SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

More documents

Recommendations

Info

Appendix D - Three-Phase Concepts & Transformations This appendix is a collection of various “basic” and “advanced” topics related to three-phase systems that appear throughout the report. Although all readers are assumed to have an electrical engineering degree, many of these basic concepts are not taught in courses outside the power systems emphasis. In addition, many modern machines and power electronics texts are nearly half a century out of date, thus many graduating students (the author included) are likely to have not been exposed to the more advanced topics. Harmonic Analysis Any periodic signal can be described in terms of a Fourier series. All ideal steady-state waveforms in nonsalient machines possess quarterwave symmetry so they can be represented using only cosine or sine terms; here, the cosine is selected in order to match the conventions for the phasor and space vector reference. Since waveforms with quarterwave symmetry also have halfwave symmetry only odd terms are present. The standard harmonic components are therefore defined by Equation (D.1). x () t X cos n t ; n1,3,5,7,9, (D.1) n n As remarked earlier, this report only considers symmetric three-phase systems. For loads, this condition means that the self impedance of each phase must be identical and the mutual impedance between any two phases must be the same regardless of which pair is considered. From Chapter 3 and Appendix B, the impedance matrix for a three-phase sinusoidal motor satisfies this requirement. For sources (a controlled three-phase source or a bEMF in a load) the restriction means that each harmonic component of the phase-neutral voltage is identical to that of the other phases but is displaced by 120°. The harmonic components defined in Equation (D.1) are for a single phase only (and since phase- A is always taken as the reference, they are for phase-A). Extending the definition to phase-B and phase-C would appear to be given by Equation (D.2). xnA() t Xncosnt n 1, 3, 5, 7,9, xnB () t Xncosnt (D.2) 120 xnC () t Xncosnt 308
However, Equation (D.2) is incorrect. Since the fundamental component of phase-B and phase-C are displaced by 120 from the fundamental component of phase-A, the harmonic components in phase-B and phase-C must be similarly displaced. That is, for each harmonic set the n th component in each phase must be displaced by 120 / 360 2 / 3 of a fundamental electrical cycle from the n th component in the other phases, not 120° of that harmonic electrical cycle as Equation (D.2) would imply. The period of the n th harmonic is 360 / n , thus the angle corresponding to a 120° delay in harmonic measure is equal to a 120 / n delay in fundamental measure. Therefore to make the phase-B and phase-C 120 harmonic delays equal to the phase- B and phase-C 120 fundamental delays, the 120 harmonic delay must be multiplied by n, as shown in Equation (D.3). In other words, there is noting special about the harmonic components themselves. If they were indeed defined by Equation (D.2) their phase offsets would all be 0, , , exactly like those of the fundamental, and they would all sum to zero just the same. It is the fact that harmonics are 120 fundamental-degrees apart that gives them different properties in a three-phase system. (Again, the reason that the harmonic components are defined to be 120 fundamental degrees apart, instead of 120 harmonic degrees, is because the harmonics of each phase are given by Equation (D.1), thus the entire set of harmonics in a phase are displaced by 120 fundamental degrees from the set of each other phase.) xnA() t Xncosnt xnB () t Xncosntn xnC () t Xncosntn xnA() t Xncosn t = xnB () t Xncosnt xnC () t Xncosnt Equation (D.3) is now evaluated for the fundamental and the first two harmonics. (D.3) 309
Page 1 and 2:
SENSORLESS FIELD ORIENTED CONTROL O
Page 3 and 4:
Table of Contents List of Figures..
Page 5 and 6:
CHAPTER 5 - Field Oriented Control
Page 7 and 8:
List of Figures Figure 2.1 - Radial
Page 9 and 10:
Figure 3.29 - Arbitrary space vecto
Page 11 and 12:
Figure 4.44 - ZS components of some
Page 13 and 14:
Figure C.13 - Single-turn full-pitc
Page 15 and 16:
List of Symbols Symbol Meaning Unit
Page 17 and 18:
Abbreviations ACIM AC induction mot
Page 19 and 20:
Superscripts * commanded value ref
Page 21 and 22:
CHAPTER 1 - Introduction Historical
Page 23 and 24:
vector theory to model a machine, w
Page 25 and 26:
publications and magazines, and Int
Page 27 and 28:
elationship between SVM and other P
Page 29 and 30:
CHAPTER 2 - Fundamentals of Electri
Page 31 and 32:
Preliminaries In reading the litera
Page 33 and 34:
Taxonomy of Motors There is any num
Page 35 and 36:
characteristics (the DC load curren
Page 37 and 38:
torque component, sturdily construc
Page 39 and 40:
Figure 2.7 - Cross sections of some
Page 41 and 42:
draw the line between the solenoida
Page 43 and 44:
These texts generally attribute the
Page 45 and 46:
N B A B Y x(t) (2.6) The induc
Page 47 and 48:
extended to the rotational case lat
Page 49 and 50:
present analysis). When this is tru
Page 51 and 52:
Figure 2.16 - Components of total f
Page 53 and 54:
As aforementioned a spatial analysi
Page 55 and 56:
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 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 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
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
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,
Page 259 and 260:
D, some of the 120° methods are ap
Page 261 and 262:
d d v Ri Ls i R dt dt The r
Page 263 and 264:
section a state-space model for the
Page 265 and 266:
Figure 6.6 - Full state feedback (o
Page 267 and 268:
State-Space Model of BPMS Motor A s
Page 269 and 270:
Figure 6.9 - FOC block diagram. The
Page 271 and 272:
loop). A second way to implement th
Page 273 and 274:
knowing the rotor position from ini
Page 275 and 276:
[11] E. Clarke, Circuit analysis of
Page 277 and 278: Control Systems, Linear Analysis [4
Page 279 and 280: [77] J.M.D. Murphy, F.G. Turnbull,
Page 281 and 282: Machine Modeling, Analysis [103] J.
Page 283 and 284: THI, SVM, SPWM [127] J.A. Houldswor
Page 285 and 286: Wisconsin-Madison. [Online]. Availa
Page 287 and 288: Angle Tracking [175] D. Morgan, “
Page 289 and 290: Appendix A - Elementary Electromagn
Page 291 and 292: L i (A.8) S SS S SR S SS SR (A.9
Page 293 and 294: First the self-flux-linkage of an i
Page 295 and 296: vAN R iA LM iA eA d v R i
Page 297 and 298: Figure B.3 - One-half of the flux p
Page 299 and 300: Equations (B.27) and (B.28) are for
Page 301 and 302: Fundamental Relationships Figure C.
Page 303 and 304: Figure C.3 - Sinusoidal winding den
Page 305 and 306: manufactured compared to the steppe
Page 307 and 308: phase winding of Figure C.4 will ha
Page 309 and 310: Distributed N Fp i 2 (C.5) Figu
Page 311 and 312: alanced machine only odd harmonics
Page 313 and 314: sinusoidal windings with a sinusoid
Page 315 and 316: 0, and since the direction of inte
Page 317 and 318: That the rotor-stator flux linkage
Page 319 and 320: Figure C.17 - Summary of rotor-stat
Page 321 and 322: Figure C.19 - Amplitudes of harmoni
Page 323 and 324: triangle with an amplitude whose nu
Page 325 and 326: Returning to Figure C.18 it is clea
Page 327: torque will describe the torque pro
Page 331 and 332: the same order as the PS set (arran
Page 333 and 334: Implications of the ZS Component Th
Page 335 and 336: Figure D.6 - Neutral voltage of loa
Page 337 and 338: 3v 3e 3v v v v v v v 3v AM BM CM A
Page 339 and 340: components of source voltage sum to
Page 341 and 342: Equation (D.25) and instantaneous q
Page 343 and 344: If a ZS component could possibly be
Page 345 and 346: xA X1cos( t) X3cos(3 t) X5cos(5 t)
Page 347 and 348: Figure D.11 - 3-D base vectors of 1
Page 349 and 350: the Clarke transform is used. Howev
Page 351 and 352: Appendix E - Park Transforms This a
Page 353 and 354: Appendix F - Useful Mathematical Re
Page 355: 335
show all

SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?