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Derivation of RMS Value of a DC Offset Sinusoid

Here's a derivation of the RMS value of a sinusoid offset by a DC value.

Vrms=  sqrt(1/T*Integral([Vdc+ Vpk*sin(t)]^2)*dt )

Where T is 2*pi

Vrms= sqrt( 1/T* Integral( Vdc^2+ 2*Vdc*Vpk*sin(t)+ Vpk^2*sin(t)^2 )*dt )

 The middle term get's dropped because the average of a sine over a period is zero.

Vrms= sqrt( 1/T* Integral( Vdc^2*dt))  + 1/T* Integral(Vpk^2*sin(t)^2 )*dt )

Vrms= sqrt( Vdc^2  + [ Vpk^2/T*(t/2- sin(2*t)/4) (from 0 to T)] )

The sin terms cancel.

Vrms= sqrt( Vdc^2 + Vpk^2/2 )

 

  

 

 

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