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L15 / Per-unit systems

Per Unit System III

Complete single-phase transformer and three-phase wye/delta normalization examples.

Available48 slides
Transformer sides and Y/Δ normalization checks

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.

Normalize and recover

z_{pu}=Z/Z_b,\qquad Z=z_{pu}Z_b

Divide physical quantities by matching bases, then multiply back to recover them. Coherent voltage, current, and power bases give |Spu|=Vpu Ipu in three-phase.

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 lecture47
  1. 1Lecture outline
  2. 2Part 1: single-phase voltage levels
  3. 3Example 1: circuit and data
  4. 4Example 1: questions
  5. 5Example 1 workflow: bases and current
  6. 6Example 1 workflow: voltages and power
  7. 7E1 Step 1: single-phase bases
  8. 8E1 Step 1: normalized data
  9. 9E1 Step 2: PU circuit
  10. 10E1 Step 2: network impedance
  11. 11E1 Step 2: total impedance
  12. 12E1 Step 2: series current
  13. 13E1 Step 3: T1 drop
  14. 14E1 Step 3: bus B2
  15. 15E1 Step 3: line drop
  16. 16E1 Step 3: bus B3
  17. 17E1 Step 3: bus B4
  18. 18E1 Step 3: voltage check
  19. 19E1 Step 3: load voltage
  20. 20E1 Step 3: actual currents
  21. 21E1 Step 3: voltage results
  22. 22E1 Step 4: source power
  23. 23E1 Step 4: absorbed powers
  24. 24E1 Step 4: power balance
  25. 25Example 1: final answer
  26. 26Part 2: three-phase Y–Δ normalization
  27. 27Example 2: circuit and data
  28. 28Example 2: questions
  29. 29Example 2 workflow: connection and bases
  30. 30Example 2 workflow: solve and restore
  31. 31E2 Step 1: Y–delta conversion
  32. 32E2 Step 1: equivalent Y load
  33. 33E2 Step 2: three-phase bases
  34. 34E2 Step 2: source phase voltage
  35. 35E2 Step 2: normalized data
  36. 36E2 Step 2: delta on the same base
  37. 37E2 Step 2: equivalent PU circuit
  38. 38E2 Step 3: line current
  39. 39E2 Step 3: equivalent phase voltage
  40. 40E2 Step 3: load line voltage
  41. 41E2 Step 4: delta branch voltages
  42. 42E2 Step 4: delta branch currents
  43. 43E2 Step 4: line-current check
  44. 44E2 Step 4: load power in pu
  45. 45E2 Step 4: Y–delta power check
  46. 46Example 2: circuit answer
  47. 47Two examples: base conventions
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.”