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Series LCR Circuit: Impedance and Phasor Relation

A series LCR circuit contains a resistor, inductor, and capacitor in series with an AC source; its total opposition to current is impedance Z = √(R2 + (XL - XC)2).

In one line: In series LCR, R stands sideways, L goes up, C goes down; only the leftover vertical part combines with R.

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Understand it

  • In a series LCR circuit, the same current flows through R, L, and C.
  • The resistor voltage VR is in phase with current, the inductor voltage VL leads current by π/2, and the capacitor voltage VC lags current by π/2.
  • Since VL and VC are in opposite directions in the phasor diagram, the effective reactive voltage is VL - VC.
  • The applied voltage is the vector sum of VR and the net reactive voltage, giving V = √(VR2 + (VL - VC)2).
  • Dividing by current gives Z = √(R2 + (XL - XC)2).
  • The phase angle is tan phi = (XL - XC)/R.

Example

If XL is greater than XC, the circuit behaves inductively and current lags the applied voltage. If XC is greater than XL, the circuit behaves capacitively and current leads the applied voltage.

Formula and rule

Z = √(R2 + (XL - XC)2), tan phi = (XL - XC)/R, I = Vrms/Z.

R, XL, XC, and Z are in ohm. phi is positive for inductive behaviour and negative for capacitive behaviour if current is taken as reference.

Use RMS values for circuit current and power calculations.

See it worked out

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

Worked example

A series LCR circuit has R = 30 ohm, XL = 80 ohm, and XC = 40 ohm.

  1. 1Net reactance = XL - XC = 40 ohm.
  2. 2Z = √(302 + 402) = √2500 = 50 ohm.
  3. 3If Vrms = 100 V, Irms = V/Z = 100/50 = 2 A. tan phi = 40/30 = 4/3, so the circuit is inductive and current lags voltage.

Method you can reuse

  1. 1Calculate XL and XC if needed.
  2. 2Find net reactance XL - XC with sign.
  3. 3Compute Z using the square-root formula.
  4. 4Use I = V/Z.
  5. 5Decide the circuit nature from the sign of XL - XC.
  6. 6Draw phasors using current as the reference.

Visual tip

The phasor diagram should show I and VR along the horizontal axis, VL upward, VC downward, net reactive voltage VL - VC, resultant supply voltage V, and phase angle phi between V and I. The circuit diagram should label R, L, C, and AC source in series.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is impedance in a series LCR circuit not simply R + XL + XC?

Think first, then show answer
Impedance is not R + XL + XC because resistance, inductive reactance, and capacitive reactance do not act in the same phase direction. The reactive parts oppose each other, so the net reactance is XL - XC and Z = √(R2 + (XL - XC)2).

Common wrong answer

Adding all three as Z = R + XL + XC gives a larger and physically wrong impedance because it ignores phase.

Don't mix this up

Students often add XL and XC directly in series LCR. They must be subtracted because inductive and capacitive voltage phasors are opposite.

Exam tip

Always start the phasor diagram with current as the reference along the horizontal direction because current is common in a series circuit.

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

Questions often ask for impedance, phase angle, current, voltage across each element, phasor diagram, or whether the circuit is inductive, capacitive, or resistive.
  1. 1

    Enough for 1 mark

    A series LCR circuit contains a resistor, inductor, and capacitor in series with an AC source; its total opposition to current is impedance Z = √(R2 + (XL - XC)2).

  2. 2

    Add for 2 marks

    Example: If XL is greater than XC, the circuit behaves inductively and current lags the applied voltage. If XC is greater than XL, the circuit behaves capacitively and current leads the applied voltage.

  3. 3

    Add for 3 marks

    In a series LCR circuit, the same current flows through R, L, and C.

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