SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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power invariant form, k c. In this report the αβ0 transform is primarily used when studying inverters so the magnitude-invariant variety is used. 1 1/2 1/2 C 0 0 3/2 3/2 ; k c1 (D.31) 1/2 1/2 1/2 2/3 1/3 1/3 k c2/3 C 0m 0 1/ 3 1/ 3 ; (D.32) (magnitude-invariant) 1/3 1/3 1/3 2/3 1/ 6 1/ 6 k 2/3 C 0 p 0 1/ 2 1/ 2 ; c 2/ 3 (D.33) 1/ 6 1/ 6 1/ 6 (power-invariant) The α axis has been defined to be coincident with the real axis and the β with the imaginary axis. However, for some reason a sizeable portion of the literature aligns the β axis in the negative direction of the imaginary axis. The order of the variables in the vector are also sometimes rearranged (for example, the 0αβ transform uses the order 0,α,β). Both of these changes affect the way the SV and Clarke transform are defined and this change obviously propagates to other definitions and expressions, thus care is required in interpreting the literature. Finally, it should be noted that that α and β axes may be labeled D,Q or d,q. The Zero Sequence Component The chapter on inverters shows that the magnitude-invariant form of the Clarke transform has direct correlation to the physical values (such as voltage). The interpretation is the same for the ZS component of the magnitude-invariant αβ0 transform. Clearly, if Equation (D.32) were used on a set of voltages, v 0 would be the same term defined by Equation (D.16). If the voltages were across an isolated-neutral wye load, the v 0 term would be the neutral voltage, in agreement with Equation (D.15). Thus the 0-component is the same as the general ZS component encountered throughout this appendix. As mentioned throughout the report, the SV (or αβ components) simply cannot encode the ZS component; this is now shown. Consider the example of phase variables (Equation D.34) that contain the first PS, ZS, and NS harmonics (which were defined by Equations D.4–D.6). Writing the phase variables in complex form (Equation D.35) and substituting them into the SV definition (Equation D.36, where k 2/3 to achieve magnitude invariance) yields Equation (D.37) which simplifies to Equation (D.38). 324
xA X1cos( t) X3cos(3 t) X5cos(5 t) xB X1cos( t) X3cos(3 t) X5cos(5 t) xC X1cos( t) X3cos(3 t) X5cos(5 t) 1 jt1 jt1 j3t1 j3t Xe 1 Xe 1 Xe 3 Xe 3 2 2 2 2 x A 1 j5t 1 j5t Xe 5 Xe 5 2 2 1 j( t) 1 j( t) 1 j(3 t) 1 j(3 t) Xe 1 Xe 1 Xe 3 Xe 3 x 2 2 2 2 B 1 j(5 t) 1 j(5 t) Xe 5 Xe 5 2 2 1 j( t) 1 j( t) 1 j(3 t) 1 j(3 t) Xe 1 Xe 1 Xe 3 Xe 3 x 2 2 C 2 2 1 j(5 t) 1 j(5 t) Xe 5 Xe 5 2 2 2 jj x xA1xBe xC e 3 jt jt j3t j3t j5t j5t Xe 1 Xe 1 Xe 3 Xe 3 Xe 5 Xe 5 1 jt j( t) j(3 t) j(3 t) j(5 t) j5t x X1e X1e X3e X3e X5e X5e 3 Xe Xe Xe Xe Xe Xe jt j( t) j(3 t) j(3 t) j(5 t) j5t 1 1 3 3 5 5 PS 0 ZS 0 0 0 NS x X e X e jt j5t 1 5 (D.34) (D.35) (D.36) (D.37) (D.38) Equation (D.37) has six vertical columns indicated by brackets. The terms in the three columns marked “0” are often encountered when SV equations are simplified by hand and they sum to zero. For each such column there is normally a corresponding column (marked PS, ZS, and NS) that does not sum to zero. However in this case it is clear that the ZS column does sum to zero. Equation (D.38) demonstrates that the SV contains the PS and NS components present in the phase variables but does not contain the ZS component. Given that the SV is of dimension two it simply cannot contain all of the information in x abc (which is of dimension three) unless 0 is true (in which case x abc can always be rewritten as dimension two). This does not mean that some quantity q of dimension two can never contain common-mode components; the basis of q could be defined such that it does. What is instead true is that the MSC and SV transforms are 325
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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
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
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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: If a ZS component could possibly be
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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