# Compact and Broadband Microstrip Band-Stop Filters with Single Rectangular Stubs
## 5 Conclusions  
In this study, we examined compact and broadband microstrip band-stop filters with single rectangular stubs. First, in order to confirm the principle of broadening the bandwidth, we focused on the characteristic impedance ratio between the stub line and the main transmission line. The change in bandwidth in rectangular stubs was examined, and we confirmed that the bandwidth widened as \(Z_{1}/Z_{0}\) became smaller. However, it was found that the transmission zero frequency shifts, due to the change in the propagation constant of the stub line, and this tendency appeared as a remarkable change in the case where \(Z_{1}/Z_{0}\) is smaller. Also, it was found that even if the impedance ratio is the same, if the width of the main transmission line was narrowed, the change of the transmission zero frequency could be suppressed to be smaller.  
Next, in order to confirm the principle of frequency adjustment and miniaturization, the case of inductance element \(L\) using a stepped impedance structure, which plays a role of lowering the transmission zero frequency, was examined, and was connected in a series to the attachment point of the stub. From this study, it was found that the transmission zero frequency decreases as \(L\) increases. However, as \(L\) became larger, the \(Q\) became larger and the bandwidth narrowed, so it has turned out that there is a trade-off relationship between miniaturization and broadband bandwidth.  
In addition, when comparing these results with the band-stop characteristics of the well-known radial stub, they were found to agree well with the circuit pattern with appropriate \(L\) inserted in the low impedance stub. Then it was confirmed that the operating principle of the radial line stub is equivalent to the circuit pattern in which the reactance was inserted into the low impedance stub. Therefore, the validity of increasing the stop bandwidth as much as possible for a single rectangular stub with frequency adjustment, by inserting a reactance element at the attachment point of the stub, was confirmed. Although not described in this report, it is possible to obtain a broader band  
Figure 10: Measured \(|S_{21}|\) and \(|S_{11}|\) results of rectangular stubs of \(Z_{1}/Z_{0}=1/4\) with a series inductance \(g=7\) mm, as shown in Figure 5d, compared with two radial stubs.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Circuit Pattern** & \(\alpha\) **[Degree]** & **\(-\)10 dB BW [GHz] (\({}^{1}\))** & \(f_{0}\) **[GHz] (\({}^{1}\))** \\ \hline (f) & \(\alpha=60^{\circ}\) & 1.20 (2.0) & 2.61 (1.0) \\ (f) & \(\alpha=90^{\circ}\) & 1.45 (2.4) & 2.81 (1.1) \\ (d) \(g=7\) & \(Z_{1}/Z_{0}=1/4\) & 2.25 (3.7) & 2.56 (1.0) \\ \hline \hline \end{tabular}  
\({}^{1}\) Indicate the magnification for \(Z_{1}/Z_{0}=1\).  
\end{table}
Table 3: Stop bandwidth in the case of radial stubs.  
characteristic than the radial line stub by using a stub with a smaller impedance ratio. In the future, based on the results of this study, we plan to investigate the broadband and miniaturization of the E-plane short-circuited stub, like the choke structure inserted in the waveguide.  
Conceptualization, methodology, writing (review and editing), investigation, validation, visualization, supervision, and project administration were contributed to by Y.K.; data curation, formal analysis, writing (original draft preparation), and validation contributed by R.I.  
This research received no external funding.  
The authors declare no conflict of interest.