This article is about the basic mathematics and principles of
three-phase electricity. For information on where, how and
why three-phase is used, see three-phase electric power.
In electrical engineering, three-phase electric power systems
have at least three conductors carrying voltage waveforms
that are 2π/3 radians (120°,1/3 of a cycle in-phase) offset in
time. In this article angles will be measured in radians
except where otherwise stated.
1 Variable setup and basic definitions
2 Balanced loads
o 2.1 Star connected systems with neutral
2.1.1 Constant power transfer
2.1.2 No neutral current
o 2.2 Star connected systems without neutral
3 Unbalanced systems
4 Revolving magnetic field
5 Conversion to other phase systems
7 See also
 Variable setup and basic definitions
One voltage cycle of a three-phase system, labeled 0 to 360°
(2 π radians) along the time axis. The plotted line represents
the variation of instantaneous voltage (or current) with
respect to time. This cycle will repeat 50 or 60 times per
second, depending on the power system frequency. The
colors of the lines represent the American color code for 120v
three-phase. That is black=VL1 red=VL2 blue=VL3
Elementary six-wire three-phase alternator, with each phase
using a separate pair of transmission wires.
Elementary three-wire three-phase alternator, showing how
the phases can share only three transmission wires.
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 VP is the peak voltage and the voltages on L1, L2 and
L3 are measured relative to the neutral.
 Balanced loads
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.
 Star connected systems with neutral
This refers to a system with a resistive load R between each
phase and neutral.
 Constant power transfer
An important property of three-phase 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.
 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 sum of
the phase current.
We define a non dimensionalized current, .
Using angle subtraction formulae
Hence also f
 Star connected systems without neutral
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 three-phase motor), as otherwise switching loads
and slight imbalances would cause large voltage
 Unbalanced systems
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.
 Revolving magnetic field
The rotating magnetic field of a three-phase motor.
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 single-phase 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
stand-still torque and will not start. The field produced by a
single-phase 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 one-third of a cycle in phase.
Unbalanced operation results in undesirable effects on
motors and generators.
 Conversion to other phase systems
Provided two voltage waveforms have at least some relative
displacement on the time axis, other than a multiple of a
half-cycle, 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 two-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. Another example is the generation of
higher-phase-order systems for large rectifier systems, to
produce a smoother DC output and to reduce the harmonic
currents in the supply.
When three-phase is needed but only single-phase is readily
available from the electricity supplier a phase converter can
be used to generate three-phase power from the single