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L13 / Per-unit systems

Per Unit System I

Choose coherent bases and normalize voltage, current, impedance, admittance, and power.

Available44 slides
Base selection and physical recovery

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.

Independent and derived bases

I_{b,3\phi}=\frac{S_b}{\sqrt3 V_{b,LL}},\quad I_{b,1\phi}=\frac{S_b}{V_b},\quad Z_b=\frac{V_b^2}{S_b}

After choosing Sb and Vb, derive other bases coherently. Three-phase uses total Sb and line voltage Vb,LL; single-phase uses terminal voltage and single-phase Sb.

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.

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 lecture38
  1. 1Lecture outline
  2. 2What is a per-unit system?
  3. 3Why use per unit?
  4. 4Per unit makes voltage levels comparable
  5. 5One network, several voltage levels
  6. 6The same equipment in per unit
  7. 7Equipment operating quantities in per unit
  8. 8Voltage, current, and power along a line
  9. 9Four coupled electrical quantities
  10. 10Circuit laws couple the four quantities
  11. 11Two independent bases determine the rest
  12. 12Convert all four quantities
  13. 13One impedance base for Z, R, and X
  14. 14Admittance and its reciprocal base
  15. 15One power base for P, Q, and S
  16. 16Coherent bases preserve circuit laws
  17. 17Two independent bases for Y and delta
  18. 18Y branch quantities and voltage references
  19. 19Y branch: calculating the bases
  20. 20Y branch: normalized circuit laws
  21. 21Delta branch quantities and current directions
  22. 22Delta branch: calculating the bases
  23. 23Delta branch: normalized circuit laws
  24. 24Checking the delta–Y relationship
  25. 25Normalization workflow
  26. 26Per-unit calculation and recovery
  27. 27Example: circuit data and specified bases
  28. 28Step 2: Derive the current base
  29. 29Step 2: Derive the impedance base
  30. 30Step 3: Normalize the given data
  31. 31The circuit ready for per-unit calculation
  32. 32Step 4: Calculate the line current
  33. 33Step 4: Calculate the load voltage
  34. 34Step 4: Calculate the circuit powers
  35. 35Step 5: Recover current and voltage
  36. 36Step 5: Recover the circuit powers
  37. 37Check: the recovered physical circuit
  38. 38Per-unit summary
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.”