SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

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v v v v v v v v v AM AN NM BM BN NM CM CN NM 1 v v v v 3v v v v 3 NM AM BM CM NM AM BM CM vAM 3vNM vBM vCM vBM 3vNM vAM vCM vCM 3vNM vAMvBM vAN vAM vNM vBN vBMvNM vCN vCMvNM 2 1 1 vAN AM BM CM 3v v v 2 2 2 1 1 vBN vAM vBM vCM 3 2 2 2 1 1 vCN vAMvBM vCM 3 2 2 vAN 2 1 1vAM 1 v BN 1 2 1 v BM vabcN vabcM 3 Q (D.45) v CN 1 1 2 v CM When there is no ZS component present, vNM 0 and it can be verified that the phase voltage vector in Equation (D.45) is the same as the pole voltage vector. An interesting result is that the phase interference matrix relates the Clarke and αβ0 transforms as shown by Equation (D.46). C QC 0 1 1/2 1/2 2 1 1 11/2 1/2 1 0 3/2 3/2 1 2 1 0 3/2 3/2 3 1/2 1/2 1/2 112 000 (D.46) As a consequence, when the phase-variable model is reduced to the stationary reference frame, the αβ0 transform can be used on the pole voltages, or, using the results of Equation (D.46), the Clarke transform can be used on the phase voltages. 330
Appendix E - Park Transforms This appendix discusses a few forms of the Park transform and the issues related to their axis and angle conventions. In Chapter 3 and Appendix D it was explained that the Clarke transform defined in this report (and found in the popular literature) is not the original Clarke transform. The original more closely resembles the αβ0 transform which contained an extra row in the matrix to give the zerosequence component. Similarly, the Park transform can be defined to transform the zero-sequence component as well. This zero-sequence form is shown in Equation (E.1). xd cos( r) sin( r) 0x x sin( ) cos( ) 0 x q r r x 0 0 0 1 x 0 By definition, the ZS component remains unchanged in the transformation. (E.1) Unlike the Clarke and αβ0 transforms, there is not only a ZS and non-ZS version, but also a completely different version of the Park transform, which is a combination of the Park and αβ0 transforms. This is the “original” Park transform is the one used by Park and those before him (although like the Clarke transform, the scaling and ordering of variables may be different) [22], [30], [32], [34], [87]. It is obtained by multiplying the ZS-Park and αβ0 transforms. After simplification the result is given by Equation (E.2). x cos( ) sin( ) 0 1 1/2 1/2 d r r x x sin( ) cos( ) 0 k 0 3 / 2 3 / 2 x q r r x0 0 0 1 1/2 1/2 1/2 x 0 xd cos( r) cos( r ) cos( r ) x x q k sin( r) sin( r ) sin( r ) x x 1/2 1/2 1/2 x 0 0 (E.2) An easier derivation that avoids the algebraic simplification can be had by using SV notation. When the real and imaginary parts of Equation (E.3) are taken and put into matrix form as usual, the result is the same as Equation (E.2) (since the SV does not transform the ZS component it must be manually added, as in the case of the αβ0 transform above. 331
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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
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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
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
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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 and 328: 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
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: the Clarke transform is used. Howev
Page 353 and 354: Appendix F - Useful Mathematical Re
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