# 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  
As mentioned previously, to create a pulse expander, the plan is to connect three different delay lines to the output ports of the splitter. These delay lines produce a time delay in the pulses, which are then summed using the combiner. At the output of the combiner, an expanded pulse is observed. These delay lines, like the pulse splitter and combiner, are created using microstrip technology on the same substrate. The complete expander can be seen in Figure 2. The delay lines are labeled Line 2, Line 3, and Line 4. These lines connect the splitter and combiner ports 2, 3, and 4, respectively. Designing the delay lines involves multiple steps--preliminary calculations, simulations, experiments, and optimization by performing simulations and experiments for validation. Preliminary calculations in Figure 4 show that a smooth and wide pulse in the expander output can be achieved by delaying a Gaussian pulse with a duration of 200 to 230 ps with a time shift between the pulses of 150 to 200 ps. The first significant challenge in designing the microstrip delay line is the frequency dependence of line parameters. Even a homogeneous microstrip line made on a dielectric substrate is a dispersive line because dielectrics are dispersive materials, and in these lines, the pulse propagation delay is a function of frequency. The frequency dependence of different dielectrics has been extensively researched, with many models of complex dielectric permittivity as a function of frequency being proposed [66]. The simplest yet most popular model is the one-pole Debye relaxation model, which has been used for many years [66; 67]. This model is widely used in permittivity measurement [67; 68] and is updated to this day [69; 70]. In microstrip line parameter calculations, generally, the effective dielectric permittivity \(\varepsilon_{\text{\,ref}}\) concept is used. Currently, \(\varepsilon_{\text{\,ref}}\) and other microstrip line parameters can be approximately calculated using equations given in [71], or software, which is based on equations given in [72; 73; 74; 75]. In this paper, custom-made software (called \"MST\") is used. MST V1.0 calculates \(\varepsilon_{\text{\,ref}}\), homogeneous line impedance \(Z_{0}\), and also other parameters, considering the dielectric frequency dependence. This software is built based on publications [72; 73; 74; 75; 76]. Pulse wavefront propagation speed calculation is the second major challenge when researching and modeling pulse propagation in dispersive media. Usually, in such situations, group velocity in line \(v_{\text{gr}}\) is calculated [30], but in engineering applications, the concept of group delay \(\tau_{\text{gr}}\) (group velocity is group delay per unit length) is more widespread [77]. Group velocity, in many cases, also determines the energy transfer speed. However, it is not the focus of this paper because only the time shift between the pulses is essential for this work. Group delay \(\tau_{\text{gr}}\) cannot be used universally, theoretically, or in real systems because it is an approximate variable dependent on frequency in dispersive systems and media. The term group delay can be used in cases when the pulse waveform, while propagating in a dispersive line, is not substantially distorted because it is considered that the group velocity of the pulse is the velocity with which the pulse waveform propagates through the line. Therefore, this concept is valid in dispersive lines when the pulse bandwidth is not very high, and the propagation distance is not long enough for the pulse waveform to be significantly distorted.