an investigation of dual stator winding induction machines

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The first component of the above equation is zero unless 1 P k = − and the second component is also zero except when 1 P k = . Hence, the final result can be written as: * ( ) ( ) ⎛ P1 ⋅ ⋅ ⎞ j 2ω1t −2P1 ωrt −P1 i−1 3C I C I e ⎧ µ r 2 0 ( α ) ⎫ r ⎪ j 2 s1 s1 ⎜ R iR1 ⎟ ⎪ ⎪ − gP1 ⎝ ⎠ ⎪ −πrl Re ⎨ ⎬ (5.84) ⎪ µ r 2 * 0 * P1 jP1 ( i−1) αr + j 2 ( 3C ⋅ )( ⋅ ) ⎪ s1 I s1 CR IiR1 e ⎪⎩ gP ⎪ 1 ⎭ Summing the torques due to all the rotor bars, the torque component becomes: 2π ∫ 0 J 1 ( θ, t) ⋅ B ( θ , t) prp ⎧ µ 0r ⎪ j ⎪ − gP1 = −πrl Re⎨ ⎪ µ 0r + j ⎪ ⎩ gP1 dθ 2 2 ⎛ ⎜3C s1⋅ I ⎝ D) The fourth term of T e1 s1 ⋅C * P1 R 2 ⎜⎛ * * 2 3Cs1 ⋅ Is1 ⋅C ⎝ ⎞ ⎟e ⎠ P1 R ⎟⎞ ⋅ ⎠ ( 2ω t −2 P ω t ) − j( P ( i−1) α ) j N 1 r ∑ i= 1 I 214 1 iR1 r ⋅ e ⋅ r ∑ i= 1 ( P ( i−1) α ) j N 1 I iR1 r ⋅ e 1 r ⎫ ⎪ ⎬ ⎪ ⎪ ⎭ (5.85) For the th i rotor loop, substituting the expressions of winding surface current distribution and the flux density into the last term, 2π ∫ 0 J 1 ( θ, t) ⋅ B ( θ, t) qrqi ⎧ ⎪ 1 ⎪ k = − rl Re⎨ 2 ⎪ ⎪ + ⎩ ⎡ ⎢ ⎣ ∑ ∫ ⎡ ⎢ ⎣ ∑ ∫ k 2π 0 dθ µ 0r j gk 2π 0 µ 0r j gk 2 2 k j ω1t + ω2t + ( k −P2 ) ωrt + k ( i−1) ( 3C ⋅ I )( C ⋅ I ) e 2 2 s1 s1 R iR2 ( α −Pθ −kθ ) * * k j( −ω1t + ω2t + ( k −P2 ) ωrt + k ( i−1) α r + P1θ −kθ ) ( 3C s1 ⋅ I s1 )( CR ⋅ I iR2 ) e r 1 ⎤ ⎫ dθ ⎥ ⎪ ⎦ ⎪ ⎬ ⎤⎪ dθ ⎥⎪ ⎦⎭ (5.86) The first component of the above equation is zero except 1 P k = − and the second component is zero unless 1 P k = . The final result of (5.86) can be written as:
* ( ) ( ) ( ) ⎛ P1 ⋅ ⋅ ⎞ j ω1t + ω2t + −P1 −P2 ωrt −P1 i−1 3C I C I e ⎧ µ r 2 0 ( α ) ⎫ r ⎪ j 2 s1 s1 ⎜ R iR2 ⎟ ⎪ ⎪ − gP1 ⎝ ⎠ ⎪ −πrl Re ⎨ ⎬ (5.87) ⎪ µ r 2 * 0 * P1 j( −ω1t + ω2t+ ( P1 −P2 ) ωrt + P1 ( i−1) αr ) + j 2 ( 3C ⋅ )( ⋅ ) ⎪ s1 I s1 CR I iR2 e ⎪⎩ gP ⎪ 1 ⎭ For all the rotor loops, the torque becomes, 2π ∫ 0 J 1 ( θ, t) ⋅ B ( θ, t) qrq ⎧ µ 0r ⎪ j ⎪ − gP1 = −πrl Re⎨ ⎪ µ 0r + j ⎪ ⎩ gP1 dθ 2 2 ⎛ ⎜3C s1⋅ I ⎝ s1 ⋅C * P1 R 2 ⎜⎛ * * 2 3Cs1 ⋅ Is1 ⋅C ⎝ ⎞ ⎟e ⎠ P1 R ⎟⎞ e ⎠ ( ω t + ω t + ( −P −P ) ω t ) − j( P ( i−1) α ) j ( −ω t + ω t + ( P −P ) ω t ) j( P ( i−1) α ) j 1 1 2 215 2 1 1 2 2 r r ⋅ N r ∑ i= 1 ⋅ N I r ∑ i= 1 iR2 I iR2 ⋅ e ⋅ e 1 1 r r ⎫ ⎪ ⎬ ⎪ ⎪ ⎭ (5.88) 5.2.2.2 Torque Components due to Currents Flowing in the XYZ winding Set. The torque component due to currents flowing in the XYZ winding set can be calculated as: T e2 = −rl 2π ∫ 0 J 2 2π ⎡ ⎢∫ J 2 = − ⎢ 0 rl ⎢ 2π ⎢+ ⎢ ∫ J ⎣ 0 ( θ, t) ⋅ B ( θ, t) g dθ 2π ( θ, t) ⋅ B ( θ , t) dθ + J ( θ , t) ⋅ B ( θ , t) 2 1 ∫ 2π ( θ, t) ⋅ B ( θ, t) dθ + J ( θ, t) ⋅ B ( θ , t) prp 0 2 ∫ 0 2 2 qrq dθ ⎤ ⎥ ⎥ ⎥ dθ ⎥ ⎥⎦ (5.89) The calculation method for the developed torque components resulting from currents flowing in the XYZ winding set is the same as the approach used for the determination of the torque generated due to currents flowing in the ABC winding set. The results are given below. A) The first term of T e2 Substituting the expressions of surface current distribution and the flux density into the first term,
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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
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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 and 252: 5.2.2 Torque Equation The calculati
Page 253: Re j( ω ) ⎧ 1t− P1θ µ 0r 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
Page 303 and 304: 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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