Modeling And Simulation Of All-electric Aircraft Power Generation And Actuation

Transcription

1 Unversty of Central Flora Electronc Theses an Dssertatons Doctoral Dssertaton (Open Access) Moelng An Smulaton Of All-electrc Arcraft Power Generaton An Actuaton 2013 Dav Wooburn Unversty of Central Flora Fn smlar works at: Unversty of Central Flora brares Part of the Electrcal an Electroncs Commons STARS Ctaton Wooburn, Dav, "Moelng An Smulaton Of All-electrc Arcraft Power Generaton An Actuaton" (2013). Electronc Theses an Dssertatons Ths Doctoral Dssertaton (Open Access) s brought to you for free an open access by STARS. It has been accepte for ncluson n Electronc Theses an Dssertatons by an authorze amnstrator of STARS. For more nformaton, please contact lee.otson@ucf.eu.

2 MODEING AND SIMUATION OF A-EECTRIC AIRCRAFT POWER GENERATION AND ACTUATION by DAVID ANDREW WOODBURN B.S. John Brown Unversty, 2004 M.S. Unversty of Central Flora, 2010 A ssertaton submtte n partal fulfllment of the requrements for the egree of Doctor of Phlosophy n the Department of Electrcal Engneerng an Computer Scence n the College of Engneerng an Computer Scence at the Unversty of Central Flora Orlano, Flora Fall Term 2013 Major Professor: Thomas X. Wu

3 2013 Dav Wooburn

4 ABSTRACT Moern arcraft, mltary an commercal, rely extensvely on hyraulc systems. However, there s great nterest n the avoncs communty to replace hyraulc systems wth electrc systems. There are physcal challenges to replacng hyraulc actuators wth electromechancal actuators (EMAs), especally for flght control surface actuaton. These nclue ynamc heat generaton an power management. Smulaton s seen as a powerful tool n makng the transton to all-electrc arcraft by prectng the ynamc heat generate an the power flow n the EMA. Chapter 2 of ths ssertaton escrbes the nonlnear, lumpe-element, ntegrate moelng of a permanent magnet (PM) motor use n an EMA. Ths moel s capable of representng transent ynamcs of an EMA, mechancally, electrcally, an thermally. Inuctance s a prmary parameter that lnks the electrcal an mechancal omans an, therefore, s of crtcal mportance to the moelng of the whole EMA. In the ynamc moe of operaton of an EMA, the nuctances are qute nonlnear. Chapter 3 etals the careful analyss of the nuctances from fnte element software an the mathematcal moelng of these nuctances for use n the overall EMA moel. Chapter 4 covers the esgn an verfcaton of a nonlnear, transent smulaton moel of a two-step synchronous generator wth three-phase rectfers. Smulaton results are shown.

5 To Go from whom all goo gfts come. v

6 ACKNOWEDGMENTS Ths ssertaton s the culmnaton of many years of research an har work. But, t was not one n a vacuum. I have benefte tremenously from the help an guance of many others. I am eeply grateful for the patent an carng guance of my avser an commttee char, Professor Thomas Xnzhang Wu. Hs ntellgence, nsght, an wsom have helpe me evelop, mature, an excel. Hs atttue towar research has bult n me an enjoyment for the work. I woul not have reache ths pont wthout hs help. I woul lke to thank my other commttee members for ther help an support: Dr. M. Georgopoulos, Dr. M. Haralambous, Dr. I. Batarseh, an Dr.. Chow. I am especally grateful for the sgnfcant help, support, an guance of Dr. Chow. I want to thank my lab mates through the years for ther nput an nsghts: Tony Camarano, Yang Hu, Shaohua n, Hanzhou u, Shan Wan, Keju Zhang, Jare Bnl, Wenell Brokaw, an Chh-Hang Wu. I also want to thank e Zhou, Yeong-Ren n, Ncholas Rolnsk, an Justn DelMar for ther help wth expermentaton. I am eeply grateful for the contnual support of my parents. They have been there for me from begnnng to en. Fnally, I want to thank my wfe for her supportve heart an unerstanng atttue as I worke to fnsh my egree. Ths materal s base on research sponsore by Ar Force Research aboratory uner agreement number FA The U.S. Government s authorze to reprouce an strbute reprnts for Governmental purposes notwthstanng any copyrght notaton thereon. v

7 TABE OF CONTENTS IST OF FIGURES... x IST OF TABES... xv IST OF ACRONYMS... xv CHAPTER ONE: INTRODUCTION...1 Objectve...1 The All-Electrc Arcraft (AEA) Concept...2 EMA Development Challenges...4 Heat Generaton...5 Smulaton...9 CHAPTER TWO: DYNAMIC HEAT GENERATION MODEING OF HIGH- PERFORMANCE EECTROMECHANICA ACTUATOR...12 Introucton...12 Moel ayout...14 Parameter Measurement an FEM Calculaton...18 Electromechancal Dynamcs...21 Thermal Component...24 Fel-orente Control...26 PWM Component...27 Inputs...29 An EMA Example an Results...31 Concluson...55 v

8 CHAPTER THREE: NOVE NONINEAR INDUCTANCE MODEING OF PERMANENT MAGNET MOTOR...56 Introucton...56 Backgroun...57 Metho of Analyss...59 Comparson wth Other Methos...95 Smplfe Moel...99 Expermental Applcaton an Comparson wth near Moelng Concluson CHAPTER FOUR: NONINEAR, TRANSIENT MODE OF SYNCHRONOUS GENERATOR Introucton DQ0 Transform Dynamcal Equatons Inuctance Moelng Full Flux nkage Dervatve Full-brge Rectfer wth Freewheelng Power Analyss Smulaton Results Concluson CHAPTER FIVE: CONCUSION APPENDIX A: PERMANENT MAGNET MACHINE DYNAMICS Voltage Equatons v

9 Dynamcal Equatons APPENDIX B: SYNCHRONOUS MACHINE DYNAMICS Introucton Dynamcal Equatons wthout Inuctances Dervatves Dynamcal Equatons wth Inuctance Dervatves Power Analyss REFERENCES v

10 IST OF FIGURES Fgure 1. Guts of an electromechancal actuator [4] showng the motor (yellow) wth ts wnngs, the gears (gray) of the gear box, the screw (blue) of the ball screw, the ball housng (re) of the ball screw, an the ro (green) nto whch the screw sles....6 Fgure 2. Block agram of an EMA system...7 Fgure 3. FEM agram of a quarter of a PM motor showng the temperature strbuton urng a smulaton....8 Fgure 4. Cross-sectonal vew of a PM (left) an magnetc flux ensty strbuton (rght)...10 Fgure 5. Moel layout showng the connecton of the four major components: felorente control, pulse-wth moulaton, electromechancal ynamcs, an the thermal component Fgure 6. Drect-axs nuctance along the axs (blue) an the nuctance value at zero current (green crcle). The nuctance s not mrrore about the quarature axs (Ths axs goes nto the page), so ts values for postve o not match those for negatve Fgure 7. Thermal lumpe element moel. The motor s efne to have a front an a back. The space between the rotor an stator s the ar gap. The rotor s compose of magnets on a shaft. The boy of the shaft s the central regon about whch the magnets are fxe. The bearngs hol the shaft n place at both of ts ens. The front an back of the motor have an alumnum cover an x

11 the ses of the motor are surroune by an alumnum case. The copper wnngs loop aroun the stator teeth concentrcally an are completely embee n epoxy. The epoxy n ron s the epoxy surrounng the wnngs an locate n the man boy of the ron stator. The en turns are the ponts on ether en where the wnngs loop aroun the stator teeth passng from one stator slot to another. An, the envronment s the surrounng atmosphere Fgure 8. The general concept of the PWM. A tralng ege carrer wave (a) s compare to a normalze snusoal phase voltage. When the normalze wave s greater than the carrer wave, the output PWM voltage (b) s turne on. Otherwse, t s off...29 Fgure 9. Actual force profle (blue) an calculate force profle (green) Fgure 10. Statc frcton analyss showng (a) stroke an (b) loa force. The green lnes on the force plot represent the mean forces to pull (-258 N) or push (425 N) the EMA ro, overcomng the frcton of the rve tran (342 N). The bas force s the weght of the EMA ro (84 N) Fgure 11. Drect-axs nuctance as a functon of rect current,, an quarature current, q. These nuctance values are average over rotor angle Fgure 12. Quarature-axs nuctance as a functon of rect current,, an quarature current, q. These nuctance values are average over rotor angle Fgure 13. Motor geometry (left) an slot measurements (rght)...37 Fgure 14. Desre (blue) an actual (green) stroke profles...39 Fgure 15. Input loa force profle (blue) an motor-generate force (green). The loa force was a sustane 0.45 kn...40 x

12 Fgure 16. DQ0 voltages wth pulse form ue to PWM Fgure 17. DQ0 currents wth pulse form ue to PWM Fgure 18. Motor powers showng (a) the total electrcal power at the termnals of the motor, (b) the power loss n the wnngs, (c) the power store n the magnetc fels, () an the mechancal power on the rotor Fgure 19. Drve tran powers showng (a) the mechancal power on the rotor, (b) the loa power, (c) the frctonal power, () the gravtatonal power, (e) an the net knetc power Fgure 20. (a) Temperature comparson an (b) agan zoome n. These plots have envronmental temperature (blue), expermentally measure temperature of stator (green), smulate temperature of stator (re), an the smulate temperature of the wnngs (cyan) Fgure 21. Temperature comparson for a half-hour run showng envronmental temperature (blue), expermentally measure temperature of stator (green), smulate temperature of stator (re), an the smulate temperature of the wnngs (cyan). Note that the smulate temperatures of the wnngs an stator are about the same, ncatng the low thermal resstance n the motor Fgure 22. Drect nuctance as a functon of an q Fgure 23. Quarature nuctance q as a functon of an q Fgure 24. Drect nuctance as a functon of rotor angle for a partcular torque angle an for varous values of current magntue Fgure 25. Quarature nuctance q as a functon of rotor angle for a partcular torque angle an for varous values of current magntue x

13 Fgure 26. Drect nuctance after beng nterpolate Fgure 27. Quarature nuctance q after beng nterpolate Fgure 28. Inuctance ervatves: (a) /, (b) / q, (c) q /, an () q / q...69 Fgure 29. as a functon of rect current where quarature current s zero. The peak s crcle n re at A Fgure 30. Inuctance ervatve / (a) before beng shfte an (b) after beng shfte. Note that between the two versons, the re an blue regons are shfte to the rght n (b). Also, note that the oman of the surface s croppe slghtly to mantan symmetry Fgure 31. Fts wth 5-k values for (a), (b) zoome n, an (c) q. The orgnal ata s shown wth sol curves, an the ftte ata s shown wth ashe curves, where each color correspons to a unque torque angle. Note the slght sparty near the peak of as seen n (b) Fgure 32. Fts wth 7-k values for (a), (b) zoome n, an (c) q. The orgnal ata s shown wth sol curves, an the ftte ata s shown wth ashe curves, where each color correspons to a unque torque angle. Note that the slght sparty seen n Fgure 31 near the peak of s far less here Fgure 33. Orgnal k 1 values over torque angle (blue) for 5-k ratonal functon ft to wth 12 th -orer polynomal ft (re) Fgure 34. Orgnal k 1 values over torque angle (blue) for 5-k ratonal functon ft to q wth 12 th -orer polynomal ft (re) x

14 Fgure 35. The (a) net an (b) max NMAD values for whole sets of k value fts (e.g. k 1 through k 5 for 5-k ratonal functon). The 5-k results are shown n blue an the 7-k results are shown n re Fgure 36. The tme requre for the three methos to calculate nuctances an ther current ervatves. The formula metho s faster Fgure 37. Comparson of NMADs for rounng lookup (blue) an formula ft (re). The percentages were obtane by smply multplyng the NMAD values by 100%. Note the lower error for the formula ft to q...87 Fgure 38. Inuctance q by (a) rounng lookup an by (b) formula ft Fgure 39. Inuctance ervatve q / q by (a) rounng lookup an by (b) formula ft Fgure 40. Absolute fference relatve to nterpolaton lookup of (a) rounng lookup an (b) formula fttng for q Fgure 41. The mean over the sx propertes of the sx normalze mean absolute fferences for the 5-k ft (blue) an the 7-k ft (re)...94 Fgure 42. Smulaton tme for 5-k (blue) an 7-k (re) tests. These results are for sngle test runs at each settng. Averagng over many test runs, these curves woul straghten Fgure 43. For current magntues up to 55 A, the reference λ flux lnkage Fgure 44. For current magntues up to 55 A, the rect flux lnkage λ (top) usng (43) an (bottom) usng (38) Fgure 45. The (a) D an (b) Q nuctances usng the smplfe ratonal functon (44) x

15 Fgure 46. Absolute fferences between reference an formula nuctances for (a) an (b) q Fgure 47. Desre stroke profle (blue) an smulate stroke profle (green). Note that the esre stroke profle was followe farly well by the control algorthm. The esre stroke profle was taken from a physcal experment wth an EMA uner actve loang Fgure 48. Smulate power losses n copper wnngs usng lnear nuctances (blue) an nonlnear nuctances (green) Fgure 49. Smulate energy losses n copper wnngs usng lnear nuctances (blue) an nonlnear nuctances (green) Fgure 50. Temperature comparson for a half-hour run showng envronmental temperature (blue), expermentally measure temperature of stator (green), smulate temperature of stator (re), an the smulate temperature of the wnngs (cyan). The noal temperatures n the EMA s electrc motor are smulate usng the nonlnear nuctances. These temperatures, whch are functons of the energy losses, match very well Fgure 51. Overall layout of generator. The excter s a 6-pole SM wthout ampers an the man s a 10-pole SM wth ampers Fgure 52. D-axs flux lnkage values over rotor angle for the PM motor of [38]. The flux lnkages are normalze per unt length. The ponts where the flux lnkage crosses through ts average are marke wth crcles xv

16 Fgure 53. Se vew of the rotor s fel flux lnkage n mwb wth three currents vare:, q, an f. Ths shape was characterstc of the rect an fel flux lnkages on both the excter an the man Fgure 54. The fel flux lnkage for the man generator as calculate by formula. The ponts chosen nclue ponts n between those use to tune the formula Fgure 55. The fel flux lnkage for the man generator as calculate by formula wth all three currents extene 20% beyon the orgnal tables Fgure 56. The D-axs magnetzng flux lnkage ( m m ) of the man as a ratonal functon of D an Q magnetzng currents Fgure 57. The D-axs magnetzng nuctance, m, of the man as a ratonal functon of D an Q magnetzng currents Fgure 58. The D-axs magnetzng flux lnkage ( m m ) of the man as a ratonal functon of D an Q magnetzng currents Fgure 59. Generalze nuctances evelope usng an exponental functon (blue) an a logstc functon (green) Fgure 60. The D-axs magnetzng flux lnkage ( m m ) of the man as a logstc functon of D an Q magnetzng currents Fgure 61. Comparson of the NMADs of for the polynomal, ratonal, an logstc methos for the man machne Fgure 62. Percent fference of M relatve to Fgure 63. D-axs magnetzng nuctance (blue) an M (green) Fgure 64. One leg of a full-wave rectfer showng voltage rops across both top (1) an bottom (2) oes xv

17 Fgure 65. Powers nto the excter Fgure 66. Powers nto the man Fgure 67. Voltages on the rotatng rectfer. The DC voltage s slghtly less than the peaks of the ABC voltages because of the forwar voltage rop an on resstance of the oes Fgure 68. oa voltage controlle to 270 V Fgure 69. Normalze mean smulaton tmes for four machne types xv

18 IST OF TABES Table 1. Smulaton States...17 Table 2. EMA Parameters...18 Table 3. EMA Parameters...32 Table 4. EMA Parameters...37 Table 5. EMA Thermal Parameters...38 Table 6. Smulaton States...58 Table 7. Smulaton States an Parameters xv

19 IST OF ACRONYMS DQ0 EMA FEM FOC PM PWM SM Drect-quarature-zero Electromechancal actuator Fnte element metho Fel orente control Permanent magnet Pulse wth moulaton Synchronous machne xv

20 CHAPTER ONE: INTRODUCTION Objectve The heat loas n an electromechancal actuator (EMA) an the electrcal power emans of an EMA system are both hghly transent; therefore, thermal an electrcal moelng esgne for tme-average, steay-state behavor are naequate. The heat loas an power emans of EMAs must be accurately known before EMAs can be truste as replacements to current hyraulc actuators n controllng flght surfaces on arcraft. Therefore, a comprehensve, nonlnear, ynamc smulaton moel nclung electromagnetcs wth non-lneartes, rotor ynamcs, power electroncs, control, an heat transfer thermoynamcs are esgne for an EMA. The secon chapter etals the comprehensve analyss of the EMA smulaton moel an nclues expermental valaton. The thr chapter focuses on how the nonlnear nuctances of the EMA s electrc motor are obtane from fnte element metho (FEM) an how a mathematcal moel s evelope to ft the tables of nuctance ata. These nuctance formulas can be use n the full EMA smulaton moel of the secon chapter. The thr chapter nclues a comparson of runnng the full smulaton usng the nonlnear nuctance formulas versus runnng the smulaton usng scalar values. Usng [1] extensvely, the rest of ths chapter etals the motvaton for ths research an an overvew of the work one. 1

21 The All-Electrc Arcraft (AEA) Concept System-level benefts Current arplanes use a combnaton of pneumatc, hyraulc, an electrc systems. It s expecte that future arplanes wll just have an electrc system. Such all-electrc arcraft (AEA) woul use electrc power exclusvely for onboar systems [2]. Some of the expecte benefts are the followng [1], [3]-[5]: Improve arcraft mantanablty o No hyraulc components o Faster arcraft turnaroun o Fewer necessary tools an spares o Better fault-agnoss through bult-n testng (BIT) Improve system avalablty an relablty o More easly reconfgurable Electrcal strbuton s more practcal an offers system reconfguraton flexblty (a capablty prevously ffcult to acheve usng hyraulcs). o Improve mean tme between falures (MTBF) Improve flght safety avong common-moe falures by usng ssmlar power supples Reuce system weght Electrc actuators woul replace not just the hyraulc actuators but ther entre hyraulc system, nclung pumps, strbuton networks (ppes an flu), an valve blocks. Reucton of arcraft operatng costs reuce fuel consumpton from reuce weght; reuce mantenance costs 2

22 Unt-level benefts As seen above, the prmary avantages of EMAs over current actuators are n the comparson of the entre systems, rather than the unts themselves. However, the EMA unt tself oes have a number of avantages: EMAs are more energy effcent than hyraulc systems n that EMAs only requre energy when force s requre [6]. Hyraulc systems must keep runnng just to mantan lne pressure. Hyraulc transmssons also ncur energy losses from pumpng [7]. EMAs lack hyraulc flu, whch can leak, s corrosve, an s a fre hazar. Wthout hyraulc flu, EMAs can be smpler to nstall an mantan. EMAs ten to be stffer than hyraulcs. Ths happens because hyraulc transmssons have an elastc qualty that EMAs lack. Precte progress In spte of the benefts of an all-electrc arcraft, there s no arcraft that fulflls ths goal yet. However, when an AEA s realze, t wll lkely happen n the followng orer [5]: Stage 1: Electro-hyrostatc actuators (EHAs) serve as auxlary to stanar hyraulc actuators. Stage 2: EHAs operate as prmary control actuators. Stage 3: Electromechancal actuator (EMA) technology mproves n the areas of thermal management, power ensty, an jam tolerance. Arcraft power generaton systems avance to support the very hgh electrcal emans. Stage 4: EMAs serve as auxlary to EHAs. Stage 5: EMAs operate as both prmary an auxlary control actuators. 3

23 Gven that the commercal arcraft Arbus A380 uses electro-hyrostatc backup actuators (EHBA) n ts flght control system, t s safe to say that the frst stage has been accomplshe. The assumpton s that wth experence of EHAs n flght control systems, EHAs wll naturally progress to beng use as prmary control actuators. Clearly, overcomng current efcences of EMA technology wll requre sgnfcant research urng the thr stage. Success at ths stage woul make t possble for EMAs to be ntrouce n commercal arcraft by the fourth stage, an subsequently to become the exclusve control actuators by the ffth stage. So, at ths pont, a prmary focus must be overcomng the efcences of EMA technology. EMA Development Challenges EMAs have been shown to match the performance of hyraulc actuators by flght tests from several research programs [8]. However, whether the current EMAs are sutable as the prmary flght control s stll ebate [4]. The evelopment of EMAs has three remanng major challenges: ncreasng power ensty, ncreasng jam tolerance, an resolvng thermal management. These ssues must each be solve before EMAs can replace hyraulcs as prmary flght control actuators. Frst, as experence wth actuators grows an esgns of power electroncs an electrc motors are enhance, the power ensty (power per unt mass) of EMAs wll certanly ncrease. Though an EMA system alreay has the potental of beng lghter than a hyraulc system, ths power ensty ncrease wll make the EMA actuators themselves more compettve wth hyraulc actuators n weght. Typcally, an EMA mechancal transmsson has ozens of gear teeth, the falure of any one of whch can cause crtcal jammng, the secon major challenge. An EMA must be able to bypass ths falure by some means, such as nclung several transmsson paths, or by removng 4

24 the gearbox altogether an usng a low-spee, hgh-torque rect rve, a currently nvestgate alternatve [9]. The thr major challenge to evelopng a hgh-power, flght worthy EMA system s thermal management. The heat generate by an EMA system can easly be enough to cause falure. Alreay, the envronment for whch the EMA system must be esgne ranges from -40 C to 125 C. Furthermore, the actuator an ts electroncs o not offer a convenent path for heat transfer. So, ether new approaches to heat transfer must be evelope or the actuator an ts surrounng structure must be esgne to operate properly at hgher temperatures. EMAs can be use on the alerons, elevators, spolers, an ruer, the prmary flght control surfaces of an arcraft. EMAs also can be use for the propulson system actuaton an seconary functons such as lanng gear an hgh-lft evces. However, thermal management makes prmary flght control surfaces, whch are n contnuous use, much more crtcal than seconary actuators, whch requre only ntermttent use allowng plenty of tme for heat to sspate. Features mportant to the mltary for ther strke arcraft, such as hgh acceleraton maneuverng loas an nverte or supersonc flght, further ncrease the emans on thermal management system esgns. If EMAs are to enter stage four an succee EHAs as the prmary actuators, heat generaton must be well unerstoo. Consequently, ynamc heat generaton n EMAs s one of the most mportant research areas n the msson of all-electrc arcraft. Heat Generaton An EMA couples an electrc motor to a loa, such as a flght control surface, usng mechancal gearng, such as a rotary gearbox, an may use a ball screw (Fgure 1) or roller screw. 5

25 Fgure 1. Guts of an electromechancal actuator [4] showng the motor (yellow) wth ts wnngs, the gears (gray) of the gear box, the screw (blue) of the ball screw, the ball housng (re) of the ball screw, an the ro (green) nto whch the screw sles. Fgure 2 shows the block agram of an EMA system (Ien, ockhee Martn Company, 2005). The EMA has three major sources of heat: the motor, the power electroncs nse the electronc unt (EU), an the gearbox. In steay-state operaton, the effcency s typcally 93% for the power electroncs, 80% for the motor, an 80% for the gearbox. It shoul be note that the actual heat generaton s hghly transent an localze n nature. Quantfyng these transent power losses an the temperature strbuton s a key objectve of ths research. 6

26 Fgure 2. Block agram of an EMA system Generally, gearboxes an ball screws are temperature nsenstve [10]. The electrc motor s probably the most temperature-senstve component. The thermal resstance of the ar gap between the stator an the rotor makes sspatng heat from the rotor ffcult. However, swtche-reluctance motors (SRM) an permanent magnet (PM) motors both lack rotor wnngs, thereby preventng heat-bulup from copper losses on the rotor altogether [10]. Motor losses manly come from the copper wnng losses an magnetc materal losses (hysteress an ey current losses). Snce ball screws are very effcent an have low frcton, even when the EMA s not movng, current must stll be supple to the motor stator to prove a torque balance for the loa an to keep the EMA from beng back-rven. Therefore, heat wll stll be generate even when nothng s movng. FEM s one on the motor esgn to analyze the heat flow an etermne thermal lumpe-element parameters (Fgure 3). 7

27 Fgure 3. FEM agram of a quarter of a PM motor showng the temperature strbuton urng a smulaton. Hgh operatng temperatures lea to the followng problems n motors [10]: The nsulaton melts, leang to electrcal shorts (a catastrophc falure) The copper resstvty ncreases, generatng more heat an further reucng effcency (a reversble problem) The magnetc permeablty ecreases (a graual, but rreversble falure), further ncreasng magnetc losses an reucng effcency. The power electroncs n the EU, controllng the recton an spee of the motor [4], generate heat mostly n the semconuctors. Semconuctors generate heat n four major ways [11]: 8

28 On or off swtchng losses. At hgh swtchng frequences, as are necessary n motor control, ths heat contrbuton becomes sgnfcant. Resstve losses of forwar conucton eakage losses n off moe. Some current from the blocke voltage gets through. Control termnal contact losses of the semconuctor The combne heat from these effects s sgnfcant [4], often of the same orer of magntue as from the motor [12], [13], an, unless remove, coul rreversbly amage the electroncs [4]. Even though electroncs can survve hgh transent temperatures, ther mean tme between falures (MTBF) s conserably reuce. Beses heat from semconuctor power evces, heat can also be generate from the copper traces an the magnetc components (wnng an core losses) on the power electroncs boar. These losses are even greater urng a transent current surge. Smulaton Accurately moelng heat generaton, nclung transents, s mportant n evelopng goo thermal management of the whole EMA system. Transent heat-generaton smulaton of an EMA system nvolves comprehensve mult-physcs, mult-scale, an mult-oman smulaton efforts. The proceure can be ve nto two steps. Frst, crtcal smulaton parameters for the EMA are obtane. Some parameters are obtane through expermentaton. For others, expermentaton woul be ether very ffcult or mpractcal. In such cases, FEM smulatons of the PM motor can prove accurate nformaton. FEM smulatons of the magnetc fel strbuton of the motor are use to fn the nonlnear self an mutual-nuctances of the three-phase wnngs. Fgure 4 shows an example PM motor n a 2D cross-sectonal vew wth the magnetc flux ensty strbuton. 9

29 Fgure 4. Cross-sectonal vew of a PM (left) an magnetc flux ensty strbuton (rght) Because FEM smulatons for nuctances are very tme-consumng, the smulatons are one on 2D cross-sectons of the motor. Ths s an accurate approach snce for a motor wth a small raus to length rato the ege effects o not sgnfcantly nfluence the smulaton results. In the system smulaton, the motor s ntegrate n a complete system that conssts of an nput voltage source, varous loa contons, a feeback control scheme, etc. Rotor ynamcs are also nclue n the system ynamcal equatons. Instea of usng emprcal formula calculatons, the flux lnkage an nuctances of the motor can be extracte from the FEM moel, where EM fel effects have been taken nto account. Therefore, the motor ynamc moel s very accurate. Transent heat transfer smulaton usng FEM s performe on the motor. Because thermal smulatons o not requre mesh resolutons nearly as etale as those of electrcal smulatons, the thermal smulatons run much faster. Therefore, the thermal smulatons are performe on 3D segments of the motor. Once, smulaton ata s obtane, correlatons between nput heat an output noal temperatures can be mae to erve thermal resstances an capactances. These values are then use n a lumpe-element thermal network wthn the full EMA smulaton moel. 10

30 Secon, a nonlnear, lumpe-element moel (NEM) s bult an the prevously obtane parameters are use n the moel. Ths approach proves the hgh accuracy of FEM wth the smulaton spee of lumpe-element moelng. Chapter 2 etals the overall smulaton moel. It explans how the moel s esgne, how many of ts parameters are obtane, an how the varous components, such as the thermal network, nteract. It also shows how the moel performs when a full smulaton s run. Chapter 3 etals the evelopment of the nonlnear nuctance formulas that are use n the overall smulaton moel of the EMA. Ths represents a large porton of the work an one of the most crtcal components of the moel. Ths chapter explans n etal how FEM s use to get tables of nuctances, how that ata s correlate wth currents, an how a formula s ftte to the ata. Chapter 4 covers the esgn an verfcaton of a nonlnear, transent smulaton moel of a two-step synchronous generator wth three-phase, full-wave rectfers. A new magnetcs moel s evelope an compare to FEM ata of a synchronous machne. Smulaton results are nclue. Fnally, Chapter 5 conclues the ssertaton. Ths work covers the prmary components nvolve n an all-electrc arcraft system. 11

31 CHAPTER TWO: DYNAMIC HEAT GENERATION MODEING OF HIGH-PERFORMANCE EECTROMECHANICA ACTUATOR Introucton The evelopment of all-electrc arcraft s a hgh prorty n the avoncs communty [1]. Current arcraft use a combnaton of hyraulc, pneumatc, an electrc systems. However, future arplanes are expecte to use a sngle, electrc system, wth electromechancal actuators (EMAs) replacng hyraulc pstons. Such a system woul reuce the cost to bul, operate, an mantan arcraft [15]. It woul also make arcraft lghter, more relable, safer, an more easly reconfgurable, mprovng the turnaroun for new technology [1]. There are two major obstacles to replacng hyraulc actuators wth EMAs: heat generaton management an power management. Frstly, unlke hyraulc actuators, EMAs o not have the nherent avantage of recrculatng flu to cool ther components [16]. Rather, wnngs n EMAs can overheat raply epenng on the emans, at whch pont materals can egrae. Neoymum-ron-boron s one of the most powerful permanent magnets for electrc motors, but t begns to emagnetze at relatvely low temperatures [17], wth an operatng range upper lmt of 120 C to 180 C. Although effectve, lqu coolng rensttutes one of the systems that are beng elmnate. Prmary flght control actuators are of partcular relevance because they are contnually engage urng flght, are requre to accelerate raply, an are often face wth hgh wn loas. As the complextes of arcraft systems exponentally grow an the eman for lghtweght compostes ncreases, thermal concerns are ncreasngly sgnfcant. Heat generaton n motor wnngs can be hghly ynamc an localze. Prectng how much heat woul be generate by an EMA s crtcal. 12

32 Seconly, EMAs have the avantage of only usng power when actve, whereas hyraulc systems requre that a pump contnuously mantan lne pressure. Ths fference shoul help to make EMA systems more effcent than hyraulc systems. However, ths avantage brngs wth t a major complcaton: the power raw from the electrcal supply s also hghly ynamc. Compounng ths problem are the hghly varable emans of all the electrcal systems of an entre arcraft rawng power from one source. Prectng how much power wll be rawn urng operaton s crtcal for szng system components an choosng the power supply type (e.g. battery-supercapactor hybr). Gven the nee for prectng ynamcally generate heat an power eman, an accurate smulaton moel s tremenously valuable n esgnng an EMA system. EMA systems are complex an testng them s an nvolve proceure. Such systems are also expensve whch scourages testng them to ther lmts, an many fferent scenaros are neee to properly test an EMA. Furthermore, there are sgnfcant etals whch cannot be practcally measure expermentally. Therefore, t s avantageous to smulate the entre EMA wth all ts ynamcs havng only measure ts key parameters. The EMA smulaton moel presente n ths chapter meets ths goal. It nclues a poston fel-orente control (FOC), a pulse-wth moulaton (PWM) component, an electromechancal ynamcs component, an a thermal component. Moel parameters are obtane by expermentaton or fnte element metho (FEM). The full smulaton moel s a nonlnear, lumpe-element moel (N-EM). The moel correspons to a lnear actuator wth a permanent magnet (PM) rotatonal motor connecte to a rotatonal-to-lnear rve tran (e.g. a ball screw or a roller screw). 13

33 The methoology presente here proves a means of prectng heat generaton an the overall power emans of an EMA. Such tools wll help n reachng the goal of an all-electrc arcraft. Smulaton results, nclung heat generaton an power eman, for a commercally avalable, test EMA are shown n the latter part of the chapter. Moel ayout The four man components of the N-EM are shown n Fgure 5. 14

34 Fgure 5. Moel layout showng the connecton of the four major components: fel-orente control, pulse-wth moulaton, electromechancal ynamcs, an the thermal component. The moel takes the EMA ro s esre moton profle translate to an electrcal reference frame of the motor s rotor, θ * me (the rotor s angle θ m multple by the number of pole pars, P/2), as an nput to the fel-orente control (FOC). Base on the present state of the rect an quarature (DQ0) currents, an q, an the actual poston of the EMA ro, θ me, the FOC algorthm calculates the reference voltages, u * an u * q, n such a way that the torque angle, ϕ, (the angle forme by the an q current vectors) s kept at ±π/2. (The power-varant DQ0 15

35 transform s use for ths work.) The PWM component takes these reference voltages as well as the present state of the DC supply voltage, u DC, an calculates the PWM voltages, u an u q. These are the actual nput voltages to the electromechancal ynamcs. Wth these an the present loa torque, τ, the electromechancal ynamcal equatons lea to the calculaton of the currents, an q, EMA poston, θ me, an ynamcally generate heat, Q. An output of the electromechancal component s the DC current of the EMA system, DC. Together wth the DC bus voltage, u DC, t represents the power emans of the EMA system at ts termnals. The heat generate n the motor s the nput to the thermal component. Wth the present value of the envronmental temperature, T env, the thermal equatons lea to the noal temperatures, T, an the upate value of resstance, R s, whch s fe back nto the electromechancal ynamcs. The noal temperatures represent the thermal behavor of the EMA system. Table 1 summarzes many of these terms. 16

36 Table 1. Smulaton States Symbol Name Unts t Tme s ϕ Torque angle ra θ m Rotor angle ra * θ m Desre rotor angle ra θ me Rotor angle θ m scale by P/2 ra * θ me Desre rotor angle θ * m scale by P/2 ra ω me Frst tme-ervatve of θ me ra/s * ω me * Frst tme-ervatve of θ me ra/s α me Secon tme-ervatve of θ me ra/s 2 τ oa torque N m τ M Motor-generate torque N m τ g Torque ue to gravty N m τ f Frctonal torque N m u Drect voltage V u q Quarature voltage V Drect current A q Quarature current A * u Desre rect voltage V * u q Desre quarature voltage V * Desre rect current A * q Desre quarature current A u DC DC bus voltage V DC DC bus current A Q Generate heat W R s Phase resstance Ω T env Envronmental temperature C T Noal temperature C All the torques are n the mechancal, angular reference frame (.e. they have not been scale by P/2). Notce that the esre poston, θ me *, the loa torque, τ, the envronmental temperature, T env, an the DC voltage, u DC, all can be varable nputs to the moel. An EMA can be run 17

37 through a moton profle n the presence of a varable loa profle whle the bus voltage an envronmental temperature change. Parameter Measurement an FEM Calculaton Table 2 shows a summary of the key EMA parameters. Some of the parameters, such as the number of slots an number of poles, are easly obtane from the actuator s ocumentaton, but others must be measure or carefully calculate. Some of the measurements are performe whle the actuator s couple wth a hyraulc press or some other actve loa. Table 2. EMA Parameters Symbol Name Unts N cr Gearng rato ra/m S Number of slots ND P Number of poles ND τ f,max Maxmum power tran frcton torque N m m Ro mass kg I Rotor moment of nerta kg m 2 /ra 2 R s,ref Phase resstance at T ref Ω T ref Reference temperature C α R Temperature coeffcent of resstance C -1 λ PM PM flux lnkage Wb (, q ) Drect nuctance H q (, q ) Quarature nuctance H Measurement Gearng rato, N cr The gearng rato s carefully calculate by countng the teeth on each gear an measurng the lnear splacement of the actuator ro for one full revoluton of the rotor. 18

38 Power tran frcton, τ f The hyraulc actuator s couple to the unpowere EMA an rves the EMA s ro up an own. The statc frcton of the EMA s rve tran s obtane by usng the force an stroke ata collecte urng ths test, an the mean forces requre to push or pull the EMA ro are calculate. The magntues of these forces are unequal ue to a bas. Ths bas exsts because of the weght of the ro tself, an the fference between ths bas an the push an pull forces s the frcton force. Ths force s then converte to a torque as seen by the motor s rotor. Ro mass, m The mass of the EMA s entre ro s erve from the frcton test. The bas n the force requre to move the EMA ro s ue to the weght of the ro. Ths translates to a certan mass. Rotor moment of nerta, I The rotor moment of nerta s obtane by measurng the mensons of the rotor an the materal enstes an calculatng the moment of nerta. Phase resstance, R s The phase resstance s measure by connectng leas to two of the three phase lnes nto the motor an measurng the open-crcut lne-to-lne resstance. Ths value s then ve by two f the motor s a wye-connecte motor. FEM Calculaton The materal composton an geometry of the motor are use n esgnng an FEM moel n ANSYS s Maxwell 2D. Values of rotor angle, torque angle, an current ampltue are vare as multple smulaton runs are performe. Many of the motor moelng researchers n the past [18], [19] an even present [20], [21] use constant parameter values n smulatons, although some use nonlnear nuctances [22], 19

39 [23], [24]. For ths chapter, FEM s use to obtan an q as nonlnear functons of an q. The ervatves of nuctance wth respect to current are numercally calculate from these tables. The nuctances an ther ervatves wth respect to DQ0 currents can be store n tables whch are then nexe urng smulaton. The FEM moel s smulate wth currents well above the motor s normal operatng range n orer to capture the behavor urng hghly transent moments. The torque angle shoul be vare from 0 to π raans to capture the full character of the nuctances. Even f the rotor of an EMA s roun, the rotor angle has an effect on the DQ0 reference frame nuctances. Therefore, the rotor angle shoul also be vare over one pero of varaton n the DQ0 nuctances. Such varaton occurs mostly at relatvely hgh current values (.e. at saturaton), so ths varaton s average out, but the calculaton of nuctance at a sngle rotor angle cannot be assume to be representatve of the average. Note that the peak nuctance values o not necessarly occur at zero current, nor are the nuctances mrrore about the quarature axs. An example of ths s n Fgure 6. Ths shoul clarfy the hghly varable nature of nuctances an the nee to use nonlnear moelng nstea of assumng constant values. 20

40 Fgure 6. Drect-axs nuctance along the axs (blue) an the nuctance value at zero current (green crcle). The nuctance s not mrrore about the quarature axs (Ths axs goes nto the page), so ts values for postve o not match those for negatve. Electromechancal Dynamcs The motor ynamcs are moele by four prmary ynamcal equatons n the DQ0 reference frame [25]. The frst two of these equatons are (1) (2) t q t qq q q = (1) G qq qq q q G q q q =. (2) G G q These come from the voltage equatons (3) an (4): 21

41 q (3) u = Rs q qqωme (3) t t q (4) uq = Rsq q qq ( λ PM ) ω me, (4) where t t (5) (6) (7) (8) = (5) q = (6) q q q = q (7) q qq = q (8) q q (9) G u R s ( q q ) ω me = (9) (10) G q = u q R s q ( λ PM ) ω me. (10) The full ervaton can be foun n Appenx A: Permanent Magnet Machne Dynamcs. These equatons carry the assumpton that the varatons n nuctances over rotor angle can be reasonably average out, that the flux lnkage from the permanent magnets appears only n the rect axs, an that the motor s balance. The voltages are prove by the PWM component; the current rates are calculate usng (1) an (2); an the currents are foun by ntegraton. In [26], there s a etale analyss of the motor-generate torque equaton. Gven the assumptons mae for (1)-(10), the motor-generate torque equaton s 3P (11) τ = [ λ ( ) ] M 4 q PM q. (11) 22

42 The thr ynamcal equaton s the torque-acceleraton equaton that lnks the motorgenerate torque to the acceleraton t prouces: ωme P 1 (12) = α = ( τ τ τ τ ) t me 2 I net M g f, (12) where I net s the combne effect of all nertas from all movng parts n the EMA translate to the mechancal, angular reference frame of the rotor. The value I net s calculate from the mass of the ro, m, an the angular moment of nerta of the rotor, I, by m (13) I net = I. (13) 2 N cr As long as velocty s zero, the frctonal torque s efne to be equal an opposte to the sum of the other component torques up to a maxmum frcton value. Ths keeps the net torque an, therefore, velocty at zero. However, when the sum of the other component torques excees ths maxmum frcton, the magntue of the frcton hols at ths maxmum value an ts sgn always opposes the recton of moton (14). (14) ( τ τ τ ) M g me M g f,max τ f = (14) sgn( ω ) τ : ω 0 τ τ τ > τ me f,max : ω me = 0 τ M τ τ g τ f,max Note that ar frcton s not nclue because EMAs use poston controlle motors, whch o not reach excessve spees where ar frcton s a practcal conseraton. Fnally, the forth ynamcal equaton s (15) θ me = ωme. t (15) Ths s smply the efnton of angular velocty an s ntegrate to get angular poston. 23

43 Thermal Component The motor losses are relate to the motor parameters. Snce losses affect temperature whch affects motor parameters whch n turn affect losses, the losses shoul be moele through the motor parameters, an not n post-processng. Copper wnng loss s a prmary power loss. The wnng loss can be calculate as (16) Q ( a b c ) R s =, (16) or equvalently 2 2 (17) Q ( q ) R s 3 = (17) 2 n the DQ0 reference frame. Ths heat s then fe nto the lumpe-element thermal moel. Some thermal moels, such as n [27], use only one temperature noe. The moel shown here uses multple noes. The temperatures at multple locatons n the motor are calculate usng a lumpe-element thermal moel (Fgure 7) of the form shown n (18): T t K (18) = Q Tk ρk σ, (18) k where T an T k are noal temperatures, Q s a heat source corresponng to that noe, K s the number of noes n the network, ρ k s a sum of recprocals of thermal resstances lnkng the noe to nearby noes, an σ s a recprocal of thermal capactance lnkng the noe to the thermal snk. Ths moel s smlar to the one evelope n [28] for another PM motor. The source of heat for the temperature calculatons s the copper power loss n the wnngs. The temperature strbuton of the motor s moele usng a lumpe-element network shown n Fgure 7. The values of the thermal resstances an capactances are evelope usng an FEM moel of the motor. 24

44 Fgure 7. Thermal lumpe element moel. The motor s efne to have a front an a back. The space between the rotor an stator s the ar gap. The rotor s compose of magnets on a shaft. The boy of the shaft s the central regon about whch the magnets are fxe. The bearngs hol the shaft n place at both of ts ens. The front an back of the motor have an alumnum cover an the ses of the motor are surroune by an alumnum case. The copper wnngs loop aroun the stator teeth concentrcally an are completely embee n epoxy. The epoxy n ron s the epoxy surrounng the wnngs an locate n the man boy of the ron stator. The en turns are the ponts on ether en where the wnngs loop aroun the stator teeth passng from one stator slot to another. An, the envronment s the surrounng atmosphere. 25

45 Ths moel s generc wth multple heat sources. Here, however, the copper loss s consere to be the preomnant heat source n the motor. Gven upate noal temperatures, the electrcal phase resstance can be upate. The relatonshp between copper resstance an temperature [29] s as follows: (19) R s = R s,ref [1 α R (T T ref )], (19) where R s,ref s the resstance of the copper wnng at the reference temperature T ref, an α R s the temperature coeffcent of resstance for the copper. Fel-orente Control A thorough analyss of ynamc motor heat generaton shoul nclue motor rve ynamcs. Ths moel s fel-orente control (FOC) translates the esre moton profle nto smulaton rve voltages. The control s an aaptve close-loop control, whch s a type of feeback lnearzaton of the spee an poston errors. In the followng ervaton, the star superscrpt s use to ncate esre values. Startng wth angular velocty an angular poston errors, a esre angular acceleraton, α * me, s evelope: * * (20) ( ωme ωme ) ( θ me θ me ) α me *, (20) = k ω t c k θ t c where Δt c s the tme step of the control loop an k ω an k θ are control coeffcents. Ths equaton tens towar changng the present angular velocty an poston to match the esre angular velocty an poston. The esre velocty must be reefne base on ths newly calculate esre acceleraton: * * (21) ω me = ωme α me tc. (21) 26

46 If the loa torque s precte by estmatng the loa torque from the prevous step an f the frctonal an gravtatonal torques are neglecte, then (12) becomes (22). (22) τ * M = 0 * 2 αme Inet τ g τ f P 0 2 3P, (22) α meinet λpmq τ g τ f P 4 ~ τ Usng (11) as a control equaton an settng * to zero yels (23) P τ * 3 M = q λpm. 4 (23) Substtutng (23) nto (22) an solvng for * q results n (24). * * 8 (24) ( αme αme ) net q = 2 q 3P λpm I (24) The esre currents an current rates are then lmte. astly, the esre voltages are calculate: (25) u * * * * * = Rs 0 q 0q ωme (25) t * * q * * (26) u R ( λ ) ω q s q q0 t * =, (26) 0 PM me where 0 an q0 are the rect an quarature nuctances at zero current, respectvely. These constants are use nstea of the nonlnear parameters for the sake of smplcty. PWM Component The PWM component of the smulaton moel uses a tralng ege moulaton carrer, as emonstrate n Fgure 8. The voltages are translate nto the ABC reference frame. The PWM algorthm s performe on those voltages. Then, these pulse voltages are translate back nto the DQ0 reference frame. 27

47 (a) 28

48 (b) Fgure 8. The general concept of the PWM. A tralng ege carrer wave (a) s compare to a normalze snusoal phase voltage. When the normalze wave s greater than the carrer wave, the output PWM voltage (b) s turne on. Otherwse, t s off. Inputs Scalng The stroke an loa force ata are recore from actual flght profles. However, these profles correspon to an actuator wth performance characterstcs that mght not match those of a test actuator. Therefore, the stroke an force profles are scale for a range whch s compatble wth the test actuator. The stroke profle s also ajuste to match the acceleraton lmts of the 29

49 actuator by stretchng the profle over tme. Ths scalng s approprate for testng the general capablty of the smulaton moel as s one n ths chapter. Force-stroke equaton In the absence of actual flght profles, a force-stroke relatonshp can be useful n smulatng the force, gven a theoretcal stroke profle. Ths force-stroke equaton comes from analyzng actual flght ata. The equaton s (27) Fˆ = C ( ρ C C 0 ) 2 ( x x ) C ( x x ) C ( x x ) 3 [ ] F cv Ma 0 C = C 1 2 : x < x 0 : x x 0 s 2, (27) where ρ s the ar ensty whch can be erve from the alttue of the arcraft, C 1, C 2, C 0, C 2, an C 3 are coeffcents, s s the arcraft spee, F 0 s the force bas ue to the weght of the aleron, c s the net frcton coeffcent, M s the effectve mass of the aleron, x s the stroke poston, x 0 s a stroke bas, v s the frst ervatve of x, an a s the secon ervatve of x. The effect of ar ensty epens on whether the flap s own or up (x < x 0 or x x 0 ). The effect of arcraft spee s moele by a Taylor seres functon of the stroke. The result of fttng ths functon to the avalable ata s shown n Fgure 9. 30

50 Fgure 9. Actual force profle (blue) an calculate force profle (green). The value of havng ths functon s that t gves the freeom to run msson profles that were never one before. An EMA Example an Results A commercally avalable EMA was measure, moele, smulate, an expermentally teste. Followng are the results. Parameters The key parameters for the test EMA are shown n Table 3. 31

51 Table 3. EMA Parameters Symbol Name Value N cr Gearng rato 1963 ra/m S Number of slots 12 P Number of poles 10 τ f,max Maxmum power tran frcton N m m Ro mass 8.5 kg I Rotor moment of nerta kg mm 2 /ra 2 R s,ref Phase resstance at T ref 1.4 Ω T ref Reference temperature 20 C α R Temperature coeffcent of resstance C -1 [29] λ PM PM flux lnkage Wb (, q ) Drect nuctance H q (, q ) Quarature nuctance H Fgure 10 shows the force-stroke ata for the frcton analyss. 32

52 (a) 33

53 (b) Fgure 10. Statc frcton analyss showng (a) stroke an (b) loa force. The green lnes on the force plot represent the mean forces to pull (-258 N) or push (425 N) the EMA ro, overcomng the frcton of the rve tran (342 N). The bas force s the weght of the EMA ro (84 N). The rect an quarature nuctances are obtane as shown n Fgure 11 an Fgure 12. The FEM moel s smulate wth a range of currents (0 to 50 A), a range of torque angles (0 to π raans), an a range of rotor angles (0 to π/3 raans). For ths motor, a range of π/3 raans captures one full cycle of varaton n nuctance values ue to rotor angle. 34

54 Fgure 11. Drect-axs nuctance as a functon of rect current,, an quarature current, q. These nuctance values are average over rotor angle. 35

55 Fgure 12. Quarature-axs nuctance as a functon of rect current,, an quarature current, q. These nuctance values are average over rotor angle. A computer renerng of the motor geometry s shown n Fgure 13. Ths geometry s use for FEM moelng. 36

56 Fgure 13. Motor geometry (left) an slot measurements (rght). Table 4 shows some of the materal propertes of the EMA motor. Table 4. EMA Parameters Part Name C p [J/kg K] a k [W/m K] b ρ [kg/m3] c Casng A amnaton Steel Wnngs Copper Epoxy (unknown) Axle Steel Magnets NFeB-30 5e a C p s the specfc heat capacty. b k s the thermal conuctvty. c ρ s the ensty. 37

57 EMA motor. Table 5 shows the thermal resstances an capactances of the thermal network for the Table 5. EMA Thermal Parameters Resstances Value [K/W] Capactances Value [J/K] R 1, C R 1, C R 2, C R 2, C 4 0 R 3, C R 4, C R 5, C R 6, C R 7, C R 8, C R 8, C R 8, C R 9, C R 9, C R 10, C R 10, C R 11, C R 11, R 12, R 13, R 13, R 14, R 14, R 14, R 15, R 17, The thermal resstances are labele accorng to the noes they lnk. The thermal capactances are labele accorng to the noes they are connecte to. 38

58 Results The followng knematc results are all n the lnear oman. The esre stroke profle an the motor's actual stroke profle are shown n Fgure 14. The motor followe the esre stroke profle very well. Fgure 14. Desre (blue) an actual (green) stroke profles The loa force profle an the motor-generate force are shown n Fgure

59 Fgure 15. Input loa force profle (blue) an motor-generate force (green). The loa force was a sustane 0.45 kn. The DQ0 voltages an currents are shown n Fgure 16 an Fgure 17, respectvely. 40

60 Fgure 16. DQ0 voltages wth pulse form ue to PWM. 41

61 Fgure 17. DQ0 currents wth pulse form ue to PWM. If the motor s treate as a system, the powers nto (postve) an out of (negatve) ths system are shown n Fgure

62 (a) 43

63 (b) 44

64 (c) 45

65 () Fgure 18. Motor powers showng (a) the total electrcal power at the termnals of the motor, (b) the power loss n the wnngs, (c) the power store n the magnetc fels, () an the mechancal power on the rotor. The electrcal power at the termnals s the power on the DC bus lne. Note that sometmes ths power s negatve, meanng that the motor s actually senng power back to the source. The reason for ths s that the wn loa s actually rvng the flght control surface, pushng aganst t as t yels to the wn pressure. If the rve tran s treate as a system, the powers nto (postve) an out of (negatve) ths system are shown n Fgure

66 (a) 47

67 (b) 48

68 (c) 49

69 () 50

70 (e) Fgure 19. Drve tran powers showng (a) the mechancal power on the rotor, (b) the loa power, (c) the frctonal power, () the gravtatonal power, (e) an the net knetc power. The comparson of the recore temperatures an the smulate temperature of the stator s shown n Fgure

71 (a) 52

72 (b) Fgure 20. (a) Temperature comparson an (b) agan zoome n. These plots have envronmental temperature (blue), expermentally measure temperature of stator (green), smulate temperature of stator (re), an the smulate temperature of the wnngs (cyan). The fference here between smulate an measure stator temperatures s actually qute small, only about 0.34 C, much less than the 1 C senstvty of the thermocouples use. The accuracy of the temperature precton s more realy observe for longer smulaton profles. Another test s shown n Fgure 21 lastng a half-hour. The smulaton of ths test matches the stator temperature very well wth a maxmum error of 0.17 C. 53

73 It s mportant to note that the power eman, power loss, an, therefore, temperatures are heavly epenent on the effcency of control. Control tunng can sgnfcantly affect the results of the smulaton even f the stroke followng s practcally entcal. Fgure 21. Temperature comparson for a half-hour run showng envronmental temperature (blue), expermentally measure temperature of stator (green), smulate temperature of stator (re), an the smulate temperature of the wnngs (cyan). Note that the smulate temperatures of the wnngs an stator are about the same, ncatng the low thermal resstance n the motor. 54

74 Concluson There s a strong motvaton to move towars all-electrc arcraft. However, the sgnfcant obstacles to full realzaton of all-electrc arcraft are the thermal management of heat loas an power management. Ths paper shows the successful mplementaton of an EMA moel wth a control algorthm, a PWM component, an electromechancal component wth nonlnear parameters, an a thermal component. The moel takes a esre poston profle, a loa force profle, an an envronmental temperature profle as nputs an prects the thermal behavor of the motor an the transent power emans of the EMA. It s shown that control plays a powerful role n how much excess heat s generate an how much the power eman spkes. It s also shown that the temperatures wthn the motor of the EMA are all wthn a farly narrow ban. Ths approach of usng fnte element metho an nonlnear, lumpe-element, ntegrate moelng gves a complete pcture of the ynamc behavor of an EMA, thereby helpng to make all-electrc arcraft a realty. 55

75 CHAPTER THREE: NOVE NONINEAR INDUCTANCE MODEING OF PERMANENT MAGNET MOTOR Introucton An all-electrc arcraft s an ambtous, hgh-prorty goal n the avoncs communty [14]. A typcal msson profle for a flght control surface electromechancal actuator (EMA) can be hghly ynamc, especally n mltary arcraft. Both ncrease relablty an effcency are ts promse mplcatons [1]. EMAs are prmary canates for substtuton of non-electrc components on arcraft. However, there are thermal [16] an power management challenges ue to ynamc msson profles an the nonlnear behavor of the nternal magnetcs of such an actuator. These challenges, couple wth the hghly flght-crtcal nature of EMAs, necesstate careful smulatons run over a we range of scenaros before EMAs can be heavly rele upon. The typcal permanent magnet (PM) electrc motors n EMAs have farly nonlnear magnetc behavor, especally as they are operate over non-steay-state msson profles. If the uty cycle were steay-state or f the ar gap between the rotor an stator were large, then lnear nuctance moelng woul be aequate because the magnetc crcuts woul never get saturate. But, the uty cycle s hghly varable an, for the sake of hgh power ensty, the ar gap s small; therefore, ynamc, nonlnear moelng s necessary. Fnte element metho (FEM) asssts n evelopng the nonlnear nuctance lookup tables that are mportant for fast an accurate lumpe-element smulaton of EMAs. Often, these tables are rectly use n smulatng an electrc machne ([22], [23], an [24]). However, such lookup tables are rough, boune, an slow. Durng smulaton, t s mportant that the states of the moel transton smoothly. Any roughness n a lookup table can cause perturbatons wth rpplng effects throughout the moel leang to reuce numercal stablty. Dervatves of such 56

76 surfaces are especally senstve to suen transtons. Furthermore, no matter how large a lookup table s, t has lmts, an smply bounng the nces to the lookup table can lea to errors ue to scontnuous graent behavor. Fnally, as the tables are mae larger to accommoate hgher bouns an prove smoother nterpolaton, the lookup routne becomes slower. Formulas whch can accurately ft the nonlnear surfaces of such lookup tables have the potental to be very smooth, to requre far less ata to efne them, an to compute quckly. It s also esrable that they extrapolate well. Ths paper emonstrates a nonlnear nuctance formula that s a functon of both rect an quarature (DQ) currents, allowng cross-couplng effects to be nclue. The process of evelopng the formula an fttng t to the reference FEM nuctance values as well as the results of the fttng an several comparsons to other methos are nclue. Fnally, a comparson of a lumpe element smulaton usng nonlnear nuctance s compare wth one usng lnear nuctance. Backgroun Table I lsts several of the varables use n ths paper. 57

77 Table 6. Smulaton States Symbol Name Unts t Tme s P Number of magnetc poles ND ϕ Torque angle ra θ m Rotor angle ra θ me Rotor angle θ m tmes P/2 ra u Drect voltage V u q Quarature voltage V Drect current A q Quarature current A R s Phase resstance Ω λ PM PM flux lnkage Wb (, q ) Drect nuctance H q (, q ) Quarature nuctance H The ynamcs of a PM electrc motor, typcal n an EMA, are moele by four prmary ynamcal equatons n the DQ0-reference frame [25]. Two of these equatons moel the electrcal behavor of the motor: (28) t qq q q = (28) G qq q G q (29) q t G G q q =, (29) qq q q where (30) (31) = (30) q = (31) q 58

78 (32) (33) q q = q (32) q qq = q (33) q q θ (34) G u Rs ( qq ) t me = (34) θ me (35) Gq = uq Rsq ( λpm ). (35) t The full ervaton can be foun n Appenx A: Permanent Magnet Machne Dynamcs. The other two ynamcal equatons moel the mechancal behavor of the motor. Equatons (28) an (29) come from the voltage equatons (36) an (37): (36) u q θme = Rs q qq (36) t t t θ q me (37) uq = Rsq q qq ( λpm ). (37) t t t These equatons carry the assumpton that the varatons n nuctances over rotor angle can be reasonably average out, that the flux lnkage from the permanent magnets appears only n the rect axs, an that the motor s balance. Though phase resstance an PM flux lnkage o vary wth current magntue, temperature, an frequency, the focus of ths paper s the sx nuctance propertes: an q an ther current ervatves /, / q, q /, an q / q. Metho of Analyss Collectng the ata The process of creatng a formula to ft nuctance values begns wth gettng the nuctance values. Usng ANSYS Maxwell 2D, an FEM software package, a PM motor s 59

79 smulate n hgh etal an the ABC-reference nuctance values for each phase are store as current magntue, torque angle, an rotor angle are vare. The torque angle s the angle prouce by the rect an quarature current vectors. The rotor angle (θ m ) s the angle of the rotor wth respect to the phase A axs. The nuctance values are then translate to the DQ0- reference frame usng the Park an Clarke transforms [30]. Averagng the nuctances over rotor angle, the rect an quarature nuctances ( an q, respectvely) are shown as functons of rect an quarature currents n Fgure 22 an Fgure 23. Fgure 22. Drect nuctance as a functon of an q. 60

80 Fgure 23. Quarature nuctance q as a functon of an q. Depenng on the maxmum current of nterest, averagng nuctance over rotor angle can be consere reasonable. The varaton n nuctance magntue over rotor angle ncreases wth current magntue. An example of ths behavor s shown n Fgure 24 an Fgure 25. In ths case, for a current magntue less than 16 A, averagng over ths varaton s reasonable. 61

81 Fgure 24. Drect nuctance as a functon of rotor angle for a partcular torque angle an for varous values of current magntue. 62

82 Fgure 25. Quarature nuctance q as a functon of rotor angle for a partcular torque angle an for varous values of current magntue. The nuctances vary smoothly over currents an angles, but the resoluton of the ata shown s somewhat low because runnng FEM smulatons s tme-consumng. In orer to make goo fts to the ata, t helps f the resoluton s hgher. Therefore, the ata s smoothly nterpolate. Interpolatng the ata The nuctances are splne nterpolate to help reuce the Runge effect urng fttng later on. Ths also helps n analyzng the ervatves of the nuctance tables wth respect to current. The rect an quarature nuctances after nterpolaton are shown n Fgure 26 an Fgure 27, respectvely. 63

83 Fgure 26. Drect nuctance after beng nterpolate. 64

84 Fgure 27. Quarature nuctance q after beng nterpolate. Wth smoothly nterpolate nuctances, ther ervatves wth respect to current can be numercally calculate usng very small steps n current. So, from the two nuctance tables, an q, four more tables are evelope: /, / q, q /, an q / q. These are shown n Fgure

85 (a) 66

86 (b) 67

87 (c) 68

88 Fgure 28. Inuctance ervatves: (a) /, (b) / q, (c) q /, an () q / q. () These ervatves are mae part of the verfcaton of goo fttng snce they are use n the motor equatons (28) an (29). Shftng the ata It s nterestng to note that the peaks of the nuctances are not naturally at the orgn. Drect nuctance versus rect current s shown n Fgure 29. The peak of ths curve s at a somewhat negatve current. 69

89 Fgure 29. as a functon of rect current where quarature current s zero. The peak s crcle n re at A. Because of the nature of the partcular fttng functon emonstrate n ths paper, the peaks of the nuctance surfaces must be at the orgn. Therefore, the nuctance surfaces are shfte pror to fttng. A shft can be ae later to get the correct values from the formula. Fgure 30 shows the before an after of / (from Fgure 28 (a)). 70

90 (a) 71

91 (b) Fgure 30. Inuctance ervatve / (a) before beng shfte an (b) after beng shfte. Note that between the two versons, the re an blue regons are shfte to the rght n (b). Also, note that the oman of the surface s croppe slghtly to mantan symmetry. Fttng along current As seen n Fgure 29, nuctance generally ecreases wth current magntue. An, ths s efntely the case after the nuctance surfaces have been shfte so that ther peaks are at the orgn. A ratonal functon s use to ft nuctance versus current usng ether 5 coeffcents (38) or 7 coeffcents (39): ( ) 1 (38) ( ) = 3 5 peak k (38) k k 1 ( k ) ( k ) 1 2 k 4 72

92 (39) ( ) k3 k1 ( k2 ) k5 k7 ( k ) ( k ) 4 6 ( k ) = peak 1, (39) 1 where s current magntue, the k varables are the tunng coeffcents, an peak s a constant. The reason for usng a ratonal functon s that t extrapolates well. Whereas polynomal functons ten to suffer from Runge s phenomenon [31], vergng wlly beyon the regon of fttng, ratonal functons ten to emonstrate asymptotc behavor along the abscssa, whch s very approprate for saturatng phenomena. Although a prmary concern n usng ratonal functons s the occurrence of sngulartes, as long as the k coeffcents are postve all the poles wll be n the negatve. Snce s always postve, these poles wll never be seen an no sngulartes wll occur. Both the (38) an (39) functons are ft to the nuctance surfaces for each value of torque angle usng an optmzaton algorthm. Upon testng, the Neler-Mea algorthm fmnsearch (from MATAB) prouces very erratc results. Though t tunes an nvual case very well, t tens to over-tune, leang to k values whch ffer greatly from one torque angle to the next. Instea, the reslent backpropagaton heurstc [32] s use. It s a graent sgn optmzaton. Ths results n very goo fts wth excellent contnuty of k values over torque angle. Usng (38), the 5-k nuctance fts for an q are shown n Fgure 31, an usng (39), the 7-k nuctance fts for an q are shown n Fgure 32. The 7-k functon tens to ft the nvual nuctance curves better than the 5-k functon. However, as s shown later, the 5-k functon gves a better overall ft. 73

93 (a) 74

94 (b) 75

95 (c) Fgure 31. Fts wth 5-k values for (a), (b) zoome n, an (c) q. The orgnal ata s shown wth sol curves, an the ftte ata s shown wth ashe curves, where each color correspons to a unque torque angle. Note the slght sparty near the peak of as seen n (b). 76

96 (a) 77

97 (b) 78

98 (c) Fgure 32. Fts wth 7-k values for (a), (b) zoome n, an (c) q. The orgnal ata s shown wth sol curves, an the ftte ata s shown wth ashe curves, where each color correspons to a unque torque angle. Note that the slght sparty seen n Fgure 31 near the peak of s far less here. Fttng along torque angle Wth the rect an quarature nuctance curves along current ftte by both 5-k an 7-k ratonal functons, the varyng k values must be correlate to the torque angle, ϕ. Because of the wely fferng shapes of these k value curves over torque angle, polynomal functons are use: P 1 (40) p kn = g p, nφ, (40) p= 0 79

99 where n s the enumeraton of the k value from 1 to N (5 or 7), g p,n s a polynomal coeffcent, p s the egree of a monomal term, an P s the number of polynomals. It s approprate to use polynomals here because ϕ s boune by 0 an π (from π to 2π s a mrror mage), so we nee not be concerne about vergence beyon the regon of fttng. In matrx form, (40) s P 1 (41) [ k k k ] = [ φ ] g (41) 1 2 N 1 φ where g s a matrx of g p,n values, P by N. Fgure 33 an Fgure 34 show two examples of the varaton of k values over torque angle an ther 12 th -orer polynomal fts. Fgure 33. Orgnal k 1 values over torque angle (blue) for 5-k ratonal functon ft to wth 12 th -orer polynomal ft (re). 80

100 Fgure 34. Orgnal k 1 values over torque angle (blue) for 5-k ratonal functon ft to q wth 12 th -orer polynomal ft (re). So, frst, the k varables are ft to the nuctance curves over current magntue, for each torque angle. Then, each k varable s ftte by a polynomal functon of torque angle. For ths fttng, torque angle s boune by 0 an π raans. In the polynomal fts, the bouns are marke by vertcal black lnes. Yet, there are k values an corresponng polynomal ft values extenng beyon these bouns as seen n Fgure 33 an Fgure 34. Snce nuctance surfaces shoul be symmetrcal about the rect axs, all fttng coeffcents shoul be symmetrcal about the rect axs, whch correspons to torque angles of 0 an π raans. Ths requres that the k curves as functons of torque angle have zero slopes at these bouns (a Neumann bounary). One metho of encouragng ths behavor n polynomal fttng s to mrror the tal ens of the curves. Hence, 81

101 the curves shown above exten slghtly beyon the state bouns by usng ths mrrorng technque. A range of polynomal fttng s one for the k values of both 5-k an 7-k ratonal functons usng from 6 to 22 polynomal terms (5 to 21 egree polynomals). For each k value ft, the normalze mean absolute fference (NMAD) s use as the error metrc. The NMAD metrc s here efne as (42) M 1 NMAD = y R y = max yˆ m m MRy m= 1, (42) { y, yˆ } mn{ y, yˆ } where M s the number of ponts, m s the nex of each pont, y m s the m th pont reference value, ŷ m s the m th pont ftte value, an R y s the range of the y an ŷ sets. The net an maxmum NMAD values for each set of k fts s calculate (Fgure 35). The net NMAD s the mean of the NMADs for all the k values. 82

102 (a) 83

103 (b) Fgure 35. The (a) net an (b) max NMAD values for whole sets of k value fts (e.g. k 1 through k 5 for 5-k ratonal functon). The 5-k results are shown n blue an the 7-k results are shown n re. As expecte, the more polynomal terms that are use the better the ft to the k values. Ths s true up to a pont, past whch the functons become over-efne. From Fgure 35, the optmum number of g coeffcents for the polynomal fts seems to be aroun 10 to 16 (9 to 15 egree polynomals) for both 5-k an 7-k ratonal functons. At ths pont, the nuctance fttng as a whole must be compare to the real nuctance surfaces an ther current ervatves. 84

104 Evaluatng the fttng In applyng a fttng formula, not only o the nuctances nee to match, but so o the current ervatves. Ths means the nuctance surfaces must match very well. To test the fttng, current magntues an torque angles are ranomly chosen. At each corresponng pont (, q ), the smoothe nuctance tables are lnearly nterpolate to prove the reference values. Ths s an accurate but slow table lookup metho. On a partcular machne runnng MATAB 2012a, ths takes about s to process a large matrx of arbtrarly chosen ponts. A smplfe lookup s also teste whereby the nearest neghbor s use an no nterpolaton s one. Ths rounng metho takes about s for the whole matrx of ponts. The optmum fttng (12 th - egree polynomals for the 5-k ratonal functon) wth the technque emonstrate n ths paper takes only s. Ths s 66% faster than the nearest-neghbor table lookup metho an 2.5 tmes faster than the lnear nterpolaton metho (Fgure 36). 85

105 Fgure 36. The tme requre for the three methos to calculate nuctances an ther current ervatves. The formula metho s faster. Beses beng fast, the formula s qute accurate. In some cases the average error from the formula s less than the error nuce by rounng a 0.5 A varaton n current urng nearestneghbor table lookup. A comparson of the NMADs of the overall formula fts versus the rounng lookup s shown n Fgure 37. The nuctances an q match the orgnal profles qute well. Ther ervatves ffer a bt more but ther errors are stll qute small. 86

106 Fgure 37. Comparson of NMADs for rounng lookup (blue) an formula ft (re). The percentages were obtane by smply multplyng the NMAD values by 100%. Note the lower error for the formula ft to q. These results are for a 12 th -orer polynomal ft for the 5-k ratonal functon. Ths uses (40) to get the k values an (38) to get the nuctance values from the k values. A comparson of some of the results s shown n Fgure 38 an Fgure

107 (a) 88

108 (b) Fgure 38. Inuctance q by (a) rounng lookup an by (b) formula ft. 89

109 (a) 90

110 (b) Fgure 39. Inuctance ervatve q / q by (a) rounng lookup an by (b) formula ft. But to show the nherent smoothness from usng a formula rather than a lookup table, a comparson s shown n Fgure 40 of the absolute fferences relatve to nterpolaton lookup of rounng lookup an formula fttng. 91

111 (a) 92

112 (b) Fgure 40. Absolute fference relatve to nterpolaton lookup of (a) rounng lookup an (b) formula fttng for q. A consequence of usng a formula rather than a lookup table s the smoothness of the results, whch can be of concern n numercal stablty. To choose the 12 th -egree polynomal ft an 5-k ratonal functon, a sute of tests s run for varous egrees of polynomals for both the 5-k an the 7-k ratonal functons. In each case, the mean of the NMADs (Fgure 37) of all sx propertes together s calculate (the net NMAD). Fgure 41 shows the results. Clearly, the 13-coeffcent (12 th -egree polynomal), 5-k ft s a very reasonable choce. Past 22 coeffcents, the errors for both fts begn to skyrocket or become numercally unstable. 93

113 The 7-k ft oes not ever show better overall performance; an past 13 coeffcents, the 5-k ft oes not show much mprovement. The valty of weghtng the sx propertes equally n calculatng the net NMADs coul be argue, but t s ffcult to say whether the nuctance or ts ervatves have a greater effect on the behavor of an electrc motor snce ths epens on how the motor s run. Fgure 41. The mean over the sx propertes of the sx normalze mean absolute fferences for the 5-k ft (blue) an the 7-k ft (re). The smulaton tmes for these tests are also shown (Fgure 42). Usng a 7-k ft takes about as long as a 5-k ft wth two atonal polynomal coeffcents. 94

114 Fgure 42. Smulaton tme for 5-k (blue) an 7-k (re) tests. These results are for sngle test runs at each settng. Averagng over many test runs, these curves woul straghten. Comparson wth Other Methos The formula use n ths paper s smlar to a ratonal functon use n [33]; however, there the egree s lower an the coeffcents are not parameterze. Also, that formula s a functon of only one current an, therefore, cannot nclue cross-couplng effects. Some moelng, such as n [34], focuses purely on the rotor angle effects on nuctance. In [35], a moel s propose where the stator phase nuctances are moele as functons of only the corresponng phase current an a Fourer seres of the rotor angle. The paper oes suggest measurements beng taken to get nuctances as functons of current magntue, rotor angle, an torque angle; however, no moel for ths s presente. In [36], a smlar moel s evelope. It 95

115 uses a sngle cosne functon wth current polynomals for the coeffcent an bas. The polynomals are functons of only the corresponng phase current. In [37], a moel that focuses on moelng the nuctances n ther ABC-reference frame s shown. The ownses to ths approach are that control coe s often alreay n the DQ0 reference frame an that ths type of formula s base on Fourer seres compostons, whch are computatonally slow. These moels may be especally helpful where the rotor angle s of nterest, such as n rotor angle estmaton. However, the cross-couplng effects of the phase currents are of much greater relevance to full, PM motor smulaton moels, such as foun n [38]. The moel shown n (38) an (39) captures these cross-couplng effects. In [39], another formula s propose to moel the DQ flux lnkages, λ an λ q, as functons of the DQ currents, thereby nclung cross-couplng effects. For both λ an λ q, the formula has the form (43) λ A3q A4q C e C e A q A q (, q ) = ( C1e C2e ) e A7q q C e C e ( ) A 8 A q A q C5e C6e e, (43) where the C an A coeffcents are tunng parameters. Though the flux lnkages use for the research n our paper are unoubtely fferent than those use n [39], the characterstc shapes shoul be the same. That paper emonstrates fts over current magntues up to 2 amperes; an for that range, the fts to the nuctances presente here are nee very goo wth NMAD values less than 0.5%. However, at hgher current values, (43) oes not seem to be able to capture the saturaton of the flux lnkages as well as (38) oes when use to calculate flux lnkages. In Fgure 43 the reference λ s shown; an, n Fgure 44, a comparson of these two methos s shown. 96

116 n ths paper. No other publshe work has been foun to use a technque smlar to the one presente Fgure 43. For current magntues up to 55 A, the reference λ flux lnkage. 97

117 (a) 98

118 (b) Fgure 44. For current magntues up to 55 A, the rect flux lnkage λ (top) usng (43) an (bottom) usng (38). Smplfe Moel It s possble to sgnfcantly smplfy the moel presente n ths work, at a slght cost to accuracy. The same form of ratonal functon shown n (38) can be use but wth the current magntue replace by the D an Q currents. The formula for both an q becomes k = (44) 1 k k k 1 (44) (, q ) k 7 2 k4 3 5 k6 q where these k values are scalars. The k 3 value proves the current shft. So, to completely escrbe the nuctances for a PM machne, only 14 coeffcents woul be requre. The NMAD 99

119 values for an q usng ths formula are 2.3% an 2.8%, respectvely. Fgure 45 shows the nuctances from usng ths formula, an Fgure 46 shows the absolute fferences from the reference nuctances. (a) 100

120 Fgure 45. The (a) D an (b) Q nuctances usng the smplfe ratonal functon (44). (b) As before, as long as the coeffcents are postve, sngulartes wll never be observe. The k 3 value s exclue from ths restrcton. It shoul also be note that the power coeffcents k 4 an k 6 shoul be greater than 1. Ths wll ensure that the peaks of the nuctances are flat an o not show saturaton even at nfntesmal currents. Ths moel forces the nuctances to be symmetrcal about the peak, so there s not as much flexblty wth (44) as wth (38) an (39). However, epenng on the nee, the greatly reuce complexty an ncrease computatonal spee may be worthwhle. For embee systems, the real number exponents mght be hghly unesrable, but each real number power can 101

121 be replace wth multple whole number powers to acheve nearly the same effect at a reuce computatonal buren. (a) 102

122 (b) Fgure 46. Absolute fferences between reference an formula nuctances for (a) an (b) q. Expermental Applcaton an Comparson wth near Moelng In [38], the nuctances scusse n ths paper are use n the smulaton of a full EMA moel whch was evaluate usng physcal experments on an EMA uner actve loang. Ths moel was also use to compare lnear (constant-value) nuctances an nonlnear (formulabase) nuctances. Because of the smulate control algorthm use, the esre stroke s followe the same regarless of whch type of nuctances are use (Fgure 47). 103

123 Fgure 47. Desre stroke profle (blue) an smulate stroke profle (green). Note that the esre stroke profle was followe farly well by the control algorthm. The esre stroke profle was taken from a physcal experment wth an EMA uner actve loang. The power loss, P Cu, n the wnngs ue to the electrcal resstance of the copper s calculate for both cases, lnear an nonlnear (Fgure 48). Ths power s ntegrate to get the cumulatve energy, E Cu, sspate as heat n the wnngs (Fgure 49). An, ths heat manfeste tself as temperature rses n the machne Fgure

124 Fgure 48. Smulate power losses n copper wnngs usng lnear nuctances (blue) an nonlnear nuctances (green). 105

125 Fgure 49. Smulate energy losses n copper wnngs usng lnear nuctances (blue) an nonlnear nuctances (green). 106

126 Fgure 50. Temperature comparson for a half-hour run showng envronmental temperature (blue), expermentally measure temperature of stator (green), smulate temperature of stator (re), an the smulate temperature of the wnngs (cyan). The noal temperatures n the EMA s electrc motor are smulate usng the nonlnear nuctances. These temperatures, whch are functons of the energy losses, match very well. ookng at the fnal values of cumulatve energy losses, there s a 19% fference after only 6 secons. So, by usng lnear nuctances, the energy sspaton s sgnfcantly overestmate. Note that these cumulatve energy loss curves verge somewhat unprectably over tme. Though ther reasons for fferng are hen n the complextes of motor control an 107

127 other ynamcs, t s lkely that ther fference wll grow wth tme. Ths comparson hghlghts the ramatcally sgnfcant fference that usng nonlnear nuctances can make. Concluson Nonlnear, lumpe-element moelng of PM motors n EMAs has value n movng towars all-electrc arcraft. At the core of ths moelng are the nonlnear nuctances. Ths paper shows an approach to fttng the nonlnear nuctances wth a ratonal functon because t extrapolates well an captures the overall characterstcs of the nuctances. The varatons n the coeffcents of the ratonal functon are moele by polynomal functons because of ther flexblty an computatonal effcency. Depenng on the machne, the formula performs faster than even smple nearestneghbor table lookup an gves farly accurate results. An ths formula nclues the crosscouplng effects that many other methos o not. Fnally, usng nonlnear nuctances nstea of lnear nuctances can result n a notceable fference n energy sspaton n the motor. So, whle nonlnear nuctances are somewhat more complex to use, they can gve ramatcally fferent results. 108

128 CHAPTER FOUR: NONINEAR, TRANSIENT MODE OF SYNCHRONOUS GENERATOR Introucton Up to ths pont, the nonlnear nuctances of the PM motor have been carefully moele an the EMA as a whole has also been moele. The semnal concern s the successful ntegraton of the EMA nto the arcraft electrcal system an ts thermal performance. In ths chapter, a generator s moele by tself, whch s the other major component n the total system. The operaton of the generator s smlar n basc concept to the motor. However, some key features stngush t. Frst, the generator s not a permanent magnet machne. The fel flux s supple by a fel current. The prmary purpose of the fel current s smply to create a lnk for the mechancal power flowng nto the generator to create electrcal power on the armature wnngs. Secon, the generator s mult-stage: t s multple synchronous machne (SM) generators n seres. The generator presente here has two stages: a 6-pole excter an a 10-pole man. Its basc layout s shown n Fgure 51. Fgure 51. Overall layout of generator. The excter s a 6-pole SM wthout ampers an the man s a 10-pole SM wth ampers. Ths work etals the methos use n moelng the flux lnkages of the generator, the moelng of the rectfers, the metho takng the full ervatve of flux lnkages, an the results of the smulaton. Because spee of smulaton s so mportant, crcut moels such as Smulnk s 109

129 SmPower Systems were avoe n favor of explct mathematcal moels, whch o ten to run sgnfcantly faster. The symbols use throughout ths work are lste n Table 7. Table 7. Smulaton States an Parameters Symbol Name Unts t Tme s θ m Rotor angle ra ω m Rotor spee ra/s θ s Synchronous flux angle ra ω s Synchronous flux spee ra/s τ Torque N m u Voltage V Current A λ Flux lnkage Wb Inuctance H u DC Inverter DC bus voltage V net Rectfer net current A p e Electrcal power W p m Mechancal power W p Q Thermal power W p H Magnetc fel power W p C Electrc fel power W P Magnetc poles ND N sr Stator-to-rotor turns rato ND R Resstance Ω U fw Doe forwar voltage V [] D-axs quantty [] q Q-axs quantty [] f Fel quantty [] k D-axs amper quantty [] kq Q-axs amper quantty [] l eakage quantty [] m Magnetzng quantty [] N Quantty scale by N sr 110

130 DQ0 Transform Because there are several versons of the rect-quarature-zero (DQ0) transform [25] an the choce of transform can affect the form of some of the equatons scusse, t s pertnent to escrbe the partcular form of DQ0 transform use n ths work. The power-nvarant, rghthane, ABC-sequence DQ0-transform matrx has the form (45) 2π 2π cos( θ s ) cos θ s cos θ s π 2π ( ) sn θ s sn θ s sn θ s, (45) where θ s s the angle of the flux fel (the rotor angle scale by the pole pars, P/2). The powernvarant DQ0 transform oes not scale the vector magntue, so the rotor values o not nee to be scale to be properly reference to the DQ0 values. By contrast, usng the power-varant transform, the rotor values must be scale by a factor of 2 / 3 to correctly reference them to the stator se. The form shown n (45) was chosen, among other reasons, because t s smply an axal rotaton an ncurs no scalng of any kn. It s also convenent that ts nverse equals ts transpose. Usng the power-nvarant transform, the electromagnetcally generate torque of a synchronous machne s P (46) τ = ( λ λ ) 2 q q. (46) The sgn of ths torque s such that the prouct of postve torque an postve shaft spee s postve mechancal power nto the machne. If the power-varant DQ0 transform were use, then torque woul be 111

131 3 P (47) τ = ( λ λ ) 2 2 q q. (47) Clearly, the form of the DQ0 transform has an effect on the form of other equatons. Dynamcal Equatons To prove context an to clarfy the choce of sgns, the ynamcal equatons for a synchronous machne wth amper wnngs an no expose neutral are (48) t λ u Rs λqω s λq u q Rsq λω s λ fn = u fn R fn fn λ kn RkN kn λkqn RkqN kqn, (48) where the varables subscrpte wth N have been multple by N sr, the effectve stator-to-rotor turns rato. Although ths form s often use n the context of motors, t s consstent wth the choce of efnng all powers as beng ponte nto the system. The full ervaton for the ynamcal equatons can be foun n Appenx B: Synchronous Machne Dynamcs. Inuctance Moelng In [38], the esgn of a permanent magnet (PM) motor s etale an the use of nonlnear moelng of ts nuctances s mentone. In [40], ths nonlnear moelng metho s explane. The fttng s very goo wth normalze mean absolute fferences (NMADs [40]) below 0.5%. It uses a ratonal functon to ft the rect (D) an quarature (Q) nuctances over the D an Q currents: 2 states. The technque s somewhat complcate, yet ths moelng task s relatvely smple compare to that of moelng the magnetcs of a synchronous machne (SM) generator wth amper wnngs: at least 5 states. 112

132 Collectng the ata The man generator s moele n Maxwell 2D, a fnte element metho (FEM) software. The establshe relatonshp between flux lnkage an nuctance [41] n an SM s (49) λ λq λ = fn λkn λkqn q fn kn kqn, (49) (50) sl 0 = m m 0 m sl mq mq fln m 0 m 0 m kln m 0 m 0 m klqn 0 mq 0 0 mq (50) Note that the nuctance matrx s sparse. Yet, the nuctance matrx that the FEM software evelops s full. Ths s partly numercal error an perhaps partly a realty of cross-saturaton. However, t woul be mmensely smplfyng to force the matrx to be sparse an to o so wthout too much loss of ata. To accomplsh ths, the FEM nuctance values are neglecte n favor of the flux lnkages, whch are then use to bul the sparse nuctance matrx. The flux lnkages are functons of each of the fve currents (, q, f, k, an kq ) an of the rotor angle, θ m. At frst glance, t woul appear that all of these parameters shoul be vare urng FEM sweeps, whch woul take a very long tme. However, some smplfyng steps can be taken. The bass of (49) s that the effects of all the currents can be groupe nto just two axes: D an Q. (For the unbalance case, the zero axs can be treate separately.) Ths leas to the evelopment of the magnetzng currents: (51) m mq = = q fn kqn kn. (51) 113

133 The magnetzng nuctances ( m an mq ) are functons of just these magnetzng currents. At ths pont, the problem s reuce to fttng over just two currents an the rotor angle. In many cases, the effects of rotor angle can be neglecte. Rather than run many FEM smulatons varyng rotor angle n orer to get goo average values, rotor angles can be foun where the flux lnkage values pass through ther own averages. Ths means that for a gven set of current values, only one smulaton nees to be run. The varaton n flux lnkage s most event at hgh saturaton. Fgure 52 shows an example of flux lnkage varaton over rotor angle wth the ponts where t crosses ts own average marke. These ponts o not vary wth current snce they are etermne mostly by the machne s geometry. Therefore, by lockng the rotor at one of these ponts, the FEM smulaton sweeps can be mae wthout any concern for varatons n rotor angle. Ths can sgnfcantly reuce the number of smulatons requre to properly analyze a machne s magnetcs. 114

134 Fgure 52. D-axs flux lnkage values over rotor angle for the PM motor of [38]. The flux lnkages are normalze per unt length. The ponts where the flux lnkage crosses through ts average are marke wth crcles. In etermnng the magnetzng nuctances, t woul be superfluous to vary the amper currents n aton to varyng the rotor s fel current. So, the FEM smulatons are run, varyng only, q, an f. Fgure 53 shows a se vew of some of the rotor flux lnkages. Each layer correspons to a specfc value of f. Fnally, three sets of flux lnkage values are use to etermne the nuctances: λ, λ q, an λ f. 115

135 Fgure 53. Se vew of the rotor s fel flux lnkage n mwb wth three currents vare:, q, an f. Ths shape was characterstc of the rect an fel flux lnkages on both the excter an the man. Moelng wth polynomals One approach to fttng the flux lnkages of the generator nvolves polynomals. Although (51) ncates that the currents can be combne nto just two magnetzng currents, the effectve stator-to-rotor turns rato, N sr, s necessary but not obvous. Wthout knowng ths, a moel of the flux lnkages or nuctances woul nee to nclue all currents, wthout combnaton. So, the followng polynomal moel s use for each of the three prmary flux lnkages (λ, λ q, an λ f ): N M c m an bn λ Gm, n f q, (52) n= 1 m= 1 (52) (, q, f ) = 116

136 where n, m, N, an M are ntegers, G s a matrx of constants, an a, b, an c are arrays of nteger exponents. The complexty of the polynomal equaton s etermne prmarly by the sze of the G matrx. In the case of the generator moele n ths work, some flux lnkages requre a G matrx as large as 7 by 14. Whle these are farly large matrces, they o lea to very low flux lnkage fttng NMADs: all less than 0.6%. Fgure 54 shows smooth flux lnkage values calculate for the man s fel. Up to the current values shown the ft s very goo an well behave. However, movng beyon the lmts of the orgnal tables by just 20%, the formula quckly evates from reasonable (Fgure 55). Fgure 54. The fel flux lnkage for the man generator as calculate by formula. The ponts chosen nclue ponts n between those use to tune the formula. 117

137 Fgure 55. The fel flux lnkage for the man generator as calculate by formula wth all three currents extene 20% beyon the orgnal tables. These excursons cause the generator smulaton to become unstable. Therefore, t s necessary to make sure that the fttng s one for values far beyon what woul ever be use urng smulaton. Ths requres gettng much more FEM smulaton results, whch s a very tme-consumng process. Moelng wth ratonal functons In [40], a smple verson of the ratonal functon nuctance moel was evelope for m an mq an s repeate here slghtly altere: k =, (53) k3 1 k k 1 (53) (, q ) k k5 q 118

138 where the k values are scalars to be tune for the partcular nuctance ( m or mq ). The last coeffcent, k 6, s not use for m. In aton to these k values, the effectve stator-to-rotor turns rato N sr an the leakage nuctances sl an fln are the coeffcents tune n orer to ft the three flux lnkage ata sets: λ, λ q, an λ f. Altogether there are 14 coeffcents. Maxwell 2D wth RMxpert has a feature to automatcally calculate the amper leakage nuctances whch are scalars an are far less crtcal. The D-axs magnetzng flux lnkage of the man as a ratonal functon of currents s shown n Fgure 56. The erve D-axs magnetzng nuctance of the man s shown n Fgure 57. The ft s farly goo; an, as scusse n [40], the ratonal functon has the beneft of avong Runge s phenomenon [31]. The prmary savantage of the ratonal functon s that t saturates too quckly, an effect whch only becomes apparent n very eep saturaton. Fgure 58 repeats the flux lnkages of Fgure 56 but up to a much hgher current. In realty, flux lnkage monotoncally ncreases wth current, but the behavor of the ratonal functon approach as shown n Fgure 58 oes not follow that pattern. If gong nto eep saturaton s a lkely possblty urng smulaton, then ths evaton becomes more than just an error n flux lnkage. It can cause a postve feeback to saturaton an nstablty of the smulaton. 119

139 Fgure 56. The D-axs magnetzng flux lnkage ( m m ) of the man as a ratonal functon of D an Q magnetzng currents. 120

140 Fgure 57. The D-axs magnetzng nuctance, m, of the man as a ratonal functon of D an Q magnetzng currents. 121

141 Fgure 58. The D-axs magnetzng flux lnkage ( m m ) of the man as a ratonal functon of D an Q magnetzng currents. Moelng wth logstc functons A few papers were foun that use exponentals to moel the magnetcs of machnes. In [42], the followng equaton s use: (54) (, θ ) = a ( θ )[ 1 [ a ( θ ) ] a ( θ ), (54) λ 1 exp 2 3 where the subscrpte a varables are coeffcents of the mechancal rotor poston θ an s the (55) phase current magntue. The same ea shows up n [43]: ψ ph s f ( θ ) ( e ph ) = ψ 1, (55) where ψ ph s the phase flux lnkage, ψ s s a constant, θ s the phase angle, f s a trgonometrc functon, an ph s the phase current magntue. 122

142 The magnetcs moelng metho use n ths work epens on a logstc functon [44] as shown here: 2 1 e (56) f ( r) = 1 (56) r (57) (58) f ( r) = k (57) m r 1 k m k4mq r k =. (58) Here, f s the logstc functon, m s the magnetzng nuctance, an r s the magntue of scale currents. Altogether, to moel the three flux lnkages, only 10 coeffcents are neee. Fgure 59 shows the fference between the logstc functon of (56) an exponental functons lke (55). The exponental functon causes the nuctance to be n saturaton even at zero current, but the logstc functon allows the nuctance to reman farly constant about the orgn. An atonal beneft s that eep saturaton usng a logstc functon s always monotoncally ncreasng (Fgure 60). No other work has been foun to use ths metho. 123

143 Fgure 59. Generalze nuctances evelope usng an exponental functon (blue) an a logstc functon (green). 124

144 Fgure 60. The D-axs magnetzng flux lnkage ( m m ) of the man as a logstc functon of D an Q magnetzng currents. Moelng comparson Fgure 61 shows a comparson of the NMADs for fttng λ, λ q, an λ f usng the polynomal, ratonal, an logstc methos. The number of coeffcents requre for each metho are 347 for polynomal, 14 for ratonal, an 10 for logstc. The ratonal an logstc methos have nearly the same error of fttng; an whle the polynomal metho oes ft better than the other two, t requres far more coeffcents. For ths reason an for ts ablty to properly moel eep-saturaton effects, the logstc metho was chosen to moel the magnetcs n ths work. 125

145 Fgure 61. Comparson of the NMADs of for the polynomal, ratonal, an logstc methos for the man machne. Full Flux nkage Dervatve The relatonshp among flux lnkage, nuctance, an current s efne n (49) an (50). Takng the full ervatve an solvng for the current ervatve gves the explct soluton t t 1 (59) = M λ (59) (60) M =, (60) 126

146 where the tme-ervatves of flux lnkage are efne n (48). Ths M matrx s full an ts analytcal nverse s prohbtvely complcate. To smplfy matters, the frst term n M s often neglecte. So, whle the nuctance s allowe to be nonlnear, t s assume that ts ervatve over current s relatvely small. Yet, ths can be a gross over-smplfcaton. Usng the logstc metho for the D-axs magnetzng flux lnkage, the percent fference s shown n Fgure 62. Ths shows M / converte to a percentage. Fgure 63 shows a comparson of M an. The ervatve of nuctance has the effect of weakenng the nuctance, an ths effect s not neglgble. Fgure 62. Percent fference of M relatve to. 127

147 Fgure 63. D-axs magnetzng nuctance (blue) an M (green). Rather than usng the explct soluton to get the full ervatve, the ervatve of nuctance s numercally calculate urng smulaton. The bass of ths approxmate soluton s the use of the last tme-step s change n nuctance n place of ths tme-step s change n nuctance: (61) λ = λ = t = t t 1 t. (61) λ t t If the tme step s very small, then ths approxmaton s qute reasonable [45]. Ths means that only the sparse matrx nees to be nverte, whch s relatvely easy. 128

148 Full-brge Rectfer wth Freewheelng The full generator moel uses a full-wave rectfer. When usng a full-wave rectfer, t s crtcal to recognze that the freewheelng conton can an oes occur easly. In ths conton, both oes n a leg are conuctng an the DC current freely flows nepenent of the phase current. To properly moel the rectfer, the problem s smplfe by usng a pecewse moel of a oe where only the forwar voltage rop, U fw, an on resstance, R on, are consere. To bul a moel of the full-wave rectfer, each leg can be consere nvually (Fgure 64). Fgure 64. One leg of a full-wave rectfer showng voltage rops across both top (1) an bottom (2) oes. The phase current a s orente nto the rectfer an the phase voltage u a has the same reference as the DC voltage, u DC. For ths ervaton, t s gven that a an u DC are known an 1 an u a are not. In the non-freewheelng conton, only one of the two oes on the leg s conuctng: oe 1 or 2. If a s postve, then u a wll be slghtly hgher than u DC ue to the voltage rop of oe 1, an 1 wll equal a. If a s negatve, then the u a wll be slghtly less than 0 ue to the 129

149 voltage rop of oe 2, an 1 wll be 0. These non-freewheelng relatonshps can be summarze n (62) an (63). (62) uc U fw Rona : a 0 u a = (62) U fw Rona : a < 0 (63) a : a 0 1 = 0 : a < 0 (63) If, however, the rectfer s n freewheelng moe, t means that both oes n a leg are conuctng. If we efne the crtcal voltage of a oe as the voltage across the oe such that t only just barely conucts, then freewheelng occurs when the DC voltage u DC has roppe far enough below zero so as to excee the crtcal voltages of both oes. However, the phase current a can srupt ths conton by ncreasng the total voltage rop across one oe such that the hearoom for the voltage rop across the other oe s too small for t to conuct. Ths means there s a crtcal current crt that the magntue of a must not excee for the freewheelng conton to exst. Usng Fgure 64 as a reference, we begn wth the followng voltage relatonshps whch are true urng the freewheelng conton: u 1 = U fw (64) u = U R ( ). (64) 2 u u 1 2 fw R = u on 1 on c 1 a If the freewheelng conton just barely exsts, then ether 1 or ( 1 a ) equals zero. Ether stuaton wll lea to the same concluson. Supposng then that t s the current through oe 2 whch s zero, we can rewrte the voltage relatonshps of (64) takng nto account that a now equals crt as 130

150 (65) u = U u 1 2 = U u u 1 2 fw fw R = u on crt c, (65) where crt must equal 1. We can then solve for crt : (66) U fw R crt on crt u = U c R on fw 2U = u fw c. (66) Ths s the crtcal current, an the magntue of a must be less than or equal to t for the freewheelng conton to exst n that leg: (67) a u 2U c fw crt. (67) Ron (68) The current through oe 1, 1, can be foun by startng wth (64) an rearrangng: u u 1 U 2 fw 2R = u R on = = 2 on 1 ( u 2U ) ( ) crt c = u U c c R a on fw 2U R fw fw on ( ) 1 R on a a a = u c We can now solve for the phase voltage u a urng the freewheelng conton: (68) (69) u a u u u u = u a a a a c = u = u = u 1 = 2 u c c c 1 ( U fw Ron1 ) ( u 2U ) U U ( u R ) c fw fw R on a on 1 u c U fw c R on 1 2 R fw on a a. (69) Therefore, n freewheelng moe, the followng are true: (70) u ( u R ) a = 2 1 c on a (70) 131

151 (71) ( ) 1 1 = crt a (71) 2 So, gven the phase current a an the DC voltage u DC, the frst step to evaluatng the fullwave rectfer s to evaluate the crtcal current (67) an check f the magntue of a s less than or equal to t. If t s not then that leg s not n freewheelng moe an (62) an (63) apply. If the magntue of a s less than crt, then the leg s freewheelng, an (70) an (71) apply. Ths logc must be apple to each leg of the rectfer an the total current from the rectfer woul be the sum of the currents comng out of each top oe. In ths moel, a capactor exsts on the DC se of the rotatng rectfer n orer to par the current source man machne wth the current source DC se of the rectfer. Power Analyss In ths work, power has been efne such that t s postve when flowng nto the machne. Multplyng (48) by the respectve currents results n the followng power equatons: (72) u u u q q fn fn 0 = R 0 = R 2 s = R 2 s q = R = R 2 knkn 2 kqnkqn λqωs λ t λωsq λqq t t λ t λ t 2 fn fn λ fn kn kn kqn kqn fn. (72) The terms on the left represent electrcal power flowng nto the system. The resstve terms represent the power losses. The secon terms on the rght of the frst two equatons represent the mechancal power flowng out of the system. The last terms n each equaton represent the power store n the magnetc fel. So, the electrcal, mechancal, thermal, an magnetc powers flowng nto the system are respectvely 132

152 (73) p e = u uqq u fn fn (73) τ P / 2 P 2 (74) p = ( λ λ ) ω τ ω ω τω m q q τ s s (75) p ( R R R R R ) Q s s q s fn s = (75) fn kn kn kqn kqn λ λq λ (76) fn λ λ kn kqn p = H q fn kn kqn. (76) t t t t t The total power balance equaton s (77) 0 = p e pq pm ph. (77) Note that the above power equatons mght ffer from some other publcatons, but ths epens on the form of the DQ0 transform use an the orentaton of power vectors. Smulaton Results The generator s esgne wth a DC-DC nverter controllng a synchronous excter whch fees a full-wave rotatng rectfer connecte to a synchronous man whch s then rectfe by a full-wave rectfer, as seen n Fgure 51. The effcency of the excter s about 55%, suggestng t s unersze for the task. The effcency of the man s about 95%. The powers of the excter an man are shown n Fgure 65 an Fgure 66, respectvely. Note, that core losses an frcton have are not nclue here. The ABC an DC voltages on the rotatng rectfer are shown n Fgure 67. The control for the generator s a smple PI loop closng on the loa voltage wth a samplng frequency of 10 khz an a swtchng frequency of 20 khz. The reference s 270 V. The loa voltage s shown n Fgure 68. The smulaton s qute fast at less than 9 real-worl secons per smulate secon on a 3.3 GHz Intel Core 5. Ths s a tremenous avantage. The tunng of the PI loop s automate, (74) 133

153 runnng through one smulaton after another whle varyng the control coeffcents proceurally. The spee of the smulaton becomes very sgnfcant when tryng to run so many smulatons n a row. An, the spee of ths coe mae the process take only a few mnutes. Fgure 65. Powers nto the excter. 134

154 Fgure 66. Powers nto the man. 135

155 Fgure 67. Voltages on the rotatng rectfer. The DC voltage s slghtly less than the peaks of the ABC voltages because of the forwar voltage rop an on resstance of the oes. 136

156 Fgure 68. oa voltage controlle to 270 V. Concluson In ths work, much emphass s place on the magnetcs moelng. There are potental rsks of postve feeback, nstablty, poor saturaton moelng even at low currents, an the requrement for excessve coeffcents to moel the magnetcs. The logstc functon metho presente here solves all these problems. Usng nonlnear nuctances suggests that the nuctances can change over tme. A full ervatve of flux lnkage shoul nclue ervatves of nuctances. The work shows a practcal metho for ong so. Choosng to moel a generator usng explct mathematcal moels means that components such as full-wave rectfers nee careful conseraton. Especally wth rotatng rectfers, the freewheelng conton can occur. The moel presente here treats that conton. 137

157 Fnally, ths generator moel s able to quckly control the output voltage to 270 V whle fully loae at 100 kw. Tunng the control s a quck, automate process gven each smulaton runs at 9 s/s. 138

158 CHAPTER FIVE: CONCUSION The work escrbe n ths ssertaton etals the esgn of nonlnear, transent moels of a permanent magnet electromechancal actuator (EMA) an a synchronous generator. Chapter 2 covers all the etals of a full EMA smulaton moel, nclung electromagnetc ynamcs, thermal networks, mechancal ynamcs (such as frcton an nerta), an control. Chapter 3 covers the methoology of bulng ratonal functon fts of nuctances for the EMA. Gven the constant rotor flux lnkage, an nuctance moel s smple enough an makes goo sense. In Chapter 4, the evelopment of a synchronous generator wth fel wnngs ntrouces a new layer of complexty. The methoology evelope n Chapter 3 s extene to a synchronous generator, an a new metho usng logstc functons s evelope. The work of Chapters 2 an 3 was, to some egree, valate by expermentaton. Ths was the culmnaton of about three years of work wth the Ar Force Research aboratory. The magnetcs moelng work of Chapter 4 n purely analytcal n nature an s valate by fttng the gven tables of ata. Altogether, ths work represents the prmary components of an electrcally powere EMA system for an arcraft. These tools can be use to test new esgns of EMAs or generators, test the nteracton of a partcular EMA wth a partcular generator, analyze the power emans of the system as a whole, evaluate the heatng of a component urng a specfc msson profle, or test a new generator or motor control scheme. All of ths leas to supportng the journey towars a more electrc arcraft. 139

159 APPENDIX A: PERMANENT MAGNET MACHINE DYNAMICS 140

160 Voltage Equatons To begn the ervaton of the ynamcal equatons of a permanent magnet (PM) machne, we start wth the voltage equatons: t * (78) u = Rs λ λqωme (78) t * (79) uq = Rsq λq λ ωme, (79) where λ * an λ q are efne as * (80) λ = λpm (80) (81) λ q = qq (81) an λ PM s the flux lnkage from the permanent magnets, whch s constant. Dynamcal Equatons Rearrangng (78) an (79) n terms of ervatves of flux lnkage an reformattng nto matrces, we get * λ u Rs λqω me (82) = * t λq uq Rsq λ ωme (82) where (83) t = t t q q λ q q t t q q q q t q q t λ * q. (83) For smplfcaton, the followng substtutons are mae: (84) (85) = (84) q = (85) q 141

161 (86) (87) q q = q (86) q qq = q. (87) q q Ths turns (83) nto (88) λ q =. t λq q qq t q (88) Settng (88) equal to (82) an makng two more substtutons, we get (89) q G =, q qq t q Gq (89) where (90) G = u Rs λqωme (90) (91) Gq = uq Rsq λ * ωme. (91) Fnally, by takng the nverse of the nuctance matrx on the left, the ynamcal equatons wth respect to current can be foun to be (92) t qq q q = (92) G qq q G q (93) t q q q =. (93) G qq q G q 142

162 APPENDIX B: SYNCHRONOUS MACHINE DYNAMICS 143

163 Introucton As a frst step, the ynamcal equatons for synchronous machnes (SM) are evelope wth a smplfyng assumpton. Much of the analyss nvolves the nverson of the nuctance matrx. After ths analyss s complete, further analyss s one to remove the smplfyng assumpton an show the full ervaton. Throughout, the ervatons are ve nto sectons for the varous types of synchronous machnes: wth a nontrval zero axs an ampers (SMZk), wth only a nontrval zero axs (SMZ), wth only ampers (SMk), or wthout ether (SM). Dynamcal Equatons wthout Inuctances Dervatves Premse The followng ervatons assume that the nuctances, though nonlnear, have neglgble tme ervatves. So, n takng the ervatve of flux lnkage, we get the ervatve of current wth an nuctance factor. To solve for the tme ervatves of currents, the nverse of the nuctances are use as shown n (94). (94) λ = λ = t t 1 = λ t t The nuctance matrx nverse 1 s the man focus throughout these ervatons. (94) Sx Dynamcs (SMZk) Begnnng wth the governng voltage equatons of a synchronous machne wth amper wnngs an a nontrval zero axs, we have (95) through (100), where varables subscrpte wth N have been reference to the stator through the stator-to-rotor turns rato, N sr. t (95) u = Rs λ λqωs (95) 144

164 t (96) uq = Rsq λ q λωs (96) (97) u z = Rsz λz (97) t t (98) u fn = R fn fn λ fn (98) (99) 0 = R kn λkn (99) kn t t (100) 0 = RkqNkqN λkqn (100) These sx equatons can be rearrange n terms of tme ervatves of flux lnkages an then combne nto a sngle matrx equaton. The zero axs oes not nee to be nclue n the matrx equaton because t s not couple wth any other axs an woul, therefore, not beneft from the matrx format. So, (95) through (100) become (101) an (102). (101) t λ λq λ fn λkn λkqn = u u Rs λqωs R sq λωs u fn R fn fn RkNkN R kqnkqn q (101) t (102) λ z = u z Rsz (102) The flux lnkages can be efne n terms of stator-reference leakage (l) an mutual (m) nuctances as shown n (103) an (104). (103) λ sl m 0 m m 0 λq 0 sl mq 0 0 mq λ = fn m 0 fln m m 0 λkn m 0 m kln m 0 λkqn 0 mq 0 0 klqn mq q fn kn kqn (103) (104) λ z = sl z (104) 145

165 146 Ths relatonshp comes from the orthogonal nature of the D an Q axes, the symmetrcal nature of the materals, an the use of the stator-to-rotor turns rato, N sr. To smplfy the ervaton, some symbolc substtutons are mae an the nuctance matrx n (103) becomes (105) ( ) ( ) ( ) ( ) ( ) = = = = = = = klqn sl kln mq fln m f c e b a b f b a e a a a a a b b c a a a c : (105) The nverse of ths matrx s (106) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = f c b cf s e c e a ce r s c b s b r c a c r ac r a r ac r e c a ce r ae s b s f b r a r ae r e a e : (106) We can further smplfy ths matrx wth another set of substtutons: (107) ( ) ( ) ( ) ( ) ( ) = = = = = = = = = = s c b I r e c a ce F r a C r c a c H s b E r ae B r ac G s f b D r e a e A I E H G C G F B E D C B A : (107) So, the ynamcal equatons for a synchronous machne wth ampers can be wrtten as (108) = kqn kqn kn kn fn fn fn s q s q s q s kqn kn fn q R R R u R u R u t ω λ ω λ 1 (108)

166 t (109) = 1 ( u R ), (109) z where sl z s z 1 s as efne n (105) through (107). Wthout Zero Axs (SMk) If the zero axs s always null (the machne s balance, there s no access to the neutral lne, or smply no current ever flows on the zero axs), then there s no reason to carry that ynamc through the smulaton an the number of ynamcs can be reuce to fve. Because the zero axs s completely uncouple wth all other axes, we can smply remove (109). Therefore, the full set of ynamcal equatons for ths machne s (110) t q fn kn kqn = 1 u u Rs λqωs R sq λωs u fn R fn fn RkNkN R kqnkqn q, (110) where 1 s as efne n (105) through (107). Wthout Dampers (SMZ) If there are no amper bars but there s a zero axs, then the number of ynamcs can be reuce to four. Ths wll reuce the nuctance matrx n (105) to (111) a = ( c a) 0 a = ( ) b = 0 c b 0 : ( ) c = a 0 a = whch nverte becomes m mq sl fln (111) 147

167 148 (112) ( ) ( ) ( ) ( ) c a c p p a c p a b c p a p a = = : (112) Ths can be further smplfe by substtuton: (113) ( ) ( ) ( ) = = = = = p a c D p a B b c C p a A D B C B A 1 : (113) Note that the varables A, B, C, an D are not efne the same as n (107). The ynamc equatons are then (114) ( ) = = z s z sl z fn fn fn s q s q s q s fn q R u t R u R u R u t 1 1 ω λ ω λ. (114) Wthout Zero Axs or Dampers (SM) If there s no zero axs an there are no amper bars, then the ynamcs reuce to three: (115) = fn fn fn s q s q s q s fn q R u R u R u t ω λ ω λ 1, (115) where 1 s as efne n (111) through (113). Smulaton Spees There s value to usng the approprate moel. Savng on states oes sgnfcantly reuce the tme to process. A test was run wheren each of the above four moels was run for 10 s of smulaton tme, three tmes each. The results are shown n Fgure 69.

168 Fgure 69. Normalze mean smulaton tmes for four machne types. Dynamcal Equatons wth Inuctance Dervatves Premse of Exact Soluton If the assumpton of neglgble tme ervatves of nuctances were no longer mae, then the ervaton n (94) woul change to (116) λ = λ = t t t λ = t t t t = 1 t λ (116) Ths s the explct ervaton of the tme ervatve of current. (Note that not nclung the effects of tme-changng nuctance can cause sgnfcant error, epenng on the current magntue.) Ths ervaton has the same concluson as (94) except that the apparent nuctance matrx has been replace by the ncremental nuctance matrx: (117) nc =. (117) 149