Alternating Quantities — RMS, Average and Form Factor

By the end you should be able to: Describe the generation of alternating voltage, define frequency, period, average value, RMS value, form factor and peak factor, and calculate them for a sinusoid.

Generation

A coil rotating in a uniform magnetic field cuts flux at a rate proportional to $\sin\theta$ , where $\theta$ is the angle between the plane of the coil and the field. With the coil turning at constant angular velocity $\omega$ , $\theta = \omega t$ and

e = E_m\sin\omega t

The sinusoid is not a convenient choice — it is what uniform rotation produces. That is worth stating, because it explains why the entire power system is sinusoidal rather than, say, triangular.

The basic quantities

Cycle — one complete set of positive and negative values.
Time period $T$ — seconds for one cycle.
Frequency $f = 1/T$ — cycles per second, hertz. India uses 50 Hz.
Angular frequency $\omega = 2\pi f$ — radians per second.
Amplitude $E_m$ — the peak value.

Average value

Over a full cycle a symmetrical sinusoid averages to zero — the negative half cancels the positive. So the average is defined over a half cycle:

V_{avg} = \frac{2V_m}{\pi} = 0.637\,V_m

Its physical relevance is limited: it matters for rectified waveforms and for moving-coil meters, and little else.

RMS value — the one that matters

Root mean square: the DC value that would dissipate the same heat in the same resistor.

That definition is the whole point, and it is the reason RMS exists. Power goes as $i^2R$ , and squaring makes the negative half cycle contribute just as much as the positive — so we square, average, and take the root:

V_{rms} = \sqrt{\frac{1}{T}\int_0^T v^2\,dt}

For a sinusoid the average of $\sin^2$ over a cycle is exactly $\tfrac12$ , giving

\boxed{\,V_{rms} = \frac{V_m}{\sqrt{2}} = 0.707\,V_m\,}

Every AC value quoted without qualification is RMS. "230 V mains" means 230 V RMS; the peak is $230\sqrt2 = 325$ V, and insulation must withstand that, not 230 V. This distinction has practical consequences — a capacitor rated 250 V DC will fail on 230 V AC.

Form factor and peak factor

\text{Form factor} = \frac{V_{rms}}{V_{avg}} = \frac{0.707V_m}{0.637V_m} = 1.11

\text{Peak factor (crest factor)} = \frac{V_m}{V_{rms}} = \frac{V_m}{0.707V_m} = 1.414

Both are shape descriptors, independent of amplitude. They exist because a moving-coil meter responds to the average of a rectified waveform while its scale is marked in RMS — the scaling factor 1.11 is built into the calibration. Feed such a meter a non-sinusoidal waveform, whose form factor differs, and it reads wrongly. That is exactly why cheap multimeters are inaccurate on the distorted waveforms drawn by electronic loads, and why "true RMS" meters cost more.

For reference: a square wave has form factor 1.00 and peak factor 1.00, a triangular wave 1.15 and 1.73.