Dynamic modes of the phase number converter based on lc circuits with a common magnetic circuit

. Phase multipliers are used in electronic equipment devices when it is not possible to connect to a power source with the required number of phases. Phase multipliers, which are used in single-phase circuits for powering three-phase consumers, have received the greatest application. From the point of view of circuitry, the simplest are multipliers of the number of phases based on nonlinear LC circuits, which have high reliability and relatively small dimensions when powering devices of low and medium power. However, phase multipliers based on LC circuits made on a common magnetic core have degraded dynamic characteristics due to the presence of magnetic bonds between phase-shifting circuits.


Introduction
The paper investigates the dynamic properties of the phase number converter under characteristic dynamic influences -switching on the device at idle, switching on the load, a sharp change in the load mode and switching on a short circuit. It is concluded that from the point of view of the duration of transients, the proposed circuit is suitable for the development of real frequency multipliers of low and medium power.
Parametric multipliers of the number of phases are used to power automation devices, electronic equipment, communications and electrotechnological devices in cases where there are no multiphase circuits nearby or their use meets technical or economic difficulties [1][2][3][4][5][6][7][8][9]. The most common devices of this type are phase multipliers, which are used in single-phase circuits to power three-phase consumers.
However, phase multipliers based on LC circuits made on a common magnetic core have degraded dynamic characteristics due to the presence of magnetic bonds between phaseshifting circuits. For this reason, the study of dynamic modes caused by abrupt changes in parameters (switching) from the point of view of the suitability of the LC circuit for real use in phase number converters is of considerable scientific interest.
Consider a parametric multiplier of the number of phases according to the scheme in Figure 1 [17], where -1 , 2 , 3 cross-sectional areas of rods; 1 , 2 , 3 -average lengths of magnetic lines; 1 , 2 , 3 -instantaneous values of magnetic fluxes in the rods of the magnetic circuit; 1 , 2 -active conductivities of primary windings 1 , 2 -the number of turns of primary windings; 1 , 2 -instantaneous current values in the primary windings;  Taking into account the approximation of the magnetization curve = ⋅ 9 for instantaneous values of induction and tension [18], after some transformations, this scheme will be described by a system of equations where the instantaneous values of currents in the branches of the circuit can be found through instantaneous inductions from the expressions (1) In this simplified model of the phase multiplier, the core magnetization characteristic model in the form of a hysteresis loop [19,20] is not used, since the extreme rods of the magnetic circuit operate in a mode close to saturation, and the width of the hysteresis loop is close to zero.
Let's transform the system (1), for which we will replace the variables We express it through the resulting expression and after the transformation we get We express 1 in accordance with (1), we get Let 's introduce the notation: Taking into account the accepted notation, we obtain the expression for We use this system as a basis for studying the dynamic modes of operation of the parametric multiplier of the number of phases.

Results and discussion
The solution of system (7) will be the induction functions b1, b2, the induction in the middle rod is the sum of the inductions in the extreme rods of the magnetic circuit, that is, 3 = 1 + 2 , however, using expressions (2) it is easy to move from them to currents in the branches of the circuit, and using the expression = = ( ), where S, W, respectively, the section of the magnetic circuit and the number of turns on its core, to the voltages in its sections. The numerical solution of system (1) was found by the fourth-order Runge-Kutta method [21,22]. Let's consider the calculation of transients using the example of the physical model of the circuit under study according to the application and find the values of the coefficients: = 0.34; = 0.34; = 2; = 2; = 0.113; = 0.00146. The coefficient M depends on the magnitude of 2 (that is, the load) and will be determined specifically for each of the types of transients.
Consider turning on the circuit at idle, at idle 2 = 1 = 0.0015Ω −1 ; = 0.101; 1 (0) = 0; 2 (0) = 0; (0) = 0. Graphs of transients are shown in Figure 2, a (the circuit turns on when the voltage u passes through zero, = 0 and in Figure 2, b (the circuit turns on at maximum voltage and phase shift = /2. It can be seen from the graphs that = 0, transients in the circuit fade out during 1-2 periods of the fundamental frequency, and = /2 -during 4 periods. The graphs in Figure 2 reflect the course of the curves , , = ( ), but lag behind in phase by the angle = /2, since according to (1) they are proportional 1 , 2 , 3 = ( 1 + 2 ) . Thus, in the circuit under the most unfavorable regime with = /2, transients fade out quite quickly. Switching on the circuit to the load. At a load close to the nominal ( 2 = 0.05 −1 ) coefficient = 3.364, the initial conditions are zero. The curves of transients = 0 and = /2 are shown respectively in Fig. 3, a, b. a) b) c) It can be seen from the graphs that connecting the load leads to an acceleration of transients and at the worst mode ( = /2) it is pumped in two periods of supply voltage fluctuations. Fig. 3, c also shows that switching on the device for short circuits causes transients that end in two periods of the fundamental frequency.
Enabling and disabling the load. The calculation was carried out for the most unfavorable conditions = /2. The initial conditions for the process of switching on the load ( 2 changes by a jump from a value of 0.0015 to a value of 0.05 −1 ) and were determined by calculating the steady-state mode at idle. At the same time, = 0.101; 1 (0) = 0.82; 2 (0) = 1.71; (0) = −2.1. The calculated curves are shown in Fig. 4, a. From the curves it can be seen that the transient process is completed in 1.5 periods of the fundamental frequency. The initial conditions for the study of transients during load shutdown ( 2 changes by a jump from a value of 0.0015 to a value of 0.05 −1 ) g2 changes by were determined from the steady state of the loaded circuit. At the same time, = 3.364; 1 (0) = 0.82; 2 (0) = 1.71; (0) = −2.1. The calculated curves are shown in Figure 4, b. From the curves it can be seen that when the load is switched off, the transient process is longer and is about four periods.

Conclusion
1. Transients in a parametric phase number converter during load activation or short-circuit activation have a shorter duration (the transition time is 1.5-2 periods of the fundamental frequency) than transients associated with abrupt load reductions or idling (the transition time is 3.5-4 periods of the fundamental frequency).
2. Abrupt changes in the modes of the parametric phase number converter on nonlinear LC elements with a common magnetic circuit practically do not affect the operating mode of the device, therefore, the circuit can be used as a phase number converter for operation in modes with sharply variable load.