Phasors

As noted earlier in this chapter, the addition of sinusoidal voltages and currents will frequently be required in the analysis of ac circuits. One lengthy but valid method of performing this operation is to place both sinusoidal waveforms on the same set of axes and add algebraically the magnitudes of each at every point along the abscissa, as shown for $c = a + b$ in [Fig. 1]. This, however, can be a long and tedious process with limited accuracy. A shorter method uses the rotating radius vector shown in [Fig. 2], During its rotational development of the sine wave, the phasor will, at the instant $t = 0$, have the positions shown in [Fig. 3(a)] for each waveform in [Fig. 3(b)]. Note in [Fig. 3(b)] that $v_2$ passes through the horizontal axis at $t = 0 s$, requiring that the radius vector in [Fig. 3(a)] be on the horizontal axis to ensure a vertical projection of zero volts at $t = 0 s$. Its length in [Fig. 3(a)] is equal to the peak value of the sinusoid as required by the radius vector in [Fig. 2]. The other sinusoid has passed through $90^\circ$ of its rotation by the time $t = 0s$ is reached and therefore has its maximum vertical projection as shown in [Fig. 3(a)].
Since the vertical projection is a maximum, the peak value of the sinusoid that it will generate is also attained at $t = 0 s$, as shown in [Fig. 3(b)]. Note also that $v_T =v_1$ at $t =0 s$ since $v_2 = 0 V $ at this instant. It can be shown see [Fig. 3(a)] using the vector algebra In other words, if we convert $v_1$ and $v_2$ to the phasor form using and add them using complex number algebra, we can find the phasor form for $v_T$ with very little difficulty. It can then be converted to the time domain and plotted on the same set of axes, as shown in [Fig. 3(b)]. [Fig. 3(a)], showing the magnitudes and relative positions of the various phasors, is called a phasor diagram. It is actually a "snapshot" of the rotating radius vectors at $t = 0 s$. In the future, therefore, if the addition of two sinusoids is required, they should first be converted to the phasor domain and the sum found using complex algebra. The result can then be converted to the time domain. The case of two sinusoidal functions having phase angles different from 0 and 90 appears in [Fig. 4]. Note again that the vertical height of the functions in [Fig. 4(b)] at $t = 0 s$ is determined by the rotational positions of the radius vectors in [Fig. 4(a)]. Since the rms, rather than the peak, values are used almost exclusively in the analysis of ac circuits, the phasor will now be redefined for the purposes of practicality and uniformity as having a magnitude equal to the rms value of the sine wave it represents. The angle associated with the phasor will remain as previously described as the phase angle.
In general, for all of the analyses to follow, the phasor form of a sinusoidal voltage or current will be where $V$ and $I$ are rms values and v is the phase angle. It should be pointed out that in phasor notation, the sine wave is always the reference, and the frequency is not represented.