Electronics Guide

The Force of an Electromagnet

A radial magnetic bearing is built from horseshoe-shaped electromagnets facing the rotor across an air gap. For a single such magnet with two gaps of length g, coil turns N, pole face area A, and current i, neglecting the reluctance of the iron and any fringing, the attractive force is proportional to the square of the current divided by the square of the gap. Two consequences fall directly out of that expression. First, the force is always attractive, whatever the sign of the current, so a single electromagnet can pull but never push, and opposed pairs are required to control an axis in both directions. Second, the force grows as the gap closes. A rotor displaced toward one pole is pulled harder toward that pole.

What Earnshaw's Theorem Forbids

Samuel Earnshaw showed in 1842 that a body acted on only by inverse-square forces cannot rest in stable equilibrium. Applied to magnetostatics, the theorem states that no fixed arrangement of permanent magnets, currents, and ferromagnetic material can hold a ferromagnetic or permanently magnetized body stably in all six degrees of freedom. At least one degree of freedom is always unstable. The theorem is a statement about static fields and about materials whose relative permeability exceeds one, and its exceptions matter: diamagnetic materials, whose permeability is slightly below one, and superconductors, which expel and pin flux, escape it entirely and can be levitated with no control at all. Electrodynamic suspension, which relies on induced currents rather than static fields, also falls outside its reach.

An active magnetic bearing does not evade Earnshaw's theorem. It accepts the unstable axis and stabilizes it with feedback, which is exactly what the theorem permits, because a controlled current is not a static field. This is why a purely passive permanent-magnet bearing can never carry a machine on its own. Passive rings can stabilize the radial axes and leave the axial one unstable, or the reverse, but something must actively control the remaining direction. Hybrid machines that use permanent magnets to carry the static load and active coils only for the unstable axis are common in small pumps and flywheels, and they save power precisely because the magnets, not the amplifiers, hold the weight.

Negative Stiffness and the Unstable Pole

Linearize the force about the centered position at a fixed current and the destabilizing effect appears as a negative stiffness, usually written as an open-loop position stiffness. Displace the rotor and the net magnetic force pushes it further, exactly as a spring with a negative constant would. Combined with the rotor mass, that negative stiffness places a real pole in the right half of the complex plane, and the rotor position diverges exponentially.

The numbers set the design tempo. A bearing carrying a 50-kilogram rotor section with an open-loop position stiffness of about 2 meganewtons per meter has an unstable pole near 32 hertz, corresponding to a time constant of roughly 5 milliseconds. A one-micrometer disturbance therefore grows to a half-millimeter gap closure in a few tens of milliseconds if the loop does nothing. Controllers accordingly run at sample rates of 10 to 20 kilohertz, giving hundreds of correction opportunities within a single time constant, and the loop must retain adequate phase margin well above the rigid-body frequencies it is stabilizing. The instability is fast enough to demand real-time digital hardware, and slow enough that ordinary digital signal processors and field-programmable gate arrays handle it comfortably.

Radial Bearings

A radial bearing is a stator of laminated electrical steel with inward-facing poles, wound so that opposed pole pairs act in the two orthogonal control directions. Eight poles is the most common count, giving two opposed pairs per axis, though four, twelve, and sixteen poles all appear. The rotor carries its own laminated sleeve, shrunk or bonded onto the shaft, because a solid rotor would dissipate heavy eddy-current losses in the alternating field. Lamination thickness of 0.1 to 0.35 millimeter is typical, following the same reasoning that governs motor cores.

Nominal radial air gaps run from a few tenths of a millimeter in small high-speed spindles to a millimeter or more in large turbomachinery. The gap is a direct design trade. A wide gap tolerates thermal growth, rotor bow, and manufacturing tolerance, and it keeps windage and the risk of contact low, but the required ampere-turns rise with the gap while the achievable force falls with its square, so a generous gap is paid for in copper, amplifier current, and bearing diameter.

Heteropolar and Homopolar Construction

In a heteropolar bearing the poles alternate in polarity around the circumference. This is the classical construction, it is simple to wind, and it packs a great deal of force into a short axial length. Its cost is rotor loss: as the shaft turns, any point on the rotor surface passes north and south poles in succession, so the flux in the rotor iron reverses several times per revolution. Hysteresis and eddy-current losses therefore scale with the product of rotation speed and pole-pair count, and they appear as heat in the rotor, which is the one part of the machine that cannot be cooled by conduction to a housing.

A homopolar bearing arranges the flux to keep the same polarity all the way around the rotor at any axial station, returning it through a second station displaced along the shaft. The bias flux is usually supplied by a permanent magnet or by a separate direct-current coil, while control coils modulate it. Because the rotor now sees a nearly constant field as it turns, rotor iron loss falls sharply, and the rotor can even be made of solid high-strength steel in some designs. Homopolar bearings are longer, more complex to manufacture, and harder to model, but they dominate where rotor heating or very high surface speed is the limiting constraint. Permanent-magnet biasing brings a second benefit, since the magnet supplies the standing flux without amplifier current and cuts the standby power the bearing draws.

The Axial Bearing

The thrust bearing is a different animal. A disc, usually integral with or shrunk onto the shaft, runs between two annular electromagnets that pull it in opposite directions along the axis. The disc must be strong, because centrifugal stress at its rim often sets the machine's speed limit, and it is normally solid steel rather than laminated, because a laminated disc of that geometry is impractical. Solid iron means eddy currents oppose any change of flux, and the axial bearing consequently has the lowest force bandwidth of the five axes, sometimes by an order of magnitude. Designers compensate with slots in the pole faces, with soft magnetic composite materials, or by arranging the machine so that the axial load is nearly steady. In a compressor the axial load follows the pressure rise across the impellers, so the thrust bearing is frequently the axis that saturates first during a process upset.

Saturation Sets the Load Capacity

The maximum pressure a magnetic pole can exert on iron across an air gap is the magnetic pressure, equal to the square of the flux density divided by twice the permeability of free space. At 1.5 tesla, a realistic working point for silicon steel, that is about 0.9 megapascal, or roughly nine times atmospheric pressure. Cobalt-iron alloys saturate near 2.3 tesla and roughly double the figure, which is why they appear in aerospace and weight-critical designs despite their cost.

Only part of the bearing bore is pole face, and current must be held in reserve for control, so the useful specific load capacity referred to the projected area of the bearing, meaning bore diameter times axial length, falls to something in the range of 0.3 to 0.5 megapascal for conventional steel. A bearing of 100-millimeter bore and 60-millimeter length therefore carries something on the order of 2 kilonewtons. That is a modest number beside a rolling bearing of the same envelope, and it is the single most important reason magnetic bearings do not displace mechanical ones in heavily loaded, slow-turning machines. Magnetic bearings win on speed, cleanliness, and controllability, never on force density.

Low-Bias and Zero-Bias Operation

Standby losses matter enormously in some applications. A flywheel that must hold energy for hours, or an implanted pump running from a battery, cannot afford to circulate a large bias current merely to keep the actuator linear. Low-bias and zero-bias strategies reduce or eliminate the standing current and accept a nonlinear actuator in exchange.

The costs are real. Near zero force the incremental gain of a zero-bias bearing approaches zero, because force varies as the square of the current, so small forces demand disproportionate current excursions and the effective bandwidth around the operating point collapses. Because only one magnet of a pair conducts at a time, a force reversal requires the current in one coil to fall to zero and the current in the other to rise, and the inductance of the coils makes that transition slow. Practical implementations therefore use a small residual bias rather than a true zero, or they apply a square-root shaping function to the current command so that the controller still sees an approximately linear actuator. Permanent-magnet biasing achieves much the same end from the other direction, supplying the standing flux at no electrical cost while keeping the linearized behavior of the differential mode.

Sensor Technologies

Eddy-current probes are the industry default for turbomachinery. A coil driven at radio frequency induces currents in the shaft surface, and the resulting change in coil impedance varies with the gap. The probes are rugged, tolerate oil, dirt, and high temperature, work against a wide range of conductive targets, and offer bandwidths in the tens of kilohertz with sub-micrometer resolution. Inductive and variable-reluctance sensors, which measure the reluctance of a magnetic circuit that includes the gap, are common in smaller machines and are inexpensive to integrate with the bearing laminations themselves. Capacitive probes offer the finest resolution and the lowest noise of the three, but they require a clean, dry gap and a well-defined target, which suits vacuum equipment and precision spindles and rules them out of a wet gas compressor. Sensors integrated into the bearing stator itself have become popular in compact drives, because they remove a separate sensor plane from the rotor layout.

Runout, Noncollocation, and Sensor Placement

Two mechanical problems dominate practice. The first is runout. The sensor measures the surface of a shaft that is never perfectly round, perfectly concentric, or magnetically uniform. Geometric runout produces a synchronous error signal that the controller interprets as motion and dutifully tries to correct, injecting a synchronous force that does not correspond to any real displacement. Electrical runout, caused by variations in permeability or residual magnetization of the target surface, produces the same effect without any geometric cause at all. Target surfaces are therefore ground, sometimes demagnetized, and the residual runout is often measured at low speed and subtracted from the signal by the controller.

The second problem is noncollocation. The sensor cannot occupy the same axial station as the bearing, because the bearing is there, so it sits beside it. For rigid-body motion the offset is harmless. For flexible modes it is not: a bending mode may move the sensor plane and the actuator plane in opposite directions, which inverts the sign of the feedback at that frequency and turns damping into excitation. Sensor placement is thus a rotordynamic decision as much as a packaging one, and placing a sensor near the node of an important mode is a classic and expensive mistake.

The Self-Sensing Question

Because the inductance of a bearing coil varies with the gap, the gap is in principle recoverable from the coil's own electrical behavior, either by demodulating the current ripple that the switching amplifier already produces or by injecting a small high-frequency probe signal. Self-sensing, sometimes called sensorless operation, would remove the sensors, their cabling, their axial length, and their failure modes at a stroke, which explains the persistent research interest.

It has not displaced dedicated sensors in industrial machines. The estimate degrades as the coil approaches saturation, exactly when the machine most needs accurate feedback; eddy currents in the iron corrupt the inductance measurement in a way that is difficult to model; and the achievable robustness of the resulting loop is fundamentally limited, a point established in the magnetic bearing literature and consistent with practical experience. Self-sensing appears today mainly in cost-sensitive or space-constrained products, and as a backup channel that lets a machine coast down safely after a primary sensor fails rather than as the normal means of control.

Decentralized PID and Its Limits

The overwhelming majority of machines in service use decentralized control: each of the five axes is treated as an independent single-input, single-output loop with its own proportional-integral-derivative controller, usually implemented as a lead compensator plus integral action. The proportional term supplies stiffness, which must exceed the open-loop negative stiffness before the loop is even stable. The derivative or lead term supplies damping and the phase lead that makes stabilization possible, and it is the term the sensor noise punishes, so it is invariably rolled off with a low-pass filter. Integral action removes the steady-state offset caused by gravity and by any constant process load, so the shaft sits at the geometric center rather than sagging toward the bottom poles.

Decentralized control works because the rigid-body dynamics of a symmetric rotor decouple reasonably well at low speed. It stops working when the rotor becomes flexible or when gyroscopic coupling becomes strong, and the remedies below address each case.

Flexible Rotor Modes

A rigid rotor has four rigid-body modes in the radial directions, two translational and two conical, and the controller sets their frequencies and damping directly. A rotor that must run above its first bending critical speed has, in addition, elastic modes whose frequencies the controller cannot choose. Those modes appear in the loop transfer function as lightly damped resonances, and unless something is done they will be excited by controller action, by unbalance, or by broadband noise.

Two approaches are used, often together. The first is to notch the controller at the modal frequency so that the loop simply does not act there, which is safe but purchases safety with phase lag below the notch and does nothing to damp the mode. The second is to damp the mode actively, shaping the controller so that the loop delivers force in phase with modal velocity at that frequency. Active damping is one of the genuine advantages of the technology, since a magnetic bearing can add damping to a bending mode that an oil film could not reach, and it is what allows some machines to run above criticals that would otherwise be prohibited. Both approaches depend on an accurate rotordynamic model, so magnetic bearing projects usually begin with a finite-element rotor model validated by modal testing, a workflow shared with the electromagnetic modeling described under Finite Element Analysis for Power Electronics.

Gyroscopic Coupling and Cross-Feedback

A spinning rotor resists tilting. The gyroscopic moment couples the two tilt axes, and the coupling grows in proportion to spin speed and to the polar moment of inertia. The consequence is visible on a Campbell diagram: the conical mode splits into a forward whirl whose frequency rises with speed and a backward whirl whose frequency falls. In a disc-shaped rotor, where the polar inertia exceeds the transverse inertia, the split is dramatic and the forward mode can climb out of the controller's effective bandwidth entirely.

Decentralized control handles this badly, because the coupling it must fight is precisely a cross-axis term it does not model. The standard remedy is cross-feedback: the tilt error measured in one plane is fed, with a gain proportional to speed, into the command of the orthogonal plane, producing a control moment that cancels the gyroscopic one. Well-implemented cross-feedback restores damping to both whirl modes across the speed range and is essential in high-speed flywheels, momentum wheels, and any machine with a large overhung disc.

Robust and Model-Based Designs

Where the machine is demanding, designers move beyond decentralized loops to multivariable methods that treat the five axes as one system. Linear-quadratic-Gaussian designs, H-infinity synthesis, and mu-synthesis all appear in the magnetic bearing literature and in commercial products. Their advantage is that they shape the closed-loop sensitivity explicitly against a stated model uncertainty, which matters here because the plant genuinely changes: bearing coefficients drift with temperature, rotor modes shift with thermal growth and with the stiffness of the foundation, and gyroscopic terms change continuously with speed. Gain scheduling against speed is common, and adaptive schemes that identify the plant online have been demonstrated. The controller structures themselves are the ones described under Control Algorithms, applied here to a plant whose defining feature is that it falls over when the computer stops.

Implementation is digital in every modern machine. Sample rates of 10 to 20 kilohertz are usual, conversion is synchronized to the amplifier carrier so that the sampled current is the average rather than a point on the ripple, and the computational and modulation delay of one to two sample periods must appear in the loop model, because at the frequencies where flexible modes live it contributes real phase lag.

Two Ways to Answer Unbalance

The first option is to insist that the geometric center of the shaft stay fixed. The controller responds to the synchronous displacement signal with full gain and generates whatever force is required to hold the shaft on the bearing axis. The rotor then turns about its geometric axis, the running clearance stays large, and the machine remains well clear of its backup bearings. The price is that the entire unbalance force passes through the bearing into the foundation, and because that force grows with the square of speed, the amplifier current required to supply it eventually saturates the bearing.

The second option is to let the rotor spin about its own principal axis of inertia. A synchronous notch filter, generalized to track speed, removes the rotational-frequency component from the feedback path, so the controller ignores the synchronous displacement and produces no synchronous force. The rotor finds its own axis, the transmitted vibration falls close to zero, and the bearing current drops sharply. This technique is known variously as unbalance force rejection control, automatic balancing, or adaptive synchronous cancellation, and it is why a magnetic-bearing machine can run astonishingly quietly at high speed. The price is that the geometric center now orbits, consuming clearance that must be available, and a rotor with substantial unbalance can orbit far enough to reach the backup bearings.

Real machines switch between the two, holding the geometric axis below a chosen speed and wherever clearance is precious, then releasing the synchronous component once the machine is above its last critical and running normally. A third variant computes the synchronous force feedforward and injects it, canceling the unbalance response without the phase lag a notch filter introduces elsewhere in the loop.

Passing Through a Critical Speed

The rigid-body critical speeds of a magnetically suspended rotor are set almost entirely by the controller, since the bearing stiffness is whatever the proportional gain makes it. Designers usually place them low, at a small fraction of operating speed, and give them heavy damping so that the machine passes through them during run-up without significant response. That freedom is not available with rolling or fluid-film bearings, whose stiffness is a property of the hardware.

Bending criticals are different, because their frequencies belong to the rotor. Passing one requires the loop to supply damping at that frequency with the correct phase, which is difficult when the mode lies near or above the loop's bandwidth. When the mode cannot be damped, the machine must accelerate through it quickly enough that the response does not build, and the acceptance procedure has to demonstrate that the transient orbit stays within the clearance. This is where the difference between an academic model and a real rotor becomes expensive, and it is why unbalance response verification and closed-loop measurement are written into the acceptance standards described below.

What a Drop Does

Delevitation has many causes: loss of supply power, an amplifier or controller fault, a sensor failure that feeds the loop a plausible lie, a mechanical shock that exceeds the bearing's force capacity, or a process event such as compressor surge that loads the rotor beyond what the amplifiers can answer. Whatever the cause, a rotor turning at full speed suddenly falls a few tenths of a millimeter onto a stationary bearing race.

What happens next depends on friction and on the dynamics of the contact. In the benign case the rotor drops, the backup bearing's inner race accelerates to shaft speed within a few revolutions, and the rotor settles into a bouncing or pendulum motion while it coasts down. In the destructive case, friction at the contact drives the rotor around the inside of the backup bearing bore in the direction opposite to its spin. This backward whirl, sometimes called dry whip, is self-sustaining, occurs at a frequency well above the spin frequency, and generates contact forces many times the rotor weight. It can wreck the backup bearings, the rotor surface, and the bearing supports in seconds. Predicting whether a given machine will enter backward whirl requires nonlinear contact modeling and is validated by deliberate drop testing, which is a standard, and nerve-wracking, part of commissioning a large machine.

Designing for the Landing

Backup bearings are usually angular-contact ball bearings, frequently with ceramic balls and a steel or polymer cage, chosen because they can accept the impulsive load and accelerate quickly. Plain bushings of bronze or a self-lubricating polymer are used in smaller machines and in vacuum equipment where a rolling bearing would outgas. The radial clearance is set to roughly half the magnetic air gap, so the rotor always lands on the backup surface before any part of it reaches the bearing laminations, and the axial clearance is set on the same principle relative to the thrust disc.

Several measures reduce the severity of a landing. Compliant, damped mounts, using tolerance rings or elastomeric elements, absorb impact energy and shift the contact dynamics away from backward whirl. Preloading the bearing prevents the balls from skidding on first contact. Some controllers implement a soft-landing or controlled-drop sequence, retaining whatever authority remains to lower the rotor gently, or deliberately holding it against one side of the clearance so that it does not orbit. Backup bearings are rated in landings rather than in hours, and manufacturers commonly specify replacement after a small number of full-speed drops, so a machine that has dropped is a machine that requires inspection before it returns to service. ISO 14839-4 addresses touchdown bearing design considerations explicitly, alongside system architecture and environmental factors, which is a fair indication of how central the subject is.

Turbomolecular and Vacuum Pumps

Turbomolecular pumps drive bladed rotors at tens of thousands of revolutions per minute with blade tip speeds approaching the thermal velocity of the gas molecules they pump. A lubricated bearing in that environment contaminates the vacuum it is supposed to create, and its vapor pressure sets a floor on the achievable pressure. Magnetically suspended pumps remove the lubricant entirely, permit any mounting orientation, tolerate higher speeds, and run with much lower vibration, which matters when the pump is bolted to an electron microscope or a surface-analysis instrument. Hybrid pumps compromise by using a permanent-magnet radial bearing at the high-vacuum end and a small ceramic ball bearing at the fore-vacuum end, where a trace of lubricant is tolerable. The equipment context appears under Semiconductor Wafer Fabrication.

Compressors, Expanders, and Subsea Machines

Process turbomachinery is the largest industrial market. A magnetically suspended compressor needs no lube oil system, no seal oil, and often no shaft seal at all, since the motor and compressor can share a single hermetically sealed casing filled with process gas. That removes an entire auxiliary skid, its pumps, coolers, filters, and alarms, and it removes the leakage path that a seal represents. Turboexpanders in natural gas processing, air separation, and organic Rankine cycle plants use magnetic bearings for the same reasons, with the added benefit that the bearing tolerates cryogenic temperatures that would defeat a lubricant.

The most demanding installation of this kind is subsea. The Åsgard field on the Norwegian continental shelf began operating the world's first subsea gas compression station in September 2015, using hermetically sealed MAN HOFIM motor-compressor units in which a seven-axis active magnetic bearing system is enclosed within the pressure casing. The logic is unanswerable: nothing on the seabed can be serviced without a vessel, so a machine with no oil system, no seal, and no wearing bearing surface is worth a great deal. MAN Energy Solutions reported in 2022 that two of the units had accumulated more than 100,000 operating hours without intervention. Pipeline compressors onshore adopt the technology for the related reason that unattended stations benefit from the absence of an oil system. The power and control electronics for such installations are treated under Downhole and Subsea Power.

Machine Tool Spindles

High-speed milling and grinding spindles use magnetic bearings to reach surface speeds that would shorten the life of a rolling bearing to hours, and to survive the thermal environment of a spindle running at full power. Two further benefits are specific to machining. The spindle measures cutting force directly from its bearing currents, which supports tool-condition monitoring and adaptive feed control, and the bearing stiffness can be varied on the fly, which allows the control system to move a chatter-prone resonance or to deliberately vary spindle behavior to break up regenerative chatter. The machine-level context appears under Computer Numerical Control (CNC) Systems. The technology has stayed a specialty rather than a default because a spindle crash is common in machining and a magnetic spindle answers a crash less gracefully than a rolling one.

Flywheel Energy Storage

A flywheel that stores energy for minutes or hours is judged on standby loss, and bearing friction is the loss that matters. Running the rotor in a vacuum housing removes windage; supporting it magnetically removes the rest. Grid-scale flywheel installations built for frequency regulation, such as the plant at Stephentown, New York, exploit exactly this, cycling continuously with none of the wear a rolling bearing would accumulate. Many designs use a passive permanent-magnet unloader to carry the static weight of the rotor with no electrical input at all, leaving the active bearings to handle only radial location and disturbance rejection, which minimizes the parasitic draw that would otherwise erode the stored energy. The converter side of such installations is covered under Energy Storage Systems.

Chillers and Oil-Free Refrigeration

Oil-free centrifugal compressors on magnetic bearings, introduced commercially for chiller duty in the early 2000s and now supplied by several manufacturers, changed the economics of large building cooling. The immediate gain is the absence of an oil system, but the larger one is that oil no longer migrates into the evaporator and condenser, where even a small film degrades heat transfer and quietly costs efficiency for the life of the machine. Because the compressor is variable speed and starts softly, part-load efficiency improves substantially, and inrush current at start is a fraction of that of a conventional machine. This is probably the application in which magnetic bearings have reached the widest ordinary commercial deployment.

Rotary Blood Pumps

An implanted ventricular assist device must run for years without maintenance and must not damage the blood passing through it. Any mechanical bearing in the blood path creates a region of high shear and stagnation where clots form. A fully magnetically levitated centrifugal rotor, as used in the Abbott HeartMate 3, removes that contact entirely and permits wide blood-flow gaps, and the pivotal MOMENTUM 3 trial reported markedly lower rates of pump thrombosis and pump replacement than the earlier axial-flow device it was compared with. The suspension in such a device must be extraordinarily efficient, because every milliwatt comes through a percutaneous cable from a battery the patient carries, which is why passive and permanent-magnet-biased arrangements dominate this application. The associated power electronics are treated under Implantable Power Systems.