# Four Subbands from Dual Mismatched Wideband Bandpass Filter for 5G/WAS/Wi-Fi/WiMAX/WLAN Applications

Authors: Omar Christian Massamba; Franck Moukanda Mbango; Désiré Lilonga-Boyenga
Journal: International Journal of RF and Microwave Computer-Aided Engineering
 (2023)

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## Abstract

Based on the simplicity of the design method, this paper presents a new approach for developing matched subbands when splitting two mismatched dual-wideband bandpass filters (BPFs) for the fifth generation (5G), wireless access systems (WAS), wireless fidelity (Wi-Fi), worldwide interoperability for microwave access (WiMAX), wireless local area network (WLAN), radar, and other communication devices. The method’s novelty involves using identical quarter-wavelength resonators terminated by alternated short-and-open stub configurations. Both configurations alternate and describe a perfect symmetry by their location from each other to make the subbands possible in the low-frequency and its harmonic (high-frequency) bandwidths (BW). A stub admittanceY1 is defined and associated with the mainline section characteristic impedance Z0 and an operating frequency f0. A quality factor Qap is connected to Y1 and f0 to approach the BPF global quality factor Qg initially fixed. The stub characteristic impedance and the mainline one differ, while electric lengths (stub and mainline section) are identical. Using the operating frequency determines physical dimensions, creates harmonic frequencies and the rejected BW, mismatches the main frequency BW, matches the subbands, and creates transmission zero (TZ). Hence, a 28.118-dB stopband that separates the two bandpasses at 9.373 GHz is made. At the same time, the unmatched dual ultrawideband (UWB) covers a large panel of communication systems. The lowest (3.146–5.431) and highest (11.891–14.749) GHz BW exhibit a minimum insertion loss (IL) of 0.656 dB and 3.027 dB. The subbands return losses (RL) are better than 28 dB and 19 dB, respectively, and a flat group delay of 0.205 ns is obtained in the upper band. All subbands adaptation methodology is read from 10 dB of the RL. In that case, the four matched subbands in its lower wideband are 3.327–3.709 GHz and 4.442–5.048 GHz, and in its higher wideband are 11.922–12.486 GHz and 14.281–14.653 GHz. The 2275/2858 MHz is the dual-wideband with a fractional BW 53.282/21.456%. The fabricated prototype has validated the EM-simulations, and Anritsu MS4642B 20 GHz vector network analyzer (VNA) has been used for experimental results by scanning the frequency range 3 GHz–15 GHz. The tested prototype is made with a 1 mm FR4 HTG-175 thickness by considering a dielectric constant of 4.4, and its overall size occupies 22.45×5.72mm2 (0.32λ0×0.082λ0mm2).


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## Body

## 1. Introduction

High and excellent wireless communications and information transfer have huge demand, creating advanced device technology systems [1]. So implementing microwave devices, such as stopband or bandpass filters (BPF) and power divider circuits, requires approaches from simple to complex structures but is easy to integrate [2]. Thereby, the filter component plays a key role in these new challenges [3, 4]. They are several kinds of filters found in the literature for electronics applications [5–7], and they may be grouped into three leading families: reflecting (constituted of lumped elements), notch (dissipate internally unneeded signals), and dissipation (produce losses through the use of material losses) filters. Filters can be designed using lumped components [8] or transmission lines [9, 10]. Initially, transfer functions were used, especially when the microwave electronics domain was underdeveloped. But researchers have adapted that principle to the radiofrequency area [11]. A large panel of studies focused on the line technique in the microwave area, terminated by open-circuit or short-circuited configurations, exist [12–14]. In addition, the filter structures can have multipath [12], use coupling lines [5, 15], have symmetrical shapes [16], etc. Those possibilities often offer the right to control [17] the bandwidth (BW) and their implementation with the rest of the integration systems [18]. Hence, the odd and even modes are the approach basis [19]. Based on the even-and-odd modes, the principle gave a vast opportunity to develop new and great BPF [20, 21]. Due to multiple controllable resonant modes, designers often use the multimode resonators (MMRs) technique. The stub-loaded resonators (SLRs) and the stepped-impedance resonators (SIRs) [22] are most commonly used in multiband filter designs [7]. The literature is rich enough in some techniques, such as SIR [23, 24], stepped-impedance ring-loaded resonator (SIRLR) [21], stub loaded with uniform impedance resonators (UIR) [6], and substrate-integrated waveguide (SIW) [25, 26]. Sometimes, the mix of techniques improves physical dimensions or electric performance [27]. Another new approach used nowadays is the defected ground structure (DGS) [28] or/and defected stub [29] technique. It finds several shapes, among them circular resonators [30]. Metamaterials are becoming an up-and-coming option, especially with high frequencies [31–33]. At the same time, some designers are developing new filter technologies to cover the terahertz area through the graphene principle as well illustrated in [34–36]. All these designs are studied and prototyped to have at least two BWs and several applications [37, 38].Above all, the T-shaped resonators, mixing identical open-and-short-circuit configurations, have not gotten enough attention nowadays. The BPF can be obtained using stubs terminated by short-circuited configurations according to the considered wavelength. Based on the mathematical formulation’s simplicity and the designed circuit facility to ease the study [39], this paper suggests a new approach, focusing on the Q-factor [40] determination when stub sections are identical and alternate the short-and-open ended configurations to split into two subbands a specific BPF bandwidth. This technique is based on the voluntary creation of mismatched bands around the rejection band to create subbands for particular applications. The excellent final goals are achieved when a transmission zero is obtained via a stopband. The design dimensions are based on the choice of the Q-factor [15], which fixes and controls the bandwidth of both main bands and four subbands. This Q-factor is closely related to the resonator’s admittance through the rejected frequency and a specific value to be found. On a wafer surface 22.45×5.72mm2, approximately 0.32λ0×0.082λ0mm2, where λ0 is the free-space wavelength of the first BW center frequency, a dual ultrawide BPF has been designed and prototyped. The proposed BPF occupies a surface area of 56.567 mm2. The BPF is measured using an Anritsu MS4642B network analyzer. We assume that the BPF BW is considered at 3 dB of the insertion loss maximum level [41] but not necessarily according to cases [42–44]. Also, we assume that the system is matched from the 10 dB of the return loss [45].


## 2. The Filter Topology and Methodology Approach

### 2.1. The Prototype Topology

The literature presents various basic concepts for creating filters. At the same time, some have proposed specific bandpass filter (monoband or multiband) topologies [46] but not some details for many bandpass filters like the one developed in this paper. This paper presented a new simple, easy topology to achieve great results for specific applications. This technique is based on the voluntary creation of mismatched bands around the rejection band to create subbands for particular applications. Hence, T-shaped stubs in short-and-open terminated configurations are put together through an alternation of these to achieve the goals. Also, all stubs and the mainline section have identical electric lengths except for their width. The design dimensions are based on the choice of Q-factor, which must be approximated when a frequency is chosen. This Q-factor fixes and controls the bandwidth of both main bands and four subbands. The excellent final goals are achieved when a transmission zero is obtained via a stopband. The bandpass filter’s structure comprises five identical resonators and four transmission line sections. All resonators are connected by a line section, as presented in Figure 1. The short circuit terminates three resonators. The metallic vias symbolize short-circuit configurations. Two resonators, ended by the open circuit configurations, are inserted between two short-circuited stubs. As described in [46], each design section can be transformed as a combination of the lumped elements. But the lumped element topology has attracted the researcher’s attention. The inductance and capacitance values are determined using the Chebyshev and/or Butterworth’s table or special calculations [11, 47–49].Figure 1
The cross-section of the prototype passband filter.The line section (constituted ofθ the electric length and Z0 the characteristic impedance) is perpendicular to stubs whose admittance is Y1 and long of θ). That gives the BPF’s shape. The normalized admittance y1 is provided as follows,
(1)y1=Y1Y0where Y0 is the line section admittance. The electric length can be defined as a frequency variation parameter and given by the following equation,
(2)θ=2πf0cl0εeff′μeff′,with εeff′ and μeff′ are the effective permittivity and permeability, respectively, c the vacuum’s light velocity, l0 the transmission line section length, and f0 the operating frequency. At the same time, the filter’s electric length θT is given below,
(3)θTf=2πfcLεeff′μeff′.The total PBF length isL, and f is the frequency covering the entire scanned frequency. Let us consider w, the stub’s width. In that case, the BPF global length is given by Equation (4),
(4)L=4l0+5w.As shown in Figure1, stubs are placed in a T-shaped and cascading position.


### 2.2. The Methodology Approach

In such a configuration, the first step consists of choosing the resonator’s number and length. This work uses two open-circuited and three short-circuited. That makes the design a five-order filter. Next, a line section is settled between two stubs that differ from their terminated configuration. It means that the short-and-open terminated stubs are mixed through their alternation position. All section lines and stubs are quarter-wavelengthλ/4 and well described in [46, 50]. The third step focuses on the Q− factor of each stub and the global Qg (also called initial quality factor Qi to fix) ones, where the subscript letter “g” indicates the global computation. It appears that using Qg and λ/4 becomes a great option [51]. The open-circuited λ/4 line leads to the stopband filter [52]. In the methodology presented in this paper, the stubs and the mainline transmission sections have identical lengths but differ in width. It means that their impedances (or admittances) are equal. Let us consider Qp and Qs, the normalized Q of parallel and series stubs as described in [46], respectively. These are given in the following equations for five resonators:
(5)Qs=8π1/y121+2y18−1/2,and
(6)Qp=π2y1y1+22161/3.In the proposed approach, three short-circuited stubs are interspersed by two open-circuited stubs, as shown in Figure1. As defined in [40], the Qap is given as follows,
(7)1Qap=1Qs+1QpSo, Equation (7) becomes as given below,
(8)Qap=2332πy1261+2y1+256y1y1+223.This equation establishes a link between theQap and the admittance of the stubs and the line sections. The fourth step needs to define the normalized admittance and apply an empirical formula to determine a value that lets the designer approximate the Qi needed through the computer-aided software.
(9)y1=2πf0Z0Xint,where Z0=50Ω, the connector impedance that is welded to the prototype input and output accesses. Finally, the designer chooses the operating frequency f0 and Qi to approximate Qap≅Qi. Both parameters define the stopband bandwidth around f0, creating two bandwidths where the high-frequency band is the harmonic of the low-frequency band. Consequently, mixing both configurations leads to the transmission zero (TZ) and a dual-band BPF with subbands in each bandwidth. The approach developed in this paper focuses on the study’s simplicity. The system must be designed with perfect symmetry to avoid input and output mismatch and those of the subbands, while the main dual-wideband is mismatched.Because of the stubs and the transmission mainline section dimensions, the rejection band shifts or slides by 1 GHz from the theoreticalf0.The BPF analysis is made from the scattering S-parameters matrix. The characteristics (return lossΓ, insertion loss IL, bandwidth Δ, cut-off frequency fc, group delay φgd) of the BPF are read from those S-parameters. The return loss (RL) is the signal’s power loss reflected by a discontinuity in a transmission line. This symbolizes the ratio of the reflected power to the incident power [53], as given below,
(10)ΓdB=−10logS112,where S11 is the reflection coefficient. Furthermore, insertion loss [8] is the amount of energy a signal loses as it travels through a component or a system [54, 55], such as the BPF device.
(11)ILdB=−10logS212,where S21 is the transmission coefficient. Whatever the BPF is, it presents a bandwidth Δ, defined from two cut-off frequencies, fh and fl, where fh>ff. In that case, the fractional bandwidth (FBW) is often called to indicate how large the bandwidth is from the cut-off frequencies. That parameter varies between 0% and 2%. The FBW is given as follows,
(12)FBW%=10021−2flΔ+2fl,where
(13)Δ=fh−fl.Reading Equation (12), the FBW depends on the bandwidth. The wider the bandwidth, the bigger the FBW. The Q is the inverse of the fractional resonator bandwidth and is determined as follows,
(14)Qap=f0Δ.Therefore, the time delay is given as shown in the following equation:(15)τ=lvp,where vp is the phase velocity. This parameter, also named the phase delay ϕpd, expressed in second, can also be defined as given [56, 57] below,
(16)ϕpd=−θTf2πf,where the electric length θf is expressed in radians. The phase delay can be expressed in nanoseconds (ns) if the frequency is in gigahertz (GHz). At the same time, the group delay φgd is obtained from the phase delay [58] through the following equation:
(17)φgd=−dθTfdf.The substitution of Equation (2) in Equation (16) gives Equation (18),
(18)ϕpd=2πcLεeff′μeff′.By subtractingθf in Equation (2) and replacing it in Equation (17), the filter’s group delay becomes as follows:
(19)φgd=Lcεeff′μeff′,which is expressed in seconds. Usually, this characteristic quantity is given in nanoseconds. Then, Equation (19) is rewritten as,
(20)φgdns=12π.dθTfdfGHz.Another important parameter that helps analyze the filter in high frequencies is the phase velocity in meters per second. From Equations (3) and (15), the substitution of the phase velocity gives,
(21)vφ=2πLθTff.Equation (21) talks about the phase velocity, which depends on the electric length behavior. That velocity decreases, stabilizes, or increases as the electric length increases or decreases.The methodology adapted can be summed up as follows: first, the short-and-open terminated stubs are mixed through their alternation position, creating the subbands. Second, all stubs and mainline sections are quarter-wavelength and calculated at the same operating frequency and the stub’s characteristic impedance. Third, at that frequency, the mainline impedance is chosen as 50Ω and differs from the stub ones. This creates the low-frequency and harmonic (high-frequency) bandwidths mismatch, depending on the Q. Fourth, an initial Qi that is fixed (chosen according to the designer’s objectives) and found must be recovered. This new Qap called approximated quality factor is linked to the structure’s impedance. Finally, the computer-aided software finds the unknown parameter leading to satisfying the condition Qap=Qi. In that case, the operating frequency belongs to the stopband, creating the transmission zero.


## 2.1. The Prototype Topology

The literature presents various basic concepts for creating filters. At the same time, some have proposed specific bandpass filter (monoband or multiband) topologies [46] but not some details for many bandpass filters like the one developed in this paper. This paper presented a new simple, easy topology to achieve great results for specific applications. This technique is based on the voluntary creation of mismatched bands around the rejection band to create subbands for particular applications. Hence, T-shaped stubs in short-and-open terminated configurations are put together through an alternation of these to achieve the goals. Also, all stubs and the mainline section have identical electric lengths except for their width. The design dimensions are based on the choice of Q-factor, which must be approximated when a frequency is chosen. This Q-factor fixes and controls the bandwidth of both main bands and four subbands. The excellent final goals are achieved when a transmission zero is obtained via a stopband. The bandpass filter’s structure comprises five identical resonators and four transmission line sections. All resonators are connected by a line section, as presented in Figure 1. The short circuit terminates three resonators. The metallic vias symbolize short-circuit configurations. Two resonators, ended by the open circuit configurations, are inserted between two short-circuited stubs. As described in [46], each design section can be transformed as a combination of the lumped elements. But the lumped element topology has attracted the researcher’s attention. The inductance and capacitance values are determined using the Chebyshev and/or Butterworth’s table or special calculations [11, 47–49].Figure 1
The cross-section of the prototype passband filter.The line section (constituted ofθ the electric length and Z0 the characteristic impedance) is perpendicular to stubs whose admittance is Y1 and long of θ). That gives the BPF’s shape. The normalized admittance y1 is provided as follows,
(1)y1=Y1Y0where Y0 is the line section admittance. The electric length can be defined as a frequency variation parameter and given by the following equation,
(2)θ=2πf0cl0εeff′μeff′,with εeff′ and μeff′ are the effective permittivity and permeability, respectively, c the vacuum’s light velocity, l0 the transmission line section length, and f0 the operating frequency. At the same time, the filter’s electric length θT is given below,
(3)θTf=2πfcLεeff′μeff′.The total PBF length isL, and f is the frequency covering the entire scanned frequency. Let us consider w, the stub’s width. In that case, the BPF global length is given by Equation (4),
(4)L=4l0+5w.As shown in Figure1, stubs are placed in a T-shaped and cascading position.


## 2.2. The Methodology Approach

In such a configuration, the first step consists of choosing the resonator’s number and length. This work uses two open-circuited and three short-circuited. That makes the design a five-order filter. Next, a line section is settled between two stubs that differ from their terminated configuration. It means that the short-and-open terminated stubs are mixed through their alternation position. All section lines and stubs are quarter-wavelengthλ/4 and well described in [46, 50]. The third step focuses on the Q− factor of each stub and the global Qg (also called initial quality factor Qi to fix) ones, where the subscript letter “g” indicates the global computation. It appears that using Qg and λ/4 becomes a great option [51]. The open-circuited λ/4 line leads to the stopband filter [52]. In the methodology presented in this paper, the stubs and the mainline transmission sections have identical lengths but differ in width. It means that their impedances (or admittances) are equal. Let us consider Qp and Qs, the normalized Q of parallel and series stubs as described in [46], respectively. These are given in the following equations for five resonators:
(5)Qs=8π1/y121+2y18−1/2,and
(6)Qp=π2y1y1+22161/3.In the proposed approach, three short-circuited stubs are interspersed by two open-circuited stubs, as shown in Figure1. As defined in [40], the Qap is given as follows,
(7)1Qap=1Qs+1QpSo, Equation (7) becomes as given below,
(8)Qap=2332πy1261+2y1+256y1y1+223.This equation establishes a link between theQap and the admittance of the stubs and the line sections. The fourth step needs to define the normalized admittance and apply an empirical formula to determine a value that lets the designer approximate the Qi needed through the computer-aided software.
(9)y1=2πf0Z0Xint,where Z0=50Ω, the connector impedance that is welded to the prototype input and output accesses. Finally, the designer chooses the operating frequency f0 and Qi to approximate Qap≅Qi. Both parameters define the stopband bandwidth around f0, creating two bandwidths where the high-frequency band is the harmonic of the low-frequency band. Consequently, mixing both configurations leads to the transmission zero (TZ) and a dual-band BPF with subbands in each bandwidth. The approach developed in this paper focuses on the study’s simplicity. The system must be designed with perfect symmetry to avoid input and output mismatch and those of the subbands, while the main dual-wideband is mismatched.Because of the stubs and the transmission mainline section dimensions, the rejection band shifts or slides by 1 GHz from the theoreticalf0.The BPF analysis is made from the scattering S-parameters matrix. The characteristics (return lossΓ, insertion loss IL, bandwidth Δ, cut-off frequency fc, group delay φgd) of the BPF are read from those S-parameters. The return loss (RL) is the signal’s power loss reflected by a discontinuity in a transmission line. This symbolizes the ratio of the reflected power to the incident power [53], as given below,
(10)ΓdB=−10logS112,where S11 is the reflection coefficient. Furthermore, insertion loss [8] is the amount of energy a signal loses as it travels through a component or a system [54, 55], such as the BPF device.
(11)ILdB=−10logS212,where S21 is the transmission coefficient. Whatever the BPF is, it presents a bandwidth Δ, defined from two cut-off frequencies, fh and fl, where fh>ff. In that case, the fractional bandwidth (FBW) is often called to indicate how large the bandwidth is from the cut-off frequencies. That parameter varies between 0% and 2%. The FBW is given as follows,
(12)FBW%=10021−2flΔ+2fl,where
(13)Δ=fh−fl.Reading Equation (12), the FBW depends on the bandwidth. The wider the bandwidth, the bigger the FBW. The Q is the inverse of the fractional resonator bandwidth and is determined as follows,
(14)Qap=f0Δ.Therefore, the time delay is given as shown in the following equation:(15)τ=lvp,where vp is the phase velocity. This parameter, also named the phase delay ϕpd, expressed in second, can also be defined as given [56, 57] below,
(16)ϕpd=−θTf2πf,where the electric length θf is expressed in radians. The phase delay can be expressed in nanoseconds (ns) if the frequency is in gigahertz (GHz). At the same time, the group delay φgd is obtained from the phase delay [58] through the following equation:
(17)φgd=−dθTfdf.The substitution of Equation (2) in Equation (16) gives Equation (18),
(18)ϕpd=2πcLεeff′μeff′.By subtractingθf in Equation (2) and replacing it in Equation (17), the filter’s group delay becomes as follows:
(19)φgd=Lcεeff′μeff′,which is expressed in seconds. Usually, this characteristic quantity is given in nanoseconds. Then, Equation (19) is rewritten as,
(20)φgdns=12π.dθTfdfGHz.Another important parameter that helps analyze the filter in high frequencies is the phase velocity in meters per second. From Equations (3) and (15), the substitution of the phase velocity gives,
(21)vφ=2πLθTff.Equation (21) talks about the phase velocity, which depends on the electric length behavior. That velocity decreases, stabilizes, or increases as the electric length increases or decreases.The methodology adapted can be summed up as follows: first, the short-and-open terminated stubs are mixed through their alternation position, creating the subbands. Second, all stubs and mainline sections are quarter-wavelength and calculated at the same operating frequency and the stub’s characteristic impedance. Third, at that frequency, the mainline impedance is chosen as 50Ω and differs from the stub ones. This creates the low-frequency and harmonic (high-frequency) bandwidths mismatch, depending on the Q. Fourth, an initial Qi that is fixed (chosen according to the designer’s objectives) and found must be recovered. This new Qap called approximated quality factor is linked to the structure’s impedance. Finally, the computer-aided software finds the unknown parameter leading to satisfying the condition Qap=Qi. In that case, the operating frequency belongs to the stopband, creating the transmission zero.


## 3. Measurement: Results and Discussion

The approach technique has been simulated, prototyped, and experimentally validated with a well-known material: FR4 HTG-175, which has 1 mm of thickness, a theoretical relative permittivity (εr), and a loss tangent (tanδ) of 4.4 and 0.02, respectively. The material intrinsic parameters are supposed to be constants during the dimensions and admittance determination. Still, according to the technique used to extract those electric parameters, the accuracy changes as long as those parameters [59, 60]. Therefore, at the operating frequency f0=10.7GHz, Qg=0.9 is the target to manufacture a dual BW and split them into two subbands. As a result, three metallic vias have been made to connect the substrate top and bottom for short-circuit stubs, as shown in Figure 2. Table 1 gives the design-specific parameters. By using Xint=252.75×10−15, the BPF dimensions obtained are depicted in Table 1.Figure 2
(a) The bandpass filter prototype, with its feedline at each access; (b) the bandpass filter feedline.
(a)(b)Table 1
The manufactured prototype parameters.
ParametersMainlineResonators 1, 3, and 5Resonators 2 and 4Impedance (Ω)5058.8558.85Width (mm)1.91191.4381.438Diameter of the metallic through hole (mm)None1.6NoneLine section’s length (mm):l03.8086


### 3.1. Extraction of the Electric Length

The high-frequency structure simulator (HFSS) software has been used for the study. The experimental results are computed after measurements with a vectorial network analyzer (VNA) Anritsu MS4642B in the frequency range of 3 GHz–15 GHz.Figure2(a) shows the entire prototype, while Figure 2(b) presents only the feedline as a transmission line. During the deembedding stage, that feedline will be removed. Finally, in Figure 3, the prototype is connected to the SMA, linked to the prototype, and the VNA through the two access ports. The prototype is 42.46 mm long, and the feedline is 20.01 mm. As shown in Figure 2, the accurate filter’s length is 22.45 mm.Figure 3
The passband filter prototype with the SMA access ports.In Equation (4) and the specifications in Table 1, the design length error can be estimated at 0.0256 mm. The measures made are plotted in Figures 4–7.Figure 4
Comparison of the simulated and measured frequency response of the dual-wideband bandpass filter.Figure 5
Frequency response measured comparison of the attenuation coefficient (AC) and insertion loss (IL).Figure 6
(a) Zoom of the frequency response (2 GHz–6 GHz) of the reflection and transmission coefficients of the BPF; (b) zoom of the frequency response (11 GHz–15 GHz) of the reflection and transmission coefficients of the BPF.
(a)(b)Figure 7
(a) Zoom of the frequency response (3 GHz–6 GHz) comparison of the attenuation coefficient and insertion loss; (b) zoom of the frequency response (11 GHz–15 GHz) of the attenuation coefficient and insertion loss.
(a)(b)The attenuation coefficient parameter (ACP), as defined in [61, 62], depends on the frequency, the RL, and the IL (see Figures 5 and 7). At the same time, when the system is well matched, the ACP equals the IL. There is no fixed matched level, as it depends on the designer and the target application domain. But, as long as the bandwidth is identified from the 10 dB of the RL, this assumption finds its importance, and the IL becomes lower than 10 dB. The filter’s physical dimensions have been calculated from a fixed relative permittivity and operating frequency. However, this parameter changes with frequency, and its accuracy depends on the material characterization technique used as developed in [63, 64]. Hence, the BW will move slightly in comparison to the target. Furthermore, physical dimensions (length and via diameters) have tolerance gaps that can change the expected results.Both wideband are separated by a stopband from 5.7 GHz to 11.57 GHz with a peak rejection at 9.373 GHz, while the transmission coefficient reaches -28.118 dB. Therefore, a return loss of 2.044 dB is achieved at that frequency. Table2 shows that lower and upper bands have 53.282% and 21.456% as FBW. Several applications are possible with this prototype, like 5G. Each wideband has two subbands, read at -10 dB of the reflection coefficient, level of system adaptation definition. Results closely conform with the concept step despite some manufacturing impacts (see Figures 4, 6, and 7). The main reason for the mismatch is the weakness of the Q-factor that has been used. As seen in Figures 8 and 9, which present the simulation results from the prototype designed on the Agilent system software (ADS), the higher the Q-factor, the more significant the stub’s width and the subbands shrink. At the same time, the isolation gets better, and the BPF becomes a dual-band or three-band bandwidths structure (according to the manufacturing constraints and accuracy) with two harmonics at its high-frequency band. The lower the quality factor, the better the adaptation of the main band. The structure is reduced to a system without a subband, and the bandwidth becomes ultrawide.Table 2
Synthesis of bandpass filter electrical performances manufactured on FR4 HTG-175, selected at 3 dB of IL.
Cut-off frequency (GHz)BW (GHz)FBW (%)Lowest IL (dB) at frequency (GHz)FirstflSecondfhBand 13.1465.4312.28553.2820.656 at 3.794Band 211.89114.7492.85821.4563.027 at 12.974Figure 8
(a) Comparison of the BPF electric simulation results whenZ0=Z1=50Ω and Z0≠Z1=58.85Ω in 1 GHz-8 GHz; (b) comparison of the BPF electric simulation results when Z0=Z1=50Ω and Z0≠Z1=58.85Ω in 8 GHz-16 GHz.
(a)(b)Figure 9
(a) Comparison of the BPF electric simulation results from the normalized quality factor variations in 1  GHz-8 GHz; (b) comparison of the BPF electric simulation results from the normalized quality factor variations in 8 GHz-16 GHz.
(a)(b)It is noticed that all simulation and experimental results have harmonic in their high-frequency band. This is because the Q-factor does not control these harmonic bands. Finally, using a high Q-factor leads stub section line behaving as capacitor elements and inductance when it becomes deficient (low). At this stage, a compromise is necessary according to the designer’s objectives. The stub width isw=3.1755mm, representing a characteristic impedance Z≈36.04Ω for Q=1.2.Each split band from the wideband is well matched to the device system, and these sidebands cover a minimum of 372 MHz. Additionally, there is a wideband and a narrowband as a subband. Therefore, when analyzing Equation (12), the biggest BW does not mean the greater FBW, as noticed in Tables 2 and 3. The electromagnetic signal propagates at a velocity that depends on the medium to cross over and the frequency range, as Equation (21) says. This gives much information about the group delay (GD), as presented in Equation (17). Figures 10 and 11 depict the phase velocity and group delay curves below.Table 3
Frequency subband characteristics of the manufactured bandpass filter prototype, considered at 10 dB of RL.
Frequency subbandCut-off frequency (GHz)Resonance freq. (GHz) at RL (dB)Bandwidth (MHz)FBW (%)LowestHighestBand 1Subband 13.3273.7093.486 at 28.74138210.858Subband 24.4425.0484.782 at 28.00160612.771Band 2Subband 311.92212.48612.146 at 19.0175644.621Subband 414.28114.65314.579 at 12.043722.571Figure 10
Experimental results of the phase velocity of the proposed bandpass filter.Figure 11
Experimental results of the group delay of the proposed bandpass filter.The linear phase response is required to avoid signal waveform distortion in communication systems [39]. Moreover, the linear phase causes a flat GD response, which is essential, especially in the microwave domain [11, 65]. Therefore, the fluctuation of the filter GD is often controlled when below 0.7 ns, and the waveform distortion can be avoided [65]. For example, Figure 11 presents a GD below 0.21 ns in the entire frequency range, with a flat response into the upper band. At the same time, the phase velocity decreases slightly. Combining Equations (3) and (21), by substituting the phase velocity when using the dielectric material, the phase velocity is inversely proportional to the effective permittivity in the propagation medium. Equation (21) becomes as given below,
(22)vφ=cεeff′.Figure10 deals well with Equation ((22)).Figure12 shows that the open-terminated resonator’s electric field distribution is high. It is noticed that the resonance frequency is better enough, as read in Figure 4. However, as the E-field distribution decreases, the filter is not matched enough in Figure 10(b), unlike Figure 10(a).Figure 12
Simulated electric field distribution of the BPF at (a) 3.486 GHz, (b) 4.782 GHz, (c) 12.146 GHz, and (d) 14.579 GHz.Table4 presents a few domains where the suggested BPF can be used. This new approach helps select some application domains and rejects the others when fixing the BW and the RL to be accepted in that range. For example, the WLAN can be inserted in the first subband while only a sub-3.6 GHz for the 5G. Finally, a study comparison is made in Table 5 with other works to highlight the great advantage of developing such a structure.Table 4
Application areas of the suggested BPF prototype.
Frequency rangeSubband frequencyApplication domainsBand 13.146–5.4313.327–3.709Worldwide interoperability for microwave access (WiMAX): 3.4 GHz–3.6 GHzFifth generation (5G): 3.4 GHz–3.8 GHzWireless access systems (WAS): 3.4 GHz–3.6 GHzWi-Fi: 3.6 GHz, 4.9 GHz, and 5.0 GHzSatellite TV: 3.7 GHz–4.2 GHzWireless local area network (WLAN): 4.9 GHz–5.0 GHz4.442–5.048Band 211.891–14.74911.922–12.486Fixed satellite service (FSS): 11.7 GHz–12.2 GHzSatellite earth station: 11.70 GHz–12.75 GHzKu-band satellite TV: 11.7 GHz–12.7 GHzMilitary: 11.7 GHz–13.25 GHzDirect broadcast satellite (DBS): 12.2 GHz–12.7 GHz14.281–14.653Table 5
Comparison with other multiwideband balanced BPFs.
Ref.TechnologyTZBands (GHz)3 dB FBW (%)IL (dB)Circuit size (mm2)εr @ tanδ @ h(mm)[3]CLB43.62/4.53/5.319.9/4.4/5.351.0/0.92/0.9625.8x10.92.54 @2×10−3 @ 0.54[6]UIR41.9/3.58/5.7517.5/12.2/8.90.25/0.37/0.933.6x20.22.2 @9×10−4 @ 0.787[12]MER62.595/5.7515/12.81.58/2.4218.19x4.93.38 @2.7×10−3 @ 0.508[27]QMSIW-CSRR23.59/5.633.3/6.40.98/0.8118.7x15.32.2 @ none @ 0.508[30]PatchNone2.74/6.011.0/4.81.6/2.335.18x35.183.66 @4×10−3 @ 0.946[37]SIR60.92/2.08/3.5940.21/10.1/16.150.41/1.39/1.9730x373.66 @4×10−3 @ 0.762[66]Lumped elements43.5/5.59.42/7.760.7/1.116.6x83.38 @2.7×10−3 @ 0.813This workIdentical resonators24.29/13.3253.282/21.4560.656/3.02722.45x5.724.4 @2×10−2 @ 1In Table5, the proposed BPF is one of the smallest compared to the others. At the same time, it has the largest 3 dB FBW, while the IL reaches a minimum of 0.656 dB and 3.027 dB in the lower and upper bands. Also, it suggests a simple technology in its feasibility during the fabrication. To improve the insertion loss, choose lossless insulator material (dielectric) with low thicknesses, reduce the filter’s size (the fabrication technology will be vital as it is limited nowadays to 100 μm), and the dielectric material with thick conductor width. In addition, the use of materials with very high conductivity is ideal (gold, etc.). Therefore, this BPF is suitable for several radars and other communication systems, as presented in Table 4.


## 3.1. Extraction of the Electric Length

The high-frequency structure simulator (HFSS) software has been used for the study. The experimental results are computed after measurements with a vectorial network analyzer (VNA) Anritsu MS4642B in the frequency range of 3 GHz–15 GHz.Figure2(a) shows the entire prototype, while Figure 2(b) presents only the feedline as a transmission line. During the deembedding stage, that feedline will be removed. Finally, in Figure 3, the prototype is connected to the SMA, linked to the prototype, and the VNA through the two access ports. The prototype is 42.46 mm long, and the feedline is 20.01 mm. As shown in Figure 2, the accurate filter’s length is 22.45 mm.Figure 3
The passband filter prototype with the SMA access ports.In Equation (4) and the specifications in Table 1, the design length error can be estimated at 0.0256 mm. The measures made are plotted in Figures 4–7.Figure 4
Comparison of the simulated and measured frequency response of the dual-wideband bandpass filter.Figure 5
Frequency response measured comparison of the attenuation coefficient (AC) and insertion loss (IL).Figure 6
(a) Zoom of the frequency response (2 GHz–6 GHz) of the reflection and transmission coefficients of the BPF; (b) zoom of the frequency response (11 GHz–15 GHz) of the reflection and transmission coefficients of the BPF.
(a)(b)Figure 7
(a) Zoom of the frequency response (3 GHz–6 GHz) comparison of the attenuation coefficient and insertion loss; (b) zoom of the frequency response (11 GHz–15 GHz) of the attenuation coefficient and insertion loss.
(a)(b)The attenuation coefficient parameter (ACP), as defined in [61, 62], depends on the frequency, the RL, and the IL (see Figures 5 and 7). At the same time, when the system is well matched, the ACP equals the IL. There is no fixed matched level, as it depends on the designer and the target application domain. But, as long as the bandwidth is identified from the 10 dB of the RL, this assumption finds its importance, and the IL becomes lower than 10 dB. The filter’s physical dimensions have been calculated from a fixed relative permittivity and operating frequency. However, this parameter changes with frequency, and its accuracy depends on the material characterization technique used as developed in [63, 64]. Hence, the BW will move slightly in comparison to the target. Furthermore, physical dimensions (length and via diameters) have tolerance gaps that can change the expected results.Both wideband are separated by a stopband from 5.7 GHz to 11.57 GHz with a peak rejection at 9.373 GHz, while the transmission coefficient reaches -28.118 dB. Therefore, a return loss of 2.044 dB is achieved at that frequency. Table2 shows that lower and upper bands have 53.282% and 21.456% as FBW. Several applications are possible with this prototype, like 5G. Each wideband has two subbands, read at -10 dB of the reflection coefficient, level of system adaptation definition. Results closely conform with the concept step despite some manufacturing impacts (see Figures 4, 6, and 7). The main reason for the mismatch is the weakness of the Q-factor that has been used. As seen in Figures 8 and 9, which present the simulation results from the prototype designed on the Agilent system software (ADS), the higher the Q-factor, the more significant the stub’s width and the subbands shrink. At the same time, the isolation gets better, and the BPF becomes a dual-band or three-band bandwidths structure (according to the manufacturing constraints and accuracy) with two harmonics at its high-frequency band. The lower the quality factor, the better the adaptation of the main band. The structure is reduced to a system without a subband, and the bandwidth becomes ultrawide.Table 2
Synthesis of bandpass filter electrical performances manufactured on FR4 HTG-175, selected at 3 dB of IL.
Cut-off frequency (GHz)BW (GHz)FBW (%)Lowest IL (dB) at frequency (GHz)FirstflSecondfhBand 13.1465.4312.28553.2820.656 at 3.794Band 211.89114.7492.85821.4563.027 at 12.974Figure 8
(a) Comparison of the BPF electric simulation results whenZ0=Z1=50Ω and Z0≠Z1=58.85Ω in 1 GHz-8 GHz; (b) comparison of the BPF electric simulation results when Z0=Z1=50Ω and Z0≠Z1=58.85Ω in 8 GHz-16 GHz.
(a)(b)Figure 9
(a) Comparison of the BPF electric simulation results from the normalized quality factor variations in 1  GHz-8 GHz; (b) comparison of the BPF electric simulation results from the normalized quality factor variations in 8 GHz-16 GHz.
(a)(b)It is noticed that all simulation and experimental results have harmonic in their high-frequency band. This is because the Q-factor does not control these harmonic bands. Finally, using a high Q-factor leads stub section line behaving as capacitor elements and inductance when it becomes deficient (low). At this stage, a compromise is necessary according to the designer’s objectives. The stub width isw=3.1755mm, representing a characteristic impedance Z≈36.04Ω for Q=1.2.Each split band from the wideband is well matched to the device system, and these sidebands cover a minimum of 372 MHz. Additionally, there is a wideband and a narrowband as a subband. Therefore, when analyzing Equation (12), the biggest BW does not mean the greater FBW, as noticed in Tables 2 and 3. The electromagnetic signal propagates at a velocity that depends on the medium to cross over and the frequency range, as Equation (21) says. This gives much information about the group delay (GD), as presented in Equation (17). Figures 10 and 11 depict the phase velocity and group delay curves below.Table 3
Frequency subband characteristics of the manufactured bandpass filter prototype, considered at 10 dB of RL.
Frequency subbandCut-off frequency (GHz)Resonance freq. (GHz) at RL (dB)Bandwidth (MHz)FBW (%)LowestHighestBand 1Subband 13.3273.7093.486 at 28.74138210.858Subband 24.4425.0484.782 at 28.00160612.771Band 2Subband 311.92212.48612.146 at 19.0175644.621Subband 414.28114.65314.579 at 12.043722.571Figure 10
Experimental results of the phase velocity of the proposed bandpass filter.Figure 11
Experimental results of the group delay of the proposed bandpass filter.The linear phase response is required to avoid signal waveform distortion in communication systems [39]. Moreover, the linear phase causes a flat GD response, which is essential, especially in the microwave domain [11, 65]. Therefore, the fluctuation of the filter GD is often controlled when below 0.7 ns, and the waveform distortion can be avoided [65]. For example, Figure 11 presents a GD below 0.21 ns in the entire frequency range, with a flat response into the upper band. At the same time, the phase velocity decreases slightly. Combining Equations (3) and (21), by substituting the phase velocity when using the dielectric material, the phase velocity is inversely proportional to the effective permittivity in the propagation medium. Equation (21) becomes as given below,
(22)vφ=cεeff′.Figure10 deals well with Equation ((22)).Figure12 shows that the open-terminated resonator’s electric field distribution is high. It is noticed that the resonance frequency is better enough, as read in Figure 4. However, as the E-field distribution decreases, the filter is not matched enough in Figure 10(b), unlike Figure 10(a).Figure 12
Simulated electric field distribution of the BPF at (a) 3.486 GHz, (b) 4.782 GHz, (c) 12.146 GHz, and (d) 14.579 GHz.Table4 presents a few domains where the suggested BPF can be used. This new approach helps select some application domains and rejects the others when fixing the BW and the RL to be accepted in that range. For example, the WLAN can be inserted in the first subband while only a sub-3.6 GHz for the 5G. Finally, a study comparison is made in Table 5 with other works to highlight the great advantage of developing such a structure.Table 4
Application areas of the suggested BPF prototype.
Frequency rangeSubband frequencyApplication domainsBand 13.146–5.4313.327–3.709Worldwide interoperability for microwave access (WiMAX): 3.4 GHz–3.6 GHzFifth generation (5G): 3.4 GHz–3.8 GHzWireless access systems (WAS): 3.4 GHz–3.6 GHzWi-Fi: 3.6 GHz, 4.9 GHz, and 5.0 GHzSatellite TV: 3.7 GHz–4.2 GHzWireless local area network (WLAN): 4.9 GHz–5.0 GHz4.442–5.048Band 211.891–14.74911.922–12.486Fixed satellite service (FSS): 11.7 GHz–12.2 GHzSatellite earth station: 11.70 GHz–12.75 GHzKu-band satellite TV: 11.7 GHz–12.7 GHzMilitary: 11.7 GHz–13.25 GHzDirect broadcast satellite (DBS): 12.2 GHz–12.7 GHz14.281–14.653Table 5
Comparison with other multiwideband balanced BPFs.
Ref.TechnologyTZBands (GHz)3 dB FBW (%)IL (dB)Circuit size (mm2)εr @ tanδ @ h(mm)[3]CLB43.62/4.53/5.319.9/4.4/5.351.0/0.92/0.9625.8x10.92.54 @2×10−3 @ 0.54[6]UIR41.9/3.58/5.7517.5/12.2/8.90.25/0.37/0.933.6x20.22.2 @9×10−4 @ 0.787[12]MER62.595/5.7515/12.81.58/2.4218.19x4.93.38 @2.7×10−3 @ 0.508[27]QMSIW-CSRR23.59/5.633.3/6.40.98/0.8118.7x15.32.2 @ none @ 0.508[30]PatchNone2.74/6.011.0/4.81.6/2.335.18x35.183.66 @4×10−3 @ 0.946[37]SIR60.92/2.08/3.5940.21/10.1/16.150.41/1.39/1.9730x373.66 @4×10−3 @ 0.762[66]Lumped elements43.5/5.59.42/7.760.7/1.116.6x83.38 @2.7×10−3 @ 0.813This workIdentical resonators24.29/13.3253.282/21.4560.656/3.02722.45x5.724.4 @2×10−2 @ 1In Table5, the proposed BPF is one of the smallest compared to the others. At the same time, it has the largest 3 dB FBW, while the IL reaches a minimum of 0.656 dB and 3.027 dB in the lower and upper bands. Also, it suggests a simple technology in its feasibility during the fabrication. To improve the insertion loss, choose lossless insulator material (dielectric) with low thicknesses, reduce the filter’s size (the fabrication technology will be vital as it is limited nowadays to 100 μm), and the dielectric material with thick conductor width. In addition, the use of materials with very high conductivity is ideal (gold, etc.). Therefore, this BPF is suitable for several radars and other communication systems, as presented in Table 4.


## 4. Conclusions

This paper described a new and straightforward approach to implementing several matched subbands from mismatched dual wideband bandpass filters. Identical resonators have created a rejection band, and two unmatched wide bandwidths (low-frequency and one harmonic) split into four subbands. The design is an alternation of quarter-wavelength resonators with short-and-open circuit terminations. There are positioned to each other symmetrically, and an approximationQap is determined from an operation frequency and compared to the initial chosen Qi. The unknown parameter allows the final results to be calculated from computer-aided software that satisfies a developed mathematical expression. The suggested technique has been validated. Furthermore, a five-order BPF with the FR4 HTG-175 (εr=4.4 and tanδ=0.02) having 1 mm of thickness has been prototyped.The BPF provides two wide unmatched BW at 2.285 GHz/2.858 GHz with 53.282%/21.456% as FBW, while its subbands have four matched BWs, 382 MHz/606 MHz/564 MHz/372 MHz obtained from 10 dB of the RL. The prototype has an IL of 0.656 dB/3.027 dB with three TZ. This device covers radar and other applications, particularly 5G, Wi-Fi, WAS, WLAN, WiMAX, FSS, DBS, etc. The newly developed approach uses a mathematical model to determine an unknown parameter value to reach the targetedQi, which controls the BPF bandwidth (dual wideband and four subbands) and the device matching when a fixed operation frequency is chosen. It has been designed, modelized, and prototyped before experimental validation with the VNA Anritsu MS4642B two-port measurements in the scanned frequency range of 3 GHz–15 GHz. The measured results have exhibited good agreement with simulated ones; slight discrepancies may have been observed due to fabrication errors. The proposed BPF has the advantages of low IL and group delay, high FBW, wide BW, small sizes, and an extended range of communication systems. A high Q provides an ample stub width and excellent isolation but shrinks the subbands.In contrast, the low Q leads to small stub width, broadens bandwidth, and mismatches the band. Both cases create harmonic in the high-frequency band. Therefore, as previously said, this prototype is a good candidate for some applications compared to others in the literature.



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*Source: 4713995-2023-03-14.xml*