A sinewave is defined by the trigonometric sine function. When plotted as voltage (V) as a function of
phase (θ), it looks similar to the figure to the right. The waveform repeats every 2
p
radians (360°), and is symmetrical about the voltage axis (when no DC offset is present). Voltage and current
exhibiting cyclic behavior is referred to as alternating; i.e., alternating current (AC). One full cycle is shown
here. The basic equation for a sinewave is as follows:
There are a number of ways in which the amplitude of a sinewave is referenced, usually as peak voltage (V
_{pk}
or V
_{p}), peak-to-peak voltage (V
_{pp} or V
_{p-p}
or V
_{pkpk} or V
_{pk-pk}), average voltage (V
_{av} or V
_{avg}), and
root-mean-square voltage (V
_{rms}). Peak voltage and peak-to-peak voltage are apparent by looking at the
above plot. Root-mean-square and average voltage are not so apparent.
Also
see
Triangle Wave Voltages and
Square Wave Voltages pages.
Root-Mean-Square Voltage (V_{rms})As the name implies, V
_{rms} is
calculated by taking the square root of the mean average of the square of the voltage in an appropriately chosen
interval. In the case of symmetrical waveforms like the sinewave, a quarter cycle faithfully represents all four
quarter cycles of the waveform. Therefore, it is acceptable to choose the first quarter cycle, which goes from 0
radians (0°) through
p/2 radians (90°).
V
_{rms} is the value indicated by the
vast majority of AC voltmeters. It is the value that, when applied across a resistance, produces that same amount
of heat that a direct current (DC) voltage of the same magnitude would produce. For example, 1 V applied across a
1 Ω resistor produces 1 W of heat. A 1 V
_{rms}
sinewave applied across a 1 Ω resistor also produces 1 W of heat. That 1 V
_{rms} sinewave has a peak
voltage of √2 V (≈1.414 V), and a peak-to-peak voltage of 2√2 V (≈2.828 V).
Since finding a full
derivation of the formulas for root-mean-square (V
_{rms}) voltage is difficult, it is done here for you.
So,
≈ 0.707 V_{pk}
=
0.70710678118654752440084436210485
Average Voltage (V_{avg})
As the name implies, V
_{avg} is calculated by taking the average of the voltage in an appropriately chosen
interval. In the case of symmetrical waveforms like the sinewave, a quarter cycle faithfully represents all four
quarter cycles of the waveform. Therefore, it is acceptable to choose the first quarter cycle, which goes from 0
radians (0°) through
p/2 radians (90°).
As with the V
_{rms}
formula, a full derivation for the V
_{avg}
formula is given here as well.
So,
≈ 0.636 V_{pk}
= 0.63661977236758134307553505349006
* I have no idea why we write "Sinewave," but not "Trianglewave" and "Squarewave."