The impedance of a series LCR circuit is –

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$\sqrt{X_{L}^{2} - X_{C}^{2} + R^{2}}$

$\sqrt{\frac{1}{X_{C}^{2}} + \frac{1}{X_{L}^{2}} + R^{2}}$

$\sqrt{R^{2} + {(X_{L} - X_{C})}^{2}}$

R + XL + XC

answer is D.

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Detailed Solution

In a series LCR circuit, the total opposition to the flow of alternating current is known as the impedance of the LCR circuit. The impedance, denoted as Z, is calculated using the following formula:

Impedance, Z = √(R² + (X_L - X_C)²)

Here are the terms involved in calculating the impedance of the LCR circuit:

R: Represents the resistance in the circuit.
X_L: Refers to the inductive reactance, calculated using the formula X_L = ωL, where ω is the angular frequency and L is the inductance.
X_C: Represents the capacitive reactance, determined by X_C = 1 / (ωC), where C is the capacitance.

The difference between inductive reactance (X_L) and capacitive reactance (X_C) determines the net reactance in the circuit. The impedance of the LCR circuit is a combination of resistance and this net reactance.

Key Points About the Impedance of LCR Circuit

The impedance of the LCR circuit depends on the resistance, inductance, and capacitance values.
At resonance (when X_L = X_C), the impedance is minimum and equals the resistance R.
The phase angle of the circuit can be derived using the relation between resistance and reactance.

By understanding the formula and components, the calculation of the impedance of the LCR circuit becomes straightforward, allowing for analysis of the circuit's behavior in different scenarios.

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