# Electrical Circuits Simulator in Null-Flux Electrodynamic Suspension Analysis
## 2 Unveiling the Inner Workings: Exploring the Principles and Model of an EDS System
### The Mathematical Model of an EDS System  
The DCT approaches an EDS system by considering space and time-dependent circuit parameters governed by differential equations. These equations are solved for time-domain currents, making the method suitable for dynamic and transient analysis through computer simulation. The electrical circuit in Figure 1 represents the simplest equivalent circuit for an 8-shaped coil interacting with a magnetic field source, such as that of the SC coils used in the JR-Maglev. The circuit features two branches in a mesh, each representing one of the loops with a resistance \(R\) and an inductance \(L\). \(M_{12}\) denotes the mutual inductance between the lower and upper loops of the 8-shaped coil. As it is a series circuit, the current in both loops is identical but flows in opposite directions, represented by \(i\). The voltage sources, \(e_{1}\) and \(e_{2}\), signify the couplings between the SC coils onboard the train and the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Physical Parameter** & **Value** & **Unit** \\ \hline Width \(\times\) height, 8-shaped coil & 0.35 \(\times\) 0.34 & m \(\times\) m \\ Number of turns, 8-shaped coil & 24 & units \\ Resistance, 8-shaped coil (1 loop) & 8.99 1 & m\(\Omega\) \\ Inductance, 8-shaped coil (1 loop) & 287.26 1 & μH \\ Mutual inducances 2 & 21.22 1 & μH \\ Distance between centers 2 & 0.42 & m \\ Pole pitch, 8-shaped coil & 0.45 & m \\ Air gap between SC and 8-shaped coil & 0.185 & m \\ \(z_{0}\) equilibrium position & 0.0366 & m \\ Width \(\times\) height, SC coil & 0.963 \(\times\) 0.5 & m \(\times\) m \\ Magnetomotive force, SC coil & 700 & kAturn \\ Inductance, SC coil & 2.7 & H \\ Pole pitch, SC coil & 1.35 & m \\ \hline \hline \end{tabular} 1
[FOOTNOTE:1]Footnote 1: These values were calculated analytically [41] and compared with those presented in the RTRI papers [24; 25]. 2  
\end{table}
Table 1: Yamanashi test track parameters [24; 25; 41].  
Figure 2: (**a**) Conventional notation used in Maglev systems. Sketch of SC and 8-shaped coils. (**b**) Dimensional drawing of the current 8-shaped and SC coils from the Yamanashi test track.  
null-flux coil loops on the track, expressed as current, displacement, and speed functions. The voltage equation for the equivalent electrical circuit is Equation (1) [36]:  
\[e_{1}-e_{2}=2 Ri+2(L-M_{12})\frac{di}{dt}, \tag{1}\]  
shows the space-time dependence of the voltages and current induced in the null-flux coil.  
The focus is on directly calculating forces. Physical quantities, such as energy, power, and forces, are expressed in their equivalent circuit parameters and spatial and temporal functions. The electromagnetic forces generated by the SC and the 8-shaped coils comprise three components. The drag force resists the train's movement, necessitating a propulsion force from the motor to overcome this force and enable displacement. The guidance force aligns the train on the track and prevents it from touching the guideway's sides. The lift force prevents the train from touching the ground and allows it to run without friction. Based on the conventional notation used in Maglev systems and illustrated in Figure 2, the drag force \(\overrightarrow{f_{x}}\), the guidance force \(\overrightarrow{f_{y}}\), and the lift force \(\overrightarrow{f_{z}}\), are given by the Equation (2) [36]:  
\[\overrightarrow{f_{j}}=Ii\bigg{(}\frac{\partial M_{S2}}{\partial j}-\frac{ \partial M_{S1}}{\partial j}\bigg{)}j, \tag{2}\]  
where \(j=x,y,z\); \(I\) represents the magnetomotive force of the SC coil, \(i\) denotes the null-flux coil current, and \(M_{S1}\) and \(M_{S2}\) are the mutual inductances between one of the ground coil loops and the SC coil. Due to these couplings, the analytical resolution of this problem relies on numerical calculation.