Magnetic Bearing Analysis
Overview
Professor! Today let us talk about magnetic bearing analysis. What is it about?
Theory: attraction and negative stiffness
Overview
Does a magnetic bearing float the shaft by magnet force alone?
The magnets supply the force, but that alone won't levitate the shaft. Electromagnets can only attract iron, and the pull strengthens as the gap narrows, so if the shaft drifts slightly to one side, that side's pull grows and it drifts further — a "negative spring" acting opposite to a normal spring. Sensors therefore measure position continuously, reducing current on the near side and increasing it on the far side to push the shaft back. Design finds how sensitive magnet force is to current and position and sets control stiffness and damping to overcome it — a coupled problem of electromagnetics, structural motion and control.
Electromagnet force and linearisation
N is turns, A one pole's area (two poles per U-core) and g the gap. Linearised for bias current I₀ in two opposing magnets with control current i added to one and subtracted from the other; iron reluctance neglected. Conditions: N = 200, A = 400 mm², g₀ = 0.5 mm, I₀ = 2 A, shaft mass 5 kg.
Equation of motion and PD control
The challenge of keeping it afloat
In 1842 Earnshaw showed that static magnets or charges alone cannot levitate an object stably (Earnshaw's theorem), which is why magnetic bearings need control. In the 1930s Beams levitated steel balls by controlling electromagnet current, and advances in electronics and sensors brought practical use. Digital control arrived in the 1980s, and today the technology is standard in turbomolecular pumps and large compressors.
Worked examples
Example 1: force coefficients and instability
| Item | Value |
|---|---|
| Current gain k_i | 161 N/A |
| Negative stiffness k_x | 6.43×10⁵ N/m (0.64 N/μm) |
| Bias flux density | 0.50 T |
| Bias force per magnet | 80 N |
| Saturation current (1.2 T) | 4.77 A |
| Maximum force (1.2 T) | 458 N |
| Uncontrolled unstable pole | 359 rad/s (57 Hz) |
| Displacement doubling time | 1.93 ms |
Example 2: PD design (closed-loop stiffness = 2|k_x|, damping ratio 0.5)
| Item | Value |
|---|---|
| Closed-loop stiffness K | 1.29×10⁶ N/m |
| Proportional gain k_p | 12,000 A/m (1.2 A per 0.1 mm) |
| Derivative gain k_d | 15.8 A·s/m |
| Natural frequency | 507 rad/s (81 Hz) |
| Current to carry gravity (49 N) | 0.31 A |
Couldn't you just set proportional gain to exactly beat the negative stiffness?
In theory it's marginally stable, but useless in practice: closed-loop stiffness K = k_i·k_p − k_x is nearly zero and small loads move the shaft a lot. Here K was set to twice the negative stiffness, 1.29×10⁶ N/m, yet gravity alone (49 N) would sag the shaft 38 μm (integral action re-centres it in practice). Raising stiffness means raising gain, limited by sensor noise, current lag and amplifier voltage. And if the shaft's own bending modes approach the control bandwidth, control can excite them. So rotor dynamics, electromagnets and control must be analysed together.
Analysis workflow
- Use electromagnetic analysis to tabulate force versus current and gap, extracting k_i and k_x.
- Check core saturation, leakage flux and force lag from eddy currents.
- Compute shaft bending modes with rotor dynamics.
- Couple magnets, shaft, sensors, amplifiers and controller for stability analysis.
- Check maximum loads against saturation and landing on backup bearings during power loss.
“It oscillated as speed rose”
A magnetic-bearing compressor levitated stably at low speed, but the shaft began vibrating at a certain speed during run-up. Control had been designed for a rigid shaft, yet the first bending mode lay within the control bandwidth and, shifting with speed through gyroscopic effects, interacted with control. Redesigning the controller with a shaft model including bending and adding a notch filter at that frequency gave stability up to rated speed.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Ignoring negative stiffness | Unstable control | Compute k_x |
| Treating the shaft as rigid | Oscillation via bending | Include bending |
| Ignoring saturation | Insufficient peak force | Check B limit |
| Ignoring current lag | Too little phase margin | Model amplifier and eddies |
| Ignoring power loss | Shaft drops, damage | Design backup bearings |
I'd like to learn related topics.
Related pages include Lorentz force, magnetostriction, piezoelectric analysis, electromagnetic forming and electrostatic MEMS.