Damping factor function in AC electrical arc models. Part 1. Heat ...

Antoni SawickiDamping Factor Function in AC Electrical Arc ModelsPart 1: Heat Process Relaxation Phenomena,their Approximations and MeasurementIntroductionTemporary changes of the columncross-section radius and the distribution ofenergy along and across an arc significantlyaffect the operation of electrotechnologicaldevices and electric appliances. They aredecisive for the possibility of breaking andre-igniting the arc. A quantity characterisingsuch possibilities in the most complete manneris a function for the factor of dampingenergetic processes in an electric arc plasma.A coarse, and yet very comfortable approximationof this function is the time constant.The knowledge of the arc time constant isnecessary for the following:• selecting economical operating conditionsof electrotechnological devices (welding,electrothermal etc.);• optimum influence on the arc in switchingdevices (breakers, switches, contactors,relays);• maximally intense influence on the arc inovercurrent and overvoltage protectivedevices (fuses, lightning protectors).The arc may stop during voltage reduction,excessive stretching of the column, excessivecurrent reduction, cooling of the gasarea (sometimes also of the electrode) or asa result of contracting the column with a diaphragm.The above phenomena trigger deionisationin the plasma column and cooling ofthe active areas in the electrodes. Dependingon the intensity and duration of these processes,re-ignition of the arc may be difficultor even impossible. For this reason, the timeconstant should be the following:• low in electric appliances so that the arccan stop relatively quickly due to disturbances;• high in electrotechnological equipment, inorder to prevent undesired terminating ofthe arc due to disturbances.Heat process relaxation phenomenain electric arcArc columns have a heat capacity, and yetthey constitute certain resistance to thermalcurrent. For this reason they have finite timesof response to forced changes of thermalstates. The amount of internal energy, accumulatedin the arc, depends on many factors:• plasma volume (radius, length and shapeof the column);• temperature distribution in the column;• pressure of the plasma-forming gas;• type of plasma-forming gas, degree ofplasma-forming gas ionisation etc.The enthalpy of the arc Q changes exponentiallyin time in accordance with the timeconstant [1]θ =dP0−dQ1dRIdQ2(1)where P 0 – dissipated power; R – arc resistance.Hence one can see that θ depends oncurrent I. In the case of alternating current,the factor θ depends on the momentary valueof current.As the electric current is characterised bythermal inertia, the changes of the thermalstate and geometrical dimensions of the col-dr hab. inż. Antoni Sawicki, professor at Częstochowa University of Technology - Facultyof Electrical EngineeringNo. 2/2013 BIULETYN INSTYTUTU SPAWALNICTWA37

umn during current rushes or changes in thecolumn length are not immediate, but proceedwith a certain time constant. That is whyarc resistance during the changes of currentr(i) alters exponentially from the stabilisedvalue r(I 0 ) to the value corresponding to newcurrent r(I 1 ). The time constant of the arc inconstant pre-set heat transfer conditions is atime after which the arc column changes itsresistance e–times after energy is no longersupplied to the column.The time constant of the arc is defined bythe column cooling conditions. In plasmatronsit is shorter by 2÷3 orders (10 -6 ÷10 -7s) if compared with ordinary free arcs. Evenat a frequency of 200 kHz the arcs of ACplasmatrons have a hysteresis, which reflectstheir low time constant. The higher the flowrate of the gas flowing around the column orthe rate of the arc motion in gas, the lowerthe time constant is. In the case of high gasflow rates, the arc time constant does not dependon the gas chemical composition or thetype of electrodes [2].When the arc is ignited or terminated theenergy accumulated in the plasma volumeunit is greater than the energy dissipatedfrom this volume. For this reason, in furnaceswith a high temperature of the atmosphereit is easy to observe the thermal breakdownof the inter-electrode gap with relatively lowvoltage. In turn, if the temperature of the atmosphereis low, the reproducing strengthof the inter-electrode gap rises quickly to acertain initial value according to a certain exponentialdependence. In order to be able toreliably re-ignite the arc, the rate of powersource voltage reproduction should be higherthan the critical value [1]. This principle isthe basis for testing and assessing the qualityof dynamic characteristics of weldingsources. The lower limit of the reproducingvoltage is defined by the arc ignition voltage.The rate of increasing reproducing strengthof the arc column is defined by the inertia ofheat processes, and especially by the arc timeconstant θ. In turn, the dynamic properties ofthe source are affected by the design and settingsof the control system as well as by thepassive conservative elements (inductanceand capacitance) of its circuits.The gas suppression ability is defined notonly by its time constant but also by its electricstrength. The reproduction of the electricstrength of the inter-electrode section stronglydepends on the falling rate of temperatureT of the plasma left after the arc column. Thiscan be roughly determined using the followingdependence [1]:ot−θ( T −T) ⋅etT = T −0 (2)otwhere T ot – ambient temperature; T 0 – temperatureon the arc axis at the beginning ofthe process; θ - arc time constant. The electricstrength of the inter-electrode gap is inverselyproportional to temperature [1]TotET= Eot(3)Twhere E T – electric strength at heightenedtemperature T; E ot – electric strength at ambienttemperature T ot .Approximating the factor of energeticprocess damping in electric arcIn the majority of simplified mathematicalmodels, very roughly approximating thephysical properties of the electric arc, thedamping factor value of transitory processes(thermal and electric) is adopted as a constantquantity (the so-called time constant).It is the proportion of two quantities; thenumerator is the heat capacity of the plasmachannel, whereas the denominator is madeup of parameters specifying the properties ofenergy dissipation [3]. More accurate models(e.g. Cassie-Mason or Lowke’s) bind proportionallythe time constant value with thecross-sectional area of the plasma column[4, 5]38 BIULETYN INSTYTUTU SPAWALNICTWANo. 2/2013

∝2(4)where p - gas density; c p – specific heat ofgas of pressure p; λ - gas heat conductionfactor; r a – arc column radius.The confirmation of this assumption wasalso attempted using the approximation ofexperimental data [6], where one should bearin mind that the column cross-sectional areadepends not only on electric current intensitybut also on the type and pressure of plasma-forminggas, the temperature of the gas inthe discharge area, the rate and direction ofgas flow, the diameter of the discharge duct,the amplitude and frequency of the magneticfield etc. In some other models of the arc thetime constant is adopted as being proportionalto the column diameter (e.g. the model byA.A. Woronin [7-10]).In relation to relatively long arcs the shortconical part at the cathode is negligible andthe shape of the whole plasma column can beassumed as cylindrical. Theoretical deliberationsand experimentation are significantlysimplified if one determines the geometricaldimensions of the DC arc. Results obtainedin this way can be adopted for low-frequencyAC arcs, assuming the necessity of maintainingwithin them the same plasma equilibrium.In welding arcs the diameter of the arccolumn is, first of all, the function of currentd a = f(I 2/3 ) [11, 12]. This function is veryclose to the empirical formula [1]≅ × , cm (5)where n = 0.6÷0.7, obtained in the case of gasflowing around the arc in a longitudinal manner.If the arc is in the air, k = 0.27 cm×A -n .If one considers a strong-current arc, e.g.in a steelmaking arc furnace with a graphitecathode, burning in air, the measured columndiameter amounts to approximately [13]:2 ⎛z0,864 0,253 ⎟ ⎞d ⎜a= rK⋅ −⎝rK⎠(6)0,5In most operation regimes the gas atmosphereof the DC arc furnace is made up bycarbon oxides. Another formula for the diameterwas provided by R.T. Jones and Q.G.Reynolds [14]:da⎡ ⎛ z ⎞⎤2 r ⎢⎜⎟K ⋅ 3,2 − 2,2exp − ⎥ (7)⎣ ⎝ 5r⎠⎦= Kwhere the cathode spot radius isrKI= , cm(8)π jKand j K = 3500 A/cm 2 – current density in thecathode spot.However, the function of the column diameterd a (i) is not monotonic in the rangeof weak currents. Plasma does not disappearalong with the momentary reduction of currentto zero. Yet, the weakening of the pincheffect may cause its expansion, accompaniedhowever by some cooling and deteriorationof electric conductance. This, in turn,depends on the conditions of heat exchangein the environment and the rate of currentchanges during polarisation alteration. Suchbehaviour is confirmed by the experimentaltests of the arc time constant, which increasessignificantly in the range of weakcurrents, below approximately 18-30 A [15,16]. The flow of current through such terminatedarc plasma is possible after applyingvoltage from an additional source [17]. Ifcurrents are weak, the arc time constants arenot only high but also strongly dependent onthe type of gas (e.g. in elgas, SF 6 , θ = 1÷2 µs,in the air θ = 100÷200 µs). In the range ofstrong currents the tendencies of time constantchanges are reverse to the changes ofcolumn diameter (cross-sectional area). Anincrease in current intensity as well as anincrease (with very weak saturation) in thecolumn diameter function (formulas (5)-(7))are accompanied by a decrease in the timeconstant, which stabilises at the lowest lev-No. 2/2013 BIULETYN INSTYTUTU SPAWALNICTWA39

ule, at such a moment this value matches themaximum damping factor value. The saidvalue can be determined by means of appropriatemeasurement methods [21] utilisingthe delay and increase in the re-ignitionvoltage. From the point of view of ensuringthe arc burning stability and the continuityof electrotechnological device operation itis justified to experimentally determine suchhighest value θ. Due to the range of weakcurrent, this constant can be used in the Mayrmodel. However, the time constant determinedin the weak-current range is sometimesalso used in the Cassie model [22], whichmay lead to discrepancies of experimentationand calculation results, especially in therange of strong currents. Therefore, if it isnecessary to precisely reproduce the coursesin circuits by means of the universal modelof strong-current arc (e.g. hybrid TWV), thewhole damping factor function θ(i) shouldbe expressed by means of dependence.Experimental methods for determiningAC arc dynamic parametersA characteristic feature of experimentalmethods for determining arc dynamic parametersis to take into account the wholerange of physical phenomena taking place inthe column, near-electrode areas and in electrodesthemselves. The separation of individualcomponents of energy processes is verydifficult but sometimes possible by means ofanalysis [11, 23, 24]. Depending on the designand the principle of operation of electrotechnologicalarc or plasma-arc devices,applied technologies and operation modes,one can observe various levels of disturbancesboth as to the amplitude and the range offrequency, which can even exceed the valuesallowed by related standards. Such disturbancesmay originate from sources which aredifficult to identify or eliminate and whichdisturb processes taking place in the arc column,electrodes, and even in the circuits ofmeasurement systems. As the quantities u(t)and i(t) are registered along with random disturbances,calculating the values of conductanceg(t) on their basis comes down to solvinga badly conditioned task. An improvementin the quality if input data can be obtainedusing appropriate methods for filtering andsmoothing time courses [25]. However, priorto undertaking such actions it is necessary tosolve the issue of recognising types of disturbancesin order to weaken only the impact ofnatural disturbances and leave disturbancestriggered on purpose.Methods for determining the dynamic parametersof the AC arc can be divided intoseveral groups:1) methods using natural periodic courses ofcurrent and voltage;2) methods introducing additional disturbancesto periodic courses, using additionalcurrent sources;3) methods introducing disturbances of thearc column length (voltage);4) methods introducing disturbances of conditionsof energy dissipation from the column[7, 21].In the case of the methods utilising naturalelectric courses it is assumed that there is anunequivocal functional relationship betweenarc parameters and current, resistance or conductance.It means that, in specified conditionsof heating and cooling the column, oneset of model parameters corresponds to onevalue of current or arc conductance. Usingproperly processed (filtered and/or smoothed)data, one can apply one of the analytical oranalytical-graphic methods (known as Amsinck,Ruppe, Asturian, Rijanto, Zuckler, Tajew,generalised etc.) in order to determinethe simple parameters of the Mayr or Cassiemodels [22]. More complex models requirethe use of numerical methods.There are also possibilities of the directdetermination of the arc time constant, notrequiring the calculations of the remainingNo. 2/2013 BIULETYN INSTYTUTU SPAWALNICTWA41

parameters of specific mathematical models.In the low-voltage arcs of sinusoidal alternatingcurrent and in the conditions ofrelatively low cooling intensity, before andafter the passage of current through zero, itis possible to observe moments at which thefirst voltage derivative, in relation to time,equals zero. Using the measurement of timet0 from the moment at which sinusoidal currentpasses through zero until the moment ofarc ignition or termination it is possible todetermine the whole time constant using thefollowing formula [21]:t 0θ =(11)2Another simple method uses the harmonicanalysis of arc voltage [26]1 ⎛ 1 ⎞θ = ⎜ − χ ⎟4ω⎝ χ ⎠(12)where χ = A 2n+1 / A 2n-1 < 1 amplitudes of theclosest harmonic odds of voltage (usu. χ =A 3 / A 1 ).A special method for testing the AC arcconsists in “placing” a properly selectedhigh-frequency current (as to the amplitudeand phase) on the current flowing through thearc [7]. For the purpose of testing air switchesthis frequency usually amounts to 20 kHz. Inthe case of switches with elgas the frequencyis much higher and equals 70 kHz. In thismanner one can trigger additional transitionprocesses in the areas of net current passagethrough zero. After registering the courses,the parameters of the Mayr model [7] can becalculated from the following dependence:= − , if =0 (13)= , if =0 (14)where P M – constant value of the dissipatedpower of the model. As the determination ofthe time constant is carried out at point i = 0,when conductance g has an indefinite value,the value of the time constant in the expression(13) is calculated from the following interpolation:( t − ∆t) + g( t + ∆ )g0 0tg = (15)2where t 0 – time instant in which i = 0 A; Δt –time interval, at which the recording of currentand voltage values takes place.According to another method, net currentdoes not have to pass through zero. In sucha case the value of power is determined usingthe formula (14), and the time constant iscalculated from the dependence below:2g ⎛ i ⎞θ = −⎜ −1⎟(16)dg⎝ gP M ⎠dtIt is also possible to determine the dampingfactor function on the basis of the reaction ofthe arc column on various length disturbances.They should also be appropriately synchronisedand shifted in the phase in relationto the course of current. In laboratory conditionsthe changes of length can be relativelyeasy to induce by means of properly selectedrotating electrodes (commutators) [7]. Theexcitation of high-frequency disturbances byelectrode vibrations is more difficult, especiallyif electrodes are massive. It also facilitatesthe sputtering of electrode material andincreases electrode erosion. A relatively highfrequency of such changes can be obtainedby the crosswise action of variable magneticfield on the arc [27]. Due to some binding ofthe column to electrode spots (especially ofthe cathode) only the central part of a longarc is the preferable area of this action.In an electric arc with stabilised currentit is possible to trigger voltage changes bymodifying the conditions of heat exchangewith the surroundings [21]. The longitudinalpulse flow of gas around the column caus-42 BIULETYN INSTYTUTU SPAWALNICTWANo. 2/2013

es momentary changes of dissipated powerand of the arc column diameter. In turn, thetransverse or slant pulse flow of gas aroundthe column causes its temporary elongationand contraction. The changing motion of thearc in relation to the gaseous environmentcan also be triggered by means of a modulatedmagnetic field properly synchronisedwith the course of discharge current. The useof parallel ring electrodes with moving arcspots enables maintaining almost the wholelength of the column.Artificially introduced arc disturbancesshould be characterised by a limited rangeof amplitude due to the strong non-linearityof static and dynamic characteristics aswell as because of discharge instability. Forthis reason, the depth of modulation usuallyamounts to a few percent. Too high an amplitudeof current disturbances changes thecharacter of arc discharge from the “dc-” to“ac-type” [18]. Also, too high a frequencyof current disturbances changes the characterof discharge from the ac-type arch dischargewith thermal plasma to the “RF-typedischarge” with non-equilibrium plasma. Forthis reason there should be an inverse proportionalitybetween the amplitude of periodicdisturbance and its frequency.It is also technically possible to carry outsynchronised disturbance of the arc columnwith two or more types of external factorsat the same time. In this manner one canobtain the deepened modulation of courses,which however may facilitate the occurrenceof discharge instability. Due to this fact suchsolutions are not applied for testing electrotechnologicaldevices. In addition, the greatercomplexity of the design and operation ofthe testing station entail the greater complexityof necessary analyses and measurements.Such inputs cannot be compensated by theimproved accuracy of obtained results. Insuch situation, the methods introducingelectric disturbances are the simplest, mostaccurate and, consequently, most popular[6, 15, 16, 18, 28].The second part of the article focuses onthe assessment of the usability of methodsused for measurements of dynamic characteristicsby simulating processes in circuitswith modified and hybrid models of the electricarc.Conclusions:1. The results of the so-far experimentationand theoretical analysis of such physicalquantities as the damping factor function andarc geometrical dimensions often do not confirmadopted assumptions, nor do they offerthe possibility of obtaining simple and directrelationships between θ and the diameter d aor the cross sectional area S of the column.2. Most of the experimental methods fordetermining the dynamic characteristicsof the electric arc enable only the determinationof the time constant in the areas ofcurrent decay. The constant constitutes themaximum value of damping function and, assuch, is predominantly useful for modellingthe joining arc.3. The pursuit of more and more precisereproduction of processes in the circuits ofwelding and electrothermal equipment withan electric arc causes the popularisation ofcomplex hybrid models increasing the usabilityof non-linear damping functions.The research work has been financedfrom the funds for sciencein 2010-2013 as research projectno. N N511 305038.No. 2/2013 BIULETYN INSTYTUTU SPAWALNICTWA43

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