# Plasma Energy Loss by Cathode Heat Conduction in a Vacuum Arc: Cathode Effective Voltage

## Abstract

**:**

_{ef}, which is weakly dependent on the current. In this paper, a physical model and a mathematical approach were developed to describe the energy dissipation due to heat conduction in the cathode body, which is heated by energy outflowed from the adjacent plasma. The arc plasma generation was considered by taking into account the kinetics of the heavy particle fluxes in the non-equilibrium layer near the vaporizing surface. The phenomena of electric sheath, heat and mass transfer at the cathode were taken into account. The self-consistent numerical analysis was performed with a system of equations for a copper cathode spot. The transient analysis starts from the spot initiation, modeled by the plasma arising at the initial time determined by the kind of arc triggering, up to spot development. The results of the calculations show that the cathode effective voltage u

_{ef}is determined by the cathode temperature as a function of spot time. The calculated evolution of the voltage u

_{ef}shows that the steady state of u

_{ef}is approximately 7 V, and it is reached when the cathode temperature reaches a steady state at approximately one microsecond. This essential result provides an explanation for the good agreement with the experimental cathode effective voltage (6–8 V) measured for the arc duration from one millisecond up to a few seconds.

## 1. Introduction

^{9}A/cm

^{2}before its initiation and a rate of current rise of 10

^{9}A/s during the EEE lifetime of approximately 1 ns. These parameters can be obtained under the relatively high electrical field of ~10

^{8}V/cm, which might be difficult to realize in developed arcs. This is due to the small Debye radius (~0.01 µm) relative to the size of irregularities (~1 µm) at the cathode surface and relatively low ion current density that determines the volume charge in the space charge layer. At the required ion current of 10

^{7}A/cm

^{2}and higher, the cathode (or the liquid-metal jet occurred by the plasma pressure) is intensely heated and evaporated by ion bombardment (i.e. not by Joule heating), and cathode plasma is generated to support the spot operation without any explosions, i.e., EEE phenomenon.

_{ef}. The significance of this parameter is due to the fact that it characterizes the heat condition of different spot types that affect the rate of cathode erosion and spot appearance depending on the time of arc development and cathode geometry. In particular, the u

_{ef}for the film cathode is significantly lower than for the bulk cathode [10].

_{arc}, which varied from a few milliseconds to a maximum of half a second. Assuming that the heat conduction and radiation losses were small, the measured maximum temperature T

_{m}was used to determine the accumulated heat as W

_{con}= m

_{d}c

_{p}T

_{m}, where m

_{d}is the disc mass, and c

_{p}is the specific heat of the disc materials. The cathode effective voltage u

_{ef}was obtained with the following equation:

_{ef}only weakly increases with the arc current. This increase is due to the increase in arc voltage with current, which can be up to a volt in a current range from approximately I = 10 A up to several hundreds of amperes. This increase is particularly noticeable in the transition region of one to a few cathode spots. For Cu, it was found that the cathode effective voltage was 5.4 V at I = 40 A and 6.2 V at I = 100 A (Table 1 of Ref. [13]).

_{ef}= 8 V [17].

_{w}is the heat capacity of water, F is the mass flow rate of the cooling water, and ΔT is the increase in the cooling water temperature at the exit port from the cathode. In the case of a 175 A arc with a mass flow rate of 36.5 g/s, the temperature near the beginning of the arc rapidly increased to approximately 8 °C above the ambient water temperature, reaching a value of 8.4 °C near the end of the arc. Using expression (2), the effective cathode voltage was calculated as u

_{ef}= W

_{w}/I, which was determined to be 6.6 V near the beginning of the arc and 7.2 V near the end of the arc. Similar results were obtained for I = 340 A. At a steady state HRAVA, the anode effective voltage was approximately 6 V.

_{ef}using a W shower-head anode with I = 200 A and a Ta one-hole anode with I = 175 A shows that, when the arc was ignited, the u

_{ef}increased to 6–7 V, where it remained for ~40 s (both anodes), while the anodes were relatively cold. For the hot anodes, the u

_{ef}increased up to ~11–12 V and reached a steady state. The experiments showed a similar dependence of u

_{ef}on time for different arc currents, while the average steady state u

_{ef}weakly increased by a few tenths of a volt, depending on F and the anode material, when I increased from 175 to 250 A. According to the experiment in [8], the calorimetric time dependence indicates that the measurements at the initial stage (up to 40 s) were related to the effective cathode voltage of u

_{ef}= 6–7 V, which agrees with such measurements in the conventional cathodic arc [13]. However, the further increasing of the heat losses in the cathode up to the steady state ~11–12 V is caused by the part of the cathode plasma jet energy dissipated in the relatively dense plasma at the anode hot stage and returned back to the cathode surface because the cathode–anode assembly is closed in the VABBA.

_{ef}= 6–8 V) are related to the measurements mainly for the long-arc existing time, namely from one millisecond up to a few seconds and more. The following question can be asked: how does the value of u

_{ef}vary during the transition period of the spot development? There is no experimental data that characterizes the energy loss due to cathode heat conduction at the spot initiation and its development. In addition, any theoretical description of the losses due to the heat conduction as a dependence on time is absent. Therefore, the main goal of the present paper is to fill this gap using the previously published kinetic model for unsteady spots [20] and develop a calculation approach allowing for the analysis of the numerical behavior of the spot transition from the arc initiation up to the steady state in case when this state can reached in unstable vacuum arc.

## 2. Physical Model and Calculation Approach

#### 2.1. Kinetic Model

_{c}, energy fluxes and the cathode erosion rate G. Below, the model is described briefly. The details of the kinetic theory for cathode vaporization and plasma generation in the spot were summarized recently in [8].

_{0}and electron n

_{e}

_{0}densities was determined by the cathode spot temperature T

_{s}. Two heavy particle fluxes (evaporated and returned) are formed in the Knudsen region. The difference between these fluxes determines the plasma velocity v

_{3}at boundary 3 and the net rate of the mass evaporation. In the electron beam relaxation region, the atoms are ionized by the electrons emitted from the cathode as well as by the plasma electrons, which were heated during the electron beam relaxation. The presence of multiple ionizations of the evaporated atoms is determined by the level of the electron temperature.

_{3}at boundary 3 of the Knudsen layer by using the quasineutrality condition. This feature of the model distinguishes it from the model of the evaporation of neutral atoms during laser irradiation of metals with moderate power, in which it is required to set the speed of sound at the Knudsen boundary [22]. At the sheath boundary 2, the plasma electrons are returned to the cathode, while the ions and emitted electrons are accelerated with energy determined by the cathode potential drop eu

_{c}. The charge particle motion and their generation due to high power dissipation are coupled self-consistently with the potential barrier. Therefore, the corresponding height of the barrier, i.e., u

_{c}, as well as the energy fluxes from the plasma to the cathode surface can be obtained by studying the equations expressing the energy and momentum conservation laws of charge particle fluxes at the above-mentioned boundaries, including the Knudsen layer and the space charge layer.

#### 2.2. Calculation Approach and Procedure

_{T}(T,t′) is the time-dependent heat flux dissipated in the cathode (in Watt), T

_{00}is the surface temperature before spot initiation, f(t′,t

_{0}) is a time-dependent function resulting from integrating the heat conduction equation in differential form, and t

_{0}(a,r

_{0}) is a parameter that indicates the heat flux concentration. This parameter is determined by the thermal diffusivity coefficient a and by the spot radius r

_{0}= I/πj, where j is the current density [8]. Equation (4) is the relationship between the transient temperature and heat conduction energy losses that, at a certain time, is determined by the cathode energy balance [23]. Using energy balance with Equation (4) in the system of equations that described the cathode plasma phenomena, the dynamics of the plasma parameters were calculated as a dependence on time. These parameters were used to determine the following time-dependent characteristics: the energy flux from the plasma (dependent on u

_{c}) to the cathode surface Q(T,t), the heat conduction loss in the cathode Q

_{T}(T,t) and spot temperature T(t), paying attention that Q

_{T}(t) was mutually determined by the character of transient T(t).

_{T}(T,t) appeared as implicit parameters in the system of equations due to the non-linear effects of the cathode and plasma phenomena in the spot. Therefore, the cathode plasma system of equations was solved numerically using an iteration method at each time step with a duration of ∆t including integration of Equation (4). In this case, the solution of the cathode system of equations will determine the dependences Q

_{T}(T

_{n},t) and f(t′,t

_{0})

_{n}for moment t(n) time and, consequently, the temperature T

_{n}(t). The temperature growth is due to its gain by ∆T at each next step ∆t, i.e., T

_{n}(t) = T

_{n}

_{−1}(t) + ∆T. These time-dependent values were also used for calculation of the heat conduction flux Q

_{T}(T

_{n}(t),∆t), which is necessary to increase the temperature from value T

_{n}

_{−1}(t) to T

_{n}(t). The initial triggered time τ was served as the first time step to determine the initial plasma parameters, which was used to reproduce a secondary plasma at the next step time ∆t by calculation of the self-consistent system of equations. This procedure was continued for each n-time step. A relatively small ∆t was chosen for the mentioned calculation so that the calculated heat conduction loss could be considered as a constant for simplicity of the integration of Equation (4) during each step ∆t. The solution continued up to time t = τ + n∆t for which the values Q

_{T}(T

_{n},t), Q

_{T}(T

_{n}(t),∆t) and T(t) as well as the cathode plasma parameters, as one of possible, can achieved a steady state.

## 3. Calculated Results

_{s}and plasma electron temperature T

_{e}, heavy-particle density n

_{0}, degree of ionization α, current density j, fraction of electron current s, electrode erosion rate and potential drop at the cathode u

_{c}. The system of equations used was recently described in detail in Chapter 17 of book [8]. The spot current was chosen as I = 10 A, the current for the fragments when the group spot was observed as I > 100 A [6], and τ = 7.5, 50 and 100 ns—modeling the arc triggering time. The cathode effective voltage was obtained using the above-mentioned parameters and according to the definition by calculation u

_{ef}(t) = Q

_{T}(T

_{n},t)/I at each n-time step and a step effective voltage u

_{st}(t) = Q

_{T}(T

_{n}(t),∆t)/I, which was determined by the difference T

_{n}(t) − T

_{n−1}(t) during the time step ∆t.

_{s}is an important parameter to describe the time behavior of the effective voltages. The calculated T

_{s}as a function of time with τ as the parameter is illustrated in Figure 2. It can be observed that the cathode temperature increased from some initial value depending on τ to a steady state temperature that did not depend on τ, i.e., on the initial conditions. The initial cathode temperature increased with τ.

_{st}(t) for different τ are presented in Figure 3. This result shows that the larger u

_{st}~ 70 V was calculated at the initial step with minimal τ = 7.5 ns, and the value of u

_{st}significantly decreased at the initial steps for lower trigger times to 18 V and 16 V at τ = 50 ns and 100 ns, respectively. It is noteworthy that the effective voltage at each step decreased with time and tended to zero at the steady state, which occurred from 0.1 to 0.5 µs depending on τ.

_{ef}are approximately 65, 20 and 16 at τ = 7.5, 50 and 100 ns, respectively. While at τ = 7.5 ns, the u

_{ef}decreased monotonically, for τ = 50 and 100 ns, a small maximum arose at the beginning of the spot development. It can be observed that u

_{ef}decreased with time asymptotically to the steady state value of approximately 7 V, which did not depend on the initial step τ. The steady state u

_{ef}was reached at approximately 2–3 µs.

_{30}=n

_{3}/n

_{0}and dimensionless velocity b

_{3}= v

_{3}/v

_{T}(v

_{T}is the thermal velocity) at the Knudsen boundary 3. The data suggest that u

_{c}strongly changes, while the current density remains mostly constant. This result follows from the kinetic model, which examines the role of the arc voltage at the moment of arc initiation and spot development. It can be observed that, when the cathode potential drop is self-consistently calculated, the spot temperature and the current density do not unlimitedly increase with time, as in the case of calculation models in which a constant u

_{c}or arc voltage was assumed [1].

_{30}= 0.67, whereas this ratio is 0.31 when the metal target evaporated in the vacuum with laser-moderated power (with free flow, i.e., with sound speed at the Knudsen layer) [22,24]. This result as well as the calculated velocity b

_{3}= 0.17 indicate that the plasma flow in the cathode spot is not free.

## 4. Discussions

_{st}) caused by temperature growth arising during a calculated time step after reaching a cathode surface temperature T

_{n}(t) at time t. Second is the total cathode energy loss at time t that is calculated using the time-dependent cathode surface temperature, which is characterized by the cathode effective voltage u

_{ef}.

_{st}and u

_{ef}is different for each τ, and these values were initially significantly larger for τ = 7.5 ns than for τ = 50 and 100 ns. In order to understand the time behavior of u

_{st}and u

_{ef}, we refer to the larger calculated cathode potential drop u

_{c}~ 100 V obtained at the initial step with minimal τ = 7.5 ns. The drop u

_{c}significantly decreased with τ as well as with time to a steady state value that agreed with the experimental data. Therefore, the time behavior of u

_{st}and u

_{ef}can be explained by taking into account their dependence on u

_{c}influenced in time through the cathode energy balance. In addition, for lower τ, the time step is very small in comparison with the time step for larger τ. As the time step is significantly small, a relatively large energy flux to the cathode is needed, according to the heat conduction equation, to reach the required cathode temperature. The required certain temperature is necessary to obtain a required cathode plasma density, which at self-consistent calculations can be supported by a relatively high u

_{c}and, as result, high u

_{st}and u

_{ef}.

_{0}) in the heat conduction Equation (4) and by the decrease of u

_{c}. At some relatively large spot time t, the calculated cathode temperature T

_{s}and the stepwise heat power reach values at which the u

_{st}and u

_{ef}weakly change with spot time by reaching a steady state.

_{n}(t) − T

_{n−1}(t) → 0. This trend of temperature change explains the calculated step effective voltage approaching u

_{st}→ 0 when u

_{st}reaches a steady state. The u

_{ef}evolution is time-dependent, and it is coupled to the cathode temperature evolution. As such, it is obvious that u

_{ef}reaches a constant value because the temperature also reaches a steady state. According to the calculations, the u

_{ef}reaches a steady state at a spot time of approximately one microsecond. This result can explain the good agreement with the experiment for which the cathode effective voltage (6–8 V) was measured for different times but always for a relatively large arc duration (see data in Introduction). The presence of a small peak of u

_{ef}near the starting time can be caused (considering the calculated plasma parameters during the step time) by different time dependence of the cathode temperature and of an integral of the function f(t′,t

_{0}) in Equation (4).

_{ef}= 2–3 V), and for refractory materials like W and Mo (high u

_{ef}= 9 V), the measured data can be explained by respectively relatively low and high cathode temperatures estimated in the developed cathode spot appearing on these materials. However, a detailed study of the transient phenomena from spot initiation to its development requires numerous and complicated calculations that are the subject for a separate study. Note that the relatively large measured energy loss by heat conduction in a cathode bulk cannot be explained with the EEE model due to the significantly lower energy loss needed for the explosion part of a spike or protrusion on the surface, which was discussed by study phenomena in film cathodes [10,30].

## 5. Conclusions

- The step energy loss by heat conduction of the cathode body is determined by the cathode energy needed to increase the cathode temperature from value T
_{n}_{−1}(t) to T_{n}(t) during step time ∆t and characterized by the effective voltage u_{st}. The time-dependent functions of u_{st}(t) decrease from large values (70, 18 and 16 V) depending on τ to zero due to T_{n}(t) − T_{n}_{−1}(t) → 0. - The transient cathode energy loss is determined by the cathode energy needed to increase the cathode spot temperature at each n-step and characterized by the time-dependent effective voltage u
_{ef}. - The time-dependent functions of u
_{ef}(t) decrease from the largest values (65, 20 and 16) depending on τ to a steady state value due to the steady state of the cathode temperature. - The predicted cathode effective voltage agrees well with the existing experimental data obtained in a range of arcing from one millisecond to a few seconds and more. This agreement is explained by the calculated cathode surface temperature that reaches a steady state value at approximately 0.1 to 0.5 µs, depending on the time of the arc or spot triggering τ.

## Funding

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## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic presentation of the physical regions according to the kinetic model of the cathode spot. The numbers indicate the boundaries of the regions.

**Figure 3.**Step effective voltage characterizing the energy needed to increase the cathode temperature from value T

_{n}

_{−1}(t) to T

_{n}(t) during step time ∆t with τ as the parameter, I = 10 A.

**Figure 4.**Cathode effective voltage u

_{ef}characterized the heat conduction loss in the cathode body at the steady state temperature with τ as the parameter, I = 10 A.

**Table 1.**Plasma parameters at spot initiation times determined with different mechanisms and at spot development time of 1 µs.

Time | u_{c}V | n_{0} × 10^{20}cm ^{−3} | j × 10^{6} cA/cm ^{2} | T_{e}eV | E × 10^{7}V | s | n_{30} | b_{3} |
---|---|---|---|---|---|---|---|---|

τ = 7.5 ns | 100 | 0.38 | 2 | 9 | 3 | 0.75 | 0.48 | 0.46 |

τ = 50 ns | 47 | 0.87 | 2.2 | 5.4 | 2.8 | 0.75 | 0.54 | 0.36 |

τ = 100 ns | 40 | 1 | 2.3 | 4.6 | 2.7 | 0.75 | 0.55 | 0.34 |

t = 1 µs | 13 | 3 | 1.2 | 1.5 | 1 | 0.73 | 0.67 | 0.17 |

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**MDPI and ACS Style**

Beilis, I.I.
Plasma Energy Loss by Cathode Heat Conduction in a Vacuum Arc: Cathode Effective Voltage. *Plasma* **2023**, *6*, 492-502.
https://doi.org/10.3390/plasma6030034

**AMA Style**

Beilis II.
Plasma Energy Loss by Cathode Heat Conduction in a Vacuum Arc: Cathode Effective Voltage. *Plasma*. 2023; 6(3):492-502.
https://doi.org/10.3390/plasma6030034

**Chicago/Turabian Style**

Beilis, Isak I.
2023. "Plasma Energy Loss by Cathode Heat Conduction in a Vacuum Arc: Cathode Effective Voltage" *Plasma* 6, no. 3: 492-502.
https://doi.org/10.3390/plasma6030034