Phasors

The mathematics of adding several alternating quantities that are out of phase with each other (for example, the separate voltage drops across a resistor, an inductor and a capacitor all carrying the same AC current) becomes much simpler if each sinusoidally-varying quantity is represented, not as a plain number, but as a rotating VECTOR of fixed length -- called a phasor.

A phasor representing a quantity like $i=i_0\sin\omega t$ (or $e=e_0\sin\omega t$) is drawn as an arrow of length equal to the quantity's own PEAK value ($i_0$ or $e_0$), anchored at the origin and rotating anticlockwise at a constant angular speed equal to $\omega$, the same angular frequency as the AC itself. At any instant, the arrow makes some angle with a fixed reference axis (conventionally the horizontal), and the INSTANTANEOUS value of the quantity is recovered as the projection of this rotating arrow onto a chosen fixed axis: for a quantity written in sine form, the projection is taken on the vertical (Y) axis; for a quantity written in cosine form, the projection is instead taken on the horizontal (X) axis. Because every phasor in a given circuit rotates at the exact same angular speed $\omega$, the ANGLE between any two phasors stays fixed throughout the motion, equal to the phase difference between the two quantities they repres …