Karakoc_Kerem_PhD_2012

Modeling and Design Optimization of Electromechanic al Brake Actuator Using Eddy Currents by Kerem Karakoc MASc, University of Victoria, 2007 BSc, Bogazici University, 2005 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in the Department of Mechanical Engineering. Kerem Karakoc, 2012 University of Victoria All rights reserved. This dissertation may not be r eproduced in whole or in part, by photocopy or other means, without the permission of the author. ii Modeling and Design Optimization of Electromechanic al Brake Actuator Using Eddy Currents by Kerem Karakoc MASc, University of Victoria, 2007 BSc, Bogazici University, 2005 Supervisory Committee Dr. Afzal Suleman, Dept. of Mechanical Engineering, University of Victoria Co[Supervisor Dr. Edward Park, Dept. of Mechanical Engineering, U niversity of Victoria Co[Supervisor Dr. Ned Djilali, Dept. of Mechanical Engineering, U niversity of Victoria Departmental Member Dr. Issa Traore, Dept. of Electrical and Computer E ngineering, University of Victoria Outside Member iii Supervisory Committee Dr. Afzal Suleman, Dept. of Mechanical Engineering, University of Victoria Co[Supervisor Dr. Edward Park, Dept. of Mechanical Engineering, U niversity of Victoria Co[Supervisor Dr. Ned Djilali, Dept. of Mechanical Engineering, U niversity of Victoria Departmental Member Dr. Issa Traore, Dept. of Electrical and Computer E ngineering, University of Victoria Outside Member Abstract A novel electromechanical brake (EMB) based on the eddy current principle is proposed for application in electrical vehicles. Th e proposed solution is a feasible replacement for the current conventional hydraulic brake (CHB) systems. Unlike CHBs, eddy current brakes (ECBs) use eddy currents and th eir interaction with an externally applied magnetic field to generate braking torque. Due to their pure electrically controllable and contact free nature, ECBs have mul tiple advantages over the current CHB systems, such as faster response, reduced weigh t and number of components, ease of implementing various controllers (e.g., antiLloc k braking), and reduced noise levels. However, the torque generated by a typical ECB at l ow speeds is insufficient to effectively and completely stop a moving vehicle. T herefore, an ECB is commonly used as an assistive brake to the CHB system in heavy ve hicles, i.e. trains and trucks. In order to overcome this shortcoming, the use of A C magnetic fields is proposed to realize a standLalone ECB system in which sufficien t braking torque can be generated at low speeds. To this end, eddy currents are modeled analytically using the governing Maxwell’s equations with the consideration of time varying field application. The analytical model was validated using finite element analysis. Results show that the braking torque increases with the application of a time varying field. Various forms of time varying fields have been stud ied. It was found that the frequencyLmodulated applied field in triangular wav eform results in the highest braking torque. Next, the design was optimized to maximize the braking torque and an optimum configuration was obtained using multiple pole proj ection areas (PPAs). Optimization results show that the braking torque significantly increases with the introduction of iv additional PPAs to the configuration, and the braki ng torque generation for an optimum fourLPPA ECB configuration exceeds the braking requ irements for current passenger vehicles. For control purposes, a dynamic model for a novel s tandLalone ECB system using AC fields for automotive applications has been success fully designed and evaluated. Also, a modelLbased predictive controller has been develope d for the optimum ECB configuration. Finally an experimental testLbed has been designed for experimentation of both DC and AC field application on ECB. Keywords: • Brake5by5wire • Eddy Current Brake • Time5varying Magnetic Fields • Analytical Modeling • Helmholtz Equations • Method of Images • Finite Element Modeling • Genetic Algorithm • Vibration Response of Human Body • Automotive Applications • Model Based Predictive Control v Table of Contents Supervisory Committee ............................. ................................................... ...................... ii Abstract .......................................... ................................................... ................................. iii Table of Contents ................................. ................................................... ............................ v List of Tables .................................... ................................................... ............................. vii List of Figures ................................... ................................................... ............................ viii Nomenclature ...................................... ................................................... ............................. x Acknowledgments.................................... ................................................... ..................... xiii Dedication ........................................ ................................................... ............................. xiv 1. INTRODUCTION ...................................... ................................................... ............. 1 1.1 Overview .......................................... ................................................... ................ 1 1.2 Objectives and contributions....................... ................................................... ..... 9 1.2.1 Contributions .................................................. .............................................. 10 1.3 Thesis Outline .................................... ................................................... ............ 11 2. ANALYTICAL MODELING OF EDDY CURRENT BRAKES WITH THE APPLICATION OF TIME VARYING MAGNETIC FIELDS ....... ................................ 13 2.1 Introduction ...................................... ................................................... .............. 13 2.2 Analytical modeling of eddy currents .............. ................................................. 1 4 2.3 Validation and preliminary results ................ ................................................... . 27 2.4 Discussion ........................................ ................................................... .............. 32 2.5 Conclusion ........................................ ................................................... ............. 35 Appendix.2A. Analytical Model: DC field application ................................................ 3 5 3. IMPROVED BRAKING TORQUE GENERATION CAPACITY OF EDDY CURRENT BRAKE WITH TIME VARYING FIELDS: A NUMERICAL STUDY ..... 44 3.1 Introduction ...................................... ................................................... .............. 44 3.2 Eddy current brake model .......................... ................................................... .... 45 3.3 Simulation results................................. ................................................... .......... 50 3.3.1 Validation of the FEM .................................................. ................................ 51 3.3.2 Time varying field application with induction effec ts ................................... 54 3.3.3 ECB configuration for automotive application ............................................. 57 3.3.4 Time varying field application and optimization wit hout induction effects .. 58 3.3.5 Effects of frequency modulation on braking torque g eneration ................... 60 3.4 Discussion ........................................ ................................................... .............. 62 3.5 Conclusion ........................................ ................................................... ............. 67 Appendix.3A. Analytical Model: DC field application ................................................ 6 7 4. OPTIMIZATION OF BRAKING TORQUE GENERATION CAPACITY OF EDDY CURRENT BRAKES WITH THE APPLICATION OF TIME VA RYING MAGNETIC FIELDS ................................... ................................................... ................. 69 4.1 Introduction ...................................... ................................................... .............. 69 4.2 Eddy current brake model .......................... ................................................... .... 70 4.3 Optimization ...................................... ................................................... ............ 74 4.4.1 Definition of the problem .................................................. ............................ 74 4.4.2 Optimization Results .................................................. ................................... 80 vi 4.4 Discussions ....................................... ................................................... ............. 82 4.5 Conclusion ........................................ ................................................... ............. 86 5. CONTROL ........................................... ................................................... .................. 87 5.1 Introduction ...................................... ................................................... .............. 87 5.2 Dynamic ECB Model ................................. ................................................... .... 89 5.3 Dynamic model of the vehicle ...................... ................................................... . 93 5.4 Design of a model based nonlinear predictive contro ller ................................. 94 5.3.1 Nonlinear model predictive control algorithm ............................................. 94 5.3.2 NMPC applied to ECB problem .................................................. ................. 98 5.5 Simulation results................................. ................................................... ........ 101 5.6 Conclusion ........................................ ................................................... ........... 105 6. EXPERIMENTAL STUDIES .............................. .................................................. 106 6.1 Introduction ...................................... ................................................... ............ 106 6.2 Experimental setup................................. ................................................... ...... 107 6.3 DC field application .............................. ................................................... ....... 109 6.4 AC field application .............................. ................................................... ....... 113 6.5 Summary ........................................... ................................................... ........... 116 7. CONCLUSIONS AND FUTURE WORK ....................... ...................................... 118 7.1 Conclusions ....................................... ................................................... ........... 118 7.1.1 Summary of Chapter 2 .................................................. .............................. 119 7.1.2 Summary of Chapter 3 .................................................. .............................. 119 7.1.3 Summary of Chapter 4 .................................................. .............................. 120 7.1.4 Summary of Chapter 5 .................................................. .............................. 120 7.1.5 Summary of Chapter 6 .................................................. .............................. 121 7.2 Future works ...................................... ................................................... .......... 121 7.3 Recommendations ................................... ................................................... ..... 124 Bibliography ...................................... ................................................... .......................... 125 Appendix .......................................... ................................................... ............................ 132 Appendix A. Vehicle dynamics ...................... ................................................... ......... 132 Appendix B. Braking torque requirements ........... ................................................... ... 136 Appendix C. Ferrite material properties and core sp ecs for AC field experiments [74] ................................................... ................................................... ............................... 138 vii List of Tables Table 1L1. Advantages and disadvantages of a CHB an d an EMB .................................... 4 Table 2L1. Specifications of the ECB configuration ................................................... ...... 29 Table 3L1. Specifications of the configuration adap ted from [29] .................................... 51 Table 3L2. Selected ECB configuration for automotiv e applications ............................... 58 Table 4L1. Design Variables and Constants for the E CB optimization problem .............. 75 Table 4L2. Optimum ECB Configurations ............. ................................................... ........ 80 Table 4L3. Optimum braking torque values for cases studied .......................................... 81 Table 5L1. Specifications of the vehicle and road c onditions ......................................... 100 Table 6L1. Experimental setup specifications ...... ................................................... ........ 110 Table 6L2. Specifications of the optimum one PPA co nfiguration from Table 4L3 ........ 115 Table AL1. Parameters for the quarter vehicle model .................................................. ... 136 Table AL2. Required braking torque values for sever al vehicles .................................... 13 7 viii List of Figures Figure 1L1. Schematic of a CHB on a passenger vehic le [1] ............................................ .. 1 Figure 1L2. Comparison of a CHB system and an elect romechanical brake (EMB) system on a passenger type vehicle [2] ................... ................................................... ..................... 3 Figure 1L3. Eddy current brake (ECB) concept ...... ................................................... ......... 6 Figure 2L1. Basic ECB domain considered for simulat ing the eddy current generation .. 16 Figure 2L2. PPA moving in externally applied field. .................................................. ...... 22 Figure 2L3. Method of images, a) electric charge, b ) magnetic dipole ............................. 25 Figure 2L4. Flowchart of the FEM implementation ... ................................................... .... 28 Figure 2L5. Convergence plot ...................... ................................................... .................. 30 Figure 2L6. Variation in braking torque with freque ncy of the externally applied magnetic flux density....................................... ................................................... .............................. 31 Figure 2L7. Variation in braking torque with angula r velocity of the rotating brake disk 32 Figure 2L8. Comparison between braking torques gene rated with AC and DC field applications ...................................... ................................................... .............................. 34 Figure 3L1. Basic ECB domain considered for simulat ing the eddy current generation .. 46 Figure 3L2. Flowchart of the proposed FEM implement ation .......................................... 50 Figure 3L3. Comparison between analytical and FEM r esults for DC field analysis ....... 52 Figure 3L4. Braking torque versus frequency at 50 r pm ................................................ ... 53 Figure 3L5. Braking torque versus rotational veloci ty at 10 Hz ....................................... . 54 Figure 3L6. Braking torque generation – (a) sinusoi dal, (b) square wave, (c) sawtooth wave, and (d) triangular wave field applications .. ................................................... ......... 55 Figure 3L7. Braking torque variation with increasin g frequency at 50 rpm with induction effects ........................................... ................................................... .................................. 56 Figure 3L8. Braking torque variation with increasin g frequency at 50 rpm without induction effects ................................. ................................................... ............................ 57 Figure 3L9. Braking torque generation at 10 Hz for triangular, sinusoidal and DC field applications ...................................... ................................................... .............................. 59 Figure 3L10. Optimum braking torque generation with frequency modulation ................ 62 Figure 3L11. Conductor brake disk with an array of PPAs .............................................. . 66 Figure 4L1. Basic ECB domain considered for simulat ing the eddy current generation .. 71 Figure 4L2. Flowchart of the FEM implementation ... ................................................... .... 73 Figure 4L3. Flowchart of the optimization procedure .................................................. ..... 79 Figure 4L4. Optimum 4LPPA configuration ........... ................................................... ........ 81 Figure 4L5. Braking torque generation at various an gular velocities ................................ 8 3 Figure 4L6. Deceleration curve .................... ................................................... .................. 84 Figure 5L1. Comparison between the estimated model and the FEM results ................... 93 Figure 5L2. Nonlinear model predictive control bloc k diagram ....................................... 9 5 Figure 5L3. NMPC Flowchart (MV L manipulated variab les, DV L disturbance variables and CV L controlled variables) .................... ................................................... ................... 98 Figure 5L4. ECB performance with NMPC ............. ................................................... .... 103 Figure 6L1. Experimental setup .................... ................................................... ................ 108 Figure 6L2. Eddy current brake (ECB) prototype with permanent magnets ................... 110 ix Figure 6L3. Comparison between the FE simulation an d experimental results for the prototyped configuration .......................... ................................................... .................... 111 Figure 6L4. Flux density distribution between two p ermanent magnets ......................... 112 Figure 6L5. AC electromagnet core configuration ... ................................................... .... 114 Figure 6L6. Selected ECB configuration for experime ntations ....................................... 11 6 Figure 7L1. Quarter vehicle shaker for testing real world performance of ECB (equipment is available at SFU, BC)........................... ................................................... .................... 122 Figure AL1. Free body diagram of a wheel .......... ................................................... ........ 133 Figure AL2. Friction coefficient versus slip ratio for several surfaces [76] .................... 135 x Nomenclature a Width of the PPA (m) Ax(i) x component of imaginary magnetic potential term (V s / m) Ay(i) y component of imaginary magnetic potential term (V s / m) Ax(p) x component of primary magnetic potential term ( V s / m) Ay(p) y component of primary magnetic potential term ( V s / m) Ax,net x component of net magnetic potential (V s / m) Ay,net y component of net magnetic potential (V s / m) A (Ax , A y) Magnetic vector potential (V s / m) b Height of the PPA (m) B Magnetic flux density vector (T) B0 Externally applied magnetic flux density (T) Bin Induced magnetic flux density (T) c,c 1,c 2 Coefficients that are used to estimate the dynamic ECB model C Coefficient function d Thickness of the conductive disk (m) dskin Skin depth (m) deff Effective thickness of the conductive disk (m) D Displacement flux density (C / m 2) ExDC x component of electric field with DC field appl ication (V / m) EyDC y component of electric field with DC field appl ication (V / m) E Electric field (V / m) f1 Carrier frequency (Hz) f2 Modulation frequency (Hz) ∆f Frequency deviation (Hz) f0 Basic coefficient fs Spaced effect coefficient F Coulomb’s force (N) G Green’s function to solve Helmholtz equation Gf Geometric Function hcg Height of the center of gravity (kg) H Magnetic field intensity (A/m) It Total moment of inertia of the wheel and the eng ine (kgm 2) J (Jx ,J y) Current density vector (A / m 2) JS Surface current density vector (A / m) Jin Induced current density vector (A) xi Kv Scaling constant lbase Wheel base (m) mw Wheel mass (kg) mv Vehicle mass (kg) pn , qm Geometric identifiers for green’s function ( n,m =1,2,3…) q Charge (C) R Radius of the conductive disk (m) r Distance between the center of conductive disk and the center of the PPA (m) rP Position vector from the center of the disk (m) Rw Wheel radius (m) sr Slip ratio dS Infinitesimal surface (m 2) t Time (s) Tb Braking torque without induction fields (Nm) Tbin Braking torque with induction effects included ( Nm) Ttime Time dependent function Tb,n Braking torque generated at the nth PPA Tb,total Total braking torque generated v Linear velocity vector (m / s) w Rotational velocity of the conductive disk (rad/s) x(i) x coordinate of imaginary magnetic potential sou rce (m) y(i) y coordinate of imaginary magnetic potential sou rce (m) φ Electric Scalar potential(V) ε Electric permittivity (F / m) µ Magnetic permeability (H / m) ρ Charge density (C / m 3) ρres Resistivity of the material (ohm m) σ Conductivity of the material (S / m) λ Wave term ϕ (ϕx ,ϕy) Source term Ω Angular frequency of the applied field (rad / s) βn Phase angles between applied fields at each PPA ( n =1,2,3…) (rad) αn Angles between PPAs on a multi PPA config. ( n =1,2,3…) (rad) Acronyms CHB Conventional Hydraulic Brake ECB Eddy Current Brake EC Eddy Currents xii EMB Electromechanical Brake MRB Magnetorheological Brake MRF Magnetorheological Fluid SR Switch Reluctance ABS AntiLlock Braking System AC Alternating Current DC Direct Current PPA Pole Projection Area FE Finite Element FEM Finite Element Method PDE Partial Differential Equation UMFPACK Unsymmetric MultiFront Method MPC Model Predictive Control RMS Root Mean Square GA Genetic Algorithm SLFEM Smoothed Finite Element Method rpm Rotation per Minute xiii Acknowledgments I would like to thank to Dr. Edward J Park and Dr. Afzal Suleman for giving me the chance to work on this project and also for their s upport, guidance, encouragement and patience. Special thanks to Sandra Makosinski who is the kind est person I have ever known. Thanks for everything you have done for me and for all of us. I cannot tell you how lucky we are to know you. Special appreciation to Art Makosinski, Barry Kent, Rodney Katz, Patrick Chang, Dorothy Burrows and all faculty from Department of Mechanical Engineering in University of Victoria. It is great to be a part of such an incredible group. I also would like to thank my friends and colleague s, Baris Ulutas, Casey Keulen, Bruno Rocha, Joana Rocha, Andre Carvalho, Ricardo P aiva, Adel Younis, Kel Sakaki, Dan Kerley, Jung Keun Lee and many more for their f riendship, support and kindness. It is not only valuable experience and knowledge but I also gained lifeLlong friends during my graduate studies. I would like to thank my family. There is no word t o describe how grateful I am to have you. My mother and father, Emine and Hasan Kar akoc, although you were not here with me, thanks for helping me be who I am today. T hank to Dr. Emre Karakoc, my brother, for always watching my back. And I would l ike to thank to my best friend and wife, Asia Karakoc, for her endless support, encour agements, love and friendship. You made this easier for me by just being a part of my life. To all of you, thank you! xiv Dedication Dedicated to my best friend and wife, Dr. Asia Karakoc . 1. INTRODUCTION 1.1 Overview Automotive industry is a dynamic and challenging fi eld and considerable technological innovations are proposed every year. Considerable resources are invested in research and development for building safer, cheape r and better performing vehicles. In particular, technological innovations such as "x by wire" solutions used to improve the existing mechanical systems have contributed signif icantly to make automobile more environmentally sustainable. For example, “brake by wire” means that the mechanical systems in the vehicles can be replaced by electrom echanical systems that are able to do the same task in a faster, more reliable and more a ccurate way when compared to the pure mechanical systems. More specifically, conventional hydraulic brakes (C HB) are currently being used in order to provide the required braking torque to sto p a vehicle. This system involves a pedal, hydraulic fluid, transfer pipes and the brak e actuator. When the driver pushes the brake pedal, the hydraulic brake fluid provides the pressure that squeezes the pads onto the rotor. The basic block diagram of this type of brake is shown in Figure 1L1. Figure 1L1. Schematic of a CHB on a passenger vehic le [1] 2 However, CHBs have a number of disadvantages. First of all, when the driver pushes the pedal, there is latency in building up the pres sure necessary to actuate the brakes. Also, since CHBs employ a highly pressurized brake fluid, there may be leakage of the brake fluid that could cause fatal accidents. Furth ermore, the fluid is harmful to the environment. CHBs employ friction between brake pads and the bra ke disk. Therefore, brake pad material wear occurs. Due to both the material wear and the friction coefficient change in high speeds, the brake performs less optimally in h ighLspeed regions as well as after a number of actuations. Thus, the brake pads must be changed periodically in order to get the optimum braking performance. Finally, ECB mecha nisms are bulky in size, considering both auxiliary components and brake act uators. In some applications such as trucks and heavy machi nery that need more braking torque, in addition to the use of CHB, drum brakes actuated by pneumatic systems are being used instead of hydraulic systems. Similar to CHBs, these types of brakes employ friction generated between two contact surfaces. In brakeLbyLwire, some of the pure mechanical compo nents of the conventional hydraulic brakes (CHBs) are replaced by electromech anical components. Such an electromechanical system will result in a number of advantages, such as faster braking response time, easy implementation of various contr ol systems and reduced number of components and wiring. Figure 1L2 (left) shows a ty pical EMB system with reduced number of components compared to the CHB system (on the right). In Table 1L1, the disadvantages of a CHB and the potential advantages of an electromechanical brake (EMB) are listed. 3 Figure 1L2. Comparison of a CHB system and an elect romechanical brake (EMB) system on a passenger type vehicle [2] 4 Table 1L1. Advantages and disadvantages of a CHB an d an EMB CHBS EMB Advantages: • High braking torque generation • Available controllers • Accepted technology with significant background Disadvantages: • Slow response due to pressure buildLup • Control requires additional electrical components • Significant weight of the overall system • Brake pad wear • Risk of environmentally hazardous fluid leakage Advantages: • Faster response time • Easy implementation of control systems • Reduced number of components and wiring • Less maintenance due to elimination of pads • Elimination of hazardous brake fluid • Simple softwareLbased brake parameter adjustment depending on the driving conditions Disadvantages: • Complexity of the system • Additional load on the battery • Considerably new technology A simple example of an EMB system is the drum brake s used in trailers where less braking torque is required. These brakes are actuat ed by an electromagnet installed in the drum brake instead of a hydraulic mechanism, attrac ting a magnetic rotating disk onto a stator. The friction generated between the stator a nd the rotor results in braking. Electric calipers developed by Delphi [2] and Continental [3 ] are examples of EMBs. In these systems, switched reluctance (SR) motors are used t o squeeze the pads onto the rotating disk. Although permanent magnet DC motors are more efficient, easy to control and cost effective, SR motors are more attractive to braking applications due to their high temperature operation capability, robustness, dynam ic bandwidth and fault tolerance [4L 6]. The high cost of the SR motor and the requireme nt for a sophisticated control system are main drawbacks of using such motors in braking applications. Another example of an EMB is the magnetorheological brake (MRB) [7L8]. These brakes employ magnetorheological fluids (MRFs), whe re the viscosity varies as a function of applied magnetic field. The MRF fills t he gap between the stator and the rotor 5 in the brake, and active control of the fluid visco sity by controlling the external magnetic field results in a retarding force in the form of a varying shear stress [9L10]. Eddy current brakes (ECBs), which are the main focu s of this thesis, are another type of EMBs. ECBs using DC fields are already being use d in heavy vehicles such as trucks, busses, trains and garbage collectors. They provide assistive braking to the main CHBs in order to decrease the braking load at high travel s peeds [11]. Eddy currents are swirlLlike electric currents gene rated on the surface of materials by means of a varying magnetic field. These currents a re formed as a reaction to the varying applied magnetic flux. The main principle behind an ECB is the conversion of the kinetic energy of a rotating disk into resistive heat, by g enerating eddy currents that oppose the varying magnetic field. A 3LD schematic diagram of an ECB concept is shown in Figure 1L3. It is comprised of a highly conductive disk (2) connected to the wh eel shaft (4) and a stationary electromagnet(s) (1) mounted to the vehicle’s chass is. When current is applied to the windings of the electromagnet (3), a magnetic field is generated between its poles and eddy currents are formed on the surface of the rota ting disk under the pole projection areas of the electromagnet. In addition to the advantages listed in Table 1L1, there are a number of additional characteristics that make ECBs a natural solution i n electrical vehicles. The main motivation of the thesis is to improve the braking performance of ECBs coupled with the following properties: contactless braking, silent o peration, no friction, natural capacity to imitate an automated brake system (ABS), which can be achieved with the replacement of the existing brake systems with ECBs. However, the current issue of the existing ECB’s limiting braking torque at low vehicle speeds needs to be addressed before realizing a wide range of practical applications as a standLalo ne ECB. Note that ECBs can readily be found in the following applications by means of ass istive braking: to slow down a high inertia actuator, to help braking of heavy vehicles such as trains and heavy trailers, to adjust the stiffness of exercise equipment (e.g., e xercise bikes), etc. Especially, slowing highLspeed trains down to speeds where conventional brakes would be more effective is one of the key applications of ECBs [12L16]. 6 Figure 1L3. Eddy current brake (ECB) concept Eddy currents generated on a rotating disk were fir st modeled by Smythe [17]. He used MaxwellLAmpere law in order to relate the curr ent flow on the surface of the conductor to the applied magnetic field. Then by us ing Faraday’s law and Gauss’ law, he obtained the induced magnetic field generated due t o the eddy currents in terms of the external field. An important conclusion he drew was that eddy currents cause demagnetizing fields on electromagnets that oppose the applied field and this results in a variation in the eddy current distribution. Similar to Smythe’s studies, Scheiber [18,19] studi ed the effects of a magnetic field on a moving conductor with low magnetic Reynolds nu mber, which means that the thickness and the speed of the conductor is conside rably small so that the skin effects (i.e. eddy currents) and the induced field effects can be neglected. In [19], using the magnetic potential of the generated eddy currents, Scheiber calculated the total eddy current power input to a rotating conductor. Then by using the co rrelation between the power and the torque, he obtained the braking torque by dividing the power by the angular velocity. Unlike the previous works by Smythe and Scheiber, W iederick [20] used another method to model the eddy currents on a moving sheet . According to this model, with the 7 help of experimental data, one can get the electric al resistance of the pole projection area (PPA). By applying Lorentz’s Force law, he describe d the eddy currents in terms of the resistance and the induced voltage due to the appli ed magnetic field. His model was then revised by Heald [21], who used Coulomb’s law to ob tain the surface charges in the PPA and the electric field generated due to these charg es. Note that the above studies were limited to slow mo ving speeds of conductors. Wouterse [22] carried out a theoretical study on th e effects of high rotational speeds in terms of the eddy current generation. His results f or the low speeds matched the results of Scheiber and Smythe, while providing an understandi ng into the underlying physics of eddy currents in the high speed region. Subsequentl y, Wouterse’s model was used by Simeu and Georges [23] and Barnes et al. [24] for t he purpose of designing ECBs. In addition, there has been a number of works that used finite element method (FEM) for studying ECBs. Burais et al. [25] looked into t he eddy current generation on a moving conductor. Peterson [26] investigated the effects of having a conducting ferromagnetic body exposed to external magnetic field, while Conr aths [27] investigated not only the eddy current generation but also the temperature ch ange within the moving conductor. More recently, Lee and Park [28L31] studied the edd y current distribution on the ECB when DC field is applied, and their model was consi derably more accurate than the previous models. This model was adapted in this wor k in order to validate our finite element model (FEM) of the ECB. Thus, the model is described in detail in Chapter 2. Apart from the above works, eddy current based brak ing has mostly been researched for railroad vehicles. Unlike the circular disk typ e ECB configuration described in this work, trains commonly employ an array of linear ECB s [12L16]. Due to the fundamental difference in the configuration, these works are no t discussed here in detail. While a common assumption made in the above previou s studies has been that the external magnetic field is time invariant (i.e. DC) , Lee and Park [28] proposed the use of sinusoidal field application (i.e. AC) to improve t he braking torque generation of ECB at low velocities. In [28], the braking torque generat ion was studied using numerical techniques on a configuration with 3 PPAs and the f requency of the applied field was optimized for better braking torque generation. For the study in [28], the induction effects were eliminated with the use of laminated magnetic core. While Lee and Park’s study 8 [28] is the first work that proposed an AC based ma gnetic field application to increase the capacity for the eddy current generation. The expl icit use of such a field for resolving an ECB’s braking torque limitation at low speeds has b een performed in this thesis and published in 2012 [32]. Here, an accurate finite e lement model (FEM) that accounts for the time varying field application was created and with the help of this model of ECB, the use of AC magnetic fields with fixed and variable f requencies in different waveforms was investigated at both low and high speeds. Results s howed that the braking torque significantly increases with the application of fre quencyLmodulated time varying fields. In order to effectively deal with the problem of re duced braking torque at low speeds, the use of the time varying (i.e. AC) magnetic fiel d is studied in this thesis. Although there is no inLdepth study on the use of AC fields for improving the ECB braking torque, eddy currents are modeled on various surfaces that are stationary in order to study the power loss in electronic components and to find def ects on conducting sheets. Yannopoulos et al. [33] modeled the eddy currents g enerated in a stationary cylindrical shell of infinite length when there is alternating magnetic field is present. They used the Maxwell’s equations and appropriate boundary condit ions to solve for the flux and the associated loss as well. He also obtained numerical solution to the problem using GaussL Laguerre method and this model was used to validate the analytical model along with experimental data. A similar work was published by Sathuvalli et al. [ 34]. In his work, they calculated the Lorentz forces on a conductive sphere placed in alt ernating (sinusoidal) magnetic field. The study expressed the eddy currents generated in terms of the source functions and it also accounted for the skin effects. In addition to the analytical model, a numerical model was also obtained and a procedure was presented to determine the magnetic pressure distribution on the surface of a liquid metal dropl et. Unlike others, in [35] Siakavellas studied the eddy current generation on the surface of conductive plates with different shapes using shape dependent factors such as equivalent resistance of the plate and the total circulation c urrent. Using their model, it is possible to estimate the eddy currents generated on the surface of the conductor plate using the shape dependent parameters he defined, however the accura cy strongly depends on the shape of the plate. Finally, he suggested improvements with the consideration of the principle of 9 minimum energy dissipation and he validated his ana lytical model with numerical analysis. Similar to Sathuvalli’s work, Lohofer [36] studied the eddy current generation as well as the vector and scalar potentials on a conducting sphere exposed to timeLvarying magnetic field. His study showed that there is no r adial component of the eddy current density in the sphere. This model can be used as a contactless inductive measurement of electrical conductivity of metallic melts, since th e impedance and the power absorption of the sphere can be obtained using his model. In one recent study [37], Sinha et al. investigated the eddy current generation on conductive plates. They investigated the eddy curre nt generation on the vacuum chambers of an accelerator. Their analytical model uses a po tential term that is used to define the field terms and using this model, they calculated t he power losses at different frequencies as well as the magnetic force acting on the plate d ue to the interaction between applied and induced fields. In the studies mentioned above, the eddy currents a re modeled on various types of surfaces that are stationary with the consideration of time varying field application. In this study, to be able to realize an ECB problem, in add ition to the effects of time varying field application, the eddy currents are modeled on a rotating circular plate. 1.2 Objectives and contributions The main objectives of this study are as follows: • To perform a literature review to gain an understan ding of the eddy current phenomenon and to gather knowledge on the stateLofL theLart on eddy current braking technologies. • To develop an analytic model of the eddy current ge neration under DC and AC magnetic field applications to a rotating conductor (brake disk) for theoretical analysis, optimization and control design purposes. • To develop accurate electromagnetic FE models for t he eddy current density generation under AC and DC field application to val idate the analytical model (and vice versa), and for design and optimization p urposes. 10 • To design an optimum ECB for higher eddy current de nsity generation with AC field application with consideration of the followi ng parameters: geometric dimensions, shape (e.g., sinusoidal or square wave) , frequency, and magnitude of the AC field to realize a standLalone ECB for autom otive applications. • To develop of a nonlinear model predictive controll er for ECB that provides improved braking performance with AC field applicat ions. • To design and setup an experimental control test be d to validate theoretical and numerical results in current and future studies. 1.2.1 Contributions The main contributions of the thesis research are a s follows: • In order to improve the performance of the ECB, the application of the AC magnetic fields is proposed. Preliminary results sh owed that this effectively increases the eddy current density generation capac ity over the surface of the ECB’s rotating/conducting disk, and hence provides an increased braking torque capacity. • From a theoretical contribution point of view, ther e is no previous work on modeling of the physics behind the eddy current gen eration on a rotating conductor due to an AC magnetic field. An accurate AC model was created and studied for analysis and control design purposes. • An accurate FEM was created to study the eddy curre nt generation over the surface of the rotating disk. This model was then u sed to validate the analytical ECB model with the consideration of the AC field ap plication. • With the help of the numerical and analytical model s, the braking torque generation is studied for various fields that are a pplied in different waveforms with consideration of frequency modulation to impro ve the results and then an optimum configuration is found using a stochastic o ptimization algorithm. Results show that the braking torque significantly increase s with the use of multiple PPAs and even exceeds the required torque generation of CHBs in passenger vehicles. 11 • A nonlinear controller is designed and the braking performance is further improved with the use of this controller. 1.3 Thesis Outline In Chapter 2, an analytical model is created for th e existing ECB problem. The braking torque is calculated using the scalar and vector po tential terms. By rearranging Maxwell’s equations, a generic potential equation can be obta ined for the electromagnetic problem and then, with the proper assumptions; the governin g equations are converted into a Helmholtz form. Next, with the help of Green functi ons and adequate boundary conditions, the magnetic vector potential term, whi ch is used to solve for the braking torque generation, is described. In order to valida te the analytical model, a FEM was created and the results show good agreement between the analytical and numerical results. In addition, the results also show increas e in the braking torque with the AC field application compared to DC field application onto t he rotating disk of the ECB. In Chapter 3, an accurate FEM was created to study effects of various types of AC fields applied. This model was also used to validat e the analytical model introduced in Chapter 2. The model accounts for electromagnetic p henomena such as skin effects as well as the displacement currents although they are considerably small in value compared to conductive current terms at low frequencies. Bra king torque generation using AC fields with sinusoidal, triangular, square and sawtooth wa veforms are studied as well as the effects of frequency modulation on the braking torq ue generation are explored. Results show significant improvements with specific wavefor ms with frequency modulation. In Chapter 4, using the outcomes of the previous se ction, an ECB is optimized using stochastic optimization algorithms. More specifical ly, a genetic algorithm is selected. The optimization problem is defined using geometric des ign variables, e.g., diameter and thickness of the disk, as well as field dependent v ariables, e.g., frequencies and phase differences, and number of PPAs. In addition, comfo rt levels are quantified and added as a constraint to the existing optimization problem f or comfortable braking performance. The results show that the braking torque significan tly increases with the introduction of an additional PPA and also with an optimum fourLPPA configuration, the braking torque exceeds the requirements for slowing down a passeng er vehicle. 12 In Chapter 5, a model based predictive controller f or the optimum fourLPPA configuration is designed. In order to do so, a dyn amical model is created with the help of the analytical model presented in Chapter 2 and the numerical FEM of the ECB from Chapter 3. Using this dynamic model, a model predic tive controller is also developed. The results show significant performance improvemen ts during a number of braking scenarios presented here. In Chapter 6, the experimental studies that have be en carried out for DC field application are summarized and the results are give n. This chapter mainly focuses on the design and development of a test bed for ECB and it is proposed that the effects of AC field application will be studied using the existin g setup with necessary modifications. Finally in Chapter 7, conclusions summarizing contr ibutions of each chapter are presented. This chapter also has the future work pl anned for the ECB research. Initially, a prototype for studying the braking torque generatio n under AC field application should be built and the braking torque generation under AC fi eld application needs to be studied. And then the controller should be implemented and t he braking performance should be studied in real time. As a further step, after the experimental studies are concluded, the regeneration of energy lost during braking by means of eddy currents need to be investigated. In addition, a number of recommendati ons are listed in this chapter for the study presented in this thesis. 13 2. ANALYTICAL MODELING OF EDDY CURRENT BRAKES WITH THE APPLICATION OF TIME VARYING MAGNETIC FIELDS 2.1 Introduction As was mentioned earlier in the previous chapter, t he main motivation behind this study on ECB is the potential performance improveme nts, such as contactless braking, silent operation, no friction, natural capacity to imitate an ABS for automotive applications, etc., that can be achieved with the r eplacement of the existing brake systems with ECBs. However, the braking torque generation o f the ECB is directly related with the rotational speeds of the conductive disk, there fore the main problem of the existing ECB’s limited braking torque at low rotational spee ds needs to be addressed before realizing a wide range of practical applications as a standLalone ECB. In terms of the ECB literature, Smythe [17] was the first one to investigate the eddy current generation on a rotating disk using governi ng Maxwell’s equations and following his studies; there has been a number of analytical (e.g., Scheiber [18,19], Wouterse [22], Lee et al. [28L31]), numerical (e,g, Burais et al. [25], Peterson [26], Conraths [27]) and empirical (e.g., Wiederick [20], Heald [21]) studie s that were published. While a common assumption made in the above previou s studies has been that the external magnetic field is time invariant (i.e. DC) , Lee and Park [28] proposed the use of sinusoidal field application (i.e. AC) to improve t he braking torque generation of ECB at low velocities. In [28], the braking torque generat ion was studied using numerical techniques on a configuration with 3 PPAs and the f requency of the applied field was optimized for better braking torque generation. For the study in [28], the induction effects were eliminated with the use of laminated magnetic core. While Lee and Park [28] is the first work that proposed that an AC based magnetic field application could increase the capacity for the eddy current generation, the expli cit use of such a field for resolving an ECB’s braking torque limitation at low speeds has b een studied and presented previously in [32]. In [32], an accurate finite element model (FEM) that accounts for the time varying field application was created and with the help of this FEM of ECB, the use of AC magnetic fields with fixed and variable frequenc ies in different waveforms was 14 investigated at both low and high speeds. Results s howed that the braking torque significantly increases with the application of fre quencyLmodulated time varying fields. In this chapter, the eddy current generated over th e surface of the rotating conductor is modeled using analytical equations in order to calc ulate the braking torque generation of an ECB configuration. Eddy currents are modeled on various surfaces, such as cylindrical shells (Yannopoulos et al. [33]), spheres (Sathuval li et al. [34], Lohofer [36]) and conductive plates (Siakavellas [35], Sinha et al. [ 37]) that are stationary. In these studies, the common assumption made was th at the surfaces where the eddy currents are generated are stationary. However, in terms of the ECB problem, in addition to the effects of time varying field application, t he eddy currents are modeled on a rotating circular plate, thus the rotational terms will be included in the ECB model to solve for the eddy current distribution over the su rface of the conductive disk. In this chapter, the main objectives are to study a nd investigate the physics behind the eddy current generation on a rotating conductor whe n there is AC field is present and to create an analytical model to be used later for con trol design purposes. The analytical model is developed using the governing Maxwell’s eq uations. The governing equations are reorganized and the field variables are redefin ed using the potential terms in this equation. With the proper boundary conditions and a ssumptions for the ECB problem, a wellLknown PDE form, Helmholtz equation, is obtaine d and it was solved using green functions. Then for validation purposes a FEM is cr eated. The analytical and numerical results show good agreement and the reasoning behin d the difference observed between these results is addressed in the discussion sectio n of this chapter. In addition, the results prove that the AC field application improves the br aking torque generation not only at low speeds but also high speeds as well for the con figuration selected in this chapter. 2.2 Analytical modeling of eddy currents In this section, the eddy currents generated on the surface of a rotating conductor is modeled using the vector and scalar potential terms . Along with the constitutive relations (Eqs. (2.6) and (2.7)) and Lorentz force law (Eq. ( 2.5)), Maxwell’s equations (Eqs. (2.1)L (2.4)) are used to describe the field terms in term s of sources as follows: 15 ∇×/g1782=/g1784+/g3105/g1778 /g3105/g3047 (2.1) ∇×/g1779=−/g2986/g1776 /g2986/g2930 (2.2) /g2744∙/g1778=/g2025 (2.3) /g2744∙/g1776=0 (2.4) /g1780=/g1869(/g1779+/g1822×/g1776) (2.5) /g1778=/g2013/g1779 (2.6) /g1776=/g2020/g1782 (2.7) where B is the magnetic flux density, H is the magnetic field intensity, E is the electric field, D is the displacement flux density, J is the current density, F is the Lorentz force, v is the linear velocity of a point on the conductor surface, q is the electric charge particle, ρ is the charge density, µ is the magnetic permeability and ε is the electric permittivity of the medium. When the disk rotates, eddy currents are generated on its surface according to the Faraday’s law (see Eq. (2.2)) due to the applied ex ternal magnetic field. Electric field will be generated when there is a change in the magnetic flux due to motional effects (rotation of the conductor) and/or transformer effects (alter nating field application). For the basic ECB problem, a domain shown in Figure 2L1 was selec ted. The outer circular shape represents the conductor disk and the inner rectang ular region is the PPA, i.e. the area that the external magnetic field is applied onto. 16 Figure 2L1. Basic ECB domain considered for simulat ing the eddy current generation Eq. (2.5) is used to solve for the Lorentz force wh en there is a particle moving in external magnetic field and electric field. Using t his relation, the eddy currents generated due to external electric fields and magnetic fields can be defined as: /g1784=/g2026(/g1779+/g1822×/g1776) (2.8) where σ is the conductivity of the rotating disk. As the n ext step, using its divergence free nature (see Eq. (2.4)), magnetic field is expressed in terms of the curl of a vector that is known as magnetic potential vector, A, i.e. /g1776=/g2744×/g1775 (2.9) Also the electric field term can be written in term s of a scalar potential (i.e. φ) as follows: /g1779=−∇/g2030−/g2986/g1775 /g2986/g3047 (2.10) 17 After the magnetic terms are defined, Eqs. (2.1) an d (2.3) are rewritten using Eq. (2.8) as well by defining the following potential terms, which are the potential representation of the Maxwell’s equations: ∇×(∇×/g1775)=/g2020/g2026/g3436−∇/g2030−/g2986/g2157 /g2986/g3047+/g1822×(∇×/g1775)/g3440+/g2020/g3105/g1778 /g3105/g3047 (2.11) ∇/g2870/g2030+/g2986∇./g1775 /g2986/g3047=−/g3096 /g3084 (2.12) After obtaining the above potential equations, it i s necessary to make a number of assumptions to be able to solve these equations to find the vector and scalar potentials. The assumptions that are used for simplifying Eqs. (2.11) and (2.12) are as follows L It is assumed that the conductive disk is made of a luminum, thus the electric permittivity ( ε) and the magnetic permeability ( µ) is assumed to be equal to the vacuum’s. L As the external magnetic flux density, a uniformly distributed sinusoidal field with a magnitude B0 is applied in perpendicular to the surface of the conductor disk: /g1776=/g1828/g2868/g1871/g1861/g1866(/g2007/g1872)/g1811 (2.13) where Ω is the angular frequency of the applied external f ield. L Since magnetic flux density follows a sine curve, t he electric field (see Eq. (2.2)) will follow a cosine curve and the magnetic potenti al (see Eq. (2.9)) will follow a sine curve: /g1779=/g1831/g3051/g1855/g1867/g1871 (/g2007/g1872 )/g1809+/g1831/g3052/g1855/g1867/g1871 (/g2007/g1872 )/g1810 (2.14) /g1775=/g1827/g3051/g1871/g1861/g1866 (/g2007/g1872 )/g1809+/g1827/g3052/g1871/g1861/g1866 (/g2007/g1872 )/g1810 (2.15) where Ex and Ey are the x and y component of the electric field an d Ax and Ay are the x and y component of the magnetic vector potent ial. L It is assumed that the current penetrates through t he whole thickness of the conductor and it is uniformly distributed across th e cross section, i.e. no skin 18 effects on the conductor. Note that this assumption makes it possible to eliminate the variation of field variables along the thicknes s and the problem can be simplified to a 2LD problem. As a result of this as sumption, Eq. (2.8) can be revised as follows: /g1784/g1793=/g2026/g1856(/g1779+/g1822×/g1776) (2.16) where JS is the surface current density (A/m), d is the thickness of the conductive disk. L Displacement currents are assumed to be negligible at low frequencies. L Coulomb gauge condition ( ∇./g1775=0 ) is adopted so that unique potentials as the solution to the above equations can be obtained. Wi th the inclusion of Coulomb’s gauge, it is assumed that the instantaneous source terms can be used to calculate the potentials that are then used to calculate the field terms through differentiation. L Since the external field is dominant over any resid ual or induced fields at low frequencies over the surface of the conductive disk , the effects of such fields are ignored. In order to realize a model with no induct ion fields, a laminated electromagnet core is utilized instead of a soft fe rromagnetic core. According to this assumption, the external flux density is the o nly source of the magnetic vector potential, i.e. /g2744×/g1775=/g1828 /g2868/g1871/g1861/g1866(/g2007/g1872)/g1811 (2.17) Eqs. (2.11) and (2.12), then, can be simplified usi ng the above assumptions and the resulting potential equations are: ∇/g2870/g1775=/g2020/g2026/g1856(∇/g2030+/g2986/g1775 /g2986/g3047−/g1822×/g1776) (2.18) ∇/g2870/g2030 =−/g3096 /g3084 (2.19) 19 Note that Eq. (2.19) does not have any timeLbased o peration (i.e. differentiation or integration). The same equation is also used to def ine the scalar potential when there is a static (DC) magnetic field applied instead of a tim e varying one. Thus, the solution of the scalar potential that is found for the DC applicati on is also the solution to the AC application, as long as the charge density defined for both cases are equal. In order to check this condition, the total charge density is c ompared with the magnitude of the charge density that exists when there is time varyi ng field. Since the displacement currents are set to be zero at low frequencies, steady current conditions are satisfied, i.e. the divergence of th e current density is equal to zero. Also, the divergence of the current density can be obtain ed using Eqs. (2.10) and (2.16) as follows: ∇∙/g1784/g1793=∇∙/g4674/g2026/g1856/g4672−∇/g2030−/g2986/g1775 /g2986/g3047+/g1822×/g1776/g4673/g4675 (2.20) Since the steady current assumption is valid, the t erm on the right hand side of Eq. (2.20) can be equated to zero and the Laplacian of the sca lar potential can be obtained using the Coulomb’s gauge and the final assumption in the abo ve list, i.e. ∇/g2870/g2030 =∇.(/g1822×/g1776) (2.21) /g2025=∇.(/g1822×/g1776) (2.22) Note that the Laplacian of the scalar potential is equal to the charge density. Similar to Eq. (2.19), Eq. (2.22) also does not hav e any timeLbased operations. Thus, it is clear that the charge density does not change in the presence of DC or AC magnetic fields. Since the magnitudes of charge densities ar e the same for both cases, it is concluded that the scalar potential that is solved for DC magnetic field application is also the solution to the AC magnetic field application u nder the listed assumptions. Thus, in this paper, in order to solve for the magnetic vect or potential, the scalar potential derived when DC magnetic field is applied is used. 20 ∇/g2870/g1775−/g2020/g2026/g1856 /g2986/g1775 /g2986/g3047=/g2020/g2026/g1856(∇/g2030−/g1822×/g1776) (2.23) Since the time dependence is known for the magnetic potential vector (see Eq. (2.15)), time derivative of this term can easily be calculat ed, which can be expressed in terms of the magnetic potential itself as shown below: /g2986/g1775 /g2986/g3047=/g3078 /g1775/g3030/g3042/g3046 (/g3078/g3047 ) /g3046/g3036/g3041 (/g3078/g3047 ) (2.24) Being able to present the derivative term in terms of the potential itself makes it possible to rewrite Eq. (2.23) in a known form that is called the inhomogeneous Helmholtz equation: ∇/g2870/g1775+/g2019/g1775=−/g2774 (2.25) where λ is the square of a constant called wave number tha t is positive and ϕ is a given vector field that is a function of the field terms and the material constants. For the ECB problem, these terms can be described as follows. /g2019 =−/g2020/g2026/g1856 /g3078/g3030/g3042/g3046 (/g3078/g3047 ) /g3046/g3036/g3041 (/g3078/g3047 ) (2.26) /g2774=/g2020/g2026/g1856(−∇/g2030+/g1822×/g1776) (2.27) Note that the inhomogeneous Helmholtz equation repr esented in Eq. (2.25) is in a vector form. This equation then can be separated into two parts: one for the x component and the other for the y component of the magnetic vector potential, i.e. ∇/g2870/g1827/g3051+/g2019/g1827/g3051=−/g2038/g3051 (2.28) ∇/g2870/g1827/g3052+/g2019/g1827/g3052=−/g2038/g3052 (2.29) where ϕx and ϕy are the first and the second ( x and y respectively) components of the vector ϕ. 21 In order to be able to solve these two Helmholtz eq uations, λ, ϕx and ϕy have to be known and in addition to the values of these variab les, proper boundary conditions have to be defined. From Eq. (2.26), it is clear that λ is a known time function and ϕ is a function of the scalar potential, velocity of the r otation and the externally applied field. The only unknown in this equation is the scalar vec tor potential; however it was previously shown that the solution for the electric potential in the presence of a DC field is also the solution to the time varying field appl ication. Thus, the results obtained in [29] are adopted here to solve the time varying magnetic vector potential. Note that the negative divergence of the scalar potential needs t o be calculated in order to solve the Helmholtz equation. From Eq. (2.10), it is clear th at when there is no time variation, the electric field is equal to the negative divergence of the scalar potential. In [29], the electric fields due to the charges acc umulated at the borders of the PPA due to the Lorentz force that separates the charged particles are derived using an analogous method to calculate the magnetic pole den sity at the surface of a permanent magnet. Below in Figure 2L2, the PPA, moving in an externally applied magnetic field, is shown. As PPA moves in the externally applied field , Lorentz force causes separation, which will result in the accumulation of charge den sities at the edges. Using these charge densities, the electric field can then be calculate d. Eqs. (2.30) and (2.31) show the x and y components of the electric field distribution withi n the PPA with respect to a frame that is located at the left bottom corner (point O) of the PPA. Figure where r is the distance between the center of the conductiv e disk and the center of the PPA (see Figure 2L1), a and ExDC=wB 4π EyDC= −wB 8πFigure 2L2. PPA moving in externally applied field. is the distance between the center of the conductiv e disk and the center of the and b are the width and the height of the PPA, respective ly, and B0 π−(2r+b)tan−1y−a x−b+(2r−b)tan−1y−a x  +(2r+b)tan−1y x−b−(2r−b)tan−1y x +a 2ln{4x2+4y2}{4a2+4x2−8ay+4y2} {(2b−2x)2+4y2}{(2a−2y)2+(2b−2x)2 B0 π−2atan−1b−x a−y+tan−1x a−y−tan−1x y−tan−1x y   +bln{(a−y)2+x2}{(a−y)2+(b−x)2} {(b−x)2+y2}{x2+y2} +2rln{x2+y2}{(a−y)2+(b−x)2} {(b−x)2+y2}{(a−y)2+x2} 22 (2.30) (2.31) is the distance between the center of the conductiv e disk and the center of the are the width and the height of the PPA, respective ly, and w }  −b y  23 is the angular velocity of the rotating disk. For t he rotating conductor, the velocity at any point can be written as follows: /g1822=−/g1875/g4672/g1877−/g3029 /g2870/g4673/g1809+/g1875(/g1876+/g1870−/g3028 /g2870)/g1810 (2.32) The electric field terms, /g1831/g3051/g3005/g3004 and /g1831/g3052/g3005/g3004 in Eqs. (2.30) and (2.31), respectively, cannot be used directly as they represent the instantaneou s field terms and do not vary with time. Faraday’s Law can be used in order to identify the variation of these field terms with time. According to Faraday’s Law, if the external m agnetic flux density is expressed using a sine function as in Eq. (2.13), then the el ectric field term has to be expressed using a cosine function as shown in Eq. (2.14). Thu s, for the current analysis, the electric field terms are used as shown below. /g1779=/g1831/g3051/g3005/g3004 /g1855/g1867/g1871 (/g2007/g1872 )/g1809+/g1831/g3052/g3005/g3004 /g1855/g1867/g1871 (/g2007/g1872 )/g1810 (2.33) Then, using Eq. (2.13), Eq. (2.32) and Eq. (2.33), ϕ term can be expressed as follows: /g2774=/g2020/g2026/g1856/g3428/g1831/g3051/g3005/g3004 /g1855/g1867/g1871 (/g2007/g1872) /g1831/g3052/g3005/g3004 /g1855/g1867/g1871 (/g2007/g1872 )+/g1875/g1828/g2868(/g1876+/g1870−/g1853/2)/g1871/g1861/g1866 (/g2007/g1872) /g1875/g1828/g2868(/g1877−/g1854/2)/g1871/g1861/g1866 (/g2007/g1872)/g3432 (2.34) As the next step, the Helmholtz equation is solved using Green’s theorem by defining boundary conditions [38]. The borders of the PPA th at defines and limits where the externally applied flux density is are the primary boundaries to the AC problem. Since it is assumed that the externally applied flux density is applied only within the PPA, we can assume that the magnetic vector potential term is z ero at the boundaries. This ensures that the flux term is zero outside the PPA. Then, the co rresponding solution to the Helmholtz equation is as follows 24 (2.35) where η and ζ are the integration variables and{…} BC represents the boundary conditions at the corresponding edges of the PPA. Also, note t hat Eq. (2.35) is solved twice to obtain the x and y components of the magnetic vector potential separa tely with the corresponding values defined for ϕ and λ. In addition, the green function G that is selected to solve the above problem is given below. /g1833 =/g2869 /g3028/g3029 ∑ ∑/g3046/g3036/g3041 (/g3043/g3289/g3051)/g3046/g3036/g3041(/g3044/g3288/g3052)/g3046/g3036/g3041(/g3043 /g3289/g3085)/g3046/g3036/g3041(/g3044/g3288/g3086) /g3043/g3289/g3118/g2878/g3044/g3288/g3118/g2879/g3090/g2998 /g3041/g2998 /g3040 (2.36) where /g1868/g3041=/g3095/g3041 /g3029 (2.37) /g1869/g3040=/g3095/g3040 /g3028 (2.38) Note that once the boundary conditions are implemen ted in Eq. (2.35), the magnetic vector potential solution can be derived as follows : /g1827/g3051(/g3052)=/g1516 /g1516/g2038/g3051(/g3052)(/g2014,/g2015)/g3029 /g2868/g3028 /g2868/g1833(/g1876,/g1877,/g2014,/g2015)/g1856/g2015/g1856/g2014 (2.39) Before deriving the final solution and carrying out braking torque calculations, the boundary of the conductive disk has to be accounted for. So far, the analytical model Ax(y)=φx(y)(ς,η)G 0b ∫ 0a ∫(x,y,ς,η)dηdς +{Ax(y)(0,y)}BC[∂ ∂ςG(x,y,ς,η)]ς=0dη 0a ∫ −{Ax(y)(b,y)}BC[∂ ∂ςG(x,y,ς,η)]ς=bdη 0a ∫ +{Ax(y)(x,a)}BC[∂ ∂ηG(x,y,ς,η)]η=adς 0b ∫ −{Ax(y)(x, 0)}BC[∂ ∂ηG(x,y,ς,η)]η=0dς 0b ∫ solves for the magnetic vector potential in a PPA o n an infinitely long conductive di and in order to account for the actual boundary at the edge of the conductive disk, the method of images is used. This method is used to re place a boundary in front of a source with an imaginary source located on the other side of th e boundary, as shown 2L3. (a) Figure 2L3 . Method of images, a) electric charge, b) magnetic dipole In Figures 4(a) and 4(b), the image sources for an electric charge and a magnetic dipole are presented, respectively. Assuming that t hese sources are placed in front of conductive surfaces, the field components can be calculated by replacing the surface with the image of the source as shown in the figure. Not e that the magnitudes of the primary source and the imaginary source and the distances b etween the conductor surface and the sources are th e same for both sources. Although they look similar , the major difference between the electric and magnetic sources in terms of the application of this method is that the normal component of the field to the surfa ce of the conductor is reversed for the magnetic source. In order to use the method of images, the cylindric al coordinates of the rotating disk needs to be transformed into a linear space using c onformal mapping techniques. In [ the same method was used to implement the boundary conditions at the edge of the conducting disk. Unlike [ solves for the magnetic vector potential in a PPA o n an infinitely long conductive di and in order to account for the actual boundary at the edge of the conductive disk, the method of images is used. This method is used to re place a boundary in front of a source an imaginary source located on the other side of th e boundary, as shown (a) (b) . Method of images, a) electric charge, b) magnetic dipole In Figures 4(a) and 4(b), the image sources for an electric charge and a magnetic dipole are presented, respectively. Assuming that t hese sources are placed in front of field components can be calculated by replacing the surface with the image of the source as shown in the figure. Not e that the magnitudes of the primary source and the imaginary source and the distances b etween the conductor surface and the e same for both sources. Although they look similar , the major difference between the electric and magnetic sources in terms of the application of this method is that the normal component of the field to the surfa ce of the conductor is reversed for the In order to use the method of images, the cylindric al coordinates of the rotating disk needs to be transformed into a linear space using c onformal mapping techniques. In [ the same method was used to implement the boundary conditions at the edge of the conducting disk. Unlike [ 29 ], where the image of the electrical charges is obt ained to find 25 solves for the magnetic vector potential in a PPA o n an infinitely long conductive di sk and in order to account for the actual boundary at the edge of the conductive disk, the method of images is used. This method is used to re place a boundary in front of a source an imaginary source located on the other side of th e boundary, as shown in Figure . Method of images, a) electric charge, b) magnetic dipole In Figures 4(a) and 4(b), the image sources for an electric charge and a magnetic dipole are presented, respectively. Assuming that t hese sources are placed in front of field components can be calculated by replacing the surface with the image of the source as shown in the figure. Not e that the magnitudes of the primary source and the imaginary source and the distances b etween the conductor surface and the e same for both sources. Although they look similar , the major difference between the electric and magnetic sources in terms of the application of this method is that the normal component of the field to the surfa ce of the conductor is reversed for the In order to use the method of images, the cylindric al coordinates of the rotating disk needs to be transformed into a linear space using c onformal mapping techniques. In [ 29], the same method was used to implement the boundary conditions at the edge of the ], where the image of the electrical charges is obt ained to find 26 the electric field distribution, the magnetic poten tial term (similar to magnetic dipoles) is the primary source in our problem and the image tha t is defined to implement the conductive disk’s boundary condition is realized ac cording to Figure 2L3 (b). A similar procedure to the one presented in [29] was carried out to find the imaginary sources herein. After carrying out the conformal mapping, t he location of the imaginary sources outside the conductive disk can be found and the ne t magnetic vector potential is calculated using the following relations: /g1827/g3051,/g3041/g3032/g3047 =/g1827/g3051(/g3043)(/g1876,/g1877)−/g1827/g3051(/g3036)(/g1876,/g1877) (2.40) /g1827/g3052,/g3041/g3032/g3047 =/g1827/g3052(/g3043)(/g1876,/g1877)−/g1827/g3052(/g3036)(/g1876,/g1877) (2.41) where /g1827/g3051(/g3036)(/g1876,/g1877)=−/g1827/g3051(/g3043)(/g1876(/g3036),/g1877(/g3036)) (2.42) /g1827/g3052(/g3036)(/g1876,/g1877)=−/g1827/g3052(/g3043)(/g1876(/g3036),/g1877(/g3036)) (2.43) and /g1876(/g3036)=/g4672/g3028 /g2870−/g1870/g4673+(/g3019/g3118(/g2879/g3029//g2870/g2878/g3045/g2878/g3051) (/g2879/g3029//g2870/g2878/g3045/g2878/g3051)/g3118/g2878(/g3052/g2879/g3028//g2870)/g3118) (2.44) /g1877(/g3036)=/g4672/g3029 /g2870/g4673+(/g3019/g3118(/g2879/g3028//g2870/g2878/g3052) (/g2879/g3029//g2870/g2878/g3045/g2878/g3051)/g3118/g2878(/g3052/g2879/g3028//g2870)/g3118) (2.45) and R is the radius of the conductive disk. Note that th e negative sign in Eqs. (2.42) and (2.43) results from the fact that the primary sourc e is a magnetic term and the normal component of the image is in the opposite direction to the normal component of the primary source. Once the magnetic vector potential is obtained usin g the equations presented above, the braking torque is calculated using the followin g formula: /g1846/g3029=∬/g1818/g2172×(/g1784/g1793×/g1776)/g3020/g1856/g1845 (2.46) 27 where Tb is the braking torque and /g1818/g2172 is the position vector and the magnitude of the position vector is r. In this equation, the magnetic flux density is kn own and the only unknown is the eddy current density. In order to fi nd the eddy current density, MaxwellL Ampere Law is used. We can replace the field term i n Eq. (2.1) with the magnetic vector potential as follows: /g2869 /g3091/g3435∇×(∇×/g1775/g2196/g2187/g2202 )/g3439=/g1784/g1793 (2.47) Also note that the displacement current term is zer o due to low frequency application assumption. Using the above relation for the eddy c urrent generated over the surface of the conductive disk, Eq. (2.46) can be rewritten an d the braking torque can be calculated for the specific externally applied magnetic flux d ensity (i.e. B). /g1846/g3029=/g2869 /g3091∬/g1818/g2172×((∇×∇×/g1775 /g2196/g2187/g2202 )×/g1776)/g3020/g1856/g1845 (2.48) 2.3 Validation and preliminary results In this section, the proposed analytical model is v alidated with our FEM presented in [32]. A commercial finite element method (FEM) soft ware package, COMSOL Multiphysics ®, is used to simulate the eddy current distribution within the ECB. A partial differential equation (PDE) is obtained by reorgani zing the Maxwell’s equations for the ECB problem. This PDE is then solved using proper b oundary conditions to obtain the eddy current distribution over the rotating disk. T he flowchart of the FEM is shown in Figure 3L2. FEM contains two solvers, one is the PD E solver that solves the eddy current distribution and the other is an electromagnetic so lver that introduces the effects of induction fields. For the validation purposes, sinc e the analytical model neglects the induction effects by assuming a laminated electroma gnet core, the induction field terms are removed from the numerical model as well. 28 Figure 2L4. Flowchart of the FEM implementation The domain selected for the problem is presented in Figure 2L1 and the configuration that is adapted here is taken from Lee and Park [29 ]. The configuration used for validation purposes is given in Table 2L1. Note tha t for the time varying field application, an externally sinusoidal field is applied with a ro ot mean square (RMS) magnitude of 0.3 T. Herein, this RMS magnitude of 0.3 T is specifica lly selected in order to make the AC field application results comparable with the DC fi eld application results presented in [29], which used a DC magnetic field of the same ma gnitude. This ensures that the same external energy input is supplied in both cases. 29 Table 2L1. Specifications of the ECB configuration Radius of the disk, R (m) 0.05 Width of PPA, a (m) 0.02 Height of PPA, b (m) 0.02 Pole location distance, r (m) 0.0323 Thickness, d (m) 0.003 Material Aluminum Conductivity, σ (S/m) 5.79E+07 Externally applied magnetic flux density, B0 (T RMS) 0.3 Before starting with the validation, there is an im portant step that needs to be carried out. In order to solve the Helmholtz equation defin ed in Eq. (2.25) for the AC field application in Sec. 2, a green’s function is select ed (see Eq. (2.36)). It is clear that the selected green’s function is an infinite sum of sin usoidal function in space dependent coordinates, x and y, as well as the integration variables, ζ and η. Therefore, it is necessary to identify the convergence of this series to a sol ution as the parameters, m and n, increase. In order to verify the convergence, a sim ple pilot study is carried out where the braking torque is calculated for cases in which the parameters gradually increased. Note that for simplicity, the values assigned to the par ameters, m and n, are kept equal (i.e. m = n). For this convergence study, the configuration in Table 2L1 is used and the frequency of the externally applied field and the angular rotati on of the conductive braking disk were kept constant at 10 Hz and 100 rpm, respectively. 30 Figure 2L5. Convergence plot In Figure 2L5, the variation of the braking torque with increasing m and n values is shown. From the figure, it is obvious that converge is achieved as the values increase. Note that the variation between the calculated brak ing torques when m and n are set to 40 versus 30 is 0.4%; however, when they are set to 50 , as compared to 40, this variation goes down to 0.1%. Also, note from computational po int of view, as the length of the series increases, the time required to obtain a sol ution increases quadratically. Therefore, for the subsequent simulations carried out in this section, the length of the series is set to 50. For the main validation study, two different time v arying field simulations are carried out. In the first case, the RMS braking torque gene ration is calculated using both the FEM and the analytical model at a constant rotational s peed and the effect of the frequency variation in braking torque generation is investiga ted. For the second case, while the frequency of the applied magnetic flux density is k ept constant, the variation of the RMS value of the braking torque generation is plotted w ith respect to varying angular speeds of the rotating disc in order to study their effect on the braking torque generation. In Figure 2L6, the variation in the braking torque generation with increasing frequencies is shown using both the analytical and FE models. It can be concluded that 31 under the assumptions mentioned in the previous sec tions, for the relatively low frequency range shown in the figure, the braking to rque does not depend on the frequency as the models do not account for the induction term s. In addition, at any given frequency, the difference between the analytical and FE models is around 10%. Note that the braking torque results presented in this section are given in terms of the RMS values. Figure 2L6. Variation in braking torque with freque ncy of the externally applied magnetic flux density Next, the variation of braking torque with increasi ng angular velocities of the rotating brake disk is presented in Figure 2L7. Under the as sumptions made previously to create the models, the braking torque increases linearly w ith increasing angular velocities. From the figure, the difference between the analytical m odel results and FEM results is found to be around 10% across all velocities as well. Figure 2L7 . Variation in braking torque with angular velocity of the rotating brak For both figures, it can be concluded that the tren d of the variation in results with increasing frequencies and increasing angular veloc ities are the same for both the analytical and FE model simulations. In addition, t he difference between these about 10% at any given frequencies or angular velocities; and, t hus, it can be concluded that the results of analytical model and FEM are in good agreement. Note that the reasoning behind the 10% difference and the nature of the variations in Figure 2L7 are di scussed in detail in the following section. 2.4 Discussion The results of the analytical model and the FEM sim ulations were given in the previous section when there is variation in freque presented in Figure 2L6 agreement in terms of the variation of the braking torque with respect to the AC field frequency and rotating disc velocity. The differenc e is calculated to be around 10% in the br aking torque generation across the frequency and ve locity ranges and various frequencies at the range investigated i n this paper. Below, a number of reasons that cause this difference are discussed in detail. . Variation in braking torque with angular velocity of the rotating brak For both figures, it can be concluded that the tren d of the variation in results with increasing frequencies and increasing angular veloc ities are the same for both the analytical and FE model simulations. In addition, t he difference between these any given frequencies or angular velocities; and, t hus, it can be concluded that the results of analytical model and FEM are in good agreement. Note that the reasoning behind the 10% difference and the nature of the variations in scussed in detail in the following section. The results of the analytical model and the FEM sim ulations were given in the previous section when there is variation in freque ncy and angular velocity. These results and Figure 2L7 show that the models are generally in good agreement in terms of the variation of the braking torque with respect to the AC field frequency and rotating disc velocity. The differenc e is calculated to be around 10% in the aking torque generation across the frequency and ve locity ranges of angular and various frequencies at the range investigated i n this paper. Below, a number of reasons that cause this difference are discussed in detail. 32 . Variation in braking torque with angular velocity of the rotating brak e disk For both figures, it can be concluded that the tren d of the variation in results with increasing frequencies and increasing angular veloc ities are the same for both the analytical and FE model simulations. In addition, t he difference between these curves is any given frequencies or angular velocities; and, t hus, it can be concluded that the results of analytical model and FEM are in good agreement. Note that the reasoning behind the 10% difference and the nature of the variations in Figure 2L6 and The results of the analytical model and the FEM sim ulations were given in the ncy and angular velocity. These results that the models are generally in good agreement in terms of the variation of the braking torque with respect to the AC field frequency and rotating disc velocity. The differenc e is calculated to be around 10% in the of angular velocities and various frequencies at the range investigated i n this paper. Below, a number of 33 In order to understand the source of this differenc e, the analytical and FE modeling procedures must be considered again. The analytical model was created using a number of assumptions that made the problem vary slightly from the one solved using the FEM. Although the effects of the displacement currents a re minimal at the low frequency region, the field equation used for FEM accounts fo r the displacement currents, whereas the analytical model assumes that they are negligib le. For instance, analytical model does not account for skin effects and the current is ass umed to be distributed uniformly along the thickness. However, at high frequency AC field application, analytical model results in higher current densities due to this assumption and the braking torque calculated will be higher than the real braking torque generation. Apart from this, the major source of the difference is the last assumption made in the analytical modeling procedure. This assumption stat es that the externally applied flux density is the only field term that exists on the s urface of the conductor, thus any field term over the surface of the conductive disk due to the applied flux density is not included in the solution. Although the magnitude is signific antly lower than the externally applied field, the residual field opposes the external flux density and therefore reduces the amount of the braking torque generation. On the other hand , FEM accounts for the residual magnetic flux term that is generated due to the app lication of the externally applied flux density. Use of this assumption explains the main r eason behind the higher braking torque results calculated using the analytical model. One last source of the difference is the numerical integration carried out to solve Eq. (2.48) for the braking torque using the analytical model. Since the equation is heavily dependent on the space coordinates, x and y, numerical integration methods are used to calculate the surface integral. The numerical metho d uses a solver that calculates the regional integrations over surfaces. However, when there is a circular area involved, it is harder for the solver to converge due to the curved nature of the edges of the domain and, therefore, the solver assigns an uncertainty value depending on the tolerances defined to identify the accuracy of the solution. Also, note t hat since this is a numerical integration, the computational effort increases with tighter tol erances. Herein, for the braking torque calculations of the ECB, the tolerances were adjust ed so that the uncertainty is no more than 1% of the result obtained. As the last step, the effect of the time varying fi eld The calculated braking torque that is generated whe n there is time varying field using the analytical model introduced here and the calculated torque generated when DC field is applied using the analytical model in [ Figure 2L8. It is clear from the figure that the braking torque in creases with the application of AC field and from the results, the c hange corresponds to over 40% increase for the given configuration assuming that the elect roma is no induction fields generated to oppose the exte rnal field on the core). Therefore, it can be concluded that the application of the time varyi ng field results in increased braking torque in both low and high speed regi potential for the ECB to be used as a stand Figure 2L8. Comparison between braking torques generated with A C Although the above results show its potential, in o rder to realize the stand the generated braking torque has to be enough to st op a vehicle. Considering the torque amount presented in Figure improved significantly in order to generate more br aking torque. This can be achi optimizing the applied field characteristics (i.e. frequency, magnitude), geometric As the last step, the effect of the time varying fi eld application is studied in this paper. The calculated braking torque that is generated whe n there is time varying field using the analytical model introduced here and the calculated torque generated when DC field is applied using the analytical model in [ 28 ] are compared at various rotational speeds in is clear from the figure that the braking torque in creases with the application of AC field and from the results, the c hange corresponds to over 40% increase for the given configuration assuming that the elect roma gnet core is laminated (thus there is no induction fields generated to oppose the exte rnal field on the core). Therefore, it can be concluded that the application of the time varyi ng field results in increased braking torque in both low and high speed regi ons and it can also be concluded that there is a potential for the ECB to be used as a stand Lalone brake actuator for certain applications. Comparison between braking torques generated with A C and DC field applications Although the above results show its potential, in o rder to realize the stand the generated braking torque has to be enough to st op a vehicle. Considering the torque Figure 2L8, it can be concluded that the brake configuration has to be improved significantly in order to generate more br aking torque. This can be achi optimizing the applied field characteristics (i.e. frequency, magnitude), geometric 34 application is studied in this paper. The calculated braking torque that is generated whe n there is time varying field using the analytical model introduced here and the calculated torque generated when DC field is ] are compared at various rotational speeds in is clear from the figure that the braking torque in creases with the application of AC field and from the results, the c hange corresponds to over 40% increase gnet core is laminated (thus there is no induction fields generated to oppose the exte rnal field on the core). Therefore, it can be concluded that the application of the time varyi ng field results in increased braking ons and it can also be concluded that there is a alone brake actuator for certain applications. and DC field applications Although the above results show its potential, in o rder to realize the stand Lalone ECB, the generated braking torque has to be enough to st op a vehicle. Considering the torque , it can be concluded that the brake configuration has to be improved significantly in order to generate more br aking torque. This can be achi eved by optimizing the applied field characteristics (i.e. frequency, magnitude), geometric 35 variables (i.e. diameter, thickness, width and heig ht of the PPA), material properties and number of PPAs. As a follow up to the current work, the braking tor que will be maximized with optimization techniques and using the presented ana lytical model, a controller will be designed and implemented for the ECB. 2.5 Conclusion In this chapter, the effects of time varying (i.e. AC) field application on the braking torque generation of an ECB are studied. In order o calculate the braking torque, an analytical model is derived, which was then validat ed using an accurate FEM. The results of this study show both models are in good agreemen t, within about 10% error. In addition, the reasoning behind this error along wit h the physical nature of the variations of the braking torque with respect to the AC frequency and the angular velocity of the rotating disc was discussed in detail. Finally, our analytical results showed the application of the time varying field indeed increase the braki ng torque generation capacity of an ECB. Appendix.2A. Analytical Model: DC field application This model is valid for determining the torque valu e for the whole range of angular velocities under DC field application. Initially, a simple case, where eddy currents are generated on a moving conductive plate, was studied . Figure 2A.1 shows the schematic of the moving plate with a velocity, v, in x direction. While the plate is moving, a cons tant magnetic field, B, is applied perpendicular to the plate surface. Si nce the plate contains charged particles ( q), they will feel a force, which is equal to accor ding to Lorentz Law, due to the applied magnetic field. This force causes the charged particles to separate. As a result of this separation, an electr ic field, E, between the upper and lower ends of the plate is formed. Bv×q 36 Figure 2A.1. Conductive plate moving with a velocit y (adapted from [29]) (A2.1) (A2.2) Eq. (A2.1) shows the total electric field generated ( ) on a moving conductive plate. Due to the presence of electric field, eddy currents are formed on the conductive plate. As Eq. (A2.2) states, eddy current density, J, is a function of the total electric field, and the conductivity of the plate, . In ECB, the conductor is circular and it rotates wi th an angular velocity, w. In this orientation, eddy current generation is limited to the PPA. The simple schematic of the actual case is shown in Figure 2A.2. The arrows swi rling in PPA represent the eddy currents generated. BvE Etot ×+= totE J⋅=σ totE totE σ 37 Figure 2A.2. Eddy currents formed on brake disk Initially, the disk is assumed to have infinite rad ius and the velocity of an arbitrary point P in the PPA is defined in terms of the angul ar velocity of the disk and the position of the selected point. Figure 2A.3 shows the posit ion and velocity of point P with respect to two different coordinate frames. One is the fra me which is positioned at the center of the brake disk, O’, and the other frame i s which is placed at the center of PPA, O. The distance between O and point ‘P’ is den oted by r d and the angle is ϕ’’ and the distance between O’ and point ‘P’ is denoted by and the angle is ϕ’. In addition to these, the relative distance between both frames is r. The horizontal and vertical components of the veloc ity are defined in Eq. (A2.3). (A2.3) (A2.4) where and are the scalar values of velocities in h orizontal and vertical directions with respect to the frame. z' y' x' zy x r' vx=vsinφ'=wr'sinφ'=wy vy=vcosφ'=wr'cosφ'=w(r+x) xvyv zy x Figure 2A.3 . Position and Velocity Representation of point ‘P’ (adapted from [ Due to the rotation, according to the Lorentz Law, a force is felt by the charged particles and this force causes separation. In charged particles in vertical and horizontal direct ions are shown individually. The figure on the left depicts the separation due to velocity in vertical direction presents the separation in horizontal direction. respectively. Surface charge densities on the edges of the PPA are defined in Eq. ( by applying a method used to calculate the surface charge density of a permanent magnet [39]. where is the surface charge density on the edge perpendic ular to in the positive (+) or negative ( of vacuum. Since field B value of the field, B0 is used to define the applied field. −+,siρ . Position and Velocity Representation of point ‘P’ (adapted from [ Due to the rotation, according to the Lorentz Law, a force is felt by the charged particles and this force causes separation. In Figure 2A.4, the separations between charged particles in vertical and horizontal direct ions are shown individually. The figure on the left depicts the separation due to velocity in vertical direction and the other one presents the separation in horizontal direction. 2a and 2 b are the height and width of PPA respectively. Surface charge densities on the edges of the PPA are defined in Eq. ( by applying a method used to calculate the surface charge density of a permanent magnet is the surface charge density on the edge perpendic ular to i in the positive (+) or negative ( L) quadrant with respect to axis i and is the permittivity is assumed to be perpendicular to the surface of th e disk, scalar is used to define the applied field. ρsx+=εw(r+a)B0 ρsx−=−εw(r−a)B0 ρsy+=εwbB0 ρsy−=−εwbB0 ε38 . Position and Velocity Representation of point ‘P’ (adapted from [ 29]) Due to the rotation, according to the Lorentz Law, a force is felt by the charged the separations between charged particles in vertical and horizontal direct ions are shown individually. The figure and the other one are the height and width of PPA respectively. Surface charge densities on the edges of the PPA are defined in Eq. ( A2.5), by applying a method used to calculate the surface charge density of a permanent magnet (A2.5) (A2.6) (A2.7) (A2.8) i axis (i.e. x or y) is the permittivity is assumed to be perpendicular to the surface of th e disk, scalar Figure 2A.4 . Surface charge density distribution due to the ve locity in From the surface charge densities, the electric fie ld between positive and nega charges is calculated using line charge charge densities over the corresponding edge. Equat ions ( horizontal and vertical components of the electric field, i.e. point P located at ( x, y ). ExDC=wB0 4π−  + +. Surface charge density distribution due to the ve locity in y direction (adapted from [29]) From the surface charge densities, the electric fie ld between positive and nega charges is calculated using line charge densities, which are the integrals of the surface charge densities over the corresponding edge. Equat ions ( A2.9) and ( A horizontal and vertical components of the electric field, i.e. ExDC and EyDC −(2r+2a)tan−1y−b x−a+(2r−2a)tan−1y−b x+a  +(2r+2a)tan−1y+b x−a−(2r−2a)tan−1y+b x+a +bln{(2x+2a)2+(2y−2b)2}{(2x+2a)2+(2y+2 {(2x−2a)2+(2y−2b)2}{(2x−2a)2+(2y+239 y direction (adapted From the surface charge densities, the electric fie ld between positive and nega tive are the integrals of the surface A2.10) show the DC , at an arbitrary (A2.9) 2b)2} 2b)2}  40 (A2.10) After defining the electric field, the eddy current density can be calculated from Eq. (A2.2). Eddy current densities in horizontal, Jx and vertical, Jy are defined as: (A2.11) (A2.12) Up to now, it is assumed that the disk has no bound aries. In order to account for the finite radius, a fictitious current density is intr oduced outside the disk that cancels the radial component of the charge density on the bound ary. Figure 2A.5 shows the boundary and the current densities; induced and the fictitio us. Induced current density is defined as the primary d ensities, denoted as Jx(1) , J y(1 ), which are defined in Eqs. (A2.11) and (A2.12). The fictit ious eddy current density is denoted as Jr’ (2) . Due to the cylindrical geometry, the primary eddy c urrent density is converted into cylindrical coordinates in Eq. (A2.13). Equation ( A2.14) shows the boundary condition on the perimeter of the disk. EyDC= −wB0 4π−4btan−1x+a y−b+tan−1x+a y+b−tan−1x−a y−b−tan−1x−a y+b    +2aln{(2x−2a)2+(2y−2b)2}{(2x+2a)2+(2y−2b)2} {(2x−2a)2+(2y+2b)2}{(2x+2a)2+(2y+2b)2} +2rln{(2x−2a)2+(2y−2b)2}{(2x+2a)2+(2y+2b)2} {(2x+2a)2+(2y−2b)2}{(2x−2a)2+(2y+2b)2}  Jx=σ(ExDC+w(r+x)B0) Jy=σ(EyDC+wyB0) Figure 2A.5 . Boundary conditions represented in spherical coor dinates where R is the radius of the rotating disk. Eddy current density on the surface of the disk at an arbitrary point P is calculated using mapping and method of images [ density as a function of primary and fictitious cur rent d In order to account for the opposing magnetic field due to the eddy current density, a lumped way was adapted. Induced field is calculated using the Ampere’s Law. According to Ampere’s Law, the line gives the current flow through the area encircled b y the closed loop. The closed loop chosen is shown in Figure . Boundary conditions represented in spherical coor dinates is the radius of the rotating disk. Eddy current density on the surface of the disk at an arbitrary point P is calculated using mapping and method of images [ 29]. Equation (A2.15 ) shows the eddy current density as a function of primary and fictitious cur rent d ensities. In order to account for the opposing magnetic field due to the eddy current density, a lumped way was adapted. Induced field is calculated using the Ampere’s Law. According to Ampere’s Law, the line integral of the magnetic field intensity around a c losed loop gives the current flow through the area encircled b y the closed loop. The closed loop Figure 2A.6 . The line integral of the intensity is equal to th e current 'sinJ'cosJ) ' , ' r ( J) 1 ( ' y) 1( ' x) 1( ' r φ φ φ + = Jr'(R,ϕ')=Jr'(1)(R,ϕ')−Jr'(2)(R,ϕ')=0 Jr'(r',ϕ')=Jr'(1)(r',ϕ')−Jr'(2)(R2 r',ϕ')41 . Boundary conditions represented in spherical coor dinates (A2.13) (A2.14) Eddy current density on the surface of the disk at an arbitrary point P is calculated ) shows the eddy current (A2.15) In order to account for the opposing magnetic field due to the eddy current density, a lumped way was adapted. Induced field is calculated using the Ampere’s Law. According integral of the magnetic field intensity around a c losed loop gives the current flow through the area encircled b y the closed loop. The closed loop . The line integral of the intensity is equal to th e current 42 enclosed in the loop. It is assumed that the permea bility of the magnet core is infinite and there is no reluctance load due to the electromagne t core. In addition, the air gap between the magnet core and the disk assumed to be negligib le. Figure 2A.6. Chosen closed loop (A2.16) where lg is the distance between the electromagnet yokes, Ir’ is the net eddy current, calculated by integrating the eddy current densitie s generated in PPA numerically (i.e. by integrating Eq. (A2.15)). Therefore the induced magnetic flux, Bi, was found to be: (A2.17) where is the permeability of the vacuum. Magnetic Reynolds number is used to relate the indu ced and the applied fields: ' 2r i g I Hl= Bin=µIr' 2lg µ 43 (A2.18) where Rm is the magnetic Reynolds number. When the angular velocity of the disk increases, ma gnetic Reynolds number increases and the net magnetic flux density in the air gap, BZ, decreases. The net magnetic flux density is written for the two extreme cases, when the angular velocity is zero, magnetic flux density is equal to the applied value and when the angular velocity is infinite, the net flux density is zero. With these two extreme cases, the net eddy current density is defined as: (A2.19) Finally the braking torque is calculated using Lore ntz’s law. (A2.20) where S denotes the surface area of PPA. Rm=Bin B0=µIr' 2lg1 B0 BZ=B0e−Rm Tbin= −dBZr'Jr'dS S∫ 44 3. IMPROVED BRAKING TORQUE GENERATION CAPACITY OF EDDY CURRENT BRAKE WITH TIME VARYING FIELDS: A NUMERICAL STUDY 3.1 Introduction As was previously mentioned in previous chapters, p otential advantages such as the contactless braking, silent operation, no friction and natural capacity to imitate ABS, make ECB an attractive substitute to conventional f riction brakes. However, ECB has a significant drawback, which is the reduction in the braking torque with decreasing velocities. This thesis proposes and extensively st udies the effect of AC field application on the braking torque generation of ECB. In the previous chapter, an accurate analytical mod el showed that the braking torque significantly increases with AC field application ( specifically sinusoidal field with constant frequency) for the configuration selected. In this chapter, in order to effectively deal with the problem of reduced braking torque at low speeds, the effects of various types of AC field applications on a selected ECB co nfiguration are studied. Therefore, the main objective of this work is the d evelopment of an accurate electromagnetic FEM that can be used to simulate an d analyze the exposing of the conductor disk to the various types of time varying fields for the purpose of improving the braking torque generation of the ECB at low speeds. Unlike the previous FEM, in this chapter, a model that accounts for magnetic phenome na like induction effects and displacement currents is developed. Herein, a commercial finite element software packag e, COMSOL Multiphysics ®, is used to carry out the simulations due to the simpli city of the 2LD ECB domain. The created model utilizes quadratic triangular element s and Lagrange interpolation functions, along with the boundary conditions and material pro perties. The resulting problem is solved using unsymmetric multifrontal method, which is an efficient direct solver for nonL symmetric geometries. The braking torque is calcula ted using the solution of this FEM. In addition, using this model, the optimum braking tor que capacity by application of time varying fields of various forms with constant and v ariable frequencies is determined. 45 The chapter includes the modeling of an electromagn etic FEM of the proposed ECB with AC field application. An analytical model is t hen used to validate our FEM for DC field application and then using the analytical mod el above, FEM is validated for AC field application. The validation results and the braking torque generation with fixedL and variableLfrequency time varying field applications in various waveforms are also presented in this chapter along with a discussion s ection where other means of further improving the braking torque generation are recomme nded. 3.2 Eddy current brake model In this section, an accurate electromagnetic FEM of the ECB, which solves for the magnetic flux density when there is an external fie ld (i.e. constant (DC) or time variant (AC)) present, is developed. The FEM is then used t o solve for the braking torque under both DC and AC field applications. In terms of AC f ield analysis for ECBs, no analytical model currently exists in the literature. Thus in o rder to validate our FEM, the analytical model presented by Lee et al. [29] for the DC field application was used instead. This analytical model [29] is summarized in Appendix 2A. As mentioned earlier, a commercial FEM software pac kage, COMSOL Multiphysics ®, was used to simulate the eddy current distributio n within the ECB. For the problem that is being considered, the governing lin ear Maxwell’s equations are as follows: /g1779+/g1822×/g1776=/g2869 /g3097/g3031 /g1784 (3.1) ∇×/g1779=−/g2986/g1776 /g2986/g2930 (3.2) ∇×/g1776=/g2020/g1784+/g3105/g1778 /g3105/g3047 (3.3) /g1778=/g2013/g1779 (3.4) where E is the electric field, D is the displacement flux density, µ is the magnetic permeability of vacuum, ε is the electric permittivity of vacuum, and d is the thickness of the conductive disk. When the disk rotates, eddy currents are generated on its surface according to the Faraday’s law due to the applied external magnetic field. Eq. (3.1) describes the 46 relationship (i.e. Lorentz law) between the angular velocity of the rotating disk and the eddy currents generated over the surface that is un der the magnetic field application. For the basic ECB problem, a 2LD domain shown in Figure 3L1 was selected. The outer circular shape represents the conductor disk and th e inner rectangular region is the PPA, the area under the poles of the electromagnet (i.e. the area that the external magnetic field is applied onto). Note that for the given problem, it is assumed that the electromagnetic vartiables are uniformly distributed over the thick ness, i.e. there are no skin effects on the conductor. Figure 3L1. Basic ECB domain considered for simulat ing the eddy current generation In order to solve for the eddy currents, the govern ing equations were then transformed into a partial differential equation (PDE). Eq. (3. 1) can be rewritten as: /g1779+v×/g1776=/g2869 /g3091/g3097/g3031 (∇×/g1776−/g2020/g2986/g1778 /g2986/g2930) (3.5) where the current density is defined in terms of th e magnetic flux density and the displacement current density using Eq. (3.3). For s implicity, in this equation, the 47 displacement current density is replaced with the e lectric field according to Eq. (3.4). Next, the circulation of the vectors on both sides of Eq. (3.5) are taken, i.e. ∇×(/g1779+/g1822×/g1776)=/g2869 /g3091/g3097/g3031 ∇×(∇×/g1776−/g2013/g2020 /g2986/g1779 /g2986/g2930) (3.6) Eq. (3.6) can be simplified using the Faraday’s Law , Eq. (3.2) as: −/g2986/g1776 /g2986/g2930+∇×(/g1822×/g1776)=/g2869 /g3091/g3097/g3031 ∇×(∇×/g1776)+/g3084 /g3097/g3031 /g2986/g3118/g1776 /g2986/g2930/g3118 (3.7) Above equation is called “the field equation”, sinc e it is only dependent on the magnetic flux density present in the medium. As the driving force of the magnetic flux density, B0 is the externally applied flux density. For the ECB problem, it is assumed that the externa lly applied field, B0, is uniformly distributed over the PPA, so that the field has no spatial dependence. Therefore, the magnetic flux term that represents the total flux d ensity in Eq. (3.7) is replaced with B + B0. Since the curl operator is distributive, Eq. (3.7 ) can be written as: (3.8) The second term on the right hand side of the equat ion and the last term on the left hand side of the equation are zero since it is assumed t hat B0 is not a function of the spatial coordinates and it is uniformly distributed over th e PPA. Note that the applied and induced magnetic fluxes, B + B 0 and Bin , have only the z components (perpendicular to the surface), while the current density and electri c field, J and E, have the x and y components. In addition, in the above equation, the first term on the left hand side and the last term on the right hand side of the equation ar e equal to zero, if no time varying field (i.e. AC) is present. −∂(B + B0) ∂t+∇× (v×B)+∇× (v×B0)=1 µσd∇×(∇×B) +1 µσd∇×(∇×B0) +ε σd∂2(B + B0) ∂t2 48 Under these conditions/assumptions, by rearranging the terms and replacing the vector terms with their magnitudes, Eq. (3.8) can be conve rted into a PDE in a coefficient form, i.e. −/g1837/g2986/g3003 /g2986/g2930−/g1863/g2986/g3118/g3003 /g2986/g2930/g3118+∇∙/g3436∇/g1828−/g3436/g1837/g1874/g3051 /g1837/g1874 /g3052/g3440/g1828−/g3436/g1837/g1874 /g3051 /g1837/g1874 /g3052/g3440/g1828/g2868/g3440=/g1837/g2986/g3003/g3116 /g2986/g2930+/g1863/g2986/g3118/g3003/g3116 /g2986/g2930/g3118 (3.9) where K = /g2020/g2026/g1856 is the constant mass coefficient, k = /g2013/g2020 is the damping coefficient, and vx and vy are the velocities of a point on the rotating cond uctor disk in the x and y directions, respectively. In the above equation, B and B0 are the magnitudes of the magnetic flux density and the constant magnetic flux density in t he PPA, respectively. COMSOL Multiphysics ® was then used to solve the above ECB problem. As t he first step, the 2LD domain in Figure 3L1 was created usin g the geometric ECB design parameters shown in Table 3L1, followed by the assi gnment of the electrical and magnetic properties of the media in the domain. The domain w as then discretized using quadratic triangular elements (6 nodes/element) and Lagrange interpolation functions. After the mesh was created, boundary conditions were defined. For the given problem, the boundary between the conductor disk and the outside environment was described using homogenous Dirichlet Boundary Condition. As the final step, considering the nonL symmetric and nonlinear nature of the problem, the timeLdependent unsymmetric MultiFrontal method (UMFPACK), which is a highly efficient direct solv er for nonlinear systems, was selected and employed. Finally, the ab solute and relative tolerances were set as 0.1 % and 0.001 % for the problem. Next, the FEM was carried out in order to calculate the magnetic flux density distribution over the defined 2LD ECB domain. Once the flux density distribution is obtained, the corresponding eddy current densities can be calculated in the x and y directions using Eq. (3.3), i.e. /g2986/g3003 /g2986/g2935=/g2020/g1836/g3051 (3.10) /g2986/g3003 /g2986/g2934=/g2020/g1836/g3052 (3.11) 49 Note that in order to obtain Eqs. (3.10) and (3.11) , it is assumed that the displacement current density is equal to zero. However, this mod el does not account for the field induced by the eddy currents. When an electromagnet with a permeable ferromagnetic core is used to apply the magnetic field onto the r otating disk, the induced magnetic field cannot be ignored if an accurate analysis required. Due to the induced field, an opposing magnetic flux will be generated within the core, an d this flux causes the net magnetic field to decrease. Such a decrease results in less eddy current generation over the surface of the conductor, thus reducing the braking torque. In order to model the eddy currents more accurately with the consideration of induction effects, an additional (secondary) electr omagnetic model was defined and coupled to the primary model in the FEM. The flowch art of our FEM is shown in Figure 3L2. The primary model, the PDE solver, is shown on the left side and the secondary model, the electromagnetic solver, is shown on the right side of the figure. The secondary model uses the same CAD model, mesh and material pr operties as the primary model; however, the underlying physics, inputs and boundar y conditions are all different between the two models. For example, electric insulation (e .g., the normal component of the electric field in the boundary is set to be zero) i s used as the boundary condition for the secondary model. In addition, it takes the results of the primary model, i.e. the current density distribution on the disk, as its input and solves for the induced magnetic flux density, i.e. Bin , and the induced current density, i.e. Jin , using the same solution method (UMFPACK). Then, they are subtracted from the appli ed field and current densities to simulate braking torque ( Tbin ) more accurately, i.e. /g1846/g3029/g3036/g3041 =/g1516/g1818/g1790×((/g1784−/g1784 /g2191/g2196 )×(/g1776+/g1776 /g2777−/g1776/g2191/g2196 ))/g1856/g1876/g1856/g1877 /g2903 (3.12) 50 Figure 3L2. Flowchart of the proposed FEM implement ation Note that in Figure 3L2, the blue arrows depict the solution process that is used to solve for the flux distribution without the induction fie lds. The results then can be used to calculate the braking torque, i.e. Tb, which does not include the induction terms. In or der to obtain the braking torque generation with the in duction effects, i.e. Tbin , the results of the primary model is run through the secondary mode l and the process is shown using the black arrows. 3.3 Simulation results In this section, the results obtained using both an alytical and FE models are presented. For benchmarking and verification of the proposed F EM, ECB configuration that was studied analytically and experimentally in [29] wit h DC field application was adapted, 51 and Table 2L1 shows the specifications of this conf iguration (note that for simplicity, the specifications of the adapted configuration is also given in Table 3L1). The analytical model of [29] is summarized in Appendix 2A. Table 3L1. Specifications of the configuration adap ted from [29 ] Radius of the disk, R (m) 0.05 Width of PPA, a (m) 0.02 Height of PPA, b (m) 0.02 Pole location distance, r (m) 0.0323 Thickness, d (m) 0.003 Material Aluminum Conductivity, σ (S/m) 5.79E+07 Externally applied magnetic flux density, B0 (T) 0.3 First, the analytical model was used to validate th e proposed FE model of the ECB for DC field application. Then, using the FEM, the brak ing torque generation capacity of various timeLvarying field applications were studie d and the frequency specifications of the applied time varying field were optimized on a configuration introduced specifically for automotive applications. Finally, for further i mprovement in the braking torque generation, the effects of variable frequency field application were studied and the frequency specifications of such fields were optimi zed on the configuration considered, in order to achieve the maximum braking torque. The re sults are presented in the corresponding subsections. 3.3.1 Validation of the FEM In order to validate the FE model, an external magn etic flux density with a magnitude of 0.3 T was applied over a wide range of rotationa l velocities, 0~1000 rpm. In Figure 3L3, the results obtained for the DC field applicat ion from the analytical model and FEM are compared. The curve with the ‘×’ data points sh ows the results of the FEM and the curve with the diamond data points shows the result s of the analytical model. These two curves, which are in good agreement with each other , clearly show the saturation of the braking torque at higher speeds due to the induced field effects. In [29], the analytical results and the corresponding experimental results were in good agreement as well, and thus we can conclude that our FEM is also validated . Figure 3L3 . Comparison between analytical and FEM results for DC field In Figure 3L3 , the curve with the triangular data points represe nts the case where the induction effects are assumed negligible. Note that , if permanent magnets or laminated electromagnet core were used as no induction effects due to the electrical properti es of the magnets. Due to the flexibility and the structure of the proposed FEM, once the sol ution is carried out, the results with and without the induction effects can easily be calculated using th e results of the secondary and preliminary models shown in The noL induction case resul However, if an electromagnet with a soft magnetic c ore is used instead, there will be a field induced in the core of the electromagnet. Thi s opposing field reduces the capacity of eddy current and braki ng torque generation in the ECB, resulting in a non linear torque speed relationship similar to the other two curves in speed region, the braking torque capacity of the no induction case – this is due to the fact that the rotational effects are dominant over the induction effects in this low In terms of the validation with in detail in the previous chapter is used. The deta ils of the model and the assumptions . Comparison between analytical and FEM results for DC field , the curve with the triangular data points represe nts the case where the induction effects are assumed negligible. Note that , if permanent magnets or laminated electromagnet core were used as the source of the external applied field, there wou ld be no induction effects due to the electrical properti es of the magnets. Due to the flexibility and the structure of the proposed FEM, once the sol ution is carried out, the results with induction effects can easily be calculated using th e results of the secondary and preliminary models shown in Figure 3L2. induction case resul ts in higher braking torque in the high However, if an electromagnet with a soft magnetic c ore is used instead, there will be a field induced in the core of the electromagnet. Thi s opposing field reduces the capacity of ng torque generation in the ECB, resulting in a non linear torque speed relationship similar to the other two curves in Figure 3L3 . Note that, in the lo speed region, the braking torque capacity of the no Linduction case matches that of the this is due to the fact that the rotational effects are dominant over the induction effects in this low Lspeed region. In terms of the validation with AC field application, the model created and describ ed in detail in the previous chapter is used. The deta ils of the model and the assumptions 52 . Comparison between analytical and FEM results for DC field analysis , the curve with the triangular data points represe nts the case where the induction effects are assumed negligible. Note that , if permanent magnets or laminated the source of the external applied field, there wou ld be no induction effects due to the electrical properti es of the magnets. Due to the flexibility and the structure of the proposed FEM, once the sol ution is carried out, the results with induction effects can easily be calculated using th e results of the ts in higher braking torque in the high Lspeed region. However, if an electromagnet with a soft magnetic c ore is used instead, there will be a field induced in the core of the electromagnet. Thi s opposing field reduces the capacity of ng torque generation in the ECB, resulting in a non linear torque L . Note that, in the lo w induction case matches that of the this is due to the fact that the rotational effects are dominant over the AC field application, the model created and describ ed in detail in the previous chapter is used. The deta ils of the model and the assumptions made to solve the problem were mentioned in the pre vious chapter. Note that the induction effects are ignore applied. Two different simulations are carried out. In the first case, the torque generation is calculated using the FEM and t he analytical model at rotational speed (50 rpm) and second case, the frequency of the applied magnetic flux density is kept constant at 10 Hz and the variation of the varying angular speeds. Figure made to solve the problem were mentioned in the pre vious chapter. Note that the induction effects are ignore d for the AC field validation an d a sinusoidal AC field is applied. Two different simulations are carried out. In the first case, the torque generation is calculated using the FEM and t he analytical model at rpm) and the effect of the frequency variation is investigat ed. For the second case, the frequency of the applied magnetic flux density is kept constant at 10 Hz and the variation of the RMS value of the braking torque generation is plotted w ith Figure 3L4. Braking torque versus frequency at 50 rpm 53 made to solve the problem were mentioned in the pre vious chapter. Note that the d a sinusoidal AC field is applied. Two different simulations are carried out. In the first case, the RMS braking torque generation is calculated using the FEM and t he analytical model at a constant the effect of the frequency variation is investigat ed. For the second case, the frequency of the applied magnetic flux density is kept constant at 10 Hz value of the braking torque generation is plotted w ith Figure 3 For both figures, it can be concluded that the tren d of the variation in results with increasing frequencies and increasing angular veloc ities are same for both analytical and FE model simulations. In addition, the error betwee n these curves is frequency or any angular velocity and thus it can b e concluded that the results of analytical model and FEM are in good agreement. Not e that th difference and the nature of the variations in detail in the discussion section of the previous chapter 3.3.2 Time varying field application with induction effec ts In this section, braking torque generation of ECB u nder time v applications with various waveforms, i.e. sinusoida l, sawtooth, triangular and square, was studied. Note that for proper comparison, the root mean square ( magnitudes of applied flux density is kept at 0.3 T . Setting a constan steady energy input for the application of various waveforms, which makes it possible to compare the results of the time varying field appli cations with those of the DC field application. Also note that the maximum field frequ ency was computational cost affordable. In addition, since E CB performs poor at low rotational 3L5. Braking torque versus rotational velocity at 10 Hz For both figures, it can be concluded that the tren d of the variation in results with increasing frequencies and increasing angular veloc ities are same for both analytical and FE model simulations. In addition, the error betwee n these curves is less than frequency or any angular velocity and thus it can b e concluded that the results of analytical model and FEM are in good agreement. Not e that th e reasoning behind the the nature of the variations in Figure 3L4 and Figure 3L 5 discussion section of the previous chapter . Time varying field application with induction effec ts In this section, braking torque generation of ECB u nder time v applications with various waveforms, i.e. sinusoida l, sawtooth, triangular and square, was studied. Note that for proper comparison, the root mean square ( RMS magnitudes of applied flux density is kept at 0.3 T . Setting a constan t magnitude ensures a steady energy input for the application of various waveforms, which makes it possible to compare the results of the time varying field appli cations with those of the DC field application. Also note that the maximum field frequ ency was set at 100 Hz to keep the computational cost affordable. In addition, since E CB performs poor at low rotational 54 Braking torque versus rotational velocity at 10 Hz For both figures, it can be concluded that the tren d of the variation in results with increasing frequencies and increasing angular veloc ities are same for both analytical and less than 10% at any frequency or any angular velocity and thus it can b e concluded that the results of e reasoning behind the 5 are discussed in In this section, braking torque generation of ECB u nder time v arying field applications with various waveforms, i.e. sinusoida l, sawtooth, triangular and square, was RMS ) value of the t magnitude ensures a steady energy input for the application of various waveforms, which makes it possible to compare the results of the time varying field appli cations with those of the DC field set at 100 Hz to keep the computational cost affordable. In addition, since E CB performs poor at low rotational velocities, the simulations are carried out for rot ational velocities ranging from 0 to 100 rpm. A sinusoidal flux density with a magnitude of cycle and a magnitude of 0.3 T, a sawtooth wave flu x density with a magnitude of 0.52 T, and a triangular wave flux density with a magnitude of 0.52 T were applied to the FEM, keeping their RMS value at 0.3 T. In orde harmonic frequencies of Fourier expansion for each waveform are used. The simulations results for a range of velocities as well as for di fferent frequencies are shown in 3L6. Figure 3L6 . Braking torque generation The above results show that the AC braking torque i s higher than the DC braking torque at the low frequencies of all four waveforms . However, all four waveforms show a decrease in the braking torque with increasing freq uencies. At 100 Hz, for AC braking torque is actually lower than that of th e DC counterpart across all velocities and waveforms. At 1 Hz, on the other hand, the brea k torque increased by 24% with the velocities, the simulations are carried out for rot ational velocities ranging from 0 to 100 A sinusoidal flux density with a magnitude of 0.4243 T, a square wave with 50% duty cycle and a magnitude of 0.3 T, a sawtooth wave flu x density with a magnitude of 0.52 T, and a triangular wave flux density with a magnitude of 0.52 T were applied to the FEM, value at 0.3 T. In orde r to model the periodic waveforms, first 10 harmonic frequencies of Fourier expansion for each waveform are used. The simulations results for a range of velocities as well as for di fferent frequencies are shown in . Braking torque generation – (a) sinusoidal, (b) square wave, (c) sawtooth wave, and (d) triangular wave field applications The above results show that the AC braking torque i s higher than the DC braking torque at the low frequencies of all four waveforms . However, all four waveforms show a decrease in the braking torque with increasing freq uencies. At 100 Hz, for AC braking torque is actually lower than that of th e DC counterpart across all velocities and waveforms. At 1 Hz, on the other hand, the brea k torque increased by 24% with the 55 velocities, the simulations are carried out for rot ational velocities ranging from 0 to 100 0.4243 T, a square wave with 50% duty cycle and a magnitude of 0.3 T, a sawtooth wave flu x density with a magnitude of 0.52 T, and a triangular wave flux density with a magnitude of 0.52 T were applied to the FEM, r to model the periodic waveforms, first 10 harmonic frequencies of Fourier expansion for each waveform are used. The simulations results for a range of velocities as well as for di fferent frequencies are shown in Figure (a) sinusoidal, (b) square wave, (c) sawtooth wave, and The above results show that the AC braking torque i s higher than the DC braking torque at the low frequencies of all four waveforms . However, all four waveforms show a decrease in the braking torque with increasing freq uencies. At 100 Hz, for example, the AC braking torque is actually lower than that of th e DC counterpart across all velocities and waveforms. At 1 Hz, on the other hand, the brea k torque increased by 24% with the sinusoidal field, 3% with the square wave field, 34 % with the sawt 39% with the triangular field application. The rela tionship between the braking torque and frequency is shown in Figure the DC braking torque. Figure 3L7 . Braking torque variation with increasing frequenc y at 50 rpm with induction effects Such behaviour of ECB under time the physics of the eddy currents. They are generate d by means of variation in the applied field, either spatial or time dependent. In the cas es that we have considered above, the eddy currents were genera variation in the externally applied field with resp ect to time. While the rotational velocity of the conductor is kept constant, increasing frequ encies will result in higher eddy current generati on on the surface of the conductor and induction cu rrents will be generated on the core of the electromagnet. Due to these currents, a n induction field is generated in the core opposing the externally applied field. In this specific ECB problem, induction will increase with increasing frequencies, causing significant reduction in the braking torque, as described by Eq. sinusoidal field, 3% with the square wave field, 34 % with the sawt ooth wave field, and 39% with the triangular field application. The rela tionship between the braking torque and Figure 3L7. Note that the black dashed line in the figure represents . Braking torque variation with increasing frequenc y at 50 rpm with induction effects of ECB under time varying (AC) field application can be explained by the physics of the eddy currents. They are generate d by means of variation in the applied field, either spatial or time dependent. In the cas es that we have considered above, the eddy currents were genera ted by means of the rotation of the conductor as we ll as the variation in the externally applied field with resp ect to time. While the rotational velocity of the conductor is kept constant, increasing frequ encies will result in higher eddy current on on the surface of the conductor and induction cu rrents will be generated on the core of the electromagnet. Due to these currents, a n induction field is generated in the core opposing the externally applied field. In this specific ECB problem, induction will increase with increasing frequencies, causing significant reduction in the braking torque, as described by Eq. (3.12). 56 ooth wave field, and 39% with the triangular field application. The rela tionship between the braking torque and that the black dashed line in the figure represents . Braking torque variation with increasing frequenc y at 50 rpm with induction effects varying (AC) field application can be explained by the physics of the eddy currents. They are generate d by means of variation in the applied field, either spatial or time dependent. In the cas es that we have considered above, the ted by means of the rotation of the conductor as we ll as the variation in the externally applied field with resp ect to time. While the rotational velocity of the conductor is kept constant, increasing frequ encies will result in higher eddy current on on the surface of the conductor and induction cu rrents will be generated on the core of the electromagnet. Due to these currents, a n induction field is generated in the core opposing the externally applied field. In this specific ECB problem, induction effects will increase with increasing frequencies, causing significant reduction in the braking In order to improve the braking to induction effects must be removed from the system b y physically replacing the soft ferromagnetic core with a laminated magnetic core. Note that the simulations in the subsequent sections were carried out witho purposes, the braking torque generation without ind uction effects was studied for various types of field applications at 50 rpm over a range of frequencies and the results are plotted in Figure 3L8 . It is clear from the figure that the braking torq ue increases with the time varying field application over the frequency range studied, compared to the DC field braking torque presented using the black dashed lin e in Figure 3L8 . Braking torque variation with increasing frequenc y at 50 rpm without induction 3.3.3 ECB configuration for aut The previous results show that the braking torque g eneration is significantly improved with the application of the time varying field, esp ecially with the triangular wave field application. Herein, an ECB configuration is introd uced spe applications. The specifications of the selected co nfiguration are given below in 3L2. This configuration is applied to In order to improve the braking to rque of the ECB with a time varying field, the induction effects must be removed from the system b y physically replacing the soft ferromagnetic core with a laminated magnetic core. Note that the simulations in the subsequent sections were carried out witho ut the induction effects. For comparison purposes, the braking torque generation without ind uction effects was studied for various types of field applications at 50 rpm over a range of frequencies and the results are plotted . It is clear from the figure that the braking torq ue increases with the time varying field application over the frequency range studied, compared to the DC field braking torque presented using the black dashed lin e in Figure 3L8. . Braking torque variation with increasing frequenc y at 50 rpm without induction effects ECB configuration for aut omotive application The previous results show that the braking torque g eneration is significantly improved with the application of the time varying field, esp ecially with the triangular wave field application. Herein, an ECB configuration is introd uced spe cifically for automotive applications. The specifications of the selected co nfiguration are given below in . This configuration is applied to all simulations that follow. 57 rque of the ECB with a time varying field, the induction effects must be removed from the system b y physically replacing the soft ferromagnetic core with a laminated magnetic core. Note that the simulations in the ut the induction effects. For comparison purposes, the braking torque generation without ind uction effects was studied for various types of field applications at 50 rpm over a range of frequencies and the results are plotted . It is clear from the figure that the braking torq ue increases with the time varying field application over the frequency range studied, compared to the DC field . Braking torque variation with increasing frequenc y at 50 rpm without induction The previous results show that the braking torque g eneration is significantly improved with the application of the time varying field, esp ecially with the triangular wave field cifically for automotive applications. The specifications of the selected co nfiguration are given below in Table 58 Table 3L2. Selected ECB configuration for automotiv e applications Radius of the disk, R (mm) 120 Width of the pole, a (mm) 30 Height of the pole, b (mm) 30 Pole location distance, r (mm) 95 Thickness, d (mm) 10 The CAD model of the FEM was updated accordingly fo r the new configuration, and the sinusoidal and triangular waveforms are used as the input sources individually. 3.3.4 Time varying field application and optimizati on without induction effects As the next step, the braking torque generation in the automotive ECB configuration in Table 3L2 is optimized. For the application of a periodic waveform such as sinusoidal or triangular, the frequency is set to be the only design variable and the resulting braking torque was plotted for various frequencies. The opt imum frequency that results in the maximum braking torque was then obtained from the p lot. Note that the geometry, properties, physics and mesh of the FEM are not cha nged for the optimization. The solution is obtained by modifying the input source term, which is a function of the frequency, and is updated at each iteration. The FE M does not require significant computational effort to obtain a result for a selec ted input source waveform at a particular frequency. The resulting ECB optimization problem does not hav e any constraints. However, the solution has to account for the skin effects due to the application of alternating flux densities. When an AC is applied to the surface of a conductor, the generated eddy current has a tendency to accumulate in the vicinity of the surface. Thus, the current density decreases with increasing depth into the material, and it diminishes after a certain depth, which is known as the skin depth. The skin depth ( dskin ) can be expressed in terms of the material properties and the frequency of the time v arying current applied, i.e. /g1856/g3046/g3038/g3036/g3041 =/g3495/g3096/g3293/g3280/g3294 /g3095/g3033/g3091 (3.13) where ρres is the resistivity and µ is the absolute magnetic permeability of the condu ctor, and f is the frequency of the current applied on the con ductor surface. As the proposed FEM does not account for the skin effects, the thic kness value given in corrected, if the skin depth is less than half of t he thickness. In such a case, the corrected thickness value (i.e. dcorr ) is set to be twice the skin depth and the simulat ions are carried out using this updated value From previous results in Sec. 3.3 braking torque values among various waveforms. Thus , here the frequencies of the triangular field and the sinusoidal field, which th e latter is served as a reference, are optimized for higher braking to space for the frequency of the applied field is def ined between 1 Hz and 100 Hz. Solving the optimization problem, it is found that the optimum frequency for both triangular and sinusoidal field ap generations of the ECB configuration at 10 Hz are s hown in applications. Figure 3L9 . Braking torque generation at 10 Hz for triangular , sinusoidal and DC field According to the results, with the DC field applica tion, the ECB configuration of 3L2 can generate 10.51 Nm at 100 rpm. However, the brak ing torque is increased by FEM does not account for the skin effects, the thic kness value given in corrected, if the skin depth is less than half of t he thickness. In such a case, the corrected ) is set to be twice the skin depth and the simulat ions are carried out using this updated value . rom previous results in Sec. 3.3 .2 , triangular wave field application gives the highe st braking torque values among various waveforms. Thus , here the frequencies of the triangular field and the sinusoidal field, which th e latter is served as a reference, are optimized for higher braking to rque generation. For the optimization problem, the design space for the frequency of the applied field is def ined between 1 Hz and 100 Hz. Solving the optimization problem, it is found that the optimum frequency for both triangular and sinusoidal field ap plication is around 10 Hz. The braking torque generations of the ECB configuration at 10 Hz are s hown in Figure 3 . Braking torque generation at 10 Hz for triangular , sinusoidal and DC field applications According to the results, with the DC field applica tion, the ECB configuration of can generate 10.51 Nm at 100 rpm. However, the brak ing torque is increased by 59 FEM does not account for the skin effects, the thic kness value given in Table 3L2 is corrected, if the skin depth is less than half of t he thickness. In such a case, the corrected ) is set to be twice the skin depth and the simulat ions are carried , triangular wave field application gives the highe st braking torque values among various waveforms. Thus , here the frequencies of the triangular field and the sinusoidal field, which th e latter is served as a reference, are rque generation. For the optimization problem, the design space for the frequency of the applied field is def ined between 1 Hz and 100 Hz. Solving the optimization problem, it is found that the optimum frequency for both plication is around 10 Hz. The braking torque 3L9 for both field . Braking torque generation at 10 Hz for triangular , sinusoidal and DC field According to the results, with the DC field applica tion, the ECB configuration of Table can generate 10.51 Nm at 100 rpm. However, the brak ing torque is increased by 60 22.5% with the application of the sinusoidal field at 10 Hz and 34.5% with the application of the triangular wave field at 10 Hz. 3.3.5 Effects of frequency modulation on braking to rque generation In this section, it is proposed that the braking to rque can be increased further with the application of a field with variable frequency onto the conductor disk. The variation of frequency of the time varying field can be defined using the frequency modulation (FM). FM is commonly used in telecommunication and signal processing to convey information over a carrier wave by varying the instantaneous fr equency of the signal. Here, the same theory is used to define a sinusoidal field that ha s varying instantaneous frequencies. The FM can be implemented as: /g1858/g2869=∆/g1858sin (2π/g1858 /g2870/g1872) (3.14) /g1828 =/g1828/g2868 sin (2π/g1858 /g2869/g1872) (3.15) where t is time, f1 is the carrier frequency, i.e. the frequency of th e applied time varying field, B0 is the peak magnitude of the applied field, f2 is the modulation frequency and ∆ f is the frequency deviation. Once the above equation s are rearranged, the modulated sinusoidal field equation becomes: /g1828 =/g1828/g2868sin (2π/g1858 /g2869/g1872−∆/g3033 /g3033/g3118cos (2π/g1858/g2870/g1872)) (3.16) Note that in order to obtain Eq. (3.16), the freque ncy variation has to be transformed into angular variation; since in reality the modulation occurs through the angular variation, not frequency. In terms of ECB braking performance, a sinusoidal w ave and a triangular wave are applied individually with variable frequencies to s tudy the improvements via FM. For comparison purposes, it is assumed that the RMS mag nitude of the applied flux densities is equal to 0.707 T (peak magnitude of 1 T for sinu soidal and 1.225 for triangular wave). The carrier frequency, frequency deviation and the modulation frequency are then optimized for higher braking torque using a gradien t free genetic algorithm (GA). For the 61 given problem, a tolerance value is defined in orde r to terminate the optimization. Although stochastic methods such as GA are powerful optimization algorithms, they are not guaranteed to converge to the global optimum. H owever, by running the algorithm multiple times for the same ECB problem, an improve d accuracy is obtained in terms of the optimum frequency. The proposed optimization pr oblem is given as follows: Maximize : /g1846/g3029=/g1516/g1818/g1787×/g3435/g1784×(/g1776+/g1776/g2777)/g3439/g1856/g1876/g1856/g1877 /g2903 (3.17) Correction : /g1856/g3032/g3033/g3033 =min /g46682/g1856/g3046/g3038/g3036/g3041 ,/g1856/g4669 (3.18) where deff is the corrected thickness value by accounting for the skin effects. Similar to the periodical wave optimization mentioned earlier in Sec 3.4, the skin effects are accounted for using the relation given in Eq. (3.18 ). The optimization results show that the sinusoidal f ield application results in the maximum braking torque when carrier frequency, modu lation frequency and frequency deviation are equal to 52, 52 and 51 Hz, respective ly. On the other hand, triangular wave application results in the maximum braking torque w hen the frequencies are 39, 39 and 38 Hz, respectively. These results are shown in Figure 3L10. Figure 3L10 . Optimum braking torque generation with frequency modulation According to the results, the braking torque is inc reased by 52.8% with the application of sinusoidal field via FM and over 60% with th reaching a maximum value of 16.85 Nm at 100 rpm (in comparison to only 10.51 Nm with the application of the DC field). 3.4 Discussion Although ECBs have a number of performance advantag es over CHBs, braking torque generation at low speeds is the braking performance of the ECB, the potential o f the time varying (AC) field application was studied. A FEM, which was verified with the analytical DC model and experime ntal results in the literature, was used to investi gate the variation in the braking torque with time varied field application. In addition to the mesh dependent analysis carried out here, there are various FE techniques that offer promising improvements traditional mesh based algorithms for specific prob lems. Mesh free algorithms, e.g., smoothed finite element method (S are examples of such numerical methods. These algor ithms may result in higher accura . Optimum braking torque generation with frequency modulation According to the results, the braking torque is inc reased by 52.8% with the application of sinusoidal field via FM and over 60% with th e application of triangular field via FM, reaching a maximum value of 16.85 Nm at 100 rpm (in comparison to only 10.51 Nm with the application of the DC field). Although ECBs have a number of performance advantag es over CHBs, braking generation at low speeds is considerably low. In this chapter , in order to improve the braking performance of the ECB, the potential o f the time varying (AC) field application was studied. A FEM, which was verified with the analytical DC model and ntal results in the literature, was used to investi gate the variation in the braking torque with time varied field application. In addition to the mesh dependent analysis carried out here, there are various FE techniques that offer promising improvements in accuracy and computation time over the traditional mesh based algorithms for specific prob lems. Mesh free algorithms, e.g., smoothed finite element method (S LFEM) [40L47 ] and isogeometric FE analyses [ are examples of such numerical methods. These algor ithms may result in higher accura 62 . Optimum braking torque generation with frequency modulation According to the results, the braking torque is inc reased by 52.8% with the application of e application of triangular field via FM, reaching a maximum value of 16.85 Nm at 100 rpm (in comparison to only 10.51 Nm Although ECBs have a number of performance advantag es over CHBs, braking , in order to improve the braking performance of the ECB, the potential o f the time varying (AC) field application was studied. A FEM, which was verified with the analytical DC model and ntal results in the literature, was used to investi gate the variation in the braking In addition to the mesh dependent analysis carried out here, there are various FE in accuracy and computation time over the traditional mesh based algorithms for specific prob lems. Mesh free algorithms, e.g., ] and isogeometric FE analyses [ 48L50], are examples of such numerical methods. These algor ithms may result in higher accura cy 63 and faster computation times for the ECB problem st udied. However, due to the relatively simplistic geometric configuration of our ECB probl em and readily available solution methods for traditional mesh dependent algorithms, a mesh based FEM was adopted and used to study the braking torque generation for the given ECB configuration. The results of the numerical analysis for different waveform applications were presented in Sec. 3.3. Note that the braking torque generated with DC field application was used as the reference for the cases studied abo ve. There are three main conclusions that can be drawn from the results obtained. First conclusion is that the braking performance of the ECB strongly depends on the natu re of the externally applied flux density. In this numerical study, the braking torque generat ion with sinusoidal, square, sawtooth and triangular wave applications was inves tigated. In all cases, a higher braking torque was obtained at veryLlow frequencies, in com parison to the DC field case. Faraday’s law of induction explains this behaviour. It states that when there exists a time varying magnetic flux, an electric field that is pr oportional to the rate of change of the flux will be generated. Having this additional elec tric field term results in variation in eddy current generation on the surface of the condu ctor. Thus, the braking torque will also vary according to the rate of change of the ap plied field. Among all applied waveforms, the triangular wave ha s the highest and steadiest rate of change and, as it was expected, the results in F igure 3L7 showed that the braking torque increased the most. Thus it is concluded that for v arious waveforms applied at the same frequency, the time rate of change of the wave defi nes the amount of the braking torque generation of ECB. In the case of the square wave a pplication, the torque variation is not significant, mainly because of the similar shape of the square wave to the DC field. In the square wave, the variation occurs only at the insta nts when the magnitude changes the sign, from negative to positive or vice versa. Sinc e the variation is significantly limited, the results are almost identical with the DC field braking torque generation. Second conclusion that can be drawn is that the bra king torque can be significantly increased with the application of time varying magn etic flux at low frequencies. Sinusoidal, sawtooth and triangular wave applicatio ns result in an average of 24%, 34% and 39% increase at 1 Hz over that of the DC applic ation, respectively. The increase in 64 the braking torque can be explained with the genera tion of additional eddy currents on the surface. At 5 Hz, the braking torque increases by 2 1% with sinusoidal wave, 16% with sawtooth wave and 31% with triangular wave applicat ion. Finally, from the results presented in the figures in Sec. 3.3, it can be concluded that with increasing frequencies, the braking torque dec reases in all waveforms. This behaviour can be explained with the induction effec ts. As the frequency increases, the eddy currents generated due to the time variation i ncrease dramatically, but with this increase, the induction currents generated on the s urface of the electromagnet core also increase and they result in an opposing field that reduces the applied field. Therefore, the amount of the braking torque decreases as a result. The variation of the braking torque with respect to the frequency at a constant rotatio nal speed of 50 rpm for all waveforms applied was shown in Figure 3L7. In terms of the practicality, it is not ideal to ha ve reduction in the braking torque with increasing frequencies. Hence, for further increase in the braking torque generation, the induction effects need to be removed by replacing t he soft ferromagnetic core of the electromagnet with a laminated one. The same plots presented in Figure 3L7 were redrawn for the case without the induction effects in Figur e 3L8. Comparing these two figures, at low frequencies, the induction effects are negligib le and the max braking torque achieved with the induction effects are comparable to the br aking torque generated without the induction effects at 1 Hz. Also, Figure 3L8 showed that the braking torque is almost constant within the frequency domain investigated. However, the detailed data shows that there exists an optimum frequency for each applied field where the braking torque is the maximum. In order to find the optimum frequency, an optimiza tion algorithm was run on an ECB configuration specific for automotive applications. Note that the optimization was only carried out for triangular and sinusoidal field app lications, since the triangular field application resulted in the highest braking torque and the sinusoidal field serves as the reference waveform. When DC field is applied, this configuration generates 10.51 Nm. It was shown in Sec. 3.3.4 that the braking torque rea ches a maximum of 12.9 Nm with the sinusoidal field application and a maximum of 14.2 Nm with the triangular field application at 10 Hz and 100 rpm. 65 Previously in [28], the braking torque generation o f an ECB model with 3 PPAs was studied. Unlike current study, this work only focus ed on sinusoidal field application and the numerical results showed that the braking torqu e increased by 13.6% at an optimum frequency of 40 Hz. In this chapter, however, it is shown that even with 1 PPA, the braking torque can be increased 22.5% with sinusoid al field and 34.5% with triangular field application at an optimum frequency of 10 Hz. Finally, it is proposed that the braking torque can be increased further by application of an AC field with varying frequency. This results in an increase in the eddy current generation and the braking torque increases accordi ngly due to the rise in the eddy current density. To model the varying frequency, FM (freque ncy modulation) is used and the braking torque is optimized on the considered ECB c onfiguration for the sinusoidal and triangular field applications. The sinusoidal field application with FM reaches a maximum braking torque of 16 Nm when the carrier fr equency, modulation frequency and frequency deviation are equal to 52 Hz, 52 Hz a nd 51 Hz, respectively. Compared to the DC field results, the braking torque increased by around 53%, and compared to the braking torque generated by the sinusoidal field ap plication with a constant carrier frequency (i.e. no FM) of 10 Hz, the torque increas ed by 24.5%. Triangular field application with FM reaches a maxi mum braking torque of 16.85 Nm when the carrier frequency, modulation frequency an d frequency deviation are equal to 39 Hz, 39 Hz and 38 Hz, respectively. Compared to the DC field torque generation results, the increase was around 60% and compared to the bra king torque generated by the triangular field application with a constant carrie r frequency of 10 Hz, the increase was around 19%. All in all, as it was originally proposed, the appl ication of time varying field resulted in the increase in the braking torque capacity for ECBs and the braking torque increases significantly not only at low speeds but also at hi gh speeds. Moreover, the braking torque of an ECB can significantly be increased with the a pplication of a time varying field (the triangular field in particular) with a variable fre quency component. According to the numerical results obtained, the ECB configuration c onsidered can generate 60% more braking torque with the application of FM triangula r wave field than the DC field application. Further improvements can also be achieved in variou s ways. Note that the simulations were carried out based on the ECB configuration pre sented in optimized for a specific automotive application tha t may introduce other dimensional and material design variables. In addition to above des ign optimization, there is a combination of approaches that can be taken in order to improve the braking performance of the ECB: (i) increasing the magnitude of the applied field, (ii) increasing the number and area of PPAs on the conductor surface (e.g., see and/or (iii) increasing the rotational speed of the conductive brake disk using gears. All these approaches result in increased eddy current g en conductive brake disk, which leads to a higher brak ing torque. In particular, having an array of PPAs on the surface can drastically improv e the braking performance. For example, a smooth but effective braking can be obta ine difference between the applied flux density of each PPA. Moreover, the relative positions of the PPAs, the shape and polarity of the applied fields, and the frequency and phase angles between each field must be optimized and highLspeed regions. Figure With the proposed modifications, further improvemen t in the braking torque generation of the ECB can be achieved. Although ECB may be able generate enough braking torque to stop the vehicle with the propose d improvements, due to its nature of Further improvements can also be achieved in variou s ways. Note that the simulations were carried out based on the ECB configuration pre sented in Table 3L 2 optimized for a specific automotive application tha t may introduce other dimensional and material design variables. In addition to above des ign optimization, there is a combination f approaches that can be taken in order to improve the braking performance of the ECB: (i) increasing the magnitude of the applied field, (ii) increasing the number and area of PPAs on the conductor surface (e.g., see Figure 3L11 where yellow areas are the PPAs) and/or (iii) increasing the rotational speed of the conductive brake disk using gears. All these approaches result in increased eddy current g en eration over the surface of the conductive brake disk, which leads to a higher brak ing torque. In particular, having an array of PPAs on the surface can drastically improv e the braking performance. For example, a smooth but effective braking can be obta ine d with the introduction of a phase difference between the applied flux density of each PPA. Moreover, the relative positions of the PPAs, the shape and polarity of the applied fields, and the frequency and phase angles between each field must be optimized to obtain the maximum braking in both low Figure 3L11. Conductor brake disk with an array of PPAs With the proposed modifications, further improvemen t in the braking torque generation of the ECB can be achieved. Although ECB may be able generate enough braking torque to stop the vehicle with the propose d improvements, due to its nature of 66 Further improvements can also be achieved in variou s ways. Note that the simulations 2, which was not optimized for a specific automotive application tha t may introduce other dimensional and material design variables. In addition to above des ign optimization, there is a combination f approaches that can be taken in order to improve the braking performance of the ECB: (i) increasing the magnitude of the applied field, (ii) increasing the number and area of where yellow areas are the PPAs) and/or (iii) increasing the rotational speed of the conductive brake disk using gears. All eration over the surface of the conductive brake disk, which leads to a higher brak ing torque. In particular, having an array of PPAs on the surface can drastically improv e the braking performance. For d with the introduction of a phase difference between the applied flux density of each PPA. Moreover, the relative positions of the PPAs, the shape and polarity of the applied fields, and the frequency and phase to obtain the maximum braking in both low With the proposed modifications, further improvemen t in the braking torque generation of the ECB can be achieved. Although ECB may be able generate enough braking torque to stop the vehicle with the propose d improvements, due to its nature of 67 consuming power to keep the vehicle parked after br aking, ECB requires an assistive braking system for parking. 3.5 Conclusion In this chapter, Eddy Current Brakes (ECBs) are pro posed as possible substitutes for the conventional hydraulic brakes (CHBs) in automot ive applications due to its unique potential for performance advantages (e.g., nonLcon tact and fast response),. As a potential solution for the limited braking torque of typical ECBs at low speeds, the application of time varying fields (i.e. AC) is proposed. A finite element model (FEM) that accounts for the effects of time varying fields on the performan ce of the ECB was developed, which was then validated using an existing analytical mod el for the DC field. It was shown that improved braking performance can be obtained when A C fields are used at both low and high velocities. Time varying fields in different w aveforms (i.e. sinusoidal, square, sawtooth and triangular waves) were applied and tri angular wave field application resulted in the highest braking torque. The numerical result s also showed that the braking torque decreases with increasing braking torque due to ind uction effects when the time varying fields are applied. In order to increase the brakin g torque generation, the induction effects were eliminated and braking torque generation was m aximized for sinusoidal and triangular field applications on a generic ECB conf iguration for automotive applications. A further improvement was subsequently proposed by ap plication of frequency modulated (FM) AC fields. The results showed that a 60% incre ase in the braking torque can be achieved with the FM compared to the DC field braki ng. Further improvements in braking torque generation of the ECB can be achieved by mea ns of increasing the magnitude of applied field, the number of PPAs and the rotationa l speed of the conductor disk using gears. Appendix.3A. Analytical Model: DC field application The DC model proposed by Lee et al. [29] uses the L orentz force law in order to explain the eddy current generation under DC field application. When a moving charged particle enters a region in which a magnetic field is present, a force, whose magnitude and 68 direction depend on the magnitude and the direction of the particle velocity and the external magnetic field, is exerted on the particle . If a conductor disk rotates under applied magnetic field, charged particles, electron s and protons that are initially in balance over the surface of the conductor, will be separated from each other by the Lorentz force. This separation results in the gener ation of surface charge densities over the PPA. Using the surface charge densities, corres ponding electric field and current densities in the PPA can be calculated. Following the work of Lee and Park [29], it is init ially assumed that the disk has infinite radius initially and the model is solved f or the electric field and the eddy current density according to this assumption. Then, the rad ial component of the eddy current density within the conductor disk at the boundary w as set to zero by introducing an imaginary current source outside the disk. By doing this, the model effectively represents a conductor disk of a finite radius. As the final s tep, this model utilizes the magnetic Reynolds number in order to solve for the induced m agnetic field that is generated by the eddy currents on the conductor. The magnetic Reynol ds number is the ratio between the induced field and the applied field. With the induc ed field computed, the torque that opposes the rotation of the disk can be calculated using the following relationship: /g1846/g3029=/g1516/g1818/g2172×(/g1784×(/g1776−/g1776 /g1809/g1814 ))/g1856/g1876/g1856/g1877 /g3020 (A3.1) 69 4. OPTIMIZATION OF BRAKING TORQUE GENERATION CAPACITY OF EDDY CURRENT BRAKES WITH THE APPLICATION OF TIME VARYING MAGNETIC FIELDS 4.1 Introduction ECBs can readily be found in the following applicat ions by means of assistive braking: to slow down a high inertia actuator, to help braki ng of heavy vehicles such as trains and heavy trailers, to adjust the stiffness of exercise equipment (e.g., exercise bikes), etc. Especially, slowing highLspeed trains down to speed s where conventional brakes would be more efficient is one of the key applications of ECBs [12L16]. Potential performance improvements, e.g., contactle ss braking, silent operation, no friction, natural capacity to imitate an ABS, etc., make ECB a very attractive alternative for CHBs. However, the main problem of the existing ECB is the limited braking torque generation at low vehicle speeds and in order to be able to realize a standLalone ECB, this problem needs to be addressed. In this thesis, use of AC fields is proposed to imp rove the braking torque generation. In Chapter 2, an analytical model was created and prel iminary results obtained showed that the braking torque generation increases with the ap plication of AC fields not only at low velocities but also high speeds. In Chapter 3, an a ccurate FEM, which was validated using 2 analytical methods for both AC and DC field appli cations, was created. Using this FEM, various forms of time varying fields applied w ith and without frequency modulation were applied and the results showed that the braking torque significantly increases with the application of AC field in the f orm of triangular waves with frequency modulation. In order to effectively deal with the problem of re duced braking torque at low speeds, all the factors (geometric, material, field depende nt and position dependent variables) that affect the braking torque generation are identified . In this chapter, the main objective of is to optimize the braking torque generation of ECBs f or configurations that consist of up to 4 PPAs with the help of an accurate electromagnetic FEM that analyzes and simulates the eddy current generation under various forms of time varying field applications with 70 variable frequencies. The optimization procedure is carried out to validate a standLalone ECB configuration that can generate sufficient amou nt of braking torque to stop a vehicle. Due to the nature of the nonlinearity of the proble m, a stochastic optimization algorithm, Genetic Algorithm (GA), is used. Stochas tic algorithms do not require gradient information and this makes such algorithms very att ractive for solving the highly nonlinear ECB optimization problem. GA searches a v ast design space very fast and efficiently with the help of evolution algorithm it employs. The previous successful design configurations are used to create the next d esign configuration. Thus, it can be said that the evolution algorithm used is the biggest ad vantage of such a technique. However at the same time, same evolution algorithm is the b iggest disadvantage, since evolution does not guarantee the best answer, GA; similar to any other stochastic algorithm, is not guaranteed to find the global optimum. In this chapter, the FEM created and described in t he previous chapter is briefly described and the braking torque generation results obtained for various waveforms are briefly presented in this chapter since they are go ing to give the reasoning behind the configuration selected to be optimized. Then, the o ptimization problem is defined for the ECB problem along with the geometric constraints an d comfort constraint. Since the braking torque is alternating due to the applicatio n of the alternating fields, the comfort levels become important in the process of braking. The variation in deceleration is converted in a quantitative comfort value and these values are kept smaller than a threshold value. Once the optimization is carried o ut, the results show that the braking torque significantly increases with the additional PPAs and braking torque generation of an optimum configuration with 4 PPA can exceed the braking torques generated by a conventional hydraulic brake. 4.2 Eddy current brake model In previous work [32], an accurate electromagnetic FEM of the ECB, which solves for the magnetic flux density when there is a time vary ing external field is present, was presented. Using the governing Maxwell’s equations, the partial differential equation (PDE) shown in Eq. (4.1) was obtained. Also, note t hat the PDE is only magnetic flux dependent. 71 −∂(/g1776+/g1776/g2777) ∂t+∇×(/g1822×/g1776)+∇×(/g1822×/g1776/g2777) =/g2869 /g3091/g3116/g3097/g3031 ∇×(∇×/g1776)+/g2869 /g3091/g3116/g3097/g3031 ∇×(∇×/g1776/g2777)+/g3084/g3116 /g3097/g3031 /g2986/g3118(/g1776/g2878/g1776/g2777) /g2986/g2930/g3118 (4.1) where J is the net eddy current density, B is the external magnetic field applied, B0 is the externally applied flux density, E is the electric field, D is the displacement flux density, µ0 is the magnetic permeability of vacuum, ε0 is the electric permittivity of vacuum, d is the thickness of the conductive disk and t is time. For the basic ECB problem, a 2LD domain shown in Fi gure 4L1 was selected. The outer circular shape represents the conductor disk and th e inner rectangular region is the PPA, the area under the poles of the electromagnet, i.e. the area that the external magnetic field is applied onto. Note that for the specific problem , it is assumed that the electromagnetic variables are uniformly distributed over the thickn ess, i.e. no skin effects on the conductor. Figure 4L1. Basic ECB domain considered for simulat ing the eddy current generation By using the PDE solver module of the COMSOL Multip hysics ®, the magnetic flux density distribution can then be solved over the ro tating conductive disk surface with the 72 help of proper boundary conditions; and by using th e magnetic flux density distribution, the eddy current densities can be calculated. In or der to account for the secondary field induced by the eddy currents, an additional (second ary) model was defined and coupled to the primary model. The secondary model takes the results of the primary model, i.e. the current density distribution on the disk, as its in put and solves for the induced magnetic flux density, Bin . Then, the braking torque, Tb, is calculated as follows: /g1846/g3029/g3036/g3041 =/g1516/g1818/g1790×((/g1784−/g1784 /g2191/g2196 )×(/g1776+/g1776 /g2777−/g1776/g2191/g2196 ))/g1856/g1876/g1856/g1877 /g2903 (4.2) where rP is the position vector that is a function of x and y coordinates, and Jin is the induced current density. The flowchart of the FEM i s shown in Figure 4L2. FEM contains two solvers, one is the PDE solver that solves the eddy current distribution and the other is an electromagnetic solver that introduces the ef fects of induction fields. 73 Figure 4L2. Flowchart of the FEM implementation In the same work published previously [32], braking torque generation of ECB under time varying field applications with various wavefo rms (i.e. sinusoidal, sawtooth, triangular and square) was studied. The results sho wed that the induction has a significant effect as the frequencies increase. Therefore, it w as recommended that laminated cores be used to remove the induction fields. Among those st udied waveforms, triangular field resulted in the highest braking torque generation. In addition, frequency modulated field application was proposed for further increase in th e braking torque generation. As a result of the preliminary analysis in [32], in the following sections, the braking torque was optimized for a configuration where the induction fields are removed by utilizing a laminated core electromagnet and where frequency modulated external magnetic field is applied. Thus, from the simulatio n point of view, the induction effects that are solved using the electromagnetic solver in Figure 4L2 are removed from the FEM 74 of ECB. Triangular waveforms are used to find the o ptimum ECB configuration since they result in highest braking torque when they are applied on the rotating conductive disk. In addition, sinusoidal fields are also studi ed and optimum ECB configurations are found, since they are the main building blocks of a ny waveforms. 4.3 Optimization 4.4.1 Definition of the problem In this section, the optimization problem for the E CB is defined and the above FEM is used to solve for the braking torque of the given c onfiguration in Table 1 and applied flux density. In setting up the optimization problem, an objective function was defined with the braking torque as a function of both the geomet ry dependent and the applied field dependent design variables. The optimization proble m of ECB is given in Eq. (4.3). The braking torque ( Tb) is maximized and the problem is subject to a numb er of constraints mentioned in detail below. Maximize: /g1846/g3029=/g1516/g1818/g1787×/g3435(/g1784)×(/g1776+/g1776/g2777)/g3439/g1856/g1876/g1856/g1877 /g2903 (4.3.a) Subject to: /g1856/g3032/g3033/g3033 =min /g34202/g3495/g3096 /g3095/g3033/g3117/g3091,/g1856/g3424 (4.3.b) /g1855/g1867/g1865/g1858/g1867/g1870/g1872 ≤0.5/g3040 /g3046/g3118(RMS) (4.3.c) /g1844 ≥/g3495(/g1870∗cos (/g2009)+/g3028 /g2870)/g2870+(/g1870∗sin (/g2009)+/g3029 /g2870)/g2870 (4.3.d) /g1870 /g1871/g1861/g1866(/g2009)≥/g1854 (4.3.e) /g1858/g2869≥∆/g1858−1 (4.3.f) The design variables are defined and listed in Tabl e 4L1 for the specific ECB optimization problem considered. Also note that the optimization was carried out at applied magnetic flux densities of 1 T and 1.225 T for the sinusoidal and triangular field application, respectively. These magnitudes satisfy a constant magnetic energy input into the system that makes the results of the two optimi zations comparable, as the RMS values of these two input fields are both 0.707 T. 75 Table 4L1. Design Variables and Constants for the E CB optimization problem Design Variables (d) Lower Boundary Upper Boundary Radius of the disk, R (m) 0.05 0.12 Thickness of the disk, d (m) 0.003 0.01 PPA width, a (m) 0.02 0.05 PPA height, b (m) 0.02 0.05 Pole location distance, r (m) 0 0.11 Position of PPAs, αn (rad) 0 π Carrier Frequency, f1 (Hz) 1 400 Modulation Frequency, f2 (Hz) 1 400 Frequency Deviation, ∆f (Hz) 0 399 Phase angle between fields at PPAs, βn (rad) 0 2π Tire Diameter (m) 0.62 Loaded Vehicle Mass (kg) 1300 Applied Flux Density – Peak Magnitude (T) 1 (Sinusoidal), 1.225 (Triangular) Rotational Velocity (rad/s (rpm)) 10.472 (100) With the introduction of multiple PPAs, the relativ e position (i.e. the geometric angle) of the PPAs and the relative phase difference (i.e. the phase angle) between the flux densities applied to each individual PPA are taken into account by means of αn and βn (i.e. n = 1, 2 and 3), respectively. The subscript “ n” represents a PPA’s coupling with a reference PPA. For example, in the case of the 2LPP A configuration, n is set to be 1. One of the PPAs is defined as the reference, thus only α1 and β1 are defined as the angle between the PPAs and phase angle between the applie d flux densities to the PPAs. For the 4 PPA configuration, n is set to be 3 and 6 variabl es are defined in total, i.e . α 1, α 2, α 3, β1, β2 and β3, to account for the relative angular position and the phase angle of the applied flux densities of each PPA with respect to the refe rence PPA. The objective function in Eq. (4.3.a) then can be r ewritten in terms of these design variables shown in Table 4L1. As the next step, the constraints are defined. There are two important constraining factors that need to be take n into consideration for improved accuracy and braking performance. The first factor is the skin effect. Since the model employs alternating flux density as the source, the eddy current generated on the surface of the conductor will also be time varying. When an alternating current (AC) is applied to 76 a conductor, the current has a tendency to accumula te in the vicinity of the surface of the conductor and, with increasing depth in the materia l; the current density decreases and diminishes after a certain depth, which is also kno wn as the skin depth. Note that the FEM introduced in [32], which is needed by the opti mization procedure, does not account for the skin effects. The skin depth ( dskin ) can be expressed in terms of the material property of the conductor and the frequency of the applied time varying current as: /g1856/g3046/g3038/g3036/g3041 =/g3495/g3096/g3293/g3280/g3294 /g3095/g3033/g3091 (4.4) where ρ is the resistivity and µ is the absolute magnetic permeability of the condu ctor, and f is the frequency of the current applied on the con ductor surface. Note that the depth is calculated for a sheet with infinite depth. Ther efore, for a sheet with finite thickness, the total skin depth will the twice the depth calcu lated to account for both the top and the bottom surfaces of the sheet. At low frequencies, the skin effects can be ignored for a thin conductor sheet. However, with increasing frequencies, the skin dept h decreases according to Eq. (4.4). Thus, in order to account for this phenomenon in th e optimization procedure, the skin depth is included as a constraint on the effective thickness ( deff ) of the rotating conductive disk, Eq. (3.b). In the optimization calculations, the effective thickness is used and its value is selected to be whichever the smallest of t he thickness ( d) the optimization algorithm selects or twice the skin depth ( dskin ) at the corresponding frequency that the solution is carried out at. Note that the magnetic field applied to the ECB is frequency modulated and does not have a fixed frequency. Thus , for simplicity, it is assumed that the frequency of the carrier field ( f1) defines the skin depth in the braking torque calculations. The second factor also stems from the alternating ( i.e. AC) nature of the problem. With alternating field applications, the computed brakin g torque will also be alternating. Having such a braking torque for a vehicle will sig nificantly affect the comfort level of the driver and passengers. Unlike the first factor, which is a magnetic phenomenon, the comfort level is a subjective term and the percepti on will vary from one person to another 77 and from one vehicle to another as well. In the lit erature, however, there are a number of studies that were conducted in order to quantify th e comfort level of drivers and passengers [51L54]. In these studies, human percept ion to vibration was measured and quantified experimentally. Although the experiments are used to quantify comfort, one’s perception of vibration also depends on cognitive e lements such as the nature of the environment where vibration occurs, expectations, r ole of the vibration and motivation [52]. In these experiments, comfort curves were plotted f or individuals. By normalizing these comfort curves, frequencyLweighting curves ar e plotted for different parts of human body. Due to complex damping effects of the human b ody and the driver/passenger seat, some parts of the body are more sensitive to the vi bration than others, such as the back and the buttocks. The vibration is defined using th e acceleration values and the RMS of the acceleration is used for the quantifying the co mfort level. The RMS acceleration is multiplied with the weighting factors and a range o f perceived vibration values are obtained. The general guidelines were required to d efine the comfort level at corresponding weighted perceived accelerations and one such guideline was suggested by Griffin and his guidelines were incorporated in Bri tish Standard 6841 [52]. According to Griffin, the resulting vibration under weighted acc eleration of 0.5 m/s 2 (RMS) or less is considered little or not uncomfortable. Between 0.5 and 1.25 m/s 2 (RMS), the vibration is defined as uncomfortable and above 1.25 m/s 2 (RMS), the vibration is considered as very uncomfortable. In the current study, the comfort level is incorpor ated into the optimization problem as an additional constraint, Eq. (4.3.c), since it can be a significant problem due to the utilization of the time varying fields as the sourc e of the eddy currents. Herein, it is assumed that in order to keep the vibrations in a t olerable/comfortable range, the weighted acceleration values are set to be lower or equal than 0.5 m/s 2 (RMS). This condition will satisfy a smooth braking curve in or der to make it comfortable enough for the driver and the passengers. In order to calculat e the deceleration during braking, a loaded vehicle that is 1300 kg is assumed to have f our ECB actuators installed, and it is also assumed that all the kinetic energy is dissipa ted in the brakes. 78 In addition to the skin effect and comfort level co nstraints, three additional constraints are defined. The first one, Eq. (4.3.d), is that th e surface defined as PPA has to be within the rotating conductor surface, since the eddy curr ents are specifically generated on the surface of the conductor. Secondly, the neighbourin g PPAs cannot have intersections, considering that electromagnet cores that generate the external fields cannot be physically overlapping each other (see Eq. (4.3.e)). Finally, since the deviation frequency ( ∆f ) cannot be higher than the carrier frequency ( f1), the value of the deviation frequency is capped b y that of the carrier frequency (see Eq. (4.3.f)). The optimization procedure was carried out for four different cases, involving sinusoidal and triangular field applications. In th e first case, the simplest 1LPPA configuration was optimized and for the subsequent cases, the number of PPAs was increased up to four. For the ECB optimization problem, a stochastic sear ch optimization algorithm, Genetic Algorithm (GA), was used. Among all available stoch astic algorithms, GA can locate good solutions in a short amount of time. However, similar to available stochastic algorithms, the result improves with time. For the specific ECB problem, a tolerance value is defined in order to terminate the optimiza tion. Although stochastic methods are powerful optimization algorithms, they are not guar anteed to converge to the global optimum. Figure 4L3 shows the flowchart of the optimization algorithm that is carried out for the specific ECB problem. Initially, GA creates the ini tial population within the user defined design space and then the thickness of the conducto r disk is adjusted using the frequency values. After the thickness has been updated, the c onstraints are checked and if they are not satisfied, the algorithm returns a penalty func tion as the result for the given configuration. But if the constraints are satisfied then the configuration is studied using the FEM [32] and the resulting braking torque is ca lculated. Once the torque value is available, the comfort level values can be calculat ed. If the comfort value is higher than the set limit, i.e. 0.5 m/s 2 (RMS), then the algorithm returns a penalty functi on for the configuration. However, if the comfort level is les s than the set limit, then the configuration is stored. Following the function eva luations, GA carries out mating, crossover and mutation between successful configura tions. Once the new population is 79 created, GA checks whether the termination criterio n (i.e. evolution stops when the best fitness value in the current population is less tha n the user defined fitness) is satisfied or not. If it is not satisfied, the algorithm repeats the whole process to find a new population. However, if the termination criterion is satisfied, then the algorithm ends and it returns the optimum configuration. Figure 4L3. Flowchart of the optimization procedure Thus, in order to increase the accuracy, the optimi zation algorithm was run multiple times for every case. The results that are shown in Table 4L2 are the optimum values for each design variable obtained after carrying out th e optimization several times. 80 Table 4L2. Optimum ECB Configurations Design Variables (d) Optimum Values Sinusoidal Triangular 1 PPA 2 PPA 3 PPA 4 PPA 1 PPA 2 PPA 3 PPA 4 PPA Radius of the disk, R (m) 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 Thickness of the disk, d (m) 0.01 0.01 0.01 0.098 0.01 0.01 0.01 0.01 PPA width, a (m) 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 PPA height, b (m) 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 Pole location distance, r (m) 0.088 0.081 0.08 0.08 0.088 0.081 0.081 0.081 Position of PPAs, αn (rad) L 0.756 0.783, 0.952 0.717, 0.781, 1.7718 L 0.8145 0.7765, 0.9097 0.709, 0.902, 0.786 Carrier Frequency, f1 (Hz) 31 100 226.65 300.2 34 209.4 241.4 214 Modulation Frequency, f2 (Hz) 31 100 226.65 300 34 209.4 241 214 Frequency Deviation, ∆f (Hz) 30 99 225 299 33 208 240 213 Phase angle between fields at PPAs, βn (rad) L 0.858 1.005, 2.404 0.923, 1.2212, 2.569 L 1.054 1.068, 2.564 1.180, 3.536, 2.357 4.4.2 Optimization Results As mentioned earlier, the optimization was carried out for four different configurations, corresponding to 1, 2, 3 or 4 PPAs, and for both sinusoidal and triangular field applications for each case. The braking torqu es generated by the optimum brake configurations are presented in Table 4L3. In addit ion to the braking torques, the comfort levels obtained for each optimum configuration are also given in the same table. Note that in order to obtain these optimization results, the rotational velocity of the conductive braking disk is kept at 100 rpm. In Figure 4L4, the optimum ECB design with 4 PPA’s is shown. Table 4 Cases 1 PPA configuration 2 PPA configuration 3 PPA configuration 4 PPA configuration One significant conclusion that can be made from the ta ble above is that the braking torque significantly increases with the introductio n of the second PPA. An optimum configuration with 1 PPA can generate up to around 35 Nm with the application of triangular flux density. However, with the second PPA is added , the braking torque significantly increases, i.e. about 8 times the bra king torque generated with 1 PPA ECB configuration. This behaviour currents gen erated under the adjacent PPAs. If the PPAs are pos itioned as far away from each other as possible (e.g., 180 torque will be roughly doubled with the introductio n of the second PPA. However, as the two PPAs get closer to each other, the interaction b etween the eddy currents increases the 4L3. Optimum braking torque values for cases studied Sinusoidal Triangular Braking Torque, Nm (RMS) Comfort level, m/s 2 (RMS) Braking Torque, Nm (RMS) 33.62 0.2454 34.77 260.21 0.4993 285.22 480.15 0.497 509.4 738.5 0.495 878.8 Figure 4L4. Optimum 4LPPA configuration significant conclusion that can be made from the ta ble above is that the braking torque significantly increases with the introductio n of the second PPA. An optimum configuration with 1 PPA can generate up to around 35 Nm with the application of flux density. However, with the second PPA is added , the braking torque significantly increases, i.e. about 8 times the bra king torque generated with 1 PPA ECB behaviour can be explained by the interaction between the edd y erated under the adjacent PPAs. If the PPAs are pos itioned as far away from each other as possible (e.g., 180 ° apart) , the interaction will be minimum and the braking torque will be roughly doubled with the introductio n of the second PPA. However, as the wo PPAs get closer to each other, the interaction b etween the eddy currents increases the 81 . Optimum braking torque values for cases studied Comfort level, m/s 2 (RMS) 0.4984 0.4726 0.4792 0.4959 significant conclusion that can be made from the ta ble above is that the braking torque significantly increases with the introductio n of the second PPA. An optimum configuration with 1 PPA can generate up to around 35 Nm with the application of flux density. However, with the second PPA is added , the braking torque significantly increases, i.e. about 8 times the bra king torque generated with 1 PPA ECB can be explained by the interaction between the edd y erated under the adjacent PPAs. If the PPAs are pos itioned as far away from , the interaction will be minimum and the braking torque will be roughly doubled with the introductio n of the second PPA. However, as the wo PPAs get closer to each other, the interaction b etween the eddy currents increases the 82 amount of braking torque generation. Note that the interaction is also strongly dependent on the phase angle of the applied flux densities on the adjacent PPAs. With different phase angles, the interaction may cause reduction i n the braking torque when the PPAs are placed close to each other. In terms of the spa cing between the PPAs, it is ideal to have them close to each other on the surface of the conductor disk. Thus, the phase angles have to be selected correctly so that the interacti on results in an overall increase in the total braking torque. Although it was not shown in Table 4L3, another int eresting result that was observed is that at specific phase angles defined between the f lux densities applied onto the PPAs, the braking torque becomes a positive value instead of a negative value. This means that instead of decelerating, the particular ECB configu ration will accelerate the rotation of the wheel. This phenomenon is also due to the interacti on between the eddy currents generated by adjacent PPAs. This phenomenon is call ed electromagnetic propulsion and it is currently used for accelerating levitated trains in a linear manner. The phase angles can be adjusted in such a manner that the poles of the PPAs are attractive to each other, which will accelerate the rotating disk. However, as our interest lies in a brake actuator, the above optimization algorithm searches for the phase angles that will generate a negative braking torque that acts to slow down the rotation of the wheel. Finally, a last comment that can be made is that th ere is still room for additional PPAs on the ECB configuration presented in Figure 4L4. T his will lead to a further increase in the braking torque, which is discussed further in t he next section, at the expense of higher complexity, increased weight, power requirement and cost. 4.4 Discussions Although ECBs have a number of performance advantag es over CHBs, their braking torque generation capacity at the low vehicular spe eds is considerably limited. In this paper, in order to improve the braking performance of the ECB, the braking torque generation of the ECB with multiple PPAs is optimiz ed, with the consideration of comfort and skin effects, as well as geometric and field de pendent factors. In previous study [32], results showed that the bra king torque increases with the application of time varying fields. In addition, am ong various waveforms, triangular field resulted in the best braking torque generation. Fur ther analysis also proved that the braking torque can be significantly increased with the applicat signal. In this paper, the ECB was optimized with t he application of frequency modulated sinusoidal and triangular external fields. In addit ion to the improvements resulted from the external fields, the performance can be improve and in order to do so, it is assumed that laminated cores are used to generate the external magnetic fields. The results of the optimization are given the braking torque significantly increases with the introduction of ad section, for discussion purposes, the configuration with the maximum braking torque output is selected for further investigation. As me ntioned earlier, the peak magnitude of the applied flux density was selected to be 1.225 T that is configuration with 0.707 T DC field systems. Thus, here both systems are compared in te rms of the braking torque generation capacity across various rotational velocities of th e conductor disk. variation of the braking torque generation with optimum 4LPPALc onfiguration. Figure 4L5 . Braki resulted in the best braking torque generation. Fur ther analysis also proved that the braking torque can be significantly increased with the applicat ion of frequency modulated signal. In this paper, the ECB was optimized with t he application of frequency modulated sinusoidal and triangular external fields. In addit ion to the improvements resulted from the external fields, the performance can be improve d by reducing the induction effects and in order to do so, it is assumed that laminated cores are used to generate the external The results of the optimization are given in Table 4L2 and Table 4L3 , which show that the braking torque significantly increases with the introduction of ad ditional PPAs. In this section, for discussion purposes, the configuration with the maximum braking torque output is selected for further investigation. As me ntioned earlier, the peak magnitude of the applied flux density was selected to be 1.225 T that is comparable to the same configuration with 0.707 T DC field application in terms of the energy input to the systems. Thus, here both systems are compared in te rms of the braking torque generation capacity across various rotational velocities of th e conductor disk. Figure variation of the braking torque generation with respect to the rotational onfiguration. . Braki ng torque generation at various angular velocities 83 resulted in the best braking torque generation. Fur ther analysis also proved that the ion of frequency modulated signal. In this paper, the ECB was optimized with t he application of frequency modulated sinusoidal and triangular external fields. In addit ion to the improvements resulted from d by reducing the induction effects and in order to do so, it is assumed that laminated cores are used to generate the external , which show that ditional PPAs. In this section, for discussion purposes, the configuration with the maximum braking torque output is selected for further investigation. As me ntioned earlier, the peak magnitude of comparable to the same in terms of the energy input to the systems. Thus, here both systems are compared in te rms of the braking torque generation Figure 4L5 shows the rotational velocities for the ng torque generation at various angular velocities The results show that the braking torque increases dras tically with the application of the frequency modulated triangular field at both hi gh and low speeds. According to the results, the braking torque ge velocities around 100 rad/s reaches as high as 17 t imes that of the DC field application. On the other hand, at higher velocities around 1000 rad/s, the the braking torque with the AC field application is 3 times that of th e DC field application. Note that the AC field generated braki ng torques in values. As the next step, the corresponding these simulations, it is assumed that the optimum 4 on each wheel of a 13L inch wheel vehicle that th e wheels are rotating at a speed of 104 rad/s (1000 rpm). Note that this rotational speed corresponds to about 120 km/h for a vehicle w ith are carried out for 8 seconds. The curve with represents the deceleration with the deceleration with the rotational velocity of each wheel results show that the braking torque increases dras tically with the application of the frequency modulated triangular field at both hi gh and low speeds. According to the results, the braking torque ge nerated with the AC field application at low rotati onal velocities around 100 rad/s reaches as high as 17 t imes that of the DC field application. On the other hand, at higher velocities around 1000 rad/s, the increase is not as dramatic; with the AC field application is 3 times that of th e DC field application. Note that the AC field generated braki ng torques in Figure Figure 4L6. Deceleration curve corresponding deceleration curve is presented in simulations, it is assumed that the optimum 4 LPPAL configuration ECB is installed inch wheel vehicle , fully loaded at 1300 kg. Initially e wheels are rotating at a speed of 104 rad/s (1000 rpm). Note that this rotational speed corresponds to about 120 km/h for a vehicle w ith the 13L inch wheels. Simulations are carried out for 8 seconds. The curve with the diamond data points in the deceleration with the DC field application and the smooth the AC field application. When only the DC field is applied of each wheel decreases down to 25 rad/s after 8 seconds 84 results show that the braking torque increases dras tically with the application of the frequency modulated triangular field at both hi gh and low speeds. According to the nerated with the AC field application at low rotati onal velocities around 100 rad/s reaches as high as 17 t imes that of the DC field application. increase is not as dramatic; with the AC field application is 3 times that of th e DC field Figure 4L5 are RMS presented in Figure 4L6. For configuration ECB is installed fully loaded at 1300 kg. Initially , it is assumed e wheels are rotating at a speed of 104 rad/s (1000 rpm). Note that this rotational inch wheels. Simulations diamond data points in Figure 4L6 the smooth curve represents DC field is applied , the after 8 seconds . However, 85 with the application of the AC field, the vehicle s tops after only about 2.7 seconds. It is evident that the deceleration performance of the ve hicle with the ACLbased ECBs is significantly better than that of the vehicle with the DCLbased ECBs. If the figure is investigated closely, unlike the s mooth DC field deceleration curve, AC field deceleration curve oscillates due to the alte rnating nature of the braking torque generated. But since the comfort level was a part o f the optimization procedure, the deceleration should not be uncomfortable for the pa ssengers. From the deceleration curves of the AC and DC based ECB configuration, the corresponding linear deceleration value can be obta ined for the vehicles themselves. Since the vehicle equipped with the ACLbased ECBs i s brought to a full stop from 120 km/h in 2.7 seconds, this corresponds to a linear d eceleration of about 12 m/s 2, which is three times more than the linear deceleration that can be achieved with the vehicle equipped with the DCLbased ECBs. This deceleration corresponds to about 1.2 g’s. For comparison, most passenger vehicles have rated dece lerations of around 0.8 to 1 g’s and some fast sport cars can achieve decelerations vary ing from 1 g to 5L6 g’s, with the use of carbon ceramic friction brakes [55]. The amount of the braking torque generated by the A CLbased ECBs is enough for slowing down regular passenger vehicles. However, f or heavier vehicles such as trucks, the required amount braking torque is significantly higher. In order to meet these higher braking torque requirements, the ECB configuration has to be further improved. The configuration that was studied employs up to 4 PPAs ; however, as mentioned in the previous section, there is room to add more PPAs. I n Table 4L2, the summation of αn’s is less than 180 ° and this implies that all 4 PPAs are located withi n the one half of the conductive braking disk as it is depicted in Figure 4L4. Introducing addition PPAs will drastically increase the braking torque in a simila r trend as shown in Table 4L3. In addition to introducing more PPAs, the ECB can be f urther optimized specifically for the heavy vehicle application. The wheel dimensions wil l be bigger for a truck than for a passenger vehicle. This will result in utilizing co nductive disks with bigger diameters compared to those for passenger vehicles. Having a bigger radius will result in higher braking torques due to the increase in the moment a rm and also there will be more room to introduce new PPAs on the surface of the conduct ive braking disks. Looking at the 86 results presented in Figure 4L5 and Figure 4L6, it can be concluded that the braking torque requirements for heavy vehicles can also be achieve d with the increased size of the conductive disks and the introduction of additional PPAs. In terms of the simulations, here genetic algorithm , a stochastic search optimization algorithm, is used to optimize the braking torque g eneration of the AC ECB. For a highly nonlinear problem, search algorithms offer higher p ossibility to converge the global optimum. However, the simulations are generally com putationally heavy and time consuming compared to gradientLbased algorithms. An other limitation in this work is the accuracy of the mathematical model of the triangula r waveform studied here. In order to model the waveform, here first 10 harmonics of the field is used. Since the problem is dependent on the first and the second time derivati ve of the field (see Eq. (3.8)), the accuracy of the derivatives decreases with increasi ng order of derivatives. 4.5 Conclusion In previous work, the use of AC fields in order to improve the braking torque generation of ECBs is investigated with the help a finite element model and results showed that the braking torque is higher when the f ield has a varying frequency instead of fixed frequency and when it is applied in triangula r waveform compared to sinusoidal, square and sawtooth waveforms. In this work, ECB wa s optimized for higher braking torque. In addition to the geometric and field depe ndent factors, skin effects due to alternating currents generated on the surface of th e conductor and the comfort level of the passengers due to the vibration during braking are accounted for in the optimization problem. Genetic Algorithm was then used to find th e optimum ECB configuration. Finally, the results showed that the braking torque significantly increases with the application of AC fields on a multiple PPA ECB conf iguration. Also the deceleration curves showed that the AC field application on the optimum configuration shown in Table 4L2 with 4 PPAs resulted in about 12 m/s 2, which is more than 20L25 % of the accelerations that can be achieved with CHBs on ave rage passenger vehicles. 87 5. CONTROL 5.1 Introduction In previous chapters, the ECB was modeled analytica lly and numerically, and then its braking torque generation capacity was optimized us ing Genetic Algorithm. It was shown that an ECB configuration with the time varying fie ld application has the potential to generate braking torques comparable to that of the conventional brake actuators. In this chapter, a model based predictive controller is des igned for improved braking performance. The main objective of this chapter is to design a c losedLloop controller that can maintain a desired ECB performance. Although there are a number of candidate control methods that can be used for the ECB problem, herei n model predictive controllers are chosen for the ECB problem due to the availability of the analytical and FE models. Model predictive control (MPC) is a control method that employs an explicit process model to estimate the future response of a system. At every interval, MPC determines a sequence of variable adjustments that optimize futu re response of the system. One the sequence is determined, then the first optimum inpu t is sent to the system and the same procedure is carried out for every control interval . The method was originally introduced for power plant and petroleum refinery control appl ications. MPC has successfully been used in numerous applicat ions ranging from chemical, food processing, automotive, aerospace, metallurgy to pu lp and paper industries [56,57], due to three important factors. Firstly, MPC uses an expli cit plant model. Therefore, it directly includes any significant process dynamics. Secondly , it determines the response over a future horizon. This helps the controller to remove any feedLforward and feedback disturbances, thus allowing the controller to follo w a closer future trajectory. Finally, MPC includes the input, state and output constraint s directly in the controller calculations. Therefore, it ensures operation within certain phys ical limits. In general, MPC is used with linear models; however , due to the nonlinearity in many realLworld applications, nonlinear MPC (NMPC) was i ntroduced. In MPC, a linear model is used to estimate the future response of the plan t. When the system is nonlinear, two 88 common approaches are used. In the first approach, MPC is linearized at multiple process points where the model is linearized individually a nd a switching algorithm is used to make the selection between these controllers depend ing on the process states. In the second approach, MPC uses a nonlinear model to dete rmine the response. The approach that is selected strictly depends on the physics of the plant. For an accurate nonlinear model, the second approach results in an accurate t rajectory [58]. Although it has a number of advantages, NMPS has a fundamental difficulty, as it requires an optimum solution of the input over a fu ture horizon for optimum response at each control action. Therefore, for realLtime contr ol, a result must be determined within the time step specified for the controller. Having simplified physics or simple linear models results in faster convergence to the optimum input values. In the literature, there are a number of studies where MPC was successfully implemented in realLtime using linear models [58, 59, 60, 61]. Also, there are a n umber of applications where the dynamics were considerably slow. In [62], an NMPC w as designed for flood prevention of a river system and in [63], it is used to normal ize the glucose levels in blood for critically ill patients. In both cases, the sample time was set to be 5 minutes. On the other hand, in [64], NMPC was applied to an oreLmilling problem in mining and a successful controller was designed with a sample time of 10 seconds. In [65], an unmanned rotorcraft was designed and NMPC was used to avoid any terrain in the path. Due to the complex nature of the model, sample rate for this application was selected to be 20 ms. As the problem gets complex, the sampling time goes even smaller due to the sudden changes in the physics of the problem. In [6 6], the sampling rate was selected to be 1 ms for a robotic hand and eye coordination sys tem. In terms of the time varying field application in E CB, the problem is significantly complicated considering the physics presented in Ch apter 2. In addition, looking at the optimum results obtained in Chapter 4, it can be co ncluded that for an accurate system the sampling rate has to be kept significantly small (e .g., if ~300 Hz sinusoidal signal is applied as the external field). Herein, for the current ECB problem, with the help of analytical and FE models, a dynamic model is obtained using model identificatio n tools. Once the braking torque generation is calculated for the optimum configurat ion selected, a quarter vehicle model is 89 used in conjunction with the ECB model to calculate the dynamics of the vehicle during braking. Since NMPC requires a discrete time nonlin ear model, the ECB model is then converted into a discrete time model and the state variables (velocities and accelerations of the wheel and the vehicle) are defined at every time step. Once the model is ready, an objective function with adequate constraints is def ined to drive the system to a desired steady state. At every time step, an optimum future response of ECB is calculated over a horizon of length 50. The first value of the input is then applied to the plant at the specific time step and the procedure is repeated until the l ast time step. Similar to [67], to solve the optimization problem, particle swarm optimizati on algorithm (PSO), a nonLgradient based algorithm, is used in this work. Simulations are carried out and the closedLloop control results of the proposed ECB controller are presented later in this chapter. This chapter is organized as follows. The ECB confi guration that is used to design the controller is introduced in Sec. 5.2. Since the mod els introduced in Chapter 2 and 3 are highly nonlinear and complex, a simplified dynamic ECB model is created using the knowledge gained in both the analytical model and F EM. In Sec. 5.3, the quarter vehicle model is introduced and this model is used to calcu late the dynamics of the vehicle during braking. In the following section, Sec. 5.4, the pr oposed NMPC algorithm is introduced and a controller is designed for the ECB problem. I n Sec. 5.5, the simulation results are presented. Finally, in Sec. 5.6, a brief conclusion is given to summarize the work carried out in this chapter. 5.2 Dynamic ECB Model The optimization results of Chapter 4 showed that t he braking torque significantly increases with the additional PPAs, with the capaci ty to exceed that of a conventional hydraulic brake. For control design purposes, the b raking torque generation model of the optimum ECB configuration is needed. In Chapter 2, an analytical model was presented; however, this model does not account for the use of multiple PPAs. Therefore, it cannot be directly used to model the braking torque genera tion for the optimum ECB configuration. In Chapter 3, an accurate FEM was in troduced; however, due the complexity and nonlinearity of the problem, this mo del is unsuitable for design of a realL time controller for the ECB. 90 Therefore, in this section, a simplified dynamic mo del for the braking torque generation of the optimum ECB configuration with 4 PPAs is derived with the help of both the analytical and FE models and nonlinear mod el identification tools. The FEM introduced in Chapter 3 was used to solve t he braking torque generation under frequency modulated sinusoidal field applicat ion on an optimum ECB configuration with 4 PPAs at various rotational spe eds. The results data is then used to create a dynamic model of the specific ECB with the help of the analytical model. Although the analytical model does not account for the frequency modulated field application nor the multiple PPA configurations, it can give important insight information about the correlation between the externally applie d field, rotational speed of the conductor brake disk and generated braking torque. The generated braking torque model for a 1LPPA conf iguration was given in Eq. (2.48). From this equation, it is clear that the br aking torque is comprised of a geometric function ( Gf), a time function ( Ttime ) and a coefficient ( C). The geometric function is simply a function of the dimensions of the conducto r disk and PPA as well as the location of their boundaries. On the other hand, the coeffic ient C is comprised of material properties such as conductivity ( σ), permittivity ( ε) and magnetic permeability ( µ) as well as the field magnitude ( Bin ) and rotational speed of the disk ( w). Finally, the time term is resulted from the alternating field application. Th e braking torque in Eq. (2.48) can then be rewritten in terms of generic geometric and time functions and the coefficient term as follows: /g1846/g3029=/g1829(/g1828/g3036/g3041 ,/g2020,/g2026,/g2013,/g1875)∙/g1846/g3047/g3036/g3040/g3032 ∙/g1833/g3033(/g1856,/g1853,/g1854,/g1844,/g1870) (5.1) where d, a, b, R , and r are the thickness of the disk, width and the heigh t of the PPA, radius of the disk and the distance pole location d istance, respectively. For a given configuration, the material properties and the geometric function will be constant and the magnitude of the applied field ( Bin ), the rotational velocity of the brake disk ( w) and the time function ( Ttime ) will be changing depending on the braking requirements. Therefore, for the ECB problem, Bin and w are the inputs to the brake system and the output will be the braking torque ( Tb). Note that it is assumed that the time 91 function is constant for the configuration and the controller alters the braking torque by varying the magnitude of the applied field. In order to come up with a realLtime dynamical mode l for the ECB, initially the proportionality between the inputs and the output i s defined. From the equations in Chapter 2 and the results presented in the same cha pter, it can be concluded that the braking torque is proportional with the square of t he magnitude of the applied field and the rotational velocity. Note that this makes the s ystem nonlinear. In addition, in Figure 4L5, the braking torque generation was plotted for an optimum configuration under the constant magnitude time varying field application. Using the analytical model and the results obtained in Chapter 4, the coefficient valu e in Eq. (5.1) can be rewritten as: /g1829 =B/g2919/g2924 /g2870∙(/g1875∙/g1855(/g2020,/g2026,/g2013)+/g1855/g2869) (5.2) where c is a coefficient that is calculated using the mate rial properties of the conductor ECB disk and c1 is an arbitrary constant. As the next step, the time function can be estimate d using the time variation term inherited from the externally applied field. Since the torque is proportional with the square of the externally applied field, the proport ionality will be similar between the torque and the sinusoidal time function that define s the applied field as well. Since the time term is a part of the externally applied field , for a sinusoidal field with frequency modulation, the time function will be as follows: /g1846/g3047/g3036/g3040/g3032 =/g4674sin/g46722π/g1858 /g2869t−∆/g3033 /g3033/g3118cos (2π/g1858/g2870t)/g4673/g4675/g2870 (5.3) where f1 is the carrier frequency, f2 is modulation frequency and ∆f is the frequency deviation. Using Eq. (5.2) and Eq. (5.3), Eq. (5.1) can be rewritten in terms of the applied field and the rotational velocities, i.e. /g1846/g3029=(B/g2919/g2924 /g2870∙(/g1875∙/g1855(/g2020,/g2026,/g2013)+/g1855/g2869))∙/g4674sin/g46722π/g1858 /g2869t−∆/g3033 /g3033/g3118cos (2π/g1858/g2870t)/g4673/g4675/g2870 ∙/g1855/g2870 (5.4) 92 where c2 is the constant value of the geometric function fo r the selected ECB configuration. Note that for the 4LPPA configuratio n, phase differences were introduced between the applied fields of each PPA. In order to account for the phase differences, the braking torque equation can be redefined as: /g1846/g3029=/g1855/g2870(B/g2919/g2924 /g2870(/g1875 /g1855(/g2020,/g2026,/g2013)+/g1855/g2869))/g4674sin/g46722π/g1858 /g2869t+/g1868−∆/g3033 /g3033/g3118cos (2π/g1858/g2870t+/g1868)/g4673/g4675/g2870 (5.5) where p is the phase difference of the applied field at th e PPA compared to a reference field. Once the correlation is defined, the dynamic ECB model can be obtained using the results of the FEM by estimating the coefficients i n Eq. (5.5) using model identification methods. Note that the frequencies for the optimum braking torque generation under the sinusoidal field application are known (see Table 4 L2), and these values are used in Eq. (5.5) to estimate the braking torque. Also, the abo ve equation is defined only for one PPA. Since the optimum configuration contains 4 PPA s, 4 different braking torque values must be defined and be summed up to estimate the to tal braking torque. /g1846/g3029,/g3041=/g1855/g2870(B/g2919/g2924 /g2870(/g1875 /g1855(/g2020,/g2026,/g2013)+/g1855/g2869))/g4674sin/g46722π/g1858 /g2869t+/g1868/g3041−∆/g3033 /g3033/g3118cos (2π/g1858/g2870t+/g1868/g3041)/g4673/g4675/g2870 (5.6) /g1846/g3029,/g3047/g3042/g3047/g3028/g3039 =/g1846/g3029,/g2869+/g1846/g3029,/g2870+/g1846/g3029,/g2871+/g1846/g3029,/g2872 (5.7) where pn is the phase difference and Tb,n is the braking torque generated at the nth PPA, and Tb,total is the total braking torque generated by the optim um 4LPPA ECB configuration. Note that the applied field to the f irst PPA is assumed to the reference field so the phase differences of the other PPAs are calc ulated in comparison to this field. Therefore p1 is equal to zero. In Figure 5L1, the FEM results (blue solid line) an d the estimated dynamic model (black solid line) are compared. The simulation is carried out for a sinusoidal variation at 100 Hz in the magnitude of the applied field, Bin (T), and decreasing rotational velocities, w (rpm). /g1828/g3036/g3041 =0.5+0.5∙sin (2/g2024100/g1872) (5.8) 93 /g1875 =1000−1000∙/g1872/0.5 (5.9) Figure 5L1. Comparison between the estimated model and the FEM results The simulation results show a good agreement betwee n the two models. Therefore, the estimated dynamic model is used in the subsequent s ections to estimate the braking torque generation of the optimum 4LPPA ECB configurations for control design purposes. 5.3 Dynamic model of the vehicle In this section, the vehicle dynamics are studied i n order to calculate the deceleration values using the braking torque generated by 4 ECB actuators of the 4LPPA configuration. The motion of a vehicle can be described using the quarter vehicle model [68]. This model uses the assumption that the weight of the ve hicle is distributed equally between the four wheels so that the calculations can be car ried out by defining the dynamics of only one wheel. Since the system is a reduction fro m the whole vehicle, the weight of the reduced system is assumed to be the 1/4 th of the total weight of the vehicle. In addition, 94 the total inertia of the system is also approximate d to a total value using inertias of wheel, engine and the brake disks. The approximated values and the calculations are presented in detail in Appendix A. After defining all the forces acting on the wheel, the Newtonian equations of motion for the wheel can be solved for deceleration values (see Eq. (A.7) and Eq. (A.8)). /g1876/g4663 =/g2879/g3091/g3281/g3034 /g2869/g2879/g3339/g3281/g3288/g3297/g3283/g3278/g3282 /g3287/g3277/g3276/g3294/g3280 /g3288/g3295 (5.10) /g2016/g4663=(−/g1846/g3029,/g3047/g3042/g3047/g3028/g3039 +/g2020/g3033/g1844/g3050/g1832/g3041−/g1844/g3050/g1832/g3045)//g1835 (5.11) where /g2016/g4663 is the deceleration, Tb,total is the total braking torque generated by the ECB ( see Eq. (5.7)), /g2020/g3033 is the friction coefficient, Rw is the radius of the wheel, Fn and Fr are the normal and the rolling resistance forces applied on the wheel and I is the total mass moment of inertia of the vehicle. (For further deta ils on the equations and the free body diagram of the wheel, see Appendix A.) 5.4 Design of a model based nonlinear predictive c ontroller 5.3.1 Nonlinear model predictive control algorithm In Figure 5L2, a block diagram of the proposed nonl inear model predictive controller is presented. As mentioned previously, NMPC utilizes a n explicit dynamic model of the plant in order to determine the optimum response in taking the system to a desired steady state. In this section, the proposed NMPC algorithm is briefly explained using a plant, which is defined in state space. 95 Figure 5L2. Nonlinear model predictive control bloc k diagram For the NMPC controller, a discrete time model of t he plant is necessary for optimum trajectory determination. Assume that the nonlinear plant can be described in state space, e.g., /g1824/g3038/g2878/g2869=/g1806(/g1824/g3038,/g1821/g3038,/g1822/g3038,/g1804/g3038) (5.12) /g1825/g3038=/g1807(/g1824/g3038)+/g1826/g3038 (5.13) where /g1821/g3038/g2035 ℜ/g3041/g3296is a vector of nu plant inputs (or manipulated variables (MVs)), /g1825/g3038/g2035 ℜ/g3041/g3300 is a vector of ny plant outputs (or controlled variables (CVs)), /g1824/g3038/g2035 ℜ/g3041/g3299is a vector of nx system state variables, /g1822/g3038/g2035 ℜ/g3041/g3297 is a vector of nv measured disturbance variables (DVs), /g1804/g3038/g2035 ℜ/g3041/g3279is a vector of nd unmeasured DVs and /g1826/g3038/g2035 ℜ/g3041/g3301is a vector of measurement noise. The control algorithm determines a sequence of inpu ts ( uk+j ) that will drive the system from the plant’s current state ( xk) to a desired steady state ( xs) according to an economic objective. The algorithm carries out a local steady state optimization in order to find the desired state. In addition to the system variables, the measured and unmeasured DVs are included in the f function defined in Eq. (5.12) and the measurement noise ( zk) is included in the output as a bias term, which compares the cu rrent output values ( yk) to the 96 measured output ( ym k). Also note that it is assumed that the bias term is constant for any given time. Once the plant is defined, as the next step, an opt imization problem is created for the plant according to the desired behaviour of the sys tem. A simple nonlinear objective function that is commonly used for NMPC application s is shown below [57]: min/g1836(/g1821/g3014)=∑/g3630/g1805/g3038/g2878/g3037/g3052/g3630 /g1791/g3285/g3044/g3017 /g3037/g2880/g2869 +/g3630/g1819/g3037/g3630/g1794/g3285/g3044+∑/g3630/g1805/g3038/g2878/g3037/g3044/g3630 /g1792/g3285/g3044/g3014/g2879/g2869 /g3037/g2880/g2868 +/g3630/g2722/g1821 /g3038/g2878/g3037/g3630/g1793/g3285/g3044 (5.14) The equality and inequality constraints are defined for a tighter control at the optimal constrained steady state of the process [58]: /g1824/g3038/g2878/g3037=/g1806/g3435/g1824/g3038/g2878/g3037/g2879/g2869,/g1821/g3038/g2878/g3037/g2879/g2869/g3439 ∀/g1862 =1,/g1842 (5.15) /g1824/g3038/g2878/g3037=/g1807/g3435/g1824/g3038/g2878/g3037/g3439+/g1802/g3038 ∀/g1862 =1,/g1842 (5.16) /g1825/g3037−/g1819/g3037≤/g1825/g3038/g2878/g3037≤/g1825/g3037+/g1819/g3037 ∀/g1862 =1,/g1842 (5.17) /g1821≤/g1821/g3038/g2878/g3037≤/g1821 ∀/g1862 =0,/g1839−1 (5.18) ∆/g1821≤∆/g1821/g3038/g2878/g3037≤∆/g1821 ∀/g1862 =0,/g1839−1 (5.19) /g1819≥0 (5.20) In Eq. (5.14), there are 4 main terms to be defined for the optimum control problem. The first term ( /g1805/g3038/g2878/g3037/g3052) is the deviation of the calculated output ( yx+j ) from the desired steady state output ( yss ) over a horizon of length P, i.e. /g1805/g3038/g2878/g3037/g3052=/g1825/g3038/g2878/g3037−/g1825/g3046/g3046 (5.21) Also, a positive definite weight matrix, ( /g1791/g3037) is defined to account for the relative importance between these 4 terms defined. The secon d term in this equation ( /g1819/g3037) is the slack variable that penalizes the output constraint violations (see Eq. (5.17)). This term also has a weight matrix ( /g1794/g3037) to define its relative importance. The third term ( /g1805/g3038/g2878/g3037/g3044) is the deviation of the calculated input term ( uk+j ) from the desired steady state input ( uss ), i.e. 97 /g1805/g3038/g2878/g3037/g3048=/g1821/g3038/g2878/g3037−/g1821/g3046/g3046 (5.22) The weight matrix defined for the input deviation t erm is /g1792/g3037. Finally, the last term in this equation penalizes the rapid input changes and the weight matrix is specified as /g1793/g3037. Note that all these deviation and constraint violation t erms are defined as norms of vectors (either L 1 or L 2 norms ( q =1, 2)). For the specific ECB problem, q is set to be 2 and the terms in Eq. (5.14) are defined as follows: /g3630/g1805/g3038/g2878/g3037/g3052/g3630 /g1791/g3285=(/g1805/g3038/g2878/g3037/g3052/g3021∙/g1791/g3037∙/g1805/g3038/g2878/g3037/g3052)/g2868./g2873 (5.23) /g3630/g1819/g3037/g3630/g1794/g3285=(/g1819/g3037/g3021∙/g1794/g3037∙/g1819/g3037)/g2868./g2873 (5.24) /g3630/g1805/g3038/g2878/g3037/g3048/g3630/g1792/g3285=(/g1805/g3038/g2878/g3037/g3048/g3021∙/g1792/g3037∙/g1805/g3038/g2878/g3037/g3048)/g2868./g2873 (5.25) /g3630/g2722/g1821 /g3038/g2878/g3037/g3630/g1793/g3285=(/g2722/g1821 /g3038/g2878/g3037/g3021∙/g1793/g3037∙/g2722/g1821 /g3038/g2878/g3037)/g2868./g2873 (5.26) In the above, the derivations were shown for a gene ric NMPC algorithm [57]. Figure 5.3 shows the flowchart of the algorithm. The reade r can find a more inLdepth description of the algorithm in [56] and [58]. 98 Figure 5L3. NMPC Flowchart (MV L manipulated variab les, DV L disturbance variables and CV L controlled variables) 5.3.2 NMPC applied to ECB problem The ECB model defined in the previous section is a continuous one and, in order to be able to design a NMPC controller, this model has to be converted into a discrete time model. Note that for the ECB problem, all the distu rbance terms are ignored ( /g1822/g3038,/g1804/g3038=0 in Eq. (5.12)) and all variables (velocity and acce leration terms for both the wheel and the vehicle) are defined as MVs. In addition, the measu rement noise is assumed to be zero 99 (/g1826/g3038=0 in Eq. (5.13)). The output from the system is defi ned to be the deceleration of the vehicle. In terms of the ECB, braking torque generation can be converted into a discrete model using the relationship below. Note that the torque is a function of the angular velocity of the wheel and the magnitude of the applied field. /g1846/g3029,/g3047/g3042/g3047/g3028/g3039 /g3038=ℎ(/g1828/g3036/g3041 /g3038,/g2016/g4662/g3038,/g1872/g3038) (5.27) where h is the dynamic model identified in Sec. 5.3.1 and tK is the time. Note that the notation in the quarter vehicle is adapted for w (i .e. the angular velocity, /g2016/g4662/g3012) in this section. As the next step, the dynamic vehicle mode l is converted into discrete equations using the relationships below. Refer to Appendix A for the full continuous set of equations used here. Also, the variables and the co nstant that are selected for the vehicle model used for the analysis is shown in Table 5L1. /g1832/g3045/g3038=/g1858/g2868+3.24/g1858 /g3046(/g1837/g3049/g1876/g4662/g3038)/g2870./g2873 (5.28) /g1832/g3041/g3038=/g1865/g3047/g1859+/g3040/g3297/g3035/g3252/g3254 /g3039/g3277/g3276/g3294/g3280 /g1876/g4663/g3038 (5.29) /g1871/g3045/g3038=/g3051/g4662/g3286/g2879/g3019/g3298/g3087/g4662/g3286 /g3051/g4662/g3286 (5.30) /g2020/g3033/g3038=/g3421/g1871/g3045/g3038/0.175 0.175 ≥/g1871/g3045/g3038≥0 1−/g2868./g2870/g3435/g3046/g3293/g3286/g2879/g2868./g2869/g2875/g2873 /g3439 /g2868./g2876/g2870/g2873 /g1871/g3045/g3038>0.175 /g1 (5.31) /g1832/g3033/g3038=/g2020/g3033/g3038/g1832/g3041/g3038 (5.32) Note that the friction coefficient is obtained from an empirical plot between the slip ration and the friction coefficient between the tire and t he road (see Figure AL2). For the purposes of current control design, it is assumed t hat the vehicle is moving on a dry surface. 100 Table 5L1. Specifications of the vehicle and road c onditions Considering that the state vector contains the spee d and acceleration terms for the wheel and the vehicle, the state in the next time s tep can be calculated using the dynamic equations and approximate integration equations as shown below. Eq. (5.33) and Eq. (5.34) are obtained using the equations of motion a nd using these two relationships, angular acceleration of the wheel and the accelerat ion of the vehicle are found (For ECB applications, accelerations are negative). Note tha t these terms not equal due to the slip between the road and the wheels. /g1876/g4663/g3038/g2878/g2869=−/g2020/g3033/g3038/g1859+/g2020/g3033/g3038/g3040/g3297/g3035/g3252/g3254 /g3039/g3277/g3276/g3294/g3280 /g3040/g3295/g1876/g4663/g3038 (5.33) /g2016/g4663/g3038/g2878/g2869=(/g2879/g3021/g3277,/g3295/g3290/g3295/g3276/g3287 /g3286/g2878/g3019/g3298/g3007/g3289/g3286/g2879/g3019/g3298/g3007/g3293/g3286) /g2893 (5.34) /g1876/g4662/g3038/g2878/g2869=/g3051/g4663/g3286/g3126/g3117/g2878/g3051/g4663/g3286 /g2870∆/g1872+/g1876/g4662/g3038 (5.35) /g2016/g4662/g3038/g2878/g2869=/g3087/g4663/g3286/g3126/g3117/g2878/g3087/g4663/g3286 /g2870∆/g1872+/g2016/g4662/g3038 (5.36) The velocity terms are then calculated using direct integration. Eq. (5.35) and Eq. (5.36) are obtained using the trapezoidal rule. Thi s approach is not ideal for cases where the noise is significant. However, since it is assu med that the noise to the system is negligible, the approach selected to calculate the velocity terms are deemed accurate. Using the above equations, the state at the next st ep can be calculated. 101 As the next step, an objective function with adequa te constraints for the ECB NMPC design. An objective function similar to Eq. (5.14) is defined, i.e. min/g1836(/g1821/g3014)=∑/g3630/g1805/g3038/g2878/g3037/g3052/g3630 /g1791/g3285/g3017 /g3037/g2880/g2869 +/g3630/g1819/g3037/g3048/g3630/g1794/g3285+/g3630/g2722/g1821 /g3038/g2878/g3037/g3630/g1793/g3285 (5.37) Deviation of the vehicle deceleration from a specif ied reference value is set to be the first penalizing factor in the objective function f or the ECB. Here, it is assumed that for comfortable braking, the deceleration reference val ue of the vehicle is set to be 6 m/s 2 (= /g1876/g4663/g3046/g3046 ). The second term is the slack term. Unlike the ge neral form, here it is defined for the input, Bin . This slack factor ensures that the control signal , magnitude of the externally applied field, is bounded as shown in Eq. (5.38). F inally, the last term is the penalty for rapid input changes in the system. In addition, wei ght factors are assigned for each penalty term. 0≤/g1821/g3038/g2878/g3037≤1.5 ∀/g1862 =1,/g1842 (5.38) Note that for the subsequent simulation analysis, t he time step is selected to be 0.1 ms in order to account for the explicit dynamics of th e system, considering that the time varying field application is at 300 Hz. While havin g a longer horizon will result in better accuracy, since the time step is significantly low, the length of the horizon for the ECB control problem is set to be 50. 5.5 Simulation results In order to carry out the controller simulations, a n optimization algorithm must be selected. There are a vast number of optimization a lgorithms readily available in the literature. These algorithms can be classified into two main categories; gradient based and nonLgradient base. GradientLbased systems are not s uccessful when there are local minima. GradientLbased techniques utilize an initia l guess and the gradient data to estimate the direction towards the minimum. If the system is highly nonlinear, the result 102 of such an algorithm may converge to a local minimu m instead of a global minimum. In terms of controller design, such results may cause stability issues in the system. For the ECB problem herein a nonLgradient based sea rch algorithm, Particle Swarm Optimization (PSO), is used to find the global mini mum. Among gradient free algorithms, PSO is a recently introduced global optimization te chnique that has been used with great success in the area of computational intelligence. It is a direct search algorithms that is used to find the global optimum in a given search s pace. Due to its simple formulation, it is easy to implement, apply, extend and hybridize t he algorithm. The algorithm is modeled on swarm intelligence, which is based on th e socialLpsychological principles exist within large groups of organisms. Communicati on or social network is critical for this algorithm, through which individual solutions passes the location of the best neighbouring solution by means of a fitness value. PSO was first introduced by Kennedy and Eberhart [69] and their study was inspired by t he social behaviour of birds. In order to find food, each bird flies with a velocity and a di rection, and these are determined by experience and the interaction of the bird with oth er birds in the flock. Every bird, particle, will be assigned with a direction and a v elocity and it will then search for the optimum in the search space and its position repres ents a potential solution. The simulations are carried out and the results obt ained are presented in Figure 5L4. For the specific simulation, the brakes are applied for a 5 second duration to slow the vehicle down from about 120 km/h. The first plot shows the variation in the control variable, B in , during the simulation. As it was bounded in the opt imization problem, the input did not exceed the 1.5 T limit specified. Also, the values are increasing as time progresses. Since the braking torque decreases with decreasing veloci ties, B in is increased to account for the decrease in the braking torque. Another constraint in the problem was the reference value defined for the deceleration of the vehicle. The va riation of the deceleration is shown in the 4 th plot in the same figure. It can be seen that the d eceleration converges rapidly to L6 m/s 2 within 0.1 s and the controller makes sure the sys tem stays at this desired steady state for the rest of the time during braking. The last constraint that was penalized in the objective function was the amount of input changes. This prohibits a significant jump in the magnitude of the applied field intensity, as sh own in the first plot. The same figure also shows the variation of the ang ular velocity of the wheel and the linear velocity of the vehicle braking (5 th plot) . The displacement is calculated using trapezoidal rule similar to Eq. (5.35) and Eq. (5.36). The controller performance can be improved by adjusting the optimization problem and by introducing additional penalty terms depending the accuracy of the controller will increase as the time step decreases and as the le The same figure also shows the variation of the ang ular velocity of the wheel and the linear velocity of the vehicle (3 rd plot) and the displacement of the vehicle during . The displacement is calculated using trapezoidal rule similar to Eq. Figure 5L4. ECB performance with NMPC controller performance can be improved by adjusting the optimization problem and by introducing additional penalty terms depending on needs. In terms of the simulations, the accuracy of the controller will increase as the time step decreases and as the le 103 The same figure also shows the variation of the ang ular velocity of the wheel (2 nd plot) and the displacement of the vehicle during . The displacement is calculated using trapezoidal rule similar to Eq. controller performance can be improved by adjusting the optimization problem and needs. In terms of the simulations, the accuracy of the controller will increase as the time step decreases and as the le ngth of 104 the horizon is increased. However, as these modific ations are made, the optimization will require longer time to determine the response over the horizon defined and it will be harder to implement the controller realLtime. Compared to a no control scenario, having such a co ntroller will result in a number of performance improvements in power consumption and i n brake parameter tuning for different braking scenarios. Power storage and cons umption are critical issues for electric cars. In terms of braking, the controller optimizes the power input to the brakes through adjusting the magnitude of the applied magnetic fie ld and this will ensure that the braking objectives, which are userLbased or environmentalLb ased, are going to be achieved with minimum power input to the system. In terms of tuning, the objective function can be d efined for various braking conditions. For the specific case presented in Figure 5L4, a re ference value is set for the deceleration and the brakes generate enough braking torque to co nverge to the set reference value. The system can be modified easily by properly defining the objective function according to needs. This makes it possible to generate optimum b raking performance for different braking conditions, i.e. braking on a slippery surf ace, emergency braking, controlled braking, etc. For emergency braking, reference valu e can be removed and the inputs that are going to result in highest braking torque at al l times can be calculated. For controlled braking, the reference value of the deceleration ca n be adjusted according to the pressure on the brake pedal applied by the driver. For slipp ery surfaces, an objective function that is defined using the relative velocities of the whe el and the vehicle can be used. Once the function is properly defined, this will result in n oLslip braking. For real road braking simulations, all these conditions can be implemente d in the controller and an adaptive or switching control algorithm can be employed to defi ne the appropriate braking parameters for the system. Although it is an advantageous algorithm, the main limitation of the above control system is the computational cost in obtaining accur ate future trajectories using optimization. The computational cost significantly depends on the nonlinearity of the problem and it directly affects the realLtime perfo rmance of the controller. In order to improve the realLtime performance, the model can be linearized around the operating point or a faster search algorithm can be used. 105 5.6 Conclusion In this chapter, a controller is designed for an op timum configuration presented in Chapter 4. Initially, a dynamic model of the system is obtained for the ECB and the vehicle to determine the state variables (speeds an d acceleration terms of the wheel and the vehicle.). Since the analytical and FE model ar e previously studied in this thesis, the application is suitable for model predictive contro l. Here a NMPC is designed for the system by defining an objective function along with constrains that will drive the plant to a desired steady state. The simulations are carried out for a scenario where the vehicle is slowed down from 120 km/h to 10 km/h in 5 seconds. The results show that the system reaches the steady state within 0.1 s and NMPC make s sure the system is behaving as requested for the rest of the time during braking. 106 6. EXPERIMENTAL STUDIES 6.1 Introduction The design of the ECB does not include any complexi ty since it is essentially composed of a conductive disk attached to the wheel shaft and the source of the applied magnetic field. Therefore, apart from all the advan tages mentioned earlier, this type of brake is very easy to prototype. Considering applic ations where DC field is applied on the conductive disk, the source of the field can be a s imple highLintensity permanent magnet. Thus, it is very easy to assemble a brake actuator using off the shelf components. Such configurations are used to slow trains before the stations when they start slowing down for a full stop at the station. Having permane nt magnets laid on the railway realizes a simple ECB configuration (the metal tires of the train are the conductive disks of the ECB configuration presented Figure 1L3. in Chapter 1). In previous chapters, it has been shown that the AC field application improves the br aking torque generation significantly. However, in order to be able to introduce the time variation in the applied field, electromagnets must be used and they become the mai n complexity of the design for a configuration where AC field is applied. For experimentation, the most important component i s to design an electromagnet that supplies the required field with minimum losses. Th ere are two major losses on an AC electromagnet [70,71], one of them is eddy current loss and the other one is the hysteresis loss. Although eddy currents are favourable in this study for braking torque generation, in terms of electromagnets, when there are eddy curren ts present, an opposing field will be created within the magnet core and this opposing fi eld will reduce the applied field significantly. Therefore, in practice, specific mat erials with high permeability and low conductivity are used to eliminate the possibility of eddy current generation on the magnet core. Ferrite cores are a good example to su ch materials. Ferrite has high magnetic permeability and they are considered nonLc onductive. In addition, they are readily available and cheap. However, the saturatio n flux density of the ferrite is significantly low compared to other options (about 0.5 T [72]). 107 Also, laminated steel is another popular alternativ e for AC electromagnet cores. High magnetically permeable and high conductive material s are stacked on each other with a layer of nonconductive bonding in between. This wil l prevent the flow of eddy currents from one strip to other, thus making the core nonco nductive with a high magnetic permeability. They are preferable for applications where higher magnetic flux densities are required for the application due to their high saturation flux density levels. However they are not readily available and in general, they are stacked according to the requirements of the specific problem. Therefore the y are a lot more expensive than ferrite cores. The other loss, hysteresis loss, is caused by the s witching direction of the applied field on the core. Due to the coercivity of the magnetic core selected for the application, some of the energy will be lost at every change of direc tion. Coercivity can be expressed as the resistance of the material to changes in magnetisat ion. The energy lost is proportional to the area between enclosed by the hysteresis curve o f the material. In power transformers, in order to be able to reduce this loss, soft coerc ive materials, such as silicon steel or soft ferrite, are used. Here in this chapter, the experimental studies that are carried out for the ECB and the setup that is designed for the experiments are expl ained in detail. In the first section, the experimental setup that was used to measure the bra king torque is presented and the equipment is introduced. In the following two secti ons, DC and AC field experiments are presented. Note that the experimental studies are o nly carried out for the DC case but the setup and ECB prototype are prepared for the AC fie ld experiments which will be carried out in future (see Chapter 7 for a list of future w orks). 6.2 Experimental setup In order to be able to measure the braking torques generated, an experimental setup is designed. A servomotor from CMC Inc. with a continu ous torque of 5.13Nm and with rotational speed of 5445 rpm is used to generate th e continuous torque. Since the torque generation capacity of the servomotor is relatively low, an ALPHA 0755LMC1L7 gear reducer (7:1) is used. The servomotor is then conne cted to a FUTEK torque sensor 108 (TRS605), which is a shaftLtoLshaft rotary torque s ensor with a torque measuring capacity up to 1000 Nm. The other end of the torque sensor i s connected to the ECB prototype. An Inertia Dynamics magnetic clutch is installed betwe en the torque sensor and the servomotor in order to release the load on the serv omotor generated by the brake. In order to connect the various components, flexible couplin gs are used. Below in Figure 6L1, the components of the experimental setup are shown and at the right end of the figure, in the metal casing, there is the ECB configuration select ed for the DC field application experiments. Figure 6L1. Experimental setup The servomotor is controlled with a proportionalLin tegralLderivative (PID) controller and the rotational speed of the brake is verified u sing the encoder embedded in the torque sensor. All input and output signals are connected to a dSPACE control board (DS1104). The control signal from the dSPACE board to the ser vo is amplified using an Advanced Motion Control Brushless PWM servo amplifier. In ad dition, a lowLpass filter circuit is implemented into the setup in order to reduce the h igh frequency noise in the torque sensor readings. During the experiments, the rotational speed is kep t constant at various values. Whenever the brake is actuated, since the system is at steady state, the relative torque measured between the shafts on each side of the tor que sensor is recorded as the braking 109 torque generated by the brake prototype. After the braking torque data is obtained, the magnetic clutch was turned on in order to release t he torque load on the servomotor. 6.3 DC field application In order to verify the DC FEM, an ECB prototype was built and the braking torque generation at various angular velocities was record ed. During the experiments, a custom conductor disk was used along with permanent magnet s positioned on each side of the disk. Figure 6L1 shows the experimental setup used. It consists of a servomotor (on the left), magnetic clutch, couplings, torque sensor an d the brake actuator (on the right). The conductor disk is mounted in between two static fix tures where the magnets are hold in place. Figure 6L2 shows a closeLup image of the ECB . The permanent magnets can easily be seen on both sides of the rotating disk. The specifications for the setup are given in Table 6L1. Figure 6L3 shows the comparison between the FE simulation and experiment al results. The nonlinear behaviour of the experimental system is evident in the figure , compared to that of the FE simulation results. The more linear behaviour of the FE simula tion is probably due to the initial assumption that the DC field does not change with r espect to the spatial coordinates and unLmodeled dynamic effects such as variations in th e magnetic property with respect to temperature. Note that since permanent magnets (ins tead of electromagnets) are used here the induction effect can be ignored. 110 Figure 6L2. Eddy current brake (ECB) prototype with permanent magnets Table 6L1. Experimental setup specifications Radius of the disk (m) 0.1524 Width of PPA (m) 0.0254 Height of PPA (m) 0.0254 Center of PPA (m) 0.0762 Thickness (m) 0.00714 Material Aluminum Alloy 6061 Conductivity (S/m) 5.79E+07 Magnet Permanent Magnets (no induction effect) Field 0.749 T (Peak) and 0.475 T(min) 111 Figure 6L3. Comparison between the FE simulation an d experimental results for the prototyped configuration In this experiment, permanent magnets were used in the ECB instead of electromagnets for quick preliminary testing. The s imulation and experimental results are in a good agreement. There are a number of importan t conclusions that can be made from Figure 6L3. First conclusion is that the braking to rque vs. disk velocity is a nonlinear behaviour. The fact that the FE simulation showed a more linear relationship is due to the fact that the FEM assumed a uniformly distributed a pplied magnetic field over the PPA on the ECB’s disk. This implies that the applied fi eld does not have a spatial dependence. Due to this assumption, the last term on the right hand side of Eq. (6.1), i.e. (6.1) becomes negligible. However, in reality, the distri bution of the field on the PPA is not uniform. When the magnetic flux leaves the core of the magnet, it bulges out of the straight linear path and curves back into the oppos ing pole. Because of this bulging effect, the field distribution in the air gap is different from that of in the core. These nonLuniform ) (1) (1) ( ) ( 0 00 0 B B Bv Bv E ×∇×∇ +×∇×∇ =××∇+××∇+×∇σd σd µ µ 112 fields are called the fringing fields, and they res ult in a spatially varying field distribution in the PPA (i.e. maximum field intensity in the cen ter of the PPA and decaying field intensity towards the edges). Another source of non linearity is the stray flux (fringing fields are considered to be a subset of the stray f lux, but for the sake of discussion, herein they assumed to be separate). Herein, the stray fie lds are defined as fields that leave the electromagnet from the core surface, not the pole f ace, and enter the opposing pole through the core surface, again not through the pol e face. These fields create a PPA that is larger than the area of the pole face, where the FE M assumes that the PPA and the area of pole face are equal. Figure 6L4 shows the flux dens ity distribution between two permanent magnets in a streamline plot and stray flux can eas ily be seen between the magnets. Figure 6L4. Flux density distribution between two p ermanent magnets In addition to the FEM assumption of uniformly dist ributed field, environmental and measurement noise in the readings as well as the ma terial property change due to the increasing temperatures cause the nonlinearity in t he experimental results presented in Figure 6L3. Another important conclusion that can be drawn from the experimental results is that the amount of braking torque generated still seems to be quite lower than what is required for a passenger vehicle, even at high velocities (s ee Appendix B for requirements for various vehicles). Some solutions to this problem h ave already been addressed earlier in Section 3.4. 113 6.4 AC field application Unlike the previous DC field application case, here for AC field application, electromagnets are required. In addition to the set up described earlier, a power source will be supplying the current to the electromagnet, whic h is the source of the applied field for the experiments. Since the field is alternating, th e power source should be able to supply alternating current to the coil of the electromagne t. For this specific purpose, a power supply (Agilent 6813B with a power rating of 1750 V A) is used. This power supply is capable of supplying currents up to 13 A, and also this specific power supply is equipped with arbitrary waveform generator which will be use ful to apply fields in numerous waveforms. From the results presented in Chapter 4, it can be said that an optimum configuration can generate over 800 Nm with 4 PPA, although the t orque sensor can measure torques up to 1000Nm, the servomotor would not be able to w ithstand such a high torque. As was mentioned earlier, the braking torque is calculated at steady state, when the brake and the servomotor are activated at the same time. With the gearbox used, the setup can generate slightly over 35 Nm. Therefore, for experimentation , a configuration which can generate braking torques less than the limiting torque the m otor can supply is selected. The dimensions of the PPA are another limiting fact or for prototyping as well. For the specific experiments, ferrite cores are obtained fr om Magnetics, Inc. (#0P49920UC). They are standard C shaped cores (specifications of the core is given in Appendix C) with one end grinded down (3.5 mm down), so that in orde r to complete an electromagnet, 2 cores are attached together and the clearance accou nts for the air gap (7 mm in total) where the conductive disk will be placed. The diagr am below shows the core of the AC electromagnet that is designed for the AC fieldLtes ting of ECB. Note that the upper part, where the cores join, are attached to each other us ing a thin layer of adhesive and they are joined right after the coil bobbin is placed in pos ition. As was mentioned earlier, ferrite has low coercivit y and high permeability and in addition it is consider a nonLconductive material. Thus it is a very good candidate for AC electromagnets. They are cheap and readily availabl e. However, the biggest problem with the material is the saturation flux density. Once t he core reaches the saturation, the relative permeability goes down to 1, making it a n onLmagnetic material; therefore the flux will be forced to find another path with high permeability (less reluctance) [ Ferrite saturates around 0.5 T thus it is not a goo d candidate f magnetic flux densities are required. Figure As the next step, the coil needs to be designed as the source of the applied magnetic field . Looking at the dimensions presented in Appendix C and the configuration shown in Figure 6L5 , the dimensions of the coil bobbin that can be pla ced on the electromagnet core can be easily calculated. For the specific problem , AWG 21 is selected since more current density can be appli Considering all the dimensions of the selected magn et cores, an ECB configuration is created and the specifications of the ECB are Table 4L3, the braking numbers of PPAs were given. Considering the limits on the supplied torque by the servomotor, a one PPA configuration with sinusoidal or triangular field application is selected. flux will be forced to find another path with high permeability (less reluctance) [ Ferrite saturates around 0.5 T thus it is not a goo d candidate f or applications where high magnetic flux densities are required. Figure 6L5. AC electromagnet core configuration the next step, the coil needs to be designed as the source of the applied magnetic . Looking at the dimensions presented in Appendix C and the configuration shown in , the dimensions of the coil bobbin that can be pla ced on the electromagnet core can be easily calculated. For the specific problem , AWG 21 is selected since more current density can be appli ed through this size of wire compared to other size s [ Considering all the dimensions of the selected magn et cores, an ECB configuration is specifications of the ECB are given in the table below. Also note that in torque generation for optimum configuratio ns with different numbers of PPAs were given. Considering the limits on the supplied torque by the servomotor, a one PPA configuration with sinusoidal or triangular field application is 114 flux will be forced to find another path with high permeability (less reluctance) [ 72]. or applications where high the next step, the coil needs to be designed as the source of the applied magnetic . Looking at the dimensions presented in Appendix C and the configuration shown in , the dimensions of the coil bobbin that can be pla ced on the electromagnet core can be easily calculated. For the specific problem , AWG 21 is selected since more ed through this size of wire compared to other size s [ 73]. Considering all the dimensions of the selected magn et cores, an ECB configuration is given in the table below. Also note that in ns with different numbers of PPAs were given. Considering the limits on the supplied torque by the servomotor, a one PPA configuration with sinusoidal or triangular field application is 115 Table 6L2. Specifications of the optimum one PPA co nfiguration from Table 4L3 Radius of the disk (m) 0.12 Width of PPA (m) 0.028 Height of PPA (m) 0.02 Center of PPA (m) 0.0762 Thickness (m) 0.005 Material Aluminum Alloy 6061 Conductivity (S/m) 5.79E+07 Magnet Ferrite core – AC electromagnet Magnet Permeability (relative) 1500 [74] Bobbin inner dimensions (m) 0.028 x 0.02 (Core cros s section) Bobbin outer diameter (m) 0.025 Coil wire gauge AWG 21 Number of turns ~2500 Applied current (A) < 2 Estimated applied flux amplitude (T) 0.5 T (peak) Estimated DC torque ~13 Nm (No induction) Estimated AC torque ~7 Nm (~12 Nm peak L No induction ) AC field Sinusoidal Table 6L2 shows the specifications of the selected configuration as well as estimated applied external field magnitude and the braking to rque generated. In order to be able to keep the applied flux high, the bobbin dimensions a re kept at their max and according to the estimates; the bobbin can accommodate around 25 00 turns on itself. This coil will be enough to apply necessary field to generate flux de nsities that are close to the saturation flux density of the ferrite core, which is around 0 .5 T [74]. Since the applied flux density and the dimensions o f the conductive disk and the PPA are known, the FEM model introduced in Chapter 3 ca n be used to estimate the amount of braking torque generated under the specified circum stances. Figure 6L6 shows the ECB configuration that is implemented into the FEM and according to the results, with DC and AC field applications, braking torque generated rea ches about 13 Nm and 8 Nm (RMS) respectively. For the AC case, the peak value reach es about 12 Nm. Note that AC braking torque is less than DC torque, because for AC field application, the peak magnitude cannot be over 0.5 T, which results in a field that has an RMS flux density of 0.35 T. Therefore the energy input for DC and AC cases are not identical. DC case takes more energy in and results in higher braking torques. Figure 6 L As it was mentioned earlier, the AC field experiments are not carried out in this study, due to time constraints and experiments are one of the future works that is goi ng to be addressed in the fo section. 6.5 Summary In this chapter, an experimental setup that was design ed for AC and DC field experiments was introduced. The setup consists of a servomotor as the source of the rotation and a torque sensor and an encoder to meas ure the speed of t of the ECB prototype, for DC field application, a s imple design with permanent magnets was selected. The experiment results showed the non linear nature of the variation in the braking torque and this nature was related to non Therefore the energy input for DC and AC cases are not identical. DC case takes more energy in and results in higher braking torques. L6. Selected ECB configuration for experimentations it was mentioned earlier, the AC field experiments are not carried out in this study, due to time constraints and the availability of equipment and material supplies. AC field experiments are one of the future works that is goi ng to be addressed in the fo this chapter, an experimental setup that was design ed for AC and DC field experiments was introduced. The setup consists of a servomotor as the source of the rotation and a torque sensor and an encoder to meas ure the speed of t he rotation. In terms of the ECB prototype, for DC field application, a s imple design with permanent magnets was selected. The experiment results showed the non linear nature of the variation in the braking torque and this nature was related to non Lmodeled physics and magnetic 116 Therefore the energy input for DC and AC cases are not identical. DC case takes more . Selected ECB configuration for experimentations it was mentioned earlier, the AC field experiments are not carried out in this study, equipment and material supplies. AC field experiments are one of the future works that is goi ng to be addressed in the fo llowing this chapter, an experimental setup that was design ed for AC and DC field experiments was introduced. The setup consists of a servomotor as the source of the he rotation. In terms of the ECB prototype, for DC field application, a s imple design with permanent magnets was selected. The experiment results showed the non linear nature of the variation in the physics and magnetic 117 phenomena such as stray flux and the assumptions ma de for numerical and analytical models. Unlike the DC field experiments, for the AC field a pplication case, the experiments were not carried out; however the ECB configuration and the electromagnet were selected and prototyped for experimentation. The main constr aint in the design is the material properties of the ferrite core. Due to its low satu ration flux density, the electromagnet won’t be able to generate more than 0.5 T (peak) fi eld. Using the specifications ECB, the braking torque generation was estimated by the FEM introduced in Chapter 3. According to the results, the brake will generate torques aro und 8 Nm RMS, when an AC field is applied. The experiments comprise the future work o n the ECB research. 118 7. CONCLUSIONS AND FUTURE WORK 7.1 Conclusions This thesis focuses on the improvements of automoti ve brakes with the use of electromechanical ECBs. ECBs have numerous advantag eous over conventional brakes and these potential advantages make ECB an attracti ve substitute to current brakes. However, the braking torque generation of the ECB s trictly dependent on the rotational speeds of the conductive disk and at low speeds, br aking torque decreases significantly. Thus in order to be able to realize a standLalone E CB actuator, this problem was addressed in this thesis. An application of alternating fields to increase th e amount of eddy currents generated is proposed and the effects of AC field application we re extensively studied. In Chapter 2, an analytical model was developed using modified Ma xwell’s equations in potential form and the model was validated using a preliminary FEM . In Chapter 3, an accurate FEM that can account for phenomena like induction and s kin effects was developed and the model was then validated using analytical models fo r AC and DC field applications individually. Then using this model, the applicatio n of various time varying fields in different waveforms with or without frequency modul ation was investigated. Moreover, in Chapter 4, ECB was optimized using the model cre ated previously. ECB problem was defined along with geometric constraints as well as driver/passenger comfort constraints. Here, the methodology of quantifying the subjective comfort was described in detail. The results show that a configuration with four PPA usi ng triangular field waveforms can result in braking torques higher than the torque ac hieved by conventional hydraulic systems. In Chapter 5, a model predictive adaptive controller was designed and for the control purposes, a dynamical model was created usi ng the analytical model and FEM mentioned in previous chapters. The results show im provements in braking for various scenarios. Finally in Chapter 6, the experimental s tudies that were carried out to validate the ECB braking performance are presented. In this chapter, experimental results for DC field application on a custom configuration were sh own. The main focus of this chapter is 119 on the design of a test bed for testing the braking performance of ECB with AC field application for validation purposes. 7.1.1 Summary of Chapter 2 The effects of time varying field application on th e braking torque generation of ECB are studied. In order to be able to calculate the b raking torque, an analytical model is introduced and presented and then this model was va lidated using an accurate FEM. It is concluded here that although there is an error calc ulated about 10% between the results of the analytical model and the FEM, they are in good agreement. In addition, the reasoning behind this error along with the nature of the vari ations of the braking torque with frequency and angular velocity was discussed in det ail. Finally, from the preliminary results, it is concluded that with the application of the time varying field, the braking torque generation of the ECB increases. 7.1.2 Summary of Chapter 3 In this chapter, Eddy Current Brakes (ECBs) are pro posed as possible substitutes for the conventional hydraulic brakes (CHBs) in automotive applications due to its unique potential for performance advantages (e.g., nonLcon tact and fast response). As a potential solution for the limited braking torque of typical ECBs at low speeds, the application of time varying fields (i.e. AC) is proposed. A finite element model (FEM) that accounts for the effects of time varying fields on the performan ce of the ECB was developed, which was then validated using an existing analytical mod el for the DC field. It was shown that improved braking performance can be obtained when A C fields are used at both low and high velocities. Time varying fields in different w aveforms (i.e. sinusoidal, square, sawtooth and triangular waves) were applied and tri angular wave field application resulted in the highest braking torque. The numerical result s also showed that the braking torque decreases with increasing braking torque due to ind uction effects when the time varying fields are applied. In order to increase the brakin g torque generation, the induction effects were eliminated and braking torque generation was m aximized for sinusoidal and triangular field applications on a generic ECB conf iguration for automotive applications. A further improvement was subsequently proposed by ap plication of frequency modulated (FM) AC fields. The results showed that a 60% incre ase in the braking torque can be 120 achieved with the FM compared to the DC field braki ng. Further improvements in braking torque generation of the ECB can be achieved by mea ns of increasing the magnitude of applied field, the number of PPAs and the rotationa l speed of the conductor disk using gears. 7.1.3 Summary of Chapter 4 The use of AC fields in order to improve the brakin g torque generation of ECBs is investigated with the help a finite element model t hat is validated with existing analytical models. Preliminary studies showed that the braking torque is higher when the field has a varying frequency instead of fixed frequency and wh en it is applied in triangular waveform compared to sinusoidal, square and sawtoot h waveforms. After the preliminary study, ECB was optimized for higher braking torque. In addition to the geometric and field dependent factors, skin effects due to altern ating currents generated on the surface of the conductor and the comfort of the passengers due to the vibration during braking are accounted for in the optimization problem. Genetic algorithm was then used to find the optimum configuration. The results showed that the braking torque significantly increases with the application of AC fields, i.e. the increas e is 1600 % compared to the DC braking torque at low speeds and 200% at high speeds. Also the deceleration curves showed that the AC field application on the optimum configurati on resulted in about 12 m/s 2, which is more than 20L25% more than the accelerations that c an be achieved with average passenger vehicles. 7.1.4 Summary of Chapter 5 In A controller was designed for an optimum configu ration presented in Chapter 4. Initially, a dynamic model of the system is obtaine d for the ECB and the vehicle to determine the state variables (speeds and accelerat ion terms of the wheel and the vehicle.). Since the analytical and FE model are pr eviously studied in this thesis, the application is suitable for model predictive contro l. Here a NMPC is designed for the system by defining an objective function along with constrains that will drive the plant to a desired steady state. The simulations are carried out for a scenario where the vehicle is slowed down from 120 km/h to 10 km/h in 5 seconds. The results show that the system 121 reaches the steady state within 0.1 s and NMPC make s sure the system is behaving as requested for the rest of the time during braking. 7.1.5 Summary of Chapter 6 An experimental setup was designed for AC and DC fi eld experiments. The setup consists of a servomotor as the source of the rotat ion and a torque sensor and an encoder to measure the speed of the rotation. In terms of t he ECB prototype, for DC field application, a simple design with permanent magnets was selected. The experiment results showed the nonlinear nature of the variatio n in the braking torque and this nature was related to nonLmodeled physics and magnetic phe nomena such as stray flux and the assumptions made for numerical and analytical model s. Unlike the DC field experiments, for the AC field a pplication case, the experiments were not carried out; however the ECB configuration and the electromagnet were selected and prototyped for experimentation. The main constr aint in the design is the material properties of the ferrite core. Due to its low satu ration flux density, the electromagnet will not be able to generate more than 0.5 T (peak) fiel d. Using the specifications of the ECB, the braking torque generation was estimated using t he FEM introduced in Chapter 3. According to the results, the brake generates torqu es around 8 Nm RMS, when an AC field is applied. These experiments are proposed to be complete as future work. 7.2 Future works In order to finalize the ECB research, the work pre sented here in this thesis needs to be carried out further in order to further validate an d evaluate the models presented and the results obtained. The future work can be divided in to four themes. In Chapter 6, an ECB prototype is designed and the test bed developed for AC field and DC field experiments is presented. As the next step , a simple ECB configuration with time varying field application needs to be adapted and the brake actuator will be designed and prototyped. The design will be strictly constra ined by the capacity and properties of the existing test bed (see Chapter 6 for details). The AC field performance of the ECB will be tested using this setup. In addition, the s ame configuration will then be tested with DC field application for comparison purposes. Later , the empirical results will be used to validate the numerical and analytical models presen ted in the first two chapters of this 122 thesis. In addition, the controller designed in Cha pter 5 needs to be implemented to improve the braking performance of the ECB. The experimental work will complete the ECB researc h; however there are three additional steps that will be taken in order to fur ther investigate the feasibility and the performance of the ECB studied. One of these steps will be to design a prototype that can be mounted on a vehicle and then to carry out road performance of the brake and the controller on the dynamometer. For the tests, it is proposed to use the Quarter Car Shaker (See Figure 7L1) available at the Simon Fraser Univ ersity in Surrey, BC. This equipment is used to test suspension modules but it will be m odified so that an ECB can be mounted on along with the necessary transducers such as enc oders, torque sensor, load cells and thermocouples. Also along with the design and proto typing, the power optimization for the ECB brake will be included as well in the propo sed study. Figure 7L1. Quarter vehicle shaker for testing real world performance of ECB (equipment is available at SFU, BC) 123 The second step will be to have an inLdepth underst anding of the ECB in terms of the performance of the brake when there are various typ es of signals applied with varying magnitude and frequency. Note that the Helmholtz eq uation that was solved is a linear equation and it can be solved using Fourier series. With the use of Fourier series, the results can be obtained as a series sum. This will help us study the performance of the brake under time varying fields in any waveform, no t just the ones that have simple mathematical expressions. The third step will be the last item to be investig ated. ECB is a magnetic brake and the kinetic energy is dissipated as heat by means of ed dy currents on the conductive disk. In theory, such a brake has the potential to be used a s a regenerative brake. These eddy currents generated can be redirected and complete a circuit that charges the battery of the vehicle. Thus, more work needs to be carried out to investigate the regenerative braking capability and methods of harvesting the currents o n a rotating disk. This third step itself will also result in further possibilities of resear ching into design, implementation, control and testing of the possible regenerative system. Note that in this study, power input to the system is not studied since the main objective is to realize a standLalone ECB that generates comp arable braking torques as CHB. The power can be calculated using the current and volta ge applied to the coil of the electromagnet. For the ECB analysis, magnetic flux densities are used as inputs and the electromagnet design that is necessary to generate such flux densities is not investigated. Since the resistance of the coils are significantly low, the voltage and current requirement for the ECB will be low as well (i.e. 42 ohms per k m for AWG 21). The requirements depend on the electromagnet design, and coil and co re properties, but the estimated requirement for the fourLPPA ECB will be between 25 0 W and 400 W and for 4 brake actuators, it will be between 1 kW and 1.2 kW. In order to calculate the required power input to t he system, an electromagnet that applies fluxes up to 1.5 T is needed. Note that as a part of the design of real vehicle ECB model, the electromagnets will be designed and opti mized for the requirements. After the detailed design, the power input to the system can be calculated for the ECB configuration. 124 7.3 Recommendations • Use of Fourier series will result in inLdepth analy tical understanding of the ECB braking torque generation using various types of pe riodic AC fields, • Analytical model does not account for any temperatu re change, thus in order to account for the temperature effect, the simulations can be carried using property values at different temperature values, • The objective is to design a standLalone ECB in thi s study, for complete analysis, not only the braking torque, but the power input, t emperature change and weight of the brake need to be included in the analysis, • In addition to designing for higher braking torque, the economics, safety, reliability, and lifetime need to be considered for the ECB design, • The focus application is selected to be automotive, however the system can be designed for various applications, • A more complete adaptive nonlinear predictive contr oller can be designed for real braking scenarios for a road vehicle, • Experimental setup can be altered for a multi PPA c onfiguration for validating the optimization results and for increased flux density application; a laminated core needs to be selected for the experimental studies. 125 Bibliography [1] Forensic Engineering Technical Services, Brake s, Portland, OR, 2007. 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Cole, Elementary vehicle dynamics, Course Notes in the Mechanical Engineering, The University of Michigan, Ann Arbor, MI, 1972. [76] C. Lee, K. Hedrick, K. Yi, RealLTime slipLbas ed estimation of maximum tireLroad friction coefficient, Mechatronics, IEEE/ASME Trans actions on,9(2), June 2004. [77] E.J. Park, D. Stoikov, L. Falcao da Luz, A. S uleman, A performance evaluationof an automotive magnetorheological brake design with a sliding modecontroller, Mechatronics, 16(2006) 405L416. [78] A. B. Will, S. Hui, S. Zak, Sliding mode whee l slip controller for an antilock braking system, Int. J. of Vehicle Design, 19(4) (1 998) 523L539. 132 Appendix Appendix A. Vehicle dynamics Here, the vehicle dynamics are studied in order to calculate the required amount of braking torque in order to stop a vehicle. Then, in the following section (Appendix B) required braking torque for several vehicles were c alculated using the dynamic vehicle model. The motion of a vehicle is described using the quar ter vehicle model [ 68 ]. This model allows us to model the vehicle in order to calculat e the required braking torque that a brake should provide. The basic assumption of this model is that the mass of the vehicle is divided equally between four wheels. This is not th e exact case. Front wheels carry much weight than the rear wheels. Usually 0.55:0.45 of t he total mass is carried by the front wheels and rear wheels respectively. As a result of the assumption, the dynamic model of a wheel is enough to determine the braking torque required to stop the rotation of the wheels. In Figure AL1, free body diagram of a wheel is shown: During braking, a torque is applied by the brake, vehicle, there would be an additional torque occurs on the wheel, moment of inertia and /g2016/g4663 is the angular acceleration. The radius of the whee l is distance traveled by the vehicle and friction force, normal force and the transfer of we ight caused by braking of the vehicle. According to the quarter vehicle model, the mass th at the calculated as: where mv is the total mass of the vehicle and the mass on the wheel, the effective mass moment of inertia must be defined as: Figure AL1. Free body diagram of a wheel During braking, a torque is applied by the brake, Tb, and due to the deceleration of the vehicle, there would be an additional torque occurs on the wheel, /g1835/g2016/g4663 /g4663is the angular acceleration. The radius of the whee l is distance traveled by the vehicle and Fr, F f, F n, and FL are the rolling resistance force, the friction force, normal force and the transfer of we ight caused by braking of the vehicle. According to the quarter vehicle model, the mass th at the wheel carries can be is the total mass of the vehicle and mw is the mass of the wheel. After defining the mass on the wheel, the effective mass moment of inertia must be defined as: mt=1 4mv+mw133 , and due to the deceleration of the /g2016/g4663, where I is the is the angular acceleration. The radius of the whee l is Rw, x is the are the rolling resistance force, the friction force, normal force and the transfer of we ight caused by braking of the vehicle. wheel carries can be (A.1) is the mass of the wheel. After defining the mass on the wheel, the effective mass moment of inertia must be defined as: 134 (A.2) where I is the total mass moment of inertia, Iw is the wheel's inertia, Ie is the engine inertial, τ is the gear ratio and Iy is the inertia of the brake disks. The inertia of the engine and the gear ratios must be considered together wit h a ratio of 0.5 in the effective mass moment of inertia. This 0.5 factor is added into th e equation because of the distribution of the inertia of the engine to each of the driving wh eels. The rolling resistance force against the motion of the wheel is defined in [68] as: (A.3) where Kv is a conversion factor that is used for converting the speed value from m/s to mph, fs and f0 are basic coefficient and the speed effect coeffic ient. This equation was developed by The Institute of Tech. in Stuttgart fo r rolling on a concrete surface [75]. The friction force acting on the wheel is defined i n terms of the normal force and the friction coefficient between the tire and the surfa ce. (A.4) (A.5) where lbase is the wheel base and hcg is the height of the center of gravity and the fri ction coefficient, µf, is a function of the slip ratio, sr, which is the relative proportion of the rolling to slipping, i.e. /g1871/g3045=/g3051/g4662/g2879/g3019/g3298/g3087/g4662 /g3051/g4662 (A.6) where /g2016/g4662 is the angular velocity of the wheel. I=Iw+1 2τ2Ie+Iy Fr=f0+3.24 fs(Kv&x)2.5 Ff=µfFn Fn=mtg−mvhcg lbase&&x=mtg−FL The correlation between the slip ratio and the coef ficient of friction is illustrated in Figure AL2. The relation is also shown for several types of sur faces. Figure AL2 . Friction coefficient versus slip ratio for severa l surfaces [ After defining all the forces that are shown in the free body diagram, the Newtonian equations of motion for the wheel can be written as : The first equation is the force equilibrium in the direction of motion and the second equation is the moment equilibrium around the axis that is perpendicular to the wheel surface. These two equations are the governing equa tions of this type of motion. And required braking torque, which stops the car, can b e found by using these two equations. I&&θThe correlation between the slip ratio and the coef ficient of friction is illustrated in The relation is also shown for several types of sur faces. . Friction coefficient versus slip ratio for severa l surfaces [ After defining all the forces that are shown in the free body diagram, the Newtonian equations of motion for the wheel can be written as : The first equation is the force equilibrium in the direction of motion and the second equation is the moment equilibrium around the axis that is perpendicular to the wheel surface. These two equations are the governing equa tions of this type of motion. And required braking torque, which stops the car, can b e found by using these two equations. mt&&x= −Ff= −µfmtg+µfmvhcg lbase&&x = −Tb+RwFf−RwFr= −Tb+µfRwFn−RwFr135 The correlation between the slip ratio and the coef ficient of friction is illustrated in . Friction coefficient versus slip ratio for severa l surfaces [ 76 ] After defining all the forces that are shown in the free body diagram, the Newtonian (A.7) (A.8) The first equation is the force equilibrium in the direction of motion and the second equation is the moment equilibrium around the axis that is perpendicular to the wheel surface. These two equations are the governing equa tions of this type of motion. And the required braking torque, which stops the car, can b e found by using these two equations. 136 Appendix B. Braking torque requirements Here in this section, required braking torque value s for various vehicles were calculated using the dynamic model introduced previ ously in Appendix A. However, in order to calculate the braking torque, there are se veral properties that have to be known. As in [77], these parameters for a loaded car were taken from the previous works of Will AB, Hui S and Zak Sh [78] and are listed in Table A L1; Table AL1. Parameters for the quarter vehicle model The required braking torque values for a sport moto rbike and a scooter were also calculated, however since the lack of detailed prop erties (i.e. f0, f s, and Kv), the braking torque requirement for these vehicles were calculat ed assuming that the whole kinetic energy of the vehicle is dissipated in the brake ac tuators. With such an assumption, the effects of rolling friction and the friction betwee n the tire and the road were omitted. In Table AL2, the assumptions made for the calculation s and the data used in the calculations are shown. 137 Table AL2. Required braking torque values for sever al vehicles This study was made in order relate the requirement s with the capacity of an EMB. Table AL2 gives an idea about how much braking torq ue must be generated by an ECB. Therefore in terms of the feasibility of such a bra ke system, the main goal must be to generate braking torques as much as the requirement s. 138 Appendix C. Ferrite material properties and core sp ecs for AC field experiments [74] 139