C
CraftExam
medium difficulty8 min

Resonance in a Series LCR Circuit

Resonance in a series LCR circuit occurs when inductive reactance equals capacitive reactance, making impedance minimum and current maximum.

In one line: At resonance, L and C cancel their reactance, leaving only R to control current.

Concept Practice Coming Soon

Understand it

  • At resonance, XL = XC.
  • Therefore the net reactance XL - XC becomes zero and the circuit behaves like a pure resistor.
  • The impedance becomes Z = R, which is its minimum value, so current reaches its maximum value Imax = Vrms/R.
  • The resonant angular frequency is ω0 = 1/√LC.
  • A sharp resonance means current changes greatly for a small change in frequency, and this sharpness is measured using the quality factor Q.

Example

Radio tuning uses resonance. A receiver selects a particular frequency when its LC circuit resonates with the frequency of the desired signal.

Formula and rule

Condition: XL = XC.

Since ω L = 1/(ω C), ω0 = 1/√LC.

Resonant frequency f0 = 1/(2 π √LC).

At resonance, Z = R, phi = 0, and power factor = 1.

Q-factor for a series circuit may be written as Q = ω0 L/R = 1/(ω0 C R).

L is in henry, C in farad, R in ohm, ω0 in rad s−1, and f0 in hertz.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

  1. 1A series LCR circuit has L = 0.5 H and C = 8 microfarad. ω0 = 1/√LC = 1/√(0.5 × 8 × 10−6) = 1/√(4 × 10−6) = 1/(2 × 10−3) = 500 rad s−1.
  2. 2If R = 50 ohm and Vrms = 100 V, current at resonance is I = V/R = 100/50 = 2 A.

Method you can reuse

  1. 1Set XL = XC.
  2. 2Substitute ω L = 1/(ω C).
  3. 3Solve for ω0.
  4. 4Convert to f0 if required.
  5. 5At resonance, take Z = R and calculate current using RMS voltage.
  6. 6Add the physical meaning: current maximum and circuit purely resistive.

Visual tip

A resonance curve of current versus frequency is useful. The x-axis is frequency, the y-axis is current, and the peak occurs at f0. A sharper peak represents larger Q. If a phasor is drawn at resonance, VL and VC are equal and opposite.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is current maximum at resonance in a series LCR circuit?

Think first, then show answer
Current is maximum at resonance because XL equals XC, so the net reactance becomes zero. The impedance reduces to R, its minimum value, and therefore I = Vrms/R becomes maximum.

Common wrong answer

Using ω0 = √LC.

Why it loses marks: The correct relation is the reciprocal square root, ω0 = 1/√LC.

Don't mix this up

Students sometimes say resonance means voltage is maximum across the source. In a series LCR circuit, resonance mainly means current is maximum because impedance is minimum.

Exam tip

In resonance questions, immediately write XL = XC, Z = R, phi = 0, and ω0 = 1/√LC. These four statements cover most scoring points.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Exams ask for resonant frequency, condition for resonance, maximum current, power factor at resonance, Q-factor meaning, or interpretation of a current-frequency graph.
  1. 1

    Enough for 1 mark

    Resonance in a series LCR circuit occurs when inductive reactance equals capacitive reactance, making impedance minimum and current maximum.

  2. 2

    Add for 2 marks

    Example: Radio tuning uses resonance. A receiver selects a particular frequency when its LC circuit resonates with the frequency of the desired signal.

  3. 3

    Add for 3 marks

    Therefore the net reactance XL - XC becomes zero and the circuit behaves like a pure resistor.

Practice

Concept practice is coming soon

Login to access concept-level MCQs and weak-concept practice as they become available.

10 MCQs5 MinutesInstant Results
Login

Help improve this page

Found something confusing, incorrect, or missing?

Open Learning Map