# Two-Wire Power Line Microwave Communication Using Low-Loss Modes
## 2 Problem Definition and Exact Solution
\[\widetilde{G}_{ez}^{11}\lx@overaccentset{{\raisebox{-1.0 pt}{ \scalebox{1.0}{$\star$}}}}{I}_{1}(\gamma)+\widetilde{G}_{ez}^{12}\lx@overaccentset{{ \raisebox{-1.0 pt}{\scalebox{1.0}{$\star$}}}}{I}_{2}(\gamma)+ \lx@overaccentset{{\raisebox{-1.0 pt}{\scalebox{1.0}{$ \star$}}}}{E}_{ez}(-d/2+a,\gamma)=-\hat{V}_{1}+z_{iw}\lx@overaccentset{{ \raisebox{-1.0 pt}{\scalebox{1.0}{$\star$}}}}{I}_{1}(\gamma) \tag{6}\]  
\[\widetilde{G}_{ez}^{21}\lx@overaccentset{{\raisebox{-1.0 pt}{ \scalebox{1.0}{$\star$}}}}{I}_{1}(\gamma)+\widetilde{G}_{ez}^{22}\lx@overaccentset{{ \raisebox{-1.0 pt}{\scalebox{1.0}{$\star$}}}}{I}_{2}(\gamma)+ \lx@overaccentset{{\raisebox{-1.0 pt}{\scalebox{1.0}{$ \star$}}}}{E}_{ez}(d/2-a,\gamma)=-\hat{V}_{2}+z_{iw}\lx@overaccentset{{ \raisebox{-1.0 pt}{\scalebox{1.0}{$\star$}}}}{I}_{2}(\gamma) \tag{7}\]  
where \(\widetilde{G}_{ez}^{11}=\widetilde{G}_{ez}^{22}=\widetilde{G}_{ez}(-d/2+a,\gamma)\) and \(\widetilde{G}_{ez}^{12}=\widetilde{G}_{ez}^{21}=\widetilde{G}_{ez}(d/2-a,\gamma)\), while \(\lx@overaccentset{{\raisebox{-1.0 pt}{\scalebox{1.0}{$\star$}}}}{E}_{ez}(-d/2 +a,h,\gamma)\) and \(\lx@overaccentset{{\raisebox{-1.0 pt}{\scalebox{1.0}{$\star$}}}}{E}_{ez }(d/2-a,h,\gamma)\) represent the axial electric field of the external source at each conductor, where \(h\) is the distance of the external source above the conductors. \(\hat{V}_{1}\) and \(\hat{V}_{2}\) represent the voltage at z = 0 of the source in series with each conductor. The solution to these equations is formally valid at any frequency for which the conductor radius a is small compared to the other dimensions and the wavelength at the frequency of interest for which the conductor is appropriately modeled by its conductance \(\sigma_{w}\). Here, (The subscript notation \(1/2\) means 1 or 2. It should not be confused with the superscript \(\frac{1}{2}\) which means square root)  
\[\widetilde{G}_{ez}(x-x_{1/2},\gamma)=\frac{1}{4\omega\varepsilon_{0}}\Big{(} \gamma^{2}-k_{0}^{2}\Big{)}H_{0}^{(2)}\bigg{(}\Big{(}k_{0}^{2}-\gamma^{2} \Big{)}^{1/2}\rho_{1/2}\bigg{)} \tag{8}\]  
where \(\rho_{1/2}=|x-x_{1/2}|\) and \(\text{Im}\big{(}k_{0}^{2}-\gamma^{2}\big{)}^{1/2}\leq 0\)  
Equations (6) and (7) can be written more compactly in matrix form as follows:  
\[ \begin{bmatrix}\widetilde{G}_{ez}^{11}-z_{iw}&\widetilde{G}_{ez}^{12}\\ \widetilde{G}_{ez}^{21}&\widetilde{G}_{ez}^{22}-z_{iw}\end{bmatrix}\begin{bmatrix} \lx@overaccentset{{\raisebox{-1.0 pt}{\scalebox{1.0}{$\star$}}}}{I}_{1} \\ \lx@overaccentset{{\raisebox{-1.0 pt}{\scalebox{1.0}{$\star$}}}}{I}_{2} \end{bmatrix} = \begin{bmatrix}-\hat{V}_{1}-E_{ez}^{1}\\ -\hat{V}_{2}-E_{ez}^{2}\end{bmatrix}  \tag{9}\]  
where (given symmetry and reciprocity) \(\widetilde{G}_{ez}^{11}-z_{iw}=\widetilde{G}_{ez}^{22}-z_{iw}\) and \(\widetilde{G}_{ez}^{12}=\widetilde{G}_{ez}^{21}\). \(\lx@overaccentset{{\raisebox{-1.0 pt}{\scalebox{1.0}{$\star$}}}}{E}_{ez}\) and \(\lx@overaccentset{{\raisebox{-1.0 pt}{\scalebox{1.0}{$\star$}}}}{E}_{ez }^{2}\) are, respectively, the axial electric fields of the external source evaluated at the bottom surfaces of conductors 1 and 2. The square matrix in (9) is \(2\times 2\). It is known that if it has 2 distinct eigenvalues, then it has two distinct eigenvectors that are orthogonal with respect to it. It is also known that any two-element vector (e.g., \(\lx@overaccentset{{\raisebox{-1.0 pt}{\scalebox{1.0}{$\star$}}}}{I}\)) can be expanded in this set of eigenvectors so that  
\[ \begin{bmatrix}\lx@overaccentset{{\raisebox{-1.0 pt}{\scalebox{1.0}{$ \star$}}}}{I}\end{bmatrix} =[\eta] \begin{bmatrix}\lx@overaccentset{{ \raisebox{-1.0 pt}{\scalebox{1.0}{$\star$}}}}{I}_{gm}\end{bmatrix}  \tag{10}\]  
where \(\lx@overaccentset{{\raisebox{-1.0 pt}{\scalebox{1.0}{$\star$}}}}{ \tilde{I}}_{gm}\) is the matrix of the \"geometric component\" amplitudes (often referred to in the power engineering literature simply as \"mode\" amplitudes), and \(|\eta|\) is the square matrix (by columns) of normalized eigenvectors of the square matrix in (9).