Superconducting maglev, 3D

What is modelled

A 20-tonne car levitates by electrodynamic suspension (EDS): superconducting magnets in the car (arranged as a Halbach array, field pattern repeating every wavelength lambda) move over short-circuited coils in the guideway. Moving flux induces currents in the coils; the currents push back. Nothing is pressed into the guideway at standstill: below take-off speed the car rolls on wheels.

Equations (lumped, Inductrack-style)

• Field at the coil face: B(y) = B0 e^(-k y), k = 2 pi / lambda.
• Magnetic pressure scale: P0 = B0^2 / (2 mu0).
• Lift: F_L = A P0 e^(-2 k g) v^2 / (v^2 + vc^2). The coil loop has resistance R and inductance L; vc = R / (k L) is the speed where the induced current has fallen to its resistive/inductive half-way point.
• Magnetic drag: F_D = F_L vc / v (loop impedance ratio R / (omega L)). Drag per unit of lift is highest at low speed and falls as 1/v.
• Aerodynamic drag: F_a = 1/2 rho CdA v^2.
• Vertical motion: m z'' = F_L - m g0 - c g', gap g = z - h(x), with an optional wavy guideway h(x). Stiff wheel contact below g = 20 mm.
• Equilibrium gap: g_eq = ln(A P0 f / (m g0)) / (2 k) with f = v^2 / (v^2 + vc^2).
• Stiffness: dF_L/dg = -2 k F_L = -2 k m g0 at equilibrium, so the vertical frequency is omega = sqrt(2 k g0) and does not depend on B0, mass or speed.
• Take-off speed: v_to = vc sqrt(f0 / (1 - f0)), f0 = m g0 / (A P0 e^(-2 k g_wheel)).
• Propulsion: a linear synchronous motor modelled as a proportional thrust, F_t = clamp(Kp (v_target - v)) with Kp = cap / 5 and a thrust limit cap.

Honest simplifications

• One lumped loop with a single time constant; no end effects, no finite coil width, no guidance (lateral) forces and no heating of the superconductor. Real systems (Inductrack, SCMaglev-type figure-eight coils) add all of these.
• An EDS suspension has almost no natural damping; the damping ratio slider stands for secondary suspension or active damping.
• Default numbers (B0 = 0.9 T at the coil face, 20 t, 400 km/h target, take-off near 107 km/h) are illustrative and are not data from any particular line. Real wheel lift-off speeds are typically higher.
• Quench: the field decays as exp(-t / 4 s), an illustrative time constant.
• Independent educational model, not affiliated with any maglev operator or manufacturer.
• The drag hump: at take-off the drag is m g0 sqrt(1/f0 - 1), often close to the car's weight, so a linear motor that is too weak never gets the car through it. That is why real systems keep wheels and a strong motor for the first minute.

The four checks

Each check integrates the equations of motion numerically (fixed 0.5 to 2 ms sub-steps) in a separate headless run and compares the result with an algebraic formula above: equilibrium gap, vertical oscillation period, take-off speed, and the steady cruise speed from the force balance. The 3D scene only draws what the model says.