Equations, Performance, Electrical Equivalent Circuits Induction Motors Equations, Performance, Electrical Equivalent Circuits

Stator Structure: Single Winding and One Turn Rotor

Induction Motor by Bullet Points Stator generates rotating, sinusoidal B-Field: This field induces current in the rotor cage loops at The stator B-Field at each rotor wire is such that 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.

Fields and Currents: Stator Field of One Winding: Three Windings – It Rotates!! Rotor Moves More Slowly than Field – “Slip” Frequency is Induces a Current in the Rotor at Slip Frequency Lorenz Force Produces a Torque: Lots of Vibration!!!

Add a Second Loop to Smooth Things Out Put another loop at right angle to the first Torque of second loop: Result is constant torque and power!! Maximum power comes with small slip. More pairs of shorted rotor turns add to torque and power directly Heat generated in rotor by induced current – use aluminum or copper bars to maximize efficiency

Formal Transformer Analogy Mutual inductance stator to rotor is time dependent The A, B, C voltages are the line voltages The rotor voltages are Given rotor frequency, calculate currents and power, subtract rotor and winding heat to get mechanical power. Shows all stator voltages and currents are at line frequency Shows magnetizing inductance per phase is Hard to get more useful results!

Simple Per-Phase Transformer Model Know that power flow is constant at constant speed (No torque variation!) Build a per-phase model with constant impedance that is a function of rotor speed Use basic single-phase transformer model with secondary impedance dependent on rotor speed Must predict proper dependence of thermal and mechanical rotor power as functions of line voltage and rotor speed Stator field is zero-slip model because no rotor current at line speed

Electrical Equivalent Circuit of Stator Alone Applies when rotor is turning at zero slip Accounts for wire loss and stator core loss Derive from DC ( ) and extrapolated zero slip ( ) conditions Leakage inductance usually larger than for a simple transformer because of air gap and slot shape Some leakage inductance designed into slot shape to limit inrush current on startup

Deriving a Rotor Model Must give thermal and mechanical rotor power correctly Sum of thermal and mechanical power is Looks like a voltage source ( ) at line frequency driving an R-L circuit where the resistor ( ) is dependent on slip? Will use an ideal transformer to match rotor resistance to the impedance at the line connection but have no way to get the turns ratio, so only fit the scaled values Model becomes:

Electrical Equivalent Circuit with Ideal Transformer S is the “slip” or Rotor inductance is Leakage inductance!

Electrical Equivalent Circuit Referred to the Stator Do not know turns ratio or rotor bar resistance directly Map rotor “components” to line side and get: Basis for calculating efficiency, start inrush, etc. Mechanical energy is electrical loss in ; all else is heat Measure remaining parameters from locked rotor, low voltage measurement

Simplify Even More! Stator magnetizing current small – does not change voltage drop in components in series with the line. Drop in those components does not change the voltage across Lprm enough to change fields significantly – approximate!!

Things Left Out! Inrush current – some leakage inductance designed into stator slots, gap size selection, etc. to limit starting current. No-load mechanical drag from cooling, bearing friction, etc. Add “windage” – subtract fixed mechanical power for bearings, cooling, etc. Design tradeoffs with cost

Single Phase Induction Motors Big advantage: No three-phase supply – 1-breaker, 2-wires, no possibility of incorrect rotation Downside: poor efficiency, low power factor Government efficiency regulation Recently – March 2015 – applies to motors from ¼ HP Minimum efficiency requires higher cost capacitor start/run designs

Try One Winding with Same Rotor Design Same sinusoidal winding – get B field of one coil: Two counter-rotating fields mean no net torque – WILL NOT START Must have a second winding to get it started Second winding at 90 electrical degrees around stator Drive stator windings with Passive phase shift problem requires different components for start/run conditions Split-phase and capacitor start motors simply disconnect start winding after speed attained. Capacitor run motors have second, permanently connected small capacitor to convert counter-rotating fields into single unidirectional field near operating point.

Single Split-phase Motor Model: General model divided by the dotted line into rotor parameters for run direction on left and opposite direction on right. Stator LPRM divided to show magnetic coupling on each side is half of B-field. The net mechanical power and torque come from the difference in energies delivered to the two rotor resistances. For approximate calculation, it is possible to move the RS and LS components past the RCORE and LPRM connections.