Question

A current element X is connected across an AC source of emf \(V = V_0\ sin\ 2πνt\). It is found that the voltage leads the current in phase by \(\frac{π}{ 2}\) radian. If element X was replaced by element Y, the voltage lags behind the current in phase by \(\frac{π}{ 2}\) radian. 
(I) Identify elements X and Y by drawing phasor diagrams.
(II) Obtain the condition of resonance when both elements X and Y are connected in series to the source and obtain expression for resonant frequency. What is the impedance value in this case?

Answer 1

Phasor Diagrams, Resonance Condition, and Resonant Frequency

(I) Phasor Diagrams

In AC circuits, the relationship between voltage and current can be represented using phasor diagrams:

For Capacitor (X):

• In a capacitor, the voltage leads the current by \( \frac{\pi}{2} \). This is because the current in a capacitor leads the voltage due to the nature of capacitive reactance.
• The phasor diagram shows that the phasor for voltage is ahead of the phasor for current.

For Inductor (Y):

• In an inductor, the voltage lags behind the current by \( \frac{\pi}{2} \). This happens because the voltage across an inductor opposes the change in current (inductive reactance).
• The phasor diagram shows that the phasor for voltage is behind the phasor for current.
  
  

(II) Condition of Resonance and Resonant Frequency

When a capacitor and an inductor are connected in series with a resistor (forming an RLC circuit), resonance occurs under specific conditions. The condition of resonance is when the inductive reactance (\( X_L \)) equals the capacitive reactance (\( X_C \)).

Condition for Resonance:

The resonance condition is given by:

\[ X_L = X_C \Rightarrow \omega L = \frac{1}{\omega C} \]

Solving for \( \omega \), we get:

\[ \omega^2 = \frac{1}{LC} \Rightarrow \omega = \frac{1}{\sqrt{LC}} \]

Resonant Frequency:

The resonant frequency \( f \) is given by:

\[ f = \frac{\omega}{2\pi} = \frac{1}{2\pi \sqrt{LC}} \]

Impedance at Resonance:

At resonance, the net reactance becomes zero because \( X_L = X_C \). Therefore, the total impedance \( Z \) in the circuit is purely resistive:

\[ Z = R \]

At Resonance:

• The impedance \( Z \) is equal to the resistance \( R \) of the circuit.
• The current is maximum and is in phase with the voltage.

Summary:

Resonance in an RLC circuit occurs when the inductive reactance equals the capacitive reactance, leading to maximum current and a purely resistive impedance. The resonant frequency is given by:

\[ f = \frac{1}{2\pi \sqrt{LC}} \]