# A Novel UWB Pulse Expander Using an Integrated Microstrip Splitter, Delay Lines, and a Combiner
## 2 Design and Modeling
### Pulse Splitter Selection and Design
\[I_{5,6}=10\lg(P_{1}/P_{5})=10\lg(P_{1}/P_{6})\, \tag{13}\]  
where \(P_{1}\) is the input power in port 1, \(P_{5}\) and \(P_{6}\) represent the power delivered to their respective ports. Another kind of isolation can also be defined between any two output ports. In this case, one of the output ports is used as the input. The other is considered the output port, while all other ports are terminated by matched loads. Complete isolation is possible only theoretically; we have the following:  
Figure 3:0  
There must also be perfect isolation between any two output ports, as follows:  
Figure 3:1  
Unfortunately, theoretical requirements (9), (11), (12), (14), and (15) can be fulfilled only at one frequency \(f_{\rm op}\). In practice, the reflection from the ports when all other ports are matched must be as low as possible, and the isolation between ports must be as high as possible. However, the isolation between the input and isolation ports may differ from the isolation between the output ports. In actual couplers, the isolated port is never completely isolated. The design process of the pulse splitter is ambiguous. Therefore, the main consideration is that the complex parameters are as similar as possible so that we have the following:  
Figure 3:2  
which means that the following conditions, i.e.,  
\[ \begin{split}|S_{21}|=|S_{31}|=|S_{41}|\ ;\\ \arg(S_{21})=\arg(S_{31})=\arg(S_{41})\end{split}  \tag{17}\]  
must be met (at least approximately). Previous calculations have already shown that the pulse splitter cannot fulfill the requirements \(|S_{11}|=|S_{51}|=|S_{61}|=0\) along with isolation between any two output ports over a wide frequency range. Therefore, the optimal design ensures similar pulses in the output ports and power losses in ports 1, 5, and 6 are as low as possible. In several simple optimization steps, the geometric sizes of the splitter were adjusted. The following requirements were set for the first step in the optimization, as follows:  
1. The absolute difference in the magnitudes of the S-parameters, \(|S_{21}|\), \(|S_{31}|\), \(|S_{41}|\), with numerical values \(\Delta|S_{i}|<0.05\) at a frequency of 2.0 GHz and \(\Delta|S_{i}|<0.1\) at a frequency of 2.5 GHz;
2. The phase difference of S-parameters \(S_{21}\), \(S_{31}\) and \(S_{41}\) within the frequency range of 0-2.5 GHz \(\Delta|\arg(S_{i})|<3^{\circ}\). If these basic requirements were fulfilled, then the following additional energy requirements were checked (second step in the optimization):
3. The pulse total energy in ports 2, 3, and 4 is \(\sum_{i=2,3,4}E_{\rm exi}\geq 0.7\cdot E_{\rm in}\), where \(E_{\rm in}\) is the input pulse energy;
4. The relative difference in pulse energies in ports 2, 3, 4 \((\max E_{\rm ex2,3,4}-\min E_{\rm ex2,3,4})/\max E_{\rm ex2,3,4}<0.1\). The following optimization step was performed if requirements (1) and (2) were fulfilled but not requirements (3) and (4). Pulse waveform differences in the ports were evaluated visually during the transient simulations because requirements (1) and (2) already state that waveforms will be approximately identical. The input pulse of the splitter is a Gaussian pulse with the duration of \(\tau_{1/2}=200\) ps, the waveform of which is described in Formula (4). This input pulse is split into three pulses, which must have as similar a waveform as possible. The spectral density of the input pulse is described in Formula (6) and is shown in Figure 3. As can be seen, most of the pulse energy is concentrated in the low-frequency region. Calculations show that 90% of the pulse energy is concentrated in the frequency range below 2.17 GHz, and 98% is concentrated below 3.09 GHz. This distribution suggests that it is sufficient to calculate the proper parameters of the pulse splitter for the frequency range up to around 2.5 GHz. In order to design a pulse splitter, a design center frequency \(f_{\text{op}}=3.95\) GHz was chosen.