# Channel Characterization and SC-FDM Modulation for PLC in High-Voltage Power Lines
## 2 HV-PLC Channel
### HF Signal Propagation Along Power Lines  
Power lines are multiconductor structures and a signal injected into one conductor will travel along all line conductors, including the ground plane [14; 20]. The way to analyze wave propagations along a power line is to first break it into homogeneous line segments, that is, into the segments between transpositions as in Figure 1. Then, the solutions of the homogeneous parts are cascaded according to the line layout and with the proper insertion of mathematical representations of the line discontinuities, in this case, of the transpositions [14; 20]. Consider a uniform power line segment. Voltages and currents along this are determined by the following line equations [20]:  
\[-\frac{d\,\mathbf{V}}{dx}=\mathbf{ZI} \tag{1}\]  
and  
\[-\frac{d\,\mathbf{I}}{dx}=\mathbf{YV}, \tag{2}\]  
where \(\mathbf{V}\) and \(\mathbf{I}\) are the respective vectors,or \(n\times 1\)-dimension column matrices, of voltages and of currents at the \(n\) line conductors. \(\mathbf{Z}\) and \(\mathbf{Y}\) are the respective \(n\times n\)-dimension matrices of line impedances and admittances in per unit of length. The general solutions of (1) and (2) are given as follows: [20]  
\[\mathbf{V}(x)=exp(-\mathbf{\Gamma}x)\mathbf{c}_{1}+exp(\mathbf{\Gamma}x) \mathbf{c}_{2} \tag{3}\]  
\[\mathbf{I}(x)=\mathbf{Y}_{c}\,exp(-\mathbf{\Gamma}x)\mathbf{c}_{1}-\mathbf{Y} _{c}\,exp(\mathbf{\Gamma}x)\mathbf{c}_{2} \tag{4}\]  
Figure 1: Diagram of an HV-PLC system. with  
\[\mathbf{\Gamma}=\sqrt{\mathbf{Z}\mathbf{Y}} \tag{5}\]  
and  
\[\mathbf{Y}_{c}=\mathbf{Z}^{-1}\mathbf{\Gamma} \tag{6}\]  
At expressions (3) and (4), \(\mathbf{c}_{1}\) and \(\mathbf{c}_{2}\) represent vectors of integration constants to be determined by the boundary conditions (i.e., terminal connections) of the line segment. As the boundary conditions of Figure 2 are introduced in (3) and (4), the following modal two-port representation is obtained for the homogeneous multiconductor line segment:  
\[ \begin{bmatrix}\mathbf{I}_{o}\\ \mathbf{I}_{l}\end{bmatrix} = \begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{B}&\mathbf{A}\end{bmatrix}\begin{bmatrix}\mathbf{V}_{o}\\ \mathbf{V}_{l}\end{bmatrix}  \tag{7}\]  
with  
\[\mathbf{A}=\mathbf{Y}_{c}\:coth(\Gamma l) \tag{8}\]  
\[\mathbf{B}=-\mathbf{Y}_{c}\:cosch(\mathbf{\Gamma}l) \tag{9}\]  
and \(l\) is the length of the line segment. The line discontinuities of most interest to analyze PLC-signal propagation are the line transpositions. Figure 3 provides an example of these for a three-conductor line. Clearly from this figure, currents and voltages at both ends of the transposition are related as follows by means of a rotation matrix \(\mathbf{R}\):  
\[\mathbf{V}^{\prime}=\mathbf{R}\mathbf{V} \tag{10}\]  
\[\mathbf{I}^{\prime}=\mathbf{R}\mathbf{I} \tag{11}\]  
with  
\[\mathbf{R}= \begin{bmatrix}0&1&0\\ 0&0&1\\ 1&0&0\end{bmatrix} ;\qquad\mathbf{V}= \begin{bmatrix}v_{1}\\ v_{2}\\ v_{3}\end{bmatrix} \]  
\[\mathbf{I}= \begin{bmatrix}i_{1}\\ i_{2}\\ i_{3}\end{bmatrix} ;\quad\mathbf{V}^{\prime}= \begin{bmatrix}v_{2}\\ v_{3}\\ v_{1}\end{bmatrix} ;\quad\mathbf{I}^{\prime}= \begin{bmatrix}i_{2}\\ i_{3}\\ i_{1}\end{bmatrix} \]  
The build-up of the nodal two-port representation of a non-uniform line, for the PLC analysis, proceeds by cascading each homogeneous section representation as in (7) with  
Figure 3: Representation of a line transposition. Figure 2: Representation of a n-conductor line segment of length \(``l\"\). its corresponding rotation matrix as in (10) and (11). The end result can be expressed as follows:  
\[ \begin{bmatrix}\mathbf{I}_{s_{1}}\\ \mathbf{I}_{s_{2}}\end{bmatrix} = \begin{bmatrix}\mathbf{A}_{1}&\mathbf{B}_{1}\\ \mathbf{B}_{2}&\mathbf{A}_{2}\end{bmatrix}\begin{bmatrix}\mathbf{V}_{s_{1}}\\ \mathbf{V}_{s_{2}}\end{bmatrix}  \tag{12}\]  
where \(\mathbf{I}_{s_{1}}\) and \(\mathbf{V}_{s_{1}}\) are the respective vectors of currents and of voltages at substation 1 (see Figure 1) and \(\mathbf{I}_{s_{2}}\) and \(\mathbf{V}_{s_{2}}\) are those vectors at substation 2.