In electrical engineering, **three-phase** electric power systems have at least three conductors carrying alternating current 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). 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 only provides one voltage magnitude, 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.^{ [1] } 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.

- 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.
- 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 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.

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

- .

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.

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^{ [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. Third harmonic currents are in-phase on each of the supply phases and therefore will add together in the neutral which can cause the neutral current in a wye system to exceed the phase currents.^{ [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. 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 be equal in magnitude, and accurately displaced one-third 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 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-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. 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 phase supply. A motor–generator is often used in factory industrial applications.

In a three-phase system, at least two transducers are required to measure power when there is no neutral, or three transducers when there is a neutral.^{ [5] } Blondel's theorem states that the number of measurement elements required is one less than the number of current-carrying conductors.^{ [6] }

In classical mechanics, a **harmonic oscillator** is a system that, when displaced from its equilibrium position, experiences a restoring force *F* proportional to the displacement *x*:

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the *pole*, and the ray from the pole in the reference direction is the *polar axis*. The distance from the pole is called the *radial coordinate*, *radial distance* or simply *radius*, and the angle is called the *angular coordinate*, *polar angle*, or *azimuth*. The radial coordinate is often denoted by *r* or *ρ*, and the angular coordinate by *φ*, *θ*, or *t*. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the *radial distance* of that point from a fixed origin, its *polar angle* measured from a fixed zenith direction, and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In mathematics and physics, **Laplace's equation** is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

In mathematics, an ** n-sphere** is a topological space that is homeomorphic to a

In electrical engineering, **electrical impedance** is the measure of the opposition that a circuit presents to a current when a voltage is applied.

An **ellipsoid** is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mathematics and physical science, **spherical harmonics** are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

In probability theory, the **Borel–Kolmogorov paradox** is a paradox relating to conditional probability with respect to an event of probability zero. It is named after Émile Borel and Andrey Kolmogorov.

In physics and engineering, a **phasor**, is a complex number representing a sinusoidal function whose amplitude (*A*), angular frequency (*ω*), and initial phase (*θ*) are time-invariant. It is related to a more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor that encapsulates the frequency and time dependence. The complex constant, which encapsulates amplitude and phase dependence, is known as **phasor**, **complex amplitude**, and **sinor** or even **complexor**.

**Etendue** or **étendue** is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics.

This is a **table of orthonormalized spherical harmonics** that employ the Condon-Shortley phase up to degree * = 10. Some of these formulas give the "Cartesian" version. This assumes **x*, *y*, *z*, and *r* are related to and through the usual spherical-to-Cartesian coordinate transformation:

In mathematics, a **multiple integral** is a definite integral of a function of several real variables, for instance, *f*(*x*, *y*) or *f*(*x*, *y*, *z*). Integrals of a function of two variables over a region in are called **double integrals**, and integrals of a function of three variables over a region in are called **triple integrals**. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.

The main **trigonometric identities** between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions.

In physics and mathematics, the **solid harmonics** are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions . There are two kinds: the *regular solid harmonics*, which vanish at the origin and the *irregular solid harmonics*, which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:

In mathematics, **vector spherical harmonics** (**VSH**) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

In fluid dynamics, the **Oseen equations** describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

**Landen's transformation** is a mapping of the parameters of an elliptic integral, useful for the efficient numerical evaluation of elliptic functions. It was originally due to John Landen and independently rediscovered by Carl Friedrich Gauss.

A **phase detector characteristic** is a function of phase difference describing the output of the phase detector.

- ↑ "Delta and Wye 3-phase circuits" (PDF). Archived (PDF) from the original on 2013-05-13. Retrieved 2012-11-21. public domain
- ↑ Keljik, Jeffrey (2008).
*Electricity 3: Power Generation and Delivery*. Clifton Park, NY: Cengage Learning/Delmar. p. 49. ISBN 978-1435400290. - ↑ Lowenstein, Michael. "The 3rd Harmonic Blocking Filter: A Well Established Approach to Harmonic Current Mitigation". IAEI Magazine. Archived from the original on 27 March 2011. Retrieved 24 November 2012.
- ↑ Enjeti, Prasad. "Harmonics in Low Voltage Three-Phase Four-Wire Electric Distribution Systems and Filtering Solutions" (PDF). Texas A&M University Power Electronics and Power Quality Laboratory. Archived (PDF) from the original on 13 June 2010. Retrieved 24 November 2012.
- ↑ "Measurement of three-phase power with the 2-wattmeter method" (PDF).
^{[ permanent dead link ]} - ↑ "THE TWO-METER WATTMETER METHOD" (PDF).

- Stevenson, William D. Jr. (1975).
*Elements of Power Systems Analysis*. McGraw-Hill Electrical and Electronic Engineering Series (3rd ed.). New York: McGraw-Hill. ISBN 0-07-061285-4.

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