Phasor Representation of a Vector

Phasor Representation of a Vector

• In general, any complex number m can be written as,

m = a + jb = r ∠ θ^o .... (9.8.1)

or m = re^jθ = r (cos θ + j sin θ) .... (9.8.2)

• In equations (9.8.1) and (9.8.2), a and b are the real and imaginary parts of complex number m. The symbol j represents complex operator. Its value is √-1. The magnitude of m is given by,

r = |m| = √a² + b²  .... (9.8.3)

• The phase angle is given by,

θ = tan-1 b / a ….. (9.8.4)

• From above discussion, it is clear that any phasor can be represented in rectangular as well as polar form represented by equations (9.8.1) to (9.8.4). Note that the phasor representation is applicable only to the sinusoidal signals. Any sinusoidal signal can be defined with the help of three parameters namely amplitude, frequency and phase. Let the applied electric field is given by,

E = E_m cos (ωt + 4)

Where         E_m = Amplitude,

ωt = Angular frequency and ϕ = Phase angle

• The complex term E_m e^jϕ 4 is called phasor. Generally it is represented by attaching suffix s to the quantity of concern, such as E_s.

• A phasor may be either scalar or vector.

• Let the vector  is time varying field which varies wit h respect of x, y, z and t. Then the phasor form of  is obtained by dropping the time factor. Let it be s which depends only on x, y and z. Then the two quantities are related to each other by the relation.

Key Point : From equations (9.8.9) and (9.8.10) it is clear that, differentiating and integrating the quantity with respect to time is equivalent to multiplying and dividing the phasor of that quantity by factor jro respectively.

Review Questions

1. What are phasors ? What is their significance ?

2. Write a note on : Phasor, it’s properties and applications.