Tutorial 2 - Extension

Why are alternating waveforms mostly sinusoidal?

The Generator Effect

The voltage of a simple generator is given by the formula:

  • N - the number of turns

  • v - the linear speed ( ms-1)

  • w - width (m)

  • B - the magnetic field strength (T)

  • E - emf (V)

 

Derivation

The diagram above is a simple alternating current generator.  It consists of a coil of N turns, radius r and length l spinning in a magnetic field of flux density B.  Its angular velocity is w radians per second.  The motion is, of course, circular.

 

We can use the fact that circular motion and simple harmonic motion are closely linked.  Simple harmonic motion describes the movement of an oscillating object, which means an object that moves to-and-fro in a period movement.  Examples include a swinging pendulum or an object bouncing up and down on a spring.   So we can use the equation for displacement:

 

x = A cos wt

 

Here the x term refers to the displacement from a fixed point.  We need to be aware of the term w.  It is the rate of rotation or the angular velocity.  In other words it is the angle turned per second.  It is measured in radians per second (rad/s).  “Radians” is a dimensionless unit, and some texts miss it out altogether.  In these notes I will always use the short-hand “rad”.

The symbol used for the Physics code, w, is omega, a Greek lower case long ‘o’ (ō).

We can make the point at which the coil is vertical the “rest position”. The maximum amplitude is r as in the diagram.

So our equation becomes x = r cos wt

Since the coil is rotating at a constant angular velocity, w, the linear speed of the edge of the coil is given as v = wr.  From SHM we can say:

v = -rw sin wt

EMF (zero current terminal voltage) and linear speed are linked by this equation (for derivation of this click here):

  • N - the number of turns

  • v - the linear speed ( ms-1)

  • w - width (m)

  • B - the magnetic field strength (T)

  • E - emf (V)

We can combine the two equations above to give:

The minus signs disappear and we can also say that A = rl.

Therefore:

This explains why the output of an AC generator is sinusoidal. The output of an AC generator is a sine wave.

 

The maximum value of the e.m.f. is when sin wt = 1.  Therefore:

E0 = BANw

The AC generator here is inefficient, but could be made more efficient by changing the shape of the magnet, and wrapping the coil around soft iron.  Practical AC generators have a rotating magnet (rotor) which passes between stationary coils (stator).  The alternating e.m.f is induced in these coils.  The machine is called an alternator.

Power station generators are massive.  They have a rotor that is connected to its own generator, called an exciter.  The stator coils are placed at 120 degrees to each other to allow 3-phase AC to be generated.  The voltage is 25 000 V, while currents of 15 000 A are common.  The whole machine is cooled by hydrogen gas, which has a particularly high specific heat capacity.  The picture above shows a power station alternator.

The generator is actually in the rectangular box on the right.  To the left is the low-pressure turbine.

Turbines and generators are so big that when the machine is off, the shaft has to be rotated slowly, otherwise it would sag and go out of shape (which is not a good idea).  It is driven by a barring motor.

Back

EMF and speed

Some revision from A-Level. 

 

Consider a wire on two rails, w metres apart, travelling a distance l metres at a velocity of v metres per second in a time of t seconds.

Faraday’s and Lenz’s Laws:

Since F = BA, we can write:

Also:

A = lw and l = vt

So we can write:

And then:

The t terms cancel out to give us:

The minus sign is there to satisfy Lenz’s Law.

 

(Questions on this involve the rather fatuous example of aeroplanes flying through the Earth’s magnetic field.  An EMF is induced due to the vertical component of the Earth’s magnetic field.  It’s no damned good to the aeroplane, which would have to fly along fixed rails to generate anything useful – a sky-train?)