active-magnetic-bearing

name	active-magnetic-bearing
description	Active magnetic bearing (AMB) — electromagnetic force model, PID/LQR control, rotor dynamics on AMB, stiffness/damping estimation, backup bearings, ISO 14839, high-speed machine applications.
metadata	{"priority":7,"promptSignals":{"phrases":["active magnetic bearing","magnetic bearing","AMB","levitation bearing","electromagnetic bearing","ISO 14839","contactless bearing"],"minScore":3}}

Active Magnetic Bearing — Complete Skill

Electromagnetic Force Model

Basic electromagnet force (one pole): F = μ₀ × N² × I² × A_pole / (4 × g²) [N; attractive; g = air gap]

For differential drive configuration (opposing electromagnets): F_net = μ₀ N² A_pole / 4 × [(I₀ + i)²/( g₀ - x)² - (I₀ - i)²/(g₀ + x)²]

I₀ = bias current; i = control current; g₀ = nominal gap; x = displacement

Linearized (small x, small i): F ≈ k_i × i - k_x × x

k_i = ∂F/∂i|{x=0} = μ₀ N² A_pole I₀ / g₀² [force-current coefficient; N/A] k_x = ∂F/∂x|{i=0} = μ₀ N² A_pole I₀² / g₀³ [negative stiffness; N/m]

Key feature: k_x is NEGATIVE (open-loop unstable) → must close loop with controller

Open-Loop Plant Transfer Function

Rigid rotor, single axis: m ẍ = F_AMB + F_disturbance = k_i × i - k_x × x + F_d

Laplace: m s² X = k_i I - k_x X + F_d X(s) = [k_i I(s) + F_d(s)] / (ms² + k_x) [note: +k_x because k_x is negative stiffness coefficient]

Plant (to control current i): G_plant(s) = k_i / (ms² - |k_x|) [unstable second-order pole]

Sensor + power amplifier: Total loop: G(s) = k_s × G_amp(s) × G_plant(s) × (1/s_notch) (with notch filters) k_s = position sensor gain [V/m]; typically eddy current sensor, 50–200 μm range

Control Design

PID Control

C(s) = K_p + K_i/s + K_d × s/(τ_d s + 1)

For AMB: primarily PD (derivative adds damping to unstable pole) K_d adds virtual damping c = k_i × K_d K_p adjusts effective stiffness k_eff = k_i × K_p - k_x [keep positive] K_i eliminates steady-state offset from constant loads (gravity)

Tuning: cross-over frequency 10× open-loop unstable pole frequency Stable if: K_p > k_x / k_i (minimum proportional gain)

LQR / State-Space Control

Full state feedback: u = -K x = -[K_p K_d] [x ẋ]^T Optimal K from LQR minimizes J = ∫(x^T Q x + u^T R u)dt Set Q and R per Bryson's rule: Q_11 = 1/x_max², R = 1/i_max²

Notch Filters (Critical)

Rotor unbalance → synchronous disturbance at running speed ω_r Notch at ω_r removes control current response to synchronous load → rotor levitates on mass center f_notch(s) = (s² + ω_r²) / (s² + 2ζ_n ω_r s + ω_r²)

Adaptive notch: automatically tracks running speed; essential for variable speed

Rotor Dynamics on AMB

Critical speeds on AMB: Rotor natural frequencies with AMB stiffness k_eff = k_i K_p - k_x: f_crit = (1/2π) √(k_eff / m)

Adjust k_eff by changing K_p → can tune critical speed

Flexible rotor: model as beam + AMB at each bearing location Modal expansion; controllers must provide adequate damping at all modes within operating range

Gyroscopic effects (high-speed): Conical mode splits into forward/backward precession modes AMB must control both → cross-coupling required in MIMO control

Stiffness and Damping (Dynamic)

AMB provides effective stiffness and damping: k_AMB(ω) = Re[G_cl(jω)] × k_i / k_s [varies with frequency] c_AMB(ω) = -Im[G_cl(jω)] × k_i / (ω × k_s)

At low frequency (< ωn_control): stiffness-dominated → k_eff ≈ K_p × k_i - k_x At high frequency: rolls off; backup bearings needed above cutoff

Backup Bearings (Touchdown Bearings)

Required for: rotor landing on failure, startup/shutdown, power-off Rolling element or sleeve bearing; large clearance (g_backup = 0.3–0.7 × g_AMB) Limit: rotor touches backup bearing at this clearance; critical condition

Backup bearing design criteria: Landing speed: depends on speed at failure; heat generated Max contact speed: 100–500 m/s (depends on backup bearing material) Number of landings: design for 10–50 events

Standard ISO 14839-4: specifies backup bearing requirements; acceptance testing

Performance Metrics and ISO 14839

ISO 14839-1 (terminology): define terms for AMB systems ISO 14839-2 (vibration): vibration level evaluation criteria ISO 14839-3 (stability margin): minimum stability requirement: μ ≥ 0.4 (maximum eigenvalue sensitivity) ISO 14839-4 (backup bearings): backup bearing function and testing

Stability margin (ISO 14839-3): Maximum singular value of sensitivity function S: ||S||∞ ≤ 2.0 (margin μ = 1/||S||∞ ≥ 0.5) Peak sensitivity < 6 dB recommended

AMB Applications

Compressors and expanders: No lubrication → clean process gas; hermetically sealed → zero emissions T range: -200°C (cryogenic expanders) to +250°C Speed: 10,000–100,000 rpm; common in LNG, petrochemical, aviation APU

Flywheel energy storage: Vacuum environment; no bearing losses → high round-trip efficiency > 90% Speed: 10,000–60,000 rpm; stored energy: 0.1–200 kWh

Machine tools: High-speed spindles (100,000+ rpm); active error compensation; vibration-free Grinding, milling; nanometer precision achievable

Medical (left ventricular assist device — LVAD): Implantable blood pump; magnetic levitation → hemolysis reduction; 5+ year life

Power Requirements

Bias current power dissipation: P_bias = 4 × (N I₀)² × R_coil / N² [W per 4-pole actuator; R_coil = total coil resistance]

Control current power: Much less than bias in most cases; peaks during transients Total AMB power: 50–500 W (industrial size); 2–20 W (miniature)

Cooling: forced air or liquid cooling for coil; copper losses dominate

Output

Provide: electromagnetic force model parameters (k_i [N/A], k_x [N/m], g₀ [mm]), PID controller gains (K_p, K_i, K_d), closed-loop poles, effective stiffness k_eff [N/m] and damping c_eff [N·s/m], critical speed on AMB [rpm], stability margin ||S||_∞ vs. ISO 14839-3 limit, backup bearing clearance and design rating, notch filter frequencies [Hz], power dissipation P [W], and applicable standard (ISO 14839-1 through 4).