Can a magnetic bearing be stable without control electronics?

Not with electromagnets acting on a ferromagnetic rotor. Earnshaw's theorem rules out stable equilibrium in every direction from static poles alone. Passive magnetic bearings using permanent magnets can be stable in some directions, but always at the cost of instability in another, so they are used as partial support alongside an active axis. What is sold as a "passive magnetic bearing" is nearly always stabilised somewhere by something active or by a mechanical constraint.

Why not simply run without a bias current?

You can, and low-bias or zero-bias schemes are used where standby loss matters. The penalty is that force becomes quadratic and one-sided again, so the effective force constant collapses near zero current exactly where the controller is working hardest. Zero-bias control needs a non-linear controller and gives up slew rate at small forces.

How big is the air gap, and why not make it smaller?

Typically 0.3–0.8 mm on industrial radial bearings. Smaller gaps give more force per ampere — but negative stiffness grows as 1/g³ while force grows as 1/g², so the control problem gets harder faster than the force gets better. The gap also has to swallow rotor growth at speed and temperature, the tolerance stack-up, and leave room for a backup bearing clearance inside it.

What happens on power failure?

The rotor lands on the backup bearings, mounted at roughly half the magnetic gap. Those are rated for a limited number of drops at speed and are inspected on a schedule. Machines that cannot tolerate an uncontrolled coast-down keep levitation alive through the run-down, either from an uninterruptible supply or by taking power from the motor acting as a generator.

Why is the rotor laminated?

In a heteropolar bearing the polarity alternates around the bore, so a point on the rotor surface sees a field that reverses once per revolution. In solid steel that would drive large eddy-current and hysteresis losses and heat the rotor. Laminating the rotor journal cuts the eddy paths. Homopolar designs, where the rotor sees a constant polarity, reduce this loss and are used where rotor heating is critical.

How do magnetic bearings compare with foil bearings?

Both are oil-free. Foil bearings are far simpler, cheaper and need no electronics, but they touch during start and stop, have fixed properties, and their load capacity falls away at low speed. Magnetic bearings never touch, work at zero speed, and let stiffness, damping and vibration behaviour be chosen — at the cost of the electronics and everything that comes with them. Below roughly 50 kW of machine, foil bearings usually win on cost; above it, and wherever active vibration control or diagnostics are wanted, magnetic bearings tend to.

Drawing No. EH–ME–008 // Mechanical Engineering

Magnetic Bearing Calculator

Reviewed August 2026

Size an active radial magnetic bearing from the pole geometry outwards: flux density at the working point, force at bias and at saturation, the force constant and the negative stiffness that comes with it, the PD gains needed to make the axis stable, and the amplifier voltage that decides how fast the bearing can actually respond.

Scope. Screening tool for an eight-pole heteropolar radial bearing running in differential driving mode, using the linearised magnetic-circuit model. Leakage, fringing, iron reluctance, eddy currents and cross-coupling between axes are not modelled. Those effects can materially change force and dynamic response, so final actuator performance should be checked with detailed electromagnetic analysis and/or calibration. Use it to choose a geometry and check feasibility, not to release a design.

01 // Bearing sizing calculator

One radial axis, two opposed horseshoe electromagnets, differential driving. Values update as you type.

Simplified view: pole geometry, coil and rotor only. Pole angle, drive and control targets use standard values. Switch to Full for the electrical and control-loop sizing.

Bearing inputs

Pole and rotor geometry

Pole face width b mm

Rotor stack length l mm

Nominal air gap g₀ mm

Rotor lamination diameter D mm

Pole pair half-angle α deg

Angle between each pole of a horseshoe and the load axis. 22.5° for a standard eight-pole heteropolar stator.

Coil and drive

Turns per horseshoe N turns

Bias current i₀ A

Peak control current i_c A

Linearised force is reported only while i₀ + i_c remains below the selected saturation current.

Coil resistance R Ω

Amplifier supply voltage U V

Sets the current slew rate, and with it the fastest force change the bearing can produce.

Material and rotor

Saturation flux density B_sat T

Rotor mass per bearing m kg

Closed-loop target

Target natural frequency f_n Hz

Target damping ratio ζ –

Check the geometry, coil, drive and rotor inputs — every value must be positive, and the pole half-angle must be below 60°.

Sizing results

Pole face area A_p

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Flux density at bias B₀

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Saturation current i_sat

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Saturation margin at peak current

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Net force at peak control current

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Force at saturation F_max

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Load capacity

—

Specific load, projected area

—

Force constant k_i

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Negative stiffness k_s

—

Coil inductance L

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Electrical time constant L/R

—

Force slew rate at bias

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Bias copper loss, both coils

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Required gain k_p

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Required gain k_d

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Closed-loop stiffness

—

Static sag under 1 g

—

Backup bearing clearance is normally set to 0.4–0.5 of the magnetic gap, here —, so the rotor lands on the backup bearing before it touches a pole face.

Force vs control current

Force vs rotor displacement at bias

02 // Worked example, step by step

The default inputs above, carried through by hand — formula, substitution and result for every step.

Brief. One radial axis of a small turbo-compressor. Rotor share per bearing 25 kg, journal diameter 100 mm, non-oriented silicon steel laminations (B_sat = 1.5 T), 150 V amplifiers. The bearing must lift the rotor with margin, hold position at 60 Hz with ζ = 0.7, and keep at least 30 % saturation margin at peak current.

Step	Formula	Substitution	Result
1 · Pole area	A_p = b × l	20 mm × 50 mm	1000 mm²
2 · Bias flux	B₀ = μ₀Ni₀/2g₀	(4π×10⁻⁷ × 200 × 2) / (2 × 0.0005)	0.503 T
3 · Saturation current	i_sat = 2g₀B_sat/(μ₀N)	(2 × 0.0005 × 1.5) / (4π×10⁻⁷ × 200)	5.97 A
4 · Margin check	1 − (i₀ + i_c)/i_sat	1 − 4 / 5.97	33 % — meets the 30 % brief
5 · Force constant	k_i = μ₀N²A_p i₀ cosα/g₀²	(μ₀ × 200² × 10⁻³ × 2 × 0.924) / 0.0005²	371.5 N/A
6 · Control force	F = k_i i_c	371.5 × 2	743 N = 3.0 × rotor weight
7 · Saturation force	F_max = B_sat²A_p cosα/μ₀	(1.5² × 10⁻³ × 0.924) / μ₀	1654 N = 6.7 × rotor weight
8 · Specific load	F_max/(D × l)	1654 / (0.1 × 0.05)	33.1 N/cm² — normal for silicon steel
9 · Negative stiffness	k_s = μ₀N²A_p i₀² cosα/g₀³	k_i × i₀/g₀ = 371.5 × 2 / 0.0005	1486 N/mm
10 · Sanity check	k_i/k_s = g₀/i₀	371.5 / 1 486 000	0.25 mm/A ✓
11 · Proportional gain	k_p = (mω_n² + k_s)/k_i	(25 × 377² + 1.486×10⁶) / 371.5	13.6 A/mm
12 · Gain split	k_s / (k_i k_p)	1.486×10⁶ / 5.04×10⁶	29 % cancels k_s, 71 % becomes stiffness
13 · Derivative gain	k_d = 2ζmω_n/k_i	(2 × 0.7 × 25 × 377) / 371.5	35.5 A·s/m
14 · Closed-loop stiffness	k_cl = k_i k_p − k_s	5.04×10⁶ − 1.486×10⁶	3553 N/mm
15 · Sag under 1 g	x = mg/k_cl	245 N / 3.553×10⁶ N/m	0.069 mm, 14 % of the gap
16 · Inductance	L = μ₀N²A_p/2g₀	(μ₀ × 40 000 × 10⁻³) / 0.001	50.3 mH
17 · Force slew	dF/dt = k_i(U − i₀R)/L	371.5 × (150 − 2.4) / 0.0503	1091 kN/s
18 · Bias loss	P = 2i₀²R	2 × 2² × 1.2	9.6 W — not thermally limited

Verdict. The design meets the brief with 33 % saturation margin, three times the rotor weight in control force, and 0.069 mm of sag. The binding constraint is not force and not heat — it is the amplifier. At 1091 kN/s, a sinusoidal force of 743 N can be produced up to roughly 230 Hz before the supply voltage runs out of slew, which sets the real disturbance-rejection bandwidth.

What to change first, and what it costs.

If you need	Change	Side effect
More force	Pole area, then bias current	Area lengthens the rotor; current eats saturation margin and burns i²R
More saturation margin	Pole area, or cobalt-iron laminations	Area costs length; CoFe costs roughly ten times the material price
Faster response	Supply voltage, or fewer turns	Fewer turns cuts k_i, so current has to rise for the same force
Less control effort	Lower bias current	k_s falls with i₀² but so does k_i; linearity and slew rate suffer
Smaller bearing	Smaller air gap	k_s grows as 1/g₀³ against force at 1/g₀² — the control problem gets harder faster than the force improves

03 // How the force is made

Reluctance force at an iron-air boundary, and the magnetic circuit that sets it.

A magnetic bearing carries load by pulling a ferromagnetic rotor towards an electromagnet. The force appears at the boundary between iron and air, where the Maxwell stress across a pole face of area A_p carrying flux density B is

σ = B² / (2μ₀) → F_face = B² A_p / (2μ₀)

At 1.5 T that stress is about 90 N/cm² of pole face — roughly nine bar. It is a hard ceiling: the only ways past it are more pole area or better iron, because B is limited by saturation, not by how much current you are willing to spend.

A horseshoe magnet presents two pole faces to the rotor and its flux crosses two air gaps in series. Neglecting the reluctance of the iron, which is two to three orders of magnitude smaller than the gaps, the flux density follows directly from the ampere-turns:

B = μ₀ N i / (2 g)

and the force resolved onto the load axis, with each pole offset from that axis by the half-angle α, is

F = μ₀ N² A_p i² cosα / (4 g²)

Two features of that expression drive every design decision that follows. Force goes with the square of current, which makes the actuator non-linear. And it goes with the inverse square of the gap, which makes the bearing stiffer the closer the rotor gets — the wrong sign for stability.

Figure 1 — Eight-pole heteropolar radial bearing. Only the upper horseshoe is shown wound and energised. Laminations, coil formers and the sensor ring are omitted for clarity.

Figure 2 — The magnetic circuit that the calculator solves. Neglecting iron reluctance is worth roughly 5–10 % of force at normal flux densities and much more once the iron approaches saturation.

04 // Bias current and differential driving

Turning a squared, one-sided actuator into something a controller can use.

A single electromagnet can only pull, and its force is quadratic in current. Near zero current it barely responds at all, which is exactly where a position controller spends most of its time. The standard answer is to place two magnets in opposition, run both at a bias current i₀, and add the control current to one while subtracting it from the other.

The quadratic terms cancel and the net force becomes proportional to the control current, with a second term proportional to displacement:

F = k_i · i_c + k_s · x

The force constant k_i is what the controller commands. The negative stiffness k_s is what it has to fight. Their ratio is fixed by the operating point alone:

k_i / k_s = g₀ / i₀

Choosing the bias is a genuine trade. High bias gives a large force constant, good linearity and fast response, but burns i₀²R continuously in both coils and eats saturation headroom. Low bias saves power at the cost of linearity and slew rate. Around half the saturation current is a common starting point; machines that idle for long periods sometimes run reduced or zero bias and accept the non-linearity.

Figure 3 — Differential driving. Setting the bias near half the saturation current is the usual compromise between linearity, loss and available peak force.

05 // Negative stiffness, and why control is not optional

Earnshaw's theorem, and what it costs in hardware.

Move the rotor a little towards the upper magnet and its gap shrinks, so the upper force rises while the lower falls. The net force pushes the rotor further off centre. That is positive feedback with a stiffness of

k_s = μ₀ N² A_p i₀² cosα / g₀³

This is not a defect of any particular design. Earnshaw's theorem states that no static arrangement of magnetic poles can hold a ferromagnetic body in stable equilibrium in all directions at once. Something must actively measure the position and modulate the field.

With proportional-derivative control the loop becomes

m·ẍ + k_i k_d ·ẋ + (k_i k_p − k_s)·x = 0

and the practical consequences fall out of it immediately. The proportional gain must first cancel k_s before it delivers any net stiffness, so a bearing with a large negative stiffness needs a fast, high-gain loop just to stand still. All damping is synthetic: there is no oil film, so if the derivative term is wrong the rotor has essentially no damping at all. And the sensor sets the floor on everything — its noise is amplified by k_p and appears as current ripple, heat and audible noise.

Design consequence. Sizing a magnetic bearing is never just a force calculation. A geometry that gives ample force but a negative stiffness beyond what the amplifier and sensor bandwidth can cover is not a usable bearing.

Figure 4 — Sensor, controller and amplifier are load-carrying elements. A magnetic bearing without its electronics is not a bearing.

06 // Load capacity, saturation and geometry

What actually limits the load, and the numbers to sanity-check a design against.

Peak force is set by the flux density the iron will carry, not by the current the coil will take. Once the pole face reaches B_sat, more ampere-turns produce heat and very little extra force:

F_max = B_sat² A_p cosα / μ₀

Two different area bases are quoted in the literature and they differ by a factor of two or more, so it is worth being explicit about which one a number refers to.

Basis	Typical value	What it means
Per pole face	90 N/cm² at 1.5 T 160 N/cm² at 2.0 T	The Maxwell stress itself, B²/2μ₀. A theoretical ceiling that no real bearing reaches over its whole bore.
Per projected rotor area (D × l)	20–40 N/cm² silicon steel up to ~65 N/cm² cobalt-iron	The number that matters when comparing against a rolling-element or fluid-film bearing in the same envelope. Pole coverage, slot space and the pole angle all take their cut.
Compared with a hydrodynamic bearing	roughly a factor of 5–10 lower	Magnetic bearings buy their advantages with size. A magnetic bearing for a given load is simply bigger.

The remaining geometry choices tend to pull against each other:

Choice	Push towards	Held back by
Smaller air gap	Force rises as 1/g² for the same coil	Rotor growth at speed and temperature, manufacturing tolerance, backup bearing clearance, and negative stiffness rising as 1/g³
More turns	Force rises as N² for the same current	Inductance also rises as N², so the bearing slows down; slot area and copper loss
Longer stack	Area, so force, rises linearly	Rotor bending modes drop into the operating range; shaft length is rarely free
Cobalt-iron laminations	Up to 2.3 T, so about double the force density	Cost per kilogram roughly an order of magnitude above silicon steel; harder to work

07 // Dynamics: stiffness, damping and the voltage limit

Why the power amplifier, not the controller, usually sets the performance ceiling.

A magnetic bearing's stiffness and damping are chosen, not inherited from a material. That freedom is the main reason to use one: the same hardware can run soft through a critical speed and stiff elsewhere, or deliberately unbalance-follow so the rotor spins about its mass axis and transmits almost nothing to the foundation.

The limit on all of it is how fast current can be changed. A coil of inductance L fed from a supply of U volts can only slew at

di/dt = (U − iR) / L → dF/dt = k_i (U − i₀R) / L

and since both k_i and L carry N², adding turns does not buy force response — it buys force at the expense of speed. The usual levers are a higher supply voltage, a smaller gap, or more pole area.

Three dynamic effects catch people out on a first design:

Effect	What happens	Usual remedy
Gyroscopic coupling	At speed, the two radial axes of one bearing stop being independent; forward and backward whirl modes separate	Cross-axis or full multi-input controllers rather than four independent loops
Flexible rotor modes	The controller can excite bending modes that a passive bearing would never reach, because it has authority at those frequencies	Notch or roll-off filters at the mode frequencies, and sensor placement away from nodes
Sensor/actuator non-collocation	Sensor and magnet planes differ, adding phase that eats stability margin	Place sensors as close to the magnet plane as the machine allows; model the offset

Rule of thumb. Closed-loop bandwidth needs to be several times the highest disturbance frequency the bearing must reject, and the amplifier needs voltage headroom to deliver the commanded current at that frequency — a current command the supply cannot slew to is just a saturated amplifier.

Figure 5 — A five-axis magnetic suspension. Backup bearings sit at a clearance of roughly half the magnetic gap.

08 // Advantages and disadvantages

Where a magnetic bearing earns its cost, and where it does not.

Advantages

Most of these follow from one fact: nothing touches.

No contact, no wearService life is set by the electronics and the backup bearings, not by a wear mechanism in the bearing itself. There is no run-in, no fatigue spalling, and no lubricant to degrade.

No lubrication systemNo oil pump, cooler, filter, seals or oil analysis. In compressors this also removes the risk of oil contaminating the process gas, which is often the deciding argument on its own.

Very low dragLosses come from windage and iron losses rather than shear, so parasitic power is typically a fraction of a fluid-film bearing's at the same speed.

Extreme environmentsWorks in vacuum, in clean rooms, at cryogenic temperatures and at process temperatures where no lubricant survives.

Adjustable stiffness and dampingSet in software, changed while running, and different at different speeds. Passing a critical speed softly and running stiff elsewhere is routine.

Built-in condition monitoringThe bearing already measures rotor position at kilohertz rates on every axis. Unbalance, rubs, cracks and load changes are visible without adding a single sensor.

Active vibration controlThe bearing can reject synchronous unbalance rather than transmit it, letting the rotor spin about its mass axis and keeping force out of the foundation.

Disadvantages

Most of these follow from the other fact: it is a control system.

Never passively stableSensors, controller and amplifiers are load-carrying parts. A failure anywhere in that chain is a bearing failure, so redundancy and functional safety analysis are part of the design, not an extra.

Low load capacity for the sizeSaturation caps force density at roughly a fifth to a tenth of a hydrodynamic bearing's, so the bearing is physically larger and adds rotor length — which lowers the bending modes it then has to control.

Backup bearings are mandatoryThey are a wear item with a rated number of drops, they need periodic inspection, and a drop at full speed is a significant event for the machine.

Continuous power and heatBias current dissipates whether the machine is loaded or not, and the heat lands in the stator next to the rotor.

Cost and complexityAmplifiers, controller, cabling, sensors and commissioning usually dominate the bearing cost. Retuning is a specialist task rather than a bearing swap.

Commissioning is control engineeringGains have to be tuned against the real rotor. Notch filters, gyroscopic coupling and non-collocation are the norm on anything fast or flexible.

Sensitive to the gapForce scales with 1/g² and negative stiffness with 1/g³, so thermal growth, tolerance stack-up and rotor expansion at speed all move the operating point.

09 // Where they are used

The pattern: no lubricant allowed, very high speed, or vibration that has to be controlled rather than tolerated.

Turbomachinery

Gas-pipeline and process compressors, turboexpanders and blowers. Removing the oil system removes contamination risk, seal gas complexity and a large part of the auxiliary skid. Subsea and unattended compressor stations are natural fits, since there is no lubricant to service.

Vacuum and semiconductor

Turbomolecular pumps run rotors at tens of thousands of rpm in vacuum, where a lubricant would outgas. Magnetic suspension also keeps particles out of the process, which matters as much as the speed.

Machine tool spindles

High-speed milling and grinding spindles use the bearing's position control to compensate tool deflection, and its sensors to detect chatter and tool breakage as they happen.

Flywheel energy storage

Low drag is the whole point: a flywheel on rolling-element bearings loses its energy to friction. Magnetic suspension in vacuum brings idle losses down to a level where hours of storage becomes practical.

Medical and life support

Ventricular assist devices suspend the pump rotor magnetically so there is no seal, no bearing to clot around, and no shear region that damages blood cells.

Test rigs and research

Because the bearing measures force and position and can inject a defined disturbance, it doubles as a rotordynamic instrument for measuring seal and impeller coefficients.

10 // Compared with conventional bearings

Where a magnetic bearing beats a rolling-element, fluid-film or foil bearing — and where it plainly does not.

Property	Rolling element	Hydrodynamic (fluid film)	Foil (air)	Active magnetic
Load capacity	High, 200–500 N/cm²	Very high, 150–350 N/cm²	Low, 5–20 N/cm²	Low, 20–40 N/cm² (65 with CoFe)
Speed limit	DN-limited; lubricant and cage bound	High, whirl-instability bound	Very high	Very high; bound by rotor stress, not the bearing
Friction / drag	Low at low speed, rises with speed	Shear loss rises steeply with speed	Low above lift-off	Windage and iron loss only
Contact	Rolling contact throughout	None above lift-off speed	Contact on start and stop	None at any speed, including zero
Lubrication	Grease or oil	Pressurised oil system	None	None
Stiffness	High, fixed by geometry	Moderate, varies with speed and load	Low, non-linear	Moderate, chosen in software and changeable while running
Damping	Almost none	High, from the film	Moderate, from foil friction	Set by the controller; zero if mis-tuned
Life limit	Fatigue, L10 rating	Effectively unlimited if the oil is clean	Coating wear at each start/stop	No wear mechanism; electronics and backup bearings set the life
Failure mode	Progressive, detectable	Loss of oil is rapid and severe	Coating wear, then seizure	Loss of power or control drops the rotor onto backup bearings
Condition monitoring	Added sensors needed	Added sensors needed	Difficult	Position and current already measured on every axis
Vibration behaviour	Transmits unbalance	Attenuates some	Attenuates some	Can reject unbalance actively and spin about the mass axis
Envelope for a given load	Compact	Compact	Moderate	Largest; adds rotor length and lowers bending modes
Auxiliaries	None to minimal	Pump, cooler, filter, tank, instrumentation	None	Amplifiers, controller, sensors, cabling, backup bearings
Installed cost	Lowest	Moderate, plus the oil skid	Low	Highest; electronics usually dominate
Commissioning	Fit and run	Fit and run	Fit and run	Control tuning against the real rotor

Choose rolling element when

Load is high and speed is moderate, cost matters, and a lubricant is acceptable. Nothing else is as compact or as cheap per newton of capacity.

Choose fluid film when

Load is high, speed is high, and damping is needed to get through criticals. The oil system is the price of admission — and the reason to look elsewhere when the process cannot see oil.

Choose magnetic when

Lubricant is prohibited, speed is extreme, drag has to be near zero, or vibration and diagnostics need to be active rather than tolerated. Expect a bigger bearing and a control system to commission.

The honest summary. A magnetic bearing is not a better bearing; it is a different kind of machine element. It trades load density, simplicity and cost for zero contact, zero lubricant, and properties you can program. When none of those three is worth paying for, a conventional bearing wins on every line of the table above.

11 // Standards and further reading

What to design and accept against.

Reference	Covers
ISO 14839-1	Vocabulary for rotating machinery equipped with active magnetic bearings
ISO 14839-2	Evaluation of vibration — displacement limits and acceptance zones
ISO 14839-3	Evaluation of stability margin, based on the peak sensitivity of the closed loop
ISO 14839-4	Technical guidelines for the control system and its design
API 617	Axial and centrifugal compressors; the magnetic bearing annex is the usual commercial baseline in oil and gas
Schweitzer & Maslen, Magnetic Bearings	The standard reference text for theory, design and application

Stability margin. ISO 14839-3 grades a magnetic bearing on the peak of its sensitivity function rather than on gain and phase margin alone, because a magnetic bearing loop is unstable open-loop and classical margins can flatter it.