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Induction Motors Equations, Performance, Electrical Equivalent Circuits Induction Motor by Bullet Points • Stator generates rotating, sinusoidal B-Field: • This field induces current in the rotor cage loops at f f f ur ur i • The stator B-Field at each rotor wire is such that B $ • Torque pushes in direction of field rotation! (That’s it!!) • Rotor currents generate triangular B-field rotating in the air gap at slip speed relative to rotor, so at line rate in reference frame! • Rotor field reduces field in stator and line current increases to maintain the stator winding voltage and the gap magnetic field • Increased line current supplies the mechanical energy and the joule heating of the rotor. ur B 32 B0 cos( t )$r S R L R Relative Torque vs Slip Angle 1 Single Rotor Coil - Resistive Current (Net Power) Reactive Torque (No Net Power) Relative Torque 0.8 Total Torque - Single Turn 0.6 0.4 0.2 0 -0.2 0 60 120 180 Slip Angle (deg.) 240 300 360 B-Field of Rotor vs. Rotor Angle as Function of Slip Cycle Angle Relative B-Field Strength (a.u.) Offset by 3 for Each Slip Angle 24 21 18 15 0 deg. 45 deg. 12 90 deg. 135 deg. 9 180 deg. 225 deg. 6 270 deg. 315 deg. 3 0 -3 0 60 120 180 240 Angle around rotor from one loop (deg.) 300 360 Shape and Rotation of the Rotor Generated B-Field in the Gap 1 0.8 Relative Magnetic Field Strength (a.u.) 0.6 0.4 0.2 0 deg. 180 deg. 0 -0.2 -0.4 -0.6 -0.8 -1 0 60 120 180 240 300 Angle arount the rotor relative to one turn of the two rotor turns (deg.) 360 Shape and Rotation of the Rotor Generated B-Field in the Gap 1 0.8 Relative Magnetic Field Strength (a.u.) 0.6 0.4 0.2 0 deg. 45 deg. 0 90 deg. 180 deg. -0.2 -0.4 -0.6 -0.8 -1 0 60 120 180 240 Angle arount the rotor relative to one turn of the two rotor turns (deg.) 300 360 Rotor Induced Triangle Wave in Stator Current Showing Waveshape withHarmonic Multiples of 3 Removed and Compared to Sinusoid 1 0.8 0.6 Relative Current (a.u.) 0.4 0.2 Triangle Wave 0 Fourier to 27th No multiples of 3 -0.2 Single Sine -0.4 -0.6 -0.8 -1 0 60 120 180 Cycle Angle (deg.) 240 300 360 Formal Transformer Analogy • Mutual inductance stator to rotor is time dependent LP LM LM LMR cos(R t ) LMR sin(R t ) iA RW vA v LM LP LM LMR cos(R t 23 ) LMR sin(R t 23 ) iB 0 B d vC LM LM LP LMR cos(R t 43 ) LMR sin(R t 43 ) iC 0 dt 2 4 LR 0 vR1 0 LMR cos(R t ) LMR cos(R t 3 ) LMR cos(R t 3 ) iR1 0 2 4 v 0 LMR sin(R t ) LMR sin(R t 3 ) LMR sin(R t 3 ) iR 2 0 0 LR R2 0 RW 0 0 0 0 0 RW 0 0 0 0 0 RR 0 0 i A 0 iB 0 iC 0 iR1 RR iR 2 • The A, B, C voltages are the line voltages • The rotor voltages are v R i • Given rotor frequency, calculate currents and power, subtract rotor and winding heat to get mechanical power. • Ugh! R R R Simple Per-Phase Transformer Model • Know that power flow is constant at constant speed (No torque variation!) • Per-phase model with constant impedance that is a function of rotor speed • Use basic single-phase transformer model with turns ratio and secondary impedance dependent on rotor speed • Stator field is zero-slip model Electrical Equivalent Circuit of Stator Alone • • • • Applies when rotor is turning at zero slip Derive from locked rotor and zero slip conditions Accounts for wire loss and stator core loss Leakage inductance usually larger than for a simple transformer because of air gap and slot shape Electrical Equivalent Circuit with Ideal Transformer • S is the “slip” or S f SLIP f NoLoad Electrical Equivalent Circuit Referred to the Stator • Basis for calculating efficiency, start inrush, etc. • Mechanical energy is loss in (1 SS )R ; all else is heat R Things Left Out! • Inrush current • No-load mechanical drag from cooling, bearing friction, etc. • Design tradeoffs with cost • Still to go: single phase operation • Government efficiency regulation • Recently – March 2015 – applies to motors to ¼ HP