DC Motor Back-EMF (Armature Equation)

A DC motor is also a generator, always, and it cannot help it. The moment the armature turns, its conductors are cutting field flux, and cutting flux generates a voltage — one that by Lenz's law opposes the supply that is driving the rotation. That opposing voltage is the back-EMF, $E_b = k\phi\omega$, and the armature equation $E_b = V - I_aR_a$ is nothing more than Kirchhoff's voltage law around the armature loop.

Everything a DC motor does that seems clever comes out of this one line. Put a load on the shaft and the motor slows; $E_b$ is proportional to speed, so the back-EMF falls; the difference $V - E_b$ grows; and more current flows, producing more torque, until the motor finds the speed at which torque matches load. Nothing commands that current. No controller measures the load. The machine self-regulates because the back-EMF is the armature reporting its own speed to the supply, and a 240 V motor drawing 25 A through a 0.6 Ω armature is running with 225 V of back-EMF and only 15 V driving current through the copper.

Multiply that split by the armature current and you have the whole electrical accounting of the machine. $E_bI_a$ is the power converted to mechanical work — 225 × 25 = 5.6 kW in the example. $I_a^2R_a$ is the copper loss, 375 W, and it leaves as heat. The ratio of the two is why armature resistance is made as small as it can be made, and why it is a fraction of an ohm on almost every machine.

The consequence people meet first is starting. At standstill the armature is not turning, so $E_b = 0$, and nothing but $R_a$ limits the current: that 240 V motor would draw 240/0.6 = 400 A across the line, sixteen times its running current. That is the entire reason for a starting resistance, a soft starter or a drive, and it is why a DC motor that stalls under power cooks its armature in seconds rather than minutes. Run the argument backwards and you get regenerative braking: drive the shaft faster than the supply alone would, $E_b$ exceeds $V$, the current reverses, and the machine pumps power back toward the source — provided the source can take it, which a rectifier front end cannot, and which is why braking resistors exist.

One practical warning about $R_a$. It is small, so a handheld meter will not measure it usefully, and the brush contact drop — a volt or two, and not really a resistance at all — is conventionally lumped into it. Take it from a locked-rotor test at reduced voltage with the machine warm, and treat any figure from a cold armature as optimistic.