SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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Returning to Figure 2.8, when an external force is applied to the rod, a voltage is induced according to Faraday’s law. Since an external resistance is connected, a current flows and dissipates heat in the resistor. To satisfy conservation of energy Equation (2.13) must be satisfied (the mechanical work input must equal the electrical energy dissipated in the resistor). F v g i (2.13) The apparatus in Figure 2.8 is an elementary, ideal linear generator with a load: mechanical energy is converted into electrical energy which is then dissipated as heat energy. Now, instead of the resistance serving as a kludge to allow the application of Lenz’s law, let the resistance represent the distributed resistance of a real coil lumped into one element and let this coil be driven by an external voltage source. The apparatus now functions as a linear motor, converting electrical energy into mechanical work. Whereas previously FAPPLIED was exerted to generate a current, here a current is made to flow and it produces FDEVELOPED, as shown in Figure 2.10. Figure 2.10 – Demonstration of the Lorentz Force law; motor. In the same way that an electrical model (Figure 2.9) was created to represent the apparatus of Figure 2.8, an electrical model (Figure 2.11) can be created to represent the apparatus of Figure 2.10. Figure 2.11 – Electrical model of coil connected to voltage source. The above discussion demonstrates that any time the flux linkage of a coil changes, a voltage is induced in the coil. The mechanism responsible for the induced voltage is the same mechanism responsible for the applied (or developed) force acting on the coil itself. This result will be 26
extended to the rotational case later. It is important to note that there is no inductance in the simple models here because the flux linkage of the coil is controlled by the rod (inductance is a measure of the flux linked by a coil due to the coil’s current; here, we assumed that the B field was fixed by an external source, thus there can be no inductance). Per-Phase Electrical Model for Generalized Brushless PM Motor The previous two sections discussed the two most fundamental concepts in electric machine operation and showed that they are inseparable. This section seeks to apply that information to develop a general electrical model of a motor with one phase winding. Since the magnetic medium is the coupling between the electrical and mechanical portions of the model, it is necessary to first introduce the basic magnetic structure of the brushless motor. From there we can obtain an expression for flux linkage and employ Faraday’s law to find the induced voltage. The resulting equation and circuit will be used to discuss the concepts of stator inductance, stator flux linkage, back-EMF, and their relationships. It is obvious that the interaction between rotor flux and the stator winding is spatial in nature (as the magnetic quantities in the motor are distributed in space), yet the voltage induced in the coil is a scalar quantity. The spatial aspect is important to understand and will be analyzed later. But in this section, attention is restricted to the scalar electrical result in order to derive the per-phase electrical model for a winding in a BPMS motor. We seek an electrical circuit model with as few mechanical and spatial dependencies as possible. It is representative of one winding in a three-phase motor and will be called the per-phase model. But it is important to note that this is not the familiar “single-phase equivalent” model found in most texts (which sometimes called “per-phase equivalent”). Single-phase equivalent analysis is concerned with the electrical phasor diagram of one phase (which is representative of the other two balanced phases) and the torque of the entire machine. In contrast, the per-phase model developed here is concerned with the time-domain representation of one phase and the torque produced per-phase. This distinction will be revisited and elaborated upon several times later on. Since real motors employ coils in a magnetic circuit, the flux linkage expression is more complicated than that of the previous section. To extend the concepts to a real motor, the coil from the previous section is now placed inside a motor and a more detailed model is derived using [42], [27], and [68] as guides. 27
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: N B A B Y x(t) (2.6) The induc
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
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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(
Page 103 and 104:
In the previous chapter the per-pha
Page 105 and 106:
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
Page 113 and 114:
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
Page 133 and 134:
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
Page 167 and 168:
to use SV theory. It may be helpful
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Figure 3.42 - Coupling between d- a
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R R d R R v Ri Li j Li dt s
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current does not influence the valu
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d v Ri L i e dt d v Ri L i e
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Figure 3.47 - Simulation diagram fo
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e je ), and (2) it was demonstrate
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Overview of Voltage Source Inverter
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In addition to measuring voltages w
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then there would be periods during
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Figure 4.9 - Gating and POLE voltag
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Figure 4.11 - Ideal voltage wavefor
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inverter is called sine-triangle co
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and it offers the fastest dynamic r
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Figure 4.18 - Fundamental gain and
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In the literature this is called th
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voltage will be below (above) the b
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Figure 4.24 - Instantaneous phase-A
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Figure 4.26 - Base SVs showing the
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Figure 4.27 - Transformed voltage w
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also correspond to six-step squarew
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was fed to the SVM inverter as a SV
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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
Page 241 and 242:
To analyze a salient machine one mu
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Figure 5.16 - Torque functions: (a)
Page 245 and 246:
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
Page 271 and 272:
loop). A second way to implement th
Page 273 and 274:
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,
Page 281 and 282:
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
Page 301 and 302:
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
Page 311 and 312:
alanced machine only odd harmonics
Page 313 and 314:
sinusoidal windings with a sinusoid
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0, and since the direction of inte
Page 317 and 318:
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
Page 329 and 330:
However, Equation (D.2) is incorrec
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
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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
Page 341 and 342:
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
Page 351 and 352:
Appendix E - Park Transforms This a
Page 353 and 354:
Appendix F - Useful Mathematical Re
Page 355:
335
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SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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