Formula & Calculator
Power Triangle Relation
Relates apparent, real, and reactive power as sides of a right triangle.
Interpretation
The power triangle relates apparent power (S), real power (P), and reactive power (Q) by Pythagoras: S² = P² + Q².
This relationship holds for sinusoidal AC circuits.
Example: S=500VA, P=400W → Q = √(500² − 400²) = √(250000−160000) = √90000 = 300 VAR.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| S | Apparent power | VA |
| P | Real (active) power | W |
| Q | Reactive power | VAR |
| θ | Power factor angle (θ = arctan(Q/P)) | ° |
| pf | Power factor (pf = cos θ = P/S) | dimensionless |
What it means
The power triangle is a right triangle that relates apparent power S, real power P, and reactive power Q. The relationship is S² = P² + Q², which is derived from the complex power equation. The triangle is a graphical representation of the power factor, with the angle θ being the phase angle between voltage and current. The power factor is cosθ = P/S, and sinθ = Q/S. The power triangle is useful for visualising the trade‑offs between real and reactive power and for calculating the required power factor correction. It is often used in design and troubleshooting. Understanding the power triangle helps engineers determine the size of capacitors needed for power factor correction and to understand the effects of inductive or capacitive loads on the power system. Example: If S=500VA and P=400W, then Q = √(500²−400²) = √(250000−160000) = √90000 = 300 VAR. The power triangle has S=500 as hypotenuse, P=400 as adjacent, Q=300 as opposite.
Worked example
Power Triangle – Practical Example
Real‑World| Parameter | Value |
|---|---|
| S | 12 kVA |
| P | 9.6 kW |
| Formula | S² = P² + Q² → Q = √(S² − P²) |
Common mistakes
- Pythagorean: S² = P² + Q² – not S = P + Q.
- Apparent power S: The hypotenuse of the power triangle.
- Units: S in VA, P in W, Q in VAR – all have the same unit dimensions but different names.
- Power factor: pf = P/S = cosθ.
- Lagging/leading: If Q>0, the power factor is lagging (inductive); if Q<0, it is leading (capacitive).
Applications
The power triangle relation S² = P² + Q² is derived from Pythagoras and relates apparent power (S), real power (P), and reactive power (Q) in sinusoidal AC circuits. This geometric relationship helps visualise the power factor and the trade‑off between real and reactive power. Engineers use it to determine the required reactive power compensation to achieve a target power factor. By understanding the power triangle, they can size capacitors or inductors for correction. It also assists in interpreting utility bills and assessing system efficiency. The power triangle is a fundamental tool in AC circuit analysis and power system design.
- Power factor analysis and correction design
- Assessment of reactive power needs in installations
- Understanding utility power factor penalties
- Design of capacitor banks for compensation
- Educational visualisation of AC power components
Frequently Asked Questions
The power triangle relation is S² = P² + Q², where S is apparent power, P is real power, and Q is reactive power.
The angle θ between S and P; cosθ is the power factor. θ = arctan(Q/P).
Q = √(S² − P²). This is useful for power factor correction sizing.
It provides a graphical and mathematical way to understand the relationships between real, reactive, and apparent power.
Q = 0, so S = P, and the triangle collapses to a line.
For inductive loads, Q is positive; for capacitive, Q is negative. The triangle is symmetric about P.
S = V_rms · I_rms. This is valid for sinusoidal signals.
It helps visualize power factor, size equipment, and understand the effect of adding capacitors.
With harmonics, S² ≠ P² + Q²; there is also a distortion power component D. The relation becomes S² = P² + Q² + D².
Common errors include: 1) confusing P and Q, 2) using the wrong sign for Q, 3) not considering harmonics, 4) using peak instead of RMS, and 5) applying to non-linear loads without measurement.