# Application of Compact Folded-Arms Square Open-Loop Resonator to Bandpass Filter Design

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Circuit Modelling of Bandpass Filter

_{1}, g

_{2}, g

_{3}, g

_{4}and g

_{5}are the filter parameters; Z

_{0}and Z

_{6}are the characteristics impedances at the input/output terminals; C

_{1}, C

_{2}, C

_{3}, C

_{4}and C

_{5}are the calculated capacitance values; L

_{1}, L

_{2}, L

_{3}, L

_{4}and L

_{5}are the calculated inductance values. The intended filter is composed with a center frequency, f

_{0}= 2.2 GHz, a fractional bandwidth, FBW = 10%, a 0.04321 dB passband ripple, and a 20 dB passband return loss.

_{2}C

_{2}and L

_{4}C

_{4}in this case) to parallel LC elements. This conversion gives rise to a BPF circuit model with equal shunt-only LC resonators and admittance inverters (that is, J-inverters), as demonstrated in Figure 3. The J-inverter values are determined using Equation (3) [1]. The J-inverters in Figure 3 are then replaced with inductor-only networks, as demonstrated in Figure 4. This relies on the transformation procedure reported in [1]. The J-inverter transformation makes it possible for the circuit model to be simulated in the circuit simulator of advanced design software (ADS), and the simulation results are described in Figure 5. The inductance values of the inductor-only networks are determined using Equation (4).

## 3. Microstrip Filter Design

#### 3.1. Resonator Dimension

_{0}, of 2.2 GHz. The substrate material employed in the design is the Rogers RT/Duroid 6010LM with a relative permittivity, ɛ

_{r}, of 10.7, a loss tangent, tan δ, of 0.0023, and a substrate thickness, h, of 1.27 mm. The microstrip dimensions, the width (w) and the guided-wavelength (λ

_{g}), are determined from [1] by means of Equations (5) and (6), respectively. The substrate material’s effective permittivity or effective dielectric constant in Equation (6) is represented by ɛ

_{eff}, while the speed of light, c

_{0}, is a constant with a value of 3 × 10

^{8}m/s. Using the technique reported in [1], the practical FASOLR dimensions were determined as shown in Figure 6.

#### 3.2. Coupling Coefficient and External Quality Factor Extraction

_{1,2}, k

_{2,3}, k

_{3,4}, k

_{4,5}) between adjacent resonators (R1, R2, R3, R4, R5) is shown in Figure 7, alongside the theoretical external quality factor (Q

_{ext}) between each port and the corresponding adjacent resonator. The theoretical coupling coefficient values are calculated using Equation (7) [1], while the theoretical external quality factor value is determined based on Equation (8) [1]. The practical filter coupling coefficients and the practical Q

_{ext}are determined using Equations (9) and (10), respectively, as well as the detailed techniques reported in [1]. The resonator layouts and the electromagnetic (EM) simulation responses for obtaining the practical coupling coefficients, k, and the practical Q

_{ext}are shown in Figure 8.

## 4. Filter Simulation

^{7}S/m for the microstrip metals (top and bottom layers). The EM simulation did not consider the thickness variation of the filter substrate material and metal surface layer roughness.

## 5. Filter Experimentation

## 6. Results Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Microstrip resonator structure evolution: (

**a**) Half-wavelength structure; (

**b**) Square open-loop structure; (

**c**) Folded-arms square open-loop structure.

**Figure 2.**Fifth-order bandpass filter transformation (g

_{0}= g

_{6}= 1.0, g

_{1}= g

_{5}= 0.9714, g

_{2}= g

_{4}= 1.3721, g

_{3}= 1.8014; Z

_{0}= Z

_{6}= 50 Ω, L

_{1}= L

_{5}= 0.3724 nH, L

_{2}= L

_{4}= 49.6310 nH, L

_{3}= 0.2008 nH; C

_{1}= C

_{5}= 14.0548 pF, C

_{2}= C

_{4}= 0.1054 pF, C

_{3}= 26.0638 pF).

**Figure 3.**Fifth-order BPF circuit model with equal LC resonators and admittance inverters (J

_{01}= 0.02, J

_{12}= 0.0168, J

_{23}= 0.0124, C = 14.0548 pF, L = 0.3724 nH).

**Figure 4.**Fifth–order BPF circuit model, with equal LC resonators and inductor-only networks replacing J-inverters (L

_{01}= 3.6172 nH, L

_{12}= 4.2989 nH, L

_{23}= 5.8542 nH).

**Figure 6.**Practical dimensions of the proposed folded-arms square open–loop resonator at 2.2 GHz and the EM simulation response (all dimensions in mm).

**Figure 7.**Fifth-order bandpass filter coupling coefficients and external quality factor arrangement (k

_{1,2}= k

_{4,5}= 0.087, k

_{2,3}= k

_{3,4}= 0.064, Q

_{ext}= 9.714).

**Figure 8.**Microstrip layouts and EM simulation responses: (

**a**) Coupling coefficient determination; (

**b**) External quality factor determination.

**Figure 9.**Fifth–order bandpass filter layout and the simulation responses for the folded-arms square open-loop resonator (all dimensions in mm).

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**MDPI and ACS Style**

Nwajana, A.O.; Obi, E.R. Application of Compact Folded-Arms Square Open-Loop Resonator to Bandpass Filter Design. *Micromachines* **2023**, *14*, 320.
https://doi.org/10.3390/mi14020320

**AMA Style**

Nwajana AO, Obi ER. Application of Compact Folded-Arms Square Open-Loop Resonator to Bandpass Filter Design. *Micromachines*. 2023; 14(2):320.
https://doi.org/10.3390/mi14020320

**Chicago/Turabian Style**

Nwajana, Augustine O., and Emenike Raymond Obi. 2023. "Application of Compact Folded-Arms Square Open-Loop Resonator to Bandpass Filter Design" *Micromachines* 14, no. 2: 320.
https://doi.org/10.3390/mi14020320