Abstract: The calculation of electro-repulsion force is of great significance for the design of low-voltage molded case circuit breakers (MCCBs). Based on the equations between current, magnetic field, and electro-repulsion force, and considering the influence of ferromagnetic materials, three-dimensional finite element nonlinear analysis is applied. A cylindrical conductive bridge model is introduced as the contact point to simulate the current contraction between contacts, and the Holm force between contacts and the Lorentz force on the moving conductive rod are calculated uniformly. The coupled circuit equations are used, and the opening time of the moving contact is determined by iterative calculation with the preload on the moving contact as a constraint. Five different MCCB structures are analyzed, and experimental studies on electro-repulsion force and contact opening time are conducted. The results show that the method is effective and can be used for the design of contact systems of new MCCB products. Keywords: Low-voltage electrical appliances, finite element, electro-repulsion force 1 Introduction When a short-circuit current is generated, the moving and stationary contacts in a low-voltage molded case circuit breaker (MCCB) separate under the action of electro-repulsion force. Driven by the operating mechanism, an electric arc is generated between the contacts, elongated, and enters the grid under the action of the arc blowing force, where it is divided into several short arcs and then extinguished. At the beginning of this process, the electrodynamic repulsion force F (including the Lorentz force FL generated by the conductive circuit and the Holm force FH generated between the contacts due to current contraction) and the pre-pressure acting on the moving contact determine the repulsion time and opening speed of the contact, thus having an important impact on the current limiting performance of the MCCB. Moreover, it can be seen from the Holm formula shown in equation (1) that the contact radius r is related to the pre-pressure FK, the Brinell hardness H of the contact material, and the contact surface contact condition (described by x, which is generally 0.3 to 0.6, usually taken as 0.45). The Holm force FH is related to r, the contact radius R, and the current magnitude i [1], so FK has a certain influence on F; on the other hand, the selection of FK is also limited by F due to the heat generation tolerance requirements. Therefore, the electrodynamic repulsion force must be calculated when designing the contact conductive circuit of the MCCB. In recent years, with the development of computer technology and the continuous progress of numerical calculation methods, computer-aided engineering has been increasingly widely used in the design of new MCCB products [2-3], which has greatly shortened the product development cycle, reduced development costs, and improved product performance. For the calculation of electro-repulsion force, scholars at home and abroad have done a lot of useful work. In the literature [4-5], since Holm's formula has been proven to be correct in calculating the electro-repulsion force between isolated contacts [6], Equation (1) is used to calculate FH, while ignoring the current contraction in the contact area. This simplification not only makes the current distribution in the current-carrying conductor very different from the actual distribution, but also further affects the distribution of the magnetic field, thus causing a certain error in the calculation result of FL. This paper uses three-dimensional finite element analysis, introduces a cylindrical conductive bridge model as the contact point, takes five different MCCB products as the object, focuses on the calculation method of electro-repulsion force and contact repulsion time, and analyzes the influence of different contact systems on electro-repulsion force. The results are compared with experimental results, showing that the simulation method is correct and can be applied to the development of contact systems for new MCCB products. Meanwhile, the calculation results can also be used as input data for further application of virtual prototype technology to simulate the MCCB breaking process. 2 Calculation method 2.1 Conductive bridge model When considering the influence of current contraction between contacts and the electric repulsion force of conductive circuit on the moving conductive rod, a reasonable calculation model must first be introduced to describe the electrical contact between contacts. When R.holm derives the analytical expression of electric repulsion force FH in formula (1), he assumes that the contact conductor is a superconducting sphere for the needs of analysis [1]. In order to be closer to the actual situation, this paper uses a cylindrical conductive bridge model located at the center of the contact to simulate the conductive spot. Its material properties are the same as those of the contact material. The radius r can be calculated by Holm formula shown in formula (1). In order to determine the conductive bridge height parameter h, an electric repulsion simulation was performed on a pair of cylindrical contacts. Figure 1 shows a schematic diagram of its cross section. The results show that under the same r, the height h has little effect on the electric repulsion force in the range of 0.1 to 0.25 mm. Table 1 and Table 2 are the simulation conditions and results, respectively. Therefore, in the following simulation, the height parameter is selected as 0.2mm. 2.2 Calculation principle For MCCB, the moving contact opens under the action of a torque M on the rotating shaft, as shown in Figure 2. For any unit i, its torque Mi on the rotating shaft O is the vector product of di and force density Fi. Then, by performing a volume integral operation on Mi in the entire moving conductive rod region, the torque acting on it relative to O can be obtained. Thus, the equivalent electrodynamic force acting on it can also be obtained, as shown in equation (2). The force density Fi can be calculated by equation (3), where Ji and Bi are the current density and magnetic flux density on unit i, respectively. Reference [7] pointed out through theoretical calculation that the eddy current in MCCB hardly affects the value and phase of the electrodynamic repulsion force. Thus, the equation of constant field can be used to calculate the distribution of current density and magnetic flux density. In the conductive region, i.e. the contact conductive circuit, the current density J satisfies the boundary conditions shown in equations (4) and (5). Where s is the conductivity of the conductor, and in this paper, the conductive rod and the contact are made of copper and silver, respectively; T is the vector potential, and I is the current flowing through the conductor. [align=center] [/align] After obtaining the distribution of the current density J, the distribution of B can be obtained in the whole field according to the relationship between the magnetic flux density B and J (6), where A is the vector magnetic potential and m is the magnetic permeability. Based on the above relationship between current-magnetic field-electric repulsion, the electric repulsion acting on the moving conductive rod and the contact can be obtained by using three-dimensional finite element analysis. Since there is a monotonically increasing relationship between this force and the short-circuit current, the electric repulsion is proportional to the square of the short-circuit current when the influence of ferromagnetic materials is not considered. Moreover, under specific short-circuit conditions, the short-circuit current and time have a certain relationship. In this paper, an oscillating circuit is used as the experimental circuit. Before the contact is repelled, the circuit equation is (7). Thus, by iteratively processing the above process, when the electric repulsion F is equal to the contact preload FK, the iteration ends, and the corresponding time is the time when the contact is repelled. In the formula, I is the effective value of the first half-wave of the current, and the frequency is the power frequency of 50Hz. 3. Analysis Model The simulation was carried out on two molded case circuit breaker products CB1 and CB2, both with a rated current of 100A. Among them, the static conductive rod of CB1 adopts a horizontal U-shape, while that of CB2 adopts a bottom-entry U-shape. In addition, without changing other shapes and dimensions of the products, the static conductive rod of CB1 was changed to a flat plate; the length of the grid leg of CB2 was shortened and a U-shaped motor slot was added; the arc-extinguishing grid of the CB1 model was removed. The change of the electric repulsion force was studied and named CB3, CB4 and CB5 respectively. [align=center] [/align] Figure 3(a), (b), (c) and (d) are the partition diagrams of CB1, CB2, CB3 and CB4 in ANSYS. Since they are all symmetrical, half of them are analyzed in the simulation process. Among them, the lower right corner of (a) and (c) is the top view of the static conductive rod in the area near the static contact and the schematic diagram of the current flow. The current distributions in the conductive loops of CB1, the moving conductive rod and the moving contact, and the conductive loop of CB3 are shown in Figures 4(a), (b), and (c), respectively. The current flow direction is consistent with that shown in Figure 3(a). From the top view of CB1 shown in Figure 4(a), it can be clearly seen that the current direction in the part shown in Figure 1 is opposite to that of the moving conductive rod due to the U-shaped bend of its static conductive loop. Furthermore, all analyses in this paper are performed relative to the coordinate system shown in this figure, where the positive directions of the x, y, and z axes conform to the right-hand rule. In this figure, the positive y-axis is perpendicular to the paper and outwards. As can be seen from Figure 4(b), due to current contraction, the maximum current density at the contact is 5 orders of magnitude higher than that on the moving conductive rod. The difference between the current distribution on the conductive loop of CB3 shown in Figure 4(c) and that in Figure 4(a) is that the current direction in the part shown in Figure 1 is the same as that of the moving conductive rod. Figure 5 shows the distribution of the average magnetic flux density Bz in the z-direction and the average electrodynamic repulsion density fy in the y-direction along the x-axis on the moving conductive rods of models CB1 and CB3, with its origin located at point A in Figure 2. It can be observed that the magnetic field and force density are relatively large in the area near the contact, and this area is far from the axis of rotation, resulting in a large lever arm. Therefore, it can be considered that the electrodynamic repulsion force of the circuit mainly depends on this part. At the same time, it is clearly seen that the magnetic field and force density of CB1 are larger than those of CB3. In the numerical analysis of the Holm force between the contacts and the Lorentz force on the moving conductive rod, it can be considered that the equivalent resultant force of the force of each unit acting on the moving conductive rod referred to point B (see Figure 2) is the circuit force, while the equivalent resultant force of the force of each unit acting on the moving contact referred to point B is the contact force, i.e., the Holm force. From the calculation data of CB1 shown in Table 3, the electrodynamic repulsion force is approximately proportional to the square of the current. When the current is 10kA, the electrodynamic repulsion force of the five models from CB1 to CB5 was calculated in detail, and the results are shown in Table 4. The radius of the conductive bridge is determined according to Equation (1) based on the design parameters of the actual products CB1 and CB2. As shown in Table 3, the electrodynamic repulsion generated by the contraction of the current between the contacts accounts for more than 70% of the total for different models under different currents. For product CB1, the electrodynamic repulsion decreases after changing the shape of its static conductive loop, indicating that the U-shaped loop in the horizontal plane can strengthen the magnetic field in the contact area by changing the current direction, thereby increasing the electrodynamic repulsion. Without considering the effect of the grid, the electrodynamic repulsion on the loop decreases by about 40%, indicating that ferromagnetic materials can effectively strengthen the magnetic field on the moving conductive rod. For product CB2, the electrodynamic repulsion also increases after changing the grid size and adding a U-shaped motor slot. At the same time, this improvement is also beneficial to fix the gas-generating material inside the motor slot and improve the breaking performance of MCCB by using the new air-blowing arc extinguishing principle. 4 Experimental Methods and Results Analysis 4.1 Experimental Methods The experiment uses a single-frequency high-voltage oscillation circuit to simulate short-circuit current. Figure 6 is the schematic diagram of the synthesized circuit. SP is the experimental prototype, Ci and Cu are the charging capacitors for the current and voltage loops, respectively, Li and Lu are the oscillating inductors for the current and voltage loops, respectively, H is a rigorously calibrated non-inductive sampling resistor with a resistance of 90mW and an error range of 0.2%, and K1 is the main closing switch. In this experiment, only the current loop is needed. Its oscillation frequency is 50Hz, and the ratio of the effective value of the first half-wave charging voltage to the effective value of the first half-wave discharging current is 46:1 (V/kA). That is, if the effective value of the first half-wave discharging current is 10kA, then the required charging voltage for the capacitor bank is 460V. Before the experiment begins, K1 is opened, and the capacitor bank Ci is charged through the rectifier. Once the required voltage is reached, the charging circuit is disconnected, and the experiment can begin. Closing K1, Ci, Li, K1, K3, and H constitutes a typical single-frequency oscillation loop. The arc current and arc voltage waveforms are measured using an oscilloscope and a high-voltage probe, respectively. A quartz tension-compression sensor is used to measure the electrodynamic repulsion force. When a short-circuit current flows through the circuit breaker, the moving contact attempts to open and generate an electric arc under the action of electro-repulsive force. In order to measure the waveform of the change in electro-repulsive force and conduct multiple experiments without fusion welding, the moving conductive rod must be pressed tightly to prevent the moving contact from being repelled. In the experiment, the sensor is mounted on the model. One end of a bakelite rod is fixed to the sensor by a thread, and the other end presses against the moving conductive rod. It can be rotated to adjust the preload acting on the moving conductive rod. 4.2 Electro-repulsive force measurement To facilitate the measurement of electro-repulsive force using the sensor, a model of product CB1 magnified 4 times was fabricated. The contact material is copper, and x can be taken as 0.45. Figure 7 shows the electro-repulsive force and short-circuit current waveform acting on the moving conductive rod when the short-circuit current peak value is 9kA, where the preload FK is 25.1N. The specific data are shown in Table 5. It can be seen from the data that the calculated value of the electro-repulsive force under the peak current and the measured force waveform peak value are very close; moreover, the phase difference between the short-circuit current and the electro-repulsive force is also very small. Considering the hysteresis caused by the response time of the tension/compression sensor and the bakelite rod itself, the influence of eddy currents on the electro-repulsive force is very small. Therefore, the introduction of the conductive bridge and the use of static analysis to calculate the electro-repulsive force is appropriate and relatively accurate. Moreover, the electro-repulsive force acting on the moving conductive rod under a current of several thousand amperes in MCCBs can generally overcome the contact preload and cause the moving contact to open. The experiment of product CB2 also illustrates this point. Thus, the static method can be used to calculate the contact opening time while ignoring the influence of eddy currents. 4.3 Contact Opening Time Experiment Taking the CB2 circuit breaker as the research object, a breaking experiment was conducted under the condition that the expected effective value of the short-circuit current is 10kA. Figure 8 shows the measured arc current and arc voltage waveforms. It can be clearly seen that when the voltage on the shunt H rises to 273mV, that is, when the short-circuit current reaches 273mV/90mW=3033A, and the time is 759ms, there is a sudden jump in the voltage between the moving and stationary contacts, indicating that the contacts open under the action of electro-repulsive force at this moment, and an arc is generated. Before this moment, the current is sinusoidal, and due to the contact resistance between the contacts, the voltage rises with the current by a very small value. After this moment, due to the existence of the arc voltage and its current limiting effect, the current waveform is no longer sinusoidal, and the arc is extinguished when the current crosses zero. The contact preload of CB2 is 4.4N, and the expected effective value of the short-circuit current is 10kA. Through iteration, the simulation result is that the current reaches 3020A and the time is 674ms. The electro-repulsive force reaches 4.4N, and the contacts begin to separate. Compared with the experimental results, it can be found that due to the influence of friction of the rotating shaft of the moving contact, the experimental value is slightly larger than the calculated value. The calculation and experimental results show that the above method is feasible. According to the relationship between current and time under different short-circuit conditions, the contact separation time can be calculated. 5 Conclusion (1) Based on the relationship between current-magnetic field-electro-repulsive force, the three-dimensional finite element nonlinear analysis is applied, and the conductive bridge model is introduced to describe the current contraction between the contacts. The electro-repulsive force acting on the moving conductive rod and the contacts can be calculated in a unified manner. Re-couple the short-circuit circuit equations, and use the contact preload as a constraint. Through an iterative process, the contact repulsion time can be calculated. Comparison with experimental results shows that the simulation method and results are correct and can be used for the contact conductive circuit design of new MCCB products. (2) The electric repulsion force is directly proportional to the square of the current. (3) The total electric repulsion force is mainly provided by the electric repulsion force of the moving contact and its surrounding area, while the electric repulsion force caused by current contraction on the moving contact accounts for more than 70% of the total electric repulsion force. (4) The arc-extinguishing grid has a significant impact on the electric repulsion force of the circuit. (5) The planar U-shaped static conductive circuit has a larger total electric repulsion force than the planar type; the electric repulsion force is larger when a short grid is added to a U-shaped motor slot than when a long grid is added. It is easier to add gas-generating materials in the structure, so the MCCB breaking performance can be improved by using the new air-blowing arc-extinguishing technology. (6) The simulation method and results can be further used for the simulation of the entire breaking process of MCCB.