Question

Differentiate between the peak value and root mean square value of an alternating current. Derive the expression for the root mean square value of alternating current, in terms of its peak value.

Hint:

The RMS value of an AC current is the effective value that produces the same power as a DC current of the same value.

Answer 1

Difference between Peak Value and RMS Value:
The peak value of an alternating current (AC) is the maximum value of the current during a cycle.
The root mean square (RMS) value of an AC is the square root of the average of the squares of the instantaneous values of current over a cycle.

Expression for RMS Value: For a sinusoidal current \( I = I_0 \sin(\omega t) \), where \( I_0 \) is the peak value of the current, the RMS value is given by: \[ I_{\text{RMS}} = \sqrt{\frac{1}{T} \int_0^T I^2 \, dt} \] Substituting \( I = I_0 \sin(\omega t) \): \[ I_{\text{RMS}} = \sqrt{\frac{1}{T} \int_0^T I_0^2 \sin^2(\omega t) \, dt} \] The average value of \( \sin^2(\omega t) \) over a full cycle is \( \frac{1}{2} \). Therefore: \[ I_{\text{RMS}} = \sqrt{\frac{1}{T} \times I_0^2 \times \frac{T}{2}} = \frac{I_0}{\sqrt{2}} \] Thus, the RMS value of the current is \( \frac{I_0}{\sqrt{2}} \), where \( I_0 \) is the peak value.