# Numerical modeling of electric arc in a low voltage breaking chamber

Arsalen Gatri, Jean-Jacques Gonzalez, Pierre Freton, Patrice Joyeux, Francis

Sambou

► To cite this version:

Arsalen Gatri, Jean-Jacques Gonzalez, Pierre Freton, Patrice Joyeux, Francis Sambou. Numerical modeling of electric arc in a low voltage breaking chamber. Plasma Physics and Technology, 2025, 12 (3), pp.190-196. 10.14311/ppt.2025.3.190. hal-05442100

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
NUMERICAL MODELING OF ELECTRIC ARC IN A LOW VOLTAGE
BREAKING CHAMBER

A. GATRI$^{a, b, *}$, J-J. GONZALEZ$^a$, P. FRETON$^a$, P. JOYEUX$^b$, F. SAMBOU$^a$

**Abstract.** This work presents the development of a MagnetoHydroDynamic (MHD) model to simulate electric arc behavior in a low-voltage circuit breaker chamber. The modeling process began with a simplified geometry to validate key phenomena such as arc displacement, segmentation, and voltage rise, showing good agreement with literature. To approach realistic conditions, contact rotation was implemented using a layering technique, improving numerical accuracy and avoiding mesh deformation. A current-limiting mechanism and electrical network coupling were also introduced, enabling dynamic current input. These modules were integrated into a 3D geometry representing a real chamber, successfully reproducing arc evolution under realistic conditions. The model captures complex arc physics and helps in future design optimization of low-voltage devices.

**Keywords:** Low-voltage circuit breaker, arc modeling, air plasma, simulation.

<footnote>a LAPLACE, UMR 5213 CNRS-UPS-INP, Université Paul Sabatier, 118 rte de Narbonne, bat3R3, 31062
Toulouse Cedex, France</footnote>

<footnote>b Hager Group, Obernai 132 Bd de l'Europe, France</footnote>

<footnote>* arsalen.gatri@laplace.univ-tlse.fr</footnote>
# 1. Introduction

Low-voltage circuit breakers (LVCBs) are essential components in both industrial and residential electrical systems, ensuring the protection of users and installations. Their critical function has motivated many research studies dedicated to improving their performance and reliability. In these devices, current interruption is achieved primarily through current-limiting mechanisms. When a current fault occurs, the contacts open and an electric arc is initiated. The design of the LVCB facilitates the movement of the arc into the breaking chamber, driven by pressure forces and the Lorentz force resulting from the current flow in the arc runners.

Inside the breaking chamber, the arc is segmented into multiple sub-arcs by metallic splitters. This segmentation leads to an increased arc voltage due to additional cathode and anode voltage drops, which helps to force the current toward zero and achieve successful interruption. To optimize the breaking capacity and improve the operational reliability of the breaker, it is crucial to understand the physical behavior of the arc throughout its evolution, from initiation and movement to quenching and extinction.

Experimental studies have traditionally investigated physical quantities such as arc velocity, arc root positioning, and contact separation speed, using high-speed imaging techniques [1–3]. In parallel, numerical simulations have enabled detailed analysis of arc dynamics in increasingly complex geometrical configurations [4–6]. However, the physical processes occurring at the plasma-electrode interface are particularly intricate and remain difficult to resolve in a global plasma flow description. Recent studies [4, 7, 8] have introduced macroscopic modeling approaches to represent these interactions, incorporating additional resistivity at the electrode-plasma interface. This allows for a more accurate estimation of the arc voltage, particularly in the presence of arc splitters.

Building on this foundation, the present work introduces a magnetohydrodynamic (MHD) model to simulate the behavior of an electric arc in a low-voltage circuit breaker. The model begins with a simplified reference geometry that validates key arc phenomena such as displacement, segmentation, and voltage evolution, with results consistent with the existing literature. To approach realistic operating conditions, the model incorporates several advanced physical mechanisms such as voltage drop due to segmentation of the arc between splitter plates, contact rotation, and a current limiting mechanism by coupling the model with the external network equations. In contrast to previous studies, which generally address these physical mechanisms separately, the present work proposes a model that integrates all of these modules into a unified environment. A new technique for contact opening is also proposed that helps reduce computational time and improve numerical stability.

In this paper, geometries, equations, hypotheses, and developed modules are presented in Section 2. The results obtained are discussed in Section 3. Lastly, a conclusion is given in Section 4.

# 2. Numerical model

This section presents the main characteristics of the model. Section 2.1 describes the governing fluid dynamics equations, while Section 2.2 introduces the electromagnetic equations. The assumptions of the model are outlined in Section 2.3. Section 2.4 discusses the implementation of the additional resistivity approach. The geometric configuration and mesh are detailed in Section 2.5. Section 2.6 highlights the model upgrades and newly developed modules. Finally, the results obtained in a real breaking chamber are presented in Section 3.

## 2.1. Fluid equations

A three-dimensional hydrodynamic model is employed using Ansys Fluent [9] to simulate the behavior of the air plasma, solving both the dynamics of the gas flow and the transport of energy. To take into account the electromagnetic phenomena associated with the arc, the model is coupled with Maxwell's equations. The coupling is achieved by introducing custom functions and additional scalar fields, which enable the simulation of current distribution and the self-induced magnetic field generated by the arc [7].

The electric arc in a LVCB is modeled as a conductive fluid using the Navier–Stokes equations. To account for the arc’s conductive and thermal effects, additional source terms are added to the governing equations. Specifically, the momentum equations include Lorentz force contributions, while the energy equation is extended to account for Joule heating, radiative heat losses, and electron enthalpy transport.

□ Mass conservation equation

$$ \frac{\partial \rho}{\partial t} + \vec{\nabla} \cdot (\rho \vec{v}) = 0 \qquad (1) $$

□ Momentum conservation equation

$$ \frac{\partial(\rho\vec{v})}{\partial t} + \vec{\nabla} \cdot (\rho\vec{v} \cdot \vec{v}) = \vec{\nabla} \cdot (\eta\vec{\nabla}\vec{v}) - \vec{\nabla}p + (\vec{j} \times \vec{B}) + s_{\nu} \qquad (2) $$
□ Energy conservation equation

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$$ \frac{\partial(\rho H)}{\partial t} + \vec{\nabla} \cdot (\rho H \vec{v}) = \frac{\partial p}{\partial t} + \vec{\nabla} \cdot \left( \frac{\lambda}{c_p} \vec{\nabla} H \right) + \sigma \vec{E}^2 - q_{\text{rad}} + S_h \quad (3) $$

$\vec{v}$ is the velocity vector, $p$ the pressure, $H$ the enthalpy, $\vec{j}$ the current density vector, $\vec{B}$ the magnetic field, $\rho$ is the mass density, $\eta$ the dynamic viscosity, $\lambda$ the thermal conductivity, $C_p$ the specific heat, $\sigma$ the electrical conductivity, $q_{\text{rad}}=4\pi\epsilon_N$, $\epsilon_N$ is the net emission coefficient calculated for a radius $R_p = 5$ mm, $S_\nu$ contains supplementary terms of viscous tensors, $S_h$ is an additional source term that is required to take into account the enthalpy flux of electrons [10].

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## 2.2. Electromagnetic equations

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The electromagnetic process is governed by Maxwell's equations, which is another complex system of differential equations [11][12]. The magnetic field and the electric potential are calculated in the whole domain. The scalar potential and the vector potential equations are used to calculate the electromagnetic fields:

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$$ \vec{B} = \vec{\nabla} \times \vec{A} \quad (4) $$

$$ \vec{E} = -\vec{\nabla}V - \frac{\partial \vec{A}}{\partial t} \quad (5) $$

$$ \vec{j} = \sigma \vec{E} \quad (6) $$

$$ \vec{\nabla} \cdot (\vec{\nabla} \vec{A}) = -\mu \vec{j} \quad (7) $$

$$ \vec{\nabla} \cdot (\sigma \vec{\nabla} V) = 0 \quad (8) $$

In the equations above, $\vec{A}$ is the vector potential, $V$ is the electric potential, and $\mu$ is the air magnetic permeability. Since the temporal variations of the electromagnetic fields are slow, inductive effects are negligible, and the system can be treated in a quasi-static regime. This assumption justifies neglecting the $\frac{\partial \vec{A}}{\partial t}$ term in Equation (5).

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To close the system for the magnetic field calculation, the Biot& Savart formulation is used at the edges of the domain as a Dirichlet boundary condition. The equation is formulated as follows [12]:

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$$ \vec{A} = \frac{\mu}{4\pi} \iiint_{\text{volume}} \frac{\vec{J}(\vec{r}')}{|\vec{r} - \vec{r}'|} dV \quad (9) $$

This equation is used to calculate the magnetic field $\vec{B}$ resulting at position $r$ and distanced of $r'$ from the electrical current in a domain of volume $V$. The resolution of the equation (9) is time consuming, however, in our case, we use a hybrid formulation which means that the magnetic field within the plasma domain is calculated using the vector potential equations to determine the self-induced magnetic field and the Biot& Savart formulation is only used as a boundary condition [12].

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## 2.3. Hypotheses

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The following assumptions have been used:

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□ The air medium plasma is assumed to be in Local Thermodynamic Equilibrium (LTE). So, only one energy equation is solved for the fluid, assuming all the species have the same temperature. This assumption is also used to calculate the plasma composition (evolution of the species densities), the transport and thermodynamic properties [13];

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□ The flow is considered laminar. The Reynolds number is assumed to be small due to the high viscosity of the arc and many research which have been performed on a simplified arc chamber show the validity of this assumption [14][15];

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□ Arc ignition is not described, and the calculation begins with an initial arbitrary temperature channel equal to 20000 K;

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□ Vapours coming from the walls and the erosion of the electrodes are not taken into account. So, copper vapours from the runners, iron vapours from the splitters or PA66 ($C_6H_{11}O_1N_1$) from lateral walls are not considered even if they may change the plasma properties and the arc motion [2, 16–18];

□ Radiation is treated using the net emission coefficient method [19]. Other methods can be adopted for the radiation as the P1 model or the DOM using mean absorption coefficients. These methods allow not only to consider the losses due to the emission of the hottest regions, but also to consider the absorption of the radiation in the surrounding plasma [20]. Certainly, the choice of the radiation model directly affects the plasma properties and so the arc motion prediction. Nevertheless, in this paper, we focus our study on the general behavior of the arc;

□ Splitters are not treated as ferromagnetic materials.

## 2.4. Additional resistivity model

Anode and cathode sheaths are non-equilibrium zones near the electrode surfaces. In these zones, the physics is complex and difficult to couple with the LTE plasma. Then, to allow the passage of the current from the solid material to the plasma, a simplified electrical approach is adopted using an additional localized electrical conductivity $\sigma_{eff}$ within these zones, defined by:

$$ \sigma_{eff} = J \frac{\Delta y}{U_s} \qquad (10) $$

Where $U_s$ is the additional voltage drop, J is the current density, and $\Delta y = 0.1$ mm is the thickness of the sheath layer. This size is predefined for the anodic and cathodic sheath regions in the Lowke model [21] and is generally used in the literature [22, 23]. Then the mesh size in the region near the electrode is equal to 0.1 mm.

The effective conductivity $\sigma_{eff}$ is strongly influenced by the nature of the plasma gas, the electrode materials, and the surface conditions. Equation (10) allows to determine an effective conductivity close to the electrode wall. It is governed by the U-J curves determined from the studies [23–25], which presented different curves (a, b, c, d) with different peak values determined experimentally for different configurations and varied materials.

Before the formation of a new arc root, an ignition voltage that corresponds to the peaks should be exceeded at low current densities [23]. The existence of ignition voltage before the formation of the arc root was observed in measurements [5][26]. For higher current densities, that is, when the current flows entirely through the arc root, the voltage drop becomes constant and independent of the current density. For a copper cathode, this voltage drop is roughly approximated by $U_s=10$ V, as reported in [27–29].

For this work, we have chosen curve (c) [30]. It represents a voltage hump (ignition voltage) before the arc root formation of $U_0 = 17.2$ V, which corresponds to the experimental value for the copper material.

The total arc voltage increases with the number of sheath voltages. Taking into account the voltage drop in these sheath layers helps improve the physical understanding of arc root movement and provides a more accurate representation of the total arc voltage.

## 2.5. Simplified geometry

Figure 1 shows a simplified geometry of the arc-breaking chamber. This initial configuration allowed the validation of essential physics of the electric arc in low-voltage breaking chamber, such as arc displacement, influence of electromagnetic and pressure forces, and segmentation by studying the influence of the number of splitter plates on the arc voltage rise. The numerical results obtained from this model showed good agreement with the literature, as the general arc behavior and the additional voltage of 20 V match numerical and experimental results reported in [7, 31–33]. This provides a solid foundation for the future integration of additional physical details and more complex geometries, ultimately aiming to replicate the full behavior of low-voltage circuit breakers in real circuit conditions.

The geometry was meshed using ANSYS ICEM CFD. A structured hexahedral mesh was selected to ensure high accuracy and better alignment with the flow and field gradients. The entire domain was discretized with a uniform cell size of 0.1 mm³, providing sufficient resolution to capture the detailed behavior of the arc.
<figure><img src="image_2.png" /><figcaption>Figure 1. Simplified arc breaking geometry with different number of splitter plates.</figcaption></figure>

## 2.6. Model upgrade

Building upon the simplified geometry, the methodology was extended to incorporate more realistic features of the LVCB. This upgraded model includes the addition of a rotating contact, represented through the layering technique, which enables an accurate simulation of mechanical movement during arc initiation. Unlike the initial model, which used a constant current or an experimental current curve as input, the new approach integrates the arc simulation with an external electrical network. This network coupling allows the current to dynamically respond to the arc's behavior and circuit conditions, providing a more realistic representation of the arc breaking via its fundamental principle: Current limiting. In Figure 2, we present the main characteristics of the model to be taken into account in a real arc breaking chamber.

<figure><img src="image_3.png" /><figcaption>Figure 2. Main real arc breaking chamber model characteristics</figcaption></figure>

### 2.6.1. Network coupling

Current-limiting behavior, achieved by increasing the arc voltage using splitter plates, is a key aspect of low-voltage breaking. By incorporating this into the model, we can more accurately represent the voltage increase and current decrease during arc extinction. For that, we need to couple our model with the external circuit, presented in Figure 3. This allows for dynamic interaction between the arc and the electrical circuit, capturing the transient current behavior.

Our experimental setup, designed to study the movement of the arc in a breaking chamber, is connected to a capacitor bench. This bench allows us to vary the current intensity while maintaining the same AC frequency, depending on the charging voltage and the (L, C) couple values.
<figure><img src="image_4.png" /><figcaption>Figure 3. Experimental setup - RLC circuit</figcaption></figure>

Based on this experimental configuration, we can define the differential equation (11) that governs the circuit in the presence of an electric arc.

$$V_c = V_{arc} + V_R + V_L \qquad (11)$$

The nonlinear differential equation (11) is discretized and solved iteratively using the Euler method as mentioned in Figure 4. The solution of the differential equation is fully developed in a User-Defined Function (UDF) in Fluent. The results of this module can be found in [34].

<figure><img src="image_5.png" /><figcaption>Figure 4. Network coupling iterative process</figcaption></figure>

### 2.6.2. Contact rotation

Reproducing the motion of the moving contact during the opening process in low-voltage arc chambers poses both physical and computational challenges. Conventional methods, such as rigid body translation of contact [35] [36] [37] [38] or dynamic remeshing [39] [40] [41], are typically limited by geometric flexibility, higher numerical instability, and large computational cost. To address these limitations, a numerical modeling approach has been adopted using the layering technique, as presented in Figure 5. This method allows to create a structured mesh with multiple layers of cells to accurately capture rapid changes in flow properties while maintaining the same number of cells. In this structure, cells of 0.1 mm are progressively introduced on one side of the domain and removed on the opposite side, following rotation of contact. This significantly reduces computational time while enhancing the precision of the solution in critical arc zones. It also enhances numerical stability and provides precise control over the opening process.
<figure><img src="image_6.png" /><figcaption>Figure 5. Opening process of moving contact</figcaption></figure>

### 2.6.3. Complex geometry

The geometrical configuration used in this study, illustrated in Figure 6, is derived from a real industrial breaking chamber and retains its essential structural complexity. This realistic geometry includes critical features that influence arc behavior, such as rotating contact, arc runners, and real-form splitters. To reduce the calculation time, certain regions, specifically the volume behind the contact system and the venting area at the outlet, were excluded from the simulation domain. Despite these simplifications, the model preserves the core physical characteristics necessary to accurately capture arc dynamics. The primary objective at this stage is to establish a robust and representative model capable of reproducing realistic arc phenomena within the breaking chamber. This also serves to evaluate the performance of the developed simulation modules in capturing key physical processes, thereby validating their reliability for future parametric studies and design optimizations.

<figure><img src="image_7.png" /><figcaption>Figure 6. LVCB real breaking chamber</figcaption></figure>

## 3. Simulation results

This section presents the results obtained from the proposed model. The current intensity varies over time and is determined based on the circuit parameters, reaching values up to 1.5 kA. Figure 7 illustrates the computed temperature distribution during the evolution of the arc within the breaking chamber, highlighting the key phases of the arc behavior:

1) arc elongation between the contacts; 2) arc commutation from the contacts to the arc runner; 3) arc displacement between the runners; 4) arc splitting process; 5) arc re-striking (back commutation).

The initial stage involving arc ignition and the onset of contact separation is not included in the simulation. Instead, the model begins at a relative time of $t_R = t_{real} + 0.5$ ms, corresponding to a current of $I = 1300$ A.

Initially, the arc forms and elongates between the contacts, driven by magnetic forces. As it extends, the hot plasma begins to interact with the upper and lower runners. At $t_R = 0.3$ ms, the arc is fully commuted from the moving contact to the upper arc runner. Then it continues its displacement toward the splitter region,
driven by both electromagnetic and pressure forces in the chamber. By $t_R = 0.5$ ms, the arc root establishes contact with the splitter plates, initiating the splitting process. Then, the arc column is progressively divided into multiple sub-arcs between the splitters, and the arc voltage rises to approximately ≈ 210 V. However, at $t_R = 0.7$ ms, a new arc root unexpectedly forms on the lower runner behind the main arc column. This results in the creation of a secondary current path, effectively replacing the previous one. This phenomenon is referred to as restrike or back-commutation. As a consequence, the segmentation process must restart from this new arc root, and the arc voltage falls to approximately ≈ 187 V. This can be detrimental to the breaking process, as it delays current interruption and increases energy dissipation in the device. Finally, the arc is divided once again between the splitters, and a total voltage of 310 V is reached.

<figure><img src="image_8.png" /><figcaption>Figure 7. Arc temperature distribution in the breaking chamber, and corresponding values of V and I.</figcaption></figure>

Although the venting regions are simplified, this simulation shows the ability of the model to capture and visualize the transient evolution of the arc in realistic circuit breaker geometry. Such results offer a valuable tool for understanding the arc behavior and serve as a solid foundation for future studies aimed at optimizing design and improving device performance.

## 4. Conclusion

This study has demonstrated the successful development and implementation of an advanced MHD model to simulate the behavior of an electric arc in a low-voltage circuit breaker. Starting from a simplified configuration, the model was progressively enhanced by incorporating key physical and computational features. An additional resistivity approach was introduced to more accurately represent voltage drops near the electrodes. The simulation was extended to handle a complex 3D geometry closely resembling a real breaking chamber. A novel layering technique was implemented to model rotating contact with variable speed, allowing an accurate representation of contact opening and reducing computational time while improving the precision of the solution in critical zones. Furthermore, the arc model was coupled to an external electrical network, ensuring realistic operating conditions. These advancements significantly enhance the model's predictive capability, deepen the understanding of arc dynamics in switching devices, and provide a solid foundation for future research focused on optimizing circuit breaker design and performance.

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