an investigation of dual stator winding induction machines

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(10.123) If the relationship between rotor mechanical speed ω rm , rotor speed of ABC winding set ω r1 and rotor speed of XYZ winding set r 2 ω = P ⋅ω r1 r 2 1 rm ω = P ⋅ω 2 rm where, 1 P and 2 ω are expressed as: 394 (10.124) P are the pole pair numbers of ABC and XYZ winding sets respectively. Then (10.123) can be rewritten in terms of rotor mechanical speed ω rm as: ˆ ω rm ⎪ ⎧ Im = −G ⋅ P1 ⋅ k⎨ ⎪⎩ + k1 − G ⋅ P ⋅ 2 = −G ⋅ P ⋅ k 1 ( 1 − k ) [ ˆ * −1 λqdr1 ⋅ ( C1) ⋅ [ p ⋅ I − A1 + K1 ⋅ C1] [ Z1 ⋅ ( ωrm − ˆ ωrm ) ] ˆ * Re λ ( C ) ⋅ [ p ⋅ I − A + K −1 ⋅ C ] [ Z ⋅ ( ω − ˆ ω ) ] ⎪ ⎧ Im ⎨ ⎪⎩ + k ⎬ [ qdr1 1 1 1 1 1 rm rm ] ⎪⎭ [ ˆ * −1 λ ( 2 ) [ 2 2 2 ] [ 2 ( ˆ qdr 2 ⋅ C ⋅ p ⋅ I − A + K ⋅ C Z ⋅ ωrm − ωrm ) ] * 1 Re[ ˆ − λ ⋅ ( C ) ⋅ [ p ⋅ I − A + K ⋅ C ] [ Z ⋅ ( ω − ˆ ω ) ] { Im[ H ] + k Re[ H ] } − G ⋅ P ⋅ ( 1 − k ) { Im[ H ] + k Re[ H ] } 1 2 1 qdr 2 where, H ˆ * −1 = λ ⋅( C ) ⋅[ p ⋅ I − A + K ⋅C ] [ Z ⋅( ω − ˆ ω ) ] i qdri i The expression of [ p I − A + K ⋅C ] is given as: H i = ˆ λ * qdri = − ˆ λ = − ˆ λ = − ˆ λ C qdri qdri qdri i ⎡ P ⎢ ⎣− P P P 2 2 2 11i ⋅ ⋅ ⋅ 22i 21i 22i ( ω − ˆ ω ) rm ( ω − ˆ ω ) rm ( ω − ˆ ω ) rm i i 1 i 2 i 2 rm 2 rm . 2 2 2 2 2 rm ⎪ ⎫ 2 rm ⎪ ⎫ ⎬ ⎪⎭ (10.125) ⋅ i i i is given in (10.74). Then the expression of H i − P12i ⎤ P ⎥ 11i ⎦ − P P 12i rm rm rm 21i Z L ⋅ D mi i L ⋅ D mi i i ( ω − ˆ ω ) rm ⎛ Lri ⎜ P Di ⋅ j ⋅ ⎝ P P 11i 12i 22i ⋅ j ⋅ P 11i 1i rm L + D − P 22i mi i 12i p − jωei ⋅ j ⋅ t + jt 2i P P 11i 21i p − jωei P − P P 12i ⎞ ⎟ ⎠ 21i (10.126)
The expressions of t1 i and t2 i are given in (10.75-10.80). Equation (10.119) can also be function expressed as: H i = − ( ω − ˆ ω ) L ( ω t + pt ) + j( pt − ω t ) 2 ˆ mi ei 1i 2i 1i ei 2i λ qdri ⋅ rm rm ⋅ ⋅ . (10.127) 2 2 Di t1i + t2i Then the imaginary and real parts of (10.127) are given as: Im Re L D 2 ( ) ˆ mi 1i H ( ˆ i = − λqdri ⋅ ωrm − ωrm ) ⋅ ⋅ 2 2 i L D pt − ω t t + t 2 ( ) ˆ mi ei 1i H ( ˆ i = − λqdri ⋅ ωrm − ωrm ) ⋅ ⋅ 2 2 i 1i 1i 2i 395 ei 2i ω t + pt t + t To simplify the analysis, the following definitions are made: f 1i f 2i L pt −ω t 2 ( ) ˆ mi 1i p = qdri ⋅ ⋅ 2 2 2i 2i (10.128) (10.129) ei 2i λ (10.130) Di t1i + t2i L pt + ω t 2 ( ) ˆ mi 2i p = qdri ⋅ ⋅ 2 2 ei 1i λ (10.131) Di t1i + t2i Then (10.125) can be rewritten as: ˆ ω rm = G ⋅ P ⋅ k 1 + G ⋅ P ⋅ 2 { f ( p) k f ( p) } ( ω ˆ 11 + 1 ⋅ 21 ⋅ rm − ωrm ) ( 1− k ) { f ( p) + k ⋅ f ( p) } ⋅ ( ω − ˆ ω ) 12 2 22 rm rm (10.132) The first idea of speed estimation is an adjustable combination of the error signals from both stator winding sets. The computed error function is expressed as: ε = k + * * { k1 Im[ ˆ λ ( ˆ ) ] ( ) [ ˆ ( ˆ qdr1 iqds1 − iqds1 + 1− k1 Re λqdr1 iqds1 − iqds1) ] } * * ( 1− k) { k Im ˆ λ ( i − iˆ ) + 1− k Re ˆ λ i − iˆ } 2 [ ] ( ) [ ( ) ] qdr 2 qds2 qds2 2 qdr 2 qds2 qds2 (10.133) where, k is used to partition the error signals from two stator windings and 0 ≤ k ≤1 ; 1 k is used to partition the active power and reactive power of ABC winding set and
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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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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
Page 383 and 384: In section 10.2, the fundamentals o
Page 385 and 386: The vector control of an induction
Page 387 and 388: possibilities for fault conditions,
Page 389 and 390: 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
Page 395 and 396: * ωr * ωr + + − ωr k p + ki s
Page 397 and 398: The value of ω 0 determines the dy
Page 399 and 400: 10.4.4 Stator D-axis Current Contro
Page 401 and 402: (a) (b) (c) (d) (e) (f) Figure 10.4
Page 403 and 404: The proposed control scheme has als
Page 405 and 406: (a) (b) (c) (d) (e) (f) (g) Figure
Page 407 and 408: 10.6 Full-order Flux Observer The c
Page 409 and 410: while the second three-phase windin
Page 411 and 412: L r L p ˆ λ ˆ ˆ (10.66) ri si r
Page 413 and 414: 373 [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( )
Page 415 and 416: t 2bi ⎛ rsiLri K iL ⎞ ⎛ ri K
Page 417 and 418: ri D i si ri ri si mi ri 2 ( K )
Page 419 and 420: esults show that the selected obser
Page 421 and 422: Figure 10.12 The poles of the 6-pol
Page 423 and 424: 383 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
Page 425 and 426: ⎧t s ⎨ ⎩ts 1 2 = t = t 1 2 Th
Page 427 and 428: Figure 10.15 The poles of the machi
Page 429 and 430: X i i = X Y = Y i i ( σ , ω) ( σ
Page 431 and 432: this error is used as the feedback.
Page 433: ˆ ω rm The output error is expres
Page 437 and 438: 397 ( ) ( ) ( ) bi ei i bi ei i ai
Page 439 and 440: The transfer function of rotor spee
Page 441 and 442: (a) (b) (c) (d) Figure 10.16 Pole-z
Page 443 and 444: Figure 10.19 Pole-zero maps with di
Page 445 and 446: Figure 10.22 Pole-zero maps with di
Page 447 and 448: should ensure the stability under a
Page 449 and 450: The boundary of the speed controlle
Page 451 and 452: 10.11 Simulation Results for Sensor
Page 453 and 454: Figure 10.28 Speed estimation for 2
Page 455 and 456: (a) (b) Figure 10.31 Speed estimati
Page 457 and 458: (a) (b) (c) (d) (e) (f) Figure 10.3
Page 459 and 460: Figure 10.34 Actual and estimated v
Page 461 and 462: CHAPTER 11 HARDWARE IMPLEMENTATION
Page 463 and 464: esistance for two phase windings. T
Page 465 and 466: 11.2.3 Short Circuit Test The short
Page 467 and 468: When the input voltages of 2-pole A
Page 469 and 470: winding set is generating and the o
Page 471 and 472: 11.4 Per Unit Model For the fixed p
Page 473 and 474: 1 = 12 2 0. 0024414 The transformat
Page 475 and 476: ase values of voltage/current. The
Page 477 and 478: Start System configuration Initiali
Page 479 and 480: CHAPTER 12 CONCLUSIONS AND FUTURE W
Page 481 and 482: 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
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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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