Electric Generators

Electric generators induce an emf by rotating a coil in a magnetic field, as briefly discussed in Induced Emf and Magnetic Flux. We will now explore generators in more detail. Consider the following example.

The emf calculated in [link] is the average over one-fourth of a revolution. What is the emf at any given instant? It varies with the angle between the magnetic field and a perpendicular to the coil. We can get an expression for emf as a function of time by considering the motional emf on a rotating rectangular coil of awidth w size 12{w} {} and height ℓ size 12{l} {} in a uniform magnetic field, as illustrated in [link].

A generator with a single rectangular coil rotated at constant angular velocity in a uniform magnetic field produces an emf that varies sinusoidally in time. Note the generator is similar to a motor, except the shaft is rotated to produce a current rather than the other way around.

Charges in the wires of the loop experience the magnetic force, because they are moving in a magnetic field. Charges in the vertical wires experience forces parallel to the wire, causing currents. But those in the top and bottom segments feel a force perpendicular to the wire, which does not cause a current. We can thus find the induced emf by considering only the side wires. Motional emf is given to be emf=Bℓv size 12{"emf"=Bℓv} {}, where the velocity v is perpendicular to the magnetic field B size 12{B} {}. Here the velocity is at an angle θ size 12{θ} {} with B size 12{B} {}, so that its component perpendicular to B size 12{B} {} is vsin θ size 12{v"sin"θ} {} (see [link]). Thus in this case the emf induced on each side is emf=Bℓvsinθ size 12{"emf"=Bℓv"sin"θ} {}, and they are in the same direction. The total emf around the loop is then

This expression is valid, but it does not give emf as a function of time. To find the time dependence of emf, we assume the coil rotates at a constant angular velocity ω size 12{ω} {}. The angle θ size 12{θ} {} is related to angular velocity by θ=ωt size 12{θ=ωt} {}, so that

Now, linear velocity v is related to angular velocity ω by v=rω size 12{v=rω} {}. Here r=w/2 size 12{r=w/2} {}, so that v=(w/2)ω size 12{v= ( w/2 ) ω} {}, and

Noting that the area of the loop is A=ℓw size 12{A=ℓw} {}, and allowing for N size 12{N} {} loops, we find that

is the emf induced in a generator coil of N size 12{N} {} turns and area A size 12{A} {} rotating at a constant angular velocity ω in a uniform magnetic field B size 12{B} {}. This can also be expressed as

where

is the maximum (peak) emf. Note that the frequency of the oscillation is f=ω/2π size 12{f=ω/2π} {}, and the period is T=1/f=2π/ω size 12{T=1/f=2π/ω} {}. [link] shows a graph of emf as a function of time, and it now seems reasonable that AC voltage is sinusoidal.

The emf of a generator is sent to a light bulb with the system of rings and brushes shown. The graph gives the emf of the generator as a function of time. emf0 size 12{"emf" rSub { size 8{0} } } {} is the peak emf. The period is T=1/f=2π/ω size 12{T=1/f=2π/ω} {}, where f size 12{f} {} is the frequency. Note that the script E stands for emf.

The fact that the peak emf, emf0=NABω size 12{"emf" rSub { size 8{0} } = ital "NAB"ω} {}, makes good sense. The greater the number of coils, the larger their area, and the stronger the field, the greater the output voltage. It is interesting that the faster the generator is spun (greater ω size 12{ω} {}), the greater the emf. This is noticeable on bicycle generators—at least the cheaper varieties. One of the authors as a juvenile found it amusing to ride his bicycle fast enough to burn out his lights, until he had to ride home lightless one dark night.

[link] shows a scheme by which a generator can be made to produce pulsed DC. More elaborate arrangements of multiple coils and split rings can produce smoother DC, although electronic rather than mechanical means are usually used to make ripple-free DC.

Split rings, called commutators, produce a pulsed DC emf output in this configuration.

In real life, electric generators look a lot different than the figures in this section, but the principles are the same. The source of mechanical energy that turns the coil can be falling water (hydropower), steam produced by the burning of fossil fuels, or the kinetic energy of wind. [link] shows a cutaway view of a steam turbine; steam moves over the blades connected to the shaft, which rotates the coil within the generator.

Steam turbine/generator. The steam produced by burning coal impacts the turbine blades, turning the shaft which is connected to the generator. (credit: Nabonaco, Wikimedia Commons)

Generators illustrated in this section look very much like the motors illustrated previously. This is not coincidental. In fact, a motor becomes a generator when its shaft rotates. Certain early automobiles used their starter motor as a generator. In Back Emf, we shall further explore the action of a motor as a generator.