## 2 Modeling of Magnetic Field
### Calibration  
The calibration of the system is mainly in two aspects. First, it is necessary to verify whether the magnetic induction calculated by the mathematical model proposed in Section 2.2 is consistent with the actually measured magnetic induction.Next, due to the processing accuracy of the core and the error in the assembly process, the magnification of each coil is different, therefore, the driven current of coils needs to be compensated. The field is measured at six positions in the workspace with \(9A\) current flowing through one coil. The resultant values are given in Table 1.  
There is a linear positive correlation between the mathematical model data and measured data. The variables \(\alpha\) and \(\beta\) can be introduced to correct the model proposed in Section 2.2\(B_{modified}=\alpha B_{model}+\beta\), where \(\alpha=1.0655\), \(\beta=0.0115\). The mathematical model we have built describes the magnetic field distribution of the coil module without the soft magnetic core. The effect of the soft magnetic core on the spatial distribution of the magnetic field is characterized by the calibration of the magnetic field. The effect of the magnetic core on the spatial magnetic field distribution is divided into two parts. First, the magnetic core has an enhanced effect on the magnetic field excited by the coil itself; the literature [17] has verified the use of high-performance soft-magnetic material in the cores. Second, the coupling relationship between electromagnets. The magnetic field excited by an electromagnet after the current is applied can magnetize the magnetic core in the adjacent electromagnet so that the magnetic field distribution is nonlinear. For the first case, we study the change of the magnetic field strength at a fixed point when the input current increases linearly for a single-coil module (coil and core are assembled together). Figure 4 shows that the magnetic core in the eight-coil modules has a linear enhancement effect on the magnetic field distribution.The abscissa is the measured magnetic field generated by a single coil module without a soft magnetic core under different currents, and the ordinate is the magnetic field generated by a single coil module under different currents when a soft magnetic core is installed. The slope of the line connecting each scattered point and the origin of the coordinate system represents the enhancement coefficient of the magnetic field of the soft magnetic core under different input current conditions. The results in the figure show that a straight line passing through the origin of the coordinate system can be used to connect the scattered points and this characterizes the effect of the linear enhancement of the magnetic core in the workspace. \begin{table}
\begin{tabular}{l c c c c c c} \hline
**Unit:\(mT\)** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \(B_{model}\) & 2.14 & 1.32 & 0.91 & 0.58 & 0.41 & 0.30 \\ \(B_{measured}\) & 2.29 & 1.44 & 0.94 & 0.65 & 0.45 & 0.33 \\ \(B_{modified}\) & 2.29 & 1.42 & 0.98 & 0.63 & 0.45 & 0.33 \\ \hline \end{tabular}
\end{table} 
Table 1: Multi-point calibration relating to the mathematical model through a given coil. Figure 4: The linear enhancement of the magnetic field distribution by the soft magnetic core in a single coil module. For the second case, the coupling effect of the magnetic field is temporarily unable to be described in accurate mathematical terms. Our research idea is to collect the magnetic field data to separate the linear change data in the magnetic field from the non-linear change data due to the coupling effect and estimate the non-linear quantities using approximate mathematical language. Therefore, we conducted the following experiment. Collecting the magnetic field intensity distribution of a single module and module assembly at a series of points. When the test input current sets the maximum current of the system (10 A), the offset of the magnetic field value generated by the coupling effect reaches the maximum.