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Phase currents formula. Three-phase AC

The currents in the generator and receiver phases are called phase currents, and the current of the linear wires - linear.

At symmetrical mode three-phase circuit and the connection of the generator windings with a star, the positive directions of the currents in the linear wires are selected from the generator to the receiver, and the currents in the neutral wire in the opposite direction. Have a symmetrical three-phase receiver or for three identical receivers connected by a star, the phase resistances are equal in absolute value , from which it follows that the linear and phase currents are numerically equal.

, (3.11)

where is the phase impedance.

Linear or phase currents are shifted in phase relative to each other by equal angles equal to 120°. The sum of the vectors of three currents forming a symmetrical triple is equal to zero, as well as the three vectors of phase voltages. Therefore, there is no current in the neutral wire: , and from neutral wire you can refuse. Such a four-wire system can be replaced by a three-wire one (Fig. 3.4).

Rice. 3.4. Diagram of a three-phase three-wire circuit

In the symmetrical mode of a three-phase circuit and the connection of the generator windings with a triangle, the positive direction of the linear currents, as in the case of a star connection, is selected from the generator to the receivers. For the positive direction of phase currents in receivers, directions from A To b, from b To With, from With To A, which is indicated by the order of the letters in the indices. For example, is the current in the phase bc, directed from the node b to node c.

The current in each linear wire is equal to the difference between the currents of the two phases that are connected to one node with this wire.

With a symmetrical voltage system and the same phase resistances of the receiver, the phase currents form a symmetrical system. The effective values ​​of the currents are the same, and in phase the currents are shifted relative to each other by angles of 120 °. Linear currents are equal to the differences of the corresponding phase currents, their effective values ​​are the same, the phase shifts between them, as well as between phase currents, are 120 °. The vectors of phase currents of adjacent phases, together with the vector of the corresponding linear current, form an isosceles triangle with angles at the base of 30° and an angle at the top of 120°. The base of the triangle defines line current.

From the triangle it turns out that:

The linear current is several times greater than the phase current in the symmetrical mode of a three-phase circuit and a delta connection. Each line current lags the corresponding phase current by 30° in phase.

Example 3.2. Generator phase voltages UF=240 V, receiver phase resistances za= ra\u003d 20 Ohm, zb= rb\u003d 8 ohm, zc= rc=50 Ohm. line resistance and neutral wires can be neglected. Determine the currents in all sections of the circuit.

Solution.

Let us direct the vector along the real positive semi-axis, i.e. we choose the zero initial phase of this voltage: IN,

voltage:

Currents in the areas.

At present, the so-called three-phase alternating current system, invented and developed in 1888 by the Russian electrical engineer Dolivo-Dobrovolsky, has received the widest distribution throughout the world. He was the first to design and build a three-phase generator, a three-phase asynchronous electric motor and a three-phase power line. This system provides the most profitable terms transmission electrical energy by wire and allows you to build simple in design and easy-to-use electric motors.

Three-phase system electrical circuits called a system consisting of three circuits in which variable EMFs of the same frequency act, shifted in phase relative to each other by 1/3 of the period (j \u003d 120 °). Each circuit of such a system is called a phase, and a system of three phase-shifted alternating currents in such circuits is called a three-phase current.

Maintaining a constant phase shift between voltage fluctuations at the output of three independent generators is a rather difficult technical problem. In practice, three-phase generators are used to obtain three phase-shifted currents. The inductor in the generator is an electromagnet, the winding of which is powered by direct current. The inductor is the rotor and the armature of the generator is the stator. Each winding of the generator is an independent current generator. By attaching wires to the ends of each of them, as shown in the figure, we would get three independent circuits, each of which could power certain receivers, for example electric lamps. In this case, to transfer all the energy that the receivers absorb, it would be required six wires. It is possible, however, to connect the generator windings together in this way three-phase current to get by with four or even three wires, that is, significantly save wiring.
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The first of these methods is called star connection. With it, all ends of the phase windings X, Y, Z are connected to a common node O (it is called the neutral or zero point of the generator), and the beginnings serve as clamps for connecting the load. The voltage between the zero point and the beginning of each phase is called phase voltage ( U f ) , and the voltage between the beginnings of the windings, that is, points A and B, B and C, C and A, called line voltage ( U l ). Wherein effective value line voltage exceeds the effective value of the phase voltage in

In the case of a uniform load of all three phases the current in the neutral wire is zero and can not be used. When not symmetrical load the current in the neutral wire is not equal to zero, but is much weaker than the current in the linear wires. Therefore, the neutral wire can be thinner than the phase wires.

windings three-phase generator can be connected in a triangle. In this case, the end of each winding is connected to the beginning of the next, so that they form a closed triangle, and the linear wires are attached to the vertices

this triangle-points A, B and C. It is easy to see that when connected by a triangle, the linear voltage of the generator is equal to its phase voltage. Therefore, to obtain the desired line voltage, each winding of the generator must be designed for a higher voltage than in the case of a star connection of the generator windings. This leads to an increase in the cost of the generator. In addition, the load is rarely perfectly symmetrical. In this regard, the generator windings, as a rule, are connected by a star.

Bibliography.

1. G. S. Landsberg "Elementary textbook of physics".

2. A. A. Pinsky "Physics-11".

The three-phase alternating current system is widespread and used all over the world. With a three-phase system, optimal conditions for the transmission of electricity over long distances by wires, the ability to create electric motors that are simple in design and easy to use.

Three-phase AC system

A system is called, consisting of three circuits with operating electromotive forces (EMF) of the same frequency. These emfs are shifted relative to each other in phase by one third. Each individual circuit in the system is called a phase. The whole system of three alternating currents shifted in phase is called a three-phase current.

Almost all generators that are installed in power plants are three-phase current generators. In the design, three are connected in one unit. The electromotive forces induced in them, as mentioned earlier, are shifted by one third of the period relative to each other.

How does a generator work

In a three-phase current generator, there are three separate armatures located on the stator of the device. They are offset by 1200 between themselves. In the center of the device rotates an inductor common to three anchors. A variable EMF of the same frequency is induced in each coil. However, the moments of passing these electromotive forces through zero in each of these coils are shifted by 1/3 of the period, since the inductor passes near each coil 1/3 of the time later than the previous one.

All windings are independent generators current and power sources. If you attach wires to the ends of each winding, then you get three independent circuits. In this case, six wires are required to transmit all the electricity. However, with other connections of the windings to each other, it is quite possible to get by with 3-4 wires, which gives a big wire savings.


Connection - star

The ends of all windings are connected at one point of the generator, the so-called zero point. Then a connection is made with consumers using four wires: three are linear wires that come from the beginning of windings 1, 2, 3, one is a neutral (neutral) wire coming from the zero point of the generator. Such a system is also called a four-wire system.


Delta connection

In this case, the end of the previous winding is connected to the beginning of the next, thereby forming a triangle. Linear wires are connected to the vertices of the triangle - points 1, 2, 3. With this connection, they coincide. Compared to a star connection, a delta connection reduces the line-to-line voltage by about 1.73 times. It is allowed only under the condition of the same load of the phases, otherwise it may increase in the windings, which is dangerous for the generator.

Separate consumers (loads), which are powered by separate pairs of wires, can be connected in the same way either in a star or in a delta. The result is a situation similar to a generator: when connected by a triangle, the loads are under linear voltage, when connected by a star, the voltage is 1.73 times less.

Three-phase power supply system - special case multiphase systems of electrical circuits in which sinusoidal EMFs of the same frequency, created by a common source, are shifted relative to each other in time by a certain phase angle. IN three-phase system this angle is 2π/3 (120°).

The multi-wire (six-wire) three-phase AC system was invented by Nikola Tesla. A significant contribution to the development of three-phase systems was made by M. O. Dolivo-Dobrovolsky, who first proposed three- and four-wire AC transmission systems, revealed a number of advantages of low-wire three-phase systems in relation to other systems, and conducted a series of experiments with an asynchronous electric motor.

Animated image of the current flow in a symmetrical three-phase circuit with a star connection

Vector diagram of phase currents. symmetrical mode.


Advantages

Possible wiring diagram three-phase network in apartment buildings

  • Profitability.
    • Cost-effective transmission of electricity over long distances.
    • Less material consumption of 3-phase transformers.
    • Less material consumption power cables, since with the same power consumption, the currents in the phases are reduced (compared to single-phase circuits).
  • System balance. This property is one of the most important, since in an unbalanced system there is an uneven mechanical load on the power generating plant, which significantly reduces its service life.
  • The ability to easily obtain a circular rotating magnetic field required for work electric motor and a number of others electrical devices. 3-phase current motors (asynchronous and synchronous) are simpler than motors direct current, single- or 2-phase, and have high efficiency rates.
  • The possibility of obtaining in one installation two operating voltages - phase and linear, and two power levels when connected to a "star" or "triangle".
  • Possibility of drastically reducing flicker and stroboscopic effect of luminaires on fluorescent lamps by placing three lamps (or groups of lamps) powered by different phases in one luminaire.

Due to these advantages, three-phase systems are the most common in today's power industry.

Connection diagrams for three-phase circuits

Star


Existing types of line voltage protection that can be found for sale in electrical stores. As required modern standards, installation takes place on a DIN rail.

A star is such a connection when the ends of the phases of the generator windings (G) are connected into one common point, called the neutral point or neutral. The ends of the phases of the windings of the receiver (M) are also connected to a common point. The wires connecting the beginning of the phases of the generator and the receiver are called linear. The wire connecting two neutrals is called neutral.


Distribution tires zero wires and ground wires when connected with a star. One of the advantages of star connection is the savings on the neutral wire, since only one wire is required from the generator to the neutral wire separation point near the consumer.

A three-phase circuit with a neutral wire is called a four-wire circuit. If there is no neutral wire - three-wire.

If the resistances Z a, Z b, Z c of the receiver are equal to each other, then such a load is called symmetrical.

Relationship between linear and phase currents and voltages.

The voltage between the line wire and the neutral (U a, U b, U c) is called phase. The voltage between two line wires (U AB, U BC, U CA) is called linear. To connect the windings with a star, with a symmetrical load, the relationship between linear and phase currents and voltages is valid:

Consequences of burnout (break) of the neutral wire in three-phase networks

With a symmetrical load in a three-phase system, supplying the consumer with linear voltage is possible even in the absence of a neutral wire. However, when supplying the load with phase voltage, when the load on the phases is not strictly symmetrical, the presence of a neutral wire is mandatory. When it breaks or a significant increase in resistance (poor contact), the so-called " phase imbalance" occurs, as a result of which the connected load, designed for the phase voltage, may be under an arbitrary voltage in the range from zero to linear ( specific meaning depends on the load distribution over the phases at the moment of break neutral wire). This is often the cause of failure of consumer electronics in apartment buildings. Since the resistance of the consumer remains constant, then, according to Ohm's law, with increasing voltage, the current passing through the consumer device will be much greater than the maximum allowable value, which will cause combustion and / or failure of the powered electrical equipment. Low voltage can also cause equipment failure. Sometimes a burnout (break) of the neutral wire at a substation can cause a fire in apartments.

The problem of harmonics that are multiples of a third

Modern technology is increasingly equipped with pulse network. A switching source without a power factor corrector draws current in narrow pulses near the peak of the supply voltage sine wave, at the moment the input rectifier capacitor is charging. A large number of such power supplies in the network creates an increased current of the third harmonic of the supply voltage. Harmonic currents that are multiples of the third, instead of mutual compensation, are mathematically summed up in the neutral conductor (even with a symmetrical load distribution) and can lead to its overload even without exceeding the permissible power consumption by phases. Such a problem exists, in particular, in office buildings with a large number of simultaneously operating office equipment.
The existing reactive power compensation installations are unable to solve this problem, since the decrease in power factor in networks with a predominance of impulse sources power supply is not associated with the introduction of a reactive component, but is due to the non-linearity of current consumption. The solution to the problem of the third harmonic is the use of a power factor corrector (passive or active) as part of the circuit of produced switching power supplies.
The requirements of IEC 1000-3-2 limit the harmonic components of the load current for devices with a power of 50 W or more. In Russia, the number of harmonic components of the load current is standardized by the standards GOST 13109-97, OST 45.188-2001.


Triangle

A triangle is such a connection when the end of the first phase is connected to the beginning of the second phase, the end of the second phase to the beginning of the third, and the end of the third phase is connected to the beginning of the first.

Relationship between line and phase currents and voltages

To connect the windings in a triangle, with a symmetrical load, the relationship between linear and phase currents and voltages is valid:

Common Voltage Standards

Marking

Conductors belonging to different phases are marked with different colors. different colors also mark neutral and protective conductors. This is done to provide adequate protection against damage. electric shock, as well as for ease of maintenance, installation and repair electrical installations And electrical equipment. IN different countries conductor marking has its own differences. However, many countries follow general principles color coding conductors set out in the IEC 60445:2010 standard of the International Electrotechnical Commission.

Phase conductor 1 Phase conductor 2 Phase conductor 3 Neutral conductor Protective conductor
USA(120/208V) Black Red Blue White or gray Green
USA(277/480V) Orange Brown Yellow White or gray Green
Canada Red Black Blue White Green
Canada(Insulated three-phase installations) Orange Brown Yellow White Green
Great Britain(since April 2006) Red (Brown) Yellow (formerly White) (Black) Blue (Gray) Black (Blue) green yellow
Europe(since April 2004) Brown Black Grey Blue green yellow
Europe(until April 2004, depending on the country) Brown or Black Black or Brown Black or Brown Blue green yellow
Europe(Tyre designation) Yellow Brown Red
Russia(THE USSR) Yellow Green Red Blue Green-yellow (black on older units)
Russia(since January 1, 2011) Brown Black Grey Blue green yellow
Australia and New Zealand Red Yellow Blue Black
South Africa Red Yellow Blue Black Green-yellow (on old installations - Green)
Malaysia Red Yellow Blue Black Green-yellow (on old installations - Green)
India Red Yellow Blue Black Green

Three-phase double-circuit transmission line

Three-phase power supply system- a special case of multi-phase systems of electrical circuits of alternating current, in which sinusoidal EMFs of the same frequency, created by a common source, are shifted relative to each other in time by a certain phase angle. In a three-phase system, this angle is 2π/3 (120°).

A multi-wire (six-wire) three-phase AC system was invented by Nikola Tesla. A significant contribution to the development of three-phase systems was made by M. O. Dolivo-Dobrovolsky, who first proposed three- and four-wire AC transmission systems, revealed a number of advantages of low-wire three-phase systems in relation to other systems, and conducted a series of experiments with an asynchronous electric motor.

Encyclopedic YouTube

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    Each of the operating EMF is in its own phase of the periodic process, therefore it is often called simply "phase". Also, "phases" are called conductors - carriers of these EMFs. In three-phase systems, the shear angle is 120 degrees. Phase conductors are designated in the Russian Federation in Latin letters L with a digital index 1 ... 3, or A, B and C.

    Common designations of phase wires:

    Russia, EU (above 1000V) Russia, EU (below 1000V) Germany Denmark
    A L1 L1 R
    B L2 L2 S
    C L3 L3 T

    Advantages

    • Profitability.
      • Cost-effective transmission of electricity over long distances.
      • Less material consumption of 3-phase transformers.
      • Less material consumption of power cables, since with the same power consumption, the currents in the phases are reduced (compared to single-phase circuits).
    • System balance. This property is one of the most important, since in an unbalanced system there is an uneven mechanical load on the power generating plant, which significantly reduces its service life.
    • The ability to easily obtain a circular rotating magnetic field, necessary for the operation of an electric motor and a number of other electrical devices. 3-phase current motors (asynchronous and synchronous) are simpler than DC motors, single or 2-phase, and have high efficiency.
    • The possibility of obtaining in one installation two operating voltages - phase and linear, and two power levels when connected to a "star" or "triangle".
    • Possibility of drastically reducing flicker and stroboscopic effect of luminaires on fluorescent lamps by placing three lamps (or groups of lamps) powered by different phases in one luminaire.

    Due to these advantages, three-phase systems are the most common in today's power industry.

    Connection diagrams for three-phase circuits

    Star

    A three-phase circuit with a neutral wire is called a four-wire circuit. If there is no neutral wire - three-wire.

    If the resistances Z a, Z b, Z c of the consumer are equal to each other, then such a load is called symmetrical.

    Linear and phase quantities

    The voltage between the line wire and the neutral (U a, U b, U c) is called phase. The voltage between two line wires (U AB, U BC, U CA) is called linear. To connect the windings with a star, with a symmetrical load, the relationship between linear and phase currents and voltages is valid:

    I L = I F ; U L = 3 × U F (\displaystyle I_(L)=I_(F);\qquad U_(L)=(\sqrt (3))\times (U_(F)))

    It is easy to show that the line voltage is phase shifted by π / 6 (\displaystyle \pi /6) regarding phase:

    U L a b = u F a − u F b = U F [ cos ⁡ (ω t) − cos ⁡ (ω t − 2 π / 3) ] = 2 U F sin ⁡ (− π / 3) sin ⁡ (ω t − π / 3) = 3 U F cos ⁡ (ω t + π − π / 3 − π / 2) (\displaystyle u_(L)^(ab)=u_(F)^(a)-u_(F)^(b )=U_(F)[\cos(\omega t)-\cos(\omega t-2\pi /3)]=2U_(F)\sin(-\pi /3)\sin(\omega t- \pi /3)=(\sqrt (3))U_(F)\cos(\omega t+\pi -\pi /3-\pi /2))

    U L = 3 U F cos ⁡ (ω t + π / 6) (\displaystyle u_(L)=(\sqrt (3))U_(F)\cos(\omega t+\pi /6))

    Three-phase current power

    To connect the windings with a star, with a symmetrical load, the power of a three-phase network is:

    P = 3 U F I F c o s φ = 3 U L 3 I L c o s φ = 3 U L I L c o s φ (\displaystyle P=3U_(F)I_(F)cos\varphi =3(\frac (U_(L))(\sqrt (3 )))I_(L)cos\varphi =(\sqrt (3))U_(L)I_(L)cos\varphi )

    Consequences of burnout (break) of the neutral wire in three-phase networks

    With a symmetrical load in a three-phase system, supplying the consumer with linear voltage is possible even in the absence of a neutral wire. However, when supplying the load with phase voltage, when the load on the phases is not strictly symmetrical, the presence of a neutral wire is mandatory. If it breaks or a significant increase in resistance (poor contact), the so-called “phase imbalance”, as a result of which the connected load, designed for the phase voltage, may be under an arbitrary voltage in the range from zero to linear (the specific value depends on the load distribution over the phases at the moment of breaking the neutral wire). This is often the cause of the failure of consumer electronics in apartment buildings, which can lead to fires. Low voltage can also cause equipment failure.

    The problem of harmonics that are multiples of a third

    Modern technology is increasingly equipped with pulse network. A pulsed source without a power factor corrector consumes current in narrow pulses near the peak of the supply voltage sinusoid, at the moment the input rectifier capacitor is charging. A large number of such power supplies in the network creates an increased current of the third harmonic of the supply voltage. Harmonic currents that are multiples of the third, instead of mutual compensation, are mathematically summed up in the neutral conductor (even with a symmetrical load distribution) and can lead to its overload even without exceeding the permissible power consumption by phases. Such a problem exists, in particular, in office buildings with a large number of simultaneously operating office equipment. The solution to the problem of the third harmonic is the use of a power factor corrector (passive or active) as part of the circuit of produced switching power supplies. The requirements of IEC 1000-3-2 limit the harmonic components of the load current for devices with a power of 50 W or more. In Russia, the number of harmonic components of the load current is standardized by the standards GOST R 54149-2010, GOST 32144-2013 (from 1.07.2014), OST 45.188-2001.