Phasor can be used to express sinusoidal easily, which is convenient method to do rather than sine or cosine functions.

Phasors is a complex number that represents the amplitude and phase of a sinusoidal.

A complex number

*z*can be expressed in rectangular form as
This time,

*x*and*y*do not refer to a coordinat or location as in analytical of two-dimensional vector but rather the real and imaginary parts of*z*in the complex plane. Though, there will be still an act to manipulate complex number with two-dimensional vector analysis method.
The complex number

*z*can also be represented in polar or exponential form like(4) |

**Addition :**

(6h) |

One time to look at Equations.(11) and (12) is to notice the plot of the

*sinor***V***e*^{jωt}=*V*_{m}*e*^{j(ωt + ø)}on the complex plane. As time increase, the sinor move rotational on a circle of radius*V*with an angular velocity_{m}*ω*in counter clockwise direction, as drawn in Fig.2(a). We may assume*v(t)*as the projection of the sinor**V***e*^{jωt}on the real axis, as drawn in Fig.2(b).
The value of sinor at time t = 0 is the phasor

Equation.(11) refers that in order to obtain sinusoid with a given phasor V, multiply the phasor with the time factor

**V**of the sinusoidal*v(t)*. The sinor can be assumed as a rotating phasor. Hence, whenever a sinusoidal is developed as a phasor, the term*e*^{jωt}is implicitly present. We need to keep in mind on the frequency*ω*of the phasor, otherwise a huge mistake can be made.Figure 2 Representation of Vejωt : (a) sinor rotating counterclockwise, (b) its projection on the real axis, as a time function |

*e*^{jωt}and take the real part. A phasor can be expressed in rectangular form, polar form, and exponential form. Phasor behaves as a vector and is written with bold font because it has magnitude and phase (direction). For example,**V**=*V*∠∅ and_{m}**I**=*I*∠∅ are graphically drawn in Figure.(3). The name of the graphical drawing is known as_{m}*phasor diagram*.
Equations.(9) through (11) show that we may first express the sinusoid in the cosine form so that the sinusoid can be developed as the real part of a complex number to get the phasor corresponding to a sinusoidal. After that, we take the time factor ejwt, and whatever is left is the phasor corresponding to the sinusoidal. This transformation method is written below :

(13) |

Figure 3. Phasor diagram of V = Vm∠∅ and I = Im∠∅ |

*v(t)*=

*V*cos(

_{m}*ωt*+ ∅), we get the corresponding phasor as

**V**=

*V*∠∅. Equation.(13) is also presented in Table.(1), where the sine function is considered in addition to the cosine function. From Equation.(13), we assume that in order to get the representation of phasor from a sinusoid, we can express it in cosine form and take the magnitude as the phasor and the argument as

_{m}*ωt*plus the phase of the phasor.

Table 1 |

*e*

^{jωt}is surpressed, and the frequency is not explicitly shown in the phasor domain representation because

*ω*is constant. Nevertheless, the response depends on

*ω*. For this reason, the phasor domain is also known as the

*frequency domain*.

From Equations.(11) and (12),

*v(t)*= Re(**V***e*^{jωt}) =*V*cos(_{m}*ωt*+ ∅), hence
The differences between

*v(t)*and**V**should be defined :*v(t)*is the*instantaneous or time domain*representation, while**V**is the*frequency or phasor domain*representation.*v(t)*is time dependent, but**V**is not.*v(t)*is always real with no complex term, but**V**is generally complex.

From above, we conclude that phasors analysis applies only when frequency is constant;it applies in manipulating two or more sinusoidal signals only if they are of the same frequency.

## Example of Phasor

For better understanding of phasors, let us review the examples below :

**1. Evaluate these complex numbers :**

(a) Using polar to rectangular transformation,

**2. Transform these sinusoids to phasors :**

(a)

*i*= 6 cos(50*t*- 40^{o}) has**3. Find the sinusoids represented by these phasors :**

(a) We rewrite it to

**4. Given**

*i*= 4 cos(_{1}(t)*ω**t*+ 30^{o}) A and*i*= 5 sin(_{2}(t)*ω**t*- 20^{o}) A, find their sum.__Solution :__

Now we see the important application of this term : to find the sum of the same frequency. Current

*i*is in the standard form. Its phasor is_{1}(t)
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