## General

Can you use the peak value of a fluctuating parameter to represent it? If the answer is no then what value best represents the parameter?

Understanding in the general terms, Root Mean Square is the statistical measure of the magnitude of a varying quantity. It is abbreviated as RMS.

## Definition

“The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean (average) of the squares of the original values (or the square of the function that defines the continuous waveform).” ^{1
}

Or

“The root mean square value of a quantity is the square root of the mean value of the squared Values of the quantity taken over an interval.”^{2
}

“In the case of a set of n values {x_{1}, x_{2}… x_{n}}, the RMS value is given by following equation;”^{3
}

X_{rms} = √ [1/n (X_{1}^{2} + X_{2}^{2} + …. + X_{n}^{2})]

## RMS Value in Electrical Engineering

In Electrical Engineering, RMS is the most common mathematical method utilized to find the effective values of voltage or current while dealing with AC circuit.

In DC Circuits, the values of Current and Voltage are constant and therefore, they are directly utilized in the calculation of Power of the DC Electrical circuits.

Whereas, while dealing with the AC Circuits, the value of AC voltage continuously changes from Zero up to the positive peak, through Zero to the negative peak and back to zero again, as depicted in figure – 1 below. “Clearly for most of the time it is less than the peak voltage, so this is not a good measure of its real effect.”^{4
}

^{}

In order to obtain the appropriate measure of current and voltage that would represent the real effect, “the RMS value is determined by carrying out the following three mathematical operations on the function representing the AC waveform;”^{5
}

1. The square of the waveform function (usually a sine wave) is determined

2. The function resulting from step (1) is averaged over time

3. The square root of the function resulting from step (2) is found

The value obtained from the above mentioned mathematical manipulations is RMS value for Current or Voltage.

“Root Mean Square Voltage (Vrms) is 0.7 of the Peak Voltage (Vpeak);”^{6
}

**V _{rms} = 0.7 x V_{peak}**

Therefore,

**V _{peak} = 1.4 x V_{rms}**

The equations can be more easily understood using the figure – 2 below;

Some facts to remember regarding RMS value / Effective value;

1. The above equations also apply to current (Alternative Current).

2. The RMS value is the effective value of a varying voltage or current.

3. It is the equivalent steady DC (constant) value which gives the same effect [e. g. a lamp connected to a 6V RMS AC supply will light with the same brightness when connected to a steady 6V DC supply]

4. AC voltmeters and ammeters usually show the RMS value of the voltage or current

## References

1. RMS Value

2. RAENG

3. Wikipedia

4. The Electronics Club

5. CIO MidMarket

6. The Electronics Club

## Sources

Wikipedia

The Royal Academy of Engineering

The Electronics Club

TechTarget