SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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PEAK RMS L-N L-L L-N L-L V DC V BUS VBUS VDC VBUS VDC VBUS VDC Table 4.4 – Magnitude of space vectors of three-phase quantities. maximum radius of SV trajectory for continuous balanced three-phase sinusoids in steady state Regular SPWM Triplen harmonic injection (THI) Fundamental of 180° six-step magnitude of discrete SV 180° base SV R T F S k=1 1.5000 1.7321 1.9099 2.0000 k=2/3 1.0000 1.1547 1.2732 1.3333 k=√(2/3) 1.2247 1.4142 1.5594 1.6330 k=1 0.7500 0.8660 0.9549 1.0000 k=2/3 0.5000 0.5774 0.6366 0.6667 k=√(2/3) 0.6124 0.7071 0.7797 0.8165 k=1 2.5981 3.0000 3.3080 3.4641 k=2/3 1.7321 2.0000 2.2053 2.3094 k=√(2/3) 2.1213 2.4495 2.7009 2.8284 k=1 1.2990 1.5000 1.6540 1.7321 k=2/3 0.8660 1.0000 1.1027 1.1547 k=√(2/3) 1.0607 1.2247 1.3505 1.4142 R T F S k=1 1.0607 1.2247 1.3505 1.4142 k=2/3 0.7071 0.8165 0.9003 0.9428 k=√(2/3) 0.8660 1.0000 1.1027 1.1547 k=1 0.5303 0.6124 0.6752 0.7071 k=2/3 0.3536 0.4082 0.4502 0.4714 k=√(2/3) 0.4330 0.5000 0.5513 0.5774 k=1 1.8371 2.1213 2.3391 2.4495 k=2/3 1.2247 1.4142 1.5594 1.6330 k=√(2/3) 1.5000 1.7321 1.9099 2.0000 k=1 0.9186 1.0607 1.1695 1.2247 k=2/3 0.6124 0.7071 0.7797 0.8165 k=√(2/3) 0.7500 0.8660 0.9549 1.0000 It should be pointed out that there is no such thing as an “RMS magnitude” of a SV, as shown in the lower right-hand corner of the table. However, once the choice for the SV definition is defined, all SVs must use the same reference, including the base SVs, thus it is listed in the table. It is not uncommon to see an RMS definition used in the literature (especially RMS, L-L, V BUS , k 2/3). Space Vector Averaging and Overmodulation Three important points from before are now summarized: the 180° inverter can be in only one of eight states (represented base SVs) at any given time (Figure 4.26), there is a precise relationship between each base SV and the pole voltages it produces (Table 4.2), and it is possible to control the inverter by commanding which state it should be in (Figure 4.25). If the reader has not already deduced, a SVM inverter rapidly switches between different inverter states (base SVs) such that the vectorial time-average of the asserted base SVs is equal to the commanded voltage (which 188
was fed to the SVM inverter as a SV). The SVM inverter is a 180° inverter with the same allowable states as a PWM inverter. SVM is simply a method of controlling the inverter as a whole rather than controlling each phase leg independently. Even though the method of control is radically different, it is still a pulse width modulator and for this reason it is sometimes called space vector PWM (SVPWM). Figure 4.28 is redrawn as Figure 4.29 without the circular trajectories shown and the sextants have been labeled. Figure 4.29 – Base SVs. The figure shows all the possible SVs but of course, only one SV can be asserted at one time. In SVM, one base SV is asserted for a period of time, another for a different period of time, and so on. There are different implementations but there are two obvious minimum requirements. First, the only way to synthesize a time-average SV is to switch between two active base SVs. Second, the only way to control the magnitude of the synthesized time-average SV is to switch between null and active base SVs. Thus the general SVM algorithm is to switch between two active and at least one null base SVs during one switching period. Like regular PWM, the switching period may be selected arbitrarily, although there are many practical limits. The SVM synthesis of a time-average SV S is shown in Figure 4.30. Simple as it is, the figure is somewhat deceiving. It shows the time-average SV S as being composed of two vectors aligned with the base SVs. But the two shorter SVs do not exist—they are time-averages as well. In reality, one active base SV is asserted, then the other active base SV, then the null base SV, and the cycle repeats (as will be shown later). Thus, the magnitude of each shorter SV represents the duty cycle of that active SV. 189
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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
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
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: also correspond to six-step squarew
Page 211 and 212: Figure 4.31 - Temporal limit of inv
Page 213 and 214: Figure 4.34 - SVM overmodulation re
Page 215 and 216: Checking sextant s 23 again for thi
Page 217 and 218: SVM Implementation The block diagra
Page 219 and 220: has been made of using THI in the S
Page 221 and 222: different triplen signal would be i
Page 223 and 224: Similarly, when the modulation inde
Page 225 and 226: Summary and Conclusion The two-leve
Page 227 and 228: CHAPTER 5 - Field Oriented Control
Page 229 and 230: torque production; this is called d
Page 231 and 232: Figure 5.3 - Torque control of sinu
Page 233 and 234: obviously a change in perspective.
Page 235 and 236: Figure 5.9 - Comparison of referenc
Page 237 and 238: variables to the stationary referen
Page 239 and 240: Figure 5.13 - Torque control of sin
Page 241 and 242: To analyze a salient machine one mu
Page 243 and 244: Figure 5.16 - Torque functions: (a)
Page 245 and 246: Figure 5.19 - Buried permanent magn
Page 247 and 248: Figure 5.21 - Flux weakening in the
Page 249 and 250: appears that this is very much rela
Page 251 and 252: Stationary and Synchronous Regulato
Page 253 and 254: stationary regulator had. For compl
Page 255 and 256: This coupling can be mitigated by m
Page 257 and 258: 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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