Question

Under what conditions the
• [(i)] impedance of the circuit is minimum?
• [(ii)] wattless current flows in the circuit?

Hint:

Remember these important AC circuit conditions: \[ X_L=X_C \quad\Longrightarrow\quad \text{Resonance} \] \[ Z=R \quad\Longrightarrow\quad \text{Minimum Impedance} \] \[ \phi=90^\circ \quad\Longrightarrow\quad \text{Wattless Current} \] These results are frequently used in board examinations and numerical problems.

Answer 1

Concept: For a series LCR circuit connected to an alternating voltage source, the impedance is given by \[ Z=\sqrt{R^2+(X_L-X_C)^2} \] where \[ X_L=\omega L \] is the inductive reactance and \[ X_C=\frac{1}{\omega C} \] is the capacitive reactance. The phase difference between voltage and current is \[ \tan\phi=\frac{X_L-X_C}{R} \] The behavior of the circuit depends upon the relative values of \(X_L\) and \(X_C\). (i) Condition for Minimum Impedance

Step 1: Examine the impedance expression The impedance of the series LCR circuit is \[ Z=\sqrt{R^2+(X_L-X_C)^2} \] Since \(R^2\) is constant for a given circuit, the value of \(Z\) will be minimum when \[ (X_L-X_C)^2 \] is minimum. The smallest possible value of a square quantity is zero. Therefore, \[ X_L-X_C=0 \] or \[ X_L=X_C \]

Step 2: Condition for resonance Substituting \[ X_L=\omega L \] and \[ X_C=\frac{1}{\omega C} \] we obtain \[ \omega L=\frac{1}{\omega C} \] Multiplying both sides by \(\omega C\), \[ \omega^2LC=1 \] Hence, \[ \boxed{\omega=\frac{1}{\sqrt{LC}}} \] or \[ \boxed{f=\frac{1}{2\pi\sqrt{LC}}} \] This condition is known as the resonance condition.

Step 3: Minimum impedance at resonance At resonance, \[ X_L=X_C \] Therefore, \[ Z=\sqrt{R^2+0} \] \[ \boxed{Z=R} \] Since \(R\) is the smallest possible value of impedance, the impedance is minimum at resonance. Result for Part (i) The impedance of a series LCR circuit is minimum when \[ \boxed{X_L=X_C} \] i.e., when the circuit is in resonance. Under this condition, \[ \boxed{Z_{\min}=R} \] (ii) Condition for Wattless Current

Step 1: Meaning of wattless current A current is said to be wattless when the average power consumed in the circuit is zero. The average power in an AC circuit is given by \[ P=V_{\rm rms}I_{\rm rms}\cos\phi \] where \(\phi\) is the phase difference between voltage and current. For wattless current, \[ P=0 \] Therefore, \[ \cos\phi=0 \] which gives \[ \phi=90^\circ \] or \[ \phi=\frac{\pi}{2} \]

Step 2: Physical interpretation When the phase difference between voltage and current is \(90^\circ\),
• Voltage and current are completely out of phase.
• Energy is alternately stored and returned to the source.
• No net energy is consumed during one complete cycle. Hence, the average power becomes zero.

Step 3: Circuits exhibiting wattless current A purely inductive circuit has \[ \phi=90^\circ \] and a purely capacitive circuit has \[ \phi=-90^\circ \] In both cases, \[ \cos\phi=0 \] and therefore \[ P=0 \] Thus, purely inductive and purely capacitive circuits draw wattless current. Result for Part (ii) Wattless current flows when \[ \boxed{\phi=90^\circ} \] so that \[ \boxed{\cos\phi=0} \] and \[ \boxed{P=V_{\rm rms}I_{\rm rms}\cos\phi=0} \] This occurs in a purely inductive or purely capacitive AC circuit. Final Answer:
• [(i)] The impedance of a series LCR circuit is minimum when \[ \boxed{X_L=X_C} \] i.e., at resonance, and \[ \boxed{Z_{\min}=R} \]
• [(ii)] Wattless current flows when \[ \boxed{\phi=90^\circ} \] so that \[ \boxed{\cos\phi=0} \] and the average power consumed is zero.