Question

(a) Does the phasor diagram of Fig. 31-26 correspond to an alternating emf source connected to a resistor, a capacitor, or an inductor? (b) If the angular speed of the phasors is increased, does the length of the current phasor increase or decrease when the scale of the diagram is maintained?

Short answer

1. The phasor diagram of Fig.31-26 corresponds to an alternating emf source connected to an inductor.
2. The length of the current phasor decreases if the angular speed of the phasors is increased when the scale of the diagram is maintained.

Step-by-step solution

The given data

Figure 31-26 shows a phasor diagram.

Understanding the concept of the phasor diagram

We can predict whether the current lead or lag the emf by observing the given phasor diagram. Phasor Diagrams are a graphical way of representing the magnitude and directional relationshipbetween two or more alternating quantities. Then inserting the formula for inductive reactance into the formula for inductive current, we can find whether the length of the current phasor increases or decreases if the angular speed of the phasor is increased when the scale of the diagram is maintained.

Formulae:

The inductive reactance of the inductor,${\mathit{X}}_{\mathbf{L}}\mathbf{=}{\mathit{\omega }}_{\mathbf{d}}\mathit{L}$ . . . (i)

The current equation for an inductor from Ohm’s law, ${\mathit{I}}_{\mathbf{L}}\mathbf{=}\frac{{\mathbf{V}}_{\mathbf{L}}}{{\mathbf{X}}_{\mathbf{L}}}$. . .(ii)

a) Calculation of the type of phasor diagram

From the given figure we can conclude that the current is lagging behind the emf.

Hence, the phasor diagram of Fig.31-26 corresponds to an alternating emf source connected to an inductor.

b) Calculation of the length of the current phasor

The current through the inductor can be given using equation (i) in equation (ii) as follows:

$I_L=\frac{V_L}{{\omega }_{d}L}$

From this, we can conclude that the length of the current phasor is decreased if the angular speed of the phasors is increased when the scale of the diagram is maintained.