A single-phase system uses a single sine wave. Three-phase systems use three similar signals 120 degrees apart, which is also third of a period. Sockets where our everyday devices are plugged into are single-phase, but some industrial equipment requires a three-phase source to operate because of its high power.
In the voltage functions, the peak voltage is 325V (230V RMS) and the angular frequency is 314, since the frequency in Europe is 50Hz. The initial phase angles distinguish the phase voltages (VL) from each other.
$ V_{L1} = 325 ⋅ sin(314 ⋅ t) $
$ V_{L2} = 325 ⋅ sin(314 ⋅ t - 120°) $
$ V_{L3} = 325 ⋅ sin(314 ⋅ t - 240°) $
The voltage measured between one of the phases and the neutral line is called the phase voltage, whose RMS voltage is 230V, while the voltage measured between the phases is the line voltage, with a 400V RMS value.
A: Three-phase consumer $ (400 V) $
B-D: Single-phase consumers $ (230 V) $
$ V_{L1} - V_{L2} = (325 \, \angle \, 0°) - (325 \, \angle -120°) $
$ V_{L1 - L2} = 325 ⋅ \sqrt{3} \, \angle -30° $
$ 230 ⋅ \sqrt{3} \approx 400 [V] $
Electricity is generated and transmitted in three phases, but its main use is industrial. Nowadays, however, it is also used in the residential sector, with the spread of photovoltaic systems and the high-power consumers (electric heating, kitchen hobs, wall chargers for electric cars) that can no longer be served by a single phase. In these cases all three phases are brought into the property instead of just one.
The two terms relate to the operation of three-phase electric motors. In star connection, the three phases meet at a common point, whereas in delta connection, the three windings of the motor are connected in a triangular shape, with no common point. The star allows for a softer motor start and the delta allows the motor to operate at higher power. When a high starting current can be dangerous for a motor, star-delta switching is used. After starting, they switch to delta connection at about 70-80% speed.