## Summary

Most of the world's electrical power is generated using electrical generators.

Generators convert rotational kinetic energy (usually powered by a turbine) into electrical energy using electromagnetic induction.

## Get the basics

#### Operation

A basic AC generator operates by the rotation of a coil within a magnetic field. This diagram shows the coil being rotated by a handle, which could represent a turbine.

The left side of this coil is moving upwards. The magnetic field direction is from north to south. Using Fleming's right hand rule, we can determine that a current would be induced in the direction away from the handle. The converse is true for the right side of the coil.

When the coil has rotated 90^{o}, the motion of the sides will be parallel to the field lines. No current is induced.

After a 180^{o} rotation, the current will again be maximised, but the direction around the coil will be opposite to its initial direction. The slip rings enable the coil to spin continuously and for this alternating current to be transmitted.

### Types of current

The current produced by the generator described above is alternating. Alternating current is defined by the repeated cyclical change of direction (not just by a change in magnitude).

Direct current, which travels in a constant direction, can be produced by a dynamo.

### Calculations

Because alternating current has equal magnitudes and equal times in opposite directions, the true mean average potential difference and current are zero. However, clearly the electrical power generated can be transmitted and used.

We therefore use another type of average, root mean square:

\(I_\text{rms}={I_0\over \sqrt2}\)

- \(I_\text{rms}\) is the root mean square current (A)
- \(I_0\) is the maximum magnitude of the current in either direction (A)

\(V_\text{rms}={V_0\over \sqrt2}\)

- \(V_\text{rms}\) is the root mean square potential difference (V)
- \(V_0\) is the maximum magnitude of the potential difference (V)

The definition of Resistance is unchanged; it is the ratio of the potential difference across a conductor to the current flowing through it at an instant:

\(R={V_0\over I_0}={V_\text{rms}\over I_\text{rms}}\)

- \(R\) is the resistance (\(\Omega\))

The electrical power produced is the product of the current and voltage. For maximum power, when the rotation of the coil is perpendicular to the magnetic field:

\(P_\text{max}=I_0V_0\)

- \(P_\text{max}\) is the maximum power (W)

The average power is the product of the root mean square current and potential difference:

\(\bar{P}={I_0\over \sqrt2}{V_0\over \sqrt2}\)

\(\Rightarrow\bar{P}={1\over 2}I_0V_0\)

- \(\bar{P}\) is the average power (W)

All of these equations are found in the data booklet and can be used to solve problems for an AC circuit.

## Stretch for 7

### Equation

#### Derivation

Although not required for the exam, it is helpful to be familiar with the equation for the EMF induced in a rotating coil. Conveniently, we have already defined the flux for a given angle:

\(\Phi=BA\cos\theta\)

Flux linkage, \(N\Phi=BAN\cos\theta\) and, for a given time, \(\theta=\omega t\)

\(\Rightarrow N\Phi=BAN\cos{\omega t}\)

The EMF is the negative of the rate of change of flux linkage, \(\varepsilon=-{\mathrm{d}N\Phi\over \mathrm{d}t}\). Mathematicians will be familiar with differentiating trigonometric functions.

\(\varepsilon=BAN\omega \sin{\omega t}\)

- \(\varepsilon\) is the EMF at a given time (V)
- \(B\) is magnetic flux density (T)
- \(A\) is the perpendicular area of the coil in the field (m
^{2}) - \(N\) is the number of turns on the coil
- \(\omega\) is angular velocity (rad s
^{-1}) - \(t\) is time, with \(t=0\) for the coil sides rotating parallel to the field

The maximum of this equation is when \(\sin{\omega t}=1\):

\(\varepsilon_0=BAN\omega\)

#### Effect of changing frequency

From the equation it is easily to identify that increasing magnetic flux density, the area of the coil and the number of turns will increase the induced EMF. But what about frequency?

Angular velocity, \(\omega\), depends on frequency, so increasing frequency has two effects:

- The size of the EMF increases
- The time between peaks decreases

### Root mean square

A root mean square quantity has the same dimension as the quantity itself. This is because the quantity is sqared at all instances (and thus becomes entirely positive) and is then rooted.

You are not expected to know the proof of the relationship, but may find this video interesting.