A Design of a Dual-Band Bandpass Filter Based on Modal Analysis for Modern Communication Systems

: A dual-band bandpass ﬁlter (BPF) composed of a coupling structure and a bent T-shaped resonator loaded by small L-shaped stubs is presented in this paper. The ﬁrst band of the proposed BPF covers 4.6 to 10.6 GHz, showing 78.9% fractional bandwidth (FBW) at 7.6 GHz, and the second passband is cantered at 11.5 GHz with a FBW of 2.34%. The bent T-shaped resonator generates two transmission zeros (TZs) near the wide passband edges, which are used to tune the bandwidth of the ﬁrst band, and the L-shaped stubs are used to create and control the narrow passband. The selectivity performance of the BPF is analyzed using the transfer function extracted from the lumped circuit model veriﬁed by a detailed even / odd mode analysis. The BPF presents a ﬂat group delay (GD) of 0.45 ns and an insertion loss (IL) less than 0.6 dB in the wide passband and a 0.92 IL in the narrow passband. A prototype of the proposed BPF is fabricated and tested, showing very good agreement between the numerically predicted and measured results.


Introduction
The development of wide-band communications has increased the application of wide-band microwave filters. Diverse design techniques and geometries have been proposed to design wide-band bandpass filters (BPFs) over the last few years. A wide-band BPF using an interdigital feed line loaded by stepped-impedance stubs was realized in [1], where a good out-of-band rejection, with a passband insertion loss (IL) and group delay (GD) of 0.57 dB and 5.91 ns, respectively were reported. A simple configuration of wide-band BPF using high impedance feed lines coupled with a bent stub was proposed in [2]. In this approach, the out-of-band suppression is controlled by the physical features of the bent stub. Another structure of BPF using stepped-impedance resonators (SIRs) was designed in [3] to achieve a 3 ns GD. Two parallel lines loaded with short and open stubs were introduced in [4] designed in [3] to achieve a 3 ns GD. Two parallel lines loaded with short and open stubs were introduced in [4] to realize a high frequency-selectivity near the passband. A wide-band BPF comprised of coupled short stubs with high selectivity was discussed in [5] that provides a 63% fractional bandwidth (FBW) and 0.6 dB of IL within its passband. A coupled feed line and opencircuited stubs were applied in [6] for designing a BPF operating over 1.22 to 2.77 GHz, presenting 6 ns GD. In [7], a spiral resonator was adopted to realize a wide-band filter, where the low and high cut-off frequencies are synchronously set. Substrate integrated waveguide (SIW) technology has widely been utilized to develop BPFs because of the associated low cost and simple topology [8][9][10][11]. The SIW filters generally present a low return loss (RL) throughout their passbands [8,9]; however, the size of the circuit can be undesirable for some applications [10,11]. In applications where highlevel immunity to noise is critical, BPF with differential configurations is used [12,13]. An interdigital structure coupled with bent rectangular stubs has been proposed for designing wide-band BPF [14]. In this approach, the interdigital structure is used to feed the SIRs. It needs to be pointed out that the application of SIRs is not limited to wide-band BPFs, and they have extensively been used in all types of microstrip filters [15][16][17]. A multi-functional response has been achieved for a BPF using a stubloaded ring [18]; however, the filter's stopband is relatively small. A modified BPF with T-shaped resonators was designed in [19] to achieve a flat GD throughout a wide passband; however, only the passband's upper edge is tunable. In another approach, BPFs were formed using the spiral lines in [20] to provide a high return loss in the passband, while the physical dimensions are not optimum, resulting in large BPFs. A simple structure composed of a feed line and rectangular resonator was discussed in [21] to provide dual-band performance. To minimize the BPF dimensions, a bent feed line was loaded by traditional stubs [22]. In a different approach, artificial intelligence has recently been introduced in the microwave and electromagnetic community and used to design a wide-band BPF [23][24][25].
In this paper, a dual-band passband filter with a modified T-shaped resonator is designed to provide a flat GD throughout a wide passband of 78.9%, where a bent T-shaped resonator is introduced to realize a tunable passband. Additionally, the bent T-shaped resonator is loaded by small L-shaped stubs to generate a second passband in the upper stopband which can be used for fixed satellite applications.

Coupling Structure
A wide-band coupling structure is designed using high impedance lines, as shown in Figure 1a. The electromagnetics simulation (EM) of the coupling system is carried out using the advanced design system (ADS) and shown in Figure 1b. According to this figure, the coupling system creates a wide passband from 5.5 up to 9.5 GHz; however, the transition bands are not sufficiently small, resulting in poor passband controllability.

Single Wide-band Response
To improve the sub-optimal filtering characteristics of the coupling structure mentioned above, a bent T-shaped resonator is added to the coupling structure, as depicted in Figure 2a. The Sparameters of the single-band BPF is presented in Figure 2b. As observed, the bent T-shaped resonator generates two TZs at 3.8 GHz (TZ1) and 11.6 GHz (TZ2). The passband bandwidth can be tuned by shifting the location of TZ1 and TZ2.

Single Wide-Band Response
To improve the sub-optimal filtering characteristics of the coupling structure mentioned above, a bent T-shaped resonator is added to the coupling structure, as depicted in Figure 2a. The S-parameters of the single-band BPF is presented in Figure 2b. As observed, the bent T-shaped resonator generates two TZs at 3.8 GHz (TZ1) and 11.6 GHz (TZ2). The passband bandwidth can be tuned by shifting the location of TZ1 and TZ2.

Single Wide-band Response
To improve the sub-optimal filtering characteristics of the coupling structure mentioned above, a bent T-shaped resonator is added to the coupling structure, as depicted in Figure 2a. The Sparameters of the single-band BPF is presented in Figure 2b. As observed, the bent T-shaped resonator generates two TZs at 3.8 GHz (TZ1) and 11.6 GHz (TZ2). The passband bandwidth can be tuned by shifting the location of TZ1 and TZ2.   One of the reliable approaches to investigate the microstrip resonators' frequency behavior is the use of lumped-element circuits (LCs) [26]. Here, a LC model is proposed for the bent T-shaped resonator and shown in Figure 3. In this model, L 1 and C 3 denote the high impedance feed line's inductance and capacitance feed lines. L 2 and C 1 represent the inductance and capacitance of the low impedance line, respectively. C 2 and L 3 , respectively, depict the capacitance and inductance of the bent Electronics 2020, 9, 1770 4 of 13 stubs. It should be mentioned that the capacitance effects of the bent section are ignored in this model. Considering the model's bent-related capacitance effects results in a more accurate model at a price of a significantly more complex model. The values of LC model parameters using the methods discussed in [26] are computed as follows: L 1 = 5.8 nH, L 2 = 1.931 nH, L 3 = 2.01 nH, C 1 = 0.446 pF, C 2 = 0.198 pF, and C 3 = 0.39 pF.
Electronics 2020, 9, x FOR PEER REVIEW 4 of 13 One of the reliable approaches to investigate the microstrip resonators' frequency behavior is the use of lumped-element circuits (LCs) [26]. Here, a LC model is proposed for the bent T-shaped resonator and shown in Figure 3. In this model, L1 and C3 denote the high impedance feed line's inductance and capacitance feed lines. L2 and C1 represent the inductance and capacitance of the low impedance line, respectively. C2 and L3, respectively, depict the capacitance and inductance of the bent stubs. It should be mentioned that the capacitance effects of the bent section are ignored in this model. Considering the model's bent-related capacitance effects results in a more accurate model at a price of a significantly more complex model. The values of LC model parameters using the methods discussed in [26] are computed as follows: L1 = 5.8 nH, L2 = 1.931 nH, L3 = 2.01 nH, C1 = 0.446 pF, C2 = 0.198 pF, and C3 = 0.39 pF. The bent T-shaped resonator's transfer function is obtained from the LC model and shown in Equation (1), where r is the 50 Ω matching impedance. The equations of TZ1 and TZ2 are extracted from the transfer function and depicted in Equations (2) and (3). As observed, TZ1 and TZ2 are related to capacitors and inductors of the bent T-shaped resonator, including C2 and L3. Variations of C2 and L3 have a direct effect on the location of TZ1 and TZ2, which in turn controls the passband by shifting the low and high cut-off frequencies. Indeed, the upper edge of the passband can be tuned by varying L3 representing the inductance effects of length d11. Figure 4a,b depict the adjustability of the passband's upper edge by varying d11 with a small effect on the lower edge of the passband. The lower edge of the passband can be adjusted by changing d12 corresponding to C2, as shown in Figure  4c,d. The bent T-shaped resonator's transfer function is obtained from the LC model and shown in Equation (1), where r is the 50 Ω matching impedance. The equations of TZ1 and TZ2 are extracted from the transfer function and depicted in Equations (2) and (3). As observed, TZ1 and TZ2 are related to capacitors and inductors of the bent T-shaped resonator, including C 2 and L 3 . Variations of C 2 and L 3 have a direct effect on the location of TZ1 and TZ2, which in turn controls the passband by shifting the low and high cut-off frequencies. Indeed, the upper edge of the passband can be tuned by varying L 3 representing the inductance effects of length d11. Figure 4a,b depict the adjustability of the passband's upper edge by varying d11 with a small effect on the lower edge of the passband. The lower edge of the passband can be adjusted by changing d12 corresponding to C 2 , as shown in Figure 4c,d. (1) Electronics 2020, 9, 1770 5 of 13 L3 have a direct effect on the location of TZ1 and TZ2, which in turn controls the passband by shifting the low and high cut-off frequencies. Indeed, the upper edge of the passband can be tuned by varying L3 representing the inductance effects of length d11. Figure 4a,b depict the adjustability of the passband's upper edge by varying d11 with a small effect on the lower edge of the passband. The lower edge of the passband can be adjusted by changing d12 corresponding to C2, as shown in Figure  4c,d.   The independent adjustability of the lower and upper edges of the passband can be investigated using the even/odd mode analysis, as explained in [27,28]. To compute the equations of the modal resonance frequency, the LC model of the bent T-shaped resonator is used, where the modal input impedances (Zine and Zino) are shown in Equations (4) and (5). In the resonance condition (Zine = Zino = 0), the Equations of even and odd mode resonance frequencies are approximately obtained by Equations (6)- (8). In this model, Cg denotes a weak coupling effect at input/output ports, and its value is 0.1 pF, approximately.
f o = 1 2π To visualize the modal resonances and verify Equations (6)-(8), the bent T-shaped resonator is simulated under a weak coupling condition. Its S-parameters are plotted in Figure 5, where adjustability of the single-band BPF response is verified through the tunability of the modal resonances. It can be seen from Figure 5a,b that d11 mainly controls the higher even mode (fe2) with a small effect on the lower even mode (fe1), and d12 controls fe1, corresponding to the lower edges of the passband of the single-band BPF, as shown in Figure 4.
The proposed structure is formed using two small L-shaped stubs added to the bent T-shaped resonator for generating the second passband, as shown in Figure 6a. Figure 6b depicts the second passband fixed at 11.5 GHz.
The L-shaped stub and its LC model are presented in Figure 7, where the high impedance open stub is modeled by C 4 and L 4 . The second passband can be tuned within a frequency window of 1270 MHz by varying L 4 and C 4 corresponding to the physical lengths of d18 and d15, respectively. Variations of the second passband versus d18 and d15 are shown in Figure 8. Another method to reshape the T-shaped resonator's passband is using a slot in the T-shaped resonator, as explained in [29], where the current distribution of the T-shaped resonator is manipulated by removing a slot from the T-shaped resonator. This method is different from our method, as it can only introduce a narrow stopband within the initial passband. The introduction of the L-shaped stub in the presented approach creates a narrow passband outside the initial passband. The proposed structure is formed using two small L-shaped stubs added to the bent T-shaped resonator for generating the second passband, as shown in Figure 6a. Figure 6b depicts the second passband fixed at 11.5 GHz.  The proposed structure is formed using two small L-shaped stubs added to the bent T-shaped resonator for generating the second passband, as shown in Figure 6a. Figure 6b depicts the second passband fixed at 11.5 GHz. The L-shaped stub and its LC model are presented in Figure 7, where the high impedance open stub is modeled by C4 and L4. The second passband can be tuned within a frequency window of 1270 MHz by varying L4 and C4 corresponding to the physical lengths of d18 and d15, respectively. MHz by varying L4 and C4 corresponding to the physical lengths of d18 and d15, respectively. Variations of the second passband versus d18 and d15 are shown in Figure 8. Another method to reshape the T-shaped resonator's passband is using a slot in the T-shaped resonator, as explained in [29], where the current distribution of the T-shaped resonator is manipulated by removing a slot from the T-shaped resonator. This method is different from our method, as it can only introduce a narrow stopband within the initial passband. The introduction of the L-shaped stub in the presented approach creates a narrow passband outside the initial passband. MHz by varying L4 and C4 corresponding to the physical lengths of d18 and d15, respectively. Variations of the second passband versus d18 and d15 are shown in Figure 8. Another method to reshape the T-shaped resonator's passband is using a slot in the T-shaped resonator, as explained in [29], where the current distribution of the T-shaped resonator is manipulated by removing a slot from the T-shaped resonator. This method is different from our method, as it can only introduce a narrow stopband within the initial passband. The introduction of the L-shaped stub in the presented approach creates a narrow passband outside the initial passband. The advantage of introducing L-shaped resonators to the bent T-shaped resonator is the addition of a second controllable passband without affecting the filter's total footprint. If such a narrow band is not required, the L-shaped resonators can be harmlessly removed from the filter configuration without affecting the primary passband performance. The advantage of introducing L-shaped resonators to the bent T-shaped resonator is the addition of a second controllable passband without affecting the filter's total footprint. If such a narrow band is not required, the L-shaped resonators can be harmlessly removed from the filter configuration without affecting the primary passband performance.

Current Distribution
Current distributions are employed in this section to further investigate the effects of different filter sections on its frequency response. The current distributions are plotted at three typical frequencies in-and out-of-band response. As shown in Figure 9a, the bent T-shaped resonator and L-shaped stubs do not allow flowing strong currents at 1.76 GHz, representing the filter's stopband. The current distributions are plotted at 9.68 and 11.5 GHz in Figure 9b,c, representing the filter's wide and narrow bands, respectively. As seen, the bent T-shaped resonator at 9.68 GHz contributes to generating the wide passband, showing a strong current. The L-shaped stubs contribute most at 11.5 GHz to create the second passband, as shown in Figure 10c.

Results
A prototype of the proposed BPF was fabricated on a 5880 substrate with εr = 2.2, h = 0.508 mm, and loss-tangent of 0.0009, as depicted in Figure 10a. The filter measurement was carried out using an Agilent N5230A network analyzer. Experimental and simulation results are demonstrated in Figure 10b, showing very good agreement. The footprint of BPF is only 18.4 × 9 mm, corresponding to 0.64 × 0.31 λg, where λg is the guided wavelength at 7.6 GHz. The filter has a large FBW of 78.9%, extending from 4.6 to 10.6 GHz with a negligible IL of 0.136 dB (simulation) and 0.6 dB (measurement) and RL performance better than 16.32 dB. The second passband is fixed at 11.5 GHz with a bandwidth of 270 MHz from 11.35 to 11.62 GHz with 0.92 IL and 2.34% FBW for fixed satellite applications. The upper out-of-band rejection is better than 10 dB, extending up to 20 GHz and the lower stopband has a rejection level of 37 dB, from dc to 3.6 GHz. Figure 10c exhibits a flat GD of 0.45 ns throughout the large passband (4.6 to 10.6 GHz). The filter demonstrates a very good isolation level between the two

Conclusions
A high-performance, dual-band BPF is proposed using double-coupled, high-impedance transmission lines and a bent T-shaped resonator loaded by L-shaped stubs. The former provides a large passband, and the latter improves the in-and-out band response of the filter. The filter's performance is investigated using LC models and their extracted transfer functions are verified by EM simulations and measured results. The filter has a FBW of 78.9% centered at 7.6 GHz with a flat GD of 0.45 ns throughout the wide passband.

Results
A prototype of the proposed BPF was fabricated on a 5880 substrate with ε r = 2.2, h = 0.508 mm, and loss-tangent of 0.0009, as depicted in Figure 10a. The filter measurement was carried out using an Agilent N5230A network analyzer. Experimental and simulation results are demonstrated in Figure 10b, showing very good agreement. The footprint of BPF is only 18.4 × 9 mm, corresponding to 0.64 × 0.31 λ g , where λ g is the guided wavelength at 7.6 GHz. The filter has a large FBW of 78.9%, extending from 4.6 to 10.6 GHz with a negligible IL of 0.136 dB (simulation) and 0.6 dB (measurement) and RL performance better than 16.32 dB. The second passband is fixed at 11.5 GHz with a bandwidth of 270 MHz from 11.35 to 11.62 GHz with 0.92 IL and 2.34% FBW for fixed satellite applications. The upper out-of-band rejection is better than 10 dB, extending up to 20 GHz and the lower stopband has a rejection level of 37 dB, from dc to 3.6 GHz. Figure 10c exhibits a flat GD of 0.45 ns throughout the large passband (4.6 to 10.6 GHz). The filter demonstrates a very good isolation level between the two passbands. According to the measurement results, an isolation level better that 22 dB has been achieved. Some of the important properties of recently published papers and the presented filter are classified and compared in Table 1.

Conclusions
A high-performance, dual-band BPF is proposed using double-coupled, high-impedance transmission lines and a bent T-shaped resonator loaded by L-shaped stubs. The former provides a large passband, and the latter improves the in-and-out band response of the filter. The filter's performance is investigated using LC models and their extracted transfer functions are verified by EM simulations and measured results. The filter has a FBW of 78.9% centered at 7.6 GHz with a flat GD of 0.45 ns throughout the wide passband.