DESCRIBING ALTERNATING CURRENT

In many countries, mains electricity is a supply of alternating current (a.c.). The first mains electricity supplies were developed towards the end of the 19th century; at that time, a great number of different voltages and frequencies were used in different places. In some places, the supply was direct current (d.c.). Nowadays, this has been standardised across much of the world, with standard voltages of $110 V$ or $230 V$ (or similar), and frequencies of $50 Hz$ or $60 Hz$.
Mains electricity is transported along many kilometres of high-voltage power lines (cables).
Transformers are used for stepping-up and stepping-down alternating voltages between the power stations and the consumers (Figure 27.1). From your prior knowledge of transformers and transmission of electrical energy, can you remember why it is necessary for power lines to use high voltage?

An alternating current can be represented by a graph such as that shown in Figure 27.2. This shows that the current varies regularly. During half of the cycle, the current is positive, and in the other half it is negative. This means that the direction of the current reverses every half cycle. Whenever you use a mains appliance, the charges (free electrons) within the wire and appliance flow backwards and forwards.
At any instant in time, the current has a particular magnitude and direction given by the graph.
The graph has the same shape as the graphs used to represent simple harmonic motion (s.h.m.) (see Chapter 18), and it can be interpreted in the same way. In a wire with a.c., the free electrons within the wire move back and forth with s.h.m. The variation of the current with time is a sine curve, so it is described as sinusoidal. (In principle, any current whose direction changes between positive and negative can be described as alternating, but we will only be concerned with those that have a regular, sinusoidal pattern.)

An equation for a.c.

As well as drawing a graph, we can write an equation to represent alternating current. This equation gives us the value of the current I at any time t:

$I = {I_0}\,sin\omega t$

where I is the current at time t, ${I_0}$ is the peak value of the alternating current and ω is the angular frequency of the supply, measured in $rad\,{s^{ - 1}}$ (radians per second). The peak value is the maximum magnitude of the current. It’s very much like the ‘amplitude’ of the alternating current, except the unit is that of current.
This is related to the frequency f in the same way as for s.h.m.:

$\omega  = 2\pi f$

and the frequency and period are related by:

$f = \frac{1}{T}$