Power Stations WORLDWIDETHE GLOBAL EDITION 01
LESSON 3 OF 6 · ORIGINAL EDUCATIONAL EXAMPLE

kW, kVA and kvar

Use the power triangle to distinguish real, apparent and reactive power without confusing them with energy.

Learning goal: Calculate apparent power and power factor from real and reactive power in a sinusoidal load.

Prepared by Power Stations Worldwide. Sources checked on 30 September 2026; not independently engineering peer reviewed.

The essentials

Three quantities, different meanings

Real power P in kW is the average rate of electrical energy transfer. Reactive power Q in kvar describes the quadrature component associated with alternating energy exchange. Apparent power S in kVA expresses RMS voltage-current loading.

A triangle, not a sum

For the sinusoidal single-phase or balanced three-phase cases here, S² = P² + Q². The hypotenuse is apparent power: kVA is not obtained by adding kW and kvar arithmetically. Complex power is P + jQ; S here denotes its magnitude.

Power is not energy

kW, kVA and kvar are power quantities. kWh is energy: a constant 40 kW load running for two hours uses 80 kWh. Under the load convention used here, inductive lagging Q is positive; capacitive leading Q is negative.

The 40–30–50 power triangle

The 40–30–50 power triangleA right triangle has horizontal real power P of 40 kilowatts, upward reactive power Q of plus 30 kvar, and apparent power S of 50 kVA on its hypotenuse. The angle phi is between P and S.P = 40 kWQ =+30 kvarS = 50 kVAφ
Original scaled triangle for the hypothetical lagging load. Positive Q is upward; leading loads have negative Q. Valid for the stated sinusoidal model.
HYPOTHETICAL VALUES · NOT PLANT RECORDS

Worked example

A load with 40 kW and +30 kvar

Given

  • Real power P = 40 kW
  • Reactive power Q = +30 kvar (lagging)
  • Sinusoidal steady state; balanced if three-phase

Formula toolbox

  • S = √(P² + Q²)
  • PF = P / S
  • P in kW, Q in kvar and S in kVA must use matching kilo prefixes

Calculate step by step

  1. S = √(40² + 30²) = √2,500 = 50 kVA.
  2. PF = 40 / 50 = 0.80 lagging.
  3. Check: 50² = 40² + 30²; 40 + 30 = 70 is not the apparent power.

50 kVA apparent power · 0.80 lagging power factor

Reactive power is not an additional 30 kW of average consumption. Distorted waveforms can invalidate this simple triangle based on fundamental reactive power.

Results are rounded only for display. Reproduce the calculations using unrounded intermediate values.

Quick quiz

Choose one answer for each question, then check your answers. This is a practice check, not a qualification. Answers stay in this page only; refreshing or leaving resets them.

1. A sinusoidal load takes 12 kW and +5 kvar. What is its apparent power?
2. Which unit measures electrical energy rather than power?

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Show answers and explanations (also works without JavaScript)
  1. 13 kVA. S = √(12² + 5²) = √169 = 13 kVA.
  2. kWh. kWh is energy. A steady kW value multiplied by elapsed hours gives kWh.

Curriculum connection: MIT 6.061 (Spring 2011). Its readings include 2007 notes; some equations use peak rather than RMS amplitudes. This lesson states its own value conventions and uses original diagrams, numbers and quiz questions. No MIT affiliation or endorsement is claimed.

Textbook references

Suggested technical background for this topic. These books are further reading, not proof that every explanation was checked against every edition. See the full bibliography and review scope.

  1. Fundamentals of Electric Circuits (7th ed.) — Alexander, C. K. & Sadiku, M. N. O., McGraw-Hill, 2021 · ISBN 9781260226409Real, reactive and apparent power in AC circuits.
  2. Electrical Machines, Drives and Power Systems (6th ed., Pearson New International Edition) — Wildi, T., Pearson, 2013 · ISBN 9781292024585Active, reactive and apparent power in power systems.