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L06 / Single-phase AC circuits

Single-Phase AC Circuits II

Calculate real, reactive, complex, and apparent power with consistent load and generator signs.

Available19 slides
Complex power, signs, and the power triangle

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.

L06 original slide 1 of 19
L06 · 1 / 19
Slide text
Loading slide text.

Figures and page order follow the student PDF for this lecture.

At 120 V RMS and 10 A RMS, voltage leads current by 50°. Why is P not 1200 W? How does source current change when pf reaches 0.95?
  • Keep RMS, cosine reference, and current direction consistent.
  • Use the current conjugate to calculate P, Q, |S|, and power factor.
  • Calculate target Q, capacitor rating, and corrected source current.
Single-phase AC circuits: concept and calculation route
Course-authored concept route; the numerical experiment follows below.

RMS and phasors

v(t)=\sqrt2 V\cos(\omega t+\phi_V)

The waveform coefficient is peak = √2×RMS, while phasor magnitude is RMS. The original L05 page retains the full conversion and phasor-addition lesson; this module connects it to power.

Conjugate and signs

S=\mathbf V\mathbf I^*=VI(\cos\Delta\phi+j\sin\Delta\phi)

Current enters the positive load terminal, using the passive convention. Δφ = φV−φI; positive Δφ means lagging current and positive Q. Calculate S = V I* before reading P and Q.

Power triangle

|S|^2=P^2+Q^2,\qquad pf=\cos\Delta\phi

P is the real-axis projection of |S| and pf = P/|S|. Negative Q means leading current; a pf number alone omits lead/lag. Instantaneous power oscillates while its average is P.

Baseline example: check each step

  1. Δφ = 50° and |S| = 1200 VA.
  2. P ≈ 771.345 W, Q ≈ 919.253 var, pf ≈ 0.643 lagging.
  3. Target Q ≈ 253.529 var; capacitor rating ≈ 665.724 var.
  4. C ≈ 122.628 μF; load current remains 10 A while source current falls to ≈ 6.766 A.
Open the L05 waveform and phasor-addition lesson →
Original slide headings for this lecture17
  1. 1Lecture outline
  2. 2Starting problem
  3. 3Starting-problem solution
  4. 4Instantaneous power
  5. 5Average real power P
  6. 6Reactive power Q
  7. 7Recap: complex and apparent power
  8. 8Power triangle
  9. 9Power factor
  10. 10Load and generator sign conventions
  11. 11Reactive power of loads and generators
  12. 12Example setup
  13. 13CE-2B Step 1
  14. 14CE-2B Step 2
  15. 15CE-2B Step 3
  16. 16CE-2B Step 4
  17. 17Complex-power summary
Cross-check the original slides

02 / EXPLORE

Change one input and explain the response

Set Δφ to −30° and explain Q and the absence of added capacitance. Edit the Python phase and check P²+Q²=|S|².

Advanced parameters / test readings

Preparing the model.

Real P—
Reactive Q—
Load pf—
Correction capacitance—
Capacitor rating—
Corrected pf—
Corrected source current—

Voltage/current phase comparison

Instantaneous and average power

Current intermediate values and numerical checks
The model assumes single-frequency sinusoidal steady state, fixed voltage, and fixed load power. Correction changes source Q and current; normalized curves compare phase only.
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
V=120 V RMS, I=10 A RMS, voltage leads current by 50°, ω=377 rad/s, target pf=0.95 lagging.
Practice uses fixed baseline inputs independently of the controls. Each field displays its tolerance.
±0.05 W
±0.05 var
±0.05 μF
  
Which operation gives load complex power under the passive convention?
Finally, explain in your own words
  1. What are the inputs, references, and main assumptions?
  2. Set Δφ to −30° and explain Q and the absence of added capacitance. Edit the Python phase and check P²+Q²=|S|².
  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.”