Phasor Representation of Alternating Quantities | Electrical Engineering

In this article we will discuss about: 1. Introduction to Alternating Quantities 2. Phasor Diagram Using RMS Values 3. Conventions.

Introduction to Alternating Quantities:

Assumed that alternating voltages and currents follow sine law and generators are designed to give emfs having sine waveform. The above said assumption makes the calculations simple. The method of representing alternating quantities by waveform or by the equations giving instantaneous values is quite cumbersome.

For solution of ac problems it is advantageous to represent a sinusoidal quantity (voltage or current) by a line of definite length rotating in counter­clockwise direction with the same angular velocity as that of the sinusoidal quantity. Such a rotating line is called the phasor.

Consider a line OA (or phasor as it is called) representing to scale the maximum value of an alternating quantity, say emf i.e., OA = E_max and rotating in counter-clockwise direction at an angular velocity ω radians/second about the point O, as shown in Fig. 3.30. An arrow head is put at the outer end of the phasor, partly to indicate which end is assumed to move and partly to indicate the precise length of the phasor when two or more phasors happen to coincide.

Figure 3.30 shows OA when it has rotated through an angle θ, being equal to ωt, from the position occupied when the emf was passing through its zero value. The projection of OA on Y-axis, OB = OA sin θ = E_max sin ωt = e, the value of the emf at that instant.

Thus the projection of OA on the vertical axis represents to scale the instantaneous value of emf.

It will be seen that the phasor OA rotating in counter-clockwise direction will represent a sinusoidal quantity (voltage or current) if:

(i) Its length is equal to the peak or maximum value of the sinusoidal voltage or current to a suitable scale.

(ii) It is in horizontal position at the instant the alternating quantity (voltage or current) is zero and increasing, and

(iii) Its angular velocity is such that it completes one revolution in the same time as taken by the alternating quantity (voltage or current) to complete one cycle.

Phasor Diagram Using RMS Values:

Since there is definite relation between maximum value and rms value (E_max = n√2 E_rms), the length of phasor OA can be taken equal to rms value if desired. But it should be noted that in such cases, the projection of the rotating phasor on the vertical axis will not give the instantaneous value of that alternating quantity.

The phasor diagram drawn in rms values of the alternating quantities helps in understanding the behaviour of the ac machines under different loading conditions.

Phase and Phase Angle:

By phase of an alternating current is meant the fraction of the time period of that alternating current that has elapsed since the current last passed through the zero position of reference. The phase angle of any quantity means the angle the phasor representing the quantity makes with the reference line (which is taken to be at zero degrees or radians). For example the phase angle of current I₂, in Fig. 3.31 is (-ɸ).

Phase Difference of Current or Voltage:

When two alternating quantities, say, two emfs, or two currents or one voltage and one current are considered simultaneously, the frequency being the same, they may not pass through a particular point at the same instant. One may pass through its maximum value at the instant when the other passes through the value other than its maximum one. These two quantities are said to have a phase difference. Phase difference is always given either in degrees or in radians.

The phase difference is measured by the angular distance between the points where, the two curves cross the base or reference line in the same direction.

The quantity ahead in phase is said to lead the other quantity while the second quantity is said to lag behind the first one. In Fig. 3.31 (b) current I₁ represented by phasor OA leads the current I₂, represented by phasor OB by ɸ or current I₂, lags behind the current I₁ by ɸ. The leading current I₁ goes through its zero and maximum values first and the current I₂ goes through its zero and maximum values after time angle ɸ. The two waves representing these two currents are shown in Fig. 3.31 (a).

If I₁ is taken as reference phasor, the two currents can be expressed as:

i₁ = I_{1 max} sin ωt

And i₂ = I₂ sin (ωt – ɸ)

The two quantities are said to be in phase with each other if they pass through zero values at the same instant and rise in the same direction, as shown in Fig. 3.32.

But the two quantities passing through zero values at the same instant but rising in opposite directions, as shown in Fig. 3.33 are said to be in phase opposition, i.e., phase difference is 180°. When the two alternating quantities have a phase difference of 90° or π/2 radians they are said to be in quadrature.

Conventions for Drawing Phasor Diagrams:

The alternating quantities (voltages and currents) in practice are represented by straight lines having definite direction and length. Such lines are called the phasors and the diagrams in which phasors represent currents, voltages and their phase difference are known as phasor diagrams.

Though phasor diagrams can be drawn to represent either maximum or effective values of voltages and currents but since effective values are of much more importance, phasor diagrams are mostly drawn to represent effective values.

In order to achieve consistent and accurate results it is essential to follow certain conventions.

Some of the common conventions in this regard are enlisted below:

1. Counter-clockwise direction of rotation of phasors is usually taken as positive direction of rotation of phasors i.e., a phasor rotated in a counter-clockwise direction from a given phasor is said to lead the given phasor while a phasor rotated in clockwise direction is said to lag the given phasor.

2. For series circuits, in which the current is common to all parts of the circuit, the current phasor is usually taken as reference phasor for other phasors in the same diagram and drawn on horizontal line.

3. In parallel circuits in which the voltage is common to all branches of the circuit, the voltage phasor is usually taken as reference phasor and drawn on the horizontal line. Other phasors are referred to the common voltage phasor.

4. It is not necessary that current and voltage phasors are drawn to the same scale; in fact it is often desirable to draw the current phasor to a larger scale than the voltage phasor when the values of currents being represented are small. However, if several voltage phasors are to be used in the same phasor diagram, they should all be drawn to the same scale. Likewise all current phasors in the same diagram should be drawn to the same scale.