Magnetic Bearings: Spinning a Rotor on Nothing

How it works: an unstable plant held by feedback

An active magnetic bearing floats a ferromagnetic shaft in a magnetic field. But a single electromagnet pulling on iron is a fundamentally unstable system: as the gap shrinks, the force grows, pulling the rotor in harder — positive feedback that ends with the shaft slammed against the stator. Earnshaw's theorem (1842) proves no arrangement of fixed magnets can statically levitate a ferromagnet in stable equilibrium. AMBs sidestep this with real-time control.

The canonical radial bearing uses eight poles in four opposing pairs (two per axis, top/bottom and left/right). A gap sensor — usually an eddy-current probe resolving position to ~0.1 µm — feeds a controller. The controller computes a correction and commands a transconductance power amplifier to set coil current. The loop runs on a DSP or FPGA at 10–20 kHz, comfortably faster than the rotor's first bending mode.

• Sense the rotor position x relative to the setpoint (bearing center).
• Compute a control force via a PID (or state-feedback / LQG) law with phase lead to overcome the plant's negative stiffness.
• Actuate by differentially driving the opposing electromagnets — increase current on the pole pulling toward the setpoint, decrease on the other.

Because the open-loop plant has a right-half-plane pole (it is open-loop unstable), the controller must supply phase lead near crossover — this is why simple integral-heavy tuning fails and why AMB design is really a controls discipline.

The force law and linearization

The attractive force from one horseshoe electromagnet acting across an air gap g is derived from magnetic co-energy:

F = (µ₀ N² i² A)/(4 g²)

where µ₀ = 4π×10⁻⁷ T·m/A, N is turns per coil, i is coil current (A), A is the pole face area (m²), and g is the air gap (m). Two things dominate: force scales with the square of current and the inverse square of gap. That i²/g² nonlinearity is exactly what makes the system unstable and hard to control directly.

The standard fix is differential driving with a bias current. Each axis uses two opposed magnets carrying i = I₀ ± iₓ, where I₀ is a fixed bias (typically 40–50% of max) and iₓ is the control current. Subtracting the two force expressions and linearizing about the centered rotor gives a clean relation:

F ≈ kᵢ·iₓ − kₓ·x

• Current stiffness kᵢ = (µ₀ N² A I₀)/g₀² — force per amp of control current (N/A), the actuator gain.
• Position stiffness kₓ = (µ₀ N² A I₀²)/g₀³ — force per unit displacement (N/m), and it is negative (destabilizing). This open-loop negative stiffness, on the order of −10⁶ to −10⁷ N/m for an industrial bearing, is the term the controller must overcome.

Bias current buys linearity and force slew rate but costs continuous ohmic and eddy-current loss; some designs use a lower bias or 'zero-bias' switching schemes to cut standby power at the expense of a nastier control problem.

Sizing: load capacity, saturation, and geometry

Load capacity is set by how much flux you can push through the iron before it saturates. The Maxwell stress (magnetic pressure) on a pole face is:

p = B²/(2µ₀)

Silicon-iron laminations saturate near B_sat ≈ 1.5–1.7 T. At B = 1.5 T the pressure is p = (1.5)²/(2·4π×10⁻⁷) ≈ 0.9 MPa (≈130 psi) — a hard ceiling on the specific load an AMB can carry per unit of projected pole area. In practice designers run at 0.35–0.6 MPa to preserve control margin.

• Estimate the load. Radial load = rotor weight share + dynamic (unbalance, process, gyroscopic) loads, with a margin — API 617 and machine builders typically want ≥ 2× peak dynamic load headroom.
• Set the projected area. A_proj = F_max / p_design. For a 5 kN bearing at 0.5 MPa you need ≈ 0.01 m² of effective pole face, which sets journal diameter and axial length.
• Pick the gap. Nominal radial gap g₀ ≈ 0.3–1.0 mm. Bigger gap tolerates more rotor excursion and thermal growth but demands far more ampere-turns (force ∝ 1/g²).
• Size ampere-turns. N·I₀ sets B via B ≈ µ₀ N I₀ / g (ignoring iron reluctance). Stay below B_sat with margin for the control swing I₀ + iₓ,max.

Rotor surface speed, not the bearing, usually caps the machine: a laminated rotor sleeve is limited by hoop stress, and beyond ~200–250 m/s you fight rotor burst, not levitation. A 100 mm journal at 30,000 rpm already sees 157 m/s.

Rotordynamics: the controller sets the stiffness and damping

In a rolling or oil-film bearing, stiffness and damping are baked into hardware. In an AMB they are software — the closed-loop transfer function defines an effective spring and dashpot. A PID controller yields, near the operating point, an effective stiffness k_eff (from proportional + integral gain, net of the −kₓ) and damping c_eff (from the derivative term). This is a double-edged sword: you can tune the bearing to place critical speeds and add damping exactly where a mode needs it — but you must guarantee stability across the whole run-up.

• Rigid-body modes (cylindrical and conical rotor whirl) sit at low frequency; the controller's proportional/integral action stiffens them and its derivative action damps them.
• Flexible bending modes lie above; the loop must not excite them. Notch filters or phase-shaping are added so the controller has gain roll-off and correct phase at each bending frequency — otherwise a lightly-damped mode goes unstable and the rotor rings itself into the touchdown bearings.
• Gyroscopic coupling splits forward/backward whirl modes as speed rises; MIMO or cross-axis control handles the coupling on high-speed, disk-heavy rotors.

A powerful trick unavailable to mechanical bearings is synchronous unbalance rejection (also called automatic balancing or force-free control): a notch at the running frequency lets the rotor spin about its mass center instead of its geometric center, so residual unbalance produces almost no force transmitted to the foundation. The machine runs quieter than any balanced conventional rotor. Acceptance criteria for rotor vibration and controller performance are codified in ISO 14839-1/-2/-3.

Hardware: amplifiers, sensors, and the axial bearing

A complete five-axis AMB system controls four radial degrees of freedom (two bearings × two axes) plus one axial (thrust) axis; rotation about the spin axis is left free. Each controlled axis needs sensing, control, and power electronics.

• Position sensors. Eddy-current or inductive probes with 1–2 kHz bandwidth and ~0.1 µm resolution, mounted near each bearing plane. Self-sensing ('sensorless') AMBs infer position from coil inductance, saving hardware but eroding robustness.
• Power amplifiers. Switching (PWM) transconductance amplifiers — essentially current-mode buck/half-bridge stages using MOSFETs or IGBTs — set coil current with high efficiency. They must supply the current slew rate to react within a control period; coil inductance L limits di/dt = V/L, so DC bus voltage (often 150–320 V) is chosen to hit the required force bandwidth.
• Laminated cores. Silicon-steel laminations (0.2–0.35 mm) on both stator and rotor sleeve cut eddy-current losses that would otherwise heat the rotor and blunt the force response.
• Axial (thrust) bearing. A solid iron thrust disk pulled by two opposed ring electromagnets; being solid, it carries more eddy loss and often runs at lower bandwidth than the radial axes.

Real deployments include Sundyne/Siemens/MAN turbo-compressors, SKF and Waukesha (Zephyr) AMB modules, turbomolecular vacuum pumps (Pfeiffer, Edwards), flywheel energy storage, and machine-tool spindles — all cases where oil-free, sealed, or ultra-high-speed operation pays back the electronics cost.

Touchdown bearings, failure modes, and limits

The Achilles' heel of levitation is loss of levitation. Every AMB machine carries backup (touchdown / auxiliary) bearings — usually angular-contact ceramic ball bearings or bushings — with a radial clearance about half the magnetic gap (e.g. 0.25 mm inside a 0.5 mm bearing gap). On power loss, sensor fault, or controller trip, the rotor drops onto these and coasts to a stop. This is a violent event: a heavy rotor dropping and spinning at full speed can cause backward whirl, high impact loads, and touchdown-bearing overheating; API 617 and ISO 14839-5 (touch-down bearings) require demonstrated de-levitation / drop tests and limit the number of allowable drops before inspection.

• Amplifier saturation. A large transient can demand more current than the bus voltage/amp can deliver, momentarily losing force authority — sized against worst-case unbalance and process upsets.
• Iron saturation. Push B past ~1.6 T and kᵢ collapses; the actuator gain drops just when you need it most.
• Control instability. An unmodeled bending mode, sensor-actuator non-collocation, or gain drift can drive a lightly-damped mode unstable — the classic AMB catastrophic failure.
• Thermal & electrical. Bias-current ohmic loss and eddy losses heat coils and rotor; loss of cooling or a shorted coil degrades a pole pair.
• Cost/complexity. Sensors, amplifiers, and a real-time controller make AMBs expensive and demand functional-safety-grade electronics with redundancy (dual sensors, UPS-backed bus) for critical service.

Best practice: collocate sensor and actuator planes, provide a ride-through UPS on the DC bus, add notch filters at every bending mode, verify stability margins per ISO 14839-3, and treat the backup bearings as safety-critical wear items with a documented drop budget.

Active magnetic bearing vs. conventional oil-film and rolling-element bearings

Attribute	Active magnetic (AMB)	Journal (oil-film)	Ball / roller
Contact / wear	None — non-contact	Hydrodynamic film, no metal contact at speed	Rolling contact, fatigue-limited
Friction loss	≈ 0.001–0.01 (windage + eddy)	µ ≈ 0.001–0.005	µ ≈ 0.0015
Max surface speed	> 200 m/s (limited by rotor stress)	~100 m/s	~25–40 m/s (dn limit)
Lubrication	None	Oil supply + seals + cooling	Grease or oil
Life / maintenance	Electronics-limited, no wear parts	Seal/oil service	L10 fatigue life
Cost & complexity	High — sensors, power amps, controller	Moderate	Low