SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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indeed defined such that common-mode components are simply not contained in the PS & NS or α & β components; there is obviously good reason for defining the transforms this way. There is a very meaningful graphical understanding of the ZS component that the interested reader can pursue [106], [105], [62, p.83], [70, p.25] but for brevity it is not presented here. However, a simplified and insightful result is shown: if the 0 axis is taken to be orthogonal to the α and β axes, we can draw a three-dimensional vector (although this does not fit our technical definition of an SV since it only has two components). From the inverter chapter, the eight base SVs of a 180° inverter were projected onto the αβ plane as shown in Figure D.10. The magnitude of each 4 SV is k2 VDC, which for k 2/3 is equal to 3 DC S V . Figure D.10 – Projection of base SVs of 180° inverter onto AB plane. If the thee-dimensional vectors were plotted they would be as shown in Figure D.11 ( k 2/3), where the cross section is that shown in Figure D.10. This shows a graphical interpretation of the ZS component of the output voltage. (Be aware that in Figure D.11, only 0 S , 1 S , 4 S , and 7 S are in the αγ plane; S2 and S3 are behind the plane in the β direction. The cross-section S but these are omitted for clarity.) should show a truncated portion of 5 S and 6 326
Figure D.11 – 3-D base vectors of 180° inverter. In inverter state S0 all legs are low and the neutral is at VDC ; in S7 all legs are high and the neutral is at VDC . In S1, A is high, B and C are low, and the neutral voltage is therefore (1 / 3) VDC . Since the pole voltage of phase-A is 1.0V DC , the line-neutral voltage is 1 ( 1/3) V (4/3) V . Figure D.10 had already shown that fact but as discussed in the DC DC inverter chapter, the SV transform of either the pole or phase voltage gives the same result (namely, Figure D.10). Figure D.11 shows graphically what was explained in the inverter chapter—that the SV (Figure D.10) could not capture the ZS component (which is shown in Figure D.11). It is known that in the active states the neutral voltage alternates between (1 / 3) VDC . This means the 3-D vector would alternately be (1 / 3) V DC below and above the αβ plane; this is evident in Figure D.11. Visualizing the ZS component in this way proves that the projection of the ABC phase variables onto the αβ plane is unique only when no ZS component is present. Likewise it clearly shows that unless the αβ0 transform is used, the ZS component will be lost when transforming back. Finally, a brief warning will be issued. It has been shown that the αβ0 transform is simply the Clarke transform (Equation D.26), augmented with terms for the ZS component. If we are working with a system in which no ZS component can possibly be present (such as current in an 327
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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
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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
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
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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
Page 345: xA X1cos( t) X3cos(3 t) X5cos(5 t)
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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