Question

In the circuit shown in the figure, $V_s = V_m \sin 2t$ and $Z = 1 - j$. The value of $C$ is chosen such that the current $I$ is in phase with $V_s$. The value of $C$ in farad is, 

• A. $1/4 \, \text{F}$
• B. $1/2 \, \text{F}$
• C. $1/8 \, \text{F}$
• D. $1/6 \, \text{F}$

Hint:

When current and voltage are in phase in an AC circuit, the net reactance (or net susceptance) must be zero. Always enforce this condition using admittance for parallel circuits.

Answer 1

The Correct Option is A

Step 1: Identify the operating condition.
The current $I$ is stated to be in phase with the source voltage $V_s$. This condition implies that the overall impedance (or equivalently, the overall admittance) of the circuit must be purely resistive.
Hence, the net imaginary part of the total admittance must be zero.
Step 2: Determine the angular frequency.
Given
\[ V_s = V_m \sin 2t \]
Therefore, the angular frequency is
\[ \omega = 2 \, \text{rad/s} \]
Step 3: Find the admittance of impedance $Z$.
The given impedance is
\[ Z = 1 - j \]
Admittance is the reciprocal of impedance:
\[ Y_Z = \frac{1}{1 - j} \]
\[ Y_Z = \frac{1 + j}{(1)^2 + (1)^2} = \frac{1 + j}{2} \]
\[ Y_Z = 0.5 + j0.5 \]
Step 4: Find the admittance of the capacitor branch.
The admittance of a capacitor is given by
\[ Y_C = j \omega C \]
Substituting $\omega = 2$:
\[ Y_C = j (2C) \]
Step 5: Apply the in-phase condition.
For current to be in phase with voltage, the imaginary part of total admittance must be zero:
\[ \text{Im}(Y_Z + Y_C) = 0 \]
\[ 0.5 - 2C = 0 \]
Step 6: Solve for $C$.
\[ 2C = 0.5 \]
\[ C = 0.25 \, \text{F} \]
Step 7: Conclusion.
The value of capacitance required so that the current is in phase with the source voltage is
\[ \boxed{C = \frac{1}{4} \, \text{F}} \]