# 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
In situations when it is appropriate to use group velocity and group delay concepts, the group velocity of one-dimensional pulses can be calculated from the dispersion equation \(v_{\text{gr}}=\text{d}\omega/\text{d}k\), where \(\omega\) is the angular frequency and \(k\) is the wave number in the line, group delay can be calculated as \(\tau_{\text{gr}}=\text{d}\phi(\omega)/\text{d}\omega\), where \(\phi(\omega)\) is the phase distortion in line transmission function. The pulse propagation in dispersive lines is still being researched. In many publications, such as [78], group velocity \(v_{\text{gr}}\) is the propagation velocity of the pulse envelope and \(v_{\text{gr}}\) and \(\tau_{\text{gr}}\) are bound by the following relationship:\[v_{\rm gr}=\frac{l}{\tau_{\rm gr}} \tag{18}\]  
where \(l\) is the transmission line length. Experimental studies [79] and simulations [80] show different approaches for measuring group delay. Following the results presented in [81], if pulses propagate in distances that are not greater than the minimum wavelength in the major part of the signal energy spectrum and the pulse shape features a distinct maximum amplitude, group velocity can be measured as the pulse waveform maximum (pulse peak) propagation speed. This assertion was validated by HFSS simulations and served as the basis of delay line calculations and design in this research. In this research, it was previously calculated that the energy spectrum region, which contains 98% of the pulse energy of a unipolar Gaussian pulse with a duration of 200 ps, is up to around 3 GHz. Using the MST software, it was calculated that at this \(f_{\rm max}\), the wavelength at the given FR-4 parameters (\(\epsilon_{r}=4.7\), \(\tan\delta=0.02\), \(h=1.6\) mm, \(W_{0}=2.90\) mm) \(\lambda_{\rm min}=c/(f_{\rm max}\cdot\epsilon_{\rm r\,eff})=52\) mm. Using the MST calculation, the effective dielectric permittivity of the microstrip line is \(\epsilon_{\rm r\,eff}=3.6\). Also, it was calculated that at the previously mentioned FR-4 parameters, the line impedance is \(Z_{0}=50\)\(\Omega\). This result was already used when calculating the final widths of the splitter port microstrip line segments \(W_{0}=2.90\) mm (see Table 2 and Figure 5). Before constructing the delay lines, the pulse summation results, as shown in Figure 4, were analyzed. Additionally, calculations were performed to assess the output pulse sensitivity to variations in the time shift between pulses and pulse durations. In practical implementation, physical elements and connectors can introduce extra attenuation and propagation delay. Therefore, pulses at the output of the splitter and delay lines exhibit slightly longer durations and different amplitudes compared to the initial calculations, shown in Figure 4. Considering this factor, it was decided to implement delay lines where the time interval between pulse peaks was 150 ps. Approximate calculations have shown that the maximum length of the delay lines will be 150 mm, which is approximately three times larger than \(\lambda_{\rm min}\). As a result of our calculations performed in MST software, where it was required that the time shift between the pulses be 150-200 ps, the approximate initial lengths of the three microstrip lines were found. HFSS simulations show that group delays in lines ranging in length from 30 to 100 mm are approximately \(\tau_{\rm gr}=6.4\) ps/mm. As can be seen in Figure 2, implementing straight delay lines is not possible in this case, and the most rational implementation is where Line 4, with impedance \(Z_{0}\), is straight, but Line 3 and Line 2 are curved. When creating the curved lines, it was decided to use smoothly curved lines to minimize radiation from the bends of the lines. The line widths at both ends match the port widths \(W_{0}\) of the splitter and combiner.