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L09 / Three-phase AC circuits

Three-Phase AC Circuits II

Use per-phase equivalents to calculate three-phase power and reconcile source, line, and load.

Available44 slides
Three-phase power and balance 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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Slide text
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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.

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 lecture39
  1. 1Lecture outline
  2. 2Identify the source, line, and load first
  3. 3Start with a balanced Y–Y network
  4. 4Extract one phase, then reconstruct three
  5. 5A Y source feeding a Δ load
  6. 6Compare Y and Δ at the same terminals
  7. 7Derive the balanced impedance relation
  8. 8Use the balanced Δ↔Y conversion
  9. 9Start from the Y–Y diagram: one-phase power
  10. 10Sum the three Y-connected load branches
  11. 11Change the load connection: Y versus Δ
  12. 12Derive both phase and line forms
  13. 13Three-phase power: phase and line forms
  14. 14From load power to feeder power balance
  15. 15Why line power uses |I_A|^2
  16. 16Source, line, and load power balance
  17. 17Example: complex power of one delta leg
  18. 18Example: power of the 13.8-kV delta
  19. 19Example A: 480-V wye feeder
  20. 20Example A: per-phase circuit
  21. 21Example A: solve the phase current
  22. 22Example A: line voltage drop
  23. 23Example A: load phase voltage and KVL
  24. 24Example A: load line-to-line voltage
  25. 25Example A: load complex power
  26. 26Example A: complete power balance
  27. 27Example A: independent checks
  28. 28Example B: 4.16-kV delta feeder
  29. 29Example B: delta-to-wye load conversion
  30. 30Example B: source reference and phase circuit
  31. 31Example B: solve the line current
  32. 32Example B: load voltage and KVL
  33. 33Example B: load terminal line voltage
  34. 34Example B: physical delta branch currents
  35. 35Example B: source complex power
  36. 36Example B: line and load absorption
  37. 37Example B: source, line, and load powers
  38. 38Example B: independent reconstruction checks
  39. 39A complete balanced three-phase solution
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.”