Peak and RMS Values of Alternating Current

Why AC needs a different kind of 'value'. A DC source such as a cell has one fixed voltage at every instant, so quoting 'the voltage' is unambiguous. An alternating voltage or current, by contrast, is CONSTANTLY changing -- so before it can be talked about sensibly, some standard, single representative number needs to be agreed on. Two such numbers are in everyday use: the peak value and the rms value.

The sinusoidal form. The alternating currents and voltages studied throughout this chapter are taken to vary sinusoidally with time:

Here $i_0$ and $v_0$ are the PEAK (or amplitude) values -- the single largest magnitude the current or voltage ever reaches in either direction -- and $\omega=2\pi f$ is the angular frequency, with $f$ the ordinary frequency in hertz (for Indian household mains, $f=50\ \text{Hz}$, so $\omega=2\pi\times50\approx314\ \text{rad/s}$).

Why a plain cycle-average is useless. The straightforward average of $i_0\sin\omega t$ over one full cycle is exactly ZERO, since the current spends exactly as much time (and reaches exactly the same peak magnitude) in the positive half-cycle as in the negative half-cycle, and the two halves cancel perfectly. A plain average is therefore useless for describing 'how much' current is really flowing, even though a very real current is clearly present at every instant.

Defining the rms value. The root-mean-square (rms) value gets around this by first SQUARING the current (which makes every value positive, whichever direction the current is actually flowing), then averaging this squared value over one cycle, and finally taking the square root to return to the original units. Carrying this out for $i=i_0\sin\omega t$, and using the fact that $\sin^2\omega t$ averages to exactly $\tfrac12$ over a whole number of cycles, gives

$I_{rms} = \sqrt{\langle i^2\rangle} = \sqrt{i_0^2\langle\sin^2\omega t\rangle} = \sqrt{i_0^2\cdot\tfrac12} = \frac{i_0}{\sqrt2} \approx 0.707\,i_0$ …