What is a phasor, and how do you find phase difference between two AC quantities?

In one line

A phasor is a rotating line whose length represents the RMS (or peak) value of a sinusoid and whose angle represents its phase, so that phase difference is simply the angle between two phasors.

Explained simply

Picture the hands of a clock. Two hands spinning at the same speed keep a fixed angle between them. Their shadows on a wall go up and down like sine waves, one always a little ahead of the other. In AC circuits all voltages and currents have the same frequency, so we can freeze the picture and just show the arrows and the angle between them.

Technically, a sinusoid \( v = V_m\sin(\omega t + \phi) \) is represented by a phasor of magnitude \( V \) (usually RMS) at angle \( \phi \), written \( V\angle\phi \). Phasors rotate anticlockwise at \( \omega \) rad/s. Phase difference is the difference of the phase angles of two quantities of the same frequency.

How it works

1. Write both quantities in the same form (both sine, both cosine).
2. Read their phase angles \( \phi_1 \) and \( \phi_2 \).
3. Phase difference \( = \phi_1 - \phi_2 \). The one with the larger angle leads; the other lags.
4. On a phasor diagram, the leading phasor is ahead in the anticlockwise direction.
5. In phase means phase difference 0°; in quadrature means 90°; anti-phase means 180°.

Key formula

\( v = V_m\sin(\omega t + \phi) \;\leftrightarrow\; \bar V = V\angle\phi, \quad V = V_m/\sqrt{2} \)

\( V_m \) = peak value (V), \( V \) = RMS magnitude of the phasor (V), \( \phi \) = phase angle (degrees or rad), \( \omega \) = angular frequency (rad/s).

Quick example

\( v_1 = 10\sin(\omega t + 30^\circ) \) and \( v_2 = 10\sin(\omega t - 15^\circ) \). Phase difference \( = 30^\circ - (-15^\circ) = 45^\circ \); \( v_1 \) leads \( v_2 \) by 45°. At 50 Hz, 45° corresponds to a time shift of \( (45/360)\times20 \) ms \( = 2.5 \) ms.

Common mistakes

• Comparing a sine and a cosine directly; first convert using \( \cos\theta = \sin(\theta + 90^\circ) \).
• Comparing quantities of different frequencies; phase difference is defined only for the same frequency.
• Saying the phasor with the smaller angle leads.

How MU asks it

Asked as "Explain phasor representation of an alternating quantity" (5 marks) or as part of a numerical: "Find the phase difference and draw the phasor diagram". For full marks, draw a rotating phasor with its projection forming the sine wave, show leading and lagging phasors with angles, and state the lead/lag clearly in words.