Power Stations WORLDWIDETHE GLOBAL EDITION 01
LESSON 5 OF 6 · ORIGINAL EDUCATIONAL EXAMPLE

Three-phase line and phase quantities

Distinguish line-to-line voltage, winding voltage and line current in balanced wye and delta connections.

Learning goal: Use √3 in the correct place and calculate line current from total three-phase kVA.

Prepared by Power Stations Worldwide. Sources checked on 30 September 2026; not independently engineering peer reviewed.

The essentials

Wye (star): voltage changes

In a balanced wye connection, each winding is connected between a line and the star point. Line-to-line voltage magnitude is √3 times winding voltage; line current equals winding current. Line-to-neutral equals winding voltage when the neutral references that star point.

Delta: current changes

In a balanced delta connection, each winding is across two lines, so winding voltage equals line-to-line voltage. Line-current magnitude is √3 times winding current. These are magnitude relationships; phasor angle relationships also matter.

Do not multiply by three twice

S = √3 VLL IL gives total three-phase apparent power. Equivalently, S = 3 Vphase Iphase using corresponding winding values for a balanced load. Do not insert a line-to-neutral voltage into the line-to-line formula.

Wye and delta quantities

Wye and delta quantitiesA wye connection shows three winding branches meeting at a star point N, with winding voltage from A to N and line voltage from A to B. A separate delta triangle shows each winding between two lines. Labels give the balanced voltage and current magnitude relationships.Wye (star)DeltaNABCVphaseVLLABCVLL = VphaseIL = √3 IphaseVLL = √3 VphaseIL = Iphase
Original connection abstractions, not wiring instructions. The star point is not necessarily earthed; protective earth, insulation and protection are not shown.
HYPOTHETICAL VALUES · NOT PLANT RECORDS

Worked example

A balanced 25 kVA wye load at 400 V line-to-line

Given

  • Total apparent power S = 25 kVA = 25,000 VA
  • VLL = 400 V RMS
  • Balanced wye-connected load

Formula toolbox

  • Balanced wye: VLL = √3 Vphase; IL = Iphase
  • Balanced delta: VLL = Vphase; IL = √3 Iphase
  • Total S = √3 VLL IL = 3 Vphase Iphase

Calculate step by step

  1. Winding voltage: Vphase = 400 / √3 ≈ 230.94 V.
  2. Line current: IL = 25,000 / (√3 × 400) ≈ 36.08 A.
  3. Wye winding current equals line current: Iphase ≈ 36.08 A.
  4. Cross-check with unrounded values: 3 × Vphase × Iphase = 25,000 VA.

230.94 V per winding · 36.08 A per line

400 V and 230 V are often nominal rounded labels; 400/√3 is about 230.94 V. Neutral current is zero only for the ideal balanced sinusoidal set—not for every real installation.

Results are rounded only for display. Reproduce the calculations using unrounded intermediate values.

Quick quiz

Choose one answer for each question, then check your answers. This is a practice check, not a qualification. Answers stay in this page only; refreshing or leaving resets them.

1. In a balanced wye load, which relationship is correct?
2. A balanced load is 10 kVA at 400 V line-to-line. What is the line current?

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Show answers and explanations (also works without JavaScript)
  1. VLL = √3 Vphase. Wye: VLL = √3 Vphase and IL = Iphase. Delta uses a different current relationship.
  2. About 14.43 A. IL = 10,000 / (√3 × 400) ≈ 14.43 A. Use VA, not kVA, with volts.

Curriculum connection: MIT 6.061 (Spring 2011). Its readings include 2007 notes; some equations use peak rather than RMS amplitudes. This lesson states its own value conventions and uses original diagrams, numbers and quiz questions. No MIT affiliation or endorsement is claimed.

Textbook references

Suggested technical background for this topic. These books are further reading, not proof that every explanation was checked against every edition. See the full bibliography and review scope.

  1. Electric Circuits (11th ed.) — Nilsson, J. W. & Riedel, S. A., Pearson, 2019 · ISBN 9780134746968Balanced three-phase circuits and phase/line relationships.
  2. Electrical Machines, Drives and Power Systems (6th ed., Pearson New International Edition) — Wildi, T., Pearson, 2013 · ISBN 9781292024585Three-phase circuits and transformer connections.