In electrical engineering, threephase electric power systems have at least three conductors carrying voltage waveforms that are 2π/3 radians (120°,1/3 of a cycle inphase) offset in time. In this article angles will be measured in radians except where otherwise stated.
Contents 
Let x be the instantaneous phase of a signal of frequency f at time t:
Using this, the waveforms for the three phases are
where V_{P} is the peak voltage and the voltages on L1, L2 and L3 are measured relative to the neutral.
Generally, in electric power systems, the loads are distributed as evenly as is practical between 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.
This refers to a system with a resistive load R between each phase and neutral.
An important property of threephase power is that the power available to a resistive load, , is constant at all times.
To simplify the math, we define a nondimensionalized power for intermediate calculations,
Using angle subtraction formula
Using the Pythagorean trigonometric identity
Hence (substituting back):
since we have eliminated x we can see that the total power does not vary with time. This is essential for keeping large generators and motors running smoothly.
For the case of equal loads on each of three phases, no net current flows in the neutral. The neutral current is the sum of the phase current.
We define a non dimensionalized current, .
Using angle subtraction formulae
Hence also f
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. In reality such connections are generally used only when the load on the three phases is part of the same piece of equipment (for example a threephase motor), as otherwise switching loads and slight imbalances would cause large voltage fluctuations.
Practical systems rarely have perfectly balanced loads, currents, voltages or 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 analyzed as the superposition of three balanced systems, each with the positive, negative or zero sequence of balanced voltages.
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. Such a revolving magnetic field makes polyphase induction motors possible. Indeed, where induction motors must run on singlephase power (such as is usually distributed in homes), the motor must contain some mechanism to produce a revolving field, otherwise the motor cannot generate any standstill torque and will not start. The field produced by a singlephase winding can provide energy to a motor already rotating, but without auxiliary mechanisms the motor will not accelerate from a stop when energized.
A rotating magnetic field of steady amplitude requires that all three phase currents are equal in magnitude and accurately displaced onethird of a cycle in phase. Unbalanced operation results in undesirable effects on motors and generators.
Provided two voltage waveforms have at least some relative displacement on the time axis, other than a multiple of a halfcycle, any other polyphase set of voltages can be obtained by an array of passive transformers. Such arrays will evenly balance the polyphase load between the phases of the source system. For example, balanced twopower can be obtained from a threephase 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 twophase system with 90° time difference between the phases. Another example is the generation of higherphaseorder systems for large rectifier systems, to produce a smoother DC output and to reduce the harmonic currents in the supply.
When threephase is needed but only singlephase is readily available from the electricity supplier a phase converter can be used to generate threephase power from the single phase supply.
