INDUCTION MOTOR MODEL AND PARAMETERS
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1 APPENDIX C INDUCTION MOTOR MODEL AND PARAMETERS C.1 Dynamic Model of the Induction Motor in Stationary Reference Frame A three phase induction machine can be represented by an equivalent two phase machine as shown in Fig.C.1, where d s - q s axes correspond to stator direct and quaature axes. The dynamic model of the induction motor in the stationary reference frame [4], used for the analysis carried out in thesis has been explained here with equivalent circuits and a series of equations (C.1) to (C.10). Fig.C.1 d-q representation of the Induction Motor 195
2 (a) D-axis circuit (b) Q-axis circuit Fig.C.2 d-q equivalent circuit of Induction Motor in Stationary Reference Frame s s d s vds = Rsi ds + ( λ ds) (C.1) s s d s vqs = Rsi qs + ( λ qs) (C.2) s s d s s v = Rri + ( λ ) +ωλ r = 0 (C.3) s s d s s vqr = Rri qr + ( λqr ) ωλ r qr = 0 (C.4) s λ s = Li s + L (i s + i ) (C.5) ds ls ds m ds s s s s qs ls qs m qs qr λ = Li + L (i + i ) (C.6) λ = Li + L (i + i ) (C.7) s s s s ls m ds s s s s qr ls qr m qs qr λ = Li + L (i + i ) (C.8) 3 s s s s Te = pl m(iqsi idsi qr ) 2 (C.9) dωm 1 dωr JT = JT = Te TL p (C.10) 196
3 C.2 Dynamic Model of the Induction Motor in Synchronously Rotating Reference Frame The induction motor equations described in C.1 are referred to the stationary reference frame. These equations can also be referred to the synchronously rotating reference aligned with d-q axes shown in Fig.2.1. (a) D-axis circuit (b) Q-axis circuit Fig.C.3 d-q equivalent circuit of Induction Motor in Synchronously rotating reference frame When equations (C.1) and (C.2) are referred to the synchronously rotating reference frame rotating at ω, d vds = Rsi ds + ( λds) ωλ qs (C.11) d vqs = Rsi qs + ( λ qs) +ωλ ds (C.12) 197
4 The last terms in equations are defined as speed emfs due to the rotation of axes. That is, when ω=0, the equations revert to the stationary reference frame form. The equations (C.3) and (C.4) for the rotor become: d v = Rri + ( λ ) ( ω ωr ) λ = 0 (C.13) d vqr = Rri qr + ( λ qr ) + ( ω ωr ) λ = 0 (C.14) All the sinusoidal variables at fundamental frequency in the stationary frame appear as dc quantities in the synchronously rotating reference frame. The flux linkage expressions can be written as: λ ds = Li ls ds + L m(ids + i ) (C.15) λ = Li + L (i + i ) (C.16) qs ls qs m qs qr λ = Li ls + L m(ids + i ) (C.17) λ = Li + L (i + i ) (C.18) qr ls qr m qs qr The torque expression of the motor can be derived as follows: 3 3 Te = pl m(iqsi idsi qr ) = p( λiqr λ qri ) (C.19) 2 2 C.3 Determination of Electrical Parameters Open circuit and short circuit tests were conducted to find out the parameters of the induction motor. The stator resistance Rs was obtained with a DC measurement. The open-circuit test was conducted by supplying the rated voltage to the stator winding while iving the induction motor at its synchronous speed using the DC motor coupled to it. When the motor runs at synchronous speed, the slip will be zero and hence the rotor current becomes zero. The conventional equivalent circuit model for this test can be reduced to the one shown in Fig.C
5 Fig.C.4 Per-phase equivalent circuit under no-load test Where, V 0 the measured open-circuit phase voltage I 0 - the measured open-circuit phase current P 0 the measured open-circuit three phase power P Total input resistance 0 R = (C.20) 0 2 3I0 Totalinput impedance 0 Z = (C.21) Total input reactance Then Rm R0 Rs 0 V I 0 (C.22) = (C.23) X = X X (C.24) m 0 ls The locked rotor test was conducted by blocking the rotor and supplying the rated current from a 50Hz ac supply. Under blocked rotor condition, as the slip is equal to 1.0, the conventional equivalent circuit with the magnetizing branch neglected, will be reduced to the one shown in Fig.C
6 Total input resistance Total input impedance Total input reactance Fig.C.5 Per phase equivalent circuit under blocked rotor test R P = (C.25) sh sh 2 3Ish V Z = = R + jx (C.26) sh sh sh sh Ish X = Z R (C.27) 2 2 sh sh sh Where, V sh - the measured short-circuit phase voltage I sh - the measured short-circuit phase current P sh - the measured short-circuit three phase input power Now the rotor resistance can be calculated as R r = R sh - R s (C.28) The total leakage inductance (L ls +L lr ) has been split into two equal halves. Various other ratios have also been investigated. It was found that other ratios (up to a factor of four) do not appreciably influence the results which are obtained by splitting of the leakage inductance equally. The total leakage inductance being smaller by a factor of nearly nine, than the mutual inductance L m, this result is expected. Xls = Xlr = X sh /2 (C.29) 200
7 C.4 Determination of Mechanical Parameters J and D of the Induction Motor Setup The J and D of the induction motor can be determined by using the coupled DC shunt motor. The following equations can be derived for the DC machine: Ea = V IaR (C.30) a P= E I (C.31) Ea a a = kω (C.32) T e = ki (C.33) a dω Te = J + Dω+ T (C.34) L T T = Dω (C.35) e L In steady-state, if the load torque shown in (C.35) is equal to zero. EI a ω a = Dω (C.36) P 2 = Dω (C.37) When the DC motor was iving the induction motor as load, the voltage, current and rotor speed were recorded and the power was calculated as in (C.31). The friction coefficient D was calculated from (C.37). Where Ea Back-emf, V - DC Voltage, I a DC motor current, ω Speed in rad/sec, T e Electromagnetic Torque, T L Load Torque, D-Friction Coefficient, J Rotor Inertia. 201
8 C kW Squirrel Cage Induction Motor In order to calculate the friction coefficient, the speed of the permanent magnet DC motor was varied by controlling the DC input voltage. The input power was plotted against ω 2, where ω is the speed in rad/sec. From the slope of the plot, the friction coefficient D was calculated to be Nm/rad/second. To calculate the total inertia J of the induction-motor-dc motor set-up, a run-down test was conducted of the DC motor. Initially, the input voltage to the DC motor was adjusted so that it runs at nearly 1200rpm, and the input power measured. The input voltage was switched off and the speed was recorded until the motor comes to rest. When the electro-magnetic torque generated by the DC motor is equal to zero, equation (C.34) can be written as: dω 0= J + Dω (C.38) J can be calculated from equation (C.38). The plot of Speed vs time for the induction motor together with the DC shunt motor obtained from a run-down test. From the plot, the combined rotor inertia of the induction motor together with the DC motor has been calculated to be kg.m 2. All the parameters of the induction motor used for control and estimation are shown in Table.C
9 Table.C.I Squirrel-Cage Induction Motor Parameters Type 3-phase Star Connected Squirrel Cage Induction Motor Rated Power 1.1kW Rated Voltage 415V Rated Current 2.77A Rated Frequency 50Hz Rated Speed 1415rpm Number of Poles 4 Stator Resistance R s 6.03Ω Rotor Leakage Inductance L lr 29.9mH Stator Leakage Inductance L ls 29.9mH Magnetizing Inductance L m 489.3mH Rotor Resistance R r 6.085Ω at 50Hz Moment of Inertia J T kgm 2 Friction Coefficient D Nm/rad/second C kW Slip-ring Induction Motor In order to calculate the friction coefficient of the slip-ring motor-dc motor set-up, the speed of the DC motor was varied by controlling the DC input voltage. The input power was plotted against ω 2, where ω is the speed in radians/second. From the slope of the plot, the friction coefficient D has been calculated to be Nm/rad/second. To determine the total inertia J of the slip ring induction motor-dc motor set-up, a run-down test was conducted as explained for the squirrel Cage Induction Motor. From the test, the combined rotor inertia of the induction motor together with the DC motor has been calculated to be kg.m
10 Table.C.II Slip-Ring Induction Motor Parameters Type 3-phase Star Connected Slip Ring Induction Motor Rated Power 3.6kW Rated Voltage 415V Rated Current 7.8A Rated Frequency 50Hz Rated Speed 1410rpm Number of Poles 4 Stator Resistance R s 1.54Ω Rotor Leakage Inductance L lr 11.65mH Stator Leakage Inductance L ls 11.65mH Magnetizing Inductance L m mH Rotor Resistance R r 2.3Ω at 50Hz Moment of Inertia J T kgm 2 Friction Coefficient D Nm/rad/second 204
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