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As a motor spins, its windings move through the magnetic field and generate a voltage that opposes the applied voltage — the back-EMF (counter-EMF). It is proportional to speed: E = Ke·ω, where Ke is the motor's speed (voltage) constant and ω is the angular speed. In a DC motor, the back-EMF is what is left of the terminal voltage after the armature resistance drop: E = V − Ia·Ra. Back-EMF is why a motor draws huge current at standstill (no back-EMF) and much less at speed.
| Quantity | Formula |
|---|---|
| Back-EMF (constant) | E = Ke × ω (ω = 2πN/60) |
| Back-EMF (terminal) | E = V − Ia × Ra |
| Speed constant | Ke = E / ω |
| Mechanical power | Pmech = E × Ia |
The product E·Ia is the mechanical power converted (before friction). The speed constant Ke (in V per rad/s, or V per 1000 rpm for BLDC motors) equals the torque constant Kt in SI units, linking voltage, speed and torque. Higher Ke means more volts per unit speed — and a lower no-load speed for a given supply.
Back-EMF (or counter-EMF) is the voltage a spinning motor generates that opposes the applied voltage. It is produced because the rotating windings move through the magnetic field, and it grows in proportion to speed.
E = Ke × ω, where Ke is the speed constant and ω is the angular speed in rad/s (ω = 2πN/60). For a DC motor you can also use E = V − Ia·Ra.
Ke is the back-EMF generated per unit speed, expressed in volts per rad/s or volts per 1000 rpm. It quantifies how much voltage the motor produces as it turns and, in SI units, equals the torque constant Kt.
The armature current is (V − E)/Ra. As the motor speeds up, E rises and opposes the supply, reducing the net voltage across the armature and hence the current. At full speed only a small current flows.
At standstill the speed is zero, so the back-EMF is zero and the only thing limiting current is the small armature resistance. This gives a very large inrush, which is why DC motors use starting resistors or controlled starters.
Kv (rpm per volt) is the inverse of the back-EMF constant. Ke in volts per 1000 rpm is about 1000/Kv. A high-Kv motor spins fast per volt and has a low back-EMF constant.
Subtract the armature resistance drop from the terminal voltage: E = V − Ia·Ra. This is the internally generated voltage that does the mechanical work.
The electrical power converted to mechanical power is P = E × Ia. This is the gross mechanical power before friction and windage losses are subtracted.
Back-EMF depends on speed, not directly on load. But adding load slows the motor slightly, which lowers E and lets more current flow to meet the extra torque, so there is an indirect link.
In consistent SI units, the back-EMF constant (V per rad/s) numerically equals the torque constant (N·m per amp). This elegant identity comes from energy conservation in the motor.
Brushless motor drives measure the back-EMF on the non-energised winding to infer the rotor position and speed, allowing commutation without a physical position sensor — except at very low speed where the EMF is too small.
Yes, if the motor is driven above its no-load speed (for example during regenerative braking or an overhauling load). The back-EMF then pushes current back toward the supply, feeding energy back.
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