Question

Under what conditions the (i) impedance of the circuit is minimum ? (ii) Wattless current flows in the circuit ?

Hint:

Wattless current is component $I_{rms} \sin \phi$ which does no useful electrical work over a full cycle because power consumed is zero.
Real choke coils are used in AC circuits to reduce current without significant energy loss because $R \approx 0$.

Answer 1

Concept:
• Impedance of an LCR series circuit is given by $Z = \sqrt{R^2 + (X_L - X_C)^2}$.

• Average power consumed in an AC circuit is $P_{avg} = V_{rms} I_{rms} \cos \phi$, where $\cos \phi = \frac{R}{Z}$ is the power factor.

• Current is said to be wattless if the power consumed in the AC circuit is zero despite current flowing through it.

Step 1: Condition for Minimum Impedance
The impedance formula is $Z = \sqrt{R^2 + (X_L - X_C)^2}$.
Since $R^2 \ge 0$ and $(X_L - X_C)^2 \ge 0$, the minimum possible value of $Z$ occurs when the term $(X_L - X_C)^2$ becomes zero.
This condition is satisfied when inductive reactance equals capacitive reactance:
\[ X_L = X_C \implies \omega L = \frac{1}{\omega C} \]
This condition is known as electrical resonance.
Under resonance, minimum impedance is equal to resistance:
\[ Z_{min} = R \]

Step 2: Condition for Wattless Current
Wattless current flows when the average power dissipation in the circuit is zero ($P_{avg} = 0$).
The average power dissipated is given by:
\[ P_{avg} = V_{rms} I_{rms} \cos \phi \]
For $P_{avg} = 0$ while $V_{rms} \neq 0$ and $I_{rms} \neq 0$, we must have:
\[ \cos \phi = 0 \implies \phi = \frac{\pi}{2} \text{ or } 90^\circ \]
From $\tan \phi = \frac{X_L - X_C}{R}$, $\phi = 90^\circ$ requires resistance $R = 0$.
Therefore, wattless current flows in a purely inductive or purely capacitive circuit containing zero resistance ($R = 0$).

Step 3: Conclusion
(i) Impedance is minimum at resonance when $X_L = X_C$ or $\omega = \frac{1}{\sqrt{LC}}$.
(ii) Wattless current flows when the circuit resistance is zero ($R = 0$), making the phase difference $\phi = \frac{\pi}{2}$.