Phasors

Addition of 2 out-of-phase sinusoidal signals is rather complicated in the time domain. An instance could be the sum of the voltages across a series connection of a resistor and an inductor.

Phasors simplify this analysis by considering the amplitude and phase components of the sine wave only. Moreover, they can be solved using complex algebra or treated vectorially by using a vector diagram.

Consider a vector quantity A in complex plane as shown in the Figure (a). Then

 
Rotating Vector Concept Let θ= ωt, therefore A= Ae^jωt

here ω is the angular velocity and t is the time. Then A is regarded as a vector rotating with an angular velocity ω. Now if 2 phasors are rotating with same velocity as shown in the Figure (b), then their relative positions are unchanged with respect to time and therefore can be added.

Two or more sinusoidal signals can be added mathematically by using phasors if they all have the same angular velocities.

Polar form is also commonly used to represent a phasor and is given by the magnitude (modulus) and phase of the signal that is

                               A =A∠θ

It may be convenient to transform the polar form into cartesian form and vice versa when adding two sinusoidal signals.

Example

Express the following as phasors and write down the corresponding polar and cartesian forms.

 

 

In above examples and all the other instances in thischapter, sinωt are used as a reference for phasor conversion. This means that a cosine signal can be converted to a sine wave before writing the phasor notation. It is valid to use cosωt as reference, but, consistency is needed in the given problem

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