Analysis and Development of a Halbach Array Motor for Application into a

Novel Delivery Drone Driving Cycle for Powertrain Optimization and

Maximization of Delivery Radius

by

Maxime Perreault

A thesis submitted in conformity with the requirements

for the degree of Master of Applied Science

Graduate Department of Applied Science and Engineering

University of Toronto

© Copyright by Maxime Perreault 2020

ii

Analysis and Development of a Halbach Array Motor for

Application into a Novel Delivery Drone Driving Cycle for

Powertrain Optimization and Maximization of Delivery Radius

Maxime Perreault

Master of Applied Science

Graduate Department of Applied Science and Engineering

University of Toronto

2020

ABSTRACT

Large companies such as Amazon and Google are currently testing deliveries using unmanned

drones, with the intent of using these drones on the market. Topics tackled in this field of research

include delivery routing optimization, object collision routing optimization, battery management

optimization, and the addition of solar panels on the drones. Little publicly available research has

been found to have been conducted on the developing methods to optimize the powertrain of the

drones to maximize their delivery radii. To achieve this end, this work puts forth a delivery drone

driving cycle simulation written in MATLAB with which to monitor their performance and fine-

tune their properties. A halbach array motor is designed and analyzed in ANSYS Maxwell, and a

data processing tool is written in MATLAB to manipulate the halbach array motor and propeller

data into useable states for the simulation. The driving cycle simulation is tested on fifty-four

drone configurations.

iii

ACKNOWLEDGMENTS

The short time I have spent in my graduate research has been a crucial moment in my personal progress,

augmented in no small part by the experience and mentorship of my supervisor, Professor Kamran

Behdinan. His care and attention to my professional development have imparted wisdoms that will

reverberate within me and from which I will keep learning as I move forward.

This work has been funded in part by the Natural Sciences and Engineering Research Council of Canada

under Grant CRDPJ 514905-17, to whom I would like to extend my appreciation for their support.

I thank Doctor Ali Hameed Radhi for his friendship and camaraderie over these last years. It was a

welcome boon to work alongside someone who is always willing to engage in stimulating discussions or

provide guidance as .I often needed.

I extend a deep gratitude to both of my parents, for the love they have shared, their stalwart devotion to

my welfare, and unwavering support for my choices in life. Without the opportunities, comforts, and

freedom to grow that they have provided, I would never have managed to be the person I am.

To my dog, Remi, whose playfulness and unappeasable desire for cuddles has helped keep my sanity in

check. I wish her many years of joy to come.

To Mengqi, my muse. You have stayed by me these many years, your ardent presence never showing any

signs of relent through both sickness and health. I only hope to be as good to you as you are to me.

iv

TABLE OF CONTENTS

List of Tables ................................................................................................................................................ v

List of Figures .............................................................................................................................................. vi

Nomenclature ............................................................................................................................................... xi

1 Introduction ............................................................................................................................................. 1

1.1 Motivation ........................................................................................................................................... 3

1.2 Outline................................................................................................................................................. 4

2 Literature Review – Halbach Array Motors............................................................................................ 5

2.1 The Halbach Array .............................................................................................................................. 5

2.2 Applications of Halbach Arrays .......................................................................................................... 8

2.3 Halbach Arrays in Motors ................................................................................................................. 10

3 Governing Equations ............................................................................................................................ 15

3.1 Electromagnetic Equations ............................................................................................................... 15

3.2 Electric Motor Characteristic Behavior ............................................................................................ 18

3.3 Drone Kinetics .................................................................................................................................. 25

4 Methodology of Halbach Motor Design ............................................................................................... 30

4.1 ANSYS Maxwell Simulations .......................................................................................................... 30

4.2 Powertrain Data Processing .............................................................................................................. 47

5 Driving Cycle of Delivery Drone .......................................................................................................... 54

6 Results of Delivery Drone Driving Cycle ............................................................................................. 58

6.1 Delivery Drone Parameters ............................................................................................................... 58

6.2 Driving Cycle Simulation Result Analysis ....................................................................................... 60

7 Conclusion ............................................................................................................................................ 75

7.1 Contributions..................................................................................................................................... 76

8 Future Work .......................................................................................................................................... 77

References ................................................................................................................................................... 78

Appendices .................................................................................................................................................. 85

Appendix A: Produced MATLAB Code ................................................................................................ 85

Appendix B: Propeller Experimental Data Curve-Fitting ....................................................................... 88

Appendix C: Detailed Powertrain Operating Point Figures .................................................................... 89

Appendix D: Detailed Ideal Pitch Angle Figures ................................................................................... 90

v

LIST OF TABLES

Table 4.1.1: General Machine Properties of the Halbach Array Motor in ANSYS Maxwell RMxprt. ...................... 31

Table 4.1.2: Circuitry Properties of the Halbach Array Motor in ANSYS Maxwell RMxprt. ................................... 32

Table 4.1.3: Material Properties of Iron in the Halbach Array Motor in ANSYS Maxwell RMxprt.......................... 32

Table 4.1.4: Stator Properties of the Halbach Array Motor in ANSYS Maxwell RMxprt. ........................................ 33

Table 4.1.5: Rotor Properties of the Halbach Array Motor in ANSYS Maxwell RMxprt. ......................................... 33

Table 4.1.6: Armature Slot Properties of the Halbach Array Motor in ANSYS Maxwell RMxprt. ........................... 34

Table 4.1.7: General Winding Properties of the Halbach Array Motor in ANSYS Maxwell RMxprt. ...................... 36

Table 4.1.8: Winding End/Insulation Properties of Halbach Array Motor in ANSYS Maxwell RMxprt. ................. 36

Table 4.1.9: Material Properties of N50M in the Halbach Array Motor in ANSYS Maxwell RMxprt. ..................... 37

Table 4.1.10: Magnet Pole Properties of the Halbach Array Motor in ANSYS Maxwell RMxprt............................. 37

Table 4.1.11: Mass Breakdown by Component of the Halbach Array Motor. ........................................................... 38

Table 4.1.12: Mesh Operations of the Halbach Array Motor in ANSYS Maxwell 2D Design. ................................. 43

Table 4.1.13: Motion Setup of the Halbach Array Motor in ANSYS Maxwell 2D Design........................................ 43

Table 6.1.1: Simulation Delivery Drone Properties. ................................................................................................... 58

Table 6.1.2: Propeller Characteristic Curves of Lift and Torque as Functions of rpm. .............................................. 59

Table 6.1.3: Simulation Powertrain Component Masses. ........................................................................................... 60

Table 6.2.1: Maximum Delivery Radius for each Propeller in Tested Configurations. .............................................. 61

Table 6.2.2: Average Ideal Pitch Angle for each Propeller in Tested Configurations. ............................................... 64

Table 6.2.3: Ideal Energy of Ascent and Descent with and without Payload for Tested Configurations. .................. 68

Table 6.2.4: Average Delivery Radii over Range of varied Cruise Heights for Drone Configurations. ..................... 71

Table 6.2.5: Average Delivery Radii over Range of varied Payload Masses for Drone Configurations. ................... 74

vi

LIST OF FIGURES

Figure 2.1.1: Examples of Halbach Arrays that are a) Linear, b) Planar, and c) Circular. ........................................... 5

Figure 2.1.2: Dipole Field Generated by Circular Halbach Array. ............................................................................... 6

Figure 2.1.3: Schematic of a Dipole Field Motor Containing no Electromagnetic Coils. ............................................ 6

Figure 2.1.4: Comparison of the Magnetic Field between a Conventional Parallel Pole Array (top) and a Halbach

Array (bottom). .............................................................................................................................................................. 7

Figure 2.1.5: Comparison of the Flux Density between a Halbach Array (left) and Conventional Parallel Pole Array

(right) in an Electrical Machine. .................................................................................................................................... 8

Figure 2.2.1: Two Linear Actuators with Halbach Arrays, one in Cylindrical Dual Halbach Configuration (left) and

one in a Flat Configuration (right). ................................................................................................................................ 9

Figure 2.2.2: Transformation of a Halbach Array Planar Grid into a Cylindrical Grid. ............................................. 10

Figure 2.3.1: Schematic of a 10-pole Radial Halbach Array Motor (left) and Axial Flux Halbach Array Motor

(right). .......................................................................................................................................................................... 11

Figure 2.3.2: Torque Density Plots of Optimized Halbach Array Motors in Configurations of Dual Array (top) and

Triple Array (bottom). ................................................................................................................................................. 12

Figure 2.3.3: Schematics of a Dual Halbach Array (left) along with a Triple Halbach Array (right). The Rotor,

which would sit between each set of Stators, is not shown. ........................................................................................ 14

Figure 3.2.1: Brushless DC Motor Electrical a) Phasor Diagram and b) Circuit Diagram. ........................................ 20

Figure 3.2.2: Gate Driver Diagram of a 3-Phase Brushless DC Motor. ..................................................................... 22

Figure 3.2.3: Pulse Width Modulation Diagram of a 3-Phase Gate Driver at 75% Throttle. ..................................... 22

Figure 3.2.4: Motor Efficiency Heatmaps as Produced by a) ANSYS Maxwell Machine Toolkit (2018) and b) Altair

Flux E-Machine Toolbox (2019). ................................................................................................................................ 23

Figure 3.2.5: Graphical Representation between a Motor’s Lamination Thickness and Stacking Factor. ................. 24

Figure 3.3.1: Quadcopter Dynamics Coordinate System. ........................................................................................... 25

Figure 3.3.2: Force Diagrams of the Drone Kinematics during a) Ascent, b) Cruising, and c) Descent. ................... 26

vii

Figure 4.1.1: Schematic of Slot Type 4 Base Model as Available in Maxwell RMxprt Detailed with all Configurable

Dimensions. ................................................................................................................................................................. 34

Figure 4.1.2: Winding Configuration Throughout the 36 Armature Slots of the Halbach Motor Generated in

Maxwell RMxprt. ........................................................................................................................................................ 35

Figure 4.1.3: Final Motor Design as Illustrated in Maxwell RMxprt. ........................................................................ 38

Figure 4.1.4: External and Internal Motor Circuitry Generated Automatically by Exporting the Maxwell RMxprt

Design to the 2D Design Environment. ....................................................................................................................... 39

Figure 4.1.5: Half Electrical Phase Motor Segment Generated Automatically by Exporting the Maxwell RMxprt

Design to the 2D Design Environment. ....................................................................................................................... 40

Figure 4.1.6: Process of Rotation and Boolean Splitting Operations to Generate the Circumferentially Aligned

Halbach Array Magnets in the Maxwell 2D Design Environment. ............................................................................. 41

Figure 4.1.7: Alternating Halbach Magnet Array Pattern with the Magnet Orientation Coded as follows: Positive

Radial Flux in Red, Negative Radial Flux in Blue, Positive Circumferential Flux in Yellow, and Negative

Circumferential Flux in Green. .................................................................................................................................... 42

Figure 4.1.8: Torque vs Time Graph for the Transient Simulation of the Halbach Array Motor at 45 Volts over a

Span of 100 milliseconds with a Time Step of 50 microseconds. ............................................................................... 44

Figure 4.1.9: Phase Winding Currents for the Transient Simulation of the Halbach Array Motor at 45 Volts over a

Span of 20 milliseconds with a Time Step of 50 microseconds. ................................................................................. 45

Figure 4.1.10: Phase Winding Induced Voltages for the Transient Simulation of the Halbach Array Motor at 45

Volts over a Span of 20 milliseconds with a Time Step of 50 microseconds. ............................................................. 45

Figure 4.1.11: Magnetic Flux Lines produced throughout the Halbach Array Motor during the Transient Simulation

at 45 Volts and 100 milliseconds. ................................................................................................................................ 46

Figure 4.2.1: Polynomial Curve Fitting of the Transient Data Points of the Halbach Array Motor Operating at 70

Volts. ........................................................................................................................................................................... 47

Figure 4.2.2: Compiled rpm-Torque Polynomial Curves of the Halbach Array Motor Operating from 5 to 70 Volts.

..................................................................................................................................................................................... 48

Figure 4.2.3: Polynomial Interpolation of the User-Defined Electric Motor Data in the Torque-rpm Plane. A small

Interpolation Density of Two is used here for Visual Clarity. ..................................................................................... 49

viii

Figure 4.2.4: Resultant Efficiency Heatmap of the Interpolated User-Defined Electric Motor Data. ........................ 50

Figure 4.2.5: Rpm-Torque and rpm-Lift Quadratic Curve-Fitting of P22x6.6 Data. ................................................. 51

Figure 4.2.6: Overlay of the P22x6.6 rpm-Torque Operating Points of the Electric Motor Efficiency Heatmap,

showing the useable Motor Operating Points of this Powertrain Configuration. ........................................................ 52

Figure 4.2.7: Overlay of the Halved P22x6.6 rpm-Torque Operating Points of the Electric Motor Efficiency

Heatmap, showing the useable Motor Operating Points of this Powertrain Configuration. ........................................ 53

Figure 5.1: Driving Cycle Breakdown in Six Segments: Ascent with Payload, Horizontal Cruise with Payload,

Descent with Payload, Ascent without Payload, Horizontal Cruise without Payload, Descent without Payload. ....... 54

Figure 5.2: Flowchart of the Delivery Drone Driving Cycle with Aim of Optimizing the Horizontal Travel Distance

of the Drone. ................................................................................................................................................................ 55

Figure 6.1: Maximum Drone Delivery Radius depending on the Propellers used, their Quantity, and the Number of

Motors attached to each. .............................................................................................................................................. 61

Figure 6.2: Ideal and Maximum Pitch Angles of Horizontal Motion for a Delivery Drone with Payload, for all 54

Tested Configurations of Powertrain Variations. ........................................................................................................ 63

Figure 6.3: Ideal and Maximum Pitch Angles of Horizontal Motion for a Delivery Drone with no Payload, for all 54

Tested Configurations of Powertrain Variations. ........................................................................................................ 63

Figure 6.4: Ideal Throttle Profile of Ascent for each of the 54 Tested Drone Powertrain Configurations while the

Drone carries a Payload. .............................................................................................................................................. 64

Figure 6.5: Ideal Throttle Profile of Ascent for each of the 54 Tested Drone Powertrain Configurations while the

Drone does not carry a Payload. .................................................................................................................................. 65

Figure 6.6: Minimum Ascent Energy for each of the 54 Tested Drone Powertrain Configurations while the Drone

carries a Payload. ......................................................................................................................................................... 66

Figure 6.7: Minimum Ascent Energy for each of the 54 Tested Drone Powertrain Configurations while the Drone

does not carry a Payload. ............................................................................................................................................. 66

Figure 6.8: Minimum Descent Energy for each of the 54 Tested Drone Powertrain Configurations while the Drone

carries a Payload. ......................................................................................................................................................... 67

Figure 6.9: Minimum Descent Energy for each of the 54 Tested Drone Powertrain Configurations while the Drone

does not carry a Payload. ............................................................................................................................................. 67

ix

Figure 6.10: Maximum Delivery Radius as a Function of Cruise Height for a Drone using 6 G29x9.5 Propellers and

either 1, 2, or 3 Motors per Propeller. .......................................................................................................................... 69

Figure 6.11: Slope of the Relation beteween Maximum Delivery Radius and Cruise Height for 18 Drone

Configurations, all using their Ideal Quantity of Propellers and either 1, 2, or 3 Motors per Propeller. ..................... 70

Figure 6.12: Average of all Maximum Delivery Radii as a Function of Cruise Height for 18 Drone Configurations,

all using their Ideal Quantity of Propellers and either 1, 2, or 3 Motors per Propeller. ............................................... 70

Figure 6.13: Maximum Delivery Radius as a Function of Payload Mass for a Drone using 6 G29x9.5 Propellers and

either 1, 2, or 3 Motors per Propeller. .......................................................................................................................... 72

Figure 6.14: Slope of the Relation beteween Maximum Delivery Radius and Payload Mass for 18 Drone

Configurations, all using their Ideal Quantity of Propellers and either 1, 2, or 3 Motors per Propeller. ..................... 73

Figure 6.15: Average of all Maximum Delivery Radii as a Function of Payload Mass for 18 Drone Configurations,

all using their Ideal Quantity of Propellers and either 1, 2, or 3 Motors per Propeller. ............................................... 73

Figure B1: Curve-Fitting of P22x6.6 Data. ................................................................................................................ 88

Figure B2: Curve-Fitting of G26x8.5 Data. ................................................................................................................ 88

Figure B3: Curve-Fitting of G27x8.8 Data. ................................................................................................................ 88

Figure B4: Curve-Fitting of G28x9.2 Data. ................................................................................................................ 88

Figure B5: Curve-Fitting of G29x9.5 Data. ................................................................................................................ 88

Figure B6: Curve-Fitting of G30x10.5 Data. .............................................................................................................. 88

Figure C1: Powertrain Operating Points of P22x6.6 Propeller Overlaid onto Motor Efficiency Heatmap. ............... 89

Figure C2: Powertrain Operating Points of G26x8.5 Propeller Overlaid onto Motor Efficiency Heatmap. .............. 89

Figure C3: Powertrain Operating Points of G27x8.8 Propeller Overlaid onto Motor Efficiency Heatmap. .............. 89

Figure C4: Powertrain Operating Points of G28x9.2 Propeller Overlaid onto Motor Efficiency Heatmap. .............. 89

Figure C5: Powertrain Operating Points of G29x9.5 Propeller Overlaid onto Motor Efficiency Heatmap. .............. 89

Figure C6: Powertrain Operating Points of G30x10.5 Propeller Overlaid onto Motor Efficiency Heatmap. ............ 89

Figure D1: Drone Configuration Ideal Pitch Angle using P22x6.6 Propellers while carrying a Payload. ................. 90

Figure D2: Drone Configuration Ideal Pitch Angle using G26x8.5 Propellers while carrying a Payload. ................. 90

Figure D3: Drone Configuration Ideal Pitch Angle using G27x8.8 Propellers while carrying a Payload. ................. 90

Figure D4: Drone Configuration Ideal Pitch Angle using G28x9.2 Propellers while carrying a Payload. ................. 90

x

Figure D5: Drone Configuration Ideal Pitch Angle using G29x9.5 Propellers while carrying a Payload. ................. 90

Figure D6: Drone Configuration Ideal Pitch Angle using G30x10.5 Propellers while carrying a Payload. ............... 90

Figure D7: Drone Configuration Ideal Pitch Angle using P22x6.6 Propellers while not carrying a Payload. ........... 91

Figure D8: Drone Configuration Ideal Pitch Angle using G26x8.5 Propellers while not carrying a Payload............ 91

Figure D9: Drone Configuration Ideal Pitch Angle using G27x8.8 Propellers while not carrying a Payload............ 91

Figure D10: Drone Configuration Ideal Pitch Angle using G28x9.2 Propellers while not carrying a Payload. ......... 91

Figure D11: Drone Configuration Ideal Pitch Angle using G29x9.5 Propellers while not carrying a Payload. ......... 91

Figure D12: Drone Configuration Ideal Pitch Angle using G30x10.5 Propellers while not carrying a Payload. ....... 91

xi

NOMENCLATURE

Symbol Parameter Unit

A Magnetic Vector Potential Teslas by Meters

A0 Boundary Magnetic Vector Potential Teslas by Meters

ag Acceleration due to Gravity Meters per Second Squared

av Vertical Acceleration Meters per Second Squared

B Flux Density Tesla

D Electric Flux Density Coulombs per Meter Squared

E Electric Field Strength Volts per Meter

Fg Force of Gravity Newtons

FHDrag Force of Horizontal Drag Newtons

FLift Force of Lift Newtons

FPropulsion Force of Propulsion Newtons

FThrust Force of Thrust Newtons

FVDrag Force of Vertical Drag Newtons

H Magnetic Field Strength Amperes per Meter

J Current Density Amperes per Meter Squared

Js Source Current Density Amperes per Meter Squared

LiftProp Propeller Lift Newtons

M Remnant Induction Magnetization Teslas

T Torque Newtons by Meters

Vs Source Voltage Volts per Meter

vH Horizontal Velocity Meters per Second

vV Vertical Velocity Meters per Second

𝝓 Magnetic Flux Webers

xii

CHDrag Horizontal Area-Drag Coefficient Meters Squared

CVDrag Vertical Area-Drag Coefficient Meters Squared

d Delivery Radius Kilometers

Ea Single-Phase Electromotive Force Volts

EAscent Electrical Energy of Ascent Joules

EDescent Electrical Energy of Descent Joules

ERemaining Electrical Energy Remaining Joules

EBattery Electrical Energy of Battery Joules

EPM Energy per Meter Joules per Meter

h Cruise Height Meters

Ia Armature Current Amperes

kl Leakage Coefficient

La Armature Inductance Henrys

M Drone Mass Kilograms

N Number of Coil Turns

PCopper Copper Power Loss Watts

PCore Core Power Loss Watts

PIn Power In Watts

PMech Mechanical Power Loss Watts

POut Power Out Watts

PWM Pulse Width Modulation

p Quantity of Pole Pairs

qpt Quantity of Powertrains

r Radial Coordinate Direction

Ra Armature Resistance Ohms

rpm Rotations per Minute Rotations per Minute

SF Safety Factor

Stacking Factor Stacking Factor

xiii

t Time Seconds

tf Ascent Time Seconds

V Electric Voltage Volts

VBattery Battery Voltage Volts

VIn Voltage In Volts

vLand Landing Speed Meters per Second

x Cartesian x-axis Direction

y Cartesian y-axis Direction

z Cartesian z-axis Direction

δLam Lamination Thickness Millimeters

ε Dielectric Permittivity Farads per Meter

η Efficiency

θ Circumferential Coordinate Dimension

θPitch Pitch Angle Degrees

μ Magnetic Permeability Henrys per Meter

υ Reluctivity Tensor Meters per Henry

ρ Density of Air Kilograms Cubed

ρv Charge Density Coulombs per Meter Cubed

σ Electrical Conductivity Siemens per Meter

ϕRoll Roll Angle Degrees

ϕGap Flux per Coil Turn Webers

ϕPhase Phase Angle Degrees

ψYaw Yaw Angle Degrees

ωe Electrical Rotational Speed Radians per Second

ωm Mechanical Rotational Speed Radians per Second

1

1 INTRODUCTION

Companies around the world are gearing up to roll out drones for use in payload delivery. Many countries

and companies have managed their first deliveries by way of drones, with flights seen in Africa,

Australia, Europe, and the United States [1], conducted by large companies such as Google [2], Amazon

[3], UPS [4], and DHL [5].

Recent academic research in the domain of delivery drones has focused toward solving the famous

traveling salesman problem, where a shortest distance is calculated between a set of points in space. The

special permutation to this scenario is that there is now a fleet of transporters around which the problem

needs to be optimized around. Kloetzer et al. have conducted tests at an indoor warehouse level,

optimizing their vehicle routing using a mathematical binary integer problem approach [6]. Kin-Ming et

al. have instead opted to tackle this problem on a larger urban scale, with a focus on balancing the

individual path lengths equally amongst all delivery drones at their disposal [7]. Their approach uses a

genetic algorithm to determine the optimal routing configuration, which can be calculated in half a

minute. Dukkanci et al. have focused their work toward delivery ports between which trucks transport

payloads and from which the drones conduct deliveries [8]. Their work was applied to realistic sample

sets and they were able to optimize both the number of delivery ports and their placement.

A hybrid delivery system that uses both drones and delivery trucks, which serve as mobile transportation

hubs, has seen a lot of progress. Research in recent years from Murray and Raj [9], Daknama and Kraus

[10], and Agatz et al. [11] have all determined that this hybrid delivery system reduces the average

delivery time and carbon footprint.

At the scale of immediate vicinity collision detecting and path planning, work has been done by Athira

Krishnan et al. which focuses on reduction of time spent in flight [12]. This small-scale path planning

involves a novel artificial potential field applicator that visualizes the drone’s environment.

2

Advancements were also achieved by Yakovlev et al. whose constraint-based path plotting algorithm was

found to be applicable in urban settings [13].

Outside of the mathematics-focused side of the field, work is also being done in improving the

operational performance of the drones. One group of researchers has been tackling the issue of battery

power management, as the battery output behavior is not linear, and it cannot precisely be known how

much energy is left for the drone to use for flight. By developing an algorithm to observe the battery’s

output, Chen et al. managed to improve the reliability of battery readings by 16% [14]. In another line of

work, looking to counteract the unbalancing effects of wind, Bannwarth et al. created wind rejection

controllers for their drone, which managed to stabilize the drone in five meter per second wind [15]. This

improved stabilization reduced the drone displacement errors by 45-66% compared to baseline results.

Both these advances have the potential to, in turn, allow for operators to fly drones with slimmer factors

of safety and increase flight time and distance.

In 2019, a drone was built and flown by researchers from the National University of Singapore, which

operated without batteries, pulling all the needed energy in real time from photovoltaic cells [16]. With a

modest drone mass of 2.6 kilograms, the solar powered technology is not yet ready to similarly power an

entire delivery drone flight, but it may soon find application in being a secondary energy source

prolonging the flights of the delivery drones.

Even with all of these advances, progress still needs to be made in modular optimization processes for the

powertrain of the delivery drones. The dynamic loading undergone by the drone’s propellers and motors

can be examined to tailor the drone’s powertrain such that the power usage decreases and the delivery

radius is maximized. Being a budding industry, the drone manufacturing process still thrives around third

party components being bought off the shelf and assembled. For regular drones used by the public, such

as camera drones or hobby drones, performance optimization revolves around maximizing flight time,

which is done by tuning the drone’s drivetrain until the hover state operates at the motors’ maximum

3

efficiency point. This method can be improved for delivery drones for two reasons. First, the delivery

drones have two distinct conditions: when they are and are not carrying parcels, which will hereon be

referred to as payloads. Both of these conditions must be balanced appropriately when tuning the

drivetrain. A motor that may be ideal for payload delivery may not be ideal for the trip returning from

delivery. Second, the delivery drone’s path and behavior before flight can be reasonably predicted with an

appropriate driving cycle.

Driving cycles are used in the automotive industry to virtually simulate or experimentally recreate the

driving conditions that vehicles are subject to on the road [17]. These driving conditions include rural

driving, urban driving, highway driving, and others. They are important tools in the industry at two

different stages of production. First, they allow for theoretical estimation of emission testing and fuel

consumption for vehicles and drivetrains before manufacturing. Second, practical data can be generated

with the fully manufactured vehicles by simulating the driving conditions on rotating barrel drums. The

driving cycles used in the latter cases are produced and used to enforce emission regulations by legislative

bodies worldwide. Existing driving cycles include the EU legislative cycles, INRETS driving cycles, BP

bus cycle, and the Worldwide Harmonised Motorcycle Emissions Certification [18]. While a staple of the

automotive industry, driving cycles can also be adopted into other industries, one such example being for

cyclists [19]. This work will attempt to do just that, by developing a driving cycle for a new field which

as of yet has no publicly available information on driving cycles, delivery drones.

1.1 MOTIVATION

While advances in this field have been fruitful, the performance of the drone’s deliveries still needs to be

linked back to the performance of the drone’s powertrains so that they may be optimized to maximize the

radius of delivery. A driving cycle has thus been developed to quickly and iteratively evaluate delivery

drone performance. Being the first of its kind, this research will explore a simpler driving cycle than the

multi-drone pathing previously discussed. The driving cycle will consist of one drone, delivering a single

4

payload to a destination, and then returning to its original starting point. This cycle will be tested on a set

of 54 drone configurations to see which outperforms the others and achieves the greatest delivery radius.

The objectives of this thesis are as follows:

• Provide a novel and robust delivery drone driving cycle written in MATLAB with which the

reader may test their own delivery drones.

• Develop a halbach array motor to use as the input motor of a quadcopter drone’s powertrain with

which to test the created driving cycle.

• Provide a data processing and interpolation code written in MATLAB with which the reader can

import electric motor and propeller powertrain data to simulate its behavior in the driving cycle.

• Analyze trends found in the results of the 54 delivery drone configurations tested.

1.2 OUTLINE

The outline of this thesis is as follows. First, a literature review on halbach array motors is conducted in

Section 2. Then, necessary governing electromagnetic equations, motor characteristic behaviors, and

drone kinetic equations are laid out in Section 3. This is followed by Section 4 where the data processing

needed to set up the simulated powertrain behavior of the drones is detailed. Next, the developed driving

cycle is explained in Section 5. Lastly, the interpretation of the results of 54 different drone configurations

that have been tested in the driving cycle is conducted.

5

2 LITERATURE REVIEW – HALBACH ARRAY MOTORS

A halbach magnet array is a permutation on the typical magnet arrangement found in electric motors. By

alternating the magnet array polarity in intervals other than 180 degrees, the magnetic field surrounding

the array can be altered such that the magnetic flux is directed to only one side. This can be very

beneficial in electric motors as will be explained in this section, starting first with the history of halbach

arrays, their applications, and their advantages in motors.

2.1 THE HALBACH ARRAY

The term halbach array refers to any arrangement of permanent magnets whose poles are not aligned

parallel to one another (in linear or planar configurations), or aligned concentrically (in circular

orientations). Figure 2.1.1 shows examples of a linear, planar, and circular halbach array.

Figure 2.1.1: Examples of Halbach Arrays that are a) Linear [20], b) Planar [21], and c) Circular [22].

The first instance of a halbach array was introduced in 1979 by Klaus Halbach who was studying the uses

of rare-earth cobalt permanent magnets (PM) [23], [24]. With this array he managed to create a near-

unidirectional magnetic field, namely a dipole field, seen in Figure 2.1.2, requiring no external power

supply. In 1986, he continued his work on these novel arrays, proposing them to be used as particle

accelerators used in medical machinery and spectrometers [25].

6

Figure 2.1.2: Dipole Field Generated by Circular Halbach Array [26].

Many other applications of these dipole fields have since been proposed by other researchers in a vast

range of fields. Some such applications are explored by Gunes et al. who studied its use for hall effect

measurements at high temperatures [27]; and Raich and Blümler, Jizhong et al., and Paulsen et al., who

have studied the integration of the dipolar array in portable nuclear magnetic resonance equipment [28]-

[30]. However, the majority of research on the dipole fields has been in the realm of synchronous electric

machinery, including energy storage [31]-[33], and electromagnetic generators and motors [34]-[37]. One

advantage of using this dipole field in motors and generators is that there are no electromagnetic coils or

permanent magnets in the rotor, only current-laden wire running perpendicular to the plane of the

magnetic field. This allows for mechanical operation with relatively little inertia [22]. This dipole field

motor is shown in Figure 2.1.3.

Figure 2.1.3: Schematic of a Dipole Field Motor Containing no Electromagnetic Coils [22].

7

While the dipole field is an interesting design that led to advances in many fields, there is another aspect

of the halbach array that has created a greater impact. That is the halbach array’s ability to efficiently

guide the magnetic field lines and allow for the permanent magnets’ flux to be used more effectively,

improving the field density by a factor of 1.4 times compared to a standard magnetic array [38]. This

phenomenon occurs because the array concentrates the magnetic field to one face, sometimes called

electromagnetic shielding, compared to a conventional magnet array which produces a balanced field as

seen in Figure 2.1.4. This allows for a localized concentration of flux in the area of interest for operation,

such as at the electromagnets in the stator of a motor. It should also be noted that the halbach array also

retains more of the unused magnetic field flow inside of the magnets rather than letting it flow out to the

surrounding environment, which generally has a higher magnetic resistance, resulting in an overall

stronger magnetic field [39]. This can be seen in Figure 2.1.5, which shows the magnetic field strength

throughout a typical motor, where the core of the rotor has a weaker field in the halbach version of the

motor compared to the standard motor, as well as a stronger field in the stator of the motor where the

electromagnets reside.

Figure 2.1.4: Comparison of the Magnetic Field between a Conventional Parallel Pole Array (top) and a

Halbach Array (bottom) [40].

8

Figure 2.1.5: Comparison of the Flux Density between a Halbach Array (left) and Conventional Parallel

Pole Array (right) in an Electrical Machine [41].

2.2 APPLICATIONS OF HALBACH ARRAYS

Halbach arrays have been found to be useful in a variety of interesting applications, ranging from single-

degree of freedom (DOF) actuators, multi-DOF actuators, electro-mechanical batteries, magnetic

levitators, and more efficient electric rotational machines.

Their use in single-DOF actuators has been studied by many researchers. The advantages of these linear

actuators compared to other forms of translational movement are that no cogging is present in their

operations along with efficient cooling [42]. Eckert et al. have shown that the actuators can be used in

both active and semi-active suspension systems, with the advantages of having lower moving mass and

ripple torque [43]. In recent research, Yan et al. have studied the use of a dual halbach arrays to enhance

flux density and improve performance [44]. Zhu et al. also investigated dual halbach arrays though

instead for the purpose of improving thrust output [45]. Wang et al. have shown how to optimize the

length, width, and magnetization direction of the halbach arrays to obtain stronger magnetic fields [46].

Jang and Choi introduced a spring into their linear actuator, proposing that the cyclical operation of the

actuator is improved when powered at the natural frequency of the spring [47]. Lastly, Hilton and

McMurry have proposed a novel adjustable linear halbach array, which has a dynamic magnetic field that

9

can be transferred through the device’s plane, essentially turning the field ‘on’ and ‘off’ as needed [48].

Figure 2.2.1 shows a comparison between two common actuator arrangements, the first being where the

magnet arrays are wrapped around a cylindrical actuator [42], [45], [49]-[51] and the second being a flat

surface with actuator placed atop [52]-[57].

Figure 2.2.1: Two Linear Actuators with Halbach Arrays, one in Cylindrical Dual Halbach Configuration

(left) [58] and one in a Flat Configuration (right) [57].

Originally, if one wanted to move an actuator on two separate axes they would simply place one linear

actuator on top of another and control them both separately. However, Compter has since proposed the

first planar motor, using a two-dimensional halbach array grid to displace the actuator [59]. On this planar

magnet grid, the actuator can in fact achieve levitation, allowing it to move freely in six degrees of

freedom without bearings. Both Peng and Zhou, and Wang et al. have since improved the performance of

the planar motor by remodeling the halbach array on which it levitates to reduce the high-order harmonics

and increase the thrust force [60], [61]. Kou et al. have introduced a computation model for planar motors

that includes the corner flux patterns of the magnet arrays, which had been, up until then, ignored [62].

Alternatively, Hao et al. were interested in a planar motor that was compromised of a halbach array

actuator sitting on an electromagnetic planar grid, for which they analyzed a conceptual design [63]. In an

attempt to reduce the normal force of the motor, Huang et al. employed a genetic algorithm, reducing it

by a factor of 4.5 [21].

10

The first researchers to use a single grid of PMs in a cylindrical linear-rotary motor, whose original design

was to combine a rotational motor with a linear actuator [64]-[67], were Krebs et al. [68]. Later, the

parallel PM grid was replaced with a halbach array grid by Jin et al. as shown in Figure 2.2.2 [69]. Jin et

al. continued on to produce the first prototype of their design and analyzed its torque and linear force

[70], subsequently proceeding to optimize the halbach array [71].

Figure 2.2.2: Transformation of a Halbach Array Planar Grid into a Cylindrical Grid [69].

While the technology of spherical motors is still in the theoretical and early prototyping stages, Xia et al.

have put forth a conceptual design that incorporates a halbach array [72], [73]. Their analyses on their

design shows that the halbach array spherical motor has greater torque compared to parallel magnet

arrays, along with a magnetic field that is more sinusoidal in shape, allowing for the torque ripple to be

suppressed more effectively [73]. Li and Li have gone on to produce a prototype spherical halbach array

motor, exploring the end-effects of the motor and showing that they account for almost half of the eddy

current losses [74].

2.3 HALBACH ARRAYS IN MOTORS

The wide variety of possible orientations of halbach arrays gives way to many approaches to motor

designs incorporating the arrays. Some interesting approaches are the dipole field method, which was

11

introduced earlier, and the addition of the halbach arrays into switched reluctance or partitioned stator

motors, which were novel for not having any permanent magnets in their original designs [75]-[79].

Nevertheless, the more common incorporation of halbach arrays is in the standard multi-pole PM motors,

in both radial and axial configurations as shown in Figure 2.3.1.

Figure 2.3.1: Schematic of a 10-pole Radial Halbach Array Motor (left) [80] and Axial Flux Halbach

Array Motor (right) [81].

The optimization of the halbach array configurations is important to achieve the best output performance

when it is incorporated into the motor. Choi and Yoo have done extensive work on evaluating different

configurations, including single, dual, and triple arrays, as well as offset arrays to determine the optimal

patterns [82]. These are shared in Figure 2.3.2 for dual array and triple array configurations. Other

researchers in Asef et al. have tackled the issue of finding the optimal halbach array, looking to maximize

air-gap flux density, output torque, and the frequency of first-order harmonics while minimizing cogging

torque [83]. Using two-dimensional finite element analysis they produced comparative analyses on six

designs revealing the strengths and weaknesses of each.

12

Figure 2.3.2: Torque Density Plots of Optimized Halbach Array Motors in Configurations of Dual Array

(top) and Triple Array (bottom) [82].

As previously mentioned, the halbach arrays can direct the bulk of the magnetic field flux onto one side

of the array in the desired location for operation. The conventional parallel-pole magnet arrangements

used in motors do not have such a property, and need a backing of iron to guide the magnetic flow back

into the desired location. Thus, by removing the need for the iron backing, halbach arrays noticeably

reduce the total mass in the motor. It should be understood that the iron backing is also a structural

13

component of the motor and, if removed, another means to withstand the stress needs to be introduced,

such as a carbon fiber shell [81]. It should also be noted that if the halbach magnets are too thin, the motor

design can still benefit from having an iron backing. Xia et al. have studied this critical magnet thickness,

showing plainly that when the critical point is passed the iron backing has no impact on the motor

performance [84]. Ofori-Tenkorang and Lang, as well as Güler et al. have also compared conventional

magnet arrangements with an iron backing to halbach arrays with an ironless backing showing that for the

same mass, a halbach array produces a higher torque beyond the critical magnet thickness [40], [85].

Using analytical design tools, Lovatt et al. have shown that the dual halbach array integrated into their

motor reduced the power loss by 20% compared to a conventional magnet arrangement of equal mass and

with an iron backing [86].

Other advantages include the increased power density in the area of operation, increased efficiency, and

reduction of eddy current loss. In their studies, Jha et al. have compared the radial halbach motor to a

conventional motor with the same dimensions, current, voltage, and magnet grade, showing an increase of

41% torque for an out-run motor and 87% for an in-run motor [87]. Through the use of finite element

analysis, Ubani et al. have shown that the addition of a halbach array in their axial flux motor

simultaneously halved the mass and increased the flux density by a factor of 2.57 [88]. By introducing a

halbach array into their axial flux motor prototype, Prasetio and Yuniarto were able to achieve an

efficiency of 92.72% with an 807 Watt power output, suitable for use in electric vehicles [89]. In

researching high acceleration machines, Dwari et al. determined that for their specific dimensions and

requirements, introducing a halbach array into their motor reduced the rotational inertia of the motor by

58% while also increasing the torque by 6.3% [90], [91]. In regards to the eddy current impact, Jun et al.

showed that incorporating a halbach array reduced the eddy current loss by 24.03%, which both increases

the efficiency of the machine while reducing the amount of heat generated [92]. Li et al. have also studied

eddy current losses, observing that there is a decreasing reduction on the losses as number of magnet

orientations per pole increases [39]. These properties, along with the lower weight due to ironless

14

backing, have made halbach array motors a prime candidate for high performance industries such as

spacecraft applications [92]-[94], vehicle racing [41], [95], and industry turbomachinery [96].

Lastly, halbach array motors have an interesting property in that they can be layered continuously. An

example of this is shown in Figure 2.3.3. While this introduces complexity into the design, especially in

the radial configuration, and more susceptibility to vibrations, unbalance, misalignment, and other

mechanical issues [97], it allows for a greater dispersion of the electromagnets in the motor. This can be

advantageous, as the increase in temperature generation in electromagnets does not scale linearly with

volume, limiting the size to which an electromagnet can realistically be produced [25].

Figure 2.3.3: Schematics of a Dual Halbach Array (left) along with a Triple Halbach Array (right) [82].

The Rotor, which would sit between each set of Stators, is not shown.

15

3 GOVERNING EQUATIONS

Throughout this work, three main systems of equations are used. In the order that they are explored in this

section, they are the electromagnetic equations governing the simulation of electric motors, the

characteristic equations governing the electrical and physical behavior of electric motors, and the kinetic

equations governing the behavior of quadcopter drones.

3.1 ELECTROMAGNETIC EQUATIONS

Many physical laws are at the core of electromagnetic finite element method (FEM) softwares. Chief

amongst them are the Maxwell Equations. This set of equations is compromised of the Extended

Ampère’s Equation, Faraday-Lenz Equation, and the two Gauss Equations of magnetic conservation and

electric conservation. These equations are laid out in that order as follows [98]:

𝛻 ⨯ 𝑯 = 𝑱 +𝛿𝑫

𝛿𝑡 (EQ 3.1.1)

𝛻 ⨯ 𝑬 = −𝛿𝑩

𝛿𝑡 (EQ 3.1.2)

𝛻 ⋅ 𝑩 = 0 (EQ 3.1.3)

𝛻 ⋅ 𝑫 = 𝜌𝑣 (EQ 3.1.4)

Five vectors are introduced here. In order of appearance they are: H magnetic field strength (amperes per

meter), J current density (amperes per meter squared), D electric flux density (coulombs per meter

squared), E electric field strength (volts per meter), and B flux density (teslas). The variable ρv is the

charge density measured in coulombs per meter cubed.

On top of these equations are the three constitutive relations. There are the magnetic relation, dielectric

relation, and Ohm’s Law, shown here in that order [98]:

16

𝑩 = 𝜇 ∗ 𝑯 + 𝑴 (EQ 3.1.5)

𝑫 = 𝜀 ∗ 𝑬 (EQ 3.1.6)

𝑱 = 𝜎 ∗ 𝑬 + 𝑱𝑠 (EQ 3.1.7)

Here, the new vectors M and Js are the remnant induction magnetization, of unit teslas, found in

permanent magnets and the source current density in amperes per meter squared found in conductors. The

variables μ, ε, and σ are the magnetic permeability (henrys per meter), dielectric permittivity (farads per

meter), and electrical conductivity (siemens per meter) respectively. The variable μ is sometimes

replaced, as will be done in this work, with the reluctivity tensor υ (meters per henry) like so [99]:

𝜇 =1

𝜐 (EQ 3.1.8)

Rather than using the field quantities of the preceding equations, FEM solvers tend to prefer solving the

equations using partial differential equations. In these, the potentials for both the magnetic and electric

fields are represented using magnetic vector potential A, measured in teslas by meters, and electrical

voltage V in volts, like this [99]:

𝑩 = 𝛻 ⨯ 𝑨 (EQ 3.1.9)

𝑬 = −𝛻 ⋅ 𝑉 (EQ 3.1.10)

Using the constitutive magnetic relation and the Extended Ampère’s Equation, the two preceding

equations can be combined and rearranged as follows [99]:

𝛻 ⨯ (𝜐 ∗ 𝛻 ⨯ 𝑨) = 𝑱𝑠 − 𝜀 ∗ 𝛻 ⋅𝛿𝑉

𝛿𝑡+ 𝜐 ∗ 𝑴 (EQ 3.1.11)

Using a calculus cross product identity, this equation can be simplified to:

𝛻(𝜐 ∗ 𝛻 ⋅ 𝑨) − 𝜐 ∗ 𝛻2𝑨 = 𝑱𝑠 − 𝜀 ∗ 𝛻 ⋅𝛿𝑉

𝛿𝑡+ 𝜐 ∗ 𝑴 (EQ 3.1.12)

17

And here, to guarantee unique solution fields, a reference Coulomb gauge is set. Naturally, this gauge will

be set in such a manner that the mathematical equations are simplified:

𝛻 ⋅ 𝑨 = 0 (EQ 3.1.13)

This gives the final three-dimensional electromagnetic equation:

𝜐 ∗ 𝛻2𝑨 = −𝑱𝑠 + 𝜀 ∗ 𝛻 ⋅𝛿𝑉

𝛿𝑡− 𝜐 ∗ 𝑴 (EQ 3.1.14)

For two-dimensional systems, this equation can be simplified even further. The magnetic vector potential

and source current density vectors would only have components in the direction perpendicular to the

plane, as such:

𝑨 = (0 0 𝐴) (EQ 3.1.15)

𝑱𝒔 = (0 0 𝐽𝑠) (EQ 3.1.16)

Also, the source current density can be broken down into its components of source voltage Vs, measured

in volts per meter, and electrical conductivity σ, measured in siemens per meter, like this:

𝑱𝒔 = 𝜎 ∗ 𝑽𝒔 (EQ 3.1.17)

Of special note when evaluating transient equations is that many properties may be dependent on physical

states. Of interest here are the remnant induction magnetization M and magnetic permeability σ, which

are dependent on temperature, and the reluctivity tensor υ which is dependent on amperage. Adding

indicators for these dependencies and the source voltage substitution mentioned prior provides the

transient three-dimensional electromagnetic equation:

𝜐(𝐴) ∗ 𝛻2𝑨 = −𝜎(𝑇) ∗ 𝑽𝒔 + 𝜀 ∗ 𝛻 ⋅𝛿𝑉

𝛿𝑡− 𝜐 ∗ 𝑴(𝑇) (EQ 3.1.18)

Following these derivations is the discretization needed for their integration into the FEM softwares.

However, these discretization equations, both in the special and temporal dimensions, along with the

18

required boundary conditions, are quite extensive and laborious, requiring too much explanation for this

work. Readers interested in these topics can explore the following sources to see the derivations and their

explanations [99], [100].

Two unique boundary conditions will be addressed here that pertain to the electric motor simulations

undertaken. The first is the magnetic boundary condition between the motor’s metal components and

surrounding air outside of the stator-rotor gap. Due to the magnetic permeability of these materials being

much greater than that of air, and that the motor’s geometry is designed to efficiently redirect the

magnetic flux within itself, it is assumed that no flux travels outside of the metals and into the

surrounding environment. The equation for this boundary relation, dubbed the Dirichlet condition, is as

follows [99]:

𝑨|Г𝐷 = 𝑨𝟎 (EQ 3.1.19)

Secondly, a special boundary condition is set when a full motor is cut down into either single electrical

phases or half electrical phases for computation reduction. In the case of the reduction to the half

electrical phase, where the magnetic flux ϕ of unit webers must be reversed, the boundary condition is as

follows [101]:

𝝓(𝑟, 𝛥𝜃, 𝑧) = 𝝓𝒎(𝑟, −𝛥𝜃, 𝑧) (EQ 3.1.20)

Here, the subscript m denotes the flux at the opposing boundary and r, θ, and z are the radial, tangential,

and axial coordinate directions.

3.2 ELECTRIC MOTOR CHARACTERISTIC BEHAVIOR

Rotating electric machine electronics all deal in the realm of alternating current, even the brushless DC

(BLDC) motors powered by DC voltage sources. This leaves the power input equation in a more

complicated state than the expected product of voltage and current that is common to DC circuits. To

begin breaking this down, the electromotive force (EMF) produced by the motor must be looked at. This

19

force, often redundantly referred to as the back EMF force, is the result of the rotating permanent magnets

inducing magnetism into the stator armatures, creating a reverse current in the motor circuitry that the

battery must overcome. The equation for this force for one phase is as follows [101]:

𝐸𝑎 = 𝜔𝑚 ∗ 𝑁 ∗ 𝑝2 ∗ 𝑘𝑙 ∗ 𝜙𝐺𝑎𝑝 (EQ 3.2.1)

In this equation, ωm is the mechanical rotating speed of the motor in radians per second, N is the number

of coil turns around each armature, p is the quantity of magnetic pole pairs present in the rotor, kl is the

ratio of flux lost in the air gap known as the leakage coefficient, and ϕGap is the flux per coil turn in

webers. This EMF is in turn used in the relation between the circuit's input voltage and armature current

like so [101]:

𝑉𝐼𝑛 = √(𝐸𝑎 + 𝐼𝑎 ∗ 𝑅𝑎)2 + (𝜔𝑒 ∗ 𝐼𝑎 ∗ 𝐿𝑎)

2 (EQ 3.2.2)

The variables used are the voltage in VIn in volts, the back EMF of a single phase Ea in volts, the armature

current Ia in amperes, the armature resistance Ra in ohms, the inductance La in henrys, and the electric

angular frequency ωe in radians per second. The latter variable is simply the product of the physical

rotational frequency ωm and quantity of pole pairs p.

This finally culminates to the electrical power PIn in watts being used by the 3-phase motor, as such

[101]:

𝑃𝐼𝑛 = 3 ∗ 𝑉𝐼𝑛 ∗ 𝐼𝑎 ∗ 𝑐𝑜𝑠𝜑𝑃ℎ𝑎𝑠𝑒 (EQ 3.2.3)

The new variable introduced, φPhase, is the angle between VIn and the EMF in the phasor diagram. This

diagram can be seen in Figure 3.2.1 along with the electrical circuit diagram.

20

a) b)

Figure 3.2.1: Brushless DC Motor Electrical a) Phasor Diagram and b) Circuit Diagram [101].

The reason PIn is of such importance and why it was examined is that it plays a central role in motor

performance. Due to the inevitable loss present inherent to all energy transfer systems, the physical power

that can be harnessed by the delivery drones will never be as great as this input power. However, it sets a

standard for how close the powertrains should aim to reach. The three main sources of loss that exist

when converting the electrical power PIn into mechanical power POut through electric motors are seen in

the following equation [101]:

𝑃𝑂𝑢𝑡 = 𝑃𝐼𝑛 − 𝑃𝐶𝑜𝑟𝑒 − 𝑃𝐶𝑜𝑝𝑝𝑒𝑟 − 𝑃𝑀𝑒𝑐ℎ (EQ 3.2.4)

In written terms, these are known as the core loss PCore, copper loss PCopper, and mechanical loss PMech

respectively, all of units watts. The core loss consists of the energy lost to eddy currents and the energy

spent orienting the magnetic flux throughout the iron of the motor, while the copper loss is the energy

spent shuttling electrons through the windings. Both of these losses generate heat in the motor which

needs to be properly dissipated lest the magnets start degrading at higher temperatures. The mechanical

loss is typically a combination of the friction in the ball bearings as well as the air friction along the

surface of the rotor. Rather than separate values, these losses are commonly joined as an efficiency ratio η

between the input and output power like so:

𝜂 =𝑃𝑂𝑢𝑡

𝑃𝐼𝑛 (EQ 3.2.5)

21

The output power is much simpler to break down than the input was, as it is simply the product of the

mechanical rotational speed ωm, and torque T in newtons by meters [101]:

𝑃𝑂𝑢𝑡 = 𝜔𝑚 ∗ 𝑻 ∗𝜋

30 (EQ 3.2.6)

While the controller logic and equations will not be elaborated on here, one of their aspects that must be

understood is the pulse width modulation (PWM). Sometimes known more commonly as throttle, PWM

is a method by which a battery can output a fraction of its voltage VBattery to a circuit [102]. As the value

for PWM is a fractional percentage value, the equation for the voltage going into the circuit is simple and

shown below in EQ 3.2.7. This value of VIn does not take into account the voltage of the back EMF it

must overcome.

𝑉𝐼𝑛 = 𝑉𝐵𝑎𝑡𝑡𝑒𝑟𝑦 ∗ 𝑃𝑊𝑀 (EQ 3.2.7)

To understand what is happening with the PWM of the voltage, the motor’s driver logic must first be

understood. As seen in Figure 3.2.2, six different gate drivers are used to choose where the current is

directed into the motor. These gates are opened and closed as needed as the motor cycles through its

electrical periods. By quickly opening and closing the top gate of a phase at a specific rate of throttle, as

can be seen in Figure 3.2.3, an equivalent percentage of voltage is sent through the gate and into the

motor. Notice how the bottom gates are never throttled, but always kept open when they are needed. This

is because the inductance energy in the motor coils will keep discharging during the modulation and must

have an avenue through which to discharge. By keeping the bottom gate open, this discharge is sent to the

flyback diodes until the top gate is activated again and a closed circuit is created.

22

Figure 3.2.2: Gate Driver Diagram of a 3-Phase Brushless DC Motor [103].

Figure 3.2.3: Pulse Width Modulation Diagram of a 3-Phase Gate Driver at 75% Throttle [104].

Through varying the throttle and load experienced by the electrical motor, a wide variety of operating

states of rotational speed and output torque can be produced. At steady state operation, each of these

states will have a set electrical power input and mechanical power output along with the related efficiency

ratio. By mapping out this efficiency for all possible operating points on a torque-rpm plane into an

efficiency heatmap, the performance of the motor can be visualized. Below in Figure 3.2.4 are example

efficiency heatmaps produced by ANSYS and Altair from their newly released software solutions in 2018

and 2019 respectively.

23

a)

b)

Figure 3.2.4: Motor Efficiency Heatmaps as Produced by a) ANSYS Maxwell Machine Toolkit (2018)

[105] and b) Altair Flux E-Machine Toolbox (2019) [106].

24

Though the entire collection of equations governing the behavior and properties of electric motors is too

great to discuss in its entirety here, one final equation that will be used in this work will be introduced.

This equation relates to the stacking factor of a motor’s stator and rotor. As mentioned earlier in EQ 3.2.4,

one of the power loss locations in a motor is in its metal core due to eddy currents. These eddy currents

are induced by the moving magnetization present during motor operation. By layering the stator and rotor

in the axial direction with insulating sheets, usually made of silicon, these eddy effects can be prevented

from moving in the axial direction, thus saving energy. This does not interfere with the magnetic flux

flow inside the motor as the flow only travels in the radial and tangential directions. When modeling

motors, the effect of the silicon laminations can be summarized by a stacking factor, calculated by the

thickness of the laminations like so [107]:

𝑆𝑡𝑎𝑐𝑘𝑖𝑛𝑔 𝐹𝑎𝑐𝑡𝑜𝑟 = 0.0425 ∗ 𝛿𝐿𝑎𝑚 + 0.6032 ∗ arctan (108.16 ∗ 𝛿𝐿𝑎𝑚) (EQ 3.2.8)

Here, δLam is the thickness of the individual laminations in millimeters. This relation is shown graphically

in Figure 3.2.5.

Figure 3.2.5: Graphical Representation between a Motor’s Lamination Thickness and Stacking Factor

[107].

25

3.3 DRONE KINETICS

To define these governing equations, the standard coordinate system used must first be specified. This

coordinate system for quad-copter drones and many other aircraft is illustrated in Figure 3.3.1 below. The

pitch, used to propel the drone in the lateral direction, is determined by the angle θPitch, which is the

rotation around the y-axis. The remaining two angles, roll φRoll and yaw ψYaw, are the rotation around the x

and z-axes respectively. Neither of these angles will be important in this study as φRoll is not needed for

one-directional motion and ψYaw is also not needed if we simply assume that the drone is already facing

the proper direction. Both of these values also come into play for stabilization purposes. The control

systems used to stabilize the drone are complex and implementing them would take away from the

efficacy of the developed driving cycle. For this, they will be negated in this research by assuming both

that the drone is properly balanced and that no outside forces, such as wind, are affecting the system. The

energy spent in stabilization can still be accounted for by the user through the safety factor defined.

Figure 3.3.1: Quadcopter Dynamics Coordinate System.

To implement a driving cycle for delivery drones, the governing equations surrounding the drone’s

operation must also be understood. The trip is split into three sections: Ascent, Cruising, and Descent. As

the standard operation of the delivery drone is to deliver a payload and return to its original destination,

each of these three segments must be completed twice, once with and once without a payload. Figure

3.3.2 shows the force diagram for these three travel segments.

26

a) b) c)

Figure 3.3.2: Force Diagrams of the Drone Kinematics during a) Ascent, b) Cruising, and c) Descent.

In mathematical notation, these diagrams are expressed respectively with the following equations [108]:

General: 𝑭𝑷𝒓𝒐𝒑𝒖𝒍𝒔𝒊𝒐𝒏 = 𝑞𝑝𝑡 ∗ 𝑳𝒊𝒇𝒕𝑷𝒓𝒐𝒑 (EQ 3.3.1)

𝑭𝑳𝒊𝒇𝒕 = 𝑭𝑷𝒓𝒐𝒑𝒖𝒍𝒔𝒊𝒐𝒏 ∗ cos (𝜃𝑃𝑖𝑡𝑐ℎ) (EQ 3.3.2)

𝑭𝑻𝒉𝒓𝒖𝒔𝒕 = 𝑭𝑷𝒓𝒐𝒑𝒖𝒍𝒔𝒊𝒐𝒏 ∗ 𝑠𝑖𝑛 (𝜃𝑃𝑖𝑡𝑐ℎ) (EQ 3.3.3)

Ascent: 𝑭𝑳𝒊𝒇𝒕 = 𝑭𝒈 + 𝑭𝑽𝑫𝒓𝒂𝒈 + 𝑀 ∗ 𝒂𝑽 (EQ 3.3.4)

Cruising: 𝑭𝑳𝒊𝒇𝒕 = 𝑭𝒈 (EQ 3.3.5)

𝑭𝑻𝒉𝒓𝒖𝒔𝒕 = 𝑭𝑯𝑫𝒓𝒂𝒈 (EQ 3.3.6)

Descent: 𝑭𝒈 = 𝑭𝑳𝒊𝒇𝒕 + 𝑭𝑽𝑫𝒓𝒂𝒈 + 𝑀 ∗ 𝒂𝑽 (EQ 3.3.7)

Where qpt is the quantity of powertrains, LiftProp is the lift generated by a single propeller in newtons,

FPropulsion is the force output of all powertrains combined in newtons, θPitch is the pitch angle of the drone

in radians, FLift is the vertical component of propulsion, FThrust is the horizontal component of propulsion,

M is the current mass of the drone in kilograms, and aV is the acceleration of the drone in the vertical

direction with units of meters per second squared. Predictably, the variables for Fg, FVDrag, and FHDrag are

27

the forces of gravity, vertical drag, and horizontal drag, all with units of newtons, and which can be

broken down as follows [109]:

𝑭𝒈 = 𝑀 ∗ 𝒂𝒈 (EQ 3.3.8)

𝑭𝑽𝑫𝒓𝒂𝒈 =𝐶𝑉𝐷𝑟𝑎𝑔∗𝜌∗𝒗𝑽

𝟐

2 (EQ 3.3.9)

𝑭𝑯𝑫𝒓𝒂𝒈 =𝐶𝐻𝐷𝑟𝑎𝑔∗𝜌∗𝒗𝑯

𝟐

2 (EQ 3.3.10)

Where ag is the acceleration due to gravity with units of meters per second squared, ρ is the density of air

in kilograms per meter cubed, CVDrag and CHDrag are the area-drag coefficients of the drone in the vertical

and horizontal directions respectively with both having units of meters squared, and vV and vH are the

vertical and horizontal velocities respectively with both having units of meters per second.

What are of real interest here, however, are the mechanical energy expenditures during these drone flight

segments. These aforementioned values can be used, along with the efficiency of the motor under the

current operating parameters, to determine the amount of electrical energy spent. First, the energy

equation used for the ascent is given:

𝐸𝐴𝑠𝑐𝑒𝑛𝑡 = 𝑞𝑝𝑡 ∗ ∫𝑃𝑂𝑢𝑡

𝜂∗ 𝑑𝑡

𝑡𝑓0

(EQ 3.3.11)

Followed by the energy equation used for landing:

𝐸𝐷𝑒𝑠𝑐𝑒𝑛𝑡 = 𝑞𝑝𝑡 ∗𝑃𝑂𝑢𝑡∗ℎ

𝜂∗𝑉𝑙𝑎𝑛𝑑 (EQ 3.3.12)

Where t is the time in seconds, tf is the time it takes to reach the cruise height in seconds, h is the cruise

height in meters, Vland is the landing speed in meters per second, POut is the mechanical output power in

watts, and η is the powertrain efficiency as a fractional percentage.

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The energy used during horizontal traveling, which has been dubbed as cruising, is slightly more

complicated. The drone is set at two different pitch angles for its two operating conditions depending on

the presence of the payload. These pitch angles are chosen to minimize the energy usage for each meter

traveled. Using the remaining energy, the safety factor, and noticing that the travel distance must be the

same to and from the delivery location, we can solve for the drone’s travel distance as follows:

𝐸𝑅𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 =𝐸𝐵𝑎𝑡𝑡𝑒𝑟𝑦

𝑆𝐹− ∑ 𝐸𝐴𝑠𝑐𝑒𝑛𝑡 − ∑ 𝐸𝐷𝑒𝑠𝑐𝑒𝑛𝑡 (EQ 3.3.13)

𝐸𝑃𝑀 = 𝑞𝑝𝑡 ∗𝑃𝑂𝑢𝑡

𝜂∗√2∗𝑀∗𝒂𝒈∗𝑡𝑎𝑛(𝜃𝑃𝑖𝑡𝑐ℎ)

𝐶𝐻𝐷𝑟𝑎𝑔∗𝜌

(EQ 3.3.14)

𝑑 =𝐸𝑅𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔

∑ 𝐸𝑃𝑀∗

𝑘𝑖𝑙𝑜𝑚𝑒𝑡𝑒𝑟

1000∗𝑚𝑒𝑡𝑒𝑟 (EQ 3.3.15)

Where EBattery, ERemaining, ΣEAscent, and ΣEDescent are the total electrical energy available, the energy

delegated to lateral cruising, and the total energy used in the two ascents and descents respectively, all

with units of joules. In turn, SF is the user-defined safety factor, EPM is the energy used to travel a meter

for the current configuration of the drone in joules per meter, and lastly d is the maximum horizontal

delivery distance in kilometers.

While not set out in rigorous equations like the preceding entries, some explanation of electric motor and

propeller powertrain behavior may be of help to the reader and will be discussed. When voltage is applied

to the controller of a BLDC motor, it will follow the behavioral rpm-torque curve of the voltage it was set

to until the torque output drops and matches the load it experiences. An unloaded motor is only affected

by the electromagnetic properties of its material components and will keep accelerating until the produced

back EMF is too great to overcome. Returning to the case of a loaded motor, changing the speed of the

motor may only be done by changing the load it experiences, or by varying the voltage supplied, usually

through means of pulse width modulation. The propellers whose drag act as the motor load have their

values of drag, rpm, and lift intrinsically tied at set angles of attack. Meaning, if one of the three values is

29

known, which in this case will be both the torque and rpm of the motor, the missing values can be found.

Finally, it should be noted that when working with quadcopter propellers, it is common to have their lift

forces denoted in their specification sheets using the unit of grams. To convert these values to newtons,

which are used throughout this research, simply follow the following equation using the acceleration due

to gravity:

𝑁𝑒𝑤𝑡𝑜𝑛 = 𝑔𝑟𝑎𝑚 ∗𝑘𝑖𝑙𝑜𝑔𝑟𝑎𝑚

1000∗𝑔𝑟𝑎𝑚∗

9.81∗𝑚𝑒𝑡𝑒𝑟𝑠

𝑠𝑒𝑐𝑜𝑛𝑑2 (EQ 3.3.16)

Likewise, battery capacity is typically measured in milliamp hours and voltage in specification sheets.

This research prefers the use of joules, which can be converted to through the use of this equation:

𝐽𝑜𝑢𝑙𝑒 = 𝑚𝑖𝑙𝑙𝑖𝑎𝑚𝑝 ℎ𝑜𝑢𝑟 ∗𝑎𝑚𝑝

1000∗𝑚𝑖𝑙𝑙𝑖𝑎𝑚𝑝∗

𝑐𝑜𝑢𝑙𝑜𝑚𝑏

𝑠𝑒𝑐𝑜𝑛𝑑

𝑎𝑚𝑝∗

3600∗𝑠𝑒𝑐𝑜𝑛𝑑

ℎ𝑜𝑢𝑟∗ 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 (EQ 3.3.17)

30

4 METHODOLOGY OF HALBACH MOTOR DESIGN

To properly model a given drone’s behavior in the driving cycle, the characteristics of the powertrain

must be known. This can prove difficult at times with the limited amount of data given by third party

suppliers and the computation time needed to fully map out a motor design’s efficiency map. ANSYS

[110] and Altair [106] are both releasing computer aided design (CAD) toolkit packages aimed at

achieving this more quickly for users, but these toolkits have yet to incorporate brushless DC motors,

which are most commonly used in the drone industry.

For these reasons, the motor performance must be manually compiled from multiple Maxwell

simulations, which will be explored in Section 4.1, to then assemble the operating points of the entire

powertrain as explained in Section 4.2.

4.1 ANSYS MAXWELL SIMULATIONS

The simulations conducted in ANSYS Maxwell to evaluate the halbach array motor start in the software’s

RMxprt tool, and are then exported to the two-dimensional design space for transient analyses.

ANSYS Maxwell is a diverse electromagnetic simulation software capable of modeling a very wide

variety of electromagnetic applications. To help in the specific evaluation of rotating electromagnetic

machine applications ANSYS has included a tool in the software titled RMxprt. This tool takes in user

input to determine the type of rotating machine to be modelled, along with its dimensions, material

compositions, and much more so as to provide the user with quick results on their design’s performance.

The limitation of this tool is that it cannot accommodate more complex geometries, like that of a halbach

array. In this research, the RMxprt tool will be used for its ease of use to create the base model which will

then be exported to the more general two-dimensional modeling software incorporated in Maxwell where

the needed modifications will be made.

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When starting up RMxprt, the first prompt allows the user to choose the machine type to be modeled,

which is chosen to be the Brushless Permanent-Magnet DC Motor. From there all of the parameters of the

motor can be input into the properties tables. These will start with the general machine parameters, and

follow up with the parameters of the circuit, stator, armature slots, windings, rotor, and magnets.

The general machine parameters set up the stator-rotor orientation, the quantity of magnet poles, the

circuit type, and the losses the motor experiences at a set reference speed. The input parameters can be

seen in Table 4.1.1. This is followed by the circuitry parameters, which are detailed in Table 4.1.2. These

include the voltage drops across the flyback diode and gate transistors, as well as the electrical trigger

pulse width and the lead angle of the electrical phases.

Table 4.1.1: General Machine Properties of the Halbach Array

Motor in ANSYS Maxwell RMxprt.

Name Value

Machine Type Brushless Permanent-Magnet

DC Motor

Number of Poles 42

Rotor Position Outer Rotor

Frictional Loss 10 W

Windage Loss 10 W

Reference Speed 10000 rpm

Control Type Direct Current

Circuit Type Delta-Type 3-Phase Winding

Next, the physical properties and dimensions of the stator and rotor are detailed. Within both the stator

and the rotor there will be laminated sheets stacked evenly in the axial direction separating the sheets of

metal. These lamination layers are used to prevent the magnetic flux from traveling in the axial directions

throughout the electric motor, reducing the loss they experience. The effect of these laminations can be

approximated by a stacking factor, which can be calculated using the formula in EQ 3.2.8. With a

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lamination thickness of 0.0242 millimeters the resultant stacking factor for both the stator and rotor is

0.728. The other parameter that will be universal to both the stator and rotor is the material used. For this

work, a standard iron has been chosen, whose properties are laid out in Table 4.1.3. The last property of

note which belongs to the stator is the skew width. This skew width spirals the armatures around the axis

in the axial direction, and can be incorporated into electric machines to reduce cogging torque. No

armature skewing will be used for this motor. The full tables of parameter values for the stator and rotor

can be found below, titled Table 4.1.4 and Table 4.1.5 respectively.

Table 4.1.2: Circuitry Properties of the Halbach Array Motor in

ANSYS Maxwell RMxprt.

Name Value

Lead Angle of Trigger 0 degrees

Trigger Pulse Width 120 degrees

Transistor Drop 2 V

Diode Drop 2 V

Table 4.1.3: Material Properties of Iron in the Halbach Array

Motor in ANSYS Maxwell RMxprt.

Name Value

Relative Permeability 4000

Bulk Conductivity 10.3 MS/m

Magnitude of Magnetic

Coercivity 0

Thermal Conductivity 79 W/m/C

Mass Density 7870 kg/m3

Specific Heat 447 J/kg/C

33

Table 4.1.4: Stator Properties of the Halbach Array Motor in

ANSYS Maxwell RMxprt.

Name Value

Outer Diameter 120 mm

Inner Diameter 90 mm

Length 5 mm

Stacking Factor 0.728

Steel Type Iron

Number of Slots 36

Slot Type 4

Skew Width 0

Table 4.1.5: Rotor Properties of the Halbach Array Motor in

ANSYS Maxwell RMxprt.

Name Value

Outer Diameter 128.25 mm

Inner Diameter 120.25 mm

Length 5 mm

Steel Type Iron

Stacking Factor 0.728

Pole Type Outrun

The slot dimensions are entered next. RMxprt provides the users with a selection of slot types from which

to base the final slot designs. As identified in Table 4.1.4, the slot type 4 is selected. An image of this slot

is shown in Figure 4.1.1. The values for its physical parameters are shown in Table 4.1.6.

34

Figure 4.1.1: Schematic of Slot Type 4 Base Model as Available in Maxwell RMxprt Detailed with all

Configurable Dimensions [111].

Table 4.1.6: Armature Slot Properties of the Halbach Array

Motor in ANSYS Maxwell RMxprt.

Name Value

Auto Design Not Selected

Parallel Tooth Not Selected

Hs0 1.11 mm

Hs1 0.63 mm

Hs2 10.0 mm

Bs0 2.37 mm

Bs1 6.32 mm

Bs2 4.73 mm

Rs 0.95 mm

The windings of the motor contain many properties that need to be specified, first regarding their general

composition followed by more minute dimensions. Starting with the general properties, a winding layer

of two is chosen, meaning each slot between armatures has two winding coils passing through them.

Double layer windings are the most common configuration in all motors, and are invariably used in DC

motors [112]. A whole-coiled winding pattern is used so that one phase can consecutively wrap around

35

two adjacent coils, and a pitch of one is selected meaning each winding wraps around only one stator arm.

Figure 4.1.2 illustrates the winding pattern, while Table 4.1.7 provides all of the general winding

properties. The specific dimensional properties of the windings are all tabulated in Table 4.1.8, and the

property explanations can be found in the ANSYS Maxwell Help Guide for reference [111].

Figure 4.1.2: Winding Configuration Throughout the 36 Armature Slots of the Halbach Motor Generated

in Maxwell RMxprt.

The magnet pole properties are to follow. To allow for the inclusion of the circumferentially aligned

halbach magnets during the two-dimensional modeling some room must be allocated between the radially

aligned magnets. This is accounted for with the embrace parameter, which takes a value of zero to one to

determine which percentage of its angular range each magnet occupies. Using an embrace of two thirds

results in the radial magnets being twice as wide as the circumferential magnets. The chosen magnet type

is a neodymium iron boron magnet of composition N50M, chosen both for its high magnetic coercivity

and ability to function at high heats [113]. Its data can be found in Table 4.1.9, followed by the simulation

magnet dimensions in Table 4.1.10.

36

Table 4.1.7: General Winding Properties of the Halbach Array

Motor in ANSYS Maxwell RMxprt.

Name Value

Winding Layers 2

Winding Type Whole-Coiled

Parallel Branches 1

Conductors per Slot 15

Coil Pitch 1

Number of Strands 1

Wire Wrap 0.01 mm

Wire Gauge 22

Wire Diameter 0.643 mm

Table 4.1.8: Winding End/Insulation Properties of Halbach

Array Motor in ANSYS Maxwell RMxprt.

Name Value

Input Half-turn Length Not Selected

End Extension 1.0 mm

Base Inner Radius 0.3 mm