# A Novel UWB Pulse Expander Using an Integrated Microstrip Splitter, Delay Lines, and a Combiner
## 2 Design and Modeling
### Design of Delay Lines and Assembly of the Pulse Expander
These requirements can be fulfilled by using delay line construction, which is described by the equation  
\[\overrightarrow{r}(u,v)=-u-u_{0}\cdot e^{-k\cdot L_{\rm delay}\cdot v^{2}} \tag{19}\]  
where by varying \(u_{0}\), \(k\), and the ranges of \(u\) and \(v\), it is possible to obtain the desired placement, curvature, and length of the delay line. For both curved delay lines--Line 2 and Line 3--\(v\) is in the range of \(\pm 0.5L_{\rm delay}\), and to achieve the desired overall width of the expander, \(k\) is set to 100,000. Such delay lines have microstrip lines with varying widths, and they become narrower toward the inflection point and increase in width toward the extreme point (see Figure 2). Calculations with MST show that the decrease in line width increases the group velocity. To implement delay lines of sufficient length, and so that these lines do not take up too much space in the transverse direction, it was calculated that Line 4 length should be \(l_{4}=50\) mm. If we denote the time interval between pulses at the output of straight delay lines--Line 3 and Line 4 as \(\Delta t_{34}\), and Line 2 and Line 3 as \(\Delta t_{23}\)--and we set a requirement that \(\Delta t_{34}=\Delta t_{23}=150\) ps, then considering the previously calculated \(\tau_{\rm gr}=6.4\) ps/mm, the length of Line 3 is \(l_{3}=73.44\) mm and the length of Line 2 is \(l_{2}=96.88\) mm. We performed simulations in which straight delay lines with constant width and curved lines with the same lengths \(l_{4}\), \(l_{3}\), and \(l_{2}\) were used. Corresponding results can be seen in Figure 9. At the input of all delay lines, the same pulse with a duration of 200 ps was used. As we can see, with curved delay lines, the time intervals \(\Delta t_{34}\) and \(\Delta t_{23}\) are noticeably smaller, indicating that the curved lines delay the pulses less than if the delay lines were straight and of the same width. Since the pulses at the output of the delay lines differ in amplitude, and the waveform is not the same, simulations of the pulse expander (see Figure 2) were performed, during which, the lengths of delay lines 3 and 4 were changed. In these simulations, the pulse waveform at the expander's output was examined. A satisfactory option was found when the length of the straight delay line was \(l_{4}=50\) mm and the lengths of the curved delay lines were \(l_{3}=78.3\) mm and \(l_{2}=102.10\) mm, respectively. The results of this simulation are shown in Figure 10.  
Figure 11 shows the pulses at the outputs of the delay lines when the input to the pulse splitter is a 200 ps duration pulse with an amplitude of 1 V.  
Mutual coupling between designed delay lines was investigated in simulations, and no significant coupling was detected. Using the previously calculated splitter and combiner sizes (see Table 2) and the above-mentioned curved lines, the UWB pulse expander model was created, as shown in Figure 2.  
Figure 10: Pulses at the expander’s input (black) and output (green). Simulation results. Figure 9: Input pulse (solid black line) in each delay line (lines 2, 3, 4), and output pulses for straight delay lines (dashed lines) and curved delay lines (solid lines). Simulation results.