# Timing Analyses in FWE Evaluation
## 2 Problem Statement and Issues
### Implementation of Circuits with Two Coupled Lines  
In this subchapter, we will present the model and the results obtained based on the simulation of a two-coupled line implemented in microstrip and stripline technologies, using FR4 and N4000-13 as s. Figure 4 presents the model of the coupled lines implemented, made in microstrip technology and FR4 substrate. The dielectric constant and loss tangent of the dielectric for the composite substrate FR4 are \(\varepsilon_{r}=4.7\) and \(tg\delta=0.02\). To obtain proper matching conditions at the ports, we determined the even and odd mode impedances and, based on these, we obtained the values of common and differential mode impedances, as presented in [1]. Using the Smith charts, presented in Figure 5a (for microstrip technology) and Figure 5b (for stripline technology), we checked if the circuits were matched by representing the \(S_{ii}\) reflection parameters. Matching is obtained when the parameters are in the middle of the Smith diagram, as shown in Figure 5, corresponding to progressive waves at the ports. Next, we implemented the same circuit and simulations for the second material. We noticed that, in Figure 6, in this case, the values for the common and differential impedances were different; thus, based on the Smith chart evaluation of the reflection coefficients, we observed that the circuit is not matched at the ports. Therefore, we optimized the values of these impedances so that the circuits were matched, as can be seen in Figure 7.  
To verify the performance of the circuit, an analysis in the time domain is performed. To do this, a pseudo-random signal source is connected at the input port, both for the circuit designed in microstrip technology and the one designed in stripline technology. Figure 8 represents a circuit implemented in microstrip technology with an FR4 substrate and pseudo-random signal source at port 1. This is a pseudo-random bit sequence with a parametrized rate, number of symbols and samples per symbol, rise and fall times, and window type. To highlight the effect of increased rate, the parameter RATE is varied between 1 and 30 GHz and the influence of sharp edges of the signals and transitions are specified by TR and TF rise and fall time. These are important for critical applications and the results are also influenced by the window type selection. TR and TF are set to 10 ps. Figure 4: FR4 substrate microstrip technology coupled lines. Figure 5: (**a**). Plotting the \(S_{ii}\) parameters on the Smith chart after matching the circuit containing two coupled microstrip lines on a uniform FR4 substrate. (**b**) Plotting the \(S_{ii}\) parameters on the Smith chart after matching the circuit containing two coupled striplines on a uniform FR4 substrate. Figure 6: (**a**). Plotting the \(S_{ii}\) parameters on the Smith chart after matching the circuit containing two coupled microstrip lines on a uniform N4000-13 substrate. (**b**) Plotting the \(S_{ii}\) parameters on the Smith chart after matching the circuit containing two coupled striplines on a uniform N4000-13 substrate. Figure 8: _Cont_. Figure 7: FR4 substrate microstrip coupled lines with a pseudo-random source. We modified all the other circuits in the same way and, based on the results obtained, we performed an analysis in the time domain using the eye diagram. First, we evaluated the voltage at port 3, and considered in this case the differential output of the circuits designed in microstrip technology for both types of substrates. Simulations for different frequency rates and line lengths were performed because we wanted to observe the behavior of the circuits as the line length increased, and also as the rate increased. The eye diagrams are graphical representations used in signal processing and telecommunications to assess the quality and integrity of digital signals. They are particularly valuable in high-speed communication systems. An eye diagram is formed by overlaying multiple signal periods on top of each other, creating a visual representation of signal quality.