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:**

i_{1} = I_{1} max sin ωt and i_{2 }= I_{2} _{max} sin (ω t – ɸ)

**The resultant sum may be expressed as:**

ADVERTISEMENTS:

I_{r} = i_{1} + i_{2} = I_{1} _{max} sin ω t + I_{2} _{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 i_{1 }and i_{2} point by point, according to the equation i_{r} = i_{1} + i_{2}. 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 I_{1} _{max} and I_{2 max }that would generate the two curves i_{1} and i_{2} and let them be in a position, as shown in Fig. 3.36 at one particular instant of time. If we now add I_{1 max} and I_{2} _{max} by completing the parallelogram as shown, the diagonal I_{r} _{max }when rotated, generate a third sine curve. It remains to be shown that this third sine curve coincides with the waveform of i_{r} obtained by adding i_{1} and i_{2} point by point.

Now the vertical component of I_{r max }is the sum of the vertical components of I_{1} _{max} and I_{2} _{max}. Therefore, the waveform of i_{r} is the graph generated by rotating I_{r} _{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 i_{1 }and i_{2} 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 I_{1} max and I_{2} 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 I_{1} _{max} a phasor is constructed parallel to I_{2} _{max} of the same magnitude and direction as the latter; then from the end point of I_{2 max} a phasor is constructed parallel to I_{3} _{max} and so on. Phasor I_{r max} from the origin of the first phasor (I_{1} _{max}) to the end point of the last phasor (I_{5 max}) represents the sum of all the phasors.

Phasors may also be subtracted by the above method. For example if phasors I_{2 max} and I_{3} _{max} are to be subtracted from phasor I_{1} _{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 I_{1} _{max} to the terminal point of the last phasor I_{3 max} gives the difference of phasors I_{2 max} and I_{3 max} from phasor I_{1 max}.