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Courses/Three-Phase Systems/Three-Phase Fundamentals

lesson · 9min · Lesson 4 of 28

Phasor representation of three-phase

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 · 9min
THREE-PHASE · 120° APARTL1 · L2 · L3STARY: Vʟ=√3·VₚΔ: Iʟ=√3·IₚDELTA
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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

Phasor representation of three-phase

Three Phase Systems · Lesson 4 · Intermediate

IntermediateReview: professional review pending

Purpose

Use phasors to represent balanced three-phase magnitude and angular relationships without confusing them with physical rotation.

Before you beginComplex numbers · RMS sine-wave quantities

Learning objectives

  • Draw a balanced phasor set
  • Choose a reference
  • Convert rectangular and polar form
  • Perform vector rather than scalar sums

A phasor compresses a steady sinusoid into an rms magnitude and phase angle at one frequency. Three balanced phase phasors are equal vectors separated by 120 degrees.

THREE-PHASE PHASOR SETTHREE-PHASE PHASOR SETL1 · 0°L2 · −120°L3 · +120°ΣV = 0equal magnitude · 120° apart

Core theory

Choose one phase as the zero-degree reference, then apply a declared positive or negative sequence consistently. Changing reference rotates the whole diagram but does not alter relative angles.

Phasors add as vectors. Three equal balanced phase-to-neutral voltage phasors sum to zero; their arithmetic magnitudes do not.

A phasor diagram represents relative sinusoidal timing in the complex plane. It is not a snapshot of electrons or necessarily a mechanical rotor drawing.

Terms, symbols, and units
TermMeaningSymbolUnit
PhasorComplex representation of sinusoidal rms magnitude and phaseNot applicableNot applicable
Reference phasorQuantity assigned zero angle for comparisonNot applicableNot applicable
Rectangular formComplex quantity expressed as real and imaginary componentsNot applicableNot applicable
Worked exampleFor a balanced set Va = 230∠0° V, Vb = 230∠−120° V, Vc = 230∠120° V, what is Va + Vb + Vc?

Assumptions: Ideal equal magnitudes; Angles use one convention.

  1. Resolve: Va = 230 + j0; Vb and Vc have equal negative real components and opposite imaginary components.
  2. Imaginary: The Vb and Vc imaginary components cancel.
  3. Real: Their combined real component is −230 V, cancelling Va.

The vector sum is 0 V.

Reasonableness check: A symmetrical three-vector set closes to form a triangle.

Common mistakes
  • Adding rms magnitudes arithmetically
  • Mixing lead and lag conventions
  • Using degrees in a radian calculator mode

Where this appears in practice

Phasors support star/delta derivations, power calculations, protection studies, and unbalance analysis.

SafetyA neat phasor diagram does not establish safe voltage, isolation, or conductor identity at a real installation.
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

What happens to the vector sum of three equal phasors spaced 120° apart?

It is zero. Their real and imaginary components cancel.

Answer: It is zero. Their real and imaginary components cancel.

Practical exercise

Convert the three balanced 230 V phasors to rectangular form and verify both component sums are zero.

Summary

  • Phasors encode magnitude and phase
  • Reference choice must be stated
  • Balanced vector sums cancel

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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