The average and effective values for the waveshaphe shown in figure are:

Options

A. `(2)/(pi)V_(m) " and " (V_(m))/(2)`
B. `(V_(m))/(pi) " and " (V_(m))/(sqrt(2))`
C. `(2)/(pi)V_(m) " and " (V_(m))/(sqrt(2))`
D. `(V_(m))/(pisqrt(2)) " and " (V_(m))/(sqrt(2))`

Answer

C

Explanation

`V_("average") =(overset(T//2)underset(0)int (V_(ac))dt)/(t//2)=(overset(T//2)underset(0)int (V_(0)sin omegat)dt)/(t//2)`
`(2V_(0))/(T) overset(T//2)underset(0)int sin (omegat)dt`
`=(2V_(0))/(T) (-cos omegat)_(0)^(T//2)`
` = (2V_(0))/(T) ("cos 0 - cos" (2pi)/(T) xx (T)/(2))`
`= (2V_(0))/(T) (1+1)= (4V_(0))/(2pi)=(2V_(0))/(pi)`
`V_(rms) = sqrt((overset(T/2)underset(0)int(V_(ac))^(2)dt)/(T//2)) = sqrt((V_(0)^(2)overset(T//2)underset(0)int (sin^(2)omegat)dt)/(T//2))`
`sqrt((V_(0)^(2)overset(T//2)underset(0)int (1-cos2omegat)dt)/(T)) = sqrt((V_(0)^(2))/(T)((T)/(2)))=(V_(0))/(sqrt(2))`