What is phasor representation of AC?

In this article, we shall see the phasor representation of alternating quantities like alternating current, alternating voltage, etc.

The phasor representation of AC quantities greatly simplifies the calculations and interpretations of calculations of alternating quantities.

Representation of Phasors

A phasor of ac quantity can be expressed in the following four forms:

• Rectangular Form
• Trigonometric Form
• Exponential Form
• Polar Form

Let us discuss each form of phasor representation of AC in detail.

(1). Rectangular Form

Let us consider three phasors I₁, I₂, and I₃ as depicted in figure-1.

The phasor I₁ can be resolved into its components along X and Y axes. Hence, it may be represented by two phasors namely I_1X and I_1Y along X-axis and Y-axis respectively. We know, the Y-axis is 90° ahead of the X-axis. Hence, we can represent Y-axis as,

Y = j X

Where the operator j is used to specify a shift of 90° in the anticlockwise direction, i.e.

`\j=1∠90°`

Therefore, the phasor I₁ can be represented as,

`\I_1=I_(1X)+jI_(1Y)…(1)`

Similarly, the phasor I₂ can be resolved into two component phasors namely I_2X and I_2Y. Where I_2X is along X-axis and I_2Y is along –Y-axis. It can be seen that the –Y-axis is 270° (i.e. 90° + 90° + 90°) ahead of the X-axis, i.e.

`\-Y=j.j.j X`

`\-Y=j^2.j X`

`\∵j^2=-1`

`\∴-Y=-jX`

Hence, the phasor I₂ can be represented as,

`\I_2=I_(2X)-jI_(2Y)…(2)`

In the same way, the phasor I₃ can be resolved along –X-axis and Y-axis as I_3X and I_3Y. Where, -X-axis is 180°, i.e. (90° + 90°) ahead of the X-axis. Thus,

`\-X=j.jX=j^2 X`

`\⇒j^2=-1`

Therefore, the phasor I₃ can be represented as,

`\I_3=-I_(3X)+jI_(3Y)…(3)`

Equations (1), (2), and (3) are called the Rectangular Form Representation of the phasors I₁, I₂, and I₃.

(2). Triangular Form

From the figure-1, it can be observed that

`\I_(1X)=I_1  cos⁡ θ_1`

`\I_(1Y)=I_1  sin⁡θ_1`

On substituting values of I_1X and I_1Y in equation (1), we get,

`\I_1=I_1  cos⁡ θ_1 + jI_1  sin θ_1`

`\∴I_1=I_1 (cos⁡ θ_1+j sin⁡ θ_1)…(4)`

Similarly,

`\I_2=I_2 [cos⁡(-θ_2 )+j sin⁡(-θ_2 )]`

`\∴I_2=I_2 (cos⁡ θ_2 - j sin⁡ θ_2)…(5)`

And

`\I_3=I_3 (cos⁡ θ_3 + j sin⁡ θ_3)…(6)`

Equations (4), (5), and (6) are called the Trigonometric Form Representation of the phasors I₁, I₂, and I₃.

(3). Exponential Form

From trigonometry, we have,

`\e^(±jθ)=cos⁡θ±sin⁡θ`

Therefore, equations, (4), (5), and (6) can be expressed as,

`\I_1=I_1 e^(jθ_1 )…(7)`

`\I_2=I_2 e^(-jθ_2 )…(8)`

`\I_3=I_3 e^(jθ_3 )…(9)`

Equations (7), (8), and (9) are called the Exponential Form Representation of the phasors I₁, I₂, and I₃.

(4). Polar Form

We can write equations (7), (8), and (9) in their more simplified forms as follows,

`\I_1=I_1∠θ_1…(10)`

`\I_2=I_2∠-θ_2…(11)`

`\I_3=I_3∠θ_3…(12)`

Equations (10), (11), and (12) are called the Phasor Form Representation of the phasors I₁, I₂, and I₃.

Important – Out of the above-mentioned four representations of phasors, the rectangular form representation and polar form representation are the most extensively used representations. The rectangular form representation of phasors is useful in the addition and subtraction of phasors, and the polar form is used in the multiplication and division of phasors.