an investigation of dual stator winding induction machines

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A) The first term of T e1 Substituting the expressions of surface current distribution and the flux density into the first term, the integration result can be written as: 2π ∫ 0 J 1 1 2 ⎡ ⎢ ⎣ µ r gP 2 * 0 * ( θ, t) ⋅ B1( θ, t) dθ = rl Re j2π ( 3 2) ( Cs1 ⋅ Cs1 )( I s1 ⋅ I s1 ) ⎥⎦ * Since both ( ) 2π ∫ 0 1 s1 s1 C C ⋅ and ⎟ * ⎜ s1 Is1 ⎝ ⎠ ( θ t) ⋅ B ( θ, t) dθ = 0 , 1 1 ⎛ I ⋅ ⎞ are real number, then: 212 ⎤ (5.74) J (5.75) B) The second term of T e1 Substituting the expressions of surface current distribution and the flux density into the first term, 2π ∫ 0 = J 2π ∫ 0 1 ( θ, t) ⋅ B ( θ, t) Re 2 dθ j( ω − θ ) ⎧ µ 1t P 0r 1 j( ω2t− P2θ ) { 2( 3C ⋅ I ) e } ⋅Re j [ 2( 3C ⋅ I ) e ] s1 s1 ⎨ ⎩ gP 2 s2 s2 ⎫ ⎬dθ ⎭ If x and y are both the complex number, then the following identity is true, ( x) ⋅ Re( y) ≠ Re( x ⋅ y) (5.76) Re (5.77) However each term in equation (5.76) can be expressed as: Re ⎧ Re⎨ ⎩ j( ω1t −P1 θ ) { 2( 3 ⋅ I ) e } = 2 3C ⋅ I cos( ω t − Pθ + θ ) Cs 1 s1 s1 s1 1 1 1 (5.78) j( ω2t− P ) ⎫ µ 0r 2 [ 2( 3C ⋅ I ) e ] = − 2 3C ⋅ I ( ω t − P θ + θ ) µ 0r θ j s 2 s2 ⎬ s2 s2 sin 2 2 2 (5.79) gP2 gP2 The multiplication of the two terms is: ⎭
Re j( ω ) ⎧ 1t− P1θ µ 0r j( ω2t− P2θ ) { 2( 3C ⋅ I ) e } ⋅Re j [ 2( 3C ⋅ I ) e ] 1 µ 0r = − 2 gP 2 s1 2 s1 2 3C s1 ⋅ I s1 3C s2 ⎨ ⎩ ⋅ I s2 gP Equation (5.76) can be written as: 2π ⎧ µ 2 0r ⎪∫ j 2 1 ⎪ gP 0 2 Re⎨ 2π 2 ⎪ µ 0r ⎪ + ∫ j 2 ⎩ gP 0 2 2 ⎡sin ⎢ ⎣+ sin s2 ( ) ( ) ⎥ ω1t + ω2t − P1θ − P2θ + θ1 + θ 2 ⎤ ω2t −ω1t − P2θ + P1θ + θ 2 −θ1 ⎦ 213 s2 j( ω1t + ω2t− P1θ −P2θ ) ( 3C ⋅ I )( 3C ⋅ I ) e 2 s1 s1 s2 s2 * * j( ω2t−ω1t + P1θ −P2θ ) ( 3Cs1 ⋅ I s1 )( 3Cs 2 ⋅ I s2 ) e ⎫ ⎬ ⎭ ⎫ dθ ⎪ ⎪ ⎬ dθ ⎪ ⎪ ⎭ (5.80) (5.81) The above equation will be zero unless the pole numbers of two stator winding meet any of the following conditions. 1 2 P P = or P1 = −P2 Since dissimilar pole numbers are chosen in this special machine, 2π ∫ 0 1 ( θ t) ⋅ B ( θ, t) dθ = 0 J (5.82) , 2 C) The third term of T e1 For the th i rotor loop, substituting the expressions of the surface current distribution and the flux density due to the ABC winding set into the third term, 2π ∫ 0 J 1 ( θ, t) ⋅ B ( θ, t) prpi ⎧ ⎪ 1 ⎪ k = − rl Re⎨ 2 ⎪ ⎪ + ⎩ ⎡ ⎢ ⎣ ∑ ∫ ⎡ ⎢ ⎣ ∑ ∫ k 2π 0 dθ µ 0r j gk 2π 0 µ 0r j gk 2 2 k j 2ω1t + ( k −P1 ) ωrt + k ( i−1) ( 3C ⋅ I )( C ⋅ I ) e 2 2 s1 s1 R iR1 ( α −Pθ −kθ ) * * k j( ( k −P1 ) ωrt + k ( i−1) α r + P1θ −kθ ) ( 3C s1 ⋅ I s1 )( CR ⋅ I R1 ) e r 1 ⎤⎫ dθ ⎥⎪ (5.83) ⎦⎪ ⎬ ⎤⎪ dθ ⎥⎪ ⎦⎭
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AN ABSTRACT OF A DISSERTAION AN INV
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coupling and interaction terms are
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CERTIFICATE OF APPROVAL OF DISSERTA
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ACKNOWLEDGEMENTS I would like to ex
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vi Page 2.2 Machine Design I.......
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viii Page 4.2.3 Self Inductances of
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x Page 7.3.4 f = 30 Hz and f = 95 H
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xii Page 10.8 D-decomposition Metho
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LIST OF FIGURES xiv Page Figure 1.1
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Figure 3.10 The simulation of the s
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Figure 3.34 Measured flux densities
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xx Page Figure 4.8 Self-inductance
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Figure 5.2. Simulation results for
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Figure 6.8. The dynamic response of
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Figure 7.22. Copper losses of the m
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Figure 8.7. The dynamic response of
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Figure 9.4 Steady state analysis, (
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Figure 9.12. The steady state wavef
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xxxiv Page Figure 10.16 Pole-zero m
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Figure 11.2 Per phase equivalent ci
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CHAPTER 1 INTRODUCTION AND LITERATU
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The last is the recently developed
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It should be noted that at any time
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A. R. Munoz and T.A. Lipo are pione
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has to be modified to adapt to thes
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circuit model of an induction machi
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ωr stator rotor 13 ωr rotor (a) (
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1.2.4 Field Analysis Method In [5.1
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separated dc source, etc. From the
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in [8.14]. In [8.15-8.17], the stud
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[8.22] and the total losses of the
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proposed. The switching frequency i
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The actual rotor mechanical speed i
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1.2.9 Induction Machine Drive---Vec
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stator windings were used as flux s
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The rotor flux can be obtained by:
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The basic idea of a closed loop flu
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estimation. It has been claimed in
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{ k [ ˆ* ( i iˆ ) ] ( k) [ ˆ* Im
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integral and a new stretch-turn ope
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The most recent overview paper is g
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CHAPTER 2 DUAL STATOR WINDING INDUC
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Table 2.1 Parameters of machine des
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A C B 2.2.2 Air Gap Flux Density X
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where, ω s1 and s2 respectively.
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There are two unknown variables in
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Since small machines typically have
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N s E = (2.18) 4. 44 fK φ 1 m Flux
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conduct away the heat produced in t
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k cs τ s = 2 2b ⎪ ⎧ ⎡ ⎤⎪
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are: Fts = H ts( ave) d s ( ave) =
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1 2 1 = H o ( ) + H o ( ) + H cr 90
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The skew of the stator winding is n
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0 bs Figure 2.5. Detailed stator sl
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y: The pole pitch at the mid point
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L lsk = L m ( α 2) ⎪⎧ ⎡sin
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2.3.4 Rotor Bar Resistance r b The
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2.3.7 Rotor Resistance Referred to
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Substituting δ = π into (2.97) an
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atio also increases. The air gap fl
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From (2.105-2.108), the ratio of th
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2.5 Conclusions In this chapter, a
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A recently developed dual stator wi
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stator winding induction machine ba
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∫ H ⋅ dl = ∫ J ⋅ ds = N ⋅
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theory itself is correct. The inacc
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( θ ) ⋅ g( θ, θ ) − H ( 0)
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where ( θ ) n A is the average of
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Since the definition of the mutual
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3Cs1 Cs1 − Cs1 − 3Cs1 6Cs1 4Cs1
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3.4.2 Mutual Inductances of the ABC
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where, ( θ ) n is the average of t
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1 0 α 1 − r 2π α − r 2π Fig
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When the skewing factor of the roto
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3.6 Calculation of Stator-Rotor Mut
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dλ v = R ⋅ i + (3.58) dt where,
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3.7.1.3 Stator flux linkage in ABC
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where, r R is the resistance matrix
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Only terms in equation (3.77) which
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119 ( ) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
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−1 −1 −1 where, λ qdr = Tr L
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3.16 and Figure 3.17. It is found f
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Figure 3.11 The simulation of the d
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Figure 3.14 The simulation of the s
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Figure 3.16 Rotor bar currents duri
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Phase Y Figure 3.20 Air gap flux de
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Figure 3.24 Total air gap flux dens
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Figure 3.27 The air gap flux densit
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Figure 3.29 Air gap flux density co
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Figure 3.33 Air gap flux density in
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All the waveforms shown above are t
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Figure 3.39 Air gap flux density pr
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Figure 3.42 Normalized spectrum of
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CHAPTER 4 FULL MODEL SIMULATION OF
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Substituting (3.28) and (3.29) into
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Figure 4.3 Self-inductance under 20
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Figure 4.4 Mutual inductance under
Page 201 and 202: that L AB = L BA , L BC = L CB and
Page 203 and 204: Figure 4.8 Self-inductance under 10
Page 205 and 206: Figure 4.10 Mutual inductance under
Page 207 and 208: the assumption that the rotor skew
Page 209 and 210: Figure 4.15 Self-inductance under 2
Page 215 and 216: 4.4 Mutual Inductances Calculation
Page 217 and 218: Figure 4.24 Stator rotor mutual ind
Page 219 and 220: Figure 4.25 Stator rotor mutual ind
Page 221 and 222: varying inductances excited by four
Page 223 and 224: Equations similar to (4.15-4.16) we
Page 225 and 226: the number of poles for the winding
Page 227 and 228: Figure 4.29 Dynamic response of dua
Page 229 and 230: (a) (b) (c) (d) Figure 4.32. Normal
Page 231 and 232: CHAPTER 5 FIELD ANALYSIS OF DUAL ST
Page 233 and 234: In the analysis that follows the fu
Page 235 and 236: where, C s1 is the number of series
Page 237 and 238: θ , ∂B1 µ 0r = ⋅ J1 θ ∂θ
Page 239 and 240: C C = k π ⋅ d s2 s2 s2 199 (5.19
Page 241 and 242: the XYZ winding set is obtained by
Page 243 and 244: u rpi 2π ' k − jk θ −( i−1)
Page 245 and 246: The corresponding flux densities in
Page 247 and 248: The first term in equation (5.55) i
Page 249 and 250: The second term in (5.64) is zero u
Page 251: 5.2.2 Torque Equation The calculati
Page 255 and 256: * ( ) ( ) ( ) ⎛ P1 ⋅ ⋅ ⎞ j
Page 257 and 258: * Since both ( ) C 2π ∫ 0 2 * Cs
Page 259 and 260: (A) Induced voltage in the ABC wind
Page 261 and 262: 5.3.3 Equation of Torque Contribute
Page 263 and 264: winding set works as a generator, w
Page 265 and 266: possible for the brushless doubly-f
Page 267 and 268: Since the number of rotor bar is n,
Page 269 and 270: 5.6 Computer Simulation and Experim
Page 271 and 272: Figure 5.2. Simulation results for
Page 273 and 274: 5.7. Conclusions In this chapter, u
Page 275 and 276: good steady-state predictions and c
Page 277 and 278: which are superimposed some space h
Page 279 and 280: (a) (b) (c) Figure 6.1: Main flux s
Page 281 and 282: (a) (b) (c) (d) (e) Figure 6.3: Fin
Page 283 and 284: (a) (b) Figure 6.5: Induced air-gap
Page 285 and 286: Substituting (6.7) into (6.4-6.6),
Page 287 and 288: L L L L L L L λ + lr1 m1 ls1 m1 lr
Page 289 and 290: 6.4 Simulation and Experimental Res
Page 291 and 292: Figure 6.7 . Simulation results for
Page 293 and 294: Figure 6.9 Experimental results for
Page 295 and 296: diminish the contribution of the 6-
Page 297 and 298: CHAPTER 7 STEADY STATE ANALYSIS OF
Page 299 and 300: where, L L = (7.11) si mi iqdri λq
Page 301 and 302: Overall efficiency of the dual stat
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Figure 7.2 Stator current speed cha
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Figure 7.6 Power factor speed chara
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Figure 7.9 Power factor speed chara
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7.3.5 f = 35Hz and f = 90 Hz abc xy
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Figure 7.16 Torque speed characteri
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this conclusion is based on the ass
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Figure 7.21. ω s1 vs s2 ω when to
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Figure 7.25 ω s1 vs s2 ω when the
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Figure 7.28. Copper losses of the m
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7.4 Conclusions Based on the steady
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applications and possible use as st
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L pλ + λ = I − = σ (8.3) ( ω
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espectively. The stator q and d axi
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* 3 * ( V I ) V Re( M I ) 3 P p = R
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the modulation index of the rectifi
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Lie derivative and relative order d
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The above algorithm requires that t
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σ = 3 1+ K ( Vqs1iqs1 + Vds1 ds1)
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Butterworth polynomial with the den
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Table 8.1 Parameters of controllers
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generally the case to get a good pe
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Table 8.2 Machine parameters for si
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(a) (b) (c) (d) (e) (f) (g) Figure
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(a) (b) (c) (d) (e) (f) (g) (h) (i)
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(a) (b) (c) (d) (e) (f) (g) (h) (i)
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CHAPTER 9 HIGH PERFORMANCE CONTROL
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The qd0 voltage equations of a dual
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317 ( ) ( ) ( ) ( ) ( ) ⎥ ⎥ ⎥
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If the rotor speed and the rotor fl
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Infeasible region (a) Vdc1 and Vdc2
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The influences of the saturation of
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Figure 9.5(a), by varying the q-axi
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Figure 9.5 Steady state analysis, (
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L1 L3 2 ( 1− K ) Vdc1 3 = ( Vqs1i
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If the PI controllers are used and
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egulation capabilities of the contr
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(a) (b) (c) (d) (e) (f) (g) (h) (k)
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(a) (b) Figure 9.11. The starting p
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9.6 Conclusions The high performanc
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induction machine are applicable to
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In section 10.2, the fundamentals o
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The vector control of an induction
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possibilities for fault conditions,
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where, where, − r r L = (10.18) (
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qsi qsi * ( I I ) σ = K ⋅ − (1
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In the proposed control scheme, rot
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* ωr * ωr + + − ωr k p + ki s
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The value of ω 0 determines the dy
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10.4.4 Stator D-axis Current Contro
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(a) (b) (c) (d) (e) (f) Figure 10.4
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The proposed control scheme has als
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(a) (b) (c) (d) (e) (f) (g) Figure
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10.6 Full-order Flux Observer The c
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while the second three-phase windin
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L r L p ˆ λ ˆ ˆ (10.66) ri si r
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373 [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( )
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t 2bi ⎛ rsiLri K iL ⎞ ⎛ ri K
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ri D i si ri ri si mi ri 2 ( K )
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esults show that the selected obser
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Figure 10.12 The poles of the 6-pol
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383 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
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⎧t s ⎨ ⎩ts 1 2 = t = t 1 2 Th
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Figure 10.15 The poles of the machi
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X i i = X Y = Y i i ( σ , ω) ( σ
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this error is used as the feedback.
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ˆ ω rm The output error is expres
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The expressions of t1 i and t2 i ar
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397 ( ) ( ) ( ) bi ei i bi ei i ai
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The transfer function of rotor spee
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(a) (b) (c) (d) Figure 10.16 Pole-z
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Figure 10.19 Pole-zero maps with di
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Figure 10.22 Pole-zero maps with di
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should ensure the stability under a
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The boundary of the speed controlle
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10.11 Simulation Results for Sensor
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Figure 10.28 Speed estimation for 2
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(a) (b) Figure 10.31 Speed estimati
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(a) (b) (c) (d) (e) (f) Figure 10.3
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Figure 10.34 Actual and estimated v
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CHAPTER 11 HARDWARE IMPLEMENTATION
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esistance for two phase windings. T
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11.2.3 Short Circuit Test The short
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When the input voltages of 2-pole A
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winding set is generating and the o
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11.4 Per Unit Model For the fixed p
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1 = 12 2 0. 0024414 The transformat
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ase values of voltage/current. The
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Start System configuration Initiali
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CHAPTER 12 CONCLUSIONS AND FUTURE W
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Using the rotating-field theory and
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each winding have been clearly show
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voltages of different frequencies,
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REFERENCES 447
Page 489 and 490:
[2.2]. P. C. Roberts, "A Study of B
Page 491 and 492:
[5.3]. R. A. McMahon, P. C. Roberts
Page 493 and 494:
[8.17]. O. Ojo, “Performance of s
Page 495 and 496:
[10.5]. J. C. Moreira, K. T. Hung,
Page 497 and 498:
[10.27]. M. Hinkkanen, “Analysis
Page 499 and 500:
[10.48]. K. Lee and F. Blaabjerg,
Page 501 and 502:
VITA Zhiqiao Wu was born in Shashi,
Page 503:
3. Zhiqiao Wu and O. Ojo, "High Per
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