SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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Examining the two windings with a squarewave motor would be more difficult because of the discontinuous function describing rotor flux but fortunately the analysis may be done intuitively, beginning again with the CFP winding. When the rotor is at zero degrees all the flux would be linked by all N turns, as shown in Figure C.14. Figure C.14 – CFP winding with ideal squarewave rotor. The peak rotor-stator flux linkage is thus found by multiplying the total flux by N. R N NB Area p D NBp Y 2 R NDY Bp (CFP winding, squarewave rotor flux) (C.18) 2 As the rotor rotates CCW the rotor-stator flux linkage changes linearly with negative slope over 0, and with positive slope over ,2 is shown as well but will not be discussed until later.) as shown in Figure C.15-a. (The position-derivative Figure C.15 – CFP winding with squarewave rotor: (a) rotor-stator flux linkage, and (b) its position-derivative. 296
That the rotor-stator flux linkage is triangular is certainly intuitive and in agreement with Figure C.14 but it seems a bit out of place, considering all of the other results (including the next case) have been squarewaves or sinusoids. An explanation will be given soon but first the sinusoidal winding with the squarewave rotor is investigated. The sinusoidal winding and squarewave rotor are shown in Figure C.16. Figure C.16 – Sinusoidal winding and squarewave rotor. There are several ways to proceed but they are all very similar to the previous derivations. Different authors take different approaches; in approximate order of formulae complexity: [69, p.6.8], [68, p.180], [26, p.462]. These proofs are not recreated here. The result is simple: the sinusoidal winding does not link any harmonics of rotor flux and the fundamental of rotor flux that it does link is only /4 of that linked by the CFP, as found earlier in the MMF section and after Equation (C.17) was derived. Since the fundamental of the squarewave is 4/ the two factors cancel and the rotor-stator flux linkage is that given by Equation (C.19). ( ) NDY B cos( ) (sine winding, squarewave rotor flux) (C.19) R r p r The rotor-stator flux linkages (or peak value) are summarized below for the four winding and rotor combinations studied. (C.14): ( ) NDY B cos( ) (CFP winding, sine rotor flux) R r p r (C.17): R( r) NDY Bpcos( r) (sine winding, sine rotor flux) 4 (C.18): R NDY Bp (CFP winding, squarewave rotor flux) 2 (C.19): ( ) NDY B cos( ) (sine winding, squarewave rotor flux) R r p r 297
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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
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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
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 and 294: First the self-flux-linkage of an i
Page 295 and 296: vAN R iA LM iA eA d v R 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: 0, and since the direction of inte
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
Page 345 and 346: xA X1cos( t) X3cos(3 t) X5cos(5 t)
Page 347 and 348: Figure D.11 - 3-D base vectors of 1
Page 349 and 350: the Clarke transform is used. Howev
Page 351 and 352: Appendix E - Park Transforms This a
Page 353 and 354: Appendix F - Useful Mathematical Re
Page 355: 335
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SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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