SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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Appendix A - Elementary Electromagnetics ............................................................................... 269 Appendix B - Phase-Variable BPMS Motor Model .................................................................... 272 Appendix C - Sinusoidal and Nonsinusoidal Motors................................................................... 280 Fundamental Relationships .....................................................................................................281 Windings .............................................................................................................................281 Concentrated Full-Pitch (CFP) .......................................................................................282 Ideal Sinusoidal...............................................................................................................282 Sinusoidal Approximations.............................................................................................284 Stator MMF.........................................................................................................................287 CFP .................................................................................................................................288 Distributed ......................................................................................................................289 Sinusoidal........................................................................................................................291 Rotor-Stator Flux Linkage ..................................................................................................292 Torque and Back-EMF............................................................................................................304 Conclusions .............................................................................................................................306 Appendix D - Three-Phase Concepts & Transformations ........................................................... 308 Harmonic Analysis ..................................................................................................................308 Implications of the ZS Component..........................................................................................313 Passive Impedance Load .....................................................................................................313 Motor Load..........................................................................................................................315 Summary .............................................................................................................................318 Method of Symmetrical Components......................................................................................320 Clarke & αβ0 Transforms........................................................................................................322 The Zero Sequence Component ..........................................................................................324 Inverse Clarke Transform....................................................................................................329 Phase Interference Matrix ...................................................................................................329 Appendix E - Park Transforms .................................................................................................... 331 Appendix F - Useful Mathematical Results................................................................................. 333 vi
List of Figures Figure 2.1 – Radial- and axial- flux motors...................................................................................11 Figure 2.2 – Outer- and inner- rotor radial flux motors. ................................................................12 Figure 2.3 – General categorization of machines. .........................................................................13 Figure 2.4 – Motors categorized by presence or absence of brushes............................................. 14 Figure 2.5 – Motors categorized by power supply type.................................................................15 Figure 2.6 – General motor taxonomy used in this report. ............................................................16 Figure 2.7 – Cross sections of some generalized synchronous motors..........................................19 Figure 2.8 – Demonstration of Faraday & Lentz laws; generator.................................................. 23 Figure 2.9 – Electrical model of ideal coil shunted by an external resistance. ..............................24 Figure 2.10 – Demonstration of the Lorentz Force law; motor. .................................................... 26 Figure 2.11 – Electrical model of coil connected to voltage source. .............................................26 Figure 2.12 – Section view of a general brushless PM motor. ......................................................28 Figure 2.13 – Developed view showing only ideal rotor flux density...........................................29 Figure 2.14 – Magnetic circuit of Figure 2.12. ..............................................................................30 Figure 2.15 – Simplified magnetic circuit of Figure 2.12..............................................................30 Figure 2.16 – Components of total flux (cf. Figure 2.15).............................................................. 31 Figure 2.17 – Per-phase electrical model of general brushless PM motor; explicit inductor. .......34 Figure 2.18 – Per-phase electrical model of general brushless PM motor; explicit inductor. .......34 Figure 2.19 – Per-phase electrical model of general brushless PM motor; implicit inductor........34 Figure 2.20 – Elementary brushless PM motors with concentrated full-pitch winding.................37 Figure 2.21 – Airgap flux density profiles for two BPMS motors.................................................37 Figure 2.22 – Torque functions of sinusoidal and arbitrary trapezoidal motors............................39 Figure 2.23 – Simulation diagram for per-phase torque function..................................................39 Figure 2.24 – Back EMF functions of a sinusoidal and arbitrary trapezoidal motor.....................40 Figure 2.25 – Simulation diagram for per-phase bEMF function..................................................41 Figure 2.26 – Rotor-stator flux linkage for a sinusoidal and arbitrary trapezoidal motor. ............44 Figure 2.27 – Simulation diagram of general motor......................................................................48 Figure 2.28 – Phase-variable simulation diagram of general motor. .............................................49 Figure 2.29 – Back-EMF and drive current for sinusoidal BPMS motor. .....................................53 Figure 2.30 – Torque production in sinusoidal BPMS motor........................................................54 Figure 2.31 – Back-EMF and drive current for trapezoidal BPMS motor.....................................55 Figure 2.32 – Torque production in trapezoidal BPMS motor. .....................................................56 vii
Page 1 and 2: SENSORLESS FIELD ORIENTED CONTROL O
Page 3 and 4: Table of Contents List of Figures..
Page 5: CHAPTER 5 - Field Oriented Control
Page 9 and 10: Figure 3.29 - Arbitrary space vecto
Page 11 and 12: Figure 4.44 - ZS components of some
Page 13 and 14: Figure C.13 - Single-turn full-pitc
Page 15 and 16: List of Symbols Symbol Meaning Unit
Page 17 and 18: Abbreviations ACIM AC induction mot
Page 19 and 20: Superscripts * commanded value ref
Page 21 and 22: CHAPTER 1 - Introduction Historical
Page 23 and 24: vector theory to model a machine, w
Page 25 and 26: publications and magazines, and Int
Page 27 and 28: elationship between SVM and other P
Page 29 and 30: CHAPTER 2 - Fundamentals of Electri
Page 31 and 32: Preliminaries In reading the litera
Page 33 and 34: Taxonomy of Motors There is any num
Page 35 and 36: characteristics (the DC load curren
Page 37 and 38: torque component, sturdily construc
Page 39 and 40: Figure 2.7 - Cross sections of some
Page 41 and 42: draw the line between the solenoida
Page 43 and 44: These texts generally attribute the
Page 45 and 46: N B A B Y x(t) (2.6) The induc
Page 47 and 48: extended to the rotational case lat
Page 49 and 50: present analysis). When this is tru
Page 51 and 52: Figure 2.16 - Components of total f
Page 53 and 54: As aforementioned a spatial analysi
Page 55 and 56: clear the meaning, but the reader i
Page 57 and 58:
Figure 2.20 - Elementary brushless
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called the per-phase torque functio
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Expressing bEMF in terms of the bEM
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Equation (2.44) is the component of
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d R( r) ke( r) kt( r) sin( r) (2.
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General Electromechanical Models Th
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Figure 2.28 - Phase-variable simula
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Trapezoidal and Sinusoidal BPMS Mot
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Figure 2.29 - Back-EMF and drive cu
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Figure 2.31 - Back-EMF and drive cu
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3 KT Ke 2 (sinusoidal) K 2 K (tra
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control scheme must keep sinusoidal
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with misconceptions. The primary is
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Part I - Sinusoidal BPMS Motors wit
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grounded-neutral wye or delta conne
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N e f A ( ) i 2 N e f B ( ) i 2
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Figure 3.7 - Developed view at zero
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Figure 3.9 - Developed view showing
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Although Equations (3.6) or (3.7) a
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direction. The contributions always
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sinusoidal electrical variable will
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Figure 3.14 - Cross section of two
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3 N e (3.6): f ( , t) I p cos(
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In the previous chapter the per-pha
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peak value is represented as an upp
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Figure 3.21 - Time relationship bet
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another and thus could be placed in
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more widely understood theory (such
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other methods, although it has clos
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The SV as a Vector The first facet
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j0 j0 j0 aˆ 1 0 1 bˆ j e e e
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j j ee 1 cos( ) (3.45) 2 3 j t x
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Notice that the amplitude of the MM
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Figure 3.27 - Equivalent MMF produc
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Ne 1 jj f , t i e i * e 2 2
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distribution that is cosinusoidal i
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3 Ne j t f Ie p 2 2 . This
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j set θ to anything. Taking the re
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epresent any physically-distributed
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lumped-parameter model in this repo
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x kCx (3.75) abc The set of varia
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phase variable axes; three axes in
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The component MMFs act away from th
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common choice is to force the power
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As an example, find the SV that cor
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the same magnitude, the projection
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Figure 3.32 - Arbitrary SV referenc
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universal and the choice of convent
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stator’s perspective and at R fr
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The potential discrepancy (p.108) w
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x cos( r) sin( r) xd x sin(
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It must be emphasized that we have
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in relative time like an oscillogra
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Part III - SV Theory Applied to Sin
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In Equation (3.135) Z has elements
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to use SV theory. It may be helpful
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Figure 3.42 - Coupling between d- a
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R R d R R v Ri Li j Li dt s
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current does not influence the valu
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d v Ri L i e dt d v Ri L i e
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Figure 3.47 - Simulation diagram fo
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e je ), and (2) it was demonstrate
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Overview of Voltage Source Inverter
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In addition to measuring voltages w
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then there would be periods during
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Figure 4.9 - Gating and POLE voltag
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Figure 4.11 - Ideal voltage wavefor
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inverter is called sine-triangle co
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and it offers the fastest dynamic r
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Figure 4.18 - Fundamental gain and
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In the literature this is called th
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voltage will be below (above) the b
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Figure 4.24 - Instantaneous phase-A
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Figure 4.26 - Base SVs showing the
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Figure 4.27 - Transformed voltage w
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also correspond to six-step squarew
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was fed to the SVM inverter as a SV
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Figure 4.31 - Temporal limit of inv
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Figure 4.34 - SVM overmodulation re
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Checking sextant s 23 again for thi
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SVM Implementation The block diagra
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has been made of using THI in the S
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different triplen signal would be i
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Similarly, when the modulation inde
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Summary and Conclusion The two-leve
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CHAPTER 5 - Field Oriented Control
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torque production; this is called d
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Figure 5.3 - Torque control of sinu
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obviously a change in perspective.
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Figure 5.9 - Comparison of referenc
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variables to the stationary referen
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Figure 5.13 - Torque control of sin
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To analyze a salient machine one mu
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Figure 5.16 - Torque functions: (a)
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Figure 5.19 - Buried permanent magn
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Figure 5.21 - Flux weakening in the
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appears that this is very much rela
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Stationary and Synchronous Regulato
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stationary regulator had. For compl
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This coupling can be mitigated by m
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educes to two independent circuits,
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D, some of the 120° methods are ap
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d d v Ri Ls i R dt dt The r
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section a state-space model for the
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Figure 6.6 - Full state feedback (o
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State-Space Model of BPMS Motor A s
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Figure 6.9 - FOC block diagram. The
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loop). A second way to implement th
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knowing the rotor position from ini
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[11] E. Clarke, Circuit analysis of
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Control Systems, Linear Analysis [4
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[77] J.M.D. Murphy, F.G. Turnbull,
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Machine Modeling, Analysis [103] J.
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THI, SVM, SPWM [127] J.A. Houldswor
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Wisconsin-Madison. [Online]. Availa
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Angle Tracking [175] D. Morgan, “
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Appendix A - Elementary Electromagn
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L i (A.8) S SS S SR S SS SR (A.9
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First the self-flux-linkage of an i
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vAN R iA LM iA eA d v R i
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Figure B.3 - One-half of the flux p
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Equations (B.27) and (B.28) are for
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Fundamental Relationships Figure C.
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Figure C.3 - Sinusoidal winding den
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manufactured compared to the steppe
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phase winding of Figure C.4 will ha
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Distributed N Fp i 2 (C.5) Figu
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alanced machine only odd harmonics
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sinusoidal windings with a sinusoid
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0, and since the direction of inte
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That the rotor-stator flux linkage
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Figure C.17 - Summary of rotor-stat
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Figure C.19 - Amplitudes of harmoni
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triangle with an amplitude whose nu
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Returning to Figure C.18 it is clea
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torque will describe the torque pro
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However, Equation (D.2) is incorrec
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the same order as the PS set (arran
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Implications of the ZS Component Th
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Figure D.6 - Neutral voltage of loa
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3v 3e 3v v v v v v v 3v AM BM CM A
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components of source voltage sum to
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Equation (D.25) and instantaneous q
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If a ZS component could possibly be
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xA X1cos( t) X3cos(3 t) X5cos(5 t)
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Figure D.11 - 3-D base vectors of 1
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the Clarke transform is used. Howev
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Appendix E - Park Transforms This a
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Appendix F - Useful Mathematical Re
Page 355:
335
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