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Courses/Three-Phase Systems/Three-Phase Power

lesson · 9min · Lesson 14 of 28

Active, reactive, and apparent power

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
POWER TRIANGLEφP: real (W)Q (VAr)S (VA)PF = P / S= cos φS² = P² + Q²
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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

Active, reactive, and apparent power

Three Phase Systems · Lesson 14 · Intermediate

IntermediateReview: professional review pending

Purpose

Calculate balanced three-phase active, reactive, and apparent power with consistent rms and power-factor conventions.

Before you beginComplex power · Star and delta relations

Learning objectives

  • Distinguish P Q and S
  • Use balanced line formulas
  • Check the power triangle
  • State sign and balance assumptions

Complex power separates energy conversion from oscillatory energy exchange. For a balanced sinusoidal three-phase load, line quantities give compact formulas independent of whether the load is star or delta.

THREE-PHASE POWER TRIANGLETHREE-PHASE POWER TRIANGLEQPSS² = P² + Q²P = √3 VL IL PF

Core theory

Apparent power magnitude is S = √3 VL IL. Active power is P = S cosφ and reactive power is Q = S sinφ under the adopted load sign convention.

The power triangle satisfies S² = P² + Q² for sinusoidal quantities. Inductive loads are commonly described as lagging and positive Q; capacitive loads as leading and negative Q, but the convention must be stated.

For unbalanced or distorted systems, sum per-phase complex power or use suitable power-quality measurements; one displacement angle may not represent true power factor.

Terms, symbols, and units
TermMeaningSymbolUnit
Active powerAverage rate of net energy transferPW
Reactive powerSigned oscillatory energy-exchange rateQvar
Apparent powerRMS voltage-current product magnitudeSVA
S = √3 VL IL; P = S cosφ; Q = S sinφ

Balanced sinusoidal three-phase power from line quantities.

VA, W, var
Worked exampleA balanced 400 V load draws 20 A at 0.80 lagging power factor. Find S, P, and Q.

Assumptions: Sinusoidal balanced load; Lagging inductive convention.

  1. Apparent: S = √3 × 400 × 20 = 13.86 kVA.
  2. Active: P = 13.86 × 0.80 = 11.09 kW.
  3. Reactive: sinφ = 0.60, so Q = 13.86 × 0.60 = 8.31 kvar.

S ≈ 13.86 kVA, P ≈ 11.09 kW, Q ≈ 8.31 kvar lagging.

Reasonableness check: 11.09² + 8.31² is approximately 13.86².

Common mistakes
  • Using phase voltage with the line formula
  • Giving Q in kW
  • Using displacement PF for distorted loads without qualification

Where this appears in practice

These quantities support supply capacity, tariffs, conductor current, generator sizing, and correction studies.

SafetyCalculated operating power does not replace fault-level, thermal, protective-device, or isolation assessment.
Local code checkConfirm nominal voltage and frequency, source/earthing arrangement, conductor and protective-device duties, meter category and connection method, harmonic/resonance conditions, capacitor-bank product and discharge provisions, and the current local installation standard. Balanced 230/400 V examples are analytical models, not approval of a site design.

Knowledge check

What unit distinguishes reactive power from active power?

var rather than W. Apparent power is expressed in VA.

Answer: var rather than W. Apparent power is expressed in VA.

Practical exercise

Reproduce the example and close the P-Q-S triangle numerically.

Summary

  • P converts net energy
  • Q represents reactive exchange
  • S bounds both in sinusoidal analysis

Sources and review

  • IEC 60038: IEC standard voltages: IEC; 2009+A1:2021; International
  • IEC 61921: Low-voltage power-factor-correction banks: IEC; 2017; International
  • Harmonics and Power Quality Analysis webinar Q&A: IET; Current online guidance; 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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