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L09b / Three-phase AC circuits

Three-Phase AC Circuits

Worked examples on phase sequence, wye/delta conversion, branch currents, and feeder power balance.

Available51 slides
Three-phase example references and checks

01 / UNDERSTAND & PREDICT

Understand the model, then predict the result

Finalized lecture slides

Open / download original PDF ↗

Follow the original explanations, diagrams, derivations, and examples in slide order, then use the companion experiment below.

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Figures and page order follow the student PDF for this lecture.

At 400 V line voltage with three identical 20+j15 Ω impedances, does changing Y to Δ make line current √3 or 3 times larger?
  • Distinguish phase/line voltage and branch/line current.
  • Check √3 and 30° using voltage differences and terminal KCL.
  • Check three-phase power and zero neutral current in a balanced set.
Three-phase AC circuits: concept and calculation route
Course-authored concept route; the numerical experiment follows below.

Sequence and reference

This module uses source VAN = Vphase∠0° relative to a reference neutral. In abc, B and C are −120° and +120°. Reversing sequence changes angles while retaining balanced-load total power.

Wye Y

V_{LL}=\sqrt3 V_{phase},\qquad I_{line}=I_{phase}

A Y branch sees VLL/√3, and branch current equals line current. For positive sequence VAB = VAN−VBN leads VAN by 30°.

Delta Δ

I_A=I_{AB}-I_{CA},\qquad Z_Y=Z_\Delta/3

A Δ branch sees line voltage. Find IAB, IBC, ICA, then IA = IAB−ICA. Balanced line-current magnitude is √3 times branch current. Equivalent Y impedance is ZΔ/3.

Three-phase power

S_{3\phi}=\sum_k\mathbf V_k\mathbf I_k^*,\quad |S|=\sqrt3 V_{LL}I_{line}

Sum source phase-voltage times line-current conjugates, or sum load branch powers. Balanced sinusoidal three-phase total instantaneous power is constant although each phase power oscillates.

Baseline example: check each step

  1. Y branch voltage = 400/√3 ≈ 230.940 V.
  2. Branch and line current ≈ 9.238 A; impedance angle ≈ 36.870°.
  3. P=5120 W, Q=3840 var, |S|=6400 VA.
  4. With the same branch impedance in Δ: branch current=16 A, line current≈27.713 A, P=15360 W.
Original slide headings for this lecture37
  1. 1Lecture map
  2. 2Balanced set and phase sequence
  3. 3Example 1: a positive-sequence voltage set
  4. 4Example 1, Q1: positive-sequence B and C voltages
  5. 5Example 1, Q2: all three voltages after a +30° shift
  6. 6Example 1, Q3: shifted voltages in rectangular form
  7. 7Example 1, Q4: balance verification after the shift
  8. 8Y connection: phase and line quantities
  9. 9Example 2: a balanced Y-connected load
  10. 10Example 2, step 1: line voltage to phase voltage
  11. 11Example 2, step 2: complete the phase-voltage set
  12. 12Example 2, step 3: solve the phase-A current
  13. 13Example 2, step 4: other line currents and neutral
  14. 14Example 2, step 5: rebuild and verify line voltages
  15. 15Delta connection: branch quantities and line currents
  16. 16Example 3: balanced 13.8-kV delta load
  17. 17Example 3, step 1: branch-voltage set
  18. 18Example 3, step 2: branch-current set
  19. 19Example 3, step 3: line current from KCL
  20. 20Example 3, step 4: complete current set and checks
  21. 21Balanced Y–Delta transformation
  22. 22Y–Delta: terminal and branch quantities
  23. 23Example 4: an equivalent delta for a Y load
  24. 24Example 4, Q1: Y-to-delta impedance conversion
  25. 25Example 4, Q2: delta branch currents
  26. 26Example 4, Q3: line-current comparison
  27. 27Example 4, Q4: delta-to-Y conversion
  28. 28Balanced three-phase power
  29. 29Source, line, and load power
  30. 30Example 5: power with line impedance
  31. 31Example 5: the source–line–load circuit
  32. 32Example 5, step 1: line current
  33. 33Example 5, step 2: source complex power
  34. 34Example 5, step 3: three-phase line absorption
  35. 35Example 5, step 4: load power by two routes
  36. 36Example 5, step 5: check the power balance
  37. 37Complete solution workflow
Cross-check the original slides

02 / EXPLORE

Change one input and explain the response

Switch to Δ and multiply both impedance components by 3. Verify the original Y current and power return. Reverse sequence and inspect VAB phase.

Advanced parameters / test readings

Preparing the model.

Load branch voltage—
Branch current—
Line current—
Total real power—
Total reactive power—
Balanced current sum—

Three phase voltages

Three line currents

Current intermediate values and numerical checks
The model uses an ideal balanced source, three identical impedances, and zero line impedance. The source neutral provides a phase reference; a delta load has no neutral conductor.
03 / EDIT & COMPUTE
Edit code to reproduce the model independently
Reproduce the baseline, then modify the parameter scan. The source contains reusable independent model functions; edit the current function and inspect numerical checks.
case is a snapshot of the controls when you press Run. Call solve(case) and assign the final solution to result to plot it.
 
Download teaching models
The first run needs internet access to download Python. Computation stays in your browser; the solver uses only the standard library.
Ready to run.
Output appears here.
Inspect and edit the model source (advanced)
Edit this module's function and run again. case.module selects the module; solve(case) returns values, plots, and checks. The parameter experiment keeps the original JavaScript reference for comparison.
04 / CHECK & EXPLAIN
Companion experiment practice and feedback
Fixed practice inputs
VLL=400 V RMS, abc sequence, Y branch Z=20+j15 Ω, VAN angle 0°.
Practice uses fixed baseline inputs independently of the controls. Each field displays its tolerance.
±0.05 V
±0.05 A
±0.05 W
  
With fixed line voltage and identical branch impedance, how much does line current increase from Y to Δ?
Finally, explain in your own words
  1. What are the inputs, references, and main assumptions?
  2. Switch to Δ and multiply both impedance components by 3. Verify the original Y current and power return. Reverse sequence and inspect VAB phase.
  3. Did your code edit change physical parameters, the method, or representation bases? Which check helps identify that?
Passing numerical and understanding checks records this lecture’s companion practice as “practice checks passed.”