# Compact and Broadband Microstrip Band-Stop Filters with Single Rectangular Stubs
## 2 Theoretical Calculation
### Principle of Miniaturization by Reactance Insertion Stub  
Figure 3 shows a circuit model of the open circuited stub with a stepped impedance structure. In the schematic view in Figure 3a, a reactance element, which has different line width to the stub line width, is connected to the stub attachment point in series connection. Figure 3b shows the TEM  
Figure 1: (a) Schematic view of a microstrip line (MSL) open-circuited stub. (b) Transmission line equivalent circuit of MSL open-circuited stub, and its input admittance when \(l_{1}<\lambda/4\).  
Figure 2: Calculated results of transmission coefficients \(\left|\,S_{21}\,\right|\) and reflection coefficients \(\left|\,S_{11}\,\right|\) for different characteristic impedance ratios between the stub line and the main transmission line (\(Z_{1}/Z_{0}\)). Three patterns show \(Z_{1}/Z_{0}\) = 1/2, 1, and 2. The stop bandwidth widens if \(Z_{1}/Z_{0}\) is smaller. Smaller impedance ratio means that the stub line width \(W_{1}\) becomes wider.  
transmission line model of the open circuited stub with a series inductance. The left side shows the equivalent circuit of the open circuited stub with series inductance attached parallel to the I/O transmission line, and the right side shows the equivalent lumped element circuit model of the open circuited stub, with a series inductance for \(l_{1}<\lambda/4\). Since the open circuited stub appears to be capacitive in the range of \(l_{1}<\lambda/4\), if the reactance is inductive, series resonance can be made with a length that is shorter than the original resonant length. Lowering the quarter-wavelength resonant frequency without increasing the resonant length leads to miniaturization. In order to represent the series inductance in the MSL, there is a method using a high impedance line with a short electric length, from the same idea as the stepped impedance structure. It is the same as making a thin constriction at the attachment point of the resonator, or as driving a rectangular wedge. In Figure 3b, the normalized input admittance \(y_{\text{in}}\) when seeing open stub from the attachment point is given by Equation (5).  
\[y_{in}=\frac{Y_{in}}{Y_{0}}=j\frac{1}{(Y_{0}/Y_{1})\cot\beta_{1}l_{1}-Y_{0} \omega L}=jb \tag{5}\]  
Figure 4 shows the calculated results of \(\mid S_{21}\mid\) and \(\mid S_{11}\mid\) for the series inductance value \(L=0\) nH, 1 nH, 10 nH, and 100 nH when \(Z_{1}/Z_{0}=1/2\). It can be seen that the transmission zero frequency decreases as \(L\) increases. However, as shown in Equation (4), as \(L\) increases, \(Q\) increases to a narrow bandwidth. In the case of \(Z_{1}/Z_{0}<1\), where the \(C\) of the resonator is large, it becomes broadband, but the size of the resonator also becomes large. On the other hand, when the \(L\) of the resonator is large, it becomes a narrow band, but the size of the resonator becomes small. Therefore, there is a trade-off relationship between miniaturization and broadband bandwidth. Radial stubs are well known as a method to achieve both compact and broadband bandwidth. In the case of radial stubs, \(C\) becomes larger with the fan-shaped plate, and the characteristic impedance of the stub line decreases as the central angle becomes larger; therefore, it is considered that the principle of broadband and miniaturization is the same as the low impedance open-circuited stub with a series inductance proposed in this study.  
Figure 3: Circuit model of an MSL open-circuited stub with inductance inserted at the attachment point to the main transmission line. (**a**) Schematic view of MSL open circuited stub with a reactive element; (**b**) equivalent circuit of MSL open-circuited stub with an inductance \(L\).