For which of the following circuit elements will the phase angle ϕ always be positive ?

Concept:

Reactance is the resistance that is offered by a capacitor or an inductor when a.c. (alternating current) flows through a capacitor or inductor respectively.

The resistance offered by an inductor when an alternating current flows through it is called inductive reactance (XL) and is given by:

\(X_L=ω L\)

With ω = 2πf, the inductive reactance becomes:

\(X_L=2\pi f L\)   ---(1)

The resistance offered by a capacitor when an alternating current flows through it is called capacitive reactance (XC) and is given by

\(X_C=\frac{1}{ω C}=\frac{1}{2 π f C}\)   ---(2)

For a series circuit, with R, L, C, the net impedance is given as:

\(Z = R + jωL + \frac{1}{j\omega C}\)

Using Equations (1) and (2), we get:

\(Z = R + jX_L-jX_C\)

\(Z = R + j(X_L-X_C)\)

And the phase angle (θ) is defined as:

\(\rm \theta = {\tan ^{ - 1}}\;\frac{{Net\;Reactance\;Part\;of\;Impedance\;}}{{Net\;Resistance\;Part\;of\;Impedance}} \)

Net Reactance is the imaginary part, i.e. X_L - X_C

Observation: For the phase angle to be positive, the net reactance, XL - XC will have to be positive.

Analysis:

Option (1):

The net impedance will be:

\(Z = R + j(X_L-X_C)\)

For this circuit configuration, the phase angle cannot always be positive, as it depends on both X_L and X_C. If X_C > X_L, the net reactance part will be negative.

Option (2):

The net impedance will be:

Z = R + jX_L - jX_C = 0 + j(0) - jX_C

Z = -jX_C

The net reactance part is negative here, so the phase angle will not be positive.

Option (3):

The net impedance will be:

\(Z = R + j(0-X_C)\)

\(Z = R - jX_C\)

The net reactance is -jX_C, which is negative. ∴ The phase angle will also be negative.

Option (4):

The net impedance will be:

\(Z = R + j(X_L-0)\)

Z = R + jX_L

For this circuit configuration, we observe that the net reactance is jX_L, which will always be positive. This will result in a phase angle to be always positive as well.