What is the mathematical expression for the average value of a rectified voltage sinusoid?

The average value of a rectified voltage sinusoid can be derived from the expression for the output of a rectifier circuit. When a sinusoidal voltage waveform is rectified, the negative portions of the waveform are clipped off, resulting in a waveform that is non-negative.

For a sinusoidal voltage of the form ( V(t) = V_m \sin(\omega t) ), where ( V_m ) is the peak voltage, the average value over one full period of the rectified waveform is calculated by integrating the positive half of the sinusoid over its period and then normalizing with respect to that period.

The average value ( V_{avg} ) for a full cycle of a rectified sinusoid can be mathematically expressed as:

[ V_{avg} = \frac{1}{T} \int_0^{T/2} V_m \sin(\omega t) , dt + \frac{1}{T} \int_{T/2}^{T} 0 , dt ]

When performing the integral over the range ( 0 ) to ( T/2 ) (which represents the positive half of the period of the sinusoid), the average value simplifies and incorporates