.7 7000 162.2 0.46 73.0
Brushed motors have specific powers between 0.12-1.01 kW/kg at up to 78.7%
efficiency. A small gas engine, the AP Yellowjacket, was analyzed in [9] and has a
specific power density of 1.1 kW/kg. For the BLDC motors tested in this thesis, the
maximum power point was interpreted from experimentally obtained data points at
full throttle.
A key point of this work was to study the ESC effect on BLDC performance.
ESC?s have a notable impact on augmenting the specific power density of the sys-
tem. For a quadcopter with 4 ESCs, oversizing the ESC can result in a substansial
reduction in mission performance associated with four times the extra mass. To
demonstrate this idea, consider the following tables which relate ESC usage to sys-
240
tem specific mass ratio:
Table 5.11: EMAX 935 KV at 7.2 V.
ESC [A] Pmax [W] Pmax/mm [kW/kg] Pmax/(mm +mESC) [kW/kg]
18 46.7 0.93 0.69
30 38.1 0.76 0.48
40 42.3 0.85 0.47
Table 5.12: EMAX 1700 KV at 7.2 V.
ESC [A] Pmax [W] Pmax/mm [kW/kg] Pmax/(mm +mESC) [kW/kg]
18 60.0 2.79 2.1
30 61.8 2.5 1.6
40 66.5 2.6 1.5
From the previous tables it is clear that improperly sizing the ESC can result
in a 30% reduction in specific system mass. The results of peak mechanical power
vs brushless motor mass for each ESC are shown on the next page in figure 5.74.
Note that increasing DC voltage increases the peak mechanical output power by
equation 2.11.
241
Figure 5.74: Maximum specific power (Pmax/mm) for all BLDC motors tested in
this thesis.
242
Note how the maximum power changes little for a given motor with a different
ESC, however, the effect of increased ESC mass can significantly reduce overall
specific power of the propulsion system. Table 5.13 recaps the power to weight
ratios of the different motors analyzed thus far:
Table 5.13: Specific powers of different small scale UAV motors.
Type Pmax/mm [kW/kg]
Brushed DC 0.12-1.01
Brushless DC 0.60-4.90
Gas Engine [9] 1.10
Engine selection makes a key impact on aircraft performance. Gas engines
do not scale down well to smaller sizes, which is evidenced by the fact that they
have mass ratios comparable to brushed motors. On the other hand, gas engines
for full size aircraft can be over 10 kW/kg, surpassing even the best BLDC motors.
The best airworthy BLDC motors are used on NASA?s X-57, and has a continuous
specific power of Pcont/mm = 3 kW/kg at 95% efficiency [7]. Although a lower
kW/kg than the motors tested using the dynamometer, the X-57?s electric motors
have a noticeable increase in efficiency compared to commercial off the shelf BLDC
motors. This reiterates the idea that engineering is a trade off, designing for higher
efficiency correlates to a lower specific mass.
Next the idea of motor variation is discussed.
243
5.5 Motor Variation Tests
Throughout this analysis, it was assumed that the characteristics of each motor
is identical to every other motor of the same type. That is, ordering two EMAX
1700 KV motors, one would expect identical performance for both motors under the
same operating conditions. However, how true is this assumption? Given the lack
of technical information provided by motor manufacturers on their products, is it
safe to assume their production methods are reliable? To answer this question, a
pack of four identical motors were ordered from a single supplier to determine the
amount of variation between the motors. Each motor was evaluated separately with
the dynamometer and their performance metrics compared.
Figure 5.75: Four EMAX 1700 KV motors used in the variation tests.
Four EMAX 1700 KV motors were tested with the dynamometer at VDC =
7.2 V, and paired with a SpiderLite 18 A ESC, MultiStar 30 A ESC, and DYS
BlHeliOpto 40 A ESC. There was no variation testing with different ESCs, as the
design space quickly becomes untenable with the number of design variables. Only
the motors were swapped, that is motor 1 was tested with the same MultiStar 30
244
A ESC as motor 2, 3, and 4. The same is true for the other ESCs. If a motor were
defective, one would expect there to be a noticeable drop in performance when the
motor is compared to the other motors. On the same note, when assigning motor
parameters to the data in post processing, it would be clear that a defective motor
would have physical parameters that would stand out, when compared to the other
motors. Results of the motor variation tests are shown in figures 5.76 to 5.81.
245
Figure 5.76: Motor variation tests - EMAX 1700 KV - 18 A ESC - 7.2 V
246
Figure 5.77: Motor variation tests - EMAX 1700 KV - 18 A ESC - 7.2 V
247
Figure 5.78: Motor variation tests - EMAX 1700 KV - 30 A ESC - 7.2 V
248
Figure 5.79: Motor variation tests - EMAX 1700 KV - 30 A ESC - 7.2 V
249
Figure 5.80: Motor variation tests - EMAX 1700 KV - 40 A ESC - 7.2 V
250
Figure 5.81: Motor variation tests - EMAX 1700 KV - 40 A ESC - 7.2 V
From the perfromance plots, the model parameters for each motor and ESC
have been extracted and are shown in tables 5.14 to 5.16.
251
Table 5.14: Motor variation test results w/ 18 A ESC.
BLDC Motor ESC
Motor ESC KT KE Io Rm C1 C0 RESC
[#] Used [A] [mNm/A] [mVs/rad] [A] [?] [-] [-] [?]
1 18 7.1498 3.8690 0.3902 0.0886 1.0002 0.2107 0.0441
2 18 7.1006 3.8852 0.5097 0.0869 0.9979 0.2103 0.0445
3 18 7.0511 3.8475 0.4155 0.0898 1.0121 0.2060 0.0426
4 18 7.2933 3.8598 0.5876 0.0885 1.0111 0.2056 0.0453
Mean 7.1487 3.8654 0.4757 0.0884 1.0053 0.2082 0.0441
Std. Dev. [%] 1.5 0.4 19.0 1.3 0.7 1.3 2.6
Table 5.15: Motor variation test results w/ 30 A ESC.
BLDC Motor ESC
Motor ESC KT KE Io Rm C1 C0 RESC
[#] Used [A] [mNm/A] [mVs/rad] [A] [?] [-] [-] [?]
1 30 7.0592 4.1228 0.5337 0.1039 0.9124 0.1763 0.0421
2 30 7.0260 4.1260 0.6364 0.1026 0.9456 0.1635 0.0375
3 30 7.0303 4.0786 0.5530 0.1043 0.9242 0.1789 0.0414
4 30 7.1328 4.1197 0.6856 0.1055 0.9279 0.1733 0.0403
Mean 7.0621 4.1118 0.6022 0.1041 0.9275 0.1730 0.0403
Std. Dev. [%] 0.7 0.5 11.8 1.1 1.5 3.9 5.0
Table 5.16: Motor variation test results w/ 40 A ESC.
BLDC Motor ESC
Motor ESC KT KE Io Rm C1 C0 RESC
[#] Used [A] [mNm/A] [mVs/rad] [A] [?] [-] [-] [?]
1 40 7.1554 4.1339 0.4759 0.0945 0.9619 0.1589 0.0353
2 40 7.1989 4.1138 0.8572 0.0986 0.9901 0.1522 0.0309
3 40 7.1390 4.0870 0.6859 0.0999 0.9633 0.1666 0.0314
4 40 7.1723 4.1748 0.7385 0.0994 0.9850 0.1537 0.0318
Mean 7.1664 4.1274 0.6894 0.0981 0.9751 0.1579 0.0324
Std. Dev. [%] 0.36 0.90 23.1 2.5 1.5 4.1 6.2
252
From tables 5.14 to 5.16, it is clear that the motors show good agreement
with one another for KT , KE, and Rm, as these parameters contained less than a
5% deviation. It is expected that KT and KE should not change between different
motors of the same type because they are set by the physical construction of the
motor, such as the size of magnets, airgap, and overall electromagnetic design.
Motor resistance could change between motors, depending on the length of wire
used to connect the ESC to the motor. All tests conducted here used the same
length of wire, and thus the grouping of Rm is within 2.5%. However, a significant
discrepancy exists for Io, as this value has a standard deviation of 19.0%, 11.8%,
and 23.1% across the different tests. A high Io means more current Irms is required
to generate the same load as a motor with a low Io and thus a lower motor efficiency
?m.
A high Io could have several explanations. From handling the motors it was
obvious that some EMAX 1700 KV motors were easier to spin than other. If the
bearing between the rotor and the stator is not clean, debris could impede the ro-
tation of the motor, accumulating in a higher current required to overcome static
friction, hence a higher Io is needed. Additionally, this hypothesis is supported
by the fact that motor 1 consistently has the lowest Io and the highest motor effi-
ciency. Motor 1 was used extensively on the dynamometer to produce the results
shown in table 5.6. During these tests debris stuck in motor 1?s bearing could have
been dislodged, whereas motors 2-4 were tested directly out of the box. Debris re-
lated to shipping the motors (dirt, packaging, etc) remained logged in the bearings,
increasing the no-load current.
253
5.6 Motor and Rotor Pairing
Electric motors and ESCs make up only half of the propulsion system on board
an aircraft: the other half is the rotor. With motors and ESCs modeled, this research
seeks to improve the design process of electric aircraft by optimally pairing motors
and rotors to reduce the total power of an aircraft. To accomplish this goal, a hover
test stand was constructed, allowing one to study the performance of a rotor, motor,
and ESC simultaneously. Before describing the hover test stand, a brief review of
rotors is required.
Rotor thrust T , rotor power P , and rotor torque Q are related to the rotor
RPM ? by [27,28]:
T = CT?A(?R)
2 (5.5)
Q = CQ?A(?R)
2R (5.6)
P = CP?A(?R)
3 (5.7)
where CT , CQ, and CP are the thrust, torque, and power coefficients, respectively.
Additionally, ?, R, and A describe the atmospheric density, rotor radius, and rotor
disk area. To determine CT , CQ, and CP from airfoil and planform properties is
beyond the scope of this thesis, but methods such as blade element momentum
theory and free vortex wake methods are used. In the context of this work, the
reader must be aware that CT , CQ, and CP change depending on airfoils used,
reference flight condition, and planform design. To study a rotor, one can predict
254
the CT , CQ, and CP from blade design, or one could extract them directly from
experimental data.
In the steady state hover condition for a given rotor, CT , CQ, and CP can be
determined from experimental results. To do so, one must record the rotor thrust
TR, rotor torque QR, and rotor RPM ?. The rotor thrust and torque both evolve
as the square of the rotor RPM, representing a unique distinction for the motor
loading condition on the dynamometer. With the use of the magnetic hysteresis
brake, the dynamometer was able to vary the load on the motor independent of the
rotor speed, creating a linear relationship between torque and RPM for the same
throttle. However, as outlined by equation 5.6, the rotor RPM and rotor torque are
linked together in a quadratic nature by the rotor design.
To change the rotor thrust, torque, and power, with a given CT , CQ, and
CP , one has to change the RPM of the rotor. For aircraft that hover with fixed
pitch blades, such as quadcopters, how is this done? The answer is that the user
must change the throttle input into the ESC. Increased voltage on the motor causes
the mechanical speed to increase. However, increasing speed increases rotor torque,
given by equation 5.6. The input to the system is the throttle setting, which increases
the rotor speed and RPM. To determine the exact mapping between throttle and
rotor performance, a hover test stand was constructed.
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5.6.1 Hover Test Stand
The goal of the hover test was to study the interaction between rotors, motors,
and ESCs. A hover test stand places a rotor on a motor, and attaches torque and
force sensors to the motor in order to record the rotor?s performance, shown in
figures 5.82. Unique to this setup is the AC power measurement setup, which allows
the user to record the AC power exiting the ESC and entering the motor. The full
suite of power and test equipment used on the dynamometer are used in the hover
test stand.
Figure 5.82: Hover test stand.
Spinning a rotor with a motor in conjugation with an ESC creates a set of
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experimental data points that are used to verify a combined momentum theory rotor
model and BLDC/ESC motor model. An EMAX 935 KV motor and a MultiStar 30
A ESC at VDC = 7.2 V configuration was selected as the drive train for the rotor.
Two 10 inch diameter commercial off the shelf rotors were used in this test, one with
2 blades (2x1045) and one with 3 blades (3x10). Both rotors are shown in figure
5.83.
Figure 5.83: Rotors used on the hover test stand.
Using these rotors on the hover test stand, the following thrust and torque informa-
tion was collected:
Figure 5.84: Hover test stand results for the 2x1045 and 3x10 rotors. Solid lines
indicates momentum theory predictions.
In figure 5.84, the momentum theory estimates for CT and CQ are drawn as
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solid lines. From the experimentally obtained information shown in figure 5.84, the
following table on rotor performance can be created:
Table 5.17: Rotor aerodynamic parameters found from hover test stand.
Rotor R [in] NB CT CQ FOM [%]
2x1045 5 2 0.0150 0.0021 62.5
3x10 5 3 0.0118 0.0018 50.4
To better understand the interaction between motors and rotors, the Q-RPM plot
of the rotor is overlaid with the Q-RPM plot of the EMAX 935 KV motor in figure
5.85.
Figure 5.85: Rotor torque requirement overlaid with motor torque-RPM curves.
Consider a nominal point with some Q, RPM, and throttle in figure 5.85. For
a motor, moving right on the Q-RPM plot requires an increase in motor voltage,
which is achieved with a higher throttle setting. Normally the load on the motor
is constant, meaning increasing voltage only increases the RPM of the motor and
258
thus new points added are on a horizontal line. However, with a rotor, increasing
RPM increases torque, meaning that as throttle increases the evolution of points is
now horizontal and upwards. Equation 5.6 describes this evolution, and thus can
be modeled.
The equivalent BLDC motor equations from Chapter 4 can be rewritten to
model the changing rotor load. As with the dynamometer, the operator?s only
input into the motor and ESC is the throttle TR, and the power required is set by
the motor parameters and the load. Similarly, we seek a relationship between input
throttle, the rotor?s rotational speed, and the required power. It is known that
changing the throttle increases the rotational speed of the motor, but how does this
pair with the increasing torque requirement? Recall the expressions for motor and
rotor torque:
Qrotor = C ?A(?R)
2
Q R
Qmotor = KT (Irms ? Io)
In a steady state condition with a direct drive configuration, both the rotor?s loads
and speeds are equal to the motor?s:
Qmotor = Qrotor ? = ?
Therefore an expression relating the rotor speed ? to the rms current Irms can be
found:
CQ?AR
3
I 2rms = ? + Io = K
2
Q/T? + Io (5.8)
KT
259
? CQ?AR
3
KQ/T =
KT
where KQ/T is a constant relating rotor to motor torque. Now that a mapping
between rotor speed and motor current exists, equation 5.8 can be combined with
the equivalent AC motor equations to determine the relationship between throttle
and rotor speed ?:
?3 VDCTR = (RESC +Rm)Irms +KE?
2?
?3 VDCTR = (R 2ESC +Rm)(KQ/T? + Io) +KE?
?2?
?KE + K2E + 4K 3Q/T (RESC +Rm)(? V2? DCTR ? Io(RESC +Rm))
? = (5.9)
2KQ/T (RESC +Rm)
Combining a momentum theory rotor model with the AC motor model reveals the
effect of throttle to rotor performance: increasing throttle TR causes a square root
increase in rotor speed ?. With an understanding on how RPM vs throttle evolves,
predictions for the rotor thrust, torque, and power can be made with equations 5.5
to 5.6. With the throttle, voltage, and load defined, the remainder of the AC motor
and ESC equations can be used to determine the motor and ESC?s performance.
As a result of dynamometer characterization, model parameters for the ESC
and BLDC motor can be simply read off table 5.5, and using equations 5.9 the
load on the motor is determined. Now that the rotor, motor, ESC parameters are
determined, the predicted results can be compared to the experimentally obtained
data, which is shown in figure 5.86.
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Figure 5.86: Predicted vs experimental results for the hover test stand.
261
Predicted results match well with experiments, indicating that this motor and
rotor pairing is valid for a given CT , CQ, and CP . Additionally, a separate RPM
and torque sensor are used on the hover test stand, indicating repeatability as the
both the dynamometer and the hover test stand results show good agreement with
one another. Using the hover test stand, rotor performance metrics were extracted
directly from the test data. However, in the future it would be better to predict
both the rotor performance and motor performance during the design phase.
The CT , CQ, and CP obtained via the test stand in hover do not represent the
rotor during forward flight or climb, meaning that a more precise aerodynamic theory
is required to describe the rotor requirements in these flight conditions. A small
1,000 gram electric helicopter is analyzed in the next section, to better understand
how the rotor?s power, RPM, and torque requirements relate to the motor and
ESC in forward flight. This demonstrates the value of combining a forward flight
aerodynamic model with the BLDC motor and ESC equations.
5.6.2 Example Helicopter Propulsive Trim
BLDC motor and ESC selection has a significant impact on electric aircraft
performance. To better understand this selection process, the analysis techniques
described in this thesis are applied to a small commercial off the shelf electric heli-
copter. Design tradeoffs between different motors, ESCs, and voltages are explored,
with this section?s trimming an electric helicopter and estimating its time of flight
using the motor and ESC equations developed in Chapter 4.
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Suppose the mission is to determine the propulsion configuration for an electric
helicopter, shown in figure 5.87, with the longest time of flight and payload. What
motor, ESC, and VDC configuration should be chosen? A higher VDC implies a lower
IDC and thus a longer time of flight. However, a heavier battery reduces the amount
of payload. Additionally, how does ESC selection change the overall helicopter?s
performance? Is it worth the increase in mass to have a lower voltage drop RESC
associated with the 40 A ESC, or will a lower rated ESC prove sufficient? These
questions are critical to a designer, and shall be addressed here. Primary objectives
of this exercise are:
1. Trim a 1,000 gram electric helicopter to determine the required amount of
rotor power.
2. Maximize the time of flight and payload.
3. Determine trade offs between motors, ESCs, and voltages combinations to the
same trimmed helicopter.
Figure 5.87: Helicopter used for trim and motor/ESC evaluation.
263
The helicopter selected for these example calculations has a single hingeless rotor
and is detailed below in table 5.18, and shown in figure 5.87.
Table 5.18: Parameters for the helicopter used in the trim analysis.
Parameter Value Parameter Value
GTO Mass, M 1000 gr Lift Slope, Cl? 5.73 rad
?1
Radius, R 355.6 mm Drag Coefficient, Cdo 0.01
Chord, c 32.6 mm Flap Frequency, ?? 1.10 rev
?1
Number of Blades, NB 2 Lock Number, ? 5.4
Solidity, ? 5.84 % Flap Hinge Location, e/R 14.3 %
Tip Speed, vtip 65 m/s Induced Power Factor, ? 1.10
Rotor RPM, ? 1750 RPM Blade Loading, CT/? 0.081
Parameter Value
Hub Location above Hub, h 127 mm
Tail location, lt 406 mm
CG Location from Hub, xCG, yCG 50.8 mm, 0 mm
Gear ratio, GR 6
Flat Plate Area Ratio, f/A 0.02
Battery Capacity, C 3000 mAH
Battery Discharge Coefficient, d 0.75
Figure of Merit, FOM 54.3%
Gross take-off (GTO) mass M of the helicopter is kept constant at 1,000 grams
for all configurations, so that different designs can be compared with the same trim
analysis, meaning that the amount of payload of the helicopter is varied. The
helicopter?s empty mass me (i.e. no payload, no battery, no motor, and no ESC) is
580 grams. Two batteries were selected from vendors, a 3000 mAh 20C 2S and a
3000 mAh 20C 3S battery, with masses of mbat,2S = 161 grams and mbat,3S = 269
grams. Motor masses mm and ESC masses mESC are provided in tables 5.1 and 5.2.
264
For a given configuration, the amount of payload is:
mpl = M ?me ?mm ?mESC ?mbat (5.10)
which changes based on battery voltage, motor, and ESC. However, the total HC
mass always sums up to 1000 grams, in order to only create one trim response for
the helicopter to keep the comparison even. Once the amount of power for a given
flight speed is determined, the time of flight can be estimated from the required
power draw.
The DC power supply for this aircraft is a liquid polymer (LiPo) battery. For
most batteries, it is common to not discharge them to 100%, as this could cause
serious damage to the chemical makeup of the battery [45]. To prevent this, a
discharge coefficient is used, which limits the effective battery capacity to:
Ceff = dC = 2250 mAH (5.11)
where d is the discharge coefficient. For a given DC current draw IDC , the amount
of time TOF to discharge the effective battery capacity is:
Ceff dC
TOF = = (5.12)
IDC IDC
which can be used to estimate Time of Hover (TOH) and Time of Forward Flight
(TOF). However, first the required DC current draw must be determined, which
requires the HC to be trimmed.
265
Trimming an aircraft requires all 6 degrees of freedom are in equilibrium
[27, 28]. That is, no net force is acting on the helicopter in any direction, and is
accomplished by iterating between control inputs and aircraft states until an equi-
librium is achieved. A discussion on the exact method and equations used to trim
a helicopter are beyond the scope of this work, but the results are shown in figure
5.88. Once the aircraft has been trimmed, the required rotor power can be found
using gear ratio relations.
Figure 5.88: Helicopter trim response in forward flight.
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Figure 5.88 shows the helicopter?s power requirement in forward flight, but
how do these relate to the motor and ESC?s requirements? As with the hover test
stand, the input throttle to the ESC determines the motor?s response. Again, a
relation between rotor loads and input throttle must be described mathematically,
and is found via the gearing of the helicopter. From the rotor response, the required
motor torque is [33]:
1 PR 1
Qmotor = = Qrotor (5.13)
GR ? GR
and the motor RPM is:
? = GR ? (5.14)
It is important to note that the rotor torque changes with forward flight, but the
rotor RPM does not. With the motor?s load and speed determined, the remaining
variable is the ESC throttle TR and is found by manipulating equations 4.35 and
4.40:
Q
Irms = + Io
KT
Irms(Rm +RESC) +KE?
TR = 3 (5.15)? V
2? DC
With the throttle evaluated, the ESC transformer relations can be used to find the
steady state DC current draw out of the battery via equation 4.58:
IDC = (C1TR + C0)Irms
There are two points of interest for this analysis: hover and the speed of minimum
267
rotor power. Speed for minimum power, also known as cruise speed vcruise, is the
forward speed where the rotor requires the least power to operate, and corresponds
to the longest endurance, or TOF, for the helicopter. Combining equations 5.13
to 5.15 and equation 4.35, one can arrive at the following plots for required input
throttle and DC current draw, shown in figure 5.89.
Figure 5.89: Samguk 2500KV motor and SpiderLite 18A ESC response in forward
flight.
Based on the rotor torque and RPM requirements, three motors were selected:
the EMAX 1700KV at VDC = 11.1 V, the EMAX 2300 KV at VDC = 7.4 V, and
the Samguk 2500 KV at VDC = 7.4 V. All motors were considered, however, most
configurations either required hover throttles above 100%, which is nonphysical, or
required the GTO mass of the helicopter to surpass 1000 grams. All three ESCs
were considered for this test, in order to explore the tradeoffs between ESC selec-
tion. Recall that the objective is to determine a configuration that maximizes the
helicopter?s payload capacity and time of flight. Therefore, to grade each motor,
268
voltage, and ESC combination, the following score is used:
S = (TOF [min])? (mpl [gr]) (5.16)
which is the product of the time of flight with the payload. The score favors a long
time of flight and a greater payload, heavier motors have less payload but a longer
time of flight, which makes the relationship interesting. Additionally, having a score
defined this way allows for comparing hover and forward flight performance. The
results for this design and trim analysis are shown in tables 5.19 and 5.20.
Table 5.19: Results for the example helicopter in hover.
Propulsion ESC mpl TOH IDC TR SH
System Used [A] [gr] [min] [A] [%] [gr-min]
EMAX 18 108 24.99 5.40 68.17 2699
1700 KV 30 96 24.25 5.57 73.47 2328
at VDC = 11.1 V 40 92 25.17 5.36 71.40 2315
EMAX 18 217 15.10 8.94 79.08 3277
2300 KV 30 205 15.71 8.59 85.69 3221
at VDC = 7.4 V 40 201 15.82 8.54 83.43 3180
Samguk 18 212 14.80 9.12 74.26 3138
2500 KV 30 200 14.66 9.21 81.12 2932
at VDC = 7.4 V 40 196 14.75 9.15 79.16 2891
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Table 5.20: Results for the example helicopter in forward flight.
Propulsion ESC mpl TOF vcruise Range IDC TR SF
System Used [A] [gr] [min] [m/s] [km] [A] [%] [gr-min]
EMAX 18 108 41.09 8.90 21.94 3.29 64.01 4438
1700 KV 30 96 40.40 8.90 21.57 3.34 68.55 3878
at VDC = 11.1 V 40 92 41.69 8.90 22.26 3.24 67.13 3835
EMAX 18 217 26.16 8.90 13.97 5.16 71.78 5678
2300 KV 30 205 27.24 8.90 14.55 4.96 78.27 5584
at VDC = 7.4 V 40 201 27.39 8.90 14.63 4.93 76.36 5505
Samguk 18 212 26.07 8.90 13.92 5.18 66.87 5527
2500 KV 30 200 25.42 8.90 13.58 5.31 73.35 5084
at VDC = 7.4 V 40 196 24.85 8.90 13.27 5.43 72.41 4871
Interestingly, the combination with the highest score for both hover and for-
ward flight is the EMAX 2300 KV motor with an 18 A ESC with VDC = 7.2 V.
Although the 40 A ESC typically requires less DC current to be drawn from the
battery, this is not enough of a performance gain to offset the loss in payload asso-
ciated with a heavier ESC mass. The 18 A and 40 A ESC have masses of 11 and 27
grams respectively, meaning that the 40 A ESC would need to reduce DC current
draw by at least 60% to stay competitive, and yet that does not occur.
Running at VDC of only 7.4 V reveals another interesting conclusion: that
higher voltage does not always translate into better performance. Battery capacity
remains constant, meaning the addition of an extra cell to a 2S battery to create
a 3S battery brings along a mass penalty. Benefits for the 3S design are found
when examining the TOF and range, as the 3S design has a 57% increase in range
and TOF when compared to the EMAX 2300 KV design at 2S. Additionally, the
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3S design is both hovering and cruising at a lower throttle, meaning more control
margin is available for climbing and maneuvering.
It is worth noting that the selection of the proper configuration is up to the
designer and is set by the mission requirements. For example, if the payload mass
had been set to 100 grams, the 3S design would be selected. However, by changing
the rules of evaluation, one can make different arguments as to the best design. For
the first time, this work allows for designers to compute time of flight estimates for
electric aircraft, determine pairing relationship between ESC and BLDC motor, and
to evaluate tradeoffs in DC voltage selection.
271
Chapter 6: Conclusions
6.1 Summary of Research
This work consisted of three major phases: development of a dynamometer
including DC, AC, and mechanical power analyzers, experimental testing of BLDC
motors and ESCs at different voltages, and determining an analytical model which
relates phyiscal motor parameters to experimental data. A custom dynamometer
was developed at the University of Maryland to characterize the performance of
BLDC motors and ESCs. By recording the DC, AC, and mechanical power through
the system, efficiency metrics can be obtained for the ESC, BLDC motor, and overall
system. 6 motors, with KV ratings between 920 KV to 2500 KV, were paired with 3
ESCs at DC voltages between 7.2 V to 14.8 V. This work represents the largest and
most comprehensive investigation of BLDC motors and ESCs to date. Key features
of the dynamometer are use of a 2 DAQ setup to record the AC power entering the
BLDC motor using the 2-wattmeter method.
A total of 34 combinations of ESCs, BLDC motors, and VDC were tested on
the custom dynamometer. From the results of these experiments, an analytical
model was developed to predict motor performance with different ESCs and BLDC
motors. Significant work to model the ESC, which is typically neglected in research,
272
shows that 2 primary power losses occur: a switching loss and a conduction loss.
Improved understanding of motor and ESC characterization is essential to enhance
electric aircraft designs in the future.
6.2 Conclusions
Several conclusions related to motor and ESC performance can be drawn from
this work.
1. A first principles analytical model was developed to understand the type of
power losses that occur within an ESC and BLDC motor system. A brushed
DC motor model was modified to include AC voltage and current quantities,
which better describe the inputs to the motor. This model was fit to the
experimental data and shows good agreement for Q, Irms, VLL,rms, PAC , and
IDC .
2. Simplifying the 3 phase AC circuit into a single phase equivalent circuit was
shown to be an effective modeling tool. Physical parameters related to the
motor?s performance can be fit to this model, allowing for motors and ESCs
to be used in a rigorous design manner. Four parameters are required to
describe the BLDC motor: KT , KE, Io, and Rm. Unique to this thesis is the
description of the three ESC parameters: C0, C1, and RESC .
3. BLDC motors tested contain specific peak power ratios of 0.94 to 4.90 kW/kg.
It was shown that each ESC does not significantly increase or decrease the
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maximum power output of the motor, thereby further reiterating the need to
not use a higher rated ESC if possible.
4. Manufacturer specified KV rating is of no practical engineering use. Com-
paring manufacturer specified KV values to the experimentally obtained KE
values show a 40% discrepancy. Furthering this argument is the fact that most
motor models use KT = KE?, however, the model presented in this thesis uses
an ideal ratio of KT/KE is 27/10 ? 1.64. Power losses associated with the
motor places the experimentally recorded value at 1.90, indicating a 15.6%
difference from the ideal ratio. This places the manufacturer given KV rating
well outside the bounds of reliable use.
5. ESC operation is a function of the user throttle. It was found that both output
rms voltage and rms current are functions of the user throttle. This allowed
for the development of an analytical ESC transformer model, which relates DC
power to AC power using the input throttle and motor rms current. Saturation
occurs in ESCs between 90% to 100% throttle, meaning that the motor never
experiences the full 0.675VDC rms voltage, reducing the maximum ?NL of the
motor by 5%.
6. ESC?s have higher efficiencies at full throttle, as the switching losses are min-
imized. Switching losses stem from the rapid turning on and off of the MOS-
FET to augment the output rms voltage and rms current. At 100% throttle,
the MOSFETs are placed in an always conducting mode, indicating that only
switches related to commutating the motor occur. Additionally, switching
274
losses increase with increasing DC voltages.
7. Maximum efficiency points for brushless DC motors occur at high RPM and
low torque. ESC efficiency generally decreases linearly with rms current, mak-
ing the overall maximum efficiency for the system occur when the motor is
operating at maximum efficiency. Selection of ESC changes the motor rms
voltage, with higher rated ESCs having a lower RESC , associated with more
expensive MOSFETs.
8. As evidenced by the hover test stand, predicting rotors, motors and ESCs
performance was achieved, allowing brushless motors to enter the design space
of electric aircraft in a more rigorous manner. For a fixed pitch direct drive
?
configuration, increasing motor throttle represents a Q ? TR increase in
rotor speed. Motor and ESC selection has a profound impact on electric
aircraft performance, as oversizing or undersizing the propulsion system could
reduce mission capabilities.
6.3 Future Work
With the growing demand for high performance electric aircraft, there are
many directions for this research to expand into, with some key areas listed.
1. Larger Dynamometers: The dynamometer outlined in this paper can serve
as a template to modify over the coming years. The active component in
studying the performance of motors is the brake, shown in figure 3.1, and
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is responsible for augmenting the load on the motor in a controlled fashion.
Studying larger motors is simple, as only the brake needs to be enhanced to a
larger maximum torque and speed. Magtrol, the company that sells the brake
used in this thesis, produces brakes rated for as large as 3500 in-oz of torque,
350 times the rating of the brake used in this work.
2. Smaller Dynamometers: When it comes to dynamometers, it also pays to
think smaller. As discussed in the beginning of chapter 4, brakes have an
inherent cogging torque, meaning that studying the performance of the mo-
tors near ?NL is not possible, where the maximum motor efficiency occurs.
By downsizing the brake used, the static 3.1 in-oz load would be eliminated,
allowing for a more complete range of operating points to be covered. Addi-
tionally, with a lower cogging torque, gearing the motors becomes an option.
Recall that higher KV motors were not tested at the higher voltage levels be-
cause of the brake?s 20,000 RPM limit. However, if the motor could be geared
such that the RPM of the brake stays below the limit, then all motors could be
tested at higher voltages. Finally, many potential ESC and BLDC motor tests
had to be turned away as a result of the cogging torque. By eliminating the
cogging torque with a smaller brake, more ESCs could be tested, expanding
the design space considerably.
3. Improved AC Power Measurements: For any future dynamometers there
is a clear need for accurate AC power measurement techniques. Refinements
in AC power techniques should include shunt resistor based current measure-
276
ments at higher voltages. Other areas for improvement are DAQs with faster
read rates, as this will be critical to studying motors with higher numbers of
pole pair (Np > 10).
4. Refined BLDC Motor and ESC Modeling: Improvements in the model
to study ESCs and predict ESC performance at higher voltages is necessary.
Future enhanced AC power measurements techniques will reduce experimen-
tal error, identifying clear trends for ESC efficiencies. From the low voltage
AC power setup, it was clear that ESC efficiency vs RPM is a straight line,
however, this result needs to be confirmed at higher voltages using accurate
AC power measurement techniques. Additionally, an improved understanding
of MOSFET characteristics, which lie at the heart of any inverter, is critical
to predicting ESC power losses.
5. Closed Loop Control: Equations from chapter 4 on BLDC motor and ESC
performance describe the steady state condition, but future work is to extend
this framework to include time varying dynamics. A state space based model
could be developed to describe the motor?s performance with time varying
throttle, current, and loading. A closed loop controller could be developed
around this framework which would be critical for improved performance on
quadcopters and other electric aircrafts. Controller goals could include RPM
tracking and optimal gain selection that minimizes DC current spikes.
6. Testing of AC Induction Motors: AC Induction Motors are favored candi-
dates for high performance manned electrical aircraft for due to their increased
277
power rating, reliability, and mature electromagnetic designs. A focus of this
work would be characterizing and modeling AC induction motors. Tradeoffs
between the scale, size, weight, and performance of AC induction motors and
BLDC motors can be explored for rigorous design use. AC induction drive
power loss and efficiency performance should also be compared to ESC per-
formance.
7. Heuristic Modeling: From the catalogs of experimental data and modeling
parameters, heuristic functions can be created to assist in aircraft sizing re-
quirements. Heuristics should relate motor torque, efficiency, and peak power
to the number of poles, motor mass, motor volume, and motor price. ESC
heuristics should relate MOSFET switching variables and ESC resistance to
the rated DC current draw, ESC mass, ESC cost, and ESC brand. For both
sets of BLDC motor and ESC heuristic equations, a large emphasis should be
placed on the cost per unit of each item, for a given amount of torque, power,
etc. Distributed electric propulsion electric aircraft require four or more sets
of ESCs and BLDC motors. Factoring monetary cost into the analysis will be
critical for mass producing a given design.
8. Optimal Motor and Rotor Pairing: With an enhanced understanding of
motor and ESC requirements, fitting specific motors and ESCs with specific
rotors can be achieved. Rotor design dictates that a certain rotor RPM and
torque are required to satisfy propulsion, thrust, and control requirements.
Once this has been achieved, the rotor requires can be paired with the BLDC
278
motor and ESC catalog from Chapter 5 to determine the configuration that
minimizes the total power draw of the system. Iteration between the rotor
design and BLDC motor/ESC selection can further reduce system power, en-
hancing time of flights for electric aircrafts.
279
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