Power Factor
Power factor is the ratio of real power (watts) to apparent power (volt-amperes) in an AC circuit, representing how effectively electrical power is being converted into useful work. It equals the cosine of the phase angle between the voltage and current waveforms, ranging from 0 (purely reactive) to 1 (purely resistive). A low power factor indicates high reactive power circulation, which increases current for a given load, causing extra losses in transmission lines, and utilities typically penalise industrial consumers for poor power factor below 0.85.
Key Formula
PF = cos(φ) = P/S = P / √(P² + Q²)
LaTeX: PF = \cos\phi = \frac{P}{S} = \frac{P}{\sqrt{P^2 + Q^2}}
| Symbol | Meaning | Unit |
|---|---|---|
| PF | Power factor | Dimensionless (0 to 1) |
| φ | Phase angle between voltage and current | degrees or radians |
| P | Real (active) power | W (watts) |
| S | Apparent power | VA (volt-amperes) |
| Q | Reactive power | VAR (volt-ampere reactive) |
Worked Example
Problem
An industrial motor draws 10 kW of real power and 7.5 kVAR of reactive power from a 230 V, 50 Hz supply. Calculate the apparent power S, power factor PF, and the required capacitor size to correct the power factor to 0.95 lagging.
Solution
Step 1: Calculate apparent power. S = √(P² + Q²) = √(10000² + 7500²) = √(10⁸ + 5.625×10⁷) = √(1.5625×10⁸) = 12,500 VA = 12.5 kVA Step 2: Calculate current power factor. PF = P/S = 10,000 / 12,500 = 0.8 lagging φ₁ = arccos(0.8) = 36.87° Step 3: Find target reactive power for PF = 0.95. φ₂ = arccos(0.95) = 18.19° Q₂ = P × tan(φ₂) = 10,000 × tan(18.19°) = 10,000 × 0.3287 = 3,287 VAR Step 4: Required capacitor reactive power. Q_C = Q₁ − Q₂ = 7,500 − 3,287 = 4,213 VAR Step 5: Capacitor size at 230 V, 50 Hz. C = Q_C / (ω × V²) = 4213 / (2π×50 × 230²) = 4213 / 16,597,300 ≈ 253.8 µF
Answer
S = 12.5 kVA, PF = 0.8 lagging; correction capacitor required: C ≈ 254 µF
Power Triangle Components and Relationships
| Quantity | Symbol | Unit | Description |
|---|---|---|---|
| Real power | P | W (watt) | Actual work done — heat, light, motion |
| Reactive power | Q | VAR | Energy stored/released by inductors and capacitors |
| Apparent power | S | VA | Vector sum: S = √(P²+Q²) |
| Power factor | PF | Dimensionless | cos(φ) = P/S, ranges 0–1 |
| Phase angle | φ | degrees | Angle between V and I phasors |
Interactive Tools
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Related Terms
Three-Phase Power
Three-phase power is an AC electrical power system using three conductors, each carrying a sinusoidal voltage of the same frequency and amplitude but displaced 120° from each other in phase. It is the dominant form of electrical power generation, transmission, and distribution worldwide because it delivers constant instantaneous power (unlike single-phase), uses conductors more efficiently, and provides self-starting capability for induction motors. Three-phase systems power virtually all industrial machinery, large HVAC systems, and the electricity grid from generating stations to distribution substations.
Signal Processing
Signal processing is the analysis, manipulation, and synthesis of signals — including audio, video, sensor data, and communications waveforms — to extract information or transform them for a desired purpose. It encompasses filtering, compression, modulation, spectral analysis, and noise reduction using both analog and digital techniques. Signal processing underpins technologies such as telecommunications, medical imaging, radar, speech recognition, and multimedia systems.
Feedback Control
Feedback control is a control strategy in which the output of a system is measured and compared to a desired reference (setpoint), and the difference (error) is used to adjust the system input to reduce that error. Negative feedback — where the output is subtracted from the reference — is the basis of stable automatic control systems in engineering, biology, and economics. Feedback control enables systems to self-correct against disturbances and parameter variations, forming the foundation of servo systems, thermostats, autopilots, and industrial process control.
The term "power factor" arose with the development of AC power systems in the 1880s–1890s, particularly in the theoretical work of Charles Proteus Steinmetz at General Electric. "Factor" derives from Latin "facere" (to make/do), indicating the fraction of apparent power that performs real work. The power factor correction concept became commercially important as AC networks expanded globally in the early twentieth century.