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The science
What is modelled
A rotor floats in four electromagnet poles (+X, −X, +Y, −Y). Each pole pulls with a force that grows with current squared and falls with gap squared, so the bare system is unstable. A PID loop reads the rotor position and shifts the pole currents around a bias current to hold it at the centre.
Equations
F_pole = k·I²/d², F_x = F(I+, g − x) − F(I−, g + x); linearised: F ≈ k_s x + k_i i with k_s = 4kI0²/g³ (negative stiffness) and k_i = 4kI0/g². Controller: F_cmd = Kp·e − Kd·v + Ki∫e dt. Unbalance force m·e·ω² rotating with the shaft. Stable static hold needs Kp > k_s.
Honest simplifications
Illustrative single radial bearing with two translational degrees of freedom (no tilt, no gyroscopic coupling, no flexible modes), first-order coil lag, saturating currents and a simple contact model at the touchdown limit; not a certified machine twin. The time axis is slowed about 30 times so the orbit is visible.
Built-in checks
The linearised stiffnesses against finite differences of the force law, the static deflection under a side load against F/(Kp − k_s), and removal of that offset by integral action.