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Formulas for power calculations on ebikes and hub motors

Published: 06 October 2010
Last updated: 06 March 2023
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Conversion of non-SI units

  • Foot = 0.3048 m
  • Mile = 1.61 km
  • Pound, lbs, lb = 0.453 kg
  • Ounce, oz = 28.35 g
  • Lbs / sf = 4.88 kg/m2
  • HP (horsepower) = 735.5 W
  • 1 tesla [T] = 104 gauss [G]

Definitions

  • m = mass [kg]
  • n = revolutions per minute [rpm]
  • v = velocity [m/s]
  • ω = angular velocity [rad/s]
  • e = back electromotive force [V]
  • F = force [N]
  • T = torque [Nm]
  • P = power [W]
  • E = energy [Ws]
  • Tf = friction torque [Nm]
  • r = radius [m]
  • Cr  = Rolling resistance [m]
  • ρ = Air density [kg/m3]
  • Cw = Drag coefficient [ ]
  • A = Reference area [m2]

Physics

  • 1 kg = 9.81 N
  • 1 calorie = 4.1868 joule
  • v [m/s] = [km/h] / 3.6
  • ω = 2 * π * n / 60 = 0.1047 * n
  • ω = v / r = kmh / (1.8 * diameter)
  • F = m * a
  • P = E / t [W] or [Nm/s]
  • E = 0.5 * m * v2

Required bicycle power

  • Proll = Cr * m * g * v
  • Pdrag = 0,5 * ρ * Cw * A * v3
  • Pslope = m * g * v * slope[%] / 100

Bicycle speed calculation

  • kmh = n * wheel circumference * 0.06

Motor torque and power

  • P = T * ω [Nm/s]

The two linear relationships of an PM motor

  • Kv = rpm/V
  • k = 30/(π * Kv) [V/rad/s]
  • k = e / ω [V/rad/s]
  • k = T / i [Nm/A]

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