# Research on New Drilling Downlink Communication Based on Anisotropic Magnetoresistive Technology
## 1 Introduction
### Mathematical Model of Magnetic Field Distribution around Downhole Drill String
According to the boundary conditions \(e_{n}\times\left(\stackrel{{\rightarrow}}{{H}}_{1}^{\rightarrow}- \stackrel{{\rightarrow}}{{H}}_{2}\right)=\stackrel{{ \rightarrow}}{{J}}_{S}^{\rightarrow}\), \(e_{n}\times\left(\stackrel{{\rightarrow}}{{B}}_{1}^{\rightarrow}- \stackrel{{\rightarrow}}{{B}}_{2}\right)=0\) and \(\stackrel{{\rightarrow}}{{B}}=\
abla\times\stackrel{{ \rightarrow}}{{A}}\) on different media, the magnetic vector \(\stackrel{{\rightarrow}}{{A}}\) on different media is given by Equations (15) and (16):  
\[e_{n}\times(\frac{1}{\mu_{1}}\
abla\times\stackrel{{\rightarrow}} {{A}}_{1}^{\rightarrow}-\frac{1}{\mu_{2}}\
abla\times\stackrel{{ \rightarrow}}{{A}}_{2}^{\rightarrow})=\stackrel{{ \rightarrow}}{{J}}_{S}^{\rightarrow}, \tag{15}\]  
\[\stackrel{{\rightarrow}}{{A}}_{1}=\stackrel{{ \rightarrow}}{{A}}_{2}^{\rightarrow}. \tag{16}\]  
For a line with length of 2 \(l\), the vector magnetic potential of the current can be represented by a simplified model as shown in Figure 5.  
The vector magnetic potential generated by the current element \(\stackrel{{\rightarrow}}{{Id}^{\prime}}\) satisfies the relationship described in Equation (17):  
Figure 5: Vector magnetic quantity of linear current. \[\stackrel{{\rightarrow}}{{A}}=e_{z}\frac{\mu_{0}Idz^{\prime}}{4\pi} \cdot\frac{1}{\sqrt{\rho^{2}+\left(z-z^{\prime}\right)^{2}}}, \tag{17}\]  
\(\mu_{0}\) is vacuum permeability, \(\mu_{0}=4\pi\times 10^{-7}N\cdot A^{-2}\). By integrating the link \(l\), Equation (18) can be obtained:  
\[\stackrel{{\rightarrow}}{{A}}=e_{z}\frac{\mu_{0}I}{4\pi}\int_{- l}^{l}\frac{dz^{\prime}}{\sqrt{\rho^{2}+\left(z-z^{\prime}\right)^{2}}}=e_{z} \frac{\mu_{0}I}{4\pi}\ln[\frac{\left(l-z\right)+\sqrt{\rho^{2}+\left(l-z\right) ^{2}}}{-\left(l+z\right)+\sqrt{\rho^{2}+\left(l+z\right)^{2}}}]. \tag{18}\]  
At that time \(l\rightarrow\infty\), Equation (18) can be simplified to Equation (19):  
\[\stackrel{{\rightarrow}}{{A}}\approx e_{z}\frac{\mu_{0}I}{4\pi} \ln[\frac{l+\sqrt{l^{2}+\rho^{2}}}{-\left(l+z\right)+\sqrt{l^{2}+\rho^{2}}}] \approx e_{z}\frac{\mu_{0}I}{4\pi}\ln\left(\frac{2l}{\rho}\right)^{2}=e_{z} \frac{\mu_{0}I}{2\pi}\ln(\frac{2l}{\rho}). \tag{19}\]  
Because the magnetoresistance sensor is close to the drill string in the information downlink communication system based on anisotropic magnetoresistance technology, it can be introduced into the solution of space magnetic field. At that time \(l\rightarrow\infty\), \(\stackrel{{\rightarrow}}{{A}}\) is infinite, that is, the vector magnetic potential of infinite linear current is infinite. In practice, infinite \(\stackrel{{\rightarrow}}{{A}}\) cannot be satisfied, the point \(\stackrel{{\rightarrow}}{{A}}=0\) (the reference point of the vector magnetic potential) is selected at \(\rho=\rho_{0}\), Equation (20) can be obtained:  
\[e_{n}\cdot\left(\stackrel{{\rightarrow}}{{B}}_{1}^{\rightarrow}- \stackrel{{\rightarrow}}{{B}}_{2}^{\rightarrow}\right)=0\ \mbox{or}\ H_{1t}=H_{2t}. \tag{13}\]0  
Since a constant vector is added to the expression of \(\stackrel{{\rightarrow}}{{A}}\), it does not affect the calculation of \(\stackrel{{\rightarrow}}{{B}}\).