Amplitude and phase relations in a two-circuit parametric circuit of ferroresonance nature

The article presented a mathematical analysis of a doublecircuit parametric circuit of ferroresonance nature at the fundamental frequency, performed by the harmonic balance method. The adjustment, current-voltage, and load characteristics of the circuit are given. The possibility of using this circuit in voltage regulators with direct current output is proved. Parametric sources of secondary power supply, in particular voltage stabilizers of ferromagnetic and ferroresonance nature, are used in autonomous vehicles (for example, in spacecraft, some types of intelligent transport systems) and in renewable energy sources due to their ability to operate in heavy environments (high and low temperatures, radiation, strong magnetic or electric fields).


Introduction
New circuit solutions combined with advances in the field of magnetic materials make stable secondary power sources based on magnetic components, ferroresonance phenomena, and parametric resonance promising. As studies show [7][8][9][10][11][12][13][14][15][16], [18,21,22], the technical and economic parameters of parametric power supplies of low-and mediumpower equipment are close to those of compensating stabilizers, and in some indicators (reliability, durability, temperature and time stability, resistance to heavy environments, low cost) exceed them. This allows us to conclude the relevance of the study of power supply circuits of parametric nature based on ferromagnetic components.
The purpose of this research is to obtain the amplitude-phase relations for a two-circuit parametric circuit of ferroresonance nature and to identify the stabilizing properties in circuit solutions based on this circuit.
The scheme is one of the models of the circuit of voltage regulator [9,10] is shown in Fig.1, where S 1 , S 2 , square cross-sections, respectively, of left and right rods magnetic core; L 1 , L 2 ,-the average length of the magnetic lines of the magnetic circuit; g1, g2 -active nonlinear conductivity of the windings of the coils NI1 and NI2; W 1 , W 2 -the number of turns of nonlinear coils NI1 and NI2; С 1 , С 2 -capacitance of capacitors; i=Im*sin(ωt+ψ i )the instant value of the supply current; u=Um*sin(ωt+ψ u ) -instant value of the supply voltage. The electrical state of the circuit can be described by a system of equations drawn up according to the first and second Kirchhoff laws for instantaneous values of electrical quantities

Materials and Methods
The solutions for the instantaneous values of the inductions in (10) - (11) are assumed in the form , for which the expressions (10) and (11) are transformed by the harmonic balance method [16,18,23,24]. Substitute the solution in (10) -(11) perform operations of differentiation and replacing the extent of harmonic functions by a sum of harmonics in the first degree and considering only members with a fundamental frequency, after transformations we get Let's introduce the notation After replacing in (12) the sines and cosines of the sum of the arguments by the products of the sines and cosines, taking into account the notation (13), we get 9 sin cos sin cos sin 1 1 1 1 1 1 2 2 1 sin cos cos cos 2 2 1 2 2 1 9 9 sin sin cos sin sin cos 2 2 1 2 2 We transform (14) by the harmonic balance method. By equating the coefficients at  sin and  cos to the left and right of the equal sign, we obtain a system of algebraic Squaring the expressions to the left and right of the equal sign in (15) and summing them, after the transformations, we get an expression describing the dependence between the amplitudes of the first harmonics of magnetic inductions in NI1 and NI2 9 2 Let's introduce the notation ; ; Given the notation (17), we multiply in (15) the upper expression by d 1 , and the lower expression by a 1 . Equating the left parts in the obtained expressions, after simple transformations, we find the value of the phase shift angle between the amplitudes of the first harmonics of the magnetic inductions in NI1 and NI2 ) ( Let's consider the main characteristics of the device on the example of a physical model with parameters: С 2 =20 µF, С 1 =15 µF, g 1 =g 2 =0.0015 Om -1 , W 1 =W 2 =400 wind; S=0,00085 m 2 , L1=L2=0.245m; Н=16.5 *В 9 , magnetic core from steel E330 (3414). The dependencies В 1m and В 2m = f (I m ) are shown on the Figure 2. The figure shows that in some range of the input current when ferroresonance racing had occurred in both circuits, set the operation mode circuits, where the first FRK (NI1 coil and capacitor C 1 ) works in inductive mode on the section a-a' and the second FRK (NI2 coil and capacitor C 2 ) operates in capacitive mode on the section b-b', the angles of these characteristics α 1 and α 2 with respect to the horizontal is approximately the same in meaning, but differ in sign (in the first case, the differential resistance is positive in the latter case, negative).
The amplitude of the first harmonic of the supply current (that is, the current I m in the unbranched part of the circuit) can be found through the parameters of any of the FKK of this circuit. Given (2), (3), in accordance with Kirchhoff's first law, for the instantaneous values of the currents in the branches of FKK1 (the currents i C1 , i g1 , i 1 , respectively, in the elements in the elements C 1 , g 1 , NI1), we will have 2 1 2 1 1 1 After performing the operations of differentiation and replacing the degree of harmonic functions with the sum of harmonics in the first degree, taking into account only the terms with the frequency of the main harmonic, after the transformations, we get Replacing in  to the left of the equal sign the sine of the sum of the arguments by the product and taking into account (19) We transform (20) (22) Dividing the lower expression in the system (21) by the upper one, we get the formula for determining the initial phase of the current ) ( After performing the differentiation operations, we get ) cos( cos ) sin(

25)
Let's introduce the notation Replacing in  to the left and right of the equal sign the sine and cosine of the sum (difference) by products and taking into account the notation (26), we get       From (23) and (29), the phase shift angle between the current vectors in the unbranched part of the circuit and the voltage applied to the circuit can be found   It can be seen from Figure 3 that the range of current changes in which ferroresonance jumps occur (a-a' in the current growth mode and b-b' in the current reduction mode) corresponds to the jumps of electromagnetic inductions shown in Fig.2. Figure 3 also shows that the circuit in this range is powered in a mode close to the current source, which allows you to stabilize the voltage on both circuits according to the principle of operation of the Bouchereau circuit. Fig. 4. shows the dependence of the rectified voltages on both circuits U d1 and U d2 , as well as the dependence of the load voltage U d , equal to the sum of U d1 and U d2 , on the supply voltage U. From  Fig. 4, it can be seen that in the range of changes in the input voltages a-b, which corresponds to a change in the voltage amplitudes from 100 to 340 V, the rectified voltage at the output of the stabilizer, determined by the expression U d = U d1 + U d2 , practically does not change (the maximum deviation is about 1%).

Results and Discussion
1. As a result of a ferroresonance jump, the oscillatory circuits of a two-circuit circuit operate in two different modes-inductive and capacitive. 2. The circuit under study is powered in a mode close to the current source, which allows you to stabilize the voltage on the supplied circuit according to the principle of operation of the Bouchereau circuit. 3. Due to the power supply of the load from two circuits through series-connected rectifiers, the influence of phase shifts of voltages on circuits operating in inductive and capacitive modes on the value of the voltage on the load is eliminated, which improves the quality of stabilization