Back electromotive force and energy balance

When the loop rotates in the magnetic field, the flux through it changes: by Faraday’s law (next chapter) an induced electromotive force $\mathcal{E}_\text{ind}$ is generated which, by Lenz’s law, opposes the imposed current. This is the motor’s back electromotive force (back e.m.f.), denoted $\mathcal{E}_c$.

In a circuit with generator $\mathcal{E}$, total resistance $R$ and back e.m.f. $\mathcal{E}_c$, the current is:

$i = \frac{\mathcal{E} - \mathcal{E}_c}{R}$

The faster the motor spins, the more $\mathcal{E}_c$ grows and the more the current drawn decreases. The energy balance per second is written by multiplying by $i$: the input electrical power splits between Joule dissipation and useful mechanical power.

$\underbrace{\mathcal{E}\,i}_{\text{pot. elettrica in}} = \underbrace{R\,i^2}_{\text{Joule}} + \underbrace{\mathcal{E}_c\,i}_{\text{pot. meccanica out}}$

Principle — Energy balance of the motor

$P_\text{el} = P_\text{Joule} + P_\text{mecc} \qquad P_\text{mecc} = \mathcal{E}_c\cdot i = M_\text{torc}\cdot\omega$ The efficiency is $\eta = P_\text{mecc}/P_\text{el} = \mathcal{E}_c/\mathcal{E}$.

This result has an important practical consequence: with the motor stopped ($\omega = 0$, hence $\mathcal{E}_c = 0$) all the electrical power is dissipated as heat, and the current drawn is maximum ($i = \mathcal{E}/R$). This is the moment of greatest risk of overheating: a motor that is mechanically jammed but still powered can burn out.

Collegamenti

Argomenti: Magnetismo Concetti: Dipolo magnetico · Forza elettromotrice · Effetto Joule

Esercizi collegati: Worked exercise — electric motor under load · Motor and dynamo · Bicycle dynamo