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Concept
Mathematics of three-phase electric power
Summary
In electrical engineering, three-phase electric power systems have at least three conductors carrying alternating voltages that are offset in time by one-third of the period. A three-phase system may be arranged in delta (∆) or star (Y) (also denoted as wye in some areas, as symbolically it is similar to the letter 'Y'). A wye system allows the use of two different voltages from all three phases, such as a 230/400 V system which provides 230 V between the neutral (centre hub) and any one of the phases, and 400 V across any two phases. A delta system arrangement provides only one voltage, but it has a greater redundancy as it may continue to operate normally with one of the three supply windings offline, albeit at 57.7% of total capacity. Harmonic current in the neutral may become very large if nonlinear loads are connected.
In a star (wye) connected topology, with rotation sequence L1 - L2 - L3, the time-varying instantaneous voltages can be calculated for each phase A,C,B respectively by:
where:
is the peak voltage,
is the phase angle in radians
is the time in seconds
is the frequency in cycles per second and
voltages L1-N, L2-N and L3-N are referenced to the star connection point.
The below images demonstrate how a system of six wires delivering three phases from an alternator may be replaced by just three. A three-phase transformer is also shown.
Hawkins Electrical Guide - 3phase Elementary 6wire.jpg|Elementary six-wire three-phase alternator, with each phase using a separate pair of transmission wires.
Hawkins Electrical Guide - 3phase Elementary 3wire.jpg|Elementary three-wire three-phase alternator, showing how the phases can share only three transmission wires.
Transformator-3-fas.png|Each phase of a three-phase transformer has its own pair of windings, with a shared core.
Generally, in electric power systems, the loads are distributed as evenly as is practical among the phases. It is usual practice to discuss a balanced system first and then describe the effects of unbalanced systems as deviations from the elementary case.
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