Question

Differentiate between inductive reactance, capacitive reactance and impedance of an ac circuit.
An ideal inductor and an ideal capacitor are connected in series across an ac voltage. Plot a graph showing variation of net reactance of the circuit with frequency of the applied ac voltage.

Hint:

In a pure LC circuit, at resonance, the impedance becomes zero, ideally leading to infinite current.

Answer 1

Step 1: Basic idea.
In an AC circuit, different components oppose the flow of current in different ways. Reactance depends on frequency, while impedance represents the total opposition offered by the circuit.
Step 2: Differentiation.

(a) Inductive Reactance \(X_L\):
Opposition offered by an inductor to AC.
\[ X_L = \omega L = 2\pi f L \]
It increases linearly with frequency.
(b) Capacitive Reactance \(X_C\):
Opposition offered by a capacitor to AC.
\[ X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} \]
It decreases as frequency increases.
(c) Impedance \(Z\):
Total effective opposition in an RLC series circuit.
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
It combines resistance and net reactance.
Step 3: Net Reactance in LC Circuit.
For a series LC circuit: \[ X = |X_L - X_C| \]
At resonance frequency \(f_r\):
\[ X_L = X_C \]
Net reactance becomes zero.
Below \(f_r\): circuit behaves capacitive.
Above \(f_r\): circuit behaves inductive.
Final Answer:
Inductive reactance \(= 2\pi f L\)
Capacitive reactance \(= \frac{1}{2\pi f C}\)
Impedance \(= \sqrt{R^2 + (X_L - X_C)^2}\)