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L17 / Three-phase transformers

Three-Phase Transformers I

Build three-phase banks and distinguish winding, line, and bank relationships.

Available38 slides
Three-phase banks: coils, terminals and angles

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.

L17 original slide 1 of 38
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Figures and page order follow the student PDF for this lecture.

A 60 MVA Y–Δ bank has winding ratio a=10 and 138 kV HV line voltage. Does its LV line voltage equal 13.8 kV? Predict the voltage and angle before using the controls.
  • Separate winding count, supply phases and bank construction.
  • Derive line voltage and current from paired winding quantities.
  • Find phase displacement from fixed dots and explicit delta coil directions.
Three-phase banks: coils, terminals and angles: baseline circuit or quantitative model illustration
Model illustration at baseline inputs. Change the parameters in the next section to explore the live diagram.

Winding and phase are different counts

A three-phase two-winding transformer has two winding sets, with three phase coils per set. Three single-phase units can form a bank; a common-core unit provides the same terminal functions.

Winding ratio versus terminal ratio

k_{LL}=a\frac{k_H}{k_L},\quad k_Y=\sqrt3,\ k_\Delta=1,\quad S_\phi=S_{3\phi}/3

a compares paired dotted-to-undotted coil voltages. Y has line voltage √3 times coil voltage; Δ has equal line and coil voltages. Bank MVA is three times phase MVA.

Delta joining order and phase displacement

\delta_{LH}=\angle V_{ab,L}-\angle V_{AB,H},\quad V_{AB,H}=\sqrt3e^{j30^\circ}V_{AN,H}

Supply sequence is positive on both sides. ABC names coil directions AB, BC, CA; ACB names AC, BA, CB. Keep the paired dots fixed. Y–Δ with low abc gives δLH=−30°; switching low joining to acb gives +30°. These joining names do not reverse the supply sequence.

Vector groups and grounding

Dyn1 indicates HV delta, LV wye, a brought-out neutral, and 30° LV lag. Dyn11 corresponds to +30° δLH. A neutral terminal being available does not specify its grounding. Y has a star point; delta has no star-point neutral, but grounding arrangements can still be provided.

Baseline example: check each step

  1. Each phase is rated at 60/3=20 MVA. EH=138/√3=79.6743 kV and EL=EH/10=7.96743 kV.
  2. H is Y and L is Δ: kLL=√3×10=17.3205. Thus Vab,L=7.96743 kV, rather than 13.8 kV.
  3. VAN,H=(138/√3)∠−30° kV. Matching dots and low abc make Vab,L=7.96743∠−30° kV.
  4. IH,line=60×1000/(√3×138)=251.022 A. IL,coil=20×1000/7.96743=2510.218 A; IL,line=√3 IL,coil=4347.826 A.
Original slide headings for this lecture38
  1. 1Three-Phase Transformers I
  2. 2Lectureoutline
  3. 3PART 1 Windings and phases
  4. 4Windingand magnetic coupling
  5. 5Two-windingand three-winding
  6. 6Phaseand phase systems
  7. 7Windingcount and phasecount
  8. 8Single-phaseand three-phase
  9. 9Bankand common core
  10. 10PART 2 Connections and phase shift
  11. 11Terminaland phase conventions
  12. 12Dotsand matching coils
  13. 13Y:line and coilquantities
  14. 14Δ:line and coilquantities
  15. 15Fromcoils to bank terminals
  16. 16Y–Y:physical connection
  17. 17Y–Y:terminal relations
  18. 18Y–Δ:physical connection
  19. 19Y–Δ:terminal relations
  20. 20Y–Δ:terminal phase shift
  21. 21Δ–Y:physical connection
  22. 22Δ–Y:terminal relations
  23. 23Δ–Y:terminal phaseshift
  24. 24Δ–Δ:physical connection
  25. 25Δ–Δ:terminal relations
  26. 26Fourconnections: summary
  27. 27Industrynotation: vectorgroups
  28. 28Y:neutral grounding
  29. 29Δ:grounding choices
  30. 30Single-linesymbols andgrounding
  31. 31PART 3 Example: a three-phase transformer
  32. 32Example: given data
  33. 33Example: find andverify
  34. 34Solution(a): ratingsand line voltages
  35. 35Solution(b): voltagephasors
  36. 36Solution(c): ratedcurrents
  37. 37Solution(d): reconnectthe coils
  38. 38Summary
Cross-check the original slides

02 / EXPLORE

Change one input and explain the response

Compare all four bank connections. Then change a delta joining order: explain the phase displacement from the paired red coils while checking unchanged voltage and current magnitudes.

Advanced parameters / test readings

Preparing the model.

LV line voltage—
H/L line-voltage ratio—
LV relative to HV (δLH)—
LV winding voltage—
Rated HV line current—
Rated LV line current—
Rated LV winding current—
Power per phase—

Voltage versus winding ratio

Rated current versus winding ratio

Current intermediate values and numerical checks
Ideal balanced positive-sequence bank, with fixed dot correspondence and no impedance, excitation or grounding-current calculation. Delta joining order follows the L17 diagrams. Diagram arrows compare normalized line-voltage directions, not magnitudes.


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
L17 example: 60 MVA, 138 kV HV line voltage, a=10, H wye, L delta abc; positive sequence and matching dots. VAB,H is the 0° reference.
Practice uses fixed baseline inputs independently of the controls. Each field displays its tolerance.


±0.02 kV
±0.1 °
±0.1 A



Switch only the LV delta joining order from abc to acb. What changes?


Finally, explain in your own words
  1. What are the inputs, references, and main assumptions?
  2. Compare all four bank connections. Then change a delta joining order: explain the phase displacement from the paired red coils while checking unchanged voltage and current magnitudes.
  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.”









    

  




  
    
    




  
  
    

      
ECE 685 · L17
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