# Effect of Fiber Wave Structure in Printed Circuit Boards on Signal Transmission Characteristics
## 3 Results and Discussion
### Impact of the Fiber Weave Effect on Impedance Fluctuations
#### 3.2.2 Effect of the Fiber Weave Effect on Differential Lines  
Differential lines are shown in Figure 7 and were made up of two halves of differential pairs (designated \"D+\" and \"D\(-\)\"). They ran over different media (resin vs fiber weave) (see correspondingly different Dk values) and had different propagation properties, especially velocity. It can be concluded that the signal propagation velocities of the two halves of the differential pairs in Figure 7 were different, which resulted in impedance fluctuations of differential lines. We can also call these impedance fluctuations differential skew.  
Figure 5: The Dk fluctuation amplitude parallel to weft direction.  
Figure 6: Effect of fiber weave style on impedance fluctuation.  
The calculation method of differential skew is shown in Equation (1) [8]. It is the two lines' propagation delay (_tpd_1 is the propagation delay of one line of the differential pairs, and _tpd_2 is another line):  
\[\text{Skew}=|tpd1-tpd2|. \tag{1}\]  
For a microstrip differential line, the propagation delay (_tdp_) can be calculated using Equation (2) [2]:  
\[tpd=85*(0.475\text{Dk}+0.67)1/2. \tag{2}\]  
In Equation (2), Dk is the medium dielectric permittivity, and _tpd_ is strongly related to Dk.  
Because the laminate was not a homogeneous medium, when a transmission line walked in a different place, the effective dielectric permittivity of the medium was different. The medium dielectric permittivity (Dk) for a specific transmission line (see Figure 8) could be calculated using Equation (3), where \(\epsilon_{\text{G}}\) is the dielectric permittivity of the fiber weave, Dk is the dielectric permittivity of the resin, and H is the total thickness of the fiber weave and resin:  
\[\begin{split}\varepsilon=\{0.5^{*}\epsilon_{\text{G}}{}^{*} \text{W}^{*}(\text{H}1+\text{H}2)^{*}0.6+\text{Dk}{}^{*}\text{W}^{*}[\text{H} -0.3^{*}(\text{H}1+\text{H}2)]\}/\text{W}/\text{H}\\ =\{0.3^{*}\epsilon_{\text{G}}{}^{*}(\text{H}1+\text{H}2)+\text{Dk} {}^{*}[\text{H}-0.3^{*}(\text{H}1+\text{H}2)]\}/\text{H}.\end{split} \tag{3}\]  
According to Equations (1)-(3), we could calculate the maximal differential skew for different styles of single fiber weaves with various differential transmission lines (see Table 2). As can be seen from the table, for some designed transmission lines, the line was big enough to have a fatal influence on the high-speed signal transmission. In addition, skew values were influenced by the designed line's width and space (W/S). It was found that theoretical line values were close to zero in special situations. This happened when (W+S) equal to X3 or Y3. This is attributed to the fact that the two lines of a differential pair had the same relative location to fiber weave bundles under this situation, and the effective Dk of the two lines was the same. Since it was difficult for designers to obtain detail data of the fiber weave and other functional design requirements, this method would be difficult to realize.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} _W/S_ \\ **(mil)** \\ \end{tabular} } & \multicolumn{8}{c}{**Differential Skew (ps/in)**} \\ \cline{2-11}  & \multicolumn{2}{c}{**1035**} & \multicolumn{2}{c}{**1080**} & \multicolumn{2}{c}{**3313**} & \multicolumn{2}{c}{**2116**} & \multicolumn{2}{c}{**1078**} \\ \cline{2-11}  & **Wett** & **Warp** & **Wett** & **Warp** & **Wett** & **Warp** & **Wett** & **Warp** & **Wett** & **Warp** \\ \hline \(4/4\) & 4.7 & 3.5 & 7.2 & 5.1 & 6.3 & 3.1 & 6.4 & 4.6 & 2.8 & 1.9 \\ \(4/8\) & 0.9 & 0.4 & 6.5 & 5.1 & 6.5 & 1.3 & 2.6 & 3.6 & 2.2 & 1.6 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Maximal differential skew for different styles of fiber weave.  
Figure 8: Photos to illustrate the Dk calculation.  
Figure 7: Photos to illustrate differential lines.