# Timing Analyses in FWE Evaluation
## 2 Problem Statement and Issues  
A transmission line is a structure consisting of two or more conductors that connect to a signal generator and along which an electromagnetic wave propagates. For this study, we will consider the model in which two parallel conductors, which are separated by a layer of dielectric material, are used to interconnect a transmitter with a receiver. Their role is to ensure the transmission of undistorted information between the system components. Next, the transmission line structure using per-unit-length parameters is presented in Figure 2.  
where:  
* \(R_{L}\) [\(\Omega/m\)] is the line resistance (total resistance per unit length of the two conductors) and represents the losses in the conductor that increase with the frequency;
* \(L_{L}\) [\(H/m\)] is the line inductance (inductance of the two-conductor system per unit length of the line);
* \(C_{L}\) [\(F/m\)] is the line capacitance (insulator capacitance between line conductors per unit length);  
Figure 2: Transmission line with per-unit-length parameters. \(G_{L}\left[S/m\right]\) is the line conductance (the conductance of the two-conductor system per unit length of the line) and represents the losses in the dielectric that increase with the frequency. FWE hinders circuit performance, particularly at speeds exceeding 3 Gbps, due to variations in the dielectric constant. This leads to differential distortion in signal lines, increasing bit error rates, especially for long high-speed transmissions. As speed, distance, and discontinuities rise, significant signal losses occur. This signal loss due to dielectric loss at higher frequencies is observed in [26]. As transmission speeds increase, multiple factors affect signal integrity. Figure 3 shows varying fiber optic styles, from loose to tight waves. An efficient evaluation of the signal integrity can be performed based on an examination of circuit performances at different frequencies, which can be achieved based on the evaluation of the scattering parameters and by time analysis of the eye diagram. For short distances and low data rates, signals travel reliably from transmitter to receiver. However, longer distances and higher speeds lead to significant signal degradation. Signal integrity assesses the quality of electric signals during transmission. PCB design depends on substrate characteristics, and a key limitation is the non-uniform distribution of resin and fibers in the substrate material. When using composite materials with an inhomogeneous structure and equal-length differential lines, differences in signal propagation times create skew. This skew results in signal asymmetry, potentially shifting from differential to common mode, leading to eye diagram closure, jitter, and distortions in rise and fall times. This phenomenon is known as the Fiber Weave Effect (FWE) [23]. The difference between the propagation times of the two differential signals can be determined using the relationship [24]:  
\[t_{skew}=l.\frac{\sqrt{\varepsilon_{r}}}{c}, \tag{1}\]  
where:  
* \(t_{skew}\) is the difference between the propagation times;
* \(l\) is the length of the transmission line or of the coupled lines in the case of differential lines;
* \(\varepsilon_{r}\) is the relative permittivity of the dielectric;  
Figure 3: Fiber weave styles. Loose weaves (**left**) create greater skew and impedance variations in a board compared to a tight weave (**right**) [15]. - \(\epsilon_{rMax}\) is the maximum relative permittivity of the dielectric;
- \(\epsilon_{rMin}\) is the minimum relative permittivity of the dielectric;
- \(l\) is the length of the transmission line or the coupled lines in the case of differential lines;
- \(c\) is the speed of light (\(c=3\cdot 10^{8}\ m/s\)). To be able to evaluate the losses in a circuit, we need the distribution parameters, also known as the \(S\) parameters. There are two types of \(S\) parameters: \(S_{ii}\) are parameters that characterize the reflection coefficient and \(S_{ij}\) are parameters that characterize the transfer coefficient.