Introduction
Dual-band filters have attracted more and more attention in many high-performance applications due to requests on the multi-channel/multi-standard RF and satellite communications, multifunctional structures, compact, lightweight multiband payloads, and efficient employment of further frequency channels. Under this trend, interference suppression becomes a pivotal challenge for circuit designs like WiFi due to enhancing the improvement of wireless circuits. The dual-band filters are critical in modern wireless communication systems because they offer two passbands that pass desired frequencies, choose channels, regulate spectrum, and suppress unwanted frequencies, which can reduce the performance of systems [Reference Abasi and Hazeri1–Reference Najafi and Hazeri3]. Consequently, various multiband filters have been investigated in recent years. So far, many dual-band bandpass filters have been presented. Dual-band bandpass filters can be classified into several classes: wideband, mid-band, and narrow bandpass filters. Another classification is in utilizing single- and multilayer microstrip, coaxial, waveguides, dielectric, and superconductors [Reference Abasi and Hazeri1–Reference Yan, Kang, Yang and Cao34]. In paper [Reference Chen, Li and Shi4], the dual-mode dielectric resonator is mounted on the bottom of the metal cavity, and in paper [Reference Tan, Lu and Chen5] differential dual-mode dielectric strip resonator by loading a pair of ground bars underneath the traditional half-wavelength has been proposed for designing a dual-bandpass filter. In paper [Reference Chen, Zhu, Lin, Wong, Li, Yang and He6] dual-mode circular spiral resonator in a single metal cavity, in paper [Reference Chang, Sheng, Cui and Lu7] slot-coupled and hairpin-line resonators, in paper [Reference Tang, Han, Deng, Qian and Luo8] using substrate-integrated defected ground structure (DGS) resonant cells, in paper [Reference Xu, Ma and Du9] suspended line platforms, in paper [Reference Zhang, Chen, Chen, Ding and Zhang10] dual-mode isosceles right-angled triangular resonators, and in paper [Reference Wei, Wu, Wang, Pan, Yang and Liu11] the double-sided spoof surface plasmon polaritons-line has been employed to fabricate a dual-bandpass filter on the multilayer substrate. However, using dielectric resonators and multilayer substrates increases the cost and complexity of designing filters. The high-temperature superconducting resonators have been widely used for the dual-mode dual-band bandpass filters [Reference Sekiya and Tsuruoka12–Reference Ren, Ma, Liu, Guan, Wang, Wen and Ohira14]. The waveguides have been utilized to design dual-band bandpass filters [Reference Zhang, Rao, Hong and Liu15–Reference Deng, Li, Sun, Guo and Ma20]. However, the waveguide filters have the drawback of many unwanted spurious resonances and poor selectivity in controlling the transmission zeros’ positions and larger circuit dimensions. Recently, a single-layer microstrip substrate has been proposed to realize dual-band bandpass filters. Methods based on the T-shaped resonator [Reference Basit, Khattak, Al-Hasan, Nebhen and Jan21], patch resonators and meandered slot lines [Reference Dong, Wang, Wu, Li, Liu and Tentzeris22], stepped impedance resonators [Reference Lu and Zhang23–Reference Tang, Liu and Yang25], open stub loaded and short-circuited stub loaded resonators [Reference Bandyopadhyay, Sarkar, Mondal and Ghatak26], self-coupled resonator consisting of asymmetrical parallel-coupled microstrip transmission lines [Reference Ma, You, Qian, Shen, Qin, Wu and He27], and cascaded capacitor-loaded coupled microstrip lines and varactors [Reference Dong, Xu and Shi28] have been widely used to design the dual-band bandpass filters. However, most planner microstrip dual-band bandpass filters have no narrowband passbands. So, they cannot be employed in narrowband applications. Recently, a few dual-narrowband bandpass filters have been introduced in papers [Reference Abasi and Hazeri1, Reference Najafi and Hazeri3, Reference Xu, Ma, Meng and Yeo29–Reference Yan, Kang, Yang and Cao34]. Three coupled asymmetrical and parallel coupled transmission lines have been utilized to present dual-narrowband bandpass filters but need a large occupied area [Reference Abasi and Hazeri1, Reference Najafi and Hazeri3]. Although they are narrowband, they do not have any exact equations for calculating scattering parameters that are very important for engineers. In paper [Reference Xu, Ma, Meng and Yeo29], a DGS has been proposed but needs extra etching on the ground side. A hybrid structure of the microstrip patch and coplanar waveguide has been introduced in paper [Reference Duan, Song, Chen and Fan30]. However, the structure has the drawback of poor operations in the first passband and last stopband. Right/left-handed transmission lines have been utilized to fabricate dual-narrowband bandpass filters but need extra etching and via [Reference Danaeian, Zarezadeh and Ghayoumi-Zadeh31]. A dual-narrowband bandpass filter has been realized by T-shaped resonators [Reference Ghaderi, Golestanifar and Shama32]. There are no exact equations for scattering parameters of previously published dual-narrowband bandpass filters [Reference Abasi and Hazeri1, Reference Najafi and Hazeri3, Reference Xu, Ma, Meng and Yeo29–Reference Ghaderi, Golestanifar and Shama32]. Furthermore, when the center frequency of the first and second passband are close together, the size or complexity of the filter is extra. In paper [Reference Firmansyah, Praptodiyono, Rifai, Alam and Paramayudha33], a dual-band BPF based on a cascaded closed ring resonator has been introduced. However, its size is large. A loop-type microstrip line loaded with four shorted stubs has been developed to present a dual-narrowband bandpass filter in paper [Reference Yan, Kang, Yang and Cao34]. However, it needs welding and via to the ground.
The motivation of this article is to present a new miniaturized dual-narrowband bandpass filter with high selectivity, compact size, close center frequencies, and wide last stopband. Furthermore, the main part is the exact formula for calculating the scattering parameters of the proposed filter with a planar structure for easy tuning, manufacturing, and assembly. Moreover, the center frequencies of the filter are adjusted by tuning the physical lengths. The proposed filter is implemented, measured, and compared with theoretical and full-wave simulation results for validation. The following sections, first, discuss the main state-of-the-art of the proposed filter, generate exact scattering parameters, and compare with simulation results; next, in the Experimental and simulation results section , the optimized filter is presented, and experimental and simulation results of the optimized filter are illustrated, and its performance is compared with the state-of-the-art designs; finally, the Conclusion section articulates conclusions.
Proposed filter
The proposed filter is analyzed in detail in this section. The schematic of the proposed filter is portrayed in Fig. 1, which consists of several open stubs with characteristic impedances 
${Z_{os}},{Z_{s2}},{Z_{s3}}$, and 
${Z_{s4}}$ along with electrical lengths 
${\theta _{os}},{\theta _{s2}},{\theta _{s3}}$, and 
${\theta _{s4}}$, respectively, parallel-coupled transmission lines with the even- and odd-mode characteristic impedance 
${Z_e}$ and 
${Z_o}$, respectively, along with electrical length 
${\theta _c}$ and two transmission lines with characteristic impedance 
${Z_T}$ and electrical length 
${\theta _T}$. Thanks to the symmetrical shape, the proposed filter may be analyzed using the even- and odd-mode excitation. As indicated in Fig. 1(b), the input impedance under even mode (
${Z_{in,e}}$) may be calculated as follows:
\begin{align}
{Z_{in,o}} &= {Z_1}_1 + \frac{{( - A{Z_1}_1^2 + B{Z_1}_1 + C)}}{{({Z_1}_1^3 + D{Z_1}_1^2 + E{Z_1}_1 + F)}} \nonumber\\
A &= {Z_1}_2^2 + {Z_1}_3^2 + {Z_1}_4^2 \nonumber\\
B &= {Z_1}_3^2{Z_1}_4 - {Z_{S3}}{Z_1}_2^2 - {Z_{S2}}{Z_1}_4^2 - {Z_{S4}}{Z_1}_2^2 - {Z_{in,eL}}{Z_1}_4^2 - {Z_{S4}}{Z_1}_3^2 \nonumber\\
&\quad- {Z_{S2}}{Z_1}_3^2 + 5{Z_1}_2{Z_1}_3{Z_1}_4 \nonumber\\
C &= {Z_1}_2^3{Z_1}_3 - {Z_1}_2^2{Z_1}_4^2 - 2{Z_1}_3^2{Z_1}_4^2 - {Z_1}_2{Z_1}_3^3 - {Z_1}_2^2{Z_1}_3^2 + {Z_1}_3^4 \nonumber\\
&\quad+ {Z_1}_4^4 - {Z_2}{Z_3}{Z_1}_4^2 - {Z_2}{Z_4}{Z_1}_3^2 + {C_1} \nonumber\\
{C_1} &= - {Z_{in,eL}}{Z_{S4}}{Z_1}_2^2 + {Z_{S2}}{Z_1}_3^2{Z_1}_4 - {Z_1}_2{Z_1}_3{Z_1}_4^2 + {Z_{S2}}{Z_1}_2{Z_1}_3{Z_1}_4 \nonumber\\
&\quad+ 2{Z_{S3}}{Z_1}_2{Z_1}_3{Z_1}_4 + 2{Z_{S4}}{Z_1}_2{Z_1}_3{Z_1}_4 \nonumber\\
D &= {Z_{S2}} + {Z_{in,eL}} + {Z_{S4}} \nonumber\\
E &= - {Z_1}_3^2 - {Z_1}_2{Z_1}_3 - {Z_1}_4^2 + {Z_2}{Z_{in,eL}} + {Z_{S2}}{Z_{S4}} + {Z_{in,eL}}{Z_{S4}} \nonumber\\
F &= {Z_1}_3^2{Z_1}_4 - {Z_{in,eL}}{Z_1}_3^2 + {Z_1}_2{Z_1}_3{Z_1}_4 - {Z_{S2}}{Z_1}_2{Z_1}_3 - {Z_{S4}}{Z_1}_4^2 \nonumber\\
&\quad+ {Z_{S2}}{Z_{in,eL}}{Z_{S4}}\end{align}(a) Dual-band bandpass filter: (b) even mode and (c) odd mode, all dimensions are in millimeters units.
where
\begin{equation}{Z_{in,eL}} = {Z_{s3}}||{Z_T}\frac{{2{Z_{OS}} + j{Z_T}\tan ({\theta _T})}}{{{Z_T} + j2{Z_{OS}}\tan ({\theta _T})}}\end{equation}
\begin{equation}{{\text{Z}}_{{\text{11}}}}{\text{ = }}{{\text{Z}}_{22}}{\text{ = }}{{\text{Z}}_{33}}{\text{ = }}{{\text{Z}}_{44}}{\text{ = - j(}}{{\text{Z}}_{\text{e}}}{\text{ + }}{{\text{Z}}_{\text{o}}}{\text{)cot(}}{\theta _c}{\text{)/2}}\end{equation}
\begin{equation}{{\text{Z}}_{{\text{12}}}}{\text{ = }}{{\text{Z}}_{21}}{\text{ = }}{{\text{Z}}_{34}}{\text{ = }}{{\text{Z}}_{43}}{\text{ = - j(}}{{\text{Z}}_{\text{e}}}{\text{ - }}{{\text{Z}}_{\text{o}}}{\text{)cot(}}{\theta _c}{\text{)/2}}\end{equation}
\begin{equation}{{\text{Z}}_{{\text{13}}}}{\text{ = }}{{\text{Z}}_{31}}{\text{ = }}{{\text{Z}}_{24}}{\text{ = }}{{\text{Z}}_{42}}{\text{ = - j(}}{{\text{Z}}_{\text{e}}}{\text{ - }}{{\text{Z}}_{\text{o}}}{\text{)csc(}}{\theta _c}{\text{)/2}}\end{equation}
\begin{equation}{{\text{Z}}_{{\text{14}}}}{\text{ = }}{{\text{Z}}_{41}}{\text{ = }}{{\text{Z}}_{23}}{\text{ = }}{{\text{Z}}_{32}}{\text{ = - j(}}{{\text{Z}}_{\text{e}}}{\text{ + }}{{\text{Z}}_{\text{o}}}{\text{)csc(}}{\theta _c}{\text{)/2}}\end{equation}Also, under odd-mode excitation, the input impedance of the proposed filter, as illustrated in Fig. 1(c), may be expressed as follows:
\begin{align}
{Z_{in,o}} &= {Z_1}_1 + \frac{{( - A{Z_1}_1^2 + B{Z_1}_1 + C)}}{{({Z_1}_1^3 + D{Z_1}_1^2 + E{Z_1}_1 + F)}} \nonumber\\
A &= {Z_1}_2^2 + {Z_1}_3^2 + {Z_1}_4^2 \nonumber\\
B &= {Z_1}_3^2{Z_1}_4 - {Z_{S3}}{Z_1}_2^2 - {Z_{S2}}{Z_1}_4^2 - {Z_{S4}}{Z_1}_2^2 - {Z_{in,oL}}{Z_1}_4^2 - {Z_{S4}}{Z_1}_3^2 \nonumber\\
&\quad- {Z_{S2}}{Z_1}_3^2 + 5{Z_1}_2{Z_1}_3{Z_1}_4 \nonumber\\
C &= {Z_1}_2^3{Z_1}_3 - {Z_1}_2^2{Z_1}_4^2 - 2{Z_1}_3^2{Z_1}_4^2 - {Z_1}_2{Z_1}_3^3 - {Z_1}_2^2{Z_1}_3^2 + {Z_1}_3^4 + {Z_1}_4^4 \nonumber\\
&\quad- {Z_2}{Z_3}{Z_1}_4^2 - {Z_2}{Z_4}{Z_1}_3^2 + {C_1} \nonumber\\
{C_1} &= - {Z_{in,oL}}{Z_{S4}}{Z_1}_2^2 + {Z_{S2}}{Z_1}_3^2{Z_1}_4 - {Z_1}_2{Z_1}_3{Z_1}_4^2 + {Z_{S2}}{Z_1}_2{Z_1}_3{Z_1}_4 \nonumber\\
&\quad+ 2{Z_{S3}}{Z_1}_2{Z_1}_3{Z_1}_4 + 2{Z_{S4}}{Z_1}_2{Z_1}_3{Z_1}_4 \nonumber\\
D &= {Z_{S2}} + {Z_{in,oL}} + {Z_{S4}} \nonumber\\
E &= - {Z_1}_3^2 - {Z_1}_2{Z_1}_3 - {Z_1}_4^2 + {Z_2}{Z_{in,oL}} + {Z_{S2}}{Z_{S4}} + {Z_{in,oL}}{Z_{S4}} \nonumber\\
F &= {Z_1}_3^2{Z_1}_4 - {Z_{in,oL}}{Z_1}_3^2 + {Z_1}_2{Z_1}_3{Z_1}_4 - {Z_{S2}}{Z_1}_2{Z_1}_3 - {Z_{S4}}{Z_1}_4^2 \nonumber\\
&\quad+ {Z_{S2}}{Z_{in,oL}}{Z_{S4}} \nonumber\\
\end{align}where
\begin{equation}{Z_{in,oL}} = {Z_{s3}}||J{Z_T}\tan ({\theta _T})\end{equation}Then, the scattering matrix under even- and odd-mode excitation can be derived as below:
\begin{equation}\begin{gathered}
{S_{11}} = \frac{{{Z_{in}} - {Z_0}}}{{{Z_{in}} + {Z_0}}} = \frac{{{S_{11,e}} + {S_{11,o}}}}{2}, \hfill \\
{S_{11,e}} = \frac{{{Z_{in,e}} - {Z_0}}}{{{Z_{ine}} + {Z_0}}}, \hfill \\
{S_{11,o}} = \frac{{{Z_{in,,o}} - {Z_0}}}{{{Z_{in,o}} + {Z_0}}}, \hfill \\
\end{gathered} \end{equation}One may rewrite equation (9) as follows:
\begin{equation}{S_{11}} = \frac{{{Z_{ine}}{Z_{ino}} - {Z_0}^2}}{{\left( {{Z_{ine}} + {Z_0}} \right)\left( {{Z_{ino}} + {Z_0}} \right)}}\end{equation} When 
$\measuredangle {Z_{ine}}{Z_{ino}} = 0$ and 
$\left| {{Z_{ine}}{Z_{ino}}} \right| = {Z_0}^2$, i.e., the image part of 
${Z_{ine}}{Z_{ino}}$ is zero and the real part of 
$\sqrt {{Z_{ine}}{Z_{ino}}} = {Z_0}$, we have the impedance matching at the input ports. Also, the insertion loss of the filter (
${S_{12}}$) can be calculated in the even- and odd-mode analyses as follows:
\begin{equation}{S_{12}} = \frac{{{S_{11e}} - {S_{11o}}}}{2}\end{equation}The scattering parameters of the proposed filter in Fig. 1 can be theoretically achieved by calculating equations (1)–(11) and compared with simulation results as illustrated in Fig. 2. The first and second center frequencies of the proposed filter in Fig. 1 are 4.37 and 6.5 GHz, respectively. The first and second 3-dB bandwidths are from 4.34 to 4.39 GHz and 6.41 to 6.57 GHz, respectively.
Scattering parameters of the dual-band bandpass filter (a) 
${S_{11}}$ and (b) 
${S_{12}}$.
Furthermore, according to the Fig. 2, the transfer function (TF) of the proposed filter may be estimated as follows:
\begin{equation}\begin{gathered}
TF(S) = \frac{{1.345E9S\left( {{S^2} + 9.991E20} \right)}}{{\left( {{S^2} + \frac{{{\omega _{0,1}}}}{{{Q_1}}}S + {\omega _{0,1}}^2} \right)\left( {{S^2} + \frac{{{\omega _{0,2}}}}{{{Q_2}}}S + {\omega _{0,2}}^2} \right)}}, \hfill \\
{\omega _{0,1}} = 2.743E10\,rad/s, \hfill \\
{\omega _{0,2}} = 4.078E10rad/s, \hfill \\
{Q_1} = 75.29, \hfill \\
{Q_2} = 41.6 \hfill \\
\end{gathered} \end{equation}and the equivalent lumped elements of equation (12) may be calculated as in Fig. 3.
The simple equivalent lumped element of the proposed filter.
As displayed in Fig. 4, the scattering parameters of the proposed filter and lumped element one are almost identical. The quality factor of the first and second passbands are 41 and 75, respectively.
Scattering parameters of the proposed filter and lumped element (a) S11 and (b) S12.
Experimental and simulation results
The optimized proposed filter is portrayed in Fig. 5. The optimized proposed filter is designed on a substrate with substrate thickness (H) = 15 mil and relative dielectric constant (
${\varepsilon _r}$) = 2.2. Also, the Momentum tool of Advanced Design Systems is employed as a simulator.
Layout and photograph of the dual-band bandpass filter.
In Fig. 6, the measurement and simulation results are portrayed. The center frequency of the first passband (FC1) is 4.554 GHz, and the center frequency of the second passbands (FC2) is 6.836 GHz. The fractional bandwidth (FBW) of the first and second passband are 1.5 and 2.5%, respectively. FC1/FC2 is 66.6%, which is small without having a large occupied area or complexity of the circuit. The loss of transmission zero between two passbands is around −20 dB. The bandwidth of the last stopband under −20 dB is from 7.62 to 20 GHz, which is very wide stopband. The occupied area of the filter is 0.12 
$ \times $ 0.096
${\lambda _g}^2$, which shows the proposed filter has a small size. Where 
${\lambda _g}$ is the guided wavelength at the center frequency of the first passband. So the optimized filter is a narrowband dual-band bandpass filter with good characteristics. Adjusting the length of the open stub with width = 1.5 mm (ZOS) moves the center frequency of the second passband independent of the first passband, as illustrated in Fig. 7, because it is shorted-circuit in odd-mode analysis. Also, Fig. 8 displays that changing the length of ZT or L11, shifts both center frequencies. Effects of the length of ZS2 i.e., L15 and L16, on S12 are demonstrated in Fig. 9. As portrayed in Fig. 9, changing the length of ZS2 moves both center frequencies. Figure 10 exhibits the effects of the length of ZS4 i.e., L7 and L8, on S12. Adjusting ZS4 changes the transmission zero between two passbands. The bandwidth of the passbands is controlled by adjusting the space between coupled lines (S2) as shown in Fig. 11. So the filter is tunable/selectable. In Table 1, the optimized filter and previously significant published papers are detailed. According to Table 1, the optimized filter has excellent features such as compact size and narrow passbands. Finally, the simulation and measurement results are in good agreement and approve the theoretical analyses.
Scattering parameters of the dual-band bandpass filter.
Variations of the length of ZOS on S12 of the dual-band bandpass filter: solid line = 2.2 mm, dot line = 1.6 mm, long-dash line = 1.7 mm.
Variations of the length of ZT on S12 of the dual-band bandpass filter: solid line = 2.3 mm, dot line = 1.2 mm, long-dash line = 0.55 mm.
Variations of the length of ZS2 on S12 of the dual-band bandpass filter: solid line: with L15 and L16, dot line: without L16, dash line: without L15 and L16.
Variations of the length of ZS4 on S12 of the dual-band bandpass filter: solid line: with L7 and L8, dot line: without L8, dash line: without L7 and L8.
Variations of the length of S2 on S12 of the dual-band bandpass filter: solid line = 0.1 mm, dot line = 0.245 mm, dash line = 0.347 mm.
Details of the optimized filter and previously significant published papers
Conclusion
In this article, a dual-narrowband bandpass filter has been introduced and analyzed. The proposed filter had good features for a dual-narrow bandpass filter, such as small fractional bandwidth in the passbands and large bandwidth in the stopbands. The equations have been exact and simple. The exact scattering matrix of the proposed filter has been calculated by utilizing the even- and odd-mode analysis. Furthermore, the proposed filter’s estimated TF and equivalent lumped element have been offered. The structure of the proposed filter provides many freedoms to design. The theoretical results, simulation results, and experimental results are confirmed together.
Acknowledgements
The authors want to thank for Kermanshah Branch, Islamic Azad University and Mr. S. Abouzari for help and support. We would like to express our special thanks of gratitude to Dr. Z. Ebrahimipour and F. Shama for taking time and support.
Ali Reza Hazeri was born in Kermanshah City, Iran, in 1984. He received the B.S. and M.S. degrees in electrical and electronic engineering from the Boroujerd and Central Tehran branch, Islamic Azad University, in 2008 and 2010, respectively, and he achieved a Ph.D. degree in microelectronic engineering from Babol Noshirvani University of Technology, one of the best universities of technology in Iran in 2019. From 2012 to 2015, he was employed by Telecommunication Company, Tehran. Since 2015, he has been a faculty member of the Kermanshah Branch, Islamic Azad University. He is the author of several articles and invited reviewer for several journals. His research interests include analog and mixed-signal design, high-frequency, microwave, and radio frequency circuits.
Masoud Najafi was born in Kermanshah, Iran, in 1988. He received his B.Sc. and M.Sc. degree in electronic engineering from Academic Jihad University and Kermanshah Branch, Islamic Azad University, Kermanshah, Iran, in 2011 and 2017, respectively. He is currently a Ph.D. student at Sanandaj Branch, Islamic Azad University, Sanandaj, Iran. His current research interest includes RF and microwave circuit design.
