Single-phase current

A single-phase current is an alternating electric current provided by means of a two-wire line. He is opposed to the polyphase currents, such as the three-phase current, for which several lines are used and out of phase between them. The single-phase current is mainly used for lighting and the heating, when the use of engines of strong power is not necessary.
The generators of alternative course produce three-phase current in which each phase is shifted of 120°. The standard frequencies are of 50 and 60 Hz.
A single-phase current can be produced starting from a three-phase current by connecting one of the three phases and the neutral, or by connecting two of the three phases. The voltages obtained vary from one area to another. North America shares the system with 120 ⁄ 208 V, which coexist with a second system (of 277 ⁄ 480 V in the United States and 347 ⁄ 600 V in Canada). Europe as for it almost exclusively uses the system with 230/400 V.
In the systems of high voltage (a few kilovolts), a transformer single-phase current can be used to generate a low tension starting from the three-phase current.
A transformer cannot produce a polyphase system starting from single-phase current.
Utility of the single-phase current
The distributions in single-phase current are generally used in the rural areas, where the cost of a three-phase network is too important and where the loads do not require such infrastructures.
The networks high voltage are almost always in three-phase current. The power of the networks in single-phase current varies considerably from one country to another, even from one area to another. In England, one can meet currents of 100A or even 125A, returning the utility of the weak three-phase current for a domestic or commercial use. The majority of the other European countries have weaker limits of current in single-phase current, involving of this fact the food of certain dwellings in three-phase current.
An asynchronous motor simple single-phase current does not produce a turning magnetic field, in practice these engines require a device of starting to produce a spinning field pattern and to generate a starting torque. Except for certain applications of transport, the asynchronous motors single-phase currents of more than 10 or 20kW are rare.
Generally one 3rd driver called ground is used to ensure the safety and to evacuate a current in the event of electric defect. Current fault to detect by the differential.

Three-phase current

Graph of the three tensions of the same frequency ⁄ amplitude and out of phase of 120°
A three-phase current is an alternating electric current composed of three sinusoidal tensions of the same frequency and generally of the same amplitude which are out of phase between them (of 120° or 2p ⁄ 3 radians in the ideal case). If the frequency is of 50Hz for example, then the three phases are delayed of 1 (50x3) seconds (either 6,7 milliseconds). When the three drivers are traversed by of the same currents effective value, the system known as is balanced.
The three-phase current makes it possible to avoid the problems of power inherent in the single-phase current (in sinusoidal mode). One can show that the three-phase current delivers an instantaneous power without component pulsated contrary to the single-phase current where the instantaneous power is a sinusoid. Moreover, it offers a better output in the alternators and less loss during the electricity transmission.
Basic definitions
Three-phase sizes
A system of three-phase sizes can be put in the form:
g1 = G1 sin (ωt + φ1)
g2 = G2 sin (ωt + φ1 - ⅔π)
g3 = G3 sin (ωt + φ1 - ⅔π)
Balanced and unbalanced three-phase systems
A system of sizes (tensions or currents) three-phase known as is balanced if the 3 sizes, functions sinusoidal of time, have the same amplitude
G1 = G2 = G3 = G
In the contrary case, the three-phase system known as is unbalanced
Direct three-phase systems and opposite
If the 3 sizes pass by value 0 in order 1,2,3,1,..., the three-phase system is known as direct. It can then be put in the form:
g1 = G1 sin (ωt + φ1)
g2 = G2 sin (ωt + φ1 - ⅔π)
g3 = G3 sin (ωt + φ1 - 1⅓π) = G3 sin (ωt + φ1 + ⅔π)
If the 3 sizes pass by value 0 in order 1,3,2,1,..., the three-phase system is known as opposite. It can then be put in the form:
g1 = G1 sin (ωt + φ1)
g2 = G2 sin (ωt + φ1 - ⅔π)
g3 = G3 sin (ωt + φ1 - 1⅓π) = G3 sin (ωt + φ1 + ⅔π)
To reverse the order of the phases, that is to say to pass from the direct order to the inverse order and reciprocally, it is necessary to reverse the connection of two phases.
Three-phase distribution
A three-phase distribution comprises 3 or 4 wire
Three drivers of phase
A driver of neutral which is not systematic but which is often distributed.
Simple tensions
The potential differences between each phase and the neutral constitute a system of three-phase tensions generally noted V (V1N, V2N, V3N) and called simple tensions, spangled tensions or tensions of phase. Mathematically, one can note:
v1 = V1√2 sin (ωt + φ1)
v2 = V2√2 sin (ωt + φ1 - ⅔π)
v3 = V3√2 sin (ωt + φ1 - 1⅓π)
VI the effective value, the pulsation,fi the phase in the beginning and T time.
In the case of balanced distributions, there is V1 = V2 = V3 = V.
Made up tensions
The potential differences between the phases constitute a system of generally noted tensions U : (U12,U23, U31) and called tensions made up or tensions of line.
uij = ui - uj = Uij√2 sin (ωt + φij)
The made up tensions constitute a system of three-phase tensions if and only if the system of simple tensions is a balanced system. The sum of the three made up tensions is always null. It results from it that the homopolar component of the interlinked voltages is always null.
In the case of balanced distributions, one has : U12 = U23 = U31 = U
Relation between simple and made up tensions

Representation of Fresnel of the simple and made up tensions for a direct balanced system
We deferred on the figure opposite the diagram of Fresnel of the simple and composed tensions delivered by a direct balanced three-phase system. While observing, for example, the isosceles triangle formed by the tensions v1, v2 and u12, we can notice that this one has two acute angles of p ⁄ 6 radians (either 30 degrees). One can thus express the effective value of the tension made up U according to the effective value of the tension simple V through the relation:
U = 2 * V * cos (π ⁄ 6)
The same applies in the case of an indirect balanced system.
Consequently, in a balanced three-phase system, the effective values of the simple and composed tensions are connected by the relation:
U = √3V
Three-phase receivers
A three-phase receiver consists of three dipoles so called rollings up or phases. If these three dipoles have the same impedance, the receiver known as is balanced.
A three-phase receiver can be connected to the food in two manners:
The anglophone literature indicates usually the couplings triangle and star by names of letters : Triangle : Delta Δ - Star: Wye Υ
A balanced receiver supplied with a balanced system of tensions will absorb three currents of line also forming a balanced three-phase system.
The currents of line or currents made up are noted I. the currents which cross the receiving elements are called current of simple phase or currents and are noted J.
Connection of a three-phase receiver
The three dipoles which constitute the three-phase receiver are connected on 6 terminals conventionally laid out as the figure indicates it below.
The advantage of this provision is to allow the realization of the two couplings with bars equal length, the distance between two contiguous terminals being constant. The apparatus is provided with three identical bars of which the length allows a horizontal or vertical wiring. One must use these bars of connection in order to carry out the desired couplings:
Coupling star
The coupling star of rollings up (the most frequent coupling) is obtained while placing two bars of following connections of the manner:
The three remaining terminals will be cabled with the three drivers of phases.
The three terminals connected together by the two bars constitute a point which will be with the potential of the neutral. This point can be connected to the neutral of the distribution, but it is not an obligation, that is even strongly disadvised for the electric machines.
In a coupling star, the currents of line and phase are the same ones, also one notes:
Coupling triangle
The coupling triangle of rollings up is obtained while placing three bars of connections in the following way:
A cable of phase is connected then to each bar. The cable of neutral is not connected.
In a coupling triangle, it is necessary to break up each current crossing the receivers. Thus, one a:
I1 = J21 - J31
I2 = J23 - J21
I3 = J23 - J31

Consumption by a three-phase receiver

Active power
The theorem of Boucherot imposes that is the sum of the consumptions by each dipole:
out of star :
P = V1I1 cosφ1 + V2I2 cosφ2 + V3I3 cosφ3
maybe in balanced mode
P = 3 * VI * cosφ (V;I)
in triangle:
P = U1J1 cosφ1 + P = U2J2 cosφ2 + P = U3J3 cosφ3
maybe in balanced mode
P = 3 * UJ * cosφ (U;J)
for the receivers balanced
P = √3 * UI * cosφ
Note: In this case, φ is not dephasing between U and I and the COSφ value is called power-factor.

Interest of the three-phase current

Interest for the electricity transmission
Transport in three-phase current makes it possible to save cable and to decrease the losses by Joule effect: three wire of phases are enough (the neutral is not transported, it is recreated on the level of the last transformer). Indeed, dephasing between each phase is such as, for a balanced system, the sum of the three currents is supposed to be null (if the three currents have the same amplitude, then cos (x) + cos (x + ⅔π) + cos (x + 1⅓π) = 0). And thus, in addition to making the saving in a cable on the long distances, one saves in premium on the Joule effects (an additional cable crossed by a current would imply additional losses).
Interest for the production of electricity
The three-phase alternator was essential right from the start (before 1900) like the best compromise.
More than 95% of electrical energy is produced by synchronous generators, electromechanical machines providing of the tensions of frequencies proportional at their number of revolutions. These machines are less expensive and have a better output than the machines with D.C. current (dynamos) which deliver continuous tensions (95% instead of 85%).
The alternators (synchronous machines) three-phase which produce electrical energy have a better output and a better weight ratio/power than an alternator of the same single-phase current power.
To cancel the fluctuating power
Let us suppose that an alternator single-phase current delivers 1000 has under a tension of 1000 V and frequency 50 Hz. The expression of the delivered power is put in the form:
P = U √2 sin (ωt) * I √2 sin (ωt + φ)
P = UI cosφ - UI cos (2ωt + φ)
Thus the delivered active power (the first term of the sum) lies between 0 and 1 MW (it depends on the power-factor of the load), but the fluctuating power (the second term of the sum) is a sinusoidal power of frequency 100 Hz and amplitude obligatorily equal to 1 MW. The turbine, because of its inertia, turns with a quasi constant mechanical speed, and thus at every moment it provides an identical power. These differences in power result in oscillations of couples which are, into major part, absorptive by the elasticity of the driveshaft and end up causing its destruction.
To remove this fluctuating power, the alternators of great power must thus necessarily produce a system of polyphase tensions: N phases (N = 2) out of phase in time should be produced suitably.
The choice which was made for the whole of the networks of the world is N = 3.

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