← All courses KANSAS STATE UNIVERSITY / ECE 685

L11 / Transformer modeling

Transformer Modeling II

Add winding resistance, leakage reactance, excitation, and checked equivalent-circuit approximations.

Available32 slides
Equivalent circuits and approximate models

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.

L11 original slide 1 of 32
L11 · 1 / 32
Slide text
Loading slide text.

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.

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.

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 lecture28
  1. 1Lecture outline
  2. 2Ideal-model limitation
  3. 3Ideal transformer equivalent circuit
  4. 4Winding resistance
  5. 5Leakage flux
  6. 6Why resistance and leakage reactance?
  7. 7Finite core permeability
  8. 8Core loss
  9. 9Why R_c and j X_m?
  10. 10Exact equivalent circuit
  11. 11Why referral and approximation?
  12. 12Referral: replace the boxed network
  13. 13Refer low-side quantities to H
  14. 14Referral derivation
  15. 15Equivalent circuit referred to H
  16. 16Excitation current
  17. 17Why the excitation branch can move
  18. 18Input-shunt approximation
  19. 19Equivalent series impedance
  20. 20Approximate circuit with excitation
  21. 21Neglecting excitation current
  22. 22Neglecting winding resistance
  23. 23Limits of the steady-state model
  24. 24Example: data and required quantities
  25. 25Example: circuit referred to H
  26. 26Example: load current
  27. 27Example: required input voltage
  28. 28Example: results and circuit check
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.”