The 3D version renders coil brightness from a single shortcut formula, cos(k·x − k·x_s). This 2D companion instead builds the guideway field the way a real LSM stator actually does: from three separate phase windings, each carrying its own sinusoidal current 120° apart in time and spatially offset 120° apart along the track. Their sum is proven — not assumed — to collapse into one clean traveling wave:
Phase currents: I_p(t) = I₀ cos(ωt − p·120°), p = A,B,C
Winding shape: W_p(x) = cos(k·x − p·120°)
Resultant field: F(x,t) = Σ_p I_p(t)·W_p(x) = 1.5·I₀ cos(k·x − ωt)
Synchronous speed: v_s = 2τf Electrical wavenumber: k = π/τ
Load angle: δ = k(x_s − x_r) Thrust: F = F_max sin(δ)
Rotor dynamics: M dv/dt = F − F_load − F_aero
The top panel plots the three individual phase contributions faintly, and their live sum as the bright traveling ribbon — the same trick used in every real three-phase induction/synchronous machine to make a rotating (here, linear) field with no moving parts in the stator itself.
The bottom panel is new physics the 3D scene never shows: a phase-plane portrait of the load angle δ against its own rate of change δ'. Differentiating the equations above twice gives a textbook synchronous-machine swing equation — mathematically the same equation as a pendulum with an applied torque:
δ' = k(v_s − v)
δ'' = −(k/M)·(F_max sin δ − F_load − F_aero)
This is exactly why a real maglev can't just switch on full running frequency from a standstill: the "pendulum" starts with far too much swing energy (huge initial slip δ') to settle into the stable well at sin δ = F_load/F_max, so it hunts back and forth (or spins through every angle) instead of locking. Real guideways solve this with a slow variable-frequency ramp — try dragging the frequency slider down, letting the train pick up speed, then raising it gradually, versus the "Start near sync" button which drops the trajectory directly into the stable well to show what a genuine lock looks like.
- The stable equilibrium sits at δ* = asin(F_load/F_max) — raise the load or lower Fmax and the well gets shallower and narrower.
- Push the load past Fmax and the well disappears entirely: sin δ can never reach load/Fmax, so the phase point can never rest — pull-out is guaranteed, not just possible.