# RMS value of Rectifier

RMS (Root mean square) value is the square root of the mean value of the squared values.

The RMS value of an alternating current is the equivalent DC value of an alternating or varying electrical quantity. RMS value of an AC current produces the same amount of heat when an equal value of DC current flows through the same resistance.

RMS value of a signal = √ Area under the curve squared / base length.

For a function f(x) the RMS value for an interval [a, b] = √ (1/b-a) a ∫b f2(x) dx.

#### RMS value of a sine wave

RMS Value = √ Area of half cycle squared / half cycle base length

The RMS value of a sine wave can be calculated by just taking the half cycle region only. Because the area of positive half cycle squared and negative half cycle squared have the same values. So the derivation will be same as it for a full wave rectifier.

The RMS Voltage of a sine wave, VRMS = Vm/ √2,   Vm – Maximum voltage or peak voltage.

##### RMS value of a Half wave rectifier

In a half wave rectifier, the negative half cycle will be removed from the output. So, the total base length(2π) should be taken from the interval 0 to 2π.

The RMS voltage, VRMS = √ Vm2/2π 0π sin2ωt dωt

= √ Vm2/2π π(1 – cos2ωt) / 2 ) dωt  =  √ Vm2/4π [ωt – sin2ωt / 2]0π

= √ Vm2/4π [ π – (sinπ) / 2 – (0 – (sin0) / 2)]  = √ Vm2/4π ( π ) = √ Vm2/ 4

Therefore the RMS voltage, VRMS  = Vm/ 2

##### RMS value of a Full wave rectifier

The RMS voltage, VRMS = √ Vm20π sin2ωt dωt

= √ Vm2π(1 – cos2ωt) / 2 ) dωt = √ Vm2/2π [ωt – sin2ωt / 2]0π

= √ Vm2/2π [ π – (sinπ) / 2 – (0 – (sin0) / 2)]  = √ Vm2/2π ( π ) = √ Vm2/ 2

RMS voltage, VRMS = Vm/ √2

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