In the most of the ac circuits, it is necessary to consider the combined action of several emfs or voltages acting in a series circuit and several currents flowing through the different branches of a parallel circuit.
Let it be required to add two currents given by the equations:
i1 = I1 max sin ωt and i2 = I2 max sin (ω t – ɸ)
The resultant sum may be expressed as:
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Ir = i1 + i2 = I1 max sin ω t + I2 max sin (ω t – ɸ) but it is too awkward and gives no idea of the peak, value and phase angle of the resultant current.
The currents may be added graphically by plotting their curves in the same system of coordinates and then adding the ordinates of i1 and i2 point by point, according to the equation ir = i1 + i2. Evidently this method is also too cumbersome and unwieldy to be practical. This is particularly so when situation arise where more than two sinusoidal quantities are to be added.
A simpler and more direct method consists in adding the sinusoidal quantities as phasors.
Consider the phasors I1 max and I2 max that would generate the two curves i1 and i2 and let them be in a position, as shown in Fig. 3.36 at one particular instant of time. If we now add I1 max and I2 max by completing the parallelogram as shown, the diagonal Ir max when rotated, generate a third sine curve. It remains to be shown that this third sine curve coincides with the waveform of ir obtained by adding i1 and i2 point by point.
Now the vertical component of Ir max is the sum of the vertical components of I1 max and I2 max. Therefore, the waveform of ir is the graph generated by rotating Ir max in counter-clockwise direction.
It follows, therefore, that two or more alternating quantities may be added in the same way as forces are added, namely by constructing parallelograms or closed polygons and either measuring or calculating the lengths of the diagonals or closing sides and the magnitude of the phase angles.
Illustrations:
The way in which the two given currents i1 and i2 can be added by the parallelogram rule of phasor addition is illustrated in Fig. 3.37 where the currents are shown as phasor drawn from the origin O of the system of coordinates. The resultant phasor is the diagonal of the parallelogram formed by the phasors I1 max and I2 max.
This method is more convenient when more than two phasors are to be added, as shown in Fig. 3.38. From the end point of I1 max a phasor is constructed parallel to I2 max of the same magnitude and direction as the latter; then from the end point of I2 max a phasor is constructed parallel to I3 max and so on. Phasor Ir max from the origin of the first phasor (I1 max) to the end point of the last phasor (I5 max) represents the sum of all the phasors.
Phasors may also be subtracted by the above method. For example if phasors I2 max and I3 max are to be subtracted from phasor I1 max each of the two phasors be reversed in direction and then added as explained above [Fig. 3.39]. This time, the phasor drawn from the origin of the first phasor I1 max to the terminal point of the last phasor I3 max gives the difference of phasors I2 max and I3 max from phasor I1 max.
