23.6 Back Emf

Back EMF in Motors and Generators

Back emf effects on motor operation

Back EMF is the voltage that a spinning motor generates against the voltage you're feeding it. It arises from electromagnetic induction: as the motor's coil rotates through a magnetic field, it induces a voltage that opposes the applied voltage. This opposition is a direct consequence of Lenz's law.

Here's how it plays out during startup and normal operation:

• At startup, the motor isn't spinning yet, so back EMF is zero. The full applied voltage drives current through the windings, which means a large current flows. That large current produces strong torque, and the rotor begins to spin.
• As the motor speeds up, the spinning coil generates increasing back EMF. This opposes the applied voltage, so the net voltage driving current drops. Less current means less torque.
• At steady state, back EMF is nearly equal to the applied voltage. Only a small current flows, just enough to overcome friction and whatever load the motor is driving. Back EMF acts as a natural speed regulator: if the motor tries to spin faster, back EMF rises and reduces the current; if the motor slows down under load, back EMF drops and more current flows to compensate.

Power calculations using back emf

Two key formulas govern motor circuits with back EMF:

Current at startup (back EMF = 0):

$I = \frac{V}{R}$

Current during normal operation (back EMF = $\varepsilon$):

$I = \frac{V - \varepsilon}{R}$

where $V$ is the applied voltage, $\varepsilon$ is the back EMF, and $R$ is the resistance of the motor windings.

Once you know the current, the power dissipated as heat in the windings is:

$P = I^2 R$

Worked example: A 12 V motor has winding resistance of 2 Ω.

1. At startup: $I = 12\text{ V} / 2\text{ Ω} = 6\text{ A}$, so $P = (6)^2 \times 2 = 72\text{ W}$
2. At normal operation (say back EMF = 10 V): $I = (12 - 10) / 2 = 1\text{ A}$, so $P = (1)^2 \times 2 = 2\text{ W}$

Notice the huge difference. This is why motors draw so much current when they first turn on and why fuses or circuit breakers sometimes trip at startup. It's also why a motor that stalls (stops spinning while voltage is still applied) can overheat: with no back EMF, the current stays at its maximum startup value.

Back emf in energy conversion

Back EMF connects directly to how energy is converted in both motors and generators.

In motors, electrical energy converts to mechanical energy. The applied voltage pushes current through the windings against the opposition of back EMF. The power delivered against the back EMF ($P_{\text{mech}} = \varepsilon \times I$) is what becomes useful mechanical output. The remainder ($I^2 R$) is lost as heat. So a motor with higher back EMF relative to applied voltage is actually more efficient, because a larger fraction of the input power goes to mechanical work rather than resistive heating.

In generators, the situation is reversed. An external force (a turbine, an engine) spins the coil through a magnetic field, inducing a voltage. That induced voltage is the back EMF, and it drives current through whatever external load is connected. Here, mechanical energy converts to electrical energy.

The magnitude of back EMF depends on two things:

• Rotational speed of the coil
• Magnetic field strength

The relationship is:

$\varepsilon = k\phi\omega$

where $k$ is a constant determined by the coil geometry (number of turns, coil area), $\phi$ is the magnetic flux through the coil, and $\omega$ is the angular velocity. Faster rotation or a stronger field produces a larger back EMF.

Electromagnetic induction in motors and generators

Electromagnetic induction is the underlying principle for both devices. In each case, a coil of wire (the armature) rotates in a magnetic field. That rotation changes the magnetic flux through the coil, which induces a voltage according to Faraday's law.

• In a motor, you supply voltage to force current through the coil. That current in the magnetic field creates torque, spinning the armature. The spinning also induces back EMF, which limits the current.
• In a generator, an external force spins the armature. The changing flux induces a voltage that drives current through an external circuit.

In DC motors and generators, a commutator acts as a mechanical switch that reverses the current direction in the armature twice per revolution. This ensures the torque always pushes in the same rotational direction (in motors) or that the output current flows in one direction (in generators).