Abstract: Low-voltage current-limiting circuit breakers are widely used in industrial and civil applications. They employ multiple-plate arc-extinguishing chambers, utilizing near-pole voltage drops to boost the arc voltage entering the chamber to a higher value, thus limiting short-circuit current while interrupting the circuit. However, it has been found that the repeated entry and exit of the arc during interruption, leading to back-side breakdown, causes a sudden drop in arc voltage, reducing interruption performance. Based on the actual interruption physical process, a back-side breakdown physical model, primarily based on thermal breakdown, was established. Using airflow fields, combined with thermal, magnetic field, and current distribution, the arc movement and back-side breakdown phenomenon during the interruption process of a low-voltage current-limiting circuit breaker were calculated and simulated. Keywords: Circuit breaker; Arc; Back-side breakdown 1 Introduction Low-voltage circuit breakers are the main switches of low-voltage power distribution branches. With the development of the power industry, the demand for low-voltage circuit breakers is increasing, as are their breaking performance requirements. However, the design of low-voltage circuit breakers has long relied on experience, determining design schemes through prototype manufacturing and extensive testing. This process consumes significant manpower and resources, and the development cycle for new products is very long, which cannot meet the needs of my country's power industry. In recent years, advancements in computer technology and achievements in the research of mathematical models for switching arcs have made it possible to perform numerical analysis of the breaking characteristics of low-voltage circuit breakers using computers. Researchers have begun to explore the establishment of dynamic models for low-voltage electrical switching arcs based on magnetohydrodynamics. Previously, some work on this model existed both domestically and internationally in describing high-voltage nozzle arcs, as nozzle arcs are axisymmetric problems with relatively simple boundary conditions. However, low-voltage switching arcs move in free space driven by magnetic fields, making the calculation conditions much more complex than those for nozzle arcs. Reports in this area only began abroad in 1996, but the calculations were limited to a simple cavity, without considering the actual circuit breaker arc-extinguishing chamber or integrating it with the entire breaking process. Furthermore, low-voltage current-limiting circuit breakers also exhibit complex physical phenomena. Unlike the arc-extinguishing chamber of a conventional circuit breaker, the arc-extinguishing chamber of a low-voltage current-limiting circuit breaker uses multiple arc-extinguishing plates. During the breaking process, the moving and stationary contacts first separate to generate an electric arc, which moves to the arc-extinguishing plates under the influence of electromagnetic, thermal, and flow fields. When the arc enters the plates, the near-polar voltage drop of the multiple short arcs causes the arc voltage to rise rapidly, thereby achieving the purpose of current limiting. However, precisely to achieve a higher arc voltage, the arc-extinguishing chamber of a current-limiting circuit breaker has more plates than that of a conventional circuit breaker, and they are arranged more closely together. When the electric arc enters the grid, there is still a certain amount of free gas in the area behind it, that is, in the arc path. Due to the sudden generation of a high arc voltage, the area behind it will break down and a new arc will appear. This new arc path short-circuit the arc in the grid, causing the arc that has entered the grid to disappear. This is called the back-of-arc breakdown phenomenon. This phenomenon will occur repeatedly during the circuit breaker breaking process. The breaking waveform of a circuit breaker with repeated back-of-arc breakdown is shown in Figure 1. It reduces the current limiting characteristic of the current limiting circuit breaker and increases the arcing time. [IMG=Typical waveform of current limiting circuit breaker]/uploadpic/THESIS/2007/12/20071226095135933811D.jpg[/IMG] Figure 1 Typical waveform of current limiting circuit breaker In 1988, Yoshiyuki Ikuma et al. of Nagoya University, Japan, first observed the back-of-arc breakdown phenomenon using a fast camera [1]. They also used microwave penetration technology to find that during the breaking process of low-voltage circuit breakers, before the arc voltage suddenly drops, the gaps that are about to undergo back-breakdown all experience a temperature rise. This is because the hot gas flow of the arc enters the corresponding area through the reflection of the back wall of the arc-extinguishing chamber. The entry of ionized gas and the temperature rise reduce the critical electric field strength in the corresponding area, which easily leads to back-breakdown. C. Fievet et al. in France also found [2] that the temperature in the area where the arc passes is still relatively high, and there is residual current, which will lead to back-breakdown in the form of thermal breakdown. Professor Manfred Lindmayer in Germany initially proposed a back-breakdown model based on thermal breakdown [3]. This model uses a thermal field to conduct a preliminary simulation of back-breakdown. This paper is based on these works to conduct more in-depth research, mainly focusing on miniature circuit breakers with single-phase current limiting. Based on magnetohydrodynamics, and integrating calculations of flow field, electromagnetic field, temperature field, etc., a dynamic mathematical model of low-voltage circuit breaker arc breaking is established. Unlike foreign studies that focus on simple cavity geometric models, this project directly takes the arc-extinguishing chamber of an actual low-voltage circuit breaker as the research object. Compared with foreign work, it makes full use of numerical calculations of electromagnetic fields and airflow fields, and considers more factors affecting breaking characteristics, making the mathematical model of the arc more realistic. Using the established arc dynamic mathematical model, the back-breakdown phenomenon affecting the breaking performance of current low-voltage circuit breakers is theoretically analyzed. This not only provides a mathematical model for the circuit breaker, but also provides a theoretical basis for the difficult problem of back-breakdown phenomenon. 2 Mathematical Model of Arc Considering Back-Breakdown Phenomenon A reasonable model is needed to simulate the breaking process of a circuit breaker with back-breakdown phenomenon. In previous studies, many simplified physical characteristics of the arc were used to calculate the arc current and voltage. When the physical characteristics of the arc are known, the current-voltage relationship of the arc can be simulated well. However, this is powerless when studying the actual physical phenomena of the arc. According to arc photographs taken by high-speed cameras [2], it can be seen that the arc is neither a simple line nor a simple column when burning. During complete combustion, the electric arc largely fills the arc-extinguishing chamber, forming a high-temperature plasma. In this case, using a field-region model to describe the arc is realistic. The calculation of back-end breakdown treats the entire circuit breaker region as a model using field-region calculations. In the calculation, it is divided into multiple small units, with areas exceeding 5000K considered arc regions. The temperature of each small unit determines its conductivity and the total resistance between the two electrodes, thus determining the current distribution in each unit region. Based on the current distribution at time tk-1, the temperature of each unit at time tk, the circuit current, and the current distribution within each unit of the circuit breaker can be obtained, and these are used as heat sources to calculate the arc parameters at time tk+1. Before the arc enters the arc-extinguishing grid, there is no near-electrode voltage drop; the overall resistance of the circuit breaker is directly obtained from the parallel and series resistances of each small unit region. The arc region has the highest temperature, and the equivalent resistance of the arc column region is much smaller than the resistance of other regions. After the arc enters the arc-extinguishing chamber, the region behind the arc includes the contact region and the arc-running region. At this point, the arc-extinguishing grid divides the arc into multiple short arcs. The near-electrode voltage drop increases the arc voltage, while the current through the arc decreases as the arc temperature decreases and the temperature of the region behind the arc increases. The arc can be considered equivalent to a variable resistor. Because the near-electrode voltage drop remains relatively high, and the current decreases, the equivalent resistance increases. The region behind the arc forms a high-temperature conductive channel with continuously decreasing resistance. As the resistance of the region behind the arc gradually decreases, the current is gradually transferred through this conductive channel, causing the temperature of this region to rise rapidly, and the resistance to decrease even more rapidly. The point with the highest temperature in this region is considered the arc center. When the arc center appears outside the arc-extinguishing grid, the arc voltage drops abruptly due to the absence of the near-electrode voltage drop, resulting in back-side breakdown. The arc belongs to low-temperature plasma. When studying its macroscopic motion, it is often treated as a fluid. However, unlike simple fluids, this fluid is composed of conductive particles that interact complexly with magnetic fields during motion. Therefore, its physical processes must be described using magnetohydrodynamics. The established arc model is a two-dimensional magnetohydrodynamic model. A cross-section of the circuit breaker is shown in Figure 2 for calculation. [IMG=Two-dimensional model of the arc]/uploadpic/THESIS/2007/12/20071226095142552472Y.jpg[/IMG] Figure 2 Two-dimensional model of the arc. The breaking arc in the circuit breaker satisfies the following equation. Mass connection equation [IMG=Mass connection equation]/uploadpic/THESIS/2007/12/20071226095148210692K.jpg[/IMG] Where: ρ is density; v＝vxi＋vyj. Momentum Conservation Equation [IMG=Momentum Conservation Equation]/uploadpic/THESIS/2007/12/2007122609515366975R.jpg[/IMG] Where: v is velocity; F is mass force F＝Fxi＋Fyj; P is pressure. Energy Equation [IMG=Energy Equation]/uploadpic/THESIS/2007/12/2007122609520092881W.jpg[/IMG] Where: ρ is density; h is enthalpy; T is temperature (K); t is time (s); K is thermal conductivity; S is the heat source term. In the calculation, the circuit breaker is treated as an entire region. Based on the temperature distribution within the current-limiting circuit breaker (including the arc region), the current distribution is calculated as the heat source of the coupled field. Regions with low resistance receive a large current, generate more heat, and experience a rapid temperature rise. The current density in each unit of each layer is given by the formula: G is the conductivity, which is determined by the temperature of element i,k in this layer. It is obtained by looking up tables and interpolating from references [4, 5]. In other words, the current distribution in the entire current-limiting circuit breaker area is determined by the resistance distribution caused by uneven temperature distribution. As the resistance of the area behind the arc gradually decreases, the current is gradually transferred by this conductive channel, causing a sudden drop in arc voltage and resulting in back-side breakdown. The arc temperature is very high, and in addition to conduction and convection, some energy changes occur through radiation. For radiation in the arc, since the arc is a low-temperature plasma, it can be regarded as being in thermal equilibrium and local thermal equilibrium, so it can be directly calculated using the radiation formula. The energy emitted by the electric arc radiation is: QR = A.ε.K.(T⁴－T⁴⁰) where: A is the surface area; ε is the emissivity; K is the Boltzmann radiation constant; T is the temperature; and T⁰ is the ambient temperature. The electric arc plasma in the magnetic field is driven by the magnetic force: F = I × B. When the electric arc plasma moves in the magnetic field, there must be an interaction between the conductive fluid and the electromagnetic field. Due to the motion of the conductive fluid relative to the magnetic field, according to Faraday's law of electromagnetic induction, an induced electric field is inevitably generated in the fluid, thus generating an induced current. This current is subjected to a force from the magnetic field, which is opposite to the direction of fluid motion, thus resisting the fluid's movement. F = V × V × B. The resistance of each small unit region in the circuit breaker is: [IMG=Resistance of each small unit region in the circuit breaker]/uploadpic/THESIS/2007/12/20071226095214777979.jpg[/IMG] Where ρi,j: resistivity of the small unit; li,j: length of the small unit, determined by the distance between the two ends of the electrode; si,j: area of ​​the small unit. The total resistance of the entire region is obtained by connecting the resistances of each small unit in parallel. The calculation assumes that the two ends of the circuit breaker are closed as the boundary condition. The simulated circuit breaker model is calculated in an LC single-frequency oscillating circuit. [IMG=LC single-frequency oscillating circuit]/uploadpic/THESIS/2007/12/2007122609522616982L.jpg[/IMG] Where L is inductance; i is current; R is arc resistance; U0 is the initial voltage of the capacitor in the oscillating circuit; C is capacitance. The expected current of the LC circuit is 3000A, and the frequency is 50Hz. When the circuit breaker opens, an electric arc is generated. The arc has a significant temperature difference with the surrounding hot gas, and due to their different conductivities, the current mainly flows through the arc. Under the influence of the strong short-circuit current and magnetic field flowing through the arc, the arc undergoes heat exchange, transferring energy through conduction, convection, and radiation, expanding itself, and heating the surrounding gas. Simultaneously, it moves forward under the influence of the magnetic field. During this process, the temperature and pressure distribution within the circuit breaker, as well as the parameters of the arc, change. These changes affect the arc's movement and airflow variations. Finally, the arc moves forward under the combined influence of the airflow and the magnetic field. The entire energy process of the arc is shown in Figure 3. [IMG=Arc Energy Process]/uploadpic/THESIS/2007/12/2007122609523658565N.jpg[/IMG] Figure 3. The solution to the arc energy process equations uses the finite difference method, the internal node method, and the ADI method (alternating direction implicit method). A staggered mesh is used in the calculation. 3. Calculation Results The circuit breaker's two ends are closed as the boundary condition in the calculation. The simulated circuit breaker model is calculated in an LC single-frequency oscillating circuit. The expected current of the LC circuit is 3000A, and the frequency is 50Hz. Figure 4 shows the voltage and current waveforms of the simulation calculation. According to experimental data, the simulated circuit breaker trips after 0.8ms. As the arc moves upwards and gradually lengthens, the arc voltage gradually increases. When the arc enters the grid, the voltage rapidly rises to a higher value, the current is limited, and it begins to decrease from its peak value. According to the back-breakdown model proposed in this paper, as the resistance of the back-breakdown region gradually decreases, current gradually flows through this conductive region, causing the temperature of this region to rise rapidly and the resistance to decrease rapidly. At 2.16 ms, the arc voltage drops, indicating back-breakdown. Figure 5 shows the breaking voltage and current waveforms obtained in the experiment. The voltage is 100 V/division, the current is 1000 A/division, and the time is 0.625 ms/division. [IMG=Simulated Back Breakdown Voltage and Current Waveforms]/uploadpic/THESIS/2007/12/2007122609524222805N.jpg[/IMG] Figure 4: Simulated Back Breakdown Voltage and Current Waveforms [IMG=Experimental Breaking Voltage and Current Waveforms]/uploadpic/THESIS/2007/12/2007122609524984586Z.jpg[/IMG] Figure 5: Experimental Breaking Voltage and Current Waveforms The sample used in the experiment was a variable-type special experimental circuit breaker with A3 steel grids, 20mm long and 12mm wide, rectangular in shape, with an approximate spacing of 1.6mm between the grids. Table 1 shows the back breakdown voltage and current waveforms obtained from the simulation and the experiment. As can be seen from Table 1, the calculations relatively accurately reflect the experimental results. When breaking in an oscillating circuit under the same expected current, the calculated and experimental peak currents and arc voltages are very consistent. Furthermore, the computational model effectively simulated the back-breakdown phenomenon. Table 1 shows a comparison between the computational and experimental results. Figure 6 shows the temperature distribution of the field at different times. If the highest temperature is taken as the arc center, Figure 6 shows the arc movement process: at 1.92 ms, the arc has entered the arc-extinguishing grid, the arc voltage rises rapidly, and the equivalent resistance of the arc remains relatively high due to the near-electrode voltage drop, while the resistance of the back-breakdown region continuously decreases. As the resistance of the back-breakdown region gradually decreases, current gradually flows through this conductive region, causing the temperature of this region to rise rapidly, the resistance to decrease rapidly, and the arc voltage to drop sharply, resulting in back-breakdown. At 2.16 ms, the arc has exited the arc-extinguishing grid. [IMG=Model-simulated arc back-breakdown phenomenon]/uploadpic/THESIS/2007/12/2007122609530743933D.jpg[/IMG] Figure 6 Model-simulated arc back-breakdown phenomenon 4 Conclusion Low-voltage current-limiting circuit breakers exhibit back-breakdown during interruption, leading to a sudden drop in arc voltage and affecting their interruption performance. Experiments show that this is related to the temperature rise in the corresponding region, the decrease in critical electric field strength, and the presence of residual current. This paper, through the analysis of back-breakdown, establishes a dynamic arc model of low-voltage electrical switch based on magnetohydrodynamics, based on the principle of thermal breakdown. Numerical calculations were performed on the current-limiting circuit breaker by combining the distribution of airflow, thermal field, magnetic field, and current. The results show that the model can well simulate the back-breakdown phenomenon in the current-limiting circuit breaker and basically matches the arc voltage drop in the back-breakdown in actual interruption. This provides a new approach for future computer numerical analysis of interruption characteristics in low-voltage circuit breakers and theoretical research on the back-breakdown phenomenon.