# Relationship of rms peak and to

### Comparing rms value and peak value of AC

The maximum value attained by an alternating quantity during one cycle is called its peak value. The average of all the instantaneous values of an alternating. RMS is not an "Average" voltage, and its mathematical relationship to peak voltage varies depending on the type of waveform. The RMS value is the square root. In statistics and its applications, the root mean square (abbreviated RMS or rms) is defined as . For a zero-mean sine wave, the relationship between RMS and peak-to-peak amplitude is: Peak-to-peak = 2 2 × R M S ≈ × R M S.

## Peak Value, Average Value and RMS Value

If we consider symmetrical waves like sinusoidal current or voltage waveform, the positive half cycle will be exactly equal to negative half cycle. Therefore, the average value over a complete cycle will be zero. The work is done by both, positive and negative cycle and hence the average value is determined without considering the signs.

So the only positive half cycle is considered to determine the average value of alternating quantities of sinusoidal waves.

Let us take an example to understand it. Divide the positive half cycle into n number of equal parts as shown in the above figure Let i1, i2, i3……. That steady current which, when flows through a resistor of known resistance for a given period of time than as a result the same quantity of heat is produced by the alternating current when flows through the same resistor for the same period of time is called R.

S or effective value of the alternating current. In other words, the R. S value is defined as the square root of means of squares of instantaneous values. Let I be the alternating current flowing through a resistor R for time t seconds, which produces the same amount of heat as produced by the direct current Ieff.

**RMS Vs Peak Values Part 2 - RMS 0.707 Proved**

In other words, we calculate the practical average value of the waveform by considering all points on the wave as positive quantities, as if the waveform looked like this: Conversely, polarity-sensitive meter movements vibrate uselessly if exposed to AC voltage or current, their needles oscillating rapidly about the zero mark, indicating the true algebraic average value of zero for a symmetrical waveform. Although the mathematics of such an amplitude measurement might not be straightforward, the utility of it is.

Consider a bandsaw and a jigsaw, two pieces of modern woodworking equipment. Both types of saws cut with a thin, toothed, motor-powered metal blade to cut wood. But while the bandsaw uses a continuous motion of the blade to cut, the jigsaw uses a back-and-forth motion.

### Measurements of AC Magnitude | Basic AC Theory | Electronics Textbook

The comparison of alternating current AC to direct current DC may be likened to the comparison of these two saw types: The problem of trying to describe the changing quantities of AC voltage or current in a single, aggregate measurement is also present in this saw analogy: A bandsaw blade moves with a constant speed, similar to the way DC voltage pushes or DC current moves with a constant magnitude. A jigsaw blade, on the other hand, moves back and forth, its blade speed constantly changing.

What is more, the back-and-forth motion of any two jigsaws may not be of the same type, depending on the mechanical design of the saws. One jigsaw might move its blade with a sine-wave motion, while another with a triangle-wave motion. To rate a jigsaw based on its peak blade speed would be quite misleading when comparing one jigsaw to another or a jigsaw with a bandsaw! Despite the fact that these different saws move their blades in different manners, they are equal in one respect: Picture a jigsaw and bandsaw side-by-side, equipped with identical blades same tooth pitch, angle, etc.

We might say that the two saws were equivalent or equal in their cutting capacity. More specifically, we would denote its voltage value as being 10 volts RMS. RMS amplitude measurement is the best way to relate AC quantities to DC quantities, or other AC quantities of differing waveform shapes, when dealing with measurements of electric power.

### AC Circuit Theory (Part 3): Peak, Average and RMS Values

For other considerations, peak or peak-to-peak measurements may be the best to employ. For instance, when determining the proper size of wire ampacity to conduct electric power from a source to a load, RMS current measurement is the best to use, because the principal concern with current is overheating of the wire, which is a function of power dissipation caused by current through the resistance of the wire.

Peak and peak-to-peak measurements are best performed with an oscilloscope, which can capture the crests of the waveform with a high degree of accuracy due to the fast action of the cathode-ray-tube in response to changes in voltage.

Because the mechanical inertia and dampening effects of an electromechanical meter movement makes the deflection of the needle naturally proportional to the average value of the AC, not the true RMS value, analog meters must be specifically calibrated or mis-calibrated, depending on how you look at it to indicate voltage or current in RMS units.

The accuracy of this calibration depends on an assumed waveshape, usually a sine wave. Electronic meters specifically designed for RMS measurement are best for the task.

Some instrument manufacturers have designed ingenious methods for determining the RMS value of any waveform. The heating effect of that resistance element is measured thermally to give a true RMS value with no mathematical calculations whatsoever, just the laws of physics in action in fulfillment of the definition of RMS.

The accuracy of this type of RMS measurement is independent of waveshape. Figure below Conversion factors for common waveforms. In addition to RMS, average, peak crestand peak-to-peak measures of an AC waveform, there are ratios expressing the proportionality between some of these fundamental measurements. The crest factor of an AC waveform, for instance, is the ratio of its peak crest value divided by its RMS value. Square-shaped waveforms always have crest and form factors equal to 1, since the peak is the same as the RMS and average values.