Editing Alternating current (section)

==Formulation{{anchor|Mathematics|Mathematics of AC voltages}}==<!--This section is linked from [[Vacuum cleaner]]-->
[[File:Sine voltage.svg|right|thumb|A sinusoidal alternating voltage.{{ordered list
| Peak,
| Peak-to-peak amplitude,
| Effective value,
| Period
}}]]
Alternating currents are accompanied (or caused) by alternating voltages. An AC voltage ''v'' can be described mathematically as a [[function (mathematics)|function]] of time by the following equation:

:<math>v(t) = V_\text{peak}\sin(\omega t)</math>,

where
* <math>V_\text{peak}</math> is the peak voltage (unit: [[volt]]),
* <math>\omega</math> is the [[angular frequency]] (unit: [[radians per second]]). {{paragraph}}The angular frequency is related to the physical frequency, <math>f</math> (unit: [[hertz]]), which represents the number of cycles per second, by the equation <math>\omega = 2\pi f</math>.
* <math>t</math> 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 <math>\sin(x)</math> is +1 and the minimum value is −1, an AC voltage swings between <math>+V_\text{peak}</math> and <math>-V_\text{peak}</math>. The peak-to-peak voltage, usually written as <math>V_\text{pp}</math> or <math>V_\text{P-P}</math>, is therefore <math>V_\text{peak} - (-V_\text{peak}) = 2 V_\text{peak}</math>.

=== Root mean square voltage{{anchor|Root mean square}} ===
{{further|RMS amplitude}}
{{broader|Root mean square voltage}}
[[File:Sine wave 2.svg|thumb|A sine wave, over one cycle (360°). The dashed line represents the [[root mean square]] (RMS) value at <math>{\sqrt {0.5}}</math> (about 0.707).|alt=A graph of sin(x) with a dashed line at y=sin(45)]]

Below an AC waveform (with no [[DC component]]) is assumed.

The RMS voltage is the square root of the [[mean of a function|mean]] over one cycle of the square of the instantaneous voltage.

{{unordered list
| For an arbitrary periodic waveform <math>v(t)</math> of period <math>T</math>:
: <math>V_\text{rms} = \sqrt{\frac{1}{T} \int_0^{T}{[v(t)]^2 dt}}.</math>
| For a sinusoidal voltage:
: <math>\begin{align}
  V_\text{rms} &= \sqrt{\frac{1}{T} \int_0^{T}[{V_\text{peak}\sin(\omega t + \phi)]^2 dt}}\\
               &= V_\text{peak}\sqrt{\frac{1}{2T} \int_0^{T}[{1 - \cos(2\omega t + 2\phi)] dt}}\\
               &= V_\text{peak}\sqrt{\frac{1}{2T} \int_0^{T}{dt}}\\
               &= \frac{V_\text{peak}}{\sqrt {2}}
\end{align}</math>

where the [[trigonometric identity]] <math>\sin^2(x) = \frac {1 - \cos(2x)}{2}</math> has been used and the factor <math>\sqrt{2}</math> is called the [[crest factor]], which varies for different waveforms.
| For a [[triangle wave]]form centered about zero
: <math>V_\text{rms} = \frac{V_\text{peak}}{\sqrt{3}}.</math>
| For a [[Square wave (waveform)|square wave]]form centered about zero
: <math>V_\text{rms} = V_\text{peak}.</math>
}}

=== Power ===
{{Main|AC power}}

The relationship between voltage and the power delivered is:
:<math>p(t) = \frac{v^2(t)}{R}</math>,

where <math>R</math> represents a load resistance.

Rather than using instantaneous power, <math>p(t)</math>, 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 <math>V_\text{rms}</math>, because
:<math>P_\text{average} = \frac{{V_\text{rms}}^2}{R}.</math>

;Power oscillation: <math>\begin{align} v(t) &= V_\text{peak}\sin(\omega t) \\ i(t) &= \frac{v(t)}{R} = \frac{V_\text{peak}}{R}\sin(\omega t) \\ p(t) &= v(t)i(t) = \frac{(V_\text{peak})^2}{R}\sin^2(\omega t) = \frac{(V_\text{peak})^2}{2 R} \ (1 - \cos(2 \omega t) ) \end{align}</math>
For this reason, AC power's waveform becomes [[Rectifier#Full-wave rectification|Full-wave rectified]] sine, and its fundamental frequency is double of the one of the voltage's.


=== Examples of alternating current ===
To illustrate these concepts, consider a 230&nbsp;V AC [[Mains power systems|mains]] supply used in [[Mains power systems#Table of mains voltages, frequencies, and plugs|many countries]] around the world. It is so called because its [[root mean square]] value is 230&nbsp;V. This means that the time-averaged power delivered <math>P_\text{average}</math> is equivalent to the power delivered by a DC voltage of 230&nbsp;V. To determine the peak voltage (amplitude), we can rearrange the above equation to:
:<math>V_\text{peak} = \sqrt{2}\ V_\text{rms}</math>
:<math>P_\text{peak} = \frac{(V_\text{rms})^2}{R}\frac{(V_\text{peak})^2}{(V_\text{rms})^2} = \text{P}_\text{average}\sqrt{2}^2 = \text{2}P_\text{average}.</math>

For 230&nbsp;V AC, the peak voltage <math>V_\text{peak}</math> is therefore <math>230\text{ V}\times\sqrt{2}</math>, which is about 325&nbsp;V, and the peak power <math>P_\text{peak}</math> is <math>230 \times R \times W \times 2</math>, that is 460&nbsp;RW. During the course of one cycle (two cycle as the power) the voltage rises from zero to 325&nbsp;V, the power from zero to 460&nbsp;RW, and both falls through zero. Next, the voltage descends to reverse direction, −325&nbsp;V, but the power ascends again to 460&nbsp;RW, and both returns to zero.