SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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common (such as stator-flux-oriented- or magnetizing-flux-oriented-). Each has various advantages and disadvantages but they are primarily used for machines other than the BPMS type. Useful references that discuss FOC of both the synchronous and induction motors include [78], [77], [87]. Field-Weakening and Salient Machines For the most efficient production of torque in a nonsalient machine the stator current SV should be completely aligned with the q axis and this is accomplished simply by controlling the dcomponent to be zero. Clearly, if the d-component is nonzero the result is a space vector in the rotor frame that is not aligned with the q axis. In motor control, using a nonzero d-component may be more appropriately called “field angle control” [78, p.203] but oftentimes it is still called FOC. Field angle control can be used to control terminal power factor; this is required in high power ratings when a load-commutated inverter is used [78, p.212] (as opposed to the hardswitched inverters considered here). Angle control can also be used to optimize some parameter (such as the minimization of copper loss, which is essentially the power factor control mentioned earlier). In BPMS motor drives there are two common reasons to use a non-zero d-component of stator current: to control torque in a salient machine or to implement something akin to field (flux) weakening. This report does not concern either topic but both will be mentioned because they highlight some of the more subtle concepts of FOC. These topics are interrelated and the author does not possess a working knowledge of them; they are presented here in an effort to aid the reader in understanding just the very basic ideas. At the end of this subsection some further technicalities are mentioned, along with some references for further study. Salient Machines In Chapter 3 the torque was found to be given by Equation (3.160). 3 (3.160): T diqqi d 2 (5.7) The regular expressions for the dq components of the stator flux linkage, Equation (5.8), were substituted in and the result simplified to Equation (5.9) as expected. d Lsid R q Lsiq (5.8) 3 T R iq (5.9) 2 220
To analyze a salient machine one must employ the “full” dq theory (the two-reaction theory used by Park and those before him). In addition to separating vectors into their dq components (as we have seen), the full dq theory assumes the effects of rotor saliency can be represented by different inductances in the d- and q- directions ( d L and q L ) instead of the synchronous inductance L s . In that case the simple SV analysis must be modified from what was shown in Chapter 3 to use these separate inductances. The only result of concern here is that Equation (5.8) must be modified to be Equation (5.10). L i q Lqiq d d d R (5.10) Substituting these flux linkages into Equation (3.160) yields the torque for a salient machine, Equation (5.11). 3 T d q q d 2 i i 3 Lii d d qRiqLii q q d 2 3 T RiqLdLqidi q 2 (5.11) In Equation (5.11) the first term describes the mutual torque produced via interaction of the stator field with the magnets and the second term describes the reluctance torque produced as the salient rotor attempts to align with the stator field in order to minimize the reluctance of the magnetic circuit. The definitions of the inductances are not given here, but in a nonsalient machine they are equal to one another and to the synchronous inductance: ( L L L ) . Thus, for a nonsalient d q s machine, Equation (5.11) simplifies to Equation (5.9) as expected. To investigate the meaning of the d- and q-axis inductances, examine the three types of rotors shown in Figure 5.15. 221
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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
Page 189 and 190: Figure 4.11 - Ideal voltage wavefor
Page 191 and 192: inverter is called sine-triangle co
Page 193 and 194: and it offers the fastest dynamic r
Page 195 and 196: Figure 4.18 - Fundamental gain and
Page 197 and 198: In the literature this is called th
Page 199 and 200: voltage will be below (above) the b
Page 201 and 202: Figure 4.24 - Instantaneous phase-A
Page 203 and 204: 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: Figure 5.13 - Torque control of sin
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
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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
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335
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