Features of calculations for arc quenching reactors with non-magnetic gaps in core of magnetic circuit

. This article presents the results of comparative studies of existing methods to calculate the inductance of dynamic and static single-phase reactors of arc extinguishing by double-rod and armored magnetic circuits with non-magnetic gaps in terms of the geometric parameters of the device. There are proposed expressions to correct the determination of the effective area of the winding of reactors by the shunting magnetic flux. On the basis of numerical calculations, the significant influence of the correct calculation of the effective area of the reactor winding by the shunt magnetic flux on the result of calculating the inductance is shown, it undoubtedly affecting the accuracy of the operation of both many digital devices of existing substations in particular, and new digital substations being introduced in general. It was also found in the work, that carrying out the correct calculation will improve both electrical characteristics of the device, and technical economic indicators of capacitive current compensation systems as part of digital substations, when setting the inverse problem of determining the geometric dimensions of the electrical part of the reactor according to the known inductance value, which depends and is calculated on the parameters of the given network, including capacitive currents.


Introduction
It is known that in ensuring the reliable operation of medium voltage high-voltage networks the role of electric reactors, called Petersen coils [1,2], is of great and fundamental importance in the fight against capacitive currents of the electric network and the consequences of single-phase earth faults. Separately, we note that a little more than 100 years ago, the first scientific works were published in the field of limiting capacitive current and suppressing the earth fault arc using an inductor [1][2][3], later called an arc suppression reactor. a) b) Fig. 1. The structure of an armored rod a) and a double rod b) a single-phase arc suppression reactor with non-magnetic gaps; 1 magnetic core, 2 magnetic rod inserts, 3 non-magnetic gaps, 4 winding.
From the scientific literature available to the authors in the field of research of electric reactors, in our opinion, one of the fundamental foreign and domestic studies reflecting the most complete calculation methodology for reactors with a steel magnetic core and gaps ( Fig. 1) are determined the works [3][4][5][6][7][8][9].
In this article, there is a made attempt to compare the ways and methods to study electric reactors proposed by various authors and applied to reactors with armored and rod magnetic circuits. The necessity for such work arose because of different interpretations by the authors of the calculation formulas and the appearance of "inconsistencies" in the presentation of the same mathematical expressions. The procedure for choosing the analysis of a specific research work is built according to the publication time and publication of the material under study.
It should be noted that here the emphasis is being done on the analysis of methods to determine the inductance from the geometric dimensions of the magnetic circuit and the reactor winding, in particular, on calculating the most effective cross-sectional area of the winding along the path of the shunting magnetic flux [6][7][8][9][10]. Let us agree to adhere to the designations in the figures and the terminology in the text proposed by E.A. Mankin, as well as in the comparative description of the works of other authors, which differ from [9] for the purpose of a uniform perception of this material.

Methods
The total flux linkage of the reactor consists of the main flux linkage passing through the steel and the shunting flux linkage passing outside the core and it is described by the equation [9,10] where w -is a number of winding turns; sh st   , -is the main and shunting magnetic flux; , , -is a magnetic induction in the core (steel), in the gap and outside the steel; -is the area of the core, the gap and the cross-sectional area of the path shunting the path of the main flow.
Following [9] for you can write is an area of a circle by the inner diameter of a round winding; st D is a core diameter; st S is a cross section (area) of the core; ( ) are sides of the core section; 3 is an area of a trapezoid equal to the area of the reactor winding at a thickness 3 b of the inner diameter; eff l is a winding circumference of the effective zone of the shunt flow section;  is a magnetic field bulge in a gap; a is a distance from winding to core; b is a radial dimension (thickness) of the winding; ob h is a winding height.
In formulas (7) and (11) in [9], in the last term in the expression sh S , in our opinion, to determine the is obtained as In this case, when calculating the area 3 b of the winding on a radial dimension, in contrast to [9], the circumference equals to ( ) l . If we apply the notation in the form b a D st + + 2 according to [9], then the thickness of the ring will be equal not 3 b , but 2 b , which significantly changes the idea of the most effective influence of the shunting magnetic flux in accordance with [9] on the thickness of the winding 3 / b ( Fig. 1) with an average circumference on the thickness of the winding 2 b . Therefore, after detailed correction In the work of Waters [6], referred by the authors of the book [7,8] in the formula (187) on page 346 when calculating the effective magnetic flux through the radial section of the winding, which is subsequently used to calculate the reactor inductance from the geometric parameters of the magnetic circuit and the reactor winding, the winding area reactor at a thickness of the inner diameter is defined as Somewhat different from [9], the method to determine the inductance of the choke with an air gap in the iron core is used in [11]. In terms of determining the cross-sectional area of the path, reduced to the full number of flux linkage, shunting the path of the main flow, as in [9], it contains an expression, as the author points out, for determining the average length of the winding and is designated as U , which corresponds eff l in this article By comparing the explanatory figure 13 and the calculated figure 16, it can be seen that the author in [1111], as well as in [9], focuses on the most effective influence of the shunt flux for the calculated value of the inductance of the choke on the thickness of the winding 3 b from the inner diameter of the winding. However, the study of numerical calculations on page 267 [11] in the form of ( ) [1212], in our opinion, there is a misprint, which consists in the absence of a coefficient 2 in front of the distance a from the core to the winding. When correcting the expression for U in [10] according to the expression for A given in [1212] and adding the missing coefficient, the length of the center line of the winding on the thickness 3 b is calculated correctly, which in its turn affects the final result of finding the inductance value L . However, analyzing papers [1111] and [1212], it should be noted that when calculating the area 1/3 S , the length of the effective midline of the trapezoid eff l is multiplied not by the value 3 b , as shown in [9], but expressed through a logarithmic function, which is not the subject of discussion in this article.
In , the expression for the length of the effective centerline of the working winding of the reactor is ( ) is obtained as In this case, the height of the trapezoid to determine the area 1/3 S is equal to 3 b . The work [1616] is devoted to the study of grounding arc suppression reactors with smooth regulation of inductance, in which there is an expression of the form 3 / b a D + + in formulas (9) and (10) to determine the length of the effective center line of the working winding of the reactor. Comparative analysis of formulas (9), (10) in Fig. 1 in [1616] shows that the quantity D contains the value of the rod diameter and two halves of the distance from the rod to the winding, i.e. in formulas (9) and (10) Both of the latest entries have the same geometrical meaning and are equal to each other.
When calculating the inductance of a double-rod reactor with gaps in formula (5) in [1818] for calculating flux linkage or self-inductance, the expression for the equivalent (given) area of the scattering channel 1/3 S is given as ( ) 4 where, according to the explanations in Fig. 2 However, when substituting this expression instead of d in (7), we obtain: , which does not correspond to the explanation of the effective scattering area on the thickness 3 / 1 of the winding from the inner diameter in accordance with [6,8,9]. Therefore, in (7) S with explanations in Fig. of   work [1818], we introduce the calculations according to [1919], which correspond to our explanations above.
In the works of recent years, for example, in [1919], the inductance of the reactor is determined through the turns, the dimensions of the turns and the rod in the same way as in (6) of this work, where is the cross-sectional area of the turns (windings) is equivalent in flux linkage. In this case, the record in the form 3 2b is valid due to the fact that the area of the circle is calculated without taking into account the equivalent area of the trapezoid, which was explained by formula (6) of this work. We also note that in the formula for calculating the inductance of the reactor in [1919], there seems to be a misprint in the choice of designation for the calculation of the gap area.
In [19], in formula (6 Before carrying out numerical calculations, we note that this work reflects the results of only some of the materials from existing scientific articles to determine the inductance of rod and armored reactors with gaps, taking into account the effective area of the shunting part of the winding from its inner diameter in order to explain the effect of the type of the effective part of the transverse area of the reactor winding on the final result of the inductance value.

Comparative verification calculations
The numerical values of the parameters of the core and the winding of the reactor are taken as in [9]    S , in which the method proposed in [6][7][8][9], is applied using the value 3 b instead of calculating the values of logarithmic functions according to .

Conclusions
Calculation L of the dependence ( ) zaz nl on for different expressions 3 / 1 S shows that the effect 3 / 1 S is noticeable with an increase in the total height (length) of the gaps, as well as/or with a decrease ob h (Fig. 5 -7). The results of the obtained calculations could be useful both in the design of new arc suppression reactors and in their study, for example, in such works as [2120 -282828].