Worked problem

Unbalanced Three-Phase Loads: Find the Neutral Current

Calculate the neutral current for phase currents of 100 A, 80 A and 60 A at unity power factor, and see why adding the magnitudes gives the wrong answer.

Electrical & Power · Last reviewed

Given

A three-phase, four-wire wye system with balanced sinusoidal line voltages. The phase currents are Ia = 100 A, Ib = 80 A and Ic = 60 A, all at unity power factor, so each current is in phase with its own phase voltage. There are no harmonics.

Find

The current in the neutral conductor.

Solution

At the neutral point the three phase currents return together, so the neutral current is their phasor sum. The phase voltages are 120° apart, and so are the currents.

  1. Ia = 100∠0°, Ib = 80∠−120°, Ic = 60∠+120°.
  2. Real parts: 100 + 80 × cos(−120°) + 60 × cos(120°) = 100 − 40 − 30 = 30 A.
  3. Imaginary parts: 0 + 80 × sin(−120°) + 60 × sin(120°) = −69.28 + 51.96 = −17.32 A.
  4. Magnitude = √(30² + 17.32²) = 34.6 A, at an angle of −30°.
  5. Check with the closed form: In = √(Ia² + Ib² + Ic² − IaIb − IbIc − IcIa) = √(10,000 + 6,400 + 3,600 − 8,000 − 4,800 − 6,000) = √1,200 = 34.6 A.

Answer

neutral current = 34.6 A (not 240 A, the sum of the magnitudes)

Why the magnitudes cannot be added

Currents are phasors with direction as well as size. Three equal currents 120° apart add to zero, so a balanced load leaves nothing in the neutral. Only the imbalance returns: here a 100 A, an 80 A and a 60 A current, which is the same as a net current of 34.6 A.

CasePhase currentsNeutral current
Balanced80 A, 80 A, 80 A0 A
This problem100 A, 80 A, 60 A34.6 A
One phase only100 A, 0, 0100 A

Common mistakes

  • Adding the three magnitudes (240 A), or averaging them.
  • Using the same formula for non-linear loads. Triplen harmonics (3rd, 9th and so on) add in the neutral instead of cancelling, so a nearly balanced system with computers or LED drivers can still carry a large neutral current.
  • Forgetting that the formula assumes the same power factor angle on all three phases. With different angles, work with the full complex values.

Related tools and labs

Calculators, labs and simulators are part of the All-Access Pass.

More worked problems

Questions

Can the neutral current be larger than a phase current?

Yes. With triplen harmonics, or a severe imbalance such as one phase loaded and the other two empty, the neutral can match or exceed a phase current.

Does this apply to a delta system?

A delta system has no neutral. The same phasor idea applies to the currents at a wye point or at a transformer with a neutral.

Does this size the neutral conductor?

No. Neutral sizing follows the code rules, including how harmonic loads are treated.

Educational material. Examples use simplified, hypothetical inputs and are not a design, a construction document or professional engineering advice. Confirm requirements with the codes and standards adopted where the work is built and with a licensed engineer.

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