Apparent Power

S = √(P² + Q²)

Quick answer

This calculator solves S = √(P² + Q²) for Apparent Power. With Real Power 22 W, Reactive Power 47 W, the result is 51.89 W. It solves in 3 directions, so you can enter the values you have and get the one you need.

Calculator

Apparent Power (W)
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Formula

S = √(P² + Q²)

Description

Apparent power is the vector sum of real power (which does useful work) and reactive power (which shuttles energy back and forth between source and load). It represents the total power that must be supplied by the source and transmitted through the wiring, even though only the real power component performs useful work. Transformers, cables, and generators must be rated for apparent power, not just real power. The power triangle relates S, P, and Q through the Pythagorean theorem, with the power factor angle between P and S.

Variables

  • S — Apparent power in volt-amperes (VA)
  • P — Real (active) power in watts (W)
  • Q — Reactive power in volt-amperes reactive (VAR)

Practical Notes

Units are deliberately different: watts for real power, VAR for reactive, VA for apparent. A 1000 VA transformer with a 0.8 power factor load delivers only 800W of real power. The remaining 600 VAR circulates as reactive power. UPS systems and generators are rated in VA to account for worst-case power factor loads.

Symbols and units

Symbol Meaning Default unit Accepted units
S Apparent Power W µW, mW, W, kW
P Real Power W µW, mW, W, kW
Q Reactive Power W µW, mW, W, kW

Step by step

  1. 1. What you know: Real Power 22 W, Reactive Power 47 W
  2. 2. The relation: S = √(P² + Q²)
  3. 3. Substitute: P = 22 W, Q = 47 W
  4. 4. Result: Apparent Power = 51.89 W

Worked examples

Minimum range

Given: Real Power 22 mW, Reactive Power 47 mW

Result: Apparent Power = 51.89 mW

Low range

Given: Real Power 0.22 W, Reactive Power 0.47 W

Result: Apparent Power = 0.5189 W

Typical

Given: Real Power 22 W, Reactive Power 47 W

Result: Apparent Power = 51.89 W

High range

Given: Real Power 2.2 kW, Reactive Power 4.7 kW

Result: Apparent Power = 5.189 kW

Maximum range

Given: Real Power 22 kW, Reactive Power 47 kW

Result: Apparent Power = 51.89 kW

Solving for each variable

Solve for Apparent Power

Real Power 22 mW, Reactive Power 47 mW → 51.89 mW

Solve for Real Power

Apparent Power 100 mW, Reactive Power 22 mW → 97.55 mW

Solve for Reactive Power

Apparent Power 100 mW, Real Power 47 mW → 88.27 mW

How Apparent Power changes with Real Power

Real Power (input) Apparent Power (result)
2.2 W 47.05 W
5.5 W 47.32 W
11 W 48.27 W
16.5 W 49.81 W
22 W 51.89 W
33 W 57.43 W
44 W 64.38 W
88 W 99.76 W
154 W 0.161 kW
0.22 kW 0.225 kW

The same result in other units

Unit Value
mW 51890
W 51.89
kW 0.05189

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Frequently asked questions

How do you calculate Apparent Power?

Use S = √(P² + Q²). With Real Power 22 W, Reactive Power 47 W, Apparent Power works out to 51.89 W.

What is the formula for Apparent Power?

The formula is S = √(P² + Q²). Apparent power is the vector sum of real power (which does useful work) and reactive power (which shuttles energy back and forth between source and load).

How do you find Real Power when you know the other values?

Rearrange for Real Power: with Apparent Power 100 mW, Reactive Power 22 mW, Real Power is 97.55 mW.

How do you find Reactive Power when you know the other values?

Rearrange for Reactive Power: with Apparent Power 100 mW, Real Power 47 mW, Reactive Power is 88.27 mW.

What units does Apparent Power use?

Apparent Power is shown in mW, W, kW. Values are held in SI base units internally, so you can mix prefixes without converting by hand.

Related Concepts

All AC Circuits formulas →
Resonant Frequency Impedance (RLC Series) Quality Factor (Bandwidth) Quality Factor (RLC) Power Factor Apparent Power Reactive Power RLC Circuit Bandwidth

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