SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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Figure 5.33 – Top-level model of rotor-oriented FOC (with decoupling). The decoupling scheme is often presented without much discussion; the author’s present understanding follows. Since the coupling exists because of the flux linkage, it is present in an R- L load as well (the bEMF offset term would not be present). Both the cross-coupling and the bEMF disturbance are multiplied by ω, which varies slowly compared to the possible rate of change of the currents (and in steady-state operation i d and i q are relatively constant). As far the as the steady-state cross-coupling of i d and i q is concerned (and as far as ω contributes to the bEMF offset), the disturbances are nearly steady-state quantities. It therefore appears that the decoupling serves to reduce the magnitude of the error (thus “unloading” the integrators) in order to improve transient response. On the other hand, transients in i d and i q feed straight through and are not steady-state disturbances, thus it appears that the decoupling also improves the transient response. At least one reference [118] seems to confirm this understanding. It is curious that most diagrams of FOC controllers do not include these decoupling terms. It is certain that by injecting the bEMF offset term its disturbance on the current loop is eliminated [158]. Since ω varies slowly compared to the bandwidth of the current loop the regulators can handle the error, but as mentioned, [118] indicates that eliminating this offset does improve response. The cross-coupling in the model is clear as well. However, if the synchronous inductance is low the cross-coupling may not be significant enough to warrant the use of decoupling. At any rate, when the decoupling is not performed and the coupling in the motor model is ignored, or when the decoupling is performed and assumed to be perfect, the BPMS model 236
educes to two independent circuits, as shown in Figure 5.34. (Again, if the bEMF is offset as in Figure 5.32, it would not be present in the q-circuit.) Figure 5.34 – Compensated or approximate model of BPMS motor in synchronous frame. Since control is executed in the synchronous frame and we have a model of the motor in the same frame, this model could be used to tune the PI controllers. As stated earlier, the decoupling is usually presented without much discussion. However, it should be helpful to examine the decoupling in terms of state space design (both feedback and feedforward). This research has stopped short of a complete understanding. The following references concerning synchronous regulation, tuning, and current regulator design should be helpful starting points for building on the fundamentals presented here: [82], [155], [156], [157], [158], [159], [160], [161], [162]. In particular, examine [125]. 237
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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
Page 205 and 206: Figure 4.27 - Transformed voltage w
Page 207 and 208: also correspond to six-step squarew
Page 209 and 210: was fed to the SVM inverter as a SV
Page 211 and 212: Figure 4.31 - Temporal limit of inv
Page 213 and 214: Figure 4.34 - SVM overmodulation re
Page 215 and 216: Checking sextant s 23 again for thi
Page 217 and 218: SVM Implementation The block diagra
Page 219 and 220: has been made of using THI in the S
Page 221 and 222: different triplen signal would be i
Page 223 and 224: Similarly, when the modulation inde
Page 225 and 226: Summary and Conclusion The two-leve
Page 227 and 228: CHAPTER 5 - Field Oriented Control
Page 229 and 230: torque production; this is called d
Page 231 and 232: Figure 5.3 - Torque control of sinu
Page 233 and 234: obviously a change in perspective.
Page 235 and 236: Figure 5.9 - Comparison of referenc
Page 237 and 238: variables to the stationary referen
Page 239 and 240: Figure 5.13 - Torque control of sin
Page 241 and 242: 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: This coupling can be mitigated by m
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 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
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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
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335
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