I. INTRODUCTION

Research involving radio frequency filters follows a growing constant as well as the development of microwave-based communication systems. Microwave circuit research and design focused on signal frequency filtering is a field of research that has continued to advance over the past few decades and gain adepts. Due to the constant advances in telecommunication systems, the demand for low cost, compact filters and increasingly specific frequency response is a tendency to be met.

Over the past few years, researchers have developed microwave planar filters based primarily on transmission line modeling [^{1}] - [^{4}] and, with dual-mode or multi-mode resonator behavior, syntheses have been made to from even and odd excitation circuit analysis to obtain expressions that contribute to resonator modeling and filter response [^{5}] - [^{12}]. The methodology based on mode analysis is also applicable in the development of bandstop structures [^{21}]. Techniques such as the use of Substrate Integrated Waveguide (SIW) structures to improve filter performance [^{22}] - [^{23}] are also valid. Much of the research involving microwave filters commonly seeks the development of performance bandpass planar circuits with one or more transmission bands and high frequency selectivity, as well as high attenuation rates in reject bands. In [^{13}], a bandpass filter is developed in microstrip using coupled lines, a common practice. A transmission line modeling is applied to obtain the coupling matrix of the circuit, and the resonance frequency is investigated as a function of the variation of the resonator key parameters. With a similar methodology, in [^{14}] a compact dual-wideband bandpass filter is proposed. The established resonator is based on the coupling of a pair of shorted grounded conductor strips parallel to the main transmission line. Using a coupling feed, in [^{6}] a hexagonal patch is proposed along with the use of hollow fractal elements. A bandpass response is obtained from an even and odd mode analysis for the resonator. Based on the fully modeled side-coupled ring resonator proposed in [^{3}] and from the use of stub-loads, in [^{2}] a bandpass filter with a high selectivity index and an increase of zero transmission frequencies is developed when compared to [^{2}], this behavior is due to the use of coupling between lines and stub-loads.

In this work, a tri-bandpass filter (TBF) is proposed. The resonator structure is inspired by the side-coupled ring geometry presented in [^{2}] due to its bandwidth, well-defined zero transmission frequencies and its fully delimited synthesis. In order to introduce new operating bands, important changes are attributed to the resonator geometry. An analysis of the proposed resonator is performed with the objective of modeling the TBF and obtaining expressions that govern its frequency response as a function of the main resonator parameters. Simulations performed in transmission line and microwave circuit analysis software are presented and a prototype filter is constructed and subjected to experimental measurement process. Section II of this paper deals with the mathematical modeling of the resonator. Simulations and analyzes are presented in section III. In item IV, the prototype design is developed and the simulation and measurement results are analyzed and compared. Finally, a conclusion is presented in item V.

II. RESONATOR MODEL

Fig. 1(a) and (b) show the planar structure resonator (TBF) proposed in this work and its transmission line modeling, respectively. The geometry of the resonator resembles the side-coupled square loop ring present in [^{2}], but its perimeter length totals about 2λ of the main resonant frequency. In [^{2}] and [^{3}], the use of the square-ring resonator with 1λ provides a frequency performance with a single operating band. In this model, a geometry of total length 2λ is proposed, which allows obtaining two additional lateral bands in the frequency response compared to [^{3}]. As can be seen, the ports are set at an angle of π/4 rad. The resonator geometry can be subdivided into three parts: two pairs of laterally coupled transmission lines of impedances *Z _{oe}, Z_{oo}* and electrical length 3λ/4, and a characteristic half-wavelength impedance centerline

*Z*and

_{L}*θ*of the TBF center frequency. Coupled pairs of lines are interconnected with each other, while impedance

_{el}*Z*makes the connection between them at the opposite vertex of the geometry. Thus, the length

_{L}*L*on either side of the square-loop ring is approximately half wavelength of the central resonant frequency of the TBF.

A. Three-port Coupled-lines Modeling

The modeling of coupled line pairs present in the proposed resonator follows the methodology presented in [^{15}]. A classic approach to coupled lines is developed and focuses on networks based on lines with an N number of ports. This paper presents an analysis of three-port coupled line networks, one of which is open circuit terminations. This approach is then used as a starting point for the resonator modeling proposed here. Fig. 2 shows the 3-port coupled line network and its equivalent circuit based on the analysis in [^{15}].

The impedance *Z _{c}* models the capacitive coupling and the losses present in the region between the conductive tapes. The coupling is modeled through the ideal transformer ratio transformer

*α*, through which the electrical signal flows without physical contact between the coupled lines. Lastly, admittance

*Y*describes the admittance of network input. Based on the analyzes made in [

_{ue}^{15}], [

^{16}] and [

^{17}], all these parameters can be written as a function of odd and even mode impedances (Z

_{oe}and Z

_{oo}):

Where the parameter *s* is being the so-called Richards variable [^{17}]. According to [^{16}], admittances *Y _{11}* and

*Y*can be respectively written as follows in (5) and (6)..

_{12}where *Y _{oo} = 1/Z_{oo}* and

*Y*. Finally, the admittance

_{oe}= 1/Z_{oe}*Y*of Fig. 2 can be written according to transformation matrix

_{ue}*A*:

B. Equivalent Circuit Assembly

From the analysis of the equivalent circuit for the three-port coupled line network presented in the previous subsection and considering *A _{L}*(8) as the transformation matrix of a characteristic impedance line

*Z*and electrical length

_{L}*θ*, it is possible to write the admittance matrix

*Y*shown in Fig. 3.

_{s}Through some algebraic operations and manipulations, it is possible to write the transformation matrix A_{s} and admittance *Y _{s}* that help describe the circuit that shapes the geometry as follows:

III. ANALYSIS AND SIMULATIONS

Based on the circuit of Fig. 1(b), which provides an equivalence to the resonator geometry, simulations were performed in Advanced Design System (ADS). Taking into account the modeling presented in the previous section, the frequency response of the 1.00 GHz principle resonance frequency f_{R} circuit was analyzed, as shown in Fig. 4. The impedance parameters employed were: *Z _{oe}* = 94 Ω,

*Z*= 49 Ω and

_{oo}*Z*= 50 Ω. Electrical lengths were determined according to a study in section II.

_{L}As can be seen, the response has a *f _{R}* main resonance frequency of 1.00 GHz and two symmetric resonances: lower transmission band (LTB) at 0.483 GHz, and upper transmission band (UTB) at 1.517 GHz. The bandwidth at −3 dB at 1 GHz equal to 54 MHz and at symmetrical transmission bands 20 MHz. The graph of Fig. 5 showing the phase variation of the spreading parameters S11 and S21 as a function of frequency contributes to the verification of the symmetries between LTB and UTB of the filter response.

It is also important to note the six symmetrical zero transmission frequencies, which contribute to TBF selectivity. Based on this, two parameters are defined that will facilitate the following analyzes: *r* and *tz*. Parameter r defines the relationship between LTB’s first resonant frequency *f*_{1} and the main frequency *f*_{R}, where *r = f _{1}/f_{R}*. As previously stated, the second resonance

*f*

_{2}is symmetrical with respect to resonance

*f*

_{1}and

*f*

_{R}. The parameter tz represents the relationship between the zero transmission frequency

*f*

_{tz}closest to the main transmission band and

*f*

_{R}, so

*tz = f*.

_{tz}/f_{R}From the established parameters *r* and *tz*, the analysis of these and the bandwidth BW of the main transmission band was performed at −3 dB as a function of the impedances that constitute the equivalent circuit, and consequently, of the resonator geometry in order to have a notion of the influence of each of the impedances on the filter response. According to Fig. 6, which brings the analysis to the previously established parameters, it is possible to make some conclusions. With respect to *r*, Fig. 6(a), the increase in impedance *Z _{L}* provides an approximation of LTB and UTB towards

*f*, but for

_{R}*Z*increase the effect is opposite. The graph in Fig. 6(b) stresses the behavior of parameter

_{oo}*tz*as a function of the variation of

*Z*and

_{L}*Z*, presenting a response similar to parameter

_{oo}*r*. With respect to the BW bandwidth of the main transmission band present in Fig. 6(c), the Zoo increase causes a sharp drop in BW, but with respect to

*Z*malfunction, the decrease in BW was much less noticeable. Also, in Fig. 7 it is possible to establish the relationship between

_{L}*r*and

*Z*for different levels of

_{L}*Z*impedance. The four curves present similar behavior, presenting a tendency of equilibrium of

_{oo}*r*for high values of

*Z*. It is important to analyze that for each value of

_{L}*Z*, different levels of constancy of

_{oo}*r*are observed.

IV. TBF FILTER DESIGN, RESULTS AND DISCUSSIONS

From the equivalent circuit modeling and the analyzes performed, a TBF model was implemented in EM simulation software. A geometry with *f*_{R} frequency response of 1.50 GHz was proposed, together with LTB and UTB centered at 0.78 and 2.30 GHz, respectively. The model with its respective dimensions is shown in Fig. 8(a). A prototype was built from a RO3006 dielectric plate with a relative permittivity of 6.15, loss tangent of 0.0024 and thickness *h* of 1.52 mm, as shown in Fig. 8(b), for measurements by means of a network analyzer and obtaining experimental results.

The Fig. 9 presents the EM simulation result for the TBF model with respect to surface current density for previously established resonant frequencies. As can be seen, the densities remain almost constant and stable for the different frequencies, presenting only sensitive alterations. The highest A/m levels are concentrated in the central parts of the coupled lines, with the centerline and supply lines having the lowest levels.

The comparison regarding the reflection and transmission parameters of the results obtained by EM simulation and measurement of the built prototype is shown in Fig. 10, showing a good agreement between the curves.

The Table I presents the summary and comparison between the results obtained via EM simulation and experimental measurement of the built prototype. As can be observed, the established and observed simulated resonant frequencies were met with the prototype implementation. Four zero transmission frequencies are observed along the analyzed spectrum, being close to 0.30, 0.96, 1.91 and 2.41 GHz, which contributes to the filter selectivity level.

Transmission Band |
f (GHz)_{sim} |
−3 dB BW (GHz)_{sim} |
Ins. Loss (dB)_{sim} |
f (GHz)_{meas} |
−3 dB BW (GHz)_{meas} |
Ins. Loss (dB)_{meas} |
---|---|---|---|---|---|---|

Main | 1.5 | 0.19 | 0.63 | 1.49 | 0.13 | 1.45 |

LTB | 0.77 | 0.04 | 1.40 | 0.78 | 0.02 | 1.48 |

UTB | 2.33 | 0.19 | 1.47 | 2.30 | 0.15 | 1.62 |

Comparisons between simulated and measured results are shown in Fig. 11. It is observed that the impedance level for the filter model decreases with frequency in both the EM simulation and the experimental results. The most important frequency ranges (LTB, Main, UTB) are precisely delimited by increases in the impedance levels, with the upper limits always being larger compared to the lower limits. Fig. 11(b) presents the Group Delay parameter, which expresses the delay of the response of a signal during its propagation along the filter structure. It is possible to observe maximum delays below 11 ns for the EM simulation and 15 ns for the signals results obtained via experimental measurement.

Table II presents the comparison of the TBF modeled and proposed here with other recent works found in the literature, thus demonstrating their equivalence, from research focused on single-band structures to triple-band structures. The resonator geometries which the base works also have similar modeling processes.

Model |
f (GHz) |
−3 dB BW(%) | N tz | Ins. Loss(dB) | Resonator |
---|---|---|---|---|---|

[2] | 2.1 | 23.7 | 6 | 1.6 | Side-coupled Square Ring / Stub-loaded |

[3] | 2 | 12 | 2 | 2.4 | Side-coupled Square Ring |

[6] | 2.4 | 5 | 2 | 0.91 | Fractally Sloted Patch |

[7] | 0.35 / 0.957 / 1.68 | 10.5 / 6.2 / 4.3 | 8 | 1.1 / 0.77 / 0.67 | Coupled Line / Stub-loaded |

[13] | 2.1 / 2.62 | 1.76 / 1.527 | 3 | 0.81 / 1.13 | Coupled Feed Lines |

[14] | 1.57 / 4.6 | 82.16 / 41.52 | 2 | 0.21 / 0.39 | Coupled Lines |

[18] | 1.8 / 2.38 / 3.5 | 4 / 4.1 / 4 | 3 | 0.07 / 0.08 / 0.11 | Embedded Stub-loaded |

[19] | 1.575 / 1.8 / 2.4 | 6.1 / 3.5 / 3.1 | 6 | 0.7 / 0.9 / 0.9 | Stub-loaded |

[20] | 3.6 / 6 / 9 | 3 / 4.7 / 3.5 | 5 | 3.16 / 2.17 / 3.16 | Double CRLH |

This work |
0.78 / 1.49 / 2.30 |
2.56 / 8.72 / 6.52 |
4 |
1.48 / 1.45 / 1.62 |
Modified Side-coupled Square Ring |

V. CONCLUSION

A triple-bandpass filter based on a modification of the side-coupled square ring geometry had its design modeled and developed in this work. Expressions were obtained that describe both the admittance matrix and the filter geometry transformation. Analyzes based on the impedances described in the modeling were developed and described in order to obtain their respective influences on the frequency response of the model. A prototype was built and measurements were made in order to allow comparisons with the results obtained by means of EM simulation, which were very coherent and approximate to each other. Finally, comparisons between the proposed model and literature works have been presented, and the approach developed here is fully in line with the latest research involving microwave filter design.