Eddy-Current Brake Calculator

Braking Force · Torque · Magnetic Flux Density · Braking Power · Deceleration

Eddy-Current Brake Calculator

Geometry and material factor; typical 1–50 (copper disc)
Permanent magnet: 0.3–1.2 T | Electromagnet: up to 2 T
Relative velocity between conductor and magnet

Formulas & Symbols

Core Formulas
Braking force (linear):
F = k × B² × v
F [N], k [N·s/(T²·m)], B [T], v [m/s]
Braking torque (rotating):
M = F × r = k × B² × v × r
M [N·m], r = effective radius [m]
Required flux density:
B = √(F / (k × v))
Required field for a desired braking force
Braking power (heat dissipated):
P = F × v = k × B² × v²
P [W] – entirely converted to heat
Deceleration:
a = F / m
a [m/s²], m = mass [kg]
Symbol Reference
FBraking force [N]
MBraking torque [N·m]
kBrake constant [N·s/(T²·m)]
BMagnetic flux density [T]
vRelative velocity [m/s]
rEffective radius [m]
PBraking power / heat [W]
aDeceleration [m/s²]
mMass [kg]
σElectrical conductivity [S/m]


Eddy-Current Brake – Fundamentals

What is an Eddy-Current Brake?

An eddy-current brake (also called a magnetic brake or retarder) is a contactless deceleration device based on the principle of electromagnetic induction. A moving electrically conductive object (e.g., a rotating metal disc) is placed in a strong magnetic field, inducing eddy currents in the conductor. By Lenz's law, these currents generate an opposing magnetic field that creates a braking force on the moving object.

The key advantage: there is no mechanical contact and therefore zero wear. All kinetic energy is converted entirely into heat in the conductor, which must be dissipated by cooling.

Advantages
  • Zero mechanical wear
  • Contactless operation
  • Continuously variable (via field strength)
  • Very fast control response
  • Low maintenance, long service life
  • No brake dust – clean operation
Disadvantages
  • No holding force at v = 0
  • Heat must be dissipated (cooling required)
  • Higher design complexity
  • Braking force strongly speed-dependent
  • Weaker effect at low velocities

Physical Principle – Lenz's Law

When a conductor moves through a magnetic field, the changing magnetic flux through it induces an EMF according to Faraday's law. This drives eddy currents, which by Lenz's law create an opposing field and a braking (drag) force.

Induced EMF (Faraday):
U_ind = −dΦ/dt = −B × l × v
Φ = magnetic flux [Wb], l = conductor length [m], v = velocity [m/s]
Braking force (Lorentz):
F = I_w × B × l = k × B² × v
k encapsulates geometry, conductivity, and effective area

Detailed Formula Derivation

1. Braking Force F
F = k × B² × v
Example: k = 8, B = 0.8 T, v = 10 m/s → F = 8 × 0.64 × 10 = 51.2 N
2. Braking Torque M
M = F × r = k × B² × v × r
Example: F = 51.2 N, r = 0.15 m → M = 7.68 N·m
3. Required Magnetic Flux Density
B = √(F / (k × v))
Example: F = 50 N, k = 8, v = 10 m/s → B = √(50/80) ≈ 0.79 T
4. Braking Power P
P = F × v = k × B² × v²
Example: k = 8, B = 0.8 T, v = 10 m/s → P = 8 × 0.64 × 100 = 512 W
5. Deceleration a
a = F / m
Example: F = 51.2 N, m = 500 kg → a = 0.102 m/s²

Practical Example: Roller Coaster Brake

Task:

A roller coaster (mass 1800 kg) must be slowed from 72 km/h (20 m/s) to 36 km/h (10 m/s). Available: aluminium fin brake with k = 12, electromagnets B = 1.0 T.

Solution at v = 20 m/s:
  • F = 12 × 1.0² × 20 = 240 N
  • a = 240 / 1800 = 0.133 m/s²
  • P = 240 × 20 = 4 800 W (heat in fins)
At v = 10 m/s (end of braking):
  • F = 12 × 1.0² × 10 = 120 N
  • a = 120 / 1800 = 0.067 m/s²
  • → Force halves at half speed! For constant deceleration: regulate B.

Frequently Asked Questions

Because F = k × B² × v is directly proportional to relative velocity v. At v = 0 there is no change in magnetic flux, no induction, no eddy currents — and therefore no braking force. This is why vehicles always combine an eddy-current retarder with a friction brake for stopping.

Materials with high electrical conductivity are ideal: copper (best effect, expensive), aluminium (excellent price-to-performance ratio, lightweight), and steel (cheap but 5–10× lower conductivity). High-performance applications often use copper- or aluminium-coated steel discs.

In electromagnet systems the excitation current is varied (I → B → F). Since F ~ B², a small current change produces a large force change. Permanent-magnet systems mechanically slide or tilt the magnets in or out of the air gap. Modern systems use electronic closed-loop control for millisecond-precise response.

High-speed trains (ICE, TGV, Shinkansen) use linear eddy-current brakes in the rail. Trucks and buses use rotary retarders as wear-free endurance brakes on downhill grades. Roller coasters and drop-tower rides use them as safe, low-maintenance speed control. Engine test benches apply them for defined load without wear.

The rotor or braking disc must transfer heat to the environment. Methods: air cooling (airflow, cooling fins), water cooling (for continuous high power), or thermal mass (for short-duration braking). At elevated temperature, conductivity σ drops, reducing braking effectiveness (fade effect).

Summary

Braking Force

F = k × B² × v
No wear, no contact

Braking Power

P = k × B² × v²
Fully converted to heat

Control

F ~ B² (quadratic)
Fast, continuously variable

Typical Applications
  • High-speed trains – Linear eddy-current rail brakes (ICE, TGV)
  • Truck retarders – Wear-free endurance braking on downhill grades
  • Roller coasters – Safe, low-maintenance speed control
  • Engine test benches – Defined load application without wear
  • Magnetic safety brakes – Elevators and hoisting equipment

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