# Design and Modelling of a Compact Triband Passband Filter for GPS, WiMAX, and Satellite Applications with Multiple Transmission Zero’s

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

_{g}× 0.24 λ

_{g}, where λ

_{g}is the guided wavelength of the material calculated at the lowest frequency.

## 1. Introduction

_{eff}× 0.77 λ

_{eff}[41].

## 2. Theoretical Analysis of the Filter

_{2}/Z

_{1}and electrical lengths θ

_{2}/θ

_{1}, respectively, as shown in Figure 2a. W

_{1}, L

_{1}and W

_{2}, L

_{2}are the physical widths and lengths of the low and high impedances. The low impedance sections are attached with 50-ohm input/output ports of the filter, and the high impedance sections are parallel coupled with metallic via at the lower corner having a radius of 0.25 mm. The first two passbands are obtained due to the ratio of the two impedances (R = Z

_{2}/Z

_{1}) of the SIR, and the third passband is obtained due to the L-shaped structure coupled to the high impedance section. The resonators are folded in a compact way to introduce magnetic and electric coupling for the generation of multiple TZs in the passbands. The electric coupling, which is a weak coupling in this case, is observed due to space S

_{1}of the open-circuited stubs of the low-impedance section of the SIR. The magnetic coupling is observed due to gap G

_{1}of the parallel short-circuited high-impedance section of the SIR. The input admittance Y

_{in1}(Y

_{in}= 1/Z

_{in}) of Figure 2a is calculated as [42,43];

_{in2}for Figure 2b is

_{2}+θ

_{3}+θ

_{4}can be calculated easily by the required values of R and K

_{x}. So, by adjusting the electrical lengths and impedance ratios, the first two passbands are obtained for GPS and WiMAX applications. Moreover, the final electrical lengths and impedances of Figure 2c on which the resultant SIR filter optimized are Z

_{0}= 50 Ω, Z

_{1}= 71 Ω, Z

_{2}= 50 Ω, θ

_{1}= 77, θ

_{2}= 78.94, θ

_{3}= 88.6, and θ

_{4}= 15.

## 3. Triple Passband Filter Architecture

_{1}is attached with the 50-ohm characteristic impedance input/outport of the filter, and the high impedance section Z

_{2}is parallel coupled with metallic via at the lower end for size compactness. Moreover, the low impedance is also coupled to the embedded L-shaped λ/2 uniform structure. The first two passbands at 1.57 GHz and 3.57 GHz are obtained due to the coupled λ/4 SIR for GPS and WiMAX wireless applications, and the third passband at 4.23 GHz is obtained due to the embedded coupled L-shaped structure for satellite communication. All the resonators are folded to introduce coupling and sharpness in the filter by exciting multiple transmission zeros between the passbands. There is a total of eight T.Z.s excited at different locations, i.e., 1.22, 1.42, 1.98, 3.18, 3.82, 3.98, 4.38, and 4.53 GHz. After simulating the proposed filter in HFSS software, it is fabricated on substrate RO-4350 using milling machine LPKF S63 ProtoLaser and tested on ZNB-20 VNA. The overall dimensions and substrate properties are listed in Table 1.

## 4. Results and Discussion

_{1}with one end open circuited and high impedance section Z

_{2,}which is folded to introduce coupling phenomena and size reduction with metallic via at lower edges. Later, a half-wavelength L-shaped resonator is further utilized for multi-passbands. The first two passbands for the applications of GPS and WiMAX are obtained due to the quarter wavelength step impedance resonator with electrical parameters Z

_{0}= 50 Ω, Z

_{1}= 71 Ω, Z

_{2}= 50 Ω, θ

_{1}= 77, θ

_{2}= 78.94, θ

_{3}= 88.6, and θ

_{4}= 15 and the resonance frequency ratio K

_{x}= f

_{2}/f

_{1}which is 2.22 in this case. The electrical lengths of the proposed SIR can be changed to observe the first and second passbands. Consider Figure 4, which shows the variation of the first and second passband for electrical stub lengths θ

_{1}, where the upper cutoff frequency of the first band decreases with an increase of stub length θ

_{1}and the upper and lower cutoff frequencies of the second band varied with no effect on third passband. Figure 5 shows that with the increase of electrical stub length θ

_{4}, only the lower cutoff frequency of the first passband and upper cutoff frequency of the second passband decreases with no effect on the third passband. This reveals that the first and second passbands are generated with the ratio of impedances and electrical stub lengths of the SIR. The third passband for satellite application is obtained due to the L-shaped structure embedded in the SIR. Thus, a triple band filter with multiple T.Z.s at different locations is obtained with the central frequencies 1.57 GHz, 3.57 GHz, and 4.2 GHz. The simulated results of the proposed prototype with and without the embedded L-shaped structure are illustrated in Figure 6 and Figure 7, where it is observed that the third passband is obtained due to the half-wavelength L-shaped resonator without affecting the first two passbands. Moreover, the third passband is controllable, and by varying the resonator length L

_{4}, the upper cut-off frequency is moved downward compared to the lower cut-off frequency without affecting the remaining passband cut-off frequencies, as shown in Figure 8, respectively.

_{1}), 1.42 (TZ

_{2}), 1.98 (TZ

_{3}), 3.18 (TZ

_{4}), 3.82 (TZ

_{5}), 3.98 (TZ

_{6}), 4.38 (TZ

_{7}), and 4.53 (TZ

_{8}) have been observed between the passbands as shown in Figure 9. Due to these T.Z.s, the roll-off rates (ξ) of the three passbands increased. A higher roll-off rate indicates better selectivity, and it can be defined by regarding −20 dB attenuation and −3 dB attenuation as two cut-off points for the passband. It is equal to the rate of 17 dB with the bandwidth of roll-off [45]. Thus, the roll-off rates of the three passbands reach up to 294 dB/GHz (ξ

_{1}), 170 dB/GHz (ξ

_{2}), 106 dB/GHz (ξ

_{3}), 242 dB/GHz (ξ

_{4}), 106 dB/GHz (ξ

_{5}), and 121 dB/GHz (ξ

_{6}), respectively. The TZ

_{1}and TZ

_{6}are generated due to the mixed-coupling of the SIR, TZ

_{2}and TZ

_{5}are generated by the SIR itself, TZ

_{3}and TZ

_{4}are produced due to the metallic hole etched to the lower end of the high-impedance section of the SIR, while TZ

_{7}and TZ

_{8}are generated by the embedded L-shaped structure coupled to the low impedance of the SIR, respectively. Moreover, the proposed tri-band BPF has a maximum roll-off rate of up to 294 dB/GHz, demonstrating its great selectivity and abrupt roll-off.

_{0}= 1/√L.C. The following design equation gives the exact values of coupling coefficients obtained in [48] for a certain asymmetrical pair of adjacent resonant circuits featuring Chebyshev frequency response.

_{−}and ω

_{+}are the lower and upper frequencies, and “w” is the relative bandwidth defined as

_{2}. The coupling coefficient decreases when the values G and S increase while the quality factor increases, as depicted in Figure 10, Figure 11 and Figure 12. Figure 13 shows the response of the three passbands’ quality factors concerning T

_{2}. The Q increases with varying values from 18 mm to 20 mm.

_{2}/f

_{1}= 2.2) because, as discussed, the first two passbands are obtained due to the ratio of the electrical lengths of the SIR. Figure 15b shows the current distribution at 4.23 GHz for the third passband, obtained by the embedded L-shaped resonator. As seen, the current is uniformly distributed only on the surface of the L-shaped resonator. Similarly, the magnitude of the electric field (E) intensity of the passbands is illustrated in Figure 16, respectively.

## 5. Fabrication and Measurement

_{11}and S

_{21}is shown in Figure 17 was obtained. The proposed filter has low signal attenuation of less than 1.2 dB and high signal reflection of better than 25 dB for the three passbands. The fractional bandwidths achieved are 2.54%, 4.2%, and 1.65% at 1.57/3.57/4.23 GHz, respectively, with rejection levels in the stopband greater than 15 dB. The area occupied by the filter on a substrate or in a circuit is 0.31 λ

_{eff}× 0.24 λ

_{eff}(0.086 λ

_{g}

^{2}), where λ

_{eff}is the effective dielectric constant of the material calculated at the lowest frequency. Table 2 listed the comparison of this work with other triband filters in the literature in terms of size, FBW, IL, R.L., and selectivity.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Placement of bandpass filter in communication systems [6].

**Figure 2.**Configuration of resonators (

**a**) Stepped impedance resonator (

**b**) dual band filter topology (

**c**) Equivalent structure of (

**b**).

**Figure 15.**Nature of the current distribution of the triband filter (

**a**) surface current density due to the frequency ratio K. (

**b**) surface current density due to L-shaped resonator at 4.23 GHz.

**Figure 16.**The magnitude of the electric field intensity (

**a**) E due to the frequency ratio K. (

**b**) E due to the L-shaped resonator at 4.23 GHz.

T_{1} | 17.61 | T_{2} | 20.88 | L_{1} | 20 |
---|---|---|---|---|---|

L_{2} | 9.5 | W | 1.35 | W_{1} | 0.8 |

W_{2} | 0.5 | L_{3} | 6.6 | L_{4} | 15 |

W_{f} | 1.6 | S | 0.1 | S_{1} | 1 |

G | 0.2 | PCB Height | 0.762 | ε_{r} | 3.66 |

Ref No. | Year | Bands (GHz) | S_{21} (dB) | S_{11} (dB) | FBW (%) | T.Z.s | Size (mm) | Size (λ_{g}^{2}) |
---|---|---|---|---|---|---|---|---|

[39] | 2020 | 2.9/5.6/11 | 3/3.1/5 | >20 | 46/18/6 | 3 | 144 × 40 | 1.864 |

[40] | 2021 | 2.4/3.5/5.25 | <1.7 | 50/33/43 | 11.6/4/6.7 | 3 | 41 × 29.5 | 0.291 |

[41] | 2020 | 1.93/2.6/3.9 | 1.5/0.6/1.8 | 15/20/20 | 5/11/3 | 5 | 17.2 × 24.5 | 0.415 |

[51] | 2022 | 3.2/3.4/3.7 | 2.65/1.95/2.69 | 18.3/17.9/19.2 | 3.57/3.77/2.07 | 5 | 75 × 51.5 | 0.469 |

[52] | 2019 | 2.45/3.5/5.25 | 0.7/0.9/0.6 | 38/32/25 | 1.2/2/1.52 | 6 | 42 × 31.5 | 0.240 |

[53] | 2018 | 0.4/0.8/1.55 | 0.7/1.8/1.6 | 10/8.2/9.3 | 55.4/20.9/10.3 | 5 | 56 × 31 | 1.72 |

[54] | 2021 | 2.05/2.55/3.1 | 1.98/2.17/2.01 | 12.3/14.9/14.1 | 4.8/7.8/8 | 3 | 72.1 × 68.5 | 1.931 |

[55] | 2022 | 4.1/6.1/14.4 | 2.1/1.3/4.08 | 19.63/22/12.1 | 5.3/8.6/2.1 | 2 | 22.42 × 7.62 | 0.094 |

[56] | 2021 | 6.28/13/19.12 | 1.6/2.5/2.2 | 22/26/21 | 9.5/6.2/4.5 | 4 | 8.5 × 15 | 0.119 |

[57] | 2022 | 4.2/7.36/9.35 | 1.38/4.86/1.27 | 15.7/17/ 38 | 5.21/3.19/9.3 | 4 | 22 × 18 | 0.803 |

[58] | 2020 | 1.52/2.0/2.36 | 5.32/4.2/6.8 | 15.4/19.1/17.3 | 3.56/8.6/2.75 | 5 | 54.9 × 28.6 | 0.248 |

[59] | 2022 | 18.2/18.7/19.1 | 2.59/2.21/2.5 | 15.9/20.9/24.8 | 0.45/1.08/0.5 | 6 | 1.1 × 1.1 × 1.2 | 1.469 |

[60] | 2022 | 3.6/4.6/5.6 | <0.78 | >30 | 11.9/11.9/11.9 | 5 | 54.34 × 22.3 | 0.911 |

This work | 2023 | 1.57/3.57/4.23 | 0.7/1.2/1.06 | 27/35/28 | 2.54/4.2/1.65 | 8 | 40.1 × 22 | 0.086 |

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**MDPI and ACS Style**

Basit, A.; Daraz, A.; Khan, M.I.; Zubir, F.; A. AlQahtani, S.; Zhang, G.
Design and Modelling of a Compact Triband Passband Filter for GPS, WiMAX, and Satellite Applications with Multiple Transmission Zero’s. *Fractal Fract.* **2023**, *7*, 511.
https://doi.org/10.3390/fractalfract7070511

**AMA Style**

Basit A, Daraz A, Khan MI, Zubir F, A. AlQahtani S, Zhang G.
Design and Modelling of a Compact Triband Passband Filter for GPS, WiMAX, and Satellite Applications with Multiple Transmission Zero’s. *Fractal and Fractional*. 2023; 7(7):511.
https://doi.org/10.3390/fractalfract7070511

**Chicago/Turabian Style**

Basit, Abdul, Amil Daraz, Muhammad Irshad Khan, Farid Zubir, Salman A. AlQahtani, and Guoqiang Zhang.
2023. "Design and Modelling of a Compact Triband Passband Filter for GPS, WiMAX, and Satellite Applications with Multiple Transmission Zero’s" *Fractal and Fractional* 7, no. 7: 511.
https://doi.org/10.3390/fractalfract7070511