Active conduction in an alternating current circuit. Conductance. Calculation formulas for a chain with parallel connection of branches. Vector diagram method

On fig. 14.14, and the same circuit elements that were considered are connected in parallel (see Fig. 14.7, a). Assume that the voltage is known for this circuit u = Um sinωt. and parameters of the circuit elements R, L, C. It is required to find the currents in the circuit and the power.

Vector diagram for a chain with parallel connection of branches. Vector diagram method

For instantaneous quantities, in accordance with the first Kirchhoff law, the current equation

Representing the current in each branch as the sum of the active and reactive components, we obtain

For effective currents, you need to write a vector equation

The numerical values ​​of the current vectors are determined by the product of the voltage and conductivity of the corresponding branch.

On fig. 14.14, b, a vector diagram is constructed that corresponds to this equation. For the initial vector is taken, as usual when calculating circuits with parallelbranch connection,vector voltage U, and then the current vectors in each branch are applied, and their directions relative to the voltage vector are chosen in accordance with the nature of the conduction of the branches. The starting point for constructing the current diagram is the point that coincides with the beginning of the voltage vector. A vector is drawn from this point l 1a active current branches I (in phase coincides with the voltage), and a vector is drawn from its end I 1p reactive current the same branch (leads the voltage by 90 °). These two vectors are components of the vector I 1 current of the first branch . Further, in the same order, the current vectors of other branches are plotted. It should be noted that the conductivity branches 3-3 active , so the reactive component of the current in this branch is zero. IN branches 4-4 and 5-5 conduction reactive , therefore, there are no active components in the composition of these currents.

Calculation formulas for a chain with parallel connection of branches. Vector diagram method

From vector diagram it can be seen that all the active components of the current vectors are directed in the same way - parallel to the voltage vector, so vector addition can be replaced by arithmetic to find the active component of the total current: I a \u003d I 1a + I 2a + I 3a .

Reactive components current vectors are perpendicular to the voltage vector, with inductive currents directed in one direction, and capacitive currents in the other. Therefore, the reactive component of the total current in the circuit is determined by their algebraic sum, in which inductive currents are considered positive, and capacitive currents are negative: I p \u003d - I 1p + I 2p - I 4p + I 5p .

The active, reactive and total current vectors of the entire circuit form a right triangle, from which it follows

Attention should be paid to possible errors in determining the total conductivity of the circuit from the known conductivities of individual branches: it is forbidden add arithmetically the conductivities of the branches if the currents in them are out of phase.

full conduction circuits are generally defined as the hypotenuse of a right-angled triangle, the legs of which are the active and reactive conductivities of the entire circuit expressed on a certain scale:

From the triangle of currents, you can also go to the triangle of powers and to determine the power, you can get the already known formulas

active power circuits can be represented as an arithmetic sum of the active powers of the branches.

Reactive power chain is equal to the algebraic sum of the cardinalities of the branches. In this case, the inductive power is taken positive, and the capacitive power is taken negative:

Calculation of the circuit without determining the conductivity of the branches

The calculation of the electrical circuit with a parallel connection of the branches can be performed without preliminary determination of active and reactive conductivities , i.e. representing the circuit elements in the equivalent circuit with their active and reactive resistances (Fig. 14.15, a).

Determine the currents in the branches according to the formula (14.4);

Where Z 1 , Z 2 etc. - total resistance of branches.

The total resistance of the branch, which includes several elements connected in series, is determined by the formula (14.5).

To build a vector diagram of currents (Fig. 14.15, b), you can determine the active and reactive components of the current of each branch using the formulas

and so on for all branches.

In this case, there is no need to determine the angles f 1 f 2 and plotting them on the drawing.

Current in the unbranched part of the circuit

In the course of general physics, for the calculation of electrical circuits, they mainly use Ohm's and Kirchhoff's laws, which include voltages, currents and resistances. However, to calculate complex electrical circuits, and especially AC circuits, it is advisable to use conductivity instead of resistance.

Conductivity in a DC circuit g is the reciprocal of resistance

The SI unit for conductivity is Siemens (in honor of the 19th-century German electrical engineer E. W. Siemens).

1 Sim is the conductivity of a conductor with a resistance of 1 ohm.

In AC circuits, as you know, there are three types of resistance: active R, reactive and total r. By analogy with this, three types of conductivities are introduced: active g, reactive b and full y. However, only the total conductivity y is the reciprocal of the total resistance:

To introduce active g and reactive b conductivities, consider an alternating current circuit of series-connected active R and inductive resistances (Fig. 1-25, a). Let's build a vector diagram for it (Fig. 1-25, b). We decompose the current in the circuit into active and reactive components, and from the resulting current triangle we move on to the resistance triangle (Fig. 1-25, c). From the latter we have:

From the vector diagram (see Fig. 1-25, b), taking into account formula (1.30), we have:

where is the active conductivity,

where is the reactive conductivity.

Now let's establish the relationship between conductivities. For the circuit under consideration, we have:

Substituting the values, respectively, from relations (1.31) and (1.32), we obtain:

where is the total conductivity of the circuit.

By analogy with the triangle of resistances (Fig. 1-25, c), we build a triangle of conductivities (Fig. 1-25, d). By analogy with inductive and capacitive resistances, inductive and capacitive conduction are distinguished.

In the case of a branched circuit (Fig. 1-26, a), the circuit can be easily converted into the so-called equivalent circuit (Fig. 1-26, b), in which two branches are replaced by one with the corresponding equivalent active and

reactive resistances. The calculation of the last resistances, as well as other circuit parameters, is easier using conductivities. Let us establish the basic regularities for conductivities in a branched circuit.

We express the total current in terms of its components or equivalent conductivities:

In turn, the active component of the total current is equal to the sum of the active components of the branch currents:

i.e., the equivalent active conductance of the branch is equal to the arithmetic sum of the active conductivities of the branches.

Since the reactive components of the branches of the circuit under consideration are in antiphase, then for the reactive component of the total current we have:

i.e., the equivalent reactive conductance of a branch is equal to the algebraic sum of the reactive conductivities of parallel branches, while it is taken with a plus sign, and with a minus sign.

Conductivity

complex conductionis the ratio of the complex current to the complex voltage

where y=1/z - the reciprocal of impedance is calledcomplete conductivity.
Complex conductivity and complex resistance are mutually inverse. The complex conductivity can be represented as

Where - the real part of the complex conductivity, is calledactive conduction; - the value of the imaginary part of the complex conductivity, is calledreactive conductivity;

From () and (3.29) it follows that for the circuit shown in Fig. 3.12, complex conductivity

Where


and are named accordingly.active, inductive and capacitive conduction.
Reactive conduction


Inductive and capacitive conductivities are arithmetic quantities, and reactive conductivity b is an algebraic quantityand can be either greater or less than zero. Reactive conduction b the branch containing only the inductance is equal to the inductive conductance, and the reactive conductivity b the branch containing only the capacitance is equal to the capacitance with the opposite sign, i.e..


The phase shift between voltage and current depends on the ratio of inductive and capacitive conductivities. For the scheme according to Fig. 3.14 presents vector diagrams for three cases, namelyWhen constructing these diagrams, the initial phase of the voltage is assumed to be zero, therefore, as follows from (3.28), are equal and opposite in sign ().
Considering the diagram in Fig. 3.12 as a whole as a passive two-terminal network, it can be seen that at a given frequency it is equivalent in the first case to a parallel connection of resistance and inductance, in the second - to resistance, and in the third - to a parallel connection of resistance and capacitance. The second case is called resonance. For given L and C ratio betweendepends on the frequency, and therefore the form of the equivalent circuit also depends on the frequency.
Note that in the diagram in Fig. 3.12 each of the parallel branches contains one element. Therefore, such a simple expression for Y was obtained, in which the conductivities of the elements are included as separate terms.
Note that the designationsare used not only for resistances and conductivities, but also for circuit elements characterized by these quantities. In such cases, the elements of the circuit are given the same names as those assigned to the quantities indicated by these letters. Complex resistances or conductivities as elements of the circuit have a symbol in the form of a rectangle (see Fig. 3.1). In the same way, reactances or conductivities are denoted if they want to note that they can be both inductive and capacitive reactances or conductivities.

. Capacitor (ideal capacitance)

The processes for the ideal capacity have a similar character. Here . Therefore, from (3) it follows that . Thus, no active power is consumed in the inductor and capacitor (P=0), since there is no irreversible conversion of energy into other forms of energy in them. Only energy circulation occurs here: electrical energy is stored in the magnetic field of the coil or the electric field of the capacitor for a quarter of the period, and during the next quarter of the period the energy returns to the network. Because of this, the inductor and the capacitor are called reactive elements, and their resistances X L and X C, in contrast to the active resistance R of the resistor, are reactive.

The intensity of energy exchange is usually characterized by the highest value of the rate of energy input into the magnetic field of the coil or the electric field of the capacitor, which is called reactive power.

In general, the expression for reactive power is:

It is positive for lagging current (inductive load-) and negative for leading current (capacitive load-). The unit of power as applied to the measurement of reactive power is called volt-ampere reactive(VAr).

In particular, for the inductor we have: , because.

.

From the latter it can be seen that the reactive power for an ideal inductor is proportional to the frequency and the maximum energy reserve in the coil. Similarly, you can get for an ideal capacitor:

.

Resistor (ideal active resistance).

Here, the voltage and current (see Fig. 2) are in phase, so the power is always positive, i.e. resistor consumes active power

25. Active, reactive and total conductivity of the circuit.

When elements are connected in parallel R, L, C(Fig. 1) the total conductivity is equal to
(1)

Where g = 1/ R - active conductivity of the circuit;

b - reactive conductance of the circuit.

The reactive conductance of the circuit is then determined by the expression
(2)

The current in the circuit is determined by the expression

(3)

The current in active conduction coincides with the voltage in phase

(4)

The current in the tank determines the voltage in phase by 90 0

(5)

The current in the inductor lags the voltage in phase by 90 0

(6)

Average activity power consumed in the circuit

(7)

Phase shift between voltage U at the circuit terminals and current I it is defined by the expressions

(8)

(9)

26. Transient processes in linear electrical circuits. Basic concepts, laws of communication.

With all changes in the electrical circuit: switching on, switching off, short circuit, fluctuations in the value of any parameter, etc. - transient processes occur in it, which cannot proceed instantly, since an instantaneous change in the energy stored in the electromagnetic field of the circuit is impossible. Thus, the transient process is due to the discrepancy between the amount of stored energy in the magnetic field of the coil and the electric field of the capacitor and its value for the new state of the circuit. During transient processes, large overvoltages, overcurrents, and electromagnetic oscillations can occur that can disrupt the operation of the device up to its failure. On the other hand, transient processes find useful practical applications, for example, in various kinds of electronic generators. All this necessitates the study of methods for analyzing non-stationary modes of operation of the circuit.

The main methods for analyzing transient processes in linear circuits:

classical method, consisting in the direct integration of differential equations describing the electromagnetic state of the circuit.

operator method, consisting in solving the system algebraic equations relative to the images of the desired variables, followed by the transition from the found images to the originals.

frequency method, based on the Fourier transform and widely used in solving synthesis problems.

Calculation method using Duhamel integral, used with a complex shape of the disturbance curve.

state variable method, which is an ordered way to determine the electromagnetic state of the circuit based on the solution of a system of differential equations of the first order, written in normal form (Cauchy form).

Switching laws

It is possible to prove the laws of commutation by contradiction: if we assume the opposite, then infinitely large values ​​are obtained And , which leads to violation of Kirchhoff's laws.

In practice, with the exception of special cases (incorrect commutations), it is permissible to use these laws in a different formulation, namely:

the first law of commutation is in the branch with the inductor current at the moment

.

the second law of commutation is voltage across the capacitor

switching retains its pre-switching value and subsequently begins to change from it: .

It should be emphasized that a more general formulation of the switching laws is the provision on the impossibility of an abrupt change at the moment of switching for circuits with an inductance coil - flux links, and for circuits with capacitors - charges on them. As an illustration of what has been said, the diagrams in Fig. 2, transient processes in which belong to the so-called incorrect switching(the name comes from the neglect of small parameters in such schemes, the correct consideration of which can lead to a significant complication of the problem).

Indeed, when translating in the diagram in Fig. 2, and the key from position 1 to position 2, the interpretation of the second switching law as the impossibility of an abrupt change in the voltage across the capacitor leads to the failure of the second Kirchhoff law . Similarly, when the key is opened in the circuit in Fig. 2, b interpretation of the first switching law as the impossibility of an abrupt change in current through the inductor leads to the failure of the first Kirchhoff law . For these schemes, based on the conservation of charge and, accordingly, flux linkage, we can write:

Dependent initial conditions are the values ​​of the remaining currents and voltages, as well as the derivatives of the desired function at the moment of switching, determined from independent initial conditions using equations compiled according to Kirchhoff's laws for . The required number of initial conditions is equal to the number of integration constants. Since it is rational to write an equation of the form (2) for a variable whose initial value refers to independent initial conditions, the problem of finding the initial conditions usually comes down to finding the values ​​of this variable and its derivatives up to (n-1) order inclusive at.

Active conduction ( G) is due to active power losses in dielectrics. Its value depends on:

leakage current through insulators (small, can be neglected);

corona power loss.

Active conduction leads to active power losses in the idling mode of the overhead transmission line. Corona power loss (  cor) are due to the ionization of the air around the wires. When the electric field strength of the wire becomes greater than the electric strength of air (21.2 kV/cm), electric discharges form on the surface of the wire. Due to surface irregularities of stranded wires, dirt and burrs, discharges first appear only at certain points of the wire - local crown. As the tension increases, the corona spreads over a large surface of the wire and eventually covers the entire wire along the entire length - common crown.

The power loss per corona depends on weather conditions. The greatest power losses to the corona occur during various atmospheric precipitation. For example, on overhead transmission lines with a voltage of 330750 kV, the core during snow increases by 14%, rain - by 47%, frost - by 107% compared with losses in good weather. Corona causes corrosion of wires, creates interference on communication lines and radio interference.

The value of power losses per corona can be calculated by the formula:

Where
coefficient taking into account barometric pressure;

U f, U cor f - respectively, the phase operating voltage of the power transmission line and the voltage at which corona occurs.

Initial tension(V good weather) at which a common corona occurs is calculated by the Pick formula:

kV/cm

Where m– coefficient of non-smoothness of the drive;

R pr - radius of the wire, cm;

coefficient that takes into account barometric pressure.

For smooth cylindrical wires, the value m= 1, for stranded wires - m= 0.820.92.

The value of δ is calculated by the formula:

,

Where R– pressure, mm Hg;

air temperature, 0 C.

At normal atmospheric pressure (760 mm Hg) and a temperature of 20 0 C= 1. For regions with a temperate climate, the average annual value is equal to 1.05.

Working tension under normal operating conditions, the power transmission line is determined by the formulas:

for unsplit phase

kV/cm

for split phase

, kV/cm

Where U ex - average operational (linear) voltage.

If the magnitude of the operating voltage is unknown, then consider that U ex = U nom.

The magnitude of the working tension on the phases is different. In the calculations, the value of the greatest tension is taken:

E max= k spread  k rasch E,

Where k rasp - coefficient taking into account the location of the wires on the support;

k rast - coefficient taking into account the construction of the phase.

For wires located at the vertices of an equilateral triangle or close to it, k dist = 1. For wires arranged horizontally or vertically, k dist = 1.05 - 1.07.

For the unsplit phase k split = 1. With a split phase design, the coefficient k rast is calculated by the formulas:

at n= 2

at n= 3

The voltage at which corona occurs is calculated by the formula:

To boost U core needs to be lowered E max. To do this, either increase the radius of the wire R pr or D cf. In the first case, it is effective to split the wires in phase. Increase D cf leads to a significant change in the dimensions of the transmission line. The action is ineffective because D cp is under the sign of the logarithm.

If E max> E 0 , then the operation of the power transmission line is uneconomical due to power losses in the corona. According to the PUE, there is no crown on the wires if the following condition is met:

E max 0.9 E 0 (m=0,82,= 1).

When designing, the selection of wire sections is carried out in such a way that there is no corona in good weather. Since an increase in the radius of the wire is the main means of reducing P cor, the minimum allowable sections are set according to the conditions of the corona: at a voltage of 110 kV - 70mm 2, at a voltage of 150 kV - 120mm 2, at a voltage of 220 kV - 240mm 2.

The value of linear active conductivity is calculated by the formula:

, cm/km.

The active conductivity of the network section is as follows:

When calculating the steady-state modes of networks with voltages up to 220 kV, active conductivity is not taken into account - an increase in the radius of the wire reduces the power loss to the corona to almost zero. At U nom 330kV increase in the radius of the wire leads to a significant increase in the cost of power transmission lines. Therefore, in such networks, the phase is split and the active conductivity is taken into account in the calculations.

In cable transmission lines, the calculation of active conductivity is carried out using the same formulas as for overhead transmission lines. The nature of active power losses is different.

In cable lines  P are caused by phenomena occurring in the cable due to the absorption current. For CLEP, dielectric losses are indicated by the manufacturer. Dielectric losses in CLEP are taken into account at U35 kV.

Reactive (capacitive conduction)

Reactive conductivity is due to the presence of capacitance between phases and between phases and ground, since any pair of wires can be considered as a capacitor.

For overhead power transmission lines, the value of linear reactive conductivity is calculated by the formulas:

for unsplit wires

, Sm/km;

for split wires

Splitting increases b 0 by 2133%.

For CLEP, the value of linear conductivity is often calculated by the formula:

b 0 =  C 0 .

Capacitance value C 0 is given in the reference literature for various brands of cable.

The reactive conductivity of the network section is calculated by the formula:

IN = b 0 l.

For overhead power lines, the value b 0 is much less than that of cable transmission lines, it is small, since D avg. overhead power lines >> D cf. KLEP.

Under the action of voltage, a capacitive current flows in the conductances (bias current or charging current):

I c= INU f.

The value of this current determines the loss of reactive power in reactive conduction or the charging power of the transmission line:

In district networks, charging currents are commensurate with operating currents. At U nom = 110 kV, value Q c is about 10% of the transmitted active power, with U nom = 220 kV - Q from ≈ 30% R. Therefore, it must be taken into account in the calculations. In a network with a rated voltage of up to 35 kV, the value Q can be neglected.

Power line equivalent circuit

So, the power line is characterized by active resistance R l, line reactance X l, active conductivity G l, reactive conductivity IN l. In calculations, a power transmission line can be represented by symmetrical U- and T-shaped circuits (Fig. 4.6).

P - figurative scheme is used more often.

Depending on the voltage class, one or another parameter of the complete equivalent circuit can be neglected (see Fig. 4.7):

VTL with voltage up to 220 kV ( R core  0);

VTL with voltage up to 35 kV ( R core  0,  Q c  0);

CLEP voltage 35 kV (reactance  0)

CLEP with a voltage of 20 kV (reactance  0, dielectric losses  0);

CLEP with voltage up to 10 kV (reactance  0, dielectric losses  0,  Q c  0).