Angular Frequency (ω)

Quick Answer

Angular frequency is the rate a sinusoid sweeps angle, in radians per second, and it is 2π times the ordinary frequency. A wave covers 2π radians in each cycle, so the two figures describe one repetition rate in different units. Reactance and phasor algebra are written in the angular form.

The conversion is one multiplication:

Nothing physical changes in it. Two pi is the number of radians in a cycle, so this is bookkeeping between two ways of counting the same turn. Mains at 50 Hz comes to about 314 rad/s, and a 60 Hz supply to about 377.

Carrying the angular form is worth the trouble because the sine function takes an angle, not a count of cycles. Write the instantaneous value of a sine wave and the argument is an angle in radians; write capacitive or inductive reactance and the factor two pi f turns up in both. Naming that product once keeps the algebra readable and keeps a stray two pi from going missing halfway through a derivation.

One trap is worth knowing about. Strip the unit names away and hertz and radians per second are both reciprocal seconds, so the bare number never tells you which one you are holding. A reactance worked out in hertz where the expression wanted radians per second is out by a factor of about 6.28, and it is the unit label rather than the arithmetic that catches it.