# Design of a Laparoscopic Robot System Based on Spherical Magnetic Field
## 4. **Mathematic Modeling**
### **The Magnetic Flux Intensity \(B\)** Produced by **Stator**
#### 4.1.1 The Magnetic Flux Intensity B Produced by Single Electromagnet  
The magnetic field produced by every electromagnet consists of two parts. One part is created by electrified coils, the other is produced by magnetized ferrite cores. On the one hand, since the magnetoconductivity of ferrite cores is very large, it can be assumed that the magnetic field produced by the ferrite cores is much larger than that of the coils, and the latter one can be ignored. On the other hand, because the ferrite cores are cylindrical and their ratio of height to diameter is very large, the ferrite cores can be considered magnetized uniformly.  
According to the laws of physics, we have:  
\[M_{q}=\chi_{m}H \tag{16}\]  
where \(M_{q}\) is the magnetization intensity of the ferrite core, \(\chi_{m}\) is the magnetic susceptibility of the ferrite core, and \(H\) is the magnetic field intensity where the iron cores are located.  
According to the Ampere circuit law, one can obtain:  
\[LH=nIL \tag{17}\]  
where \(n\) represents the turns of coil on unit length. \(L\) indicates the total length of coil. \(I\) indicates the electric current through coil.  
By combining Equations (16) and (17), one can obtain:  
\[M_{q}=\chi_{m}nI \tag{18}\]  
As the ferrite cores are magnetized uniformly, the magnetizing current only exists in the surface of them [16], those magnetized iron cores can be seen as concentrated electromagnets distributed in the surface of the magnetized iron cores, and their surface current density can be expressed as:  
\[i=M_{q}\times n_{n} \tag{19}\]  
where \(n_{n}\) is the normal vector of the surface of iron core.  
According to [16], when the center of magnetic dipole is taken as the coordinate origin \(O\), the magnetic vector potential created by single magnetic dipole at an arbitrary point \(P\) in space is:  
\[A_{p}=\frac{\mu_{0}a^{2}l\,sin\,\,\theta_{d}}{4r^{2}} \tag{20}\]  
In the above equation, \(\mu_{0}\) indicates the magnetoconductivity in vacuum, \(a\) is the diameter of the magnetic dipole, \(I_{d}\) represents the electric current through the magnetic dipole, \(r\) indicates the vector length from the coordinate origin to point \(P\), and \(\theta\) indicates the angle between vector \(\overset{\rightarrow}{\text{OP}}\) and normal vector of magnetic dipole.  
Therefore, the magnetic vector potential produced by a single iron core is:  
\[A=\int_{l}^{L+l}\frac{\mu_{0}a^{2}i\,sin\,\,\theta}{4r^{2}}dh \tag{21}\]  
where \(l\) is the distance between the target point and coil. \(L\) is the height of coil. According to [16], we have:  
\[B=\
abla\times A \tag{22}\]  
By combining Equations (18)-(22), one can get the relationship between magnetic flux density \(B\) created by every single coil to the electric current \(I\) through it:  
\[B=B^{u}(\chi_{m})I \tag{23}\]where \(I\) is the electric current through electromagnets, \(B^{u}(\chi_{m})\) is the magnetic flux intensity by unit electric current, which concerns the unknown variable \(\chi_{m}\).  
By fitting the unknown variables in Equation (23) with the exact solution obtained by finite element simulation, the approximate solution of \(X_{m}\) can be obtained [12], therefore, the magnetic flux density of every coil \(B_{i}\) can be expressed by the input electric current \(I_{i}\).