SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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There are also the harmonics associated with the carrier frequency f tri . These are the carrier harmonics and their sidebands, which we will lump together and call the switching harmonics. The spectrum of switching harmonics is governed by the frequency modulation ratio defined by Equation (4.3) [67], [75]. In this report, “harmonics” is generally used to refer to the baseband harmonics; switching harmonics are not discussed further. m f tri f (4.3) f1 At the most basic level (shown in Figure 4.14) the sine-triangle PWM it is an open-loop scheme; in other words, it functions as a controlled voltage source. The usual way to make it closed-loop is to generate the voltage command by closing the loop around current; closed-loop current control of voltage converters is the most commonly used method [124]. The result is called a current-regulated PWM (CRPWM) inverter and it functions as a controlled current source. There are essentially three groups of methods that can be used to make a CRPWM [78, p.322]. The first method is to use some sort of ramp-comparison controller. The simplest example of this type is shown in Figure 4.15. When the reference and feedback signals are phase variables, this method has the disadvantage that the current ripple is fed back into the command [125] and usually use filtering [162] which would limit the bandwidth and stability [47]. Further the tuning of the regulator is dependent on the frequency of operation. However, these problems are limited can be eliminated by executing control in the synchronous reference frame (discussed in a later section). Figure 4.15 – Ramp-comparison current regulator for one phase. The second method to make a CRPWM is to use a bang-bang controller with hysteresis (also called tolerance band controller) [75, p.241], [61, p.243], [80, p.252]. In this method the carrier is eliminated and the switching signals are generated directly by feeding the current error to a comparator with hysteresis, as shown in Figure 4.16. The advantages of this method are that it is simple, it inherently compensates for dead-time distortion, it has excellent disturbance rejection, 172
and it offers the fastest dynamic response of any current controller [62, pp.30-31,93]. The chief disadvantage is that the switching frequency is not constant throughout an electrical cycle it varies depending on the load type and the shape of the reference waveform. Another disadvantage is that independent control of the three phases is not possible (Appendix D) thus “phaseinterference minimization” control strategies are required to prevent errant behavior [132], which can include limit cycles [78, p.326]. There are methods to mitigate switching frequency variation ([62, p.31], [43, p.752]) or achieve constant switching frequency [132]. Hysteresis regulation is in use in some areas [78, p.324] but at this time hysteresis control is not widely in brushless motor control applications. A third class of CRPWMs are the predictive (and/or dead-beat) methods [43, p.751] that rely on a load model. As far as this report is concerned, those methods will be lumped with on-line optimization methods; neither are discussed here. Figure 4.16 – Hysteresis CRPWM. In the sine-triangle inverter it is possible for the reference value to be greater than peak value of the carrier; this causes the inverter to saturate over the peaks of the reference, as shown in Figure 4.17. 31 This causes harmonics to be produced as the average waveform deviates from sinusoidal. At the same time, the amplitude of the fundamental of the average waveform produced becomes larger than the amplitude of the sinusoidal output when no harmonics are generated (normal operation, when the reference amplitude is less than that of the carrier). In a properly designed SPWM controller the reference would be clamped so that saturation would not occur and harmonics would not be generated. However if the harmonics can be tolerated in exchange for the larger fundamental amplitude, the inverter could be designed to operate in this mode, which is 31 This can happen in the hysteresis CRPWM as well but in that case it is an uncontrollable saturation, whereas the sine-triangle PWM the controller can prevent this from occurring, or command it to occur if desired. 173
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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
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: inverter is called sine-triangle co
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
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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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