Phasor representation of sinusoidally varying alternating quantities

PHASOR REPRESENTATION OF SINUSOIDALLY VARYINGALTERNATING QUANTITIES

Consider a phasor OP = E_m E_m is the maximum value of alternating voltage, which varies sinusoidally. Let this phasor rotate in the anti-clockwise direction at a speed of a radians/sec.

The phasor and the voltage wave forms are shown in the figures 1.145 (a) and 1.145 (b) respectively.

Any alternating sinusoidal quantity can be represented by a rotating phasor, if it satisfies the following conditions.

(a) The magnitude of rotating phasor should be equal to the maximum value of the quantity.

(b) The rotating phasor should start initially at zero and then move in positive direction (anti- clockwise direction).

(c) The speed of the rotating phasor should be in such a way that during its one revolution the alternating quantity completes one cycle.

Note:

(i) Generally the effective values (RMS values) are used to represent the phasors.

(ii) For sinusoidal alternating quantities, effective value = RMS value = 0.707 × maximum value.

Let us consider the following two sinusoidal equations.

1. e = 200 sin ωt

2. i = 10 sin (ωt + π/3)

It is required to show the above equations as phasors. The following steps are used for this. Only RMS values are considered.

Step 1: Find RMS value for each.

Thus, Irms = 0.707 × 10 = 7.07A

and Erms = 0.707 × 200 = 141.4V

Step 2: Take quantity having the angle as oot along positive x-axis. Here, voltage has got the angle ωt (no plus or no minus angle other than ot). Select a voltage scale and draw a line OA to represent 141.4V.

Step 3: The rms value of current 7.07A. Take a suitable current scale and draw a line OB, at an angle π/3, in anti-clockwise direction, from voltage OA. OB represents current phasor.

Thus the voltage phasor and the current phasor are shown for the given sinusoidally varying voltage and currents.

Note:

1. If both equations are sinusoidally varying with the same angle ot, both phases are shown along positive x-axis. Such quantities attain their maximum (or minimum) values simultaneously and are said to be in phase.

Example, e = E_m sin ωt

I = I_m sin ωt

Phasor representation

OA represents E and OB represents I.

2. The alternating quantities which vary sinusoidally, but having different angles, are said to be out of phase or there is said to be existence of phase difference.

The quantity having angle only ωt is taken as reference. The other quantity is compared with the reference quantity. The quantity with ωt + some angle is said to be leading the other one.

Let e = E_m sin ωt

i = I_m sin (ωt + α)

Here, angle of i is a more than that of e. So we, say that current leads the voltage by angle a. The leading quantity is shown making an angle a with E, in the anticlockwise direction. It is shown below.

An alternating quantity with negative angle (other than wt) is said to be lagging.

Consider the following two equations

e = E_m sin ωt

i = I_m sin (ωt -β)

The current phasor has got an angle of - ẞ other than ot. It means that current lags behind the voltage by B. The phasors are as shown in the figure.

In the above phasor diagram, an arrow is given in anti-clockwise direction. It shows the direction of the rotation of the phasors. Leading quantity is shown on arrow side and lagging quantity on tail side.