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Evgenii Konkin
Evgenii Konkin

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The Engineering Math Behind Capacitor Sizing for Power Factor Correction: Deriving the kVAR Formula Step by Step

#capacitorsizing #powerfactorcorrection #kvarcalculation #reactivepowercompensation

A 500 kW motor running at 0.75 power factor draws 444 kVAR of reactive power from the grid. Correcting that to 0.95 requires 248 kVAR of capacitors — roughly the size of a small car. That reactive power doesn't do useful work, but it heats cables, burdens transformers, and triggers utility penalties.

The Formula

The capacitor sizing formula is:

Qc = P × (tan(arccos(PF_initial)) - tan(arccos(PF_target)))
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Where:

  • Qc = capacitor size in kVAR
  • P = real power in kW (the actual work done)
  • PF_initial = existing power factor (decimal, e.g. 0.75)
  • PF_target = desired power factor (decimal, e.g. 0.95)

Why each term?

  • P scales the compensation: a larger load needs more kVAR for the same PF improvement.
  • tan(arccos(PF)) is the ratio of reactive to real power (kVAR/kW). The difference tan(φ1) - tan(φ2) is the reduction in reactive power per kW needed to shift from initial to target PF.
  • The formula comes from the power triangle: S² = P² + Q², where S is apparent power. Power factor = P/S = cos φ, so Q = P × tan φ.

Why not just Q = P × (tan φ1)? Because the load still needs some reactive power after correction (unless target PF = 1.0). The capacitor only supplies the difference.

Worked Example 1

Scenario: Factory with 800 kW load, initial PF = 0.65, target PF = 0.92.

  1. Compute φ1 = arccos(0.65) = 49.46°
  2. tan φ1 = tan(49.46°) = 1.169
  3. Compute φ2 = arccos(0.92) = 23.07°
  4. tan φ2 = tan(23.07°) = 0.426
  5. Qc = 800 × (1.169 - 0.426) = 800 × 0.743 = 594.4 kVAR

Result: 594 kVAR capacitor bank needed.

Worked Example 2

Scenario: Commercial building with 250 kW load, initial PF = 0.88, target PF = 0.98.

  1. φ1 = arccos(0.88) = 28.36° → tan = 0.540
  2. φ2 = arccos(0.98) = 11.48° → tan = 0.203
  3. Qc = 250 × (0.540 - 0.203) = 250 × 0.337 = 84.25 kVAR

Result: 84 kVAR capacitor bank.

What Engineers Often Miss

  1. Harmonic resonance: Adding capacitors shifts the system's resonant frequency. If harmonic sources (VFDs, rectifiers) are present, resonance can amplify voltage distortion. Always check harmonic impedance before installing fixed capacitors.

  2. Load variation: Fixed capacitors at a constant kVAR can cause overcorrection (leading PF) during light load periods. For variable loads, consider automatic power factor correction with staged switching.

  3. Utility tariff structure: Many utilities charge for reactive demand (kVAR) or have PF penalties. Calculate payback using actual tariff rates, not just kVAR savings. Sometimes correcting to 0.95 is optimal, not unity.

Try the Calculator

Estimate your capacitor size quickly with the Capacitor Sizing for Power Factor Correction calculator.

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