SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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Clarke & αβ0 Transforms The matrix transformations used in electrical circuit and machine analysis ([32], [34], [22], [35]) have evolved over the years, becoming rearranged, scaled, expanded, reduced, and reordered to serve various purposes; in the process they also acquire various names. The Clarke, Park, and MSC transformations given in this report are just one particular form of each. To compare the various versions is important for several reasons: to understand the significance of the ZS (0) term, to get an idea of the varieties in use, and to understand that there are so many variants that one should never take a sentence at its word—such as, “the Park transform accomplishes…”— until the matrix transformation in question is verified. This section examines the Clarke and αβ0 transforms; the “Park,” “original Park,” and dq0 transforms are investigated in Appendix E. The primary difference between the Clarke and αβ0 transforms is the presence or absence of the ZS component. It has been shown numerous times that the SV transform is the complex-valued version of the Clarke transform, and that neither can contain the ZS component. This is very much worth a closer look. In the simplest case when we know the ABC components do not contain a ZS component (such as the measurement of currents in a wye connection with isolated neutral), we are free to not have to calculate it, as in Equation (D.26). This is the “raw” Clarke transformation matrix defined in Chapter 3. It is sometimes called a phase transformation matrix, which is a fitting name because it implies a stationary transform that effectively functions as a 3to-2-phase “Scott-T” transformer connection. x A x 1 1/2 1/2 k x B x 0 3/2 3/2 x C (D.26) We previously acknowledged that since 0 it would be more convenient if we made that simplification in order to obtain more direct relationships between the components. Rearranging the 0 condition and substituting gives the form of either Equation (D.27) or (D.28). x A x 3/2 0 0 k x B x 0 3/2 3/2 x C x A x 3/2 0 0 k x B x 3/2 3 0 x C (D.27) (D.28) 322
If a ZS component could possibly be present in the ABC quantities (such as the measurement of voltages in a three-phase system), it could be calculated by adding a row to the matrix in Equation (D.26), as in Equation (D.29). This is the original transformation used by Edith Clarke [11, p.308], although it had been presented and used by different authors prior to that, at least as early as 1917 [11, p.310]. The transform was used in a fashion similar to that in the MSC (which is not what we have been using it for here). More information about the original Clarke transform and its application to synchronous machines can be found in [11], [12]. x 1 1/2 1/2 xA x k 0 3/2 3/2 x B x 0 1 1 1 x C (D.29) In the popular literature it is common to call Equation (D.26) (or any of its 0 variants) the Clarke transform and this report adopts that usage. The academic literature is more technical and generalized, studying circuits in which a ZS component is assumed to be present. Equation (D.29) or a variant (discussed below) is used, which in this report is called the αβ0 transform. These names are used interchangeably in the literature and it may also be called a Concordia transform (after the pioneer Charles Concordia ([9], [10]). In addition, the axis along which the ZS component acts may be called the γ- or Z- axis. Some variations will now be discussed. We have seen the usage of the scaling constant k in regards to the αβ components. The same remarks apply to the ZS component, meaning that there is a choice of k to make the ZS part of the transform in Equation (D.29) be magnitude- or power- invariant just like the αβ part. Since the ZS coefficients are scaled by k, one would expect to find three matrices that match the three varieties of the Clarke transform. However, no such standardization exists because authors often choose scaling between the αβ and 0 components to be different. This is handled in this report by introducing a separate constant c to describe the ZS component scaling. The general αβ0 transform is given by Equation (D.30) (hence, the original Clarke transform used c 2 ). x 1 1/2 1/2 k xA x k 0 3/2 3/2 x B x0 c 1/2 1/2 1/2 x C (D.30) Since the αβ0 transform is the Clarke transform plus the ZS (0) component, the matrix will be denoted C 0 . The “raw,” magnitude-invariant, and power-invariant αβ0 transform matrices are given by Equations (D.31), (D.32), and (D.33), respectively [87, p.34], [88, p.87]. Note that in the 323
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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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Page 293 and 294: First the self-flux-linkage of an i
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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
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Page 309 and 310: Distributed N Fp i 2 (C.5) Figu
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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
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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: Equation (D.25) and instantaneous q
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
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