# Timing Analyses in FWE Evaluation
## 2 Problem Statement and Issues
### Implementation of Circuits with Two Coupled Lines
The resulting pattern typically resembles an \"eye,\" hence the name. The horizontal axis represents time, while the vertical axis represents signal amplitude. Key aspects of eye diagrams include opening width, rise and fall times, jitter, and noise. The width of the \"eye\" opening indicates signal timing and the potential for timing errors. Rise and fall times, often measured at the edges of the eye, represent the speed at which the signal transitions between its high and low states. Deviations from the ideal eye shape indicate the presence of jitter (timing variations) and noise, both of which can affect signal reliability. The clearer and wider the eye opening, the better the signal quality. Engineers use eye diagrams to quickly assess and troubleshoot signal integrity issues, making them a valuable tool in optimizing communication systems, especially those operating at high data rates. As shown in Figure 9, the simulations were performed for two values of line lengths, more precisely, for \(L\) = 60 mm and \(L\) = 250 mm, respectively. Also, we used three data rates, namely, {1 GHz, 10 GHz, 30 GHz}. In the eye diagrams, shown in Figure 9, blue is the result obtained for the circuit with substrate FR4 and pink is the result corresponding to the circuit with substrate N4000-13. Analyzing the results, we noticed how the eye closes as the line length increased and as the rate increased. At low rates and small lengths of the transmission line, the eye is open. We noticed that for a length of 250 mm and a rate of 30 GHz, the eye diagram associated with the FR4 substrate circuit was completely distorted. Figure 8: (**a**) Eye diagram for two coupled microstrip lines with L = 60 mm and Rate = 1 GHz. (**b**) Eye diagram for two coupled microstrip lines with L = 250 mm and Rate = 1 GHz. (**c**) Eye diagram for two coupled microstrip lines with L = 60 mm and Rate = 10 GHz. (**d**) Eye diagram for two coupled microstrip lines with L = 250 mm and Rate = 10 GHz. (**e**) Eye diagram for two coupled microstrip lines with L = 60 mm and Rate = 30 GHz. (**f**) Eye diagram for two coupled microstrip lines with L = 250 mm and Rate = 30 GHz. Based on the eye diagram representations, we extracted details about the rise and fall time and jitter. We simulated these three parameters on separate diagrams for the two substrates used. The simulations were performed for a length of 250 mm at two frequency rates: 1 GHz and 10 GHz. Figure 10a,b show the results for the FR4 substrate and Figure 10c,d show the results for the N4000-13 substrate. In all four figures, the rising time is marked with light blue, the falling time is marked with green, and the corners of the eye are marked with orange. At a rate of 1 GHz, we noticed that we have a single line for the rise time and one for the fall time, which suggests that the transition is made through a single point, leading to the minimum value achieved for jitter. With the increase of the frequency rate to 10 GHz, we noticed that the transitions were made through several points, both for the rise time and for the fall time, with more green and light blue lines appearing. Figure 9: (**a**) Rise time, fall time, and corners on the eye diagram for two coupled microstrip lines and FR4 substrate for L = 250 mm and Rate = 1 GHz. (**b**) Rise time, fall time, and corners on the eye diagram for two coupled microstrip lines and FR4 substrate for L = 250 mm and Rate = 10 GHz. (**c**) Rise time, fall time, and corners on the eye diagram for two coupled microstrip lines and FR4 substrate for L = 250 mm and Rate = 1 GHz. (**d**) Rise time, fall time, and corners on the eye diagram for two coupled microstrip lines and FR4 substrate for L = 250 mm and Rate = 10 GHz. Figure 10: (**a**) Eye diagram for two coupled lines, stripline technology for L = 60 mm and Rate = 1 GHz. (**b**) Eye diagram for two coupled lines, stripline technology for L = 250 mm and Rate = 1 GHz. (**c**) Eye diagram for two coupled lines, stripline technology for L = 60 mm and Rate = 10 GHz.