Define rms value of alternating current
The effective or RMS (root mean square) value of an alternating current is the value of a direct current that would generate the same amount of heat in a resistor over the same period of time as the alternating current does.
Formula of rms value of AC current
\[I_{\text{rms}} = \frac{I_m}{\sqrt{2}}\]
For voltage:
\[V_{\text{rms}} = \frac{V_m}{\sqrt{2}}\]
Derivation of RMS value of alternating current
Let the instantaneous value of alternating current be:
\[i(t) = I_m \sin(\omega t)\]
where:
\( i(t) \) is the instantaneous current,
\( I_m \) is the peak (maximum) current,
\( \omega \) is the angular frequency,
\( t \) is the time.
The RMS (Root Mean Square) value is defined as:
\[I_{\text{rms}} = \sqrt{\frac{1}{T} \int_0^T i^2(t) \, dt}\]
Substitute \( i(t) = I_m \sin(\omega t) \) into the formula:
\[I_{\text{rms}} = \sqrt{\frac{1}{T} \int_0^T I_m^2 \sin^2(\omega t) \, dt}\]
Factor out the constant \( I_m^2 \):
\[I_{\text{rms}} = \sqrt{\frac{I_m^2}{T} \int_0^T \sin^2(\omega t) \, dt}\]
Use the trigonometric identity:
\[\sin^2(\omega t) = \frac{1 - \cos(2\omega t)}{2}\]
Substitute into the integral:
\[I_{\text{rms}} = \sqrt{\frac{I_m^2}{T} \int_0^T \frac{1 - \cos(2\omega t)}{2} \, dt}\]
Simplify:
\[I_{\text{rms}} = \sqrt{\frac{I_m^2}{2T} \int_0^T \left(1 - \cos(2\omega t)\right) \, dt}\]
Integrate term by term:
\[\int_0^T \left(1 - \cos(2\omega t)\right) \, dt = \left[t - \frac{\sin(2\omega t)}{2\omega}\right]_0^T\]
Since \( \sin(2\omega T) \) over a full cycle \( T \) is 0:
\[\int_0^T \left(1 - \cos(2\omega t)\right) \, dt = T\]
Substitute back:
\[I_{\text{rms}} = \sqrt{\frac{I_m^2}{2T} \cdot T} = \sqrt{\frac{I_m^2}{2}} = \frac{I_m}{\sqrt{2}}\]
$\textbf{Therefore,}$
\[\boxed{I_{\text{rms}} = \frac{I_m}{\sqrt{2}}}\]