Equations, Performance, Electrical Equivalent Circuits 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 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.

Stator Structure: Single Winding and One Turn 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 effiency

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 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? Try a fit: Need ideal transformer to match the actual rotor resistance to the impedance at the line connection Adding more loop pairs changes rotor parameter values but not the form of the equations

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

Simplify! Do not know turns ratio or rotor bar resistance directly Map rotor “components” to line side and get:

Electrical Equivalent Circuit Referred to the Stator Basis for calculating efficiency, start inrush, etc. Mechanical energy is loss in ; all else is heat Measure remaining parameters from locked rotor, low voltage measusrment

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