15.3.1 Generator

While the discovery of $\displaystyle F=BIL$ led to the invention of the electric motor, the discovery of motional emf led to the invention of electric generator.

Below is the schematic of a very basic electric generator: a rectangular coil (length $\displaystyle L=PQ$ and width $\displaystyle W=QR$ ) of N turns, rotating in a uniform magnetic field B at constant angular velocity w.

To determine the direction of the induced emf, it is easier to use the “flux cutting” approach. At this instant, segments PQ and SR are moving vertically upward and downward respectively. Using FRHR, we can deduce that induced current is going to be clockwise (PQRS) in direction.

To obtain the magnitude of the induced emf at each orientation, it is actually easier to switch to the “flux changing” approach.

First, we note that the flux linkage of the coil is zero when the coil is horizontal, and maximum when it is vertical. In fact, the flux linkage $\displaystyle \displaystyle \Phi$ varies sinusoidally with time because the area of the coil facing B directly varies sinusoidally with time. So we can express $\displaystyle \displaystyle \Phi$ as

$\displaystyle \displaystyle \Phi =NBA\sin \theta =NBA\sin \omega t$

By considering the rate of change of flux linkage, we obtain

$\displaystyle \varepsilon =-\frac{{d\Phi }}{{dt}}=-\frac{d}{{dt}}(NBA\sin \omega t)=-NBA\frac{d}{{dt}}\sin \omega t=-\omega NBA\cos \omega t$

A few things to note:

Since the flux linkage varies sinusoidally, the induced emf also varies sinusoidally. The frequency of the induced emf follows the frequency of rotation the coil.
The induced emf and the flux linkage are out-of-phase by a quarter of a cycle. The induced emf is zero when the flux linkage peaks, and peaks when the flux linkage is zero.
The maximum value or amplitude of the induced emf is given by $\displaystyle {{\varepsilon }_{{\max }}}=NBA\omega$ . Do you understand why ω has an effect on ε_max? Rotating the coil at a higher angular frequency ω increases the rate of change of flux linkage, and thus leads to higher $\displaystyle {{\varepsilon }_{{\max }}}$ .