Electromagnetic FEM Analysis of Induction Motors
Theory: slip creates current and torque
Overview
Starting torque from the induction-motor equivalent circuit came out nearly 30% below measurement. What's missing?
Probably the 'deep-bar effect' — the redistribution of current inside rotor bars — is missing. An induction motor produces torque from currents induced in the rotor by the difference between the rotating field and rotor speed (slip). At rated running slip is a few percent and rotor current frequency is slow, about 1–2 Hz; at start slip is 1 and the rotor carries 50 Hz current. Skin effect then crowds current toward the top of the bars, raising resistance and torque. Holding the circuit resistance constant underestimates starting torque.
Equivalent circuit and torque
The per-phase equivalent circuit. $R_1$, $X_1$ are stator; $R_2'$, $X_2'$ rotor referred to the stator; $X_m$ magnetising reactance; $\omega_s$ synchronous angular speed. Power dissipated in $R_2'/s$ corresponds to torque.
Deep-bar effect
From the ratio $\xi$ of bar depth $h$ to skin depth $\delta$ at rotor frequency $sf$, the resistance factor $K_R$ follows. At small slip the frequency is low and $K_R$ is nearly 1.
Tesla's rotating field
The induction motor began around 1888, when Tesla and the Italian Ferraris independently showed how polyphase AC creates a rotating magnetic field. Needing neither brushes nor commutator, its simple construction made it the workhorse of factory power. Deep or double cage bars to raise starting torque emerged in the early 20th century. Today inverter-fed variable-frequency operation is standard, and induction motors also drive electric vehicles. FEM analysis handles such bar shapes and magnetic saturation directly.
Worked example: a 4-pole, 50 Hz, 400 V motor
Per phase: R1 = 0.5, X1 = 1.0, R2' = 0.4, X2' = 1.0, Xm = 30 Ω (star connection). Synchronous speed 1,500 rpm; aluminium rotor bars 20 mm deep.
| Item | Constant R2' | With deep-bar effect |
|---|---|---|
| Torque at 3% slip (1,455 rpm) | 65.4 N·m | 65.4 N·m |
| Current at 3% slip | 18.0 A | 18.0 A |
| Peak torque (slip) | 190.9 N·m (0.197) | 190.9 N·m (0.201) |
| Starting torque | 81.9 N·m | 110.1 N·m |
| Starting current | 107.0 A | 103.4 A |
Output at 3% slip is about 10.0 kW. Bar skin depth is 12.7 mm at slip 1 (50 Hz), 40 mm at slip 0.1 (5 Hz) and 74 mm at slip 0.03 (1.5 Hz), with resistance factors 1.44, 1.01 and 1.00.
Identical near rated, yet starting torque differs by 34%.
Right — the deep-bar effect acts only at large slip. At start the skin depth is just 12.7 mm against 20 mm bars, so current crowds upward, resistance rises 1.44 times, torque increases and current drops slightly. For simplicity this calculation includes only the resistance change, omitting the drop in leakage reactance, but the trend is clear. Real design solves bar cross-sections (deep bar, double cage, trapezoidal) with FEM to get the torque curve from start to rated. A single set of circuit constants misjudges starting performance.
Analysis workflow
- Get approximate torque–slip, efficiency and power factor from the equivalent circuit.
- Solve with 2D FEM (time-harmonic or transient) including saturation and bar current distribution.
- Vary slip to get torque and current from start to rated, updating circuit constants.
- Add core and stray losses for efficiency and pass to thermal analysis.
- Compare with no-load and locked-rotor tests.
“Hotter on an inverter”
An induction motor designed for mains supply ran with a hotter rotor on an inverter at the same load. Inverter voltage contains high-frequency components, for which rotor bars carry high-frequency currents; by the same mechanism as the deep-bar effect, current crowds toward the bar tops and loss rises. Transient FEM with the actual inverter voltage waveform reproduced the extra loss. Remember that changing frequency changes current distribution in the bars.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Constant rotor resistance | Starting torque underestimated | Include deep-bar effect |
| Ignoring saturation | Starting current wrong | Nonlinear B–H curve |
| Sinusoidal supply only | Inverter losses missed | Transient with real waveform |
| Ignoring end-ring resistance | Rotor resistance too low | Add end rings via circuit |
| Verifying at one slip | Curve shape misjudged | Compare across slip |
I'd like to try related calculations.
Try the induction motor simulator and motor starting current calculator. Related motor analyses include efficiency maps, motor thermal analysis, PMSM design and torque ripple.
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