# Research on New Drilling Downlink Communication Based on Anisotropic Magnetoresistive Technology
## 1 Introduction
### Mathematical Model of Downhole Drill String Current Distribution  
In order to study a new type of downlink communication technology based on anisotropic magnetoresistance technology, the current distribution of downhole drill string is modeled. According to the study, when verylowfrequency (VLF) alternating current (AC) is injected into the electrode, the distribution of electric current field in rock and soil layer can be approximately regarded as the electric field distribution of electrostatic field dipole [28]. According to the electric field line and potential distribution of electric dipole electrostatic field, the electric current field distribution of a pair of electrodes in rock and soil layer can be similar. The rock and soil layer belongs to semiconductor medium. Under the known dielectric constant of the rock and soil layer, when an AC voltage \(U\) varying with time is applied between the electrodes, an alternating current will flow along the electric field line between the two electrodes. The relationship between the current and voltage of the whole system can be equivalent to that of the circuit system composed of multiple capacitors and resistors in series and parallel relationships. Before establishing the rock and soil layer's current model, the volume of the rock and soil layer is infinite, with the conductivity \(\sigma\) and the dielectric constant \(\varepsilon\). Then, a pair of electrodes with applied voltage \(U\) and distance \(L\) can be equivalent to the model shown in Figure 2.  
The resistivity of the drill string is far less than that of rock and soil layer. To facilitate subsequent analysis, assumingthat the drill string is a straight line with uniform and charged quantity \(q\). Then, the charge distribution law of the drill string can be expressed as \(\lambda=q/H\), \(H\) is the length of the downhole drill string. The potential function of the electric field generated by the point charge \(q\) at point \(r\) can be expressed as Equation (1):  
\[\varphi(r)=q/(4\pi\varepsilon\left|\overset{\rightarrow}{r}-r^{\overset{ \rightarrow}{1}}\right|). \tag{1}\]  
Figure 2: Equivalent circuit diagram. According to the model shown in Figure 3, the potential generated by an arbitrary point charge dy at point \(-q\) with the distance \(r\) can be expressed as Equation (2):  
\[\varphi(a)=q/4\pi\epsilon H\sqrt{L^{2}+a^{2}}. \tag{2}\]  
In practice, the drill string is usually surrounded by drilling fluid, and its conductivity is much lower than that of common metals. According to the boundary conditions and voltage relations, it can be concluded that the tangential components of the two media in the electrode are continuous, as shown in Equation (3):  
\[\sigma_{1}E_{1}=\sigma_{2}E_{2}. \tag{3}\]  
Assuming that the electric field in the two media is uniform, the voltage can be obtained by Equation (4):  
\[E_{1}d_{1}+E_{2}d_{2}=U, \tag{4}\]  
where \(d_{1}\) is the width of drilling fluid, \(\sigma_{1}\) is the conductivity of drilling fluid, \(d_{2}\) is the width of rock and soil layer, \(\sigma_{2}\) is the conductivity of rock and soil layer. Since \(d_{1}\ll d_{2}\), the influence of drilling fluid on field strength can be ignored. According to the proportional relationship between the constant electric field and the static electric field, the voltage expression of any current element flowing through the rock and soil layer is given in Equation (5):  
\[U(a)=I/4\pi\sigma H\sqrt{L^{2}+a^{2}}, \tag{5}\]  
where, \(I\) is the transmitting current of the transmitter. The impedance of the conductor and electrode is much less than that of rock and soil, so the impedance of the conductor and electrode can be ignored, and the input impedance \(Z\) is equivalent to the impedance of the rock and soil layer.