an investigation of dual stator winding induction machines

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L rr µ 0rl = g 0 µ 0rl = g 0 ∫ ∫ 2π 0 2π 0 N ⎛ ⎜ni ⎝ ( θ ) ⋅ N ( θ ) ( θ ) α r ⎞ − ⎟ dθ 2π ⎠ 2 µ ⎛ ⎞ 0rl β α r = ⎜ ⎜α − − ⎟ r g0 ⎝ 3 2π ⎠ i i dθ 2 108 (3.54) The winding functions of the adjacent rotor loops will overlap each other when the rotor is skewed, such that the mutual inductances between th i and from th i and ( ) th i + k ∈[ 2, n −1] i 1 k , where n is the number of rotor bar. The mutual inductance between th i and L rm1 2 µ ⎛ ⎞ 0rl β α r = ⎜ − ⎟ g0 ⎝ 6 2π ⎠ th i + 1 rotor loop is: th + will be different 2 ⎧ θ ⎫ i −β ⎛ α ⎞ θi r ⎛ α r ⎞⎡ 1 α r ⎤ ⎪∫ ⎜− ⎟ dθ + ∫ ⎜− ⎟⎢ ( θ −θ i + β ) − ⎥dθ ⎪ 0 ⎪ ⎝ 2π θi −β ⎠ ⎝ 2π ⎠⎣ β 2π ⎦ ⎪ ⎪ θ ⎪ i + αr −β ⎛ α ⎞ ⎛ α ⎞ θi + 1+ αr −β r r ⎛ α r ⎞ ⎛ α r ⎞ ⎪+ ∫ ⎜− ⎟⋅ ⎜1− ⎟dθ + ∫ ⎜− ⎟⋅ ⎜1− ⎟dθ ⎪ θ ⎪ i −β + µ ⎝ 2π ⎠ ⎝ 2π θi 1 ⎠ ⎝ 2π ⎠ ⎝ 2π 0rl ⎠ ⎪ = ⎨ ⎬ g θ + α 0 i r ⎪ ⎡ 1 α r ⎤ ⎡ 1 α r ⎤ + ( ) ⋅ ( ) ⎪ ⎪ ∫ ⎢ θ −θ i + β − ⎥ ⎢ θi + α r −θ − + − ⎥dθ θi α β ⎣β 2π ⎦ ⎣β 2π ⎦ ⎪ ⎪ ⎪ 2 ⎪ θi + 1+ αr ⎛ α ⎞⎡ 1 α ⎤ 2π ⎛ α ⎞ ⎪ r r r ⎪+ ∫ ⎜− ⎟⎢ ( θi + α r −θ ) − ⎥dθ + ∫ ⎜− ⎟ dθ ⎪ θi + 1+ αr −β + + ⎩ ⎝ 2π ⎠⎣ β 2π θi 1 αr ⎦ ⎝ 2π ⎠ ⎭ All the mutual inductance between th i and it can be calculated by: L rm2 µ 0rl = g 0 µ 0rl = g 0 ∫ ∫ 2π 2π ⎛ ⎜ni ⎝ ⎡ ⎢ni ⎣ 2 µ ⎛ ⎞ 0rl α r = ⎜ ⎜− ⎟ g0 ⎝ 2π ⎠ 0 0 α ⎞ ⎛ ⎟ ⎜ 2π ⎠ ⎝ r ( θ ) − ⋅ n ( θ ) α r ⎞ − ⎟dθ 2π ⎠ r ( θ ) n ( θ ) − ( n ( θ ) + n ( θ ) ) i+ k i+ k α 2π i (3.55) th i + k rotor loop have the same value and i+ k 2 α ⎤ r − ⎥dθ 2 4π ⎦ (3.56)
3.6 Calculation of Stator-Rotor Mutual Inductances Both the stator and the rotor loop winding functions are represented by piecewise linear equations while the position of the th i rotor loop depends on the rotor angle. Using the expression for the mutual inductance in (3.57), the mutual inductances between th i rotor loop and the stator winding set are shown in Figure 3.8 and Figure 3.9. 2π 1 Lij = µ 0rl ∫ ⋅ ni j g 0 0 ( θ ) ⋅ N ( θ ) ⋅ dθ where, i = A, B, C and j = 1, 2, L, n , n is the rotor bar number. 109 (3.57) The results shown above are for the mutual inductance between th i rotor loop and the stator winding set. The mutual inductances between exactly the same shape, except there is a phase shift angle α r . th i + 1 rotor loop and stator have Figure 3.8 Stator rotor mutual inductance in the ABC winding set
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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
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
Page 107 and 108: 0 bs Figure 2.5. Detailed stator sl
Page 109 and 110: y: The pole pitch at the mid point
Page 111 and 112: L lsk = L m ( α 2) ⎪⎧ ⎡sin
Page 113 and 114: 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
Page 121 and 122: From (2.105-2.108), the ratio of th
Page 123 and 124: 2.5 Conclusions In this chapter, a
Page 125 and 126: A recently developed dual stator wi
Page 127 and 128: stator winding induction machine ba
Page 129 and 130: ∫ H ⋅ dl = ∫ J ⋅ ds = N ⋅
Page 131 and 132: theory itself is correct. The inacc
Page 133 and 134: ( θ ) ⋅ g( θ, θ ) − H ( 0)
Page 135 and 136: 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
Page 145 and 146: 1 0 α 1 − r 2π α − r 2π Fig
Page 147: When the skewing factor of the roto
Page 151 and 152: dλ v = R ⋅ i + (3.58) dt where,
Page 153 and 154: 3.7.1.3 Stator flux linkage in ABC
Page 155 and 156: where, r R is the resistance matrix
Page 157 and 158: Only terms in equation (3.77) which
Page 159 and 160: 119 ( ) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Page 161 and 162: −1 −1 −1 where, λ qdr = Tr L
Page 163 and 164: 3.16 and Figure 3.17. It is found f
Page 165 and 166: Figure 3.11 The simulation of the d
Page 167 and 168: Figure 3.14 The simulation of the s
Page 169 and 170: Figure 3.16 Rotor bar currents duri
Page 171 and 172: Phase Y Figure 3.20 Air gap flux de
Page 173 and 174: Figure 3.24 Total air gap flux dens
Page 175 and 176: Figure 3.27 The air gap flux densit
Page 177 and 178: Figure 3.29 Air gap flux density co
Page 179 and 180: Figure 3.33 Air gap flux density in
Page 181 and 182: All the waveforms shown above are t
Page 183 and 184: Figure 3.39 Air gap flux density pr
Page 185 and 186: Figure 3.42 Normalized spectrum of
Page 193 and 194: CHAPTER 4 FULL MODEL SIMULATION OF
Page 195 and 196: Substituting (3.28) and (3.29) into
Page 197 and 198: 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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Figure 4.8 Self-inductance under 10
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the assumption that the rotor skew
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Figure 4.15 Self-inductance under 2
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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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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
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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
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6.4 Simulation and Experimental Res
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Figure 6.7 . Simulation results for
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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
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where, L L = (7.11) si mi iqdri λq
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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
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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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