SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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Figure 3.12 – Simplified developed view of a four pole motor. This notion leads to the development of two systems of angular measure. The relationship between electrical and mechanical measurement is shown in Equation (3.8), where P indicates the number of poles. 16 Taking the time derivative yields Equation (3.9), which is the relationship between electrical and mechanical velocities; ωe is usually called the synchronous speed. To avoid having to memorize the formulas, reconsider the gear ratio analogy, where P/2 is the ratio. A free rotating piece of steel or magnet situated inside the stator can do nothing other than follow the rotating field tied to the currents’ electrical position and velocity, thus the mechanical velocity can never be faster than the electrical velocity. This means the ratio has a lower limit of unity, hence the ratio is always multiplied by the mechanical measure. P 2 elec mech P 2 elec mech It is important to make the distinction between a mechanical or electrical angle (that is, an angle describing a mechanical or electrical displacement) and an angle measured in mechanical or electrical degrees. For example, in Equation (3.4) and Equation (3.6) t is an electrical angle and θ is a mechanical angle but both are measured in electrical degrees. The derivative of a 16 Some texts use P to indicate pairs of poles to eliminate the factor of two in equations. However, industry specifies motors almost exclusively by the number of poles (not pole pairs) so this report uses P to indicate the number of poles. The reader is cautioned that many authors use a lowercase p to indicate poles (or pole pairs) which can also be used to denote the time derivative operator d/dt. The two are used interchangeably without distinction, often within the same formula. This is unfortunate, as the correct meaning in an equation cannot always be discerned from context without studying the derivation, which is often not given. This report uses rho ( ) to denote the time derivative operator d/dt. (3.8) (3.9) 76
sinusoidal electrical variable will produce a factor of ωe while the derivative of a rotor position will produce ωr. In this report both electrical position and mechanical angles are generally specified in electrical degrees, thus the velocities would be ωe,elec and ωr,elec. Since this report concerns only synchronous machines there would then be no difference between electrical and mechanical quantities so the notational distinction between ωe & ωr and the ‘elec’ & ‘mech’ subscripts are unnecessary. 17 Reworking the units in Equation (3.9) yields Equation (3.10), where f e is the electrical frequency expressed in [Hz] and Nm is the mechanical rotational speed expressed in [rpm]; solving for f e yields Equation (3.11). 120 fe Nrpm , f e in [Hz] P (3.10) PNrpm fe , f e in [Hz] 120 (3.11) Torque Production As discussed in the previous chapter, non-salient machines produce torque through the interaction of the magnetic field associated with the stator and that associated with the rotor. Previously the airgap MMF due to stator current was developed. Now the magnetic field of the rotor will be introduced in order to examine torque production. A magnetic field may be described by the distribution of its MMF, field strength, flux, or flux density. Using the assumptions presented earlier, all of these distributions are equivalent and only differ by constants. In reality this is not true due to several factors but it is a good enough approximation to explain motor operation [1, p.149], [27, p.117]. Therefore, the torque described by the interaction of stator and rotor fields may be expressed as various combinations of stator and rotor field quantities. The MMF produced by the stator is easily understood and is a reasonable choice because it is very simply related to the current by the winding function. But the influence of the rotor magnets is often thought of in terms of flux density rather than MMF. In P or its inverse will not appear in formulas. This is fitting since for simplicity only 2-pole machines are considered thus the factor is also excluded from the derivations and discussions. The full formulas can be found in most machine texts. 17 An important consequence of doing this is that the factor /2 77
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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
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: direction. The contributions always
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
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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
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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
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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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