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Courses/Three-Phase Systems/Delta Connection

lesson · 9min · Lesson 11 of 28

Line current = √3 × phase current: 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 · 9min
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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 current = √3 × phase current: proof

Three Phase Systems · Lesson 11 · Intermediate

IntermediateReview: professional review pending

Purpose

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

Before you beginDelta topology · Phasor subtraction

Learning objectives

  • Form a line-current equation
  • Derive the magnitude ratio
  • State angular displacement
  • Apply the relation only to balanced sets

At a delta terminal, line current is the vector difference between the two branch currents meeting there. Equal branch currents separated by 120 degrees produce a resultant √3 times one branch current.

DELTA VOLTAGE AND CURRENT PATHSDELTA VOLTAGE AND CURRENT PATHSL1L2L3Vph = VLIL = √3 Iph · 30° shift

Core theory

With declared current directions, a representative relation is IA = IAB − ICA. The exact signs and 30-degree lead/lag statement depend on the adopted sequence and reference convention.

For a balanced positive-sequence delta, |IL| = √3|Iph|. The line current is displaced by 30 degrees from its associated phase current under the chosen convention.

The shortcut does not directly solve unbalanced impedances, missing branches, harmonic components, or distorted currents; those require complex branch analysis and KCL.

Terms, symbols, and units
TermMeaningSymbolUnit
Line currentCurrent delivered through one line terminalILA
Branch currentCurrent within one delta sideIphA
Current displacementAngular difference between associated line and branch currentNot applicableNot applicable
|IL| = √3 × |Iph|

Balanced delta line-current magnitude is root-three times phase-current magnitude.

A
Worked exampleA balanced delta branch current is 12 A. Find line-current magnitude.

Assumptions: Balanced sinusoidal currents.

  1. Formula: IL = √3 Iph.
  2. Substitute: IL = 1.732 × 12 A.
  3. Calculate: IL ≈ 20.8 A.

Line-current magnitude is approximately 20.8 A.

Reasonableness check: The vector result is greater than 12 A but less than the scalar sum 24 A.

Common mistakes
  • Multiplying voltage by √3 in delta
  • Omitting the 30° relationship
  • Using the balanced formula after a branch opens

Where this appears in practice

The relationship supports delta load, winding, protective-device, and conductor calculations.

SafetyA correct current calculation does not establish conductor capacity, fault protection, isolation, or terminal safety.
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

Why is delta line current not twice branch current?

The two branch currents combine as phasors, not in-phase scalars. Their vector difference produces the √3 ratio.

Answer: The two branch currents combine as phasors, not in-phase scalars. Their vector difference produces the √3 ratio.

Practical exercise

Resolve two 12 A branch-current phasors into rectangular components and reproduce 20.8 A.

Summary

  • Line current is a phasor difference
  • Balanced magnitude ratio is √3
  • Unbalance needs full 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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