AFFINITY OF CONTACT MATERIALS TO FORM THE ELECTRIC DRAWN ARCS ""France Pavlovčič, ^Janez Nastran ^ Ministry for environment and spatial planning, Environmental Agency of the Republic of Slovenia ^University of Ljubljana, Faculty of electrical engineering, Ljubljana, Slovenia Key words: drawn electric arc, arc formation, sliding contact, commutator, arc ignition voltage, contact material. Abstract: Electro mechanic contacts are widely used in several appliances, because they are cheap, but also very robust, capable to withstand overvolt-age and overcurrent surges, and when used in electric relays they enable galvanic separation of electric circuits. Especially they are necessary with commutators of universal electric motors for they are enabling commutation. These motors achieve high power per a unit of volume due to their high shaft speed, and they are produced at relatively low costs for a unit of power So, they are economicaily very important fortheirnumerous production, and they use sliding contacts for their operation. The sliding contacts are mechanical contacts, as switching contacts are, and so electric arcs ignite between their contact members during commutation. Therefore it is very important to determine contact materials with a small tendency to form the electric arcs. The arcs between electrodes of the sliding contacts are usually drawn arcs, so that affinity of the contact material to form the drawn arcs is very significant attribute of determining the right contact mate-rials. Conclusions made for the sliding contacts are also useful for the switching contacts. Težnja kontaktnih materialov k tvorjenju električnih potegnjenih oblokov Kjučne besede: potegnjeni električni oblek, nastanek obloka, drsni kontakt, komutator nape-tost vžiga obleka, kontaktni material. Izvleček: Elektromehanski kontakti so množično uporabljeni v številnih napravah, ker so poceni, vendar robustni, zmožni vzdržati prenapetostne in preto-kovne preobremenitve in kadar so uporabljeni v električnih relejih omogočajo galvansko ločitev tokokrogov. Posebno so nepogrešljivi pri komutatonih univerzalnih električnih motoriev, kajti omogočajo komutacijo. Ti motorji razvijejo veliko moč na enoto volumna zaradi velike hitrosti vrtenja in njihovi proizvodni stroški na enoto moči so relativno majhni. Zaradi njihove velikoserijske proizvodnje so ekonomsko zelo pomembni. Za svoje obratovanje pa uporabljajo drsne kontakte. Drsni kontakti so mehanski kontakti, kakor so tudi stikalni kontakti in zato se med njihovimi kontakti pri komutaciji vžgejo električni obleki. Zato je zelo pomembno določiti kontaktne materiale z majhno tendenco tvorjenja električnih oblokov Obloki med drsnimi kontakti so ponavadi potegnjeni obloki, tako da je težnja kontaktnih materialov k tvonenju potegnjenih oblokov zelo pomembna lastnost pri določanju pravega kontaktnega materiala. Zaključki, narejeni za drsne kontakte, so uporabni tudi za stikalne kontakte. 1. Introduction Talking about affinity of the contact materials to form the electric drav\/n arcs, we must have in mind that it is not a physical quantity, but rather a conception, which is numerically estimated by one or more physical quantities. To begin with, we consider the voltage - current (Ul) stationary electric arc characteristics, for instance of the copper contacts, shown in Fig. 1 as hyperbolae, in relation to the pure resistive load, shown in the same figure - the line marked with the label "break" /1 /■ This line illustrates the load characteristic of the electric arc at the breaking of an electric resistive circuit by the axial switching contacts. This drawn arc ignites at the voltage of 13.1 V, which is slightly over the voltage of the asymptote of the hyperbolae, and at the current of 1.95 A. The voltage of the asymptote is defined as the minimal arc voltage Um by Holm /1/ and it is the property of the material. But we name it as the infimum arc voltage because it is the greatest lower limit voltage. The arc burns until its length is 0.8 mm and is extinguished at the voltage of 48.7 V and at the current of 0.73 A. tunnai offoct Fig. 1: The Ul stationay and load arc characteristics. Legend: arc stationary arc characteristics, comm ioad arc characteristic at resistance commutation, breal< load arc characteristic at breaking resistive load. If another contact material is taken into consideration then the hyperbolic characteristics are the same curves, but voltage {Um) and the current {Im) the asymptotes have different values. The current asymptote is defined as the infi-mum arc current and is also the property of the material. If the value of the Um - asymptote is higher than the one of copper the arc ignites at a lower current, and of course, at a higher voltage, and its length is shorter at the point of the extinction. The values of the arc voltage and the arc current at its extinction are also changed. Because the ignition and the length of the arc are depended on the Um -asymptote, we decide that the infimum arc voltage is just the proper quantity to define affinity of the contact material to form the electric drawn arc. It seems very simple to define affinity of the contact materials to form the drawn arcs between the axial switching contacts, but in the case of the sliding contacts, such as they are in the commutators, is a much more difficult task. Namely, the drawn arc ignites and burns at a macroscopic geometrical separation of the axial switching contacts, but researching the arc phenomenon with the sliding contacts it must be taken into account, that the drawn arc ignites and burns between the overlapping electrodes with no macroscopic geometrical separation ofthem /2/,/3/. Ifwe consider only the Ul stationary arc characteristics and the load characteristic of the arc during the resistance commutation, shown in Fig. 1 - the diagram marked with the label "comm", it is not selfevident that the infimum arc voltage is the measure of this kind of affinity. The arc ignites at the infimum value of the arc voltage and at the full value of the current through the contact. It burns at the voltage, which slightly rises from the infimum arc volt-age of 13 V up to 15.2 V for copper, and is extinguished at the latter value of the voltage and at the current of 0.59 A. Its length is in the range of some nanometres throughout its burning, so it is a short arc. It is exactly the same with another contact material: the ignition occurs at the infimum arc voltage and at the full current, the burning is at its length of some nanometres and at the corresponding voltage, and extinction is nearly at the current asymptote, which value is the infimum arc current. Therefore we have to establish the mathematical model of electric current conduction through the gap of some nanometres between the commutator bar and the brush during the commutation to make a clear definition of affinity of the contact materials to form the drawn arcs. When the commutator bar moves over the brush, the thickness of the gap between them varies, and so the way of the current conduction alters from tunnel effect to arc conducting mode - Fig. 2, If the thickness of the gap is so large, that its corresponding voltage drop equals to or is greater than the infimum arc voltage Um, the arc ignites. With the same thickness of the gap, some contact materials form the drawn arcs, but some not, depending on their infimum arc voltage. As this voltage is the property of the contact material, it fully determines affinity of the contact material to form the electric drawn arc. fully ionized layer partly ionized layer free atoms layer Fig. 2: The formation of the eiectric drawn arc during the commutation. There are many physical constants and variables in this paper that are not explained in the text, so their definitions are present in the chapter of used symbols at the end of the paper. We also deal with many chemical elements, although they could not be used as pure materials in contact technique. The direct current case is considered by this model. 2. The mathematical model of the drawn arc formation With the sliding contacts of the commutator, a very thin insulating film arises on the contact surface /1 /,/4/. This film is the collector film. Beside it, there is also the gap between the contacts. The gap is filled with atoms and molecules of an external medium, and also with the atoms of the contact materials. The gap and the film thickness are up to a few nanometres, so that no ionization occurred. The electric current flows due to tunnel effect. One contact member is a cathode, the other one is an anode. The charge carriers are free electrons in the electric field of the gap, so they are accelerated when moving along the path through the gap. They are emitted from the cathode by three ways: by the field emission, called also the cold emission; bythe thermionic emission including also the Schottky effect, which is depended on the electric field, but its contribution is not included in the cold emission; by the emission due to the different work functions of the cathode and anode matehals. Considering the cathode is still cold, only the field emission is taken into account, and the current density is defined bythe Fowler - Nordheim equation /5/,/6/ and /7/, which is equal to Eq. (1) for the electron emission from the smooth and unstained metal cathode in vacuum: 2 S-lt'^l-ml'e V^J'' ^S ml-e^ -1/2 Je^- e' E ■n —;--- --^i_?__J/ 3A e' 3/ie„ (1). Experiments show that the electric field /8/, the work function /Q/ JQ/, which is defined as the voltage Vtk , and the effective electron mass /10/,/11/ changed due to a contamination of the cathode, but the roughness of the cathode surface effects only the electric field intensity. So the following substitutions must take place to descride exploat-ation condition of the contacts: E' = V y* m„ = X) ß>l D >1 ^>1 (2), (3), (4). The collector film and the moleculas of the air gases in the very thin gap are considered as the contanimation over the cathode surface. Therefore the equation, describing the current density due to the cold emission from the cathode, overlayed with the collector film and being in the air, is: ^'■"8-71-A' K 8.71 u (5). The current density is known in the most cases, so that the electric field intensity is calculated from Eq. (5) by some iterative method. The commutator bar moves over the brush, so the thickness of the gap increases, as it is demonstrated in Fig. 2. Presuming the electric field is homogeneous the acceleration of the electrons in the gap is constant. The kinetic energy of the electron is /11/: and the rootmeansquare value or, as it is also called, the effective value is: 1-- m„ ■ c 2- 1 + m„ + m-c (9). The motion of the electron is accelerated, so there is a constant force on the electron. This is the force of the electric field on the electron, and the vectors of the force and of the electric field are colinear: F = e-E (10). This force is also the function of the electron mass and his acceleration /11/, and the vectors of the force and of the acceleration are colinear: F = - m. dv dt 1- (11). A path the electron must move over to get the sufficient velocity for the ionization of the atom is derived from Eqs (10) and (11). This path is: • c s = e-E N, m^-c' 1 + ^ m,. (12). y m. ■ c r - m„ ■ c' 1- (6). C When the electron achieves such a velocity, that his kinetic energy is: / \ W N. 1 + m. (7), the ionization of the atoms in the gap begins. The energy Wii is the first ionization energy of one mole of the gaseous element in the gap. Two values of the electron velocity are derived from Eqs (6) and (7). The first one is the average value of the linear motion of the electron /2/, and the second one is the rootmeansquare value needed for the ionization when collision between the electron and the atom is not centric. The average value of the electron velocity is: V = r ■ 1-- N„ 1 + m„ m,. + m„ -c (8), " J Because the electric field is homogeneous, the voltage over this path is equal to: U = 772 „ N^i m^ ■ c 1 + m ^ -1 m,. " y (13). y These two equations (12) and (13) are simplified according to the following rule of small numbers: Vl + 2-8 «1 + 5 <= 5 «1. So, the path and the corresponding voltage are: s = - e-E-N^, W U = ■ e-N.^ (14), (15). These are the basic equations that determine the path of the electron flow and the voltage over this path, which is sufficient for the ionization. If the gap is shorter than this path the current flows due to tunnel effect. The ionization means the end of tunnel effect, and it is also the beginning of the formation of the drawn arc. Comparing the drawn and the discharging arc, the ionization is present at the formation of them both, but for the latter one, the overvolt- age between the contact members and the breakdown of the gap medium are the cause of the ignition. In the case of the drawn arc, the current conduction smoothly turns from tunnel effect to arc conducting mode, when the ionization begins. While the tunnel conduction has only one kind of the charge carriers, which are the electrons emitted from the cathode, the drawn arc conduction has the following charge carriers: the electrons passed from the cathode by the field emission; the electrons passed from the cathode by the thermionic emission at the higher value of the cathode temperature as with tunnel effect; the cathode and the anode ions, which resulted from the ionization of the free atoms of the cathode and the anode materials in the conducting volume of the gap. When the thickness of the gap sufficiently increases the ionization begins, but not the ionization of the cathode and the anode atoms at the same time. If the first ionization energy of the cathode material is lower as the first ionization energy of the anode material the ionization of the cathode atoms takes place, while the anode atoms are still not ionized. Further increase in the gap thickness - Fig. 2, causes that the electrons get the higher kinetic energy, sufficient to ionize the free anode atoms. So, there are three layers within the arc illustrated in Fig. 3: the first one, where the kinetic energy of the electrons is not sufficient to ionize any free atoms in this layer; the free cathode and anode atoms only are present there, therefore this one is named the free atoms layer: the second one, where, in the discussed case, the cathode atoms are ionized, but the anode atoms are still neutral; this layer is the partly ionized layer, the third one, where all atoms are ionized is called the fully ionized layer. The free atoms layer is in literature IM J A! defined as a cathode layer, but only the cathode atoms and ions are taken into account with its definition. To distinguish between our arc model, which considers the cathode and the anode atoms and ions, and the other arc model from the literature, we define the arc layers in this special way. The difference between the free atoms layer and the cathode layer, as defined in the literature, occurs when the first ionization energy of the cathode material is higher than that of the anode material. When the fully ionized layer is established the drawn arc ignites. The question arises about the length of the path through the fully ionized layer. It must be so large that the ions, while moving along this path, get the sufficient kinetic energy to heat the cathode to such amount that the cathode atoms pass from the cathode to the gap by vaporization or by sublimation, and that the thermionic emission of partly ionized layer fully ionized layer free cathode and anode atoms <1—@ recombtned catfiode and anode atoms with constant velocity Q-». cathode and anode ions and electrons accelerated by E Fig. 3: The arc layers. the electrons is established. Further on, travelling toward the cathode, they are recombined, and their collisions against the cathode are nonperpendicular. There must also be equilibrium of the charge carriers, so that the arc is neutral to the external medium. The mathematical model of the drawn arc formation has the following presumptions: there is only the field emission with tunnel effect; the electric field intensity is constant while the current conduction passes from tunnel effect to arc conducting mode; the electric field is also homogeneous; the contact current is constant while the current conduction passes from tunnel effect to arc conducting mode; the motion of the electrons in the gap between the cathode and the anode is linear; the collisions of the particles are not centric, northey are perpendicular; the ions are recombined before they collide against the cathode; the arc is externally neutral; the arc temperature is equal to the cathode temperature at the instant of the arc formation; the ratio between the number of the free cathode atoms and ions and the number of all free atoms and ions in the conducting volume is calculated as the cathode and the anode temperatures are equal; the external medium is air under the normal pressure, or is optionally vacuum; the total pressure of gases in the conducting volume between the cathode and the anode is equal to the pressure of the external medium, unless it is determined by the thermionic emission to be higher. There are the cathode and the anode atoms in the gap, which are ionized by the collisions with the electrons. The first ionization energy of the cathode atoms is not equal to the one of the anode atoms. Therefore the electrons that ionize the cathode atoms have the different average and effective values of the velocity than the electrons that ionize the anode atoms. The ratio between the number of the free cathode atoms and ions and the number of all free atoms and ions in the conducting volume must be determined to calculate the overall average velocity of the electrons. This ratio is according to the ideal gas law: (16). In Eq. (16), there are the saturated vapour pressures of the cathode and the anode atoms and ions. They are determined according to the Clapeyron - Clausius equations /11/: dpM) ' V I _ .i'^. dT^ r-T:- K (17), y d T R-T:- Y ^il^L -0 V. j-iva (18). The Lvk and Lva are the values of the latent heat of vaporization of the cathode and the anode materials depended on the cathode temperature. Their functions are presented as the linear interpolated equations between the boiling and the critical temperatures: KK (^k ) ~ KbK ■ T CK T ■ ^ CK (19), T ca -TK T ^ ca -T,a (20). The ratios between the volume of one mole of the material in liquid state and the one in gaseous state are neglected in Eqs (17) and (18) as = 0 a = 0. So, the satu- V... V... Hvic ' jiva rated vapour pressures for the cathode and the anode materials are: Ate P.AT.)= P.,-e r.- Tu. T. T,„ ■/;„ (21), (22), where = 101.325 [kPa] is the saturated vapour pressure of any material at the boiling temperature by the definition of the boiling point. Presuming the current density is known, the density of the electrons is calculated: J (23), where the Vek_avg and Vea^avg are the average values of the velocities of the electrons that are to ionize the cathode and the anode atoms respectively. According to the ideal gas law we get the following equation: a-N, (24). Fig. 4: The layer energy in comparison to ttie ionization and dissociation energy of nitrogen and oxygen. Legend: Watm+Wa+i(C) iayer energy of free atoms layer and partly ionized layer together -anode is carbon, Wion(C) layer energy of fully ionized layer - anode is carbon, WfA/g the first ionization energy of nitrogen molecules, \N(N A/^J the first ionization energy of nitrogen atoms, W(N2 2N) dissociation energy of nitrogen molecules, W('02 02^)the first ionization energy of oxygen molecules, WfO —> O^j the first ionization energy of oxygen atoms, W(02 —> 20) dissociation energy of oxygen molecules. The ionization coefficient a is less than 1 with the whole conducting volume of the gap. The cathode temperature is calculated from this Eq. (24) by some iterative method. The pressure of the external medium is = 101.325 [kPa] when it is air, but zero with vacuum. The question is, why the external medium is so important. The external medium is air, so it contains the nitrogen molecules (78%) and the oxygen molecules (21 %). If the saturated vapour pressure of the catode and the anode atoms and ions together is less than the pressure of external medium the molecules of the external gas are also present in the gap, but their partial pressure is less than the external pressure. There are also the collisions between the electrons and the nitrogen and the oxygen molecules. This collisions are elastic, unless the kinetic energy of the elec-tons is sufficient to ionize the nitrogen and the oxygen moleculas, or to dissociate them and further on, to ionize the gaseous atoms. Though the collisions are elastic, the velocity vector of the electrons changes by impacts, and cosequently the final kinetic energy of the elecrons changes. The sufficient kinetic energy of the electrons for full ionization is gained after they have moved along the path through some layer, so we define this energy to be the layer energy. According to Eq. (14) the layer energy is not depended on the particles, which are either the electrons or the ions. The layer energy of the free atoms layer and the partly ionized layer together and the layer energy of the fully ionized layer for several cathode materials, while the anode is carbon, are compared with the first ionization energy of the molecules, the dissociation energy and the first ionization energy of the atoms of nitrogen and of oxygen. The results are shown in Fig. 4. The dissociation energy of the nitrogen and the oxygen molecules is achieved with some cathode materials in the first two layers of the arc. But no ionization energy of the nitrogen and the oxygen particles is ever achieved. The elastic collisions and the dissociation of the nitrogen and the oxygen molecules waste the kinetic energy of the electrons, so that the total ionization of the cathode and the anode atoms is not attained in the fully ionized layer. To establish the arc, the cathode temperature is increasing, so that the total saturated vapour pressure of the cathode and the anode atoms and ions forces the nitrogen and the oxygen particles out of the conducting volume of the gap. Then the conditions for the arc formation exist. The current density due to the thermionic emission is defined by the Richardson - Dushman equation /5/: a-t;. e-ßE (25). The presumptions have the following effects on the total current density and on the crosssection of the conducting path between the cathode and the anode, that is the cross-section of the arc: E = const J = JE+J7 I - const Arc = Ad Je (26), (27). Je + JT The current density increases, but the cross-section constricts. A narrow ionized conducting channel, something similar as a pilot streamer at discharges /12/, arises. It is a hot flow of the electrons, the cathode and the anode ions and atoms. The absence of the ions of the air gases of the external medium is the main difference between the drawn and the discharge arcs. There is a loop in this procedure: Eq. (26) effects Eq. (23), but the algorithm converges. The results of the temperature calculations for the electrodes, both of the same metal, are presented in Fig. 5. The comparison is carried out between the air medium and the vaccum, and further on, the cathode temperatures toward the melting and the boiling temperatures. The cathode temperatures are somehow between the melting and the boiling points, but there are some exceptions. Fig. 5: The cathode temperatures, melting and boiling temperatures. Legend: Tm melting temperature, Tb boiling temperature, Tk-air cathode temperature in air, Tk-vac cathode temperature in vacuum. So far we presume that both, the cathode and the anode atoms and ions are present in the conducting volume of the gap. The number of the cathode atoms and ions Na+ik and the number of the anode atoms and ions Na+ia are natural numbers unequal to zero. But it could happen, especially with refractory materials, that either one of them is zero. It means, that it is possible with the great mathematical confidence, that the atoms and ions of one material are absent in the tiny conducting volume of the gap. The Dirac and Heaviside functions are introduced to include such cases into the mathematical model. The used Dirac function is symmetrical, but the Heaviside function is asymmetrical: V^{x<0)=0 (28). While the electron moves along the sufficient path, it achieves the energy to ionize the cathode atom, which has the corresponding voltage drop, derived from Eq. (15), and equals to: U^. = W:, e-N, (29). The corresponding voltage for the ionization of the anode atoms is: = _ " ila e-N.. (30). The voltage drop of the free atoms layer is: _ ■ (S )+1)■ (5 )+1)) (31). The voltage of the partly ionized layer is: ^ ■ (iV,,.,) W,^ ■ (yV,,.„ )) (32). e-N, All atoms are ionized at the far end of the partly ionized layer, looking from the cathode. Then the full ionized layer begins. In this layer, the ions are accelerated toward the cathode by the electric field. But, when they leave this layer, they are recombined, because they are not immune to the collisions from the electrons. From this point on, the recombined atoms move toward the cathode uniformly with the constant velocity, hence there is no force on them due to the electric field, for they are neutral. The velocity of the ions is gained in the fully ionized layer, and the corresponding kinetic energy must be sufficient to heat the cathode to cause the vaporization or the sublimation of the cathode material, and to cause also the thermionic emission of the electrons. The average kinetic energy of the ion, either it is of the cathode or of anode material, is: There is one parameter in the calculation of the cathode temperature - Eq. (24), which has not been fully defined. It is the ionization coefficient of the whole conducting volume. The volume occupied only by the ions is the volume of the fully ionized layer. There are some ions also in the partly ionized layer, but we neglect them when estimating the ionization coefficient. The density of the ions and atoms together is constant throughout the whole conducting volume, so the coefficient is the ratio of the volume of the fully ionized layer and the whole conducting volume. Presuming the crosssection area is uniform the coefficient is the ratio of the thickness of the fully ionized layer and the arc path. Because the electric field is homogeneous the ionization coefficient becomes: a = U,. _ ion U.. (37). So, this equation leads to another loop in the mathematical model - Eq. (24), but the convergence still exists. The results of this mathematical model are shown in Fig. 6. The infimum arc voltage directly depends on the energy of the first ionization of the cathode and the anode material, on the latent heat of vaporization of the cathode material, and on the energy of the work function of the cathode. These quantities are the properties of the contact materials. But, this voltage also depends on the cathode temperature indirectly through the ratio between the number of the free cathode atoms and ions and the number of all free atoms and ions. The whole model is recalculated at the contact current / = 1 A, at the crosssection of the conducting path Asd = 5x10 ® m^, and the coefficients b, u and IJ being unit. W = ionk N, V (fK (33), Because the energy of the work function is: = e ■ (34) the voltage drop of the full ionized layer, derived from Eq. (15), is: e-N, e-v (35). The infimum arc voltage is the sum of the layers voltages -Eqs (31), (32) and (35), and it is: e-N, + (36). e-N^ e-v Fig. 6: The infimum arc voltage with its influential quantities depended on the cathode material against the anode of the same material. Legend: Wi1 k energy of ionization of cathode material, Lvk latent heat of vaporization of cathode material, (|)k energy of work function. Um infimum arc voltage. It has to be emphasized that the cathode temperature depends mainly on the latent heat of vaporization of the cathode and the anode material, their boiling and criticai temperatures, and the external pressure. 3. The comparison of the results The infimum arc voltage is determined not only by the cathode material, but also by the anode material. The diagrams in Fig. 7 show/ the infimum arc voltage depended on the cathode material when facing the anode made by the same material, and the one of tungsten and of carbon. The next diagrams in Fig. 8 present the comparison of the infimum arc voltage achieved by our model and the values of the minimal arc voltages according to Holm, Fink and Gaulrapp /1/, which are considered as experimental values. But, as stated in the literature /1 / and presented in Fig. 8, several researchers got very different results of their experimental determination of the minimal arc voltage, although they are determined by the contact electrodes of the same material. These authors considered that minimal arc voltage is essentially determined only by the cathode material, nevertheless the contact electrodes are made of the different material. That is the significant difference from our model. 2835 K by the model and up to 2273 K by their estimation of the experiment /2/. The comparison between the model results and the experimental results by several researchers/1/ is illustrated in Fig. 8. Having in mind uncertainties of the experimental results, and the fact that the results of the model are presented without any contamination of the electrodes and with no roughness of the contact surfaces, the results are estimated as very good and useful. 4. Conclusions The evaluation of affinity of the contact materials to form the drawn electric arc carried out by defining the infimum arc voltage for several combination of the contact materials is only one aspect of determining the proper contact materials. The second parameter to be taken into account is the cathode temperature, which is also obtainable by this model, especially, when reducing it under the calculated value by heat transfer to the external medium, and so avoiding the arc. Fig. 7: The comparison of the infimum arc voitage of several combinations of materials. Legend: Um infimum arc voltage with cathode and anode of the same material, Um(W) infimum arc voltage with anode of tungsten, Um(C) infimum arc voltage with anode of carbon. Some researcher of the commutation phenomenon /2/,/3/ discovered by their experiments, that there was practically no difference in the infimum arc voltage whether the cathode was made of carbon and the anode of copper, or just the opposite, the cathode was of copper and the anode of carbon. And their observations are confirmed by the model: the cathode against the anode C-Cu 12.5 V, Cu-C 15.6 V by the model, C-Cu or Cu-C from 14 Vto 20 V by their measurements /2/, and C-Cu 20 V and Cu-C 13 V by Holm /1/, and further on, the cathode temperature Fig. 8: The comparison of the infimum arc voltage and minimal arc voltage with the electrodes of the same material according to several researchers /1/. Legend: model infimum arc voltage by our model. Holm minimal arc voltage by Holm 111, Fink minimal arc voltage by Fink /f/, Gaulrapp minimal arc voltage by Gaulrapp /7/. There are many mechanical properties, for instance elasticity, plasticity and hardness and the frictional wear of the materials and the electric properties as it is the contact and the bulk resistance, which must be considered when determining the contact materials for an application. Nevertheless, this model is a good mathematical tool to determine the contact material from the viewpoint of the drawn arc formation. The model is applicable to the switching contacts, and these conclusions are also valid for them. 5. Used symbols a coefficient of ionization ß enhancement factor of electric field intensity 5 Dirac function V decreasing factor of work function rik ratio between number of free cathode atoms and ions and number of all free atoms and ions work function energy in vacuum eo dielectric constant in vacuum jj factor between rest and effective electron mass Aarc cross-section of arc Asel cross-section of path of current conducting by t unnel effect c light velocity in vacuum e elementary charge £ electric field intensity £* enhanced electric field intensity e" exponential function F force h Planck constant I contact current / contact current density /£ current density due to cold emission It current density due to thermionic emission k Boltzmann constant Lvk molar latent heat of vaporization of cathode material Lva molar latent heat of vaporization of anode material Lvbk molar latent heat of vaporization of cathode material at boiling temperature Lvba molar latent heat of vaporization of anode material at boiling temperature /77a rest atom mass rrie rest electron mass me effektive electron mass Na Avogadro number Na-nk number of free cathode atoms and ions Na+ia number of free anode atoms and ions He electron density Pvk saturated vapour pressure of cathode atoms and ions Pva saturated vapour pressure of anode atoms and ions Pvb saturated vapour pressure of boiling point R general gaseous constant s path through layer Tk cathode temperatrure Tbk cathode boiling temperature Tba anode boiling temperature Tek cathode critical temperature Tea anode critical temperature U layer voltage Ua-hi partly ionized layer voltage Uaim free atoms layer voltage Uion fully ionized layer voltage Urn infimum arc voltage v+ asymmetrical Heaviside unit function V particle velocity average electron velocity average electron velocity up to ionization of cathode atom VeoLsi/gaverage electron velocity up to ionization of anode atom Ve_ms effective (root-mean-squere) electron velocity Vfik work function voltage V0k' decreaced work function voltage Vijia mole volume of anode material in liquid state ViJik molar volume of cathode material in liquid state molar volume of anode material in vapour state V^vk molar volume of cathode material in vapour state V\lek electron kinetic energy \Nii molar energy of 1 ionization Wiik molar energy of 1 ionization of cathode atoms \Nna molar energy of 1 ionization of anode atoms References /1/ Holm, R.: Electric contacts, theory and application, fourth completely rewritten edition. Springer - Verlag, Berlin - Heidelberg - New York, 1979. /2/ Rejec, J.: Vpliv l