Magnetic Levitation (2D) — PID-Controlled Electromagnet & Ball

Controller

Setpoint gap 12 mm

Proportional gain Kp 0.55

Derivative gain Kd 0.012

Ball load 55 g

Perturb

Tip: click/tap directly on the ball in the canvas to drag it and let go — the controller has to catch it.

Live Readout

Gap x

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Error e = x−x₀

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Coil current I

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Magnetic force

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Ball velocity

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Status

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How it works

A single-axis maglev: a ferromagnetic ball hangs below an electromagnet along a vertical track. Gravity pulls it down; the coil's field pulls it up. The attractive force from a coil falls off with the square of the gap and rises with the square of the current:

F_mag(I, x) = k * I^2 / x^2
m * x'' = m*g − F_mag(I, x)

This is why an electromagnet alone can never hold the ball still: at the setpoint gap the force exactly balances gravity, but nudge the ball a hair closer and F_mag grows faster than the restoring need — it snaps to the coil. Nudge it a hair farther and F_mag collapses — it drops. The equilibrium is open-loop unstable, the textbook example used to teach feedback control.

A gap sensor and a PID controller close the loop: measure the error between the actual gap and the setpoint, and drive the coil current to cancel it before the ball can run away:

e(t) = x(t) − x0
I_cmd = clamp(I_bias + Kp*e + Kd*(de/dt), 0, I_max)

Raise Kp and the loop reacts harder to any error — good for stiffness, but push it too far relative to Kd and the ball starts to overshoot and ring (visible as growing oscillation). Kd damps that ringing by reacting to the ball's velocity, not just its position. Heavier loads need more steady-state current just to hold position, which eats into the headroom the controller has left for correcting disturbances — overload it enough and the coil saturates at I_max and the ball falls.

Coil field (glow = current) Ferromagnetic ball Setpoint gap

drag the ball · click Nudge to disturb it