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L12 / Transformer modeling

Transformer Modeling III

Infer transformer parameters from open- and short-circuit tests; calculate regulation and efficiency.

Available39 slides
Test parameters, losses, and regulation

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.

L12 original slide 1 of 39
L12 · 1 / 39
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.

Open- and short-circuit tests

R_c=V_{oc}^2/P_{oc},\quad X_m=V_{oc}/I_m,\quad X_{eq}=\sqrt{|Z_{eq}|^2-R_{eq}^2}

The open test identifies excitation: Ic=Poc/Voc and Im=√(Ioc²−Ic²). The short test identifies series terms: |Zeq|=Vsc/Isc and Req=Psc/Isc². Readings require P≤VI and parameters need their test side.

Nonideal model and approximations

VR\approx\lambda(r_{pu}\cos\phi+x_{pu}\sin\phi)

Resistance gives copper loss, leakage reactance gives series drop, and excitation represents magnetization and core loss. Here rated-base rpu, xpu and loading λ estimate regulation at lagging pf.

Power and efficiency

\eta=\frac{P_{out}}{P_{out}+P_{cu}+P_{core}}

Real output is Srated×λ×pf. Copper loss varies with λ² and core loss is approximately constant at fixed voltage. Check ideal bank ratios separately from loss-based efficiency.

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 lecture35
  1. 1Lecture outline
  2. 2Determining model parameters
  3. 3The model established in L11
  4. 4Parameters to determine
  5. 5Open-circuit condition
  6. 6Why energize LV for the OC test?
  7. 7Open-circuit test
  8. 8Meter readings and RMS phasors
  9. 9Inferring the current phase
  10. 10OC formulas: resolve the measured current
  11. 11OC formulas: identify the core-loss branch
  12. 12OC formulas: identify the magnetizing branch
  13. 13Example: OC measurements
  14. 14Example: excitation-current components
  15. 15Example: core-loss resistance
  16. 16Example: magnetizing reactance
  17. 17Short-circuit condition
  18. 18Why energize HV for the SC test?
  19. 19Short-circuit test
  20. 20SC formulas: identify impedance and resistance
  21. 21SC formulas: identify leakage reactance
  22. 22Example: SC measurements
  23. 23Example: SC current phasor
  24. 24Example: series impedance magnitude
  25. 25Example: series resistance
  26. 26Example: leakage reactance
  27. 27Putting both test results on HV
  28. 28The L11 model with measured parameters
  29. 29Operation at rated load
  30. 30Voltage regulation: no-load and full-load
  31. 31Voltage regulation: series voltage drop
  32. 32Example: voltage regulation
  33. 33Losses in the operating circuit
  34. 34Efficiency: real-power balance
  35. 35Summary: tests, regulation, and efficiency
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.”