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.
Key Formula
P_total = √3 × V_L × I_L × cos(φ)
LaTeX: P_{total} = \sqrt{3}\, V_L\, I_L\, \cos\phi
| Symbol | Meaning | Unit |
|---|---|---|
| P_total | Total three-phase real power | W (watts) |
| V_L | Line-to-line (line) voltage | V (volts) |
| I_L | Line current | A (amperes) |
| cos(φ) | Power factor of the load | Dimensionless |
| √3 | Factor arising from 120° phase displacement | Dimensionless ≈ 1.732 |
Worked Example
Problem
A three-phase industrial motor is connected to a 415 V (line-to-line), 50 Hz supply and draws a line current of 50 A at a power factor of 0.85 lagging. Calculate the total real power consumed, the apparent power, and the reactive power.
Solution
Step 1: Calculate total real power. P = √3 × V_L × I_L × cos(φ) P = 1.732 × 415 × 50 × 0.85 P = 1.732 × 415 × 42.5 P = 1.732 × 17,637.5 P ≈ 30,527 W ≈ 30.5 kW Step 2: Calculate apparent power. S = √3 × V_L × I_L = 1.732 × 415 × 50 = 35,929 VA ≈ 35.9 kVA Step 3: Calculate reactive power. Q = √(S² − P²) = √(35929² − 30527²) Q = √(1.29×10⁹ − 9.32×10⁸) = √(3.59×10⁸) ≈ 18,952 VAR ≈ 18.95 kVAR
Answer
P ≈ 30.5 kW, S ≈ 35.9 kVA, Q ≈ 19.0 kVAR
Star (Y) vs Delta (Δ) Three-Phase Connections
| Parameter | Star (Y) Connection | Delta (Δ) Connection |
|---|---|---|
| Line voltage V_L | V_L = √3 × V_phase | V_L = V_phase |
| Line current I_L | I_L = I_phase | I_L = √3 × I_phase |
| Neutral wire | Available (4-wire system) | Not available |
| Phase voltage (415V sys) | 240 V | 415 V |
| Use case | Distribution, motors, unbalanced loads | High-power motors, balanced loads |
Interactive Tools
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Related Terms
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.
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 three-phase system was invented by Nikola Tesla and independently by Mikhail Dolivo-Dobrovolsky in the late 1880s. Dolivo-Dobrovolsky demonstrated the first complete three-phase AC power system at the 1891 Frankfurt International Electrotechnical Exhibition, transmitting 175 hp of power over 175 km. "Phase" derives from Greek "phasis" (appearance), originally used in astronomy, later applied to the timing of oscillating waves.