The Russian Academy of Sciences Siberian Branch A.P. ERSHOV INSTITUTE OF INFORMATICS SYSTEMS S.N. KHARIN MATHEMATICAL MODELS OF PHENOMENA IN ELECTRICAL CONTACTS Novosibirsk 2017 UDС 517.58 LBC22.1 K 42 Kharin S.N. Mathematical models of phenomena in electrical contacts: Мonograph . / A.P. Ershov Institute of Informatics system , Siberian Branch of RAS, 2017. 193 p. ISBN 978-601-06-4797-8 The different methods of solution of mathematical models describing phenomena in electrical contact systems by arcing are presented in the book. All phenomena at contact opening and closure are considered as consecutive stages including the initial stage of Joule’s heating, appearance and explosion of a liquidmetallic bridge, ignition, evolution and extinction of the electrical arc in metallic and gaseous phases. Step by step the mechanism of the arc erosion due to vaporization, as well as the new mechanisms of erosion due to thermo-capillary convection and erosion in solid state due to thermo- elastic stresses are introduced and considered. Mathematical models of arc floating, bouncing and welding, arcto-glow transition have practical applications in design of vacuum circuit breakers and other electrical contact systems. Most of above models are based on the boundary-value problems for the systems of partial differential equations of parabolic and elliptic types in domains with free boundaries, and included the phase transformations such as melting and evaporation. For their solution were elaborated, in particular, the method of majoring functions, methods with using integral error functions, heat polynomials, dual integral equations. Results of the modeling can be used for development of new circuit breakers and other electrical contact systems. Researches, results of which are presented in the book, are supported by the Grant 5133/ГФ4 – «Methods of special functions for the solution of free boundary problems and their applications for modeling of heat and mass transfer at electrical arc phenomena» awarded by the Science Committee of the Ministry of Education and Sciences of Kazakhstan. Established for publishing by Scientific Council Of A.P. Ershov Institute of Informatics system SB RAS ISBN 978-601-06-4797-8 UDС 517.58 © Kharin S.N., 2017 © A.P.Ershov Institute of Informatics Systems SB RAS, 2017 Preface 5 Chapter 1 STATIONARY CONTACTS. 1 THE CONTACT SURFACE 1.1 Modeling of contact surface. Contact resistance. 1.2 Dynamics of contact spot when current passing 2 PHENOMENA IN CLOSED CONTACTS. 2.1 Joule heating 2.2 Thomson effect 2.3 Peltier effect 2.4 Kohler effect 2.5 Galvano-magnetic and thermo-magnetic phenomena 3 MODELING OF CONTACT HEATING 3.1 S Spherical contact model 3.2 Cylindrical model of ideal and non-ideal contacts 4 ELECTROMAGNETIC FIELD IN CONTACTS 4.1 Mathematical model. 4.2 Electric potential and constriction resistance. 4.3 The model of a contact with a short constriction 4.4 Influence of the skin effect. 5 STATIONARY CONJUGATED FIELDS OF THE TEMPERATURE AND ELECTRICAL POTENTIAL 5.1 Ideal symmetric contact. 5.2 Bimetallic contacts 5.3 Heating of ideal contacts with Thomson effect 5.4 Special cases 5.5 The limit of maximal current 5.6 Non-ideal contact 6 NON-STATIONARY TEMPERATURE FIELD 6.1 Transient model of Joule contact heating. 6.2 Special cases. 6.3 The role of the arc flux and Joule heating in the erosion of electrical contacts 7 BOUNCING AND WELDING OF ELECTRICAL CONTACTS 7.1 Static welding. 7.2 Dynamic bouncing and welding during contact closure 7.3 The model of blow-off phenomena CHAPTER 2 MODELING OF THE BRIDGE TRANSFER 1 CYLINDRICAL MODEL OF A BRIDGE 1.1 Introduction 3 6 6 6 9 11 11 12 13 14 16 17 17 17 22 22 24 25 27 30 30 36 39 44 47 51 57 57 59 60 69 69 80 88 103 103 103 1.2 Pre-arcing stage. 2 GENERAL MODEL OF BRIDGE TRANSFER 2.1 Bridge phenomena. 2.2. Mathematical model 2.3 Results of solution. 2.4 The mechanism of bridge rupture. 2.5 Criteria of bridging. 2.6 Self-restrained bridge erosion. CHAPTER 3 ELECTRICAL CONTACT ARC 1 GENERAL MODEL OF ELECTRICAL ARC IN THE FRAME OF MHD-THEORY 1.1 Review of main theoretical results. 1.2 Phenomena in the arc column. 1.3 Phenomena in near-electrode zones. 1.4 Intra-electrode phenomena. 2 MODELS OF ELECTRICAL ARC EROSION 2.1 Introduction. 2.2 Vaporization. 2.3 Liquid droplets erosion 2.4 Thermo-capillary mechanism of arc erosion. 2.5 Solid state type of erosion 2.6 Arc-to-glow transition 2.7 Dependence of the arc erosion on the current frequency 3 ARC DURATION AND CONDITIONS OF ARC INSTABILITY 3.1 Mathematical model of arc temperature and conductivity at metallic arc phase 3.2 Transition from metallic arc phase to gaseous arc phase 3.3 Phenomena in thermo-chemical cathodes 3.4 Arc to glow transition 3.5 Dependence of the arc erosion on the current frequency REFERENCES 4 104 113 113 114 118 120 121 122 124 124 124 124 124 124 124 124 125 132 134 136 138 140 140 141 148 150 154 155 156 Preface One of the main tendencies in the progress of modern low voltage apparatus is the increase of their performance. The demand for creation of apparatus which operate in micro and nanosecond ranges increases more and more. However experimental investigations in this direction are faced with the problem of a short duration of these processes. Besides it is possible often to obtain the required information about the final result of a process but not about its dynamics. In this case the mathematical modeling only is capable to get an idea about evolution of a process in time which should be checked then experimentally. The basic theory for the development of modeling phenomena occurring in electrical contact is presented in various fundamental books The stationary models of contact heating are considered and then generalized taking into account the influence of Thomson and Kohler effects on contact heating. However the demand to use extra-thin films in modern nanotechnology generates the need to develop non-stationary models of contact heating where the Kohler effect plays the main role. Static welding of closed electrical contacts has been investigated in details It was found that the probability of welding is proportional to the current pulse duration. The explanation of this dependence can be given using the model of the ring-shaped welding. The theory of the bridge erosion presented early should be developed for the high velocity of the contact opening. The investigation of transient bridge phenomena and elaboration of non-stationary bridge models are extremely important not only for the calculation of the fine bridge transfer but also for the estimation of the initial contact gap of the arc ignition which defines the duration of the arc metallic phase Modeling of phenomena in the electrical arc is considered in many papers. It should be noted that mathematical models are different for the short and for the long arcs. A model for the long arc should be based on the equations of magnetic gas dynamics describing phenomena in arc column, while the model for a short arc describes processes in ionization zone, sheath and solid domains of anode and cathode . Mathematical modeling is very helpful to make clear dynamics of many arc phenomena: - transition from the metallic arc phase to the gaseous phase]; - various types of arc erosion such as evaporation of material from the contact surface, the ejection of droplets of molten metal due to metallic vapour or magnetic pressure, thermo-capillary mechanism of ejection of liquid metal, ejection of solid particles of contact material when thermoelastic stresses overcome the breaking point; - transformation arc-to-glow discharge]; 5 - the influence of metallic vapours pressure on the blow-off repulsion of closed contacts and on floating or bouncing with final welding at contact closure; - phenomenon of electrochemical electron emission - non-monotonic dependence of arc erosion on the current frequency ]. The special analytical and numerical methods for the solution of these problems are elaborated which are based on integral equations, special functions, majorant functions etc. They can be used for practical calculations. CHAPTER 1. STATIONARY CONTACTS 1.1 THE CONTACT SURFACE 1.1 Modeling of contact surface. Contact resistance. Electrical contact between two current currying elements is formed as a result of compressed mechanical force Pc called contact load. One of main important characteristics of closed electrical contacts is the contact area between them. It is well-known that whole geometrical surface of a contact is so-called apparent (or nominal) contact area, while the true contact area occurs only on separate spots, which total area are much less than apparent area. Microscopic asperities, elevations and depressions on a surface with the deep depressions remaining as voids in the contact stipulate it. Those usually much spread voids have a negligible influence on the contact resistance. When two surfaces are brought together, mechanical contact occurs only in those regions where the surface asperities meet. Most metal surfaces are covered with an oxide film, which must be ruptured before true metal-metal contact takes place. As the force increases, the number and the area of these small metal-metal contact spots will increase as a result of oxide film rupture and metal extrusion through the ruptures. These spots, termed a -spots, are small cold welds providing the only conducting paths for the transfer of electrical current. Current passing across a contact interface is therefore constricted to flow through these a -spots. Hence, electrical resistance of the contact due to this constricted flow of current is called "constriction resistance" and is related to basic properties of metal such as hardness and electrical resistivity. The current, constricted to flow through narrow conducting path, will raise the temperature of these paths Ts above that of bulk Tb . The contact spot temperature (so-called "super temperature”) is related to the voltage drop across the contact interface U as [1] Ts 2 Tb 2 6 U2 , 4L (1.1.1) where L is the Wiedemann - Franz - Lorenz constant with the value 2.45 108V K 1 . Hence, a relatively small increasing in the contact voltage drop U can considerably raise the contact spot temperature sufficiently to produce basic metallurgical changes such as softening and melting of the conducting areas. The total contact resistance R of an interface where there are many conducting spots is given by R. Holm [1] as R 2na , 2 (1.1.2) where the first term refers to the constriction resistance of n spots with an average radius a , while the second term is additional resistance due to interaction between spots forming a cluster, calculated theoretically by J.A. Greenwood [2] as 1 aa ai i j , 2 i i , j sij 2 (1.1.3) where ai and a j are the radii of individual spots and sij is the distance between them. In order to account for long-term degradation of stationary contacts, R. Malucci [3] developed a model by introducing a third level of multi-spot constrictions on the surface of each asperity. This model was used to simulate contact degradation by fretting. It was shown that asperity deformation and film growth are important variables in delaying fretting degradation. The model was also used to provide statistical data on change in resistance as contact degrades. However, the uncertainty of the parameter determining the fraction of the oxidized contact region limits wider application of this model. The constriction resistance and distribution of the asperities in the contact zone are best described by the model of J.A. Greenwood and J.B.P. Williamson [4]. This model relates the contact deformation to the topography of the surface and establishes the criterion for distinguishing surfaces undergoing elastic or plastic deformation. Based on this approach M.D. Bryant [5] used polynomial and Gaussian probability density function to describe the degradation of copper contact by oxidation. Using fractal surface modeling and applying the analogy between thermal and electrical constriction, M.T. Singer and K. Kshonze [6] developed a model for the constriction resistance of random rough contacting surfaces. This model correlates well with that of J.A. Greenwood and J.B.P. Williamson. The analogy between thermal and electrical contact resistance was also used by R.A. Burton and R.G. Burton [7] to describe the co-operative interaction of asperities in sliding contacts. It is worth mentioned that computer modeling, based on finite element analysis, was also used to determine the contact resistance [8], [9]. However, 7 these methods, although very useful for solving particular practical problems, are inadequate to describe the dynamic processes occurring in the contact zone. Bearing in mind the above discussion, we shall keep the approach given in [4] - [5], using, however a new modification. Hence, the total contact resistance can be expressed as (1.1.4) R Rc Rm R f , where Rc , Rm , R f are respectively the constriction resistance, the resistance of asperities and the resistance of a film in a quasi-metallic part of a contact spot. These resistances can be determined in terms of electrical conductivities c , m , f as Rc 1 1 1 , Rm , Rf , c S m S fS (1.1.5) is the nominal contact area. The total loud Pc acting on a unit area of the contact surface is calculated in the Appendix 1: where S z z2 4 NE rm 3/ 2 Pc ( z d ) exp 2 2 dz 3 (1 2 ) d 3/ 2 2 rm 3/ 2 4 NE (i erfc i 3/2 erfc * ) . 3 (1 ) 2 (1.1.6) Hence, knowing the applied load Pc , the values for , and d can be derived from this transcendental equation. The above-described model was used to calculate the initial contact resistance of an aluminum-copper bolted connection with a nominal contact area of S 33,33 106 m2 . The contact load was Pc 16 103 N . The surfaces of aluminum and copper bus bars were first machined and then brushed with a brush having a density of 1 106 wires m2 [11]. Typical surface profile and height distribution is shown in Fig. 1-1. 40 20 0 -20 0 -40 0,5 1 1,5 2 -60 -80 -100 Fig. 1.1. Typical surface profile of a brushed surface.Vert. scale = 25.786 A , Horiz. scale unit = 9.960 m . 8 The asperity heights are measured from an arbitrary reference plane. The results of topographic analysis of several surface profiles yielded the following values for r0 5.8 105 m , N 16 , 1.6 105 m , z 2 105 m . By inserting these values in equation (1.1.18) and solving it with respect to , we obtain 0.6 10-5 m , and d 1.4 105 m . Using these values and equations (1.1.10), (1.1.13), (1.1.15) and (1.1.5), the following values for the constriction resistance Rc , asperity resistance Rm and film resistance R f are obtained R f 4.5 106 . Rc 2.3 106 , Rm 0.1 106 , The total contact resistance is then R 6.9 106 . This value agrees well with the experimentally observed initial contact resistance of a bolted aluminum-copper connection before being subjected to current-cycling [11]. 1.2 Dynamics of a contact spot when current passing The expressions for the contact resistance obtained above correspond to the elastic mechanism of deformations, and they can be used in a range of relatively small contact loading, current and temperature. In the case of plastic or elasticplastic deformations the size of a contact spot is defined using the concept of the contact hardness introduced by R. Holm [1]. It is equal numerically to the average pressure in contact asperities (filaments), that is approximately three times more then the pressure corresponding to the beginning of the plastic deformation. It is necessary to take into account also decreasing of contact hardness when the temperature of a contact surface increases when current is passing. In this case the formula (1.1.6) must be replaced by the expression Pc ( n ) n H B ( )dS , (1.1.7) Sn where Pc is the contact load applied to a single contact spot Sn , and n is the coefficient of compressibility in a neighborhood of a spot. This coefficient takes into account the degree of a contact surface treatment , which is responsible for its elastic-plastic properties. For the plastic deformation n 1 , while for elasticplastic deformation 0.3 n 1 . If the current is absent, then the summary value of the elastic-plastic forces Pep , which is balanced by the total contact load Pc , can be found by summation of (1.1.20) with respect to n : (n) m Pep n H B ( )dS , n 1 where m is the number of contact spots. 9 Sn (1.1.8) It will be shown below, that a contact spot S n is an equipotential and isothermal surface for stationary phenomena, therefore one can derive from (1.1.7) H B ( )S ( ) const . (1.1.9) This relationship agrees well with experimental data. Fig. 1-2 shows the temperature dependence of the hardness for copper, silver and silver-nickel all H B , 10 7 N / m 2 100 1 80 3 60 40 2 20 0 0 200 400 600 800 , 0C Fig. 1.2 Temperature dependency of hardness : 1 Cu, 2 Ag, 3 AgNi (0.1%Ni ) One can see that a hardness falls sharply just on reaching of the softening temperature s (about 200o C ). It decreases two-three times in the temperature interval 200o C 500o C . Accordingly to the formula (1.1.9) the increasing of the radius of the contact spot r at the temperature 500o C in comparison with its value at the temperature 200o C is equal to the ratio r (500) r (200) H B (200) . H B (500) For copper 1.4 , for silver 1.65 , and for silver-nickel alloy 1.45 , that agrees well with the experimental data obtained by N.E. Lyssov [12], who found that this ratio is equal to 1.5 for all these materials. Increasing of the size of a contact spot in the process of heating is accompanied by a rapprochement of contacts until elastic-plastic force is compensated again with the contact load Pc . This motion is described by the equation d 2x Pep Pc , dt 2 where m is the mass of moving contact, and x x(t ) is the coordinate of a point m hard placed on it (Fig. 1.2). The current area of the spot S (t ) r02 (t ) can be expressed in term of x(t ) by the formula S (t ) S0 [1 2 r0 ( x x0 )], S0 S0 S (0) 2 r0 ( z d ), 10 x0 x(0) . (1.1.10) I the range of a high current ( 103 104 A ) it is necessary to take into account the electrodynamic force stipulated by interaction of electric and magnetic fields in the current constriction region : 1 Ped 0 E HdV , (1.1.11) D where is the relative magnetic permeability, 0 1.257 106 H / m is the magnetic permeability of vacuum, is electric resistivity of the contact material, E and H are vectors of electric and magnetic fields, D is the volume of the contact. With due regard for this phenomena the equation of the contact motion can be written in the form: m d 2x P Pep Pc , dt 2 (1.1.12) where P is the component of electrodynamic force normal to the contact surface. Beside phenomena of contact material’s softening and electrodynamic force other factors can influence on the change of a contact spot’s size, such as fritting and additional repulsion force stipulated by explosive evaporation of contact material at very high temperature on the contact spot. These phenomena will be considered below. 1.2 PHENOMENA IN CLOSED CONTACTS 2.1 Joule heating Heating of an electrical contact during passing current is a result of interaction of multifarious phenomena. The most important factor of a contact heating is undoubtedly the Joule sources in the constriction region. Specific output q produced by these sources in the unit volume is defined by the formula q 1 E2 (1.2.1) The vector of electric field strength E can be expressed in terms of electric potential and current density j using expression E grad j (1.2.2) Side by side with Joule sources thermoelectric effects of Thomson, Peltier and Kohler stipulated by the relationship between electric and electromagnetic fields can influence on the contact heating. It is concluded in numerous papers devoted to these effects that their contribution in contact heating either negligible [13] or can be important only for low current liquid contact bridges [1], [14], [15]. It will be shown below that such conclusion made on the base of rough estimations is not quite correct. The definition of the conditions when 11 either effect is essential can be obtained only in the frame of an adequate mathematical model. 2.2 Thomson effect Thomson effect becomes apparent in the case of uneven heated electrical conductor. It consists in the additional heat transfer by electrical carriers from more heated to less heated regions. This effect is positive if heat generation occurs in a conductor, when the directions of current and heat flow are the same, and negative when they are different. In other words the positive Thomson effect acts as though heat transfer is proceed by the positive current carriers. Otherwise the effect is considered as negative. The estimation of Thomson effect is often based on the assumption that the total quantity of the Thomson heat QT which was produced in a conductor with the passing current I during the time t is proportional to the difference of the temperatures at the ends of a conductor , if this difference is not too large [16]: QT T I t . (1.2.3) The sign of the Thomson coefficient T coincides with the sign of the effect. Experimental data show that the value of the Thomson coefficient T depends on the temperature, and this dependence is linear for most of metals. The temperature dependence of T is presented in Fig. 1.3 [1]. T , 10 6 V / K 40 20 1 2 3 0 -20 4 -40 -60 5 6 -80 -100 7 0 2 1 3 , 10 3 0 K Fig. 1-3 The temperature dependence of the Thomson coefficient T [1]. 1 – Ag, 2 – Cu, 3 – Au, 4 – Mo, 5 – Pt, 6 – W, 7 – Pd Applying this law to the element l of a conductor and dividing both sides of (1.2.3) by the value of elementary volume S l , where S is the area of the cross-section of a conductor, one can obtain the formula qT T j, l (1.2.4) where qT is the specific power produced by the Thomson heat sources per unit volume, and j is the current density. 12 Generalization of the formula (1.2.4) leads to the conclusion that qT is proportional to the scalar product of the temperature gradient and electric potential gradient qT T grad grad T E grad . (1.2.5) The difference between the formula (1.2.4), which is usually applied for an estimation of the Thomson effect, and the exact formula (1.2.5) is evident. Comparing the average value of the Thomson component of the electric field for the whole constriction region ET T with the ohmic component E j one l can find that ET E for all real range of considered parameters. Thus according to the formula (1.2.4) the Thomson heating component should be negligible in comparison with the Joule heating component. On the contrary the formula (1.2.5) shows that small local zones may exist, where Thomson heating is essential, if the temperature gradient into these zones is sufficiently large. This situation is typical for non-stationary phenomena in the case, when a power of high intensity is produced inside a small surface zone during very short time, especially for materials with large Thomson coefficient. It will be shown in the Chapter 4 that the Thomson heating may surpass the Joule heating at electrical arcing phenomena between tungsten contacts for a current about 100 A. 2.3 Peltier effect Peltier effect lies in the fact, that in the case of an electrical contact between two different materials special heating or cooling takes place in addition to the Joule heating. This thermo-electric effect can be explained by the contact voltage at the interface between two different metals that decreases or increases the average power of electrons crossing the contact. Contrary to the Joule and Thomson heat sources, which act into a volume, the Peltier sources are surface. Their specific power per unit area of the contact spot P is proportional to the current density j P uP j . (1.2.6) The Peltier coefficient u P is positive for heating and negative for cooling produced by Peltier sources. Similar to Thomson coefficient the Peltier coefficient depends linearly on the temperature u P k S . The coefficient of proportionality k S (Seebeck coefficient) is defined by the expression kS k n1 ln . e n2 13 where k is the Boltzmann constant, e is the electron charge, n1 and n2 are the number of free electrons per the unit volume of each contact metal. The values of Seebeck coefficients for some contact pairs are given in the Table 1.1. Table 1.1 Seebeck coefficient k S ( 10 3V K 1 ) for some contact pairs [17] Metal Contact pair with Cu Fe M Mo 4.4 6.5 W 0.4 11.0 Zn Cu 0.1 0 11.0 11.4 Ir 1.1 12.0 Ta 2.6 14.5 Ni Al 24.0 3.2 34.0 - g 3.5 - 2.4 Kohler effect The tunnel effect plays an important role in many fields of engineering applications and modern technologies such as electric vacuum devices, semiconducting materials, super-conducting contacts, technology of thin films etc. Various problems of tunnel conductivity, magnitude of tunnel current, tunnel voltage across a contact film as well as properties of varied films have been investigated and much progress made in the understanding of these characteristics. The problem of contact heating owing to tunnel effect is one of them, and it was M. Kohler who discovered theoretically that an analogue to the well-known Wiedemann - Franz' law holds for the electric and thermal currents through the film [1]. It enables to estimate a possible temperature difference between cathode and anode. Now this phenomenon of electrical contact overheating owing to tunnel effect is known in the literature as Kohler effect. Let us consider a circular contact spot between cathode and anode covered with a thin (a few A ) chemisorbed or adhesive films that are penetrated by the conduction electrons by means of tunnel effect. The electrons don't alter their energy level when tunneling. Since they land in anode with a lower negative potential than at the cathode, they have a surplus kinetic energy there. The increment of kinetic energy is given off as heat creating the source on the interface between anode and film with specific capacity j u f j 2 f , (1.2.7) where j is current density in contact spot, u f is the tunnel voltage across the film, f is the electric tunnel resistivity per unit area of the film. A portion of liberated heat with specific capacity c f , W (1.2.8) where f is the temperature difference and W is the specific thermal resistance across the film, flows back to the cathode through the film whereas the remainder 14 a f W (1.2.9) flows into the anode (Fig. 1.4) . 1 (r , z 0, ) r 2 (r , z 0, ) d Cathode D1 : z 0 Anode D2 : 0 z f a c z 0 f r Fig. 1.4. Cathode and anode heat flux ' across the contact film Overheating of anode by tunnelling electrons is called Kohler effect. The magnitude of tunnel resistance f depends on the contact materials, particularly on work function , radius of contact spot f and thickness of the film d . Specific thermal resistance of the film, W , is given by W d f , (1.2.10) where f is the significant parameter of Kohler effect . However this formula is of little use for practical calculations because of very poor information concerning thermal conductivity of the film f . But combining Kohler and Wiedemann - Franz' laws f f LT , where f and are electrical resistivity of the film and the metal respectively, is the thermal conductivity of the contact metal, and using the relation f f d we can obtain instead of (4) the expression W f , (1.2.11) that is much more efficient. Further investigations carried out by R.Holm concerning the influence of thin films on contact superconductivity [1] as well as results of other investigators (I.Dietrich , P. Kisliuk, W. Meissner et al. ) led to the conclusion that the contribution of tunnel effect to contact heating is visible only at low current and small contact loading. It has to be noted that such estimations are based on the integral balance of heat transfer between anode and cathode. So they are rather rough and may be used successfully at the limited range of current. 15 The modern tendency to use extra-low range of the current (about 10 6 A and less) in many fields of engineering leads to the situation when investigators are faced with new serious problems and phenomena that remain obscured at ordinary current range but come to the forefront if the range is extra-low. The Kohler effect is one of them. The overheating of anode in comparison with cathode due to Kohler effect, which is not very important at ordinary conditions, is essential at extra-low current, as it will be shown below. It leads to the thermal asymmetry in the microscopic molten bridge that appears between two electrodes during their opening just before arc ignition. If cathode and anode are of the same material the point of the bridge with maximum temperature is displaced toward anode, and when the temperature rises to the boiling point the bridge breaks at this point, and molten metal transfers from anode to cathode. After certain cycles of opening and closing operations one can see the formation of very thin pips (or spires) on the cathode and craters on the anode. This phenomenon, called bridge erosion, is very dangerous for micro-relays. Reducing of the current diminishes the volumes of the spires but not their height. Even the very thin spires are able to make sensitive micro-relays unreliable. The hypothesis that Kohler effect may be responsible for the bridge erosion of electrical contacts was proposed by E. Justi and H. Schultz [18]. New aspects of this theory and corresponding mathematical models are developed in the papers [19] - [20]. and some methods for solution of the problem ( mainly numerical ) are elaborated. Mathematical modeling is very important for the understanding and calculation of relative contributions of Joule and Kohler components of contact heating and bridge erosion because experimental study of this phenomena is very difficult, sometimes on account of microscopic size of a bridge ( 10 6 10 8 m ) at extra-low current. 2.5 Galvano-magnetic and thermo-magnetic phenomena If electrical contacts are placed in a sufficiently strong external magnetic field H , then following galvano-magnetic and thermo-magnetic phenomena should be estimated. Hall and Nernst effects. These effects create additional transverse electrical fields Ei H R0 Ei H , i N Ei Qi 0 grad i H , where R and Qi are Hall and Nernst constants. Ettingshausen and Righi- Leduc effects. 16 (1.2.12) (1.2.13) These effects cause additional temperature gradients in transverse direction grad E i Pi 0 Ei H , i R grad i S 0 H grad i , (1.2.14) (1.2.15) where Pi and S are Ettingshausen and Righi-Leduc constants. In addition electrical contact resistance may be changed also. The problem appears now: is it possible that tangible galvano-magnetic and thermo-magnetic phenomena can be emerged due to self-magnetic field creating by current passing through contact? This problem with regard to electrical arc phenomena has been discussed in some papers, for example, in [21] and [13]. It was found that considered effects should be taken into account for medium containing addition of bismuth or antimony. However self-magnetic field of closed electrical contacts creating by passing current is not sufficient to provoke a visible influence of considered effects on the process of heat transfer. It can be explained by convincing circumstance that for electric contact materials Hall constant R is multiple of 1010 m3 / A sec , Nernst constant Qi is multiple of 108 m2 / sec K , Ettingshausen constant Pi is multiple of 108 K m2 / V A sec and Righi-Leduc constant is multiple of 103 m2 / V sec [22]. Electrical contact resistance can be appreciably changed in strong magnetic fields only which have a strength of several tens or hundreds oersted [22]. Thus the influence of galvano-magnetic and thermo-magnetic phenomena on the heating of closed electrical contacts can be neglected. 3. MODELING OF CONTACT HEATING 3.1 Spherical contact model See Appendix 2. 3.2 Cylindrical model of ideal and non-ideal contacts The mathematical equations describing non-stationary heat transfer phenomena in electrical contacts have to be considered taking into account all above mentioned effects. Owing to Joule and Thomson heat sources acting inside the volume of electrodes, which are described by the formulas (1.2.1), (1.2.5), the heat equation for each electrode can be written in the form ci i i 1 div(i grad i ) Ti Ei grad i Ei 2 , t i i (1.3.1) where index i 1 is used for the cathode, which occupies the region D1 bounded by the surface S1 , index i 2 corresponds to the anode occupying the region D2 17 with the boundary S 2 , ci is the thermal capacity, i is the density, i is the thermal conductivity, i is the electrical resistivity, Ti is the Thomson coefficient, i is the temperature, Ei is the electrical field, and t is the time. If the Thomson effect is negligible, then simplified equation can be used instead of (1.3.1): ci i i 1 div(i grad i ) Ei 2 . t i (1.3.2) The system of Maxwell’s equations for electric field Ei and magnetic field H i has to be added to the equation (1.3.1) or (1.3.2): curl Ei 0 i curl H i 1 i H i , t Ei , (1.3.3) (1.3.4) div H i 0 . (1.3.5) The equations (1.3.1) – (1.3.5) are a bounded system because i depends on the temperature i . The temperature dependence of magnetic permeability i is very small for diamagnetic and paramagnetic contact materials. The electrical field Ei can be eliminated from the system (1.3.2) – (1.3.3), and it converts into the form: ci i i div(i grad i ) i curl 2 H i , t H i 0 i curl( i curl H i ) . t Sometimes it is more convenient to introduce the electrical potential i instead of Ei using formula (1.2.2). Applying the divergence operator to the equation (1.2.12) we obtain instead of (1.3.1) – (1.3.4) the following system of the equations for variables i and i : ci i i 1 div(i grad i ) Ti grad i grad i grad 2 i , t i i 1 div grad i 0 . i (1.3.6) (1.3.7) This general system of equations can be concretized and simplified in dependence on the geometry of a contact spot. If we assume that the contact spot is a circle with the radius r0 , then the axial symmetry takes place, and use of cylindrical coordinates is most convenient. In this case the equations (1.2.14) and (1.2.15) take the form: 18 ci i i 1 i i Ti i i i i r i + t r r r z z i r r z z 2 1 i i 2 ( ) , + i r z 1 1 i 1 i r 0. r r i r z i z (1.3.8) (1.3.9) In the case of along constriction [1] characterized by the inequality f R0 , where R0 is the radius of the contact cross-section, electrodes can be considered as two half-spaces. The cathode region (i 1) is D1 ( z 0, 0 r ) , the anode region (i 2) is D2 (0 z , 0 r ) , and the contact spot is S ( z 0, 0 r r0 ) (Fig. 1.6). For a quasi-metallic contact with a film of very small thickness d (d r0 ) we can apply the model of non-ideal heat contact through the spot S with the temperature difference f across the film like the difference between left-hand limit for 2 (r, z, t ) and right-hand limit for 1 (r, z, t ) as z 0 and 0 r r0 : f 2 (r,0, t ) 1 (r,0, t ) . 1 (r , z 0, t ) Cathod D1 : e z r 2 (r , z 0, t ) Anod d 0 D2 e: 0 z r0 a c z 0 r0 r Fig. 1.6 The axial section of a contact with a film Let us formulate now the boundary conditions. If i is the supertemperature of an electrode, that means the difference between the electrode temperature and surroundings temperature, the at the initial time t 0 (the moment of current switching) we get i t 0 0 . (1.3.10) More general initial condition i t 0 F ( M i ) , M i Di may be used for consecutive stages of contact heating during commutations. 19 Due to Kohler and Peltier effects the surface heat source appears on the anode side of the film with specific power f P (1.3.11) where components on the right side are defined by the expressions (1.2.6) and (1.2.8). A portion of this heat with specific power c f Wf , (1.3.12) where W f is the specific thermal resistance, flows back to the cathode through the film whereas the reminder a f (1.3.13) Wf enters the anode. Thus the boundary conditions for the temperature on the contact spot are z 0, 1 0 r r0 , 2 1 ( r, 0, t ) 2 ( r, 0, t ) 1 ( r, 0, t ) , z Wf 2 ( r, 0, t ) ( r, 0, t ) 1( r, 0, t) . 2 z Wf (1.3.14) (1.3.15) The boundary conditions for the potential at the same place are z 0, 2 ( r, 0) 1( r, 0) u f uP , 1 1 ( r, 0) 1 2 ( r, 0) . 1 (1 ) z 2 (2 ) z 0 r r0 (1.3.16) (1.3.17) The condition (1.2.22) describes the potential difference across the film, while (1.2.23) corresponds to the continuity of the current density passing through the contact spot. There is no heat and current transfer outside the contact spot, hence the conditions of thermal and electrical insulation must be written z 0, 1 ( r, 0, t ) 2 ( r, 0, t ) 0, 0 z z 1 ( r, 0) 2 ( r, 0) 0, 0. z z r0 r (1.3.18) (1.3.19) Finally it has to be assumed that far from the contact spot z or 1 0, r 20 2 0 (1.3.20) 1 uc , 2 2 uc . 2 (1.3.21) As a rule u f uP uc , therefore significance of Kohler and Peltier thermoelectric effects as volumetric heat sources is negligible in comparison with Joule heating, however, Kohler effect as a surface heat source may be very important. Sometimes this effect is responsible for for tunnel overheating and thermal asymmetry of the constriction zone. Specific thermal resistance of the film Wf d f , (1.3.23) where f is the thermal conductivity of the film, is the significant parameter of Kohler effect. However the formula (1.2.28) is of little use for practical calculations because of very poor information concerning f . But combining Kohler and Wiedemann-Franz laws f f LT , where f is electrical resistivity of the film, and are thermal conductivity and electrical resistivity of the electrode material, on which surface film is formed, and using the relation f f d we can obtain instead of (1.2.28) the expression Wf f , (1.3.24) that is much more convenient for practical calculations. If we want to take into account Thomson effect not only in the metallic constriction region, but inside the film itself, considering the film as a conductor with the length d and the cross-section S f 2 , then we have to add the special heat flux due to Thomson effect T T j (2 1 ) z 0, 0 r r0 additionally to the main heat flux c entering the cathode due to heat conduction. It seems as though Thomson effect increases the heat conductivity of the film up to the value Ti j . Thus to take into account Thomson effect inside the film we have to introduce the concept of equivalent heat resistivity Wf 1 / f T j (1.3.25) instead of the previous expression (1.2.29). It has to be noted that this correction is very small as a rule. Al observed thermoelectric effects are significant mainly for low current contacts ( I 10 A) , when the radius of a contact spot is small (r0 10 5 m) because 21 of a small contact load. Particularly this case is typical for liquid metal bridges appearing during contact separation. The same thermoelectric effects occur on the interface between liquid bridge and solid electrode as on the interface between two different metals. The boundary conditions (1.2.20), (1.2.21) show that anode temperature and cathode temperature are different even material of both electrodes is the same, if there exists a film exiting tunnel effect between contacts. Thus above formulated heat conduction problem is always asymmetrical about contact plane. It is very important for the substantiation thermal theory of bridge erosion which will be considered in the Chapter 3. 4 ELECTROMAGNETIC FIELD IN CONTACT S 4.1 Mathematical model Mathematical models describing electromagnetic field in electrical contacts are based as a rule on the hypothesis of constant current density in a contact spot. It enables to reduce the problem of determination of components of electric field to the solution of the Neumann’s problem for the Laplace equation which can be found easily [15]. Such a model is approximate and can be used for determination of contact resistance or average temperature in the contact constriction domain only. However essential local overheating of periphery of contact spot experimentally observed at high current at welding can be explained only by nonuniformity of current density in contact spot. It is stipulated by three factors, namely: 1) due to nature of current constriction, 2) due to influence of cross-section of radius of conductor, 3) due to skin effect in the case of alternative current. To estimate each of above factors we consider first the case when two R, semi-cylinders of the radius and D1 (0 r R, z 0) D2 (0 r R, 0 z ) contact each other by a current-carrying circle spot 2 r 2 z 2 r0 . The axisymmetric vector of electrical field E has two components, radial Er and axial Ez , while the vector of magnetic field H has the only angular component H . It follows from the Maxwell equations (1.2.11)- (1.2.13) that H satisfies the equation H t (i ) (i ) 1 H i (i ) (rH ) . i 0 z z r r r (1.4.1) Index i 1 relates to cathode and index i 2 relates to anode. The boundary conditions can be written in the form H (r , z,0) 0 , (i ) 22 (1.4.2) H (i ) 0, z (1.4.3) z 0 0 r r0 H (i ) H (i ) z 0 r0 r R r H z I 2 r , (1.4.4) 0, (1.4.5) 0. (1.4.6) r R (i ) z The conditions (1.4.2) - (1.4.6) are evident. In particular the condition (1.4.4) describes Ampere's circuital law. The electrical field can be derived from magnetic field using Maxwell equations: I 2 i qi ( r, z, r0 ) i ( jz jr ) [2(2 2 s)( r02 2 )]1 2 2 2 or qi ( r, z, r0 ) I 2 i q0 ( r, z, r0 ) , 16 2 r0 2 (1.4.7) where 1 z 2 ( r r0 )2 z 2 ( r r0 )2 (1.4.8) q0 ( r, z, r0 ) 2 2 . r z ( r r0 )2 z 2 ( r r0 )2 Near by the axis r 0 another representation of q0 (r, z, r0 ) is more convenient q0 ( r, z, r0 ) 16r02 1 z 2 ( r r0 )2 z 2 ( r r0 )2 [ z 2 ( r r0 )2 z 2 ( r r0 )2 . (1.4.9) It follows from the above expressions that the maximal current density and power of Joule sources are placed on the edge of a contact spot z 0 , r r0 , where they have an integrable singularity. Such contact model enable to explain the phenomena of ring-shaped welding at high currents, while a model based on a priori given constant current density in a contact spot is useless in this case. The components of current density on the contact plane z 0 are determined by the expressions jr ( r,0, r0 ) 0 , jz ( r,0, r0 ) I 2 r0 r02 r 2 , if r r0 . (1.4.10) They can be derived from the relations 2 2 r r0 , lim z 0 0, 23 if r r0 if r r0 (1.4.11) 0, r lim , z 0 r r2 r 2 0 if r r0 if r r0 r0 2 2 , r0 r lim z 0 z 0, if r r0 if r r0 Electrical field in the constriction region corresponding to the cylindrical model is presented in Fig.1.8. Accordingly to well-known relation [1] the electrical field and the temperature field for symmetric contacts in the stationary regime are the same. The circle contact spot of radius f is an infinitely thin disk conducting a current and a heat flow into a half-space electrode. In contrast to the spherical model equipotential and isothermal surfaces (1) are not spheres, but ellipsoids defined by the equation (1.4.14), while electrical and temperature gradient lines form a family of confocal paraboloids of revolution. z 2 1 r r0 0 r0 r Fig. 1.8 Electrical (temperature) field in the contact constriction region. 1 – equipotential (isothermal) surfaces; 2 – electrical (thermal) current lines 4.2 Electric potential and constriction resistance To get more detail information about phenomena in closed contacts it is useful to find also the distribution of electrical potential i ( r , z ) and constriction resistance Rc . Taking into account the conditions (1.2.27) one can get 1 ( r, z ) 2 ( r, z ) z z uc u I 1 u I 1 Ez (1)dz c 1 2 dz c arctan , 1.4.12) 2 2 2 2 r0 z 2 2 r0 2 r0 uc u I 1 u I 2 Ez (2)dz c 2 2 dz c arctan 2 f 2 z 2 2 z r0 z 2 2 f 2 The constriction resistance of a half-space contact is formula [22] : Rc ( i ) 2 ( Er ( i ) H ) rdr . z 0 I 2 0 From the relations (1.4.16), (1.4.22) one can derive 24 . (1.4.13) defined by the Er (i ) z 0 0 if I i , if 2 r r 2 r 2 0 I i , if r r0 , 2 r0 r0 2 r 2 0, if r r0 r r0 r r0 , Ez (i ) z 0 (1.4.14) therefore using (1.4.4) Rc (i ) i dr i . 2 2 2 r r r r0 4r0 0 Constriction resistance of whole contact is Rc Rc (1) Rc (2) 1 2 4r0 . (1.4.15) For homogeneous contacts 1 2 and Rc 2r0 . (1.4.16) This formula enables to get the relation between current and contact voltage and to express a potential in any point in their terms. Using conditions (1.2.22), (1.2.23) and expressions (1.4.23), (1.4.24) we get uc u f uP uc i ( r, z ) ( 1)i 2 I ( 1 2 ) , 4r0 (1.4.17) (uc u f uP ) arctan , i 1,2 . 1 2 r0 2 2 i (1.4.18) It has to be noted that if contacts are homogeneous ( 1 2 ) and thermoelectric effects are negligible ( u f uP 0 ), then formula (1.4.29) in this special case coincides with well-known expression for the contact potential [1] i (r , z ) (1) i uc arctan r0 . (1.4.19) Finally we note that the expressions (1.4.17), (1.4.18) enables to conclude about identity of current density in each contact Di , while specific power of Joule sources qi is different and it is greater into electrode with greater value of i . Kohler and Peltier phenomena don’t influence on the symmetrical expression for qi about the contact plane z 0 . 25 4.3 The model of a contact with a short constriction Let us consider now the case of a short constriction, when the cross-section radius of cylindrical contacting members R0 can not be negligible, and a constriction is limited to a distance comparable with the radius of a spot f (Fig. 1.9). R 2 r0 R0 r0 0 2 r0 R0 Fig. 1.9 The model of a contact with a short constriction. We can accept as a first approach that the expressions (1.4.7), (1.4.8), (1.4.13), (1.4.14) describe the electromagnetic field only inside the region bounded by the contact plane z 0 , 0 r R0 and based on it ellipsoid r2 z2 1 . Outside this constriction region the magnetic field is radial, R0 2 R0 2 r0 2 and the electric field is axial. To check how well such model is working, let us compare the expression for the axial component of electrodynamic repulsion force P calculated by use of this model with the exact expression presented in [1]: P 0 I 2 R0 ln . 4 r0 (1.4.20) Formula (1.1.24) applying to above model gives that coincides with (1.4.31) If it requires more detail investigation of the influence of R0 on the distortion of electromagnetic field, then one has to consider the equation (1.4.9) with the conditions (1.4.3) – (1.4.6) in a bounded cylinder. Herewith the magnetic field can be presented in the form of the Fourier-Bessel series: 26 H ( r , z ) I 2 C n 1 n z r exp n J 1 n , r0 r0 (1.4.21) where n are the roots of the equation J1 (n ) 0 , and Cn r0 , R0 can be found from the solution of the dual series equations: C J n 1 n 1 n r 1 , r0 r C J n 1 n n 1 n r 0, r0 if r0 r R0 , if 0 r r0 . (1.4.22) Dual series equations of such type is considered in the paper [23]. Using these results one can conclude that the coefficients C n can be determined by the formula Cn 2 2 2 J 3/ 2 (n ) n 1 1 4 (1.368 0.216 2 ) (1.4.23) with the error not exceeding . If R0 , then the series (1.4.32) can be summarized and reduced to the expression (1.4.13). 5 4.4 Influence of the skin effect Finally we consider the case of alternating periodic current J (t ) J 0 exp(it ) . Let us assume for simplicity that contacts are homogeneous and symmetrical, const , R0 .The equation (1.4.1) is reduced in this case to the form 2 H 1 2 rH , t r r r z H where 2 0 2 (1.4.24) . If a periodical current passes through contacts over a sufficiently long period of time, then the equation (1.4.1*) has to be considered without the initial condition (1.4.2) (it is assumed that t ), only for the boundary conditions (1.4.3) – (1.4.6) with R0 . The solution of this problem can be found in the form H ( r , z , t ) J0 exp(it ) exp( 2 k 2 J 1 ( r ) ( )d , 2 0 27 (1.4.25) where k 2 i 2 , and ( ) is unknown function determined from the conditions (1.4.3), (1.4.4) , that gives the dual integral equations 2 k 2 J 1 ( r ) ( )d 0 , if 0 r r0 , (1.4.26) 0 1 J ( r ) ( )d r , if r0 r . 1 (1.4.27) 0 If we represent a solution of the equations (1.4.36), (1.4.37) in the form sin t cos t dt , t f ( ) J 0 ( f ) (t ) 0 (1.4.28) then the equation (1.4.37) is satisfied for any (t ) due to the well-known property of discontinuity of Weber’s integral. Replacing ( ) in the equation (1.4.36) by the expression (1.4.38) and using the formula r 2 J 1 ( r ) d 2 [r J 2 ( r )] dr we get d 2 k2 sin t r ( r ) r 2 J 2 ( r ) d (t ) cos t dt , dr 0 t 0 f 2 where (r) 0 2 k2 J 1 ( r ) d . Integrating both sides with respect to r from 0 to r , then interchanging the procedure of integration and calculating the inner integral, we obtain the equation r t 2 (t )dt r2 t2 0 f r r ( r )dr K0 ( r, t ) (t )dt , 2 0 (1.4.29) 0 where 2 k2 1 0 K0 ( r, t ) r 2 sin t cos t d . J 2 ( r ) t (1.4.30). The left side of (1.4.39) is an integral operator of the Abel’s type. Reversion of this operator leads to the equation f ( r ) K ( r, t ) (t )dt F ( r ) , 0 where 28 (1.4.31) 2 d K0 (u, t ) r2 u2 du 2 r dr 0 u r K ( r, t ) (1.4.32) and F (r ) after simplification is F (r) 2 1 d 2 k2 d . J 0 ( f )r 5/ 2 J 5/ 2 ( r ) 2 r dr 0 (1.4.33) The equation (1.4.41) belongs to the type of Fredholm integral equations of the second kind with the kernel K ( r , t ) permissible for the representation of the solution in the form of the series: ( r ) n ( r ) , 0 ( r ) F ( r ) , n 1 (1.4.34) f n 1 ( r ) F ( r ) K ( r, t ) n (t )dt . 0 Thus, the solution of the equation (1.4.1*) for the periodic alternating current with the boundary conditions (1.4.3) – (1.4.6) is determined by the expressions (1.4.35), (1.4.38) – (1.4.44). In the case of a faint skin effect kf 1, (1.4.35) which often corresponds to a real conditions for electrical contacts, it is more convenient to represent the solution in another form: ( ) sin f sin t (t ) cos t dt . f t 0 f Taking into account that in this case g ( ) k2 2 2 and calculating corresponding integrals one can find the following description for the magnetic field: H ( r , z , t ) I 0 exp(it ) 1 z k 2z 1 (2 2 2 z z 2 r 2 ) . 2 r 6r (1.4.36) If 0 , then this expression transforms into the formula (1.4.13). Replacing H in the formulas (1.4.7), (1.4.8) by the representation (1.4.46) one can find the components of electrical field, hence, the current density. Particularly, the current density on the contact spot due to expressions (1.4.22) is 1 I 0 exp(i t ) k 2 f 2 3r 2 jz ( r,0, t ) [rH ( r,0, t )] 1 1 . r r 3 2 f 2 2 f f 2 r 2 (1.4.37) The second term in square brackets of the right side corresponds to the component of additional increasing of the current density at the edge of a contact spot due to skin effect. Let us estimate now the range of the correctness of the condition (1.4.45). For non-magnetic materials 1 . Taking into consideration that 29 ~107 108 m , 0 1.257 107 H m1 we conclude that the thickness of skin- layer is 1 c k 0 20 where c~0.03 0.10 . For the current frequency 50Hz and f ~108 104 m one can find that kf ~ 7 107 3 102 , thus the condition (1.4.45) is accomplished. The same conclusion is true for ferromagnetics with the temperature above the Curie point. Thus, influence of skin effect on the electromagnetic field can be important in exceptional cases only, for example, for high-frequency current ( 103 Hz ) or for extra-high contact loads ( Pc ~104 N ) providing considerable size of a contact spot ( f ~102 m ) . Herewith one can conclude that an unevenness of current density on a contact spot is stipulated mainly by the constriction phenomena, and skin effect is of no importance for real conditions of electrical contacts commutation. 5 STATIONARY CONJUGATED FIELDS OF THE TEMPERATURE AND ELECTRICAL POTENTIAL. 5.1 Ideal symmetric contacts Stationary temperature and electromagnetic fields in symmetric electrical contacts can be described by the differential equations div( grad ) 1 grad 2 0 , 1 div grad 0 , where T T0 is the difference between current contact temperature T and initial contact temperature T0 before heating, , , are electrical potential, heat conductance and electrical resistivity respectively. In cylindrical coordinates these equations can be written as 1 1 r r r r z z 2 2 0 , r z 1 1 1 r 0. r r r z z 30 (1.5.1) (1.5.2) The boundary conditions are z z 0, 0 0 r r0 z or r 0, (1.5.3) z 0 z z 0, r0 r 0, z or uc / 2 . 0, r (1.5.4) (1.5.5) It has to be mentioned that this problem is essentially non-linear due to temperature dependence of thermal conductivity ( ) and electrical conductivity ( ) . The method of the solution can be obtained from the suggestion that the identity of equipotential and isothermal surfaces in contacts, which is correct for stationary fields in linear case, keeps safe for non-linear case as well. In linear case these surfaces are ellipsoids of revolution r2 z2 1 . 2 r0 2 2 (1.5.6) Therefore if (r, z) const is required equipotential and isothermal surface, then electromagnetic and temperature fields can be characterized by the only one coordinate instead of two independent variables r and z . That is the motivation for an attempt to find the solution of the equations (1.5.1) and (1.5.2) in the form (1.5.7) ( ) , ( ) , (r, z) . Putting this expression in the equations (1.5.1)- (1.5.2) we get [ ( ) ( ) ( ) ( )2 ( ) ( ) ( )2 ] ( )2 ( ) ( ) ( )2 0 , (1.5.8) ( ) ( ) ( ) ( ) ( ) ( )2 ( ) ( )2 0 , (1.5.9) 2 1 2 ( )2 , 2 2 . r r r z 2 r z (1.5.10) where 2 2 Hereby, if a solution in the form (1.5.7) exists, then the ratio 2 /( )2 must be not dependent on the variables r and z , it should depend on the only variable . Indeed, using the relationship (1.5.6) it is easy to check that 2 2 2 . 2 ( ) r0 2 31 (1.5.11) This result is expected because a temperature dependence of the potential at any point of electromagnetic field may change from physical point of view a magnitude only, but not the configuration of equipotential surface, which remains the same for both linear and non-linear problems. Using (1.5.11) one can rewrite the equations (1.5.8) and (1.5.9) in the form 2 1 ( ) ( ) 2 ( ) 2 0 , 2 ( ) r0 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 0 . r0 ( ) ( ) ( ) ( ) 2 (1.5.12) (1.5.13) To write the boundary conditions in term of variable note that ( ) , z z ( ) . z z (1.5.14) Taking into account that z = z=0 r0 if r0 r 2 2 0 r r0 and z if =0 r0 r z=0 we can conclude that all boundary conditions (1.5.3)-(1.5.5) are satisfied if we put: for 0 for (0) 0 , (0) 0 , () 0 , ( ) uc 2 . (1.5.15) (1.5.16) (1.5.17) (1.5.18) Thus, the primary problem is reduced to the system of equations (1.5.12) – (1.5.13) with the boundary conditions (1.5.15) –(1.5.18). We begin its solution from the equation (1.5.13) rewritten in the form ( ) 2 ( ) 2 0. ( ) r0 2 Integration of this equation gives the relationship between and : 32 ln[ ( )] ln{[ ( )]} ln( r0 2 ) ln C , 2 thus ( ) C ( ) . 2 r0 2 (1.5.19) The constant C can be defined using Ohm’s Law. The equipotential surface (5.6) approaches the sphere of the radius as . If the potentials of two such spheres with the radii and d are equal correspondingly to and d , then accordingly to the Ohm’s Law d I ( ) thus ( ) I ( ) 2 2 d 2 2 . as Comparing this expression with (1.5.19) and calculating the limit as , we can find C I 2 , (1.5.20) thus ( ) I ( ) . 2 2 (r0 2 ) (1.5.21) Using this expression, we can transform the equation (5.12) to the form ( ) ( ) 2 2 I ( ) 0 ( ) 2 2 r0 2 4 2 (r0 2 ) to solve this equation we use the substitution arctan T ( ) ( ) , r0 (1.5.22) reducing the functions domain to the domain / 2 / 2 and the heat equation to the form (T )T ( ) (T )T ( ) 2 I 2 (T ) 0. 2 4 2 r0 The derivative T ( ) can be expressed from this equation as dT [2 (T ) (T )dT ]1/ 2 d (T ) T ( ) T (0) where 33 , (1.5.23) 2 I2 . 2 4 2 r0 (1.5.24) This equation can be solved with respect to the variable (T ) T (0) 1 2 T (0) (T1 )[ T (T2 ) (T2 )dT2 ]1/ 2 dT1 . (1.5.25) T1 The temperature T ( ) can be found from this equation as an inverse function, if the functions (T ) and (T ) are given. Equating the heat flux passing through isothermal surface and specific power generating by Joule heating we get dT 1 d j , d d where j is current density. Thus d d d d dT dT and integration yields well-known Kohlrausch’s temperature-potential relation 1/ 2 T0 2 dT T , T0 T (0) . (1.5.26) However, the formula (1.5.25) has an advantage over (1.5.26) because it enables one to get the information about the space distribution of temperature field. Special cases. 1. Thermal conductivity and electrical resistivity are constant. (T ) 0 const , (T ) 0 const In this case the expression (1.5.25) becomes the form (T ) 1 2 [T (0) T ] and the maximum temperature on the contact spot T (0) can be found from the condition 34 T 0 if 2 , that gives well-known expression [16] T (0) I 2 0 . 2 32r0 0 The expressions for spatial distribution of temperature and potential are 4 I 2 4 T T0 1 2 2 1 2 arctan , r0 32r0 I arctan . 2 r0 r0 (1.5.27) (1.5.28) 2. Thermal conductivity is constant, electrical resistivity is a linear function of the temperature (T ) 0 (1 T ) , (T ) 0 const After integration of (1.5.25) we get (T ) 0 0 1 T arccos 1 T0 1 and the condition (1.5.26) gives the expression T ( ) 1 cos( 0 / 0 ) [ 1] , cos( / ) 0 0 2 (1.5.29) which has been presented in [17]. The maximum temperature in this case is T (0) 1 sec( 0 / 0 ) 1 . 2 (1.5.30) The expression for electrical potential can be obtained by integration of the equation (A5.21) using substitution (1.5.22). It gives ( ) 00 sin( 0 / 0 ) . cos( / ) 0 0 2 35 3.Thermal conductivity and electrical resistivity are linear functions of the temperature. (T ) 0 (1 T ) , (T ) 0 (1 T ) . (1.5.31) (1.5.32) In this case to the second-order infinitesimal (T ) (T ) 00[1 ( )T ] and the expression (1.5.25) can be integrated (T ) 0 ( ) 0 1 1 ( )T arccos [1 ( )T0 ]2 [1 ( )T ]2 1 ( )T0 (1.5.33) The expression for the temperature T ( ) can be obtained by the solving of the equation (1.5.33) using the condition (0) / 2 for determination of T0 . 5.2 Ideal asymmetric contacts In the case of bimetallic contacts the equations (1.5.12) and (1.5.13) should be considered for each contact separately. In term of variable these equations can be written as i (T )Ti( ) i(T )Ti( )2 2i i 0 , i ( ) i , where i 1 if / 2 0 and i 2 if 0 / 2 . (1.5.34) (1.5.35) The boundary conditions on the contact spot (1.5.15)-(1.5.16) have to be replaced by the conditions of continuity of heat flux and current density (1.5.36) T1 (0) T2 (0) , 1T1 (0) 2T2 (0) , 1 (0) 2 (0) , 1 1 1 (0) 1 2 2 (0) . (1.5.37) The conditions (1.5.17)-(1.5.18) should be changed for the new conditions T1 ( / 2) 0 , T2 ( / 2) 0 , 1 ( / 2) UC / 2 , 1 ( / 2) UC / 2 . (1.5.38) In asymmetric contacts the temperature maximum Tm is displaced from the contact surface having temperature T0 toward the contact which material has 36 greater value of i i . Let us assign the index (Fig. 5.2) i 2 for this contact material T Tm T0 T1 T2 2 2 0 m Fig. 5.2 Temperature distribution in asymmetric contacts The solution of the problem (1.5.34)-(1.5.38) is: For the domain / 2 0 1 T0 T0 Tm 1/ 2 1 (2 11dT 2 2 2 ) dT ; T T (1.5.39) T0 T0 T 0 U 1 ( ) C (2 1 1dT )1/ 2 (2 1 1dT )1/ 2 2 0 T for the domain 0 m 1 Tm T (2 dT ) 2 T0 2 1/ 2 2 dT ; T 2 ( ) (1.5.40) T T m m UC (2 2 2 dT )1/ 2 (2 2 2 dT )1/ 2 2 0 T and for the domain m / 2 m 1 Tm Tm (2 dT ) 2 T T 2 2 1/ 2 dT ; (1.5.41) T T m m U 2 ( ) C (2 2 2 dT )1/ 2 (2 2 2 dT )1/ 2 . 2 0 T The values T0 and Tm can be found from the system of equations 2 T0 T0 Tm 0 T T0 1 (2 1 1dT 2 2 2 dT ) 1/ 2 dT 37 (1.5.42) Tm Tm T0 T 2 (2 2 2 dT ) 1/ 2 dT 2 Tm Tm 0 0 (1.5.43) 2 (2 2 2 dT ) 1/ 2 dT The temperature maximum occurs at the point m 1 Tm Tm (2 dT ) 2 2 T0 1/ 2 2 (1.5.44) dT T In terms of cylindrical coordinates ( r , z ) the axial displacement of temperature maximum in axial direction at r 0 is (1.5.45) zm r0 tan m . In the case of linear temperature dependence of i and i (1.5.31), (1.5.32) integration of the expressions (1.5.39) - (1.5.41) yields for the domain / 2 0 1 A1 (arcsin 10 arcsin 1 ) s s (1.5.46) B1 ( s 10 s ) 2 2 2 2 1 for the domain 0 m 2 [ A2 (arcsin 2 arcsin 20 ) 2 m 2 m (1.5.47) B2 ( 2 m 2 2 m 20 ) 2 2 2 2 and for the domain m / 2 m 2 ( A2 arccos 2 2 2 B2 2 m 2 ) 2 m (1.5.48) where i i 1 (i i )T , i 0 1 (i i )T0 , im 1 (i i )Tm , s 10 2 20 20 1 1 2 2 (2 m 20 ) , 10 10 2 2 38 Ai i i i , 1 Bi i 0 , i 0 (i i ) i i i . 5.3 Heating of ideal contacts with Thomson effect Let us consider the problem of stationary temperature and electromagnetic fields in closed heterogeneous contacts taking into account the Thomson effect. We assume that specific electrical resistivity i and Thomson coefficient Ti are linearly dependent on the temperature i : i i 0 (1 ii ) , Ti Ti 0 (1 ii ) (1.5.49) but thermal conductivity i is constant, because its temperature dependence is not so essential and can be taken as average for considered temperature interval. The system of equations for temperature i and potential i can be written in the form i i Ti 1 i i 2i 0 , i i div( 1 i i ) 0 (1.5.50) In the cylindrical coordinate system these equations take the form (1 ii )( i 1 i 2i i i i 2 ) Ti 0 (1 ii )( i ) 2 r r r z i i 0 r r z z [( i )2 ( i )2 ] 0 i 0 r z (1.5.51) 1 (1 ii )( 2i 1 i 2i 2 ) i ( i i i i ) 0 , i 1,2 . 2 r r r z r r z z (1.5.52) The index i 1 corresponds to the cathode region D1 (0 r , z 0) and the index i 2 corresponds to the anode region D2 (0 r ,0 z ) . The boundary conditions for electrical potential and temperature can be written as 1 2 (1.5.55) uc , 2 uc , 2 if if r2 z2 , r2 z2 , z0 z0 (1.5.54) 1 (r, 0) 2 (r, 0) 1 1 2 10 [1 11 ( r, 0)] z 20 [1 22 ( r, 0)] z 1 1 ( r, 0) 2 ( r, 0) z z i 0 , (1.5.53) if r2 z2 , 39 (1.5.56) (1.5.57) i 1,2 (1.5.58) 1 (r,0) 2 (r,0) , 1 r r0 1 (r ,0) (r ,0) 2 2 , z z 1 (r ,0) 0, z (1.5.59) r r0 2 (r ,0) 0, z (1.5.60) r r0 (1.5.61) It has to be mentioned that this problem is essentially nonlinear because of equations (1.5.51), (1.5.52), (1.5.56). In terms of variable we get (1 ii )i Ti 0 1 2 (1 ii )ii (i )2 (1 ii )i 2 0 i i 0 i i 0 f 2 (1 ii )i iii (1 ii )i 2 0 f 2 (1.5.63) 2 To write the boundary conditions in term of variable (1.5.62) note that i z f 2r 2 i i ( ) ( 1) i ( ) [1 2 ] z z ( f 2 )2 f 2r 2 (1.5.64) Taking into account the relations (1.4.22) we can conclude that the conditions (1.5.13) and similarly (1.5.9) are satisfied automatically at such choice of the function (r, z ) . The relation (1.5.22) at z 0 , r f takes the form i z i (0) z 0 f f 2 r , thus the condition (1.5.12) and similarly the condition (1.5.8) are reduced to the form 1 10 [1 11 (0)] 1 (0) 1 20 [1 22 (0)] 11 (0) 22(0) 2 (0) (1.5.65) (1.5.67) Finally, instead of conditions (1.5.5), (1.5.6), (1.5.7), (1.5.10), and (1.5.11) we may write the corresponding conditions in term of : 40 1 ( ) uc 2 (1.5.68) uc 2 1 (0) 2 (0) 2 ( ) (1.5.69) (1.5.70) 1 () 2 () 0 (1.5.71) 1 (0) 2 (0) (1.5.72) Thus, the primary problem is reduced to the system of equations (1.5.20) – (1.5.21) with the boundary conditions (1.5.23) – (1.5.29). We begin its solution from the equation (1.5.21) rewritten in the form i i ii 2 2 0 1 ii f 2 Integration of this equation gives the relationship between i and i : i ( ) Ci (1 ii ) f 2 2 (1.5.73) The constants C i can be defined from the following consideration. The equipotential surface (1.5.18) approaches the sphere of the radius as . If the potentials of two such spheres with the radii and d are equal correspondingly to i and i di , then accordingly to the Ohm Law di I i 0 (1 ii ) thus i ( ) d 2 I i 0 (1 ii ) 2 2 as Comparing this expression with (1.5.30) and calculating the limit as , we can find Ci I i 0 2 thus i ( ) (1.5.74) I i 0 1 ii 2 f 2 2 It has to be noted that such definition of a constant automatic satisfiability of the condition (1.5.23). (1.5.75) Ci involves an Now the equation (1.5.20) can be reduced using (1.5.32) to the form 41 i I Ti 0 I 2 1 ii 2 (1 ii ) 2 i 2 2 i 0 2 0 2 2 2 2 i 2i f 4 i ( f ) f Let us try to find the solution of this equation in a form i ( ) i ( ) , arctan 1 i (1.5.76) f i ( ) are Then the equations concerning i i 2 (i ) 2 i (1 i i ) i 0 (1.5.77) where i I 2f i0 i , i i I Ti0 4i f (1.5.78) It will be shown below that Thomson effect is not so important for heating of closed contacts in comparison with Joule effect, thus the coefficient i can be considered as a small parameter. An approximate solution of the nonlinear differential equation (1.5.34) can be found in this case using the method of Krylov-Bogolubov [24] : i ( ) 1 i exp( i )( Ai cos i Bi sin i ) where i i (1 The constants Ai and (1.5.24), (1.5.28), (1.5.29): Ai Bi (1.5.79) i ) i are determined from the boundary conditions 1 r1 r 1 1 [ exp( 1 ) 2 exp( 2 ) ( 1)i ( ) 3i 1 2 2 2 1 2 1 i i Bi ( 1)i { exp[( 1)i i ]cos ec Ai cot } i 2 2 2 (1.5.80) where i ri [cos 1 2 , ri ii sin 3i 2 i i ( 1)i i sin ], 2 i 2 The potentials i ( ) can be found now by integration of the equation (1.5.32) with the boundary conditions (1.5,25), (1.5.26): 42 u I 1 11 ( ) u I 1 ( ) c 01 d c 10 2 2 2 2 f 2 2 f ( [1 11 ( )]d / 2 1 uc I 101 1 {( A1 1 B1 )[exp( 1 ) cos 1 exp( 1 ) cos 2 2 1 2 2 f 1 2 2 (1.5.81) 1 1 A1 B1 )[exp( 1 )sin 1 exp( 1 )sin ]} 1 2 2 Similarly 2 ( ) uc I 20 2 2 2 {( A2 2 B2 )[exp( 2 ) cos 2 2 2 2 f 2 2 2 2 exp( 2 ) cos 2 ) ( (1.5.82) 2 2 A2 B2 )[exp( 2 )sin exp( 2 )sin 2 ]} 2 2 2 where arctan . f (1.5.83) Using the expressions (1.5.34) and Ohm Low it is not difficult to check that the functions (1.5.38) and (1.5.39) satisfy the condition (1.5.27). Thus, an approximate solution of the problem (1.5.3) – (1.5.13) is determined by the formulas (1.5.30) – (1.5.40), where is given by the expression (1.5.18). It has to be noted that the temperature maximum is displaced from the contact plane to anode or cathode due to heterogeneous properties of electrode materials and due to thermoelectric effects. Such a displacement can have an influence on a shape of contact liquid bridge, zones of plasticity and contact welding, as well as on other phenomena accompanied with structural phase changes, particularly, melting of contact materials which arises inside of an electrode. If the position of isotherm 0 where the temperature reaches its maximum is found, then one can analyze parameters determining this position. Thereby, it enables to predict the course of contact heating phenomena and to operate it in a certain degree by a corresponding choice of contact materials. The value 0 can be found from the extremum test for the function (1.5.36). It gives the formula 0 1 i i arctan Ai i Bi , i Ai Bi (1.5.84) where i 1 i / i and the index i 1 or i 2 has to be chosen corresponding to the material with the greater value i . 43 The constriction resistance Rc is determined by the expression 10 0 1 11 20 1 2 2 Rc f 2 2 d 2 0 f 2 2 d 2 10 20 [ A1 sin 1 B1 (cos 1 1)] [ A2 sin 2 B2 (cos 2 1)] 2 f 1 2 2 2 f 2 2 2 (1.5.85) 5.4 Special cases Let us consider some special cases when it is available to get the exact solution of the problem. 1. Thomson coefficient is constant. (1.5.34) takes the form i 2 i i i (i 1 i In this case i 0 , the equation )0 (1.5.86) and its solution is determined by the same expressions (1.5.33) – (1.5.34) for the temperature and (1.5.38) – (1.5.39) for the potentials, in which i and i must be replaced with i and i 1 Ti 0 4 i 0i i respectively. 2. Thomson effect is negligible. In this case Ti 0 0 , i 0 and expressions (1.5.36) – (1.5.39) converts into the formulas 1 i ( ) where Ai Bi i Ai cos i Bi sin i , i 1,2 (1.5.87) 1 11 1 1 [ sin 2 2 2 sin 1 ( 1)i ( ) 3i , 1 2 2 2 1 2 3i3i [( 1 1 1 1 1 cos 2 cos 2 cos 1 ], 2 2 2 1 2 2 2 1 2 , i ii cos i sin 3i 1 ) cos 2 and 1 ( ) 2 ( ) 2 uc 1011 [ A1 (sin 1 sin 1 ) B1(cos 1 cos 1)] 2 2 2 uc 2022 [ A2 (sin 2 sin 2 ) B2 (cos 2 cos 2 )] 2 2 2 These formulas become more convenient for calculation, if we introduce into consideration the temperature and potential in the contact spot: 44 11 1 22 2 tan tan 4 2 4 0 1 (0) 2 (0) 1 1 2 11 cot 22 cot 2 (1.5.88) 0 1 (0) 2 (0) 2 1 uc 2 1011 ( 0 ) tan 1 2 4 2 u 2 c 20 22 ( 0 ) tan 2 2 4 In this case i ( ) 1 i (1 i0 )sin {i [ 2 ( 1)i ]} sin [i ( 1)i ] i sin i (1.5.89) (1.5.90) 2 i 1,2 i ( ) 0 i 0ii [ Ai sin i Bi (1 cos i )] (1.5.91) The constriction resistance of the region bounded by the contact surface and the isotherm is determined from the expression ( 1)i Ri ( ) [i ( ) 0 ] , I i 1,2 (1.5.92) 3.Homogeneous contacts. Thomson effect is negligible. In this case 10 20 0 , 1 2 , 1 2 , 1 2 , Ti 0 and the expressions (1.5.44) have the simplest form ( ) 1 cos ( 1) sin uc 0 1 (sin sin ) 2 cos 2 2 u 1 2 ( ) c 0 (sin sin ) 2 cos 2 2 1 ( ) (1.5.93) (1.5.94) (1.5.95) These expressions coincide with well-known formulas presented in the paper [25]. The constriction resistance bounded by the contact surface and the equipotential surface is determined in this case as 45 R( ) 0 sin . 2 f cos (1.5.96) 2 The constriction resistance of one contact member ( / 2 ) is R( / 2) 0 tan , 2 f 2 (1.5.97) and the total resistance is double. 4. Heterogeneous contacts. resistivity is constant. In this case Ti 0 , i ( ) i 0 . Thomson effect is negligible. 20 10 If I 2 i 0 2 Ai Bi , 2 2 8 i f Electrical , then R 1 2 , 2 (1.5.98) where I 2 20 10 Ai ( ), 1 2 32 f 2 2 1 2 i 0 I 2 ( 201 102 ) B1 B2 32 f 2 (1 2 ) ( 201 102 ) 4 20 (1 2 ) The greatest displacement of the temperature maximum isotherm 0 does not exceed the radius f of the contact spot, because 0 4 If the temperature interval is relatively small (or constriction resistance R can be written in the form R 10 20 2f 1 where i 0 Ai i . i is small), then the 2 10110 20220 1012 202 1 (20 10 ) 3 2f 4 f (1 2 ) (1.5.99) is the value of the temperature on the contact spot. 5. Heterogeneous contacts with Thomson effect. Electrical resistivity is constant. In this case i 0 , and the equation (1.5.36) transforms to i 2i (1 i i )i i 0 where 46 (1.5.100) i I 2 i 0 4 2 f 2i The substitution i ( ) yi ( xi ) (1.5.101) where xi yi ( xi ) exp{ i [1 i i ( )]d } , xi Pi , Pi ( i i i )1/ 3 (1.5.102) 0 reduces this equation to the form d 1 d2 { [ yi ( xi )]} 1 0 , dxi yi ( xi ) dxi 2 2 P1 x1 0 , 0 x2 2 P2 The general solution of the last equation can be expressed in terms of the Bessel functions: 2 2 3/ 2 yi ( xi ) vi [Ci J 1/ 3 ( vi 3/ 2 ) DY )] , i 1/ 3 ( vi 3 3 vi xi Ei (1.5.103) The constants Ci , Di , Ei are determined using conditions (1.5.24), (1.5.28), (1.5.29) which can be written due to equations (1.5.30), (1.5.58), (1.5.59) in the form 1P12 2 P2 2 [ y (0) y1 (0)] , 11 1 2 2 1 P1 1 P [ y1 (0) 1] [ 2 y2 (0) 1] 1 1 2 2 (1.5.104) y1( 2 y2 ( P2 ) 2 P2 2 P1 ) 1P1 , with two additional conditions yi (0) 1 , i 1,2 (1.5.105) arising from the expression (1.5.59). The potentials i are determined by the previous expressions u I i ( ) {( 1) c i 0 2 2 f i [1 i i ( )]d (1.5.106) ( 1)i / 2 5.5 The limit of maximal current It has to be noted that all above derived formulas make sense only if i 1 , in other words I i 0 2f i 0i (1.5.107) 47 Such a rather unusual constraint can be explained by a temperature dependence of electrical resistivity , which can stipulate the increasing of the contact voltage uc even to infinity, but it leads to unlimited increasing of the temperature and the constriction resistance while the value of the current remains limited by the inequality (1.5.64). Let us prove this inequality. We consider for simplicity the case of symmetrical homogeneous contacts without Thomson effect. The equation (1.5.20) for the temperature ( ) both for the cathode as well as for the anode can be written in this case as (1 ) 1 0 2 (1 ) 2 0 f 2 (1.5.108) 2 The solution of this equation is given by the expressions (1.5.50), (1.5.40): ( ) 1 cos ( arctan [ cos f ) 1] (1.5.109) 2 where 0 I 2f The expression (1.5.67) for it ( ) (1.5.110) was found using the equation (1.5.30), that C (1 ) f 2 2 (1.5.111) and the constant C was found from the physical consideration using the formula (1.5.34). Let us find it now directly from the boundary conditions for the potential, which can be written in this case due to the contact symmetry as (0) 0 , ( ) uc . 2 (1.5.112) Putting the expression (1.5.68) in the equation (1.5.65) and solving this equation we obtain the same formula (1.5.66), where C f 0 Integration of the equation (1.5.68) using formulas (1.5.66) and (1.5.69) gives the expression for the potential ( ) C f sin ( arctan cos 2 f ) 0 sin ( arctan cos Thus ( ) therefore 48 uc 0 tan , 2 2 2 f . 2 arctan( ) , 0 uc 2 (1.5.113) and C f 0 2 u arctan ( c 2 ) 0 (1.5.114) It can be derived directly from the expression (1.5.70) that 1 , therefore the inequality (1.5.64) is proved. In that way there exists the maximum value of the current for every given radius of the contact spot, which is defined by the properties of the contact material. However this value never reaches because it corresponds to infinite temperature, while increase of the radius of the contact spot occurs already at the softening temperature and the more so at the melting temperature. The ratio of the current and diameter of the contact spot corresponding to the maximum value I 2 f 0 max 0 as well as corresponding to the melting temperature m I 2f 1 m 2 arccos 0 1 m m is given for some materials in the Table 1-1. The values 0 and m could be really rather greater because of a heat emission from the contact surface. It has to be noted that if the temperature dependence of electrical resistance is ignored the current limitation will not occur. This conclusion is confirmed by the formula ( ) I 2 0 4 (1 arctan 2 ) 2 32 f f (1.5.115) that can be derived from 1.5.66) as limit for 0 . This fact testifies to a limited applicability of the formula (1.5.72). It means also that the correct mathematical statement of the contact heat problem with temperature dependence of contact resistance involves the voltage as input instead of the current. TABLE 1-1. Limit of maximal current 0 Material Al Cr Fe Ni 10 8 m 2.9 20.0 10.0 8.0 10 3 K 1 1 10 W m K 2 4.0 2.0 6.5 5.0 2.10 0.67 0.60 0.70 49 1 0 m 106 A m 1 106 A m 1 4.23 1.29 0.96 1.32 3.42 1.09 0.90 1.21 Cu Zn Mo Ag Cd Sn Ta Au W Pt Pd Pb 1.75 6.1 5.8 1.65 7.5 12.0 14.0 2.3 5.5 11.0 10.8 21.0 4.0 3.7 4.5 4.0 4.0 4.5 3.0 4.0 5.0 3.8 3.3 4.0 3.80 1.10 1.40 4.18 0.90 0.64 0.54 3.10 1.90 0.7 0.7 0.35 7.33 2.19 2.30 7.92 1.72 1.08 1.13 5.78 2.62 1.29 1.39 0.64 6.45 1.63 1.59 6.87 1.23 0.73 1.05 5.07 2.53 1.18 1.25 0.46 It is appropriate to mention here that the solution of the problem (1.5.3) (1.5.13) could be obtained using a formal change of the cylindrical co-ordinates (r , z ) for the ellipsoidal co-ordinates ( , ) , which are bounded up by the relationship Fig. Ellipsoidal coordinates r2 z2 1 2 f 2 2 r2 2 (1.5.116) z2 1 f 2 2 (1.5.117) 50 The ellipsoidal co-ordinates are determined by the family of isothermal ellipsoids of revolution and orthogonal family of hyperboloids of one sheet that are given by the equations (1.5.73) and (1.5.74). From these equations one can derive the inverse relationship r f 2 2 , f - , z f 2 2 , f . 0 f The contact spot given in cylindrical co-ordinates by z 0, 0 r f is replaced in ellipsoidal co-ordinates by 0, f , while the remaining part of the contact plane z 0, r f is replaced by f , 0 . The solution of the problem can be obtained after such change of co-ordinate systems using method of separation of variables. One of the most important theoretical results of this section is the conclusion that the temperature of an ideal contact spot (even nonhomogeneous) is the same at stationary regime for the all area of the contact spot z 0, 0 r f . It can be derived from the expressions (1.5.29)-(1.5.36) at 0 . It will be shown below that this conclusion is not valid for a non-stationary regime of contact heating, when temperature maximum is displaced to the edge of the contact spot, where current density is maximal. The temperature distribution along contact spot is also not uniform in the case of a non-ideal contact. 5.6 Non-ideal contact The temperature distribution into cathode 1 (r, z ) and into anode 2 (r, z ) occupying the cylindrical half-spaces D1 ( z 0,0 r ) and D2 (0 z ,0 r ) respectively are described by the heat equations 2 i 1 i 2 i j 2 0 r 2 r r z 2 (1.5.118) with i 1 for D1 and i 2 for D2 . The current density j is determined by the expression [5] j 2 jr j z 2 G 2 I2 1 , G G (G G ) 2 z (r f ) 2 2 (1.5.119) 2 Because of very small thickness of contact film we apply the model of nonideal contact through the circular spot S 0 ( z 0,0 r f ) with the temperature difference f across the film like the difference between left-hand limit for 2 and right-hand limit for 1 when z 0 and 0 r f : f 2 (r,0) 1 (r,0) (1.5.120) 51 The boundary conditions for such non-ideal contact may be obtained from (2), (3), (6) as z 0,0 r f : 1 (r ,0) 2 (r ,0) z W 2 (r ,0) (r ,0) 1 (r ,0) 2 2 z W 1 (1.5.121) (1.5.122) There is no heat exchange outside the contact spot, hence z 0, f r : 1 ( r,0) 2 (r ,0) 0 z z (1.5.123) i (r , z ) denotes the difference between contact temperature and If surroundings temperature , then far from contact spot z , or z , or r : (1.5.124) 1 2 0 Solution. To solve the equations (6) with the conditions (8) - (12) we introduce new unknown functions u(r, z ) and v(r , z ) instead of previous 1 (r , z ) and 2 (r, z ) by substitution (1.5.125) u ( r , z ) 2 ( r , z ) 1 ( r , z ) , v ( r , z ) 2 ( r , z ) 1 ( r , z ) where z 0 . If we change z for z in the equation (6) for i 1 , afterwards add and subtract left and right parts of these equations for i 1 and i 2 and take into account the equality j 2 (r , z ) j 2 (r , z ) , then for u (r , z ) and v(r , z ) we obtain equations : 2 u 1 u 2 u 2 j 2 0 , r 0, z 0 r 2 r r z 2 2 v 1 v 2 v 0 , r 0, z 0 r 2 r r z 2 After addition and subtraction the boundary conditions reduced to the conditions for u(r, z ) and v(r , z ) : u (r ,0) if z 0 if (1.5.126) (1.5.127) (9) - (12) are r f r f (1.5.128) 2 v(r ,0) v(r,0) if r f W z if r f 0 uv0 , if r or 52 z (1.5.129) Hence we obtain two independent boundary problems (14) , (16) , (18) for u(r, z ) and (15) , (17) , (18) for v(r , z ) . The solution of first of them is already known [5] : u (r , z ) f exp( xz ) J 0 ( xr ) J 1 ( xf ) 0 dx x I 4 (1 2 arctan 2 ) 2 f 16 f 2 where 1 2 (1.5.130) s s2 4 f 2z2 , s z2 r2 f 2 The solution of the second problem may be represented in the form v(r , z ) A( x) exp( xz ) J 0 ( xr )dx (1.5.131) 0 To satisfy the boundary condition (17) we get the system of dual integral equations for unknown function A(x) : 2 ( x W ) A( x) J 0 ( xr )dx 0 , 0r f 2 (1.5.132) A( x) J 0 f r ( xr ) xdx 0 , 0 After substitution f A( x) (t ) cos( xt )dt 0 system (21) can be reduced to the Fredholm integral equation for (t ) : 2 (t ) f 2 t2 f 4 ln W 0 f 2 t2 f 2 t1 t 2 t1 2 2 (t1 )dt1 (1.5.133) The kernel of this integral equation has a week singularity at t t1 , therefore it may be solved by the standard Picard's method. It is available at the frame of this approach to get the analytical expression for the Kohler overheating in the center of contact spot. From (20) at z 0 and 0 0 v(r ,0) A( x) J 0 ( xr )dx J 0 ( xr )dx (t ) cos( xt )dt from (22) we get r 0 hence 0 (t ) 1 r2 t2 v(0,0) (0) 2 dt 0 (rx) 1 x2 dx . 53 To find (0) we rewrite the equation (22) in the form (t ) 2 f 2 t2 (1.5.134) 4f 1 2 1 2 x2 (tx )dx ln W 0 1 x2 Since lim 1 ( t 0 ) ln 2 1 2 x2 1 x 0 t 0 we obtain (0) 2f 2 v(0,0) k 2 hence from (23) at 1 1 2 f /( W ) and where (tx )dx (0) f k 1 , (1.5.135) 2f W (1.5.136) Then from (19) we find I 2 u (0,0) k 2 J 16 f 2 f (1.5.137) where I 2 J 32 f 2 (1.5.138) Finally from (13) we can obtain the temperature in the center of contact spot on the cathode and on the anode k , 2 1 2 2 (0,0) J k 2 1 1 (0,0) J (1.5.139) Thus temperature difference across the contact film in the center of the contact spot is f (0) 2 (0,0) 1 (0,0) k 1 (1.5.140) It has to be noted that the magnitude k is equal to the temperature that the cathode should be heated by tunnel mechanism only without Joule sources and heat transfer to the cathode. In contrary the magnitude J is the maximum temperature at the center of contact spot at Joule mechanism of heating only without tunnel effect. 54 Criterion of tunnel heating. To estimate the influence of Kohler effect on the contact heating we introduce the new criterion Ko , called Kohler criterion, that is the ratio of film resistance R f f /(f 2 ) and contact resistance without film ( so called constriction resistance ) Rc that is determined by the wellknown formula [2] Rc . 2f Hence we introduce the Kohler criterion as Ko Rf Rc 2 f (1.5.141) f It defines the level of additional heating of anode contact surface owing to tunnel effect in comparison with the Joule heating of constriction zone. In terms of criterion Ko the formula (1) can be written as where I is the current, hence k Therefore Ko f 16 I 2 f 2f 4 Ko J k 16 J Further from (5) , (30) , (25) W Ko , (1.5.142) (1.5.143) f , 4 1 2 Ko (1.5.144) and using the formulas (28) we get 1 (0,0) J [1 1 ( Ko)] 2 (0,0) J [1 2 ( Ko)] (1.5.145) where 1 ( Ko) 32 1 1 4 /( Ko) 16 1 2 /( Ko) 2 ( Ko) Ko 1 4 /( Ko) 2 (1.5.146) The tunnel overheating is 1 2 f (0) 2 (0,0) 1 (0,0) J Ko 1 ( Ko) (1.5.147) The functions 1 ( Ko) and 2 ( Ko) are the ratio of tunnel and Joule components for cathode and anode heating respectively. It is easy to calculate that if Ko 0.02 , then tunnel heating of the anode is less 10% compared with Joule heating ; if Ko 0.28 they are equal ; if Ko 1 , then tunnel component of anode temperature is 10 times more than Joule component. 55 It is interesting to note that this ratio is limited for the cathode by the value 32 / 2 3.03 even if Ko . It equals to 0.1 and 1 at Ko 0.04 and Ko 0.57 respectively . Comparison with experimental data . Experimental data were obtained by R.Holm and I.Dietrich [2] for Pt contacts at the conditions : I 10 A , f 0.5 10 5 m , 60W m 1 K 1 , 9.5 10 7 Ohm m , f 1.3 10 12 Ohm m 2 The measured temperature difference across the film was f 176 o C . Formulas (30) - (35) give J 2000 o C , k 1770 o C , 7.34 , Ko 0.177 1 (0,0) 2780o C , 2 (0,0) 2970o C , f 190 o C One can see that coincidence of theoretical and experimental data is satisfactory. As it may be seen the tunnel overheating at this current is not so much. Unfortunately other experimental data concerning temperature measurements are still absent in the literature. But information about measured values of film resistance enables us to conclude that at low current ( all the more at extra-low current ) with small contact load the Kohler effect may plays main part in contact heating. Experimental data for the measured values of contact resistance for Au contacts [2] , given in the Table 1 (columns 1 - 4 ) , and corresponding calculated values of Ko criterion ( 5-th column ) lead to the conclusion that at low current ( corresponding to low contact load ) tunnel heating is more much than Joule one. Table 1. Dependence of constriction resistance Rc , film resistance R f , and Ko criterion on the contact load N . f N 10 3 kg 10 400 35 11 1.15 8.6 5.0 6.1 4.6 13 Ohm m Rc 2 4 Rf Ko 4 10 Ohm 10 Ohm 1.16 3.94 7.0 22.0 0.28 1.86 7.2 52.0 56 0.24 0.47 1.08 2.36 6 NON-STATIONARY TEMPERATURE FIELD 6.1 Transient model of Joule contact heating For the temperature averaged values of and the equations (9) and (10) take the form T 1 a 2 ( T grad 2 ) t 0 (1.6.1) (1..6.2) where a 2 / C is the thermal diffusivity of contact material, C is the thermal capacity, and is the density. In cylindrical coordinates r, z 2T 1 T 2T r 2 r r z 2 , grad 2 ( )2 ( )2 T r z If a contact junction is symmetrical, then the domain occupied by one contact member is cylindrical semi-space 0 r , 0 z . Since the boundary conditions at this stage remain the same like (A.1.3) - (A.1.5) in the Appendix 1, the expression for potential (r, z) remains the same as well. Thus it can be defined by the expression (A.1.28), where the current should be now timedependent: ( r, z ) I (t ) arctan 2 r0 r0 (1.6.3) where (r, z, r0 ) is the expression obtained from the equation (1.6.4) ( r, z, r0 ) 1 s s 2 4r0 z , s 2 r 2 z 2 r02 2 (1.6.4) In principle substitution the expression (A..3.3) in the equation (A.3.1) enables one to get a standard solution of linear heat equation with the boundary conditions (A.1.3)-(A.1.5) and zero initial condition in the form t T ( r, z, t ) 2 dt1 dz1 G( r, r1, z z1, t t1 ) grad 2 ( r1, z1 )rdr 1 1 C 0 0 where G(r, r1 , z, t ) is the Green’s function defined by the expression 57 (1.6.5) G( r, r1 , z, t ) 1 4a 3 r 2 r12 z 2 rr exp( ) I 0 ( 12 ) 2 3 4a t 2a t t (1.6.6) 2 However integrand grad in (A.3.5) has a singularity at the point z 0, r r0 , and it creates a problem at the numerical calculation of the triple integral (A.3.5). From physical point of view this singularity means that current j (r , z ) 1 grad 2 on the contact spot z 0 approaches infinity as r r0 . density That is follows from the expression j ( r,0) I 2 r0 r0 2 r 2 , r r0 (1.6.7) which can be easily derived from (A.1.3) and (A.1.4). To avoid this singularity we use the relation grad 2 1 2 2 which is valid for any harmonic function, in particular for . Then the substitution u T 1 2 2 (1.6.8) enables one to reduce the heat equation (A.3.1) to the form u 1 2 a 2 u t 2C t (1.6.9) which has no longer singularity. The equation (A.3.9) at corresponding boundary conditions can be solved using Laplace transform. Finally we can write for T (r, z, t ) the expression T ( r, z, t ) ( r, z, t ) 0 dz1 G ( r, r1 , z z1 , t ) ( r1 , z1 , t ) r1dr1 t 0 0 2 dt1 dz1 G ( r, r1 , z z1 , t t1 ) (1.6.10) ( r1 , z1 , t1 )r1dr t1 where (r, z, t ) is the corresponding quasi-stationary solution (stationary solution with t as parameter) defined by the expression 58 ( r, z, t ) I 2 (t ) 4 ( r, z, r0 ) [1 2 arctan 2 ]. 2 32r0 r0 (1.6.11) ( r, z, r0 ) and G( r, r1, z, t ) are defined by the expressions (A.3.4), (A.3.6). The another equivalent form of this solution T ( r, z, t ) ( r, z, r0 ) I 2 (t ) arctan 2 2 2 8 r0 r0 I 2 (t ) 2 (r , z , r ) dz1 0 G(r, r1, z z1, t ) arctan 1 r01 0 r1dr1 8 2 r0 t 16r0 2 0 [ I 2 (t ) I 2 (0)] 0 (1.6.12) I (t1 ) I (t1 ) dt1 16r0 2 dz1 G ( r, r1 , z z1 , t t1 ) arctan 2 ( r1 , z1 , r0 ) r0 r1dr can be derived from (A.3.1) by integration using the formula dz G(r, r , z z , t )rdr 1 1 1 1 1 1 0 In particular the temperature at the centre of the contact spot is I 2 (t ) T (0,0, t ) 2 2 dz exp( r 2 z 2 ) 4 r0 0 arctan 2 t 0 (2a tr, 2a t z, r0 ) r0 rdr 16r0 2 [ I 2 (t ) I 2 (0)] (1.6.13) I (t1 ) I (t1 ) dt1 dz exp( r 2 z 2 ) 16r0 2 0 arctan 2 (2a t t1 r, 2a t t1 z, r0 ) r0 rdr The integrals in (A.3.10)-(A.3.13) have no singularity now and can be calculated easily using MathCad. 6.2 Special cases. 1. I (t ) I const . In this case the formula (A.3.11) gives T ( r, z, t ) 1 2 1 2 2 ( ) (1.6.14) dz G(r, r , z z , t ) 1 1 1 0 59 2 [ ( r1 , z1 , r0 )]r1dr1 and from (A.3.13) T (0,0, t ) arctan I 2 (t ) dz exp( r 2 z 2 ) 2 2 r0 2 0 0 2 (2a tr , 2a t z , r0 ) r0 (1.6.15) rdr If I (0) 0 , then arctan 2 I (t1 ) I (t1 )dt1 dz exp( r 2 z 2 ) 2 r0 5/ 2 0 0 0 t T (0,0, t ) (2a t t1 r, 2a t t1 z, r0 ) r0 rdr (1.6.16) If I (t ) I 0 sin t , then arctan I 0 2 sin 2t1dt1 dz exp( r 2 z 2 ) 2r0 2 5/ 2 0 0 0 t T (0,0, t ) 2 (2a t t1 r , 2a t t1 z , r0 ) r0 (1.6.17) rdr 6.3 The role of the arc flux and Joule heating in the erosion of electrical contacts Let us consider now the dynamics of heating of opening contacts by volume and surface thermal sources. In this case power losses consumed for the phase transformations (melting and evaporation) will be neglected. The temperature and electromagnetic fields will be described by the system of equations 1 a 2 ( grad 2 ) t 0 (1.6.18) (1.6.19) with the initial and boundary conditions (r, z,0) 0 (r , 0, t ) r P(r , t ) r 2 z 2 0 (1.6.20) 1.6.21) (1.6.22) (r , 0) 0, 0 r r0 (r , 0) 0, z r0 r (1.6.23) Here we use the same notation as the previous part, with the averaged values of , and a2 /(c ) , c is the heat capacity, is the density of the contact material. 60 We represent the temperature in the form of two components (r, z, t ) 1 (r, z, t ) 2 (r, z, t ) (1.6.24) where 1 (r, z, t ) is the temperature component corresponding to the volume Joule heating, and 2 (r, z, t ) is the surface temperature component. The function 1 (r, z, t ) is the solution of the problem (23)-(28) for P(r, t ) 0 , and 2 (r, z, t ) is the solution of the same problem (23)-(28) for grad 0 . Since (r, z) does not depend on the temperature in this case it can be defined by the formula (14), i.e. (r , z ) I 2 arctan 2 r0 r (1.6.25) This function satisfies the Laplace equation (24), i.e. it is a harmonic function for which 1 2 2 1 (r , z, t ) introduce the new unknown grad 2 Thus if we instead u (r , z, t ) 1 (r , z, t ) 1 2 function u(r , z, t ) 2 (r , z ) (1.6.26) Then we get for u(r , z, t ) the problem 2u 1 u 2u u a2 2 t r r z 2 r u (r , z,0) 1 2 (1.6.27) 2 (r, z) (1.6.28) u (r , 0, z ) 0 z I 2 u r 2 z 2 u 2 32ro (1.6.29) (1.6.30) The last formula coincides with (12). The boundary conditions (34)-(35) can be obtained using the formulas r0 2 2 , lim r0 r r r0 z 0 z 0, u( r,0, t ) 1 ( r,0) ( r,0) z z r 2 r 2 , 0 lim z 0 0, r r0 0, ( r,0) I r 2 r0 2 arctan , r0 2 r0 I ( r,0) 2 r 0 z 0, 1 r0 r 2 2 , r r0 61 r r0 r r0 r r0 r r0 r r0 The solution of the problem (32) – (35) is u( r, z, t ) u 1 4a 3 t 3/ 2 1 2 dz1 0 2 (r1, z1 ) u r 2 r12 ( z z1 )2 rr I 0 12 exp r1dr1 4a 2 t 2a t Taking into account that r 2 r12 ( z z1 )2 rr1 3 3/ 2 dz I exp r1dr1 4a t 1 0 0 2a2t 4a 2 t we obtain u( r, z, t ) 1 8a 3 t 3/ 2 0 2 dz1 (r1, z1 ) r r ( z z1 )2 rr I 0 12 exp r1dr1 4a 2 t 2a t 2 2 1 Thus finally 1 (r , z, t ) 1 2 2 (r , z ) 8a 3 1 2 dz1 0 (r1, z1 ) t 3/ 2 (1.6.31) r r ( z z1 ) rr I 0 12 exp r1dr1 4a 2 t 2a t 2 2 1 2 For small values of t this formula can be rewritten in more convenient form 1 (r , z, t ) 8a 3 1 2 2 dz1 0 (r1, z1 ) (r, z) t 3/ 2 r 2 r12 ( z z1 ) 2 rr I 0 12 exp r1dr1 4a 2t 2a t (1.6.32) It is not difficult to check that 1 (r , z, ) u 1 2 2 (r , z ) I 2 4 2 1 2 arctan 2 r0 32r0 (1.6.33) that coincides with the obtained above formula (13). Thus the solution for large t can be represented as the small deviation from the stationary solution 62 1 (r , z, t ) 1 (r , z, ) u 8a 3 1 t 3/ 2 dz1 2 (r1 , z1 ) 0 r 2 r12 ( z z1 ) 2 rr1 I 0 2 exp r1dr1 4a 2t 2a t u (1.6.34) I 2 1 (0, 0, ) 2 32r0 As is well known, at the initial stage of Joule heating the temperature maximum occurs at the edge of the contact spot r r0 , where the current density is maximal, then it moves toward the centre of the spot, and finally at the stationary regime whole contact spot becomes isothermal. Let us investigate this motion of the isotherm r rm (t ) on the contact spot z 0, 0 r r0 where the temperature reaches its maximum: 1 (rm (t ), 0, t ) max (r, 0, t ) 0 r r0 We have 1 (r , 0, t ) 1 8a 3 t 3/ 2 2 dz1 (r1, z1 ) (1.6.35) 0 r r z rr I 0 12 exp r1dr1 4a t 2a t 2 2 1 2 2 1 Thus the temperature at the centre and at the edge of the spot are I 2 1 (0, 0, t ) 2 32 5/ 2 a3 r0 t 3/ 2 dz arctan 2 1 0 1 r0 r12 z12 exp r1dr1 2 4a t 1 (r0 , 0, t ) I 2 2 32 5/ 2 a 3 r0 t 3/ 2 dz1 arctan 2 0 1 r0 r r z rr I 0 0 21 exp 0 r1dr1 4a t 2a t 2 2 1 2 2 1 where 1 1 2 2 2 S1 S1 4r0 z1 , 2 S1 z1 r1 r0 2 2 2 If we use the substitution of variables in the integrals r1 r0 x, z1 r0 y then they can be represented in the form 1 (0, 0, t ) I 2 ( ) 2 1 32 r0 1 (r0 , 0, t ) I 2 1( ) 2 32 r0 63 (1.6.36) (1.6.37) where 1 ( ) 8 5.2 2 ( x dy arctan x, y, e 1( ) 2 y2 ) xdx, 0 8 5/ 2 dy arctan 2 x, y, 0 e 2 x I 0 (2 x ) exp ( x ) 2 y 2 xdx, 1 S ( x, y , ) S 2 ( x , y , ) y 2 2 x2 y 2 S ( x, y , ) 2 2 1 ( x, y , ) The graph f the functions 1 ( ) and 1 ( ) is represented in Fig. 3 and Fig. 4 Fig.3 Graph of the function 1 ( ) Fig.4 Graph of the function 1 ( ) The desired isotherm r rm (t ) is determined from the maximum condition 64 r r ( t ) r r (t ) r (t ) r 2 dz1 0 (r1, z1 ) 2a12t I1 m2a 2t 1 2ma 2t I 0 m2a 2t 1 r 2 (t ) r12 z12 exp m r1dr1 0 4a 2t After substitution x r1 , 2a t y z1 2a t we get I ( ) dy 2 ( x, y ) xI1 (2vx ) vI 0 (2vx ) (1.6.38) 0 exp v ( ) x y xdx 0 2 2 2 where 2a t , v( ) rm (t ) . 2a t Expanding the left side of (41) with respect to and equating the coefficients at the same degree we get for small values of v( ) r0 2a o(1) (1.6.39) where can be found from the transcendent equation lim 0 dI 0 d (1.6.40) Taking into account the asymptotic formulas I 0 (2vx ) exp( 2vx ), I1 (2vx) exp( 2vx) 1 2 vx , if 0 we get lim I ( ) lim 0 0 dy ( (v x ), y ) 2 1 0 x1 (v x1 ) exp x12 y 2 dx1 0 2 v(v x1 ) One can derived from (42) rm (t ) r0 t o( t ) (1.6.41) From the formula (40) we can obtain the expressions for the temperature at the edge ( r r0 ) and in the centre) of the contact spot ( r 0 ) if the time if sufficiently small: 1 (0,0, t ) 2 u , 2 2 4 u 1 (r0 , 0, t ) 3/ 2 , r I 2 0 , u 2 32r0 2a t 65 (1.6.42) (1.6.43) (1.6.44) These formulas can be applied for 1, or in the terms of Fourier criteria Fo a 2t 1 . 2 r0 The comparison the graph for 1 ( ) (Fig. 3) with the asymptotic formula (45) enables us to conclude that they both give the same value for 2 , 1 (2) 0.05 . For the values 2 the formula (45) gives more exact result than the graph in Fig. 3 due to an inaccuracy of calculation of the integral. Thus it is recommended to use the graph only for 0 2 , and the asymptotic formula (45) for 2 . The magnified graph of 1 ( ) from Fig. 3 is represented in Fig. 5. The same conclusion remains correct for the asymptotic formula (46) which should be applied for 2 < while for the range 0 2 one can use the graph. Fig. 5 The graph of 1 ( ) for 0 2 It should be noted that in the range of very high current, opening velocity and contact forces the temperature maximum at the edge of a contact spot may reach the melting point with further formation of a hollow liquid bridge and specific arc [9] – [10]. The asymptotic formula (46) may be very useful to estimate this phenomenon. Let us consider now the second temperature component 2 (r, z, t ) conditioned by the arc surface flux. It satisfied the heat equation (23), where grad 0 , and the conditions (25) – (27). The solution of this problem can be written in the form [11] – [12] 2 (r , z, t ) t 1 2a e P(r1 , t1 ) (t t1 )3/ 2 0 dt1 0 r 2 r12 z 2 4a 2 rr ( t t1 ) I0 2 1 r1dr1 2a (t t1 ) (1.6.45) We consider two most typical radial distribution of the arc heat flux: 66 a) the constant flux along the contact spot P0 const , P(r , t ) 0, 0 r r0 (1.6.46) r r0 b) the flux with the normal Gaussian distribution P(r , t ) Pa (t ) e r2 ra 2 ( t ) (1.6.47) where ra (t ) is the effective radius of he entering heat flux. Sometimes it can be identified with the arc radius. For the constant heat flux (49) we have the solution r dt1 0 2 (r , z, t ) 3/ 2 e 2a 0 t1 0 t P0 r 2 r12 z 2 4 a 2t1 rr I 0 21 r1dr1 2a t1 In particular, 2 (0,0, t ) P0r0 2 ( ), 2 ( ) 1 1 i erfc (1.6.48) Fig.5 Graph of the function 2 ( ) Similarly, using the formula 1 xe 0 x 2 I 0 (2 x)dx 1 e e I 0 (2 ) 4 we get 2 (r0 , 0, t ) P0 r0 2 ( ), (1.6.49) 2 2 2 x 1 e I 0 2 dx x (1.6.50) where 2 ( ) 1 2 1/ 0 The graph of the function 2 ( ) is presented in Fig. 6 67 Fig.6 Graph of the function 2 ( ) For 1 we get 2 (1/ ) 1 1 1 2 and 2 (r0 , 0, t ) 1 o 2 2 P0 r0 1 (1.6.51) Similarly for the heat flux (50) we get 2 (r , z, t ) 1 2a t Pa (t1 )dt1 (t t ) 3/ 2 1 0 2 r r r1 z 2 rr1 exp I 0 2 r1dr1 2 0 ra (t1 ) 4a (t t1 ) 2a (t t1 ) 2 1 2 2 (1.6.52) Integrating and using formula e 0 r12 2 1 4 I 0 ( r )rdr e 2 we obtain 2 ( r, z, t ) a t 0 Pa (t1 ) ra 2 (t1 ) (t t1 )1/ 2 z r2 exp 2 2 2 4a (t t1 ) ra (t1 ) 4a (t t1 ) dt 1 ra 2 (t1 ) 4a 2 (t t1 ) 2 68 (1.6.53) In particular, 2 (0,0, t ) a t (t t ) P (t1 ) ra 2 (t1 )dt1 ra 2 (t1 ) 4a 2 (t t1 ) a 1/ 2 0 1 and if Pa (t ) Pa const , ra (t ) r0 const then 2 (0,0, t ) P0 ra arctan 1 (1.6.54) It is interesting to note that for 1 we get here the same asymptotic like (54). 7 BOUNCING AND WELDING OF ELECTRICAL CONTACTS 7.1 Static welding Stationary model of static welding. Symmetric contacts. Welding of electrical contacts is a reason of failure in switchers, connectors, relays and other electrical equipment. The problem of optimal choice of contact parameters with low weldability is very important to enhance endurance and reliability of contact systems. Analysis of investigations in this field shows that the formation of a weld is a complicated function of current, voltage, contact force, parameters of contact material, arc duration etc. The variation of the weld strength may be of a wide range even at the same current. That is a reason why many experimental data are presented in a form of cumulative probability [1]-[3]. As a rule experimental investigation of welding phenomena is focused on the resulting values of welded area and welding force because of very short duration of this phenomenon, which embarrasses measurements in dynamics. Nevertheless evolution of contact surface melting and its further welding is very important for understanding and explanation of welding characteristics. Mathematical modelling is very helpful in this situation. There are two types of welding in electrical contacts. The first type is static welding, which occurs in closed stationary contacts with a short circuit current. The second one is dynamic welding stipulating by electric arc in closing contacts with bounce. Simple expressions for approximate estimation of welding forces are given in [4]. Some models and theoretical criteria of static weldability are presented in [5] for the steady heat state and in [6] for the transient heat state. Later this approach was developed for asymmetric contact pairs [7]-[8]. However simplifying assumptions concerning adiabatic conditions of heating, neglect of phase transformation, and dynamical increase of contact radius during melting restrict significantly the range of application of these models. The radial temperature distribution on the contact area is also important for specificity of welding, especially in the range of high current and high contact loading force. 69 If that’s the case, then the welding area on a surface of closed contacts has ringshaped form due to displacement of current density and temperature maximum hardly from the centre of contact spot to its edge .This phenomenon was observed experimentally in [9], [10] and [29]. It occurs at the initial stage of non-stationary contact heating. That is the reason why for a low range of current and contact force such displacement can be observed because of very short duration of transition to stationary temperature field in contacts, therefore welding area in low current contacts is observed usually in the form of a circle. Thus mathematical models for static welding in closed contacts should be chosen different for low and high current. Investigation of dynamic welding occurring at contact closure with arcing is more complicated due to difficulties in mathematical description of arc evolution. If welding occurs at the first contact touch after melting of a microasperity, then the welding force is weak as a rule due to very short arc duration. However the problem becomes more serious with welding at contact bouncing. It is interesting to note that strong contact welds may be formed in case of single very short bounce arcs as opposed to long and multi-bounce arcs [11]. The influence of asymmetry of contact material and polarity on anti-welding properties and erosion is considered in [12]. Publications concerning mathematical modelling of dynamic welding are very rare. The model presented in [8], [13] is based on the thermionic electron emission at arcing and heat equation. However dynamic parameters of bounce length and duration, time dependence and space distribution of welding characteristics, dynamics of phase transformation remain out of deliberation. An attempt to involve these aspects into consideration is made in [14]-[15]. This paper presents a generalization of static welding model for non-linear case of contact characteristics and for ac current. Further development of the model of dynamic welding in dependence on bouncing parameters is considered as well. The fundamental potential-temperature relation discovered by Kohlrausch (Appendix 1, formula (A.1.26)) and proved mathematically in [18] and [19] is a theoretical base for explanation of many stationary phenomena occurring in closed electrical contacts such as contact resistance, softening, melting and boiling potentials, thermoelectric effects in contacts, fritting phenomena etc. [16], [4]. This relation is very important for study of welding in closed stationary contacts. The threshold welding current IW related to attainment of melting temperature on the contact spot and the criterion of weldability W in symmetric stationary contacts for different contact materials are presented in [5] in the form 70 IW r0 2 2W W TW TW (1.7.1) (T )[ (T ) (T )dT ] 1/ 2 1 0 2 2 2 dT1 (1.7.2) T1 The radius of the contact spot r0 in this formula was calculated taking into account temperature dependence of contact hardness H H (T ) r0 F H (TW ) (1.7.3) where F is the contact force and TW is melting temperature of contact material. It has to be noted that the expression (3) should be corrected due to electrodynamic repulsion force, thus r0 should be found as the solution of the equation F 107 IW ln 2 r0 H (TW ) r0 R (1.7.4) In the range of high current this correction is very important [4]. Results presents in [5] are in a good agreement with experimental data. However the problem of spatial temperature distribution, which is important as well for welding characteristics, especially for welding area and force in asymmetric contacts, was kept a mystery. Such information can be obtained by the direct solution of the differential equations describing temperature and potential distribution inside contacts rather than by use of Kohlrausch’s law. This approach is presented in the Appendix 1. The threshold welding current IW can be found from the equations (A.1.24) - (A.1.25) at the values T (0) TW , (0) / 2 exactly at the same form (1). In the case of linear temperature dependence of electrical resistivity and thermal conductivity the formula for welding current can be derived from the expression (A.1.33) at T 0, T0 TW , / 2 IW 4r0 0 ( ) 1 { arccos 0 1 ( )TW [1 ( )TW ]2 1]} 71 (1.7.5) It is interesting to note that if we put here 0 , then calculating the limit as 0 , using for r0 the expression (7.1.3) and replacing 0 by the averaged value 2 3 0 (1 TW ) [16] we get the well-known Hilgarth’s formula [20] 0TW F IW 4 2 (1.7.6) 2 H 0 (1 TW ) 3 The relation between the radius of welding area r0 and current can be found by the solving of the system of equation (7.1.1), (7.1.4). Fig. 1 depicts results of calculation of the diameter d of welding area as a function of current [21]. The curve 1 is calculated using Hilgarth’s formula (7.1.6), which gives redundant temperature, the curve 2 relates to the formula (7.1.3), and the curve 3 corresponds to the expression (7.1.4), which is mostly inside the dashed area of experimental data. d , 103 m 2 1 3 3 2 1 0 8 1 6 2 4 3 2 I , kA Fig.1 Diameter of welding zone as a function of current [20] The weld force FW can be estimated approximately as FW r0 , 2 where is the tensile strength of contact material. The dependence of weld force on current is shown in Fig.2. 72 (1.7.7) FW , N 3 10 1 100 3 2 10 1 4 5 6 7 8 9 10 I , kA Fig. 2 Weld force as a function of current The area between curves 1 and 2 corresponds to experimental data [21], the curve 3 is calculated One can see a satisfactorily agreement of calculated values and experimental data. Asymmetric contacts. Welding phenomenon in asymmetric contacts is characterized by a displacement of temperature maximum inside the contact member with greater value of . The values of this maximum Tm and corresponding voltage m can be obtained by modelling on the base of temperature-potential relation at the assumption that m UC / 2 [22]-[24]. However this assumption can be accepted as an approximation only. Sometimes it fails, especially if materials used for contact junction have quite different values of and . Furthermore, the information about the value of maximum displacement, which is very important for welding parameters, can not be achieved using this model. However it is available if we apply the description of temperature and electrical fields by differential equations similar like in previous section. Such model is presented in the end of Appendix I. The criterion of weldability should be ascribed now not to temperature maximum Tm like in the expression (3) but to the temperature on the contact surface T0 where welding occurs (Fig. 3). 73 r Tm , m T0 , 0 2 0 2 m T1 , 1 T2 , 2 r Fig.3 Distribution of temperature and potential in asymmetric contacts It can be defined from the equation (1.7.6) as T0 T0 Tm W 1 (2 1 1dT 2 2 2 dT ) 1/ 2 dT . 0 T (1.7.8) T0 The values of T0 , Tm and displacement m can be calculated by the numerical solving of the system of equations (A.2.9)-(A.2.11). Then the threshold welding current and the weld force can be found from the same expressions (7.1.1), ( 7.1.4) and (7.1.7). Fig. 4 depicts the axial temperature distribution between Ag and W contacts with radius of contact area r0 3.17 104 m at the current 2 kA calculated using the expressions of Appendix II. T ( z ), 0C 1500 1400 1300 1200 1100 1000 900 800 700 600 500 -0.4 -0.3 -0.2 -0.1 0 0.1 zm 0.3 0.4 z, mm 74 Fig. 4 Axial temperature distribution between Ag and W contacts One can see that the temperature on the contact spot is T0 963 0C , thus welding should occur. The calculation for symmetric Ag contacts using expressions of Appendix I yields at the same conditions the value T0 233 0C , while for symmetric W contacts the temperature should be T0 2788 0C . The temperature maximum T0 1262 0C is displaced from the contact plane z 0 to the point zm 1.79 104 m . The potential distribution is shown in Fig. 5. i , V 0.1 2 0 0.1 1 0.2 0.3 -0.4 -0.3 -0.2 -0.1 0 0.1 zm 0.3 0.4 z, mm Fig.5 Potential distribution in Ag against W contact junction The total contact voltage in this case is UC 0.513 V , the potential of contact spot is 0 0.168 V . The potential at the point zm 1.79 104 m , where temperature maximum occurs, is m 0.029 V . Thus the assumption that m 0 [23] is sometimes not correct. Transient model of static welding. In the case of ac current or dc pulse current of a short duration the above considered stationary model ceases to be valid because of time dependence of all parameters responsible for heating and welding of closed contacts including temperature, voltage, current, contact area etc. The stationary mathematical model describing contact heating at the stage up to attainment of melting temperature should be replaced by the nonstationary heat equation C 1 div( grad ) grad 2 t while the equation for electric potential remains the same 75 (1.7.9) 1 div( grad ) 0 (1.7.10) The simplified adiabatic model of transient contact heating is presented in [6] based on the assumption that div( grad ) 0 and that current density is averaged over contact cross-section. A criterion of weldability as a function of melting temperature, heat capacity, resistivity and hardness was introduced. Later this approach was extended for asymmetric contact junctions [8]. However man-made assumptions, restrict significantly incidence of application of this model. Neglect of thermal conductivity is correct only for a heating due to current pulse of very short duration. From the other hand averaging-out of current density for phenomena of a short duration is a problematical operation, especially in the range of high current and contact load. For example, as it will be shown below, the phenomenon of ring-shaped welding occurs due to radial nonuniformity of current density on the contact spot, which maximum is displaced to the edge of contact area. The model based on the general heat equation (7.1.9) is presented in the Appendix 3. The fulltime of contact Joule heating is divided in two stages, premelting and melting. At the first stage no phase transformation occurs, thus the boundary conditions are described now as before by the equations (A.1.3)(A.1.5). To get an analytical solution for examination of heat transfer dynamics, electrical conductivity and thermal conductivity are averaged over temperature accordingly to recommendations advised in [16]. It is important at this stage to take into account increase of contact area at the attainment of softening temperature and the action of electromagnetic force using the expression (7.1.4). Results of calculation using the model presented in the Appendix 3 are shown in Fig. 6 and 7 T , 0C 1000 900 800 700 600 500 400 300 200 100 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t , ms Fig. 6 Dynamics of contact temperature at pre-melting stage 76 They depict dynamics of the temperature at the centre of the contact spot and dynamics of contact radius during pre-melting stage for the following data: contact material is Ag-Ni (20%), current is half-wave pulse of 1.4 kA in a peak and 15 ms of duration, contact force is 1.4 N . These parameters were used in experiments described in [25]. r0 (t ), m . 40 35 30 25 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t , ms Fig. 7 Dynamics of contact radius at pre-melting stage One can see increase of contact radius when contact temperature reaches softening temperature and corresponding deceleration of temperature rate in spite of current rise. The second stage of the Joule heating begins when the temperature on the contact spot riches the melting value. The main problem in modelling of transient welding is estimation of melted area. It should be noted that due to nonuniformity of the current density on the contact spot, which is expressed by the formula (3.7) , the temperature maximum is displaced from the centre of a spot to its edge. That is a reason why melting always starts from the circumference of contact circle and expands toward its centre. After some time the whole contact spot may be heated up to approximately the same temperature and stationary regime comes. The time required for the transition from nonstationary to stationary temperature field in contact area should be estimated using Fourier criterion (dimensionless time) Fom a 2tm / r02 , where tm is the time from the initial current passing to attainment of melting temperature Tm at the hottest place z 0, r r0 . This time can be calculated from the equation T (r0 ,0, tm ) Tm (1.7.11) If Fom is much greater than 1, then stationary regime becomes very quickly and simple stationary model presented in Appendix 1 can be successfully applied. However if Fom is much less than 1, then the melted zone has the shape of a anchor ring on the periphery of the contact spot. In any case this phenomenon of the liquid phase formation away from the centre of spot should 77 be taken into consideration at the modelling of the initial stage of welding. In this case in addition to the above mentioned boundary conditions the new conditions on the interface z (r, t ) between melted and solid zones should be considered. These conditions, so-called Stefan conditions, can be written in the form [26] T (r, (r, t ), t ) Tm [1 ( 2 T ) ] r z ( r ,t ) z ( r ,t ) (1.7.12) j 2dz Lm 0 t (1.7.13) where Lm is the latent heat of fusion. The solution of this problem is based on the theory of heat potentials with following reduction of free boundary problem to non-linear integral equations. To obtain approximate solution the new integral method is proposed. As provided by this method the temperature rise is suggested being non-zero in the most heated initially region 0 z Z1 (r, t ) with a moving boundary S (t ) and then it expands in both directions according to the equation The temperature T (r, z, t ) and surface of phase transformation z (r, t ) are represented in the form z 2 (r , t ) 2 T (r , z, t ) Tm (1 )2 , 2 2 Z1 (r , t ) (r , t ) (r , t ) (t ){1 [r r0 (t )]2 1/ 2 } (t )2 The functions S (t ), (t ), r0 (t ) are defined from the system of ordinary differential equations. Results of numerical calculation for Cu contacts (the amplitude of ac current is 70 kA, the contact force is 3500 N) are represented in Fig. 8. They enable to explain the experimentally observed [10], [27] a ring-shape zone of most intensive welding 1 (t ) r 2 (t ) corresponding to this maximum and two zones of partial welding: internal 0 r 1 (t ) and external 2 (t ) r 3 (t ) . 78 (t ), 103 m 3 (t ) 3 2 (t ) 2 1 1 (t ) 0 Iˆ 0 2 4 6 8 1 t , 103 s 0 Fig.8 Evolution of welding zones Fig. 9 depicts expansion of the melted zones from the edge of the contact spot ( r r / r0 1 ) to its centre ( r 0 ). Isothermal surfaces of melting temperature Ti , i 1, 2,...,5 correspond to increasing Fourier criteria Foi a 2ti from the start r0 (ti )2 of Joule heating ( i 1, Fo1 0.01 ) to reaching of stationary temperature ( i 5, Fo5 2.5 ), when the temperature on the contact spot is constant along its radius. One can see that ring-shape contact heating and welding occurs only if Fo 2 , i.e. for sufficiently large values of the radius of a contact spot r0 , in other words for large contact forces used in high current equipment. Otherwise this non- stationary duration of the ring-shaped heating passes very quickly and can not be observed at the ordinary conditions. For example it is equal about 2 µs for the Cu-Cr contacts at current 20 A and contact force 200 N. z 1.0 0.8 T5 0.6 T4 0.4 0.2 T3 T2 T1 0 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 r Fig. 9 Dynamics of temperature zones 79 2.00 7.2 Dynamic bouncing and welding during contact closure Introduction. Phenomena occurring at closure of electrical contacts in vacuum circuit breakers are very important for understanding of arc evolution, mechanism of contact erosion and welding. Dynamics of contact closure should be considered as a consecutive chain of interrelated stages, and parameters at the end of each preceding stage should be assigned as the initial conditions for the following stage. Electrical breakdown, micro-arcing, melting and evaporation of micro-asperities, compression (penetration) have to be considered at contact approaching, and then stages at contact opening (restitution, bridging and arcing) should be investigated as well. Electrical arc igniting at make and break operations due to thermal ionization or electrical breakdown is the main source of failure in vacuum circuit breakers [1]. Formation, evolution and explosive rupture of molten contact bridge at the pre-arcing stage of contact opening are very important for further dynamics of electrical arc [2], [3]. Contact bouncing accompanying by welding is another cause of failure in low voltage electrical apparatus [6]-[8]. It may be reduced by decrease of contact force and deceleration of closure process; however another problem appears in this case. Gas pressure in contact gap during arcing at closure may produce contact quakes and phenomena of contact floating which increases arc duration and contact erosion [4]. In the case of short-circuit current it may provoke even contact repulsion [5]. The main goal of this paper is to interlink all above mentioned phenomena at contact closure and opening in dynamics on the base of a general model. Experimental set up. A standard vacuum circuit breaker used for experiments. The contact system was mounted inside vacuum bottle on a massive support. Two laser sensors and special equipment were used to measure displacement of each contact member and to calculate the distance between contact surfaces by Difference Path Method (DPM), which is very convenient for analysis of contact gap dynamics, compression, restitution and fluctuations of contact force. [4]. The current variation from 20 A to 2 kA (dc or half-wave ac) was measured by non-inductive resistance. For voltage measurement, which variation is very large, 3 voltage dividers were applied. An ordinary RC–divider was used to measure high-range voltage up to 400 V, isolation amplifier (optical coupling) was applied for voltage in the range from 4 V to 40 V, and a clamp diode device with non-linear characteristic was inserted inside to measure voltage across closed contact, which range varies from 0 to 4 V. All other parameters are the same like described in the paper [4]. Experimental results of contact closure dynamics are presented in Fig. 1. 80 Fig. 1 Dynamics of contact closure Arc ignition and current start at t0 0 , when the contact gap is S0 63 m , and the first contact touch in molten metal occurs at FW rm (t ) 2 (the point A1). Thereby both contact pieces, fixed and movable, get an impulse and bounce in opposite directions due to the pressure of expanding vapour and molten metal until the next touch in solid ground (the point A2) at t2 480 s . Then the last bounce occurs, however without contact opening, and the final touch (the point A3) is accompanied by contact welding at t3 860 s . All consecutive interrelated stages of contact closure should be considered step by step. Initial stage of contact closure. Electrical contact at closure can be established due to breakdown by strong electrical field or due to touch and explosion of micro-asperities with thermal ionization. Breakdown of the gap between approaching contacts in vacuum circuit breakers occurs at electrical field [1]. It means that the critical contact gap required for breakdown in low voltage vacuum circuit breakers ( U 400 V ) 81 should be a few microns. This situation can be observed sometimes for a slow contact speed (not greater than 0.1 m / s ). On the contrary, if closure speed is 0.5 m / s or greater, then the duration of micro-arc is insufficient to melt and evaporate a micro-asperity, thus the most apparent mechanism of final electrical contact in this case is the mechanical touch of micro-asperities on contact surface. Fig. 2 depicts the initial stage of contact closure before the first touch (A1) with a greater time resolution. Fig. 2 Contact gap, voltage and current between movable contact piece (mcp) and fixed contact piece (fcp). Cu (60%) – Cr (40%) , V 0.6 m / s One can see that the arc indeed ignites at the contact gap 63 m , that is an evidence of the ignition mechanism by the explosion of micro-asperities. The micro-arc of 35 s duration (stage A in Fig. 2) is the cause of explosive evaporation of contact material accompanying by rise of the pressure in contact gap and backward contact motion. However this pressure decreases very quickly because of vacuum condition, contacts reverse and approach again until the touch of a new micro-asperity with new backward and forward steps of motion (stage B). This phenomenon called contact floating is very dangerous for vacuum circuit breakers [4]. The next touch occurs in a molten area on the contact surface. The current density is not sufficient now for explosion of contact material and arc ignition therefore only a small floating jump arises due 82 to extension of liquid metal and evaporation (stage C). Further expansion of contact area leads finally to the full closure with contact solidification, compression, restitution and following mechanical bounce if the repulsion force is greater than the force of micro-welding. Let us consider the model of contact dynamics at the initial stage before mechanical bounce. Three zones in contact area should be considered to find temperature dynamics and pressure of metal vapours in contact gap, including the zone of evaporated micro-asperity D [ l z 0] with adjoining evaporated area inside 0 contact D1 [0 z b ( r , t )] ,melted D3 [ m (r , t ) z ] D2 [ b (r , t ) z m ( r , t )] zone and solid zone (Fig. 3). r rm (t ) S ( z) D1 rb (t ) D2 D3 h (t ) l T0 T2 T1 D0 0 zb T3 z zm z m (r , t ) z b (r , t ) r Fig. 3 Temperature contact zones. The process of Joule heating and evaporation of a micro-asperity and adjoining zone of contact material should be considered in dynamics step by step beginning from the time of contact touch. The initial stage of heating and melting of a micro-asperity up to boiling temperature can be described by the bridge model, which is presented in [2]. This liquid bridge can be considered as a bar with variable cross-section S ( z ) . The heat equation for the temperature field T ( z, t ) in such domain should be written in the form 0 c0 0 T0 t 1 S ( z ) z [ S ( z )0 T0 z I 0 2 ] 2 S ( z) (1.7.14) where c0 , 0 , 0 , 0 are heat capacity, density, heat conduction, electrical resistivity of molten contact material respectively, and I is the current. The appropriate shape of the cross-section S ( z ) should be chosen from the analysis 83 of a Talyrond trace (profilogram) of the contact surface. In this paper it was identified with a paraboloid having the altitude l and the radius of the base h . If z (t ) corresponds to the cross-section of boiling temperature T , then the boundary condition on this surface after arc ignition is b T0 0 z ( t ) T0 z Tb (1.7.15) d P0 Lb 0 (1.7.16) dt z ( t ) where P0 is the arc power density, and Lb is the latent heat of evaporation. Heat equations for the domains D1 , D2 and D3 can be written in standard form ci i Ti t Ti j i , 2 i 1, 2, 3 (1.7.17) where j is the current density. The boundary conditions should be written separately for each stage. Initially, when molten area is the micro-asperity only, the condition on the boundary T ( r , b ( r , t ), t ) Tb is T0 When the molten area expands into zone z m (r , t ) 0 (7.2.5) T1 D1 T0 z 1 T1 z (1.7.18) the conditions on the boundary are 2 T2 n 3 T3 n Lm m T0 T1 m t (1.7.19) (1.7.20) Finally for the last stage when evaporation boundary goes inside contact the condition on the boundary z b (r , t ) is T T Lb 2 b (1.7.21) 1 1 n 2 2 t n 84 the last boundary condition away from the contact area ( z or r ) is T3 0 (1.7.22) The problem (1) – (10) is solved using method of majorant functions [9]. The interface of evaporation z b (r , t ) is found from the equation T ( r , b ( r , t ), t ) Tb Thus the volume of exploded vaporized zone V1 2 3 D1 (1.7.23) is rb 2 zb (1.7.24) where rb and zb are roots of the equations b ( rb , t ) 0 and b (0, t ) zb Dynamics of the arc temperature TA ( r , z , t ) is calculated using model presented in [10]. The pressure of metallic vapour p (t ) in the arc can be estimated now by gas law p (t ) kTA (t ) (1.7.25) V1 (t ) and corresponding force is F (t ) rb (t ) p(t ) 2 (1.7.26) Fig. 4 depicts dynamics of the vapour force during first and second touches of micro-asperities. F (t ), N 200 160 120 80 40 0 0 20 40 60 80 100 Fig. 4 Forces of metallic vapour pressure 85 To verify model experimental data of contact gap measurement compared with values X c (t ) calculated from the motion equation mX c (t ) F (t ) Fs where Fs X e (t ) are (1.7.27) is the force of contact spring Fs kX c (t ) The results of comparison are shown in Fig. 5 X e (t ), X c (t ) m . 150 120 90 1 60 2 30 0 0 25 50 75 t, s 100 Fig. 5 Dynamics of contact gap 1 – measured, 2 – calculated One can see a satisfactory agreement of calculated data with measured values. IV. BOUNCING The next contact touch at the point A1 occurs in molten metal. Current density j (t ) I (t ) rb (t ) 2 (1.7.28) at this point decreases very quickly in spite of current rise due to rise of contact spot as well generating micro-welding and compression stage of contact closure begins. This stage is described in details in the paper [4]. Contact motion can be described by the equation mX (t ) Fs (t ) Fep Fed Fw 86 (1.7.29) where Fep , Fed and FW are the forces of elastic-plastic deformation, electromagnetic repulsion and welding respectively. Calculation show that elastic-plastic force Fep is much greater than other terms in the right side of the equation (7.2.15), therefore contact opening occurs at the restitution, which accompanies by the arc ignition. Again explosive evaporation of contact material provokes the contact repulsion, which can be modeled and described by the same equations like at the initial stage. After the second bounce at the point A2 the contact voltage decreases up to the value U c 1.5 V (Fig. 6) which is sufficient for boiling and melting of contact material, but not sufficient to maintain arcing. Fig. 6 Dynamics of contact voltage and resistance at bouncing Actually it is a contact backlash rather then opening. Calculating the radius of molten zone rm (t3 ) at the point of the last touch A3 ( t3 860 s ) and then the welding force 87 FW rm (t3 ) where FW 8 2 10 N / m Fep 2 2 is the tensile strength for Cu one can conclude that , thus welding occurs in considered case. 7.3 The model of blow-off phenomena Introduction. Dynamics of contact blow-open forces is very important to provide desired opening conditions during a short circuit current for reduction of arc duration and welding probability. It is suggested as a rule [1] that two forces appearing in closed contacts at inrush of short circuit current are responsible mainly for a character of contact separation at blow-off process. The first of them is electromagnetic force and the second one is arc plasma pressure force. Influence of magnetic force on contact repulsion is investigated in details experimentally [2] and by numerical simulation [3]-[5]. Dynamics of blow-open contact repulsion is explained usually in the following way [6], [7], [8]. Increasing electrical current produces magnetic force, which reduces the load of contact spring and the radius of contact spot. It results in rise of magnetic force. When magnetic force becomes equal to contact spring force contact separation starts, which accompanied by increasing of current density and temperature up to the melting point with following liquid bridge formation. The arc igniting after bridge rupture produces intensive evaporation of contact material and gas pressure in the contact gap; this creates a repulsion impulse and blow-open process. The main problem here is the evaluation of plasma pressure that can be done using some indirect methods. The method employed for this aim in the paper [6] is based on the dependence of arc electrical conductivity on the arc temperature, which is different for different pressure. It enables to estimate average values of blow-open force using their excellent experimental data and Yos’s theory for arc conductivity-temperature relationship [7]. However, as it was mentioned by authors, a more accurate method for calculating the arc dynamics pressure needs to be developed taking into account anode and cathode phenomena, and time dependence of all parameters. The method used in [8] deals with the relationship between plasma emission coefficient and plasma pressure. The analytical model, which is based on the force balance equation, was elaborated to calculate the dynamics of contact gap. Some noticeable discrepancy between results of calculation and experimental data due to simplification of this model indicate the necessity of more detailed time-dependent description of all phenomena at blow-open process. In particular the basic assumption in both papers that 70% of arc power is consumed for arc radiation should be replaced by the more detailed arc power balance. All parameters of blow-open repulsion including arc and contact temperature, phase transformation, forces interaction have to be considered and modelled as time-dependent characteristics. 2 Experimental set up and measurements methods This investigation was concentrated on pre-breaking phenomena occurring at short-circuit conditions in the double-break contact system of low-voltage three-pole circuit breaker for motor protection. Circuit breakers of high breaking capacity are able to clear off short-circuit within the first AC half-period of power frequency. Therefore also the experimental equipment was set up in order to provide facilities for simulation of such conditions. Attention was focused on the pure blow-open phenomena of contacts, therefore no operating means for switching of contacts have been involved. 88 Blow-open experiments were conducted on double-break asymmetric pairs of AgC-AgNi contacts and arc chutes of commercial switching device. The radius of contact cross-section was R0 1.75 mm . Movable contact piece was firmly cemented into contact holder on whom a helical spring of determined characteristics was attached in order to provide the required contact force directly on contacts. No other driving means for switching or loading of contact was attached. A lightweight screen was mounted on contact holder to provide feasible measurement of contact displacement. Test circuit breaker was connected into test circuit by screw terminals, which permitted to connect the stranded conductor of effective cross-section 6 mm2. Test circuit includes the current source in series with make switch, multitap air core reactor, precision shunt resistor, and tested device as shown in Fig. 1. Fig. 1 Shows test circuit diagram A capacitor bank of 10 electrolytic capacitors 2200 F (-10/+30%) 350 V with charging accessory was used as the supply of test current. The discharge was initiated by closing the make switch and by this way an aperiodic current waveform of exponential time rise and fall characteristics was obtained. Air core reactor is applied in order to adjust time-to-peak of test current with waveform in its rising part and around its maximum essentially similar to the AC halfwave of power supply 50 Hz. The maximal achieved peak value was 2 kA. All time-variable quantities are measured as voltage drops by 100 MHz digital CRT oscilloscope over the period of the duration of test current wave. Current trace was picking up by CRT probe on precision shunt resistor 120 mV/250 mA, ± 0,5 %. The terminals of CRT probe for registration of contact voltage drop (and arc voltage) are connected onto power connection terminals of fixed contact pieces. The measuring error due to the additional voltage drop on 89 contact pieces can be disregarded, as the probe terminals were placed as close to the contact sites as possible and the rest of current-carrying parts are short and of considerable cross-section. Two-probe method was used to eliminate other parasitic voltage drops. By this way the measured voltage represents the sum of voltage drops of double-break contact pairs. The position of movable contact bridge was measured by optical sensor. A screen firmly attached on the holder of movable contact bridge was placed in a gap between source of laser beam, which emits 1 mm thick and 10 mm wide beam, and laser-light sensor with orifice of the same cross-section as a receiver for laser beam. The sensor's orifice was partially shadowed by movable screen regarding the instantaneous position of movable contact piece. The output of sensor unit is directly proportional to the unscreened area of orifice with the rate 1,02 V/mm of contact travel. The frequency response and linearity of laser emitter-sensor system was confirmed by measurements as adequate to follow movements at velocity of more than 8 m/s without distortion of its response. It should be noted, that the total voltage drop measured as the sum of both contact voltage drops in series does not give us any information of asymmetry of double break. By measurement of contact movement only translation was recorded although in the case of double-break contacts the presence of rotational component is reported [2], [6]. The rotational component is remarkably reduced but not eliminated by guiding of contact holder as per actual design of regular CB type. The actual asymmetry was indicated at the end of experiment by dismantling of contact system. 3 Experimental results Measurements were carried out in test circuit with circuit parameters maintained at constant values. In fact ohmic resistance, which was actually less than 0, 3, could vary to a certain extent due to variable contact resistance of test device. When the capacitor battery loaded on voltage UC was discharged, a waveform of transient current was obtained, which simulates AC current halfwave. The magnitude of test current wave was adjusted for experiment by charge voltage in the range up to 350 V. Contact system was loaded by tensile force of helical spring. It is the only external force determining the dynamics of movable contact due to negligible gravitation force. The initial value of this force in closed contacts is F0 1.4 N with spring constant k 0.15 N / mm . Fig. 2 depicts typical oscillograms of current, voltage and contact gap at blow-open repulsion during half-wave current with 1.5 kA peak. 90 U (t ), V 60 50 40 I (t ), kA 1.8 1.5 1.2 I (t ) 30 20 10 0 0.9 0.6 0.3 0 U (t ) x(t ) x(t ), mm 1.2 1.0 0.8 0.6 0.4 0.2 0 -0.2 0 2 4 6 8 1 0 t , ms D A BC 12 1 4 1 6 -0.4 -0.6 E F G Fig. 2 Dynamics of voltage U (t ) , current I (t ) , and contact displacement x(t ) Trace of contact displacement starts from the beginning of blow-open to its maximum and returns back into the position of reclose. As it can be seen from graphs, the reclose position does not coincide with the initial close position due to certain burnout of contact tips during blow-open cycle. Analysis of repulsion dynamics enables to conclude that blow-open process should be divided into several consecutive stages. At the first stage of contact separation AB, which duration is t1 1.1 ms , voltage increases from 0 to its boiling value 0.75 V. It is caused due to increase of current and contact temperature. The magnetic force, which should reduce contact load force and increase current density, contributes to this voltage increase as well, however its role is less significant due to relatively small current at this time. Transition to the next stage at the point B (Step 1) lasts from t1 1.1 ms to t2 1.2 ms . It is accompanied by abrupt voltage rise to the value 13.5 V required for arc ignition on the contact M 1 (Fig.1), while the second contact M 2 remains still in a good electric conductance. This voltage is approximately constant for whole arc duration BC in the contact M 1 from the time of arc ignition t2 1.2 ms to the time t3 1.7 ms . The power of this transition is consumed for melting and vaporisation of a zone in the constriction region and for cathode fall formation required providing minimum arc voltage. It will be shown below that this phenomenon occurs as explosion due to Joule heating, however a pressure impulse of generated vapours is not sufficiently high yet to initiate a contact motion. Next voltage jump from 13.5 V to 24 V (Step 2) occurring at the point C from t3 1.7 ms to t4 1.8 ms indicates the similar phenomena of Joule explosion and arc formation on the second contact of bridge pair M 2 . Now the resulting force of vapour pressure and magnetic repulsion in sum on both contacts plus magnetic force becomes to exceed spring force, and contacts start to move. 91 Arc temperature and heat fluxes into contacts become greater provoking increase of evaporation rate, rise of gas pressure and acceleration of contact motion at the beginning of the next stage of separation. However after some time gas pressure decreases due to increasing of contact gap and decreasing of current, and when it becomes equal to spring pressure at the point D ( t5 9.1 ms ) the direction of contact motion change to reverse. The voltage also decreases to the point E, at which it steps down from t6 13.92 ms to t7 14.1 ms indicating arc extinguishing and contact closure of one contact, while the second contact is still arcing. It closes later at the point F stepping down from t8 14.8 ms to t9 14.88 ms . A bounce may be observed sometimes at the final stage of contact closure DE from t9 14.88 ms to t10 16 ms . The object of this paper is the clarification of the mechanism of open-blow process, especially the dynamics of forces at contact opening. Therefore the information about dynamics of current, voltage and displacement will be used directly from the experimental data, rather than derived from the equations to estimate each force component during separation. The mathematical model should describe dynamics of blow-open phenomena as a chain of consecutive stages before and after arc ignition, including pre-arcing stage of contact separation, which continues from the start of opening to arc ignition at the first contact pair M 1 , transition stage up to arc ignition at the second contact pair M 2 , and arcing stage. Each stage should be considered separately. 4. Pre-arcing stage of contact separation The duration of the pre-arcing stage of contact separation can be obtained from the oscillogram (the zone AB in Fig. 2). This stage should be divided into two periods. The first period lasts from the start of current passing to attainment of the melting temperature on the contact surface. The second period continues from melting temperature to arc ignition at M 1 . The total force F (t ) acting the contacts at the pre-arcing stage can be represented as the sum of components F (t ) Fc (t ) Fep (t ) Fed (t ) Fp (t ) Fg (t ) (1.7.30) The force of contact spring Fc (t ) is defined by the expression Fc (t ) F0 kx(t ) (1.7.31) where F0 is the initial value of this force in closed contacts, and k is the spring constant. In the case of double-break bridge contacts the force pressing on each contact is Pc (t ) 0.5 Fc (t ) ; 92 (1.7.32) The force of elastic-plastic deformation of contact zones Fep (t ) acting in both contacts, M 1 and M 2 , which compensates the spring force in state of closed contact (after melting it becomes to equal zero), is defined by the formula Fep (t ) 2 rC (t ) H B 2 (1.7.33) where rC (t ) is the radius of contact spot, is the coefficient of surface treatment (in considered case 1 ), and H B is the contact hardness [1] depending on the temperature. The electrodynamic magnetic force Fed (t ) acting initially in both two contacts is defined as [1] Fed (t ) 2 I 2 (t ) ln R0 rC (t ) (1.7.34) where 107 N / A2 , I (t ) is the current, R0 is radius of contact cross-section. The force Fp (t ) appearing due to pinch pressure from the electrode jet can be estimated from the expression [15] Fp (t ) 107 I 2 (t ) rC 2 (t ) (1.7.35) The gravitation force Fg (t ) is equal to mg for vertical orientation of the contact system and to 0 for horizontal orientation. In considered case the force components Fp (t ) and Fg (t ) are negligible in comparison with the other force components in the equation (1). Electrodynamic force Fed (t ) , which appears in both contact constriction zones at the first period simultaneously with rising current, reduces the contact force Fc (t ) and develops a micro-motion of contacts. Strictly speaking it is decompression of elastic-plastic deformation in constriction zone rather than motion of contacts. This displacement occurs only away from the contact zone, and laser sensors mounted on contact member record it, while the contact plane remains unmoved in axial direction. The reduced contact load Fc (t ) Fed (t ) is counterbalanced by elastic-plastic force Fep (t ) . The dependence of Fc (t ) on displacement x(t ) at this micro-motion is negligible ( Fc (t ) kx(t ) ), thus one can derive from (2), (4) and (5) the equation rC (t ) Pc (t ) I 2 (t ) ln[ R0 / rC (t )] H B (1.7.36) which should be solved with respect to rC (t ) taking into account the dependence of hardness H B on temperature. 93 The temperature field due to volumetric Joule heating in closed contacts TJ (r , z, t , rC ) is given in the Appendix 1 by the expressions (A.6) or (A.9). At the conditions corresponding to Fig.1 ( a2 104 m2 / s, t 103 s, rC 2 1010 m2 ) one can conclude that Fourier criterion Fo a 2t 103 is very large, thus quasi-stationary 2 rC approach (A.9) is correct. Attainment of the melting temperature Tm at the centre of the contact spot z 0, r 0 occurs at the time t tm1 that can be calculated from the equation (1.7.37) TJ (0,0, tm1 , rC (tm1 )) Tm giving the value tm1 0.686 ms The second period of pre-arcing stage consists of two steps. The first one is melting of contact spot, which starts from t tm1 and continues up to the time of boiling attainment tb1 1.1 ms that can be found from Fig. 2. The second step is boiling with duration from tb1 to the time of arc ignition ta1 1.2 ms at the contact M 1 , which can be easily found as well from oscillograms. It is interesting to note that the solution of the equation TJ (0,0, ta1 , rC (ta1 )) Tb (1.7.38) gives the same value of ta1 , that confirms correctness of applied model. The temperature distribution in liquid and solid zones at melting step can be represented by the expressions (A.16) and (A.19). The contact radius rC (t ) and the melting isotherm m (t ) can be found from the conditions (A.7) and (A.8). All three zones Ds , Dm and Db should be considered at the boiling step. The temperature fields in liquid and solid zones are described by the equations (A.22) and (A.19). The boiling isotherm can be found by the solution of the equation (A.5). The main problem at modelling is to find the temperature distribution in vaporised zone Db because of insufficient information about electrical and thermal conductivities of contact material in vapour state. Therefore it is reasonable to suggest that the spatial temperature -profile is a parabola with a top maximum at the contact spot 0 , which increases in time from boiling temperature Tb to the temperature of metallic vapour ionisation Tvi , while the temperature at the boundary b is equal to boiling value: T0 ( , t ) T0 (0, t ) 2 / b (t )[T0 (0, t ) Tb ] 2 (1.7.39) The duration of boiling stage is very short (a few hundred microsecond), thus we may assume the linear rise of temperature maximum in time 94 T0 (0, t ) Tb t tb [Tvi Tb ] tvi tb (1.7.40) Dynamics of the contact radius rC (t ) at the pre-arcing stage at the conditions of Fig.2 is presented in Fig. 3. Corresponding temperature at the centre of the contact spot M 1 is shown in Fig. 4 rC (t ), m 140 120 100 80 60 40 20 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 t , ms Fig. 3 Dynamics of contact radius rC (t ) at the pre-arcing stage TJ (0,0, t ) 0C 4500 4000 3500 3000 2500 2000 1500 1000 500 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 t , ms Fig.4 Dynamics of contact spot temperature due to Joule heating at pre-arcing stage The oscillogram shows that no motion of the contact M 1 occurs up to the time ta 2 1.8 ms because the contact force is counterbalanced this time by the elastic force in the contact pair M 2 . The radius of the contact spot decreases at the pre-softening stage from t 0 to tS 0.35 ms due to electrodynamic force Fed (t ) . However this decreasing is very small (from 23 m to 22.8 m ) and can be neglected. At the next stages one can see increasing rate of contact radius due 95 to reduction of contact hardness H B accompanying by corresponding decreasing rate of contact temperature at the softening point tS 0.35 ms and melting point tm1 0.66 ms . The axial distribution of temperature just before arc ignition is shown in Fig. 5. TJ (0, z, ta1 ), 0C 4500 4000 3500 3000 Db 2500 2000 1500 1000 500 0 Dm Ds 20 40 60 80 100 120 140 160 180 200 z, m Fig. 5 Axial temperature distribution at the time of arc ignition in vaporised zone Db , melted zone Dm and solid zone Ds The high temperature gradient in the vaporised zone indicates importance of heat transfer inside contact that should be taken into account in the power balance at arcing. 5. Transition stage. The transition stage of contact separation continues from the time ta1 1.2 of arc ignition at the first contact pair M 1 to the time ta 2 1.8 ms of arc ignition at the second contact pair M 2 . The temperature of vaporised zone Db increases at this stage from the boiling point Tb 2193 0C to the threshold temperature of metallic vapour ionisation Tvi 4300 0C . Dynamics of metallic vapour pressure and force due to explosive Joule heating is very important to clear up the mechanism of contact repulsion. The partial metallic vapour pressure can be estimated using formula pm (t ) RT0 a (t ) b (t ) (1.7.41) where R is absolute gas constant, is atomic weight of contact metal, b (t ) is density of metallic vapour and T0 a (t ) is volume-averaged vapour temperature (10), i.e. 96 2 1 T0 a (t ) T0 (0, t ) Tb 3 3 (1.7.42) To find b (t ) we use the law of power dependence of vapour electrical conductivity b on vapour density b at isochoric heating of non-ionised metallic vapour [14] b Ab b (1.7.43) where Ab 8.3 ohm1m1 and 0.56 for silver if b is measured in kg / m3 . Therefore, if 1 and 2 are the values of vapour density, 1 and 2 are the values of electrical conductivity of vaporised zone Db at respective values of temperature T1 and T2 , then 1 2( 1 1/ ) 2 (1.7.44) The law (15) remains to be correct for very high values of temperature T2 and pressure, when the values 2 and 2 for metallic vapour approach to corresponding values for solid metal, therefore the expression (15) can be written in the form 1 s ( 1 1/ ) s (1.7.45) where s and s are density and electrical conductivity of the solid metal. In terms of electrical resistance of vaporised zone Db this expression can be represented in the form b (tb ) s [ R1 (tb ) 1/ ] R1 (ta1 ) (1.7.46) where R1 (tb ) and R1 (ta1 ) are the values of electrical resistance of the zone Db before and after vaporisation. These values are determined from experimental data for power and resistance, which can be calculated as product and ratio between measured voltage and current. Results are presented in Fig. 5. 97 W1 (t ), 103W R1 (t ), m 40 35 30 R1 (t ) 25 20 15 10 W1 (t ) 5 0 0 0.5 1 1.5 2 2.5 t , ms 3 Fig. 5 Resistance R1 (t ) and power W1 (t ) at the transition steps At both transition steps, step 1 ( 1.1 ms t 1.2 ms ) and step 2 ( 1.7 ms t 1.8 ms ) power and resistance can be considered linearly dependent on the temperature. Then metallic vapour pressure pm (t ) can be evaluated using formula (12). Corresponding force Fm (t ) can be represented in the form Fm (t ) pm (t )Sb (t ) (1.7.47) where Sb (t ) is the surface area of vaporised zone which is the semi-ellipsoid of revolution about z-axis (See the equation (A3) in Appendix) r2 z2 1 b 2 rC 2 b 2 This area can be calculated by the standard formula for a surface area of revolution of the curve r ( z ) b rC 2 2 1 z2 2 rC about z-axes: 2 dr Sb (t ) 2 r ( z ) 1 dz , dz 0 Evaluating the integral we get where rC 1 1 2 2 Sb (t ) rC 1 2 2 1 2 ln . (1.7.48) tan b . The calculation of vapour force at the arc ignition using expressions (12), (19) gives the value Fb (ta1 ) 0.653 N . The electrodynamic force at this time is Fed (ta1 ) 107 I 2 (t ) ln R0 0.105 N rC (ta1 ) 98 (1.7.49) Since the sum Fb (ta1 ) Fed (ta1 ) 0.758 N is less than the spring force Fc (ta1 ) 1.4 N , no contact motion occurs still at this time. The two next steps of the transition stage are the melting of the second contact pair M 2 from ta1 1.2 s to tm2 1.7 s and its boiling from tm2 1.7 s to tb 2 1.8 s . All characteristics for these steps can be calculated similarly like for the contact pair M 1 above. The contact M 1 is in the arcing state this time and its temperature should be calculated as the sum of two components (A.23) taking into account both causes of heating, to volumetric Joule sources and surface source by arc heat flux. The Fourier criterion at this time ceases to be small due to increasing of contact radius at arcing. Therefore the quasi-stationary solution of heat equation should be replaced by non-stationary solution with components defined by the expressions (A.10) and (A.26). The main peculiarity of this model is the expansion of the contact radius rC (t ) , i.e. arc root radius, during arcing. It is reasonable to suggest [9] that the arc root is attached to the expanding vaporised zone Db in the course of further arc evolution. Thus it is identified with the radius of boiling isotherm rb (t ) , which can be found as above from the conditions (A.4) – (A.5). Calculation shows that when arc ignites, the contact radius increases abruptly from 122 m to 570 m , thus the current density at the contact M 1 decreases significantly (Fig. 6). j (t ), A / m2 12 10 11 10 10 10 9 10 8 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t , ms Fig. 6 Dynamic of current density at transition stage It explains the appearance of high vapour pressure at explosion in contact zone due to Joule heating, which is however is not sufficient still to initiate contact motion. But estimation shows that for a current density greater than 1011 A / m2 contact repulsion may start even at this first step. The heat flux Pc (t ) entering contact from arc should be calculated taking into account positive components due to arc radiation, electron (or ion) bombardment of anode (cathode) contact surface, inverse electrons from the arc column, and negative components due to power losses for evaporation, radiation, electron emission cooling and heat conduction inside the contact body. The expressions for all these components can be found in the paper [11] and 99 [12]. However the model in considered case can be simplified because the information about current, voltage and displacement is available from experiment. Therefore it is more convenient to use the arc power balance equation CA dTA dp WA Wc A dt dt (1.7.50) Here WA is the total power generated by arc in a unit volume, which can be calculated directly from the measured values of arc voltage U A (t ) , arc current I A (t ) and contact displacement x(t ) as WA I A (t ) U A (t ) rA2 (t ) x(t ) (1.7.51) The last term in the right side of (21) is small and can be neglect. The force due to arc plasma pressure FA (t ) consists of two components during arcing FA (t ) Fm (t ) Fa (t ) (1.7.52) The first component on the right side is caused by partial pressure of metallic vapours in plasma, while the second one appears due to partial pressure of heated gas (air). The relation (14) is not valid for plasma, thus Clapeyron equation should be used to calculate both components Fm (t ) M c (t ) RTA (t ) M (t ) RTA (t ) rC 2 (t ) c mcVA (t ) mc x(t ) Fa (t ) M a (t ) RTA (t ) M (t ) RTA (t ) rA2 (t ) a maVA (t ) ma x(t ) (1.7.53) (1.7.54) Here R is the gas constant, TA (t ) is the volume averaged gas temperature, mc , ma and M c (t ) , M a (t ) are the atomic weights and mass of evaporated metal (index c) and gas (index a) in the arc column, and VA (t ) rA2 (t ) x(t ) is the arc volume. It should be noted that the force component due to gas pressure is absent for metallic arc phase, when the arc temperature is less than temperature of gas ionisation (for air it is 5000 0C ) or for a vacuum arc. The mass of evaporated metal M c (t ) in the expressions (25) should be calculated from its volume, which can be identified with the region of evaporated zone between contact surface z 0 and isothermal surface of boiling temperature z b (r, t ) (Fig. A.1). The calculation of all parameters for consecutive phenomena, which occur on the contact pair M 2 at the second step from t 1.7 ms to t 1.8 ms , can be 100 performed similarly like for the contact pair M 1 above. It should be noted that in considered case no liquid metal bridge exists between contacts at the start of contact separation because of formed vaporised zones. However in the range of more high current the contact repulsion may be initiated by electromagnetic force. In this case liquid bridge appears at the initial stage of repulsion [8] and its final length should be added to the length of contact gap. This bridge length lbr can be found as the product of bridge duration and opening velocity. The bridge duration may be calculated as the time corresponding to the boiling temperature. Alternatively it may be found directly from an oscillogram. Above described mathematical model can be applied in this case as well if the contact gap x(t ) in the expressions (25) and (26) is replaced by the sum x(t ) lbr . One can see that the metallic plasma force Fm (t ) appearing due to Joule heating at the transition stage is much greater than the magnetic force Fed (t ) . It becomes equal to the spring force due to Joule explosion in the second contact pair M 2 at t 1.8 ms , and then contacts begin to move. However the magnetic force being relatively small contributes as well into start of contact repulsion because the plasma force alone is not sufficient to overcome the spring force. 6. Arcing stage. Gas pressure due to arc heating Further extension of contact gap and current rise increase the arc heat flux entering contact, rate of evaporation and gas pressure. Calculation of contact temperature using expression (23) shows that now arc component TA (r , z, t ) becomes much greater than Joule component TJ (r, z, t ) and motive power is caused already by the gas plasma force, which is predominant in comparison with magnetic force (Fig. 8 A). It is interesting to note that in contrast to results published in [6] and [7] heat flux component due to electron and ion bombardment amounts to 65%, while flux component due to arc radiation is 35%. It seems suggestion concerning 70% portion of arc heat flux for radiation may be correct only for high current which was used in above referred papers. At the time when current ceases to increase, gas pressure becomes to decrease and at critical time tcr 3.1 ms it is equalised with spring force. However reverse motion begins later at tr 9.1 ms (the point D in Fig. 3) due to inertia of contact motion. 101 F (t ), N 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 F (t ), N 1.6 Fc (t ) 1.55 1.5 FA (t ) FA (t ) Fed (t ) 1.45 Fm (t ) 1.4 Fc (t ) 1.35 FA (t ) 1.3 Fed (t ) 1.25 1.2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t , ms t , ms Fig. 8 Dynamics of forces for the whole duration of blow-off repulsion 5. Experimental verification of forces dynamics modelling. Verification of above considered modelling of force dynamics could be achieved by comparison of calculated contact motion due to resulting force with measured values of contact displacement. The resulting motive force Fres (t ) can be obtained from the expression Fres (t ) Fg (t ) Fed (t ) Fc (t ) Corresponding acceleration W (t ) (1.7.55) 1 Fres (t ) is presented in Fig. 9. m W (t ), mm / s 2 0.1 0.05 0 0.05 0.1 0.15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t , ms Fig. 9 The acceleration W (t ), mm / ms 2 , versus time t , ms The contact displacement x(t ) can be calculated by the formula t x(t ) (t )W ( )d (1.7.56) 0 Results of calculation using MathCad and data in Fig. 9 in comparison with experimental data from oscillogram in Fig. 2 are presented in Fig. 10 102 x(t ), mm 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t , ms Fig. 10 Measured (solid) and calculated (dashed) contact displacement One can see a good agreement of measured and calculated values of contact displacement. It should be noted that presented above mathematical model may be simplified significantly if we use the experimental information about contact displacement for direct calculation of acceleration and total force rather than for verification of the model. CHAPTER 2. MODELING OF THE BRIDGE TRANSFER 1 CYLINDRICAL MODEL OF A BRIDGE 1.1 Introduction Phenomena at the initial stage of electrical arc depend on preceding conditions of separation and heating of electrical contacts. The sequences of voltage evolution before and after appearance of melting liquid bridge are described in the paper [1]. According to this observed voltage waveform one can distinguish 4 sequences during opening: heating of solid material (I), fast adiabatic melting (II), quasi-stationary extended melting (III), quasi-adiabatic pinch and heating (I\/). Experimental voltage-time characteristics enable authors to establish duration of each sequence, change in slope and magnitude of voltage and to get some information about bridge phenomena. However more detailed estimations [2] show that it is necessary to ground the hypothesis about adiabatic melting and quasi-adiabatic heating. It seems such an assumption may be correct only for limited range of opening extent. It is well known those heat conduction phenomena's influences on the temperature increasing in contacts as well as on the bridge and arc duration. This conclusion will corroborate with the observed solidification of bridges just after it's melting. The experiments T. Takagi with co-authors [3]--[4] show considerable decreasing of bridge and arc duration (in metallic phase) when electrodes are heated preliminary. 103 Mathematical model mentioned below is an attempt to take into account thorough kinds of heat transfer in power balance of solid contacts, bridges and electrical arc that is more simple comparing with previous models [5] -- [8]. 1.2 Pre-arcing stage Models of contact heating. This stage lasts from the start of contact separation up to ignition of electrical arc. It includes phenomena in solid electrodes, contact melting, appearance, evolution and rupture of liquid contact bridge. There exist two models describing voltage and temperature waveforms during contact separation. One of them is based on the classical Holm model with constriction resistance Rc / 2ro , where is electric resistivity and r0 is radius of contact spot. Well-known Holm relations [9] between voltage and temperature, current and radius of spot when temperature rises to melting point describe satisfactorily quasi-stationary contact phenomena for a slow separation velocity. This model was extended concerning the non-stationary phenomena and non-ideal electrical contact [5], [7], when contact opening is so rapid that Holm relationship will no longer be valid. Another approach [1], [10] is associated with a filament or roughness between the electrodes, connecting the contact with the resistance R f 2 / ro ,where 2 is the length of filament, that has to be added to the constriction resistance Rc . Further heating of this filament leads to its melting, creating of a molten bridge and arc ignition after bridge boiling. However the assumptions about adiabatic heating and melting of filament [1] or about constant temperature of the ends of filament [10] don't enable to take into account real quantity of the power dissipation caused by heat conduction from filament to electrodes that is very important as it is noted above. Clearly model [1] yields overstated temperature in filament while model [10] understates it. To correct both these models we consider the axial contact of thin semifilament ( z 0) with electrode (0 r, z ) through the circle z 0,0 r r0 . It is the symmetrical half of whole filament 2 z 0 . HEATING OF SOLID CONTACTS. At the first period 0 t t m of nonstationary heating the temperature fields of a solid filament 1 ( z, t ) and solid electrode 2 (r, z, t ) can be found from the solution of the heat contact problem [11] in the form: 104 1 ( z, t ) 1 (e kt 1) (t ) z [3( 1) 2 1] 6 (2.1.1) e k (t ) ( )d 0 2 t 2 (r, z, t ) 21(r, z, t ) 22 (r, z, t ), (2.1.2) where (t ) 0 (1 e t ), is the heat flux entering electrode from the filament, the first term in the right side of (2) corresponds to the heating of the electrode by the heat flux (t ) from the filament, while the second one is responsible for the Joule heating inside electrode. The constants are defined by formulas: 0 0 j 2 [cot() r0 ] k , 0 ,j r c cos 0 4 where 0 , , c, , are electrical resistivity, its temperature coefficient, thermal capacity, mass density and thermal conductivity respectively, j I r0 2 is the current density in the filament. Expressions for components 21 (r, z, t ) and 22 (r, z, t ) are presented in the paper [11]. The model considered in [1] may be obtained from the expression (1) if the heat conduction from filament to electrodes is neglected ( (t ) 0) . When t is small we get adiabatic heating 1 ( z, t ) kt 0 j12 t. c (2.1.3) Let us compare the results of calculation corresponding to three models. 1) Adiabatic model [1], formula (3); 2) Model with the constant ambient temperature 0 on the interface of filament with electrodes [10]; 3) General model given by the expression (8). These results for the data given in Table 1 are presented in Fig.1. Table 1. Data for the calculation of the temperature in filament c 3.9 10 3 J kg 1 K 1 8.9 103 kg m 3 3.8 10 2 W m 3 K 1 0 1.55 10 8 Ohm m 4 10 3 K 1 J 25 A , t t m 1.4 10 5 s 105 1 ( z , t m ) , o C 1 1000 3 800 600 2 400 200 l 0 z Fig.1 Temperature distribution in the Cu half-filament, 25 A, r0 0.78 10 5 m, 1.44 10 5 m [1] ; 1- equation (3), 2 – paper [10], 3 – equation [1]. The time t m required for the melting of filament according to adiabatic model given in [1] (curve 1) is equal to 1.4 10 5 s . At this time the temperature corresponding to the model given in [10] is 1 (, t ) 342o C in the center of the filament and ambient temperature 0 20o C on its edge. According to the general model the real temperature is 1 (, t ) 7400 C , 1 (0, t ) 3360 C . It is not so difficult to calculate that only 54% of dissipated energy W J U t m is spent for the heating of filament, while 46% is passed through interface of electrode owing to heat conduction. Therefore the radius r0 and the length of the filament calculated in [1] using adiabatic energy balance has to be corrected. MELTING OF CONTACT CONSTRICTION REGION. This period 6 5 (t m t t1 ) is of very short duration (10 s 10 s). It is identified in [1] with the time required for the melting of a cylindrical filament or conducting roughness that is transformed into molten bridge . From measured values of voltage amplitude U and duration t 0 (Fig.2) it is possible to calculate geometrical extent of bridge and current density. 106 U,V 1.0 0.8 2 25A 0.6 2 0.4 12A 0.2 Fig. 2 Fast bridge melting , Cu -electrodes. Melting voltage 0.41 V. Molten (1) and solidified (2) bridges [1] Let b (t ) be the boundary between liquid part of filament z b (t ) and its solid part b (t ) z 0 , so b (tm ) , b (t1 ) 0 . It may be found from the equation 1 ( b (t ), t ) m , (2.1.4) where 1 ( z, t ) is given by the formula (1) and m is melting temperature. The energy balance of the liquid and solid parts of filament at the time t is given by the expression W Wd WL Wc (2.1.5) where Wd r0 t 2 ( )d (2.1.6) tm is the energy dissipated to the electrode through interface z 0 , 2 WL L r0 [ b (t ) ] , (2.1.7) ( L is latent heat melting ) is the energy required for melting of the liquid region z b (t ) , and Wc r0 c 2 0 ( x, t )dx (2.1.8) 1 b (t ) is the energy for the heating of the solid region b (t ) z 0 . Taking into account W 0.5U J t , U J 0 (1 1 ) ro 2 (2.1.9) we can calculate from (23) - (28) , r0 ,WL ,Wd ,Wc .The calculations show that Wd (t1 ) 0.23 10 5 J , when t t1 4.7 106 s , then WL (t1 ) 0.21 105 J , 107 Wc (t1 ) 0.06 105 J , so the radius and the length of the bridge are more small comparing with adiabatic model : r0 4.6 106 m , 8.2 10 6 m The temperature and potential b, b described by more simple equations c b for the bridge z 0 b J 2 2 (b b ) T b 2 b4 t z z rb r0 are (2.1.10) d 1 d b ( ) 0, dz b dz (2.1.11) where T is total heat emission owing to both radiation and convection [12] . The boundary conditions take into account continuity of temperature and heat fluxes with phase transformation on the interfaces. The results of calculation using model (10) - (13) are given in the Table 2. Table 2 Comparison of models ( I = 25 A ) W 10 5 J Wc V 5 15 10 1 2 3 0.5 0.5 - J 0 0.28 - 10 m 3 2.74 0.84 0.39 - 1.54 r 6 6 j 10 m 10 m 1011 A m 2 7.8 6.1 -7 14.4 7.2 5- 10 1.3 2.6 2 - 3.5 1 - calculations using model [1] 2 - calculations using model (10) - (13) 3 - experimental data [1] One may resume that model (10) - (13) is much closer to experimental data in comparison with the adiabatic model. QUASI – STATIONARY EXTENSION AND HEATING OF MOLTEN BRIDGE. Experimental data [1] concerning first period ( extension of molten bridge ) are given in Table 3. If extension velocity of molten bridge is low, quasi - stationary model may be used successfully, but it has to be corrected taking into account the change of bridge shape during contact separating. Under the action of the forces of surface tension, gravitation, pinch - effect the bridge takes shape of revolution surface round z – axis. 108 TABLE 3 Characteristics of bridge’s extension v 0.1m s 1 Duration , 10 6 s Contact initial gap, 10 6 s final Voltage, initial V final Voltage velocity, v 0.01m s 1 10 1 2 0.5 - 0.8 1.0 100 4 5 0.5 1.0 5- 8 0.4-0.5 5 5 dU 10 4 V s 1 dt Voltage gradient, dU , 105V m 1 dz Mathematical model describing temperature and electromagnetic fields is based on the one dimensional heat equation for the bridge with variable crosssection [7] d 2 b 2 y b J 2 b 0 dz 2 y z z 2 b y 4 1 d b J 2 b dz y (2.1.12) (2.1.13) Other equations for 1 , 2 , 1 , 2 are the same as at preceding stage. The solutions of the equations (14) , (15) are found in the form : (16) where z 2 y ( vt, t ) 0 dz , y ( z, t ) 2 r0 2 and constants A and B are chosen to satisfy the boundary conditions [7], [11]. The free boundary (the shape of the bridge) r y( z, t ) is determined from the variation principle declaring that free energy of bridge must be minimum for the really observed bridge shape, i.e. it is necessary to find minimum of functional F ( y) Fk ( y) FH ( y) Fg ( y) where Fk ( y), FH ( y), Fg ( y) are components of surface tension, electromagnetic and gravitation energy respectively described in details in [7], [11]. 109 When temperature in narrow cross-section of bridge rises to the point corresponding to the abrupt decreasing of surface tension coefficient, quasi stationary period of this stage finishes and next heating period bridge starts. Experimental data for the calculations at this period with J 50 A and v 0.1m s 1 are given in the Table 4. Table 4 Voltage in the rupturing bridge Duration , 6 0.05 0.10 10 s - Initial gap , 6 2 10 m Voltage , initial 1 10 U final - 20 Voltage velocity , 1 - 10 dU , dt 108 V s 1 The results of calculation according to above mathematical model are given on Fig. 5. , 10 3 0 C U, V 2 2 1 1 5 t , 10 3 s 0 Fig. 5 Voltage U (1) and temperature b (0, t ) (2) in Cu - contacts. I 50A, v 10 2 m / s 110 The comparison of calculations with experimental curves [1] shows a good coincidence. INFLUENCE OF BRIDGING ON THE ARC DURATION. Transition from bridging to arcing is accompanied by a voltage drop. A typical oscillogram of voltage and contact gap corresponding to consecutive periods of contact opening is shown in Fig. 6 l(t) u uA 2 lcr u ign lmin umin lb 1 tb t cr t Fig. 6 Voltage and contact gap of opening contacts. 1- contact gap, 2- voltage, t a - bridge lifetime, lb - bridge length, t cr - critical time, lcr -critical gap, uig -voltage of arc ignition , umin - min voltage, uarc - voltage of cathode arc One can see following typical stages : 1). The stage of a liquid molten bridge with the length lb and the lifetime tb ; 2). The transition stage of an anode dominated arc phase ( tb t tcr ) with the linear decreasing voltage umin u uig in the region lb r lmin and increasing voltage umin u uarc in the region lmin r lcr ; voltage umin appearing on contacts just after the rupture of a bridge bears a direct relation to the cathode fall of the material. ; 3). The stage of a diffusive cathode dominated arc ( t tcr ) with constant voltage uarc . At the beginning of the second stage, just after bridge rupture and arc ignition, the distance between the cathode with single cathode spots on the surface and the anode is very small , therefore the power capacity from the arc to the anode is large. As a result the anode undergoes powerful local overheating and becomes a source of vapors influencing on the next arc stage. Owing to the plasma from the cathode region being formed as truncated cone with the greater 111 base on the anode, there is a certain critical gap lcr , which becomes too large to support the power capacity required for the further melting and evaporation of the anode. The another reason for the appearance of the critical gap is a cooling effect due to increasing of evaporation surface on the anode . The time of the beginning of the third stage depends on the duration of two previous stages, particularly, on the bridge time life and bridge length. If the length of a bridge lb exceeds the critical gap distance lcr , then the second transition stage could be not appearing at all. On the contrary, if the bridge length is negligible, or if bridge does not appear at all , then duration of the transition anode dominated arc stage ta tcr tb is maximal. The dependence of t a on the bridge time-life tb for the Co and Mo electrodes [13] is given in Fig. 7 . ta ,10-4 sec 16 1 12 3 8 4 4 0 2 4 2 6 8 -4 t b ,10 sec Fig.7 The dependence of the anode arc duration t a on the bridge lifetime tb ; 1- Cu, I 650 A ; 2 - Cu, I 270 A 3- Mo , I 650 A; 4- Mo , I 450 A Experimental observed values of anode arc duration have considerable deviations ( Fig. 8 ). It is a result of deviations of bridge lifetimes. For cases when bridges don't appear at all (hatched column in Fig. 8 the anode arc duration is maximal. 112 W 0.25 0.20 Mo 0.15 0.1 0.05 0 10 20 30 40 50 60 70 80 l -6 b ,10 m Fig. 8 The histogram of statistical distribution of frequency of a bridge appearance W in dependence on the bridge length lb If the velocity of contact opening V0 0. 05m sec1 , then the transition anode dominated arc stage is not observed in the case of Cu -electrodes for the bridge length lb 40 106 m . For the velocity range V0 0. 2m sec1 this threshold of the bridge length is lb 85106 m . This phenomena can be explained by suggestion, that the mechanism of bridge generation for such materials as Cu and Ni is stipulated by stretching of a liquid metal from the electrode rather than by melting of a surface filament. Bridges on the Mo -electrodes are stable and generated accordingly to the normal length distribution low with the maximum magnitude lb,max 25 106 m . The analysis of experimental data [13] enables to conclude that for the Cu electrodes tb ta , thus in this case the bridge is the main factor predicting the following transition arc stage or its absence . On the contrary , for the Mo electrodes tb ta , and one can observe relative self-dependence of the transition arc phenomena. 2 GENERAL MODEL OF BRIDGE TRANSFER 2.1Bridge phenomena A molten metal bridge which is formed between electrical c2ontacts in the last stage of contact opening is a cause of unreliable performance of low current relays and control systems working with milli- and micro amperes. Duration and other characteristics of an electrical arc ignited in vapours of ruptured bridge depend also on the parameters of a bridge [1]. Problems of experimental and theoretical investigations of bridge phenomena have been considered in many papers. One can find the detailed review concerning this matter in the book [2]. However the problem of mathematical modeling of bridging remains still one of the main problems in the modern theory of electrical contacts. 113 The mathematical model describing the dynamics of the temperature field at the consecutive stages of contact opening [3] enables to conclude , that the hottest section of the bridge is displaced from the centre because of thermoelectric effects. It was shown further in the papers [4] , [5], that if contacts are homogeneous, then the tunnel effect is most important, and it is responsible for the rupture of a bridge at its anode side. This rupture is accompanied by metal transfer from the anode to the cathode. During next commutation contact surfaces consist of the same material as the anode, therefore the material transfer occurs again from the anode to the cathode due to the same tunnel overheating. As a result of such transfer a thin pip of the anode material, called spire, builds up on the cathode, while a crater appears on the anode . The process of material transfer is quite different if one could succeed in choosing of a contact pairs composition to provide the negative materials transfer from the cathode to the anode at the first opening operation. In this case the materials transfer is positive , it comes from the anode to the cathode. This transfer continues as long as all transferred during first opening operation material is replaced again to the cathode, and after that negative transfer occurs again. Very scanty thickness of the cathode material has a share in the transfer at this case, thus the reliability of the contacts could be sufficiently improved. Empirical choice of such contact pairs composition , that provides above self-restrained bridge transfer, was fulfilled by W.G. Pfann [6] , who has discovered the contact pairs:Au - Pt , Ag - Pt , Au - Pt Ir (20%), Au - Pd , Pd Pt with the first component as anode . M.A.Razumikhin [7] has confirmed the principle of choice of contact pairs with initial negative transfer by using alloys Pt Ni ( 5%) as anode and Pd Ag (40%) as cathode . He found that this composition provides very small layer of cathode material , which has more resistivity, performing in the transfer even during long working . 2.2 Mathematical model The regions of bridging . A mathematical model of contact bridging transfer must describe the dynamics of formation, evolution and rupture of a molten liquid bridge . Many investigators proceed from the assumption that visible part of a bridge has the shape of a cylinder with the axis parallel to the contact plane. Of course, such assumption is true in the case of high current bridge at high velocity of contact opening. It enables to simplify the theoretical investigation of bridging phenomena in a large measure. All the same time it is necessary to take into account the shape of the bridge discrepant from cylindrical if contact opening duration is commensurable with the time of bridge formation . However no investigations have been carried to determine the bridges geometry and its influence on contact opening phenomena , except 114 classical results obtained by P.Davidson as early as 1954 , who has considered bridges as nodoids and cathenoids formed due to surface tension force . We assume in general case, that under the action of forces of surface tension, gravitation, and pinch effect the bridge takes the shape of a surface of revolution around z - axis : r y( z, t ) r1 (t ) r (t ) , for ( t ) z 0 , and z 10 (r , t ) for where (t ) is the contact gap at the time t . Geometry of such bridge is presented on Fig. 1 . It occupies the region D2 D3 where D2 {0 r y( z, t ); (t ) z 0} is the visible part of the bridge , and D3 {10 (r , t ) z 1 (r , t ) , if r1 (t ) r (t ) ; 0 z 1 (r , t ) , if 0 r r1 (t )} is the part of the bridge embedded inside solid electrode occupying the D4 {1 (r , t ) z , if 0 r (t ) ; 0 z if (t ) r } . region The bridge is generated from the electrode, which material is more fusible, if contact materials are heterogeneous , or from the anode in the case of homogeneous material due to tunnel extra heating at the previous stage . The region D1 in Fig. 1 is more refractory and fuses later than D3 . z 2 (r , t ) D1 (t ) 2 r0 2 (t ) D2 r r 0 3 D3 r1 ( t ) (t ) z 1 ( r , t ) z 10 ( r , t ) D4 z Fig.1 Geometry of a bridge and adjoining regions Differential equations and boundary conditions. The general mathematical model describing electromagnetic and temperature fields in the bridge and adjoining regions by the system of MHD - equations is presented in the papers [ 8 ] - [ 9 ] . However this model is very complicated and requires special computer programmes . There is shown in the paper [ 9 ] , that if velocity of contact opening is not high , precisely d / dt 4m sec1 , then bridging process can be considered as quasi-steady and described by equations, where t is parameter only. 115 For this case we present rather simple model which enables to analyze an influence of different factors on the bridging process. In accordance with this model equations i i 2i Ti i i (i ) 2 0 ( 1 ), curl( 1 i i ) 0 , i=1,3,4 (2.1.15) where i is thermal conductivity , i i 0 (1 i i ) is specific electrical resistance , i is its temperature coefficient , and Ti is Thomson coefficient , describe the temperature and electromagnetic fields i and i in each region Di ( i = 1, 3, 4 ) . To choose a simple equation for the bridge we use the result of F.L. Jones [10] who showed that temperature gradient in a bridge cross-section is negligible . In this case a bridge can be observed as a bar with variable crosssection y 2 ( z, t ) carrying only the axial component of heat flux . The equations of quasi-steady heat transfer and potential distribution in the visible part D2 of a bridge in this model are : d 2 2 2 yz 2 I T 2 d 2 I 2 20 (1 2 2 ) 0, 22 y 4 dz 2 y z 2 y 2 dz d 2 20 (1 2 ) dz y 2 (2.1.16) (2.1.17) where I is the current . The boundary conditions are : a). For the boundaries of the visible part of the bridge z 0 , if 0 r r1 (t ) , and z (t ) , if 0 r r0 i 3 for z (t ) ) ( i 1 for z 0 , and we can write the conditions of continuity of temperature, heat fluxes, electrical potentials, and current densities : i i 1 , i i i 1 i , z z i i 1 , 1 i 1 i 1 i z i 1 z (2.1.18) b). On the lateral surface of the bridge as well as on the contact surfaces outside the bridge z 0 and z (t ) the heat and electrical transfer can be neglected : i 0, n i 0 n (2.1.19) c). In regions faraway from the contact zone electrical potentials are given , and the temperature is constant ( this constant we identify as 0 ) : 1 0 , 4 0 , 1 2 1 uc , 1 2 4 uc z 0 , r 2 z2 , if , if 116 z 0 , r 2 z2 (2.1.20) (2.1.21) d). On the surface z 1 (r , t ) between liquid and solid zones ( D1 and D4 ) the temperature is equal to the melting point : 3 4 m , if z (r , t ) (2.1.22) Equations for free boundaries . The boundaries z 1 (r , t ) , r y( z, t ) , z 10 (r , t ) are free and have to be determined from the additional conditions. The interface z 1 (r , t ) between solid and liquid zones can be found from the Stefan condition, that transforms in the stationary case into simple equation 3 3 4 4 n n (2.1.23) The free unknown boundaries r y( z, t ) and z 10 (r , t ) for the bridge shape can be derived from the Euler equation, which can be reduced to the form ( see Appendix 1 ) ; yy zz yz 1 0 I 2 20 k 2 , ( t ) z 0 , 2 2 y (1 yz ) 3/ 2 8 2 y 2 2 r ( 10 ) rr ( 10 ) r [1 ( 10 ) r ] 0 I 2 1 1 20 2 k2 , r1 (t ) r (t ) , 2 3/ 2 2 2 8 r r[1 ( 10 ) r ] rm 2 where k2 0 I 2 k3 , 2 8 2 r1 ( t ) k3 ( 20 2 3 ) z 0 , (2.1.24) (2.1.25) (2.1.26) 20 is the surface tension between the bridges molten metal and surroundings (air), 0 1. 257 106 H m1 is magnetic permeability of vacuum, rm rm (t ) is the point of maximum of the function z 10 (r , t ) in the domain r1 (t ) r (t ) . It is necessary to put boundary conditions to obtain the unique solutions of the equations ( 9 ) and ( 10 ). The first of them for the equation ( 9 ) is evident : y( (t ), t ) r0 (2.1.27) and the second can be obtained from Young formula for a boundary wetting angle 2 : cos 2 [1 yz ( ( t ), t )]1/ 2 2 10 21 20 (2.1.28) where ij is the coefficient of the surface tension between the domains Di and D j ( index 0 corresponds to the surroundings ). The boundary conditions for the equation ( 10 ) are : 10 ( (t ), t ) 0, ( 14 ) cos 3 1 [( 1 ) r ( 10 ) r ]r (t ) [ 1 ( 1 ) r 1 ( 10 ) r ]r (t ) 2 2 117 40 34 30 (2.1.29) 2.3 Results of solution. The equations ( 1 ) and ( 2 ) can be reduced to the ordinary differential equations by appropriate substitutions, and their solutions , as well as the solutions of the equations ( 3 ) , ( 4 ) , and the surface of phase transformation z 1 (r , t ) , are found in analytical form ( Appendix 2 ). Similar solutions are presented also in the paper [5]. To find the free surfaces y ( z, t ) and 1 (r , t ) we shall use the Ritz method. In addition to the boundary conditions ( 12 ) - ( 15 ) we must take into account r1 (t ) r (t ) , the balance of volumes of the bridge meniscus Dm : 0 z 10 (r , t ) and its visible part D2 , thus 0 2 dz ( t ) y ( z ,t ) (t ) 0 r1 ( t ) (1 2 )rdr 2 10 (r , t )rdr , (2.1.30) where is the thermal expansion coefficient . Besides that the conditions of smooth conjugation on the interface z 0 , r r1 have to be provided : y (0, t ) 10 (r1 , t ) 1. z r 10 (r1 , t ) 0 , y (0, t ) r1 , (2.1.31) In accordance with the Ritz method the boundary problem for equations (9) - ( 10 ) to a variational problem for minimum of certain functional F , constructed in [8], [9] , which is equal to the free energy of a bridge : F Fg FH F (2.1.32) where the terms on the right side are equal to gravitational component, electromagnetic component, and surface tension component of the bridge energy respectively. The term Fg is negligible, and for a low current ( less than 500 A ) FH 0 . In this case F F 2 0 (t ) r0 ( t ) 0 0 2 2 20 y 1 y z dz 2 34 1 ( 1 ) r rdr 2 12 rdr (2.1.33) Unknown y ( z, t ) , and 10 (r , t ) are represented in the form y( z, t ) z( z ) 2 C z 2 ( z )( 2 10 ( r , t ) r1 r0 cot 2 ) z2 2 ( r1 r0 ) r1 , ( r r1 )( r ) ( r r1 ) 2 ( r ) cot 3 C ( r1 ) 2 ( r )2 (2.1.34) (2.1.35) It is not difficult to verify that the conditions ( 12 ) - ( 15 ) , ( 17 ) are satisfied. Parameter C is determined from ( 16 ). Minimizing the functional F by parameter r1 , we obtain the last equation F 0 r1 (2.1.36) Thus the bridging problem is reduced to the system of equations ( 8 ), ( 16 ), ( 22 ) with the additional equation 118 0 0 2r0 ( t ) dz y 2 ( z, t ) (2.1.37) for parameters 0 , m , C , r1 . This system was solved using method of half-dividing intervals. The results of calculations of the bridge rupture voltage Vr are given in the Table 1 in comparison with experimental data which review is adduced in [10]. Apparently, formula Vr V3 (m ) V2 (0) used for calculation gives satisfactory coincidence for all metals except nickel, apparently because of oxidizing reactions , that can not be described in the frame of presented model. Table 1. The voltage Vr of bridge rupture ( volts ) M Voltage, experiment (V) etal Voltage, calculated (V) 0.8 1 Cu 0.64-0.68 (1,5,f) 1.0-1.5 (1,4,8,d) 0.44 (1,6,f) 1.0-1.5 (1,4,f) 0.8 (2,4,b) 0.8 (1,4,7,d) Au 0.84 (1,5,g) 0.85 (1,4,d) 0.88 (1,4,g) 1.0 (2,4,b) 2 1.14 (1,6,g) 1.21 (1,5,g) 1.44 (1,4,g) 1.44 (1,4,a) 2 1.0 (1,6,f) 1.22 (1,5,f) 1.26 (1,6,f) 1.06 (1,6,g) 1.2 (1,4,f) 1.45 (1,5,f) 0.9 (1,6,f) 1.02 (1,5,f) 1.52 (1,4,d) 1.05 (1,5,h) 0.96 (1,4,d) 1.4 (2,4,b) 1.06 (1,6,f) 1.29 (1,4,a) 1.34 1.0 (1,6,f) 1.0 (1,5,f) 1.1 (1,4,f) 1.08 (1,6,g) 1.32 (1,4,g) 1.6 1.12 (1,5,g) 1.32 (1,4,d) 1.6 1.16 (1,5,f) 1.32 (1,4,e) 1.35 Ir Fe M o Ni Pd Pt 1.1 1,4,d) 1.20 (1,5,h) ( 1.18 (1,5,g) 1.4 (1,4,h) 0.9 1.1 1.0 3 1.1 3 119 0.4 1 1.3 1.4 1 Ag W (1,4,f) (2,4,b) (2,4,c) (1,4,b) 0.35 (1,4,d) 0.35 (1,4,d) 1.5 (1,6,f) 0.35 (1,4,d) 0.35 (1,4,d) 1.5 (1,6,f) 0.35 (1,4,d) 0.35 (1,4,d) 1.5 (1,6,f) 0.35 (1,4,d) 0.35 (1,4,d) 1.5 (1,6,f) 0.5 8 1.4 2 1 - quasi static method, 2 - oscillographic method, 3 - method is unknown, 4 - in air, 5 - in vacuum, 6 - in vacuum with cleaned electrodes, 7,8 - different bridges are given a - A.Fairweather, b - J.Lander and L.Germer, c - J,Warham, d - L.Jones, e - R.Holm and E.Holm, f - C.Jones, G - R.James, h - F.Llevellin Jones and M.Price 2.4 The mechanism of bridge rupture. The process of a bridge rupture may be caused by mechanical or thermal forces dependently on criteria considered hereafter. Mechanical rupture occurs when surface tension force is less than electromagnetic and gravitation forces. The bridge rupture begins at its narrowest cross-section z zr which can be determined from the equation y ( zr , tr ) 0 z (2.1.38) Thermal mechanism of a bridge rupture can be consisted of one or two stages. It is necessary to consider two different cases separately. Case 1. The melting temperature m1 of the electrode, occupied the region D1 is more than the boiling temperature of the bridge : m1 b . In this case the bridge rupture occurs when the temperature reaches the boiling point in any cross-section z zb at the time t tb . Many experiments show that the bridge rupture is accompanied by its explosion. It confirms the fact, that the surface tension force can not be a cause of the bridge rupture and can influence only on the location of the cross-section with maximal temperature where explosion occurs during boiling. Thus, unknown values zb and tb can be found from the equations 2 0, z 2 b for z zb , t tb that transforms in this case to the expressions 2b 1 22 22 2 A2 2 2 B2 2 arctan 2 2 2 2 2 A2 2 B2 120 (2.1.39) b 1 exp ( 2 2b )[ A2 cos( 2b 2 2 ) B2 sin( 2b 2 2 )] , 2 2 2 2 2 where z 2b 2r 0 b dz 2 (t ) y ( z, t b ) b . Thus rb y ( zb , tb ) . The metal transfer due to bridging is directed then to the region D1 , on which side the portion zb y 2 ( z, t b )dz of the bridge is displaced. The portion ( tb ) 0 of the bridge y 2 ( z, t b )dz remains on the other electrode. zb If m1 b , the melting of the second electrode begins at some time t tm2 before the bridge rupture. Three new zones instead of one D1 appear during further opening process : z (t ) , if D11 : z 2 (r , t ) , if 0 r 2 (t ) , and Case 2. 2 (t ) r 2 (r , t ) z (t ) D12 : , 0 r 2 (t ) ; D13 : (t ) z 1 (t ) , 0 r y( z, t ) The function 1 (t ) is determined by viscosity and thermal parameters of liquid metal. It can be taken as 1 (t ) 0.5[(t ) (tm2 )] for a first approximation. The equation of heat transfer in the forms ( 1 ) or ( 3 ) remains correct for each of these zones. The functions i (r1 , z, tm2 ) are the initial temperature distribution in this system ( i = 1, 2, 3, 4 ), but additional boundary conditions are the same. The cross-section z zb and the time of explosion tb are determined as usually from ( 25 ), however, in this case cross-section z zb may be placed in zone D13 or in D2 because of essential temperature gradient along the bridge. The mechanism of the rupture for quasi-static bridges is thermal as a rule. It was established, that the diameter of a bridge increases with current as linear function for the current range 10 A - 100 A ( both for the minimal diameter d min 2rb and for the maximal diameter d max 2(t ) ). 2.5 Criteria of bridging The analysis obtained results enables us to estimate the influence of different factors on the bridging process before the solution of the problem. They are determined by following criteria : 121 a). Surface tension criterion ( Weber criterion ) We 103 , then We 0 . If g 0 2 ( t ) r1 2 , the curvature of a bridge surface is large , and surface rb tension is very important during rupture process. b). The inverse value of We is Bond criterion Bo We 1 , which determines the value of gravitation force with respect to the surface tension force. It is not essential in the current range up to 100 A . c). Pinch effect criterion Pi 0 I 2(t ) is very important if its value is 102 2 4 2 r0 0 and more. If I 103 A , then Pi 103 , and a bridge may be squeezed and crushed by the pinch effect only. d). Thermal criteria. Pomerantsev criterion Po criterion Th I 2 2 ( t ) , 2 rb 2 2 Thomson 2 f I T 2 , Kohler criterion Ko , where f is tunnel resistivity, 2 rb 2 r1 are responsible for the thermal mechanism of bridge rupture due to the Joule heating and thermoelectric effects. 2.6 Self-restrained bridge erosion. The analysis of the expressions describing the temperature field in a molten bridge enables us to obtain the theoretical conditions for self-restrained erosion. They are : 1). The anode begins to melt first, while the cathode melts later, i.e. m1 m2 . (2.1.40) 2).The temperature along the bridge , generating mainly from the anode material, increases monotonously and reaches the maximum on the contact surface with the anode. The inequalities 1 4 , 1 4 , 2 1 c2 2 c1 1 (2.1.41) enable to satisfy this suggestion. 3).The melting point of the cathode material is less than the boiling point of the anode material : m2 b1, (2.1.42) i.e. the cathode melts before the bridge rupture . 4). The anode is more fusible than the cathode : b2 b1 , (2.1.43) but the maximum of the bridge temperature is placed on the cathode part of the bridge , and the boiling temperature at this place reaches early than the boiling temperature of the anode part. Thus , the inequalities ( 26 ) - ( 29 ) provide the conditions for optimal self-restrained bridge transfer. 122 These conditions can be verified for the found by W.G.Pfann [ 6 ] selfrestrained contact pairs given in the Table 2. Table 2. Contact pairs with self-restrained erosion No 1 2 3 4 5 Material Ag – anode, Pdcathode Ag – anode, Pt - cathode Au – anode, Pt - cathode Au – anode, Pd - cathode Ag – anode, Pt-Ir20% - cathode i / ci i i i 8 10 m 1.65 10.5 1.65 11.0 2.30 11.0 2.30 10.5 1.65 32.0 watt / m K 418 71 418 70 310 70 310 71 418 8.92 10 14 4 m K j 0.676 4.00 0.676 3.93 0.92 3.93 0.92 4.00 0.676 8.92 bi mi 3 10 C 0 0.960 1.554 0.960 1.773 1.063 1.773 1.063 1.554 0.960 1.830 103 C 0 2.000 4.000 2.000 4.400 2.970 4.400 2.970 4.000 2.000 4.400 It has to be noted, that alloys give more ample opportunity for a choice of selfrestrained contact compositions in comparison with pure metals. The best result from all proposed by W.G. Pfann pairs gives the pair No 5. The obtained above criteria enables us to find new contact compositions with self-restrained bridge erosion. They are : 1). Ag Au (10%) - anode 2). Ag Pd (5%) - anode Pd Ag ( 40%) - cathode Pt Ir ( 20%) cathode This contact composition were tested in the Lab. of Electrical Apparatus of Kharkov Polytechnic Institute, and it was found, that bridge erosion after 350,000 operations, for the current I 1 A and the voltage uc 8volts , is very small ( 2 1013 51014 cm3 / A2 ), especially for the second contact composition because of the self-restrained mechanism of the process. 123 CHAPTER 3. ELECTRICAL CONTACT ARC 1. GENERAL MODEL OF ELECTRICAL ARC IN THE FRAME OF MHD-THEORY 1.1 Review of main theoretical results. 1.2 Phenomena in the arc column. 1.3 Phenomena in near-electrode zones. 1.4 Intra-electrode phenomena (See Appendix 7) 2. MODELS OF ELECTRICAL ARC EROSION 2.1 Introduction The arc erosion in opening electrical contacts depend on many factors such as the range of current and voltage, opening velocity, properties of contact materials and surroundings, parameters of electrical circuit etc. Erosion of electrical contacts at low current due to vaporization of material was considered in many papers [1] – [4]. It was found that direction of transfer of contact material depends on the mechanism of electrical conductivity in the arc column and changes from anode to cathode during the transition from the metallic arc phase to the gaseous arc phase [5] – [9]. It is very important to know information about duration of each phase to provide minimum of arc erosion due to vaporization. Experimental investigation of vaporization in dynamics is rather difficult because of very short of phenomena life time. Therefore mathematical models describing this mechanisms of erosion seems to be very important. Such models of phenomena accompanying transition from metallic to gaseous arc are presented in [10] – [11]. However they are complicated for engineering application and need to be simplified. The mechanism of erosion stipulated by gaseous and electromagnetic pressure with the effect of these and other forces in a liquid melt on the surface has been considered in [12], but no detailed analysis or method of calculation were provided. Incomplete estimates of surface tension, as well as convective and electrodynamic forces were given in [13] – [14]. A rigorous substantiation of the hypothesis of the thermo-capillary mechanism of contact erosion during arcing was given in [13]. Mechanism of contact erosion in the solid state was treated for zirconium carbide in [16] and that for tungsten in [17]. It was confirmed that eroded particles have distinct crystalline cleavage faces. Some approach for modelling of this phenomena was given in [18] – [19]. 124 This paper is an approach of further development of previous models describing different types of contact erosion (vapors, droplets, solid particles) as well as removal mechanism of contact material. 2.2 Vaporization Mathematical model. Mathematical description of electrical contact erosion in dynamics due to vaporization has to take into account such phenomena as ion and electron bombardment of electrode surfaces, electron emission from the cathode, inverse electron flux and radiation from the arc column, time-dependent electromagnetic and temperature fields with heat conduction, melting and vaporization. Dynamics of these phenomena has to be investigated in dependence on given current and voltage, opening velocity, inductance of circuit, properties of contact material and surroundings. A general theoretical model describing dynamics of electric arc was presented in [10]. However it is rather difficult to apply this model for practice, therefore simplified approach has to be retrieved from the general model without losses of main properties of arc dynamics. It can be obtained by replacing of differential equations for the arc column, sheath , ionization zone, near-anode region with more simple balance energy equations, while differential equations remain only for electrodes with given heat fluxes from the arc column. As a result of arc heat flux a pool of liquid metal forms on the contact surface. Fig.1 depicts the axial symmetric region D1 ( h (r, t ) z hm (r, t ), 0 r (t ) ) occupied by melted metal and the region D2 ( ( z 0, r 0) \ D2 ) occupied by solid zone of electrode. The heat equation for both these regions can be represented in the form Ci i Ti 2 div (i Ti ) i ji t (3.2.1) where Ci , i , i , i , ji , Ti , and t are thermal capacity, density, thermal conductivity, electrical resistivity, current density, temperature, and time respectively, h (r , t ) and h (r , t ) are the isotherms of evaporation and melting, and h0 are the radius and the depth of the melt, f is the radius of arc root, index i 1 is related to the melted zone, while i 2 - to the solid zone. At the isothermal surface of vaporization z h (r, t ) the equilibrium of heat fluxes given as 125 1 h T1 Q f L 1 n t (3.2.2) rA (t ) QA z h (r , t ) r t t 0 r D1 z hm r , t D2 z Fig. 1 Melted ( D1 ) and solid ( D2 ) regions of electrode with heat flux from the arc column Q A q 0 (t ) exp( r2 rA (t ) 2 ) (3.2.3) where q0 (t ), L , rA (t ), and n are magnitude of heat flux at the center of contact spot, heat of vaporization, effective radius of normal distribution, and surface normal respectively. The rate of vaporization can be described by the Langmuir law 1 h B exp( A ) , t T1 T1 (3.2.4) where (2RT / M )1/ 2 , RT is the gas constant, M is the vapour molecular weight, A and B are the constants of vaporization. If we suggest that portion of entering the electrode heat flux Q (r , t ) consumed for vaporization has normal distribution along radius Q q (t ) exp( r2 r (t ) 2 ) (3.2.5) where q (t ) and r (t ) are magnitude and effective radius of vaporization, then instead of Langmuir Law (4) one can consider only boundary condition (4) corrected by the subtraction of the expression (5). Effective radius of vaporization in this case can be found from the equation 126 T1 ( r (t ), 0, t ) Tb (3.2.6) where Tb is the boiling temperature. The magnitudes q0 (t ) and q (t ) are determined from the energy balance equation for arc column, anode and cathode surfaces. At the interface surface z hm (r, t ) between melt and solid states the Stefan conditions are applicable 1 h T1 T 2 2 Lm 1 m n n t (3.2.7) T1 T2 Tm (3.2.8) are the latent heat of melting and melting temperature where Lm and Tm respectively. For closing of this system of equations it is necessary to put additionally the condition of axial symmetry T1 0, r if r 0 (3.2.9) T1 (0,0,0) Tm (3.2.10) and initial conditions for the temperature T2 (r, z,0) T0 (r, z), where Tm is the melting temperature. The expression for T0 (r , z) can be found from the solution of heat equation at the period preceding melting [11]. Results of calculations. Copper electrodes. The calculations are carried out for the Cu electrodes with the current I 100 A, 300 A, 500 A and opening velocity V0 0.5 m / sec, 5 m / sec, 20 m / sec . The results are presented in Fig. 2 – 4. 127 Ta , Tc 10 3 0 C 6,00 1 2 3 6 5 5,00 4,00 4 3,00 2,00 1,00 0 0,2 0,4 0,6 0,8 1 1,2 t, msec Fig. 2 The temperature Ta (0, t ) and Tc (0, t ) at the center of the anode and the cathode spots respectively as functions of t . Cu - contacts. 1 - I 100 A, 2 - I 300 A, 3 - I 500 A . For anode: For cathode: 4 - I 100 A, 5 - I 300 A, 6 - I 500 A Critical time: t cr 0.4 m sec (300 A), 0.65m sec (500 A), 0.8m sec (500 A) Just after bridge rupture the temperature at the center of the anode spot Ta (0, t ) ( Fig. 2 ) jumping up to the value 6 103 K decreases then during the time less then 1103 sec down to the values comparable with the cathode group temperature ( Tc 4 103 K ). Herewith the anode ceases active evaporation, and the arc transforms from the metallic phase into the gaseous phase at the critical points t cr of intersection of the anode and cathode temperatures. The current range is very important because the lesser life time of the anode arc corresponds to the lesser current . Decreasing of the temperature at the center of the anode spot is a result both of the anode heat flux decreasing owing to increasing of a contact gap and of evaporation cooling . Therefore conditions of appearance and evolution of the gaseous arc stage depend on properties of a contact material such as melting and boiling temperature , specific heat of vaporization etc., that have to be taken into account in a mathematical model of a short arc . Vaporization of material from the arc roots is also very important for the evolution of a short arc . The effective radius of evaporation from the anode spot r (t ) increases with current increasing ( Fig. 3 ) , while rv (t ) for the cathode changes very slowly . 128 r t , 10 6 m 3 3 2,5 2 2 1,5 1 6 5 4 1 0,5 0 t , m sec 0 0,2 0,4 0,6 0,8 1 1,2 Fig. 3 Dynamics of the effective radius of evaporation rev (t ) from the anode spot ( 1 - I = 100 A , 2 - I = 300 A , 3 - I = 500 A) and from the cathode spot ( 4 - I - 100 A , 5 - I = 300 A , 6 - I = 500 A ) . ( Co - contacts) The anode rate of increasing drev changes from 0.1m / sec to 0. 2m / sec and dt approach 0 to the end of the anode arc stage . The curves of the anode evaporation losses Pev (t ) presented on the Fig. 4 rises rapidly for the first time up to the maximum at t 0.1 - 0.5 msec and then falls down to zero intersecting the cathode curves at the critical points . Q a t , Q c t 10 3 W 10 8 6 6 3 4 2 2 5 1 4 0 t , m sec 0 0,2 0,4 0,6 0,8 1 Pev ( t ) ( 1 - I = 100 A , 2 - I = Fig. 4 Anode evaporation losses 300A, 3 - I = 500 A) and cathode evaporation losses Qev (t ) ( 4 - I = 100 A , 5 Co - contacts - I = 300 A, 6 - I = 500 A ) , V0 0. 2m / sec . 129 The maximum point of the anode vaporization is displaced slightly when the current increases to the greater time . This phenomena can be explained by the changing of the relationship between the temperature rate and vaporization intensity . At the initial stage vaporization energy rises owing to the sharp increasing of the anode local temperature .This time rapidly increasing radius rv (t ) stipulates an intensive vaporization . After maximum point the temperature decreases as a result of both vaporization cooling effect and decreasing of heat flux from the arc due to increasing of contact gap. The vaporization intensity is very sensible to the change of the temperature because it of exponential dependence ( 4 ) as well as to a current value . One can conclude from the Fig.4 that the anode evaporation comes to an end at t = 5 msec for the current I = 100 A , while at the same time evaporation for the current I = 500 A is maximum. The comparison of the results of these calculations with experimental data [20] showed satisfactory conformity with the error not more than 20%. Erosion of AgMeO electrodes. Similar calculations were carried out for silver metal oxide AgCdO and AgCdO at the conditions : DC I 20 A and voltage U 14 V , opening velocity V0 0.2 m / sec , inductance I 40 A , L 0 and L 50 mH . These conditions are typical for interruption of current flow in automobile area as well as in telecommunication. The results of calculations are presented in Fig. 5 – 7. 130 Mass variation (ng/op) 400 1 2 200 0 -200 1 10 3 4 100 1000 10000 -400 Mass variation (ng/op) Arc length ( m) 100 1 2 0 1 10 3 -100 100 1000 10000 4 -200 Arc length ( m ) Mass variation (ng/op) Fig. 5 AgCdO mass variation versus arc length. L 0 mass variation versus arc length. L 0 . 1 - I 40 A ( cathode ), 2 - I 20 A ( cathode ) . ( cathode ), 2 - I 20 A ( cathode ) 3 - I 20 A ( anode ), 4 - I 40 A ( anode ). ( anode ), 4 - I 40 A ( anode ). 400 200 0 -200 1 -400 -600 -800 -1000 Fig. 6 AgSnO2 1 - I 40 A 3 - I 20 A 3 1 10 2 100 1000 10000 4 Arc length ( m ) Fig. 7 AgCdO and AgSnO2 mass variation versus arc length. L 50 mH , I 40 A 1 - AgCdO cathode , 2 - AgCdO 3 - AgSnO2 anode , 4 - AgSnO2 anode, cathode These results are in a good agreement with experimental data[9], [22]. Its analysis enables to conclude that in metallic arc phase (short arc length) , which 131 is characterized by material transfer from the anode to the cathode, the erosion of AgSnO2 contacts is considerably small than erosion of AgCdO contacts both for resistive and inductive circuits, while in gaseous arc phase (long arc length) with opposite material transfer the rate of erosion depends on the inductance. If the inductance L 0 , then AgSnO2 contacts have smaller erosion in comparison with AgCdO contacts, however for inductive circuits situation is quite different, thus use of AgCdO contacts in the case of long arcs burning in gaseous phase is more preferable. 2.3 Liquid droplets erosion Side by side with vaporization the another mechanism of contact erosion in the form of ejected liquid droplets occurs in the range of moderate and high current. It is stipulated by interaction of various forces in the region of liquid metal pool, such as electromagnetic and gas-kinetic pressure, surface tension, explosion and spraying of gaseous inclusions inside molten metal etc. Mathematical model (1) – (10) presented above has to be corrected to take into account heat and mass transfer in a thin boundary layer on the molten surface. The motion equation for the liquid metal in the region D1 can be written as V 1 V V P V F t 1 (3.2.11) and the continuity equation is V 0 (3.2.12) The heat equation (1) for i 1 has to replaced by the energy equation C1 1 ( T1 2 V T1 ) div (1T1 ) 1 j1 t (3.2.13) while for i 2 it remains the same. Here V (Vr , V z ) is the velocity of molten metal, P is the gas-kinetic pressure, F ( 0 / 1 ) j H is the electromagnetic force, and 0 are viscosity and magnetic permeability. Additionally to the boundary conditions for the temperature similar conditions must be given for the velocity Vr 0 z on the surface 132 z h (r , t ) (3.2.14) Vr V z 0 on the surface z hm ( r , t ) (3.2.15) To simplify this problem we can use the radial distribution of pressure as P P0 (1 r 2 / 2 ) (3.2.16) Using the theory of similarity one can derive that the characteristic time t h for the heat transfer phenomena is much greater in comparison with the t for the hydrodynamic phenomena. Hence the characteristic time hydrodynamic parts of the above considered equations can be solved separately using quasi-stationary heat approximation, i.e. assuming that T1 , hm and h are constants for a given time t . Applying the law of energy conservation 1 1 V 2 t 2 2V E H E M (3.2.17) where the terms on the left hand side correspond to kinetic and potential energy, while terms on the right-hand side represent the action of hydrodynamic and electromagnetic forces. Using functional analysis methods the following approach can be obtained tA 1 V 2 V 2 dt WM WV (3.2.18) 0 where t A is the arcing time, and the expressions for E M , E H , WM , WV are given in the paper [15]. The relationship (18) allows the hydrodynamic and electromagnetic forces to be estimated without solving the differential equations but comparing each term on the right side of (18). This equation becomes more effective by replacing region D1 with the boundary layer near the vaporization surface z h (r, t ) . Here the temperature and velocity fields at vaporization can be described by the system of equations C1 1 T1 T 2 (1 1 ) 1 j r , z t z 2 Vr Vr , t z 2 Vr Vr V z 0, r r z V z 2V r 0 1 F jr H 1 z 1 t z 2 (3.2.19) that can be solved by numerical methods. The melting isotherms T of a copper cathode under the action of arc heat flux Q A and Joule sources with current density j are shown in the Fig. 5 for the parameters 133 Q A Q0 exp( i r 2 ) , Q0 2.2 10 7 W / m 2 , i 3.18 10 4 m 2 j 0 1.9 10 6 A / m 2 , j 1.3 10 2 m 2 j j 0 exp( j r 2 ) , The dynamics of melting isotherms during arcing is shown in Fig. 8 0 1 2 3 1 4 2 5 3 6 7 8 r , 10 3 m 4 1 2 3 4 z , 10 4 m Fig. 8 Dynamics of melting isotherms of convective heat transfer during arcing. I 100 A , 1 t 0.2 m sec , 2 t 0.4 m sec , 3 t 0.6 m sec , 4 t 0.8 m sec It enables to estimate erosion in the form of liquid droplets which kinetic energy is greater than the energy of surface tension. 2.4 Thermo-capillary mechanism of erosion Some times it is impossible to explain measured values of erosion in the forms of droplets due to motion of liquid metal from the center to periphery by the action of electromagnetic and gas-kinetic pressure, especially for such refractory materials as tungsten, molybdenum, zirconium etc. In this case erosion phenomena can be explained by the influence of thermo-capillary Marangoni effect which provokes an intensive convective flow in a narrow surface layer of melting zone owing to temperature dependence of surface tension of liquid metal. To take into account this effect it is necessary to correct the above presented model of convective heat and mass transfer by replacing of the condition (14) with the special boundary condition for thermo-capillary forces causing radial stresses on the melted surface: Vr d T1 , z dT1 r z h (r , t ) (3.2.20) where is the dynamic viscosity and is the surface tension. Beside that, additional terms E and W responsible for the thermo-capillary effect must be added in the right side of the equations (17) and (18). 134 In this case, the analog of the thermo-capillary Reynolds number Re W10 h0 1 / , W10 d Tm T0 , the Prandtl number dT1 Pr 1 and the Marangoni number Ma Re Pr play key role. The melting isotherms of melting tungsten for the same above given conditions are shown in Fig. 9 taking into account thermo-capillary convection 0 1 2 3 1 4 2 5 3 6 7 8 r , 10 3 m 4 1 2 3 4 z , 10 4 m Fig. 9 I 100 A , Melting Ma 2 10 isotherms of thermo-capillary heat transfer. 3 It was shown in [15] that for tungsten with current density j 6.45 10 7 A / m 2 and heat flux Q0 3.2 10 8 W / m 2 the Marangoni number is Ma 1.13 10 2 , while the rate of thermo-capillary convection V r at the molten surface reaches 13 m / sec , thus causing the ejection of metal from the molten pool by the thermo-capillary forces. Fig. 10 depicts a typical picture of thermo-capillary waves moving from the center of molten pool to its periphery. c is the cathode material density . Fig.10 Thermo-capillary waves [21] 135 2.5 Solid state type of erosion Brittle fraction of metallic surface layer caused by thermal stresses generated by high power current pulses of short duration and concentrated heat sources from the arc discharge ( i 10 7 m 2 ) can also significantly contribute to the overall erosion of contacts as the electric arc takes the effect. The nature of thermal stresses is strongly dependent on the current density in the area of the arc. At the current density 10 8 A / m 2 and higher, the temperature rise is due to localized heating in the region under the arc spot, while at the density 10 7 A / m 2 and lower, the surface heat sources prevail. The occurrence of thermo-elastic stresses due to Joule heat sources can be described by the model based on the same heat equation like equation (1): c T div (T ) j 2 , t r 0, z0 (3.2.21) The current density in the case of a pulse can be calculated by solving of the equation for electric potential distribution in the form [11]: j (r , z, t ) t 1 ( ) 4fr (3.2.22) where is the rate of the current growth, f f (t ) is the arc spot radius growing over time, and [ z 2 (r f ) 2 z (r f ) 2 2 ]1 / 4 The solution of the equation (21) with the boundary conditions in the form of heat flux as P0 t entering the contact from the arc is well-known [11]. Using this solution and thermo-elastic potential, it is possible to find stress components. Numerical calculation show that thermo-elastic stresses according to this model become important only when the rate of the current growth is very high ( 10 7 A / sec for tungsten and 10 9 A / sec for copper ). In this case the stress is highest along the outer rim of the arc spot ( z 0, r 0 ), where densities of current and heat sources are also highest, whereas in the arc spot center the stresses are considerably lower as seen from the equation (22). When the surface sources of heat are considerable, the stress distribution is opposite. In this case, it is essential to use a simple spherical Holm’s model describing the temperature field in the electrode rather than a cylindrical one. Hence, a heat-receiving spot with radius f has to be replaced with a hemisphere 136 having an ideal conductivity with radius b f 2 , for which entering heat flux is equal to the value T (b, t ) P0 t r (3.2.23) The principal components of the stress tensor can be determined using the quasi-static approximation from the expressions rr 2 E 0 1 1 r 3 r 2 x T ( x, t )dx , b E 0 1 [ 1 r 3 r x 2 T ( x, t )dx T (r , t )] (3.2.24) b Solution of the equation (21) at the condition (23) is T (r , t ) bP0 t [erf ( ) 2( ) ierfc ( )] (3.2.25) 1 4 exp{( ) [( ) ( (]} 2 2 2 where b , 2a t r , ( z ) exp( z 2 ) erfc z 2a t By inserting formula (25) in (24), we find the stress tensor components. If y is the yield strength of contact material, the condition excluding the fracture by temperature in its simplest form is rr y , i.e. E 0 P0 t 0 b 4 1 4 2 [1 (1 / 2 )] y (3.2.26) 1 where t 0 is the pulse duration. Let us calculate as an example the thermo-elastic stresses generated in tungsten contacts as described in [17]. Assuming I t 0 1000 A , then b 2.55 10 4 m , 2.2 10 9 kg / m 2 , that t 0 2 10 6 sec , (1 / 2 ) 0.94 exceeds the f 2 5 10 4 m , , yield and stress consequently of tungsten y 1.3 10 9 kg / m 2 . The experimentally observed fracture of contact material confirmed the validity of the inequality (26). Using the expression y rr we can calculate the characteristic size l 0 of the region exposed to the action of attenuating thermo-elastic wave that causes the ejection of contact material. 137 rr , 10 8 kg / m 2 6 4 y 1.3 10 8 kg / m 2 2 0 0.5 0.7 0.9 1.1 1.3 Fig. 12 Variation of stress tensor rr As seen from Fig. 12 the inequality 0.65 with y rr is valid if r b 1.3a t . Hence the fracture of electrode occurs at the i.e. depth l 0 1.3a t 0 1.67 10 5 m It appears that the inequality (26) can be very useful for calculating the thermal stresses in contacts operating in a pulsing regime. 2.6 Arc to glow transition For some contact materials at certain circuit parameters (low current and high inductance) the arc instability may lead to the transformation of arc to glow discharge rather than to the arc extinction. Fig. 14 depicts such transformation for all stages of arc root immobility and arc running. t t=43ms tG tA i(t) 10ms u(t) 1A UG 5 ms, d0,6mm 13 ms, d1,6mm IG 29 ms, d3,5mm 250V 30 ms, d3,6mm 32 ms, d3,8mm Fig. 14 The unstable arc to glow transition when nickel was used. This phenomenon was described in detail in [13]. It was found that such transition appears in low current inductive circuits and accompanied by a step of 138 spasmodic voltage increase and current decrease with duration 107 106 s. (Fig. 15). UA, V IA, A 0.25 500 400 UA 0.2 0.15 300 IA 200 0.1 0.05 100 0 0 1 2 3 4 t, s Fig. 15 Transition voltage and current. AgCdO contacts, I 0.22 A Dashed- calculation, solid- experiment cr At certain conditions arc stage duration becomes much smaller than glow stage duration. The problem is to find criteria and optimal choice of interdependent parameters (material properties, current, voltage, resistance, inductance, pressure, opening velocity etc.) providing arc instability and controlled arc-to-glow transformation. Such information is very important, because new resources for diminution of failure and for enhancement of time life and reliability of electrical contacts may be found due to reduction of arc duration at the expense of enlarging of glow duration, which burns practically without erosion. The conditions of arc instability from electrical point of view in terms of circuit parameters are discussed in [13]. 2.7 Dependence of the arc erosion on the current frequency The rate of the arc erosion in opening electrical contacts at ac current has non-monotonic dependence on the frequency f . It increases in the range 50Hz f 500Hz and then decreases for f 500Hz . This phenomenon can be explained by the redistribution of the components of the arc heat flux between anode and cathode during the transition from the metallic arc phase to the gaseous arc phase [34], [48]. Shifting of the arc temperature maximum in time entails the change of the time duration t when the arc temperature is greater 139 than the temperature of gas ionization ( 50000 K ) at which evaporation is more intensive (Table 2). Table 2. Dependence of t on the current frequency f , Hz t , ms 50 49 250 67 500 78 750 62 1000 36 The table 3 gives the information about relationship between current frequency and evaporated domain. Table 3. Dependence of erosion on the current frequency Frequency Hz Mass, mg 50 0.54 250 0.73 500 0.75 750 0.48 1000 0.23 The comparison of these values with the experimental data [34] confirms the conclusion about the non-monotonic dependence of arc erosion on the current frequency. 3 ARC DURATION AND CONDITIONS OF ARC INSTABILITY Investigation of dynamical arc phenomena in opening electrical contacts is very important for performance build-up of circuit breakers by means of decrease of arc duration and erosion. Mayr’s and Cassie’s models [1] and their generalization [2] based on the power balance method are not applicable to describe arc temperature field at the initial arc stage just after arc ignition. Elenbaas-Heller equation gives information about radial distribution of the arc temperature however it is correct for stationary arcs only [3]. Arc dynamics should be described by transient heat equation taking into account nonlinear arc characteristic. It is the first intent of this paper. The second one is to device a method for calculation of arc erosion in dynamics. \ 140 3.1 Mathematical model of arc temperature and conductivity at metallic arc phase Equation for the temperature. The arc temperature (r , t ) in opening contacts just after ignition is less than the threshold value required for gas ionization, ig , however it is sufficient to ionize metallic vapours in the contact gap, which takes place at the temperature im : im ig This initial stage, called metallic arc phase, has very short duration and occurs in a small contact gap. Therefore the arc takes the form of a disk, which thickness is much less than radius, and the axial temperature component can be neglected in comparison with radial component. In this case the heat equation for the arc should be written in the form C 1 ( r ) E 2 Wr t r r r (3.3.1) where C and are thermal capacity and density, and are heat and electrical conductivities of the arc plasma, E is electrical field and Wr is power loss due to arc radiation and heat conduction from arc column to electrodes. The initial temperature distribution along radius (r,0) f (r ) (3.3.2) can be found from the solution of the heat equation for metallic vapours at the pre-arcing stage [4]-[5]. We can approximate the function f (r ) 0 J 0 (1r / rA ) by parabola f (r ) 0 (1 r2 ) rA2 or by the Bessel function f (r ) 0 J 0 (1r / rA ) (3.3.3) where 2.405 is the first root of the Bessel function and 0 is the temperature maximum at the centre of arc disc. The temperature on the interface r rA between ambient air and arc plasma should be equal to threshold of metal ionization 1 (rA , t ) mi (3.3.4) It should be noted that thermal and electrical plasma conductivities, and , depend essentially on the temperature and this dependence can not be averaged. 141 In contrast the arc radiation Wr can be neglected for metallic arc phase, which temperature is relatively low : im < ig 5000 0C (See Fig. 1). , C ,Wr 6 5 1 4 2 3 2 1 0 3 5 6 7 8 9 10 T , 103 K Fig,1 Temperature dependence of , and Wr 1- , Wm1K 1 ; 2- C, 102 Jm3 K 1 3- Wr , 1011 Wm3 [6] Equation for electrical conductivity . To solve the heat equation (1) we use the Kirchhoff’s substitution ( )d S ( ) (3.3.5) mi Then the equation (1) transforms to C S 1 S (r ) E 2 Wr t r r r (3.3.6) Solving the equation (5) with respect to we get mi g (S ) (3.3.7) Since the function ( ) is given (See Fig. 1), we can write this function in term of using (6), i.e. (S ) . Linearization of this function gives the expression (See Fig. 2) g (3.3.8) bS , b tan S gi where g is given electrical conductivity at the transition from metallic arc phase to gaseous arc phase, when gi , and S gi gi ( )d mi 142 (3.3.9) (S ) g S gi 0 S Fig. 2 Linear approximation of (S ) Substituting (8) in (7) and using notation r x E b , C E 2b (3.3.10) we can write the equation with respect to 2 1 2 t x x x (3.3.11) It should be noted that can be considered as constant because the thermal diffusivity a2 C / is approximately constant (See Fig. 1). The domain for this equation is 0 x x0 , where x0 rA E b . The boundary conditions (2)-(4) transform to the type (3.3.12) ( x,0) F ( x) , with f (x/ E b) F ( x) b ( )d mi ( x0 , t ) 0 (3.3.13) The solution of the problem (11)-(13) can be found in the form of FourierBessel series ( x, t ) Cn exp[(kn 2 1)t / ]J 0 (k0 x) n 1 where 143 (3.3.14) Cn x0 2 x0 2 J12 ( n ) 0 F ( x) J 0 (kn x) xdx , kn n / x0 , and n are roots of the Bessel function: J 0 (n ) 0, n 1, 2,3,... For approximation (3) ( x,0) 0 J 0 (1 x / x0 ) and the solution (14) takes the simple form ( x, t ) 0 exp[( 12 x0 2 1)t / ]J 0 ( 1 x / x0 ) Taking into account (10) we get finally the expression for arc electrical conductivity in the form a 2t (r , t ) 0 exp[( E brA ) 2 ]J 0 ( 1r / rA ) rA 2 1 2 2 (3.3.15) The arc temperature can be found now from the expressions (5) and (8). Let us introduce the criterion of arcing E 2brA2 12 (3.3.16) We should distinguish three cases (Fig. 3): (0, t ) 0 0 0 0 t Fig. 3 Evolution of arc conductivity 1). 0 . Rise of arc conductivity, power and temperature due to Joule heating. 2). 0 . Maximum value of arc conductivity and power 3). 0 . Arc conductivity, power and temperature decrease, thus the arc should extinguish. 144 Interaction between arc and contact surface. At the first stage of contact opening 0 and then changes the sign. To find the critical point 0 we need to know the dynamics of arc radius rA , which expands during arcing. Then using formula E I rA2 (3.3.17) and the expression (16) we can find the critical time t tcr at which 0 . For this purpose we consider the region DA occupied by arc interacting with contact surface (Fig. 4). This interaction results into phase transformations of contact material and formation of three zones: 1) The zone of evaporated material Db : 0 r rb (t ), 0 z b (r, t ) , Dm : b (r , t ) z m (r , t ), 2) The zone of melted material if 0 r rb (t ), and 0 z m (r , t ) if rb (t ) r rm (t ) 3) The solid zone Ds : m (r , t ) z , if 0 r rm (t ), and 0 z if rm (t ) r x(t) rm (t ) z m (r , t ) Ds Dm Db rb (t ) rA (t ) 0 zb (t ) zm (t ) z z b (r , t ) Fixed contact DA Movable contact r Fig. 4 The arc and contacts geometry: arc region DA , evaporated zone Db , melted zone Dm and solid zone Ds The contact temperature TC (r, z, t ) can be presented as the sum TC (r, z, t ) TJ (r, z, t ) TS (r, z, t ) 145 (3.3.18) where TJ (r, z, t ) and TS (r , z, t ) are the temperature components due to volumetric Joule heating and due to surface arc flux heating respectively. The expression for calculating of the first component is given above. It can be shown that the Joule component TJ (r, z, t ) is important at the pre-arcing stage only, and it can be neglected after arc ignition. The expression for the second component can be found similarly in the form t 0 0 TS (r , z, t ) dt1 [ Pc (r1 ,t1 ) Pb (r1 , t1 ) (3.3.19) Pm (r1 , t1 )]G (r , r1 , z, t t1 )r1dr1 Here Pc (r , t ) is the total heat flux (power per unit area) entering the contact surface during arcing, Pb (r , t ) and Pm (r , t ) are portions of this flux consumed for evaporation and melting of contact material, which can be found by the expressions Pb (r , t ) Lb b (r , t ) (r , t ) , Pm (r , t ) Lm m t t (3.3.20) where Lb and Lm are specific heat for evaporation and melting, is density of contact material, It reasonable to assume that the isothermal surfaces z b (r, t ) and z m (r , t ) are ellipsoids of revolution that can be found from the equations r2 z2 1 rb (t )2 zb (t ) 2 r2 z2 1, rm (t )2 zm (t )2 in other words b (r , t ) zb (t ) 1 r 2 / rb (t )2 m (r , t ) zm (t ) 1 r 2 / rm (t )2 (3.3.21), The functions rb (t ) , zb (t ) , rm (t ) , and zm (t ) should be found from the equations TC (rb (t ),0, t ) Tb , TC (0, zb (t ), t ) Tb TC (rm (t ),0, t ) Tm , TC (0, zm (t ), t ) Tm (3.3.22) where Tm is the melting temperature of the contact material. If the heat fluxes Pc (r , t ) Pb (r , t ) , Pm (r , t ) obeys the normal Gauss’s radial distribution Pc (r , t ) Pc (t ) exp( 146 r2 ) rA (t )2 Pb (r , t ) Pb (t ) exp( r2 ) rA (t )2 Pm (r , t ) Pm (t ) exp( (3.3.23) r2 ) rA (t )2 then the integral with respect to r in the formula (19) can be calculated and the expression for the contact temperature becomes more simple form TS (r , z , t ) [ Pc (t1 ) Pb (t1 ) Pm (t1 )]rA (t1 ) 2 0 [rA (t1 )2 4a 2 (t t1 )] t t1 t a z2 r2 exp[ 2 ]d 4a (t t1 ) rA (t1 ) 2 4a 2 (t t1 ) (3.3.24) The heat flux Pc (t ) should be calculated taking into account positive components due to arc radiation, electron (or ion) bombardment of anode (cathode) contact surface, inverse electrons from the arc column, and negative components due to power losses for evaporation, radiation, electron emission cooling and heat conduction inside the contact body. The expressions for all these components can be found in the paper [7]. However the model in considered case can be simplified because the information about current, voltage and displacement is available from experiment. Therefore it is more convenient to calculate power generated by arc WA directly from the measured values of arc voltage U A (t ) , arc current I A (t ) and then arc heat flux entering contact is Pc (t ) I A (t ) U A (t ) P (t ) A2 2 2 rA (t ) 2 rA (t ) (3.3.25) This expression is the final equation, which enables in the aggregate with other cited above equations to calculate dynamics of contact melting, evaporation, arc radius rA (t ) and arc power PA (t ) . Fig.5 and Fig. 6 depict dynamics of arc power and temperature for AgCdO contacts calculated using above considered model at the conditions: supplied voltage U 0 14 V , current I 0 20 A , inductance L 47.5 mH , opening velocity V 0.2 m / s [2] PA (t ), W 400 350 300 250 200 150 100 50 0 Fig. 5 Dynamics of arc power PA (t ) 0 5 10 15 20 25 30 35 40 t , ms 147 TA , 103 K 8 7 6 5 2 4 1 3 1 0.1 10 t , ms 100 Fig. 6 Arc temperature. 1 – Experimental data [2], 2 – calculation One can see that critical time in this case is tcr 10 ms , however the maximum of arc temperature occurs a little bit later, at t 15 ms due to thermal inertia. 3.2 Transition from metallic arc phase to gaseous arc phase Temperature field and erosion. The duration of metallic phase is very short, therefore the arc thickness is still small and above considered model can be applied to describe the transition from metallic to gaseous phase if we replace all parameters of metallic vapours by parameters of gaseous vapours. Dynamics of this transitions is represented in Fig. 7. One can see that at the first stage of arcing, when the contact gap does not exceed 20 m , anode temperature rises very sharp in comparison with cathode temperature. Ta , Tc , 103 K 4 Ta 3 Tc 2 1 0 0 0.1 0.2 0.3 0.4 0.5 t , ms Fig. 7 Dynamics of anode and cathode temperature at the centre of arc root. It can be explained by the fact that in a short arc, which length is comparable with the length of ionization zone, electron temperature Te is much greater than ion temperature Ti , therefore kinetic energy of electrons 148 bombarding anode, 3 kTe je , exceeds significantly kinetic energy of ions entering 2 e 3 kTi ji . Moreover, calculation shows that in this range of contact gap 2 e electron component of current density je is much greater than ion component ji , cathode, that is an additional reason for anode overheating and material transfer from anode to cathode. However intensive evaporation from anode and increasing of anode arc spot radius, that entails decreasing of current density, cause anode cooling and decreasing of its temperature, while cathode temperature continues to increase. The point of intersection of anode and cathode temperature occurring at tac 0.15 ms corresponds to change the direction of material transfer for inverse and to beginning of compensation arc stage, which continues up to t1 1.8 ms and transforms then into cathodic stage (Fig. 8). Calculation enables to conclude that cathodic arc stage begins in metallic phase with temperature about 4700 K, that is less than threshold ionisation, however transition to gaseous phase occurs just at t1 2 ms . Results of calculated erosion given in Fig. 8 are evidence of the fact, that the main portion of erosion in inductive circuits occurs in gaseous phase. Calculated values for metallic phase are slightly greater than experimental data. It can be explained by recycling phenomenon, i.e. re-deposition of evaporated material, which is ignored in the mathematical model. M ea , M ec 107 g 4 5 3 0 2 -5 1 -10 -15 -20 -25 -30 0.001 0.0 1 0. 1 1 10 t, ms 100 Fig. 8 Anode and cathode mass transfer. From bottom to top: 1 - M ec (calculated), 2 - M ec (experiment [2]), 3 - M ea (experiment [2]), 4 - M ea (calculated) Influence of inductance on arc duration. Similar calculations were carried out for different values of inductance in the range from 1 mH to 400 mH. It was found that arc duration increases proportionally inductance and depends on current at relatively small values of inductance (Fig. 9). However for inductance 149 greater than 10 mH this dependence becomes negligible. This result correlates with experimental data observed in [2]. tA , s 100000 10000 100 100 1 3 1 2 1 0. 0.001 0.01 0. 1 1 100 L, mH Fig. 9 Arc duration versus inductance 1 - I 0 0.6 A , 2 - I 0 1 A , 3 - I 0 20 A Increasing of arc duration with inductance occurs on account of enlarging of gaseous arc phase, while variation of metallic phase is relatively small. The same conclusion may be proposed for increasing of erosion. However further increasing of inductance up to a few hundred millihenry in the range of low current leads to decrease arc duration and erosion due to arc-to-glow transformation, which is considered below. 3.3 Phenomena in thermo-chemical cathodes The current density at the cathode is one of important characteristics responsible for the evolution of electrical arc between opening contacts. In the case of refractory materials such as tungsten and carbon, the normal cathode spot reaches temperatures of about 3000 o K – 4000 o K, which is sufficient for thermionic emission of observed current density, maintaining of evaporation and arcing. The problem appears when this model is applied for explanation of the rate of erosion for Cu , Zr , Hf and other materials used for thermo-chemical cathodes of plasma generators. In this case it is not possible to explain mechanism of erosion, because the basic Richardson-Dushman formula even with Schottky correction gives too small values for the current density of electron emission in comparison with experimental observed values. The reason of such discrepancy is stipulated by the too large value of eff which varies from 3 eV to 4 eV . Thus, some other mechanism has to be involved to explain required level of current density or decreasing of the value of effective work function eff . Many attempts to ground the probable diminution of the work function were undertaken, such like electron affinity of the semiconductor [1], electrical field enhancing factor due to asperity peaks on the cathode surface [45], I-effect [46] or even by redoubling of Richardson constant [47]. However they could not explain observed experimental data. High values of current density for such materials like Cu, Fe, Ni similarly as for thermo-chemical cathodes Zr, Hf etc can be 150 explained by the phenomenon of electrochemical emission of electrons [31], [33]. This phenomenon was observed first in experiments with thermo-chemical zirconium cathode (TCC) used for plasma generators [47]. It is caused by the formation of a solid or liquid electrolyte on the contact surface after its melting. The anode oxidation of the material occurs on the outer side of the surface with electrochemical ejection of electrons, while the cathode reduction to the initial material takes place on the inner side of the surface. If the rates of electrochemical oxidation and reduction are equal, then the constancy of electron emission is provided. Herewith the energy required for electron emission is equivalent to the energy of chemical reaction spent for the generation of an electrolytic film. Thus, heat energy of chemical reactions spent for generation of oxides is compensated by electrical energy for their destruction. Fig 21 depicts clearly outlined zones formed in TCC. External surface (zone 1) consists of zirconium dioxide. The next (in depth ) zone 2 contains smaller amount of oxygen, and this amount decreases even more in zone 3 due to appearance of metallic zirconium globules in it. Zone 4 is identified as zirconium nitride. Oxides of zones 2 and 3 are amorphous, that testifies to a liquid aggregate state during operations, while the nitrides in zone 4 have crystalline structure. Below oxide layer (zone 5) there is metallic zirconium having no phase transformation during arcing because oxygen does not penetrate into this zone. Fig 22. Cross-section of zirconium cathode. 1 – crystalline zirconium dioxide ZrO2 2 – amorphous molten layer of ZrO2 Zr3O 3 – eutectic of ZrO2 Zr3O and globules of metallic zirconium 4 – zirconium nitride ZrN 5 – zirconium Zr 151 A general model describing interdependent phenomena in electric arc column, anode and cathode ionization zones sheathes and bulks during opening of electrical contacts was presented in the papers [31], [33]. However chemical reactions and diffusive phenomena inside cathode bulk were neglected in this model. To take them into consideration the corresponding subsystem of equations for cathode bulk has to be modified and presented in the form 1 div ( ) 0 (3.3.26) (CvT ) 1 (3.3.27) div (T ) FT (C , T , X ) ( ) 2 t C (3.3.28) div ( DC ) FC (C , T , X ) , t where , T , C , and X X 1 ,..., X n are electrical potential, temperature inside cathode, concentration of diffusing gas, and the quantity of the substance of each i -phase X i (r , z, t ) (mol / m 3 ) respectively, n FT (C , T , X ) riT i 1 n FC (C , T , X ) riC i 1 X i t X i t and are heat and concentration sources density of chemical and riC are specific heat of reaction and stechiometric reactions, riT coefficients, , Cv , , and D are electrical resistivity, thermal capacity, thermal conductivity, and diffusion coefficient. In particular, for above considered Zr O system n 5 (See Fig. 22). Dynamics of X i quantities is describing by the system of differential equations dX i K i ( X i X eq ,i )i (3.3.29) dt the balance quantities of the i -th phase determined from a where X eq ,i respective state diagram. Similar phenomena were observed on the copper cathode of plasma generator with arc root on the oxidized surface, because the electron work function from this place is less then from the pure copper. The copper oxides have strong polar covalent coupling and properties of a semi-conductor. They are able to furnish the ion electrical conductivity in thin films for providing of the emission current. The oxidation on the heated copper takes place accordingly to the scheme 2Cu O2 2CuO , 4Cu O2 2Cu 2 O Since isobaric-isothermal potential g for Cu 2 O is equal to 146.2 103 J / mol , while for CuO it is equal to 118.7 103 J / mol , so generation of CuO is more preferable from thermodynamic point of view. From the other side electrochemical reduction of copper oxides to metal can occur on the interface between cathode surface and plasma accordingly to the scheme: 152 2CuO 4e 2Cu 0 O2 4e , 2Cu 2 O 4e 4Cu 0 O2 4e It means that relay-race transmission of electrons from the cathode to the plasma occurs at the electrochemical reduction of copper oxides to metal. Thus side by side with thermionic component of electron current density j em there exists electrochemical component jec , and the total current density jet is defined by the formula jet jem jec (3.3.30) Consequently the mathematical model presented in the paper [8] has to be corrected by replacing the expressions for energy of emitted electrons 1 3 2 meVem kTc eU c , 2 2 for electron current density jem nem eVem , as well as Richardson-Dushman formula for thermionic emission jem ATc exp( 2 e eff kTc ) (3.3.31) with corresponding new expressions taking into account also electrochemical mechanism of electron emission: 1 3 2 meVem kTc eU c e ec , 2 2 jet nem eVem nec eVec , (3.3.32) (3.3.33) and with formula (3.43) for total current density. The electrochemical component of current density jec can be evaluated using Faraday’s law and Arrhenius law, which in simplest case is defined as K K 0 exp( Ea ). RTc Then jec jec exp( 0 Ea ), RTc (3.3.34) where jec 0 ZFCK 0 rc 2 (3.3.35) It is possible to simplify the expressions (3.43)-(3.48) if we suggest that activation energy E a is equivalent to the energy required for destruction of oxides: 153 Ea e p where p g is the potential required for the electrochemical mechanism of ZF electron emission.. Using the value g one can estimate the corresponding value of this potential At the standard conditions p 0.53 V for CuO and p 0.76 V for Cu 2 O . Taking into account the theory of mirror image for the electron emission from the cathode surface to plasma these values must be doubled. Thus finally p 1.06 V for CuO and p 1.52 V for Cu 2 O . The analysis of the solution shows that this potential p can be considered as correction of effective work function in Richardson-Dushman formula (3.44) to take into account electrochemical mechanism of electron emission: jet ATc exp[ 2 e( eff p ) RTc ] (3.3.36) In this case the interval of corrected potential of electron emission et eff p for Cu contacts is 3.0 3.5 V , and that is enough to maintain arc and explain experimental observed electron current density. 3.4 Arc to glow transition For some contact materials at certain circuit parameters (low current and high inductance) the arc instability may lead to the transformation of arc to glow discharge rather than to the arc extinction. Fig. 14 depicts such transformation for all stages of arc root immobility and arc running. t t=43ms tG tA i(t) 10ms u(t) 1A UG 5 ms, d0,6mm 13 ms, d1,6mm IG 29 ms, d3,5mm 250V 30 ms, d3,6mm 32 ms, d3,8mm Fig. 14 The unstable arc to glow transition when nickel was used. This phenomenon was described in detail in [13]. It was found that such transition appears in low current inductive circuits and accompanied by a step of spasmodic voltage increase and current decrease with duration 107 106 s. (Fig. 154 15). UA, V IA, A 0.25 500 400 UA 0.2 0.15 300 IA 200 0.1 0.05 100 0 1 2 3 4 0 t, s Fig. 15 Transition voltage and current. AgCdO contacts, I 0.22 A Dashed- calculation, solid- experiment cr At certain conditions arc stage duration becomes much smaller than glow stage duration. The problem is to find criteria and optimal choice of interdependent parameters (material properties, current, voltage, resistance, inductance, pressure, opening velocity etc.) providing arc instability and controlled arc-to-glow transformation. Such information is very important, because new resources for diminution of failure and for enhancement of time life and reliability of electrical contacts may be found due to reduction of arc duration at the expense of enlarging of glow duration, which burns practically without erosion. The conditions of arc instability from electrical point of view in terms of circuit parameters are discussed in [13]. 3.5 Dependence of the arc erosion on the current frequency The rate of the arc erosion in opening electrical contacts at ac current has nonmonotonic dependence on the frequency f . It increases in the range 50Hz f 500Hz and then decreases for f 500Hz . This phenomenon can be explained by the redistribution of the components of the arc heat flux between anode and cathode during the transition from the metallic arc phase to the gaseous arc phase [34], [48]. 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Doremieux, “ Phenomena Preceding Arc Ignition Between Opening Contacts: Experimental Study and Theoretical Approach”. Proc. 36-th Holm Conf. on Electrical Contacts, Montreal, 1990, p.p. 543 - 549 2. S.N. Kharin, “Dynamics of the Transition of the Arc Metallic Phase into Gaseous Phase in Opening Electrical Contacts”. Proc. 17-th Int. Conf. on Electrical Contacts, Nagoya, Japan, 1994, p.p. 817 - 824 3. K. Sato, T. Sato, H. Sone, T. Takagi, “Relationship Between Bridge Energy and Metallic Phase Arc Duration in Electrical Contacts”. Japanese Journal of Applied Physics, vol. 226, 4, 1987, p.p. L261 - L263. 4. T.Takagi. “Relationship Between Contact Operating Conditions Contact Phenomena in Ag and Pd Breaking Contacts”. Proc. 36-th Holm Conf. on Electrical Contacts, Montreal, 1990, p.p. 1 - 7. 5. S.N. Kharin. “ Models for Investigation of Heat and Mass Transfer in Electrical Contacts”. Proc. 8-th Int. Conf. on Electrical Contacts Phenomena, Tokyo, Japan, 1976, p.p. 553 - 558. 6. S.N. 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Chen and K. Sawa, “Particle sputtering and deposition mechanism for material transfer in breaking arcs”, J. Appl. Phys., Vol. 76, pp. 3326- 3331, 1994 164 APPENDIX 1. MODELLING AND CALCULATION OF THE CONTACT RESISTANCE COMPONENTS A1.1 Constriction resistance. The contact resistance components are defined in terms of electrical conductivity components c , m , f by the expression (1.1.5). To calculate these conductivity components, we use to consider both contact surfaces as planes covered with a multitude of spherical asperities with the same radius rm and the variances of height distributions 1 and 2 for each of the contacting surface ( Fig. A-1.1). z r0 rm z d x rm d rm z 0 Fig A-1.1 Simplified schematic model of a micro-asperity The contact between such rough surfaces can be replaced by the contact of plane surface with model rough surface, if the variance of height distribution of the model surface is chosen as 12 2 2 165 (A.1.1) n of contact spots per unit area of the nominal Assuming Gaussian asperity height distribution, the number contact surface can be expressed as d 2 , z n N 2 z z 2 exp( 2 2 )dz d N (erfc erfc ) , 2 ( A.1.2 ) 2 where N is the total number of asperities defined by the function of the surface finish, d is the distance ( gap ) between the reference planes, z is the maximal asperity height of the model plane . The special function in the right side is determined by the expression 2 exp( z erfc 2 )dz If the spots are uniformly distributed over the contact surface, and the distance 2r between two individual spots is much greater than the radius of a spot r0 , then the radius r of the constriction for each individual spot is given as r 1 2 n [2 N (erfc erfc ]1/ 2 (A.1.3) and the constriction resistance can be expressed as [1]: Rc ( z ) ( 1 2 ) / 2 , 1 where Thus and 2 N c 2 From Fig. A-1.1 one can derive arctan r arctan r0 r0 , (A.1.4) are electrical resistivity of the contacting members. z 1 z exp( 2 )dz 2 c ( z) R d 1 r0 ( z d )(2rm z d ) , and since r r0 , z d 2rm , then r and r0 2 r0 2rm ( z d ) (A.1.5) Therefore, we obtain for the constriction conductivity the following expression c 4 N 2rm 3 ( 2)1/ 2 ( ) exp[( )2 ] o( ) (A.1.6) A1.2 Micro-asperity resistance. Bearing in mind that the asperity radius at the point x (Fig. A-1.1) is h (rm x)1/ 2 (rm x)1/ 2 , the resistance Rm ( z ) of an asperity of the height z is given as Rm ( z ) rm z d rm z z (2rm z d ) dx ln (rm x)(rm x) 2 rm ( z d )(2rm z ) Hence, the conductivity of asperities m is given as 166 (A.1.7) m z N d 1 z2 exp( 2 )dz Rm ( z ) 2 This integral can be approximated if (A.1.8) ( z d ) d . Then the expression (A.1.8) becomes 0 2 (2 0 )( ) exp( 2 ) li ( ) m 4 N (2 0 ) 0 2 where 0 rm 2 and dt ln t 0 li ( x) is the integral logarithmic function . The contact resistance of an oxide film for a single spot Rf ft r0 2 R( z ) is structure- Cu2O this resistance can be expressed as: dependent. For conductive oxide films, such as f (A.1.9) x A1.3 Film resistance. where , ft 2 rm ( z d ) o( 1 ), zd (A.1.10) t is its thickness. Hence, the oxide film conductivity is is the resistivity of oxide film and z Nr 2 N 2 rm z2 f ( z d ) exp( )dz m (ierfc ierfc * ) , (A.1.11) 2 2 ft 2 f t d where ierfc i erfc 1 2 ( z ) exp( z 2 )dz e 2 erfc , and in general i erfc 2 1 ( z ) exp( z 2 )dz ( 1) for any real . In the case of thin (adhesive or passivating) films whereby the conduction occurs via tunneling, the film conductivity can be expressed as f N rm 2 f ( ierfc ierfc ) , (A.1.12) where f is the tunnel resistivity. For practical use of the above described model it is essential to know the values for and d . The variance of the height distribution of asperities can be determined from the surface profile-metric traces, while d can be derived from the expression relating deformation z d of the i th asperity with the initial height z by load Pci [10]: E rm 4 , Pci ( z d )3/ 2 3 1 2 (A.1.13) where E is the elastic modulus, is the Poisson ratio. It has to be noted that this formula combined with the formula (A.1.5) gives well-known Hertz formula for a plane-spherical contact: r0 1.2 3 Pc rm (1 2 ) E 167 Hence, the total load Pc acting on a unit area of the contacting surface can be derived by integration of (A.1.13) with Gaussian distribution density: 3/ 2 z 2 rm 3/ 2 4 NE rm z2 4 NE 3/ 2 Pc ( z d ) exp( ) dz (i erfc -i3/2erfc * ) 2 2 2 3 (1 ) d 2 3 (1 ) (A.1.14) The graph of the function f ( ) i erfc is given on the Fig. A-1. 3/ 2 f ( ) 0.4 0.3 0.2 0.1 0 0.4 0.6 0.8 Fig. A-1 The graph of the function Hence if the applied contact force 1.0 1.2 f ( ) i3/ 2erfc Pc is given, then the values for , and d can be derived from this tra 168 Scientific publication Stanislav Nikolaevich Kharin APPENDIX 2. SOLUTION IN SPHERICAL MODEL A.2.1 Radial temperature and potential distribution. The current density in the spherical model is j I , 2 r 2 (A.2.1) thus Joule heat power per unit volume is 1 grad 2 j 2 I 2 4 2 r 4 The equation (1.2.22) for a stationary temperature distribution ( (A.2.2) 0 ) in spherical co-ordinates taking into t account expression (A.2.2) can be written in the form 1 d 2 d I 2 ( r ) 0 r 2 dr dr 4 2 r 4 (A.2.3) The boundary conditions for the equation (A.2.3) are very simple. The surface of ideal sphere r b is a natural adiabatic boundary between two contact pieces due temperature symmetry, therefore d (b) (A.2.4) 0 dr while outside the constriction contact zone ( r ) the super-temperature (i.e. the real temperature minus temperature of the surroundings) remains the same like before heating: () 0 (A.2.5) If we assume the linear temperature dependence of resistivity 0 (1 ) (A.2.6) then the solution of the equation (A.2.3) with the boundary conditions (A.2.4) and (A.2.5) is (r ) 1 cos 2 [ cos (1 b / r ) 1] (A.2.7) 2 where 0 b I (A.2.8) 2 169 Using this formula the gradient of potential can be represented as I d 0 (1 ) I 0 2 2 dr 2 r 2 r cos 2 cos (1 b / r ) 2 and after integration we obtain the expression for the potential in the form 0 I sin 2 (1 b / r ) (r ) 2 b cos (A.2.9) 2 A.2.2 Simplified models. In the range of relatively small temperature electrical resistivity can be averaged and considered as constant: ( ) const . Then corresponding expressions for the temperature 0 (r ) and the potential : 0 (r ) can be derived from (A.2.7) and (A.2.9) using limit as I 2 1 b (1 ) 2 4 b r 2r I b 0 (r ) (1 ) 2 b r 0 (r ) The value of and replacing 0 by (A.2.10) (A.2.11) can be obtained by comparison of the real constriction resistance with modelled constriction resistance for the constant averaged The resistance 0 . R(r ) of a spherical shell between two hemispheres with radii b and r respectively is R(r ) (r ) I 0 sin 2 (1 b / r ) 2 b cos (A.2.12) 2 thus the resistance of one contact member R ( ) 0 2b tan 2 (A.2.13) and the total constriction resistance Rc 2 R() Similar values for the averaged 2 0 2b tan 2 (A.2.14) can be calculated from (A.2.11) as R0 () 0 ( ) I 2 b 170 (A.2.15) 2 0 () I b R0c 2 R0 () (A.2.16) Equating (A.2.14) and (A.2.16) we get the expression for the replaced averaged resistivity 1 2 0 If (hence tan in the form tan 2 (A.2.17) 2 1 1 ( / 2) 2 and one can derive from (A.2.8), (A.2.10) / 2 3 as well) is small, then the well known formula [1] 2 3 2 0 [1 0 (b)] (A.2.18) However much better result, especially for high current, can be achieved by replacing 2 3 3 0 [1 (b)] 0 (b) for (b) : (A.2.19) It should be noted that temperature calculation using expressions (A.2.7) and (A.2.10) gives very similar results, especially in the range of not too high temperature. A.2.3 Kohlraush’s law If (r ) and (r ) are the temperature and electrical potential corresponding to a hemisphere of the radius r , then equating power of heat flow passing through the surface of hemisphere and electrical power generating this heat flow we obtain: 2 r 2 d I (r ) dr This balance formula can be checked also by direct substitution of the expressions (A.2.7) - (A.2.9). Multiplying both sides by the factor ( ) 2 r 2 and taking into account that Integration from b to I ( ) d we get 2 r 2 dr d d dr dr r gives well-known Kohlraush’s law (b ) d (r ) 2 2 (r ) Since super-temperature (r ) T (r ) T0 , (A.2.20) T (r ) and T0 are contact and ambient temperature where respectively, we get T (b ) dT (r ) 2 T (r ) Using Wiedemann – Franz law conclude that LT , where 2 L 2.4 108V 2 K 2 is the Lorenz constant, one can L[T (b)2 T (r )2 ] (r )2 171 (A.2.21) In particular for the whole constriction zone of one contact member ( r , T () T0 , () U k / 2 ) the final formula, so-called the temperature – potential relationship, takes the form [1] L[T (b)2 T0 ] U c / 4 2 2 (A.2.22) It will be shown later that this formula keeps to be valid not for the spherical model but in general case as well. APPENDIX 3. Reducing the system of dual integral equation (21) to a Fredholm equation. System (21) is : 2 0 ( x W ) A( x) J 0 ( xr )dx 2 , 0 r f (A.1) A( x) J 0 ( xr ) xdx 0 , f r 0 After substitution A( x) x C ( x) this system is represented in the form [1 g ( x)] C ( x) J 0 ( xr )dx 0 0r f , (A.2) C ( x) J 0 ( xr )dx 0 , f r 0 where g ( x) 2 1 W x Let be f C ( x) x (t ) cos( xt )dt (A.3) 0 Then the second equation of the system (A.2) is satisfied for every differentiable function (t ) . Indeed , f 0 0 C ( x) J 0 ( xr )dx x J 0 ( xr )dx (t ) cos( xt )dt J 0 ( xr ) [ ( f ) sin( fx) 0 0 f f o 0 0 0 '(t ) sin( xt )dt ]dx ( f ) J 0 ( xr ) sin( fx)dx '(t )dt J 0 ( xt ) sin( xt )dx 0 if r f , in accordance with the formula J 0 ( xr ) sin( xt )dx 0 , if 0 t r , 0 J 1 , if 0 r t t r2 The first equation of the system (A.2) may be written in the form 0 ( xr ) sin( xt )dx 2 0 172 C ( x) J 0 ( xr )dx 0 C ( x) g ( x) J o ( xr )dx , 0 0r f This equation together with the second equation enables to write the Bessel integral C ( x) J 0 ( xr )dx F (r ) 0 where F (r ) C ( x) g ( x) J 0 ( xr )dx , if 0 r f 0 if F (r ) 0 , Converting this Bessel integral we obtain (A.4) f r f C ( x) x F ( ) J 0 ( x ) d 0 Using the formula 2 cos( xt ) J 0 ( x ) 0 2 t2 we get f f 2x F ( ) C ( x) cos( xt )dt d 0 2 t2 t and after comparison this expression for C (x) with (A.3) one can write (t ) 2 F ( ) f 2 t2 t d or taking into account (A.4) f f 2 d 2 (t ) K (t , t1 ) (t1 )dt1 t 2 t 2 0 (A.5) where f K (t , t1 ) t 2 W f t d 2 t2 d 2 t2 x g ( x) J J 0 ( x ) cos( xt1 )dx 0 0 ( x ) cos( xt1 )dx 0 Since J 0 ( x ) cos( xt1 )dx 2 t1 , if t1 2 0 173 J 0 ( x ) cos( xt1 )dx 0 , if t , 0 therefore K (t , t1 ) 2 W d f t 2 t1 2 2 t 2 ln W f 2 t2 K (t , t1 ) 2 W 2 ln W f 2 t1 t 2 t1 d t 2 t1 2 2 t1 f 2 t1 2 , if t t1 2 2 f 2 2 , if f 2 t2 t t1 t1 t 2 2 hence f 2 t1 f 2 t2 2 2 K (t , t1 ) ln W t1 t 2 2 and equation (A.5) finally is written in the form : 2 (t ) 4 W t ln f 2 t2 f 2 t2 0 t 2 t1 f 2 t1 2 2 (t1 )dt1 , (22) APPENDIX 4. GENERAL MHD ARC MODEL A. Equations for the cathode region. 1. Cathode surface ( z 0) Energy balance equation qi qie qem q qr qev ( A.1 ) consisting of : - Ion bombardment component qi ji ( uc i eff where ji is the ion current density , uc is the cathode fall , 174 5 kTc ) , (A.2) 2 e i is the ionization voltage , eff is the effective work function , Tc is the cathode spot temperature , ( Tc c (r , z , t ) at z 0 , 0 r rc (t ) ) k is the Boltzmann constant , e is the electron charge . - Back-diffusion electrons component qie jie ( eff where 5 kTe ) , ( A.3 ) 2 e jie is the back-diffusion electron current density , Te is the electron temperature . - Electron emission component qem jem ( eff where jem is the electron emission current density . -Heat conduction energy flux component q c where 5 kTc ) , ( A.4 ) 2 e c c z at z 0 , ( A.5 ) is the thermal conductivity of the cathode material . -Arc column radiation component where T 4 4 qr T ( A Tc ) , ( A.6 ) is the Boltzmann's constant . -Evaporation energy flux component qev could be evaluated using simple de Boer equation [1,p.430] or the equation proposed by M.H.Kogan and H.K.Makashev [21] . We shall use the Frenkel kinetic theory of evaporation . Accordingly to this theory the rate of evaporation Vev in front of the cathode depends on the cathode temperature Tc as Vev Vo exp ( ) , ( A.7 ) where V0 0.82 c Tc 0 ( r / ma ) 1/ 3 , c c k , o is the corrected frequency of the normal lattice oscillations , ma is the atom mass of the cathode material , c is the atom bond energy . Thus the evaporation energy flux component is qev Lc cVem , ( A.8 ) where Lc is the latent heat of vaporization , c of the cathode material , is the cathode material density . 175 2. Sheath (0 z d1 ) Equation for the electrical potential U : e U 0 ( ni nem nie ) , (A.9 ) where U 2U 1 U 2U r 2 r r z 2 for a two-dimensional model , and d 2U U dz 2 for a one-dimensional model Boundary conditions : U 0 at z 0 , U uc at z d1 ( A.10 ) Equations for the number densities of emitted electrons nem , ions ni , and back-diffusion inverse electrons nie . At the point z inside the sheath with the corresponding potential U above mentioned equations are : ji 2 3 [ ( euc eU kTc )]1/ 2 ( A.11 ) 2 e mi j 2 3 ( A.12 ) nem em [ ( eU kTc )]1/ 2 2 e me j 2 3 nie ie [ ( kTie euc eU )]1/ 2 (A.13) e me 2 ni These expressions are obtained from the equations : nem jem j j , ni i , nie ie ( A.14 ) eVem eVi eVie where the velocities Vem , Vi , and Vie for emitted electrons, ions , and back-diffusion inverse electrons can be evaluated using energy balance equations at the position z , U : 1 3 2 meVem kTc eU ( A.15 ) 2 2 1 2 2 3 mi V i kTc e( uc U ) ( A.16 ) 2 2 1 3 2 meVie kTie e( uc U ) ( A.17 ) 2 2 Integration of the one-dimensional differential equation (A.9) gives the expression for the electric field in the sheath 176 dU E dz 2 z d1 2 z 0 ( u c u1 u1 ) 4e 0 [( j ie e ji e mi j em 2e e me ) 2e me ( u 2 u 2 u c )] 2e ( A.18 ) where 3 kTe 3 kTie me jie 2 , u2 ( ) , 2 e 2 e 2e enie 0 u1 (A.19 ) nie is the number density of the back-diffusion inverse electrons on the boundary z d1 given below . 0 3). Boundary between the sheath and the ionization zone ( z d1 ) . Equations for the components of the current density : j ji jem jie ( A.19 ) with components - Ion current density ji 1 0 1 0 8kTc eni vi eni , (A.20 ) 4 4 mi 0 where ni is the ion number density , mi is the ion mass - Back-diffusion electrons current density - jie 1 0 8 kTe enie , ( A.21 ) 4 me 0 where nie is the back-diffusion electron number density , me is the electron mass . - Electron emission current density is determined by the Richardson-Dushman equation jem Ac Tc exp ( 2 e eff kTc ) ( A.22 ) with the Schottky correction eff 0 and Ac Where 0 is the work function of the cathode Ec is the electric field strength at the eE c , ( A.23 ) 4 0 4 k 2 eme h3 cathode spot , 177 0 is the dielectric constant , me is the electron mass , h is Plank's constant ; It has to be noted that theoretical value of A is Ac 120 104 A m2 K 2 , however , the practical value of A is different for different materials . Particularly , for Cu Ac 60 A m2 K 2 , and for Mo Ac 55 A m2 K 2 . Equations for the particles density at z d1 . 0 - Electron emission number density nem at z 0 can be obtained from the equation ( A.11) by substitution U uc : nem 0 jem 2 3 [ ( euc kTc )]1/ 2 (A.24) 2 e me - Back-diffusion electron number density nie ( ni nem ) exp ( 0 0 euc ) (A.25) kTe 4).Ionization zone (d1 z d2 ) Motion equations for electrons ( index "e"),ions ( index "i" ) and neutral atoms ( index "g" ) me dVe 1 dpe eE dz ne dz M eg eg (Ve Vg ) M ei ei (Ve Vi ) ( A.26 ) mV i i dVi 1 dpi eE dz ni dz Mie ie (Vi Ve ) Mig ig (Vi Vg ) ( A.27 ) mgVg dVg dz 1 dpg ng dz M ge ge (Vg Ve ) M gi gi (Vg Vi ) ( A.28 ) Here and Meg M ge , Mie Mie , M ge Meg eg ge , ie ei ge eg are reduced masses and frequencies of collisions respectively ; 178 pe , pi , pg and Ve , Vi , Vg are partial pressure and velocities of electrons , ions , and neutral atoms respectively . Energy equations for particles - For electrons : d dT 3 d ( e e ) ( e i kTe ) ( neVe ) 2 dz dz dz 3 3 dV ne k es es ( Te Ts ) ne kTe e 2 2 dz s i , g ne es s i , g me ms ( e s ) 2 2 2 me ms - For ions ( s i ) and for atoms ( ( A.29 ) s g) : d dT 3 ( s s ) ei ns es k ( Te Ts ) dz dz 2 ( A.30 ) where es 2me ms 2 2 me ms is the portion of the s - particle's energy transferred during collision with electrons ; e , s are thermal conductivity . Ionization and recombination equations : d 3 ( eneVe ) ne ng ne ( A.31 ) dz d 3 ( eniVi ) ne ng ne dz where and ( A.32 ) are the ionization and recombination rates Condition of quasi-neutral plasma equilibrium ne ni ng (A.33 ) Dalton's Low for the pressure p: p ne kTe ni Ti ng Tg (A.34 ) Equation for the current density : j eneVe enV i i const ( A.35 ) Modified Saha equation ne ni 2 Zi ( 2 me k A ) 3/ 2 i exp ( i ) 3 ng Z0 h kTe 179 ( A.36 ) where the lowering of the ionization energy is given by i e2 4 0 [ e2 ne (1 Te 1/ 2 )] , ( A.37 ) Tc where Zi is the statistical ion state function , Z0 is the statistical atom function . 4. The boundary between the ionization zone and the arc column ( z d2 ) Energy flux W into plasma : 5 kTc 5 kTc ) ji ( i ) 2 e 2 e , 5 kTe jie ( uc ) 2 e Wa jem ( uc ( A.38 ) and if the plasma is almost fully ionized, then [14] Wa kTe m 2 [ 0 5 e ( i 5)] j K A 2e mi k A ( A.39 ) where 0 is the thermal diffusion factor , K is the thermal conductivity of the plasma gas 5. Cathode bulk ( z 0) . Equation for the temperature c ( r , z , t ) c 1 ( cr c ) t r r r , ( A.40 ) 2 c Jc (r , z , t ) ( ) z c z c (c ) c where c is the cathode thermal capacity, Jc (r , z, t ) is the current density in the cc c cathode bulk , . c (c ) is the cathode electrical conductivity depending on the Equation for the electrical potential c c 1 c c [ r c (c ) ] [ c (c ) ]0 r r r z z ( A.41 ) Relationship between current density and potential 180 Jc (r , z ) c (c ) ( c / r ) 2 ( c / z ) 2 ( A.42 ) Initial condition for the temperature c (r , z, tb ) gc (r , z) (A.43 ) is determined as the solution of the same equation ( A.40 ) at the time of the bridge rupture Boundary conditions for the temperature a) At z 0 c q , if r rc (t ) , ( A.44-1 ) z c 0 , if r rc (t ) ( A.44-2 ) z where rc is the cathode spot radius . c z b) At where c T0 , T0 is the ambient temperature . Boundary conditions for the electrical potential a). At z 0 c 0 , if a 0 z b). At , if r rc (t ) ( A.45-1 ) r rc (t ) ( A.45-2 ) z a U c , ( A.46 ) U where c is the voltage in the cathode bulk . B. Equations for the arc Heat equation cA A A A 1 ( Ar A ) t r r r 1 j [ q p ] l (t ) , 1 2 ( A.47 ) where c A is the arc plasma thermal capacity , A A A is the arc plasma density , is the arc plasma thermal conductivity , is the arc plasma electric conductivity , 181 tb . q is the heat flux into cathode , p is the heat flux into anode . The dependencies of cA , A , A , A on the temperature A are given usually by graphs . Heat fluxes q and p can be evaluated from the formulas ( A.1 ) and ( A.54 ) Boundary conditions A 0 r A T where at r0 at r rA ( A.49 ) ( A.48 ) T determined from the equation A (T ) 0 , ( A.50 ) 1 rA ( t ) [ rc ( t ) ra ( t )] , 2 I . rc ( t ) j ( t ) ( A.51 ) ( A.52 ) Ohm's low I where 2 l (t ) rA ( t ) ( A )rdr , ( A.53 ) 0 l ( t ) is the arc length . Equations of quasi-neutral plasma condition , Dalton's Low , current density , Saha equation are quite the same as for the ionization zone given by the expressions (A.33)- ( A.37 ) . C. Equation for the anode region . 1) . Anode surface ( z l (t )) . Energy balance equation pe pr pev p ( A.54 ) consisting of - Electron energy flux component pe je ( ua 0 where ua is the anode fall ; - Heat conduction energy flux component p a a z at 5 kTe ) , ( A.55 ) 2 e z l (t ) , 0 r ra (t ) ( A.56 ) a where is the thermal conductivity of the anode material ; - Arc column radiation energy flux component 182 4 4 pr T ( A Ta ) , ( A.57 ) where Ta a (r , z, t ) at z l (t ) , 0 r ra (t ) - Evaporation energy flux component pev La aVev , Vev V0 exp ( a Ta ) (A.58 ) 2) Anode bulk (l (t ) z ) Equation for the temperature a ( r , z , t ) a 1 ( ar a ) t r r r 2 J (r , z, t ) ( a a ) a z z a (a ) ca a ( A.59 ) where ca is the anode thermal capacity, Ja (r , z, t ) is the current density in the a (a ) anode bulk , is the anode electrical conductivity depending on a , Equation for the electrical potential a 1 a a [ r a (a ) ] [ a (a ) ]0 r r r z z ( A.60 ) Relationship between current density and potential Ja (r , z) a (a ) ( a / r ) 2 ( a / z) 2 ( A.61 ) Initial condition for the temperature a (r , z, tb ) ga (r , z) ( A.62 ) is determined similarly like for the cathode , Boundary conditions for the temperature z l (t ) a a p , if r ra ( A.63 - 1 ) z a 0 , if r ra , ( A.63 - 2 ) z where ra is the anode spot radius . a). At b). At z a T0 ( A.64 ) 183 Boundary conditions for the electrical potential z l (t ) a). At a 0 , if a 0 , if z r ra (t ) ( A 65 - 1 ) r ra (t ) ( A.65 - 2 ) z b). At a U a , ( A 66 ) where U a is the voltage in the anode bulk . 3). Simplified model Equations for the heat fluxes r2 ] , ( A.67 ) rs ( t ) 2 r2 0 pm ( r , t ) pm ( t ) exp[ ] , ( A.68 ) rm ( t ) 2 where index s means i , ie, r , em, ev , qs ( r , t ) qs ( t ) exp[ 0 and index m means e, r , ev . Equation for the current densities me 4 jie 8kTc jem [ ( 4 ji mi 8kTc 2 3 eu ( euc kTc )]1/ 2 ) exp ( c ) me 2 kTc ( A.69 ) Equation for the total current rc ( t ) 2p т [ ji (r , t ) + jem (r , t ) - jie (r , t )]rdr = I 0 Simplified equation for the temperature 2 a 1 a 2 2 aa ( 2a ) t r r r z 2 where aa 2 ( A. 71 ) a ca a Equation for the electric potential 184 ( A.70 ) 2 a 1 a 2 a 0 r 2 r r z 2 ( A.72 ) Analytical solutions a (r , z, t ) arctan[ I 1 2 ra ( t ) ca a a 1 ra ( t ) 2 s s2 4ra ( t )( z l 2 ( t )) ] 2 ( A.73 ) where 2 s r 2 ra (t ) [ z l (t )]2 ( A.74 ) a ( r , z , t ) a1 ( r , z , t ) a 2 ( r , z , t ) a 3 ( r , z , t ) ( A.75 ) where t ra ( t ) 0 0 a1 (r , z, t ) dt1 [G( z l (t ), r, r , t t ) 1 1 1 ( A.76 ) G ( z l (t1 ), r , r1 , t t1 )] p (r1 , t1 )r1dr1 is the component of the anode temperature due to arc heat flux , t a 2 (r , z, t ) dt1 0 dz1 a J a (r , z ) l ( t1 ) 1 2 0 [G ( z z1 , r , r1 , t t1 ) G ( z z1 2l (t1 ), r , r1 , t t1 )]r1dr1 ( A.77 ) is the Joule sources component of the anode temperature , a 3 (r , z, t ) l (t ) dz1 [G ( z z1 , r , r1 , t ) 0 G ( z z1 2l (t ), r , r1 , t )]g a (r1 , t1 )r1dr1 ( A.78 ) is the component of the anode temperature caused by the initial condition ( A.62 ) G ( z , r , r1 , t ) 1 4aa a t 3 ( A.79 ) z 2 r 2 r1 rr exp ( ) I0 ( 1 ) 2 2 aa t 4 aa t and I 0 is the modified Bessel function . 2 The expression for the anode current density Ja (r , z ) is given by the equation ( A.61 ) and the contact gap ( arc length ) l ( t ) is l (t ) lb V0t , ( A80 ) where V0 is the contact opening velocity . 185 The temperature at the centre of the anode and the cathode spots ( r 0 ) : t Ta (0, t ) = pm (t )rm (t ) dt 2 rm (t ) + 4a (t - t ) t - t те m 0 ( A81 ) 2 q ( )rs ( ) d Tc (0, t ) 2 s 2 t 0 s rs ( ) 4a (t ) 2 t where and m s electrode , and sign ( A.82 ) are considered as algebraic sums with sign " " for the flux components entering the " " for the flux components leaving the electrode . Equation for the average arc temperature c A Al (t ) rA 2 (t )[ Iu A 2 rA ( t ) dTA 2 (TA T )] dt rA (t ) ( A.83 ) [ q ( r , t ) p ( r , t )]rdr 0 where TA (t ) 2 2 rA (t ) rA ( t ) A (r , t )rdr ( A.84 ) 0 rA is the effective arc radius , u A is the arc voltage , T is given by the formulas ( A.49 ) -( A.50 ). Equations for approximate effective radii of heat flux components rs (t ) rc (t ) , s ev ( A.85 ) rm (t ) ra (t ) , m ev ( A.86 ) Tc (rev , t ) Tb for cathode rev ( t ) ( A.87 ) Ta (rev , t ) Tb for anode rev ( t ) , ( A.88 ) where Tb is the boiling temperature of the contact material Equation for the critical time t cr Ta (0, tcr ) Tc (0, tcr ) Appendix 5 . ( A.89 The temperature field in electrical contacts Each member of a contact pair can be considered as a body occupying the cylindrical halfspace region D(0 r , 0 z ) . 186 The region occupied by arc at blow-off repulsion can be considered as a cylindrical disk DA : 0 r rb (t ), 0 z x(t ) interacting with contact surface, which radius rb (t ) is much greater than contact gap x(t ) (Fig. A.1). This interaction results into phase transformations of contact material and formation of three zones: 4) The zone of evaporated material Db : 0 r rb (t ), 0 z b (r , t ) , 5) The zone of melted material Dm : b (r, t ) z m (r, t ), if 0 r rb (t ), and 0 z m (r, t ) if rb (t ) r rm (t ) 6) The solid zone Ds : m (r, t ) z , if 0 r rm (t ), and 0 z if rm (t ) r r x(t) Ds rm (t ) z m (r , t ) Dm Db rb (t ) 2rA0 0 zb (t ) zm (t ) z b (r , t ) z DA Movable contact Fixed contact r Fig. A.1 The arc and contacts geometry: arc region DA , evaporated zone Db , melted zone Dm and solid zone Ds The dynamics of contact heating can be described by the heat equations for temperature Ti (indexes i 0,1 and 2 correspond to the zones Db , Dm and Ds relatively): Ci Ti 1 div(i gradTi ) grad 2i t i (A.1) where Ci , i , and i are thermal capacitance, thermal conductivity and electrical resistivity respectively. The electrical potential i (r , z, t , rC ) can be expressed as [5], [12] 187 i (r , z, t , rC ) I (t ) i (r , z, rC ) arctan 2 rC rC (A.2) where I (t ) is the electrical current, and rC is the current conducting contact radius. The function (r , z, rC ) can be found from the equation r2 z2 1 2 2 rC 2 i.e. (r , z, rC ) 1 2 s s 2 4rC z 2 , 2 s r 2 z 2 rC 2 (A.3) The boundary conditions on the interface z b (r , t ) can be expressed as T0 z b ( r ,t ) T1 z b ( r ,t ) Tb 1 (A.4) T1 z PC (r , t ) Pb (r , t ) (A.5) z b ( r ,t ) Here Tb is the boiling temperature of contact material, PC (r , t ) is the total heat flux (arc power per unit contact area) entering the surface z b (r , t ) from heated vapour. It consists of many components, such as heating due to ion bombardment, electron emission cooling, radiation etc. which are different for anode and for cathode. They are described and defined in [12]. To a first approximation one can suggest that total arc power should be divided into I (t )U (t ) equal parts for anode and for cathode [1]. In this case PC (r , t ) 2 . The flux component 2 rC Pb (r , t ) is the portion of total flux consumed for evaporation of contact material, which can be found by the expression Pb (r , t ) Lb b (r , t ) t (A.6) where Lb and is specific heat for evaporation, is density of contact material. The boundary conditions on the interface z m (r , t ) are similar if we replace boiling index b by melting index m : T1 z m ( r ,t ) T2 z m ( r ,t ) Tm (A.7) 1 T1 z 2 z m ( r ,t ) T2 z Pm (r , t ) (A.8) z m ( r ,t ) with heat flux consumed for melting Pm (r , t ) Lm m (r , t ) t Each stage of contact separation should be considered separately. 188 (A.9) 1. Initial stage of contact separation. At this stage zones Db and Dm vanish, no arcing occurs, thus the right side of (A.5) is equal to zero. The solution of the equation (A.1) for averaged values of C and can be presented for this stage in the form t TJ (r , z, t , rC ) dt1 dz1 G(r , r1 , z z1 , t t1 ) 0 0 1 [( / r1 ) 2 ( / z1 ) 2 ]r1dr1 (A.10) where G(r , r1 , z, t ) is the Green’s function defining by the formula G(r , r1 , z, t ) r 2 r1 z 2 rr ) I 0 ( 12 ) 2 4a t 2a t 2 1 2a t 3 exp( (A.11) and a 2 / C is the thermal diffusivity of contact material. In a time range, for which dimensionless time (Fourier criterion) Fo a 2t 2 is sufficiently rC large, Fo 1 , the quasi-stationary model for Joule heating can be applied, and then the formula (A.6) transforms into more simple expression [5], [10] TJ (r , z, t , rC ) T2 ( , t , rC ) cos[2 (t , rC ) ] 1} cos[2 (t , rC ) / 2] 1 { (A.13) where arctan (r , z, rC ) rC (A.14) 2 (t , rC ) I (t ) 2 rC 20 2 2 (A.15) 20 is electrical resistivity before heating and 2 is its temperature coefficient for the solid material. It should be noted that the introduction of the new variable (A.14) transforms the half-infinite region 0 into segment 0 / 2 . The initial stage comes to the end at the time t tm when the temperature at the contact surface reaches the melting value Tm . This time can be found from the equation T2 (0, tm , rC (tm )) Tm or from oscillograms as the time corresponding to the melting voltage. The estimation of Fourier criterion shows that quasi-stationary model can be applied as well for following contact separation up to arc ignition. 2. Melting. The melting stage of contact separation is characterised by appearance of two zones, melted Dm and solid Ds . The quasi-stationary solution of the equation (1) gives the temperature distribution in the melted zone Dm in the form T1 ( , t , rC ) cos[1 (t , rC ) ] 1Tm 1} 1 cos[1 (t , rC ) m ] 1 { (A.16) 189 This function satisfies the stationary heat equation and the conditions T1 0 (A.17) T1 Tm m z 0 (A.18) Similarly the expression T2 ( , t , rC ) (1 2Tm )sin[2 ( / 2 )] sin[2 ( m )] sin[ 2 ( / 2 m )] 2 sin[2 ( / 2 m )] (A.19) satisfies the stationary heat equation for the solid zone Ds and the conditions T2 Tm (A.20) m T1 / 2 0 (A.21) The melting isotherm m (t ) can be found from the equation (A.8) by substitution of the expressions (A.16) and (A.19) and replacing of m (r , t ) by m (t ) . 3. Evaporation. The temperature distribution in melted zone after the time t tb , when the temperature at the contact spot 0 becomes equal to boiling value Tb , becomes the form T1 ( , t , rC ) [1 1 (Tb Tm )] sin[1 ( m )] sin[1 ( b )] 1 Tm 1 sin[1 ( m b )] 1 (A.22) while the temperature in the solid zone describes by the previous expression (A.19). The boiling isotherm should be calculated using the equation (A.5). 4. Arcing. The Fourier criterion at arcing is not sufficiently large due to sharp increase of current conducting radius, therefore the quasi- stationary model ceases to remain correct and non-stationary solution of the equation (A.1) should be found. It can be represented in the form TC (r, z, t , rC ) TJ (r, z, t , rC ) TS (r, z, t , rC ) (A.23) where TJ (r , z, t , rC ) and TS (r , z, t , rC ) are the temperature components due to volumetric Joule heating and due to heating of contact surface by arc heat flux respectively. The first term in the right side is defined by the expressions (A.19) and (A.22) for solid and liquid zones, while the second term can be written as t 0 0 TS (r , z, t , rC ) dt1 G(r , r1 , z, t1 ) PC (r1 , t1 )r1dr1 190 (A.24) with the Green’s function G(r , r1 , z, t ) defined by the expression (A.11). Calculation shows that the role of contact heating by flux is more significant. If the heat fluxes Pc (r , t ) Pb (r , t ) , Pm (r , t ) obeys the normal Gauss’s radial distribution r2 r2 r2 Pc (r , t ) Pc (t ) exp( ), Pb (r , t ) Pb (t ) exp( ) , Pm (r , t ) Pm (t ) exp( ), rA (t )2 rA (t )2 rA (t )2 (A.25) then the integral with respect to r in the formula (A.24) can be calculated and the expression for the contact temperature becomes more simple form t [ Pc (t1 ) Pb (t1 ) Pm (t1 )]rA (t1 )2 a z2 r2 TS (r , z, t ) exp[ ]d 4a 2 (t t1 ) rA (t1 )2 4a 2 (t t1 ) 0 [rA (t1 )2 4a 2 (t t1 )] t t1 (A.26) 191 Scientific publication Kharin Stanislav Nikolaevich MATHEMATICAL MODELS OF PHENOMENA IN ELECTRICAL CONTACTS Monograph Editor Z.V. Skok ________________________________________________________________ Signed in print 07. 11.2017 Scope 11,9 pub. sheet-copies. Paper Size 60 × 84 1/16 Circulation of 300 Printed in CJSC RIC «PRice-courier» 630090, Novosibirsk, Akad. Lavrentiev str., 6, Ph. 330-72-02 Order No. 61 192 Научное издание Харин Станислав Николаевич МАТЕМАТИЧЕСКИЕ МОДЕЛИ ПРОЦЕССОВ В ЭЛЕКТРИЧЕСКИХ КОНТАКТАХ Монография Редактор З.В. Скок ________________________________________________________________ Подписано в печать 07. 11.2017 Объем 11,9 уч.-изд.л. Формат бумаги 60 × 84 1/16 Тираж 300 экз. Отпечатано в ЗАО РИЦ «Прайс-курьер» 630090, г. Новосибирск, пр. ак. Лаврентьева, 6, тел. 330-72-02 Заказ № 61