SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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In a symmetrical machine the self inductance of each phase is equal and the mutual inductance between any two phases is equal, LAA LBB LCC L LAB LBA LAC LCA LBC LCB M thus the set of flux linkages can be written as Equation (B.8). A LiA MiB MiC AR B MiA LiB MiC BR Mi Mi Li C A B C CR (B.8) When the phase resistance is considered the voltage equation can be written for each phase as Equation (B.9). d vAN RiA A dt d vBN RiB B (B.9) dt d vCN RiC C dt Substituting in Equation (B.8) yields the full voltage equations, Equation (B.10). Using the bEMF and matrix form this is given as Equation (B.11). vAN RiA Li A Mi B Mi C AR vBN RiBMi A Li B Mi C BR v Ri Mi Mi Li CN C A B C CR vAN R iA L M M iA eA d v R i M L M i e BN B B B dt v CN R i C M M L i C e C (B.10) (B.11) This report considers only isolated neutral motors thus the currents must sum to zero. The linear dependence of the currents is expressed by 0 the expression; this can be arranged in the three different ways shown in Equation (B.12). i i i i i i i i i B C A A C B A B C (B.12) Writing out Equation (B.11) by hand, substituting Equation (B.12) and simplifying gives Equation (B.13). The result is that although the mutual inductances still exists, the isolation of the neutral decouples the circuits. 274
vAN R iA LM iA eA d v R i LM i e BN B B B dt vCN R i C LM i C e C (B.13) However, the voltages are true line-neutral voltages (by definition). If a grounded neutral were used these voltages would simply be the terminal voltages ( vAM, vBM, v CM) because the neutral would be at zero potential. The same is true when common-mode/zero-sequence component is not present in the applied voltages (see Appendix D). However, if ZS components are present the terminal voltages must be used instead of the line-neutral voltages by using the “phase interference matrix” (Appendix D). Now this result will be specialized for the sinusoidally distributed winding which this report assumes. From this point on only the matrix of inductances will be manipulated to save space. Including the magnetizing inductance (Equation B.6) yields Equation (B.14). ( Lmag L ) M ( Lmag L) M ( Lmag L ) M (B.14) If the windings are sinusoidally distributed the mutual inductance between is given by Equation (B.15) [28, pp.251,654], [69, p.6.8]. M L cos( ) (B.15) mag For a three-phase machine 120 thus the mutual inductance is given by Equation (B.16). M Lmag cos( 120 ) 1 (B.16) M Lmag 2 It is worth noting that using 0 in Equation (B.15) gives the airgap portion of the self inductance of a winding. This emphasizes that the mutual inductance is a result of magnetizing flux. 44 Substituting the result of Equation (B.16) into Equation (B.14) yields Equation (B.17). 44 There is also a slot leakage component of mutual inductance but it is usually negligible [28, p.251]. 275
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SENSORLESS FIELD ORIENTED CONTROL O
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Table of Contents List of Figures..
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CHAPTER 5 - Field Oriented Control
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List of Figures Figure 2.1 - Radial
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Figure 3.29 - Arbitrary space vecto
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Figure 4.44 - ZS components of some
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Figure C.13 - Single-turn full-pitc
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List of Symbols Symbol Meaning Unit
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Abbreviations ACIM AC induction mot
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Superscripts * commanded value ref
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CHAPTER 1 - Introduction Historical
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vector theory to model a machine, w
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publications and magazines, and Int
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elationship between SVM and other P
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CHAPTER 2 - Fundamentals of Electri
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Preliminaries In reading the litera
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Taxonomy of Motors There is any num
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characteristics (the DC load curren
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torque component, sturdily construc
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Figure 2.7 - Cross sections of some
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draw the line between the solenoida
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These texts generally attribute the
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N B A B Y x(t) (2.6) The induc
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extended to the rotational case lat
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present analysis). When this is tru
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Figure 2.16 - Components of total f
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As aforementioned a spatial analysi
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clear the meaning, but the reader i
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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
Page 243 and 244: Figure 5.16 - Torque functions: (a)
Page 245 and 246: Figure 5.19 - Buried permanent magn
Page 247 and 248: Figure 5.21 - Flux weakening in the
Page 249 and 250: appears that this is very much rela
Page 251 and 252: Stationary and Synchronous Regulato
Page 253 and 254: stationary regulator had. For compl
Page 255 and 256: This coupling can be mitigated by m
Page 257 and 258: educes to two independent circuits,
Page 259 and 260: D, some of the 120° methods are ap
Page 261 and 262: d d v Ri Ls i R dt dt The r
Page 263 and 264: section a state-space model for the
Page 265 and 266: Figure 6.6 - Full state feedback (o
Page 267 and 268: State-Space Model of BPMS Motor A s
Page 269 and 270: Figure 6.9 - FOC block diagram. The
Page 271 and 272: loop). A second way to implement th
Page 273 and 274: knowing the rotor position from ini
Page 275 and 276: [11] E. Clarke, Circuit analysis of
Page 277 and 278: Control Systems, Linear Analysis [4
Page 279 and 280: [77] J.M.D. Murphy, F.G. Turnbull,
Page 281 and 282: Machine Modeling, Analysis [103] J.
Page 283 and 284: THI, SVM, SPWM [127] J.A. Houldswor
Page 285 and 286: Wisconsin-Madison. [Online]. Availa
Page 287 and 288: Angle Tracking [175] D. Morgan, “
Page 289 and 290: Appendix A - Elementary Electromagn
Page 291 and 292: L i (A.8) S SS S SR S SS SR (A.9
Page 293: First the self-flux-linkage of an i
Page 297 and 298: Figure B.3 - One-half of the flux p
Page 299 and 300: Equations (B.27) and (B.28) are for
Page 301 and 302: Fundamental Relationships Figure C.
Page 303 and 304: Figure C.3 - Sinusoidal winding den
Page 305 and 306: manufactured compared to the steppe
Page 307 and 308: phase winding of Figure C.4 will ha
Page 309 and 310: Distributed N Fp i 2 (C.5) Figu
Page 311 and 312: alanced machine only odd harmonics
Page 313 and 314: sinusoidal windings with a sinusoid
Page 315 and 316: 0, and since the direction of inte
Page 317 and 318: That the rotor-stator flux linkage
Page 319 and 320: Figure C.17 - Summary of rotor-stat
Page 321 and 322: Figure C.19 - Amplitudes of harmoni
Page 323 and 324: triangle with an amplitude whose nu
Page 325 and 326: Returning to Figure C.18 it is clea
Page 327 and 328: torque will describe the torque pro
Page 329 and 330: However, Equation (D.2) is incorrec
Page 331 and 332: the same order as the PS set (arran
Page 333 and 334: Implications of the ZS Component Th
Page 335 and 336: Figure D.6 - Neutral voltage of loa
Page 337 and 338: 3v 3e 3v v v v v v v 3v AM BM CM A
Page 339 and 340: components of source voltage sum to
Page 341 and 342: Equation (D.25) and instantaneous q
Page 343 and 344: 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
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335
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