Phasors and the Phasor-Diagram Method

Why phasors are needed. When two sinusoidal quantities of the same frequency -- such as an applied voltage and the current it drives -- are not perfectly in step with each other, describing the gap between them by writing out full trigonometric expressions every time quickly becomes cumbersome. The phasor-diagram method gives a purely geometric shortcut: it represents each alternating quantity by a rotating arrow (a phasor) instead of by an explicit sine function.

What a phasor is. A phasor is a vector-like arrow, drawn on a two-dimensional diagram, whose LENGTH equals the peak value of the quantity it represents (e.g. $v_0$ for a voltage), and which is imagined to rotate anticlockwise about its own tail, at an angular speed equal to the quantity's own angular frequency $\omega$. At any chosen instant, the actual, instantaneous value of the quantity is simply the phasor's PROJECTION onto a fixed reference axis (conventionally the vertical axis, or equivalently, its component along whichever axis the sine is measured from).

Recovering the sine wave. As the phasor sweeps steadily around a full circle, this projection rises smoothly from zero to the peak value, back through zero, down to the negative peak, and back to zero again -- tracing out exactly the familiar sine curve, one full cycle per full rotation of the phasor (the figure for this section shows this correspondence directly, side by side). …