Question

A sinusoidal voltage is applied to an electric circuit containing a circuit element `X' in which the current leads the voltage by \(\pi/2\).
(a) Identify the circuit element `X' in the circuit:
(b) Write the formula for its reactance:
(c) Show graphically the variation of this reactance with frequency of ac voltage:
To illustrate the relationship graphically:

Hint:

Remember, capacitors store electrical energy temporarily and can affect circuits differently based on whether the source is AC or DC, primarily due to their frequency-dependent reactance.

Answer 1

(a) Identify the circuit element ‘X’ in the circuit:

The circuit element ‘X’ where the current leads the voltage by \( \frac{\pi}{2} \) is the capacitor. In capacitive circuits, the charging current for the capacitor leads the voltage across the capacitor.

(b) Write the formula for its reactance:

The reactance \( X_C \) of a capacitor is given by:

\[ X_C = \frac{1}{\omega C} \]

where \( \omega \) is the angular frequency (\( \omega = 2 \pi f \), with \( f \) being the frequency of the AC supply), and \( C \) is the capacitance in farads.

(c) Show graphically the variation of this reactance with frequency of AC voltage:

The capacitive reactance \( X_C \), for a capacitor in an AC circuit is defined by the formula:

\[ X_C = \frac{1}{\omega C} \]

where \( \omega = 2 \pi f \) represents the angular frequency, and \( C \) is the capacitance. This formula shows that \( X_C \) is inversely proportional to the frequency \( f \).

Graphical Representation:

To illustrate the relationship graphically:

1. Axes Setup: Plot frequency (\( f \)) on the horizontal axis and capacitive reactance (\( X_C \)) on the vertical axis.
2. Curve Behavior: The graph will display a hyperbolic decay as \( f \) increases. At very low frequencies, the reactance is very high, representing nearly an open circuit. As \( f \) increases, \( X_C \) decreases, suggesting that the capacitor offers less impedance to higher frequency AC signals.
3. Key Points:
   • At \( f = 0 \, Hz} \), \( X_C \) approaches infinity, which can be represented by the curve starting from the top of the y-axis.
   • As \( f \) increases, \( X_C \) decreases sharply initially and then more gradually, asymptotically approaching zero but never actually reaching it.

(d) Explain the behaviour of this element when it is used in:

(i) an AC circuit:

In an AC circuit, a capacitor impedes the flow of the current depending on the frequency of the AC supply. Higher frequencies reduce the reactance, allowing more current to pass through, demonstrating a characteristic called capacitive reactance.

(ii) a DC circuit:

In a DC circuit, a capacitor initially conducts as it charges, but once fully charged, it acts as an open circuit. Thus, after the initial charge period, no current flows through the capacitor in a steady-state DC circuit.