It has already been pointed out that it is 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 counter-clockwise (It is a standard convention that the phasor is rotated in counter-clockwise direction – a convention that is in harmony with the general use of polar coordinates.) direction with the same angular velocity as that of the sinusoidal quantity. Such a rotating line is called the phasor. (Strictly speaking, sinusoidal quantities are scalar quantities varying periodically with time and, according to the definition of vector quantities, they are not true vectors. Voltage is simply energy or work per coulomb and cannot be classified as a vector. Current is also not a vector quantity because it is merely the flow of electrons through a wire. But keeping in mind that any sinusoidal quantity at a given frequency is completely specified by its amplitude and phase angle, its similarity to a vector quantity is evident, since the amplitude may be considered as the magnitude and the phase angle as the direction of a vector. To account for the difference the term phasor has been adopted, instead of the term vector, for representing graphically the magnitude and phase of a sinusoidal current or voltage.)
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 = Emax and rotating in counter-clockwise direction at an angular velocity ω radians/second about the point O, as shown in Fig. 1. 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 1 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 = Emax 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 counterclockwise direction will represent a sinusoidal quantity (voltage or current) if
- its length is equal to the peak or maximum value of the sinusoidal voltage or current to a suitable scale.
- it is in horizontal position at the instant the alternating quantity (voltage or current) is zero and increasing and
- 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 , 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 behavior 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. 2 is (Φ).
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. 2 (b) current I1 represented by phasor OA leads the current I2 represented by phasor OB by Φ or current I2 lags behind the current I1 by Φ. 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 Φ. The two waves representing these two currents are shown in Fig. 2 (a). If I1 is taken as reference phasor, the two currents can be expressed as
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. But the two quantities passing through zero values at the same instant but rising in opposite directions, as shown in Fig. 4 are said to be in phase opposition, i.e. phase difference is 1800. When the two alternating quantities have a phase difference of 900 or radians they are said to be in quadrature.
Conventions for Drawing Phasor Diagrams
As already mentioned 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:
- 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.
- 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.
- 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.
- 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.
Example 1: Calculate (a) the maximum value and (b) the root mean square values of the following quantities:
(i) 40 sin ωt (ii) B sin (iii) 10 sin ωt – 17.3 cos ωt. Draw the phasors showing the phase difference with respect to A sin
(i) For 40 sin ωt
(a) Maximum value = Coefficient of the sine of the time angle = 40 Ans.
(b ) Ans.
(ii) For B sin
(a) Maximum value = B Ans.
(iii) For 10 sin ωt-17.3 cos ωt, which may be written as
(a ) Maximum value = 20
(b ) Ans.
Phasors showing the phase difference with respect to A sin are shown in Fig. 5.
Example 2. Three sinusoidal alternating current of rms value 5, 7.5, and 10 A are having same frequency with phase angle of 300, -600 and 450.
(i) Find their average values, (ii) Write equations for their instantaneous values, (iii) Draw waveform and phasor diagrams taking first current as the references, (iv) Find their instantaneous value at 100 ms from the original reference.
(i) Average value of first current,
Average value of second current,
Average value of third current,
(ii) Instantaneous value of a sinusoidal current is given by
Thus, instantaneous values of current i1, i2 and i3 are given by
First current is the reference phasor. From the expression of instantaneous values of currents i1, i2 and i3, second current I2 lags behind first current by (600 + 300) i.e. by 900 and third current leads the first current by (450 – 300) i.e. by 150. Waveforms and phasor diagrams are given in Figs. 6(a) and (b) respectively.
(iv) The period of 50 Hz ac quantity is second i.e. it completes one cycle in 20 ms and therefore 5 cycles in 100 ms. Original reference is the starting point required for this purpose. Thus, the instantaneous value of currents i1, i2 and i3 at 100 ms from the reference and given respectively by