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medium difficulty10 min

Power Factor and Average Power in AC Circuits

Average power consumed in an AC circuit is P = Vrms Irms cos phi, where cos phi is the power factor and phi is the phase difference between voltage and current.

In one line: Only the current component in step with voltage earns power: I cos phi does the work.

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Understand it

  • In AC, instantaneous power changes with time because voltage and current vary continuously.
  • The useful average power over a complete cycle depends on how much of the current is in phase with voltage.
  • The factor cos phi measures this in-phase part.
  • A pure resistor has phi = 0, so cos phi = 1 and power is maximum for given Vrms and Irms.
  • In a pure inductor or pure capacitor, phi = π/2, so cos phi = 0 and average power is zero; this is called wattless current because energy is alternately stored and returned rather than consumed over a full cycle.

Example

If Vrms = 200 V, Irms = 5 A, and power factor = 0.8, the average power is P = 200 × 5 × 0.8 = 800 W.

Formula and rule

Average power: P = Vrms Irms cos phi.

Power factor: cos phi = R/Z for a series LCR circuit.

P is in watt, Vrms in volt, Irms in ampere, and phi is the phase angle between voltage and current.

Use RMS values, not peak values, unless converting them first.

See it worked out

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

Worked example

A series AC circuit has Vrms = 240 V, Irms = 3 A, and phase angle phi = 60 degrees.

  1. 1Average power P = Vrms Irms cos phi = 240 × 3 x cos 60 degrees = 240 × 3 × 0.5 = 360 W.
  2. 2The apparent power is 720 VA, but only 360 W is consumed as average power.

Method you can reuse

  1. 1Identify RMS voltage and RMS current.
  2. 2Find cos phi from the given phase angle or from R/Z.
  3. 3Substitute in P = Vrms Irms cos phi.
  4. 4Interpret whether power is maximum, reduced, or zero depending on phi.

Visual tip

A power-factor phasor may show current I, voltage V, phase angle phi, and the in-phase component I cos phi. For pure L or C, the V-I phase difference is 90 degrees, making average power zero.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What is wattless current in an AC circuit?

Think first, then show answer
Wattless current refers to the current in a purely reactive AC circuit where the phase difference is π/2. Since cos phi = 0, the average power P = Vrms Irms cos phi is zero even though current flows.

Common wrong answer

Writing P = V0 I0 cos phi directly without averaging is incomplete. If peak values are used, average power is P = (V0 I0/2) cos phi.

Don't mix this up

Students often calculate AC power as only VI without multiplying by cos phi. That is correct only when voltage and current are in phase.

Exam tip

Whenever an AC circuit contains L or C, check the phase angle before calculating average power. Write the power factor clearly.

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 ask for average power, power factor, wattless current, comparison of pure R/L/C power, or why power factor correction is useful in AC systems.
  1. 1

    Enough for 1 mark

    Average power consumed in an AC circuit is P = Vrms Irms cos phi, where cos phi is the power factor and phi is the phase difference between voltage and current.

  2. 2

    Add for 2 marks

    Example: If Vrms = 200 V, Irms = 5 A, and power factor = 0.8, the average power is P = 200 × 5 × 0.8 = 800 W.

  3. 3

    Add for 3 marks

    In AC, instantaneous power changes with time because voltage and current vary continuously.

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