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Courses/Three-Phase Systems/Star (Wye) Connection

lesson · 10min · Lesson 7 of 28

Line voltage = √3 × phase voltage: proof

Course syllabusCourse overview
01Three-Phase Fundamentals
  1. ReadWhy three-phase?: advantages over single-phase
  2. ReadGeneration of three-phase voltages
  3. ReadPhase sequence: L1, L2, L3
  4. ReadPhasor representation of three-phase
  5. quizPhase fundamentals quiz
02Star (Wye) Connection
  1. ReadStar connection topology
  2. ReadLine voltage = √3 × phase voltage: proof
  3. ReadNeutral current in star systems
  4. exerciseStar circuit analysis problems
03Delta Connection
  1. ReadDelta connection topology
  2. ReadLine current = √3 × phase current: proof
  3. ReadCirculating currents in delta
  4. exerciseDelta circuit analysis problems
04Three-Phase Power
  1. ReadActive, reactive, and apparent power
  2. ReadPower factor in three-phase
  3. ReadTwo-wattmeter method
  4. ReadPower correction capacitor sizing
  5. quizThree-phase power quiz
05Three-Phase Induction Motors
  1. ReadHow induction motors work
  2. ReadMotor nameplate data and efficiency classes
  3. ReadStarting currents and starting methods
  4. ReadDOL starters: design and wiring
  5. ReadStar-delta starters: wiring and timing
06Transformers
  1. ReadTransformer construction and principles
  2. ReadTurns ratio and voltage/current transformation
  3. ReadThree-phase transformer connections
  4. quizTransformer quiz
  5. quizFinal assessment
Lesson · 10min
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In this lesson

PurposeCore theoryWorked exampleKnowledge checkSources

In this lesson

PurposeCore theoryWorked exampleKnowledge checkSources
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ElectraCore lesson handout

Line voltage = √3 × phase voltage: proof

Three Phase Systems · Lesson 7 · Intermediate

IntermediateReview: professional review pending

Purpose

Derive the √3 magnitude and 30-degree angular relationship between balanced star line and phase voltages.

Before you beginStar topology · Phasor subtraction

Learning objectives

  • Form a line voltage
  • Derive √3 geometrically
  • State the 30° displacement
  • Apply nominal-voltage caveats

A line-to-line voltage is the phasor difference between two line-to-neutral voltages. Because those phase phasors are 120 degrees apart, subtraction produces a magnitude √3 times one phase voltage.

STAR VOLTAGE AND CURRENT PATHSSTAR VOLTAGE AND CURRENT PATHSL1 · 0°L2 · −120°L3 · +120°VL = √3 VphIL = Iph · IN = phasor sum

Core theory

For positive sequence with Va as reference, Vab = Va − Vb. Subtraction means adding the opposite of Vb, not adding scalar rms values.

The resulting balanced magnitude is VL = √3 Vph, and the line voltage is displaced 30 degrees relative to the corresponding phase voltage under the stated labelling convention.

Thus an ideal 230 V phase-to-neutral system has about 398 V line-to-line, commonly described by a nominal 230/400 V system. Actual measured voltage and permitted tolerances require current supply data.

Terms, symbols, and units
TermMeaningSymbolUnit
Line voltageVoltage measured between two line conductorsVLV
Phase voltageVoltage from line to star point/neutralVphV
Phasor subtractionVector difference defining voltage between two nodesNot applicableNot applicable
VL = √3 × Vph

For a balanced star system, line voltage magnitude is root-three times phase voltage.

V
Worked exampleCalculate ideal line voltage for Vph = 230 V.

Assumptions: Balanced sinusoidal star set.

  1. Formula: VL = √3 Vph.
  2. Substitute: VL = 1.732 × 230 V.
  3. Calculate: VL ≈ 398 V, conventionally associated with a nominal 400 V system.

The ideal calculated line voltage is approximately 398 V.

Reasonableness check: It must be greater than 230 V but less than the scalar sum 460 V.

Common mistakes
  • Adding 230 + 230
  • Using √3 for an unbalanced set without analysis
  • Ignoring the 30° phase relation

Where this appears in practice

The relationship underpins equipment voltage selection, measurements, transformer connections, and power calculations.

SafetyLine-to-line voltage is higher than line-to-neutral voltage. Instruments and procedures must be rated for the actual measurement environment.
Local code checkConfirm the nominal system voltage/frequency, earthing and neutral arrangement, phase-sequence convention, conductor and protective-device data, harmonic assessment, isolation method, and current local installation standard. The 230/400 V examples are instructional values, not site measurements or universal supply values.

Knowledge check

Why is line voltage not twice phase voltage?

The phase voltages are phasors 120° apart, not in-phase scalars. Their vector difference has magnitude √3 times one phase value.

Answer: The phase voltages are phasors 120° apart, not in-phase scalars. Their vector difference has magnitude √3 times one phase value.

Practical exercise

Derive Vab from rectangular Va and Vb components and identify its magnitude and angle.

Summary

  • Line voltage is a phasor difference
  • Balanced magnitude ratio is √3
  • A 30° displacement accompanies the ratio

Sources and review

  • IEC 60038: IEC standard voltages: IEC; 2009+A1:2021; International
  • Broken PEN: IET Wiring Matters; Issue 84, 2021; United Kingdom
  • Minimizing unnecessary live testing for initial verification: IET Wiring Matters; Issue 105, 2025; United Kingdom

Editorial review date: 2026-08-22. Professional electrical review is pending.

Educational material for learning and preliminary checks. Verify current local requirements and exact equipment instructions. This lesson does not replace competent professional work.

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