SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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Ne d( ) Ipsin( ) cos( t) sin( ) cos( t) sin( ) cos( t) 2 N 1 sin( t) sin( t) sin( t ) sin( t) e I p 2 2 sin( t) sin( t) 3 Ne d( ) Ip sin( t) (3.65) 2 2 It is clear that the effective number of amp-turns is a function around the stator and rotates according to the electrical position of the currents. Setting 0 the amp-turn distribution becomes Equation (3.66). 3 Ne d( ) Ipsin( t) (3.66) 2 2 This can be rewritten in terms of the current space vector as in Equation (3.67). 3 Ne d( ) Ipsin( t) 2 2 3 Ne Ipcos( t / 2) 2 2 Ne 3j( t/2) Re Ie p 2 2 Ne j/2 3 jt Ree Ipe 2 2 Ne Reji 2 N e d ( ji) 2 (3.67) It is seen that if the rotating amp-turn distribution were described by a space vector it would be 90° ahead of the current SV ([42, p.181], [166]), and since we know that a sine-distributed winding produces a cosine-distributed MMF this should not come as a surprise. Likewise Equation (3.67) is very similar to the MMF expression (Equation 3.42), which is reproduced here as Equation (3.68). (3.42): Ne f i 2 (3.68) The MMF and current SVs are related by only a scalar constant of proportionality thus their “distributions” must be the same. But Equation (3.67) showed that the actual amp-turn (or current density) distribution is not equal to the “distribution” of the current SV (and we really should not expect this to be true anyway). This leads us to conclude that the current space vector does not 112
epresent any physically-distributed quantity; this conclusion holds for all other space vectors except MMF. What its “distribution” instead represents is the meaning of reference frame theory discussed in the previous subsection; that interpretation is ultimately the most useful view of space vector theory. The meaning of the current SV will be further examined using Equation (3.68). This direct phase relationship between current and MMF has already been encountered: the sinusoidal case was examined in Part I, and both the sinusoidal and arbitrary cases have been examined here in Part II. In Part I when the MMF wave was developed mathematically (Equation 3.6, θ=0), it was obvious that the angular position of the stator MMF is identical to the electrical position (ωt) of the balanced sinusoidal currents (refer again to Figure 3.9). In Part II when the SV was developed (for the case of balanced sinusoidal currents) it was shown that the MMF SV describes the same MMF with the peak again at ωt and that this is shown very clearly in the SV itself, as in Equation (3.49) for example. 3 Ne j t (3.49): f Ie p 2 2 When the currents are arbitrary the SV theory gains an advantage over the real-valued descriptions of MMF since it is not restricted to balanced sinusoidal current. As discussed before, the MMF always retains its cosinusoidal distribution. Even in the arbitrary case, the SV of stator MMF is still cophasal with the SV of stator current, per Equation (3.42). Thus the angle ξ of the MMF SV represents the angular position of the peak of the MMF distribution, as measured from the phase-A axis and this is the same angle of the current SV. Further, the magnitude ˆ F of the physical MMF SV is proportional to the amplitude of the MMF wave and proportional to the magnitude of the current SV. Again from Part II, this was given by Equation (3.50). Ne (3.50): ˆ j ˆ j f Ie Fe 2 When Equation (3.50) was developed it was shown that the stator MMF SV was cophasal with the current SV in Figure 3.27. But at that time the primary interest was the addition of the SVs and the distributions were ignored. The figure is now redrawn as Figure 3.28 and the equivalent sinusoidal amp-turn distribution is shown. This leads to a better interpretation of the current SV. When Figure 3.27 was discussed it was said that the resulting MMF could be thought of as being produced by a coil whose magnetic axis was oriented along the direction of the total MMF produced by the three phase currents; this was compared to the CRT example of Figure 3.24. The situation is the same here except now we can say that the MMF could be thought of as being 113
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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
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 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: j set θ to anything. Taking the re
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(
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
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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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