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L14 / Per-unit systems

Per Unit System II

Propagate bases through transformers and apply change of base across a multi-voltage network.

Available38 slides
Corresponding bases and change of base

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.

For the same 1+j4 Ω impedance, what happens when power base changes from 100 to 50 MVA? Does physical impedance change?
  • Choose power and voltage bases before deriving current and impedance bases.
  • Distinguish single-phase terminal bases from three-phase line-voltage/total-power bases.
  • Verify invariant per-unit impedance on corresponding transformer bases.
Per-unit systems: concept and calculation route
Course-authored concept route; the numerical experiment follows below.

Across the transformer

V_{b,H}=aV_{b,L},\quad Z_{b,H}=a^2Z_{b,L}

Keep the same power base and propagate voltage bases through the corresponding ideal ratio. Here both connection factors match, so the voltage ratio is a. Physical impedance and its base both scale by a², retaining per-unit impedance.

Change of base

z_{pu,new}=z_{pu,old}\frac{S_{b,new}}{S_{b,old}}\left(\frac{V_{b,old}}{V_{b,new}}\right)^2

Keep physical Z fixed and choose new Sb and Vb. Per-unit numbers change with the base; do not interpret that change as modified equipment.

Baseline example: check each step

  1. For three-phase Sb=100 MVA, Vb,H=138 kV and a=10, Vb,L=13.8 kV.
  2. Zb,L=1.9044 Ω and Ib,L≈4183.698 A.
  3. 1+j4 Ω gives zpu≈0.525100+j2.100399.
  4. Referring to HV gives 100+j400 Ω with Zb,H=190.44 Ω and the same zpu.
  5. Halving the power base at fixed voltage bases halves zpu, while recovery with the new Zb retains physical Z.
Original slide headings for this lecture34
  1. 1Lecture outline
  2. 2The ideal transformer and local bases
  3. 3Transformer per-unit base rules
  4. 4The current and impedance bases on each side
  5. 5Voltage normalization cancels the turns ratio
  6. 6Current normalization cancels the inverse ratio
  7. 7The ideal per-unit circuit
  8. 8Approximate circuit referred to H
  9. 9Equivalent impedance before and after normalization
  10. 10The series per-unit circuit
  11. 11The complete physical circuit
  12. 12Normalization of the excitation branch
  13. 13The complete per-unit circuit
  14. 14Transformer normalization: summary
  15. 15The goal: one complete per-unit circuit
  16. 16The supplied component bases are incompatible
  17. 17System bases for every component
  18. 18Change of base
  19. 19Voltage and current
  20. 20Impedance
  21. 21Admittance
  22. 22Power
  23. 23System application
  24. 24Example circuit
  25. 25Step 1: select corresponding voltage bases
  26. 26Step 2: calculate both impedance bases
  27. 27Step 3: normalize directly on the 120-V side
  28. 28Step 4: verify from the 480-V side
  29. 29The per-unit result on both sides
  30. 30The transformer series circuit in per unit
  31. 31Recovering quantities on the correct side
  32. 32The same transformer on new system bases
  33. 33The new bases preserve the physical impedance
  34. 34Summary
Cross-check the original slides

02 / EXPLORE

Change one input and explain the response

Change only Sb and check physical recovery. Change Vb,H and inspect the square dependence. Edit the Python base formula and use recovery error to test it.

Advanced parameters / test readings

Preparing the model.

LV voltage base—
LV current base—
LV impedance base—
Re(Zpu)—
Im(Zpu)—
Recovered physical resistance—
Current per unit—
Physical apparent power—

Base changes and per-unit impedance

Recovery of the same physical impedance

Current intermediate values and numerical checks
This lab uses corresponding voltage ratios, one power base, and a fixed physical impedance. Switching single/three-phase changes quantity interpretation; labels state voltage and total-power conventions.
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
Three-phase Sb=100 MVA, Vb,H=138 kV, a=10, physical LV Z=1+j4 Ω.
Practice uses fixed baseline inputs independently of the controls. Each field displays its tolerance.
±0.005 Ω
±0.005 pu
±0.05 A
  
If only power base changes while voltage base and physical impedance stay fixed, what remains unchanged?
Finally, explain in your own words
  1. What are the inputs, references, and main assumptions?
  2. Change only Sb and check physical recovery. Change Vb,H and inspect the square dependence. Edit the Python base formula and use recovery error to test it.
  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.”