# Analytical Solution and Experimental Validation of the Electromagnetic Field in an IPT System
## 3 Analytical Solution of the Time-Harmonic Electromagnetic Field around One Current-Carrying Coil
### Analytical Solution of the Electric Field  
The following relationships exist in the cylindrical coordinate system:  
\[\
abla\cdot\mathbf{P}=\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho P_{ \rho}\right)+\frac{1}{\rho}\left(\frac{\partial P_{\phi}}{\partial\phi} \right)+\frac{\partial P_{z}}{\partial z}, \tag{15}\]  
\[\
abla\times\mathbf{P}=(1/\rho)\cdot\left[\frac{\partial P_{z}}{\partial\phi}- \frac{\partial\left[\rho P_{\phi}\right]}{\partial z}\right]\mathbf{e_{\rho}}+ \left[\frac{\partial P_{\rho}}{\partial z}-\frac{\partial P_{z}}{\partial \rho}\right]\mathbf{e_{\phi}}+(1/\rho)\left[\frac{\partial\left[\rho P_{\phi} \right]}{\partial\rho}-\frac{\partial P_{\rho}}{\partial\phi}\right]\mathbf{e_{z}}, \tag{16}\]  
\[\
abla\Psi=\frac{\partial\Psi}{\partial\rho}\mathbf{e_{\rho}}+\frac{1}{\rho}\frac {\partial\Psi}{\partial\phi}\mathbf{e_{\phi}}+\frac{\partial\Psi}{\partial z}\mathbf{ e_{z}}, \tag{17}\]  
\[\
abla^{2}\Psi=\
abla\cdot(\
abla\Psi)=\frac{\partial^{2}\Psi}{\partial\rho^ {2}}+\frac{1}{\rho}\frac{\partial\Psi}{\partial\rho}+\frac{1}{\rho^{2}}\frac {\partial^{2}\Psi}{\partial\phi^{2}}+\frac{\partial^{2}\Psi}{\partial z^{2}}, \tag{18}\]  
where \(\mathbf{P}\) represents a space vector and \(\Psi\) is a scalar value. By combining Equations (6), (10)-(12), and by rearranging Equation (7), then  
\[\
abla^{2}\mathbf{E}+k^{2}\mathbf{E}=j\omega\mu\left[\mathbf{J}_{S}+\frac{1}{k^{2}}\
abla[ \
abla\cdot\mathbf{J}_{S}]\right], \tag{19}\]  
where \(\mathbf{k}^{2}=-\mathrm{j}\omega\mu(\sigma+\mathrm{j}\omega\varepsilon)\). Meanwhile, \(\mathbf{J}_{S}\cdot\mathbf{e_{\rho}}=\mathbf{J}_{S\rho}=\mathbf{J}_{S}\cdot\mathbf{e_{z}}=\mathbf{J}_ {Sz}=0\) and \(\partial J_{S\phi}/\partial\phi=0\), so the divergence of the current density is  
\[\
abla\cdot\mathbf{J}_{S}=0, \tag{20}\]  
Therefore, Equation (19) can be written as  
\[\
abla^{2}\mathbf{E}+k^{2}\mathbf{E}=j\omega\mu\mathbf{J}_{S}, \tag{21}\]  
If there is no free current density, then Equation (19) can be expressed as  
\[\
abla^{2}\mathbf{E}+k^{2}\mathbf{E}=0, \tag{22}\]  
Replacing the space vector \(\mathbf{P}\) by \(\mathbf{E}_{i}\) in Equation (15) yields  
\[\
abla\cdot\mathbf{E}_{i}=\frac{1}{\rho}E_{i\rho}+\frac{\partial E_{i\rho}}{ \partial\rho}+\frac{1}{\rho}\frac{\partial E_{i\phi}}{\partial\phi}+\frac{ \partial E_{iz}}{\partial z}, \tag{23}\]  
According to Equation (34), Equation (22) can be written as  
\[\frac{\partial^{2}E_{i\phi}}{\partial\rho^{2}}+\frac{1}{\rho}\frac{\partial E_{i \phi}}{\partial\rho}+\frac{\partial^{2}E_{i\phi}}{\partial z^{2}}+\bigg{(}k^{2 }-\frac{1}{\rho^{2}}\bigg{)}E_{i\phi}=0, \tag{36}\]  
Defining \(E_{i\phi}=R(\rho)Z(z)\) and using the method of separation of variables in Equation (36) yields  
\[\
abla\times\mathbf{P}=(1/\rho)\cdot\left[\frac{\partial P_{z}}{\partial\phi}- \frac{\partial\left[\rho P_{\phi}\right]}{\partial z}\right]\mathbf{e_{\rho}}+ \left[\frac{\partial P_{\rho}}{\partial z}-\frac{\partial P_{z}}{\partial \rho}\right]\mathbf{e_{\phi}}+(1/\rho)\left[\frac{\partial\left[\rho P_{\phi} \right]}{\partial\rho}-\frac{\partial P_{\rho}}{\partial\phi}\right]\mathbf{e_{z}}, \tag{16}\]0  
By defining the first and second parts of the left side of Equation (37) as \(H(\rho)\) and \(Z(z)\), respectively, it can be easily found that \(H(\rho)\) and \(Z(z)\) are linearly independent.