# Electrical Circuits Simulator in Null-Flux Electrodynamic Suspension Analysis
## 2 Unveiling the Inner Workings: Exploring the Principles and Model of an EDS System
### The Electromagnetic Forces  
The displacement of a single SC coil in front of an 8-shaped coil was simulated to calculate the electromagnetic forces. The drag, guidance, and levitation forces, according to Equation (2), depend on the derivatives of the mutual inductance concerning the \(x\)-, \(y\)-, and \(z\)-axis, respectively. These data were also calculated numerically. The centered derivative method was used to calculate the derivative of mutual inductances. For the drag force, the mutual inductance was calculated twice for all 3201 points along the 32 m, offset  
Figure 6: Circuit implemented in Ansys® Twin Builder software. The indices 1, 2, and sc refer to the upper and lower loops of the 8-shaped and the SC coil, respectively. Data: data tables with mutual inductances \(M_{s1}\) and \(M_{s2}\). Figure 7: Electric current induced in the 8-shaped coil fed by one SC coil for five different speeds. by \(-1\) and \(+1\) mm, \(\Delta x\), concerning the longitudinal axis. The drag force was determined by Equation (4):  
\[\overrightarrow{f_{x}}=I\dot{i}\left(\frac{M_{s,1}[x+\Delta x]-M_{s,1}[x-\Delta x ]}{2\Delta x}-\frac{M_{s,2}[x+\Delta x]-M_{s,2}[x-\Delta x]}{2\Delta x}\right) \hat{t} \tag{4}\]  
and the same procedure was performed for the guidance and lift forces, with their equations being Equations (5) and (6):  
\[\overrightarrow{f_{y}}=I\dot{i}\left(\frac{M_{s,2}[y+\Delta y]-M_{s,2}[y- \Delta y]}{2\Delta y}-\frac{M_{s,1}[y+\Delta y]-M_{s,1}[y-\Delta y]}{2\Delta y }\right)\hat{t} \tag{5}\]  
\[\overrightarrow{f_{z}}=I\dot{i}\left(\frac{M_{s,2}[z+\Delta z]-M_{s,2}[z- \Delta z]}{2\Delta z}-\frac{M_{s,1}[z+\Delta z]-M_{s,1}[z-\Delta z]}{2\Delta z }\right)\hat{k} \tag{6}\]  
where the differences between (4), (5) and (6) were due to the positioning of the coils relative to the coordinate system. The imported tables for the circuit simulator were generated similarly to those for the mutual inductance between the upper and lower loops of the null-flux coil and the SC coil as a function of time for each speed, using Wolfram(r)'s Mathematica software. Figure 8 displays the output to Ansys(r) Twin Builder user for the calculated forces at a movement speed of 600 km/h, showcasing the circuit with the plot of these forces. The results were consistent with those presented in various classical works in the literature, demonstrating that the method is suitable for the study of EDS systems [21, 36, 45]. Figure 9 shows the drag force, Figure 10 the guidance force, and Figure 11 the lift force generated in this system under different operational speeds of the Maglev vehicle. Figure 7 demonstrates that the induced currents became positive with increasing speed. As the leading edge of the SC coil approached the 8-shaped coil, the induced current increased. Conversely, it decayed and turned negative as the trailing edge of the SC coil moved away from the 8-shaped coil. This alternation of the current signal induced by the same SC coil implied that it received a repulsive impulse when approaching and an attractive impulse when moving away. Consequently, the lift force oscillated between positive and negative values at one period. As the speed increased, the effect of the oscillating lift force ceased, as seen in Figure 11. The current induced in the 8-shaped coil no longer changed sign, and the lift force remained consistently positive. Figure 8: The electrical circuit implemented in the Ansys® Twin Builder software, along with the respective force curves generated as a function of time for a cruising speed of \(v=600\) km/h. The red, green, and blue lines represent drag, lift, and guidance forces, respectively. The change in the behavior of the induced electric current was due to the longitudinal force component \(f_{x}\), which consists of a conservative part and a dissipative part [13].