Design and Voltage Supply of High-Speed Induction - Aaltodoc

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130 A. CALCULATION OF ELECTROMECHANIC CHARACTERISTICS The calculation of electromagnetic characteristics, including the electromagnetic torque, is based on the finite element method, FEM. Only the basic idea and the equations for losses are presented here. The electromagnetic model is based on Maxwell’s equations. Even in high-speed machines, the frequencies of the electromagnetic quantities are low enough to use the quasi-static formulation. There, the displacement current term ∂D /∂t is omitted from the equations. In the quasi-static case, the magnetic vector potential A is ( ∇ × A ) = J ∇ × ν r , (A1) where νr is reluctivity of a material and J is a current density. The current density can be expressed using the magnetic vector potential and scalar potential φ of an electric field: ∂A J = −γ − γ∇φ , (A2) ∂t where γ is electric conductivity. In a 2D case, A and J only have components normal to the 2D- plane, which is selected to be the cross-section of the motor. Eq. A2 can be substituted in A1, assuming that the electric potential difference u between the ends of the conductor is known. The substitution yields ∂A γ ∇ × ( ν r∇ × A) + γ = ue z , (A3) ∂t l where l is the length of a conductor. Integrating the current density in Eq. A2 over the cross section of the conductor, a relation between the u and the current i of the conductor becomes: ∂A u = Ri + R∫ γ ⋅ dS , (A4) ∂t S j where R is a DC-resistance. The end winding impedance of the stator windings and the supply voltage are taken into account using Kirchoff laws for electric circuits together with Eq. A4. The end ring of the squirrel cage is modeled with circuit equations. The resistance is calculated from the conductivity and the dimensions of the end ring. The end ring inductance is modeled with a separate 2D FEM model. For the solid iron, there is no end ring impedance modeled. Thus, the
currents in the solid iron flow more freely as they would in the real case. The total impedance of the rotor is underestimated and the torque produced by these currents is slightly overestimated as a function of a slip. In the case of a coated rotor, the coating is divided into bars and the end ring sections. The third dimension, the length of the machine is defined by the effective motor length leff, which takes into account the axial flux fringing at the air gap region. The flux fringing and the determination of the effective length is modeled with a separate 2D FEM model. An electromagnetic torque is calculated using Maxwell’s stress tensor, integrating the tensor over the air gap region Sag. l T = S , (A5) e ( ) ∫ eff rBrBϕ d 0 rs − r Sag where µ0 is the permeability of vacuum and rs and rr are outer and inner radii of air gap, respectively. Br and Bϕ are radial and tangential components of magnetic flux density, respectively. The computation is done using either a time-harmonic formulation or a time discretization. The solution of the equations yields the magnetic vector potential, the currents for the stator windings and the voltages for the rotor bars. In time-harmonic formulation, the rotation of the rotor is approximated by multiplying the conductivities of the rotor conductors with the value of the slip. The non-linearity of iron is taken into account by using effective reluctivity. In the time stepping formulation, the time is discretized into time steps using the Crank-Nicholson method. At each time step, the rotor is rotated by an appropriate angle corresponding to its rotational speed. Non-linearity of the magnetic materials is taken into account and the system of the non-linear equations is solved with the Newton-Raphson iteration method. With the time-stepping method, any supply waveforms can be modeled. This is a basic requirement in order to model the inverter fed machines. The time stepping method has been used in the study to obtain the results for the calculations. However, the harmonic analysis is needed to give an initial starting point for the time-stepping method. The initial point given by the harmonic analysis is improved by computing a DC field from the instantaneous values of the currents solved. Thus, the initial starting point includes the instantaneous currents from the time harmonic solution and the DC-field based on those currents. 131
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El 108 ACTA POLYTECHNICA SCANDINAVI
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PREFACE This thesis is the result o
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5 Comparison of rotors for 60 kW 60
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LIST OF SYMBOLS a Number of paralle
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Rn Resistance of a parallel conduct
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1 INTRODUCTION The need for high ro
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Conventional (high-speed) drive Hig
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Chapter 5 reports the comparison of
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Some penetration depth values are c
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2.2 Stator windings for high-speeds
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⎛ π ⎞ sin⎜ν ⎟ ⎛ π ⎞
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Table 2.3 Air gap characteristics o
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Cooling of the stator could need ax
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f sw N p = (2.9) fs The switching f
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3 ROTOR CONSTRUCTIONS Starting from
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and eddy-currents traveling on the
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support the squirrel cage winding,
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3.1.3 Rotor loss and friction loss
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Table 3.1 High-speed induction moto
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critical speeds are passed relative
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Table 3.2 The free-free natural fre
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Table 3.3 The free-free natural fre
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4 COMPARISON OF ROTORS FOR 65 KW 30
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Temperature [ºC] 110 109 108 107 1
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Temperature rise [K] 120 100 80 60
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improve the laminated design, more
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The copper coated solid steel rotor
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As reported in Lähteenmäki et al.
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Temperature rise [K] 100 80 60 40 2
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6. The rest of the loss is defined
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Temperature rise [K] 100 80 60 40 2
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5.2.7 About the mechanical robustne
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6.1 Comparison of PAM and PWM for 6
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h d = , (6.2) I I h, PAM where Ih,P
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Fig. 6.4 Phase-to-phase voltage and
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Machines loss [W] 12000 10000 8000
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considerably higher than expected.
Page 79 and 80: In order to improve the performance
Page 81 and 82: l = + , (7.2) sλs 2leλ e l wλ w
Page 83 and 84: gap of 1 mm would yield an effectiv
Page 85 and 86: Table 7.1 Calculated effective leng
Page 87 and 88: Input current [A] 170 160 150 140 1
Page 89 and 90: 8 CIRCULATORY CURRENTS IN A STATOR
Page 91 and 92: Table 8.1 Circulatory current loss
Page 93 and 94: Fig. 8.4 Strand currents in the two
Page 95 and 96: 8.2 Impedance of the circulatory cu
Page 97 and 98: Because Φ l is approximately at th
Page 99 and 100: Table 8.2 Circulatory current loss
Page 101 and 102: Power loss [W] 3500 3000 2500 2000
Page 103 and 104: k cc 600 500 400 300 200 100 0 0 50
Page 105 and 106: 9 OPTIMIZATION OF SOLID STEEL ROTOR
Page 107 and 108: whole solution space determined by
Page 109 and 110: the average time saving is 39 %. Th
Page 111 and 112: Fig. 9.3 Cross sections of the 16,
Page 113 and 114: Height of bar slot [mm] 8 6 4 2 0 R
Page 115 and 116: The different rotor topologies were
Page 117 and 118: possible. The coating practically v
Page 119 and 120: Table 9.3. Optimization results for
Page 121 and 122: Fig. 9.11 Initial (left) and opt. (
Page 123 and 124: high-speed induction machine. Accor
Page 125 and 126: Foelsch K. 1936. Magnetfeld und Ind
Page 127 and 128: Pyrhönen J. and Kurronen P. 1994.
Page 129: APPENDIX 129
Page 133 and 134: where the C sw E is the loss coeffi
Page 135 and 136: B. CALCULATION OF THERMAL CHARACTER
Page 137 and 138: Axial cooling duct Frame R fr3 Stat
Page 139 and 140: Linear equalities Ax = b, (C3) wher
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Design and Voltage Supply of High-Speed Induction - Aaltodoc

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