an investigation of dual stator winding induction machines

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8.3 Steady State Analysis With the derivative terms in (8.1-8.9) set to be equal to zero, the system equations are used for the steady state analysis. If q- and d-axis stator and rotor fluxes are chosen as the state variables, the voltage equations for 2-pole ABC winding set are expressed in complex variable-form as : V qds1 M qds1 = ⋅Vdc 2 ⎛ rs1L r1 ⎞ = ⎜ − jωe1 ⋅λ L ⎟ ⎝ δ1 ⎠ qds1 rs1L − L m1 δ1 λ qdr1 288 (8.11) ⎛ rr1L ⎞ s1 rr1Lm1 Vqdr1 = 0 = ⎜ − jωe1 ⋅ λqdr1 − λqds1 L ⎟ (8.12) ⎝ δ 1 ⎠ Lδ 1 Similar equations are obtained for the 6-pole XYZ winding set, V qds2 M qds2 = ⋅Vdc 2 ⎛ rs 2L ⎞ r 2 = ⎜ − jωe2 ⋅ λ L ⎟ ⎝ δ 2 ⎠ qds2 rs 2L − L m2 δ1 λ qdr 2 (8.13) ⎛ rr 2 Ls 2 ⎞ rr 2 Lm2 Vqdr2 = 0 = ⎜ − jωe 2 ⋅λ qdr2 − λqds2 L ⎟ (8.14) ⎝ δ 2 ⎠ Lδ 2 where, Lδ = L L − L and L = L L − L as: 1 s1 r1 2 m1 δ 2 s2 r 2 2 m2 The dc capacitor voltage equation in steady state can be expressed in complex format V R dc L * * ( M I + M I ) 3 = − Re qds1 qds1 qds 2 qds 2 (8.15) 2 If the output powers of the 2-pole ABC winding set and the 6-pole XYZ winding set are defined as: .
* 3 * ( V I ) V Re( M I ) 3 P p = Re qds1 qds1 = dc qds1 qds1 (8.16) 2 4 * 3 * ( V I ) V Re( M I ) 3 P q = Re qds2 qds 2 = dc qds 2 qds 2 (8.17) 2 4 Then the output power ratio coefficient K between the 2-pole ABC winding set and the 6-pole XYZ winding set is expressed as: * ( M qds2I qds2 ) * ( M I ) Pq Re Pp 1 K = = , = = α P Re P + P 1+ K p qds1 qds1 p q 289 (8.18) The rotor flux terms are eliminated in equation (8.11), and the equation expressed in terms of the stator fluxes is substituted in (8.15). This results in an equation that expresses the required excitation of each winding set in terms of the magnetizing flux linkage using the magnitude of the modulation index as a measure of the excitation. M M 2 1 2 2 − 2 = 3R L ( 1+ K ) − 2K = 3R L ( 1+ K ) ⎧ ⎡ ⎪ ⎢ s1 ⎪ Re⎨2 ⎣ ⎪ ⎡ Lr ⎪ ⎢ ⎩ ⎣ L ⎧ ⎡ ⎪ ⎢ s ⎪ Re⎨2 ⎣ ⎪ ⎡ L ⎪ ⎢ ⎩ ⎣ L ( A − jω ) 1 δ1 + L e1 δ1 ( A − jω ) 2 r 2 δ 2 + L e2 δ 2 B ⎤ ⎫ s1Br1 − ⎥ ⎪ Ar1 − jωs1 ⎦ ⎪ ⎬ L B ⎤ ( ) ⎪ ⎪ m1 r1 ⎥ A − jω r1 s1 ⎦ ⎭ B ⎤ ⎫ s2B r 2 − ⎥ ⎪ Ar 2 − jωs 2 ⎦ ⎪ ⎬ L B ⎤ ( ) ⎪ ⎪ m2 r 2 ⎥ A − jω where, ω is the slip frequency of the 2-pole ABC winding set and s1 s2 r 2 s2 ⎦ ⎭ (8.19) (8.20) ω is the slip frequency of the 6-pole XYZ winding set. The other variables are defined as follows: A r L si ri si = , Lδ i A r L ri si ri = , Lδ i B − r L i mi si = , Lδ i B − r L ri mi ri = , 1 or 2 Lδ i i = . Magnetizing flux linkage saturation reflected in the values of the magnetizing inductances of the machine (Figure 8.2) has an influence on the converter excitation
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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
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The second term in (5.64) is zero u
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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
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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
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Since the number of rotor bar is n,
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5.6 Computer Simulation and Experim
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Figure 5.2. Simulation results for
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5.7. Conclusions In this chapter, u
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good steady-state predictions and c
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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
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: espectively. The stator q and d axi
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)
Page 351 and 352: (a) (b) (c) (d) (e) (f) (g) (h) (i)
Page 353 and 354: CHAPTER 9 HIGH PERFORMANCE CONTROL
Page 355 and 356: The qd0 voltage equations of a dual
Page 357 and 358: 317 ( ) ( ) ( ) ( ) ( ) ⎥ ⎥ ⎥
Page 359 and 360: If the rotor speed and the rotor fl
Page 361 and 362: 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
Page 367 and 368: Figure 9.5 Steady state analysis, (
Page 369 and 370: L1 L3 2 ( 1− K ) Vdc1 3 = ( Vqs1i
Page 371 and 372: If the PI controllers are used and
Page 373 and 374: egulation capabilities of the contr
Page 375 and 376: (a) (b) (c) (d) (e) (f) (g) (h) (k)
Page 377 and 378: (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,
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[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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