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.
Definitions
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.
Diagrams
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.
Balanced loads
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.
Constant power transfer
An important property of three-phase power is that the instantaneous power available to a resistive load, , is constant at all times. Indeed, let
To simplify the mathematics, we define a nondimensionalized power for intermediate calculations,
Hence (substituting back):
Since we have eliminated we can see that the total power does not vary with time. This is essential for keeping large generators and motors running smoothly.
Notice also that using the root mean square voltage , the expression for 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.
Again, using the root mean square voltage , can be written in the usual form
.
No neutral current
For the case of equal loads on each of three phases, no net current flows in the neutral. The neutral current is the inverted vector sum of the line currents. See Kirchhoff's circuit laws.
We define a non-dimensionalized current, :
Since we have shown that the neutral current is zero we can see that removing the neutral core will have no effect on the circuit, provided the system is balanced. Such connections are generally used only when the load on the three phases is part of the same piece of equipment (for example a three-phase motor), as otherwise switching loads and slight imbalances would cause large voltage fluctuations.
Unbalanced systems
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. An unbalanced system is analysed as the superposition of three balanced systems, each with the positive, negative or zero sequence of balanced voltages.
When specifying wiring sizes in a three-phase system, we only need to know the magnitude of the phase and neutral currents. The neutral current can be determined by adding the three phase currents together as complex numbers and then converting from rectangular to polar co-ordinates. If the three-phase root mean square (RMS) currents are , , and , 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