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Electricity

The three-phase system

The three-phase is a system of three sinusoidal voltages of the same frequency which are out of phase with each other (by 120 ° or? Π radians in the ideal case). If the frequency is 50 Hz for example, then the three phases are delayed by? 50/3 seconds (or 6.7 milliseconds). When the three conductors are traversed by currents of the same rms value, the system is said to be balanced.

Animation d'un alternateur triphasé
Animation of a three-phase alternator

Animation of a three-phase alternator

Basic definitions

Three-phase quantities

A three-phase system of quantities can take the form:

g_1 = G_1\sin\left( \omega t+\varphi_1\right)
g_2 = G_2\sin\left( \omega t+\varphi_1 - \tfrac23\pi \right)
g_3 = G_3\sin\left( \omega t+\varphi_1 + \tfrac23\pi\right)

Balanced and unbalanced three-phase systems

A three-phase system of quantities (voltages or currents) is said to be balanced if the 3 quantities, sinusoidal functions of time, have the same amplitude: G1 = G2 = G3 = G

Otherwise, the three-phase system is said to be unbalanced

Three-phase direct and indirect systems

If the 3 quantities pass through the value 0 in the order 1, 2, 3, 1, …, the three-phase system is said to be direct. It can then be put in the form:

g_1 = G_1\sin( \omega t+\varphi_1)
g_2 = G_2\sin\left( \omega t+\varphi_1 - \tfrac23\pi \right)
g_3 = G_3\sin\left( \omega t+\varphi_1 - \tfrac43\pi\right) = G_3\sin\left( \omega t+\varphi_1 + \tfrac23\pi\right)

If the 3 quantities pass through the value 0 in the order 1, 3, 2, 1, …, the three-phase system is said to be indirect. It can then be put in the form :

g_1 = G_1\sin( \omega t+\varphi_1)
g_2 = G_2\sin\left( \omega t+\varphi_1 + \tfrac23\pi \right)
g_3 = G_3\sin\left( \omega t+\varphi_1 + \tfrac43\pi\right) = G_3\sin\left( \omega t+\varphi_1 - \tfrac23\pi\right)

Three-phase distribution

A three-phase distribution has 3 or 4 wires

  • Three phase conductors
  • A neutral conductor which is not systematic but which is often distributed.

Simple voltages

The potential differences between each of the phases and the neutral constitute a system of three-phase voltages generally denoted V (V1N, V2N, V3N) and called phase-to-phase voltages or phase voltages. Mathematically, we can note:

v_1 = V_1\sqrt 2\sin( \omega t+\varphi_1)
v_2 = V_2\sqrt 2\sin\left( \omega t+\varphi_1 - \tfrac23\pi \right)
v_3 = V_3\sqrt 2\sin\left( \omega t+\varphi_1 - \tfrac43\pi\right)

Vi the effective value, ω the pulsation, φi the phase at the origin and t the time.

In the case of balanced distributions, we have V1 = V2 = V3 = V.

Compound voltages

The potential differences between the phases constitute a system of voltages generally denoted U: (U12, U23, U31) and called phase-to-phase voltages or line voltages.

u_{ij} = v_i - v_j = U_{ij}\sqrt 2\sin( \omega t+\varphi_{ij})

Phase-to-phase voltages constitute a three-phase voltage system if and only if the phase-to-phase voltage system is a balanced system. The sum of the three phase-to-phase voltages is always zero. As a result, the zero sequence component of the phase-to-phase voltages is always zero (see Fortescue transformation below)

In the case of balanced distributions, we have: U12 = U23 = U31 = U

Relationship between phase-to-phase and phase-to-phase voltages

In the case of balanced distribution, we have:

U =  \sqrt 3\cdot V

Intensities

Line currents are noted I and currents flowing through a receiver are noted J (sometimes called phase currents). In a so-called ‘star’ coupling I = J.

In a so-called ‘triangle’ coupling, it is necessary to break down each current passing through the receivers. So we have: I1 = J21 + J31I2 = J23 – J21I3 = – J23 – J31

Three-phase receivers

A three-phase receiver consists of 3 dipoles. If these 3 dipoles are absolutely identical, the receiver is said to be balanced.

A three-phase receiver can be connected to the power supply in 2 ways:

Three-phase receivers

A balanced receiver fed by a balanced voltage system will absorb 3 line currents also forming a balanced three phase system.

Connection of a three-phase receiver

The three dipoles which constitute the three-phase receiver are connected to 6 terminals conventionally arranged as shown in the figure below.

three-phase receiver

The advantage of this arrangement is that it allows two couplings to be made with strips of equal length, the distance between two contiguous terminals being constant. The device is supplied with three identical strips, the length of which allows horizontal or vertical wiring. These connection strips must be used in order to achieve the desired couplings:

Star coupling

The star coupling of the windings (the most frequent coupling) is obtained by placing 2 connection strips as follows:

Star coupling

The remaining 3 terminals will be wired with the 3 phase conductors.

The 3 terminals connected together by the two bars constitute a point which will be at neutral potential. This point can be connected to the neutral of the distribution, but this is not an obligation, it is even strongly discouraged for electrical machines.

Triangle coupling

The triangle coupling of the windings is obtained by placing 3 connection strips as follows:

Triangle coupling

A phase cable is then connected to each strip. The neutral wire is not connected.

Nameplates for three-phase receivers

The nameplate of a three-phase receiver specifies the value of the two phase-to-phase voltages used to supply it: Example Water heater: 230/400:

  • The first value is the phase-to-phase voltage required to wire the receiver in delta
  • The second value is the phase-to-phase voltage required to wire the receiver in star

Power consumed by a three-phase receiver

Active power

Boucherot’s theorem requires that this be the sum of the powers consumed by each of the dipoles:

  • star: either, in balanced mode: [ref. necessary]
  • in triangle: either, in balanced diet: [ref. necessary]
  • For balanced receivers and whatever the coupling, we can write

Note : In this case, is not the phase shift between and

Interest of three-phase

Interest in the transmission of electricity

\cos(x) + \cos(x+\tfrac23\pi) + \cos(x+\tfrac43\pi)=0

Three-phase transport makes it possible to save cable and reduce losses by Joule effect: 3 phase wires are sufficient (the neutral is not transported, it is “recreated” at the last transformer). Indeed, the phase shift between each phase is such that, for a balanced system, the sum of the three currents is assumed to be zero (if the three currents have the same amplitude, then). And therefore, in addition to saving a cable over long distances, we save as a bonus on Joule effects (an additional cable crossed by a current would imply additional losses). We can already see great interest in having chosen these phase shifts!

Interest in the production of electricity

Better alternators

The three-phase alternator has established itself from the outset (before 1900) as the best compromise.

More than 95% of electrical energy is produced by synchronous alternators, electromechanical machines providing voltages of frequencies proportional to their speed of rotation. These machines are less expensive and have a better efficiency than direct current machines (dynamos) which deliver continuous voltages (95% instead of 85%).

Three-phase alternators (synchronous machines) that produce electrical energy have better efficiency and a better weight / power ratio than a single-phase alternator of the same power.

Cancel fluctuating power

Suppose that a single-phase alternator delivers 1000 A at a voltage of 1000 V and a frequency of 50 Hz. The expression of the power delivered is in the form:

P = U\sqrt 2\sin( \omega t) \cdot I\sqrt 2\sin( \omega t+\varphi)
P = UI\cos \varphi - UI\cos( 2\omega t+\varphi)

So the active power delivered (the first term of the sum) is 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 with an amplitude necessarily equal to 1 MW. The turbine, due to its inertia, rotates with an almost constant mechanical speed, and therefore at all times it provides identical power. These differences in power result in oscillations of torque which are, for the most part, absorbed by the elasticity of the driveshaft and eventually destroy it.

To suppress this fluctuating power, high power alternators must therefore necessarily produce a polyphase voltage system: it is necessary to produce n phases (n ≥ 2) suitably phase-shifted in time.

For example in two-phase:

P = U\sqrt 2\sin( \omega t) \cdot I\sqrt 2\sin( \omega t+\varphi)+U\sqrt 2 \cos( \omega t) \cdot I\sqrt 2 \cos( \omega t+\varphi)
P = UI \cos \varphi - UI \cos( 2\omega t+\varphi)+UI \cos \varphi + UI \cos( 2\omega t+\varphi)
P = 2UI\cos \varphi

The fluctuating power has been canceled.

The choice made for all the networks in the world is n = 3.

Fortescue transformation

Any unbalanced three-phase system of quantities can be put in the form of the sum of three balanced systems:

  • A direct balanced system noted Gd.
  • An inverse balanced system denoted Gi.
  • A zero-sequence voltage system noted Go (in reality a single-phase quantity that is divided into 3 for the matrix calculation).

Three-phase homopolar systems

As explained previously, it is not really a three-phase system because it corresponds to a system of 3 phase voltages:

g_o = G_o\sin( \omega t+\varphi_o)
g_o = G_o\sin( \omega t+\varphi_o)
g_o = G_o\sin( \omega t+\varphi_o)

The interest of this decomposition is to facilitate the matrix writing of the transformation of Fortescue.

Transformation matrix

The goal is to find the values of Gd, Gi and Go from G1, G2 and G3.

Calculation of GB

As the sum of the three quantities of a balanced system is zero, we have to:

3 G_o\sin( \omega t+\varphi_o) = G_1\sin( \omega t+\varphi_1)+G_2\sin( \omega t+\varphi_2)+G_3\sin( \omega t+\varphi_3)

Rotation operator: a

Note: An underlined quantity represents the complex number associated with the considered sinusoidal quantity.

\tfrac23\pi
\underline a = e^{j\frac23\pi}

It is a complex number of modulus 1 and argument:

\tfrac23\pi
\tfrac23\pi

The result of its multiplication by the complex number associated with a quantity corresponds to another quantity of the same amplitude and out of phase with respect to the initial quantity. It corresponds to a rotation of in the Fresnel plane.

Fortescue matrix

\begin{bmatrix} \underline G_d\\  \underline G_i\\ \underline G_o \end{bmatrix} = \frac13 \begin{bmatrix} 1 & \underline a & \underline a^2  \\ 1 & \underline a^2 & \underline a  \\ 1 & 1 & 1 \end{bmatrix}  \begin{bmatrix} \underline G_1\\  \underline G_2\\ \underline G_3  \end{bmatrix}

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