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Active Magnetic Bearing

Active Magnetic Bearing Rotor Stiffness Simulator

Design Active Magnetic Bearings (AMB) that levitate a rotor without contact. From rotor mass, RPM, air gap, coil turns and bias current, this tool computes per-pole attractive force, bearing stiffness, natural frequency, critical-speed ratio and power consumption in real time, so you can size the levitation system of turbomolecular pumps, centrifugal compressors and flywheels.

Parameters
Bearing arrangement
Compare radial / radial+thrust / 5DOF conical layouts
Rotor mass m
kg
Rotor speed N_rot
RPM
Air gap g
mm
Pole-to-rotor clearance. Stiffness scales as 1/g³
Coil turns N
turns
Bias current I_b
A
Midpoint current of the differential drive. Trades stiffness for power
Control current I_c
A
Pole face area A_p
cm²
Results
—
Pole force F (N)
—
F / weight (g-equiv)
—
Bearing stiffness (N/mm)
—
Natural freq. (RPM)
—
Critical-speed ratio
—
Power (W)
—
AMB cross-section — rotor levitation & 4-pole control

A central rotor is differentially driven by four electromagnets, with a position sensor closing the feedback loop. Pole colour reflects current direction and amplitude.

Attractive force vs air gap — F ∝ 1/g²
Load, stiffness and power by arrangement
Theory & Key Formulas

$$F = \frac{\mu_0 N^2 I^2 A}{4 g^2}, \qquad k_{AMB} = \frac{k_i\, i_c}{g} - k_s$$

Pole force F and bearing stiffness k_AMB. N: coil turns, I: total current (bias + control), A: pole area, g: air gap. The current stiffness k_i provides the positive support that cancels the negative stiffness k_s.

$$k_i = \frac{\mu_0 N^2 I_b A}{g^2}, \qquad k_s = \frac{\mu_0 N^2 I_b^2 A}{g^3}$$

Current stiffness k_i and displacement (negative) stiffness k_s. k_s scales with 1/g³ and is why losing the controller drops the rotor instantly.

$$\omega_n = \sqrt{\frac{k_{AMB}}{m}}, \qquad \text{ratio} = \frac{\omega_{rotor}}{\omega_n}$$

Rotor natural frequency ω_n and critical-speed ratio. The 0.8–1.2 band is a resonance-amplification danger zone.

Active Magnetic Bearing AMB Rotor Stiffness — Non-contact Rotating Machinery

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A "magnetic bearing" is that thing where the rotor literally floats in mid-air with magnets, right? What makes it different from a regular ball bearing?
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Right — it's the trick used inside turbomolecular pumps and flywheels where the shaft spins without touching anything. A normal rolling bearing transmits load through steel balls and races, but an AMB pulls the rotor up with four electromagnets and keeps it floating in mid-air. No oil, no grease, no wear. That's why it's standard above 100,000 RPM and in vacuum / aseptic / cryogenic environments where you can't even bring lubricant into the chamber.
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If electromagnets only pull, won't the rotor stick to the closest magnet the moment it drifts a little?
🎓
That's exactly the catch. F = μ₀N²I²A/(4g²) grows with 1/g², so the closer the rotor gets, the harder it's pulled — that's the "negative stiffness". Left alone, it would snap to a pole in milliseconds. So an AMB is always paired with feedback control: position sensors measure g, and the controller adjusts the control current i_c in real time to pull harder on the opposite pole, giving the rotor an apparent positive stiffness. Lose control and the rotor falls instantly. It's literally a bearing that only exists while the controller is running.
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If it falls when control stops, isn't that terrifying? What happens at 60,000 RPM when the power goes out?
🎓
That's what "touchdown bearings" are for. Concentric with the AMB, but with a slightly larger clearance, sit a pair of rolling bearings — usually ceramic-ball ones. When levitation collapses, the rotor catches on these and spins down safely. A typical semiconductor TMP touches down maybe once a year, and the backup bearings absorb it without destroying the unit. The design rule of thumb is "survive at least 20 touchdowns".
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If I set the "critical-speed ratio" near 1.0 the tool says NG. What's actually happening there?
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When the operating speed ω hits the rotor's natural frequency ω_n (set by bearing stiffness and rotor mass), whirl amplitudes blow up — that's the critical speed. The 0.8–1.2 band is a resonance-amplification zone where the shaft can rub or even touch down. Real machines either accelerate through it quickly (subcritical) or run well above it (supercritical). Beacon Power's flywheels and LNG compressors are classic supercritical designs whose controllers are tuned to sweep cleanly through the 1st and 2nd criticals on every spin-up.
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So if I just keep raising the bias current, both stiffness and critical speed go up, problem solved?
🎓
In theory, yes. In practice power loss scales as I_total², the coils heat up, and once they saturate the stiffness actually drops. So the trade is "keep bias at 2–5 A, and make up the missing stiffness with high-frequency controller gain (H∞ or μ-synthesis)". Push the bias slider up in the chart and you'll see stiffness rise but power explode. In a battery-powered HeartMate III artificial heart that power budget fight literally drives half the design.

Frequently Asked Questions

The attractive force of a single solenoid pole is F = μ0·N²·I²·A/(4g²), where μ0 is the permeability of free space, N is the coil turns, I is the coil current (bias plus control), A is the pole face area and g is the air gap. The key is that the force scales with 1/g², so halving the gap quadruples the force. A differentially driven 4-pole radial bearing nets out two opposing pole pairs, and the total bearing load capacity is the sum of pole forces evaluated about the operating point.
Because magnetic force grows like 1/g², any displacement of the rotor towards a pole increases that pole's pull even more, making the system inherently unstable. Linearized about the operating point this is the negative stiffness k_s = μ0·N²·I_b²·A/g³, which acts opposite to a mechanical spring. The AMB uses feedback control (PID, H∞, μ-synthesis) to inject a control current i_c, producing a current stiffness k_i·i_c/g that cancels k_s and yields an apparent positive stiffness k_AMB. If control fails the rotor immediately touches down — AMB is fundamentally a "bearing that only exists while the controller is running".
Pick AMB when any of these apply: speeds above ~100,000 RPM, vacuum / cryogenic / aseptic / oil-free environments, or maintenance-free long life. Semiconductor turbomolecular pumps, LNG centrifugal compressors, flywheel energy storage, HeartMate III artificial hearts and LIGO's helium recompressors all run on AMB. For low-speed, heavy-load, cost-sensitive industrial machines, rolling-element or hydrostatic oil bearings are usually more economical. This tool also reports the power-loss difference against a notional rolling bearing so you can see the operating envelope where AMB wins.
Critical speed corresponds to the natural frequency of the rotor system (set by bearing stiffness and rotor mass). When the operating RPM hits it, whirl amplitude grows rapidly. The 0.8–1.2 ratio band is a resonance-amplification zone where shaft rub and touchdown become likely, so designs either pass through it quickly (subcritical) or run well above it (supercritical operation). This tool flags ratios between 0.8 and 1.2 as NG for exactly that reason — fix it by raising the bias current to gain stiffness, or shifting the operating speed.

Real-world applications

Semiconductor turbomolecular pumps: Edwards, Pfeiffer and Shimadzu TMPs spin rotors at 24,000–90,000 RPM in vacuum where no oil-lubricated bearing would survive. AMB suspends the rotor fully without contact so there is no oil backflow, enabling the clean ultra-high vacuum (10⁻⁸ Pa class) that drives semiconductor lithography and deposition yield.

Large centrifugal compressors and LNG liquefaction: AMB compressors from Atlas Copco and MAN Diesel are deployed in LNG trains, CO₂ re-injection and natural gas pipelines. Removing oil lubrication eliminates the entire oil console, coolers and seals, dramatically lowering capex and opex for offshore platforms and remote sites. Multi-megawatt units up to 20 MW are in service.

Flywheel energy storage: Beacon Power's grid-frequency-regulation plants run 200 flywheels, each levitated by AMB inside a vacuum vessel at 16,000 RPM. AMB keeps standing losses under about 0.1%, ideal for UPS and grid power-quality services. Plants are operational in Denmark and California.

Artificial hearts and circulatory assist devices: The HeartMate III and Berlin Heart EXCOR centrifugal blood pumps levitate the impeller in blood with AMB. Eliminating seals and contact bearings minimises thrombus formation and haemolysis, lifting average device life past 10 years. On battery power, the entire pump must run under 8 W.

Common misconceptions and pitfalls

The first trap is believing that more stiffness always means a more stable AMB. Raising the bias current does increase k_i, but the negative stiffness k_s grows with I_b² — even faster. Past a balance point the apparent stiffness collapses and the control loop saturates, destabilising the whole system. Real designs do not maximise k_AMB; they aim for an optimum where rotordynamic modes (1st and 2nd bending) and controller robustness coexist. Bias is typically held at 2–5 A and the rest of the stiffness comes from H∞ or μ-synthesis loop gain.

The second is treating touchdown bearings as emergency-only afterthoughts. In reality, grid fluctuations, sensor channel faults and rapid eccentric load changes cause several touchdowns per year. Ceramic-ball touchdown bearings typically have a fatigue life of 20–50 events. Exceed it and the next anomaly destroys the backup itself, slamming the rotor into the poles for a multi-million-dollar repair. Logging touchdown counts and scheduling preventive replacement is mandatory.

The third is assuming that staying just outside ratio 1.0 is enough to avoid resonance. This tool flags 0.8–1.2 as NG, but that target only addresses the 1st rigid-body mode. A real rotor has higher bending modes (2nd, 3rd) that the gyroscopic effect splits into forward and backward whirl branches at speed. Careful Campbell-diagram analysis and active damping of those higher modes (μ-synthesis, notch filters) are essential — clearing the rigid-body critical alone does not guarantee safe operation.

How to Use

  1. Enter rotor mass (kg) between 2–50 kg; typical industrial spindle rotors range 5–15 kg.
  2. Set operating speed (RPM); compressors and turbomachinery typically run 3000–15000 RPM.
  3. Define air gap (mm), normally 0.5–1.5 mm; smaller gaps increase pole force but reduce stability margin.
  4. Input coil turns per pole (50–500 turns); higher turns increase electromagnetic stiffness at fixed current.
  5. Click simulate to calculate pole force, bearing stiffness, natural frequency, and critical-speed ratio.
  6. Verify critical-speed ratio exceeds 2.5 to avoid resonance during run-up and shutdown.

Worked Example

A 10 kg turbopump rotor operating at 8000 RPM with 1.0 mm air gap and 200 coil turns per pole: Pole force calculates to approximately 450 N (4.59 g-equivalent load). Bearing stiffness reaches 180 N/mm, yielding natural frequency of 6750 RPM. Critical-speed ratio = 6750/8000 = 0.84, indicating proximity to critical speed—reduce air gap to 0.8 mm or increase turns to 250 to raise stiffness to 280 N/mm and natural frequency to 8400 RPM (ratio 1.05, safer). Estimated control power consumption is 85 W.

Practical Notes

  1. Air gap of 0.6–0.8 mm suits high-speed spindles (12000+ RPM); below 0.5 mm risks mechanical contact during transients.
  2. Stiffness asymptotes with coil turns; diminishing returns appear beyond 350 turns due to saturation in ferrous cores.
  3. Critical-speed ratio below 1.5 requires active damping or controller gains (typically 50–200 A/mm) to suppress vibration.
  4. Power dissipation scales as I²R in windings; monitor thermal limits in confined bearing housings.