an investigation of dual stator winding induction machines

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associated with rotor misalignment, rotor shaft bending, weak bearings and manufacturing tolerances. The behavior and the performance of induction machines under rotor eccentricity conditions have been a subject of recent investigations. Generally speaking, the air gap eccentricity can be classified as: static eccentricity, dynamic eccentricity and mixed eccentricity [4.4]. The diagram of different eccentricity conditions are shown in Figure 1.2. In the uniform air gap condition, the center of the rotor and the center of the stator have the same location as shown in Figure 1.2(a). However, the location of the centers of the rotor and stator are not the same under all eccentricity conditions. The differences between the eccentricity conditions are explained as: under static eccentricity condition, the rotor still rotates around its own center axis and the air gap is not uniform as shown in Figure 1.2(b). The rotor rotates around the stator’s center axis under dynamic eccentricity condition. The central axis shown in Figure 1.2(c) is the one for the stator. For mixed eccentricity condition, the rotor will still rotate around the rotor’s central axis while the rotor’s central axis rotates around stator central axis. Both the central axis of the rotor and stator are shown in Figure 1.2(d) and the possible rotating behaviors are represented by arrows. Some of the previous works focus on the definitions of the air gap and the approximation of the inverse of the air gap, the determination of the harmonic components present in the machine variables (currents, torques etc.) due to rotor eccentricity, the computer simulation of the machine under eccentricity and the evaluation of the inductances [3.7-3.9, 4.1-4.6]. Based on the winding function method and coupled-circuit model simulation, the modeling methodology of the induction machine under eccentricity conditions has been proposed in [3.7]. 12
ωr stator rotor 13 ωr rotor (a) (b) ωr rotor stator ωr rotor (c) (d) stator stator Figure 1.2 The diagrams of uniform air gap and air gap eccentricity conditions. (a) uniform air gap condition, (b) static eccentricity condition, (c) dynamic eccentricity condition, (d) mixed eccentricity condition However, the winding function definition under eccentricity conditions is different from the one under uniform air gap length condition, which is an important issue that was ignored by previous authors. As a result the inductance calculation and simulation results have questionable credibility. The proposed methodology for the study of eccentricity conditions has been adopted for the future work. The same methodology has been applied for a synchronous reluctance machine as shown in [3.8]. An extension of the winding
Page 1 and 2: AN ABSTRACT OF A DISSERTAION AN INV
Page 3 and 4: coupling and interaction terms are
Page 5 and 6: CERTIFICATE OF APPROVAL OF DISSERTA
Page 7 and 8: ACKNOWLEDGEMENTS I would like to ex
Page 9 and 10: vi Page 2.2 Machine Design I.......
Page 11 and 12: viii Page 4.2.3 Self Inductances of
Page 13 and 14: x Page 7.3.4 f = 30 Hz and f = 95 H
Page 15 and 16: xii Page 10.8 D-decomposition Metho
Page 17 and 18: LIST OF FIGURES xiv Page Figure 1.1
Page 19 and 20: Figure 3.10 The simulation of the s
Page 21 and 22: Figure 3.34 Measured flux densities
Page 23 and 24: xx Page Figure 4.8 Self-inductance
Page 25 and 26: Figure 5.2. Simulation results for
Page 27 and 28: Figure 6.8. The dynamic response of
Page 29 and 30: Figure 7.22. Copper losses of the m
Page 31 and 32: Figure 8.7. The dynamic response of
Page 33 and 34: Figure 9.4 Steady state analysis, (
Page 35 and 36: Figure 9.12. The steady state wavef
Page 37 and 38: xxxiv Page Figure 10.16 Pole-zero m
Page 39 and 40: Figure 11.2 Per phase equivalent ci
Page 41 and 42: CHAPTER 1 INTRODUCTION AND LITERATU
Page 43 and 44: The last is the recently developed
Page 45 and 46: It should be noted that at any time
Page 47 and 48: A. R. Munoz and T.A. Lipo are pione
Page 49 and 50: has to be modified to adapt to thes
Page 51: circuit model of an induction machi
Page 55 and 56: 1.2.4 Field Analysis Method In [5.1
Page 57 and 58: separated dc source, etc. From the
Page 59 and 60: in [8.14]. In [8.15-8.17], the stud
Page 61 and 62: [8.22] and the total losses of the
Page 63 and 64: proposed. The switching frequency i
Page 65 and 66: The actual rotor mechanical speed i
Page 67 and 68: 1.2.9 Induction Machine Drive---Vec
Page 69 and 70: stator windings were used as flux s
Page 71 and 72: The rotor flux can be obtained by:
Page 73 and 74: The basic idea of a closed loop flu
Page 75 and 76: estimation. It has been claimed in
Page 77 and 78: { k [ ˆ* ( i iˆ ) ] ( k) [ ˆ* Im
Page 79 and 80: integral and a new stretch-turn ope
Page 81 and 82: The most recent overview paper is g
Page 83 and 84: CHAPTER 2 DUAL STATOR WINDING INDUC
Page 85 and 86: Table 2.1 Parameters of machine des
Page 87 and 88: A C B 2.2.2 Air Gap Flux Density X
Page 89 and 90: where, ω s1 and s2 respectively.
Page 91 and 92: There are two unknown variables in
Page 93 and 94: Since small machines typically have
Page 95 and 96: N s E = (2.18) 4. 44 fK φ 1 m Flux
Page 97 and 98: conduct away the heat produced in t
Page 99 and 100: k cs τ s = 2 2b ⎪ ⎧ ⎡ ⎤⎪
Page 101 and 102: are: Fts = H ts( ave) d s ( ave) =
Page 103 and 104:
1 2 1 = H o ( ) + H o ( ) + H cr 90
Page 105 and 106:
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
Page 115 and 116:
2.3.7 Rotor Resistance Referred to
Page 117 and 118:
Substituting δ = π into (2.97) an
Page 119 and 120:
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
Page 125 and 126:
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
Page 137 and 138:
Since the definition of the mutual
Page 139 and 140:
3Cs1 Cs1 − Cs1 − 3Cs1 6Cs1 4Cs1
Page 141 and 142:
3.4.2 Mutual Inductances of the ABC
Page 143 and 144:
where, ( θ ) n is the average of t
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1 0 α 1 − r 2π α − r 2π Fig
Page 147 and 148:
When the skewing factor of the roto
Page 149 and 150:
3.6 Calculation of Stator-Rotor Mut
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dλ v = R ⋅ i + (3.58) dt where,
Page 153 and 154:
3.7.1.3 Stator flux linkage in ABC
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where, r R is the resistance matrix
Page 157 and 158:
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
Page 165 and 166:
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
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
CHAPTER 4 FULL MODEL SIMULATION OF
Page 195 and 196:
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:
Page 205 and 206:
Page 207 and 208:
the assumption that the rotor skew
Page 209 and 210:
Page 211 and 212:
Page 213 and 214:
Page 215 and 216:
4.4 Mutual Inductances Calculation
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Figure 4.24 Stator rotor mutual ind
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Figure 4.25 Stator rotor mutual ind
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varying inductances excited by four
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Equations similar to (4.15-4.16) we
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the number of poles for the winding
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Figure 4.29 Dynamic response of dua
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(a) (b) (c) (d) Figure 4.32. Normal
Page 231 and 232:
CHAPTER 5 FIELD ANALYSIS OF DUAL ST
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In the analysis that follows the fu
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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
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Re j( ω ) ⎧ 1t− P1θ µ 0r j(
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* ( ) ( ) ( ) ⎛ P1 ⋅ ⋅ ⎞ j
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* Since both ( ) C 2π ∫ 0 2 * Cs
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(A) Induced voltage in the ABC wind
Page 261 and 262:
5.3.3 Equation of Torque Contribute
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winding set works as a generator, w
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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
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good steady-state predictions and c
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which are superimposed some space h
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(a) (b) (c) Figure 6.1: Main flux s
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(a) (b) (c) (d) (e) Figure 6.3: Fin
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(a) (b) Figure 6.5: Induced air-gap
Page 285 and 286:
Substituting (6.7) into (6.4-6.6),
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L L L L L L L λ + lr1 m1 ls1 m1 lr
Page 289 and 290:
6.4 Simulation and Experimental Res
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Figure 6.7 . Simulation results for
Page 293 and 294:
Figure 6.9 Experimental results for
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diminish the contribution of the 6-
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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
Page 317 and 318:
Figure 7.25 ω s1 vs s2 ω when the
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Figure 7.28. Copper losses of the m
Page 321 and 322:
7.4 Conclusions Based on the steady
Page 323 and 324:
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
Page 331 and 332:
the modulation index of the rectifi
Page 333 and 334:
Lie derivative and relative order d
Page 335 and 336:
The above algorithm requires that t
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σ = 3 1+ K ( Vqs1iqs1 + Vds1 ds1)
Page 339 and 340:
Butterworth polynomial with the den
Page 341 and 342:
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
Page 357 and 358:
317 ( ) ( ) ( ) ( ) ( ) ⎥ ⎥ ⎥
Page 359 and 360:
If the rotor speed and the rotor fl
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Infeasible region (a) Vdc1 and Vdc2
Page 363 and 364:
The influences of the saturation of
Page 365 and 366:
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
Page 383 and 384:
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) (
Page 391 and 392:
qsi qsi * ( I I ) σ = K ⋅ − (1
Page 393 and 394:
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
Page 483 and 484:
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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an investigation of dual stator winding induction machines

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