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Reminder sheet

AC power

Bastaya

In AC, power has three faces: active (useful), reactive (exchanged) and apparent (what the cable carries). cos φ tells which share is truly useful.

In AC we distinguish the apparent power S = U·I (VA), the active power P = U·I·cos φ (W, the only one producing work or heat) and the reactive power Q = U·I·sin φ (var, exchanged with coils and capacitors). They form a triangle: S² = P² + Q².

The power factor cos φ = P/S measures the efficiency of the installation: a low cos φ requires more current for the same useful power, hence more line losses. It is corrected by adding capacitors in parallel.

Formulas

The three powersS = U · I
P = U · I · cos φ
Q = U · I · sin φ
QuantitySymbolUnit
Apparent powerSVA
Active powerPW
Reactive powerQvar
Power triangleS² = P² + Q²
cos φ = PS
QuantitySymbolUnit
Power factorcos φ-
Power factor correctionC = P · (tan φ₁ − tan φ₂)U² · ω
QuantitySymbolUnit
Capacitance to addCF
Phase shifts before / afterφ₁, φ₂rad

Worked example

A single-phase motor draws 5 A at 230 V with cos φ = 0.8. Find S, P and Q. What capacitance should be placed in parallel to raise the power factor to 0.95 (50 Hz)?

S = U · I = 230 × 5 = 1 150 VA; P = S · cos φ = 1 150 × 0.8 = 920 W; Q = S · sin φ = 1 150 × 0.6 = 690 var (check: 920² + 690² = 1 150²).

tan φ₁ = 0.75 (cos φ = 0.8); tan φ₂ = 0.329 (cos φ = 0.95).

C = P · (tan φ₁ − tan φ₂) / (U² · ω) = 920 × 0.421 / (230² × 314) = 387 / 16.6 × 10⁶ = 23 µF.

Common mistakes

  • Mixing the units: W for active power, VA for apparent, var for reactive; they do not add up, they form a triangle.
  • Computing cos φ = P / Q: the power factor is P / S.
  • Believing the correction capacitor consumes power: it only exchanges reactive power with the motor, P does not change and the line current decreases.

IEC symbols

WWattmeter
Capacitor

Illustration

P (W)Q (var)S (VA)φ

Power triangle: S² = P² + Q², cos φ = P / S.

Frequently asked questions

Why does the utility penalise a poor cos φ?
For the same useful power P, a low cos φ requires a larger current (I = P / (U·cos φ)), hence line losses and oversized cables the utility pays for. Beyond a threshold it bills the reactive power.
What does the electricity meter measure?
The active energy, in kWh: the integral of P over time. A reactive energy meter (kvarh) is only installed at large consumers.

15 exercises in this topic

This chapter is covered by the paths: General electricity, ISL-oriented exercises

All sheets · All formulas · Glossary of electrical quantities · Standard electrical symbols (IEC 60617)

Practise this topic
Simulated time00:00:00
Q = I·t-
W = P·t-

Workbench

Circuit diagram CEI 60617

How does it work?

The perfboard: each big hole is a circuit node. You push a single leg of a component (resistor, lamp...) into it. The 4 small holes around it are connected to it: that is where chips and leads plug in.

  1. Drag a component onto the board: it snaps onto the grid. R rotates it.
  2. To join neighbouring big holes, click Jumper chip in the palette (or C) then drag on the board: a line of chips is laid at once. Bridged holes become a single node.
  3. For long links and for the power supply (placed next to the board), drag a lead from one big hole to another (or to a terminal).
  4. As soon as a loop is closed, the circuit is solved in real time (Ohm + Kirchhoff) and the diagram follows.
  5. Zoom with the wheel (or + / −, two-finger pinch); drag the background to pan, 0 fits the board. The + buttons around the board add a row or a column.

Del delete · R rotate · C chip brush · Esc cancel · wheel zoom · drag the background to pan

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