Three-phase power is the standard method for generating, transmitting, and using large amounts of electrical energy in industry. This cheat sheet helps students connect circuit diagrams, line values, phase values, and power formulas in one clear reference. It is especially useful for analyzing motors, generators, transformers, and balanced three-phase loads.
Students need these relationships because small mistakes in voltage, current, or power factor can cause large errors in engineering calculations.
The core ideas are the difference between line and phase quantities, the difference between wye and delta connections, and the role of power factor. In balanced three-phase systems, the three voltages or currents are equal in magnitude and separated by 120 degrees. Total real power is usually found with P = sqrt(3) V_L I_L PF.
Apparent power, reactive power, and phase angle help describe how efficiently electrical power is being delivered.
Key Facts
- In a balanced three-phase system, the three phase voltages have equal magnitude and are separated by 120 degrees.
- For a wye connection, line voltage and phase voltage are related by V_L = sqrt(3) V_Ph.
- For a wye connection, line current and phase current are equal, so I_L = I_Ph.
- For a delta connection, line voltage and phase voltage are equal, so V_L = V_Ph.
- For a delta connection, line current and phase current are related by I_L = sqrt(3) I_Ph.
- Total three-phase real power is P = sqrt(3) V_L I_L PF for a balanced load.
- Total three-phase apparent power is S = sqrt(3) V_L I_L, measured in volt-amperes.
- Power factor is PF = cos(theta), where theta is the phase angle between voltage and current.
Vocabulary
- Three-phase power
- An AC power system that uses three sinusoidal voltages or currents separated by 120 degrees.
- Line voltage
- The voltage measured between any two line conductors in a three-phase system.
- Phase voltage
- The voltage across one individual phase winding or load element.
- Wye connection
- A three-phase connection in which one end of each phase is joined at a common neutral point.
- Delta connection
- A three-phase connection in which the three phase elements are connected end to end in a closed loop.
- Power factor
- The ratio of real power to apparent power, equal to cos(theta) for sinusoidal voltage and current.
Common Mistakes to Avoid
- Using phase voltage when the formula requires line voltage is wrong because P = sqrt(3) V_L I_L PF assumes line-to-line voltage and line current.
- Applying wye relationships to a delta load is wrong because wye has V_L = sqrt(3) V_Ph, while delta has V_L = V_Ph.
- Forgetting the square root of 3 factor is wrong because total balanced three-phase power is three single-phase powers combined through line quantities.
- Treating apparent power and real power as the same is wrong because real power is P = S PF, so a power factor below 1 reduces usable power.
- Ignoring units is wrong because real power is measured in watts, apparent power in volt-amperes, and reactive power in vars.
Practice Questions
- 1 A balanced three-phase motor draws I_L = 18 A from a 480 V line-to-line supply at PF = 0.85. Find the real power using P = sqrt(3) V_L I_L PF.
- 2 A wye-connected load has phase voltage V_Ph = 120 V. What is the line voltage V_L?
- 3 A delta-connected load has phase current I_Ph = 10 A. What is the line current I_L?
- 4 Why does a low power factor cause a three-phase system to carry more current for the same real power?