Speed Constant⚠ unverified
Electrical / Motors · Compute the motor speed constant from the back-EMF constant
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Ke | Ke | V.s/rad | 1.0 | Back-EMF constant |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Kn | — | Speed constant, in radians per volt-second (rad/(V.s)). Returns 0.0 when ``Ke`` is not positive |
The science & history
Understanding the Parameters
- Back‑EMF constant $K_e$ — the volts generated per rad/s (see Back Emf).
- $K_n$ — its inverse: the speed produced per applied volt. A "high‑$K_v$" motor spins fast at low voltage but produces little torque per amp (since $K_t = K_e = 1/K_n$).
Derivation (Approaching a Proof)
At no load and neglecting resistive drop, the applied voltage is balanced entirely by the back‑EMF, $V \approx E = K_e\,\omega$. Solving for speed,
$$\omega \approx \frac{V}{K_e} = K_n V, \qquad K_n \equiv \frac{1}{K_e}.$$
So the speed constant is defined as the reciprocal of the back‑EMF constant. Because $K_e = K_t$ in SI (see Torque Constant), a large $K_n$ (fast, low‑torque motor) and a small $K_n$ (slow, high‑torque motor) are two ends of the same trade‑off — you cannot independently raise speed‑per‑volt and torque‑per‑amp.
History
The speed/velocity constant became a standard catalogue figure with small permanent‑magnet DC and, later, brushless motors, where "$K_v$ in rpm/V" is the headline spec. It is simply the inverse framing of Faraday's back‑EMF relation, chosen because designers usually think in "how fast per volt."
Related Concepts: Back Emf, Torque Constant, Motor Current
Notes: Registry calculator speed-constant (unverified; the unit label is muddled — $K_n$ is the
reciprocal of $K_e$, units rad·s⁻¹·V⁻¹). Idealised no‑load relation.