7.3 Representation of AC Current and Voltage by Rotating Vectors — Phasors

In the previous section, we learnt that the current through a resistor is in phase with the ac voltage. But this is not so in the case of an inductor, a capacitor or a combination of these circuit elements. In order to show phase relationship between voltage and current phase relationship between voltage and current in an ac circuit, we use the notion of phasors. The analysis of an ac circuit is facilitated by the use of a phasor diagram. A phasor* is a vector which rotates about the origin with angular speed \(\omega\), as shown in Fig. 7.4. The vertical components of phasors \(\mathbf{V}\) and \(\mathbf{I}\) represent the sinusoidally varying quantities \(v\) and \(i\). The magnitudes of phasors \(\mathbf{V}\) and \(I\) represent the amplitudes or the peak values \(v_m\) and \(i_m\) of these oscillating quantities. Figure 7.4(a) shows the voltage and current phasors and their relationship at time \(t_1\) for the case of an ac source connected to a resistor i.e., corresponding to the circuit shown in Fig. 7.1. The projection of voltage and current phasors on vertical axis, i.e., \(v_m \sin \omega t\) and \(i_m \sin \omega t\), respectively represent the value of voltage and current at that instant. As they rotate with frequency \(\omega\), curves in Fig. 7.4(b) are generated. From Fig. 7.4(a) we see that phasors \(\mathbf{V}\) and \(I\) for the case of a resistor are in the same direction. This is so for all times. This means that the phase angle between the voltage and the current is zero.

*Though voltage and current in ac circuit are represented by phasors – rotating vectors, they are not vectors themselves. They are scalar quantities. It so happens that the amplitudes and phases of harmonically varying scalars combine mathematically in the same way as do the projections of rotating vectors of corresponding magnitudes and directions. The rotating vectors that represent harmonically varying scalar quantities are introduced only to provide us with a simple way of adding these quantities using a rule that we already know.