an investigation of dual stator winding induction machines

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The control parameters of a PI controller have to be properly selected to regulate the error function of the Model Reference Adaptive Scheme (MRAS), yield accurate estimated rotor speed and maintain the stability of the whole motor drive. The stability of the system in the whole variable speed range is the first concern and the condition required to achieve stability is that all the poles of the transfer function should be in the left half s-plane, which also implies that all the poles should have negative real parts. In the real applications, from the system noise point of view, the real parts of the poles cannot be too large. The zeroes of the transfer function also need to have negative real parts to ensure that the system is the minimum phase system. Several approaches have been proposed to select the control parameters and ensure the stability of the system. The Routh criteria is one of those methods, in which the coefficients of the transfer function are used to build up a Routh table, and the stability conditions can be found from the Routh table. However, because of the complexity of the coefficients, the Routh criteria is not applicable in this case. The D-decomposition is applied to select the parameters of the PI controller. Basically, the boundary of the stability region is found in the control parameters plane and the parameters within the stability region are applied to the system. There are two critical conditions that need to be examined. The first case is the condition in which the poles located at the origin of the s- plane, which can be expressed as: p = 0 (10.143) The second condition is that the poles are located at the imaginary axis of s-plane. The expression for this case is: p = ± jω (10.144) 0 398
The transfer function of rotor speed is given as: ˆ ω ω rm rm 4 3 2 A4i p + A3i p + A2i p + A1i p + A0i = 5 4 3 2 p + B p + B p + B p + B p + B 4i 3i The condition that satisfies the first condition is: B 0i = 0 ⇒ a − ω ⋅ t ei 2bi 0i = 0 ⇒ + k ⋅ω ⋅ t i ei 1bi = 0 2i 1i 0i 399 (10.145) Equation (10.145) is a function of the observer gains, machine parameters and the error function partition coefficient of active and reactive power. Since the observer gains have been determined in the previous section and the machine parameters are all known and assumed to be constant here, (10.138) becomes the function of error function partition coefficients ( k , 1 k and 2 k ). Substituting (10.144) into the denominator of (10.140), the expression satisfying the second condition is given as: 4 2 ⎧ ω0 − B3i ⋅ω0 + B1i = 0 ⎨ 4 2 ⎩B4i ⋅ω0 − B2i ⋅ω0 + B0i = 0 (10.146) Since all the B coefficients in (10.146) are functions of the controller parameters G pi and G ii , the boundary of the controller parameters can be obtained when ω 0 is changed. The boundaries of the controller parameters should be found under different slip conditions and different values of factor k i so that the controller parameters chosen guarantee stability under all operating conditions. The stability issue using only active power has been studied by plotting the zeros and poles of the transfer function for speed estimation. The observer gains and parameters of the speed estimator are kept to be constant and factor k i is set equal to zero, the poles 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
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In section 10.2, the fundamentals o
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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 and 434: ˆ ω rm The output error is expres
Page 435 and 436: The expressions of t1 i and t2 i ar
Page 437: 397 ( ) ( ) ( ) bi ei i bi ei i ai
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
Page 485 and 486: voltages of different frequencies,
Page 487 and 488: REFERENCES 447
Page 489 and 490:
[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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