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L10 / Transformer modeling

Transformer Modeling I

Derive ideal voltage and current ratios, power conservation, and impedance referral.

Available38 slides
Ideal transformers and referral side

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.

L10 original slide 1 of 38
L10 · 1 / 38
Slide text
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Figures and page order follow the student PDF for this lecture.

Does a turns ratio a=10 always convert 138 kV high-side line voltage to 13.8 kV low-side line voltage? Compare Y–Y and Δ–Y before studying tests and losses.
  • Distinguish winding turns ratio from three-phase line-voltage ratio.
  • Infer equivalent parameters from open/short tests with consistent referred sides.
  • Estimate regulation, copper loss, and efficiency and check approximation conditions.
Three-phase transformer modeling: concept and calculation route
Course-authored concept route; the numerical experiment follows below.

Ideal winding and referral

V_H=aV_L,\quad I_H=I_L/a,\quad Z_H=a^2Z_L

a=NH/NL=VH,winding/VL,winding. Ideal winding-current magnitudes scale inversely and complex power is conserved. Refer low-side impedance by a², voltage by a, and current by 1/a.

Baseline example: check each step

  1. Bank teaching case: 138 kV, a=10, Y–Y, 30 MVA, loading 0.8, pf=0.9.
  2. No-load LV line voltage is 13.8 kV; ideal LV line current ≈ 1004.087 A.
  3. rpu=0.01 and xpu=0.08 give ≈ 3.510% regulation; copper and 30 kW core loss give ≈ 98.983% efficiency.
  4. The independent L12 test case is a 10 kVA, 2400/240 V single-phase unit. OC gives Rc,L=480 Ω and Xm,L≈123.935 Ω.
  5. SC gives Req,H≈10.368 Ω and Xeq,H≈26.869 Ω. Refer excitation parameters to HV using the test transformer’s fixed ratio 10².
Original slide headings for this lecture34
  1. 1Lecture outline
  2. 2Transformers in the power system
  3. 3Transformers in Power Systems
  4. 4How a transformer works
  5. 5Winding variables and units
  6. 6Core material and geometry
  7. 7Magnetizing current
  8. 8Zero winding resistance Ideal assumption 1
  9. 9Zero leakage flux Ideal assumption 2
  10. 10Infinite core permeability Ideal assumption 3
  11. 11Zero core loss Ideal assumption 4
  12. 12Ideal model and reference directions
  13. 13Flux linkage in an ideal winding
  14. 14Equal flux per turn
  15. 15Induced winding voltage
  16. 16Voltage ratio
  17. 17Ampere-turn balance
  18. 18Current ratio
  19. 19Power quantities and units
  20. 20Complex-power conservation
  21. 21Load impedance referred to H
  22. 22Impedance referred to L
  23. 23Low-voltage-side equivalent circuit
  24. 24Equivalent circuit referred to H
  25. 25Example setup
  26. 26Example: Step 1
  27. 27Example: Step 2
  28. 28Example: Step 3
  29. 29Example: Step 4
  30. 30Example: Step 5
  31. 31Example: Step 6
  32. 32Direct high-side impedance
  33. 33Apparent-power check
  34. 34Ideal-transformer summary
Cross-check the original slides

02 / EXPLORE

Change one input and explain the response

Switch to Δ–Y and check line ratio a/√3. Raise OC watts above VI and inspect the consistency feedback; explain why invalid readings cannot determine parameters.

Advanced parameters / test readings

Preparing the model.

LV no-load line voltage—
H/L line-voltage ratio—
Ideal LV line current—
Approx. regulation—
Loss-based efficiency—
L12 Rc (LV)—
L12 Req (HV)—
L12 Xeq (HV)—

Approximate regulation versus loading

Efficiency versus loading

Current intermediate values and numerical checks
The bank uses ideal winding magnitudes and first-order small-drop approximations. L12 tests concern a separate fixed 10 kVA single-phase unit; they do not parameterize this bank. L20–L21 are the post-Exam-1 connection extension.
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
Bank: VH,LL=138 kV, a=10, Y–Y. Independent L12 test unit: 10 kVA, 2400/240 V; OC(LV) 240 V/2 A/120 W; SC(HV) 120 V/4.1667 A/180 W.
Practice uses fixed baseline inputs independently of the controls. Each field displays its tolerance.
±0.05 kV
±0.05 Ω
±0.05 Ω
  
Does the label Δ–Y alone uniquely determine +30° or −30° low-side line displacement?
Finally, explain in your own words
  1. What are the inputs, references, and main assumptions?
  2. Switch to Δ–Y and check line ratio a/√3. Raise OC watts above VI and inspect the consistency feedback; explain why invalid readings cannot determine parameters.
  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.”