Average and RMS Values of Mixed Waves

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.

`\"Average value"=("Area of the wave")/"Base"`

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 i₁, i₂, i₃, … i_n, the RMS value is given by,

`\I_(RMS)=\sqrt{(i_1+i_2+⋯+i_n)/n}`

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,

`\i=2+4 sin⁡ωt+8 sin⁡(2ωt-30°)+10 sin⁡(3ωt+45°)`

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,

`\I_(av)=2 "A"`

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,

`\I_(RMS)=\sqrt{2^2+(4/\sqrt{2})^2+(8/\sqrt{2})^2+(10/\sqrt{2})^2 }`

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

`\i=i_0+i_(m1)  sin⁡ωt+i_(m2)  sin⁡(2ωt-θ_1 )+i_(m3)  sin⁡(3ωt+θ_2 )`

Then, we have,

The average value of this complex wave is,

`\I_(av)=i_0`

The RMS value of the complex wave is,

`\I_(RMS)=\sqrt{i_0^2+(i_(m1)/\sqrt{2})^2+(i_(m2)/\sqrt{2})^2+(i_(m3)/\sqrt{2})^2 }`

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