The calculate of current transformers with combined magnetic cores

The manufacture of current transformers is associated with many design constraints, which include the restricted dimensions of its installation places, requirements for the accuracy class, as well as the limitation on the safety factor or the minimum factor of the accuracy limit factor. In a design sense of current transformers, these requirements are contradictory, which resulting in a flexible approach to the manufacture of current transformers. The use of combined magnetic cores can be referred to one of these areas, in other words magnetic cores consisting of materials with different magnetic properties, which allows to obtain unique magnetic properties that are inaccessible when using magnetic cores of the same type (material), thus providing a rational model of production. The article presents a feature of calculating of current transformer with combined magnetic cores.


Introduction
Current transformers became widely used for current measurement in high-voltage installations. The standards [1,2] indicate the basic requirements for current transformers. In the present article, special attention is given to the approach to the calculating of the current and angular error, as well as the safety factor for the measuring windings in the case of the combined execution of the magnetic core on the current transformer. Particular attention is given to the appliance of nanocrystalline materials for magnetic cores [3][4][5].
The appliance actuality of combined magnetic cores for current transformers consists in the possibility of obtaining measuring transformers with a high accuracy class at a minimum cost, mass and, accordingly, dimension [6]. The latter requirement is often determinative due to the limited space for installing a set of current transformers at the outputs of breakers, supply transformers, and the like devices.
Also, no less important limitation in the design of measuring transformers is the requirement for the safety factor, that is, the multiplicity of the secondary current of the transformer while an increase of the primary current from overloads and short circuits.
Thus, the designer is in search of the optimal mass and dimension parameters of the magnetic core, at which the required accuracy class is ensured (usually this is most actual for accuracy classes 0.2S and 0.5S intended for fiscal electricity metering), and the safety

Materials and methods
The structural design of a current transformer depends on the requirements that are imposed on it concerning the installation place, the required accuracy class, primary current, voltage level in the primary winding, etc. Let's highlight the dominant requirements that have a direct impact on the calculation of the error of the current transformer, as well as the safety factor.
The following data serves as the initial data for the calculation while designing of current transformers: 1) the number of taps and their nominal primary currents I1 nom, A; 2) nominal current of secondary winding I2 nom, A; 3) frequency f, Hz; 4) required accuracy class; 5) nominal load S2 nom, ВА; 6) load power factor cosφ2; 7) length of taps of conductors, m 8) limiting overall dimensions: inner and outer diameters, height. The overwhelming majority of internal current transformers are performed for the case of one turn of the primary winding w1. Generically the number of turns of the secondary winding w2 is calculated by the formula .
In accordance with GOST 7746-2015 the power coefficient of nominal secondary load is set equal to cosφ2 = 0,8. The active RH and reactive XH resistance of secondary load can be determined from the formulas The resistance of the secondary winding wire is added to the resistance of the measuring instruments, which is calculated by the formulas where ρobm -volume resistivity of secondary winding wire, ohmmeter; dout and din -outer and inner diameters of the magnetic core, taking into account insulation, m; hi -the height of the magnetic core, taking into account insulation, m; kobm -the height of the magnetic core, taking into account insulation, m; d2 -diameter of the secondary winding wire, м 2 ; fcurrent frequency, Hz. The resistance of the offtake wires is added to the resistance of the coil winding отв отв отв отв where µ0 -vacuum permeability equal to 4π · 10 −7 H/m; ρotv -volume resistivity of tap leads, ohmmeter; lotv -the length of the offtake wires, m; rotv -cross-sectional view radius of tap leads, m. The phase displacement angle, taking into account the load, secondary winding and tap leads, is determined from the formula The induction in the magnetic core at the nominal operating conditions is determined from the formula Dependences of the magnetic permeability of the magnetic core µ and the magnetic loss angle on the induction Bm is shown in Fig. 1. Specified characteristics available from experience with the help of measuring the magnetic properties in laboratory conditions. Thus, the values of a function that is set table-like can be found using interpolation. The method of piecewise linear approximation according to the method of cubic spline was used in this search.
A feature of the calculation of current transformers with combined magnetic cores is that the magnetic behavior of materials makes its contribution in direct ratio to the fraction of the cross-section that it occupy in the magnetic core (Fig. 2). It can be seen from the figure that the magnetic cores can be combined in height or diameter. As practice shows, option (b) of the combination in height is the most technologically simple from the point of view of production, therefore, we will consider the calculation of equivalent characteristics in this case.   where h1 and h2 -heights of the magnetic cores of 1 and 2 types, respectively, m; 1(Bm) and 2(Bm) -dependence of magnetic loss angle on magnetic induction for the magnetic cores of 1 and 2 types, respectively, grad; µ1(Bm) and µ2(Bm) -dependence of relative magnetic permeability on magnetic induction for the magnetic cores of 1 and 2 types, respectively. The presented calculation method was verified in laboratory conditions with the use of the KNT-05 comparator and showed good convergence of the calculation results and measurement results. The characteristics of the magnetic cores were taken using an FT-01 ferrotest. Applicable for stationary calculations. To calculate the dynamics of transient processes, the complicating of the calculation model is required, as shown in [7,8].

Results and discussion
Let's consider current transformer design with an accuracy class of 0.2S with the following required characteristics: nominal primary current I1 nom -400 А and 600 А; nominal current of secondary winding I2 nom -5 А; nominal load S2 nom -20 ВА (for tap 400 А) and 40 ВА (for tap 600 А); the maximum safety factor -5; the minimum internal diameter din -320 mm; the maximum outside diameter dout -510 mm; maximum height hi -100 mm; the length of the taps wires -8.5 m.
Let's calculate the current transformer with the combined magnetic core of АМЕТ-5В:82В type 1:1. The split 1:1 shows that the cross-sectional areas of the components of the magnetic core are equal. The size of the magnetic core din × dout × h is as follows: 345 × 445 × 40 mm. With the same diameters, the heights h1 and h2 will be 20 mm each.
The calculation is carried out according to the formulas presented in Section 2. At the same time, it was count of the fact that the current error can be easily corrected with the help of the turns correction technology.
Turns correction is carried out by winding operation of an additional winding with the number of turns one less than that of the main one. Changing in cross-section of the winding of the turns correction allows to obtain a fractional turns correction on the equivalent winding (not an integer number of turns). This technology makes it possible to reduce the magnetic core section without detriment to the measurement accuracy; therefore, it was automatically applied in the calculation.
The results of calculating the current and angular errors are summarized in Table 1. For comparison, the table also shows the results of calculating a current transformer using magnetic cores of the АМЕТ-5В and АМЕТ-82В types.