CIRCUIT SIMULATION OF ELECTRICAL BREAKDOWN IN AIR USING KIND ’ S EQUAL-AREA

Запропонована спрощена схемотехнічна модель електричного пробою повітряного проміжку з використанням критерію рівної площі Кінда. Моделювання виконане за допомогою демонстраційної версії програми Micro-Cap 11. Модель дозволяє передбачити вольт-секундну характеристику різних повітряних проміжків. Модель розглядається як допоміжний засіб для студентів, що вивчають курс техніки високих напруг. Ключові слова: схемотехнічне моделювання, техніка високих напруг, електричний пробій, вольт-секундна характеристика, критерій рівної площі. Trotsenko Ye., Brzhezitsky V., Masluchenko I.


Introduction
In recent years, the role of virtual experiment in the high voltage engineering has been continuously increasing.Physical experiments in high-voltage laboratory require considerable work for preliminary preparation, including moving large-sized equipment (high-voltage generators, current and voltage transformers, insulators, electrostatic voltmeters, voltage dividers, etc.).In addition, a large area of the high-voltage laboratory is necessary.Preliminary simulation on the personal computer allows to reduce the time to prepare the experiment in a high-voltage laboratory to a certain extent, and in some cases completely replace it.For these reasons, immediately after their appearance, various circuit simulation programs began to be used both in education and scientific activity.With the increase in the performance of personal computers and the capabilities of circuit simulation programs, the possibilities of a virtual experiment are also expanding.
The field of research in high voltage engineering is very broad.To solve some, now «classical» problems, circuit simulation programs are successfully used in education.This includes, for example, simulation of pulse current generators, high-voltage multipliers, etc.At the same time, a number of equally important problems remain practically unaffected (for example, simulation of the electrical breakdown of an air gap).The possibility of determining the discharge voltage of insulation gaps using models is important for design of high-voltage installations with air insulation.This includes a variety of electrical installations, including overhead transmission lines, the main insulation of which is atmospheric air.

The object of research and its technological audit
The object of research is the circuit simulation model of a streamer breakdown of a rod-rod air gap.In this case, it is necessary to solve the problem of determining the time interval of the streamer propagation.The lower bound of this interval corresponds to the beginning of the streamer propagation, and the upper bound corresponds to the time when the streamer reaches the opposite electrode.
To create such model, it is not enough to take into account only the functional relationship between the breakdown voltage and the spacing between the electrodes.In this article, the authors propose to solve the posed problem using Kind's equal-area criterion [1,2].

The aim and objectives of research
The main aim of the article is to demonstrate the potential possibility for modeling the electrical breakdown of air gaps using circuit simulation programs.
The following objectives were set to accomplish this aim: 1. Creation of a simplified model of electrical breakdown of a rod-rod air gap using Kind's equal-area criterion as one of the criteria for streamer breakdown in air.
Determination of the discharge times with the help of the obtained model under the action of voltage pulses of positive polarity with different amplitudes and a given shape.
3. Comparison between simulation and experimental results.

Research of existing solutions of the problem
The importance and applicability of circuit simulation programs for high voltage engineering was stressed long time ago [3].Traditionally, such programs are used to explain the «classical» issues of high voltage engineering.This applies mainly to the simulation of the pulse voltage generator [4][5][6][7][8], the simulation of the pulse current generator [9] and the study of voltage multipliers [3].A list of these important tasks can be expanded.In addition to these issues, circuit simulation programs can be used to simulate electrical breakdown of air gaps [10].Obviously, considering how difficult this phenomenon is, it can talk only about simplified modeling [10].Separately, it should be noted the trend towards the creation of virtual high-voltage laboratories [11].In any case, if it deal with the electrical breakdown of an air gap with a highly non-uniform electric field, the volt-time characteristic of the gap must be taken into account [2,10,12].In this paper, as will be shown below, this is taken into TECHNOLOGY AUDIT AND PRODUCTION RESERVES -№ 3/1(35), 2017 ISSN 2226-3780 account with the help of an equal-area criterion [1,2], which can be relatively easily taken into account in simulation modeling.

Materials and methods of research
To achieve objectives that were set such research method was applied: circuit simulation on personal computer.The main material in this research is the circuit simulation model of the electrical breakdown of a rod-rod air gap.

Research results
It was shown in [1,2] that to determine the breakdown time of an air gap when an impulse voltage is applied to it, it is necessary to consider the area of the volt-time characteristic.For a specific electrode configuration, we can write: where u t ( ) -applied voltage; U b -reference voltage or breakdown voltage under prolonged application of voltage; A -constant area of the volt-time characteristic or the formation area; t 1 -time corresponding to the beginning of the streamer propagation; t d -time at which the streamer reaches the opposite electrode.
Since the subsequent formation of the discharge channel occurs much faster, it is believed that time t d is almost the same as the instant of voltage cutoff.The area A depends on the geometric dimensions of the electrode system and increases with increasing spacing between the electrodes s .For convenience of calculations, the concept of the relative area of the volt-time characteristic A s is used.Table 1 [13] shows the values of A s for some electrode configurations.Using expression (1), it is possible to construct a simplified model for calculating the breakdown time of an air gap when an impulse voltage is applied.To estimate the error of the results obtained in this model, it is necessary to compare the results of the simulation with the experimental data.As such experimental data, the volt-time characteristic of the rod-rod gap for a 1.5/40 μs wave of positive polarity, described by the following formula [14], can be used: where s -spacing between electrodes, cm; t d -discharge (breakdown) time, μs.In this case, the voltage U (kV) in formula ( 2) is determined as follows.If the discharge occurs at the tail of the wave, then the points of the volt-time characteristic relate the discharge time t d and the pulse amplitude U m .If the discharge occurs at front of the wave, then these points relate the discharge time t d to the pulse voltage at the time of discharge U t d ( ).To describe the voltage pulse by the shape 1.5/40 μs and the amplitude U m , authors proposed the expression (3).(3) Let's consider the rod-rod gap configuration under the action of pulses with positive polarity, which corresponds to row 3 of Table 1.The spacing between the electrodes s is assumed equal to 1 m.Let's determine the reference voltage U b for such gap.To do this, substituting the values of s = 100 cm and t d = ∞ in the formula (2), let's obtain that U b = 495 kV.From Table 1, for s = 1 m, let's obtain that A = ⋅ 0 62 . .V s The electrical breakdown model of the air gap is shown in Fig. 1.Simulation was performed using the evaluation version of the Micro-Cap 11 circuit simulator, developed by Spectrum Software [15].In Fig. 1 E 1 , E 2 , E 3 -functional voltage sources, the values of which are separately listed in Table 2; R 1damping resistor; X 1 -voltage comparator; S 1 -voltage controlled switch.

Table 2
The values of the functional voltage sources in the circuit in Fig. 1 according to the Micro-Cap syntax [15] No. Component Value The circuit in Fig. 1 works as follows.The voltage source E1 simulates the voltage pulse applied to the interelectrode gap.The pulse amplitude is specified by the «.DEFINE» directive.For example, «.DEFINE Um 600e3» corresponds to the value of U m = 600 kV.
Source E2 simulates part of equation ( 1) to the left of the equal sign.In turn, the source E3 simulates part of the equation (1) to the right of the equal sign.Let's assume that the damping resistor R 1 has a typical resistance ISSN 2226-3780 value of 1000 Ω.The circuit uses a voltage comparator X 1 .Switch S 1 is used as a discharge channel model.The switch is controlled by voltage and operates in a hysteresis mode.The switch resistance is 100 MΩ in open state and is 1 Ω in closed one, which determines the resistance of the discharge channel in this model.
In the normal state, the switch S 1 is opened.The voltage across the gap in this state is: As can be seen from equation ( 4), it can assume that all the voltage in the circuit is applied to the air gap.This is done in order to avoid modeling the pulse voltage generator when studying the breakdown process, which is not critical for this article.
The moment of activation of the switch determines the breakdown of the air gap.The switch S 1 closes when a voltage appears at the output of the comparator, that is, between nodes 3 and 0, (the grounded nodes are numbered 0, which are not shown in the circuit diagram).In turn, this can happen only when the voltage between nodes 2 and 0 becomes greater than the voltage between nodes 1 and 0. This will happen only when the curve E2 intersects with the curve E3, that is, when the Kind's equal-area criterion (1) is fulfilled.In order to determine the source E1, it is necessary to preliminarily determine the parameter t1 (Table 2) corresponding to the time at which the voltage curve across the gap intersects the straight line U b = 495 kV.This can be done directly in the Micro-Cap as follows.Let's demonstrate it with the example of a pulse with U m = 600 kV.
At first, the moment of time t 1 is unknown, therefore it is assumed that its initial value is zero.Therefore, at the first iteration let's assume that the source E2 is described by the expression «SUM(v(4,0)-495e3,t,0)».After the first run of the transient analysis mode («Transient…») this parameter will be determined and can be viewed in the «Measurements» window (Fig. 2, a).Now substitute this value t 1 in the last expression -«SUM(v(4,0)-495e3,t,1.026e-6)» and start the transient analysis mode («Tran-sient…») again.After this iteration, the discharge (breakdown) time t d becomes known, which is also displayed in the «Measurements» window (Fig. 2, b).
In Fig. 1 the first line with the «.MEASURE» directive is intended for t 1 calculation, and the second line is for t d calculation.The virtual oscillograms of the breakdown voltage, obtained in this way for different pulses, are shown in Fig. 3, 4. In Fig. 3 ISSN 2226-3780 Table 3 shows the values of the source E2 corresponding to each curve in Fig. 3, 4, now with a calculated value of the parameter t1 (calculated after the second iteration).

Table 3
The values of the functional voltage source E2 in the circuit in Fig. 2 according to the Micro-Cap syntax [15] No. Thus, the curves in Fig. 3, 4 predict individual points of the volt-time characteristic of the air gap.This volt-time characteristic can be called simulated volt-time characteristic.At the same time, as was indicated above, for the given electrode configuration an experimental volt-time characte ristic is known.Table 4 shows the comparison between simulated and experimental volt-time characteristics.From the curves in Fig. 3, 4, the breakdown time is determined and listed in Table 4 (on virtual oscillograms, the breakdown time corresponds to an abrupt fall of the voltage curve to zero value).

Amplitude
Simulation relative error in Table 4 is calculated using the following formula:  4

in it).
As can be seen from Table 4, the simulation error makes it possible to use this model in the educational process for visual demonstration of the breakdown criterion of an air gap under the influence of voltage pulses with a given shape.The fulfilled verification of the proposed method has shown the possibility of its application for other spacing bet ween the electrodes of the rod-rod configuration.

SWOT analysis of research results
Strengths.The strengths of the proposed approach are: -the possibility of modeling the breakdown processes with the help of circuit simulation programs.This is important, since a virtual experiment in technical education is necessary; -all complex calculations are performed by the program, not by the user.The user needs to know only two parameters: reference voltage U b and formation area A for considered electrode configuration.
Weaknesses.The weaknesses of the proposed approach are: -in the proposed model, the calculation of the start time of the streamer formation t 1 and the discharge (breakdown) time t d is determined not for one execution of the program, but for two iterations; -the model gives somewhat underestimated values compared to the actual values of the discharge voltage of the air gap.
Opportunities.The opportunities of the proposed approach are: -determination of the breakdown moments of various air gaps during the action of voltage pulses of all typical shapes and amplitudes, which are used in high voltage engineering; -use of this model to determine the flashover voltage of complex insulating structures, for example, for suspension insulator strings on overhead transmission lines during lightning surges.
Threats.The model is intended for use in scientific work or in the educational process as a tool for visual demonstration of breakdown conditions of long air gaps.Unlike steady state or transient processes, circuit simulation of electric breakdown processes for high voltage engineering is a relatively new direction.At the moment, the results obtained on simplified models are only approximately equal to the results of a real physical experiment in a high-voltage laboratory.

Conclusions
1.The model of the electrical breakdown of the rod-rod air gap is created with a help of the evaluation version of Micro-Cap 11 circuit simulator.The model uses Kind's equal-area criterion, as one of the criteria for streamer breakdown in air.
2. Using the model, the breakdown time of the air gap is calculated under the action of voltage pulses of positive polarity with different amplitudes and shape 1 5 40 ./ μs. 3. Simulation results are compared with experimental data.As the latter, the experimentally obtained expression for the volt-time characteristic of the rod-rod air gap is used.The simulation relative error does not exceed 10 %, which makes it possible to use this model in the educational process to visually demonstrate the conditions for the breakdown of an air gap when subjected to voltage pulses of various shapes.

Introduction
The wide use of thermoelectric devices as sources of electric energy, coolers and measuring equipment is due to their reliability, high service life, ability to work under extreme conditions and ecological purity.Thermoelectric devices are used in space and defense technology, metrology, electronics, medicine, etc.
The basis of most thermoelectric devices is a semiconductor thermocouple element, which, along with positive qualities, has drawbacks that prevent further progress in the development of thermoelectricity.The principal shortcomings of the thermocouple element include the ineffective participation of its volume in thermoelectric energy conversion, switching problems, and limitations on the Q-factor of the used materials.Therefore, the further development of thermoelectricity by searching for semiconductor materials with a higher quality factor has already almost exhausted itself.
Progress in thermoelectricity is possible with the introduction of new thermoelements (TE), which are dominated by the possibilities of thermocouple elements.Among the new promising TEs there are vortex TEs with unique properties, which are based on the use of the effect of vortex thermoelectric currents (VTC).It is possible to indicate, in particular, spiral vortex thermoelements, in which the problem of commutation and obtaining the necessary stresses is solved [1,2].
However, the elemental base of thermoelectricity needs replenishment due to the creation of new types of thermoelements based on known thermoelectric phenomena and phenomena that have not been sufficiently studied.To known phenomena, the mechanisms of their origin have been studied, but on the basis of which TE is created, for example, the volume effects of Peltier and Bridgman can be attributed.Studies of the feasibility of creating anisotropic optical thermoelements (AOTE) on these effects on the basis of existing and hypothetical materials, as well as studying the depth of their cooling, are of current interest.Such studies could lead to the creation of AOTE for cooling or stabilizing the temperature of various kinds of microsensors and microelectronic devices, and are distinguished by a simpler design than the known ones.To the insufficiently studied phenomena from the point of view of practical application it is possible to attribute umkehr effect, VTC effect, the effect of the anisotropy of thermal conductivity on the transverse thermoelectric power.
An in-depth study of the influence of the anisotropy of thermal conductivity on the temperature field and the transverse thermoelectric power of electrically insulated AOTE and its energy would lead to a refinement of the

Fig. 1 .
Fig. 1.Simulation of electrical breakdown of the air gap using Kind's equal-area criterion in the evaluation version of Micro-Cap 11

Fig. 2 .
Fig. 2. Calculation results of t 1 and t d at U m = 600 kV: a -t 1 is calculated after the first iteration; b -t d is calculated after the second iteration

Fig. 3 .
Fig. 3. Breakdown of the air gap at the front of wave, under the impulse voltage with the shape of 1.5/40 μs and amplitude of: a -1000 kV; b -975 kV; c -950 kV; d -925 kV defined by experimental volt-time characteristic (determined by formula (2), substituting simulated breakdown time from Table

Fig. 4 .
Fig. 4. Breakdown of the air gap at the tail of wave, under the impulse voltage with the shape of 1.5/40 μs and amplitude of: a -675 kV; b -650 kV; c -625 kV; d -600 kV