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# Wave nature of a.c.

## Wave nature of a.c.

The variation of the voltage and current of alternating current with respect to time can be expressed as a periodic wave.

Alternating current must change direction periodically.

Most a.c. sources, such as those from mains power sockets are sinusoidal.

However, a.c. may take a broad variety of forms, including square waves. Some equations for a.c. quantities apply to all types of alternating current. Others only apply to sinusoidal a.c. and it is important to distinguish them.

Like other types of waves, a.c. has a frequency $f$, period $T$ and an amplitude $A$.

Alternating current usually comes in the form of sinusoidal waves.

The graph for the voltage of an alternating current shows that very briefly during each period, the voltage and current becomes zero. This means that an electrical appliance connected to an alternating current switches off periodically during its operation.

However, the frequency is so high ($\sim 50\text{ Hz}$) that we would be unable to notice it.

A street lamp flickers so rapidly that the human eye sees the light as being constantly on!

## Peak current and voltage

The amplitude of an alternating current is known as its peak (or maximum) value. The symbols for the peak voltage and current are $V_{0}$ and $I_{0}$ respectively.

Peak current is, for example, an important value for fuses. Fuses prevent excessive current flow through wires and appliances. The ratings of fuses are defined in terms of the peak current that the fuse wire allows to pass through before melting and breaking the circuit.

The maximum power loss in a resistor when a.c. flows through it is called the peak power loss. The peak power loss $P_{0}$ in a resistor of resistance $R$ connected to an a.c. source is: $$P_{0}=I_{0}^{2}R=V_{0}^{2}/R$$

The peak current is the amplitude of the graph for current against time.

## Sinusoidal a.c. equations

The equations for the voltage $V(t)$ and current $I(t)$ of a sinusoidal a.c. as a function of time and peak voltage $V_0$ or peak current $I_0$ are: \begin{align*}V(t) &=V_{0}\sin(\omega t)\\I(t) &=I_{0}\sin(\omega t)\end{align*}

$\omega$ is the angular frequency (measured in radians per second). Note that this quantity cannot be measured and has no physical significance. It is simply part of the argument $\omega t$ of the sinusoidal function and gives the rate of change the argument.

It is related to the frequency $f$ (or the period $T$) by: $$\omega=2\pi f = \frac{2\pi}{T}$$This value, when substituted into the general equation for a sinusoidal curve, gives a particular solution for the voltage function of an a.c.