# Miniaturized Quadband High-Temperature Superconducting Bandpass Filter with a Wide Frequency Ratio

Authors: Liejun Shen; Jiaqi Ju; Zhaojiang Shang
Journal: International Journal of RF and Microwave Computer-Aided Engineering
 (2023)

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## Abstract

This paper presents a miniaturized quadband high-temperature superconducting (HTS) bandpass filter based on a multistub-loaded resonator. Owing to the symmetrical structure of the proposed resonator, the odd- and even-mode analysis method can be used to derive its resonance frequencies. First, the frequency ratio between passbands may be significantly changed by adjusting the impedance ratio and fold spacing of the resonator. Moreover, the proposed filter exhibits well-controlled center frequencies and bandwidths for the four passbands, resulting in a considerably enhanced design possibility of quadband bandpass filters. A low insertion loss within each band is obtained using HTS technology. Additionally, the proposed filter was etched on an MgO substrate with a thickness of 0.5 mm and combined with YBCO thin films for demonstration. The results obtained were highly consistent with those of the simulation, validating the effectiveness and viability of the proposed method.


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## Body

## 1. Introduction

Bandpass filters (BPFs), which are located in radio frequency front ends, are key parts of transmitters and receivers [1]. With significant progress in wireless communication, multiband microstrip BPFs have played a crucial role in the integration of multiple communication modes into a single system, and different frequency bands fulfill the requirements of different standards [2].Owing to the advances in high-temperature superconducting (HTS) materials and processing technology (in terms of stability), HTS microstrip filters and their design have gained significance. Recently, BPFs with multiple passbands have been extensively studied [3–11]. A miniaturized quint-band HTS BPF was developed based on a multimode resonator with a single perturbation [3]. A stepped-impedance stub-loaded stepped-impedance resonator (SIR) was proposed to design a dual-band BPF [4]. A compacted tri-band HTS BPF with controllable feeding/coupling structures based on crossed resonators was developed [5]. A stepped-impedance stub-loaded interdigital capacitor resonator for designing a dual-band BPF with a large bandwidth ratio was also developed [6]. As mentioned, all the multiband filters in References [3–6] were implemented using HTS-based technologies. These studies have summarized the advantages and limitations of HTS BPFs using two modes and bands [7]. Despite the widespread use of HTS, few attempts to design multiband filters using HTS technology have been reported [8–11].This study proposes a quadmode T-shaped resonator and developed its equivalent circuit to clarify the resonance characteristics of the proposed resonator. Subsequently, two resonators were employed to design a quadband HTS BPF. Herein, both frequencies and bandwidths of the four bands are controllable. Additionally, the filter achieves a high frequency ratio. The results obtained are highly consistent with the simulation results.


## 2. Resonator Analysis and Filter Design

### 2.1. T-Shaped Resonator Mode Analysis and Frequency Control

The schematic of the proposed resonator, comprising a2/λ uniform impedance resonator (UIR) (Z2 and θ2) with a two-end-open midloaded stub (Z3 and θ3) and four open stubs (Z1 and θ1), is shown in Figure 1(a). The resonance characteristics of this resonator can be analyzed using the even–odd mode method because the characteristics are symmetrical along the T-T’ plane. The odd- and even-mode equivalent circuits are shown in Figures 1(b) and 1(c), respectively.Figure 1
(a) Transmission line model of the proposed resonator. (b) Odd-mode equivalent circuit. (c) Even-mode equivalent circuit.
(a)(b)(c)Hence, the input admittance of the proposed resonator under even- and odd-mode excitation can be expressed as follows:(1)Yin−odd=−2jY1Y2cotθ2−2Y1tanθ12Y1+Y2tanθ1cotθ2,Yin−even=2jY1Y2Y2tanθ2+Y3tanθ3+2Y1tanθ1Y2−Y3tanθ2tanθ32Y1Y2−Y3tanθ2tanθ3−Y2tanθ1Y2tanθ2+Y3tanθ3.The model contained various variables, and the two initial conditions were assumed to beθ1=θ2=θ and Y2=Y3 for convenience of analysis. From a transmission line model analysis, the odd- and even-mode resonance conditions are expressed as follows:
(2)2tan2θ=R,(3)2cot1+αθ=−Rcotθ.Equations (2) and (3) yield the resonance electrical lengths of odd and even-modes, respectively. Herein, R=Y2/Y1 and α=θ3/θ. From Equations (2) and (3), the central stubs (Y3 and θ3) influence the even-mode resonant frequency but not that of the odd-mode. Therefore, the resonator’s resonant frequency can be precisely controlled. Based on Equations (2) and (3), the variables θ1 and θ2 influence the odd- and even-mode frequencies. Therefore, the curious mode frequency can be adjusted by changing (Y3 and θ3) to adjust the even-mode frequency.Note thatθ has different solutions for different values of R and α. The design curves of the first two resonant frequencies of the resonator are shown in Figure 2. Fodd and Feven denote the odd- and even-mode frequencies, respectively. As the 2/λ UIR (Z2 and θ2) part will be folded with equal spacing in the following sections, the resonant frequency positions of Feven2 and Fodd2 will significantly change; conversely, those of Feven1 and Fodd1 will have minimal change. Therefore, the resonant frequencies of Feven1 and Fodd1 are mainly analyzed here. When R and α assume typical values, the achievable frequency ratio ranges from 1.2 to 3.4. The frequency ratios for different values of α when R=0.5,1,1.5,2,and3 are shown in Figure 2, according to which Feven1/Fodd1 decreases with increasing R and α. Therefore, based on the aforementioned analysis, the desired specifications can be identified from Figure 2 by choosing different combinations of R and α. The admittance ratio of the resonator can be varied to obtain a wide range of frequency ratios. Compared with the traditional SIR, the multistub-loaded resonator can obtain a wider range of frequency ratios, assuming the same admittance ratio.Figure 2
Frequency ratios ofFeven1/Fodd1 with different values of α when R=0.5,1,1.5,2,and3.For demonstration, the proposed T-shaped resonator is utilized to implement a quadband HTS BPF. The2/λ UIR (Z2 and θ2) part is folded with equal spacing to easily realize the quadband design. The microstrip structure of a quadmode multistub-loaded resonator is shown in Figure 3(a). A Sonnet EM-simulated resonance response under a weak-coupling condition is shown in Figure 3(b). Realization of a quadband design of the unfolded resonator is challenging; however, the folded resonator significantly reduces the frequency difference between two sets of odd–even modes, simplifying the quadband design. Using the resonator depicted in Figure 3(a) as an example, before the resonator folds, Feven2/Fodd1=3.775 and Fodd2/Feven1=2.188. After folding the resonator, Feven2/Fodd1=3.212 and Fodd2/Feven1=1.84. Therefore, the frequency difference between the two sets of odd–even modes can be reduced using a folding resonator.Figure 3
(a) Proposed T-shaped multimode resonator. (b) EM simulation results under weak-coupling condition.
(a)(b)Using a full-wave simulation software Sonnet, the resonator’s frequency response was analyzed under weak-coupling conditions. Figure4 depicts the modal resonance characteristics at different microstrip line lengths of the resonator, namely, L1, L2, L4, d1, and W2. In particular, the microstrip line lengths L1 and L4 affect both odd- and even-mode frequencies. The even-mode frequency can be controlled by the microstrip line length L2 alone. The microstrip line lengths d1 and W2 can change the frequency difference between the two sets of odd–even modes. Keeping the other variables constant, when the microstrip line length d1 is 0.28, 0.36, and 0.44 mm, Feven2/Fodd1 and Fodd2/Feven1 correspond to 3.106, 3.212, 3.301 and 1.828, 1.84, and 1.854, respectively. Therefore, the frequency difference between the two sets of odd–even mode increases with increasing d1. Conversely, keeping the other variables constant, when the microstrip line length W2 is 0.16, 0.24, and 0.32 mm, Feven2/Fodd1 and Fodd2/Feven1 correspond to 3.279, 3.212, 3.189 and 1.929, 1.84, and 1.787, respectively. Therefore, the frequency difference between the two sets of odd–even modes decreases with increasing W2. This trend is consistent with previous studies.Figure 4
Modal resonant characteristics with varied microstrip line lengths. Varying (a)L2, (b) L1 and L4, (c) d1, and (d) W2 from Figure 3.
(a)(b)(c)(d)In summary, the initial structure of the resonator can be determined as follows. First, the impedance ratio of each stub is determined before the resonator is folded. Next, the length ofd1 is adjusted with equal spacing in the folded resonator to ensure that the frequency difference between the two sets of odd–even modes is in a relatively suitable range. Additionally, the impedance at the fold is slightly tuned to determine the frequency difference between the two sets of odd–even modes. Thus, the lengths of L1 and L4 are fine-tuned to achieve the desired fundamental frequency. Finally, L2 is adjusted to change the position of the two sets of even-mode frequencies.


### 2.2. Resonator Coupling Analysis and Bandwidth Control

After determining the initial structure of the resonator, the main challenge is to investigate the coupling control mechanism to meet the required design parameters, including the internal coupling coefficientsk and external quality factor Qe for the prescribed specifications in the four passbands.The coupling configuration of the proposed resonator is shown in Figure5. According to the method reported in Reference [12], the coupling coefficients at four resonant frequencies can be extracted using the following formula:
(4)k=f22−f12f22+f12,where f1 and f2 are the lower and higher coupling resonant frequencies, respectively.Figure 5
Configuration of the two coupled resonators.The initial distances areS=0.4mm, h2=3.04mm, h3=1.04mm, and h4=1.32mm. Changing one parameter and keeping the others constant, the coupling coefficients as a function of coupling distances S, h2, h3, and h4 are shown in Figure 6.Figure 6
Coupling coefficients as a function of coupling distance. Varying (a)S, (b) h2, (c) h3, and (d) h4 from Figure 5.
(a)(b)(c)(d)The coupling coefficientsk1 (Fodd1) of the odd-mode frequency Fodd1, k2 (Feven1) of the even-mode frequency Feven1, and k3 (Fodd2) of the odd-mode frequency Fodd2 are mainly adjusted using the coupling distance S. The coupling coefficients decrease with an increase in S. Further, h2, h3, and h4 have little influence on k1 (Fodd1); moreover, the coupling strength increases before h2 decreases, increases monotonously as h3 increases, and then decreases continuously as h4 decreases.Similarly,k4 (Feven2) of Feven2 is adjusted using S, h2, h3, and h4. This coupling coefficient decreases with an increase in S, increases with an increase in h2 and h3, and then decreases continuously as h4 decreases.In summary, the circuit consisted ofh2, h3, and h4 that mainly affected k4 (Feven2) and had little influence on k1 (Fodd1), k2 (Feven1), and k3 (Fodd2). First, k1 (Fodd1) and k2 (Feven1) are adjusted. Subsequently, k3 (Fodd2) is determined. Finally, because the microstrip circuit composed of h2, h3, and h4 is not loaded, the return loss of band IV is not very good; thus, the circuit composed of h2, h3, and h4 is used for slight tuning.


### 2.3. External Coupling Structure

The dual-feed external coupling configuration of the proposed resonator coupled with a 50Ω microstrip feedline is shown in Figure 7. Two open-circuited-coupled feedlines were used to obtain the required Qe of the four passbands. Based on this dual-feed coupling structure, there are two main degrees of freedom, that is, the coupled gap and its length, denoted as g and L3, respectively. The external quality factors can be extracted as follows:
(5)Qe=πf0τ21f0,where f0 represents the resonant frequency and τ21 (f0) is the group delay of S21 at resonance.Figure 7
Feeding structure of the proposed resonator.The initial distances areg=0.12mm and L3=3.16mm. One parameter is changed, and the others are kept constant when extracting the external quality factor Qe. The graphs of the external quality factors at four resonant frequencies as functions of different geometrical parameters of the external coupling structure, namely, g and L3, are shown in Figure 8.Figure 8
Simulated external quality factors of the four passbands with different (a) positions (g) and (b) lengths (L3) of the coupled feedline.
(a)(b)As shown in Figure8(a), Qe at Fodd2 and Feven2 decreases, whereas Qe at Fodd1 and Feven1 increases with the increase in the structural parameter g. The external quality factors under the structural parameter L3, which dominantly influences external coupling at Fodd2 and Feven2, are shown in Figure 8(b).First, the coupled gap position parameterg is adjusted to achieve the desired Qe for odd-mode frequency Fodd1 and even-mode frequency Feven1. Subsequently, it is varied to meet the Qe requirement for odd-mode frequency Fodd2. Finally, the Qe at even-mode frequency Feven1 is realized by adjusting parameter L3.The change in one length parameter within a certain range slightly affects other external couplings at four resonant frequencies. Therefore, the adjustment process should be repeated several times. Finally, the required external couplings of the four bandwidths can be met simultaneously.


## 2.1. T-Shaped Resonator Mode Analysis and Frequency Control

The schematic of the proposed resonator, comprising a2/λ uniform impedance resonator (UIR) (Z2 and θ2) with a two-end-open midloaded stub (Z3 and θ3) and four open stubs (Z1 and θ1), is shown in Figure 1(a). The resonance characteristics of this resonator can be analyzed using the even–odd mode method because the characteristics are symmetrical along the T-T’ plane. The odd- and even-mode equivalent circuits are shown in Figures 1(b) and 1(c), respectively.Figure 1
(a) Transmission line model of the proposed resonator. (b) Odd-mode equivalent circuit. (c) Even-mode equivalent circuit.
(a)(b)(c)Hence, the input admittance of the proposed resonator under even- and odd-mode excitation can be expressed as follows:(1)Yin−odd=−2jY1Y2cotθ2−2Y1tanθ12Y1+Y2tanθ1cotθ2,Yin−even=2jY1Y2Y2tanθ2+Y3tanθ3+2Y1tanθ1Y2−Y3tanθ2tanθ32Y1Y2−Y3tanθ2tanθ3−Y2tanθ1Y2tanθ2+Y3tanθ3.The model contained various variables, and the two initial conditions were assumed to beθ1=θ2=θ and Y2=Y3 for convenience of analysis. From a transmission line model analysis, the odd- and even-mode resonance conditions are expressed as follows:
(2)2tan2θ=R,(3)2cot1+αθ=−Rcotθ.Equations (2) and (3) yield the resonance electrical lengths of odd and even-modes, respectively. Herein, R=Y2/Y1 and α=θ3/θ. From Equations (2) and (3), the central stubs (Y3 and θ3) influence the even-mode resonant frequency but not that of the odd-mode. Therefore, the resonator’s resonant frequency can be precisely controlled. Based on Equations (2) and (3), the variables θ1 and θ2 influence the odd- and even-mode frequencies. Therefore, the curious mode frequency can be adjusted by changing (Y3 and θ3) to adjust the even-mode frequency.Note thatθ has different solutions for different values of R and α. The design curves of the first two resonant frequencies of the resonator are shown in Figure 2. Fodd and Feven denote the odd- and even-mode frequencies, respectively. As the 2/λ UIR (Z2 and θ2) part will be folded with equal spacing in the following sections, the resonant frequency positions of Feven2 and Fodd2 will significantly change; conversely, those of Feven1 and Fodd1 will have minimal change. Therefore, the resonant frequencies of Feven1 and Fodd1 are mainly analyzed here. When R and α assume typical values, the achievable frequency ratio ranges from 1.2 to 3.4. The frequency ratios for different values of α when R=0.5,1,1.5,2,and3 are shown in Figure 2, according to which Feven1/Fodd1 decreases with increasing R and α. Therefore, based on the aforementioned analysis, the desired specifications can be identified from Figure 2 by choosing different combinations of R and α. The admittance ratio of the resonator can be varied to obtain a wide range of frequency ratios. Compared with the traditional SIR, the multistub-loaded resonator can obtain a wider range of frequency ratios, assuming the same admittance ratio.Figure 2
Frequency ratios ofFeven1/Fodd1 with different values of α when R=0.5,1,1.5,2,and3.For demonstration, the proposed T-shaped resonator is utilized to implement a quadband HTS BPF. The2/λ UIR (Z2 and θ2) part is folded with equal spacing to easily realize the quadband design. The microstrip structure of a quadmode multistub-loaded resonator is shown in Figure 3(a). A Sonnet EM-simulated resonance response under a weak-coupling condition is shown in Figure 3(b). Realization of a quadband design of the unfolded resonator is challenging; however, the folded resonator significantly reduces the frequency difference between two sets of odd–even modes, simplifying the quadband design. Using the resonator depicted in Figure 3(a) as an example, before the resonator folds, Feven2/Fodd1=3.775 and Fodd2/Feven1=2.188. After folding the resonator, Feven2/Fodd1=3.212 and Fodd2/Feven1=1.84. Therefore, the frequency difference between the two sets of odd–even modes can be reduced using a folding resonator.Figure 3
(a) Proposed T-shaped multimode resonator. (b) EM simulation results under weak-coupling condition.
(a)(b)Using a full-wave simulation software Sonnet, the resonator’s frequency response was analyzed under weak-coupling conditions. Figure4 depicts the modal resonance characteristics at different microstrip line lengths of the resonator, namely, L1, L2, L4, d1, and W2. In particular, the microstrip line lengths L1 and L4 affect both odd- and even-mode frequencies. The even-mode frequency can be controlled by the microstrip line length L2 alone. The microstrip line lengths d1 and W2 can change the frequency difference between the two sets of odd–even modes. Keeping the other variables constant, when the microstrip line length d1 is 0.28, 0.36, and 0.44 mm, Feven2/Fodd1 and Fodd2/Feven1 correspond to 3.106, 3.212, 3.301 and 1.828, 1.84, and 1.854, respectively. Therefore, the frequency difference between the two sets of odd–even mode increases with increasing d1. Conversely, keeping the other variables constant, when the microstrip line length W2 is 0.16, 0.24, and 0.32 mm, Feven2/Fodd1 and Fodd2/Feven1 correspond to 3.279, 3.212, 3.189 and 1.929, 1.84, and 1.787, respectively. Therefore, the frequency difference between the two sets of odd–even modes decreases with increasing W2. This trend is consistent with previous studies.Figure 4
Modal resonant characteristics with varied microstrip line lengths. Varying (a)L2, (b) L1 and L4, (c) d1, and (d) W2 from Figure 3.
(a)(b)(c)(d)In summary, the initial structure of the resonator can be determined as follows. First, the impedance ratio of each stub is determined before the resonator is folded. Next, the length ofd1 is adjusted with equal spacing in the folded resonator to ensure that the frequency difference between the two sets of odd–even modes is in a relatively suitable range. Additionally, the impedance at the fold is slightly tuned to determine the frequency difference between the two sets of odd–even modes. Thus, the lengths of L1 and L4 are fine-tuned to achieve the desired fundamental frequency. Finally, L2 is adjusted to change the position of the two sets of even-mode frequencies.


## 2.2. Resonator Coupling Analysis and Bandwidth Control

After determining the initial structure of the resonator, the main challenge is to investigate the coupling control mechanism to meet the required design parameters, including the internal coupling coefficientsk and external quality factor Qe for the prescribed specifications in the four passbands.The coupling configuration of the proposed resonator is shown in Figure5. According to the method reported in Reference [12], the coupling coefficients at four resonant frequencies can be extracted using the following formula:
(4)k=f22−f12f22+f12,where f1 and f2 are the lower and higher coupling resonant frequencies, respectively.Figure 5
Configuration of the two coupled resonators.The initial distances areS=0.4mm, h2=3.04mm, h3=1.04mm, and h4=1.32mm. Changing one parameter and keeping the others constant, the coupling coefficients as a function of coupling distances S, h2, h3, and h4 are shown in Figure 6.Figure 6
Coupling coefficients as a function of coupling distance. Varying (a)S, (b) h2, (c) h3, and (d) h4 from Figure 5.
(a)(b)(c)(d)The coupling coefficientsk1 (Fodd1) of the odd-mode frequency Fodd1, k2 (Feven1) of the even-mode frequency Feven1, and k3 (Fodd2) of the odd-mode frequency Fodd2 are mainly adjusted using the coupling distance S. The coupling coefficients decrease with an increase in S. Further, h2, h3, and h4 have little influence on k1 (Fodd1); moreover, the coupling strength increases before h2 decreases, increases monotonously as h3 increases, and then decreases continuously as h4 decreases.Similarly,k4 (Feven2) of Feven2 is adjusted using S, h2, h3, and h4. This coupling coefficient decreases with an increase in S, increases with an increase in h2 and h3, and then decreases continuously as h4 decreases.In summary, the circuit consisted ofh2, h3, and h4 that mainly affected k4 (Feven2) and had little influence on k1 (Fodd1), k2 (Feven1), and k3 (Fodd2). First, k1 (Fodd1) and k2 (Feven1) are adjusted. Subsequently, k3 (Fodd2) is determined. Finally, because the microstrip circuit composed of h2, h3, and h4 is not loaded, the return loss of band IV is not very good; thus, the circuit composed of h2, h3, and h4 is used for slight tuning.


## 2.3. External Coupling Structure

The dual-feed external coupling configuration of the proposed resonator coupled with a 50Ω microstrip feedline is shown in Figure 7. Two open-circuited-coupled feedlines were used to obtain the required Qe of the four passbands. Based on this dual-feed coupling structure, there are two main degrees of freedom, that is, the coupled gap and its length, denoted as g and L3, respectively. The external quality factors can be extracted as follows:
(5)Qe=πf0τ21f0,where f0 represents the resonant frequency and τ21 (f0) is the group delay of S21 at resonance.Figure 7
Feeding structure of the proposed resonator.The initial distances areg=0.12mm and L3=3.16mm. One parameter is changed, and the others are kept constant when extracting the external quality factor Qe. The graphs of the external quality factors at four resonant frequencies as functions of different geometrical parameters of the external coupling structure, namely, g and L3, are shown in Figure 8.Figure 8
Simulated external quality factors of the four passbands with different (a) positions (g) and (b) lengths (L3) of the coupled feedline.
(a)(b)As shown in Figure8(a), Qe at Fodd2 and Feven2 decreases, whereas Qe at Fodd1 and Feven1 increases with the increase in the structural parameter g. The external quality factors under the structural parameter L3, which dominantly influences external coupling at Fodd2 and Feven2, are shown in Figure 8(b).First, the coupled gap position parameterg is adjusted to achieve the desired Qe for odd-mode frequency Fodd1 and even-mode frequency Feven1. Subsequently, it is varied to meet the Qe requirement for odd-mode frequency Fodd2. Finally, the Qe at even-mode frequency Feven1 is realized by adjusting parameter L3.The change in one length parameter within a certain range slightly affects other external couplings at four resonant frequencies. Therefore, the adjustment process should be repeated several times. Finally, the required external couplings of the four bandwidths can be met simultaneously.


## 3. Fabrication and Measurement

The proposed HTS quadband BPF is shown in Figure9(a). The dimensions of the filter (in mm) were as follows: L1=1.32, L2=2.56, L3=1.84, L4=1.84, W1=0.2, W2=0.24, W3=0.08, W4=0.32, h1=1.6, h2=3.04, h3=1.04, h4=1.32, k=0.56, d1=0.36, and S=0.4. The proposed filter was developed on YBCO/MgO HTS thin films (20mm×12mm and 0.438λg×0.263λg, where λg denotes the guided wavelength corresponding to the line with an impedance of 50 Ω at the center frequency of passband no.1). Its thickness and ε were 0.5 mm and 9.8, respectively. The HTS film was patterned onto the filter circuit on one side using standard ion etching and photolithography and grounded on the other side. Additionally, the filter was packed into a shield box. An image of the proposed filter is shown in Figure 9(b).Figure 9
(a) Layout of the designed two-order quadband high-temperature superconducting (HTS) bandpass filter (BPF). (b) Photograph of the fabricated high-temperature superconducting (HTS) bandpass filter (BPF) with connectors.
(a)(b)The filter was cooled to 77 K. The measurement and simulation results are shown in Figure10. The four passbands were observed at 2.57, 4.29, 7.40, and 8.26 GHz, respectively, with 3 dB fractional bandwidths (FBWs) of 4.05%, 2.64%, 0.72%, and 0.23%, respectively. As indicated, measurement and simulation results slightly deviated, particularly in the fourth passband. This could be attributed to the slightly varying length of the stub, which was mounted on a box. The experimentally measured return losses were below 24.37 dB for all bands, whereas the experimentally measured insertion losses were 0.08, 0.09, 0.18, and 0.35 dB, owing to the use of HTS thin films. Performance comparison of the proposed filter with some reported quadband BPFs is listed in Table 1. The proposed HTS quadband filter exhibits low insertion losses, good return losses, and large upper/lower bandwidth ratios.Figure 10
Simulated and measured results of the quadband high-temperature superconducting (HTS) bandpass filter (BPF).Table 1
Comparison of the proposed filter and previously reported filters.
Referencesf0 (GHz)FBW (%)TZsIL (dB)RL (dB)Size (λg×λg)[13]1.5/2.5/3.5/5.51.3/1.9/0.5/0.982.6/2.1/2.9/2.1>120.2×0.18[14]1.55/2.2/3.45/5.319.3/9.1/8.69/3.7730.5/1/2.4/2.910/18/13/160.64×0.45[15]2.42/3.54/5.22/5.837.6/8.2/2/3.93<0.87>80.238×0.135[16]1.66/1.83/2.26/2.6N72.5/1.8/2.53/1.413/21/10/9N[12] I0.99/1.15/1.29/1.441.02/1.02/1.04/0.93N0.11/0.10/0.12/0.1423.74/25.89/18.90/26.510.33×0.17[12] II0.99/1.15/1.29/1.440.99/0.93/0.74/0.6610.11/0.11/0.24/0.1826.64/21.27/12.22/20.010.33×0.17[17]2.4/3.5/5.2/5.86.87/6.1/3.1/1.471.8/1.5/2.1/2.918/30/28.7/27.60.23×0.23[18]2.25/2.76/4.6/7.457.8/0.72/16.5/2.630.67/1.5/0.3/2.119.5/11.8/20.8/14.80.24×0.19TS2.57/4.29/7.40/8.264.05/2.64/0.72/0.2310.08/0.09/0.18/0.3524.93/24.37/28.35/37.650.438×0.263Abbreviations: TS: this study; FBW: fractional bandwidth; TZs: transmission zeros.


## 4. Conclusion

This study designed an HTS quadband filter to regulate center frequencies and bandwidths of the four passbands by adjusting the relationship between the parameters of the T-shaped resonator. A theoretical analysis of the even–odd mode was performed to enhance the design reliability. A wider range of frequency ratios and clear design structure exhibited advantages. Subsequently, the HTS quadband filter could be designed with an enhanced degree of freedom. The circuit design method was validated through simulation and measured results. The results indicated that the proposed filter has a high filtering performance. Finally, the excellent suitability of the compact quadband filter can be used in multiband and multiservice communication applications.



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