2.3 Root mean square current and voltage

  Alternating Current

rms value of alternating current is defined as the square root of the average of $I^2$ during a complete cycle.

Mathematically it is given by, $${\left\langle {{i^2}} \right\rangle _{one\,cycle}} = \frac{{\int\limits_0^T {{i^2}dt} }}{{\int\limits_0^T {dt} }}$$$${\left\langle {{i^2}} \right\rangle _{one\,cycle}} = \frac{{\int\limits_0^T {i_0^2{{\sin }^2}\omega tdt} }}{{\int\limits_0^T {dt} }}$$$${\left\langle {{i^2}} \right\rangle _{one\,cycle}} = \frac{{i_0^2}}{2}\frac{{\int\limits_0^T {\left( {1 - \cos 2\omega t} \right)dt} }}{{\int\limits_0^T {dt} }}$$$${\left\langle {{i^2}} \right\rangle _{one\,cycle}} = \frac{{i_0^2}}{2}\frac{{\left[ {t - \frac{{\cos 2\omega t}}{{2\omega }}} \right]_0^T}}{{\left[ t \right]_0^T}}$$$${\left\langle {{i^2}} \right\rangle _{one\,cycle}} = \frac{{i_0^2}}{2}\frac{{\left[ {\left( {T - 0} \right) - \left( {\frac{{\cos 2\omega \times \frac{{2\pi }}{\omega }}}{{2\omega }} - \frac{1}{{2\omega }}} \right)} \right]}}{{\left[ {T - 0} \right]}}$$$${\left\langle {{i^2}} \right\rangle _{one\,cycle}} = \frac{{i_0^2}}{2}$$ As, ${i_{rms}} = \sqrt {{{\left\langle {{i^2}} \right\rangle }_{one\,cycle}}} $, $${i_{rms}} = \frac{{{i_0}}}{{\sqrt 2 }} = 0.707{i_0}$$

Similarly, $${V_{rms}} = \frac{{{V_0}}}{{\sqrt 2 }} = 0.707{V_0}$$

Note:

* rms value is also known as virtual value or effective value.

* All A.C. instrument measure rms value.

* In India our household power supply is 220 $V$ A.C. This means, $${V_{rms}} = 220\,V$$

So, voltage amplitude $\left( {{V_0}} \right)$ is given by, $${V_{rms}} = \frac{{{V_0}}}{{\sqrt 2 }}$$

So, $${V_0} = \sqrt 2 {V_{rms}}$$$${V_0} = \sqrt 2 \times 220\,V$$