Question

In a series LCR circuit, establish the conditions under which:
[(i)] impedance is minimum and
[(ii)] wattless current flows in the circuit.

Hint:

- Impedance is minimum when the inductive and capacitive reactances cancel each other out at resonance. - Wattless current flows when the phase difference between voltage and current is \( \frac{\pi}{2} \), which happens at resonance.

Answer 1

(i) Impedance is minimum:
 The total impedance \( Z_{\text{total}} \) of a series LCR circuit is given by: \[ Z_{\text{total}} = \sqrt{R^2 + \left(X_C - X_L \right)^2} \] where:
\( R \) is the resistance,
\( X_C = \frac{1}{\omega C} \) is the capacitive reactance,
\( X_L = \omega L \) is the inductive reactance,
\( \omega = 2\pi f \) is the angular frequency.
For impedance to be minimum, the capacitive reactance \( X_C \) and inductive reactance \( X_L \) must cancel each other out. This occurs when:
\[ X_C = X_L \] Thus, the condition for minimum impedance is when:
\[ \frac{1}{\omega C} = \omega L \quad \Rightarrow \quad \omega^2 = \frac{1}{LC} \] or \[ \omega = \frac{1}{\sqrt{LC}} \] At this frequency, the impedance becomes:
\[ Z_{\text{total}} = R \] Hence, the impedance is minimum at the resonant frequency \( f_0 = \frac{1}{2\pi\sqrt{LC}} \). 

(ii) Wattless current flows in the circuit:
For wattless current, the power consumed in the AC circuit is zero. The average power \( P \) in an AC circuit over a cycle is given by:
\[ P = VI \cos \phi \] where \( \phi \) is the phase difference between the voltage and current. For wattless current, \( P = 0 \), which occurs when: \[ \cos \phi = 0 \] Since \( V \neq 0 \) and \( I \neq 0 \), we must have: \[ \phi = \frac{\pi}{2} \] Thus, wattless current flows when the phase difference between the voltage and current is \( \frac{\pi}{2} \), which happens at resonance when \( X_C = X_L \). 

 Correct Answer:} 
For impedance to be minimum: \( X_C = X_L \). 
For wattless current to flow: \( \phi = \frac{\pi}{2} \), which occurs at resonance.