Phasors

13.4 Phasors

The study of AC circuits is greatly simplified if the alternating current and the alternating emf are represented as rotating vectors, with the angle between them equal to the phase difference between the current and the emf. Such rotating vectors are called phasors.

Definition

A rotating vector that represents a quantity varying sinusoidally with time is called a phasor, and the diagram representing it is called a phasor diagram.

How a Phasor Diagram Is Drawn

• The phasors for the alternating emf and the alternating current are inclined to the horizontal axis at angles $\omega t$ and $\omega t + \alpha$ respectively, and rotate in the anticlockwise direction.
• The length of the arrow represents the maximum (peak) value of the quantity, $i_0$ or $e_0$.
• The projection of the phasor on a fixed axis gives the instantaneous value of the alternating current or emf.
• In the sine form, $i = i_0\sin\omega t$ or $e = e_0\sin\omega t$, the projection is taken on the Y-axis.
• In the cosine form, $i = i_0\cos\omega t$ or $e = e_0\cos\omega t$, the projection is taken on the X-axis.

Why Phasors Are Used

Representing the harmonically varying quantities as rotating vectors enables us to use the laws of vector addition for adding these quantities. This is exactly what makes the series LCR analysis possible: the three voltages, which differ in phase, can be combined as vectors instead of being added algebraically.

Standard Results

• Resistor: current phasor and emf phasor point in the same direction — phase angle zero.
• Pure inductor: the current phasor is turned 90° clockwise from the emf phasor — current lags.
• Ideal capacitor: the current phasor is turned 90° anticlockwise from the emf phasor — current leads.
• Series LCR: $e_R$ along the current, $e_L$ along $+$Y, $e_C$ along $-$Y; their resultant is the applied emf.