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Impedance and resonance in LCR circuits: Class 12 practice

In a series LCR circuit X_L = 2πfL, X_C = 1/(2πfC), Z = √(R² + (X_L − X_C)²) and I = V/Z. Read the method, follow the three worked examples, then try the six questions.

What you should know first

  • Square roots
  • Scientific notation

The idea behind impedance and resonance in lcr circuits

In a series LCR circuit X_L = 2πfL, X_C = 1/(2πfC), Z = √(R² + (X_L − X_C)²) and I = V/Z. Resonance occurs at f₀ = 1/(2π√(LC)), where Z = R.

Step-by-step method

  1. Compute X_L and X_C.
  2. Z = √(R² + (X_L − X_C)²).
  3. I = V/Z; at resonance Z = R.

Worked example 1 (Basic)

A series LCR circuit has R = 5 Ω, X_L = 20 Ω, X_C = 8 Ω. Find Z.

  1. Z = √(5² + (20 − 8)²)
  2. = 13 Ω
  3. Answer: 13 Ω

Worked example 2 (Standard)

The circuit (R = 5, X_L = 20, X_C = 8 Ω) is on 78 V rms. Find the rms current and power factor.

  1. Z = 13 Ω, I = 78/13 = 6 A
  2. cos φ = R/Z = 5/13
  3. Answer: 6 A, cos φ = 5/13

Worked example 3 (Challenge)

Find the resonant frequency for L = 0.2 H and C = 54 μF.

  1. f₀ = 1/(2π√(LC))
  2. = 48.43 Hz
  3. Answer: 48.43 Hz

Common mistakes and misconceptions

  • Adding R, X_L, X_C directly
  • Using peak and rms values interchangeably

Practice questions

  1. (Basic) A series LCR circuit has R = 6 Ω, X_L = 12 Ω, X_C = 4 Ω. Find Z.
  2. (Basic) A series LCR circuit has R = 3 Ω, X_L = 8 Ω, X_C = 4 Ω. Find Z.
  3. (Standard) The circuit (R = 12, X_L = 10, X_C = 15 Ω) is on 65 V rms. Find the rms current and power factor.
  4. (Standard) The circuit (R = 12, X_L = 10, X_C = 15 Ω) is on 52 V rms. Find the rms current and power factor.
  5. (Challenge) Find the resonant frequency for L = 0.8 H and C = 90 μF.
  6. (Challenge) Find the resonant frequency for L = 0.8 H and C = 39 μF.

Answer key with full working

  • Q1: Z = √(6² + (12 − 4)²) → = 10 Ω → Answer: 10 Ω
  • Q2: Z = √(3² + (8 − 4)²) → = 5 Ω → Answer: 5 Ω
  • Q3: Z = 13 Ω, I = 65/13 = 5 A → cos φ = R/Z = 12/13 → Answer: 5 A, cos φ = 12/13
  • Q4: Z = 13 Ω, I = 52/13 = 4 A → cos φ = R/Z = 12/13 → Answer: 4 A, cos φ = 12/13
  • Q5: f₀ = 1/(2π√(LC)) → = 18.76 Hz → Answer: 18.76 Hz
  • Q6: f₀ = 1/(2π√(LC)) → = 28.49 Hz → Answer: 28.49 Hz

Syllabus note

This skill is listed under "Alternating Current" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

Frequently asked questions

Which chapter is impedance and resonance in lcr circuits in?

"Alternating Current", Class 12 Physics (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

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