an investigation of dual stator winding induction machines

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* ( i ) 3 Pi Tei = Im λqdsi qdsi (7.17) 2 2 where, * represents complex conjugation of variables. The total electromagnetic torque of the dual stator winding induction machine is the sum of the torques due to the two stator winding sets, which can be expressed as: * 3 P2 * ( λ i ) + Im( i ) 3 P = (7.18) 2 2 2 2 1 Te Te1 + Te2 = Im qds1 qds1 λqds2 qds2 Then by substituting (7.15) and (7.16) into (7.18), the torque equation can be expressed in terms of the input voltages, input frequencies and machine parameters as: T e = 3 P1 L 2 2 D m1 1 3 P2 L + 2 2 D m2 2 ( ) ( ( ) ) ( ( ) ) ( ( ) ( ) ) ( ( ) ) 2 2 2 2 Ar1ω s1Vs1 Br1 + ωs1 2 2 2 2 2 2 Cs1 Br1 + ωs1 − As1Ar 1Br1 + ωe1 Br1 + ωs1 + As1A r1ωs1 C 2 2 2 Ar 2ωs 2Vs 2 Br 2 + ωs 2 2 2 2 2 2 B + ω − A A B + ω B + ω + A A ω s2 r 2 s2 s2 r 2 r 2 260 e2 r 2 s2 s2 r 2 s2 (7.19) If the machine parameters are assumed to be constant at any operating condition and the constant V/Hz control is applied to the machine, the variables in equation (7.19) are electromagnetic torque T e , rotor mechanical speed ω rm , electric speed of the ABC winding set ω e1 and electric speed of the XYZ winding set e2 ω . It should be noticed that only three of them are independent, which means that if any three of them are known, the last one is also fixed. The complex power equation of the machine is given by * ( V I ) 3 S i = qdsi qdsi (7.20) 2 The power factor of the dual stator induction machine can then be expressed as: ( Si ) ( S ) Re Power Factor ( PF) = (7.21) Abs i
Overall efficiency of the dual stator induction machine is given by, Te ⋅ωrm η = (7.22) Re S ( S ) + Re( ) 1 2 The copper loss of dual stator winding induction machine is given as: 3 2 3 2 P copper = rs1 iqds1 + rs 2 iqds2 (7.23) 2 2 The copper loss equation can also be expressed in terms of the stator voltages as 2 2 ⎡⎛ L ⎞ ⎛ ⎞ ⎤ r1 Lm1 Ar1Br 1 Lm1 Ar1ω s1 ⎢ ⎥ 2 2 2 2 3 ⎢ ⎜ − ( ) ⎟ + ⎜ ( ) ⎟ V 1 1 1 1 1 1 1 ⎣⎝ D D Br + ωs ⎠ ⎝ D Br + ωs ⎠ ⎥ P ⎦ copper = rs1 2 2 2 ⎛ A ⎞ ⎛ ⎞ s1Ar 1Br1 As1 Ar1ω s1 ⎜ ⎜C 1 2 2 ⎟ ⎜ 1 2 2 ⎟ s − + ωe + ⎝ Br1 + ωs1 ⎠ ⎝ Br1 + ωs1 ⎠ 2 2 ⎡⎛ L ⎞ ⎛ ⎞ ⎤ r 2 Lm2 Ar 2Br 2 Lm2 Ar 2ωs 2 2 ⎢ + ⎥ 2 2 2 2 2 3 ⎢ ⎜ − 2 2 ( 2 2 ) ⎟ ⎜ 2 ( 2 2 ) ⎟ Vs ⎣⎝ D D Br + ωs ⎠ ⎝ D Br + ωs ⎠ ⎥ + r ⎦ s2 2 2 2 ⎛ A ⎞ ⎛ ⎞ s2 Ar 2Br 2 As 2 Ar 2ωs 2 ⎜ ⎜C 2 − + 2 2 ⎟ ⎜ 2 + 2 2 ⎟ s ωe ⎝ Br 2 + ωs 2 ⎠ ⎝ Br 2 + ωs 2 ⎠ 7.3 Steady State Analysis 261 2 s1 (7.24) In the first part of steady state analysis, four typical steady state curves of the dual stator winding induction machine---the output electromagnetic torque vs speed curve, the input stator current vs speed curve, the input power factor vs speed curve and the efficiency vs rotor speed curve, are given. The constant V/Hz control is applied to the machine and the rated V/Hz value of 2.99V/Hz is used in the steady state analysis. Since the pole ratio between the two winding sets of the dual stator winding induction machine is 3, the input frequency ratio is normally kept to be 3 to avoid extreme losses. In the following analysis, different values of the input voltage frequency ratio of the dual stator
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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
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the assumption that the rotor skew
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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
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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
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θ , ∂B1 µ 0r = ⋅ J1 θ ∂θ
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C C = k π ⋅ d s2 s2 s2 199 (5.19
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the XYZ winding set is obtained by
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u rpi 2π ' k − jk θ −( i−1)
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The corresponding flux densities in
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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 and 254: Re j( ω ) ⎧ 1t− P1θ µ 0r j(
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: where, L L = (7.11) si mi iqdri λq
Page 303 and 304: Figure 7.2 Stator current speed cha
Page 305 and 306: Figure 7.6 Power factor speed chara
Page 307 and 308: Figure 7.9 Power factor speed chara
Page 309 and 310: 7.3.5 f = 35Hz and f = 90 Hz abc xy
Page 311 and 312: Figure 7.16 Torque speed characteri
Page 313 and 314: this conclusion is based on the ass
Page 315 and 316: Figure 7.21. ω s1 vs s2 ω when to
Page 317 and 318: Figure 7.25 ω s1 vs s2 ω when the
Page 319 and 320: 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
Page 325 and 326: L pλ + λ = I − = σ (8.3) ( ω
Page 327 and 328: espectively. The stator q and d axi
Page 329 and 330: * 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
Page 337 and 338: 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
Page 343 and 344: generally the case to get a good pe
Page 345 and 346: Table 8.2 Machine parameters for si
Page 347 and 348: (a) (b) (c) (d) (e) (f) (g) Figure
Page 349 and 350: (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
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[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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