The idea of effective value arises from the need to measure the effectiveness of a voltage or current source in delivering power to a resistive load.
The effective value of a periodic current is the dc current that delivers the same average power to a resistor as the periodic current.
The effective value of a periodic signal is its root mean square (rms) value.
\(\boxed{X_\textrm{rms} = \sqrt{\frac{1}{T}\int_{0}^{T} x^2\,dt}}\)
For the sinusoid \(i(t) = I_m\cos(\omega t)\), the effective or rms value is
\(\displaystyle I_\textrm{rms} = \sqrt{\frac{1}{T}\int_{0}^{T} I_{m}^{2}\cos^{2}(\omega t)\,dt} = \sqrt{\frac{I_{m}^2}{T}\int_{0}^{T} \frac{1}{2}\left[1 + \cos(2\omega t)\right]\,dt}\)
\(\boxed{I_\textrm{rms} = \frac{I_m}{\sqrt{2}}}\quad\textrm{(only valid for sinusoidal signals)}\)
Similarly, for \(v(t) = V_m\cos(\omega t)\),
\(\boxed{V_\textrm{rms} = \frac{V_m}{\sqrt{2}}}\quad\textrm{(only valid for sinusoidal signals)}\)
The average power can be written in terms of the rms values.
\(\displaystyle P = \frac{1}{2} V_m I_m \cos(\theta_v - \theta_i) = \frac{V_m}{\sqrt{2}}\frac{I_m}{\sqrt{2}}\cos(\theta_v - \theta_i)\)
\(\boxed{P = V_\textrm{rms}I_\textrm{rms}\cos(\theta_v - \theta_i)}\)
Similarly, the average power absorbed by a resistor \(R\) can be wrtten as
\(\boxed{P = I_\textrm{rms}^2 R = \frac{V_\textrm{rms}^2}{R}}\)
When a sinusoidal voltage or current is specified, it is often in terms of its maximum (or peak) value or its rms value, since its average value is zero. The power industries specify phasor magnitudes in terms of their rms values rather than peak values. It is convenient in power analysis to express voltage and current in their rms values.