SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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imparted by the definition of the inverse transform. This is made clear when an attempt is made to take the real part of the phasor only to find that it yields a stationary quantity: Re X Re j X e X cos . p p In contrast, the inverse of the phasor transform is defined as 1 P X Re j jt X e e Re jt X e p j t and it is seen that the e term is appended to the phasor as a part of the definition of the inverse transform—the phasor does not contain any “frequency” information itself. The trajectory of the j t phasor (coupled to e ) is the same as the trajectory of the SV discussed above for the case of balanced sinusoidal quantities in steady-state. In contrast, the “frequency” information of the SV is contained in the SV itself. The SV can represent any 0 combination of the instantaneous phase variables, whether those variables are changing, stationary, or switching between discrete values. It is the instantaneous nature of the SV (along with its ability to capture each phase’s contribution) that make it superior to the phasor for dynamic analysis. 142
Part III – SV Theory Applied to Sinusoidal BPMS Motor In Part III the SV models of a sinusoidal BPMS motor are derived. In the first section, the phasevariable electrical model of the synchronous machine is reduced to the two-phase model in the stationary reference frame (both the real-valued vector-matrix version and the complex-valued space vector version are given). Then the model is transformed to the rotor reference frame. The second section examines torque production in both frames. Finally, the third section gives the electromechanical simulation model in both frames. Electrical Models The phase-variable model of the brushless permanent magnet motor will be transformed into the stationary and rotor reference frames. Stationary Frame From Chapter 2 and Appendix B, the stator voltage equations are given by Equation (3.123), where the voltages are the line-neutral voltages. Equation (3.124) is the vector-matrix form. Since the inverter can apply voltages that contain a ZS component, we should rewrite the equations using the pole voltages. Although, when the Clarke (or SV) transform is used to express the variables in the stationary frame, the ZS component cannot be represented anyhow and the ZS component cannot drive any current so it seems there is no problem. However, this also means that the three circuits represented by Equation (3.123) are not independent, and this is a problem. It turns out that the math takes care of this—performing the derivation in terms of pole voltages (which requires use of the phase interference matrix) will yield the same results as those given here. Since using the terminal voltages is simpler, it is presented here and the explanation is deferred to Appendix D. vAN R iA Ls iA eA d v R i L i e BN B s B B dt v CN R i C L s i C e C (3.123) d vabc Riabc Lsiabc e abc (3.124) dt It is convenient to use the time derivative operator d / dt to combine the resistance and inductance matrices and eliminate the derivative of the current vector. In traditional general 143
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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
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(
Page 159 and 160: It must be emphasized that we have
Page 161: in relative time like an oscillogra
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
Page 201 and 202: Figure 4.24 - Instantaneous phase-A
Page 203 and 204: Figure 4.26 - Base SVs showing the
Page 205 and 206: Figure 4.27 - Transformed voltage w
Page 207 and 208: also correspond to six-step squarew
Page 209 and 210: was fed to the SVM inverter as a SV
Page 211 and 212: 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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