# Triangular Sierpinski Microwave Band-Stop Resonators for K-Band Filtering

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## Abstract

**:**

## 1. Introduction

## 2. Spectral Considerations of Sierpinski Resonators

_{resonance}for equilateral full triangles (or simple geometries such as isosceles and right-angled triangles) was exactly calculated, considering the TM (Transverse Magnetic) mode configuration for the equivalent cavity model. The following formula gives the resonance frequency F

_{resonance}for the main mode of the equilateral triangle [12,15,16,17], as it was also used in [9]:

_{effective}) and the relative dielectric constant ε (ε

_{effective}) might be introduced, as in [12], to account for the needed phenomenological corrections typically adopted for planar high-frequency geometries. It must be remembered that, according to the results in [12], only the effective value of the side length (a

_{effective}) of the triangle is necessary to have an agreement between the experimental value of the frequency of resonance and the expected one for the main mode given by Equation (1).

_{resonance}, it is expected to be in a ratio of 4/3 using a triangle with the same edge length as a square. At the same time, the area reduction is in the order of 0.4, which is a clear advantage when integrating the resonators into a planar array. Circles are easier to design but difficult to integrate using a side-by-side coupling. The disadvantage with triangles is the necessity for an accurate design accounting for the discontinuities on the corners.

**Figure 1.**Full equilateral triangle (C0) excited by a microstrip placed in the center of the side and its meshing used for the simulation. The number “1” is the reference port of the simulation, while the external small triangle indicates that the port is grounded.

_{11}, is plotted on a dB scale.

_{11}are plotted.

## 3. Design of the Single 20 GHz Resonators

_{21}of the symmetric structure; i.e., when the peak of the band-stop filter is deeper.

_{resonance}for the case of the CPW-fed resonators, even considering that the CPW has been manufactured onto an oxidized high resistivity Silicon wafer, and accounting for a dielectric constant value of approximately $\epsilon =5.5$ for a wide frequency range, as obtained in [21]. For this reason, we tried values of a busing electromagnetic simulations to obtain operating frequencies close to the desired ones, choosing $a\left(20\mathrm{G}\mathrm{H}\mathrm{z}\right)=6\mathrm{m}\mathrm{m}$ and $a\left(26\mathrm{G}\mathrm{H}\mathrm{z}\right)=4\mathrm{m}\mathrm{m}$. As a result, the C0 configuration was simulated almost in agreement with the desired value of F

_{resonance}, while we saw that higher frequencies needed further elaboration to improve both the prediction of the resonance and the selectivity of the resonating structure. An additional comment must be made regarding the symmetry of the boundary conditions. As has been shown in [9], the separation between the resonator and the central conductor of the CPW, as well as the distance between the edges of the triangle and the ground plane, must be equal to provide a symmetric boundary for the resonator, favoring a simpler design procedure.

_{resonance}, from the simulations, is given in Table 3, from which, as well as from Figure 7, it is not clear to obtain a law for F

_{resonance}as a function of the internal complexity.

_{resonance}is now in agreement with the aim of our work to have resonance frequencies close to 20 GHz.

## 4. Design of the 20 GHz Coupled Hexagonal Resonators

## 5. Design of the 26 GHz Single Resonators

## 6. Design of the 26 GHz Coupled Hexagonal Resonators

_{21}of the single resonators is shown, the C1 structure is not necessarily the best one, but it is characterized by a sharp peak around 27 GHz, which is almost the same for the hexagonal resonator but with an enlarged band of operation. An additional optimization for most of the studied structures should concern the electrical matching of the I/O lines.

## 7. Experimental Results and Discussion

#### 7.1. Manufacturing of the Devices

#### 7.2. Measurements

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**First-order Sierpinski triangle excited by a microstrip and the meshing used for the simulation. The number “1” and the downward external small triangle are the port number for the simulation and its ground reference, respectively.

**Figure 3.**Spectral response of the triangles C0 and C1 excited by a short microstrip connected to the edge of the triangles. The dark curve is for C0 and the red one is for C1.

**Figure 4.**Excitation by microstrip of the C0 configuration using a corner of the triangle (

**a**), and its spectral response using the reflection parameter S

_{11}(

**b**). The number “1” in (

**a**) is the port number for the simulation and the triangle is the ground reference, respectively. The meshing used for the simulation is also shown in (

**a**).

**Figure 5.**CPW excited C0 resonator (

**a**) and its microwave response in reflection (S

_{11}parameter) (

**b**).

**Figure 6.**Symmetric (

**a**) and asymmetric (

**b**) configurations of a triangular patch resonator in the CPW configuration, and comparison (

**c**) between the expected performances of the transmission S

_{21}parameter of both structures in dB scale.

**Figure 7.**The Sierpinski structures ((

**a**–

**c**) above) and the simulated transmission parameters ((

**d**), S

_{21}in dB scale) for the C1, C2, and C3 resonators (below), compared with the response of the reference structure C0.

**Figure 8.**Configuration C0C1 (

**a**), made using a C0 triangle interfaced with a C1 one, and its predicted performance (

**b**) compared to the pure C0 configuration. Both transmission (S

_{21}) and reflection (S

_{11}) parameters are drawn to give evidence for the expected electrical matching of the resonator.

**Figure 9.**Hexagonal resonators obtained by combining three coupled triangles mirrored with respect to the central conductor of the CPW. All the possible internal complexity levels were designed up to the C3 configuration.

**Figure 10.**Expected response of the hexagonal resonators, simulated by Microwave Office. The S

_{21}parameter is plotted in the dB scale, and the internal complexity is given for each structure with the definition “hex” to indicate that the resonators C0, C1, C2, and C3 were organized in a hexagonal arrangement.

**Figure 11.**Simulated response of the transmission parameter S21 in dB scale for the small resonators C0, C1, C2, and C3.

**Figure 12.**Comparison between the footprint of the big and the small triangular resonators, originally minded for 20 GHz and 26 GHz operations. The side lengths of the triangles are 6 mm and 4 mm, respectively.

**Figure 13.**Simulated performance of the hexagonal resonators for operation around 26 GHz. The configuration based on the C1 resonator is the best one in terms of the depth of the notch.

**Figure 14.**Theoretical and experimental response for the C0 resonator. A significant shift in the expected frequency of resonance and an enlarged bandwidth were obtained.

**Figure 16.**Experimental response for the transmission parameter S

_{21}(in dB scale) of the C0, C1, C2, and C3 resonators for the operation at approximately 26 GHz. In addition, the response of the mixed configuration C2C3 was also measured and was shown and compared with the performance of the other resonators.

**Figure 17.**(

**a**) C2C3 structure, with the geometry C2 on the top and C3 on the bottom of the resonator, and (

**b**) its simulated response for the S

_{21}parameter compared with the experimental measurement.

**Figure 18.**C1 (

**left**side) and C3 (

**right**side) structures arranged in the hexagonal configuration. The photo is for the 26 GHz resonators, but the geometry is the same as the 20 GHz structure.

**Figure 20.**Simulated and experimental responses for the small hexagonal resonator based on the C1 Sierpinski triangles. The label “hex” defines the hexagonal arrangement of the C1 structures. The multi-peak expected response was experimentally measured with a shift in the frequency of resonance of approximately 500 MHz.

**Figure 21.**Measured and simulated responses of the small hexagonal resonator based on the C3 Sierpinski configuration for operating frequencies around 26 GHz. In this case, the expected frequency of resonance is around 25 GHz, but the actual frequency is close to 26 GHz. Additional expected modes were not recorded in the experimental measurements.

**Table 1.**Frequencies of resonance, in GHz, for the C0 configuration, comparing the experimental results of [12] with the simulations obtained using CADENCE Microwave Office (MWOffice) release 22.1.

Resonance Frequency for C0 (MWOffice) [GHz] | Resonance Frequency for C0 from Ref. [12] [GHz] | % Error on the C0 Frequency of Resonance |
---|---|---|

1.290 | 1.280 | 0.008 |

2.255 | 2.242 | 0.006 |

2.615 | 2.550 | 0.025 |

3.440 | 3.400 | 0.012 |

3.850 | 3.824 | 0.007 |

**Table 2.**Frequencies of resonance, in GHz, simulated by Microwave Office for the C0 configuration, and the corresponding resonances obtained for the CPW-fed structure. Only an approximate agreement can be obtained.

Frequency of Resonance for C0 (Microstrip) [GHz] | Frequency of Resonance for C0 (CPW) [GHz] |
---|---|

1.290 | 0.970 |

2.255 | 1.930 |

2.615 | 2.915 |

3.440 | ? |

3.850 | 3.925 |

4.660 | 4.870 |

Frequency of Resonance (F_{resonance}) [GHz] | |||
---|---|---|---|

C0 | C1 | C2 | C3 |

20.160 | 20.950 | 19.670 | 19.610 |

**Table 4.**Comparison among the simulated frequencies of resonances and the 3-dB bandwidth for the configurations C0 and C0C1. Both quantities are in GHz.

C0 | C0C1 | |
---|---|---|

F_{resonance} [GHz] | 20.16 | 20.95 |

Bandwidth [GHz] | 0.096 | 0.178 |

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**MDPI and ACS Style**

Marcelli, R.; Sardi, G.M.; Proietti, E.; Capoccia, G.; Iannacci, J.; Tagliapietra, G.; Giacomozzi, F.
Triangular Sierpinski Microwave Band-Stop Resonators for K-Band Filtering. *Sensors* **2023**, *23*, 8125.
https://doi.org/10.3390/s23198125

**AMA Style**

Marcelli R, Sardi GM, Proietti E, Capoccia G, Iannacci J, Tagliapietra G, Giacomozzi F.
Triangular Sierpinski Microwave Band-Stop Resonators for K-Band Filtering. *Sensors*. 2023; 23(19):8125.
https://doi.org/10.3390/s23198125

**Chicago/Turabian Style**

Marcelli, Romolo, Giovanni Maria Sardi, Emanuela Proietti, Giovanni Capoccia, Jacopo Iannacci, Girolamo Tagliapietra, and Flavio Giacomozzi.
2023. "Triangular Sierpinski Microwave Band-Stop Resonators for K-Band Filtering" *Sensors* 23, no. 19: 8125.
https://doi.org/10.3390/s23198125