Introduction:

A series LCR circuit consists of an inductor (L), a capacitor (C), and a resistor (R) connected in series to an alternating current (AC) source. The analysis of such a circuit is crucial for understanding the behavior of AC circuits in practical applications, including resonance, impedance, and phase relationships. This study note provides a comprehensive explanation of the response of a series LCR circuit to an AC voltage.

Understanding the Series LCR Circuit:

A series LCR circuit is driven by an alternating voltage source given by:

V = V₀ sin(ωt)

where V₀ is the peak voltage, and ω is the angular frequency of the AC source.

Each component (R, L, and C) has a different behavior in the presence of AC, affecting the circuit's total impedance and phase angle.

Impedance of the Series LCR Circuit:

The total impedance (Z) of the circuit is given by:

Z = √(R² + (X_L - X_C)²)

where:

- X_L = ωL (Inductive reactance)

- X_C = 1/(ωC) (Capacitive reactance)

The impedance determines the overall resistance to the AC current and affects the circuit's current flow.

Phasor Diagram and Phase Angle:

The phase angle (ϕ) between the applied voltage and the current is given by:

tan ϕ = (X_L - X_C) / R

A phasor diagram represents the relationship between the voltage and current in the circuit and helps in analyzing phase shifts.

Resonance in Series LCR Circuit:

Resonance occurs when the inductive reactance equals the capacitive reactance:

X_L = X_C → ω₀L = 1/(ω₀C)

Solving for ω₀, we get the resonance frequency:

ω₀ = 1/√(LC)

At resonance:

- Impedance is minimum and equals R.

- Current is maximum.

- Voltage across L and C are equal and opposite, canceling out.

Power and Power Factor in LCR Circuit:

The average power (P) in the circuit is given by:

P = V_rms I_rms cosϕ

where cosϕ is the power factor, determining the efficiency of power transfer. At resonance, the power factor is 1, meaning all power is consumed by the resistor.

Applications of Series LCR Circuit:

Radio Tuning: LCR circuits are used in radio receivers for frequency selection.

Filters in Communication Systems: These circuits help in signal filtering.

Power Distribution: Used in AC power systems to manage voltage levels and phase angles.

Resonant Transformers: Used in induction heating and wireless power transfer.

Conclusion:

A series LCR circuit plays a fundamental role in AC circuit analysis, demonstrating crucial concepts like resonance, impedance, and phase relationships. Understanding this circuit is essential for students in Class 12 Physics, as it has numerous practical applications in electrical and communication systems.