The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a series of equally spaced samples. Additionally, the RMS value of various waveforms can also be determined without calculus.
RMS of common waveforms
Waveform Equation RMS
Sine wave
Square wave
Modified square wave
Sawtooth wave
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This wave pattern occurs often in nature, including ocean waves, sound waves, and light waves. Also, a rough sinusoidal pattern can be seen in plotting average daily temperatures for each day of the year, although the graph may resemble an inverted cosine wave.
Graphing the voltage of an alternating current gives a sine wave pattern.
A cosine wave is said to be "sinusoidal", because cos(x) = sin(x + π / 2), which is also a sine wave with a phase-shift of π/2. Because of this "head start", it is often said that the cosine function leads the sine function or the sine lags the cosine.
The human ear can recognize single sine waves as sounding clear because sine waves are representations of a single frequency with no harmonics; some sounds that approximate a pure sine wave are whistling, a crystal glass set to vibrate by running a wet finger around its rim, and the sound made by a tuning fork.
To the human ear, a sound that is made up of more than one sine wave will either sound "noisy" or will have detectable harmonics; this may be described as a different timbre.
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An ideal square wave requires that the signal changes from the high to the low state cleanly and instantaneously. This is impossible to achieve in real-world systems, as it would require infinite bandwidth.
Animation of the additive synthesis of a square wave with an increasing number of harmonicsReal-world square-waves have only finite bandwidth, and often exhibit ringing effects similar to those of the Gibbs phenomenon, or ripple effects similar to those of the σ-approximation.
For a reasonable approximation to the square-wave shape, at least the fundamental and third harmonic need to be present, with the fifth harmonic being desirable. These bandwidth requirements are important in digital electronics, where finite-bandwidth analog approximations to square-wave-like waveforms are used. (The ringing transients are an important electronic consideration here, as they may go beyond the electrical rating limits of a circuit or cause a badly positioned threshold to be crossed multiple times.)
The ratio of the high period to the total period of a square wave is called the duty cycle. A true square wave has a 50% duty cycle - equal high and low periods. The average level of a square wave is also given by the duty cycle, so by varying the on and off periods and then averaging, it is possible to represent any value between the two limiting levels. This is the basis of pulse width modulation.
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This waveform is a compromise between the sine wave and the square wave. The positive and negative pulses of the square wave are thinned, separated and made taller, so the peak voltage is much closer to that of a sine wave, and the overall shape of the wave more closely resembles that of a sine wave. At the same time, the cost of the circuitry to produce a modified square wave output is much closer to the cost of a square wave's circuitry than that of a sine wave unit. (In fact, you can create a modified square wave by adding together two square waves that are shifted in phase slightly from each other.) Many fewer pieces of equipment have problems with modified square wave power than with straight square wave. Modified square wave output is used on many lower- to middle-range UPSes, and is also sometimes called "stepped approximation to a sine wave", "pulse-width modified square wave", or even "modified sine wave". The last term is marketing cutesy-speak, since the output form isn't really a sine wave, modified or otherwise.
Schematic representation of one cycle of a sine wave, square wave, and modified square wave
output. The area under each curve is the same, so they each carry the same amount of power.
In practical terms, for a home PC UPS modified square wave output is fine. It will power a PC, monitor and similar equipment without any trouble. Of course, also remember that less expensive UPSes are normally running off line power anyway, and it is only when operating off the battery that the output waveform comes into play.
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The sawtooth and square waves are the most common starting points used to create sounds with subtractive analog and virtual analog music synthesizers.
The sawtooth wave is the form of the vertical and horizontal deflection signals used to generate a raster on CRT-based television or monitor screens. Oscilloscopes also use a sawtooth wave for their horizontal deflection, though they typically use electrostatic deflection.
On the wave's "ramp", the magnetic field produced by the deflection yoke drags the electron beam across the face of the CRT, creating a scan line.
On the wave's "cliff", the magnetic field suddenly collapses, causing the electron beam to return to its resting position as quickly as possible.
The voltage applied to the deflection yoke is adjusted by various means (transformers, capacitors, center-tapped windings) so that the half-way voltage on the sawtooth's cliff is at the zero mark, meaning that a negative voltage will cause deflection in one direction, and a positive voltage deflection in the other; thus, a center-mounted deflection yoke can use the whole screen area to depict a trace. Frequency is 15.734 kHz on NTSC, 15.625 kHz for PAL and SECAM)
The vertical deflection system operates the same way as the horizontal, though at a much lower frequency (59.94 Hz on NTSC, 50 Hz for PAL and SECAM).
The ramp portion of the wave must appear as a straight line. If otherwise, it indicates that the voltage isn't increasing linearly, and therefore that the magnetic field produced by the deflection yoke is not linear. As a result, the electron beam will accelerate during the non-linear portions. This would result in a television image "squished" in the direction of the non-linearity. Extreme cases will show marked brightness increases, since the electron beam spends more time on that side of the picture.
The first television receivers had controls allowing users to adjust the picture's vertical or horizontal linearity. Such controls were not present on later sets as the stability of electronic components had improved.
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The amplitude of an AC waveform is its height as depicted on a graph over time. An amplitude measurement can take the form of peak, peak-to-peak, average, or RMS quantity.
Peak amplitude is the height of an AC waveform as measured from the zero mark to the highest positive or lowest negative point on a graph. Also known as the crest amplitude of a wave.
Peak-to-peak amplitude is the total height of an AC waveform as measured from maximum positive to maximum negative peaks on a graph. Often abbreviated as "P-P".
Average amplitude is the mathematical "mean" of all a waveform's points over the period of one cycle. Technically, the average amplitude of any waveform with equal-area portions above and below the "zero" line on a graph is zero. However, as a practical measure of amplitude, a waveform's average value is often calculated as the mathematical mean of all the points' absolute values (taking all the negative values and considering them as positive). For a sine wave, the average value so calculated is approximately 0.637 of its peak value.
"RMS" stands for Root Mean Square, and is a way of expressing an AC quantity of voltage or current in terms functionally equivalent to DC. For example, 10 volts AC RMS is the amount of voltage that would produce the same amount of heat dissipation across a resistor of given value as a 10 volt DC power supply. Also known as the "equivalent" or "DC equivalent" value of an AC voltage or current. For a sine wave, the RMS value is approximately 0.707 of its peak value.
The crest factor of an AC waveform is the ratio of its peak (crest) to its RMS value.
The form factor of an AC waveform is the ratio of its peak (crest) value to its average value.
Analog, electromechanical meter movements respond proportionally to the average value of an AC voltage or current. When RMS indication is desired, the meter's calibration must be "skewed" accordingly. This means that the accuracy of an electromechanical meter's RMS indication is dependent on the purity of the waveform: whether it is the exact same waveshape as the waveform used in calibrating.
Just a lil reading info for ya new guys.........
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