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Mechatronics / Rotating

Active Magnetic Bearing (AMB) Radial Load Simulator

Design an Active Magnetic Bearing that levitates a spinning rotor with electromagnets. Adjust rotor mass, speed, air gap, bias current, pole area and PID gain to watch the peak magnetic force, current and position stiffness, levitation stability and coil power loss update in real time — and size AMBs for turbomolecular pumps, LVAD heart pumps and flywheel rotors.

Parameters
Rotor mass m
kg
Rotational speed N
rpm
Air gap g_0
mm
Single-side air gap between rotor and electromagnet. Narrow = strong but risky.
Bias current i_0
A
Steady current for linearisation. Higher k_i but more dissipation.
Coil turns N_c
turns
Pole face area A
mm²
Iron-core area per pole. Larger = more headroom before saturation.
Control gain K_p
A/m
PID proportional gain that compensates the negative k_s.
Results
—
Peak force F_max (N)
—
Current stiffness k_i (N/A)
—
Position stiffness |k_s| (N/m)
—
Rotor weight (N)
—
Safety factor F_max/F_total
—
Coil power loss (W)
—
AMB cross-section — 4-pole electromagnets and rotor

The rotor in the centre is surrounded by 4 electromagnets. Differential currents (top vs bottom, left vs right) cancel gravity and unbalance force while the rotor spins. Colour shows the safety factor (green=margin, orange=tight, red=saturated).

Peak force F_max vs air gap g_0
Levitation stability — PID gain K_p vs position stiffness |k_s|
Theory & Key Formulas

$$F = \frac{\mu_0 N^2 i^2 A}{4 g^2},\quad k_i = \frac{2\,\mu_0 N^2 i_0 A}{g_0^2},\quad k_s = -\frac{2\,\mu_0 N^2 i_0^2 A}{g_0^3}$$

F: magnetic pull per pole (μ_0 = 4π×10⁻⁷ H/m). k_i: current stiffness. k_s: position stiffness (negative). Differential capacity is 2F. The closed loop is stable when K_p·k_i − |k_s| > 0.

$$F_{\text{unb}} = m\,e\,\omega^2,\quad \omega = \frac{2\pi N_{\text{rpm}}}{60}$$

Unbalance force. e is the residual mass eccentricity (≈ 1 µm for high-precision balancing). ω is the angular speed; the force grows as the square of rotational speed.

Active Magnetic Bearing (AMB) Radial Load Control — High-Speed Rotor Suspension

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So an "active magnetic bearing" floats the rotor on magnets? Can magnets alone really hold a heavy spinning shaft?
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Yes — Active Magnetic Bearing, AMB for short. You put four (or eight) electromagnets symmetrically around the rotor and let a controller adjust the currents in real time. If the rotor starts to drop, pull harder from the top; if it drifts right, pull harder from the left. The basic law is F = μ_0·N²·i²·A/(4·g²), where μ_0 is vacuum permeability, N is coil turns, A is pole area and g is the air gap. With the defaults (N=200, i=5 A, g=0.5 mm, A=1000 mm²) you get about 1.26 kN per pole, or 2.5 kN in differential operation. That easily lifts the 491 N weight of a 50 kg rotor.
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Got it. But the panel says "position stiffness k_s is negative". A negative spring constant sounds creepy — what does that actually mean?
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That's the heart of AMB design. The closer the rotor gets to an electromagnet, the stronger the pull (F ∝ 1/g²). So when the rotor drifts up, the top magnet sucks it up even harder. The force pulls in the direction of displacement, which means the equivalent spring constant is negative — the system is inherently unstable in open loop. With the defaults, k_s comes out around −2×10⁷ N/m, meaning "displace 1 mm and you get pulled with 20,000 N (2 tonnes)". That's why you must wrap a PID controller around it that drives current opposite to the displacement and synthesises an effective positive stiffness.
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So K_p in the panel does that compensation. How large does it need to be?
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The rule is simple: K_p·k_i − |k_s| > 0. Current stiffness k_i tells you "force per amp", which is 2010 N/A with the defaults. So K_p = 5×10⁵ A/m gives 5×10⁵·2010 ≈ 10⁹ N/m of positive stiffness, which overpowers the |k_s| of 2×10⁷ by more than 50×. The catch is that high K_p makes the loop sensitive to noise and delays. Real AMBs use a 3-DOF PID with derivative damping K_d and integral K_i, sampled at 10 kHz or faster, plus notch filters at the rotor bending modes. If the loop delay hits a resonance, the rotor touches down instantly — that's the hardest part of magnetic bearing design.
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Unbalance force shows up too, and at the default speed it's already 494 N — about the same as the weight.
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Good catch. F_unb = m·e·ω² explodes with speed squared. With the defaults (30,000 rpm, ω ≈ 3140 rad/s, e = 1 µm, m = 50 kg) you get 494 N of synchronous rotating load on top of gravity. Take the same rotor to 60,000 rpm and the unbalance jumps to ~2 kN; at 100,000 rpm it's around 5.5 kN and eats the entire AMB capacity. That's why high-speed AMB rotors need both ISO G0.4-class balancing and matching dynamic capacity, plus auto-balancing notch filters in the controller that let the rotor spin about its mass-centre axis instead of the geometric axis.
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Where do people actually use these complicated bearings in real machines?
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The classic is the turbomolecular pump (TMP) used in semiconductor fabs. They spin at 90,000+ rpm to pump a vacuum, no lubricant allowed — only magnetic bearings work. Pfeiffer, Edwards and Shimadzu are the big TMP makers. Next come implantable LVAD heart pumps such as HeartMate3 and HVAD, where AMBs float the impeller for years without touching anything. Then flywheel energy storage (Beacon Power), semiconductor lithography stages, cryogenic refrigerator compressors, and large oil-and-gas motors from Calnetix, Synchrony Magnetic Bearings and SKF Magnetic Mechatronics. The common thread is "can't lubricate, can't maintain, has to spin extremely fast".

Frequently Asked Questions

The peak force per axis of a differential AMB is F_max = 2·(μ_0·N²·i_0²·A)/(4·g_0²), where μ_0 is vacuum permeability (4π×10⁻⁷ H/m), N is coil turns, i_0 is bias current, A is pole face area and g_0 is the air gap. The factor of 2 accounts for the two opposing electromagnets in differential operation. In practice the load is also limited by core saturation (around 1.6 T for electrical steel), so design for a flux density B = μ_0·N·i/g under about 1 T.
They are the two basic coefficients of the linearised AMB model. Current stiffness k_i = 2μ_0·N²·i_0·A/g_0² [N/A] is the force per amp of control current and acts as the actuator gain. Position stiffness k_s = -2μ_0·N²·i_0²·A/g_0³ [N/m] pulls the rotor further off-centre as it drifts, and is always negative. Because of this negative stiffness an AMB is unstable in open loop and must use a PD/PID controller to build a positive closed-loop stiffness K_p·k_i − |k_s| > 0.
Unbalance force F_unb = m·e·ω² grows with the square of speed. For a 50 kg rotor at 30,000 rpm (ω ≈ 3142 rad/s) with a high-precision balance e = 1 µm, F_unb = 50·1×10⁻⁶·3142² ≈ 494 N — about the same as the rotor weight of 491 N. Doubling the speed quadruples the force, so high-speed AMBs are governed jointly by balance quality (ISO G0.4 class) and AMB dynamic capacity.
An AMB levitates the rotor contact-free with electromagnets, so there is zero friction, no lubricant, no wear, speeds above 100,000 rpm and vacuum compatibility. The trade-offs are continuous bias-current power (tens of W to kW), backup touchdown bearings for power loss events, and a fast (≥ kHz) PID controller to overcome the negative stiffness k_s. AMBs are the go-to choice for turbomolecular pumps, LVAD heart pumps and flywheel energy storage where maintenance, speed or lubrication constraints rule out conventional bearings.

Real-world applications

Turbomolecular pumps (TMP): Semiconductor fab vacuum pumps spin rotors at 24,000-90,000 rpm to knock gas molecules out of the chamber. Oil lubricant would contaminate the vacuum, so magnetic bearings are the only option. Pfeiffer Vacuum, Edwards and Shimadzu dominate the market, and every MOCVD, lithography, FIB and SEM tool includes one. Typical design points fall in the 0.3-0.5 mm air gap, 3-6 A bias current range — the values you can match in this tool.

Ventricular assist devices (LVAD): HeartMate3 and HVAD float the impeller of a centrifugal blood pump on AMBs and run for years contact-free. With no rubbing seals or bearings, thrombus formation is dramatically reduced and patient outcomes improve. Design constraints unique to medical devices include levitation power below 5-10 W (battery operation), graceful touchdown behaviour on power loss, and corrosion resistance against blood components.

Flywheel energy storage and high-speed motors: Beacon Power's 20-MW-class frequency-regulation flywheels combine AMBs with a vacuum chamber to cut self-discharge to a tenth of conventional designs. Calnetix and Synchrony Magnetic Bearings supply AMBs for oil-and-gas turbo-expanders, and Waukesha Bearings outfits industrial gas turbines. System-level vibration and loss design extends the lumped-parameter calculations of this tool with FEM modal analysis.

Where this tool sits in the CAE workflow: Detailed AMB design proceeds in three stages: first a lumped-parameter model (this tool) to size turns, gap and bias current; then a magnetic FEM solver (COMSOL, JMAG, Ansys Maxwell) to check core saturation and stray flux; and finally a coupled MBD + controller co-simulation to tune rotor dynamics and PID gains. This calculator gives you the order-of-magnitude force and stiffness needed to enter stage one with confidence.

Common misconceptions and pitfalls

The biggest trap is ignoring core saturation when sizing the load. The formula F = μ_0·N²·i²·A/(4·g²) assumes infinite core permeability, but real electrical steel saturates around 1.6 T. With the default numbers (N=200, i=5 A, g=0.5 mm) the flux density is roughly 2.5 T — deeply saturated. A real machine drops the bias current to 3-4 A or widens A to keep B near 1.2 T. Always cross-check this tool's linear estimate against the actual B-H curve before fixing the design.

Next, treating touchdown bearings as an afterthought. Every AMB will lose levitation at some point — power loss, controller fault or large disturbance. If a 90,000 rpm rotor hits the iron stator directly it destroys million-dollar laminations in one event. So a ceramic angular-contact ball bearing (the "touchdown" or "auxiliary" bearing) is mounted at half the magnetic gap (e.g. 0.25 mm) to catch the rotor. Designs that optimise AMB capacity but defer the touchdown bearing rating typically wreck their entire bearing set on the first commissioning run. Decide AMB gap g_0, touchdown clearance g_TD and the maximum allowable displacement together from day one.

Finally, the myth that "higher PID gain = higher stiffness". Statically, K_p·k_i − |k_s| does scale linearly with K_p. But high K_p also excites the rotor's first and second bending modes (typically a few kHz), and a delay of even one sample at those frequencies destabilises the loop. Practical AMB design predicts the bending modes with rotor FEM, places notch filters at each peak, and only then pushes K_p hard. The "closed-loop stiffness > 0" criterion in this tool is the rigid-body condition only — bending-mode stability is usually the binding constraint in real machines. Bleuler / Maslen / Schweitzer "Magnetic Bearings: Theory, Design, and Application to Rotating Machinery" (Springer, 2009) is the standard reference text.

How to Use

  1. Enter rotor mass (1–50 kg) and rotation speed (5,000–50,000 rpm) to define baseline operating conditions.
  2. Adjust air gap (0.5–2.0 mm) and bias current (1–10 A) to control electromagnetic stiffness and levitation force.
  3. Monitor Peak force F_max, position stiffness |k_s|, and Safety factor to ensure F_max exceeds rotor weight and margin exceeds 1.5.
  4. Validate coil power loss stays below thermal budget; reduce bias current if loss exceeds cooling capacity.

Worked Example

A 15 kg turbomolecular pump rotor spinning at 35,000 rpm requires levitation with 147 N weight support. Set air gap to 0.8 mm and bias current to 6 A. Simulator yields Peak force F_max = 220 N, position stiffness |k_s| = 18,500 N/m, current stiffness k_i = 36.7 N/A, and coil power loss = 42 W. Safety factor = 220/147 = 1.50, meeting minimum threshold. Radial disturbance at 35 kHz frequency induces deflection δ = 0.012 mm, well within bearing clearance.

Practical Notes

  1. Reduce air gap below 0.6 mm only for precision compressors; magnetic saturation in laminated poles increases nonlinearity above 2.0 A/mm².
  2. Bias current above 8 A demands active cooling (forced convection); copper coil losses scale as I²R, reaching 80 W at 10 A.
  3. Position stiffness must exceed 10,000 N/m for rotor stability under 1g horizontal shock; verify against actual unbalance force profile.
  4. Verify Safety factor ≥ 1.8 in critical applications (aerospace turbochargers); margin accounts for sensor noise and control loop lag.