How are active reactive and admittance determined? Calculation of circuits with parallel connections of branches. Active, reactive and admittance of the circuit
Active conductivity ( G) is caused by active power losses in dielectrics. Its value depends on:
leakage current through insulators (small, can be neglected);
power losses to the corona.
Active conduction leads to losses of active power in the no-load mode of overhead power lines. Corona power loss ( cor) are caused by ionization of the air around the wires. When the electric field strength of a wire becomes greater than the electrical strength of air (21.2 kV/cm), electrical discharges are formed on the surface of the wire. Due to uneven surfaces of stranded wires, dirt and burrs, discharges initially appear only at individual points of the wire - local crown. As the voltage increases, the corona spreads over a larger surface of the wire and ultimately covers the entire length of the wire - common crown.
Corona power losses depend on weather conditions. The greatest power losses to the corona occur under various atmospheric precipitations. For example, on overhead power lines with a voltage of 330750kV, the corr during snow increases by 14%, rain - by 47%, frost - by 107% compared to losses in good weather. Corona causes corrosion of wires, creates interference on communication lines and radio interference.
The amount of power loss to the corona can be calculated using 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(in good weather), in which the overall corona occurs is calculated using the Peak formula:
kV/cm
Where m– drive roughness coefficient;
R pr – wire radius, cm;
coefficient taking into account barometric pressure.
For smooth cylindrical wires the value m= 1, for stranded wires – m= 0.820.92.
The value δ is calculated using the formula:
,
Where R– pressure, mmHg;
air temperature, 0 C.
At normal atmospheric pressure (760 mm Hg) and temperature 20 0 C = 1. For areas with a temperate climate, the average annual value is 1.05.
Work tension under normal operating conditions, the power line is determined by the formulas:
for unsplit phase
kV/cm
for split phase
, kV/cm
Where U ex – average operating (linear) voltage.
If the value of the operating stress is unknown, then it is assumed that U ex = U nom.
The amount of operating tension in the phases is different. In the calculations, the value of the highest tension is taken:
E max = k disp k dist E,
Where k ras – coefficient taking into account the location of the wires on the support;
k dist – coefficient taking into account the phase design.
For wires located at the vertices of an equilateral triangle or close to it, k distribution = 1. For wires located horizontally or vertically, k disp = 1.05 – 1.07.
For the unsplit phase k split = 1. With a split phase design, the coefficient k dist is calculated using the formulas:
at n= 2
at n= 3
The voltage at which corona occurs is calculated by the formula:
To increase U core needs to be reduced E max. To do this, you need to increase either the radius of the wire R pr or D avg. 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 power lines. The event is ineffective because D cp is under the sign of the logarithm.
If E max > E 0, then the operation of the power line is uneconomical due to power losses due to the corona. According to the PUE, there is no corona 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 increasing the radius of the wire is the main means of reducing P core, the minimum permissible cross-sections for corona conditions have been established: at a voltage of 110 kV - 70 mm 2, at a voltage of 150 kV - 120 mm 2, at a voltage of 220 kV - 240 mm 2.
The value of linear active conductivity is calculated by the formula:
, Sm/km.
The active conductance of a network section is found as follows:
When calculating steady-state modes of networks with voltages up to 220 kV, active conductivity is not taken into account - increasing the radius of the wire reduces power losses to the corona to almost zero. At U rated 330 kV, an increase in the radius of the wire leads to a significant increase in the cost of power lines. Therefore, in such networks the phase is split and active conductivity is taken into account in the calculations.
In cable power lines, active conductivity is calculated using the same formulas as for overhead power lines. The nature of active power losses is different.
In cable lines P are caused by phenomena occurring in the cable due to absorption current. For CLEP, dielectric losses are indicated by the manufacturer. Dielectric losses in CLEP are taken into account at U35 kV.
Reactive (capacitive conductivity)
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 lines, the value of linear reactive conductivity is calculated using the formulas:
for unsplit wires
, S/km;
for split wires
Cleavage increases b 0 by 2133%.
For CLEP, the linear conductivity value is often calculated using the formula:
b 0 = C 0 .
Capacity size C 0 is given in the reference literature for various brands of cable.
The reactive conductivity of a network section is calculated using the formula:
IN = b 0 l.
Overhead power lines have a meaning b 0 is significantly less than that of cable power lines, it is small, since D Wed overhead power lines >> D cf CLEP.
Under the influence of voltage, a capacitive current flows in the conductivities (bias current or charging current):
I c = INU f.
The magnitude of this current determines the loss of reactive power in reactive conductivity or the charging power of the power line:
In regional networks, charging currents are comparable to operating currents. At U nom = 110 kV, value Q c is about 10% of the transmitted active power, with U rated = 220 kV – Q with ≈ 30% R. Therefore, it must be taken into account in calculations. In a network with a rated voltage of up to 35 kV Q c can be neglected.
Power line equivalent circuit
So, power lines are characterized by active resistance R l, line reactance X l, active conductivity G l, reactive conductivity IN l. In calculations, power lines can be represented by symmetrical U- and T-shaped circuits (Fig. 4.6).
The U-shaped scheme is used more often.
Depending on the voltage class, certain parameters of the complete equivalent circuit can be neglected (see Fig. 4.7):
Overhead power lines with voltage up to 220 kV ( R core 0);
Overhead power lines with voltage up to 35 kV ( R cor 0, Q c 0);
CLEP voltage 35 kV (reactance 0)
CLEP with a voltage of 20 kV (reactance 0, dielectric losses 0);
CLPP with voltage up to 10 kV (reactance 0, dielectric losses 0, Q c 0).
Answer: R=r o ·l, where r o is linear active resistance (Ohm/km). The active resistance of overhead and cable lines is determined by the material of the current-carrying conductors and their cross-section. To some extent, it depends on the temperature of the conductors and the frequency of the alternating current flowing through them. However, this influence is small and is usually not taken into account when calculating electrical networks. Therefore, resistance values r 0 for each brand of wire or cable are usually taken according to tables corresponding to direct current transmission and temperature +20ºС. r 0 t = r 0 20 ·(1+α(t-20)), where α is the temperature coefficient; r 0 20 – resistance at 20 ºС. When making estimation calculations for conductors made of non-ferrous metals, the active resistance can be determined by the formula: r 0 =ρ/F, where ρ is the resistivity (Ohm mm 2 /km), F is the cross-section of the conductor (mm 2).
G=g 0 ·l, where g 0 is specific active conductivity (S/km). The conductivity due to corona losses is highly variable and depends on air humidity and other meteorological conditions. The average value of active conductivity for a year is obtained through the average losses to the corona ΔP k: ; , where ΔP kud - specific average annual losses per corona (kW/km).
Corona power losses are taken into account for overhead lines with Unom 330 kV and above. In 110-220 kV overhead lines, these losses can be ignored, because The PUE has established minimum wire cross-sections to reduce ΔP to acceptable levels. For 110 kV overhead line - AS 70/11, 220 kV overhead line - AS 240/32.
The most radical means of reducing power losses to the crown is to increase the diameter of the wire. X=x 0 ·l, where x 0 is the linear inductive reactance (Ohm/km).
Inductive reactance is caused by the magnetic field that arises around and inside the wires and cores of cables, which induces an electromotive force of self-induction in each conductor. Inductive reactance depends on the relative position of the conductors, their diameter and magnetic permeability and frequency of alternating current. For overhead lines with aluminum and steel-aluminum wires, the resistance per 1 km is calculated: x 0 = 0.144 lg(2 D av / d) + 0.0156, where D av is the geometric mean distance between the phase wires, mm, d is the wire diameter, mm .
D avg depends on the type of support arrangement and U nom D av = 
, where D А B, D BC, D CA - the distance between the wires of the corresponding phases.
For overhead lines, the value x 0 is given in the reference table depending on D cf or voltage and type of wire. The inductive reactance of cable lines is influenced by the design features of the cables. When making calculations, use the factory data about x 0 given in the reference book. The reactive conductivity of the line is caused by the capacitances between the wires of different phases and the wire-to-ground capacitance. It is determined by the formula: , B=b 0 ·l, where b 0 is the specific reactive (capacitive) conductivity, Ohm/km. For overhead lines, the capacitive conductivity can be found as 
or determined from reference tables depending on the type of wire and the geometric mean distance between the wires or nom. voltage. The capacitive conductivity of cable lines depends on the cable design and is indicated by the manufacturer, but for approximate calculations it can be estimated using the formula. It is obvious that the value of b 0 for cable lines is much greater than for overhead lines due to the lower values of Dav.
Conductivity
Complex conductivity is the ratio of complex current to complex voltage
where y=1/z is the reciprocal of impedance, called admittance.
Complex conductivity and complex resistance are mutually inverse. Complex conductivity can be represented as
where is the real part of the complex conductivity, called active conductivity; - the value of the imaginary part of the complex conductivity is called reactive conductivity;
From (3.30) and (3.29) it follows that for the circuit shown in Fig. 3.12, complex conductivity
and are called active, inductive and capacitive conductance, respectively.
Reactive conductivity
Inductive and capacitive conductivity are arithmetic quantities, and reactive conductivity b is an algebraic quantity and can be either greater or less than zero. Reactive conductance b of a branch containing only inductance is equal to inductive conductance , and reactive conductance b of a branch containing only capacitance is equal to capacitive conductance with the opposite sign, i.e.
The phase shift between voltage and current depends on the ratio of inductive and capacitive conductivity. For the circuit according to Fig. 3.12 in Fig. Figure 3.14 shows vector diagrams for three cases, namely, When 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 ().
Looking at the diagram in Fig. 3.12 as a whole as a passive two-terminal network, it can be noted 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, the relationship between depends on the frequency, and therefore the type of equivalent circuit also depends on the frequency.
Let us pay attention to the fact that in the diagram of Fig. 3.12 each of the parallel branches contains one element. Therefore, we got such a simple expression for Y, in which the conductivities of the elements are included as separate terms.
Note that the notation is used not only for resistance and conductivity, but also for circuit elements characterized by these quantities. In such cases, the elements of the diagram are given the same names as those assigned to the quantities denoted by these letters. Complex resistances or conductivities as circuit elements have a symbol in the form of a rectangle (see Fig. 3.1). In the same way, they denote reactance or conductance if they want to note that they can be either inductive or capacitive reactance or conductivity.
Conductivity
When novice radio amateurs see the equation for calculating the total resistance of a parallel circuit, a natural question arises:,"Where did it come from?" In this article we will try to answer this question.
Due to the fact that electrons, colliding with particles of a conductor, overcome some resistance to movement, it is customary to say that conductors have electrical resistance . Resistance is designated by the letter "R" and is measured in Ohms. However, any conductor can be characterized not only by its resistance, but also by the so-called conductivity - ability to conduct electric current. Conductivity is the reciprocal of resistance:
The greater the resistance, the less the conductivity and vice versa. Resistance and conductivity are opposite ways of referring to the same electrical property of materials.If, when comparing the resistances of two components, it turns out that the resistance of component "A" is half that of component "B", then we can alternatively express this relationship by saying that the conductivity of component "A" is twice that of component "B". If the resistance of component "A" is one-third that of component "B", then component "A" can be said to be three times more conductive than component "B", and so on.
Conductivity is designated by the letter "G", and its unit of measurement was originally "Mo", that is, "Ohm" written backwards. But, despite the relevance of this unit, it was later replaced by “Siemens” (abbreviated as Sm or S).
Now let's go back to our parallel circuit example. When viewed from a resistance perspective, having multiple paths (branches) for electron flow reduces the overall resistance of the circuit because it is easier for electrons to flow through multiple paths than through one path that has some resistance. If we consider a circuit from the point of view of conductivity, then multiple paths for the flow of electrons, on the contrary, increase the conductivity of the circuit.
The total resistance of a parallel circuit is less than any of its individual resistances because multiple parallel branches create less obstruction to the flow of electrons than each resistor individually:
The total conductivity of a parallel circuit is greater than the conductivity of any of its individual branches, since parallel-connected resistors conduct electric current better than each resistor individually:
It would be more accurate to say that the total conductivity of a parallel circuit is equal to the sum of its individual conductances:
Knowing that conductivity is equal to 1/R, we can transform this formula into the following form:
From this formula it can be seen that the total resistance of the parallel circuit will be equal to:
Well, we have found the answer to the question posed at the beginning of the article! You should be aware that conductivity is very rarely used in practice, and therefore this article is purely educational.
Short review:
- Conductivity is the opposite value of resistance.
- Conductivity is designated by the letter "G" and is measured in Mo or Siemens.
- Mathematically, conductivity is the inverse of resistance: G=1/R