# A Novel UWB Pulse Expander Using an Integrated Microstrip Splitter, Delay Lines, and a Combiner
## 2 Design and Modeling
### Pulse Splitter Selection and Design
Initially, the dimensions of the splitter transmission lines were chosen to ensure the necessary power splitter S-parameter ratio (9) assuming isolation port resistor values of \(R_{5,6}=50\)\(\Omega\). The chosen \(f_{\text{op}}\) was larger than the input pulse essential frequency range because the goal is that--at any frequency in the essential frequency range (0 to 2.5 GHz)--the reflection coefficient of the pulse splitter must be as similar as possible and not significantly smaller than at any other frequency, which will ensure there is little distortion of the input pulse waveform at the input of the pulse splitter. The initial values of the splitter construction (see Figure 5) are listed in Table 2. These values were chosen, considering that the design will be implemented on the FR-4 substrate with the following parameters: dielectric constant \(\epsilon_{\text{r}}=4.7\), loss tangent \(\tan\delta=0.02\), substrate thickness \(h=1.6\) mm, and copper laminate thickness \(H=0.035\) mm. During the modeling, microstrip line segment parameters (width \(w_{i}\) and distances \(L_{1}\) and \(L_{2}\)) were adjusted so that requirements (1), (2), (3), and (4) were met. Multiple research papers state that when constructing similar branch-line dividers, optimal \(R_{\text{iso}}\) values can be found [61; 62]. In many implementations of the branch-line divider, optimal \(R_{\text{iso}}\) values are not the same as found in the initial calculations. For example, patent [63] states that the terminating impedances were initially calculated to be 46 \(\Omega\) but were changed to 30 \(\Omega\) due to the optimization process. In this research, multiple isolation resistor values \(R_{\text{iso}}=35,50,70,100\), and 120 \(\Omega\) were tested. In the design and simulations performed in this research, minimal coupling between output ports 2, 3, and 4 was not required, as in the case of an ideal power divider (15). Simulations show that previously described optimization requirements (1) to (4) are met even better if \(R_{\text{iso}}\) is increased and the best results are achieved at \(R_{\text{iso}}=1\) M\(\Omega\). These results suggest that the most optimal pulse splitter construction is achieved when isolation resistors \(R_{\text{iso}}\) are absent. Figure 6 shows the simulated S-parameter magnitude \(|S_{21}|\), \(|S_{31}|\), \(|S_{41}|\), and \(S_{21}\), \(S_{31}\), \(S_{41}\) phase curves. It can be seen that requirement (1), set for determining appropriate geometric sizes of the splitter elements, is fulfilled when the frequency is less than 2 GHz. The S-parameter \(S_{21}\) and \(S_{41}\) curves match because ports 2 and 4 are symmetrical. For ease of comparison, Figure 6 shows cases where \(R_{\text{iso}}=100\)\(\Omega\) and \(R_{\text{iso}}=1\) M\(\Omega\). It can be seen that requirements (1) and (2) are met more closely when \(R_{\text{iso}}=1\) M\(\Omega\); also, the reflection coefficient \(|S_{11}|\) is much smaller in this case. In the design used for the practical implementation of the pulse splitter (see Figure 5), when isolation resistors in ports 5 and 6 were not connected (\(R_{\text{iso}5,6}=\infty\)), the lengths of the microstrip line segments in ports 5 and 6 \(L_{3}\) were adjusted. However, the results are close to the simulated results: \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline
**Parameter** & \(W_{0}\) & \(W_{1}\) & \(W_{2}\) & \(W_{3}\) & \(W_{4}\) & \(L_{1}\) & \(L_{2}\) & \(L_{3}\) \\ \hline
**Initial value, mm** & 2.94 & 2.40 & 2.00 & 1.20 & 0.60 & 11.0 & 10.0 & — \\ \hline
**Final value, mm** & 2.90 & 1.17 & 1.17 & 0.94 & 0.38 & 9.45 & 9.24 & 2.69 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometrical parameters of a coupled branch-line 3-way pulse splitter. Table 3 shows the calculated pulse energies at \(R_{\rm iso}=100\)\(\Omega\) and \(R_{\rm iso}=1\)\(\rm M\Omega\).