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 I_{1}, I_{2},
and I_{3} as depicted in figure-1.

The phasor I_{1} 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.

Therefore, the phasor I_{1} can be represented as,

Similarly, the phasor I_{2} 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 I_{2} can be
represented as,

In the same way, the phasor I_{3}
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 I_{3} can
be represented as,

Equations (1), (2), and (3) are called
the **Rectangular Form Representation**
of the phasors I_{1}, I_{2}, and I_{3}.

# (2). Triangular Form

From the figure-1, it can be observed that

`\I_(1X)=I_1 cos Î¸_1`

`\I_(1Y)=I_1 sinÎ¸_1`

On substituting values of I_{1X}
and I_{1Y} in equation (1), we get,

`\I_1=I_1 cos Î¸_1 + jI_1 sin Î¸_1`

`\∴I_1=I_1 (cos Î¸_1+j sin Î¸_1)…(4)`

Similarly,

`\I_2=I_2 [cos(-Î¸_2 )+j sin(-Î¸_2 )]`

Equations (4), (5), and (6) are called
the **Trigonometric Form Representation**
of the phasors I_{1}, I_{2}, and I_{3}.

# (3). Exponential Form

From trigonometry, we have,

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

`\I_1=I_1 e^(jÎ¸_1 )…(7)`

`\I_2=I_2 e^(-jÎ¸_2 )…(8)`

`\I_3=I_3 e^(jÎ¸_3 )…(9)`

Equations (7), (8), and (9) are called
the **Exponential Form Representation**
of the phasors I_{1}, I_{2}, and I_{3}.

# (4). Polar Form

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

`\I_1=I_1∠Î¸_1…(10)`

`\I_2=I_2∠-Î¸_2…(11)`

`\I_3=I_3∠Î¸_3…(12)`

Equations (10), (11), and (12) are called
the **Phasor Form Representation** of
the phasors I_{1}, I_{2}, and I_{3}.

**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.