Complete Knowledge database of Electricity and Electrical Technology : January 2014

In Alternating current (AC), the flow of electric charge periodically reverses direction.The usual waveform of an AC power circuit is a sine wave. In certain applications, different waveforms are used, such as triangular or square waves. Audio and radio signals carried on electrical wires are also examples of alternating current.

Alternating Current (green curve)

AC voltage may be increased or decreased with a transformer. Use of a higher voltage leads to significantly more efficient transmission of power. The power losses in a conductor are a product of the square of the current and the resistance of the conductor, described by the formula

$P_{\rm L} = I^2 R \, .$

This means that when transmitting a fixed power on a given wire, if the current is doubled, the power loss will be four times greater. The power transmitted is equal to the product of the current and the voltage (assuming no phase difference); that is,

$P_{\rm T} = IV \, .$

Thus, the same amount of power can be transmitted with a lower current by increasing the voltage. It is therefore advantageous when transmitting large amounts of power to distribute the power with high voltages (often hundreds of kilovolts).

AC power supply frequencies:

The frequency of the AC system varies by country; most electric power is generated at either 50 or 60 hertz.
A low frequency eases the design of electric motors, particularly for hoisting, crushing and rolling applications, and commutator-type traction motors for applications such as railways. However, low frequency also causes noticeable flicker in arc lamps and incandescent light bulbs. The use of lower frequencies also provided the advantage of lower impedance losses, which are proportional to frequency.

At very high frequencies the current no longer flows in the wire, but effectively flows on the surface of the wire, within a thickness of a few skin depths because of skin effect. The skin depth is the thickness at which the current density is reduced by 63%. Even at relatively low frequencies used for power transmission (50–60 Hz), non-uniform distribution of current still occurs in sufficiently thick conductors. For example, the skin depth of a copper conductor is approximately 8.57 mm at 60 Hz, so high current conductors are usually hollow to reduce their mass and cost.

Mathematics of AC voltages:

Alternating currents are accompanied (or caused) by alternating voltages. An AC voltage v can be described mathematically as a function of time by the following equation:

$v(t)=V_\mathrm{peak}\cdot\sin(\omega t)$ ,

where

$\displaystyle V_{\rm peak}$ is the peak voltage (unit: volt),
$\displaystyle\omega$ is the angular frequency (unit: radians per second)
- The angular frequency is related to the physical frequency, $\displaystyle f$ (unit = hertz), which represents the number of cycles per second, by the equation $\displaystyle\omega = 2\pi f$ .
$\displaystyle t$ is the time (unit: second).

The peak-to-peak value of an AC voltage is defined as the difference between its positive peak and its negative peak. Since the maximum value of $\sin(x)$ is +1 and the minimum value is −1, an AC voltage swings between $+V_{\rm peak}$ and $-V_{\rm peak}$ . The peak-to-peak voltage, usually written as $V_{\rm pp}$ or $V_{\rm P-P}$ , is therefore $V_{\rm peak} - (-V_{\rm peak}) = 2 V_{\rm peak}$ .

Power and root mean square

Main article: AC power

The relationship between voltage and the power delivered is

$p(t) = \frac{v^2(t)}{R}$ where $R$ represents a load resistance.

Rather than using instantaneous power, $p(t)$ , it is more practical to use a time averaged power (where the averaging is performed over any integer number of cycles). Therefore, AC voltage is often expressed as a root mean square (RMS) value, written as $V_{\rm rms}$ , because

$P_{\rm time~averaged} = \frac{{V^2}_{\rm rms}}{R}.$

For a sinusoidal voltage:

$\begin{align} V_\mathrm{rms} &=\sqrt{\frac{1}{T} \int_0^{T}[{V_{pk}\sin( \omega t+\phi)]^2 dt}}\\ &=V_{pk}\sqrt{\frac{1}{2T} \int_0^{T}[{1-\cos(2\omega t+2\phi)] dt}}\\ &=V_{pk}\sqrt{\frac{1}{2T} \int_0^{T}{ dt}}\\ &=\frac{V_{pk}}{\sqrt {2}} \end{align}$

The factor $\sqrt{2}$ is called the crest factor, which varies for different waveforms.

For a triangle waveform centered about zero

$V_\mathrm{rms}=\frac{V_\mathrm{peak}}{\sqrt{3}}.$

For a square waveform centered about zero

$\displaystyle V_\mathrm{rms}=V_\mathrm{peak}.$

For an arbitrary periodic waveform $v(t)$ of period $T$ :

$V_\mathrm{rms}=\sqrt{\frac{1}{T} \int_0^{T}{v^2(t) dt}}.$

Example:

To illustrate these concepts, consider a 230 V AC mains supply used in many countries around the world. It is so called because its root mean square value is 230 V. This means that the time-averaged power delivered is equivalent to the power delivered by a DC voltage of 230 V. To determine the peak voltage (amplitude), we can rearrange the above equation to:

$V_\mathrm{peak}=\sqrt{2}\ V_\mathrm{rms}.$

For 230 V AC, the peak voltage $\scriptstyle V_\mathrm{peak}$ is therefore $\scriptstyle 230 V \times\sqrt{2}$ , which is about 325 V. The peak-to-peak value $\scriptstyle V_\mathrm{P-P}$ of the 230 V AC is double that, at about 650 V.

Alternating current

AC power supply frequencies:

Mathematics of AC voltages:

Power and root mean square

Example:

Direct current

Applications: