magnetic-bearing

name	magnetic-bearing
description	Active magnetic bearing (AMB) design — electromagnetic force, bias flux, control current, position stiffness (negative), current stiffness, PID/LQR control, rotor dynamics (critical speed, stability), auxiliary (backup) bearings, ISO 14839, API 617 AMB, power amplifier sizing, contact-free operation for high-speed/high-temperature applications.
metadata	{"priority":7,"promptSignals":{"phrases":["magnetic bearing","active magnetic bearing","AMB","electromagnetic bearing","contact-free bearing","magnetic levitation bearing"],"minScore":3}}

Active Magnetic Bearing (AMB) Design — Complete Skill

Electromagnetic Force Fundamentals

Single Electromagnet Force

Magnetic force on ferromagnetic rotor (simplified): F = μ₀ × N² × I² × A_gap / (2 × g²) [N; μ₀ = 4π×10⁻⁷ H/m; N = turns; I = current; A_gap = pole face area [m²]; g = air gap [m]]

More precisely (flux-based): F = B² × A_gap / (2 × μ₀) [B = flux density at gap [T]; target B = 0.8–1.5 T for silicon steel]

Iron saturation limit: B_sat ≈ 1.7–2.0 T for silicon steel; operating B < 1.5 T to avoid saturation

Coil inductance: L = μ₀ × N² × A_gap / (2g) [H; changes with gap g → nonlinear!]

Opposing Actuator Pair (Differential Mode)

Standard AMB uses two opposing magnets:

• Bias current I₀ provides steady levitation
• Control current i_c superimposed on bias for position control

Force linearization around equilibrium gap g₀: F_net = K_i × i_c - K_x × x [linearized; K_i = current stiffness; K_x = position stiffness; x = displacement from center]

Current stiffness: K_i = ∂F/∂i |_(g₀,I₀) = μ₀ × N² × I₀ × A_gap / g₀² [N/A; force per unit control current at nominal gap g₀ and bias current I₀]

Position stiffness (negative — destabilizing): K_x = ∂F/∂x |_(g₀,I₀) = μ₀ × N² × I₀² × A_gap / g₀³ = F₀/g₀ × 2 [N/m; always positive number; force increases as gap decreases → unstable equilibrium] K_x > 0 (destabilizing): open-loop magnetic bearing is inherently unstable; control REQUIRED

Stable equation of motion: mẍ = K_i × i_c - K_x × x [unstable without control; closed-loop stabilizes]

Control law (PID example): i_c = K_p × x + K_d × ẋ + K_int × ∫x dt [position feedback]

AMB Design Parameters

Sizing

Load capacity: F_max = K_i × (I₀ + I_max_control) - K_x × x_max [maximum net force; I_max_control = peak control current] Design rule: F_max ≥ 3–5 × F_static (static load) to allow dynamic capability

Bias current selection: I₀ = (F_static × 2 × g₀²) / (μ₀ × N² × A_gap)^0.5 [sets quiescent operating point at mid-current range] Operating current range: I₀ ± I_control where I_control_max ≤ I₀ (stays positive → no coil deactivation)

Nominal air gap (g₀): Too small: tight tolerances; risk of touchdown on vibration Too large: lower force density; more ampere-turns needed Typical: g₀ = 0.3–1.0 mm (industrial); 0.1–0.5 mm (high-precision)

Power consumption: P_coil = I₀² × R_coil × N_magnets [W; R_coil = coil resistance; N_magnets = 2 per axis] Copper losses + iron losses (eddy current + hysteresis) Iron losses: P_iron = k_h × f × B^1.6 + k_e × f² × B² × t_lamination² [Steinmetz]

Radial AMB Geometry

Heteropolar vs. homopolar: Heteropolar: N-S poles alternate around circumference; most common; each coil energized for one direction Homopolar: bias flux from external DC winding; control flux from radial coils → lower eddy current losses → better high-frequency performance

Number of poles: 8 poles (4 pairs): most common for radial AMB; compact; good force uniformity 12 poles: smoother force; more complex coil winding

Lamination thickness: For f_control ≤ 1 kHz: 0.35 mm silicon steel laminations (same as motor) For f_control > 2 kHz: 0.1–0.2 mm laminations (reduce eddy current losses; better dynamic response)

Thrust (Axial) AMB

Disk-type axial AMB: Two magnets on either side of thrust disk (collar on shaft) Same force equations; gap = axial gap; control = axial direction Design: smaller force required (axial load typically < radial); bias current lower

Control System Design

Sensor

Position sensor: Eddy current proximity probe: inductance changes with gap → voltage output; resolution 0.1–5 μm Range: g₀ ± 50–75% (before touching auxiliary bearing) Output: 4–20 mA or ±10 V analog; digital (IQ demodulation) for modern systems Sensor bandwidth: > 5× maximum control bandwidth required (typically > 5 kHz)

Differential (differential sensor pair): two probes on opposite sides → average = position; subtract = common-mode noise rejection

PID Controller

Transfer function: C(s) = K_p + K_d × s + K_i/s [continuous; often implemented as lead-lag in digital]

Stability condition: |G_open_loop(jω)| = 1 and ∠G_open_loop(jω) > -180° at crossover Gain margin > 6 dB; Phase margin > 30° (AMB typically needs 40–60°)

PD controller (minimum for stability): u = -K_p × x - K_d × ẋ [eliminates instability if K_d is sufficient] K_p_min: K_p × K_i > K_x (position stiffness overcome) K_d_min: provides adequate damping (ζ > 0.3 for closed-loop)

Cross-coupling: Tilting modes create coupling between x-y axes and angular modes → need off-diagonal terms in gain matrix → MIMO control for AMB

LQR for AMB (see LQR skill)

State feedback: x_state = [x, ẋ, y, ẏ, θ_x, θ̇_x, θ_y, θ̇_y]ᵀ (for rigid rotor; 8 states) LQR gain K → u = -K × x_state → optimal stability and performance

Rotor Dynamics in AMB

Critical Speeds

Rigid rotor critical speeds: ω_cr = √(K_p × K_i / m) [radians/s; m = rotor mass; K_p = proportional gain; K_i = current stiffness] AMB can tune effective bearing stiffness by changing K_p → shift critical speed → move away from operating range

Flexible rotor (bending modes): Higher-order bending modes must not coincide with operating speed AMB adds damping → allows operation through critical speeds (unlike fluid bearings which avoid) Notch filters at bending mode frequencies → prevent actuator exciting flexible modes (not just passing through)

Cross-over speed: gyroscopic effects couple forward and backward whirl modes → precession frequencies change with speed AMB must provide adequate damping for all modes at all speeds in operating range

Stability — ISO 14839 Assessment

ISO 14839-3: stability assessment for AMB-supported rotors Log decrement δ ≥ 0.10 (target); measured from rotordynamic analysis Minimum sensitivity peak M_s: maximum value of |(1 + G_ol(jω))⁻¹| ≤ 2 (for ±6 dB gain margin)

Auxiliary (Backup/Touchdown) Bearings

Purpose: support rotor during power loss, AMB overload, or malfunction; emergency only Clearance: r_aux ≈ 0.5 × g₀ (rotor drops half the AMB gap before hitting auxiliary) Type: angular contact ball bearings or roller bearings (dry; no lubrication required — intermittent only) Material: steel (outer) on rotor; Haynes 25 or Al-bronze (inner race if rotor is steel; hardness mismatch for minimal wear)

Design requirement: Survive: N_touchdown touchdowns at rated speed × 2 (shock) Typical: 10–20 touchdowns at full speed before inspection ISO 14839-4: specification for auxiliary bearings; defines clearance, load capacity, contact stress

Touchdown dynamics: Rotor impact → shock load on auxiliary bearing → rotor bounces → re-levitate if AMB functional AMB must re-levitate within 0.5–2 s after touchdown; controller must detect touchdown (large gap) and manage re-levitation

Power Amplifier

Amplifier requirements: Bandwidth: > 5–10× control bandwidth; for 1 kHz control → amplifier bandwidth > 5 kHz Current slew rate: sufficient to provide force at maximum rate of change (dynamic load) Peak current: I₀ + I_control_max; continuous current: I₀

Switching amplifier (PWM): DC bus voltage V_bus = 100–400 V DC Switching frequency: 10–50 kHz (above audio; below core loss significant range) Current ripple: ΔI = V_bus × Δt / L_coil ≤ 5% of I₀ [must be controlled]

Analog linear amplifier: Lower bandwidth limit; lower EMI; higher power dissipation; used in precision applications

Applications

Compressors (API 617 and 619): AMB eliminates oil system complexity; enables hermetic compressors (no shaft seal needed) High-speed (up to 100,000 rpm for small turbocompressors); oil-free (pharmaceutical, food, semiconductor) Condition monitoring: AMB position sensors provide real-time rotor trajectory → health monitoring built in

High-temperature gas turbines: AMB can operate at 200–400°C (bearing electromagnets outside hot zone; rotor targets inside) Eliminates high-temperature lubrication challenges

Flywheel energy storage: Frictionless magnetic levitation + vacuum enclosure → very low standby losses Vertical axis: single thrust AMB + radial AMB at bottom; passive magnetic at top (permanent magnet)

Machine tool spindles: Ultra-precision: stiffness and damping programmable → active compensation of thermal growth Speed: 30,000–100,000 rpm (ceramic-tipped tools)

Standards

Standard	Scope
ISO 14839-1	Vocabulary for AMB systems
ISO 14839-2	Evaluation of vibration performance
ISO 14839-3	Stability evaluation (rotordynamic)
ISO 14839-4	Technical requirements for auxiliary bearings
API 617	Axial/centrifugal compressors (includes AMB section)
API 619	Rotary-type compressors (AMB option)

Output

Provide: rotor mass m [kg] and rated speed [rpm], required load capacity F [N] (radial and axial), AMB geometry (heteropolar/homopolar, N_poles, A_gap [cm²], g₀ [mm]), bias current I₀ [A] and maximum control current I_control [A], current stiffness K_i [N/A] and position stiffness K_x [N/mm], power amplifier specifications (V_bus [V], bandwidth [kHz], peak current [A]), controller type (PID/LQR; gains K_p, K_d, K_int), closed-loop stability margins (gain margin [dB] > 6, phase margin [°] > 30), critical speed separation from operating speed [%], auxiliary bearing clearance r_aux [mm] and type, touchdown load capacity [kN], ISO 14839-3 log decrement δ (≥ 0.10), and applicable standard (ISO 14839-1/-2/-3/-4, API 617).