APPLICATION OF VYSHNEGRADKY'S DIAGRAMS FOR TRANSIENT ANALYSIS IN ELECTRIC DISCHARGE INSTALLATIONS WITH STOCHASTIC LOAD.
Introduction. In the electric discharge installations (EDI) with reservoir capacitors, in particular in the semiconductor (thyristor) installations for volumetric electro-spark dispersion (VESD) of the metals, the oscillatory discharge of capacitor with a small reverse recharge (less than 30 % in voltage) is the most efficient technologically and energetically mode of discharge through electric spark load [1-6]. In this case, there is a fast natural locking of the discharge semiconductor switch, which makes it possible to quickly carry out the subsequent charge of the capacitor and further its discharge trough the load [1, 4-6]. Thus, we can realize a high frequency of charge-discharge cycles and stability of the duration of discharge currents in the EDI load.
At the same time the resistance of such load as a metal granular layer can stochastically increase several times during discharge. As a result, a so-called idle discharge trough the load, i.e., a long-term discharge with a low current without sparking can occur [1, 4, 6-8]. Since the increase in active resistance of load decreases the 0-factor of the discharge circuit, then the oscillatory capacitor discharge transient can become aperiodic one, and discharge duration can increase many times. Because of such long capacitor discharges, we can not to realize high frequency and stability of charge-discharge cycles, and thus the yield of spark-eroded powders.
To reduce the discharge pulse duration in such EDI, we have proposed to connect an additional shunt chain [VT.sub.2]-[L.sub.2]-[R.sub.2] in parallel to the capacitor at a certain time [t.sub.1] as shown in Fig. 1. The parameters of the additional chain must be chosen from the condition for avoiding of a periodic capacitor discharge.
The purpose of this paper is to analyze the transient processes in the discharge circuit of reservoir capacitor of electric discharge installations at a change in the circuit configuration during the discharge as well as to determine the appropriate circuit parameters for which the discharge process described by third-order differential equation remains a damped oscillatory process.
Transient analysis of capacitor discharge through the load when the capacitor is shunted by the RL-chain. As an example, we have performed the transient analysis of the capacitor discharge trough the load in the thyristor installation for VESD with an additional parallel active-inductive chain. In the installation for VESD, whose electrical circuit is shown in Fig. 1, the capacitor C is charged to a voltage [U.sub.0] from a shaper of direct voltage (SDV). Then, after switching on the discharge thyristor [VT.sub.1], the capacitor C is discharged through the load with the resistance [R.sub.load] and discharge circuit inductance [L.sub.1], which is usually 1-5 [micro]H.
We have assumed that the resistance [R.sub.load] (that take into account not only the resistance of the electric spark load, but the active resistance of the circuit wires) remains unchanged during the discharge, but could change discontinuously between the discharges. It has been also assumed that the thyristor [VT.sub.2] was locked until the time [t.sub.1], and the discharge process was aperiodic, that is, the Q-factor of the C-[VT.sub.1]-[R.sub.load]-L1-C discharge circuit [Q.sub.1] < 0.5.
During the discharge transient analysis, we have believed that the thyristors [VT.sub.1] and [VT.sub.2] were ideal switches, that is, the commutation occurred instantaneously and without power loss.
Expressions for the voltage of the capacitor [u.sub.C](t) and the current i(t) in the discharge circuit are :
[mathematical expression not reproducible] (1)
[mathematical expression not reproducible], (2)
where [U.sub.0] is the initial capacitor voltage; [p.sub.1] and [p.sub.2] are the roots of the characteristic equation of this circuit:
[mathematical expression not reproducible]
At point in time t = [t.sub.1], when the current in the circuit is equal to a certain value i([t.sub.1]) = [I.sub.1], and the capacitor voltage has a certain value [u.sub.C]([t.sub.1]) = [U.sub.1], the thyristor [VT.sub.2] unlocks and an additional [L.sub.2][R.sub.2]-chain is connected to the circuit, that is, the circuit changes its configuration.
In new transient process, which started at t [greater than or equal to] [t.sub.1] in the circuit with the changed configuration, the following system of equations is valid according to the second Kirchhoffs law:
[mathematical expression not reproducible] (3)
As [mathematical expression not reproducible], then system (3) can be written as:
[mathematical expression not reproducible] (4)
Since according to first Kirchhoffs law i = [i.sub.1] + [i.sub.2], and the current i flowing through the capacitor is i = C [du.sub.C/]dt, we can write the following expression:
[du.sub.C]/dt = [i.sub.1]/C + [i.sub.2] /C. (5)
Let us perform the differentiation of the system (4):
[mathematical expression not reproducible] (6)
After substituting (5) in (6) and performing the transformations, we obtain the system:
[mathematical expression not reproducible] (7)
Let us perform the differentiation of second equation of system (7) once, and then twice:
[mathematical expression not reproducible] (8)
[mathematical expression not reproducible] (9)
Substituting (8), (9) and the second equation of system (7) into the first equation of this system and performing the transformations, we get
[mathematical expression not reproducible]
After integrating this expression, we have
[mathematical expression not reproducible] (10)
where [A.sub.i] is a constant of integration, which we define from the final conditions.
Since at t = [infinity] the capacitor is discharged to zero and all currents in the circuit (as well as their derivatives) are 0, then [A.sub.i] = 0, and equation (10) takes the form
[mathematical expression not reproducible] (11)
Thus, we have obtained a third-order differential equation whose characteristic equation can be written as
[mathematical expression not reproducible] (12)
For delimitation of areas with different types of transients, which are described by the third-order differential equations, in many cases it is expedient to use Vyshnegradsky's diagrams . Vyshnegradsky's criterion and its graphic representation in the form of diagrams allow us to judge the influence of parameters of third-order system on its stability without solving the differential equation.
Bringing the equation (12) to a normalized form and introducing a new variable
q = p x 3 [square root of [a.sub.0]/[a.sub.3], (13)
we obtain, as a result, the normalized equation
[q.sup.3] + [Aq.sup.2] + Bq +1 = 0, (14)
where A = [a.sub.1]/3[square root of [a.sub.3] and [a.sup.2.sub.0] and B = [a.sub.2]/3 [square root of [a.sub.0][a.sup.2.sub.3]]| coefficients are called the Vyshnegradsky's parameters.
On the plane of A and B parameters we can plot a Vyshnegradsky's diagram that display the regions of stable and unstable operation of the system described by a third-order differential equation whose characteristic equation has the form (12).
The stability conditions for the third-order system, formulated by Vyshnegradsky, are
A > 0, B > 0, and AB > 1. (15)
The equation for oscillatory stability threshold is AB = 1 at A > 0 and B > 0.
This is an equilateral hyperbola, for which the coordinate axes are the asymptotes (Fig. 2). The region of system stability according to conditions (15) lies above this curve.
The stability region can be divided into separate parts corresponding to different combinations of the roots of the characteristic equation. It should be noted that at the point D, where A = 3 and B = 3, the characteristic equation (14) takes the form [(q + 1).sup.3] = 0. Consequently, at this point all three roots are equal [q.sub.1] = [q.sub.2] = [q.sub.3] = -1. In this case, for the initial equation (13), we obtain [p.sub.1] = [p.sub.2] = [p.sub.3] = -3 [square root of [a.sub.3]/[a.sub.0]] .
In the general case, two options are possible: 1) all three roots are real; 2) one root is real and two are complex. The boundary between these two cases is determined by the vanishing discriminant of the thirddegree equation (14), which can be received, for example, from the Cardano's formula for solving the cubic equation:
[A.sup.2][B.sup.2] - 4([a.sup.3] + [B.sup.3])+18AB - 27 = 0.
This equation gives two curves in the plane of the A and B parameters: DE-curve and DF-one (Fig. 2). Inside of EDF region, the discriminant is positive. Consequently, in this region there are three real roots (region I). In the remaining part of the plane, the discriminant is negative, which corresponds to the presence of a pair of complex roots (region II).
In region I, where all roots are real, an aperiodic transient process takes place, and in region II, where there are one real and two complex roots, the transient process is oscillatory.
Calculating the value of Vyshnegradsky's parameters at changing the parameters of the discharge circuit (parameters of [R.sub.2][L.sub.2] -chain connected to the capacitor), we can immediately conclude whether they are in the stability region of the system and if this is the case, then in which part of the region they are located (aperiodic discharge region I or oscillatory one II).
Hence, when the load resistance increases stochastically during the discharge of the capacitor we can easy choose the necessary parameters [R.sub.2][L.sub.2]-chain for connecting to the capacitor in order to prevent a long-term discharge with a low current without sparking in the load.
The investigations carried out in the installation for the volumetric electro-spark dispersion of aluminum in water with the following parameters: [L.sub.1] = 5 [micro]H, C = 100 [micro]F, showed that resistance of the load, which is a layer of aluminum granules located between the electrodes, can vary within [R.sub.load] = 0.2 - 5 Ohm. Therefore, the Q-factor of the discharge circuit C-VT1-[R.sub.load]-L1-C: [Q.sub.1] can be in the range of 1.118 - 0.045. That is, the discharge of the capacitor with certain changes in the load resistance can be aperiodic ([Q.sub.1] <0.5). That's why, it is necessary to connect an additional active-inductive chain in order to change the nature of the discharge process. The resistance [R.sub.2] of such a chain takes into account the active resistances of both the wires of the inductive coil [L.sub.2], and the wires connecting this coil to the discharge circuit. This value is about 0.001 Ohm.
Fig. 3 shows the values of the Vyshnegradsky's parameters calculated using the software package Mathcad 12 for the discharge circuit C-[VT.sub.1] - [R.sub.load]-[L.sub.1]-C with the parameters C = [10.sup.-4] F, [L.sub.1] = 5 x [10.sup.-6] H, [R.sub.load] = 5 Ohm, 2.5 Ohm, and 1 Ohm. The initial Q-factors are, respectively, [Q.sub.1] = 0.045, 0.089, and 0.224, i.e. the capacitor discharge is aperiodic and for changing the discharge character it is necessary to connect an additional active-inductive chain. The resistance of the additional chain was assumed to be [R.sub.2] = 0,001 Ohm, and the inductance value varied in the range [L.sub.2] = [10.sup.-7] / 14 x [10.sup.-4] H.
According to Fig. 2 in zones defined by conditions, for example [mathematical expression not reproducible], the discharge of the capacitor when the additional active-inductive circuit is connected becomes an oscillatory. Further, according to the dependences shown in Fig. 3, the inductance values [L.sub.2] satisfying the above conditions are determined.
The results of the analysis of the value ranges of the additional inductance [L.sub.2], which are required for the realization of the oscillatory discharge of the capacitor in the circuit with different load resistance, are given in Table. 1.
According to the proposed procedure, Vyshnegradsky's diagrams can be used to estimate the transient processes in the circuits of electrical discharge installations with different parameters and configuration.
Since the oscillatory discharge duration is proportional to the circuit inductance value, then in order to ensure short-term discharges, the appropriate values of [L.sub.2] should be minimum values from corresponding ranges: 103 [micro]H, 48 [micro]H, 15 [micro[H
1. In the discharge circuit of the capacitor of electric discharge installations whose load resistance can increase randomly, a low-current discharge (so-called idle discharge) through the load may occur. In order to transfer such a discharge into the required high-current and quickly damped discharge, we have connected the additional inductance to the discharge circuit.
2. To determine the value of the additional inductance, it is advisable to apply Vyshnegradsky's criteria and their graphic representations in the form of diagrams. This approach allows us to determine the range of values of such inductance for various load resistances in order to transfer aperiodic long-term capacitor discharge through the load in a short-term oscillatory discharge without solving a third-order differential equation.
3. As an example, we have performed a transient analysis in the discharge circuit of installation for volumetric electrospark dispersion of the metals in a liquid with parameters C = [10.sup.-4] F, [L.sub.1] = 5 x [10.sup.-6] H, [R.sub.2] = 0.001 Ohm. We have calculated the Vyshnegradsky's parameters for load resistance of the installation [R.sub.load] = 5 Ohm, 2.5 Ohm, and 1 Ohm. Using the Vyshnegradsky's diagram, we have determined the ranges of the values of the additional inductance [L.sub.2] for the realization of the oscillatory discharge of capacitor of the installation with a change in its stochastic load resistance.
Appropriate values of [L.sub.2] are the minimum values from the corresponding ranges: 103 [micro]H, 48 [micro]H, 15 [micro]H.
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N.I. Suprunovska (1), Doctor of Technical Science, M.A. Shcherba (1), Candidate of Technical Science,
(1) The Institute of Electrodynamics of the NAS of Ukraine, 56, prospekt Peremogy, Kiev, 03057, Ukraine, phone +380 44 3662493, e-mail: firstname.lastname@example.org, email@example.com
Caption: Fig. 1. Electric schematic diagram of EDI with additional RL-chain shunting the capacitor
Caption: Fig. 2. Vyshnegradsky's diagram for the system, described by third-order differential equation
Caption: Fig. 3. The calculated values of the Vyshnegradsky's parameters for a discharge circuit with [L.sub.1] = 5 x [10.sup.-6][degrees]H, C = [10.sup.-4] F, [R.sub.load] = 5 Ohm, 2.5 Ohm, 1 Ohm
Table 1 Value ranges of additional inductance [L.sub.2] for different load resistance [R.sub.load] [R.sub.load], [mathematical [mathematical Ohm expression not expression not reproducible] reproducible] 5 103-567 103-1240 2.5 48-281 48-617 1 15-109 55-242
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|Author:||Suprunovska, N.I.; Shcherba, M.A.|
|Publication:||Electrical Engineering & Electromechanics|
|Date:||Jun 1, 2018|
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