Field-to-Wire Coupling Analysis of Arbitrarily Configured Wires Using the BLT Equation and Diakoptics Techniques

Modifications to the extended Baum-Liu-Tesche Equation (BLT; F. M. Tesche and C. M. Butler, Interaction Notes, Note 588) are made in order to determine the field to wire coupling of a transient electromagnetic wave incident to a cylindrical wire. This technique implements the transmission line method (TLM) in order to determine the voltage/current across the loads at the wire terminals. The Lorentz Reciprocity Theorem is used to describe the source/coupling matrix resulting in the current and voltage waves traveling along the line. This work looks to further test the robustness of the BLT equation to provide fast and accurate coupling analysis of complex arbitrarily configured wires. In doing so, further changes have been made to the coupling matrix, where the electric field vector E is expressed as a 3 × 1 column vector. The complex shaped wire is analyzed using a diakoptics method that results in the wire being split into vector segments, resulting in the 3 × N matrix D. The electric field vector is transposed <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(E^{T})$</tex> making it a 1 × 3 matrix (row vector), thus making it suitable to solve for the dot product <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(E^{T}\cdot \mathbf{D})$</tex>. This dot product results in the 1 × N scalar matrix V, which contains the voltage across each segment. The current scalar matrix I is then calculated by multiplying the N × 1 admittance scalar matrix Y with the V scalar matrix. The dimensions of I are N×N, however the correct current of each segment is within the main diagonal of matrix I <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(I_{11}-I_{nn})$</tex>. The forward and reverse current vectors of the segments are each summed at the appropriate terminals of the line, using their propagation functions. This results in a 2 × 1 matrix, which is used as the source matrix for the BLT equation. This modification is applied to the example of a single transmission line above a ground plane, where they are connected by two loads on either end of the line. A plane wave at various angles are solved for, where the reflected electric field vector is calculated using image theory by placing an image source below the ground plane. The incident and reflected E-fields <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$E^{inc}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$E^{ref}$</tex> respectively, are stored in a single 3 × 2 matrix. If the above methodology is used, the voltage scalar matrix is then a 2 × N matrix. This matrix is reduced to a 1 × N matrix by taking the sum of the row elements. In order to determine the accuracy of this technique, multiple examples are simulated using CST's Cable Studio. Along with this, a ground plane has been fabricated and each wire configuration in conjunction with the ground plane is tested inside an anechoic chamber. The testing in the chamber is conducted by taking the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$S_{21}$</tex> measurement, allowing for the gain pattern of the transmission line to be measured relative to that of an antenna with a known gain pattern, thus determining the voltage at the antenna's terminal. Both azimuth and elevations measurements were taken. Analysis of this technique is conducted through the comparisons of measurements and simulations. The results of these tests allow for electromagnetic wave analysis to be performed on electronic devices, to protect them from unintentional and intentional electromagnetic interference.