Back Emf⚠ unverified
Electrical / Motors · Compute the back electromotive force of a motor
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| K | K | V.s/rad | 1.0 | Back-EMF constant |
| omega | ω | rad/s | 1.0 | Angular speed of the shaft |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | E | V | Back EMF, in volts (V) |
The science & history
Understanding the Parameters
-
Back‑EMF constant $K_e$ — a machine constant set by the magnetic flux and winding geometry; larger flux or more turns gives more volts per rad/s. In SI it is numerically equal to the torque constant $K_t$ (see Torque Constant).
-
Angular speed $\omega$ — in rad/s. Back‑EMF is zero at standstill and rises linearly with speed.
- $E$ — the internally generated voltage that subtracts from the supply.
Derivation (Approaching a Proof)
By Faraday's law of induction, a conductor moving through a magnetic field develops an EMF equal to the rate of change of flux linkage. In a motor the windings sweep through the field flux $\Phi$ as the rotor turns, so the induced voltage is proportional to how fast the flux linkage changes — i.e. to the angular speed:
$$E = \frac{d\lambda}{dt} \propto \Phi\,\omega \equiv K_e\,\omega,$$
with the constant of proportionality $K_e$ lumping the flux and winding turns. By Lenz's law this EMF opposes the applied voltage (hence "back"). Combined with the armature circuit it sets the running current, $I = (V - E)/R$ (see Motor Current): as the motor speeds up, $E$ rises, current falls, and the motor settles at the speed where torque matches load.
History
Back‑EMF is a direct manifestation of Faraday's 1831 law of induction and Lenz's law (1834). Its recognition explained why a motor draws a large inrush current at start (no back‑EMF yet) and much less when running — foundational to motor control and protection ever since.
Related Concepts: Motor Current, Speed Constant, Torque Constant, Watt's Law
Notes: Registry calculator back-emf (unverified). Ideal linear machine; saturation and armature
reaction make $K_e$ mildly non‑constant in real motors.