Three coils, three phases
A power-station generator spins a magnet inside three fixed coils spaced 120° around the stator. Each coil gives a sine voltage, and because the coils are 120° apart, so are their voltages: phase B lags A by 120°, and C lags B by another 120°.
va = Vp·sin(ωt) vb = Vp·sin(ωt − 120°) vc = Vp·sin(ωt + 120°)
Phase voltage and line voltage
Phase voltage is measured from a line to neutral; line voltage between two lines. The line voltage is the difference of two phase phasors 120° apart, which is √3 times longer and leads by 30°. That is why the Thai low-voltage supply is quoted as 230/400 V.
V_L = √3 · V_ph 230 V · √3 ≈ 400 V
Star (Y) and delta (Δ)
In star, each load connects from a line to a common star point tied to neutral: it sees the phase voltage, and the line current is its own current. In delta, each load connects between two lines: it sees the line voltage, and each line current is the difference of two load currents, √3 times larger when balanced.
- Star: V_load = V_ph, I_line = I_load, has a neutral
- Delta: V_load = V_L, I_line = √3·I_load, no neutral
- Same three loads in delta draw 3× the power of star
Why grids use three phases
In a balanced system the three currents add to zero at every instant, so the neutral carries nothing and three wires do the work of six. And the total power v·i summed over the phases is constant, not pulsing at twice the supply frequency as single phase does, which is why three-phase motors run smoothly. Unbalance the load and both benefits fade: current returns through the neutral and the power ripples.
P = √3 · V_L · I_L · cos θ = 3 · V_ph · I_ph · cos θ
Worked example (the simulator’s default values)
- Star-connected supply, 230 V phase, 50 Hz; each load R = 20 Ω, X = 10 Ω (inductive).
- V_L = √3·230 = 398 V; |Z| = √(20² + 10²) = 22.4 Ω; I = 230/22.4 = 10.3 A, lagging by 26.6°.
- PF = cos 26.6° = 0.894; P = 3·10.3²·20 = 6.35 kW; Q = 3.17 kvar; neutral current 0 A.
- Reconnect the same loads in delta: each sees 398 V, I_load = 17.8 A, I_line = 30.9 A, P = 19.0 kW (three times).
Common mistakes
- Adding phase voltages arithmetically. Two 230 V phases 120° apart give 400 V between them, not 460 V.
- Expecting neutral current to be three times one phase current. Balanced, it is zero.
- Forgetting which voltage the load sees: phase voltage in star, line voltage in delta.
Common questions
Why is line voltage √3 times phase voltage?
The line voltage is the difference between two phase voltages 120° apart. Geometrically, that difference is √3 times as long as each phasor and leads by 30°.
What is the difference between star and delta connection?
In star each load sits between a line and neutral (phase voltage, line current = load current). In delta each load sits between two lines (line voltage, line current = √3 × load current), with no neutral.
Why is the neutral current zero in a balanced three-phase system?
Three equal currents 120° apart sum to zero at every instant, like three equal arrows at 120° adding to nothing. Any unbalance leaves a remainder, which returns through the neutral.