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, phasor diagrams illustrate the relationship between voltage and current.

For Capacitor (X):

• In a capacitor, voltage leads current by \( \frac{\pi}{2} \). This is due to capacitive reactance, where current leads voltage.
• The phasor diagram depicts the voltage phasor preceding the current phasor.

For Inductor (Y):

• In an inductor, voltage lags current by \( \frac{\pi}{2} \). This occurs because inductive reactance opposes changes in current, causing voltage to lead.
• The phasor diagram shows the voltage phasor trailing the current phasor.
  
  

(II) Condition of Resonance and Resonant Frequency

Resonance in a series RLC circuit occurs when inductive reactance (\( X_L \)) equals capacitive reactance (\( X_C \)).

Condition for Resonance:

The resonance condition is defined as:

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

Solving for \( \omega \) yields:

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

Resonant Frequency:

The resonant frequency \( f \) is calculated as:

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

Impedance at Resonance:

At resonance, \( X_L = X_C \), resulting in zero net reactance. Consequently, the total impedance \( Z \) becomes purely resistive:

\[ Z = R \]

At Resonance:

• The impedance \( Z \) equals the circuit's resistance \( R \).
• Current reaches its maximum value and is in phase with the voltage.

Summary:

Resonance in an RLC circuit is achieved when inductive reactance matches capacitive reactance. This condition leads to maximum current and a purely resistive impedance. The resonant frequency is given by:

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