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 2p 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}, where = 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,} where = 0.63661977236758134307553505349006 * I have no idea why we write "Sinewave," but not "Trianglewave" and "Squarewave." |