# What is RMS Value?

The RMS or active value of a sinusoidal waveform gives the same heating effect of an equivalent DC resource

In our tutorial on AC Waveform, we briefly looked at the **RMS** value of a sinusoidal waveform and said that this RMS value gives the same effect as an equivalent DC power, and in this tutorial we will expand this theory by examining it a little more.

The term "RMS" means "Root-Mean-Square".Most books describe it as "the amount of AC power that produces the same effect as equivalent DC power" or something similar to these lines, but an RMS value is more than that.The RMS value is the square root of the average value of the squared function of instantaneous values.Symbols used to define an RMS value are V_{RMS} or I_{RMS.}

The term RMS refers only to sinusoidal voltages, currents or complex waveforms that vary over time, is the magnitude of changes of the waveform over time and is not used in DC circuit analysis or calculations, the magnitude is always constant. When used as an equivalent DC circuit to compare the equivalent RMS voltage value of an alternative sinusoidal waveform that provides the same electrical power to a particular load, the RMS value is called the "active value" and is usually presented as follows: V_{eff} or I_{eff}.

In other words, the active value is an equivalent DC value that tells you how many volts or amperage DC a sinusoidal waveform changes over time in terms of the ability to generate the same power.

For example, take an area with a local grid feed of 240Vac.This value is assumed to indicate the effective value "240 Volt rms".This means that the sinesoidal rms voltage from the wall sockets of a house in the UK can produce the same average positive power as the constant DC voltage of 240 volts, as shown below.

### RMS Voltage Equivalent

The RMS voltage of a sinusoid or complex waveform can be determined by two basic methods:

- Graphical Method – Can be used to find the RMS value of any non-sinusoidal waveform by drawing a series of medium coordinates to the waveform.
- Analytical Method – It is a mathematical procedure to find the effective or RMS value of any periodic voltage or current using calculus(advanced mathematics).

## RMS Voltage Graphics Method

Although the calculation method is the same for both halves of an AC waveform, in this example we will only consider the positive semi-cycle.The effective or rms value of a waveform can be found with a reasonable amount of accuracy by taking equally spaced instantaneous values throughout the waveform.

The positive half of the waveform is divided into any number of "n" equal parts or middle ordinates, and the result is so accurate based on the excess number of ordinates drawn along the waveform. Therefore, the width of each middle ordinate will not be a degree, and the height of each middle ordinate will be equal to the instantaneous value of the current waveform along the x-axis of the waveform.

### Chart Method

Each medium coordinate value of a waveform (in this case, the voltage waveform) is multiplied by itself (square) and added to the next.This method gives us the "square" part of the RMS voltage expression.Then this square value is divided by the number of middle coordinates used to give us the **Average** part of the RMS voltage expression, and the number of middle coordinates used in our simple example above is twelve (12).Finally, the square root of the previous result gives us the **Root** part of the RMS voltage.

We can then define the term used to describe an rms voltage (V_{RMS)}as "square root of the square of the middle ordinates of the voltage waveform", and this is given as follows:

and for our simple example above, the RMS voltage will be calculated as follows:

So, let's say that an alternative voltage has a peak voltage of 20 volts (V_{pk),} and when a medium coordinate value of 10 is taken, it changes over a half-cycle as follows:

Voltage | 6.2V | 11.8V | 16.2V | 19.0V | 20.0V | 19.0V | 16.2V | 11.8V | 6.2V | 0V |

angle | 18 ^{O} | 36 ^{O} | 54 ^{O} | 72 ^{o} | ^{90 O} | 108 ^{o} | 126 ^{O} | 144 ^{O} | 162 ^{O} | 180 ^{O} |

Therefore, **the RMS voltage** is calculated as follows:

Then, using the graphics method, the RMS Voltage value is given as follows: 14.14 Volts .

## RMS Voltage Analytical Method

The graphical method above is a very good way to find the active or RMS voltage (or current) of an alternative waveform that is not inherently symmetrical or sinusoidal.In other words, the waveform shape is similar to a complex waveform.However, when dealing with pure sinusoidal waveforms, we can make operations a little easier by using an analytical or mathematical way to find the value of RMS.

Periodic sinusoidal voltage is constant and can be defined as V _{(t)} = V _{max} *cos(ωt) with T period.Then we can calculate the **square-mean-squared** (rms) value of a sinusoidal voltage (V _{(t)} as follows:

When integrated with limits from 0 to 360 ^{o}or "T", the period returns:

Where: VM is the peak or maximum value of the waveform. If we divide further in ω = 2π/T, the complex equation above is eventually reduced:

### RMS Voltage Equation

The RMS voltage (VRMS) of a sinusoidal waveform is then determined by multiplying the peak voltage value by 0.7071. The RMS voltage, which can also be called an active value, depends on the size of the waveform and is not a function of the frequency or phase angle of waveforms.

From the graphical example above, the peak voltage ( V_{pk} ) of the waveform is given as 20 Volts.Using the analytical method just described, we can calculate the RMS voltage as follows:

V _{RMS} = V _{pk} * 0.7071 = 20 x 0.7071 = 14.14V

Note that this value of 14.14 volts is the same as the previous graphical method.Next, we can use the graphical method of the middle coordinates or the analytical calculation method to find the RMS voltage or current values of a sinusoidal waveform.

Note that multiplying the peak or maximum value by a constant of 0.7071 applies only to sinusoidal waveforms. Graphic method should be used for non-sinusoidal waveforms.

But in addition to using the peak or maximum value of sinusoids, we can also use _{top-to-bottom (V P-P)} or average (V _{AVG)} to find the equivalent root mean square value of sinusoids, as shown:

### Sinusoidal RMS Values

## RMS Value Summary

Then let's summarize it this way: When dealing with alternating voltages (or currents), we face the problem of how to represent a voltage or signal size. An easy way is to use peak values for waveform. Another common method is to use a more common expression of the Root Average Square, or the active value, also known only by the RMS value.

The average square of a sinusoid, the RMS value, is not the same as the average of all instantaneous values.The ratio of voltage to maximum voltage value of RMS is the same as the ratio of current RMS to maximum current value.

Most multimeters, whether voltmeter or amperemeter, measure RMS by assuming pure sinusoidal waveform.A " True RMSMultimeter" is required to find the RMS value of the non-sinusoidal waveform.

The RMS value of a sinusoidal waveform gives the same heating effect as a DC current of the same value.So if a direct current passes through an R ohm resistance, the DC power consumed by the resistance as heat will therefore be I ^{2} R watts.If the alternating current passes i = I_{max} *sinε from the same resistance, the AC power converted to heat: I will be ^{2 }rms *R _{watts.}

Then, when dealing with alternating voltages and currents, they should be treated as RMS values unless otherwise stated.Therefore, an alternating current of 10 amps will have the same heating effect as a direct current of 10 amps and a maximum value of 14.14 amps.

After determining the RMS value of an alternate voltage (or current) waveform, in the next tutorial, we will look at calculating the Average value of an alternate voltage, V_{AVG,}and finally compare the two.