In this article, we will discuss the determination of the Average and RMS values of complex (mixed) waves. Let us start with the basic introduction of the average value and RMS value of the AC wave.

## Average Value:

The average of all instantaneous values of the ac quantity
over one cycle is known as the **average value**
of the ac quantity. The average value is generally denoted by the capital letter
with the subscript “av”.

Mathematically, the average value of an ac wave is defined
as the ratio of the area of the wave to the base, i.e.

## RMS Value:

The **RMS
(Root Mean Square) value** of an ac wave (alternating current) is equal to
that dc wave (direct current) which when flowing through a resistor for a given
time interval generates the same amount of heat as generated by the alternating
current when flowing through the same resistor for the same duration of time.

For an alternating current wave with
instantaneous values i_{1}, i_{2}, i_{3}, … i_{n},
the RMS value is given by,

After getting insights into the basics of
average value and RMS value. Let us now discuss the average and RMS values of
complex waves.

Sometimes, we have AC quantities that are
the combination of DC components and sinusoidally varying AC components of
different frequencies. Such ac waves are called **complex waves**. For example,

This current equation is a complex
current wave containing dc components as well as ac components of different frequencies.

# Average Value of Complex Wave

As we know, the average value of a pure
sine wave over one cycle is zero. Therefore, the average value of the complex
wave is equal to its DC component, i.e.

For the above given complex wave, the
average value is,

# RMS Value of Complex Wave

For a complex ac wave, the RMS value is
equal to the square root of the sum of the squares of individual RMS values of
different components.

Thus, the RMS value of the above complex
wave is,

In **generalized
form**, if we have a complex current wave given by,

Then, we have,

The average value of this complex wave
is,

The RMS value of the complex wave is,

Hence, this is all about the average and
RMS values of complex waves.