Root Mean Square Value of Current
The root mean square value of any current is defined as that value of steady current, which would generate the same amount of heat in a given resistance in a given time as generated by actual current passing through the same resistance for the same given time.
The Root Mean Square value is known as effective value or virtual value of current. It is denoted by IV.
\(I_{V}^{2}R({{t}_{2}}-{{t}_{1}})=\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left[ I{{(t)}^{2}} \right]}Rdt\).
\(\Rightarrow \,\,{{I}_{V}}=\sqrt{\frac{\int\limits_{{{t}_{1}}}^{{{t}_{2}}}{\left[ I{{(t)}^{2}} \right]}dt}{({{t}_{2}}-{{t}_{1}})}}\).
We square the instantaneous current I, take the average value of I², and finally take the square root of that average. This procedure defines the root mean square current, denoted as Irms or IV. Even when I is negative, I² is always positive, so IV is never zero (unless I is zero at every instant).
Root mean square value of current (I) = I₀ sinωt.
Over first half cycle: For this t₁ = 0, \({{t}_{2}}=\frac{T}{2}=\frac{\pi }{\omega }\).
\(I_{V}^{2}\left( \frac{T}{2}-0 \right)=\int\limits_{0}^{\frac{T}{2}}{I_{0}^{2}{{\sin }^{2}}\omega tdt=I_{0}^{2}\,\int\limits_{0}^{\frac{T}{2}}{\left( \frac{1-\cos 2\omega t}{2} \right)}}dt\).
\(\Rightarrow \,\,{{I}_{V}}\,\,=\,\,\frac{{{I}_{0}}}{\sqrt{2}}\,\,=\,\,0.707{{I}_{0}}.\)Over full cycle: For this t₁ = 0, \({{t}_{2}}=T=\frac{2\pi }{\omega }\).
\(I_{V}^{2}\left( T-0 \right)=\int\limits_{0}^{T}{I_{0}^{2}{{\sin }^{2}}\omega tdt=I_{0}^{2}\,\int\limits_{0}^{T}{\left( \frac{1-\cos 2\omega t}{2} \right)}}dt\).
\(\Rightarrow \,\,{{I}_{V}}\,\,=\,\,\frac{{{I}_{0}}}{\sqrt{2}}\,\,=\,\,0.707{{I}_{0}}\).
The Root Mean Square of ac over half cycle or full cycle is same. This is also valid over a long period of time.