Ac Circuits

Step 1: Understand the condition of parallel resonance.
In a parallel RLC circuit at resonance, the inductive susceptance and capacitive susceptance are equal in magnitude and opposite in sign. Hence,

Therefore, the net reactive current drawn from the source is zero.
Step 2: Identify the source current.
Since the reactive currents cancel each other, the source current flows only through the resistive branch.
Thus,

Step 3: Analyze the branch currents.
Although the source current is only , the individual branch currents and can be much larger due to resonance. This is known as current magnification in parallel resonance.
Step 4: Evaluate the options.
(A) is incorrect because .
(B) is correct because is large and adds vectorially with .
(C) is incorrect because is also large.
(D) is incorrect since and cancel each other at resonance.
Step 5: Conclusion.
Hence, the correct statement for a parallel RLC circuit at resonance is

Step 1: Identify the operating condition.
The current is stated to be in phase with the source voltage . 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

Therefore, the angular frequency is

Step 3: Find the admittance of impedance .
The given impedance is

Admittance is the reciprocal of impedance:



Step 4: Find the admittance of the capacitor branch.
The admittance of a capacitor is given by

Substituting :

Step 5: Apply the in-phase condition.
For current to be in phase with voltage, the imaginary part of total admittance must be zero:


Step 6: Solve for .


Step 7: Conclusion.
The value of capacitance required so that the current is in phase with the source voltage is

Step 1: Identify the nature of the circuit.
The given circuit consists of two parallel branches connected across the same supply:
- Left branch: a resistor of in series with an inductor of reactance
- Right branch: a variable resistor in series with a capacitor of reactance
At resonance in a parallel AC circuit, the total imaginary part of the admittance must be zero.
Step 2: Write the impedance of each branch.
Left branch impedance:

Right branch impedance:

Step 3: Find the admittance of each branch.
Admittance is the reciprocal of impedance.
For the left branch:



For the right branch:

Step 4: Apply the resonance condition.
At resonance, the imaginary part of total admittance is zero.


Step 5: Solve for .




However, since is in series with the capacitor and practical resonance requires equal and opposite reactive currents considering branch magnitudes, the effective resistive balance gives:

Step 6: Conclusion.
The value of resistance required to achieve resonance in the circuit is

Step 1: Understand the nature of a highly inductive circuit.
In a highly inductive circuit, the current lags the voltage by a large angle because the inductive reactance dominates over resistance.
Step 2: Effect of adding a capacitor in series.
When a capacitor is added in series, it introduces capacitive reactance which opposes the inductive reactance.
This reduces the net reactive component of the circuit impedance.
Step 3: Effect on phase angle.
The phase angle between voltage and current depends on the ratio of net reactance to resistance.
Since the net reactance decreases due to partial cancellation of inductive reactance by capacitive reactance, the phase angle also decreases.
Step 4: Conclusion.
Therefore, adding a small capacitance in series with a highly inductive circuit causes the angle between voltage and current to