an investigation of dual stator winding induction machines

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This expression shows that an additional electric field will be induced by a time varying magnetic field and consequently a voltage is produced in a closed coil placed in the magnetic field. If the Electro-Motive Force (EMF) in equation (3.12) is replaced by the negative of the voltage − v , a conventional expression will be found as: dψ v = (3.13) dt 3.3 General Winding Function Expression The winding function theory which is used to calculate the inductances of induction machines was developed in the 1960's. The winding function theory has been used with success to calculate the inductances of induction and synchronous machines with constant or variable air gap length with or without rotor eccentricities [3.6-3.8]. It can be found from previously published papers that the winding function is only considered after the average of the turn function is removed from itself. This winding function definition yields correct results when the air gap of the machine is constant around the inner surface of the stator. However, if the same definition of the winding function is applied under rotor eccentricity conditions, in which the air gap length of the machine is no longer constant, the equality of L 12 = L21 = L cannot be observed by using the winding function definition for inductance calculation under rotor eccentricity conditions. This inequality, which does not agree with the equality that we can find in the linear magnetic circuit, was first reported in [3.7]. It is very difficult to find a proper explanation for this inequality. As a result, a method called the modified winding function approach and extension of winding function theory were proposed in [3.8, 3.9]. The winding function 90
theory itself is correct. The inaccuracy under rotor eccentricity conditions comes from some simplification or assumption during the equation derivation. Hence the name “extension of the winding function theory” may not be proper. The derivation of the general winding function from the fundamental relationships is the subject of this section. With all the assumptions or simplifications removed, a general equation for the winding function of a winding distribution can be obtained. A general diagram of an electric machine is given in Figure 3.1, in which the air gap length may not be constant. A closed area which includes the stator core, the air gap and the rotor core can be found and its boundary is shown as a dash line in the figure, where AB is in the stator core; O is the central point of stator; OA and OB go through the rotor, the air gap and the stator core and can be considered to be orthogonal to the inner surface of the stator. Based on the Ampere's Law, the magnetic field (H) of the boundary AOB can be expressed in terms of the current density J, the differential length ∂l and the area ∂s as: ∫ C ∫ H ⋅ dl = J ⋅ ds (3.14) S If the turn function of an arbitrary winding A is ( θ ) the inner surface of the stator, then equation (3.14) can be written as: ( ) iA 91 n A , where θ is the angle around ∫ H ⋅ dl = nA θ ⋅ C (3.15) There are two assumptions that need to be clarified before the next step. The first one is: The air gap is so small compared to the stator or the rotor core that the magnetic field in the air gap can be considered to be orthogonal to the inner surface of the stator. This is a very fundamental assumption in electric machine analysis. The second one is:
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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
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
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: ∫ H ⋅ dl = ∫ J ⋅ ds = N ⋅
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 and 148: When the skewing factor of the roto
Page 149 and 150: 3.6 Calculation of Stator-Rotor Mut
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
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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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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,
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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