SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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values could be interpreted as the maximum modulation index referenced to the SPWM linear limit, defined in terms of whichever row is selected. These values of maximum modulation indices are summarized in Table 4.5. (As mentioned, k 2/3 is used in this report when working with inverters.) To be clear, these are the maximum values of the modulation indices. Table 4.5 – Maximum ma referenced to SPWM linear limit (R). peak RMS L-N L-L L-N L-L (R) SPWM linear limit VDC 1.0000 1.7321 0.7071 1.2247 VBUS 0.5000 0.8660 0.3536 0.6124 (T) THI linear limit VDC 1.1547 2.0000 0.8165 1.4142 VBUS 0.5774 1.0000 0.4082 0.7071 (F) fundamental of squarewave VDC 1.2732 2.2053 0.9003 1.5594 VBUS 0.6366 1.1027 0.4502 0.7797 Many of the values in Table 4.5 are used in the literature when discussing inverter performance but references often do not define the definition of the modulation index they use; in those cases the table can be used as a guide. For example, an article may state that the maximum modulation index corresponding to six-step square wave operation is 0.78. Searching the (F) entries for 0.78 will show that the modulation index was defined using line-line RMS quantities referenced to the full bus voltage VBUS. When the modulation index is defined with reference to the THI linear limit the maximum modulation indices are those shown in Table 4.6. Table 4.6 – Maximum ma referenced to THI linear limit (T). L-N L-L peak RMS peak RMS (R) SPWM linear limit VDC 0.8660 1.5000 0.6124 1.0607 VBUS 0.4330 0.7500 0.3062 0.5303 (T) THI linear limit VDC 1.0000 1.7321 0.7071 1.2247 VBUS 0.5000 0.8660 0.3536 0.6124 (F) fundamental of squarewave VDC 1.1027 1.9099 0.7797 1.3505 VBUS 0.5513 0.9549 0.3898 0.6752 202
Similarly, when the modulation index is defined with reference to the fundamental obtained in squarewave mode the maximum modulation indices are those shown in Table 4.7. Table 4.7 – Maximum ma referenced to fundamental obtained in squarewave mode (F). L-N L-L peak RMS peak RMS (R) SPWM linear limit VDC 0.7854 1.3603 0.5554 0.9619 VBUS 0.3927 0.6802 0.2777 0.4810 (T) THI linear limit VDC 0.9069 1.5708 0.6413 1.1107 VBUS 0.4534 0.7854 0.3206 0.5554 (F) fundamental of squarewave VDC 1.0000 1.7321 0.7071 1.2247 VBUS 0.5000 0.8660 0.3536 0.6124 Finally, when the modulation index is defined with reference to the magnitude of a base SV the maximum modulation indices are those shown in Table 4.8. This reference is used in the literature so it is presented here, but some caution is due in its interpretation. The modulation index is the ratio of [the amplitude of the fundamental component of the output voltage (equivalently, the magnitude of a magnitude-invariant SV that represents the combined action of the same)] to [the amplitude of the fundamental component of some other important waveform (or the magnitude of its SV)]. But the six-step basis space vector represents a fixed (and imbalanced) state of the inverter; it is not a space vector that represents a sinusoidal set, thus it is somewhat abstract to use its magnitude as the reference in the definition of modulation index. However, it is a valid concept and is frequently encountered in the literature. Table 4.8 – Maximum ma referenced to magnitude of six-step basis space vector (S). peak RMS L-N L-L L-N L-L (R) SPWM linear limit VDC 0.7500 1.2990 0.5303 0.9186 VBUS 0.3750 0.6495 0.2652 0.4593 (T) THI linear limit VDC 0.8660 1.5000 0.6124 1.0607 VBUS 0.4330 0.7500 0.3062 0.5303 (F) fundamental of squarewave VDC 0.9549 1.6540 0.6752 1.1695 VBUS 0.4775 0.8270 0.3376 0.5848 203
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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
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x cos( r) sin( r) xd x sin(
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It must be emphasized that we have
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in relative time like an oscillogra
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Part III - SV Theory Applied to Sin
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In Equation (3.135) Z has elements
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to use SV theory. It may be helpful
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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: different triplen signal would be i
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
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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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