Power Triangle (S, P, Q)
S² = P² + Q²
Quick answer
This calculator solves S² = P² + Q², or S = √(P² + Q²) for Apparent Power (VA). With Real Power (W) 10 W, Reactive Power (VAR) 1.1, the result is 10.06. It solves in 3 directions, so you can enter the values you have and get the one you need.
Calculator
Formula
Description
The power triangle relates the three types of power in an AC circuit: real power P (watts, the useful power that does work), reactive power Q (volt-amperes reactive, the power that oscillates between source and reactive components), and apparent power S (volt-amperes, the total power delivered by the source). Real and reactive power are orthogonal (90° apart), so they combine as a Pythagorean sum to give apparent power. The power factor is cos(φ) = P/S. Utilities charge industrial customers for low power factor because high reactive power increases the apparent power (and thus the current) that must be delivered.
Variables
- S — Apparent power (VA)
- P — Real (active) power (W)
- Q — Reactive power (VAR)
Practical Notes
The power factor is PF = P/S = cos(arctan(Q/P)). Capacitive loads have leading (negative) Q, inductive loads have lagging (positive) Q. A PF of 0.85 means 15% of the apparent power is reactive. To reduce Q and improve PF, add PFC capacitors (for inductive loads) or PFC inductors (for capacitive loads, rare). Most utility penalties begin below PF = 0.9. Unity PF (PF = 1.0) means S = P and Q = 0.
Symbols and units
| Symbol | Meaning | Default unit | Accepted units |
|---|---|---|---|
| S | Apparent Power (VA) | — | — |
| P | Real Power (W) | W | µW, mW, W, kW |
| Q | Reactive Power (VAR) | — | — |
Step by step
- 1. What you know: Real Power (W) 10 W, Reactive Power (VAR) 1.1
- 2. The relation: S² = P² + Q², or S = √(P² + Q²)
- 3. Substitute: P = 10 W, Q = 1.1
- 4. Result: Apparent Power (VA) = 10.06
Worked examples
Minimum range
Given: Real Power (W) 10 mW, Reactive Power (VAR) 0.055
Result: Apparent Power (VA) = 0.0559
Low range
Given: Real Power (W) 100 mW, Reactive Power (VAR) 0.55
Result: Apparent Power (VA) = 0.559
Typical
Given: Real Power (W) 10 W, Reactive Power (VAR) 1.1
Result: Apparent Power (VA) = 10.06
High range
Given: Real Power (W) 1 kW, Reactive Power (VAR) 1.76
Result: Apparent Power (VA) = 1000
Maximum range
Given: Real Power (W) 10 kW, Reactive Power (VAR) 17.6
Result: Apparent Power (VA) = 10000
Solving for each variable
Solve for Apparent Power (VA)
Real Power (W) 10 mW, Reactive Power (VAR) 0.055 → 0.0559
Solve for Real Power (W)
Apparent Power (VA) 0.25, Reactive Power (VAR) 0.118 → 0.2204 W
Solve for Reactive Power (VAR)
Apparent Power (VA) 0.025, Real Power (W) 10 mW → 0.02291
How Apparent Power (VA) changes with Real Power (W)
| Real Power (W) (input) | Apparent Power (VA) (result) |
|---|---|
| 1 W | 1.487 |
| 2.5 W | 2.731 |
| 5 W | 5.12 |
| 7.5 W | 7.58 |
| 10 W | 10.06 |
| 15 W | 15.04 |
| 20 W | 20.03 |
| 40 W | 40.02 |
| 70 W | 70.01 |
| 100 W | 100 |
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<p><a href="https://vbatpower.com/formulas/power-triangle/">Power Triangle (S, P, Q)</a> Calculator by vbatpower.com</p>Frequently asked questions
How do you calculate Power Triangle (S, P, Q)?
Use S² = P² + Q², or S = √(P² + Q²). With Real Power (W) 10 W, Reactive Power (VAR) 1.1, Apparent Power (VA) works out to 10.06.
What is the formula for Power Triangle (S, P, Q)?
The formula is S² = P² + Q², or S = √(P² + Q²). The power triangle relates the three types of power in an AC circuit: real power P (watts, the useful power that does work), reactive power Q (volt-amperes reactive, the power that oscillates between source and reactive components), and apparent power S (volt-amperes, the total power delivered by the source).
How do you find Real Power (W) when you know the other values?
Rearrange for Real Power (W): with Apparent Power (VA) 0.25, Reactive Power (VAR) 0.118, Real Power (W) is 0.2204 W.
How do you find Reactive Power (VAR) when you know the other values?
Rearrange for Reactive Power (VAR): with Apparent Power (VA) 0.025, Real Power (W) 10 mW, Reactive Power (VAR) is 0.02291.
Related Concepts
All AC Circuits formulas →Need more features?
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