Mathematics of three-phase electric power
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 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.
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: The below images demonstrate how a system of six wires delivering three phases from an alternator may be replaced by just three.
Generally, in electric power systems, the loads are distributed as evenly as is practical among the phases.
above takes the following more classic form: The load need not be resistive for achieving a constant instantaneous power since, as long as it is balanced or the same for all phases, it may be written as so that the peak current is for all phases and the instantaneous currents are Now the instantaneous powers in the phases are Using angle subtraction formulae: which add up for a total instantaneous power Since the three terms enclosed in square brackets are a three-phase system, they add up to zero and the total power becomes or showing the above contention.
can be written in the usual form For the case of equal loads on each of three phases, no net current flows in the neutral.
In practice, systems rarely have perfectly balanced loads, currents, voltages and impedances in all three phases.
The analysis of unbalanced cases is greatly simplified by the use of the techniques of symmetrical components.
When specifying wiring sizes in a three-phase system, we only need to know the magnitude of the phase and neutral currents.
, the neutral RMS current is: which resolves to The polar magnitude of this is the square root of the sum of the squares of the real and imaginary parts, which reduces to[2] With linear loads, the neutral only carries the current due to imbalance between the phases.
Devices that utilize rectifier-capacitor front ends (such as switch-mode power supplies for computers, office equipment and the like) introduce third order harmonics.
[3][4] Any polyphase system, by virtue of the time displacement of the currents in the phases, makes it possible to easily generate a magnetic field that revolves at the line frequency.
Unbalanced operation results in undesirable effects on motors and generators.
Such arrays will evenly balance the polyphase load between the phases of the source system.
For example, balanced two-phase power can be obtained from a three-phase network by using two specially constructed transformers, with taps at 50% and 86.6% of the primary voltage.
This Scott T connection produces a true two-phase system with 90° time difference between the phases.
