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

Peak Value vs. RMS Value: The peak value represents the maximum current in an alternating current (AC) cycle. The root mean square (RMS) value is the square root of the average of the squared instantaneous current values over a cycle.

RMS Value Formula: For a sinusoidal current \( I = I_0 \sin(\omega t) \), where \( I_0 \) is the peak current, the RMS value is calculated as: \[ I_{\text{RMS}} = \sqrt{\frac{1}{T} \int_0^T I^2 \, dt} \] Substituting the current equation: \[ I_{\text{RMS}} = \sqrt{\frac{1}{T} \int_0^T I_0^2 \sin^2(\omega t) \, dt} \] Since the average of \( \sin^2(\omega t) \) over a full cycle is \( \frac{1}{2} \): \[ I_{\text{RMS}} = \sqrt{\frac{1}{T} \times I_0^2 \times \frac{T}{2}} = \frac{I_0}{\sqrt{2}} \] Consequently, the RMS value of the current is \( \frac{I_0}{\sqrt{2}} \), with \( I_0 \) being the peak value.