SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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Figure 3.23 – Time relationships between flux linkages and induced voltages; lagging PF. These relationships in Figure 3.23 are given by Equations (3.29), (3.30), and (3.31). a S, a R, a LI s a R, a G a j a jL I j jL IE a R, a s a R, a s a (3.29) (3.30) E j (3.31) Once again, the stator flux linkage is composed of two components: the rotor-stator flux linkage and the self flux linkage. The bEMF is the time derivative of the former and the inductive voltage drop is that of the latter. The total induced stator voltage is then the sum of the inductive voltage drop and the bEMF. In reality, the flux due to stator currents (which produces the self flux linkage) and the flux due to the rotor (which produces the bEMF) add together in the airgap to produce a resultant flux (with corresponding flux density, MMF, and field strength), and the time derivative of this resultant flux linkage is the total induced stator voltage ( a G ). That the rotor- stator flux linkage induces the bEMF is familiar. That an inductor produces flux linkage with itself as a result of the current through it is familiar. That the two act independently of one 88
another and thus could be placed in series to model the circuit is reasonable and familiar. But the key to understanding this vital aspect is not evident in the circuit theory approach: the flux produced by the stator winding exists in the same airgap as the rotor flux. As natural as the rotor- produced airgap flux linkage induces the bEMF 90° ahead, the stator-produced airgap flux linkage induces a voltage 90°—the standard terminal characteristic of the inductor. This discussion should cement the meaning of flux linkage provided in the previous chapter: the flux linkage is the expression that describes the total flux linked by a winding (phase-A in this case). This should demonstrate why the inductor can be considered an electric circuit element separate from the bEMF, but both the inductance and bEMF are a result of the resultant airgap flux and for this reason there are the two distinct (but equivalent) circuit models. Finally, this reinforces the dependence of bEMF on rotor position. 89
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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
Page 57 and 58: Figure 2.20 - Elementary brushless
Page 59 and 60: called the per-phase torque functio
Page 61 and 62: Expressing bEMF in terms of the bEM
Page 63 and 64: Equation (2.44) is the component of
Page 65 and 66: d R( r) ke( r) kt( r) sin( r) (2.
Page 67 and 68: General Electromechanical Models Th
Page 69 and 70: Figure 2.28 - Phase-variable simula
Page 71 and 72: Trapezoidal and Sinusoidal BPMS Mot
Page 73 and 74: Figure 2.29 - Back-EMF and drive cu
Page 75 and 76: Figure 2.31 - Back-EMF and drive cu
Page 77 and 78: 3 KT Ke 2 (sinusoidal) K 2 K (tra
Page 79 and 80: control scheme must keep sinusoidal
Page 81 and 82: with misconceptions. The primary is
Page 83 and 84: Part I - Sinusoidal BPMS Motors wit
Page 85 and 86: grounded-neutral wye or delta conne
Page 87 and 88: N e f A ( ) i 2 N e f B ( ) i 2
Page 89 and 90: Figure 3.7 - Developed view at zero
Page 91 and 92: Figure 3.9 - Developed view showing
Page 93 and 94: Although Equations (3.6) or (3.7) a
Page 95 and 96: direction. The contributions always
Page 97 and 98: sinusoidal electrical variable will
Page 99 and 100: Figure 3.14 - Cross section of two
Page 101 and 102: 3 N e (3.6): f ( , t) I p cos(
Page 103 and 104: In the previous chapter the per-pha
Page 105 and 106: peak value is represented as an upp
Page 107: Figure 3.21 - Time relationship bet
Page 111 and 112: more widely understood theory (such
Page 113 and 114: other methods, although it has clos
Page 115 and 116: The SV as a Vector The first facet
Page 117 and 118: j0 j0 j0 aˆ 1 0 1 bˆ j e e e
Page 119 and 120: j j ee 1 cos( ) (3.45) 2 3 j t x
Page 121 and 122: Notice that the amplitude of the MM
Page 123 and 124: Figure 3.27 - Equivalent MMF produc
Page 125 and 126: Ne 1 jj f , t i e i * e 2 2
Page 127 and 128: distribution that is cosinusoidal i
Page 129 and 130: 3 Ne j t f Ie p 2 2 . This
Page 131 and 132: j set θ to anything. Taking the re
Page 133 and 134: epresent any physically-distributed
Page 135 and 136: lumped-parameter model in this repo
Page 137 and 138: x kCx (3.75) abc The set of varia
Page 139 and 140: phase variable axes; three axes in
Page 141 and 142: The component MMFs act away from th
Page 143 and 144: common choice is to force the power
Page 145 and 146: As an example, find the SV that cor
Page 147 and 148: the same magnitude, the projection
Page 149 and 150: Figure 3.32 - Arbitrary SV referenc
Page 151 and 152: universal and the choice of convent
Page 153 and 154: stator’s perspective and at R fr
Page 155 and 156: The potential discrepancy (p.108) w
Page 157 and 158: 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
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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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