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A cogging-torque-assisted motor drive for internal combustion engine valves Reinholz, Bradley 2016

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A Cogging-Torque-Assisted Motor Drive for Internal Combustion Engine Valves  by  Bradley Reinholz  B.A.Sc., The University of British Columbia, 2014  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  in  THE COLLEGE OF GRADUATE STUDIES  (Electrical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA   (Okanagan)  March 2016   © Bradley Reinholz, 2016         The undersigned certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis entitled:         A Cogging-Torque-Assisted Motor Drive for Internal Combustion Engine Valves  Submitted by                     Bradley Reinholz                    in partial fulfillment of the requirements of   The degree of       Master of Applied Science                                              .  Dr. Rudolf Seethaler, School of Engineering Supervisor, Professor (please print name and faculty/school above the line)  Dr. Wilson Eberle, School of Engineering Supervisory Committee Member, Professor (please print name and faculty/school in the line above)  Dr. Yang Cao, School of Engineering Supervisory Committee Member, Professor (please print name and faculty/school in the line above)  Dr. Homayoun Najjaran, School of Engineering University Examiner, Professor (please print name and faculty/school in the line above)   External Examiner, Professor (please print name and university in the line above)   March 23, 2016 (Date submitted to Grad Studies)       Abstract  Internal combustion engine valve trains form the interface between the intake and exhaust systems and largely contribute to the overall engine performance, emissions and efficiency. Most modern engines use a camshaft to operate the valve train, but suffer from suboptimal performance since valve events cannot be dynamically altered during transient engine operation. Electromechanical valve actuation is a type of variable valve actuation that uses electromechanical actuators to replace the camshaft in an internal combustion engine. Electromechanical valve actuation promises to improve engine performance, reduce fuel consumption and lower harmful emissions by allowing for fully-independent control of the intake and exhaust valves. A major goal of electromechanical valve actuation is to achieve fully-independent valve control while minimizing the impact on concomitant systems, such as the charging and cooling systems. A new type of electromagnetic actuator is presented in this thesis, which uses cogging torque to recover kinetic energy in the form of a magnetic field. Cogging torque allows the presented actuator to be much more efficient and compact compared to other electromechanical valve actuators that use external mechanical spring systems. To utilize cogging torque effectively, design motivations are initially established and used to conceptualize a practical and efficient design. The proposed design is first simulated to predict its performance and later is experimentally validated using a fabricated prototype. The results of the experiments reveal a highly efficient and fast actuator design compared to other electromechanical actuators found in literature. The energy loss is further reduced by generating an optimal kinematic trajectory using the Nelder-Mead algorithm. The optimal kinematic trajectory enabled the proposed actuator design to be the most efficient electromechanical valve actuator found in literature. The results presented show the novel actuator design reduced losses by over 40% when compared with the most efficient electromechanical valve actuator published in literature and by over 70% when compared with a conventional camshaft. The conclusions of this thesis suggest a cogging-torque-assisted actuator could be feasibly retrofitted into an existing engine with only minor modifications due to its compact and highly efficient nature.    ii Preface  All work presented in this thesis was performed under the supervision of Dr. Rudolf Seethaler at the School of Engineering within the University of British Columbia.   Chapters 2, 3 and 4 are primarily based off the work published in B. Reinholz and R. Seethaler, “A cogging torque assisted motor driven valve actuation system for internal combustion engines,” presented at the Int. Des. Eng. Tech. Conf. Comput. Inf. Eng. Conf., Portland, OR, USA, 2013. For this paper, I worked collaboratively with Dr. Seethaler to derive the actuator model and Simulink simulation. I also created the FEA and wrote the first draft. Lastly, I presented this paper at the ASME conference in Portland, Oregon.   Chapter 5 and chapter 6 includes published works from B.A. Reinholz, R.J. Seethaler, "Experimental Validation of a Cogging Torque Assisted Valve Actuation System for Internal Combustion Engines," Mechatronics, IEEE/ASME Transactions on, vol.PP, no.99, pp.1,1, May. 2015. In this journal article I fabricated the actuator and performed all of the experiments. I also wrote the first draft of the article.   Section 6.1 and chapter 7 are based off the works in B. A. Reinholz, L. Reinholz and R. J. Seethaler, “Optimal Trajectory Planning for a Cogging Torque Assisted Motor Driven Valve Actuator for Internal Combustion Engines,” Industrial Electronics, IEEE Transactions on, Submitted Sept. 10. In this journal article, I applied the Nelder-Mead algorithm through the fminsearch function built in MATLAB to generate optimal kinematic trajectories. Lindsey Reinholz, a PhD candidate in Pure Mathmatics at the University of British Columbia, contributed to this paper by determining the application criteria for the Nelder-Mead algorithm. She also composed a subsection describing the mathematics behind the Nelder-Mead algorithm. I composed all other sections in the paper.       iii Table of Contents  Abstract .................................................................................................................................... ii Preface ..................................................................................................................................... iii Table of Contents ................................................................................................................... iv List of Tables .......................................................................................................................... vi List of Figures ........................................................................................................................ vii List of Symbols ........................................................................................................................ x List of Abbreviations ............................................................................................................ xii Acknowledgements .............................................................................................................. xiii Dedication ............................................................................................................................. xiv Chapter 1 Introduction........................................................................................................... 1 1.1 Background and Motivations ................................................................................................ 1 1.2 Electromechanical Valve Actuation ...................................................................................... 3 1.2.1 Spring-assisted Solenoid Drive ......................................................................................... 3 1.2.2 Motor Drive ...................................................................................................................... 4 1.2.3 Spring-assisted Motor Drive ............................................................................................. 5 1.2.4 Spring-assisted Voice Coil ............................................................................................... 6 1.3 Scope of Thesis ..................................................................................................................... 7 Chapter 2 Principles of Cogging Torque for EVA .............................................................. 9 2.1 Motor Cogging Torque ......................................................................................................... 9 2.2 Strategically Utilizing Cogging Torque for Electromechanical Valve Actuation .............. 10 Chapter 3 Design of a Cogging-Torque-Assisted Motor Drive ........................................ 14 3.1 Electromechanical Valve Actuation Constraints ................................................................ 14 3.2 Actuator Design .................................................................................................................. 14 3.2.1 Rotor and Excenter Arm Design ..................................................................................... 15 3.2.2 Stator Design .................................................................................................................. 18 3.2.3 Winding Design .............................................................................................................. 20 3.3 Finite Element Analysis ...................................................................................................... 22  iv 3.3.1 Material Specification ..................................................................................................... 22 3.3.2 Finite Element Analysis Results ..................................................................................... 23 3.4 Predicted Specifications of the Proposed CTAMD Design ................................................ 24 Chapter 4 Simulation of a CTAMD .................................................................................... 27 4.1 Current Controller ............................................................................................................... 29 4.2 Position Controller .............................................................................................................. 30 4.3 Valve Dynamics and Exhaust Pressure Modeling .............................................................. 31 4.4 Initial Trajectory Design ..................................................................................................... 32 4.5 Simulated Position Control ................................................................................................. 35 Chapter 5 Fabrication and Characterization of a Prototype CTAMD System .............. 40 5.1 Fabrication of a Prototype CTAMD Actuator and Testbed ................................................ 40 5.1.1 Fabrication of a CTAMD Actuator ................................................................................. 40 5.1.2 Testbed and Data Acquisition Hardware ........................................................................ 42 5.2 Actuator Characterization ................................................................................................... 43 5.2.1 Open-Circuit Analysis .................................................................................................... 44 5.2.2 Closed-Circuit Analysis .................................................................................................. 46 5.3 Prototype CTAMD Specifications ...................................................................................... 49 Chapter 6 Experimental Validation of the CTAMD System ............................................ 51 6.1 Position Control Experiments ............................................................................................. 51 6.2 Position Control Experiments using Heuristic Current-Shaping Techniques ..................... 55 Chapter 7 Kinematic Trajectory Optimization ................................................................. 60 7.1 Cost Function Development and Application of the Nelder-Mead Algorithm ................... 60 7.2 Optimized Reference Trajectories....................................................................................... 62 7.3 Experimental Results Using the Optimized Trajectory....................................................... 65 Chapter 8 Conclusion ........................................................................................................... 70 References .............................................................................................................................. 72 Appendices ............................................................................................................................. 79 Appendix A: Relationship between Number of Magnetic Poles, Motor Efficiency and Maximum Valve Acceleration ........................................................................................................................... 79   v List of Tables  Table 3.1 Magnetic Pole Selection ......................................................................................... 17 Table 3.2 Predicted CTAMD System Specifications ............................................................. 26   Table 5.1 Simulated and Experimental Specifications Comparison ....................................... 49  vi List of Figures  Figure 1.1 Diagram of a SASD [14]. ........................................................................................ 4 Figure 1.2 Diagram of a MD [14]. ............................................................................................ 5 Figure 1.3 A diagram of a SAMD [18]. .................................................................................... 6 Figure 1.4 Diagram of a SAVC [22]. ........................................................................................ 7  Figure 2.1 A 360 electrical degree region of a single-phase permanent magnet motor with an equal number of magnetic poles and stator teeth. .....................................10 Figure 2.2 Output torque generation resulting from the summation of cogging torque and electrical torque over 180 electrical degrees. The motor snapshots below the graph show the location of the green permanent magnet at various points during the transition. ..............................................................................................12  Figure 3.1 Parameters used for modeling the valve dynamics of the CTAMD [26]. ............. 16 Figure 3.2 Proposed rotor geometry. ...................................................................................... 18 Figure 3.3 Proposed stator design. .......................................................................................... 19 Figure 3.4 Flux density predicted by Maxwell SV when 100A are applied to the windings [26]. ........................................................................................................24 Figure 3.5 Geometry of the proposed CTAMD [35]. ............................................................. 25  Figure 4.1 Cascaded controller. .............................................................................................. 28 Figure 4.2 Electrical model used to describe the CTAMD. .................................................... 29 Figure 4.3 The reference angular acceleration, velocity and position reference trajectories for a 3.5ms transition time [14], [26]. .................................................34 Figure 4.4 The position tracking performance of the simulated CTAMD system when opened against exhaust pressure [26]. ...................................................................35 Figure 4.5 The ohmic energy loss incurred by the CTAMD when compensating for exhaust pressure for a 3.5ms transition with 8mm of valve lift [26]. ....................36  vii Figure 4.6 A comparison of ohmic energy loss over a range of transitions times with various EVA systems published in literature in the absence of exhaust pressure [14], [21], [22], [26], [28]. .......................................................................38 Figure 4.7 Ohmic power loss comparison when only considering the intake valves of a 4-cylinder, 16-valve IC engine at 6000RPM [14], [21], [22], [26], [28]. ..............39  Figure 5.1 Diagram of the wound CTAMD............................................................................ 41 Figure 5.2 Finished CTAMD prototype [37]. ......................................................................... 42 Figure 5.3 Diagram of the experimental setup........................................................................ 43 Figure 5.4 A comparison of the modeled and experimental actuator generator constant [39]. ........................................................................................................................45 Figure 5.5 A comparison between the modeled and experimental cogging torque [39]. ....... 46 Figure 5.6 Measured and averaged resistance of the prototype CTAMD. ............................. 47 Figure 5.7 The measured and averaged inductance of the prototype CTAMD. ..................... 48 Figure 5.8 Ohmic power loss comparison when only considering the intake valves of a 4-cylinder, 16-valve IC engine at 6000RPM [14], [21], [22], [26], [28], [39]. .....50  Figure 6.1 Prototype CTAMD position tracking performance for a 3.5ms transition time. The vertical dashed lines indicate 5% of total valve lift and 95% of total valve lift [35]. ................................................................................................52 Figure 6.2 Prototype CTAMD velocity tracking performance for a 3.5ms transition time. The vertical dashed lines indicate 5% of total valve lift and 95% of total valve lift [35]. ................................................................................................53 Figure 6.3 Prototype CTAMD current tracking performance for a 3.5ms transition time. The vertical dashed lines indicate 5% of total valve lift and 95% of total valve lift [35]..........................................................................................................54 Figure 6.4 Ohmic power loss comparison when only considering the intake valves of a 4-cylinder, 16-valve IC engine at 6000RPM [14], [21], [22], [26], [28], [35], [39]. ........................................................................................................................55 Figure 6.5 Current response of the prototype CTAMD after heuristic current-shaping for a 3.5ms transition time [39]..............................................................................56  viii Figure 6.6 Position tracking performance of the prototype CTAMD after heuristic current-shaping for a 3.5ms transition time [39]. ..................................................57 Figure 6.7 A comparison of the CTAMD ohmic loss with various EVA systems in literature over a range of transition times [14], [21], [22], [26], [28], [35], [39]. ........................................................................................................................58 Figure 6.8 Torque profiles of the simulated and prototype CTAMD. .....................................59  Figure 7.1 Optimized valve trajectory using the Nelder-Mead algorithm with 21 points. ......62 Figure 7.2 A comparison of the optimized and unoptimized trajectories for a 3.5ms transition time [35]. ................................................................................................63 Figure 7.3 Optimal current profile for a 3.5ms transition time [35]. .......................................64 Figure 7.4 Reference position tracking performance of the prototype CTAMD using an optimized kinematic trajectory for a 3.5ms transition time. The vertical dashed lines indicate 5% of total valve lift and 95% of total valve lift [35]. .........65 Figure 7.5 Reference velocity tracking performance of the prototype CTAMD using an optimized kinematic trajectory for a 3.5ms transition time. The vertical dashed lines indicate 5% of total valve lift and 95% of total valve lift [35]. .........66 Figure 7.6 Reference and measured current profiles using optimal 3.5ms kinematic trajectories [35]. .....................................................................................................67 Figure 7.7 A comparison of the CTAMD power loss when using the optimized and unoptimized trajectories on a 4-cylinder, 16-valve IC engine operating at 6000RPM when only considering the intake valves [35], [39]. ............................68 Figure 7.8 Ohmic power loss comparison when only considering the intake valves of a 4-cylinder, 16-valve IC engine at 6000RPM when only considering the intake valves [14], [21], [22], [28], [35]. ...............................................................69        ix List of Symbols  Symbol  Description 𝐴𝑔𝑔   Area of the winding gap  𝐴𝑐   Cross-sectional area of the air gap 𝐴𝑠   Stator slot area 𝐵𝑚𝑚   Magnet residual flux density 𝐵𝑠𝑎𝑡   Saturation flux density 𝐶𝑓   Winding fill factor 𝐸   Energy loss per transition 𝐸𝑚𝑚   Mechanical energy 𝐹𝑔𝑔   Force created by exhaust back-pressure 𝐾𝐾𝑐   Cogging torque constant  𝐾𝐾𝑔𝑔   Generator constant  𝐾𝐾𝑖𝑖   Current controller integral gain 𝐾𝐾𝑖𝑝   Current controller proportional gain 𝐾𝐾𝑝𝑑   Position controller derivative gain 𝐾𝐾𝑝𝑝   Position controller proportional gain 𝐾𝐾𝑡   Electrical torque constant 𝐼   Winding current 𝐼𝑟𝑒𝑓   Reference winding current 𝐽𝐿   Inertia of the load 𝐽𝑟   Inertia of the rotor 𝐿𝐿   Winding inductance 𝑙   Length of the rotor 𝑙𝑐   Length of the connecting rod 𝑙𝑔𝑔   Length of the air gap 𝑙ℎ𝑡   Winding half-turn length 𝑚   Mass of the valve 𝑁   Number of windings per magnetic pole pair  x 𝑁𝑃𝑃𝑅   Number of pulses per revolution for an encoder 𝑛𝑝   Number of magnetic poles on a rotor 𝑃   Ohmic power loss 𝑃𝑚𝑚   Mechanical power 𝑅𝑅   Winding resistance  𝑟   Rotor outer radius 𝑟𝑎   Excenter arm length  𝑇𝑐   Cogging torque 𝑇𝑒   Electrical torque 𝑇𝑔𝑔   Exhaust pressure disturbance torque 𝑇𝑜   Output torque 𝑇𝑟𝑒𝑞   Required torque 𝑡   Time 𝑡𝑓   Total transition time 𝑈𝑈   Supply voltage 𝑦   Valve displacement  𝜃𝜃𝑒   Rotor position in electrical degrees  𝜃𝜃𝑚𝑚   Rotor position in mechanical degrees 𝜃𝜃𝑟𝑒𝑓   Reference angular position in mechanical degrees  𝜌   Density of steel 𝜌𝑐   Resistivity of copper 𝜇   Relative permeability 𝜔𝑖   Current controller bandwidth 𝜔𝑝   Position controller bandwidth 𝜉𝑖   Current controller damping ratio 𝜉𝑝   Position controller damping ratio     xi List of Abbreviations  Abbreviation  Definition CTAMD  Cogging-torque-assisted motor drive EVA   Electromechanical valve actuation  IC   Internal combustion MD   Motor drive PD   Proportional-plus-derivative PI   Proportional-plus-integral PWM   Pulse width modulation SAMD   Spring-assisted motor drive SASD   Spring-assisted solenoid drive SAVC   Spring-assisted voice coil VVA   Variable valve actuation   xii Acknowledgements  I would like to offer my sincerest gratitude to my supervisor, Dr. Rudolf Seethaler, for his guidance, encouragement and support. He has been vital to my success and has imparted me with invaluable knowledge and experiences throughout my research. He has been truly influential in my life and I am profoundly grateful for the opportunities he has provided me with.   I extend my thanks to my committee members, Dr. Wilson Eberle, and Dr. Yang Cao for their time, effort and instruction. I would also like to thank Dr. Homayoun Najjaran, Dr. Jahangir Hossain, Dr. Richard Klukas, Dr. Yannacopoulos and Malcolm Metcalfe for fostering my interest in engineering and contributing to my success.   I would like to thank NSERC for offering me USRAs and a CGSM. The financial support of the awards enabled me to focus on my research, and motivated me to continue my academic pursuits.    I extend my thanks to my friends and colleagues that I have met as a student and I am grateful for the positive experiences they have provided me with. Finally, I would like to thank my family who has supported me throughout my entire life. Their encouragement and guidance have shaped my life and career.       xiii Dedication   This thesis is dedicated to my family   xiv Chapter 1 Introduction The aspiration and expiration of the combustion chambers in an internal combustion (IC) engine are controlled by intake and exhaust valves. In most overhead cam engines, the intake and exhaust valves are driven by a camshaft that is coupled to the crankshaft using a timing belt or chain. The coupled nature of the camshaft and crankshaft permit only a single timing for the opening and closing valve events. Furthermore, conventional camshafts have a fixed cam lobe geometry that allows only a single lift profile to be obtained. These constraints only enable the timing and lift profiles of the intake and exhaust valves to be optimized for a specific load and speed. Typically, IC engines perform through a transient range of conditions and therefore engine designers must compromise between performance, fuel economy and emissions when designing the valve train.  Electromechanical valve actuation (EVA) is an established research area that aims to replace the function of a camshaft with electromagnetic actuators. Over the past few decades, progress has been made that improves the feasibility of EVA systems being integrated into mass-produced IC engines. This thesis proposes the use of a novel cogging-torque-assisted motor drive (CTAMD) to address issues encountered by previously developed EVA actuators. The following section conducts a review of advances and challenges encountered throughout the developmental history of EVA systems as well as motivations for current research.   1.1 Background and Motivations The transportation of freight and people in developed societies continues to be dominantly facilitated by conventional IC engines. Despite a growing interest in alternative fuel vehicles, IC engines continue to thrive due to superior robustness, fuelling infrastructure, and convenience. However, increasingly strict emissions standards and demands for improved fuel efficiency are driving the evolution of IC engines. The valve train of an IC engine forms the interface between the intake and exhaust systems and plays a critical role in emissions, performance and fuel efficiency. Unfortunately, the synchronization of the crankshaft with a fixed-geometry camshaft leads to substantial losses in automotive engines, which perform over a range of speeds and loading conditions. Intuitively, allowing variable valve event  1 timings and lift profiles would enable optimal valve actuation to be achieved throughout the entire range of transient engine conditions.   Variable valve actuation (VVA) is a general term used to describe valve trains that have the capability to alter the opening and closing events of engine valves. The degree of variability ranges from basic switchable two-profile camshafts to fully-independent valve control. Currently, economic and robust fully-independent valve control has yet to be demonstrated and implemented in a mass-produced IC engine. Some automobile companies have developed and marketed VVA systems with limited capabilities, such as BMW’s Vanos and Valvetronic valve trains or Honda’s VTEC valve train [1]-[3]. While limited VVA techniques, such as cam phasing or continuously adjustable valve lift, can be combined to offer more comprehensive VVA, overall these systems are expensive and still lack many of the capabilities and benefits an independent valve control system would exhibit. Furthermore, limited VVA systems still require a camshaft, and therefore still incur the frictional losses associated with it. Lastly, it is important to note that these limited systems react to transient engine conditions slowly, and therefore are only effective at improving performance when the engine is operating at steady state.   Methods for achieving fully-independent valve control use isolated actuators on each engine valve. Typically, the actuators can either be classified as electrohydraulic, electropneumatic or electromechanical. Independent valve actuation systems have been investigated by companies such as Sturman and Freevalve [4], [5]. While these commercially developed systems have been tested in working engines, they have been unable to enter production due to numerous challenges including cost and a large valve train energy consumption [6].   Despite considerable challenges, the development of independent valve control techniques continues to be researched due to the anticipated advantages. If fully-independent valve control is achieved, the camshaft, cam sensor, throttle body and timing belt or chain could be removed from a conventional engine. Independent valve control would reduce throttling losses and allow new operating and combustion strategies to be implemented, such as advanced cylinder deactivation and homogeneous charge compression ignition [7]. These  2 improvements would enable IC engines to benefit from increased performance, reduced emissions and up to a 20% reduction in fuel consumption [8], [9].   1.2 Electromechanical Valve Actuation EVA is a developing area of VVA that promises to enable independent valve control. Typical EVA systems propose to use an individual electromagnetic actuator for each intake and exhaust valve. In literature, the electromagnetic actuators are often solenoids, motors, and voice coils. While many EVA systems have been demonstrated and validated in literature, these systems have various issues that make them impractical or infeasible for implementation in a modern IC engine. The following subsections review various EVA systems shown in literature while noting the benefits and disadvantages of each.   1.2.1 Spring-assisted Solenoid Drive Spring-assisted solenoid drives (SASD) have been investigated numerous times in literature [10]-[13]. SASDs typically use two solenoids that are individually operated to pull the valve open and closed. Furthermore, valve springs are often attached to the valve which provide a method for kinetic energy recovery and increased initial opening and closing accelerations. A common SASD geometry is shown in Fig. 1.1. SASD designs often boast high valve transition energy efficiencies and often improve upon the frictional losses of a conventional camshaft [13]. Furthermore, SASD systems have a proven capability to operate against exhaust gas pressure, making them practical for both intake and exhaust valves [13].  While SASD systems have notable advantageous attributes, they also have many disadvantages and challenges that make them impractical for typical IC engines. SASDs have nonlinear force constants that make them inherently difficult to control [12]. Furthermore, because the solenoids are only able to pull the armature, if the armature moves too quickly, the solenoid will not be able to slow the armature down effectively [13]. In addition to control difficulties, SASDs expend a large amount of energy when holding the valve in the open or closed position, since electromagnetic force must counteract the valve spring force [13]. Lastly, SASDs often have trouble for compensating for valve wear as many designs require the valve lash adjuster to be removed [13].  3  Figure 1.1 Diagram of a SASD [14].   1.2.2 Motor Drive Motor drives (MD) are electromagnetic motors that use a rocker arm or excenter arm to convert rotary armature motion into linear valve motion. MD setups often have no valve springs or use the traditional valve springs in a manner that does not directly assist the actuator [14]-[16]. MDs are usually permanent magnet brushless DC motors due to high energy efficiency and torque [14], [15]. A diagram of a typical MD geometry is shown in Fig. 1.2. MDs are often mechanically simpler than other types of EVA actuators as they do not require additional external spring setups. The mechanical simplicity enables MDs to be compact in size, allowing them to fit more elegantly under the valve cover of an internal combustion engine. Furthermore, Zhao and Seethaler proved that a carefully selected off-the-shelf motor could be used for EVA without requiring a specialized custom design [14].  Since most MD designs lack kinetic energy recovery valve springs, they often have large energy consumptions that make them impractical compared to a camshaft [14]. Furthermore, robust exhaust pressure compensation has yet to be demonstrated by a MD [14], [15]. Without a spring-assist, MD’s require large electromagnetic torque to be generated quickly, which is challenging since winding inductance often leads to a sluggish acceleration [14].   Opening Spring Armature Opening Magnet Closing Spring Valve Cylinder Head Closing Magnet  4   Figure 1.2 Diagram of a MD [14].  1.2.3 Spring-assisted Motor Drive Spring-assisted motor drives (SAMD) have been developed in an effort to improve upon the energy efficiency of MDs, while providing improved controllability compared to SASDs [17]-[21]. SAMDs typically use a motor driven disc cam to transform rotary armature motion into linear valve motion [18]. SAMDs also use external spring systems that strategically assist the actuator open and close [18]. A diagram of a typical SAMD geometry is seen in Fig. 1.3. SAMDs typically boast the highest energy efficiency out of all other EVA actuator types [21]. Furthermore, the roller slot in the disk cam can be strategically cut to allow the valve to lock in its open and closed positions without requiring continuous energy expenditure unlike SASDs [18].   While SAMDs boast high energy efficiencies, their application is inevitably limited by their size. The need for a disk cam and external spring setup make the overall size of the SAMD too large to feasibly fit under the valve cover of an internal combustion engine [21]. Furthermore, compared to a MD or SASD actuator, SAMDs are mechanically complex, which make them more difficult to design and fabricate [21].  Valve Cylinder Head BLDC Motor  Excenter Arm  Connecting Rod   5   Figure 1.3 A diagram of a SAMD [18].   1.2.4 Spring-assisted Voice Coil Spring-assisted voice coils (SAVC) is a recently proposed EVA method [22]. Voice coil actuators are known for their small moving mass and inductance, allowing them to be very fast and ideal for applications such as audio speakers [22]. Unlike solenoids, voice coils have permanent magnets and are able to both push and pull on an armature. Naturally, voice coils have improved controllability over solenoids and much faster response times [22]. While voice coil actuators do not require springs, a complex spring setup was added to the design in [22] to allow for kinetic energy recovery. The specialized spring setup also allows the valve to be held in the open and closed positions unstably with no energy expenditure. A diagram of the SAVC used in [22] is shown in Fig. 1.4. The SAVC system not only demonstrated some of the fasted transition times in literature, but was able to handle exhaust gas pressure while operating on a single cylinder engine [22].   While the SAVC demonstrated very robust EVA actuation, it has one of the worst energy efficiencies in literature [22]. Even when compared with a MD, which has no kinetic energy recovery, the SAVC produced substantially more ohmic loss [14], [22]. The substantial ohmic loss also created serious heating issues that required the SAVC to use forced-air  Opening Spring Closing Spring Valve Cylinder Head    Motor Driven Disc Cam Roller  6 cooling by having the actuator plumbed in-line with the intake manifold [22]. Naturally, this would heat the air up entering the engine, reducing the air density and degrading engine performance. Due to the excessive ohmic loss, the SAVC is impractical to implement into a conventional internal combustion engine.    Figure 1.4 Diagram of a SAVC [22].  1.3 Scope of Thesis This thesis presents a novel CTAMD system. The following chapters address the modeling, design, fabrication, characterization and optimization of the CTAMD. In chapter 2, the principle of cogging torque is discussed including an explanation of how cogging torque is useful for EVA. In chapter 3, design constraints are noted and a feasible CTAMD design is generated using analytic and finite element analysis modeling techniques. In chapter 4, a Simulink simulation is developed to simulate the expected performance of the proposed CTAMD design. In chapter 5, the fabrication of the proposed actuator and an experimental test bed are discussed. Also in chapter 5, characterization experiments are designed and the results of these experiments are used validate the accuracy of the analytic and FEA models.  7 In chapter 6, the results of experimental tests using the finished prototype CTAMD system are discussed and compared to the simulated system and other EVA methods in literature. In chapter 7, an optimized kinematic trajectory for a CTAMD is generated using the Nelder-Mead algorithm and the experimental CTAMD prototype is retested using the new trajectory. In chapter 8, the findings presented in this thesis are summarized and directions for future research are established.                            8 Chapter 2 Principles of Cogging Torque for EVA This chapter defines cogging torque and explains how it can be utilized for EVA by enabling the kinetic energy of engine valves to be recovered. The discussion presented in the following sections highlights the superior performance of a cogging-torque-assist compared to a spring-assist.   2.1 Motor Cogging Torque Cogging torque is a characteristic parameter for permanent magnet motors that describes the force of attraction between the metal stator teeth and the permanent magnets on the rotor. Cogging torque is approximately sinusoidal in nature and varies with angular rotor position. For a single-phase motor with equal numbers of magnetic poles and stator teeth, the cogging torque will have a period of 180 electrical degrees. Electrical degrees, 𝜃𝜃𝑒, are related to mechanical degrees, 𝜃𝜃𝑚𝑚, using (1), where 𝑛𝑝 is the number of magnetic poles.  𝜃𝜃𝑚𝑚 = 2𝑛𝑝 𝜃𝜃𝑒          (1)  360 electrical degrees of a motor as shown in Fig. 2.1, or translating two stator teeth, corresponds to the angular period of the electrical torque, which is twice as long as the angular period of the cogging torque. Therefore, cogging torque, 𝑇𝑐, has twice the angular frequency of the electrical torque,  𝑇𝑒, which can be noted in eq. 2 and 3 where 𝐾𝐾𝑐 is the cogging torque constant, 𝐾𝐾𝑡 is the electrical torque constant and 𝐼 is the current being supplied to the motor.   𝑇𝑐 = 𝐾𝐾𝑐 sin(2𝜃𝜃𝑒)         (2)  𝑇𝑒 = 𝐾𝐾𝑡𝐼 sin(𝜃𝜃𝑒)         (3)  Intuitively the output torque of the motor, 𝑇𝑜, is the summation of the electrical and cogging torques as indicated in eq. 4.    9 𝑇𝑜 = 𝑇𝑐 + 𝑇𝑒          (4)   Figure 2.1 A 360 electrical degree region of a single-phase permanent magnet motor with an equal number of magnetic poles and stator teeth.          2.2 Strategically Utilizing Cogging Torque for Electromechanical Valve Actuation When designing a motor for most applications, cogging torque is almost unanimously viewed as a determent to performance. The presence of cogging torque leads to vibrations, noise and control complications, all of which are not ideal for typical motor applications. Therefore, when motors are designed, techniques like stator lamination skewing or using electrical torque to suppress cogging torque ripple are implemented to negate the effect of cogging torque [23]-[25] . However, for EVA applications, MDs are operated as stepper motors and often rotate through small angles [14]. By examining Fig. 2.2 it can be proposed that designing a motor to operate through a small angle that is exactly 180 electrical degrees could benefit from higher peak output torque due to the presence of cogging torque. The N S S N Permanent Magnet Rotor Windings Stator Tooth Stator Slot  10 potential to use cogging torque to achieve larger opening and closing valve accelerations was previously noted by Mercorelli in [15]. However, Mercorelli did not recognize the potential to use cogging torque as the sole kinetic energy recovery mechanism that would remove the need for external valve springs. If designed properly, cogging torque can emulate the behaviour of mechanical spring systems used in the SASD, SAMD and SAVC designs by allowing for kinetic valve energy to be recovered in the form of a magnetic field. Using cogging torque instead of mechanical springs allows for more efficient energy recovery and a much more compact actuator size.    11  Figure 2.2 Output torque generation resulting from the summation of cogging torque and electrical torque over 180 electrical degrees. The motor snapshots below the graph show the location of the green permanent magnet at various points during the transition.   The characteristics of the torque profiles shown in Fig 2.2 are ideal for a CTAMD. To obtain these torque profiles, the rotor must start in the position relative to the stator shown in Fig. 2.1. In this position the magnets on the rotor are aligned with the midpoint of the stator slots. Start - Valve Closed Finish - Valve Open  12 This corresponds to a position of maximum electrical torque and minimum cogging torque. Having maximum electrical torque at the open and closed positions allows for fast accelerations and decelerations that improve exhaust pressure compensation and valve seating velocities. Furthermore, the starting position shown in Fig. 2.1 corresponds to a position of minimum cogging torque that allows the valve to be held in unstable equilibrium in its open and closed positions without any continuous energy expenditure.   Ideally, a CTAMD would have a large cogging torque capable of providing nearly all of the acceleration required to achieve a specified transition time. To obtain a large cogging torque, a motor should be designed with strong and thick magnets, thin and straight stator teeth, no laminate skew and be as long as possible [26]. However, given size constraints and electrical torque considerations, often design compromises need to be made. Theoretically, a CTAMD only requires a small amount of electrical torque to start a valve transition and compensate for losses if a standard valve lift and transition time are desired. However, if reduced valve lift, or very slow valve transitions are desired, the electrical torque must always be capable of exceeding the cogging torque. If the cogging torque ever exceeds the electrical torque, the CTAMD would enter a minimal-control state where the application of electrical torque would have minimal influence over a valve trajectory. Furthermore, in this minimal-control state, the rotor magnets have a risk of becoming trapped underneath the stator teeth in a zero-torque position.             13 Chapter 3 Design of a Cogging-Torque-Assisted Motor Drive This chapter proposes a practical CTAMD design generated using analytic modeling and FEA analysis. The constraints, modeling techniques, and design considerations are discussed in the sections below.   3.1 Electromechanical Valve Actuation Constraints In order for an EVA system to be considered both practical and feasible for application in a modern IC engine, it must at minimum exhibit performance characteristics that are comparable to an IC engine with a camshaft. Small passenger vehicles often are powered by a four-cylinder, four-stroke, 16-valve engine [9], [14]. These engines typically have a redline engine speed of 6000RPM while performing 8mm valve lifts [14]. If a minimum 6000 RPM redline engine speed is adhered to as a constraint, it will be necessary to design an EVA system capable of performing valve transitions under 3.5ms measured between 5% and 95% of total valve lift [18], [27]. Furthermore, exhaust valves will need to achieve a 3.5ms transition time in the presence of exhaust pressure which is often 7bar at full load [28]. Even if acceptable transition times are attained, in order for an EVA system to be considered practical, it will need to achieve comparable loss characteristics similar to a camshaft valve train. Typically, at redline engine speed, camshaft valve trains experience mechanical power losses in the range of 2-3kW [18]. Acoustic emissions and wear characteristics must also be considered when designing an EVA. Small valve seating velocities are required to reduce engine noise and avoid excessive valve wear. Typical camshaft engines have valve seating velocities of 0.3m/s at redline speed and 0.05m/s at idle [29], [30]. Lastly, it is important when designing an EVA system to consider ambient constraints that include engine temperature and limited space underneath the valve cover.   3.2 Actuator Design The following subsections propose a highly energy efficient actuator design for a CTAMD, while maintaining elements of practicality allowing a prototype to be fabricated with relative ease. The subsections are organized in a sequential procedure used to generate the actuator design.    14 3.2.1 Rotor and Excenter Arm Design When designing the rotor and excenter arm for a CTAMD it is important to first establish the number of phases to be used. Single-phase motors are the best choice for a compact CTAMD because they offer the highest peak torque for a given size actuator. The high peak torque allows the CTAMD to open against gas forces and accelerate the valve quickly from rest. Furthermore, single-phase drivers are simpler and more cost effective than three-phase drivers. However, unlike three-phase motors, single-phase motors have the inherent inability to self-start if the rotor magnets are aligned with the stator teeth and often require additional auxiliary coils to overcome this limitation [31]. Regardless of this limitation, a single-phase motor is selected for the proposed design.   After choosing how many phases the actuator will have, it is important to select a coupling mechanism that converts the rotary motion of the motor into linear valve motion. For the proposed design, a similar excenter arm to the one shown in Fig. 1.2 will be used. This particular excenter arm design is desirable due to its simplicity and because its connecting linkage can be designed to buckle in the event a CTAMD fails and a valve-piston collision occurs [14]. The dynamics of the excenter arm are provided in (5) where the parameters are defined in Fig. 3.1 [26].  𝑦 = −𝑙𝑐 + 𝑟𝑎 sin(𝜃𝜃𝑚𝑚) + �𝑙𝑐2 − 𝑟𝑎2�1 − 𝑐𝑜𝑠(𝜃𝜃𝑚𝑚)�2     (5)  If the rotor only rotates through a small angle when performing a full valve transitions, the following assumption can be made:  𝑙𝑐 ≈ �𝑙𝑐2 − 𝑟𝑎2�1 − 𝑐𝑜𝑠(𝜃𝜃𝑚𝑚)�2       (6)  This approximation allows the dynamics in (5) to be greatly simplified into (7).  𝑦 = 𝑟𝑎 sin(𝜃𝜃𝑚𝑚)          (7)   15  Figure 3.1 Parameters used for modeling the valve dynamics of the CTAMD [26].   If (1) is substituted into (7) and π electrical radians are desired, then (8) results.  𝑦 = 𝑟𝑎 sin �2𝜋𝑛𝑝�          (8)  Clearly, one must first select the number of magnetic poles to place on the rotor before the excenter arm length can be defined for an 8mm valve lift. It is seen in (1) that the more magnetic poles a rotor has, the smaller the mechanical angle the rotor will rotate through if it is constrained to rotate through 180 electrical degrees. Therefore, in order to invoke the small angle approximation used in (6), it is advantageous to design a motor with many poles. Furthermore appendix A shows that increasing the number of poles increases the maximum valve acceleration without having to make the stator physically larger. However, appendix A also shows that increasing the number of poles adversely impacts CTAMD efficiency. Furthermore, a motor with many poles is typically more difficult to fabricate than a motor  16 with fewer poles. Table 3.1 presents some examples of feasible magnetic pole and excenter arm combinations that result in 8mm valve lifts [26]. For the proposed design, 10 poles were selected as it enabled the motor to remain compact and efficient, while still providing adequate electrical torque to accelerate the valve.   Table 3.1 Magnetic Pole Selection np ra [mm] θmax [mech. deg] 6 8 ± 30 8 10.5 ± 22.5 10 12.9 ± 18 12 15.5 ± 15  Once the length of the excenter arm is determined, the radius of the rotor can be calculated. When designing a CTAMD for efficiency, it is important to match the inertia of the rotor, 𝐽𝑟 with the load, 𝐽𝐿. The load of the CTAMD system is the engine valve, which is approximated to be a 40g point-load, 𝑚, on the excenter arm. By matching inertias, the rotor radius, 𝑟, is approximated using (12) where 𝜌 is the density of the rotor and 𝑙 is the length of the rotor [26].  𝐽𝐿 = 𝑚𝑟𝑎2          (9)  𝐽𝑟 = 12  𝜌𝜋𝑟4𝑙          (10)  𝐽𝐿 = 𝐽𝑟           (11)  𝑟 = �2𝑚𝑚𝑟𝑎2𝜌𝜋𝑙4          (12)  For the proposed design, the rotor length, excluding output shafts, is prescribed to be 25mm, which corresponds to the diameter of a typical valve head and feasibly addresses motor size constraints. Using the aforementioned parameters and setting 𝜌 to be the density of steel, a radius of 12.1mm can be approximated. However, the rotor radius was rounded to 13mm to account for the mass of the excenter arm and connecting linkage. Using Maxwell SV, it was  17 iteratively found that thick magnets produced a larger cogging torque and electrical torque and therefore the magnet spans 5mm of the 13mm radius. A CAD model of the proposed rotor design is shown in Fig 3.2.   Figure 3.2 Proposed rotor geometry.   3.2.2 Stator Design Once the outer diameter of the rotor is specified, the inner diameter of the stator can then be chosen. For the proposed design, a relatively wide air gap of 0.75mm between the permanent magnets and the stator was selected. The air gap dictates the inner diameter of the stator to be 27.5mm. The large air gap allows for large fabrication tolerances at the expense of decreased performance. The length of the rotor also dictates the laminated length of the stator which is set to 25mm.  The outside dimensions of the stator are primarily limited by size constraints. For the proposed design, an unconventional square shape is used as it enables the stator teeth to be much more easily aligned with the valve for proper 180 electrical degree transitions. The square-shape also allows special holes to be drilled for dowel pins and bolts without Excenter Arm Output Shaft Permanent Magnets Rotary Encoder Output Shaft  18 interfering with the stator slots. To keep the actuator compact, while allowing adequate room for windings, the stator was constrained to a 60mm x 60mm square.   The stator slots and stator teeth were designed almost exclusively using Maxwell SV. To design the stator slots properly one should attempt to maximize the slot area and allow for thick windings, while ensuring the stator teeth and perimeter thicknesses remain large enough to carry peak magnetic flux without saturating. Furthermore, the stator teeth should be designed to maximize cogging torque using the principles aforementioned in chapter 2, which include using no skewing of the teeth and having thin tips on the teeth. Using these principles, an initial guess for optimum geometry was drawn in Maxwell SV. Afterwards, small iterative adjustments were made to the slot and tooth geometries until the cogging torque was maximized while maintaining adequate electromagnetic torque. The stator lamination geometry for the proposed design is shown in Fig. 3.3.   Figure 3.3 Proposed stator design.   19 3.2.3 Winding Design Before windings can be designed, it is important to establish maximum supply voltage and current constraints. 12V is the most common voltage in modern vehicles although 42V has been suggested as a future voltage standard [32]. When looking at the model for an inductor shown in (13), it is seen that a large voltage, 𝑈𝑈, is preferable when trying to achieve a fast current response, 𝐼,̇  for a given inductance, 𝐿𝐿.   𝑈𝑈 = 𝐿𝐿𝐼 ̇          (13)  While large voltages can be obtained through a vehicle charging system redesign or a solid state power electronic transformer, in order to maintain an elegant and cost effective implementation, the 42V future standard is used for the proposed design. Furthermore, to allow the use of cost-effective current driver and sensor hardware, a 100A current limit is proposed.   Before selecting the number of windings to use in an actuator, it is important to understand the effect windings have on the resistance, inductance, magnetic saturation and electrical torque of the motor. The electrical torque constant of a CTAMD is defined in (14) where 𝑁 is the number of windings per coil, 𝐵𝑚𝑚 is the residual flux density of the permanent magnets and 𝐴𝑔𝑔 is the area of the winding gap [33].  𝐾𝐾𝑡 = 𝑁𝐵𝑚𝑚𝐴𝑔𝑔𝑛𝑝         (14)  Substituting (3) into (14) yields (15).  𝑇𝑒 = 𝑁𝐵𝑚𝑚𝐴𝑔𝑔𝑛𝑝𝐼 sin(𝜃𝜃𝑒)        (15)  If the ohmic power loss, 𝑃, of a CTAMD is defined by (16), where 𝑅𝑅 is the winding resistance, then it is advantageous to maximize the electrical torque while minimizing power loss.    20 𝑃 = 𝐼2𝑅𝑅          (16)  Combining (15) and (16) yields (17).  𝑃 = 𝑇𝑒2𝑁2𝐵𝑚2𝐴𝑔2𝑛𝑝2 sin2(𝜃𝑒)𝑅𝑅        (17)  Examining (17) suggests that increasing the number of windings reduces power loss. However, since the stator has a fixed slot area, if the number of windings is increased, the resistance must increase to accommodate more windings as shown in (18) where 𝜌𝑐 is the resistivity of copper, 𝑙ℎ𝑡 is the half-turn length and 𝐶𝑓 is the winding fill factor [26].    𝑅𝑅 = 𝜌𝑐(𝑙+𝑙ℎ𝑡)𝑁2𝑛𝑝𝐴𝑠𝐶𝑓         (18)  If (18) is substituted into (17), it is seen that the number of windings has no impact on the ohmic power loss.  𝑃 = 𝑇𝑒2𝜌𝑐(𝑙+𝑙ℎ𝑡)𝐵𝑚2𝐴𝑔2𝑛𝑝 sin2(𝜃𝑒)𝐴𝑠𝐶𝑓        (19)  Concluding that the number of windings has no effect on ohmic energy loss allows the winding optimization to be simplified. Only magnetic saturation, peak torque, and inductance need to be considered. Ideally, the inductance of an actuator is minimized so that is has a very fast torque response. Low inductance is critical for EVA since short transition times require a very fast torque response.  The inductance of a coil is modeled in (20) where 𝐴𝑐 is the cross-sectional area of the air gap, 𝑙𝑔𝑔 is the length of the air gap and 𝜇 is the relative permeability [33].  𝐿𝐿 = 𝑁2𝜇𝐴𝑐𝑙𝑔          (20)   21 Clearly, the number of windings has a substantial effect on inductance and should be minimized while still being able to reach the point of saturation at peak current. The saturation magnetic flux density, 𝐵𝑠𝑎𝑡, is defined by (21) and is often 1.2T for amorphous silicon steel [33], [34].  𝐵𝑠𝑎𝑡 = 𝜇𝑁𝐼2𝑙𝑔           (21)  If a peak of 100A is applied to the windings with an air gap of 0.75mm assuming a 1.2T saturation flux density, then 15 windings are required to reach the point of saturation.  When selecting the gauge of wire to wind the stator with, it is important to first pick a winding fill factor. For the proposed design it was estimated that a winding fill factor of 70% would be possible. To achieve a fill factor of 70% for a slot area of approximately 125mm2, it was found that 15 windings of 12 gauge wire should be used. Furthermore, it was estimated that the half-turn length of the coils would be 12.5mm. Using (18), a winding resistance of 22mΩ was estimated.   3.3 Finite Element Analysis Predicting important performance specifications of a motor, such as inductance or the cogging torque, is often most accurately estimated using FEA. Analytical techniques often have difficulty accounting for complex geometries or high-order effects such as magnetic fringing. Therefore, many of the performance specifications for the proposed design are established using Maxwell SV. The following subsections discuss the proposed CTAMD material selection used in Maxwell SV and the motor specifications found from the FEA.   3.3.1 Material Specification FEA requires all components discussed in section 3.2 to have materials assigned to them. For the rotor, AISI 1018 steel was specified due to its low cost, machinability and moderate flux saturation properties. The permanent magnets were specified to be N42 neodymium magnets due to their large residual flux density and coercivity compared to other magnet types such as samarium-cobalt. The large residual flux density enables the CTAMD to have larger cogging  22 and electrical torques while the large coercivity prevents the magnet from demagnetizing if subjected to a large external magnetic field. For the stator, amorphous M19 silicon steel was specified as the material. Silicon steel was selected as its magnetic saturation density is large compared to standard carbon steel. Amorphous silicon steel was selected instead of crystalline silicon steel since it is preferable for motor stators that require the flux to arc for most of the flux path. Furthermore, a laminated stator design was chosen to suppress eddy current formation that would degrade motor performance. The stator windings were then chosen to be copper and occupy the equivalent area of 15 windings of 12 gauge wire.   3.3.2 Finite Element Analysis Results The magnetostatic solver in Maxwell SV was used to calculate the inductance, cogging torque constant and electrical torque constant. To solve for the electrical torque constant, the rotor was placed at -90 electrical degrees so that the electrical torque would be at its maximum while the cogging torque would be zero. A current of 100A was applied to the windings and the output torque, inductance and stator flux density were measured. Since -90 electrical degrees corresponds to a position of zero cogging torque, the measured output torque is equivalent to the electrical torque. The measured output torque was then divided by 100A to determine an electrical torque constant of 35.2mN∙m. The inductance was found to be 85𝜇H and the resulting stator flux density is shown in Fig. 3.4.   To find the cogging torque constant, the rotor was then rotated to -45 electrical degrees, which is the position of peak cogging torque. For this FEA simulation, the current was set to zero so that the output torque would only consist of cogging torque. After executing the FEA, a peak cogging torque of 1.31N∙m was measured which corresponds to the magnitude of the cogging torque constant.   23  Figure 3.4 Flux density predicted by Maxwell SV when 100A are applied to the windings [26].   3.4 Predicted Specifications of the Proposed CTAMD Design A CAD model of the proposed CTAMD design is displayed in Fig. 3.5. Clearly, the CTAMD resembles the simplistic design of the MD shown in Fig. 1.2. While similar, the presence of cogging torque allows the CTAMD to be physically smaller than a MD. Using analytical techniques and FEA software, the important performance specifications of the proposed CTAMD design are summarized in Table 3.2. These parameters are later validated with a physical prototype in Chapter 5.     24  Figure 3.5 Geometry of the proposed CTAMD [35].           Valve Connecting Rod  Excenter Arm  Rotor  Stator   25 Table 3.2 Predicted CTAMD System Specifications Parameter Quantity Number of Magnetic Poles 10 Resistance 22mΩ Inductance 85μH Electrical Torque Constant  35.3mN∙m/A Cogging Torque Constant 1.31N∙m Supply Voltage 42V Peak Current 100A Number of Windings Per Pole Pair 15 Rotor Length 25mm Rotor Radius 13mm Rotor Inertia 8.75kg∙mm2 Stator Length 25mm Stator Width 60mm Stator Height 60mm Excenter Arm Length 12.9mm Valve Mass 40g              26 Chapter 4 Simulation of a CTAMD The following chapter designs a simulation to predict the expected performance of the proposed CTAMD system. The chapter discusses the development of a cascaded PD-position controller, PI-current controller shown in Fig. 4.1. The cascaded controller is then augmented with a basic gas force model to emulate exhaust pressure acting against the exhaust valves. In the latter portion of the chapter, the results of the simulation are provided and discussed. 27  Figure 4.1 Cascaded controller.  28 4.1 Current Controller The proposed CTAMD current controller uses a proportional-plus-integral control law. The current controller forms the inner loop of the global cascaded control structure and is solved for first. The voltage equation (22) for the CTAMD is described by applying Kirchhoff’s voltage law to the circuit modeled in Fig. 4.2. It is important to note that the generator constant, 𝐾𝐾𝑔𝑔, is numerically identical to the electrical torque constant of the CTAMD.   𝑈𝑈 = 𝑅𝑅𝐼 + 𝐿𝐿𝐼̇ + 𝐾𝐾𝑔𝑔 sin(𝜃𝜃𝑒) ?̇?𝜃𝑚𝑚       (22)   Figure 4.2 Electrical model used to describe the CTAMD.   If the generator voltage is neglected, a second-order current controller can be expressed using (23) where 𝐾𝐾𝑖𝑝 and 𝐾𝐾𝑖𝑖 are the proportional and integral gains of the current controller respectively.   𝑈𝑈 = 𝐾𝐾𝑖𝑝�𝐼𝑟𝑒𝑓 − 𝐼� + 𝐾𝐾𝑖𝑖 ∫�𝐼𝑟𝑒𝑓 − 𝐼�𝑑𝑡      (23)  Equations (24) and (25) show how the current controller gains are calculated noting that 𝜔𝑖 is the current controller bandwidth and 𝜉𝑖 is the current controller damping ratio.  𝐾𝐾𝑖𝑝 = 2𝜉𝑖𝐿𝐿 − 𝑅𝑅         (24)  𝐾𝐾𝑖𝑖 = 𝜔𝑖2𝐿𝐿          (25) 𝑈𝑈 + − 𝑅𝑅 𝐿𝐿 𝐾𝐾𝑔𝑔 ?̇?𝜃𝑚𝑚   29 After manual tuning, the current controller was found to perform well using a bandwidth of 2000Hz while being critically damped.   4.2 Position Controller The proposed CTAMD position controller uses a proportional-plus-derivative control law with an additional torque feedforward term to account for the cogging torque. To derive the control law, the mechanical model of a conventional brushless DC motor, given in (26), is required where 𝑇𝑟𝑒𝑞 is the torque required to move the inertia of the system at a given reference angular acceleration, ?̈?𝜃𝑟𝑒𝑓.   𝑇𝑟𝑒𝑞 = (𝐽𝐿 + 𝐽𝑟)?̈?𝜃𝑟𝑒𝑓         (26)  Equating the required torque with the output torque in (4) allows the necessary electrical torque to be solved for:  𝑇𝑒 = (𝐽𝐿 + 𝐽𝑟)?̈?𝜃𝑟𝑒𝑓 − 𝑇𝑐        (27)  The required current, 𝐼𝑟𝑒𝑞, is then found by substituting (1)-(3), into (27):  𝐼𝑟𝑒𝑞 = (𝐽𝐿+𝐽𝑟)?̈?𝑟𝑒𝑓−𝐾𝑐 sin�𝑛𝑝𝜃𝑟𝑒𝑓�𝐾𝑡 sin�𝑛𝑝2𝜃𝑟𝑒𝑓�       (28)  Adding error dynamics to (28) allows a second-order PD position controller to be described by (29) where 𝐼𝑟𝑒𝑓 is the reference input current for the PI current controller and 𝐾𝐾𝑝𝑝 and 𝐾𝐾𝑝𝑑 are proportional and derivative position controller gains respectively.  𝐼𝑟𝑒𝑓 = 𝐾𝑝𝑝�𝜃𝑟𝑒𝑓−𝜃𝑚�+𝐾𝑝𝑑�?̇?𝑟𝑒𝑓−?̇?𝑚�+(𝐽𝐿+𝐽𝑟)?̈?𝑟𝑒𝑓−𝐾𝑐 sin�𝑛𝑝𝜃𝑟𝑒𝑓�𝐾𝑡 sin�𝑛𝑝2𝜃𝑟𝑒𝑓�    (29)  The controller gains in (29) are calculated using (30) and (31) where 𝜔𝑝 is the controller bandwidth and 𝜉𝑝 is the controller damping ratio.  30 𝐾𝐾𝑝𝑝 = 𝜔𝑝2(𝐽𝐿 + 𝐽𝑟)         (30)  𝐾𝐾𝑝𝑑 = 2𝜔𝑝𝜉𝑝(𝐽𝐿 + 𝐽𝑟)        (31)  After manual tuning, the position controller was found to perform well using a bandwidth of 250Hz while being critically damped. It is important to note that the bandwidth of the position controller is eight times slower than the bandwidth of the current controller which is an adequate difference to ensure the cascaded structure remains controllable.   4.3 Valve Dynamics and Exhaust Pressure Modeling The CTAMD model used in the simulation receives the voltage output from the current controller. If the inductance is assumed to be small relative to the resistance, (3) can be rewritten as:  𝑇𝑒 = 𝐾𝑡𝑈𝑠𝑖𝑛(𝜃𝑒)𝑅           (32)  If (1), (2) and (32) are substituted in (27), the angular acceleration is found for the case that considers an intake valve.   ?̈?𝜃𝑚𝑚 = 𝐾𝑡𝑈𝑠𝑖𝑛�𝜃𝑚𝑛𝑝2 �𝑅 +𝐾𝑐𝑠𝑖𝑛�𝜃𝑚𝑛𝑝�(𝐽𝐿+𝐽𝑟)        (33)  The angular acceleration is then twice integrated to find the rotor position which is related to the valve position through the excenter arm relationship described in (7).  𝑦 = 𝑟𝑎𝑠𝑖𝑛 �����𝐾𝑡𝑈𝑠𝑖𝑛�𝜃𝑚𝑛𝑝2 �𝑅 +𝐾𝑐𝑠𝑖𝑛�𝜃𝑚𝑛𝑝�(𝐽𝐿+𝐽𝑟) 𝑑𝑡�𝑑𝑡��    (34)   31 If exhaust valves are considered, an additional exhaust pressure disturbance torque, 𝑇𝑔𝑔, needs to be subtracted from (33).  ?̈?𝜃𝑚𝑚 = 𝐾𝑡𝑈𝑠𝑖𝑛�𝜃𝑚𝑛𝑝2 �𝑅 +𝐾𝑐𝑠𝑖𝑛�𝜃𝑚𝑛𝑝�−𝑇𝑔(𝐽𝐿+𝐽𝑟)        (35)  The torque caused by the exhaust pressure is related to the linear upward force acting on the valve, 𝐹𝑔𝑔, through the excenter arm. Furthermore, (35) can be twice integrated to find the exhaust valve displacement model for the CTAMD.  𝑦 = 𝑟𝑎𝑠𝑖𝑛 �����𝐾𝑡𝑈𝑠𝑖𝑛�𝜃𝑚𝑛𝑝2 �𝑅 +𝐾𝑐𝑠𝑖𝑛�𝜃𝑚𝑛𝑝�−𝐹𝑔𝑟𝑎 sin(𝜃𝑚)(𝐽𝐿+𝐽𝑟) 𝑑𝑡�𝑑𝑡��  (36)  For this simulation, a linear decay gas force model is used to describe the exhaust pressure acting on the exhaust valve [36], [37]. The function given in (37) describes the linear gas force decay for a valve displacement from -18 to +18 mechanical degrees.  𝐹𝑔𝑔 = 𝐹𝑔𝑔𝑚𝑎𝑥 �− 136 𝜃𝜃𝑚𝑚 + 12�        (37)  4.4 Initial Trajectory Design To simulate the position control of the proposed CTAMD system, reference kinematic trajectories must be predefined. While an arbitrary trajectory can be supplied to the controller, ideally one should select a minimum energy trajectory to reduce the losses of the CTAMD. Unfortunately, due to the presence of cogging torque and sinusoidal electrical torque, generating an optimal CTAMD trajectory requires a complicated nonlinear optimization technique. To avoid performing the complicated optimization, it was assumed that even though the CTAMD has cogging torque and a sinusoidal electrical torque, that the optimized trajectory of a constant torque MD would provide an approximate optimal trajectory for a CTAMD. Therefore, the method used to generate optimal kinematic  32 trajectories for the MD in [14] is applied to the CTAMD system. Equation (38) describes the optimal acceleration trajectory which is twice integrated and given to the cascaded controller, noting that 𝑡𝑓 is the total transition time [14], [38].   ?̈?𝜃𝑚𝑚(𝑡) = ?̈?𝜃𝑚𝑚𝑚𝑚𝑎𝑥 �1 − 2𝑡𝑡𝑓�        (38)  The kinematic trajectories for a 3.5ms valve transition measured between 5% - 95% of the total valve lift are displayed in Fig. 4.3. It is important to note that (38) requires infinite jerk and assumes the actuator has no inductance or is supplied with infinite voltage [14]. Intuitively, these assumptions are not physically possible and the error produced by them is left to be compensated for by the cascaded controller.     33  Figure 4.3 The reference angular acceleration, velocity and position reference trajectories for a 3.5ms transition time [14], [26].      34 4.5 Simulated Position Control Using the reference trajectories shown in Fig. 4.3, the actuator constitutive equation (36) and the control law (29), the simulated position tracking performance of the proposed CTAMD is displayed in Fig. 4.4. For the case of an intake valve where exhaust pressure is absent, it is seen that the simulated CTAMD tracks the reference exceptionally well. However, when exhaust pressure is added, the CTAMD begins to lag the reference trajectory. For the 200N gas force case, the CTAMD is only marginally able to maintain 5% position undershoot.    Figure 4.4 The position tracking performance of the simulated CTAMD system when opened against exhaust pressure [26].  The effect exhaust pressure has on energy loss of the CTAMD is presented in Fig. 4.5. Naturally, as exhaust force on the valve increases, more electrical torque is required to  35 augment the cogging torque. The increased electrical torque leads to an exponential increase in power loss which is described in (17) and evident in Fig. 4.5.    Figure 4.5 The ohmic energy loss incurred by the CTAMD when compensating for exhaust pressure for a 3.5ms transition with 8mm of valve lift [26].  In addition to handling exhaust pressure, the simulated CTAMD system shows a robust ability to operate over a wide range of transition times as seen in Fig. 4.6. While a 3.5ms transition time corresponds to an engine speed of 6000RPM, the simulation suggests that the proposed CTAMD design would be capable of driving intake valves using a 2.1ms transition time. A 2.1ms transition time would correspond to an engine speed of 10,000RPM, making it one of the fastest EVA designs in literature. Furthermore, when using a 3.5ms transition time, the CTAMD is simulated to have an energy loss of about 0.13J per transition. By examining Fig. 4.6 it is seen that 0.13J is the lowest energy loss shown in literature. The small loss is  36 attributable to the highly efficient kinetic energy recovery the cogging torque provides. For faster transition times, the energy losses begins to exponentially increase since increased electrical torque is required to supplement the cogging torque to achieve faster accelerations. Interestingly, if substantially slower transition times than 3.5ms are desired, the CTAMD begins to produce increased losses. This is caused by the cogging torque producing a fixed-magnitude sinusoidal torque which attempts to move the rotor with a given acceleration. If a slow transition time is required, electrical torque must be applied to oppose the cogging torque since the cogging torque will attempt to accelerate the rotor too quickly. Intuitively, for slow-speed transitions, cogging torque becomes detrimental. Therefore, to operate a CTAMD system more efficiently at low engine speeds, one should use a faster transition time than required. For example, to operate an engine at 1000RPM, one could use a maximum transition time of 21ms. However, for the CTAMD, it would be more efficient to use a 4.4ms transition time rather than attempt to use a 21ms transition time.    37  Figure 4.6 A comparison of ohmic energy loss over a range of transitions times with various EVA systems published in literature in the absence of exhaust pressure [14], [21], [22], [26], [28].  If the energy loss of the CTAMD is extrapolated for the intake valves of a four-cylinder, 16-valve engine, operating at 6000RPM, it would require 104W as shown in Fig. 4.7. This figure highlights the remarkably low ohmic power loss experienced by the CTAMD compared to other valve trains when only considering intake valves. Compared to a modern low friction camshaft, the simulated CTAMD is estimated to reduce power losses by over 85%. Furthermore, the simulated CTAMD suggests it would offer 70% fewer losses than the SAMD, which is one of the most efficient EVAs published in literature.    38  Figure 4.7 Ohmic power loss comparison when only considering the intake valves of a 4-cylinder, 16-valve IC engine at 6000RPM [14], [21], [22], [26], [28].               05001000150020002500SimulatedCTAMDSAMD SASD Low FrictionCamshaftMD SAVCPower Loss [W]  39 Chapter 5 Fabrication and Characterization of a Prototype CTAMD System This chapter describes the fabrication of a prototype CTAMD actuator and testbed. The last section of the chapter characterizes the prototype CTAMD system and compares its performance specifications with the simulated CTAMD.  5.1 Fabrication of a Prototype CTAMD Actuator and Testbed The following subsections describe the fabrication of a CTAMD prototype and testbed that will enable position control experiments to be performed. The prototype CTAMD attempts to achieve the proposed design specifications established in chapter 4.   5.1.1 Fabrication of a CTAMD Actuator The fabrication of the motor began with the construction of the rotor. The rotor was made out of AISI 1018 steel and was machined to have a diameter of 26mm. The rotor had a 9mm and 10mm diameter output shaft to attach the excenter arm and a rotary encoder on. Overall the rotor length was 87mm including the output shafts and 25mm excluding the output shafts. Rectangular 5mm x 5mm x 25mm N42 neodymium magnets were attached to the rotor using Loctite and epoxy.   The stator laminations were cut out of amorphous M19 silicon sheet steel. Two dowel pins were fabricated to be inserted into the stator holes for lamination alignment. The laminations were then stacked on the dowel pins and TIG welded at the corners to produce a stator length of 25mm. Initially, each stator tooth was attempted to be wound with 15 windings of 12 gauge wire. Unfortunately, 12 gauge wire, as suggested in the proposed design, proved to be too thick to wind by hand without breaking the wire insulation. Instead, five parallel strands of 22 gauge wire were used to create the 15 windings. Inevitably, the winding fill factor was reduced by approximately a factor of three. As a result, it was expected that the resistance and ohmic loss would be about three times larger than predicted in the simulation. A picture of the wound stator with the rotor inserted is presented in Fig. 5.1.    40  Figure 5.1 Diagram of the wound CTAMD.  Once the rotor and stator were fabricated, an excenter arm was fabricated out of aluminum. The excenter arm design includes two set screws used to clamp it to the output shaft of the rotor. A small plastic bushing was inserted into the outer hole of the excenter arm to hold the connecting rod. The connecting rod was made out of a thin piece of steel piano wire and was attached to the valve using a threaded aluminum cap. Two aluminum end caps were then fabricated to house and protect the winding half-turns and were fastened to the stator using two bolts. Excluding the length of the output shafts, the finished actuator size is 60mm x 60mm x 55mm. The finished prototype CTAMD actuator is seen in Fig. 5.2.   Stator Windings Dowel Pin Permanent Magnet Rotor Winding Retainer Ball Bearing  41   Figure 5.2 Finished CTAMD prototype [37].  5.1.2 Testbed and Data Acquisition Hardware The prototype CTAMD was mounted on the head of a small four-cylinder Honda engine. The square shape of the actuator allowed it to be easily mounted on a piece of angle iron using bolts that screwed into the end caps. An intake valve was used since the testbed does not have a pressurized combustion chamber below the head, and therefore is unable to emulate exhaust pressure.   The controller built in chapter 4 was loaded onto a dSpace control board. The dSpace board has eight analog-to-digital converters and a designated encoder port which allowed a current sensor and rotary encoder to be easily integrated into the controller. Furthermore, the dSpace is capable of sampling at 40kHz, which is ample for a CTAMD system operating with 3.5ms transition times. The dSpace also has a designated pulse width modulation (PWM) output port that allowed the controller output to be specified by the duty cycle of a 5V signal.   The CTAMD was driven using a MACCON SWM current driver. The MACCON driver is capable of delivering and measuring pulse currents in excess of 100A and therefore meets the constraints of the CTAMD system. The MACCON driver was powered by a variable-voltage Connecting Rod Rotor Valve Rotary Encoder Stator Excenter Arm  42 power source also capable of providing pulse currents over 100A as shown in Fig. 5.3. The PWM signal supplied by the dSpace board was received by the MACCON and was modulated into a 42V output signal with an equivalent duty cycle.   To measure the kinematics of the valve and rotor, a rotary encoder was attached to the rear output shaft of the CTAMD. A Quantum QR12 rotary encoder was selected which has 20,000 pulses per revolution and a frequency response of 500kHz. By applying (39) the sensitivity of the encoder is found to be 0.0045 degrees, where 𝑁𝑃𝑃𝑅 is the number of pulses per revolution. Since the proposed CTAMD system operates through 36 mechanical degrees, the QR12 introduces less than 0.02% position error and therefore is sufficient for the CTAMD system.  ∆𝜃𝜃𝑚𝑚 = 3604𝑁𝑃𝑃𝑅          (39)   Figure 5.3 Diagram of the experimental setup.   5.2 Actuator Characterization After fabricating the actuator and testbed, characterization experiments were performed to validate the characteristic parameters modeled in chapter 3. In the subsections below, details of open-circuit and closed-circuit experiments are provided. These experiments determine the   Computer dSpace   Current Driver   CTAMD   Rotary Encoder   Current Sensor  Power Supply  43 cogging torque constant, electrical torque constant, resistance and inductance, which are then compared to the FEA results. For the experiments, the excenter arm and connecting rod were detached from the rotor to reduce friction and simplify the inertia of the system.  5.2.1 Open-Circuit Analysis  The CTAMD was disconnected from the current driver to situate the CTAMD in an open-circuit state. An open-circuit state dictates the current and its derivative to be zero. This allows (22) to be greatly simplified into (40).  𝐾𝐾𝑔𝑔 = 𝑈sin(𝜃𝑒) ?̇?𝑚         (40)  The generator constant in (40) is obtainable after using the rotary encoder to measure angular position and velocity and an oscilloscope to measure voltage. To perform the open-circuit experiments, the rotor of the CTAMD was manually placed in the -90 electrical degree position and then gently nudged from rest. Once nudged, cogging torque began to accelerate the rotor which would then oscillate freely. While oscillating, the uncontrolled dynamic response of the rotor was measured by the rotary encoder and oscilloscope, allowing the generator constant to be calculated. From the experiments, the generator constant was found to be approximately 34.5mV∙s/rad as shown in Fig. 5.4. Recalling that the electrical torque constant is numerically equivalent to the generator constant reveals an electrical torque constant of 34.5N∙m/A.  44  Figure 5.4 A comparison of the modeled and experimental actuator generator constant [39].  The open-circuit experiments also allow the cogging torque constant to be isolated and found. Rearranging (27) and substituting in (2) and (3) produces (41).   𝐾𝐾𝑐 = (𝐽𝐿+𝐽𝑟)?̈?𝑚−𝐾𝑡𝐼 sin(𝜃𝑒)sin(2𝜃𝑒)         (41)  Noting that the current is zero with an open-circuit and that the excenter arm was detached from the rotor allows (41) to be simplified:  𝐾𝐾𝑐 = 𝐽𝑟?̈?𝑚sin(2𝜃𝑒)          (42)   45 Assuming that the rotor inertia retains its modeled value of 8.75kg∙mm2, the cogging torque can be plotted as seen in Fig. 5.5. A cogging torque constant of 1.4N∙m was estimated from the Fig.5.5.   Figure 5.5 A comparison between the modeled and experimental cogging torque [39].   5.2.2 Closed-Circuit Analysis After completing the open-circuit experiments, the actuator was reconnected to the current driver and closed circuit experiments were performed. For the closed-circuit experiments, the rotor of the CTAMD was placed at zero electrical degrees, which corresponds to a position of no cogging torque or electrical torque. In this position, if current is applied to the windings, the rotor remains stationary allowing (22) to be simplified into (43).   𝑈𝑈 = 𝑅𝑅𝐼 + 𝐿𝐿𝐼 ̇          (43)   46 The resistance is determined by applying a small voltage to the actuator and letting the current reach steady-state so that the derivative of the current is zero. This allows the current to be easily calculated using Ohm’s law:  𝑅𝑅 = 𝑈𝐼           (44)  After applying a small voltage to the windings, the terminal voltage of the actuator was measured with an oscilloscope and the steady-state current was measured with the current sensor allowing Fig. 5.6 to be generated. The averaged resistance in the figure was calculated by averaging all data points in the window of the plot which corresponds to 67.8mΩ.    Figure 5.6 Measured and averaged resistance of the prototype CTAMD.    47 Once the resistance is known, the inductance can be measured using a rearranged form of (43):  𝐿𝐿 = 𝑈−𝑅𝐼𝐼̇          (45)  After applying a small voltage to the CTAMD, the current and its derivative were measured using the current sensor, and the terminal voltage was measured with an oscilloscope. To avoid a division by zero in (45), the current must be in a transient state. Therefore, the inductance was only measured for a very brief time after the voltage was applied and is seen in Fig. 5.7. The averaged inductance was calculated by averaging the measured inductance over the window of the plot which was found to be 124μH.   Figure 5.7 The measured and averaged inductance of the prototype CTAMD.   48 5.3 Prototype CTAMD Specifications The results of the characterization experiments are listed and compared in Table 5.1.  The experimental electrical torque constant and cogging torque constant were found to be in close agreement with the values predicted by the FEA model. The experimental inductance was found to differ slightly from the modeled inductance, which is likely due to unaccounted for three-dimensional fringing and mutual flux linkages in the FEA. However, the experimental resistance was found to be approximately three times larger than the modeled resistance. This discrepancy was predicted in section 5.1 since the prototype winding fill factor was reduced by approximately a factor of three compared to the modeled CTAMD.   Table 5.1 Simulated and Experimental Specifications Comparison Parameter Modeled Experimental Resistance 22mΩ 67.8mΩ Inductance 85μH 124μH Electrical Torque Constant  35.3mN∙m/A 34.5mN∙m/A Cogging Torque Constant 1.31N∙m 1.4N∙m  To better analyze the usefulness of the Simulink simulation presented in chapter 4, the simulated CTAMD performance specifications were updated to correspond with the experimentally obtained results for the prototype actuator. By updating the simulation, the performance tracking of the simulated and prototype CTAMDs can be more easily compared to determine the validity of the simulation and its effectiveness as a tool to predict CTAMD performance. While the updated parameters led to no noticeable difference in terms of trajectory tracking, there was a large increase in energy loss per transition due to the poor winding fill factor. In the updated simulation, an energy loss of 0.307J per transition was found for a 3.5ms transition time. By extrapolating the energy loss for the intake valves of a four-cylinder IC engine allows Fig. 5.8 to be produced. Even though the power loss is substantially worse than indicated in the original simulation, it is seen that the updated CTAMD simulation still offers a substantial improvement when referenced to other EVA systems or a conventional camshaft.    49  Figure 5.8 Ohmic power loss comparison when only considering the intake valves of a 4-cylinder, 16-valve IC engine at 6000RPM [14], [21], [22], [26], [28], [39].               05001000150020002500OriginalSimulationUpdatedSimulationSAMD SASD Low FrictionCamshaftMD SAVCPower Loss [W]  50 Chapter 6 Experimental Validation of the CTAMD System After fabricating and characterizing the prototype CTAMD actuator, position control experiments were performed using the reference trajectories provided in Fig. 4.3. This chapter starts by providing the results of the prototype CTAMD position control experiments using 3.5ms transition times. The latter part of the chapter discusses heuristic current-shaping strategies used to improve the energy efficiency of the CTAMD.   6.1 Position Control Experiments The position tracking performance of the prototype CTAMD was tested using a 3.5ms transition time as shown in Fig. 6.1. From the figure it is seen that the CTAMD was able to track the reference trajectory well while maintaining less than 1% final position undershoot. Furthermore, Fig. 6.2 shows the CTAMD is able to track its reference velocity relatively well, with some error during the midpoint of the transition. This is attributed to the small electrical torque available during the midpoint of the transition which is evident in (3). However, the seating velocity was 0.06m/s and is a large improvement over a typical camshaft engine operating at redline speed [29], [30].     51   Figure 6.1 Prototype CTAMD position tracking performance for a 3.5ms transition time. The vertical dashed lines indicate 5% of total valve lift and 95% of total valve lift [35].   52  Figure 6.2 Prototype CTAMD velocity tracking performance for a 3.5ms transition time. The vertical dashed lines indicate 5% of total valve lift and 95% of total valve lift [35].   The current response of the prototype CTAMD is displayed in Fig. 6.3. The figure shows a slight controller delay at the beginning of the trajectory, but overall tracks its reference quite well. It is important to note the current controller was given a nonlinear constraint to set the current to zero near the midpoint of the transition. This constraint was required to avoid a division by zero in the control law established in (29). It was also noted that the kinematic trajectories shown in Fig 4.3 are not optimal for the CTAMD due to the large negative current spikes in Fig. 6.3. These negative current spikes imply the cogging torque is accelerating or decelerating the armature too quickly and therefore the controller is compensating by applying an opposing electrical torque. Furthermore, the negative current spikes lead to large energy losses since the unnecessary application of electrical torque leads to losses and the kinetic energy recovery of the cogging torque is squandered. These findings establish a clear motivation to generate an optimal kinematic trajectory for the CTAMD.   53  Figure 6.3 Prototype CTAMD current tracking performance for a 3.5ms transition time. The vertical dashed lines indicate 5% of total valve lift and 95% of total valve lift [35].   When attempting faster transition times, it was noted that the CTAMD began to draw current near its 100A limit, but the acceleration did not proportionally increase. This led to the finding the CTAMD suffered from premature flux saturation which began at approximately 60A to 70A. It was observed that increasing the current above 70A produced a negligible increase in electrical torque, which is indicative of saturation as opposed eddy currents or other nonlinear effects. Naturally, if additional current above 70A is applied to the windings, substantial losses will amount with negligible useful work being performed.   For the 3.5ms transition time, it was found that the energy loss per transition was 0.49J. When examining Fig. 6.4, it is evident that the prototype CTAMD exhibited appreciably larger losses than the updated simulation. Furthermore, using the unoptimized trajectory, the CTAMD was found to approximately match the losses of the SAMD.   54  Figure 6.4 Ohmic power loss comparison when only considering the intake valves of a 4-cylinder, 16-valve IC engine at 6000RPM [14], [21], [22], [26], [28], [35], [39].  6.2 Position Control Experiments using Heuristic Current-Shaping Techniques  In an effort to mitigate the losses spurred by the suboptimal nature of the kinematic trajectory and premature flux saturation, nonlinear constraints were added to the controller. These nonlinear constraints attempted to shape the current heuristically to avoid flux saturation and negative current spikes. To avoid flux saturation, the current limit given to the controller was reduced from 100A to 60A. Furthermore, near the midpoint of the transition, a much larger region was constrained to zero compared to Section 6.1. This constraint attempted to minimize the negative current spikes in Fig. 6.3, while avoiding the region where little electrical torque is produced even if large current is applied as dictated by (3).  Fig. 6.5 shows the current response for a 3.5ms valve transition time after adding the nonlinear constraints. When compared to Fig. 6.3, it is seen that the negative current spikes are almost halved and the peak current is smaller. These nonlinear constraints allowed for a drastic improvement in energy consumption as the energy loss per valve transition was 05001000150020002500UpdatedSimulationUnoptimizedMeasuredCTAMDSAMD SASD Low FrictionCamshaftMD SAVCPower Loss [W]  55 reduced from 0.49J to 0.33J. It can be noted that the heuristic current-shaping allowed the prototype CTAMD loss to be comparable to the simulated loss of 0.307J per transition.    Figure 6.5 Current response of the prototype CTAMD after heuristic current-shaping for a 3.5ms transition time [39].  Despite noticeable changes to the current response, it is seen in Fig. 6.6 that the position tracking performance remained similar with only a slightly larger final position undershoot. Furthermore, the prototype system was capable of tracking a reference position trajectory within 5% position undershoot for a range of transition times up to 2.5ms as shown in Fig. 6.7. While this is slower than the 2.1ms transition time predicted in the simulation, the prototype CTAMD is still one of the fastest EVA systems published in literature. Also, in Fig. 6.7 it can be noted that ohmic loss quickly increases after the benchmark 3.5ms transition time. Clearly, other actuators in literature, such as the SAMD, offer reduced losses for faster transition times compared to the CTAMD. However, it is important to note that the  56 spring systems connected to the actuators do not allow them to be slowed down without an energy penalty, making the CTAMD the most efficient actuator for a 6000RPM redline engine speed. As predicted in section 4.5, the CTAMD also incurs an energy penalty if slow transition times are required. It was experimentally found that the minimum energy transition occurred at approximately 4.7ms and therefore, in order to minimize losses, the CTAMD should not be operated slower even if slower transition times are possible.    Figure 6.6 Position tracking performance of the prototype CTAMD after heuristic current-shaping for a 3.5ms transition time [39].   57  Figure 6.7 A comparison of the CTAMD ohmic loss with various EVA systems in literature over a range of transition times [14], [21], [22], [26], [28], [35], [39].  While the effects of saturation are avoidable using heuristic current shaping for a 3.5ms transition time, saturation ultimately degrades the peak performance characteristics of the CTAMD. Fig. 6.8 shows the prototype CTAMD torque profiles assuming the actuator saturates with a current of 70A. Unfortunately, Fig. 6.8 shows the peak experimental electrical torque is less than the cogging torque from approximately -30 to +30 electrical degrees, which was not predicted in the simulation. This leads to a considerable complication as the rotor will become trapped in the -30 to +30 electrical degree region unless it is able to enter the region with adequate momentum. Although resonating the rotor out of this position may be possible, this is undoubtedly a disadvantage compared to the simulated electrical torque, which is always equal or greater than the cogging torque. Because the experimental electrical torque does not allow the rotor to be controllably held within the -30 to +30  58 electrical degree region, the following theoretical variable lift capabilities are established in (46) for an 8mm max valve lift.  Variable lift capabilities = �−4𝑚𝑚 ≤ 𝑦 < −1.3𝑚𝑚1.3𝑚𝑚 < 𝑦 ≤ 4𝑚𝑚       (46)  Even though the prototype CTAMD does not offer fully-variable lift like some EVA methods, from (46) it is seen that the prototype CTAMD is still able to perform a reduced range of variable lifts despite prematurely saturating. Lastly, due to the reduced peak electrical torque, it is speculated that the prototype CTAMD would be incapable of handling exhaust gas pressure.   Figure 6.8 Torque profiles of the simulated and prototype CTAMD.    59 Chapter 7 Kinematic Trajectory Optimization After observing the current profiles in Fig. 6.3 and Fig. 6.5, it was evident that optimized kinematic trajectories would offer a further reduction to ohmic energy loss. Chapter 7 describes the application of the Nelder-Mead algorithm used to generate optimal kinematic trajectories. In the latter portion of the chapter, the optimal trajectories are applied to the prototype CTAMD system and new position control experiment results are provided.   7.1 Cost Function Development and Application of the Nelder-Mead Algorithm Before an optimization algorithm is selected, it is important to establish the cost function to be optimized, since the cost function dictates the nature of optimization algorithm required. For the CTAMD a cost function needs to relate energy loss to angular rotor position. Combining (16) and (28) allows an expression to be formulated that describes actuator power loss as a function of angular rotor position.  𝑃 = �(𝐽𝐿+𝐽𝑟)?̈?𝑟𝑒𝑓−𝐾𝑐 sin�𝑛𝑝𝜃𝑟𝑒𝑓�𝐾𝑡 sin�𝑛𝑝2𝜃𝑟𝑒𝑓��2𝑅𝑅       (47)  The cost function is then established by integrating (47) with respect to time where 𝐸 is the energy loss and 𝑡𝑓 is the total transition time.  𝐸 = � �(𝐽𝐿+𝐽𝑟)?̈?𝑟𝑒𝑓−𝐾𝑐 sin�𝑛𝑝𝜃𝑟𝑒𝑓�𝐾𝑡 sin�𝑛𝑝2𝜃𝑟𝑒𝑓��2𝑅𝑅𝑡𝑓0 𝑑𝑡       (48)  The nonlinear nature of (48) is evident and therefore simple derivative optimization techniques cannot be applied. The complexity of the optimization is attributable to the cogging torque since most published optimization techniques for servo applications cannot account for cogging torque [40]-[45]. In literature, it was noted that the Nelder-Mead algorithm allowed an energy-optimal trajectory to be solved for a nonlinear EVA solenoid [46]. This application of the Nelder-Mead algorithm suggests it could be applied to the nonlinear prototype CTAMD system.   60 The Nelder-Mead algorithm is a derivative-free, simplex direct search method that was establish in 1965 [47], [48]. Furthermore, the Nelder-Mead algorithm is useful for performing unconstrained, multidimensional, nonlinear optimizations, which makes it applicable for a CTAMD [49]. It is important to note the Nelder-Mead algorithm does not have a proven convergence result for all dimensions [48]. However, the energy equation described in (48) is in a one-dimensional space for which the Nelder-Mead algorithm has been proven to converge.   After ensuring the CTAMD met the criteria to apply the Nelder-Mead algorithm, the fminsearch function in Matlab was used to optimize (48). When applying the Nelder-Mead algorithm it is important to select the number of points on the position trajectory to optimize and to provide a reasonable initial guess for the points. Choosing too few points results in an optimized trajectory that is prone to large interpolation error. However, choosing too many points can cause the algorithm to converge on a local minimum solution rather than the global minimum. Using the unoptimized trajectory as an initial guess for the points to optimize, the algorithm was executed multiple times for a range of 17-21 points. It was found that all of the executions produced approximately the same solution regardless of the number of points and therefore 21 points was selected as it provided a reference position profile with good resolution as shown in Fig. 7.1.   61  Figure 7.1 Optimized valve trajectory using the Nelder-Mead algorithm with 21 points.   7.2 Optimized Reference Trajectories The optimized kinematic trajectories are displayed in Fig. 7.2 and compared with the unoptimized trajectories in Fig 4.3. From this figure, it is seen that the unoptimized and optimized position and velocity trajectories show similar characteristics throughout the transition. However, the optimized acceleration greatly differs from the unoptimized trajectory. The effects of the electrical torque and cogging torque are clearly visible in the optimized acceleration trajectory as there are two maxima that are visible during both positive and negative accelerations. The two maxima are intuitive since the peak electrical torque and the peak cogging torque do not occur concurrently with respect to position.     62  Figure 7.2 A comparison of the optimized and unoptimized trajectories for a 3.5ms transition time [35].     63 The optimal current profile shown in Fig. 7.3 is found by substituting the kinematic trajectories into (28). Naturally, the current profile suggests minimal energy losses are attained by applying current early and late in the transition where the electrical angle is large. Furthermore, Fig. 7.3 shows that only positive current is required to drive the valve unlike Figures 6.3 and 6.5 where large negative current spikes were observed. Removing the negative current spikes implies that the electrical torque and cogging torque add constructively throughout the entire transition as expected. It can also be noted that the peak current required for a 3.5ms transition time is below 60A, which is distant from the 100A limit and below the premature saturation limit. Since the peak currents are below the saturation limit, the optimal kinematic trajectory is acceptable for the CTAMD and therefore additional nonlinear constraints do not need to be added to the Nelder-Mead optimization. Integrating (48), while using the kinematic trajectories, reveals a theoretical energy loss of 0.23J per transition for a 3.5ms transition time.    Figure 7.3 Optimal current profile for a 3.5ms transition time [35].   64 7.3 Experimental Results Using the Optimized Trajectory Once the optimized kinematic trajectories were obtained, the position control experiments performed in section 6.1 were repeated using the new trajectories. Fig. 7.4 shows the position tracking performance of the prototype CTAMD using the optimized trajectory. Even though there is a slight undershoot throughout the trajectory, the final position undershoot is less than 1%. In Fig. 7.5 it was observed that the seating velocity was 0.06m/s and is equivalent to the seating velocity found using the unoptimized trajectory. The error during the midpoint of the transition can be attributed to the small electrical torque available when the electrical angle is small.    Figure 7.4 Reference position tracking performance of the prototype CTAMD using an optimized kinematic trajectory for a 3.5ms transition time. The vertical dashed lines indicate 5% of total valve lift and 95% of total valve lift [35].    65  Figure 7.5 Reference velocity tracking performance of the prototype CTAMD using an optimized kinematic trajectory for a 3.5ms transition time. The vertical dashed lines indicate 5% of total valve lift and 95% of total valve lift [35].  The reference and measured current profiles produced using the optimized kinematic trajectories are displayed in Fig. 7.6. From the figure, it is seen that the negative current spikes observed in Fig. 6.3 and Fig. 6.5 were eliminated. When compared with Fig. 7.3, it is seen that Fig. 7.6 deviates slightly from the U-shape. However, these deviations spur from disturbances including friction, controller delay and inductance. Regardless, by having only positive current, it is evident that the optimized kinematic trajectories constructively utilized electrical torque and cogging torque over the entire transition.    66  Figure 7.6 Reference and measured current profiles using optimal 3.5ms kinematic trajectories [35].   The ohmic power loss, dictated by the measured current in Fig. 7.6, was integrated over 5ms to determine the total energy loss. After averaging the energy loss for an opening and closing transition, an energy loss of 0.29J per transition was calculated. Fig. 7.7 compares theoretical, simulated and measured power loss of the CTAMD for a 4-cylinder, 16-valve IC engine operating at 6000RPM. It can be noted that the prototype CTAMD, using the optimized trajectory, produced about 26% more loss than theoretically expected. However, the measured loss using the optimized trajectory was over 40% smaller than using the unoptimized trajectory. Furthermore, the optimized trajectory reduced the power loss of the prototype CTAMD by 12% when compared with the heuristic current-shaping results.    67  Figure 7.7 A comparison of the CTAMD power loss when using the optimized and unoptimized trajectories on a 4-cylinder, 16-valve IC engine operating at 6000RPM when only considering the intake valves [35], [39].  Fig. 7.8 compares the extrapolated power loss between the prototype CTAMD and other valve trains when operating a 4-cylinder 16-valve IC engine at 6000RPM. Clearly, the CTAMD offers a 70% reduction in losses compared to a conventional low friction camshaft. The figure also shows that the CTAMD is able to improve upon all other EVA systems found in literature by an energy savings in excess of 40%. 050100150200250300350400450OptimizedTheoreticalOptimizedMeasuredUnoptimizedSimulatedHeuristic Current-Shaping MeasuredUnoptimizedMeasuredPower Loss [W]  68  Figure 7.8 Ohmic power loss comparison when only considering the intake valves of a 4-cylinder, 16-valve IC engine at 6000RPM when only considering the intake valves [14], [21], [22], [28], [35].            05001000150020002500CTAMDOptimizedMeasuredSAMD SASD Low FrictionCamshaftMD SAVCPower Loss [W]  69 Chapter 8 Conclusion In this thesis, a novel EVA design and prototype was presented. The goal of this thesis was to prove cogging torque is a superior energy recovery method compared to traditional EVA methods that used mechanical springs. By utilizing cogging torque, the CTAMD was able to ascertain its status as the most energy efficient EVA found in literature. The CTAMD reduced energy loss by over 70% compared to a conventional camshaft while maintaining a compact size.  The modeling established in this thesis was validated by performing characterization experiments on a fabricated prototype CTAMD. Furthermore, the simulation was found to be effective at predicting the position tracking performance and energy loss of the physical prototype for transition times over 3.5ms. Naturally, the success of the modeling and simulation suggest they could be useful to develop CTAMD systems for larger application including large commercial transport vehicles.   Through experimentation, it was found that the CTAMD performed suboptimally using kinematic trajectories derived for a constant-torque actuator. The Nelder-Mead algorithm proved to be an effective method to handle the nonlinear cost function describing the CTAMD, and allowed a minimum-energy trajectory to be established. The optimized trajectory enabled the CTAMD to constructively use sinusoidal electrical torque and cogging torque. Furthermore, the optimized trajectory was found to drastically reduce the energy loss of the CTAMD, allowing it to have 40% fewer losses than the most efficient EVA published in literature.   The compact size and highly efficient nature of the CTAMD provides many practical advantages for integration into a mass-produced IC engine. The compact size allows the CTAMD to be feasibly installed under a valve cover of a typical engine. The remarkably low energy loss produces little heat and therefore reduces or negates the need for active cooling of the actuators. The highly efficient nature of the CTAMD also mitigates its impact on the charging system of a vehicle. These practical advantages suggest the CTAMD could feasibly replace the function of a camshaft while providing the benefits of independent valve control.   70 While this thesis fully develops and validates the concept of a CTAMD, future work would allow for various performance improvements. The Nelder-Mead optimization could be improved by adding a nonlinear constraint to account for a finite voltage supply. One of the most notable areas for improvement would be to advance the flux modeling within the actuator. Premature saturation reduced the ability of the CTAMD to perform variable lifts, handle exhaust gas forces and self-start. This work would likely involve using more advanced FEA software to diagnose the regions of saturation, followed by a redesign of these regions. Another area for future work would be to improve the winding fill factor. If a larger fill factor can be obtained, the ohmic loss could be proportionally reduced. 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Optim., vol. 9, 1998.                           78 Appendices Appendix A: Relationship between Number of Magnetic Poles, Motor Efficiency and Maximum Valve Acceleration  This appendix derives relationships between number of magnetic poles, the power loss and the maximum valve acceleration of the CTAMD. Equation (26) is used as a starting point for the derivation and is equated with the electrical torque to produce (A1).   𝑇𝑒 = (𝐽𝐿 + 𝐽𝑟)?̈?𝜃𝑚𝑚         (A1)  Since the rotor inertia and load inertia are matched, (A1) can be combined with (9) and (11) to produce (A2).  𝑇𝑒 = 2𝑚𝑟𝑎2?̈?𝜃𝑚𝑚         (A2)  Employing a small angle approximation allows the valve motion equation in (7) to be simplified into (A3).  𝑦 = 𝑟𝑎𝜃𝜃𝑚𝑚           (A3)  The second derivative of (A3) with respect to time is shown in (A4).  ?̈? = 𝑟𝑎?̈?𝜃𝑚𝑚           (A4)  (A2) can then be rewritten as follows:  𝑇𝑒 = 2𝑚𝑟𝑎?̈?          (A5)  (A5) is substituted into (19) to produce (A6).   79 𝑃 = 4𝑚𝑚2𝑟𝑎2?̈?2𝜌𝑐(𝑙+𝑙ℎ𝑡)𝐵𝑚2𝐴𝑔2𝑛𝑝 sin2(𝜃𝑒)𝐴𝑠𝐶𝑓        (A6)  After examining (A6) it appears increasing the number of poles reduces ohmic loss. However, this is not the case for the unique design of the CTAMD since many of the variables in (A6) are related to the number of poles. In (1), it is clear than the rotor displacement is inversely proportional to the number of poles. When examining (A3), it is seen that for a fixed valve displacement, that the excenter arm length and rotor displacement share an inversely proportional relationship. Therefore, if it is assumed that the number of poles is scaled by a factor of 𝛿 as shown in (A7), then the excenter arm length would be scaled by the same factor as seen in (A8).  𝑛𝑝 = 𝑛𝑝∗𝛿          (A7)  𝑟𝑎 =  𝑟𝑎∗𝛿          (A8)  Since the excenter arm is influenced by the number of poles, the reflected inertia of the valve experienced by the rotor will change, which is evident in (9). If the rotor and load inertias are to remain matched, (A8) can be substituted into (12) to reveal that the rotor radius must increase if more poles are desired.   𝑟 = �2𝑚𝑚𝑟𝑎∗2𝜌𝜋𝑙4√𝛿          (A9)  If the winding area is decomposed as shown in (A10), it becomes evident that the winding area is a function of rotor radius. Therefore, the winding area is also influenced by the number of poles as seen in (A11) and (A12).     𝐴𝑔𝑔 = 2𝑟𝑙          (A10)  𝐴𝑔𝑔 = �2𝑚𝑚𝑟𝑎∗2𝜌𝜋𝑙4 2𝑙√𝛿         (A11)  80 𝐴𝑔𝑔 = 𝐴𝑔𝑔∗√𝛿          (A12)  Furthermore, for a given stator size, if the number of poles is increased, the number of stator teeth would also increase, reducing the available space for windings. Therefore, the scaling factor can be related to stator slot area as shown in (A13).  𝐴𝑠 = 𝐴𝑠∗𝛿           (A13)  Substituting (A7), (A8), (A12) and (A13) into (A6) yields (A14).  𝑃 = 4𝑚𝑚2𝑟𝑎∗2?̈?2𝜌𝑐(𝑙+𝑙ℎ𝑡)𝛿𝐵𝑚2𝐴𝑔∗2𝑛𝑝∗ sin2(𝜃𝑒)𝐴𝑠∗𝐶𝑓        (A14)  (A14) clearly shows that the ohmic power loss increases as the number of poles increase through the relationship given in (A15).  𝑃 = 𝑃∗𝛿          (A15)  Even though increasing the number of poles reduces the efficiency of a CTAMD, it can be proven that the maximum valve acceleration increases with more poles. Substituting (A4) into (A2) under maximum conditions produces (A16).  ?̈?𝑚𝑚𝑎𝑥 = 𝑇𝑒𝑚𝑎𝑥2𝑚𝑚𝑟𝑎           (A16)  The relationship between the number of poles and electrical torque is obtained by substituting (A7), (A12) into (15) as shown below.   𝑇𝑒𝑚𝑚𝑎𝑥 = 𝑁𝐵𝑚𝑚𝐴𝑔𝑔∗𝑛𝑝∗𝐼𝑚𝑚𝑎𝑥 sin(𝜃𝜃𝑒)√𝛿3      (A17)  𝑇𝑒𝑚𝑚𝑎𝑥 = 𝑇𝑒𝑚𝑚𝑎𝑥∗√𝛿3         (A18)  81 Finally, substituting (A8) and (A18) into (A16) allows (A19) and (A20) to be found.   ?̈?𝑚𝑚𝑎𝑥 = 𝑇𝑒𝑚𝑎𝑥∗2𝑚𝑚𝑟𝑎∗ √𝛿         (A19)  ?̈?𝑚𝑚𝑎𝑥 = ?̈?𝑚𝑚𝑎𝑥∗√𝛿         (A20)  Clearly, increasing the number of poles enables a CTAMD to be designed with more electrical torque which allows for faster valve accelerations.     82 

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