SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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variables are independent, but the resulting axes (A&B, A&C, or B&C) are not orthogonal. Thus the purpose of the Clarke transform is to form an orthogonal basis such that a SV is easily defined by components along the α and β axes as shown in Figure 3.33. This also allows other reference frames to be developed more easily as shown next. Figure 3.33 – Space vector in stationary reference frame showing phase axes. When a quantity is described as a SV ( x ) or in terms of the αβ plane ( x ) it is said to be in the stationary or stator reference frame because the angle of the SV is measured from the real (α) axis fixed to the stator. A vector can be described in different reference frames and there are a number of them defined in the literature. It has been seen that SV theory can be used to describe purely electrical quantities or physical quantities such as those present in a machine; it follows that a reference frame could be defined relative to an electrical or physical reference. Examples include the voltage at a power grid bus, the rotor position in a motor, the rotor flux (which is not coincident with the rotor position in induction motors), or a flux linkage. The only difference between these various reference frames is how they are defined with respect to the stationary reference frame, thus many texts base discussions on the general reference frame; the position of this frame (call it θΔ) could then be specialized to whatever frame is appropriate [87], [35]. Since in the synchronous machine the rotor flux and rotor-stator flux linkage SVs are aligned with the rotor position, only the stationary reference frame and the rotor reference frame will be used in this report. The rotor reference frame is defined using the position of the rotor ( r ) as shown in Figure 3.34. The direct axis (d) is defined as being in the center of the rotor’s North pole and the quadrature axis (q) axis is defined to be 90° electrical ahead of the direct axis in the positive direction of rotation. The rotor position is measured from α to d. These choices are not 130
universal and the choice of conventions is very important (see Appendix E). For clarity the phase axes will no longer be drawn but they are understood to be present. Figure 3.34 – Definition of rotor reference frame. Figure 3.35 shows a SV and its projection onto the stator and rotor reference frames. Figure 3.35 – SV in the stator and rotor reference frames. The SV x S is defined in the stator reference frame as ˆ j x Xe x or x x as usual. There are several ways to obtain the components in the rotor reference frame but one of the easiest is to use a transformation matrix. Since we have a transform of the form xNEW Tx OLD the 131
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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
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 and 108: Figure 3.21 - Time relationship bet
Page 109 and 110: another and thus could be placed in
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: Figure 3.32 - Arbitrary SV referenc
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(
Page 159 and 160: It must be emphasized that we have
Page 161 and 162: in relative time like an oscillogra
Page 163 and 164: Part III - SV Theory Applied to Sin
Page 165 and 166: In Equation (3.135) Z has elements
Page 167 and 168: to use SV theory. It may be helpful
Page 169 and 170: Figure 3.42 - Coupling between d- a
Page 171 and 172: R R d R R v Ri Li j Li dt s
Page 173 and 174: current does not influence the valu
Page 175 and 176: d v Ri L i e dt d v Ri L i e
Page 177 and 178: Figure 3.47 - Simulation diagram fo
Page 179 and 180: e je ), and (2) it was demonstrate
Page 181 and 182: Overview of Voltage Source Inverter
Page 183 and 184: In addition to measuring voltages w
Page 185 and 186: then there would be periods during
Page 187 and 188: 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
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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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