Magnetization modes modelling and analysis for controlled reactors with a rotating field

Electromagnetic reactors are used to carry out practical tasks of automatic control of power lines operation modes, distribution grids and energy supply systems of the industrial enterprises. The reactors’ characteristics significantly depend on the harmonic’s manifestation magnetic field saturation, which is different in the magnetization modes of reactors: forced, free and symmetrical. For the purpose of their analysis, a generalized mathematical model of the cross-sectional magnetic circuits of static ferromagnetic devices with rotating magnetic field has been developed on the base of nonlinear magnetic circuits theories. It was found that the best mode is symmetric magnetization, characterized by the following advantages: an increased range of reactive power control, enhanced stabilizing effect on the controlled current, reduced losses in steel, increased speed and the absence of “shaking” vibrations of the reactor magnetic core. It is shown that the free magnetization mode is close in its merits to the symmetric magnetization mode.


Introduction
In the second decade of the twentieth century, the outlines of one of the problems of high-voltage power transmission technology were outlined, consisting in the fact that at large distances the capacitive conductivity of the line starts to have a significant effect and the capacitive current significantly decreases, which reduces the line capacity. This phenomenon was accompanied by an unacceptable increase in voltage.
In the early 40-ies of XX century R. Rüdenberg (Germany) proposed to solve this problem using a strong magnetic saturation of electrical steel, pointing to the need to eliminate the higher harmonic. R. Rüdenberg's ideas were developed in the works of E. Friedlander (GEC, England). GEC company has manufactured and installed more than 50 ferromagnetic reactors at power from tens to hundreds of Mvar in various countries [1][2][3].
In Russia, a group of organizations led by M.A. Bryantsev mastered the production of a series of high-voltage controllable reactor 25, 32, 63, 100, 180 MV·A for the 500 kV grids. The series of arcquenching magnetically controlled reactors with oil cooling with a capacity from 190 to 1520 kV·A and a voltage 6-10 kV were developed and massively introduced [4][5][6].
As a means of regulating reactive power, reactors are necessary to control the modes of electric power systems in order to solve problems: compensation of excess charging power of power lines (transmission lines) and increase their capacity, limit switching overvoltages and short-circuit currents, reduce voltage fluctuations, rational distribution of voltage and current, etc [7][8][9].
Controlled reactor (CR) is a static energy nonlinear device, which operation is based on the phenomenon of electromagnetic induction. The active part of the reactor contains one or more windings and a magnetic core. The reactive power it consumes is regulated by changing the saturation of the magnetic circuit. The operating range of magnetic induction values is located behind the "knee" of the magnetization curve. This is due to the appearance of harmonics of the magnetic field saturation, which affect the characteristics of the reactors.
There are three modes of magnetization, which are determined by the manifestation of these harmonics: the forced magnetization (FM) mode in which harmonics are present in the induction of the magnetic field; the mode of free magnetization by the 2nd harmonic (MH2) in which this harmonics are manifested mainly in the magnetic field strength; the mode of symmetric magnetization (SM) in which, when magnetizing the magnetic circuit, even harmonics are absent in the induction and in the magnetic field strength [3,8,10].
The purpose of this research was to analyze the listed magnetization modes, their influence on the properties of the reactors and ascertain the best mode of magnetization, a positive effect on the characteristics of the reactors.

Method of research
The process of researching devices should be preceded by the implementation of the model-algorithmprogram triad, so the device is replaced with its model, which is then analyzed using PC experimentation using computational logic algorithms.
The mathematical model, based on the theory of nonlinear magnetic circuits, takes into account the discrete structure of the tooth-slot layer, the non-magnetic gap between the stator and rotor, the spatialtemporal spectrum of saturation harmonics, and the MMF values of the reactor windings. The derivation of a mathematical model of the magnetization regimes of reactors in general form is given in [8,3,10]. For the analysis of the free magnetization mode by the 2nd harmonic, the model is presented in the form (1), and for the symmetric magnetization mode - (2). On the basis of the developed generalized mathematical model of the phenomena arising at magnetization of a magnetic circuit, the best modes of magnetization are established.

Forced magnetization mode
The forced magnetization (FM) mode under the higher harmonics by the saturation of the magnetic field is considered as a mode in which these harmonics appear in the induction of the magnetic field but are practically absent in the field strength.
In order to explore the influence of the field higher harmonics on the operating modes of a controllable reactor (CR), the graphs, reflected the mutual influence of the main harmonics of a rotating field and a bias field are given. The nature of the change in the amplitudes of higher harmonics of saturation from various factors are also to be deduced. For that, the results of PC calculations of a nonlinear transcendental equation that simulates a magnetic circuit of a reactor operating in various magnetization modes, including the FM mode, are used [8].
Later in the document there are the calculated curves for the reactor, which correspond to rational values of the quantities defined in [6]: the ratio of the transverse geometric dimensions of the magnetic circuit * = 0.22 , * = 0.6 ; the number of pole pairs of the three-phase working winding 1 = 1; the relative magnitude of the nonmagnetic gap δ * = 50.
Dependencies 1 * = ( 0 * ) at 1 * = , named IO curves, are shown in figure 1. They reflect the relationship between the amplitude of the magnetomotive force (MMF) of the three-phase winding 1 * and MMF of the control winding 0 * , which is practically linear in the working range. Figure 2 displays the dependence 0 * = ( * ) at 1 * = and 1 * = . As can be seen from the graph, with an increase in the gap * for the given 1 * и 1 * the MMF value of the control ring winding necessary to ensure a certain level of MMF 1 * , decreases. This is natural, since an increase in the gap in itself leads to an increase in the MMF 1 * . However, it is not advisable to increase the gap in the reactor in order to reduce the 0 * at 1 * = and 1 * = or in order to reduce the MMF 1 * at 0 * = and 1 * = , since this drastically limits adjustment capability of the reactor. Figure 3а shows the curves 1 * = ( 1 * ) at 0 * = , which should be used in the reactor calculation. It's seen from the figure, that the reactor working inductions should be taken from the part of the magnetization curve within 6 ≤ 1 * ≤ 7. More precise recommendations on the choice of 1 * are given in [3,8]. The curved 1 * = ( 0 * ) at 1 * = , shown in figure 3b are conveniently used for the calculation when the reactor is switched on with a load [3].   The curves, shown on the figure 4, illustrate a decrease in the induction of a constant field with an increase in the amplitude of the main harmonic of induction 1 * (a) and a change in the constant induction component ∆ 0 * in the non-magnetized yoke (b) [8]. Next, we find out the dependence of the amplitudes of higher harmonics of saturation on the transverse geometry of the magnetic circuit and other factors.

Free magnetization mode
The free magnetization mode by the higher saturation harmonics is a mode in which these harmonics present in the magnetic field strength, and practically absent in magnetic induction. The controlled reactor in the unforced magnetization mode at the 2nd harmonic of saturation (UM2) has, in addition to the 2-pole winding and the control winding, also a 4-pole antivibration winding.
In order to determine the characteristics of the reactor in the MH2 mode, the non-linear equation of the magnetic circuit was calculated on a PC [8].
The graphs of the amplitude change of the MMF of the anti-vibration winding 2 * = ( 0 * ) at 1 * = and 2 * = ( 1 * ) at 0 * = , which are used in the design of the reactor, are shown in figure 6. With working inductions 1 * = 6. . .7 and 0 * = 12000 the MMF value 2 * is approximately 25% of the MDS of the working winding 1 * . Summing up the analysis of the free magnetization mode on the 2 nd harmonic, we note that the latter favorably differs from the forced magnetization mode by a significant decrease in the vibrations of the magnetic circuit and an extended range of operating current control, as evidenced experimental data [3]. a b Figure 6. The dependence of the amplitude of the MMF 2 * in MH2 mode: afrom the value of the MMF 0 * ; bfrom the amplitude of induction 1 * .
The ratio of the radii of the average circles of yokes is = 0.475. The length of the gap is * = 50.
To illustrate the fact of the absence of even harmonics with symmetric magnetization the figure 7 shows dependences 2,3,4 * = ( 0 * , 1 * ) at 0 * = 8000. The first quadrant corresponds to the consonant action of the MMFs of the control windings 0 * and 0 * with respect to the contour of the rotating field. Therefore, the even harmonics of the same name, generated respectively by the outer and inner yokes, coincide in phase, and the amplitudes of the resulting even harmonics have the greatest value. With a decrease in the MMF, 0 * the resulting even harmonics also decrease [3]. Since there are no even harmonics with symmetric magnetization, there is no "magnetic rectification" in the yokes [8]. Therefore, in the SM mode, the magnitudes of the constant magnetic inductions 0 * and 0 * take their natural values, determined respectively by the MMF 0 * and 0 * , the amplitude 1 * and the magnetic state of the system. The curves 0 * , 0 * = ( 0 * , 1 * ) at 0 * = 8000 are shown on the figure 8. If 1 * = 0 and 0 * = , then the induction value 0 * = 7.8 remains constant, and the value 0 * changes, and the graph 0 * = ( 0 * ) is located in the 1st and 3rd quadrants symmetrically relative to the origin of the axes of coordinates. When 1 * > 0 and 0 * = in the general case even harmonics appear in the field and "magnetic rectification" occurs in the yokesthese are additional constant induction components ∆ 0 * and ∆ 0 * , due to the imposition of even and odd saturation harmonics in them. With variations of 1 * and 0 * , the saturation harmonics change, and hence the values ∆ 0 * and ∆ 0 * . Therefore, with a change in the MMF 0 * within −8000 < 0 * ≤ 10000 , the induction 0 * decreases with the transition from the 1 st quadrant to the 2 nd . When 0 * = −8000 there are no even harmonics in the field, and ∆ 0 * = 0.  For the values of MMF | 0 * | > 8000 (2 nd quadrant) the phase of the resulting even harmonics and the sign ∆ 0 * change to opposite ones. As for the 0 * = ( 0 * , 1 * ≠ 0) curves at 0 * = 8000, their symmetry relative to the origin becomes broken due to the "magnetic rectification". When MMF 7 0 * = 0, the presence of a constant induction in the inner yoke is determined only by "magnetic rectification", 0 * = ∆ 0 * . In order to avoid a constant induction in the inner yoke, it is necessary to apply some negative value of the MMF 0 * . For example, for 1 * = 8 and 0 * = 8000 the induction 0 * = 0, if 0 * = − 1750. The difference between the induction 0 * at 1 * = , 0 * = 0 * = 8000 and 0 * = − 0 * = −8000 is caused by "magnetic rectification". An interesting fact is that the symmetric magnetization mode affects the 3 rd harmonic of magnetic induction qualitatively the same as the mode of free magnetization by the 2 nd harmonic ( figure 5 and 9). In [3], experimental analogs of the calculated dependences shown in figure 9b.
Dependencies 0 * , 0 * = ( 0 * = |− 0 * |, 1 * ) in SM mode, when 0 * = |− 0 * |, and in FM mode, are shown in figure 10. Curves must be used when calculating the loss in steel. As follows from figure 10, induction 0 * and 0 * in the SM mode are always less than those in the FM mode. Therefore, the yoke losses in the SM mode will be less than in the FM mode.   The dependencies 1 * = ( 1 * , 0 * = |− 0 * |), illustrating the increase in the range of current control in the SM mode, are shown on the figure 11. These graphs used in calculation of the reactor.
The mode of SM favorably differs from the mode of FM by elimination of the "shaking" vibrations, by the extended control range of the operating current, reduced losses in steel, enhanced stabilizing effect over current and increased speed, which is confirmed by the calculation and experiment [3]. The photo ( figure 12) shows a prototype of a reactor, which was used study of the dependence spectrum of the induction of a rotating field in forced and symmetric magnetization modes on the amplitude of the first harmonic of field and control windings.
The photograph (figure 13) shows reactor with symmetrical magnetization, intended for use as a regulating element of the plasmatron power source air plasma cutting of metals [3].

Conclusions
• It is shown that the optimal, in terms of maximum current gain, is a reactor in the symmetric magnetization mode, which has a three-phase 2-pole working winding and transverse geometry of the magnetic circuit, corresponding to the stator plates of standard 4-pole AC electrical machines.
• It was found that the reactor in the symmetric magnetization mode favorably distinguishes the following qualities: increased control range of reactive power, enhanced stabilizing effect on current, reduced steel losses, increased speed and the absence of "shaking" vibrations of the magnetic circuit.