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

2. 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.

3. Stator Structure: Single Winding and One Turn Rotor

4. 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!!!

6. 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

12. 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!

13. 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

14. 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

15. 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

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

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

18. 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

19. 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