Analysis of dynamic circuits of contactless switching devices

The main content of the study is the analysis of theoretical and virtual-experimental studies and methods of analysis of transients in semiconductor nonlinear dynamic circuits of contactless switching devices, presents transient graphs constructed using a virtual computer model. In addition, presents solutions of differential equations of the state of such circuits by the numerical Euler method.


Introduction
In connection with the development of reliable high-quality contactless switching devices on the basis of semiconductor elements, nonlinear dynamic circuits are widely used in various fields of automation, electronics, computer technology and power supply systems. When creating powerful contactless semiconductors of switching devices, non-autonomous nonlinear dynamic circuits consisting of nonlinear resistive semiconductors can be used in the control circuit of power thyristors [1][2][3][4].

Analysis of diode dynamic circuits
In the control circuit of thyristor switching devices, various dynamic circuits can be used as a pulse generator of the control signal, connected both in series and in parallel. Currently, various methods of analysing such circuits are widely used. Nonlinear dynamic circuits consisting of a diode, active resistance and capacitance are shown in figure 1 [2][3].
We take the diode characteristics ideal and assume that = sin . Then from the moment t = 0 to t1 the diode is open and the circuit equation has the following form: Considering, that = we have: where uccapacitance voltages. The solution of equation (2) according to Euler looks as follows: Until the moment t = t1, the voltage across the capacitance is determined by (3), taking into account the initial conditions. From the moment t = t1, the diode opens and until the moment t2 the voltage across the capacitance remains at the voltage level for the moment t1, from the moment t2 the diode opens again and the voltage across the capacitance is again described by the dependence (2 and 3) with a different initial condition.
In figure 1b shows the curve of the voltage change across the capacitance, obtained by solving equation (3) on a computer. It is assumed that Um = 100 V, R = 300 Ohm, C = 200 μF.
In figure 3 shows the dependence of the change in the voltage across the capacitor on time for different values of the active resistance R.
As can be seen from this figure, a change in the value of the active resistance R leads to a change in the charging time of the capacitor. Thus, by changing the parameters of the circuit, it is possible to regulate the time of the steady-state voltage across the capacitor and its value.     Figure 5 shows a circuit consisting of a diode and a parallel connected resistor and capacitance. Consider a numerical method for solving the equation of state of the chain for this case. Considering = sin and taking the characteristic of the diode to be ideal, it can be assumed that from the moment t = 0 to t1, the diode is open, and the voltage across the capacitance changes according to a sinusoidal law. From the moment t = t1, the diode opens and the capacitor begins to discharge to the resistor. To determine the law of change in the voltage across the capacitance, it is necessary to solve the following equation of state of the circuit: Let us determine the value of for various points from t1 to t2 by setting the integration step h.
The value t2 is determined by the following ratio 2 = − g( ) Figure 5. The investigated scheme. Figure 6. The form of voltage across the capacitance.
In figure 5 a graph of the voltage across the capacitance obtained by solving equation (4) on a computer for the parameters of the circuit C = 6.25 μF is presented; R = 100 ohms. Thus, the analysis of dynamic circuits can be successfully carried out by numerically solving the equations of state on a computer.

Analysis of thyristor dynamic circuits
We can also carry out a theoretical analysis of a non-autonomous dynamic circuit consisting of a thyristor connected in series with a parallel circuit containing a capacitance and an active resistance, which is influenced by an external sinusoidal voltage (figure 7).  Let us assume that the voltage of the power supply changes according to the sinusoidal law and the thyristor has an ideal characteristic. It is obvious that until the moment t = t1 the thyristor will be closed, the voltage across the capacitance C will be equal to zero. At the moment t = t1, the thyristor abruptly opens and a voltage will be applied to the capacitance C from time t2, the current through the thyristor takes a zero value. Write the expression of the current flowing through the thyristor where = − g .
Current will turn to zero at + = 0 i.e. at 2 = − g At time t2, the voltage across the capacitor C will be equal to the voltage of the source, i.e. = sin 2 and the thyristor VT is closed, therefore, the capacitor is discharged to the resistance Rn.
To determine the law of voltage change across the capacitance, it is necessary to solve the following equation of state of the circuit: Let us determine the value of uc for various points from t2 to t3 by setting the integration step h. Figure 8 the graph of the voltage across the capacitance, obtained by numerical solution of equations (7), is presented.

Analysis of thyristor power circuits
The need to increase labour productivity and the steady complication of technical processes determines the widespread use of power semiconductor devices at industrial enterprises, which allow for a smooth voltage sweep across loads, which helps to limit shock currents when the load is turned on and reduce the negative impact on the network; limit short-circuit currents, reduce over voltages during the switching period.
Let us consider the operating mode of a circuit consisting of a series-connected thyristor, an inductive coil and an active resistance (figure 9).  The equation for this chain is as follows:  5 We take the characteristic of the thyristor ideal for the open state of the thyristor, while equation (8) will take the form: For different values of t, setting the integration step h, we have: Figure 10 shows the curves of voltage and current at the terminals of elements L and R and current, constructed by solving equation (11) by the numerical method. As can be seen from this figure, the current gradually increases and the moment of termination of the current relative ratio of the transition of the phase voltage through the zero value is delayed. It should be noted that the shape of the current curve depends on the ratio of the parameters of the circuit L and R.

Conclusion
Thus, the analysis of nonlinear semiconductor dynamic circuits can be successfully carried out by numerically solving the equations of state of the circuit using a computer. Theoretical and virtualexperimental studies have shown that the proposed technique allows for a qualitative analysis of steadystate modes and transient processes in semiconductor circuits with various variations of parameters.