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In a series LCR circuit at resonance, the phase difference between applied voltage and resulting current is:

0 radians (in phase). At resonance, inductive and capacitive reactances cancel (X_L = X_C), making tan φ = (X_L - X_C)/R = 0, so φ = 0.

Physics · Class 12 · JEE / NEET / AP Physics

Alternating Current (AC) & Series LCR Circuit

Alternating Current (AC)periodically reverses direction and changes magnitude continuously with time. In aSeries LCR Circuit, inductive and capacitive reactances combine with resistance to dictate circuit impedance and sharp resonance.

11. Visual Overview & Intuitive Discovery: Alternating Current (AC) & Series LCR Circuit

In the late 1880s, the fierce "War of the Currents" pitted Thomas Edison's Direct Current (DC) against Nikola Tesla's Alternating Current (AC). Edison argued that DC was safer, but Tesla understood an unbeatable mathematical reality: alternating current could be stepped up to hundreds of thousands of volts using transformers, traveling hundreds of miles across copper lines with almost zero heating loss, and then stepped down safely into homes. Tesla's victory powers our modern civilization.

Series LCR Phasor Diagram & Resonance Curve: RMS Value, Total Impedance (Z), Resonant Frequency, Power Factor — Series LCR Phasor Diagram & Resonance Curve
Impedance triangle with inductive/capacitive reactance and resonance curve showing peak current at f₀ = 1 / [2π √(LC)].
1RMS Value — V_rms = V₀ / √2 ≈ 0.707 V₀, representing equivalent DC heating power
2Total Impedance (Z) — Z = √[ R² + (X_L - X_C)² ]
3Resonant Frequency — f₀ = 1 / [2π √(LC)] where Z = R and current is maximized
4Power Factor — cos(φ) = R / Z, defining average active power P_avg = V_rms I_rms cos(φ)

Interactive Study Prompts (Click to Explore)

Clickable

AI Deep Dive 🪄

22. Physical Mechanism Flowchart: Alternating Current (AC) & Series LCR Circuit

Alternating Current (AC) & Series LCR Circuit Step-by-Step Flowchart

1
Initial State
Initial parameters and boundary conditions are established.
2
Fundamental Force/Interaction
Governing physical laws act continuously across the system.
3
Energy & Vector Conservation
Conservation principles restrict permitted trajectory states.
4
Dynamic Equilibrium
Final observable state satisfies equilibrium equations.

Follow the numbered pipeline above. In physics examinations, sketching this mental flowchart ensures you never miss intermediate steps in complex multi-concept numericals.

33. Rigorous Mathematical Proofs & Governing Equations

Derivation — Derivation of Governing Formula for Alternating Current (AC) & Series LCR Circuit

1
$F_{\text{net}} = m a$
Starting from Newton's foundational law
2
$\int F \, dx = \Delta K$
Integrating over displacement

Quality Factor (Q-Factor) of Resonance

Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}} = \frac{\omega_0}{2\Delta\omega}

Q = Sharpness of resonance (higher Q means sharper tuning)
$2\Delta\omega$ = Bandwidth of the circuit

Core Physical Variables

Variable	Symbol	SI Unit
Energy	E	Joules (J)
Power	P	Watts (W)

Energy
Symbol
E
SI Unit
Joules (J)
Power
Symbol
P
SI Unit
Watts (W)

44. Engineering Deep Dive: Radio Tuning and High-Q Resonance

Every time you tune a radio to 98.3 FM or your smartphone locks onto a 5G wireless carrier frequency, you are using series LCR electrical resonance! Thousands of electromagnetic radio waves at different frequencies constantly wash over the antenna, inducing tiny voltages across it. By adjusting a variable capacitor inside the tuning circuit, you alter the resonant frequency $f_0 = 1 / [2\pi\sqrt{LC}]$ . At resonance, the circuit's impedance drops to pure resistance $R$ , amplifying the target frequency current by hundreds of times while completely suppressing all other radio stations!

Common mistake: Misconception: "An AC voltage of 220 V is equivalent in shock hazard to a DC voltage of 220 V." In reality, household 220 V AC is the Root Mean Square (RMS) value! The actual peak instantaneous voltage swings up to

V_0 = \sqrt{2} \times 220\,\text{V} \approx 311\,\text{Volts}

. A 220 V AC line is significantly more hazardous than a 220 V DC source.

Exam tip: Exam Formula Shortcut: In a series LCR circuit at resonance, the voltage across the inductor

V_L

and capacitor

V_C

can actually exceed the applied source voltage by a factor equal to the Quality Factor (

Q = \frac{\omega_0 L}{R}

). Do not panic if

V_L > V_{\text{source}}

—this is the classic resonant voltage amplification phenomenon!

55. Solved Standard & Advanced Exam Problems

Exam question · JEE Standard

Sample Question

Exam question · JEE Main & NEET Standard

A series LCR circuit contains R = 10 Ω, L = 100 mH, and C = 10 μF connected across an AC supply of 220 V, 50 Hz. At what frequency does the current in the circuit reach its maximum value?

A.159.1 Hz
B.50.0 Hz
C.100.0 Hz
D.318.2 Hz

Remember

Household AC voltage quoted as 220 V is the RMS value; its peak voltage is ~311 V.
At resonance in a series LCR circuit: Impedance is minimum (Z = R), current is maximum, and power factor is unity (cos φ = 1).
Transformers operate solely on alternating flux (AC) and cannot step up or step down DC voltages.

Check yourself

Question 1
In a series LCR circuit at resonance, the phase difference between applied voltage and resulting current is: