# The Electromagnetic Field Analytical Solution Analysis to Downhole Current Injection [PERSON]
## 2 Signal Model and Electromagnetic Field Distribution Analysis
### The Secondary Electric Potential Analysis  
Under the quasi-stationary field assumption, the secondary electric potential satisfies the homogeneous Laplace equation, which can be represented as follows:  
\[\
abla^{2}V_{2}=0. \tag{7}\]  
By utilizing the cylindrical coordinate system, the Laplace equation can be expanded as follows:  
\[\dfrac{1}{\rho}\dfrac{\partial}{\partial\rho}\bigg{(}\rho\dfrac{\partial V_{2 }}{\partial\rho}\bigg{)}+\dfrac{1}{\rho^{2}}\dfrac{\partial^{2}V_{2}}{ \partial\varphi^{2}}+\dfrac{\partial^{2}V_{2}}{\partial z^{2}}=0. \tag{8}\]  
Assume that the solution is separable, substituting \(V_{2}=R(\rho)\Phi(\varphi)Z(z)\) into Equation (8) yields  
\[\
abla^{2}V_{2}=\dfrac{1}{\rho}\dfrac{\partial}{\partial\rho}\bigg{(}\rho \dfrac{\partial V_{2}}{\partial\rho}\bigg{)}+\dfrac{1}{\rho^{2}}\dfrac{ \partial^{2}V_{2}}{\partial\varphi^{2}}+\dfrac{\partial^{2}V_{2}}{\partial z ^{2}}=\dfrac{1}{\rho}\big{(}R^{\prime}\Phi Z+\rho R^{\prime\prime}\Phi Z\big{)} +\dfrac{1}{\rho^{2}}R\Phi^{\prime\prime}Z+R\Phi Z^{\prime\prime}=0, \tag{9}\]  
where the superscripts / and \({}^{\prime\prime}\) represent the first and second derivatives of a function. Let \(\dfrac{eR^{\prime}\Phi+\rho^{2}R^{\prime\prime}\Phi+R\Phi\sigma^{\prime\prime }}{\rho^{2}R\Phi}=-\frac{Z^{\prime\prime}}{Z}=\lambda^{2}\), where \(\lambda\) is an introduced variable. Separate Equation (9) into the following two equations:  
\[-\frac{Z^{\prime\prime}}{Z}=\lambda^{2}, \tag{10}\]  
\[\dfrac{\rho R^{\prime}\Phi+\rho^{2}R^{\prime\prime}\Phi+R\Phi^{\prime\prime}} {\rho^{2}R\Phi}=\lambda^{2}. \tag{11}\]  
The solution to Equation (10) is \(Z(z)=e^{\pm i\lambda z}\). To solve Equation (11), it also needs to be separated into two equations as \(\dfrac{e^{2}R^{\prime\prime}+\rho R^{\prime}-\lambda^{2}\rho^{2}R}{R}=-\frac{ \Phi^{\prime\prime}}{\Phi}=m^{2}\). Obviously, the solution to \(-\frac{\Phi^{\prime\prime}}{\Phi}=m^{2}\) is \(\Phi(\varphi)=e^{\pm im\varphi}\), while the equation \(\frac{\rho^{2}R^{\prime\prime}+\rho R^{\prime}-\lambda^{2}\rho^{2}R}{R}=m^{2}\) needs to be further analyzed. Here, the variable transformation is used. Let \(x=\lambda\rho\), \(\frac{\rho^{2}R^{\prime\prime}+\rho R^{\prime}-\lambda^{2}\rho^{2}R}{R}=m^{2}\) can be transformed into  
\[x^{2}\dfrac{\partial^{2}R\big{(}\frac{x}{\lambda}\big{)}}{\partial x^{2}}+x \dfrac{\partial R\big{(}\frac{x}{\lambda}\big{)}}{\partial x}-\Big{(}x^{2}+m^ {2}\Big{)}R=0, \tag{12}\]  
which is similar to the modified Bessel equation. The standard Bessel function and the modified Bessel function are the solutions of the standard Bessel equation and the modified Bessel equation, respectively, and this distinction should be made. The only difference between the standard Bessel equation and the modified Bessel equation is the opposite sign of the non-derivative term. Since they have numerous solutions, they have been classified accordingly. More details can be seen in [33]. Equation (12) has exactly the same form as the Bessel equation, with the only difference being that the independent variable in \(R\big{(}\frac{x}{\lambda}\big{)}\) is \(\frac{\pi}{\lambda}\) rather than \(x\). Hence, it is concluded that the solution to Equation (12) is a Bessel function that can be expressed as follows:  
\[R(\rho)=AI_{0}(\lambda\rho)+BK_{0}(\lambda\rho), \tag{13}\]  
where \(A\) and \(B\) are boundary coefficients. Since the solution is a periodic function of the azimuth angle, \(\mathbf{m}\) is an integer. Additionally, the eigenvalue of Equation (12) can take values over the entire real number range.